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elec_mag_suscep.mcd 1

T h e o r yo f

D i e l e c t r i c s, D i a m a g n e t s, P a r a m a g n e t s, a n d F e r r o m a g n e t s,i n c l u d i n g t h e

C a l c u l a t i o n o f E l e c t r i c a n d M a g n e t i c S u s c e p t i b i l i t i e s

byRonald W. Satz, Ph.D.*Transpower Corporation

Abstract

This paper presents the derivation of the equations for the properties of dielectrics, diamagnets, paramagnets, andferromagnets according to the Reciprocal System of physical theory developed by D. B. Larson. The factors include:atomic electric and magnetic rotational displacements, electric rotational vibration frequency, magnetic rotational vibrationfrequency, Planck's constant, Rydberg's constant, temperature, and molecular or crystal structure.

keywords: scalar motion, dielectrics, diamgagnets, paramagnets, ferromagnets, gravitational force, electrical force,magnetic force, electric susceptibility, magnetic susceptibility, RLC circuits, capacitors, permittivity, permeability, index ofrefraction, magnetic charge, magnetic saturation, Curie temperature, magnetic hysteresis, magnetic energy product

*The author is president of Transpower Corporation, a commercial and custom software manufacturing company and engineering/physicsconsultancy. Mailing address: P. O. Box 7132, Penndel, PA 19047. He is a full member of ASME, SAE, INFORMS, ISUS, and SIAM.


Introduction and Literature Survey

Thousands of scientists and engineers have worked on the experimental and theoretical aspects of electricity andmagnetism, but D. B. Larson was the first and only one to reduce all electric, magnetic, and gravitational concepts andequations to space-time terms alone. This paper translates Larson's work, in Ref. [1] (particularly pp. 133-260), Ref. [2] (pp.50-89, first ed.) , Ref. [3] (pp. 153-190), Ref. [4], and Ref. [5], into the language of mathematical physics. It is runnable as aMathcad program, so it is a highly computational paper.

Conventional physical theory can be found in any university physics textbook, such as Ref. [6], or at a somewhat higher level,in the nine volume Encyclopaedic Dictionary edited by Thewlis, Ref. [7].

The properties of matter are discussed, at an elementary level, in Ref. [8] and Ref. [9], and at a higher level in Ref. [10] andRef. [11].

Tables of material data can be found in a variety of handbooks, such as Ref. [12] - [17].

Specific dielectric properties and equations are discussed in detail in Von Hippel's three works, Ref. [18], Ref. [19], and Ref.[20].

Electric circuits, specfically DC RC, DC RL, DC RLC, AC RC, AC RL, and AC RLC, are conventionally analyzed in Ref. [21].

The standard work in Quantum Mechanics for the calculation of electric and magnetic susceptibilities is by Van Vleck, Ref.[22].

A fairly recent work in the Quantum Mechanics of magnetism in solids is by Martin, Ref. [23].

The atomic/molecular beam experiments, used to determine atomic and "nuclear" electric and magnetic moments, aredescribed by Fraser, Ref. [24], and Ramsey, Ref. [25].

Two major works in magnetochemistry are those by Bhatnagar and Mathur, Ref. [26], and Selwood, Ref. [27].

General works in magnetism include Ref. [28], Ref. [29], and Ref. [30].

Applied magnetism is the focus of Ref. [31] and Ref. [32].


Field theory is described and applied in Ref. [33] - [42]. As will be explained, the Reciprocal System is compatible with fieldtheory calculations, but not with the conventional physical interpretations.

Computer animation can be helpful in understanding electrical and magnetic phenomena. Prof. Goodstein's series of DVD's,The Mechanical Universe and Beyond, Ref. [43], is excellent in this regard, though it's naturally biased toward conventionaltheory. On the Web, P. Falstad, Ref. [44], has displays of numerous applets showing many different kinds of fields in action.There is not, as of yet, much in the way of animations of various aspects of the Reciprocal System, but the 17 figures in theauthor's first book, The Unmysterious Universe, Ref. [45], should be helpful to newcomers of the theory. Also, some of theauthor's previous papers, Ref. [46]-[54], may be of assistance in understanding this paper (but where there is a differencebetween the previous papers and this one, this one supercedes).

Last, but not least, mention must be made of a pioneer experimenter and theoretician, Sir James A. Ewing (1855-1935). Hismajor work in ferromagnetism is now online, Ref. [55]. It was Ewing who conceived of atoms as acting like bar magnets,somewhat like the picture we get from the Reciprocal System! His understanding of ferromagnetic hysteresis was quiteaccurate, as we shall see.

Additional references are given, as needed, following Ref. [55], at the end of this paper.


Nomenclature

a = general symbol for acceleration

au = natural Reciprocal System unit of acceleration, stated in cm/sec2

au_cgs = natural cgs unit of acceleration = 1 cm/sec2

B = general symbol for magnetic flux density (context determines whether units are cgs, SI, or Reciprocal Sys.)

Bext = external magnetic flux density impressed on solid, tesla = webers/m2

Bext_c = external magnetic flux density at coercivity point in hysteresis cycle, tesla

Bext_max = maximum external magentic flux density in hysteresis cycle, tesla

BH = magnetic energy product, J/m3 (subscript max = maximum, subscript may have mat.; calc =calculated, obs=observed)

Bint = internal magnetic flux density (that within the material), tesla

Bnat_SI = natural Reciprocal System unit of magnetic flux density (time-space region), stated in tesla

Br = remanence internal magnetic flux density (context determines whether at 0 K or temperature T), tesla

Bs = saturation internal magnetic flux density at 0 K, tesla (at room temperature for hysteresis curves); subscript has material

Bs_T = saturation internal magnetic flux density at temperature T, tesla

C = capacitance, farads

C = Curie constant in Curie-Weiss magnetic susceptibility equation, K if using non-dimensional


CM = Curie constant on molar basis

Cu = natural Reciprocal System unit of capacitance, stated in farads

c = velocity of light (subscript has units)

cu = natural Reciprocal System unit of velocity = 1 unit of space / 1 unit of time

ccgs = speed of light expressed as cm/sec

cSI = speed of light expressed as m/sec

c0, c1, c2 = constants in electrical circuit differential equations

convAtomm = factor to convert angstroms to mm

convcaltoev = factor to convert calories to electron-volts

convcmtofarad = factor to convert cm to farads

convCV = factor to convert CV to joules

convfaradtocm = factor to convert farads to cm

conviC = conversion constant for RLC circuit

convjoulestoev = factor to convert joules to electron-volts

convRCV = factor to convert RC/V to seconds

convseccmtojoule = factor to convert sec/cm to joules


convvolttoseccm2 = factor to convert volts to sec/cm2

D = diameter of atom, m (subscript has element symbol)

Du = natural Reciprocal System unit of atomic diameter, stated in m

d = number of natural Reciprocal System units of density

du = natural Reciprocal System unit of density = 1 unit of mass / 1 unit of volume, stated in g/cm3

dcgs = density in cgs units

du_cgs = natural cgs unit of density = 1 g/cm3

dBdz = gradient of external magnetic flux density, Bext, with respect to z-direction, tesla/m

E = energy from voltage source, joules

EC = energy stored in capacitor, joules

EL = energy stored in inductor, joules

EI_atom_elec = ionization energy of atom-electron pair, eV

EI_pos_mole = ionization energy for atoms (not including the electrons) in a solid, kJ/mole

ER = energy dissipated in resistor, joules

E0 = energy of incident photons on medium, eV

E = energy of photons after travelling through x cm of medium, eV


edgea = edge_a of the crystal volume unit cell (could be in angstroms, cm, or m, by context)

edgeb = edge_b of the crystal volume unit cell (could be in angstroms, cm, or m, by context)

edgec = edge_c of the crystal volume unit cell (could be in angstroms, cm, or m, by context)

edgeuc = general symbol for any of the three edges of the crystal volume unit cell (could be in angstroms, cm, or m, by context)

F = general symbol for force (subscript has direction)

FE = electrostatic force (subscript has units)

FM = magnetostatic force (subscript has units)

FG = gravitational force (subscript has units)

F = refraction vibration factor (dimensionless)

f = frequency, cycles/sec

G = gravitational constant (actually dimensionless whether stated in cgs or SI)

h = Planck's constant, eV-sec

I = steady-state current in electrical circuit, amps

IM = transmission ratio for internal magnetic flux density (dimensionless)

i = instantanous current in electrical circuit, amps

iRLC = function to compute current for RLC circuit, amps

ic = complementary solution for instantaneous current, amps


ieff = effective current for AC circuit, amps

ip = particular solution for instantaneous current, amps

iu = natural Reciprocal System unit of current, stated in amps

j = imaginary unit (square root of -1)

k = extinction coefficient (dimensionless)

k0, k1, ... = constants for use in the electric circuit equation

kB = Boltzmann's Constant, joule/K; kB_ev = Boltzmann's Constant, eV/K

kc = factor relating coercivity of ferromagnet to the coercivity of its predominant ferromagnetic material (dimensionless)

kdpm = geometric coefficient for magnetic dipole length (dimensionless)

kG = crystal unit cell volume geometric factor (dimensionless)

kr = refraction constant (dimensionless) (subscript has element or compound symbol)

kr0 = number of 1/9 initial units for index of refaction equation (dimensionless) (subscript has element symbol)

L = inductance in inductor, henries

Lu = natural Reciprocal System unit of inductance, stated in henries

M1 or M2 = magnetic charge of body (subscript has units)

MP = magnetic polarization (dipole moments per unit volume), webers/m2

elec_mag_suscep.mcd 9P

MP_nat_t = natural Reciprocal System unit of magnetic polarization (time region), stated in tesla = webers/m2

Mu = natural Reciprocal System unit of magnetic charge (subscript has units)

m = general symbol for mass (if atomic, subscript has element symbol)

m1 or m2 = mass of body (subscript has units)

mu = natural Reciprocal System unit of mass, stated in sec3/cm3

mu_g = natural Reciprocal System unit of mass, stated in g

n = index of refraction (dimensionless) or real component of complex index of refraction

n* = complex index of refraction (dimensionless)

n1 = net number of atoms of element 1 of compound in unit cell

n2 = net number of atoms of element 2 of compound in unit cell

na = number of atoms of element in formula molecule (subscript has element symbol)

nf = fraction of maximum possible magnetic dipole moments induced in crystal volume unit cell, 0 to 1

nM = number of magnetic charges on each rotational system of an atom, 0 to tp (subscript may have material)

nM_r = number of magnetic charges on each rotational system of an atom at remanence (subscript may have material)

nr = factor relating remanence of ferromagnet to the remanence of its predominant ferromagnetic base material (dimensionless)

elec_mag_suscep.mcd 10nuc = net total number of atoms in unit cell

p = instantaneous power from voltage source, watts

pC = instantaneous power supplied to capacitor, watts

pL = instantaneous power supplied to inductor, watts

pR = instantaneous power supplied to resistor, watts

Q1 or Q2 = electric charge of body (subscript has units)

Qu = natural Reciprocal System unit of electric charge (subscript has units)

Qu_cgs = natural cgs unit of charge= 1 esu

R = resistance of resistor, ohms

R = Rydberg frequency (hydrogen), cycles/sec

Ru = natural Reciprocal System unit of resistance, stated in ohms

R = Rydberg frequency (hydrogen), half-cycles/sec

s = space dimension, m in SI or cm in cgs (subscript has units)

s0 = interatomic distance (subscript has units)

sd = deflection of vapor atoms from centerline, m

stot_sep = total separation between traces in Stern-Gerlach experiment, m

st_u = natural Reciprocal System unit of space in time region, stated in cm


su = natural Reciprocal System unit of space in time-space region, stated in cm

su_cgs = natural cgs unit of space = 1 cm

T = time period of AC voltage cycle, sec

T = temperature of substance, K (subscript has element symbol)

T0 = zero-point temperature, K (Ref. [1], p. 84)

T1 = temperature at end of first specific heat line segment, K (Ref. [1], p. 86)

Tc = Curie temperature, K (actual transition from ferromagnet to paramagnet takes place over a large zone of temperature)

te = electric rotational displacement of an atom or sub-atom (dimensionless)

te_mod = modified electric rotational displacement of an atom to account for neutrinos within and thus for isotopic differences inproperties (dimensionless)

teff = effective atomic rotational displacement (both magnetic and electric) for use in refraction calculations (dimensionless)

tp = principal (or primary) magnetic rotational displacement of an atom or sub-atom (dimensionless)

tr = equivalent net added time to motion of photons or electrons through atoms (natural units of time)

ts = subordinate (or secondary) magnetic rotational displacement of an atom or sub-atom (dimensionless)

Note: tp_eff, ts_eff, te_eff are listed in three column headings of magnetic susceptibility calculations in Table V (from Excel)--theoretically they should be identical to tp, ts, te

Tsc = superconducting temperature, K

Tt_u = natural Reciprocal System unit of temperature in the time region (solid, liquid states), stated in K


Tv_u = natural Reciprocal System unit of temperature at boundary between time and time-space regions (vapor state), K

t = time dimension, sec

tu = natural Reciprocal System unit of time, stated in sec

Ucohes_molec_mole = cohesive energy of atomic or molecular solid, kJ/mole

Uionic_atom_pair = Coulombic potential energy of pair of ions, J

V = DC voltage of electric circuit source, volts

Vmax = amplitude of AC voltage, volts

Vu = natural Reciprocal System unit of voltage, states in volts

Vuc = volume of crystal unit cell (subscript has units)

v = velocity of body or light (subscript has units or element symbol)

vC = instantaneous voltage across capacitor, volts

velec = atomic electric rotational velocity, natural units

vL = instantanous voltage across inductor, volts

vmag = atomic magnetic rotational velocity, natural units

vR = instantaneous voltage across resistor, volts

vu = velocity of body or light in natural units

elec_mag_suscep.mcd 13WH = magnetic hysteresis energy loss, J/m3 (subscript may have material name; calc=calculated, obs=observed)

w = atomic weight, amu (atomic mass units)

x = distance traveled by light within medium (cm for cgs, m for SI) (subscript has units)

x = T/Tc for graph show variation of saturation internal magnetic flux density versus temperature

y = length of path of vapor atoms through poles of electromagnet, m

Zatno = atomic number (subscript has element symbol)

Zuc = number of formula molecules (atoms for an element) in a crystal unit cell

= absorption coefficient (cm-1 for cgs, m-1 for SI) (subscript has units)

, = statistical parameters for argument of erf for magnetic hysteresis curves (subscript i for initial magnetization)

= Weiss constant in denominator of Curie-Weiss Law, K

= dielectric loss angle, rad

= permittivity of medium separating electric charges (electrostatics) or electrons (currents), units should be in s2/t but are notin SI or cgs

' = real component of complex permittivity

'' = imaginary component of complex permittivity

r* = complex relative permittivity (dimensionless)

r' = complex relative permittivity coefficient (dimensionless)


o = permittivity of free space, unity in both the Reciprocal System (s2/t) and cgs, but incorrect dimensions in both SI and cgs

([Ref. [1], p. 172, p. 184) (subscript has units)

r = relative permittivity or dielectric constant (dimensionless)

r1 = real component of complex relative permittivity (dimensionless)

r2 = imaginary component of complex relative permittivity (dimensionless)

= wavelength of light within medium (cm in cgs, m in SI) (subscript has units)

0 = wavelength of light incident to medium (cm in cgs, m in SI) (subscript has units)

0_n = wavelength of light incident to medium, natural units

= permeability of medium separating magnetic charges (magnetostatics), abhenry/cm for cgs and henry/m in SI (subscripthas units)

0 = permeability of free space, unity in both the Reciprocal System and cgs, abhenry/cm for cgs and henry/m in SI (subscripthas units; 0_SI must be used for magnetic equations expressed in SI)

adpm = atomic magnetic dipole moment, weber-m (subscript has element symbol)

adpm_el_som = Sommerfeld atomic magnetic dipole moment, joule/tesla, for element "el", joule/tesla

adpm_u = natural Reciprocal System unit for atomic magnetic dipole moment, weber-m

B = Bohr magneton for Quantum Mechanics

dpm = magnetic dipole moment, weber-meter

elec_mag_suscep.mcd 15r = relative permeability (dimensionless); r_0 = natural relative permeability of free space = 1

r_avg = average relative permeability of ferromagnet in second quadrant of hysteresis curve (dimensionless)

r* = complex relative permeability (dimensionless)

r' = complex relative permeability real coefficient (dimensionless)

r'' = complex relative permeability imaginary coefficient (dimensionless)

W = Weiss magneton

= frequency (natural units)

M_0 = physical "zero" for the rotational vibration frequency of magnetic charge, cycles/sec

M_n = rotational vibration frequency of an n magnetic charge unit, cycles/sec

phot = frequency of photons used in photomagnetization experiments, cycles/sec

e = electrical resistivity of medium, ohm-cm for cgs and ohm-m for SI (subscript has units)

C = time constant for capacitor, sec (subscript "legacy" for conventional value)

L = time constant for inductor, sec (subscript "legacy" for conventional value)

= magnetic loss angle, rad

ac = phase angle for AC voltage, rad

= general symbol for magnetic susceptibility (dimensionless, subscript has compound name)


cgs_mass = magnetic susceptibility, stated in cm3/g

cgs_mol = magnetic susceptibility, stated in cm3/mole

obs = observed value of magnetic suscepibility (dimensionless, subscript indicates source or compound name)

SI_mass = magnetic susceptibility, stated in m3/kg

SI_mol = magnetic susceptibility, stated in m3/mole

u_SI = natural Reciprocal System unit of magnetic susceptibility, computed using SI value of c

= angular frequency, rad/sec

Note: A black square in the upper right of an equation means that the equation is disabled from running in Mathcad. This is done because not allvariables in the equation have, as yet, been given numerical values at that point in the program. In a few cases, equations (like those with apostrophes)have to be given as text because Mathcad cannot represent the symbols properly; in this case, there is no black square, but the equation is notcomputational.


Gcgs 6.6656 108

(dimensionless!) (Ref. 1, pp. 162-163)

GSI Gcgs 103

GSI 6.6656 1011

(dimensionless!)

su_cgs 1 cm au_cgs 1 cm/sec2 mu_cgs 1 g Qu_cgs 1 esu

mu_SI 1 kg Qu_SI 1 Csu_SI 1 m au_SI 1 m/sec2

Reciprocal System Physical Constants (from Ref. [1]-[2], or derived therefrom)

su 4.558816 106

cm tu 1.520655 1016

sec au

su

tu2

au 1.971472 1026

cm/sec2

IR 156.4444 (inter-regional ratio)

tu_cgs 1 sec tu_SI 1 sec

st_u

su

IR st_u 2.914 10

8 cm

mu

tu3

su3

mu 3.7114 1032

sec3/cm3 mu_g 1.65979 1024

g

Gcgs 3mu

mu_g

1

1.00639


emu

du_cgs 1 g/cm3 Mu_weber Mu_emu 299.7925 Mu_weber 4.8029 1018

weber

(magnetic charge)

j 1 convfaradtocm 8.98758 1011

convvolttoseccm2 7.85794 1015

convseccmtojoul 4.472162

convcmtofarad 1.112647 1012

convCV convfaradtocm convvolttoseccm2 convseccmtojoul IR

convCV 4.9412 (so if V = 5, .5 x C x V x convCV = .5 x C x V2

approx.!)

Ru 8.83834 1011

Vu 9.31146 108

ohms volts Cu st_u convcmtofarad Cu 3.2423 1020

farads

Qu_esu 4.80287 1010

esu (electric charge)

Qu_coul 1.602062 1019

coulombs (electric charge)

R 6.576115 1015

half-cycles/sec (Rydberg frequency for H) RR

2 R 3.2881 10

15 cycles/sec

ccgs 2.997925 1010

cm/sec cSI 2.997925 108

m/sec cu 1

Mcgs 1 emu MSI 1 weber Mu_emu

Qu_esu

ccgs Mu_emu 1.6021 10

20


Mnat_SI Vu tu Mnat_SI 1.416 107

webers r_0 1 0_SI 4 107

henry/m

Bnat_SI

Mnat_SI

su 102

2

Bnat_SI 6.8131 107

webers/m2 (time-space region)

Tt_u 510.8 K kB 1.38 1023

J/K (Boltzmann Constant) convcaltoev 2.613 1019

kB_ev 8.617 105

eV/Kconvjoulestoev 6.242 10

18

(see Ref. [1], p. 111, comparing su and st_u)

Lu Ru tu Lu 0.0001 henries iu 1.05353 103

amps

convRCV

Ru Cu

Vu tu convRCV 0.2024 (for use in RC time constant calculations)

conviC iu Cu Vu conviC 3.1806 1014

(conversion constant for RLC circuit)

convAtomm 107

converting angstroms to mm for dielectric strength calculations)

Av 6.02486 1023

molecules/g-mole h 4.14 1015

eV sec (Planck's constant)


1. Force Definitions and Calculations

a. Inertial Force

Force, not mass, is defined by Newton's Second Law of Motion:

F m a (1a)

In the Reciprocal System, there are no so-called "fundamental forces"; rather, motion or space-time in the most generalsense is what is "fundamental." Thus force is merely a property of motion, not the other way around. All physicalquantities in the Reciprocal System therefore reduce to expressions involving space, s, and time, t, only. In space-timeterms, Eq. (1a) is

t

s2

t3

s3

s

t2

(1b)

All forces, with the exception of magnetomotive force (for reasons to be discussed later), have the dimensions t/s2. Forcemay be scalar or vectorial and may be positive or negative.

b. Scalar vs. Vectorial Motions; Reference Systems

The "fundamental motions" in the Reciprocal System are scalar and are either linear or rotational, either uniform orvibrational, and either one- , two- , or three- dimensional. Mass is comprised of one- and two- dimensional scalarrotational motions, totalling three dimensions, so the correct dimensions of mass are t3/s3. As can be inferred from these

elec_mag_suscep.mcd 21rotational motions, totalling three dimensions, so the correct dimensions of mass are t /s As can be inferred from thesedimensions, mass resists a change of motion in any dimension--this is the property of "inertia." But the same rotationalmotion which gives rise to mass also gives rise to gravitation, and so inertial and gravitational mass are precisely equal,which shows that the Reciprocal System derives the "Principle of Equivalence" easily and naturally.

Vectorial motion cannot occur until a gravitationally-bound system comes into existence; this establishes an inertialreference frame. Therefore, scalar motion precedes, and is thus more fundamental, than vectorial motion. By its nature,the scalar motion involved in gravitation or electrostatics or magnetostatics is mutual; when we couple such motion to thestandard Cartesian reference system (with three coordinates of space and one of time and with a specific origin), themotion may be mis-attributed to one or the other of the (apparently) interacting entities. This standard reference systemcan display only one scalar dimension of motion (with three space coordinates), even though the motion may actually betwo- or three- dimensional. Such multi-dimensional motion does have indirect effects which can be detected, as we willsee later. The methods of field theory may be used to calculate the results of gravitational, electric, and magnetic forces,but there is nothing "physical" in the field--just coordinate space (which is, itself, generated by clock space). Each of the(apparently) interacting entities is pursuing its own course relative to its surrounding space-time locations. There is no"action at a distance" and there are no "gravitons" or "virtual photons" being exchanged. Mass, electric charge, andmagnetic charge have the effect of "concentrating" space-time locations and thus "redirecting" scalar motionautomatically and instantly. There are no explicit time terms in the three basic scalar motion equations given in the nextsection.

Vectorial motion (kinetic energy) may be added to the scalar motion, and can be properly displayed in the conventionalreference system. Vectorial motion is not a mutual motion.


b. Gravitational Force

According to Eq. (1a), force is proportional to the first power of mass. Therefore, Newton's gravitational equation, whichhas the gravitational force equal to the second power of mass, has to be "re-dimensionalized" to be made correct. Thegravitational constant, G, to which dimensions have traditionally been attached, needs to be changed to a pure number orratio, for the same reason that permittivity and permeability are, respectively, in the electrostatic and magnetostaticequations of the Reciprocal System. Also, there needs to be an acceleration term, but this is numerically equal to unity inthe system of units used to represent the force and distance. Finally, because of the distribution of scalar motion over thesurface of an imaginary sphere enveloping the mass (or electric charge or magnetic charge) there must be an inversesquare distance term, but this must be non-dimensional, like the acceleration term. Therefore, using cgs units,

(2a)FG_dynes

m1_g

m2_g

mu_cgs au_cgs

1

Gcgs

scm

su_cgs

2

where the inverse of the gravitational constant has been put in the denominator of the expression to be consistent withthe other scalar force equations. (A negative sign is preprended to the RHS to indicate that the force is attractive.) Indimensional terms, as expected:

(2b)t

s2

t3

s3

s

t2


But, since in cgs units, mu_cgs and au_gcs and su_cgs are all unity, by definition, this equation numerically reduces to

FG_dynes

m1_g m2_g

1

Gcgsscm

2

(2c)

This is Newton's gravitational equation, of course, but now we understand it much more profoundly!

In SI units, the equation is

FG_N

m2_kg m2_kg

1

GSIsm

2

(2d)

Ref. [46] shows that at a high relative speed of the masses, the gravitational equation (in SI units) is

FG_N 1vSI

2

cSI2

m2_kg m2_kg

1

GSIsm

2

(2e)

elec_mag_suscep.mcd 24Therefore, as velocity increases, the effective force declines, with the masses remaining constant!

Note: Newton's gravitational equation is not as "universal" as traditionally claimed--beyond a certain distance, thespace-time progression begins to dominate. For details, see one of the author's papers, Ref. [60].

c. Electrostatic Force

Whereas mass is uniform three-dimensional rotational motion, electric charge is vibrational one-dimensional rotationalmotion. Its dimensions are therefore t/s. To be made correct, Coulomb's equation for electrostatics needs to be"re-dimensionalized" as follows.

(3a)FE_dynes

Q1_esu

Q2_esu

Qu_cgs

1

su_cgs

r

scm

su_cgs

2

where the symbols have their usual meanings, and Q1 and Q2 may be positive or negative. In dimensional terms, asexpected,

t

s2

t

s

1

s

(3b)

But, since in cgs units, Qu_cgs and su_cgs are unity, by definition, this equation numerically reduces to


FE_dynes

Q1_esu Q2_esu

r scm2

(3c)

This is Coulomb's electrostatic equation. In air, the relative permittivity r = 1.006 and is often taken to be 1. In SI units, a

40_SI factor is necessary due to the way coulombs (= 3x109 esu) and 0_SIare defined:

FE_N

Q1_C Q2_C

4 0_SI r sm2

(3d)

Of course, at a high relative speed of the electric charges, Eq. (3d) becomes

FE_N 1vSI

2

cSI2

Q1_C Q2_C

4 0_SI r sm2

(3e)

with the charges remaining constant. In contrast to the SI system of units, both the cgs system and the Reciprocal Systemhave the permittivity of free space equal to one:

0_cgs 1 0_u 1(4a)

0_SI 8.85 1012

'farad/m' (4b)

elec_mag_suscep.mcd 26But in the cgs system of units, permittivity (not just relative permittivity) is dimensionless, and in the SI system the wrongunits, farad/m, are used (which from the perspective of the Reciprocal System means dimensionless, as well). Thecorrect dimensions of permittivity are s2/t. Nonetheless, it's clear that the cgs system of units is closer to the naturalsystem of units of the Reciprocal System. The relative permittivity, also termed the dielectric constant, is defined as

r

0

(5)

This is a basic property of matter and its calculation will be given later in this paper. r is dimensionless, of course.

The natural unit of charge is that of the charged electron. There are no fractional charges in the Reciprocal Systemand therefore no quarks!

d. Magnetostatic Force

Whereas mass is uniform three-dimensional rotational motion and electric charge is vibrational one-dimensional rotationalmotion, magnetic charge is vibrational two-dimensional rotational motion. Its dimensions are therefore t2/s2. To be madecorrect, Coulomb's equation for magnetostatics needs to be "re-dimensionalized" as follows.

(6a)FM_dynes

M1_emu

M2_emu

Mu_cgs

1

tu_cgs

r

scm

su_cgs

2

where the symbols have their usual meanings, and M1 and M2 may be positive (north pole) or negative (south pole). In

elec_mag_suscep.mcd 27where the symbols have their usual meanings, and M1 and M2 may be positive (north pole) or negative (south pole). Indimensional terms, as expected,

t

s2

t2

s2

1

t

(6b)

A magnetic charge really is two-dimensional! But, since in cgs units, Mu_cgs and su_cgs are unity, by definition, thisequation numerically reduces to

FM_dynes

M1_emu M2_emu

r scm2

(6c)

This is Coulomb's magnetostatic equation. In air, the relative permeability r = 1.00000037 and is almost always taken

to be 1. In SI units, a 40_SI factor is necessary due to the way webers (= 108 / 4 emu) and 0_SIare defined:

FM_N

M1_weber M2_weber

4 0_SI r sm2

(6d)

Of course, at a high relative speed of the magnetic charges, Eq. (6d) becomes

FM_N 1vSI

2

cSI2

M1_weber M2_weber

4 0_SI r sm2

(6e)

with the charges remaining constant.

elec_mag_suscep.mcd 28See McCaig (Chapter 2 of Ref. [31]) for a thorough discussion of magnetic units and definitions; some authorsactually put the permeability in the numerator! In contrast to the SI system of units, both the cgs system and theReciprocal System have the permeability of free space equal to one:

0_cgs 1 0_u 1 (although, in cgs, the units should be abhenry/cm)(7a)

0_SI 4 107

henry/m (7b)

It's clear, then, that once again, the cgs system of units is closer to the natural system of units of the ReciprocalSystem. The relative permeability is defined as

r

0 (8)

This is a basic property of matter and its calculation will be given later in this paper. r is obviously dimensionless.

The natural unit of magnetic charge, Mu_weber = 4.80287 1018

weber, should be that of amagnetically-charged subatom with a single rotational system, like a proton (an independent charge or"monocharge"). But, in the case of the proton, the measurements seem to be all based on the hydrogen inwater, so apparently we don't have a direct particle confirmation. (Hydrogen has two rotating systems.) Themacroscopic verification will be given later.


Therefore:

FM_dynes 1.235 1029

FM_dynes

M1_emu M2_emu

r scm2

r 1Mu_emu 1.6021 1020

M2_emu Mu_emuM1_emu Mu_emu

FE_dynes 1.1099 108

FE_dynes

Q1_esu Q2_esu

r scm2

Qu_esu 4.80287 1010

Q2_esu Qu_esuQ1_esu Qu_esur 1

FG_dynes 8.8357 1045

FG_dynes

m1_g m2_g

1

Gcgsscm

2

scm sum2_g mu_gm1_g mu_gGcgs 6.6656 108

mu_g 1.6598 1024

Sample Calculation Comparing Gravitational, Electrical, and Magnetic Forces

Assuming natural unit masses, unit electric charges (opposite signs), and unit magnetic charges (opposite signs),separated by one natural unit of distance in vacuum, the following values apply, in cgs:

elec_mag_suscep.mcd 30FG_dynes

FE_dynes7.9605 10

37 FG_dynes

FM_dynes7.1546 10

16

FM_dynes

FE_dynes1.1126 10

21 (as expected, 1/ccgs

2)

2. The Speed of Light, Permittivity, Permeability, and Index of Refraction

Maxwell's famous formula relating the speed of light, permittivity, and permeability, all in free space, is

c1

0 0 (9a)

In Reciprocal System space-time terms:

s

t

1

s2

t

t3

s4

(9b)

This equation is dimensionally correct. The speed of light is the natural unit of velocity in the Reciprocal System--and thiscomports with the deduction that the natural unit of permittivity (in free space) is one and the natural unit of permeability (infree space) is one:


11

1 1 (9c)

Ref. [56], pp. 118-119, tabulates the "rationalized natural" system of units in which c = 1 , 0 = 1, and 0 = 1, andpresents Maxwell's electromagnetic equations with these values. This is as close as other authors have come to thesame conclusions as the Reciprocal System.

Incidentally, and unlike conventional physicists, we don't conclude from Eq. (9a) that light is "electromagnetic radiation"with fluctuating electric and magnetic fields. Energy changes in the radiation occur only at the time of emission orabsorption of the photons, not during transit. Therefore, Eq. (9a) is really just a spatio-temporal relation.

According to Ref. [7], Vol. 5, p. 342, "The terms 'permittivity' and 'refractive index' represent merely alternative ways ofattaching numerical values to one and the same property of a dielectric medium. The customary use of the former inelectrical problems and of the latter in optical ones arises only from convenience, and the historical development of thesubject. Permittivity is a somewhat simpler concept in problems involving wave-lengths large compared with thedimensions of the medium concerned, so that the spatial change of the electric gradient can be ignored, whilerefractive index becomes simpler for problems having the converse relation, of a wave-length very small comparedwith the extent of the medium."

The speed of light "slows down" in matter with a relative permittivity of r and a relative permeability of r; the speed is

vc

r r (10)

(assuming no losses). By definition, the index of refraction is


where k is the extinction coefficient. The article in Ref. [7] given above goes on to point out that for anisotropic media,

(14)(Von Hippel's notation)n* = n (1 - j k )or(n*)2 := r' r' (1 - j tan() ) (1 - j tan() )

Therefore, the complex refractive index is

(13)(Von Hippel's notation)with tan() = r''/r'r := r' - j r''orr

:= r' (1 - j tan() )

Likewise, the complex permeability becomes

(12)(Von Hippel's notation)with tan() = r''/r'r* := r' - j r''orr* := r'(1 - j tan() )

But all real media are dispersive, and possess a corresponding dielectric loss angle, and magnetic loss angle, .Letting j be the imaginary unit (rather than i), the complex permittivity becomes

(11c)r n2

n r

In many situations involving dielectrics, the relative permeability can be neglected (set equal to 0, which is 1 in cgs orReciprocal System units), so Eq. (11b) becomes

(11b)n r r

so, for a loss-free medium,

(11a)nc

v

elec_mag_suscep.mcd 33where k is the extinction coefficient. The article in Ref. [7] given above goes on to point out that for anisotropic media,there would be three principal values of r* and r*, not necessarily with coincident electric and magnetic axes, or withequal values of dispersion. "The most general linear case therefore requires 12 constants and two direction cosines for itsdescription. Magnetic properties, however, are in general non-linear, and there may also be interaction effects betweenthe magnetic and optical constants, so that an even higher number of constants may be involved."

Ref. [10] provides tables of values for tan() for various dielectric materials. Unfortunately, there do not seem to be readilyavailable tables of values for tan() for various magnetic materials

Where we can neglect the permeability but not the dielectric losses, Eq. (14) becomes

(n*)2 := r' (1 - j tan() ) (15)

Other references, like Ref. [13], p. 6-118, and Ref. [16], p. 12-145, define the index of refraction and permittivitysomewhat differently, so that care must be used when consulting various data sources. Suppose a light beam ofenergy E , in eV, and wavelength 0, in cm, impinges on a dielectric. The beam loses energy as it traverses a lengthx (cm) of material. Let be the absorption coefficient and k be the extinction coefficient. Then

(16)E E0 e

cgs xcgs eV

(17)0_cgs

1.2398 104

E0 cm

kcgs 0_cgs

4 (18)

r1 n2

k2

(19)


(23c)k0_SI cSI

4 e_SI n

so it's dimensionally correct. In most practical situations, SI = 0_SI, so

(23b)k

t3

s4

ss

t

t2

s2

1

Note the n in the denominator. In space-time terms:

(23a)kSI 0_SI cSI

4 e_SI n

Ref. [57] provides an equation in SI for the extinction coefficient, k, in terms of the electrical resistivity, e, of the material:

r* := n2 - k2 - 2 j n kr* := (n*)2r* := rj r2(22)

(21)(which is different from Von Hippel; the k here = n x Von Hippel's k)n* := n - j k

(20)r2 2 n k


The author's previous paper on electrical resistivity, Ref. [53], can be used to determine e_SI. Note that the valuesgiven there for metals will have to be multiplied by the factor 10-8, and the values given there for semiconductors will

have to be multiplied by 10-2, to convert to ohm-m, for use in Eq. (23c).

Eq. (22) now becomes

r

* := n2

- (0_SI cSI

4 e_SI n)2

- 2 j n (0_SI cSI

4 e_SI n) (24)

To use Eq. (24), we need to determine n, the component of the index of refraction not involved with losses. We willuse Larson's method described in Ref. [2], 1st ed., pp. 125-130, to calculate n.

Larson says, p. 125: "Since matter is a time displacement, the space-time ratio involved in the passage ofradiation through matter or matter through radiation is not unity, but a modified value resulting from the addition ofthe time displacement to the time component of the original unit velocity. Addition of more time to the ratio s/tdecreases the numerical value and the apparent velocity of radiation in matter is therefore less than unity."

Larson continues, p. 126: "...on this basis the index of refraction relative to a vacuum is equal to the total timeassociated with unit space in the motion of the radiation. For present purposes we will be interested in the timedisplacement, rather than the total time and since the time per unit space in undisplaced space-time is unity, thedisplacement is n - 1. The displacement due to the presence of any specific atom or quantity of matter isindependent of temperature but there is a temperature variation in the refractive index due to the accompanyingchange in density. We may eliminate this effect by dividing each displacement by the corresponding density,obtaining a temperature independent quantity (n - 1) / d."

Continuing, p. 126: "The refractive displacement is the sum of two components, one due to the motion of matterthrough radiation (the apparent translatory motion of the radiation) and the other due to the vibratory motion of theradiation through matter. In the first of these we have a simple motion at unit velocity in the time region. We havepreviously determined that the three-dimensional distribution of motion in the time region reduces the component


.7700BCC

.7071HCP

.7071FCC

Crystal Type kG

Here's a table of values of kG for ideal crystal types (calculated as the cube root of the values given on p. 98 of Ref. [1]).

(26)Vuc_cgs kG s0_cgs

3

This equation must be modified to account for the geometry of the crystal unit cells. The volume of a unit cell is equalto the cube of the nearest neighbor distance multiplied by the geometric factor, kG:

(natural units)n 1( )

d

du

kr

16

(25)

previously determined that the three-dimensional distribution of motion in the time region reduces the componentparallel to one-dimensional time-space region motion to 1/8 of the total [1/23], and the vibratory nature of the motionof matter in the time region introduces an additional factor of 1/2. We therefore find the displacement on a time-space region basis to be 1/16 of the effective time region displacement units.

Continuing, p. 126: "If the magnetic rotational displacement is unity the refractive displacement is also unity, butwhere the rotational displacement is teff [our notation] the radiation travels through only one of the teff displacementunits and the effective displacement is 1/teff. If we represent the average value of 1/teff as kr, the refractivedisplacement due to the translatory motion is"


(translation)n 1( )

dcgs

du_cgs

.6588 kr(29)

then

kG .7071

If we have, as is the usual case,

(28)n 1( )

dcgs

du_cgs

.9317 kG kr

This value may be inverted and placed on the RHS of Eq. (25) and then multiplied by kG:

(27)mu_g

st_u3

0.0671dcgs

du_cgs

d

du

mu_g

st_u3

Eq. (25) is expressed in natural units; to convert to cgs, the density must be multiplied by the cgs values of unit density:

These values are for atomic crystals only--not molecular crystals, like ice. For molecular crystals, one has to attemptto figure out the distance between molecular force centers, not individual atoms.

Diamond (ZnS) Cube 1.5396

Simple (NaCl) Cube 1.00

elec_mag_suscep.mcd 38which is, in essence, Eq. (148) of Ref. [2], first ed., but with a more explicit derivation of the coefficient!

Larson continues, Ref. [2], first ed., pp. 126-127: "In the second refractive component, that due to the vibratorymotion of the radiation, we are not dealing with unit velocity but with a lower velocity (frequency), and the refractivecomponent is reduced by the ratio /1. This component is also modified by the geometrical relationship betweenthe path of the radiation and the structure of the material medium through which it passes. We will call thismodifying factor the vibration factor, F. The vibrational refraction is then"

n 1( )

dcgs

du_cgs

.9317 kG F kr (vibration) (30)

Adding Eq. (28) and Eq. (30) we obtain

n 1( )

dcgs

du_cgs

.9317 kG kr 1 F (translation and vibration) (31a)

But 1

0_n (where we are using natural units) so

(31b)n 1( )

dcgs

du_cgs

.9317 kG kr 11

0_nF

or


ndcgs

du_cgs.9317 kG kr 1

1

0_nF

1 (31c)

The refraction constant, kr, remains to be determined. This is the reciprocal of the effective time displacement minusthe initial level in 1/9 increments.

kr1

teff1 kr0

1

9

(32a)

Ref. [53], the author's paper on electrical resistivity in the Reciprocal System, shows that

teff

te_mod

tp2

ts 1

3

2

(33)

Eq. (32a) now becomes

kr

tp2

ts 2

3

te_mod2

1 kr01

9

(32b)

(Larson simply used teff = tp or ts for H, C, and O in the published work, but it's necessary to use the full expression foraccuracy, because the time involved must be the same or close to that used for the electrical resistivity calculations.Appendix B of Ref. [2], first ed., p. 218, states, "The omitted portions [of Section XXXIV] include a discussion of the


The most commonly used wavelength for refraction measurements is that of the sodium D line, 5893 x 10-8 cm. So,usually,

(35)kr_compoundi

na_i Zatno_i kr_i

i

na_i Zatno_i

To get the average index of refraction for a compound or an alloy we must weight the number of atoms for eachelement, na_i, by its atomic number, Zatno_i. The atomic number is the net number of equivalent electric timedisplacement units of the atom, and is thus is more appropriate to use here, rather than atomic weight (which is whatconventional physics and chemistry use).

(same as for the specific heat patterns) but there may be deviations, as shown in Larson's tabulations of (n-1)/d forvarious organic compounds, pp. 128-129. (Atoms of the same element in a compound may have different valuesof kr0!)

(34b)(for electronegative elements, usually)kr0 0orkr0 1orkr0 2

(34a)(for electropositive elements, usually)kr0 2

Appendix B of Ref. [2], first ed., p. 218, states, "The omitted portions [of Section XXXIV] include a discussion of themore complex refraction patterns and numerical calculations of both refraction and dispersion for approximately 500substances." Unfortunately, this material has, to date, not been found, so we are on our own.)

Larson says on p. 30, Ref. [2], first ed., "This initial unit is distributed over three dimensions and the one-third unit in thedimension parallel to the space-time progression is again distributed over the three dimensions of the time region.The resultant is 1/9 unit in each magnetic dimension, a total of 2/9 units." Therefore,


(36)0_n

ccgs

5893 108

R 0_n 0.0774 (natural units)

Therefore, for this wavelength,

E01.2398 10

4

5893 108

E0 2.1039 eV (36)

Also, for many substances, F .75 (Ref. [2], p. 127)

The theoretical justification for this factor is as follows. There are two two-dimensional rotational systems of theatom. Each has two dimensions or directions, four total, for the radiation to go. But one of these must be collinear,because there are only three dimensions available. Therefore, the ratio of available paths to total paths is 3/4 =.75. By this argument, other allowed values of Fv would be .5 and .25. This deduction needs more confirmation, ofcourse.


teff_O 1teff_O

te_mod_O

tp_O2

ts_O 1

3

2

ts_O 2tp_O 2te_mod_O 22-2-(2)O:

(but could be .5, if D isgiven at. no. 1)

Zatno_H 1na_H 2kr_H 1.9599kr_H1

teff_H1 kr0

1

9

(for both H and O)kr0 2

teff_H 0.3969teff_H

te_mod_H

tp_H2

ts_H 1

3

2

ts_H 1tp_H 2te_mod_H 12-1-(1)H:

Sample Calculation for the Simple Index of Refraction of Water or Ice, H2O


kr_O1

teff_O1 kr0

1

9

kr_O 0.7778 na_O 1 Zatno_O 8

kr_H2O

na_H Zatno_H kr_H na_O Zatno_O kr_O

na_H Zatno_H na_O Zatno_O kr_H2O 1.0142

dH2O 1 g/cm3 (approx.) 0_n1

.0774 F .75

The value of kG is somewhat difficult to determine, because we're dealing with molecules, not atoms. There are fourmolecules of H2O per unit hexagonal cell. What is needed is the average separation of the molecules from oneanother, not the distance between individual atoms of hydrogen or oxygen. From Ref. [58], p. H-190, for ice Ih, theobserved a- , b-, and c-edge lengths of the cell (at 0o C) are 4.5135 A, 4.5135 A, and 7.3521 A. The computed unitcell volume at this temperature is 129.709 A3. If we assume the ice or water molecules are at an averageseparation of the c-edge, 7.3521 A (as opposed to the closest O atoms which are 4.5135 A apart), which seemsreasonable, then

kG_H2O129.709

7.3521( )3

kG_H2O 0.3264

ndH2O

du_cgs.9317 kG_H2O kr_H2O 1

1

0_nF

1

n 1.3263

elec_mag_suscep.mcd 44Ref. [17], p. 1.95, gives the measured value as 1.333 for temperatures ranging from 0o C to 25o C. So, perhaps,the actual effective molecular separation is a little less than the c-edge.

Sample Calculations for the Complex Index of Refraction and Permittivity of Semiconductor Elements

Use of the above equations in an Excel spread sheet results in the following table, with the observed values of n and k from Ref.[16] appended. The values of kG come from the Reciprocal system Data Base, so there are some deviations from ideality.

Element Rot. Displ. tp ts te te_mod_rho te_mod t_eff kr0 kr Density kG Fv n n_obs rho_e k k_obsB 2-2-(5) 2 2 -5 -7 -7 12.2500 0 0.0816 2.288 2.013 0.75 1.3706 8.7443E+04 0.0003

C 2-2-(4) 2 2 -4 -4 -4 4.0000 0 0.2500 3.515 1.558 0.75 2.3496 2.4150 8.1040E+94 0.0000 0.0000

Si 3-2-(4) 3 2 -4 -4 -3 1.3104 0 0.7631 2.329 1.570 0.75 3.7508 3.9690 2.8853E-01 27.7014 0.3000

S 3-2-(2) 3 2 -2 -2 -3 1.3104 0 0.7631 2.115 2.566 0.75 5.0828 3.0706E+12 0.0000

Ge 3-3-(4) 3 3 -4 -1 -4 1.7778 0 0.5625 5.321 1.471 0.75 5.3402 5.7480 4.7318E-01 11.8642 1.6340Te 4-3-(2) 4 3 -2 -7 -6 2.7257 0 0.3669 6.267 1.461 0.75 4.3114 4.6700 2.9580E-03 2350.7191 4.6700

Table I. Complex Index of Refracftion and Permittivity of Semiconductor Elements

The agreement is fair for n, but not for k. The column labeled te_mod_rho comes from Ref. [53]--these are the values ofte_mod used in the calculation for resistivity (which includes the isotopic effect); the column labeld te_mod includes thevalues used here; they are nearly the same except for Ge. All values of kr0 for these electronegative semiconductor elementsare zero. Ref. [16] does not have observed values for B or S. The extinction coefficients are not in agreement, but then theresistivities for these elements are in dispute--see Ref. [53] for a particular discussion about Si. Also, Ref. [16] uses


e := ''

Note 2: Von Hippel ran the Laboratory for Insulation Research at MIT for many years and generated much data ondielectrics. (And he lived to be 105!) His equation for dielectric conductivity (Ref. [18], p. 5) is

The Reciprocal System Data Base calculates the index of refraction and permittivity for elements, compounds, andalloy/mixtures. The anisotropric aspect of this property is contained in kG.

tr

dcgs

du_cgs.9317 kG

tp2

ts 2

3

te_mod2

1 kr01

9

11

0_nF

n 1 trn

1

1

1

1 tr

ncu

vu

Note 1: Here is an alternative derivation for n, using natural units:

n*Te = 4.3114 - j 2350.7191r*_Te = -5.5259E6 - j 2027E4

n*Ge = 5.3402- j 11.8642r*_Ge = -112.2415 - j 126.7144

n*S = 5.0828 - j 0r*_S = 25.0828- j 0

n*Si = 1.570 - j 27.7014r*_Si = -764.9027 - j 86.9824

r*_C = 5.5206 - j 0n*C = 2.3496 - j 0

r*_B = 1.8785 - j 8.2236E-04n*B = 1.3706 - j .0003

resistivities for these elements are in dispute--see Ref. [53] for a particular discussion about Si. Also, Ref. [16] usesinconsistent units for the extinction coefficient, so there's a lot to disagree with there. Von Hippel (Ref. [19], p. 302) says thatfor sulfur "tan() < 5"; no other elements are discussed there. Using Eq. (21) and Eq. (22) we have for these elements:


(Von Hippel's notation)tan

0_SI

e_SI 2 cSI

0_SI n2

tan() := '' / '

(Von Hippel's notation)* := 0_SI n2 - j 0_SI / ( e_SI 2 cSI )* := ' - j ''

And so

' := 0_SI n2

For ', Von Hippel has

which is zero for all intents and purposes. Von Hippel's equations should be used if you're using his data or his laboratory's.

'' := 1.0189 1028

ohm-me_SI 3.0706 1012

m0_SI 5893 1010

For the sodium D line and sulfur:

'' := 0_SI / ( e_SI 2 cSI )

Solving for '':

e_SI-1 := 2 cSI '' / 0_SI

Converting to our symbolism and assuming that dielectric conductivity = electrical conductivity:


This will help with the symbolic differentiation and integration of the expressions. All variables in this section are inthe usual SI units: volts, amps, ohms, so no subscripts for units will be used here. However, where there is a

=

Dielectrics are used in capacitors. In the Reciprocal System, an ordinary electric current is comprised of massless,uncharged electrons, not charged electrons, and therefore, our analysis of electric circuits with capacitors is different from thatof conventional physics and electrical engineering. Before continuing with the derivation of the electric and magneticsusceptibility equations, we'll now take a little detour to analyze the transient and steady state operation of six electric circuits,three DC and three AC, as shown in the Figure 1, (a)-(f). In what follows, we will often make use of Mathcad's symbolic equalsign,

3. Dielectrics and Electric Circuits

(Von Hippel's notation)n*S := 5.0828 (1 - j 0)n*S := nS (1 - j k)

(as before)k 0

For sulfur,

k1

2 tan 2 2 1 tan 2

1

2

which givestan 2 k

1 k2

=

Von Hippel (Ref. [18], p. 27) defines the extinction coefficient (for negligible magnetic loss) in terms of tan():

tan 4.4562 1019

tan

0_SI

e_SI 2 cSI

0_SI nS2

nS 5.0828

For sulfur,

elec_mag_suscep.mcd 48the usual SI units: volts, amps, ohms, so no subscripts for units will be used here. However, where there is adifference between the Reciprocal System equations and the conventional ones, we will use the subscript "legacy"to denote the conventional equations.


Figure 1. DC and AC Circuits. As is customary in the Reciprocal System ,

actual electron flows are shown, rather than assumed positive charge flow .

V

RvR

vC C

e

V

RvR

vL L

e

V

RvR

vL L

e

vC

C

v RvR

vC C

e

v RvR

vL L

e

vRvR

vL L

e

vC

C

(a)

(b)

(c)

(d)

(e)

(f)


(38c)

t1

d vCR C

V vC V

d=

This is a simple first order differential equation, so we can separate variables and integrate:

(38b)tvC

d

dV vC V

R C=

or

VC

V tvC

d

d

R vC=(38a)

Using Kirchhoff's voltage law around the circuit:

(37b)(therefore, correct dimensions)

s

t

s

t

s2

1

t

t

s2

=s

t

s

t=

Dimensionally:

(but see discussion of units below)iC

V tvC

d

d

=(37a)

The electric current through the resistor and capacitor, according to the Reciprocal System, is

a. DC RC Series Circuit (Fig. 1(a))


and, using Mathcad's symbolic integration routine, we get:

t1

d t vCR C

V vC V

d 3288057500000000. ln V 1. vC C

V

(38d)

without the constant of integration. But we'll solve for vC first:

ln V vC RC

V t=

(38e)

vC V exp tV

R C

(38f)

But, at t = 0 the capacitor is uncharged, vC = 0, so we must prefix the exponential term with V, which iseffectively the constant of integration:

vC V V exp tV

R C

(38g)

which simplifies to

vC V 1 exp tV

R C

(38h)

However, we now have an issue: RC/V is not recognized as time in conventional physics, so we need to convertV/RC to natural units and then divide by the natural value of unit time expressed in seconds; this factor convRCV isgiven in the section on Reciprocal System constants. With this factor, Eq. (38h) becomes


vC V 1 exp tV

R C convRCV

(38i)

The current is then

iC

V tvC

d

d

= (39a)

iV

Rexp t

V

R C convRCV

(39b)

According to conventional physics, as given in Ref. [21], p. 245:

vC_legacy V 1 expt

R C

ilegacyV

Rexp

t

R C

The capacitor time constant in the Reciprocal System is, by inspection,

CR C

V convRCV

t2

s3

s

t

s2

t (therefore, correct dimensions) (40)


versus that for conventional theory:

C_legacy R C

Sample DC RC Circuit

Now let's plot the results for a specific example from Ref. [21], p. 255, problem 16.5. Here,

V 100 volts R 5000 ohms C 20 106

farads

The capacitor has no initial "charge." At t = 0, the switch is closed.


0 0.01 0.02 0.03 0.040

20

40

60

80

100

V 1 exp tV

R C convRCV

V 1 exp t1

R C

t

0 0.01 0.02 0.03 0.040

0.005

0.01

0.015

0.02

V

Rexp t

V

R C convRCV

V

Rexp t

1

R C

t

Reciprocal System curves arein red; conventional theorycurves are in blue.

Figure 2. vC and i Graphsfor DC RC Series Circuit


The conventional theory graphs actually look silly compared with those calculated from the Reciprocal System.Obviously, an experimental physicist or electrical engineer should very easily be able to confirm or disconfirm thispoint! Tuma, in Ref. [59], p. 223, says, in regards to RC circuits, "Theoretically, after an infinite time the current ibecomes zero. In practical cases, it is a matter of seconds." For the above circuit, it's a matter of milliseconds.Real capacitors perform much faster than the conventional physicists and electrical engineers expect them to!

CR C

V convRCV C 0.0049 sec

C_legacy R C C_legacy 0.1 sec

The instantaneous power supplied to the capacitor is defined as the product of the instantaneous voltage and current:

pC V 1 exp tV

R C convRCV

V

Rexp t

V

R C convRCV

(41)

The conventional expression is

pC_legacy V 1 exp t1

R C

V

Rexp t

1

R C


Here are the plots for the example problem:

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

V 1 exp tV

R C convRCV

V

Rexp t

V

R C convRCV

V 1 exp t1

R C

V

Rexp t

1

R C

t

Reciprocal System curve is in red;conventional theory curve is in blue.

Figure 3. pC

Graphs for DC RCSeries Circuit.


The energy can also be directly calculated from the integral, using Mathcad.

joulesEC 0.0049

(42b)EC .5 C convfaradtocm V convvolttoseccm2 convseccmtojoul IR

To get to joules for the Reciprocal System, we have to use Eq. (4b) of Ref. [52]. Here "conv" means conversionfactor. IR is the inter-regional ratio and is necessary to use here because we're dealing with the time region.

joulesEC_legacy 0.1

(see below for units)EC 0.001

For the sample problem

farads x volts2 = (sec2 x coul2 /(kg x m2)) x (kg x m2 / (sec2 x coul))2 = joulesEC_legacy .5 C V2

(which is the same expression as derived in the author's paper "Theory of the Capacitor," Ref. [52]).Conventional theory (Ref. [21], p. 246) gives

(42a)(but farads x volts is not recognized as energy; see below)EC .5 C V

The total energy supplied to the capacitor is the integral of the power from t = 0 to t = infinity. Carrying out theintegration (by hand), we obtain


0

tV 1 exp tV

R C convRCV

V

Rexp t

V

R C convRCV

d 4.9411533071847302248 10-3

joules

(For symbolic calculations, Mathcad uses double precision.)

This is 4.9% of the conventional value, and makes sense: we are storing uncharged, massless electrons, notcharged electrons!

The voltage across the resitor is

vR V exp tV

R C convRCV

(43)

The instantaous power supplied to the resistor is

pR vRV

Rexp t

V

R C convRCV

(44)

The energy delivered to the resistor from t = 0 to t = infinity is

ER

0

tvRV

Rexp t

V

R C convRCV

d (45)

elec_mag_suscep.mcd 59Here are the voltage and power plots for the resistor, both for the Reciprocal System and conventional theory:

0 0.01 0.02 0.03 0.040

20

40

60

80

100

V exp tV

R C convRCV

V exp t1

R C

t

Reciprocal System curves arein red; conventional theorycurves are in blue.

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

V exp tV

R C convRCV

V

Rexp t

V

R C convRCV

V exp t1

R C

V

Rexp t

1

R C

t

Figure 4. vR and pR Graphsfor DC RC Series Circuit


For the Reciprocal System, the total energy delivered to the resistor (or load) is

0

tV exp tV

R C convRCV

V

Rexp t

V

R C convRCV

d 0.0049 joules

For conventional theory, the total energy delivered to the resistor (or load) is

0

tV exp t1

R C

V

Rexp t

1

R C

d 0.1 joules

It's evident that an equal amount of energy goes to the capacitor and resistor according to both theories. The total energysupplied by the DC voltage source is

0

tVV

Rexp t

V

R C convRCV

d 0.0099 joules for the Reciprocal System

0

tVV

Rexp t

1

R C

d 0.2 joules for the conventional theory


i c0 expR

Lt

V

R=

(47c)

This is a first order, linear differential equation and its solution is:

DR

L

iV

L=

(47b)

or, letting D be the differential operator, d/dt

V R i Lti

d

d

=(47a)


(46b)(therefore, correct dimensions)

s

t

1

t3

s3

t

s2

t=s

t

s

t=

Dimensionally:

i1

LtvL

d=(46a)

Our derivation here is the same as that in Ref. [21], pp.242-244. The electric current through the resistor and inductor is

b. DC RL Series Circuit (Fig. 1(b))


(49)(therefore, correct dimensions)

t3

s3

t2

s3

tLL

R

The inductor time constant is, in both conventional theory and the Reciprocal System, by inspection,

vL V expR

Lt

=vL Lti

d

d

=(48)

The voltage across the inductor is then

(47e)iV

R1 exp

R

Lt

=oriV

Rexp

R

Lt

V

R=

Eq. (44c) becomes

(47d)c0V

R

which can be verified by substitution, and where c0 is a constant. At t = 0, i = 0, so


Sample DC RL Circuit


V 100 volts R 50 ohms L 10 henries

The switch is closed at t = 0.


0 0.2 0.4 0.6 0.80

20

40

60

80

100

V expR

Lt

tReciprocal System andconventional theory are inagreement here.

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

V

R1 exp

R

Lt

t

Figure 5. vL and i Graphsfor DC RL Series Circuit


LL

R L 0.2 sec

The instantaneous power supplied to the inductor is defined as the product of the instantaneous voltage and current:

pL V expR

Lt

V

R1 exp

R

Lt

(50)

Here is the plot for the example problem:

0 0.2 0.4 0.6 0.80

20

40

60

80

100

V expR

Lt

V

R1 exp

R

Lt

t

Reciprocal Systemand conventionaltheory are inagreement here.

Figure 6. pL Graphfor DC RL SeriesCircuit


Ref. [1], p. 220: "It follows that m (mass) and L (inductance) are equivalent....Just as inertia resists any change in speed orvelocity, inductance resists any change in the electric current."

Ref. [2], first ed., pp. 76-77: "An equivalent mass L, moving with a velocity I must have a kinetic energy of .5 LI2 and wefind experimentally when a current I flowing in an inductance L is destroyed an amount of energy .5 LI2 does make itsapperance. The explanation on the basis of existing theory is that this energy is 'stored in the electromagnetic field', butthe dimensional clarification shows that it is actually the kinetic energy of the moving electrons."

Here, the Reciprocal System agrees numerically with the conventional theory. But the interpretation of "where" thisenergy is stored is very different. Quoting Larson from two of his books:

joules

0

tV expR

Lt

V

R1 exp

R

Lt

d 20

We can also use the integral:

(52)joulesEL 20EL .5 L I2

ampsI 2IV

R

For the sample problem

where I is the steady current.

(51)EL .5 L I

2

The total energy supplied to the inductor is the integral of the power from t = 0 to t = infinity. Carrying out theintegration (by hand), we obtain


The voltage across the resitor is

vR V 1 expR

Lt

(53)

The instantaneous power supplied to the resistor is

pR vRV

R1 exp

R

Lt

(54)

Here's the plot:

0 1 20

200

V 1 expR

Lt

V

R1 exp

R

Lt

t

Figure 7. pR Graph forDC RL Series Circuit

The power to the resistor gets to its constant value, 200 watts, in approx. 1 second. A larger value of L wouldcause a longer delay.

elec_mag_suscep.mcd 68c. DC RLC Circuit (Fig. 1(c))

For this circuit we will have to use natural units from the start. The electric current through the resistor, inductor, andcapacitor, according to the Reciprocal System, is

(55)i

iu

C

Cu

V

Vu

t

vC

Vu

d

d

= (t should be divided by tu here, but for convenience that willbe done later)


(56a)V

Vu

C

Cu

V

Vu

t

vC

Vu

d

d

R

Ru

L

Lu

C

Cu

V

Vu

t t

vC

Vu

d

d

d

d

vC

Vu=

L

Lu

C

Cu

V

Vu

2t

vC

Vu

d

d

2

C

Cu

R

Ru

V

Vu

t

vC

Vu

d

d

vC

Vu

V

Vu 0=

or

2t

vC

Vu

d

d

2

R

Ru

L

Lu

t

vC

Vu

d

d

V

Vu

vC

Vu

L

Lu

C

Cu

V2

Vu2

L

Lu

C

Cu

0= (56b)


Let's simplify this equation before continuing, using k0, k1, etc.

k0

R

Ru

L

Lu

k1

V

Vu

L

Lu

C

Cu

k2

V2

Vu

L

Lu

C

Cu

2t

vCd

d

2k0

tvC

d

d

k1 vC k2 0= (56c)

This is a second order differential equation, and we'll use Laplace Transforms to solve it; see Ref. [61] for a fineexposition of this method. Using Laplace Transforms on Eq. (51b) and assuming vC(0) = 0 and d/dt(vC(0) = 0 weobtain

s2

vC s( ) k0 s vC s( ) k1 vC s( )k2

s 0= (56d)

Solving for VC(s) we get

vC s( )k2

s s2

k0 s k1 = (56e)


(57)i iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

The current is then (after carrying out the differentiation in Eq. (55))

(The constants k5 and k6 are in units of 1 divided by natural units of time, so that's why we need to divide by tu.)

(56f)vC k2 k3 exp k5t

tu

k11 cos k6t

tu

k10 sin k6t

tu

k13 k5 k10 k11 k6k12 k5 k11 k10 k6

k11 k8 k9k10

k3 k7 k0

k4k9

k3 k02

k4k8

4

k4

k7 4 k1 k02

1

2k6

1

24 k1 k0

2

1

2k5

1

2k0k4 4 k1 k0

2k3

1

k1

Taking the inverse Laplace Transform of Eq. (56e) and carrying out a great deal of algebra with some additionalconstants we finally get


Now we have to differentiate Eq. (57) and multiply by L to get the value of the voltage to the inductor.

vL Lt

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

d

d

(58)

But the algebra is too messy to display symbolically, so we'll just graph it for a sample problem.

The voltage across the resistor is simply

vR R iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

(59)

Obviously, we should obtain

V vR vL vC=


k13 k5 k10 k11 k6k12 k5 k11 k10 k6

k11 k8 k9k10

k3 k7 k0

k4k9

k3 k02

k4k8

4

k4

k7 4 k1 k02

1

2k6

1

24 k1 k0

2

1

2k5

1

2k0k4 4 k1 k0

2k3

1

k1

k2

V2

Vu

L

Lu

C

Cu

k1

V

Vu

L

Lu

C

Cu

k0

R

Ru

L

Lu

The switch is closed at t = 0.

faradsC 50 106

henriesL .1ohmsR 50voltsV 100


Sample DC RLC Circuit


Here is the plot of vC:

0 0.002 0.004 0.006 0.008100

0

100

200

k2 k3 exp k5t

tu

k11 cos k6t

tu

k10 sin k6t

tu

t

(The reference doesn't provide an equation for the voltage.)

Figure 8. vC Graph for DCRLC Series Circuit


Here are the plots for current for both the Reciprocal System and the equation from conventional theory given in the abovereference:

0 0.005 0.01 0.015

1

0

1

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

exp 250 t( ) 2.7 sin 371 t( )

t

Reciprocal System curve is in red;conventional theory curve is in blue. Figure 9. i Graphs for DC RLC Series

Circuit

The Reciprocal System current is lower and more oscillatory.


Now here's the plot for vL.:

0 0.002 0.004 0.006100

50

0

50

100

Lt

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

d

d

t

Note: The differentiation is numerical, so the graph is not as smooth as it would be if we used the symbolic expression.

Figure 10. vL for RLC Series Circuit


And here's the plot for the voltage across the resistor:

0 0.002 0.004 0.006

20

10

0

10

20

R iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

t

Figure 11. vR for RLC Series Circuit


The sum of the voltages across the resistor, inductor, and capacitor should equal V:

0 0.002 0.004 0.006

0

50

100

150

k2 k3 exp k5t

tu

k11 cos k6t

tu

k10 sin k6t

tu

Lt

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

d

d

R iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

t

Figure 12. vR+vL+vC for DC RLC Series Circuit

As stated above, the plot is slightly ragged due to the numerical differentiation, but that's understandable--the derivativeof i is too messy to give explicitly here, but we have computed it. Obviously, the sum of the voltages = V, as required.The reference for this problem does not give the individual voltages or their sum.


Now we'll plot the instanteous power supplied to the resistor, inductor, and capacitor.

pR vR i (60)

pL vL i (61)

pC vC i (62)

(The graph has the equations expanded out. Because of the length of the equations, the graph spills over to another page.)


R iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin

exp k5t

tu

k11 sin k6t

tu

k6 k10

Lt

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

d

d iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin

exp k5t

tu

k11 sin k6t

tu

k6 k10

k2 k3 exp k5t

tu

k11 cos k6t

tu

k10 sin k6t

tu

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6 k10 cos k6t

tu

k6

Figure 13. pR, pL, pC Graphs for DC RLC Series Circuit

Obviously, the voltage and power to the inductor and resistor go to zero; the capacitor then stays at the voltage of thesource, and there is no further current in the circuit. The energy stored in the capacitor is then


EC

0

tk2 k3 exp k5t

tu

k11 cos k6t

tu

k10 sin k6t

tu

iu

C

Cu

V k2 k5 exp k5

t

tu

k11 cos k6t

tu

k10 sin k6t

tu

exp k5t

tu

k11 sin k6t

tu

k6

k10 cos k6t

tu

k6

d

EC 0.0123 joules (63)

Using our alternative equation:

.5 V C convCV 0.0124 joules

Very close!


(67)0

t

tv i

d

0

t

tvR i

d

0

t

tvC i

d=

Energy conservation then requires that

(66)

v i vR i vC i=

The current is the same around the circuit, so the instantaenous power is

(65)v vR vC=


We cannot simply put Eq. (64b) in Eq. (37a) because a singularity would result. Also, we cannot use the conventionalexpressions for "impedance" and "reactance" for the capacitor, because the dimensions are wrong! So we will use acompletely new method which depends on our knowledge of the energy stored in the capacitor.

(64b)v Vmax sin t

By starting the source at time t = 0 (when the switch is closed), we can set ac = 0, and so Eq. (64a) can be simplified to

where ac is the phase angle and can take on values from 0 to 2 rad/sec.

(64a)v Vmax sin t ac

The voltage from the source is

d. AC RC Circuit (Fig. 1(d))


For an alternating electric voltage we define the time period of a cycle in terms of the angular frequency:

T2

f

1

T (68)

(If f = 60 cycles per second, T = 1/60 = seconds, so = 120 x rad/sec.)

It's best to work with quarter cycles here to avoid difficulties with the integration. The other quarters could be analyzed in asimilar fashion (except that at the beginning of the second quarter there would be an initial capacitor "charge", etc).

0

.25 T

tVmax sin t ieff

d

0

.25 T

tvR ieff

d

0

.25 T

tvC ieff

d= (69)

where ieff is the equivalent effective (or measured) constant current. We already know the value of the energy term for thecapacitor:

EC C vC_avg convCV joules (70)

The average value of the source voltage over one-quarter cycle is

vavg1

.25 T0

.25 T

tVmax sin t

d volts (71a)


vavg

4 Vmax

T 1 cos .25 T volts (71b)

The average voltage across the resistor is, of course,

vR_avg R ieff volts (72)

Therefore,

vC_avg vavg vR_avg (73)

Eq. (69) now becomes

0

.25 T

tVmax sin t ieff

d

0

.25 T

tR ieff2

d C4 Vmax

T 1 cos .25 T R ieff

convCV=

(74)

Using Mathcad's variable solver we obtain the equation for ieff:


voltsvR_avg 3.9321vR_avg ieff R

voltsvavg 159.1549vavg

4 Vmax

T 1 cos .25 T

ampsieff 0.0393ieff 4 CconvCV

T

secT 0.0126T2

ac 0rad/sec 500voltsVmax 250

The capacitor has no initial "charge." At t = 0, ac = 0, the switch is closed.

faradsC 25 106

ohmsR 100voltsV 250 sin 500 t 0( )

Now let's compute the results for a specific example from Ref. [21], p. 259, problem 16.14. Here,

Sample AC RC Circuit

(75)(discarding the solution in which there would be no vC_avg)ampsieff 4. CconvCV

T


vC_avg vavg vR_avg vC_avg 155.2229 volts

vavg ieff .25 T 0.0197 joules

vR_avg ieff .25 T 0.0005 joules

EC C vC_avg convCV EC 0.0192 joules vavg ieff .25 T vR_avg ieff .25 T 0.0192

So the energies sum correctly. Clearly, most of the energy contributed by the source is stored in thecapacitor and not wasted in the resistor.

According to conventional theory, as given in the reference above, the instantaneous current is:

ilegacy expt

R C

Vmax

Rsin ac

Vmax

R2 1

C

2

sin ac atan1

C R

Vmax

R2 1

C

2

sin t ac atan1

C R

amps

Plotting this over one-quarter cycle:


0 0.001 0.002 0.0032

1

0

1

2

expt

R C

Vmax

Rsin ac

Vmax

R2 1

C

2

sin ac atan1

C R

Vmax

R2 1

C

2

sin t ac atan1

C R

t

Figure 14. ilegacyilegacy_max

Vmax

R2 1

C

2

ilegacy_max 1.9522 amps

According to Ref. [59], p. 207, or any electrical engineering handbook, the conventional value for ieff is

ieff_legacy

ilegacy_max

2 ieff_legacy 1.3804 amps


The conventional expression for the voltage across the capacitor for one-quarter cycle is

vC_legacy1

C

0

.25 T

texpt

R C

Vmax

Rsin ac

Vmax

R2 1

C

2

sin ac atan1

C R

Vmax

R2 1

C

2

sin t ac atan1

C R

d

vC_legacy 132.2695 volts

The conventional expression for energy stored at the end of the one-quarter cycle is then

EC_legacy1

2C vC_legacy

2 EC_legacy 0.2187 'joules'

The ratio of the results from the two theories is

EC

EC_legacy0.0877

So, for this example, the capacitor in the Reciprocal System stores 8.77 % of the energy of that of thecapacitor in conventional theory. The ratio of effective currents is


ieff

ieff_legacy0.0285

Quite an amazing difference. This is understandable,however, when one considers that conventional theorysays that charges are being stored, whereas the Reciprocal System says "No, it's uncharged electrons." Also,to be quite honest, we don't see how or why there would be a current of 1.38 amps through the capacitor! It's adielectric, after all!

But what is the instantaneous current? To derive an expression for this, we must return to Eq. (74) and replace thelimits of the integral with ti and tf. so as to be able to calculate i step-by-step or discretely over small time intervals.We will also assume that the average voltage for the resistor is Ri/2.

ti

tf

tVmax sin t i

dti

tf

tR i2

d CVmax

tf ti 1 cos tf ti R

i

2

convCV= (76)

i1

4 R

2 cos ti Vmax 2 cos tf Vmax C convCV R 4 cos ti 2 Vmax2

8 cos ti Vmax2

cos tf

(77)

Putting this into a function so as to make it easy to graph:

Putting this into a function:

iRC R C Vmax convCV ti tf 1

4 R

2 cos ti Vmax 2 cos tf Vmax C convCV R 4 cos ti 2 Vmax


Let ti = t and tf = ti + .0001. Here's the plot:

0 5 104

0.001 0.0015 0.002 0.0025 0.003

0

0.05

0.1

iRC R C Vmax convCV t t .0001

t

Figure 15. iRC for AC RC Circuit

The calculated effective value of the current seems reasonable based on this graph:ieff 0.0393 amps

Note how iRC decreases with time, unlike ilegacy, and is more in keeping with the DC RC graph.

Knowing iRC we can compute the voltage across the capacitor and compare with the voltage from the source and theresistor.


0 5 104

0.001 0.0015 0.002 0.0025 0.003

0

50

100

150

200

250

Vmax sin t

iRC R C Vmax convCV t t .0001 R

Vmax sin t iRC R C Vmax convCV t t .0001 R

t

Figure 16. v, vR, vC for AC RC Circuit

Also, knowing iRC enables us to plot the instantaneous power from the source and delivered to the resistor and capacitor.


0 5 104

0.001 0.0015

0

2

4

6

8

Vmax sin t iRC R C Vmax convCV t t .0001

iRC R C Vmax convCV t t .0001 2 R

Vmax sin t iRC R C Vmax convCV t t .0001 R iRC R C Vmax convCV t t .0001

t

Figure 17. p, pR, pC for AC RC Circuit

Clearly, a good approximation for the vC curve is the source voltage sine wave with a slightly lower value of amplitudeBecause there is no inductor, there is no inertia in this circuit. At this time, we do not have a more precise equation for vC.


e. AC RL Circuit (Fig. 1(e))

Here the Reciprocal System and conventional theory are in agreement. Therefore we can use the treatment give in Ref.[21], pp. 249-250. Applying Kirchhoff's voltage law around the circuit:

Vmax sin t ac R i Lti

d

d

= (78)

In Reciprocal System space-time terms, the dimensions are

t

s2

t2

s3

s

t

t3

s3

s

t

1

t= so each term has the dimensions of voltage

The complementary solution is

ic c0 expR

Lt

(79)

The particular solution is

ip expR

Lt

texpR

Lt

Vmax

L sin t ac

d (80a)

orip

Vmax

R2

2

L2

sin t ac atan L

R

(80b)


The complete solution is then

i c0 expR

Lt

Vmax

R2

2

L2

sin t ac atan L

R

(81)

As is well known, the inductance prevents any sudden change in the current. Before the switch was closed, the currentwas zero, so i0 = 0. Then at t = 0

(82a)i0 0=

i0 c0 expR

L0

Vmax

R2

2

L2

sin 0 ac atan L

R

= (82b)

Solving for c0:

c0

Vmax

R2

2

L2

sin ac atan L

R

(83)

Substituting this expression for c0 into Eq. (81):

i expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

(84)


With this expression for i, we know the voltages for the resistor and inductor:

vR R expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

(85)

vL Lt

expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

d

d

(86a)

orvL L

R

Lexp

R

Lt

Vmax

R2

2

L2

1

2

sin ac atan L

R

Vmax

R2

2

L2

1

2

cos t ac atan L

R

(86b)

The instantaneous power to the circuit is

p Vmax sin t ac expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

(87)


pR R expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

2

(88)

pL LR

Lexp

R

Lt

Vmax

R2

2

L2

1

2

sin ac atan L

R

Vmax

R2

2

L2

1

2

cos t ac atan L

R

expR

Lt

(89)

The energy over the first quarter cycle of the voltage is:

E

0

.25 T

Vmax sin t ac expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan

(90)

ER

0

.25 T

tR expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

2

d

(91)


EL

0

.25 T

LR

Lexp

R

Lt

Vmax

R2

2

L2

1

2

sin ac atan L

R

Vmax

R2

2

L2

1

2

cos t ac atan L

R

exp

(92)


Figure 18. i for AC LR Circuit

0 0.001 0.002 0.0030

0.5

1

1.5

expR

Lt

Vmax

R2

2L

2

sin ac atan L

R

Vmax

R2

2L

2

sin t ac atan L

R

t

Here are all the plots over the first quarter cycle:

secT 0.0126T2

Vmax 150

rad/sec 500ac 0henriesL .2ohmsR 50voltsv 150 sin 500 t ac

We'll use the problem (16.12) given in Ref. [21], pp. 258-259.

Sample AC RL Circuit


0 0.001 0.002 0.0030

50

100

150

Vmax sin t ac

R expR

Lt

Vmax

R2

2L

2

sin ac atan L

R

Vmax

R2

2L

2

sin t ac atan L

R

LR

Lexp

R

Lt

Vmax

R2

2L

2

1

2

sin ac atan L

R

Vmax

R2

2L

2

1

2

cos t ac atan L

R

t

Figure 19. v, vR, vL for AC LR Circuit


Vmax sin t ac expR

Lt

Vmax

R2

2L

2

sin ac atan L

R

Vmax

R2

2L

2

sin t ac atan L

R

R expR

Lt

Vmax

R2

2L

2

sin ac atan L

R

Vmax

R2

2L

2

sin t ac atan L

R

2

LR

Lexp

R

Lt

Vmax

R2

2L

2

1

2

sin ac atan L

R

Vmax

R2

2L

2

1

2

cos t ac atan L

R

expR

Lt

Vmax

R2

2L

2

sin ac atan

Figure 20. p, pR, pL for AC LR Circuit


E

0

.25 T

Vmax sin t ac expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan

E 0.1837 joules

ER

0

.25 T

tR expR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

2

d

ER 0.0521 joules

EL

0

.25 T

LR

Lexp

R

Lt

Vmax

R2

2

L2

1

2

sin ac atan L

R

Vmax

R2

2

L2

1

2

cos t ac atan L

R

exp

EL 0.1316 joules

ER EL 0.1837 in agreement with E


f. AC RLC Circuit (Fig. 1(f))

The series RLC circuit shown in Fig. 1(f) has a sinusoidal voltage applied when the switch is closed. The resultingequation, according to conventional theory is

Vmax sin t ac R i Lti

d

d

1

Cti

d=

However, as pointed out previously, the capacitor term is actually dimensionless, not a voltage, and thus is wrong.Therefore we will need to replace the capacitor term.

(93)Vmax sin t ac R i L

ti

d

d

vC=

The instantaneous power is

Vmax sin t ac i R i2

Lti

d

d

i vC i= (94)

Integrating over the first quarter of the voltage cycle gives us the energy for that period:

0

.25 T

tVmax sin t ac i

d0

.25 T

tR i2

d

0

.25 T

tLti

d

d

i

d

0

.25 T

tvC i

d= (95a)


It would appear, at least at first glance, that

1

s

ts

t

s2

(the correct dimensions for voltage)vC

conviC

iavg C

Multiplying by the instantaneous value of the current, i, gives us the power.

0

.25 T

tVmax sin t ac i

d0

.25 T

tR i2

d

0

.25 T

tLti

d

d

i

dconviC

C

0

.25 T

ti

iavg

d= (95b)

We can now replace the capacitor term with EC, which we do know for sure because we can compute the averagevoltage..

0

.25 T

tVmax sin t ac i

d0

.25 T

tR i2

d

0

.25 T

tLti

d

d

i

d EC= (95c)

(This also gets rid of our uncertainty in the dynamic value of vC.) For simplicity, we'll assume ac = 0 and we will also nowassume an effective constant current, ieff. With this, it's easy to transform the remaining terms to the correct energyexpressions.

Vmax

1 cos .25 T ieff R ieff

2 .25 T .5 L ieff

2 EC= (95d)


Solving for ieff , we obtain:

ieff1

2 R T 2 L 4 Vmax cos

1

4T

4 Vmax 4 Vmax2

cos1

4T

2

2 Vmax2

cos1

4T

Vmax2

(96)We already know vavg:

vavg

4 Vmax

T 1 cos .25 T

We know vR_avg:

vR_avg R ieff

vL_avg is:

(97)vL_avg L

2 ieff

T

And so:

vC_avg vavg vR_avg vL_avg (98)

EC C VC_avg convCV


EC C vavg vR_avg vL_avg convCV(99a)

EC C4 Vmax

T 1 cos .25 T R ieff L

2 ieff

T

convCV (99b)

Putting this expression for EC in Eq. 96, we get:

ieff1

2 R T 2 L 4 Vmax cos

1

4T

4 Vmax 4 Vmax2

cos1

4T

2

2 Vmax2

cos1

4T

Vmax2

=

(99c)

Solving ieff, we get:

ieff 4. CconvCV

T (99d)

which is, interestingly enough, the same as Eq. (75), and shows that ieff is not directly influenced by L.


voltsvavg 63.662vavg

4 Vmax

T 1 cos .25 T

joulesEC 0.1447EC C vC_avg convCV

voltsvC_avg 58.5669vC_avg

4 Vmax


2 ieff

T

ampsieff 0.3932ieff 4. CconvCV

T

First we'll find ieff and then the voltages and energies and powers:

secT 0.0251T2

faradsC 500 10

6henriesL .1ohmsR 5

rad/sec 250ac 0voltsVmax 100voltsv 100 sin 250 t ac

We'll use the problem (16.42) given in Ref. [21], p 264.

Sample AC RLC Circuit


wattspR_avg 0.7731pR_avg vR_avg ieff

wattspavg 25.0322pavg vavg ieff

The average power of this circuit is

The voltages and the energies sum correctly.

ER EL EC 0.1573

E 0.1573E vavg ieff .25 T

joules

ER 0.0049ER R ieff2

.25 T

joules

EL 0.0077EL .5 L ieff2

vR_avg vL_avg vC_avg 63.662

vL_avg 3.129vL_avg

L 2 ieff

T

voltsvR_avg 1.966vR_avg R ieff


(103)vC_avg vavg vR_avg vL_avg

(102)(approximately)vL_avgL i

t

(101)(approximately)vR_avg Ri

2

(100)vavg

Vmax

t1 cos t

0

.25 T

tVmax sin t i

d0

.25 T

tR i2

d

0

.25 T

tLti

d

d

i

d EC=

But what is the instantaneous current? To compute that we will first need to compute the instantaneous averagevalues of voltage.

The powers sum correctly, of course.

wattspR_avg pL_avg pC_avg 25.0322

wattspC_avg 23.0288pC_avg vC_avg ieff

wattspL_avg 1.2303pL_avg vL_avg ieff


0

t

tVmax sin t i

d0

t

tR i2

d

0

t

tLti

d

d

i

d C convCVVmax

t1 cos t R

i

2 L

i

t

=

iRLC Vmax R L C t 1

4 t R t2 Vmax t 2 C convCV L 2 cos t Vmax t C convCV R t 4 Vmax

2t2

(104)

0 0.001 0.002 0.003 0.004 0.005 0.0060

0.5

1

1.5

iRLC Vmax R L C t exp 25 t( ) 5.42 cos 139 t( ) 1.89 sin 139 t( )( )

5.65 sin 250 t 73.6 deg( )

t

Reciprocal System curve (approximate) isin red; conventional theory curve is in blue.

Figure 21. iRLC vs. conventional current


Clearly, there is a huge difference in the calculation of RLC current between the two theories. Any experimental physicistor electrical engineer should have no trouble proving or disproving the Reciprocal System on this point. Our value for ieff

also seems to be correct based on the above graph. A precise expression for vC and thus i will have to await future work.

The six electrical circuits discussed in this section represent just a tiny fraction of the circuits in use; see Ref. [62] for acatalog of over 3600 circuits! The essense, here, is that whenever and wherever capacitors are used, the equationsmodelling them must change to conform to the new understanding provided by the Reciprocal System.


4. Dielectric Constants and Electric Susceptibilities

a. Elements

As quoted previously, from Ref. [7], Vol. 5, p. 342, "The terms 'permittivity' and 'refractive index' represent merelyalternative ways of attaching numerical values to one and the same property of a dielectric medium." The question is:Why? The answer from the Reciprocal System is simple: both photons and (uncharged, massless) electrons movethrough the atoms and not (merely) through the interstices. The increase in space and time, due to the rotationaldisplacements of the atoms, is the cause for both permittivity and refraction. The space-time factor is the same as thatused in the electrical resistivity calculations, Ref. [53], and that used above in the index of refraction calculations. Here'sthe equation, repeated (from the note) for convenience:

n 1 tr tr

dcgs

du_cgs.9317 kG

tp2

ts 2

3

te_mod2

1 kr01

9

11

0_nF

(105a)

For noble elements--those without rotational electric displacement--the equation is obvously modified to

n 1 tr tr

dcgs

du_cgs.9317 kG tp

2ts

2

31 kr0

1

9

11

0_nF

(105b)

(Of course, if the noble element has one or more gravitational charges--the cause for the different isotopes--thenthere would be an effective value of te_mod, and so Eq. (105a) would then be used.) Eq. (11c) gives the relativepermittivity, or dielectric constant, if losses are excluded and if magnetic permeability can be neglected,


r n2

(Ref. [52] uses the symbol K for this constant. Both symbols are common in the literature.) The electricsusceptibility is, by definition, then

e r 1(106)

Notice that our definition has nothing to do with "free" or "bound" charges as in conventional theory. The relativepermittivities or dielectric constants and electric susceptibilities are usually used in regard to capacitor materials,rather than with, say, Van De Graaf generators, where we are dealing with actual electric charges. Therefore, thecalculations in this section are specifically for use in describing ordinary dielectrics.

For static or low frequency calculations, we will set F/n to zero, otherwise we will assume that the to-and-frocycling of an electron in the dielectric is physically similar to the transverse oscillation of a photon and set n

accordingly. Conventional theory makes this same assumption tacitly, right or wrong. Regardless, most of thefrequency-dependent part of the dielectric constant is due to the loss term, which we are neglecting here.

Our method of attack will be to calculate the index of refraction and then the relative permittivity or dielectricconstant and compare with observed data from Ref. [14], which is probably the most authoritative of the materialreferences given at the end of this paper. We will also tabulate the electric susceptibility. For the elements, withthe exception of Te, we don't have data for the different crystalline directions, so there would be no point intabulating anisotropic values other than for Te. Differences due to physical state, solid or liquid, are not great,because the statistical average of the geometric coefficient, kG, for the liquid is close to that of the solid, as wouldbe expected. Gases are a separate issue, and will have to be left for later treatment. The following is the table ofcalculations, from Excel:


Table II. Dielectric Constants and Electrical Susceptibilities of the Elements (No Loss)

Correl = .9918

1.116 1.2459 1.225 0.2251.266 1.6027 1.048 0.048

2.276 5.1783 5.680 4.680

2.276 5.1783 5.910 4.910

1.16 1.345 1.450 0.450

1.22 1.4877 1.505 0.5051.324 1.752 1.518 0.518

3.311 10.963 11.700 10.700

3.311 10.963 12.000 11.000

1.459 2.1284 2.150 1.150

1.069 1.1418 1.516 0.516

4.192 17.57 16.000 15.0004.192 17.57 16.000 15.000

3.543 12.551 11.200 10.200

2.818 7.9396 8.500 7.500

4.751 22.572 24.000 23.000

2.083 4.3386 5.000 4.000

1.592 2.533 2.200 1.200

H (liquid) 1 2-1-(1) 2 1 -1 0.090 0.7071 2 0He (liquid, 3.15 K) 2 2-1-0 2 1 1 0.206 0.7071 2 0

C 4 2-2-(4) 2 2 -4 3.515 1.5580 0 0

C--1 MHz 4 2-2-(4) 2 2 -4 3.515 1.5580 0 1.41E-10

N (liquid, 74.8 K) 7 2-2-(3) 2 2 -4 0.970 0.7071 0 0

O (liquid, 81 K) 8 2-2-(2) 2 2 -4 1.334 0.7071 0 0F (liquid, 3.2 K) 9 2-2-(1) 2 2 -4 1.965 0.7071 0 0

Si 14 3-2-(4) 3 2 -3 2.329 1.5700 1 0

Si--1 MHz 14 3-2-(4) 3 2 -3 2.329 1.5700 1 1.41E-10

Cl (liquid, 213 K) 17 3-2-(1) 2 2 7 1.979 3.9200 2 0

Ar (liquid, 89 K) 18 3-2-0 2 2 8 1.672 0.7040 0 0

Ge 32 3-3-(4) 3 3 -4 5.323 1.4710 2 0Ge--1 MHz 32 3-3-(4) 3 3 -4 5.323 1.4710 2 1.41E-10

As--optical 33 3-3-(3) 3 3 -5 5.777 1.5660 2 0.0774

Se--3.3 cm 34 3-3-(2) 3 3 -3.5 4.836 0.7060 2 1.04E-06

Sn--1 MHz 50 4-3-(4) 4 3 -5 5.777 1.4840 1 1.14E-10

Te--par. to c 52 4-3-(2) 4 3 -9 6.273 1.4610 2 0

Te--per. to c 52 4-3-(2) 4 3 -9 6.273 0.6207 0 0

er_obsr_calcnF/0_nkr0kGd_cgste_modtstpRot. Dis.At. No.Element


Table Notes:

1) C, N, O, F all have values of te_mod = -4 apparently for the same reason that they have equal rotational speeds forthe determination of interatomic distance, 3-3-10 or 3-3-1: the additional equivalent electric time displacement unitsdo not change the result. Ref. [1], p.22.

2) The non-static calculations have neglible differences with the static calculations--but this is because we areneglecting the dielectic loss, which is the main frequency dependent term.

3) The only anisotropic calculation is for Te, where the ratio of the cube of the a-edge to the c-edge has been taken tomultiply the normal kG value to calculate the value perpendicular to the c-edge.

4) For the resistivity calculation (and for the calculation given in Table I above) of Te, we use -6 for te_mod, whereashere we have to use -9. But the empirical data for n given in Ref. [16] and Ref. [14] do not agree either! Also, thesame Ref. [14], p. 92, says that the dielectric constant of Si is 11.7, but on p. 830, it says that it's 11.0-12.0. Ourcalculation for Si, in Table II, gives 10.93.

5) The correlation between the calculated and observed values of r is very high, but obviously more work needs to bedone to explain the rather wide devations between te_mod and te; most of this can be explained as isotopicdifferences, but probably not all. Also, the densities used in the calculations may be somewhat different from theactual densities used in the experiments or observations.

6) Se can be used both as a conductor and as a dielectric: for the resistivity calculations given in Ref. [53] we usedthe electropositive form of Se's rotational displacement; here, for the dielectric calculation, we use theelectronegative form. This gives us confidence that we're on the right track.

7) On the other hand, we have found it necessary to use the electropositive form of the rotational displacement for Cland Ar here. Clearly, more theoretical work and experimental work are necessary. Further, the loss calculations aretoo uncertain, at this time, to be included here.


Zatno_O 8na_O 2kr_O 0.7778kr_O1

teff_O1 kr0_O

1

9

(same as in H2O)kr0_O 2

teff_O 1teff_O

te_mod_O

tp_O2

ts_O 1

3

2

te_mod_O 2ts_O 2tp_O 22-2-(2)O:

Zatno_Si 14na_Si 1kr_Si 0.3339kr_Si1

teff_Si1 kr0_Si

1

9

kr0_Si 2

teff_Si 2.3295teff_Si

te_mod_Si

tp_Si2

ts_Si 1

3

2

te_mod_Si 4ts_Si 2tp_Si 33-2-(4)Si:

SIO2

Chapter 4.4, "Dielectrics and Electrooptics," in Ref. [14] is, as stated above, probably the most authoritativereference for data on dielectric compounds, so that's what we'll use here for comparing our calculations withobservations of the static dielectric constants (but using the sodium D-line to determine n), as well as Ref. [16],pp.10-246 to 10-249 for the index of refraction. A sampling of calculations follows; the Reciprocal System DataBase will have the calculations for all appropriate compounds.

b. Compounds


e_SiO2 0.8369e_SiO2 r_SiO2 1r_SiO2 1.8369r_SiO2 nSiO22

nSiO2 1.3553nSiO2

dSiO2

du_cgs.9317 kG_SiO2 kr_SiO2 1

1

0_nF

1

F

0_n0.058kG_SiO2 0.2386kG_SiO2

Vuc_SiO2

edge_cSiO2 3 Zuc_SiO2

Aedge_cSiO2 5.4046Zuc_SiO2 3A3Vuc_SiO2 112.979

Now we come to the difficult issue of the calculation of kG. Ref. [14] apparently uses the c-edge of thehexagonal unit cell here, and there are 3 molecules for this kind of SiO2 per unit cell as determined by itsdensity.

(from the Reciprocal System Data Base, for tridymite, to match the value from Ref. [14])g/cm3dSiO2 2.648

kr_SiO2 0.5706kr_SiO2

na_Si Zatno_Si kr_Si na_O Zatno_O kr_O

na_Si Zatno_Si na_O Zatno_O


teff_F 4teff_F

te_mod_F

tp_F2

ts_F 1

3

2

te_mod_F 4ts_F 2tp_F 22-2-(1)F:

Zatno_Ca 20na_Ca 1kr_Ca 1.3355kr_Ca1

teff_Ca1 kr0_Ca

1

9

kr0_Ca 2

teff_Ca 0.5824teff_Ca

te_mod_Ca

tp_Ca2

ts_Ca 1

3

2

te_mod_Ca 2ts_Ca 2tp_Ca 33-2-2Ca:

CaF2

Ref. [14] says that the static r_SiO2 = 3.5 (for the principal 11 direction) and that nSiO2 (at .5461 m wavelength,not too different from .5893 m) is 1.46. But 1.462 = 2.1316. So even using the experimental value of nSiO2

doesn't get us to the value of r_SiO2 given in Ref. [14]. If we use the value of r_SiO2 = 3.5 from Ref. [14], thennSiO2 would equal 1.8708. As shown in the table below, n calculated for edge_a and edge_b is 1.473, close tothe observed; the average for the threee principle axes is 1.44, very close to that observed.


e_CaF2 1.6336e_CaF2 r_CaF2 1r_CaF2 2.6336r_CaF2 nCaF22

nCaF2 1.6228

nCaF2

dCaF2

du_cgs.9317 kG_CaF2 kr_CaF2 1

1

0_nF

1

kG_CaF2 0.25kG_CaF2

Vuc_CaF2

edgeCaF2 3 Zuc_CaF2

AedgeCaF2 5.4630Zuc_CaF2 4A3Vuc_CaF2 163.035

The crystal unit cell is cubic, with 5.460 A per edge. There are 4 molecules per cell, and the volume is 163.035 A3.

(from the Reciprocal System Data Base; this matches the value from Ref. [14])g/cm3dCaF2 3.179

kr_CaF2 0.795kr_CaF2

na_Ca Zatno_Ca kr_Ca na_F Zatno_F kr_F

na_Ca Zatno_Ca na_F Zatno_F

Zatno_F 9na_F 2kr_F 0.1944kr_F1

teff_F1 kr0_F

1

9

kr0_F 2


kr_MgO 0.7kr_MgO

na_Mg Zatno_Mg kr_Mg na_O Zatno_O kr_O

na_Mg Zatno_Mg na_O Zatno_O

Zatno_O 8na_O 1kr_O 0.25kr_O1

teff_O1 kr0_O

1

9

kr0_O 0

teff_O 4teff_O

te_mod_O

tp_O2

ts_O 1

3

2

te_mod_O 4ts_O 2tp_O 22-2-(2)O:

Zatno_Mg 12na_Mg 1kr_Mg 1kr_Mg1

teff_Mg1 kr0_Mg

1

9

kr0_Mg 0

teff_Mg 1teff_Mg

te_mod_Mg

tp_Mg2

ts_Mg 1

3

2

te_mod_Mg 2ts_Mg 2tp_Mg 2Mg: 2-2-2

MgO

Ref. [14] says the dielectric constant is 7.4, considerably higher than our calculation. But, both Ref. [14] and Ref.[16] say that the index of refraction is 1.4388, which is somewhat lower than with our calculation. So: noconclusion can be drawn here.


Ref. [14] says that r = 9.7, which is way higher than our calculation. But: it also says that the refractive index is1.7217, which is a bit higher than our calculation.

e_MgO 1.6155e_MgO r_MgO 1r_MgO 2.6155r_MgO nMgO2

nMgO 1.6172

nMgO

dMgO

du_cgs.9317 kG_MgO kr_MgO 1

1

0_nF

1

kG_MgO 0.25kG_MgO

Vuc_MgO

edgeMgO 3 Zuc_MgO

AedgeMgO 4.2130Zuc_MgO 4A3Vuc_MgO 74.778

MgO is cubic, with an edge lenth of 4.2130 A, and 4 molecules per cell.

(from the Reciprocal System Data Base)g/cm3dMgO 3.578


(from the Reciprocal System Data Base)g/cm3dKCl 1.986

kr_KCl 0.7631kr_KCl

na_K Zatno_K kr_K na_Cl Zatno_Cl kr_Cl

na_K Zatno_K na_Cl Zatno_Cl

Zatno_Cl 17na_Cl 1kr_Cl 0.7631kr_Cl1

teff_Cl1 kr0_Cl

1

9

kr0_Cl 0

teff_Cl 1.3104teff_Cl

te_mod_Cl

tp_Cl2

ts_Cl 1

3

2

te_mod_Cl 3ts_Cl 2tp_Cl 33-2-(1)Cl:

Zatno_K 19na_K 1kr_K 0.7631kr_K1

teff_K1 kr0_K

1

9

kr0_K 0

teff_K 1.3104teff_K

te_mod_K

tp_K2

ts_K 1

3

2

te_mod_K 3ts_K 2tp_K 3K: 3-2-1

KCl


KCl is cubic, with an edge lenth of 6.2929 A, and 4 molecules per cell.

Vuc_KCl 249.203 A3 Zuc_KCl 4 edgeKCl 6.2929 A

kG_KCl

Vuc_KCl

edgeKCl 3 Zuc_KCl kG_KCl 0.25

nKCl

dKCl

du_cgs.9317 kG_KCl kr_KCl 1

1

0_nF

1

nKCl 1.3735

r_KCl nKCl2

r_KCl 1.8865 e_KCl r_MgO 1 e_KCl 1.6155

Ref. [14] says that n is 1.4792 and r is 4.6. Again, not too close to our calculations. Tables, from Excel, follow for themajor binary compounds. Included are those with cubic, hexagonal, and tetragonal unit cells.


Compound Crystal Density Vuc Zuc tp_1 ts_1 te_mod_1 teff_1 kr0_1 kr_1 na_1 Zatno_1 tp_2 ts_2 te_mod_2 teff_2 kr0_2 kr_2SiO2 Hex. 2.648 112.979 3 3 2 -4 2.3295 2 0.3339 1 14 2 2 -2 1.0000 2 0.7778

CaF2 Cubic 3.179 163.035 4 3 2 2 0.5824 2 1.3355 1 20 2 2 -4 4.0000 2 0.1944

MgO Cubic 3.578 74.778 4 2 2 2 1.0000 0 1.0000 1 12 2 2 -4 4.0000 0 0.2500

KCl Cubic 1.986 249.203 4 3 2 3 1.3104 0 0.7631 1 19 3 2 -3 1.3104 0 0.7631

NaCl Cubic 2.162 179.425 4 2 2 1 0.2500 2 3.1111 1 11 3 2 -4 2.3295 2 0.3339SrF2 Cubic 4.288 194.507 4 3 3 3 1.0000 2 0.7778 1 38 2 2 -4 4.0000 2 0.1944

CdTe Hex. 5.889 135.290 2 4 3 6.5 3.1989 2 0.2431 1 48 4 3 -4 1.2114 2 0.6420

GaAs Cubic 5.314 180.717 4 3 3 2 0.4444 2 1.7500 1 31 3 3 -2 0.4444 2 1.7500

ZnSe Cubic 5.253 182.477 4 3 3 -6 4.0000 2 0.1944 1 30 3 3 -2 0.4444 2 1.7500

ZnTe Cubic 5.643 227.093 4 3 3 -3 1.0000 2 0.7778 1 30 4 3 -3 0.6814 2 1.1414

GaP Cubic 4.137 161.611 4 3 3 2 0.4444 2 1.7500 1 31 3 2 -1 0.1456 2 5.3420BeO Hex. 3.007 27.611 2 2 1 1 0.3969 0 2.5198 1 4 2 2 -4 4.0000 0 0.2500

CdS Hex. 4.820 99.503 2 4 3 3 0.6814 2 1.1414 1 48 3 2 -2 0.5824 2 1.3355

SiC Cubic 3.239 82.199 4 3 2 -2 0.5824 0 1.7171 1 14 2 2 -2 1.0000 0 1.0000

ZnS Cubic 4.089 158.279 4 3 3 -3 1.0000 0 1.0000 1 30 3 2 -2 0.5824 0 1.7171

ZnO Hex. 5.736 47.114 2 3 3 -6 4.0000 2 0.1944 1 30 2 2 -2 1.0000 2 0.7778

GaN Hex. 6.144 45.242 2 3 3 -5 2.7778 2 0.2800 1 31 2 2 -4 4.0000 2 0.1944TiO2 Tetrag. 4.247 62.423 2 3 2 4 2.3295 2 0.3339 1 22 2 2 -4 4.0000 2 0.1944

TeO2 Tetrag. 6.125 86.499 2 4 3 -7 3.7100 2 0.2096 1 52 2 2 -4 4.0000 2 0.1944


CompoundSiO2

CaF2

MgO

KCl

NaClSrF2

CdTe

GaAs

ZnSe

ZnTe

GaPBeO

CdS

SiC

ZnS

ZnO

GaNTiO2

TeO2

na_2 Zatno_2 kr edge_a edge_b edge_c kG_a kG_b kG_c n_obs n_a n_b n_c2 8 0.5706 4.9130 4.9130 5.4046 0.3176 0.3176 0.2386 1.4601 1.4730 1.4730 1.3553

2 9 0.7950 5.4630 5.4630 5.4630 0.2500 0.2500 0.2500 1.4338 1.6228 1.6228 1.6228

1 8 0.7000 4.2130 4.2130 4.2130 0.2500 0.2500 0.2500 1.7217 1.6172 1.6172 1.6172

1 17 0.7631 6.2929 6.2929 6.2929 0.2500 0.2500 0.2500 1.4792 1.3735 1.3735 1.3735

1 17 1.4249 5.6402 5.6402 5.6402 0.2500 0.2500 0.2500 1.5313 1.7592 1.7592 1.75922 9 0.5903 5.7940 5.7940 5.7940 0.2500 0.2500 0.2500 1.4380 1.6238 1.6238 1.6238

1 52 0.4506 4.5700 4.5700 7.4800 0.7087 0.7087 0.1616 2.6930 2.8538 2.8538 1.4228

1 33 1.7500 5.6537 5.6537 5.6537 0.2500 0.2500 0.2500 3.5072 3.2917 3.2917 3.2917

1 34 1.0208 5.6720 5.6720 5.6729 0.2500 0.2500 0.2499 2.4800 2.3215 2.3215 2.3209

1 52 1.0084 6.1010 6.1010 6.1010 0.2500 0.2500 0.2500 2.6900 2.4023 2.4023 2.4023

1 15 2.9213 5.4470 5.4470 5.4470 0.2500 0.2500 0.2500 3.5072 3.9783 3.9783 3.97831 8 1.0066 2.6980 2.6980 4.3800 0.7030 0.7030 0.1643 1.7055 3.0974 3.0974 1.4902

1 16 1.1899 4.1364 4.1364 6.7152 0.7030 0.7030 0.1643 2.2120 4.9744 4.9744 1.9289

1 6 1.5019 4.3480 4.3480 4.3480 0.2500 0.2500 0.2500 2.5830 2.1989 2.1989 2.1989

1 16 1.2494 5.4093 5.4093 5.4093 0.2500 0.2500 0.2500 2.2130 2.2590 2.2590 2.2590

1 8 0.3173 3.2420 3.2420 5.1760 0.6913 0.6913 0.1699 1.9390 2.2401 2.2401 1.3047

1 7 0.2642 3.1800 3.1800 5.1660 0.7034 0.7034 0.1641 2.3300 2.1257 2.1257 1.26262 8 0.2752 4.5937 4.5937 2.9581 0.3220 0.3220 1.2058 2.4851 1.3709 1.3709 2.3890

2 8 0.2061 4.7900 4.7900 3.7700 0.3935 0.3935 0.8072 2.2005 1.4896 1.4896 2.0042


(r_a) (r_b) (r_c)

CompoundSiO2

CaF2

MgO

KCl

NaClSrF2

CdTe

GaAs

ZnSe

ZnTe

GaPBeO

CdS

SiC

ZnS

ZnO

GaNTiO2

TeO2

er_a er_b er_c xe_a xe_b xe_c2.1697 2.1697 1.8369 1.1697 1.1697 0.8369

2.6335 2.6335 2.6335 1.6335 1.6335 1.6335

2.6154 2.6154 2.6154 1.6154 1.6154 1.6154

1.8865 1.8865 1.8865 0.8865 0.8865 0.8865

3.0948 3.0948 3.0948 2.0948 2.0948 2.09482.6366 2.6366 2.6366 1.6366 1.6366 1.6366

8.1439 8.1439 2.0242 7.1439 7.1439 1.0242

10.8354 10.8354 10.8354 9.8354 9.8354 9.8354

5.3893 5.3893 5.3864 4.3893 4.3893 4.3864

5.7709 5.7709 5.7709 4.7709 4.7709 4.7709

15.8266 15.8266 15.8266 14.8266 14.8266 14.82669.5940 9.5940 2.2207 8.5940 8.5940 1.2207

24.7444 24.7444 3.7206 23.7444 23.7444 2.7206

4.8350 4.8350 4.8350 3.8350 3.8350 3.8350

5.1031 5.1031 5.1031 4.1031 4.1031 4.1031

5.0180 5.0180 1.7023 4.0180 4.0180 0.7023

4.5188 4.5188 1.5941 3.5188 3.5188 0.59411.8794 1.8794 5.7075 0.8794 0.8794 4.7075

2.2190 2.2190 4.0170 1.2190 1.2190 3.0170

Table III. Dielectric Constants and Electrical Susceptibilities of Selected Compounds (No Loss)


The average of the ratios of observed to calculated values of the index of refraction for the c_edge and a_edge is

.9127 1.0780

20.9954

which seems to verify our calculations. The same value of F/0 is used in each calculation (.058), and in most cases thevalue of kr_0 for both elements of the compound is 2. It's important to keep in mind that we are using the edges of the unit cellto calculate the principal values of the geometric factor kG; differences between observed and calculated values of the index ofrefraction and, hence, the dielectric constants and electric susceptibilities, may be due to slight inaccuracies in these lengths.

The experimental values of the dielectric constants are much more uncertain than those for the index of refraction. A table ofobserved values follow, from Excel, with the calculation of the effective exponents of the indexes of refraction to get thedielectric constants.

(r_obs)

Compound n_obs er_obs effec. Exp.SiO2 1.4601 3.50 3.3098

CaF2 1.4338 7.40 5.5546

MgO 1.7217 9.70 4.1820

KCl 1.4792 4.60 3.8980

NaCl 1.5313 5.90 4.1654SrF2 1.4380 7.69 5.6157

CdTe 2.6930 11.00 2.4205

GaAs 3.5072 12.95 2.0410

ZnSe 2.4800 9.12 2.4337

ZnTe 2.6900 10.10 2.3370

GaP 3.5072 11.10 1.9182BeO 1.7055 6.82 3.5962

CdS 2.2120 8.67 2.7206

SiC 2.5830 6.65 1.9965

ZnS 2.2130 3.22 1.4710

ZnO 1.9390 8.33 3.2014

GaN 2.3300 9.50 2.6615TiO2 2.4851 85.00 4.8804

TeO2 2.2005 22.70 3.9590

3.2822

Table IV. Calculation of Effective Exponent Relating n and r


It's curious that the average effective experimental exponent comes out to 3.2822, when it should be precisely 2! Of course,some of the discrepancy may be due to the use of different principal directions, but still.... Again, more work is needed here.

c. Dielectric Strength

If, due to very high voltages, the dielectric breaks down, what seems to happen is this: the ("free") electrons becomenegatively charged and the atoms become positively charged. The electrons repel themselves to the surface of thedielectric, and the positively-charged atoms force themselves apart, rupturing the solid. Of course, this simple answer fromthe Reciprocal System cannot be used in conventional theory, because there the electrons and atoms are already charged!(Note: Some of the surface atoms become neutralized, again, from the charged electrons coming from within the solid, andalso some of the newly-charged electrons leave the surface.)

Unfortunately, there do not exist experimental values of dielectric strength for most of the elements and compounds discussedabove. However, Ref. [16], p. 15-44, has a table which includes NaCl, so we'll use this compound for a tentative, samplecalculation.

From Eqs. (9a) and (9b) of Ref. [48] the energy required to inonize an atom-electron pair is

EI_atom_elec 2 hR

2

c

vmag or EI_atom_elec 2 h

R

2

c

velec1 eV/atom-elec.

pair(107)

(where R/2 is the electric rotational vibrational frequency of the charged electron; the factor 2 is due to the 2 charges created).For NaCl, the Na atoms become positively-charged, and the single "free" electron (not bound in the Reciprocal System!)becomes negatively-charged. Nothing happens to the Cl atoms, because they (ordinarily) cannot take a positive charge. ForNa, c/velec = 2 (because te = 1), so


(110)kJ/moleUionic_Na_mole 174.3193Uionic_Na_mole Uionic_Na_pairAv

2

1

1000

For one gram-mole, there is one-half of Avogadro's number of pairs of atoms,

(109)JUionic_Na_pair 5.7867 1019

Uionic_Na_pair

Q1_C Q2_C

4 0_SI sNa

(108)Q2_C Qu_coulQ1_C Qu_coul

The Coulombic potential energy of two positively-charged atoms is (from Ref. [64], p. 551)

msNa 3.9882 1010

According to the Reciprocal System Data Base, the separation of the Na atoms (at room temperature) is

(Higher values of work function, and thus of ionization energy, are possible, though less probable.) Ref. [63], p. 537, doessay that "Dielectric breakdown is caused by an enormous increase in the number of charge carriers...." Charged electronscan move through both space and time; there is less resistance in moving through the interstices than through the atoms, sothese charged electrons make their way to the surface. The first group neutralizes the surface atoms; this is followed by asecond group of electrons, which should make the surface, if not the whole solid, negative. The positively charged atomsnow move apart, rupturing the solid.

eV/atom-elec. pairEI_Na_elec 4.34EI_Na_elec 4.34 2 1


One might think that it would be necessary for these three values to be exactly equal. Not so. Undoubtedly the energyapplied to the dielectric would be uneven, so that in some "weak areas" of the solid, the Coulombic energy would begeater than the cohesive energy and the substance would rupture there and spread--hence causing the interesting patterns("Lichtenberg figures") photographed in Von Hippel's books.

Further theoretical work and experimental work are certainly needed here, but it's time to move on to the ReciprocalSystem theory of magnetism.

kJ/mole209.4vs.kJ/mole184.6vs.kJ/mole174.3

So we have

(111)kJ/moleEI_pos_mole 209.4446EI_pos_mole4.34

21.602 10

19

Av

1000

The energy per mole to create the positive charges is half that of the ionization energy:

kJ/moleUcohes_NaCl_mole 184.558

Also, according to the Reciprocal System Data Base, the cohesive energy of solid NaCl is


5. Diamagnets and Magnetic Susceptibilities

a. Elements

In conventional theoretical physics, the charged electrons in an ordinary capacitor make it "polarized." In theReciprocal System, the electrons in an ordinary capacitor are uncharged, and therefore the capacitor is not"polarized"; electric permittivity is due to the added space and time of the atoms of the dielectric. But in diamagnetsand paramagnets, the theories switch sides. Here, conventional theory says that magnetic charges and poles do notreally exist; ultimately it's the Coulombic electric charges of the moving electrons that are the cause of all magneticphenomena, including that involved in magnetostatics. In the Reciprocal System, there are actual magnetic chargesand these account for all magnetic phenomena, except, of course, for electromagnetism (which results from themotion of uncharged or charged electrons or other subatoms). Chapter 21 of Ref. [1] discusses electromagnetism inthe Reciprocal System quite thoroughly and so there need be no further discussion of that topic here. The focus nowwill be on the calculation of the magnetic susceptibility of the crytalline elements and compounds, something that hasnot been done before in the Reciprocal System. (Larson did some work on the magnetic susceptibility of liquidorganic compounds, Ref. [1], pp. 247-249.)

Each atom in the Reciprocal System has two orthogonal rotational systems. When magnetically charged, an atomtakes two charges, one on each rotational system. A magnetic charge is a two-dimensional rotational vibration,whereas an electric charge is a one-dimensonal rotational vibration. From Ref. [47], the physical "zero" of therotational vibration frequency is

RR

2 (here R means Rydberg frequency, not resistance)

M_0R

cycles/sec (112)

The magnetic charge is a space displacement, hence the value of the frequency increases with each unit increase ofcharge:

(113a)M_n nM 1 R

cycles/sec


where nM is the number of charges (same for both rotational systems). Properly speaking, what we have is a"multiple-charge" unit, rather than separate charges per se. The possible range for nM is

(114)0 nM tp

where tp is the principal magnetic rotational displacement of the atom. Charge saturation occurs when

nM_sat tp= (115)

One of the two magnetic charges on an atom may be designated the "north pole" or N; the other charge would then be the"south pole" or S. Monopoles do not exist for atoms and have not been observed. Subatoms, with one rotational system,may take just one charge, so it's possible to have a "monocharge." But a two-dimensional rotational vibration appearsclockwise from one direction and counterclockwise from the other direction, so even here it would probably not be correctto speak of a "monopole."

Ref. [48] provides an expression for the photomagnetization energy of magnetic charges; here it is:

h phot h M_nvmag

c

B

Bnat_SI (116)

where vmag is the magnetic rotational displacement (a pure number) and the other symbols have their usual meanings. Forthe science of magnetic materials, the most important matter property is the magnetic susceptibility. Our focus here will be oncrystalline matter, rather than gases, so no further discussion of photomagnetization will be considered presently.

In what follows we will use the natural units of the Reciprocal System, as well as cgs and SI. Unfortunately, there isconsiderable confusion in magnetic units and dimensions in both cgs and SI. Jiles, Ref. [30], pp. 15, 40, 4, explains:


(117c)(SI))r

0_SI(Bint r Bext

(although, technically, r is not defined or used in cgs)

(cgs))r =(Bint Bext(117b)

(117a)(Reciprocal System)Bint

Bnat_SIr

Bext

Bnat_SI

)r =(

But the magnetic charges in the Reciprocal System are physically real, and so, as stated in a previous section of this paper,the Reciprocal System is closer to the cgs system than to SI. Unfortunately, however, the H vector (the so-called "magneticfield intensity") in cgs has different dimensions than the H vector in SI. Cgs uses the same dimensions for H and B, but usesdifferent units (oersted and gauss). (Larson discusses the issue of the H vector on pp. 224, ff., in Ref. [1].) For our purposes,it seems best to avoid H altogether and use B, the magnetic flux density, in all the equations which follow.

The internal magnetic flux density of a material is related to the external magnetic flux density by the relative permeability, r:

"The cgs and SI systems of magnetic units have different philosophies. The cgs system took an approach based onmagnetostatics and the concept of the 'magnetic pole', while the SI system takes an electrodynamic approach to magnetismbased on electric currents.

"As far as is known, neither magnetic poles nor bound currents are real in a physical sense. Instead they are both merelymathematical artifacts, or approximations, which allow calculation of magnetic fields and magnetic moments in a wide rangeof different situations.

"It can be shown that the description of fields due to magnetic poles or Amperian currents are mathematically equivalent solong as the calculation is made at field points which are sufficiently distant from the poles or the current sources. Therefore ingeneral neither approach is better or worse than the other....the two formalisms are mathematically equivalent at points farfrom the field source."


m3/kgSI_mass

SI

SI

(120b)

(120a)cm3/gcgs_mass

cgs

cgs

Unfortunately, the values of listed in the data compilations are not the pure numbers given above. They are either massvalues or molar values, computed as follows:

SI 4 cgs(119)

From the above, it's seen that

(SI, dimensionless)r_SI 1 SI(118c)

(118b)(cgs, dimensionless)cgs 1 4 cgs

(118a)(Reciprocal System, dimensionless)r 1

The relative permeability (simple permeability in cgs) is defined, as follows, in the three systems:


(121a)cgs_mol cgs Av

Vuc

Zuc cm3/mol (all in cgs)

(121b)SI_mol SI Av

Vuc

Zuc m3/mol (all in SI)

where Av is Avogadro's number, Vuc is the volume of the crystal unit cell and Zuc is the number of atoms in a unit cell. TheCRC tabulation (Ref. [16], pp. 4-142 to 4-47) is given in cm3/mol and so must be multiplied by 4to obtain SI-defined values(but can still be expressed in cgs units). The Springer tabulation (Ref. [14], pp. 54 to 158 for the elements, lists both the massand mole values using cgs units but with the SI definition. For Table V, which follows, the observed values of magneticsusceptibilities will be taken to be the mean value of the pure (dimensionless) values calculated from the CRC and Springertabulations.

obs_Springer cgs_mass cgs (but using SI definition of already in the Springer tabulation)

obs_CRC cgs_mol

Zuc

Av Vuc 4 (using SI definition of by multipling CRC values by 4)

obs .5 obs_Springer obs_CRC (SI and Reciprocal System definition of dimensionless)


Magnetic susceptibility is defined as the sum of the magnetic dipole moments induced per unit volume per unit externalmagnetic flux density. The Reciprocal System uses the Kennelly expressions for "pole strength" and "dipole moment." InSI units, the dipole moment is therefore expressed as

(122)dpm nM Mu_weber edgeuc weber-meter

(dpm should not be confused with r.) Mu_weber is the Reciprocal System value of the unit magnetic charge and edgeuc isthe length of the appropriate edge of the crystal unit cell. In non-isotropic cells, the c-edge is longest and therefore isfavored in diamagnets--the greater the dipole moment, the greater the opposition to the impressed magnetic field.Therefore in what follows, the c-edge for all the diamagnetic elements will be used. (To get the theoretical value ofmagnetic susceptibility for edge_a or edge_b, simply multiply the calculations by the ratio edge_a/edge_c oredge_b/edge_c.) The "magnetic polarization" is the sum of the magnetic dipole moments divided by the appropriatevolume.

(123)webers/m2MP

nf

Zuc

2 dpm

Vuc_SI

where nf is a fraction (0 to 1, rational for a unit cell, possibly irrational for bulk matter). Because there are Zuc atoms in a unitcell, there can be a maximum of Zuc/2 dipoles in a unit cell (in which case nf is 1). Now can be expressed as

MP

Bext

(124)

We can determine the saturation value for MP without difficulty, but it's difficult to determine the value of Bext at which thatoccurs, which then prevents us from directly calculating .


Fortunately we know the natural unit value of and MP. From Larson, Ref. [1], p. 241:

"....the numerical factor relating the magnitudes of quantities differing by one scalar dimension, in terms of cgs units, is3 x 1010. The corresponding factor applicable to the interaction between a ferromagnetic charge and an internalmagnetic charge is the square root of the product of 1 and 3 x 1010, which amounts to 1.73 x 105. The internal

magnetic effects are thus weaker than those due to ferromagnetism by about 105."

In this part of this section we are using SI units, so here:

u_SI

1m

sec

cSIm

sec 1

(125)u_SI 5.7755 10

5

MP_nat_t

1 Mu_weber st_u 102

st_u 102

3

webers/m2 (time region) (126)

MP_nat_t 56.561 webers/m2 (time region)

Originally it was thought by workers in the Reciprocal System that the magnetic susceptibility would be proportional only to1/tp or 1/ts, but not both. However, assuming that the magnetic charge is on the principle magnetic rotational displacement,it must be subject to all three atomic rotational displacements--principal (or primary) magnetic, subordinate (or secondary)magnetic, and electric. Now we can form a ratio of the magnetic susceptibility to the natural value:


(127a)

u_SI

nf floorZuc

2

nM Mu_weber edgec

Vuc_SI

Bext

Mp_nat_t

Bext

1

tp

1

ts

1

te

But the Bext cancel out, and we are finally left with

(127b)

u_SI

nf floorZuc

2

nM Mu_weber edgec

Vuc_SI

Mp_nat_t

1

tp

1

ts

1

te

or

u_SI

nf floorZuc

2

nM Mu_weber edgec

Vuc_SI

Mp_nat_t

1

tp

1

ts

1

te

(127c)

Using Eq. (127c) in Excel, Table V results. (Vuc in the table is in units of cm3 x 10-24 and converted to m3 in Excel; likewise,edge_c is in angstrom units and converted to m in Excel. Crystal data values come from the Reciprocal System Data Base.)


ElementAtomic No. Rot. Displ. tp ts te Zuc Vuc edge_c x k_dpmH 1 2-1-(1) 2 1 -1 4 74.532 6.1200

He 2 2-1-0 2 1 0 2 64.348 5.8300

Li 3 2-1-1 2 1 1 2 43.218 3.5093

Be 4 2-1-2 2 2 -6 2 16.011 3.5942

B 5 2-1-3 2 2 -5 50 391.422 5.0880C 62-1-4 / 2-2-(4) 2 2 -4 8 45.377 3.5668

N 7 2-2-(3) 2 2 -3 4 94.233 6.6700

O 8 2-2-(2) 2 1 6 6 106.606 11.2560

F 9 2-2-(1) 2 2 -1 8 128.395 5.5000

Ne 10 2-2-0 2 2 0 4 87.884 4.4460

Na 11 2-2-1 2 2 1 2 78.987 4.2906Mg 12 2-2-2 2 2 2 2 46.193 5.1998

Al 13 2-2-3 2 2 3 4 66.410 4.0496

Si 142-2-4 / 3-2-(4) 3 2 -4 8 160.134 5.4304

P 15 3-2-(3) 3 2 -3 84 1829.671 21.9100

S 16 3-2-(2) 3 2 -2 18 433.779 4.2800

Cl 17 3-2-(1) 3 2 -1 8 230.910 4.4800Ar 18 3-2-0 3 2 0 4 149.790 5.3108

K 19 3-2-1 3 2 1 2 152.616 5.3440

Ca 20 3-2-2 3 2 2 4 173.926 5.5820

Sc 21 3-2-3 3 2 3 2 50.004 5.2733

Ti 22 3-2-4 3 2 4 2 35.713 4.7290V 23 3-2-5 3 2 5 2 28.092 3.0399

Cr 24 3-2-6 3 2 6 2 24.011 2.8850

Mn 25 3-2-7 3 2 7 4 57.647 3.8630

Fe 26 3-2-8 3 2 8 2 23.554 2.8665

Co 273-2-9 / 3-3-(9) 3 2 9 2 22.147 4.0686

Ni 28 3-3-(8) 3 3 -8 4 43.758 3.5239Cu 29 3-3-(7) 3 3 -7 4 47.253 3.6153

Zn 30 3-3-(6) 3 3 -6 2 30.421 4.9467

Ga 31 3-3-(5) 3 3 -5 8 155.751 11.4672

Ge 32 3-3-(4) 3 3 -4 8 181.067 5.6574

As 33 3-3-(3) 3 3 -3 6 129.145 10.5480

Se 34 3-3-(2) 3 3 -2 3 81.301 6.1869Br 35 3-3-(1) 3 3 -1 4 260.568 17.9200

Kr 36 3-3-0 3 3 0 4 187.247 5.7210

Rb 37 3-3-1 3 3 1 2 174.209 5.5850

Sr 38 3-3-2 3 3 2 4 225.277 6.0847

Y 39 3-3-3 3 3 3 2 66.023 5.7306Zr 40 3-3-4 3 3 4 2 46.562 5.1470

Nb 41 3-3-5 3 3 5 2 35.950 3.3004

Mo 42 3-3-6 3 3 6 2 31.173 3.1472

Tc 43 3-3-7 3 3 7 2 26.670 4.4000

Ru 44 3-3-8 3 3 8 2 27.135 4.2803

Rh 453-3-9 / 4-3-(9) 3 3 9 4 55.015 3.8033

CRC_susc Spr_susc-4.4664E-06 -2.2500E-06

-1.3095E-06 -1.2154E-06

-2.3449E-05 -2.4284E-05

-1.7851E-05-2.1695E-05 -2.1688E-05

-1.0624E-05 -5.3298E-06

-1.2515E-05

-6.6073E-06 -6.4008E-06

8.5008E-061.1880E-05

2.1306E-05

-3.2511E-06 -4.1922E-06

-1.9889E-05

-1.2896E-05 -1.3447E-05

-2.9194E-05 -1.4681E-05-3.1191E+00 -1.0909E-05

5.6950E-06

2.1420E-05

2.6259E-04

1.7849E-043.7812E-04

3.1627E-04

8.3853E-04

-9.6402E-06 -9.6512E-06

-1.2547E-05 -9.8518E-06

-2.3141E-05 -2.3225E-05

-1.0690E-05 -7.0689E-06

-2.2481E-05 -2.2538E-05

-1.9241E-05 -1.9344E-05-1.8058E-05 -2.2600E-05

-1.2921E-05 -1.2835E-05

3.6155E-06

3.4096E-05

1.2069E-041.1096E-04

2.3681E-04

1.2262E-04

3.5142E-04

6.6405E-05

1.6888E-04

Avg_CRC_SPR

-2.3866E-05

-1.7851E-05-2.1692E-05

-7.9771E-06

-3.7216E-06

-1.9889E-05

-1.3171E-05

-2.1937E-05

-9.6457E-06

-1.1199E-05

-2.3183E-05

-8.8794E-06

-2.2510E-05

-1.9293E-05-2.0329E-05


Y 39 3-3-3 3 3 3 2 66.023 5.7306Zr 40 3-3-4 3 3 4 2 46.562 5.1470

Nb 41 3-3-5 3 3 5 2 35.950 3.3004

Mo 42 3-3-6 3 3 6 2 31.173 3.1472

Tc 43 3-3-7 3 3 7 2 26.670 4.4000

Ru 44 3-3-8 3 3 8 2 27.135 4.2803

Rh 453-3-9 / 4-3-(9) 3 3 9 4 55.015 3.8033Pd 46 4-3-(8) 3 3 10 4 58.873 3.8902

Ag 47 4-3-(7) 4 3 -7 4 68.227 4.0862

Cd 48 4-3-(6) 4 3 -6 2 42.927 5.6058

In 49 4-3-(5) 4 3 -5 2 52.333 4.9455

Sn 50 4-3-(4) 4 3 -4 8 272.854 9.7290

Sb 51 4-3-(3) 4 3 -3 6 181.231 22.5486Te 52 4-3-(2) 4 3 -2 3 101.287 11.8298

I 53 4-3-(1) 4 3 -1 8 341.046 14.3700

Xe 54 4-3-0 4 3 0 4 237.982 6.1970

Cs 55 4-3-1 4 3 1 2 220.897 6.0450

Ba 56 4-3-2 4 3 2 2 125.000 5.0000La 57 4-3-3 4 3 3 4 148.540 5.2960

Ce 58 4-3-4 4 3 4 4 137.484 5.1612

Pr 59 4-3-5 4 3 5 4 137.468 5.1610

Nd 60 4-3-6 4 3 6 2 68.356 5.9020

Pm 61 4-3-7 4 3 7 2 68.356 5.9020

Sm 62 4-3-8 4 3 8 2 67.381 5.9340Eu 63 4-3-9 4 3 9 2 94.258 4.5510

Gd 64 4-3-10 4 3 10 2 66.105 5.7960

Tb 65 4-3-11 4 3 11 2 63.939 5.6936

Dy 66 4-3-12 4 3 12 2 63.052 5.6680

Ho 67 4-3-13 4 3 13 2 62.238 5.6158Er 68 4-3-14 4 3 14 2 61.010 5.5900

Tm 69 4-3-15 4 3 15 2 60.197 5.5546

Yb 704-3-16 / 4-4-(16) 4 3 16 2 165.126 5.4862

Lu 71 4-4-(15) 4 3 17 2 58.993 5.5090

Hf 72 4-4-(14) 4 3 18 2 44.642 5.0510

Ta 73 4-4-(13) 4 3 19 2 36.127 3.3058W 74 4-4-(12) 4 3 20 2 31.695 3.1647

1.2069E-041.1096E-04

2.3681E-04

1.2262E-04

3.5142E-04

6.6405E-05

1.6888E-048.0400E-04

-2.3845E-05 -2.3833E-05

-1.9144E-05 -1.9200E-05

-8.1305E-06 -1.0196E-05

-2.2871E-05 -1.9077E-05

-6.8362E-05 -6.6930E-05-2.3475E-05 -2.4465E-05

-4.4033E-05 -2.1793E-05

-1.5951E-05 -1.5385E-05

5.4765E-06 5.2948E-06

6.8746E-06 6.7602E-065.3864E-05 6.8299E-05

1.5171E-03 1.4682E-03

3.3562E-03 3.2256E-03

3.6189E-03 3.4325E-03

0.0000E+00

7.9120E-04 7.8525E-041.3675E-02 1.4671E-02

1.1674E-01 1.1603E-01

1.1091E-01 1.2624E-01

6.4837E-02 6.7240E-02

4.8861E-02 4.8296E-023.2820E-02 2.9357E-02

1.7117E-02 1.7700E-02

1.6926E-05 1.7047E-05

1.2933E-04 1.2800E-04

6.6345E-05 7.0358E-05

1.7782E-04 1.7795E-046.9755E-05 7.7032E-05

-2.3839E-05

-1.9172E-05

-9.1634E-06

-2.0974E-05

-6.7646E-05-2.3970E-05

-3.2913E-05


Re 75 4-4-(11) 4 3 21 2 29.251 4.4930Os 76 4-4-(10) 4 3 22 2 27.983 4.3190

Ir 77 4-4-(9) 4 3 23 4 56.597 3.8394

Pt 78 4-4-(8) 4 3 24 4 60.407 3.9237

Au 79 4-4-(7) 4 4 -7 4 67.866 8.1580

Hg 80 4-4-(6) 4 4 -6 2 45.087 4.2375Tl 81 4-4-(5) 4 4 -5 2 56.074 16.4340

Pb 82 4-4-(4) 4 4 -4 4 121.258 7.4244

Bi 83 4-4-(3) 4 4 -3 6 210.126 94.4520

Po 84 4-4-(2) 4 4 -2 3 109.781 4.9160

At 85 4-4-(1) 4 4 -1 3 109.781 4.9160

Rn 86 4-4-0 4 4 0Fr 87 4-4-1 4 4 1

Ra 88 4-4-2 4 4 2 2 136.432 5.1480

Ac 89 4-4-3 4 4 3 4 149.806 5.3110

Th 90 4-4-4 4 4 4 4 131.461 5.0847

Pa 91 4-4-5 4 4 5 4 100.233 3.2140

U 92 4-4-6 4 4 6 2 40.708 3.4400Np 93 4-4-7 4 4 7 2 43.614 3.5200

Pu 94 4-4-8 4 4 8 4 99.704 4.6370

Am 95 4-4-9 4 4 9 4 117.217 4.8940

Cm 96 4-4-10 4 4 10 2 59.967 5.6655

Bk 97 4-4-11 4 4 11 2 55.930 5.5345Cf 98 4-4-12 4 4 12

Es 99 4-4-13 4 4 13

Fm 100 4-4-14 4 4 14

Md 101 4-4-15 4 4 15

No 1024-4-16 / 5-4-(16) 4 4 16

Lw 103 5-4-(15) 5 4 -15Rf 104 5-4-(14) 5 4 -14

Db 105 5-4-(13) 5 4 -13

Sg 106 5-4-(12) 5 4 -12

Bn 107 5-4-(11) 5 4 -11

Hs 108 5-4-(10) 5 4 -10

Mt 109 5-4-(9) 5 4 -9Ds 110 5-4-(8) 5 4 -8

Rg 111 5-4-(7) 5 4 -7

9.5549E-05 9.6362E-051.6398E-05 1.4666E-05

3.6853E-05 3.7665E-05

2.6656E-04 2.7871E-04

-3.4421E-05 -3.4310E-05

-2.2298E-05-3.7196E-05 -3.5967E-05

-1.5825E-05 -1.5763E-05

-1.6682E-04 -1.5999E-04

6.1560E-05 8.4391E-05

0.0000E+00

4.1912E-04 4.1928E-045.4996E-04

4.3931E-04 5.3842E-04

6.8825E-04

-3.4365E-05

-1.1149E-05-3.6582E-05

-1.5794E-05

-1.6340E-04

elec_mag_suscep.mcd 140Element Atomic No. Rot. Displ.

H 1 2-1-(1)

He 2 2-1-0

Li 3 2-1-1

Be 4 2-1-2

B 5 2-1-3C 6 2-1-4 / 2-2-(4)

N 7 2-2-(3)

O 8 2-2-(2)

F 9 2-2-(1)

Ne 10 2-2-0

Na 11 2-2-1Mg 12 2-2-2

Al 13 2-2-3

Si 14 2-2-4 / 3-2-(4)

P 15 3-2-(3)

S 16 3-2-(2)

Cl 17 3-2-(1)Ar 18 3-2-0

K 19 3-2-1

Ca 20 3-2-2

Sc 21 3-2-3

Ti 22 3-2-4V 23 3-2-5

Cr 24 3-2-6

Mn 25 3-2-7

Fe 26 3-2-8

Co 27 3-2-9 / 3-3-(9)

Ni 28 3-3-(8)Cu 29 3-3-(7)

Zn 30 3-3-(6)

Ga 31 3-3-(5)

Ge 32 3-3-(4)

As 33 3-3-(3)

Se 34 3-3-(2)Br 35 3-3-(1)

Kr 36 3-3-0

Rb 37 3-3-1

Sr 38 3-3-2

Y 39 3-3-3Zr 40 3-3-4

Nb 41 3-3-5

Mo 42 3-3-6

Tc 43 3-3-7

Ru 44 3-3-8

Rh 45 3-3-9 / 4-3-(9)Pd 46 4-3-(8)

n_M tp_eff ts_eff te_eff susc_calc

2 2 2 -2 -2.7523E-05

2 2 2 -4 -1.9922E-052 2 2 -4 -1.9275E-05

2 2 2 -4 -8.6784E-06

2 3 2 -6 -3.6958E-06

2 3 2 -4 -2.0555E-05

2 3 2 -1 -1.4517E-05

3 3 2 -1 -1.9030E-05

2 3 3 -2 -8.3383E-06

3 3 3 -2 -1.3291E-05

3 3 3 -2 -2.4072E-05

3 3 3 -2 -1.0216E-05

3 3 3 -2 -2.0028E-05

3 3 3 -1 -1.8660E-053 3 3 -1 -2.2485E-05

susc calc/obs

1.153

1.1160.889

1.088

0.993

0.964

1.102

0.867

0.864

1.187

1.038

1.150

0.890

0.9671.106


Zr 40 3-3-4

Nb 41 3-3-5

Mo 42 3-3-6

Tc 43 3-3-7

Ru 44 3-3-8

Rh 45 3-3-9 / 4-3-(9)Pd 46 4-3-(8)

Ag 47 4-3-(7)

Cd 48 4-3-(6)

In 49 4-3-(5)

Sn 50 4-3-(4)

Sb 51 4-3-(3)Te 52 4-3-(2)

I 53 4-3-(1)

Xe 54 4-3-0

Cs 55 4-3-1

Ba 56 4-3-2La 57 4-3-3

Ce 58 4-3-4

Pr 59 4-3-5

Nd 60 4-3-6

Pm 61 4-3-7

Sm 62 4-3-8Eu 63 4-3-9

Gd 64 4-3-10

Tb 65 4-3-11

Dy 66 4-3-12

Ho 67 4-3-13Er 68 4-3-14

Tm 69 4-3-15

Yb 704-3-16 / 4-4-(16)

Lu 71 4-4-(15)

Hf 72 4-4-(14)

Ta 73 4-4-(13)W 74 4-4-(12)

4 4 3 -1 -1.9582E-05

4 4 3 -1 -2.1348E-05

4 4 3 -2 -7.7243E-06

4 4 3 -1 -2.3316E-05

4 4 3 -1 -6.1019E-054 4 3 -1 -2.8640E-05

4 4 3 -1 -2.7552E-05

0.821

1.114

0.843

1.112

0.9021.195

0.837

Re 75 4-4-(11)Os 76 4-4-(10)

Ir 77 4-4-(9)

elec_mag_suscep.mcd 142Re 75 4-4-(11)Os 76 4-4-(10)

Ir 77 4-4-(9)

Pt 78 4-4-(8)

Au 79 4-4-(7)

Hg 80 4-4-(6)Tl 81 4-4-(5)

Pb 82 4-4-(4)

Bi 83 4-4-(3)

Po 84 4-4-(2)

At 85 4-4-(1)

Rn 86 4-4-0Fr 87 4-4-1

Ra 88 4-4-2

Ac 89 4-4-3

Th 90 4-4-4

Pa 91 4-4-5

U 92 4-4-6Np 93 4-4-7

Pu 94 4-4-8

Am 95 4-4-9

Cm 96 4-4-10

Bk 97 4-4-11Cf 98 4-4-12

Es 99 4-4-13

Fm 100 4-4-14

Md 101 4-4-15

No 1024-4-16 / 5-4-(16)

Lw 103 5-4-(15)Rf 104 5-4-(14)

Db 105 5-4-(13)

Sg 106 5-4-(12)

Bn 107 5-4-(11)

Hs 108 5-4-(10)

Mt 109 5-4-(9)Ds 110 5-4-(8)

Rg 111 5-4-(7)

4 4 4 -1 -2.9477E-05

4 4 4 -1 -1.1523E-054 4 4 -1 -3.5933E-05

4 4 4 -1 -1.5014E-05

4 4 4 -1 -1.6534E-04

-3.4365E-05 0.858

-1.1149E-05 1.034-3.6582E-05 0.982

-1.5794E-05 0.951

-1.6340E-04 1.012

0.996 1.001

Correl. Avg. Ratio

Table V. Calculated Values of Diamagnetic Susceptibility for the Elements (Crystalline, No Loss)


Notes:

1) The diamagnetic susceptibilites are negative because te is negative. Electronegative elements arediamagnetic, whereas electropositive elements are paramagnetic. Obviously some nominally electronegativeelements use their alternative electropositive rotational displacements and so are paramagnetic, and viceversa.

2) All diamagnetic elements have nf = 1.

3) Generally, the value of nM = tp, meaning that saturation has been obtained. If nM < tp, then saturation has notbeen obtained. It's clear that nM cancels tp or ts in many of the elements, which explains why in early work on theReciprocal System it was thought that susceptibility was proportional to 1/tp or 1/ts only.

4) The correlation of the effective values of tp, ts, and te, with the actual rotational displacements is quiteamazing!

5) The average ratio of calc/obs = 1.001, which means that the calculated and observed values are within1.5% of each other, and therefore within the experimental error. The correlation is .996.

6) The values given above are those most probable, given the experimental conditions.

7) Except for elements like Hg, the calculations are for values of the crystal unit cell at room temperature. Otherthan very small changes of edge_c and Vuc with temperature, diamagnetic susceptibility is not dependent ontemperature. Crystal unit cell edge_c has been used in most of the calculations. In a few, a factor kdpm

multiplies the value of edge_c; the maximum of this factor is 8, for Bi--this explains the remarkablediamagnetism of this element. The proof of this extended magnetic dipole length can be seen in the spiral,stair-step nature of the macroscopic crystal--which shows the misalignment of the unit cells (and allows thedipole length to go across unit cells). The diamagnetic compounds, unlike the elements, generally have kdpm >1.

8) Red phosphorus and gray/white tin are the forms of these elements used in the calculations.

9) Non-integer values cannot be used for te here, because the isotopic neutrinos have no affect on atomicmagnetic charges. While it's true that te_eff is not quite equal to te for many of the elements, the seriesrelationships are quite closely in accord: as atomic number increases in a division, te_eff decreases, as itshould. Isotopes of a given element should not vary in magnetic susceptibility.


10) The orthogonal charge of each of the two atoms involved in a magnetic dipole has no direct effect on themagnetic susceptibility; however, they can change the dimensions of edge_a or edge_b and thus slightlychange the volume of the unit cell. The dipole itself causes a small decrease of edge_c; the calculations arestraightforward but beyond the scope of this paper because they involve the interatomic distance equations.See the article on "magnetostriction" in Thewlis, Ref. [7], vol. 4, pp. 480-482; the equations there areapproximate macroscopic ones; to do the calculations correctly requires detailed knowledge of the atomicforces involved.

11) As should be clear, the Reciprocal System has no use for the "Bohr magneton," the "Weiss magneton," orthe "nuclear magneton."

12) In diamagnets and paramagnets there is no hysteresis and therefore no energy loss. In thermodynamicterms, the process is reversible. When the horizontal electromagnet in a Gouy Balance is turned on, magneticcharges are induced in the specimen, and the diamagnet sample in the vertical test cylinder repels upward;when the electromagnet is turned off, the diamagnet sample drops back down to its equilibrium position, andvice versa for a paramagnet sample. Note that the horizontal force of the north and south poles of theelectromagnet cancel, so that they cause no horizontal motion of the diamagnet or paramagnet. Butmagnetism is two-dimensional, according to the Reciprocal System, so vertical motion of the diamagnet orparamagnet is the result when the electromagnet is turned on. Conventional physics has no conceptualexplanation for this effect!


b. Compounds

Letting sdpm = the average magnetic dipole length, nuc = net total number of atoms in a volume unit cell, n1 = net totalnumber of atoms of element 1, and n2 = net total number of atoms of element 2, we can generalize Eq. (127c) forbinary compounds:

u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(128)

(Further generalization to more complex compounds is obvious.) We will apply Eq. (128) to all the dielectric binarycompounds previously considered which are diamagnetic. SI units are used throughout, except to convert observedvalues to the dimensionless value of for use with comparison to the calculated values. The CRC Handbook, Ref. [16],pp. 4-142 to 4-147 is used for the observed values, except for a few from Ref. [15], pp. 730, ff. Unfortunately and oddly,the Springer compilation, Ref. [14], does not have the data, a real lacuna, because it would be helpful to average theresults as before.

In what follows, the following parameter equations obtain:

nf 1 nM min tp1 tp2 = (In very weak fields, the values would begin smaller and becomelarger as the field increases until getting to these values.)

Also, a nominally electropositive element assumes its alternative equivalent electronegative rotational displacement.Otherewise, therer are no modifications to the values of tp, ts, or te from their values in the Reciprocal System periodictable.


As explained above, the semiconductor compounds have kdpm >= 1 for use with Eq. (129d) or (129c). Wyckoff'svolumes, Ref. [65], were consulted in order to get an "intuitive feel" for how the dipoles would actually look. His latticediagrams and packing drawings are very helpful for figuring out sdpm.

(129e)kdpm1

2

1

2 1

5

4

7

4 2 3

=

In Eq. (129d), kdpm can have the following values (for the compounds considered here), based on geometry of the cell:

(129d)(a multiple of the diagonal formed between s0 and an edge)sdpm kdpm s02

edge2

(129c)(a multiple of one of the edges of unit cell, kdpm = (1, 2, 3))sdpm kdpm edge

(an average of interatomic distances) (129b)sdpm

s0_1 s0_2

2

(129a)(a multiple of shortest atomic distance, kdpm >= 1)sdpm kdpm s0

In Eq. (128), essentially all the values are fixed, except for sdpm. A diamagnetic crystal will orient itself so as tomaximally oppose the impressed field. This means that nature "chooses" the particular subset of all possible paralleldipole moments which has the highest average length. In many cases, this length will be a diagonal. In the case ofsemiconductor compounds, the lengths can actually span multiple unit cells--because of the vacancies of atoms atopposite corners; semiconductors have long-range order, not short-range order. The possible values of sdpm are asfollows:


In this compound, the average dipole length = the length of edge_a.

SiO2

obs_SI_SiO21.0118SiO2 1.6587 10

5

SiO2 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

sdpm edgea

te2 2ts2 2tp2 2n2 6n2 3 22-2-(2)O:

te1 4ts1 2tp1 3n1 3 1

obs_SI_SiO2 29.6 106

4 3

Av 112.979 1024

SiO2 obs_SI_SiO2 1.6394 105

nuc 9nf 1 Zuc 3 nM 2 edgec 5.4046 1010

m Vuc_SI 112.979 1030

m3

edgea 4.9130 1010

m s0 1.6109 1010

m

Si: 3-2-(4)


In this compound, the average dipole length = the shortest interatomic distance.

CaF2

obs_SI_CaF21.007CaF2 1.4429 10

5

CaF2 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

sdpm s0

te2 1ts2 2tp2 2n2 8n2 4 22-2-(1)F:

(alternative electronegative equivalent)te1 16ts1 3tp1 3

CaF2 obs_SI_CaF2 28.0 106

4 4

Av 163.035 1024

obs_SI_CaF2 1.4328 105

nf 1 Zuc 4 nM 2 edgea 5.4360 1010

m Vuc_SI 163.035 1030

m3 nuc 12

(all edges same)

s0 2.3655 1010

m

Ca: 3-2-2 n1 4


n1 4 tp1 3 ts1 2 te1 6 (alternative electronegative equivalent)

O: 2-2-(2) n2 4 tp2 2 ts2 2 te2 2

sdpm

s0_1 s0_2

2 The two interatomic distances are averaged.

MgO u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

MgO 1.1533 105

MgO

obs_SI_MgO1.0134

In this compound, the average dipole length = the average of two interatomic distances.

MgO obs_SI_MgO 10.2 106

4 4

Av 74.778 1024

obs_SI_MgO 1.138 10

5

nf 1 Zuc 4 nM 2 Vuc_SI 74.778 1030

m3 nuc 8edgea 4.2130 1010

(all edges same)

s0_1 2.1065 1010

m s0_2 3.6486 1010

m

Mg: 2-2-2


In this compound, the average dipole length = the length of edge_a.

KCl

obs_SI_KCl0.9908KCl 1.287 10

5

KCl u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

sdpm edgea

te2 1ts2 2tp2 3n2 43-2-(1)Cl:

(alternative electronegative equivalent)te1 17ts1 3tp1 3n1 43-2-1K:

ms0 3.1465 1010

(all edges same)

nuc 8m3Vuc_SI 249.203 1030

medgea 6.2929 1010

nM 3Zuc 4nf 1

obs_SI_KCl 1.299 105

obs_SI_KCl 38.8 106

4 4

Av 249.203 1024

KCl


In this compound, the average dipole length = the diagonal formed between edge_a and s0, reduced by 1/SQR(2).

NaCl

obs_SI_NaCl0.9919NaCl 1.3929 10

5

NaCl u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

cubic, with octahedral coordinationsdpm1

2s0

2edgea

2

te2 1ts2 2tp2 3n2 43-2-(1)Cl:

(alternative electronegative equivalent)te1 7ts1 2tp1 3

NaCl obs_SI_NaCl 30.2 106

4 4

Av 179.425 1024

obs_SI_NaCl 1.4043 10

5

nf 1 Zuc 4 nM 3 edgea 5.6402 1010

m Vuc_SI 179.425 1030

m3 nuc 8

(all edges)

s0 2.8201 1010

m

Na: 2-2-1 n1 4


In this compound, the average dipole length = 1/2 the diagonal formed between edge_a and s0.

SrF2

obs_SI_SrF21.0081SrF2 1.6086 10

5

SrF2 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

sdpm1

2s0

2edgea

2

te2 1ts2 2tp2 2n2 82-2-(1)F:

(alternative electronegative equivalent)te1 16ts1 3tp1 4n1 43-3-2Sr:

ms0 2.5089 1010

nuc 12m3Vuc_SI 194.507 1030

medgea 5.7940 1010

nM 2Zuc 4nf 1

obs_SI_SrF2 1.5956 105

obs_SI_SrF2 37.2 106

4 4

Av 194.507 1024

SrF2


In this compound, the average dipole length = 3 x edge_a.

GaAs

obs_SI_GaAs1.0656GaAs 1.6366 10

5

GaAs u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be a semiconductor characteristic)sdpm 3 edgea

te2 3ts2 3tp2 3n2 43-3-(3)As:

te1 5ts1 3tp1 3n1 43-3-(5)Ga:

ms0 2.4481 1010

nuc 8m3Vuc_SI 180.717 1030

(all edges)medgea 5.6537 1010

nM 3Zuc 4nf 1

obs_SI_GaAs 1.5359 105

obs_SI_GaAs .230 106

4 5.314GaAs


In this compound, the average dipole length = 2 x edge_a.

GaP

obs_SI_GaP0.9958GaP 1.5428 10

5

GaP u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be semiconductor characteristic)sdpm 2 edgea

te2 3ts2 2tp2 3n2 43-2-(3)P:

te1 5ts1 3tp1 3n1 43-3-(5)Ga:

ms0 2.3586 1010

(all edges same)

nuc 8m3Vuc_SI 161.611 1030

medgea 5.4470 1010

nM 3Zuc 4nf 1

obs_SI_GaP 1.5492 105

obs_SI_GaP .298 106

4 4.137GaP


In this compound, the average dipole length = the diagonal formed between s0 and edge_a.

BeO

obs_SI_BeO1.0424BeO 1.8742 10

5

BeO u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

4 2

1

tp1

1

ts1

1

te1

21

tp2

1

ts2

1

te2

(must be a semiconductor characteristic)sdpm s02

edgea2

te2 2ts2 2tp2 2n2 22-2-(2)O:

te1 6ts1 2tp1 2n1 2

BeO obs_SI_BeO 11.9 106

4 2

Av 27.611 1024

obs_SI_BeO 1.7979 10

5

nuc 4nf 1 Zuc 2 nM 2 edgea 2.6980 1010

m edgec 4.3800 1010

m Vuc_SI 27.611 1030

m3

s0 1.6556 1010

m

Be: 2-1-2


In this compound the average dipole length = 3 x the diagonal formed between s0 and edge_a.

CdS

obs_SI_CdS0.9963CdS 2.0885 10

5

CdS u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be a semiconductor characteristic)sdpm 3 s02

edgea2

te2 2ts2 2tp2 3n2 23-2-(2)S:

te1 6ts1 3tp1 4n1 2

CdS obs_SI_CdS 50.0 106

4 2

Av 99.503 1024

obs_SI_CdS 2.0962 10

5

nf 1 Zuc 2 nM 3 edgea 4.1364 1010

mVuc_SI 99.503 10

30 nuc 4m3

edgec 6.7152 1010

m

s0 2.5182 1010

m

Cd: 4-3-(6)


In this compound the average dipole length = 5/4 x the diagonal formed between s0 and edge_a.

SiC

obs_SI_SiC1.1333SiC 1.4723 10

5

SiC u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be semiconductor characteristic)sdpm5

4s0

2edgea

2

te2 4ts2 2tp2 2n2 42-2-(4)C:

te1 4ts1 2tp1 3n1 43-2-(4)Si:

ms0 1.8827 1010

(all edges same)m3 nuc 8Vuc_SI 82.199 10

30

medgea 4.3480 1010

nM 2Zuc 4nf 1

obs_SI_SiC 1.2992 105

obs_SI_SiC 12.8 106

4 4

Av 82.199 1024

SiC


In this compound the average dipole length = 7/4 x the diagonal formed between s0 and edge_a.

ZnS

obs_SI_ZnS1.0106ZnS 1.3318 10

5

ZnS u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be semiconductor characteristic)sdpm7

4s0

2edgea

2

te2 4ts2 2tp2 3n2 43-2-(4)S:

te1 6ts1 2tp1 3n1 43-2-(6)Zn:

ms0 2.3423 1010

(all edges same)m3 nuc 8Vuc_SI 158.279 10

30

medgea 5.4093 1010

nM 3Zuc 4nf 1

obs_SI_ZnS 1.3178 105

obs_SI_ZnS 25.0 106

4 4

Av 158.279 1024

ZnS



ZnO

obs_SI_ZnO1.0014ZnO 2.4116 10

5

ZnO u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be semiconductor characteristic)sdpm 2 s02

edgea2

te2 2ts2 2tp2 2n2 22-2-(2)O:

te1 6ts1 2tp1 3n1 2

ZnO obs_SI_ZnO 27.2 106

4 2

Av 47.114 1024

obs_SI_ZnO 2.4083 105

nf 1 Zuc 2 nM 2 edgea 3.2420 1010

mVuc_SI 47.114 10

30 nuc 4m3

edgec 5.1760 1010

m

s0 1.9651 1010

m

Zn: 3-2-(6)



GaN

obs_SI_GaN0.9973GaN 2.5564 10

5

GaN u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

1

tp1

1

ts1

1

te1

n21

tp2

1

ts2

1

te2

(must be semiconductor characteristic)sdpm 3 s02

edgea2

te2 3ts2 2tp2 2n2 22-2-(3)N:

te1 5ts1 3tp1 3n1 2

GaN obs_SI_GaN .332 106

4 6.144 obs_SI_GaN 2.5633 105

nf 1 Zuc 2 nM 2 edgea 3.1800 1010

m Vuc_SI 45.242 1030

m3 nuc 4

edgec 5.1660 1010

m

s0 1.9373 1010

m

Ga: 3-3-(5)


The average of the ratios of calculated to observed values of is:

SiO2

obs_SI_SiO2

CaF2

obs_SI_CaF2

MgO

obs_SI_MgO

KCl

obs_SI_KCl

NaCl

obs_SI_NaCl

SrF2

obs_SI_SrF2

GaAs

obs_SI_GaAs

GaP

obs_SI_GaP

BeO

obs_SI_BeO

CdS

obs_SI_CdS

SiC

obs_SI_SiC

ZnS

obs_SI_ZnS

ZnO

obs_SI_ZnO

GaN

obs_SI_GaN

141.019

Therefore, we may conclude that the calculations agree with the observed values to within the experimental error.

Note: Superconductors have a diamagnetic susceptibility of -1 (Ref. 30, p. 104), meaning that r = 0, and this is easilyexplained by the Reciprocal System. The massless, uncharged electrons (space units) comprising the electric currenthave rotational displacement 1-0-(1), which shows that the one-dimensional electric displacement is -1. Thisone-dimensional motion combined with the one-dimensional translation provides two-dimensional electromagnetism inopposition to the impressed field--diamagnetism--and there is no reduction factor. No magnetic charges are induced inthe matter of the conductor.


6. Paramagnets and Magnetic Susceptibilities

a. Elements

Previous to this paper there has been little work done on the calculation of the magnetic susceptibilities of the solidparamagnetic substances; therefore the following is somewhat provisional. In the first edition of Ref. [2], p. 84,Larson says:

"From this new theoretical viewpoint it would appear that there should be a class of paramagnetic substancescorresponding to the diamagnetic group with relatively small positive susceptibilities independent of temperature.When we examine the experimanetal susceptibilities of the electropositive elements this is just what wefind....these small positive susceptibilities show little or no temperature variation....Where any temperaturevariation does exist it is usually an increase in the susceptibility at the higher temperatures, ..., and suggests thatsecondary causes may be responsible.

"The remaining electropositive elements have much higher susceptibilities which are stronglytemperature-dependent, and while there is a very large difference between the susceptibilities of the ferromagneticelements and the other elements of this group, it would seem that they all belong in the class of inductiveparamagnetics." Chapter 26 of the Russian data compilation, Ref. [15], illustrates these conclusions graphically.

We will call the temperature-independent paramagnets Class I and the the temperature-dependent paramagnetsClass II. Some of the Class II elements, like Cr, Mn, and V, can become ferromagnetic under somecircumstances. The rare earth elements will be considered to be "low-end" ferromagnets, rather than as "high-endparamagnets" and will be treated as such in section 7 of this paper.

Eq. (127c) is modified for electropositive elements by inverting (naturally!) the rotational displacements terms:

u_SI

nf floorZuc

2

nM Mu_weber edgec

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

Class I Paramagnets (130)


Here, te is always positive, and so the susceptibilities are always positive. This equation applies to the lowerelements of the various paramagnetic elements in each group in the following table, Table VI. For the higherelements of each such group, Eq. (130) needs a temperature term added (with w = atomic mass, T = temperature,Tt_u = natural unit of temperature for the solid state). Provisionally:

u_SI

nf floorZuc

2

nM Mu_weber edgec

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

(131)

Class II Paramagnets

The second term is properly non-dimensional, like the first. Note that the dimensions of w/T are

[t2/s2] = [t3/s3]/[t/s] (132)

These are the dimensions of magnetic charge. What appears to be happening is this:

1) Unlike the situtation with gases, the solid inter-atomic forces and thermal vibrations initially prevent most of theatoms from obtaining internal magnetic charges, but some do.

2) The induced internal magnetic charges then cause the magnetic susceptibility given by Eq. (130) or the first term ofEq. (131). nf increases from 0 to 1 and nM increases from 0 to tp, with temperature (up to a certain limit).

3) Thermal energy now produces translational vibrations of the atoms with magnetic charges, which creates anadditional magnetic effect, similar to the effect of moving charged electrons in a magnetic field. (Incidentally, thismotion should induce a small electric current in a nearby electrical conductor, although this might be hard to detect.)The magnetic effect is proportional to the mass of the atom and inversely proportional to the temperature--the lower thetemperature, the less the disorienting effect of the thermal motion.


In the electropositive elements, the thermal motion is translational vibration is in space, just as the magnetic charge isrotational vibration in space; therefore the effects are additive. This is in contrast with the electronegative elements, inwhich the thermal motion is in time. Hence the diamagnets do not have a temperature term in their susceptibilityequation.

The only "adjustable" parameter in Eq. (130) and Eq. (131) is nf, the fraction of the crystal unit cells which aremagnetised and oriented in the direction of the magnetic field. Whereas in the diamagnetic elements, nf is normallyequal to 1, this is not the case with the paramagnetic elements--because of the stronger solid inter-atomic forces andthe spatial thermal vibrations. For use in Table VI, we will therefore use the "best-fit" values--these should beconsidered to be predictions, to be verified by experiment later. For paramagnetic gases, we would expect the value of to be roughly 1/nf times that of the solid values. Unfortunately, empirical data verifying this deduction are hard to comeby--none of the references have data comparing the values of for the same paramagnetic substances in the solid,liquid, and gas states.

However, the literature does refer to the paramagnetic metals as being "feebly paramagnetic." Both the classical theoryfrom Curie and Weiss and the Quantum Mechanics theories of Van Vleck and his associates focus mostly on theparamagnetic gases. Van Vleck spends just six pages out of 384 on the topic of solid metallic paramagnetism (Ref.[22], pp. 347-353). Terms such as "quenching" and "exchange demagnetization" and "Pauli paramagnetism" arebandied about. The "standard" Curie-Weiss Law is expressed as

C

T

where C is the Curie constant and is the Weiss constant. But Van Vleck says (Ref. [22], p. 303), "Whole pagescould be devoted to recording the values of C and reported by different investigators, not always in overly goodaccord with each other." So, Van Vleck doesn't devote any pages to this data--it's that unreliable! (A search on theInternet has revealed that the situation hasn't changed in the intervening years.) Furthermore, negative values aresometimes given for ! We have to conclude that the "modern" theory of magnetic susceptibility is not very tenable.

In the following table, T has been set to room temperature, 293.15 K, for the Class II Paramagnets. nM has beenset to tp at this temperature. Other data are in the same units as the previous table.


Element At. No. At. Weight Rot. Displ. Zuc Vuc edge_a edge_c CRC_susc Spr_susc avg CRC Spr n_M tp_eff ts_eff te_effH 1 1.00790 2-1-(1) 4 74.532 3.7500 6.1200

He 2 4.00260 2-1-0 2 64.348 3.5700 5.8300

Li 3 6.94100 2-1-1 2 43.218 3.5093 3.5093

Be 4 9.01202 2-1-2 2 16.011 2.2680 3.5942

B 5 10.81000 2-1-3 50 391.422 8.7710 5.0880C 6 12.01100 2-1-4 / 2-2-(4) 8 45.377 3.5668 3.5668

N 7 14.00670 2-2-(3) 4 94.233 4.0390 6.6700

O 8 15.99940 2-2-(2) 6 106.606 3.3070 11.2560

F 9 18.99840 2-2-(1) 8 128.395 7.2840 5.5000

Ne 10 20.17900 2-2-0 4 87.884 4.4460 4.4460

Na 11 22.98980 2-2-1 2 78.987 4.2906 4.2906 8.4500E-06 8.5008E-06 8.4754E-06 2 2 2 1Mg 12 24.30500 2-2-2 2 46.193 3.2028 5.1998 1.1830E-05 1.1880E-05 1.1855E-05 2 2 2 2

Al 13 26.98150 2-2-3 4 66.410 4.0496 4.0496 2.0729E-05 2.1306E-05 2.1018E-05 2 2 2 3

Si 14 28.08550 2-2-4 / 3-2-(4) 8 160.134 5.4304 5.4304

P 15 30.97380 3-2-(3) 84 1829.671 9.1500 21.9100

S 16 32.06000 3-2-(2) 18 433.779 10.8180 4.2800

Cl 17 35.45300 3-2-(1) 8 230.910 6.2400 4.4800Ar 18 39.94800 3-2-0 4 149.790 5.3108 5.3108

K 19 39.09830 3-2-1 2 152.616 5.3440 5.3440 5.6853E-06 5.6950E-06 5.6902E-06 3 3 2 1

Ca 20 40.08000 3-2-2 4 173.926 5.5820 5.5820 1.9187E-05 2.1420E-05 2.0304E-05 3 3 2 2

Sc 21 44.95590 3-2-3 2 50.004 3.3090 5.2733 2.4627E-04 2.6259E-04 2.5443E-04 3 3 2 3

Ti 22 47.88000 3-2-4 2 35.713 2.9530 4.7290 1.7638E-04 1.7849E-04 1.7743E-04 3 3 2 4V 23 50.94150 3-2-5 2 28.092 3.0399 3.0399 4.2321E-04 3.7812E-04 4.0066E-04 3 3 2 5

Cr 24 51.99600 3-2-6 2 24.011 2.8850 2.8850 2.9013E-04 3.1627E-04 3.0320E-04 3 3 2 6

Mn 25 54.93800 3-2-7 4 57.647 3.8630 3.8630 7.3955E-04 8.3853E-04 7.8904E-04 3 3 2 7

Fe 26 55.84700 3-2-8 2 23.554 2.8665 2.8665

Co 27 58.93320 3-2-9 / 3-3-(9) 2 22.147 2.5071 4.0686

Ni 28 58.69000 3-3-(8) 4 43.758 3.5239 3.5239Cu 29 63.54600 3-3-(7) 4 47.253 3.6153 3.6153

Zn 30 65.39000 3-3-(6) 2 30.421 2.6648 4.9467

Ga 31 69.72000 3-3-(5) 8 155.751 4.5107 7.6448

Ge 32 72.59000 3-3-(4) 8 181.067 5.6574 5.6574

As 33 74.91260 3-3-(3) 6 129.145 3.7600 10.5480

Se 34 78.96000 3-3-(2) 3 81.301 4.3552 4.9495Br 35 79.90400 3-3-(1) 4 260.568 6.6700 4.4800

Kr 36 83.80000 3-3-0 4 187.247 5.7210 5.7210

Rb 37 85.46780 3-3-1 2 174.209 5.5850 5.5850 4.0707E-06 3.6155E-06 3.8431E-06 3 3 3 1

Sr 38 87.62000 3-3-2 4 225.277 6.0847 6.0847 3.4072E-05 3.4096E-05 3.4084E-05 3 3 3 2

Y 39 88.90590 3-3-3 2 66.023 3.6474 5.7306 4.2332E-05 1.2069E-04 8.1511E-05 3 3 3 3


Zr 40 91.22400 3-3-4 2 46.562 3.2320 5.1470 1.0751E-04 1.1096E-04 1.0924E-04 3 3 3 4Nb 41 92.90640 3-3-5 2 35.950 3.3004 3.3004 2.4136E-04 2.3681E-04 2.3908E-04 3 3 3 5

Mo 42 95.94000 3-3-6 2 31.173 3.1472 3.1472 9.6349E-05 1.2262E-04 1.0948E-04 3 3 3 6

Tc 43 97.90720 3-3-7 2 26.670 2.7430 4.4000 1.7987E-04 3.5142E-04 2.6564E-04 3 3 3 7

Ru 44 101.07000 3-3-8 2 27.135 2.7056 4.2803 5.9955E-05 6.6405E-05 6.3180E-05 3 3 3 8

Rh 45 102.90600 3-3-9 / 4-3-(9) 4 55.015 3.8033 3.8033 1.5468E-04 1.6888E-04 1.6178E-04 3 3 3 9Pd 46 106.42000 4-3-(8) 4 58.873 3.8902 3.8902 7.6524E-04 8.0400E-04 7.8462E-04 3 3 3 10

Ag 47 107.86800 4-3-(7) 4 68.227 4.0862 4.0862

Cd 48 112.41000 4-3-(6) 2 42.927 2.9737 5.6058

In 49 114.82000 4-3-(5) 2 52.333 3.2530 4.9455

Sn 50 118.71000 4-3-(4) 8 272.854 6.4860 6.4860

Sb 51 121.75000 4-3-(3) 6 181.231 4.3083 11.2743Te 52 127.60000 4-3-(2) 3 101.287 4.4467 5.9149

I 53 126.90500 4-3-(1) 8 341.046 7.2701 4.7900

Xe 54 131.29000 4-3-0 4 237.982 6.1970 6.1970

Cs 55 132.90500 4-3-1 2 220.897 6.0450 6.0450 5.4765E-06 5.2948E-06 5.3856E-06 4 4 3 1

Ba 56 137.33000 4-3-2 2 125.000 5.0000 5.0000 6.8746E-06 6.7602E-06 6.8174E-06 4 4 3 2

La 57 138.90600 4-3-3 4 148.540 5.2960 5.2960 5.3864E-05 6.8299E-05 6.1081E-05 4 4 3 3Ce 58 140.12000 4-3-4 4 137.484 5.1612 5.1612 1.5171E-03 1.4682E-03 1.4927E-03 4 4 3 4

Pr 59 140.90800 4-3-5 4 137.468 5.1610 5.1610 3.3562E-03 3.2256E-03 3.2909E-03 4 4 3 5

Nd 60 144.24000 4-3-6 2 68.356 3.6570 5.9020 3.6189E-03 3.4325E-03 3.5257E-03 4 4 3 6

Pm 61 145.00000 4-3-7 2 68.356 3.6570 5.9020

Sm 62 150.36000 4-3-8 2 67.381 3.6210 5.9340 7.9120E-04 7.8525E-04 7.8822E-04 4 4 3 8Eu 63 151.96000 4-3-9 2 94.258 4.5510 4.5510 1.3675E-02 1.4671E-02 1.4173E-02

Gd 64 157.25000 4-3-10 2 66.105 3.6290 5.7960 1.1674E-01 1.1603E-01 1.1638E-01

Tb 65 158.92540 4-3-11 2 63.939 3.6010 5.6936 1.1091E-01 1.2624E-01 1.1858E-01

Dy 66 162.50000 4-3-12 2 63.052 3.5840 5.6680 6.4837E-02 6.7240E-02 6.6038E-02

Ho 67 164.93000 4-3-13 2 62.238 3.5773 5.6158 4.8861E-02 4.8296E-02 4.8578E-02

Er 68 167.26000 4-3-14 2 61.010 3.5500 5.5900 3.2820E-02 2.9357E-02 3.1088E-02Tm 69 168.93400 4-3-15 2 60.197 3.5375 5.5546 1.7117E-02 1.7700E-02 1.7408E-02

Yb 70 173.040004-3-16 / 4-4-(16) 2 165.126 5.4862 5.4862 1.6926E-05 1.7047E-05 1.6987E-05 4 4 3 16

Lu 71 174.96700 4-4-(15) 2 58.993 3.5031 5.5090 1.2933E-04 1.2800E-04 1.2867E-04 4 4 3 17

Hf 72 178.49000 4-4-(14) 2 44.642 3.1946 5.0510 6.6345E-05 7.0358E-05 6.8351E-05 4 4 3 18

Ta 73 180.94800 4-4-(13) 2 36.127 3.3058 3.3058 1.7782E-04 1.7795E-04 1.7789E-04

W 74 183.85000 4-4-(12) 2 31.695 3.1647 3.1647 6.9755E-05 7.7032E-05 7.3394E-05 4 4 3 20Re 75 186.20700 4-4-(11) 2 29.251 2.7553 4.4930 9.5549E-05 9.6362E-05 9.5956E-05 4 4 3 21

Os 76 190.20000 4-4-(10) 2 27.983 2.7352 4.3190 1.6398E-05 1.4666E-05 1.5532E-05 4 4 3 22

Ir 77 192.22000 4-4-(9) 4 56.597 3.8394 3.8394 3.6853E-05 3.7665E-05 3.7259E-05 4 4 3 23

Pt 78 195.08000 4-4-(8) 4 60.407 3.9237 3.9237 2.6656E-04 2.7871E-04 2.7263E-04 4 4 3 24

Au 79 196.96700 4-4-(7) 4 67.866 4.0790 4.0790Hg 80 200.59000 4-4-(6) 2 45.087 3.9950 2.8250

Tl 81 204.38300 4-4-(5) 2 56.074 3.4380 5.4780

Pb 82 207.20000 4-4-(4) 4 121.258 4.9496 4.9496

Bi 83 208.98000 4-4-(3) 6 210.126 4.5333 11.8065

Po 84 209.00000 4-4-(2) 3 109.781 5.0780 4.9160


Pt 78 195.08000 4-4-(8) 4 60.407 3.9237 3.9237 2.6656E-04 2.7871E-04 2.7263E-04 4 4 3 24

Au 79 196.96700 4-4-(7) 4 67.866 4.0790 4.0790Hg 80 200.59000 4-4-(6) 2 45.087 3.9950 2.8250

Tl 81 204.38300 4-4-(5) 2 56.074 3.4380 5.4780

Pb 82 207.20000 4-4-(4) 4 121.258 4.9496 4.9496

Bi 83 208.98000 4-4-(3) 6 210.126 4.5333 11.8065

Po 84 209.00000 4-4-(2) 3 109.781 5.0780 4.9160

At 85 210.00000 4-4-(1) 3 109.781 5.0780 4.9160Rn 86 222.00000 4-4-0

Fr 87 223.00000 4-4-1

Ra 88 226.02500 4-4-2 2 136.432 5.1480 5.1480

Ac 89 227.02800 4-4-3 4 149.806 5.3110 5.3110

Th 90 232.03800 4-4-4 4 131.461 5.0847 5.0847 6.1560E-05 8.4391E-05 7.2976E-05 4 4 4 4

Pa 91 231.03600 4-4-5 4 100.233 5.5660 3.2140U 92 238.02900 4-4-6 2 40.708 3.4400 3.4400 4.1912E-04 4.1928E-04 4.1920E-04 4 4 4 6

Np 93 237.04800 4-4-7 2 43.614 3.5200 3.5200

Pu 94 244.00000 4-4-8 4 99.704 4.6370 4.6370 4.3931E-04 5.3842E-04 4.8887E-04 4 4 4 8

Am 95 243.00000 4-4-9 4 117.217 4.8940 4.8940

Cm 96 247.00000 4-4-10 2 59.967 3.4960 5.6655Bk 97 247.00000 4-4-11 2 55.930 3.4160 5.5345

Cf 98 252.10000 4-4-12

Es 99 252.00000 4-4-13

Fm 100 257.00000 4-4-14

Md 101 258.00000 4-4-15

No 102 259.000004-4-16 / 5-4-(16)Lw 103 260.00000 5-4-(15)

Rf 104 261.10880 5-4-(14)

Db 105 262.11410 5-4-(13)

Sg 106 266.12190 5-4-(12)

Bn 107 264.12000 5-4-(11)Hs 108 277.00000 5-4-(10)

Mt 109 268.13880 5-4-(9)

Ds 110 271.00000 5-4-(8)

Rg 111 272.15350 5-4-(7)


Element At. No.H 1

He 2

Li 3

Be 4

B 5C 6

N 7

O 8

F 9

Ne 10

Na 11Mg 12

Al 13

Si 14

P 15

S 16

Cl 17Ar 18

K 19

Ca 20

Sc 21

Ti 22V 23

Cr 24

Mn 25

Fe 26

Co 27

Ni 28Cu 29

Zn 30

Ga 31

Ge 32

As 33

Se 34Br 35

Kr 36

Rb 37

Sr 38

Y 39

At. Weight Rot. Displ. Zuc Vuc edge_a edge_c1.00790 2-1-(1) 4 74.532 3.7500 6.1200

4.00260 2-1-0 2 64.348 3.5700 5.8300

6.94100 2-1-1 2 43.218 3.5093 3.5093

9.01202 2-1-2 2 16.011 2.2680 3.5942

10.81000 2-1-3 50 391.422 8.7710 5.088012.01100 2-1-4 / 2-2-(4) 8 45.377 3.5668 3.5668

14.00670 2-2-(3) 4 94.233 4.0390 6.6700

15.99940 2-2-(2) 6 106.606 3.3070 11.2560

18.99840 2-2-(1) 8 128.395 7.2840 5.5000

20.17900 2-2-0 4 87.884 4.4460 4.4460

22.98980 2-2-1 2 78.987 4.2906 4.290624.30500 2-2-2 2 46.193 3.2028 5.1998

26.98150 2-2-3 4 66.410 4.0496 4.0496

28.08550 2-2-4 / 3-2-(4) 8 160.134 5.4304 5.4304

30.97380 3-2-(3) 84 1829.671 9.1500 21.9100

32.06000 3-2-(2) 18 433.779 10.8180 4.2800

35.45300 3-2-(1) 8 230.910 6.2400 4.480039.94800 3-2-0 4 149.790 5.3108 5.3108

39.09830 3-2-1 2 152.616 5.3440 5.3440

40.08000 3-2-2 4 173.926 5.5820 5.5820

44.95590 3-2-3 2 50.004 3.3090 5.2733

47.88000 3-2-4 2 35.713 2.9530 4.729050.94150 3-2-5 2 28.092 3.0399 3.0399

51.99600 3-2-6 2 24.011 2.8850 2.8850

54.93800 3-2-7 4 57.647 3.8630 3.8630

55.84700 3-2-8 2 23.554 2.8665 2.8665

58.93320 3-2-9 / 3-3-(9) 2 22.147 2.5071 4.0686

58.69000 3-3-(8) 4 43.758 3.5239 3.523963.54600 3-3-(7) 4 47.253 3.6153 3.6153

65.39000 3-3-(6) 2 30.421 2.6648 4.9467

69.72000 3-3-(5) 8 155.751 4.5107 7.6448

72.59000 3-3-(4) 8 181.067 5.6574 5.6574

74.91260 3-3-(3) 6 129.145 3.7600 10.5480

78.96000 3-3-(2) 3 81.301 4.3552 4.949579.90400 3-3-(1) 4 260.568 6.6700 4.4800

83.80000 3-3-0 4 187.247 5.7210 5.7210

85.46780 3-3-1 2 174.209 5.5850 5.5850

87.62000 3-3-2 4 225.277 6.0847 6.0847

88.90590 3-3-3 2 66.023 3.6474 5.7306

avg CRC Spr n_M tp_eff ts_eff te_eff T_flag T_factor

8.4754E-06 2 2 2 1 N 0.0000E+001.1855E-05 2 2 2 2 N 0.0000E+00

2.1018E-05 2 2 2 3 Y 4.0468E-03

5.6902E-06 3 3 2 1 N 0.0000E+00

2.0304E-05 3 3 2 2 N 0.0000E+00

2.5443E-04 3 3 2 3 N 0.0000E+00

1.7743E-04 3 3 2 4 N 0.0000E+004.0066E-04 3 3 2 5 Y 9.0311E-03

3.0320E-04 3 3 2 6 N 0.0000E+00

7.8904E-04 3 3 2 7 Y 9.4924E-03

3.8431E-06 3 3 3 1 N 0.0000E+00

3.4084E-05 3 3 3 2 N 0.0000E+00

8.1511E-05 3 3 3 3 N 0.0000E+00


Zr 40Nb 41

Mo 42

Tc 43

Ru 44

Rh 45Pd 46

Ag 47

Cd 48

In 49

Sn 50

Sb 51Te 52

I 53

Xe 54

Cs 55

Ba 56

La 57Ce 58

Pr 59

Nd 60

Pm 61

Sm 62Eu 63

Gd 64

Tb 65

Dy 66

Ho 67

Er 68Tm 69

Yb 70

Lu 71

Hf 72

Ta 73

W 74Re 75

Os 76

Ir 77

Pt 78

Au 79Hg 80

Tl 81

Pb 82

Bi 83

91.22400 3-3-4 2 46.562 3.2320 5.1470 1.0924E-0492.90640 3-3-5 2 35.950 3.3004 3.3004 2.3908E-04

95.94000 3-3-6 2 31.173 3.1472 3.1472 1.0948E-04

97.90720 3-3-7 2 28.670 2.7430 4.4000 2.6564E-04

101.07000 3-3-8 2 27.135 2.7056 4.2803 6.3180E-05

102.90600 3-3-9 / 4-3-(9) 4 55.015 3.8033 3.8033 1.6178E-04106.42000 4-3-(8) 4 58.873 3.8902 3.8902 7.8462E-04

107.86800 4-3-(7) 4 68.227 4.0862 4.0862

112.41000 4-3-(6) 2 42.927 2.9737 5.6058

114.82000 4-3-(5) 2 52.333 3.2530 4.9455

118.71000 4-3-(4) 8 272.854 6.4860 6.4860

121.75000 4-3-(3) 6 181.231 4.3083 11.2743127.60000 4-3-(2) 3 101.287 4.4467 5.9149

126.90500 4-3-(1) 8 341.046 7.2701 4.7900

131.29000 4-3-0 4 237.982 6.1970 6.1970

132.90500 4-3-1 2 220.897 6.0450 6.0450 5.3856E-06

137.33000 4-3-2 2 125.000 5.0000 5.0000 6.8174E-06

138.90600 4-3-3 4 148.540 5.2960 5.2960 6.1081E-05140.12000 4-3-4 4 137.484 5.1612 5.1612 1.4927E-03

140.90800 4-3-5 4 137.468 5.1610 5.1610 3.2909E-03

144.24000 4-3-6 2 68.356 3.6570 5.9020 3.5257E-03

145.00000 4-3-7 2 68.356 3.6570 5.9020

150.36000 4-3-8 2 67.381 3.6210 5.9340 7.8822E-04151.96000 4-3-9 2 94.258 4.5510 4.5510 1.4173E-02

157.25000 4-3-10 2 66.105 3.6290 5.7960 1.1638E-01

158.92540 4-3-11 2 63.939 3.6010 5.6936 1.1858E-01

162.50000 4-3-12 2 63.052 3.5840 5.6680 6.6038E-02

164.93000 4-3-13 2 62.238 3.5773 5.6158 4.8578E-02

167.26000 4-3-14 2 61.010 3.5500 5.5900 3.1088E-02168.93400 4-3-15 2 60.197 3.5375 5.5546 1.7408E-02

173.040004-3-16 / 4-4-(16) 2 165.126 5.4862 5.4862 1.6987E-05

174.96700 4-4-(15) 2 58.993 3.5031 5.5090 1.2867E-04

178.49000 4-4-(14) 2 44.642 3.1946 5.0510 6.8351E-05

180.94800 4-4-(13) 2 36.127 3.3058 3.3058 1.7789E-04

183.85000 4-4-(12) 2 31.695 3.1647 3.1647 7.3394E-05186.20700 4-4-(11) 2 29.251 2.7553 4.4930 9.5956E-05

190.20000 4-4-(10) 2 27.983 2.7352 4.3190 1.5532E-05

192.22000 4-4-(9) 4 56.597 3.8394 3.8394 3.7259E-05

195.08000 4-4-(8) 4 60.407 3.9237 3.9237 2.7263E-04

196.96700 4-4-(7) 4 67.866 4.0790 4.0790200.59000 4-4-(6) 2 45.087 3.9950 2.8250

204.38300 4-4-(5) 2 56.074 3.4380 5.4780

207.20000 4-4-(4) 4 121.258 4.9496 4.9496

208.98000 4-4-(3) 6 210.126 4.5333 11.8065

3 3 3 4 N 0.0000E+003 3 3 5 N 0.0000E+00

3 3 3 6 N 0.0000E+00

3 3 3 7 N 0.0000E+00

3 3 3 8 N 0.0000E+00

3 3 3 9 N 0.0000E+003 3 3 10 Y 1.8005E-02

4 4 3 1 N 0.0000E+00

4 4 3 2 N 0.0000E+00

4 4 3 3 N 0.0000E+004 4 3 4 Y 1.0151E-02

4 4 3 5 Y 1.0210E-02

4 4 3 6 Y 1.0509E-02

4 4 3 8 Y 1.1113E-024 4 3 9 Y 8.0290E-03

4 4 3 16 N 0.0000E+00

4 4 3 17 N 0.0000E+00

4 4 3 18 N 0.0000E+00

4 4 3 20 N 0.0000E+004 4 3 21 N 0.0000E+00

4 4 3 22 N 0.0000E+00

4 4 3 23 N 0.0000E+00

4 4 3 24 N 0.0000E+00


Ir 77

Pt 78

Au 79Hg 80

Tl 81

Pb 82

Bi 83

Po 84

At 85Rn 86

Fr 87

Ra 88

Ac 89

Th 90

Pa 91U 92

Np 93

Pu 94

Am 95

Cm 96Bk 97

Cf 98

Es 99

Fm 100

Md 101

No 102Lw 103

Rf 104

Db 105

Sg 106

Bn 107Hs 108

Mt 109

Ds 110

Rg 111

192.22000 4-4-(9) 4 56.597 3.8394 3.8394 3.7259E-05

195.08000 4-4-(8) 4 60.407 3.9237 3.9237 2.7263E-04

196.96700 4-4-(7) 4 67.866 4.0790 4.0790200.59000 4-4-(6) 2 45.087 3.9950 2.8250

204.38300 4-4-(5) 2 56.074 3.4380 5.4780

207.20000 4-4-(4) 4 121.258 4.9496 4.9496

208.98000 4-4-(3) 6 210.126 4.5333 11.8065

209.00000 4-4-(2) 3 109.781 5.0780 4.9160

210.00000 4-4-(1) 3 109.781 5.0780 4.9160222.00000 4-4-0

223.00000 4-4-1

226.02500 4-4-2 2 136.432 5.1480 5.1480

227.02800 4-4-3 4 149.806 5.3110 5.3110

232.03800 4-4-4 4 131.461 5.0847 5.0847 7.2976E-05

231.03600 4-4-5 4 100.233 5.5660 3.2140238.02900 4-4-6 2 40.708 3.4400 3.4400 4.1920E-04

237.04800 4-4-7 2 43.614 3.5200 3.5200

244.00000 4-4-8 4 99.704 4.6370 4.6370 4.8887E-04

243.00000 4-4-9 4 117.217 4.8940 4.8940

247.00000 4-4-10 2 59.967 3.4960 5.6655247.00000 4-4-11 2 55.930 3.4160 5.5345

252.10000 4-4-12

252.00000 4-4-13

257.00000 4-4-14

258.00000 4-4-15

259.000004-4-16 / 5-4-(16)260.00000 5-4-(15)

261.10880 5-4-(14)

262.11410 5-4-(13)

266.12190 5-4-(12)

264.12000 5-4-(11)277.00000 5-4-(10)

268.13880 5-4-(9)

271.00000 5-4-(8)

272.15350 5-4-(7)

4 4 3 23 N 0.0000E+00

4 4 3 24 N 0.0000E+00

4 4 4 4 N 0.0000E+00

4 4 4 6 N 0.0000E+00

4 4 4 8 N 0.0000E+00


Element At. No.H 1

He 2

Li 3

Be 4

B 5C 6

N 7

O 8

F 9

Ne 10

Na 11Mg 12

Al 13

Si 14

P 15

S 16

Cl 17Ar 18

K 19

Ca 20

Sc 21

Ti 22V 23

Cr 24

Mn 25

Fe 26

Co 27

Ni 28Cu 29

Zn 30

Ga 31

Ge 32

As 33

Se 34Br 35

Kr 36

Rb 37

Sr 38

Y 39

susc_wo_nf nf_calc susc_calc

2.1312E-04 0.0398 8.4754E-068.8330E-04 0.0134 1.1855E-05

5.4823E-03 0.0038 2.1018E-05

3.0911E-04 0.0184 5.6902E-06

1.1333E-03 0.0179 2.0304E-05

2.7928E-03 0.0911 2.5443E-04

4.6758E-03 0.0379 1.7743E-041.3807E-02 0.0290 4.0066E-04

6.3641E-03 0.0476 3.0320E-04

1.7774E-02 0.0444 7.8904E-04

4.2451E-04 0.0091 3.8431E-06

1.4306E-03 0.0238 3.4084E-05

3.4480E-03 0.0236 8.1511E-05


Zr 40Nb 41

Mo 42

Tc 43

Ru 44

Rh 45Pd 46

Ag 47

Cd 48

In 49

Sn 50

Sb 51Te 52

I 53

Xe 54

Cs 55

Ba 56

La 57Ce 58

Pr 59

Nd 60

Pm 61

Sm 62Eu 63

Gd 64

Tb 65

Dy 66

Ho 67

Er 68Tm 69

Yb 70

Lu 71

Hf 72

Ta 73

W 74Re 75

Os 76

Ir 77

Pt 78

Au 79Hg 80

Tl 81

Pb 82

Bi 83

Po 84

5.8549E-03 0.0187 1.0924E-046.0782E-03 0.0393 2.3908E-04

8.0212E-03 0.0136 1.0948E-04

1.5292E-02 0.0174 2.6564E-04

1.6710E-02 0.0038 6.3180E-05

1.6478E-02 0.0098 1.6178E-043.5504E-02 0.0221 7.8462E-04

6.4420E-04 0.0084 5.3856E-06

1.8832E-03 0.0036 6.8174E-06

5.0359E-03 0.0121 6.1081E-051.7221E-02 0.0867 1.4927E-03

1.9048E-02 0.1728 3.2909E-03

2.2704E-02 0.1553 3.5257E-03

2.7698E-02 0.0285 7.8822E-041.8258E-02 0.7763 1.4173E-02

1.2514E-02 0.0014 1.6987E-05

3.7371E-02 0.0034 1.2867E-04

4.7943E-02 0.0014 6.8351E-05

4.7010E-02 0.0016 7.3394E-057.5933E-02 0.0013 9.5956E-05

7.9933E-02 0.0002 1.5532E-05

7.3459E-02 0.0005 3.7259E-05

7.3395E-02 0.0037 2.7263E-04


Pt 78

Au 79Hg 80

Tl 81

Pb 82

Bi 83

Po 84

At 85Rn 86

Fr 87

Ra 88

Ac 89

Th 90

Pa 91U 92

Np 93

Pu 94

Am 95

Cm 96Bk 97

Cf 98

Es 99

Fm 100

Md 101

No 102Lw 103

Rf 104

Db 105

Sg 106

Bn 107Hs 108

Mt 109

Ds 110

Rg 111

7.3395E-02 0.0037 2.7263E-04

9.7121E-03 0.0075 7.2976E-05

1.5914E-02 0.0263 4.1920E-04

2.3356E-02 0.0209 4.8887E-04

Avg. n_f 0.0471

Table VI. Susceptibilities of Paramagnetic Elements (Crystalline, No Loss)


Table Notes:

1) The column labeled "T_flag" indicates whether the paramagnet is Class I ("N") or Class II ("Y").

2) The column labeled T_factor shows the results for the second term of Eq. (131).

3) Notice that some nominally electronegative elements assume their alternative electropositive rotations and so have apositive magnetic susceptibility.

4) As with the diamagnetic elements, the two sources for observed magnetic susceptibilities are the CRC compilation andthe Springer compilation, averaged.

5) Edge_c of the crystal unit cell has been used for all calculations, rather than edge_a, and there is no need for k_dpm.

6) The crystal unit cell values come from the Reciprocal System Data Base (which in turn makes use of the data in Ref. [58]and Ref. [65]).

7) The average ratio of observed to calculated values = 1, with a correlation of 1.000--due to the "best-fit" values of nf. Theaverage value of nf = .0471. So only approx. 5% of the atoms have the induced magnetic charges.

8) No parameter other than nf has been "manipulated." The nf values should be considered as predictions. Presumably, thegas values will be roughly 1/nf or approximately 21 times the values given here.

9) Eq. (131) should work down to T = T1, the temperature at the end of the first specific heat line segment (which begins atT0, which is itself 1/2 x Tsc, the superconducting temperature). See Ref. [1], pp. 84-86.


obs_O 0.0107obs_O .0102Zuc

Av Vuc_SI 106

4

The observed value comes from the CRC handbook (Ref. [16], p. 4-145), and converted to a pure and SI:

O 0.0107

O u_SI

nf

Zuc

2 nM Mu_weber edgec

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

.

w 15.9994KT 54nM 2(all edges same)

te 6ts 1tp 2medgec 6.830 1010

m3Vuc_SI 318.612 1030

Zuc 16nf .83

Note 1: The Special Case of Oxygen

At first glance, both the conventional theory and the Reciprocal System would categorize oxygen as diamagnetic, notparamagnetic. Conventional theory says that only atoms with unpaired electrons or an "odd number" of electrons can beparamagnetic. But oxygen is claimed to have 16 electrons, an even number. As for the Reciprocal System, oxygen has theatomic rotational displacements 2-2-(2) and should therefore be diamagnetic. Even Larson says (Ref. [1], p. 242), "Just whyoxygen is the element in the negative division in which the polarity reversal takes place is not yet known."

The alternative equivalent electropositve rotational displacements of oxygen are: 2-1-6. Here is the calculation for solid oxygen(cubic-FCC) at T = 54 K, assuming 83% complete saturation (and data from Reciprocal System Data Base):


The Springer compilation has the same value. At complete saturation, nf = 1, and O = .0907This should be testable witha strong enough field. As it stands, we do not have a means of verifying that nf = .83 for the experimental setup(s), and soit will have to be treated as a prediction.

Note 2: Antiparamagnetic or "Antiferromagnetic" Substances

If the magnetic dipoles in a crystal unit cell are oriented in opposite directions, then the first term of Eq. (131) cancels out,leaving just the second term:

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

(133)

Unfortunately, the magnetochemists, like Selwood (Ref. [27], Chapter 13, pp. 326-350), define as "antiferromagnetic"substances which have the following magnetic susceptibibility vs. temperature curve (p.327, Fig. 108): the first leg is alinear increase of suceptibility with temperature up to a maximum (the Neel point); the second leg is paramagnetic, withsusceptibility decreasing as C/T (Curie) or C/(T+) (Curie-Weiss).

The Reciprocal System interpretation is: the first leg is not temperature dependent--it's just the first term of Eq. (131), butwith nf increasing from 0 to its maximum and with nM increasing from 0 to its maximum (therefore, the dipoles are notneutralized). At the Neel point, the second term kicks in, but nf and nM remain constant (at the maximum values), apparently.One would think that all paramagnetic elements would have these two legs. Note: this interpretation is provisional, simplybecause we don't have enough empirical data to make a final determination; most of the tests do not go below 100 K, sothe first leg is not seen. It's possible that the Neel point coincides with T1, the end of the first segment of the specific heatcurve (Ref. [1], p. 86).


b. Compounds

As Larson shows in the second edtion of Ref. [2], pp. 219-284, all chemical compounds are combinations ofelectropositive and electronegative components. In Ref. [1], p. 242, he says, "The presence of any significant amount ofmotion in time (space displacement) in a molecular structure prevents establishment of the positive magneticorientation. All compounds, except those that are ferromagnetic, or heavily weighted with paramagnetic elements, aretherefore diamagnetic. The overwhelming preference for diamagnetism in the compounds is probably what led to thecurrently accepted hypothesis of a universal diamagnetism." Perusal of the table of magnetic susceptibilities of theelements and inorganic compounds in the CRC Handbook (Ref. [16], pp. 4-142 to 4-147) confirms that diamagneticcompounds far outnumber paramagnetic compounds. However, there are a sufficient number (five) to make an attemptat a comparison of theory with observation.


Mn: 3-2-7 n1 4 tp1 3 ts1 2 te1 7

O: 2-1-6 n2 4 tp2 2 ts2 1 te2 6 (alternative electropositive equivalent)

sdpm edgea

MnO u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

MnO 0.0045MnO

obs_SI_MnO0.9922

In this compound the average dipole length = edge_a. Selwood, Ref. [27], p. 332, says that this compound is"antiferromagnetic" with a "sharp antiferromagnetic Curie point at -151oC" and follows the Curie-Weiss law with "5.9

Bohr magnetons and a Weiss constant of about 610o."

MnO obs_SI_MnO 4850 106

4 4

Av 89.735 1024

obs_SI_MnO 0.0045

nf .34 Zuc 4 nM 2 edgea 4.4770 1010

m Vuc_SI 89.735 1030

m3 nuc 8

(by iteration) (all edges same)

s0 2.2385 1010

m w 70.9370 T 293.15 K


For the graphs below, the starting point will be at the Neel point (-151 + 273.15 = 122.15 K):

(which does agree with observation atroom temperature)

MnO_conv 0.0044MnO_conv

CM

T 4

Zuc

Av Vuc_SI 106

CM 4.3146CM

W2

14.062

Curie Constant on molar basis

W 29.205W B 4.95Weiss magneton

Then:

(empirically determined) 610Weiss constantB 5.9Bohr magnetonGiven:

Using the conventional equations in the literature (which are semi-empirical and semi-theoretical):


200 3000.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

CM

TT 4

Zuc

Av Vuc_SI 106

u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

TT

1

Vuc_SI

st_u 102

3

TT

Figure 22. Paramagnetic Susceptibility of MnO (regular ReciprocalSystem curve is in blue; conventional theory is in red; starting point isSelwood's supposed Curie point, 122.15 K)

Neither Selwood nor Van Vleck nor Bhatnagar nor Stoner provides a theoretical calculation for B or for MnO; it seemsodd that should be so large and why B should be 5.9. In the Reciprocal System, nM = 2, an integer, for MnO. On theother hand, it's not clear why we have nf = .34; however, from T0_MnO to 122.15 K, nf probably goes from 0 to .34 and nM

goes from 0 to 1 to tp, which accounts for the increase in susceptibility up to 122.15 K. The peak Reciprocal Systemsusceptibility value for MnO is .0082 at the supposed Curie [Neel] point (but we don't have the empirical data to confirm it).


T 293.15 K

Cr: 3-2-6 n1 2 tp1 3 ts1 2 te1 6

Cl: 3-2-(1) n2 4 tp2 2 ts2 2 te2 7 (alternative electropositive equivalent)

sdpm edgeb (longest edge would be the most likely dipole length)

CrCl2 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

CrCl2 0.0022CrCl2

obs_SI_CrCl21.0028

CrCl2 obs_SI_CrCl2 7230 106

4 2

Av 138.026 1024

obs_SI_CrCl2 0.0022

nf .25 Zuc 2 nM 2 edgea 5.9740 1010

m Vuc_SI 138.026 1030

nuc 6m3

(by iteration) edgeb 6.6240 1010

m (orthorhombic crystal cell)

edgec 3.4880 1010

m

s0 2.9169 1010

m w 122.9020


In this compound the average dipole length = edge_b (longest side). Selwood, Ref. [27], p. 332, says that thiscompound is "antiferromagnetic" and that "the susceptibility is a complicated function of temperature." No parametersfor the Curie-Weiss law are provided. Van Vleck (Ref. [22], p. 304) covers CrCl3, not CrCl2. Stoner says (Ref. [28], p.325) that the effective value of the Bohr magneton for this compound is 4.81, for use in the Curie (not Curie-Weiss) law.Here:

(M = molar value)

M_CrCl2_conv4.81

2

2.8392

1

T M_CrCl2_conv 0.0098 (based on pp. 282, 325 of Stoner)

CrCl2_conv M_CrCl2_conv 4 Zuc

Av Vuc_SI 106

CrCl2_conv 0.003

This is quite a bit larger than the currently accepted observed value.


200 3000.002

0.003

0.004

0.005

0.006

0.007

4.812

2.8392

1

TT 4

Zuc

Av Vuc_SI 106

u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

TT

1

Vuc_SI

st_u 102

3

TT

Figure 23. Paramagnetic Susceptibility of CrCl2 (Reciprocal System curve is in blue; conventional theory is in red)


Cu: 3-3-(7) n1 4 tp1 3 ts1 2 te1 11 (alternative electropositive equivalent)

O: 2-2-(2) n2 4 tp2 2 ts2 1 te2 6 (alternative electropositive equivalent)

sdpm edgec (longest side) T 293.15 K

CuO u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

CuO 0.00026CuO

obs_SI_CuO1.0304

CuO obs_SI_CuO 238 106

4 4

Av 79.940 1024

obs_SI_CuO 0.00025

nf .013 Zuc 4 nM 2 edgea 4.6530 1010

m Vuc_SI 79.940 1030

nuc 8m3

(by iteration) edgeb 3.4100 1010

m (monoclinic crystal cell)

edgec 5.1080 1010

m

s0 1.9472 1010

m w 79.5450 T 293.15 K


Selwood (Ref. [27], p. 335 says, "Cupric oxide has a very low paramagnetism. There seems little doubt that it exhibitsantiferromagnetism, with a Curie [Neel] point at about 150o C." He does not supply values for the Curie or Curie-Weisslaws. And neither do Bhatnagar, Stoner, or Van Vleck. Presumably, at

T 150 273.15 T 423.15 K nf 1 (maximum value, might be lower)

CuO_sat u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

CuO_sat 0.0166

The references do not supply an observed value of CuO_sat at this temperature.


4-3-15 n1 32 tp1 4 ts1 3 te1 15


sdpm edgec T 293.15 K

Tm2O3 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

Tm2O3 0.01497Tm2O3

obs_SI_Tm2O31.0058

None of the references have empirical values for the Curie or Curie-Weiss laws.

Tm2O3 obs_SI_Tm2O3 51444 106

4 16

Av 1153.661 1024

obs_SI_Tm2O3 0.01488

nf .36 Zuc 16 nM 2 edgec 10.4880 1010

m Vuc_SI 1153.661 1030

nuc 80m3

(by iteration) (all edges same)

s0 2.2462 1010

m w 385.8660 T 293.15 K

Tm:


Cr: 3-2-6 n1 12 tp1 3 ts1 2 te1 6


sdpm edgec T 293.15 K

Cr2O3 u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

T

1

Vuc_SI

st_u 102

3

Cr2O3 0.00084Cr2O3

obs_SI_Cr2O30.9937

Cr2O3 obs_SI_Cr2O3 1960 106

4 6

Av 288.716 1024

obs_SI_Cr2O3 0.00085

nf .037 Zuc 6 nM 2 edgea 4.9540 1010

m Vuc_SI 288.716 1030

nuc 30m3

(by iteration)edgec 13.5840 10

10 m

s0 2.0130 1010

m w 151.9900 T 293.15 K


nf .056

For the graphs below, the starting point will be at the Neel point (50 + 273.15 = 323.15 K); also, by iteration, at thattemperature:

MnO_conv 0.0012MnO_conv

CM

T 4

Zuc

Av Vuc_SI 106

CM 1.7898CM

W2

14.062

Curie Constant on molar basis

T 323.15T 50 273.15W 18.81W B 4.95Weiss magneton

Then:

("well over 300") 305Weiss constantB 3.8Bohr magnetonGiven:

Using the conventional equations in the literature (which are semi-empirical and semi-theoretical):

According to Selwood (Ref. [27], p. 330): "Chromia shows a very weakly defined antiferromagnetic Curie [Neel] point in theneighborhood of 45 to 55o C; the susceptibility at liquid air temperature is definitely lower thant at room temperature. Abovethe Curie [Neel] point, it has an approximately normal moment of about 3.8 Bohr magnetons, but a Weiss constant of well over

300o."


350 4000.0011

0.00112

0.00114

0.00116

0.00118

0.0012

0.00122

CM

TT 4

Zuc

Av Vuc_SI 106

u_SI

nf floornuc

2

nM Mu_weber sdpm

Vuc_SI

MP_nat_t

1

nuc n1

tp1

1

ts1

1

te1

1

n2

tp2

1

ts2

1

te2

1

u_SI nf Zucw

1

Tt_u

TT

1

Vuc_SI

st_u 102

3

TT

Figure 24. Paramagnetic Susceptibility of Cr2O3 (ReciprocalSystem curve is in blue; conventional theory is in red; starting point isSelwood's supposed Curie point, 323.15 K)

Neither Selwood nor Van Vleck nor Bhatnagar nor Stoner provides a theoretical calculation for B or for Cr2O3; it seemsodd that should be so large and why B should be 3.8. In the Reciprocal System, nM = 2, an integer, for Cr2O3. On theother hand, it's not clear why we have nf = .056 (at and above the Neel point); however, from T0_Cr2O3 to 323.15 K, nf

probably goes from 0 to .056 and nM goes from 0 to 1 to tp, which accounts for the increase in susceptibility up to 323.15 K.The peak Reciprocal System susceptibility value for Cr2O3 is .0012 at the supposed Neel point (but we don't have theempirical data to confirm it).


That's about it for solid paramagnatic compounds--as explained above, practically all compounds are diamagnetic. Fourof these paramagnetic compounds contain oxygen, itself oddly paramagnetic, which probably explains why thesecompounds are paramagnetic. The other compound contains Cl--it's interesting that we needed to use the electropositiverotations of this element for the dielectric constant calculation, as well.


Supplement: The Stern-Gerlach Experiment

Before leaving the discussion of diamagnetism and paramagnetism, it's important to consider the Stern-Gerlachexperiment. Here we are dealing with individual atoms in the vapor state, rather than with atoms in a crystal unit cell of thesolid state. The experiment is commonly claimed to have disproved Classical Mechanics and proved Quantum Mechanics;while we can agree that it disproved Classical Mechanics, because the results show that magnetic effects are quantizedrather than continuous, we cannot agree that it has proved Quantum Mechanics, because the Reciprocal System is alsoconsistent with the results, as will be shown.

A diagram of the experiment is shown in Figure 25 (adapted from Selwood, Ref. [27], p. 139, Fig. 64). Other references forthe experiment include Ref. [18], pp. 215-218; Ref. [24], pp. 114-146; Ref. [25], pp. 100-102; Ref. [26], pp. 178-182; Ref.[28], pp. 213-228; Ref. [29], pp. 12-20; Ref. [30], pp. 278-280; and Ref. [66], pp. 120-123. Magnetic moments for 20elements were obtained using the methods of Stern and Gerlach over the 1920's and 1930's. After that, experimentalphysicists switched to resonance techniques, and focused on "nuclear" moments rather than on total atomic moments, asdescribed by Ramsey in Ref. [25].

A small quantity of the substance under investigation is placed in a 200 watt, 2.5 amp electric filament cyclindrical oven,measuring 20 mm long and 7 mm diameter. The oven aperature is a slit, and there are two additional collimating slits,exactly aligned by telescope. The electromagnet pole pieces (the first being a wedge having a 70o angle and the secondbeing a channel with a height of 3-4 mm and a depth of 3 cm, the point of the wedge being 2.5 mm from the center of theopening of the channel) are usually 60 to 100 mm long, and the usual inhomegenous field gradient is in the range of 60tesla/m (measured, and presumably even across the height of the front of the channel); a homogeneous field would notcause any deflection of the atoms. The detector is usually of the condensation type, either glass or metal. The oventemperature must be high enough to vaporize (sublimate) a sufficient number of atoms of the substance to obtain visibletraces on the detector. The interaction of the inhomogenous field with the atoms' magnetic moments deflects the atoms.From the measured separation of the traces the magnetic moment of the atoms can be inferred.

Whereas, the Quantum Mechanics interpretation of the experiment is that it reveals the permanent magnetic moment (ifany) of an atom, the Reciprocal System interpretation is that the magnetic field induces two magnetic charges (or twomultiple charge units) in the atoms--one north and one south.

If we assume that the stronger pole face (the wedge) is north and the weaker pole face (the channel) is south, an atom withit's south pole to the wedge side will be pulled toward the wedge, whereas an atom with its north pole to the wedge side willbe pushed to the channel side, and vice versa, by Coulomb's magnetic law. Hence there will be two traces, if all atoms havethe same value of nM. If the atoms have 2 or 3 or 4 different values of nM, then there will be twice that number of tracesdisplayed. Regardless, by probability, half of the atoms will deflect one way and half the other way.


Side View of Wedge and Channel Pole Pieces--Creates Inhomogeneous Magnetic Field

Top View

Screen Detector of Traces

Pole Pieces

Collimating Slit

Collimating Slit

Source--Oven

Figure 25. Diagram of Stern-Gerlach Apparatus


weber-meteradpm_u 9.3175 1037

(135)adpm_u

1m

sec

cSIm

sec 1

Tv_u

Tv_u nM Mu_weber Du

(134)nM 1mDu 3.359 1015

For the electrical resistivity calculations, involving the high speed electron gas, we used Tu (see Ref. [53]). For the solid stateparamagnetic calculations, we used Tt_u. Now for the Stern-Gerlach experiment calculations, we must use Tv_u as theappropriate natural unit.

The length of the atomic magnetic dipole moment is simply the diameter of the atom. This size, in the Reciprocal System,corresponds to the diameter of the "nucleus" in conventional theory. It's calculated in the Reciprocal System Data Base (anddepends on the atomic magnetic rotational speeds, rather than directly on the rotational displacements), and the values havealready been used in Ref. [53]. The natural unit for atomic diameter is 3.359 x 10-15 m. Elements below Na, which have

incomplete rotational dimensions, have diameters in the range of 2 x 10-15 m.

At vapor temperatures, no atom can be in a ferromagnetic state, so the reduction factor previously used for the natural unit ofsusceptibility must be used for the natural unit of atomic magnetic dipole moment. Putting all this together we have

(time region, for solids and liquids)KTt_u 510.8

(borderline between time-space region and time region, for vapors)KTv_u 3.5978 109

(time-space region, for gases, including electron gases)KTu 7.20423 1012

Now we'll compute the Reciprocal System natural unit of atomic magnetic dipole moment, which involves both magneticvibration and thermal vibration. As Larson discusses in Ref. [1], pp. 59-60, there are three natual units of temperature in theReciprocal System:


(checks)kgm

3

coul sec

coul2

kg m m

2 coul

sec

joule/tesla=1

0_SI

xweber-mIn conventional base units:

joule/teslaadpm_Ag_som 1.0142 1023

adpm_Ag_som

adpm_Ag

0_SI

The Reciprocal System uses the Kennelly magnetic system, whereas Quantum Mechanics uses the Sommerfeld magneticsystem. It's evident from Jiles, Ref. [30], p. 16, Table 1.2, that we can convert from the Kennelly system to the Sommerfeldsystem by dividing the Kennelly value by the SI permeability of free space. So:

weber-meter for thisvalue of nM

adpm_Ag 1.2744 1029

adpm_Ag adpm_u

Tv_u

TAg

nM_Ag

1

DAg

Du

(This value, for silver, 4-3-(7), can theoretically range from 1 to 4.)nM_Ag 1

(This is somewhat of an estimate, because in many cases the experimenters didn'treally know what the vapor temperature was! Fraser uses this value in his examplecalculation, Ref. [24], p. 116. This is below the melting point, so there must havebeen sublimation.)

KTAg 1000

(from Reciprocal System Data Base)mDAg 1.277 1014

For silver, which was used in the original experiment of 1921, the parameters are:

(136)weber-meteradpm adpm_u

Tv_u

T

nM

1

D

Du


In space-time terms:

t2

s

s4

t3

1

ts

3 (checks)

In Quantum Mechanics, the Bohr mageton has the value

B_SI 9.2741 1024

joule/tesla

Therefore, in terms of the Bohr magneton,

adpm_Ag_som

B_SI1.0935

From Quantum Mechanics, the predicted value for the experiment with silver is +/- 1 Bohr magneton, due to the up/downspin of the alleged one unpaired electron (and zero total orbital angular momentum). The experimental error has beenestimated to be 10% (Ref. [24], p.138). Both the Reciprocal System and Quantum Mechanics therefore agree with theexperimental value. However, the Reciprocal System predicts that if much stonger fields could be used, we could get 2, 3,or even 4 times this value. Thus, it would be desirable for this or a similar experiment to be run with much stronger fields inorder to distinguish between the two theories. Both the Reciprocal System and Quantum Mechanics predict changes inatomic magnetic moment with temperature, but with opposite effects, so that would be another test.

Table VII, from Excel, shows the calculations for the 20 elements which have been investigated, together with the observedresults, in Bohr magnetons, for convenience. Three of these, Co, N, and Bi have observed values which are too uncertain tobe included in the observation column. The shaded cells in the table show the most probably value of nM. The T columnvalues are estimates obtained (as best as we can) from the references listed above. Eight of the zero values are somewhatdubious, because the Maxwellian distribution of velocities smudged the traces so badly that determination of the separationof the traces was difficult. For the remainder of the 17 elements, the Reciprocal System is in agreement with theexperimental values.


Element At. No. Rot. Displ. n_M max D (fm) T_mp (K) T (K) mu if n_M=0 mu if n_M=1 mu if n_M=2 mu if n_M=3 mu if n_M=4 mu_obsH 1 2-1-(1) 2 2.000 14.15 171 0 1.002 2.003 1.000

Li 3 2-1-1 2 2.000 453.16 400 0 0.428 0.856 1.000

N 7 2-2-(3) 2 2.000 63.15 100 0 1.713 3.426

O 8 2-2-(2) 2 2.000 55.15 100 0 1.713 3.426 1.670

Na 11 2-2-1 2 4.048 370.96 371 0 0.934 1.869 1.000

K 19 3-2-1 3 8.366 336.33 715 0 1.002 2.004 3.006 1.000

Fe 26 3-2-8 3 8.366 1808.16 1800 0 0.398 0.796 1.194 0.000Co 27 3-2-9/3-3-(9) 3 8.366 1765.16 1800 0 0.398 0.796 1.194

Ni 28 3-3-(8) 3 8.366 1726.16 1800 0 0.398 0.796 1.194 1.100

Cu 29 3-3-(7) 3 8.366 1356.16 1500 0 0.478 0.955 1.433 1.000

Zn 30 3-3-(6) 3 10.624 693.16 900 0 1.011 2.022 3.033 0.000

Pd 46 4-3-(8) 4 11.660 1823.16 1950 0 0.512 1.024 1.536 2.048 0.000

Ag 47 4-3-(7) 4 12.646 1234.16 1000 0 1.083 2.166 3.249 4.332 1.000

Cd 48 4-3-(6) 4 14.922 594.16 1200 0 1.065 2.130 3.195 4.260 0.000

Sn 50 4-3-(4) 4 12.646 505.16 1080 0 1.003 2.006 3.008 4.011 0.000Au 79 4-4-(7) 4 13.942 1336.16 1200 0 0.995 1.990 2.985 3.980 1.000

Hg 80 4-4-(6) 4 11.660 234.26 990 0 1.009 2.017 3.026 4.035 0.000

Tl 81 4-4-(5) 4 13.942 577.16 700 0 1.706 3.411 5.117 6.823 0.333

Pb 82 4-4-(4) 4 13.942 600.16 700 0 1.706 3.411 5.117 6.823 0.000

Bi 83 4-4-(3) 4 13.942 544.16 700 0 1.706 3.411 5.117 6.823

Table VII. Stern-Gerlach Experiment Calculations and Observations

The only difficulty here for the Reciprocal System is the claimed atomic magnetic moment of Tl, 1/3 B. Given that thenearby elements of Hg and Pb have zero moment (no induced magnetic charges), it seems rather odd that Tl would have anon-zero value. Quantum Mechanics explains this value as consistent with spectroscopy. More precise experiments shouldbe conducted to resolve this issue.


(total separation of traces)mstot_sep 0.000126stot_sep 2 sd

msd 6.2992 105

(deflection from center line, in x-direction)sd1

2

1

mAg

adpm_Ag

0_SI dBdz

y2

vAg2

(length of path in y-direction through magnet poles)my .1

(z-force on atom causing deflection in perpendicular direction, x)NFz 6.085 1022

Fz

adpm_Ag

0_SIdBdz

(magnetic field gradient in z-direction of inhomogeneous field)tesla/mdBdz 60

m/secvAg 519.4539vAg

3.5 kB TAg

mAg

(effective mean velocity according to Sternaverage energy = (3kBT + 4kBT) / 2

(estimated temperature of vapor)KTAg 1000

(mass of individual silver atom)kgmAg 1.79 1025

based on Ref. [66], p. 123; Ref. [29], pp. 14-15

Sample Calculation of Trace Separation for Silver


Ref. [67] is a YouTube video showing the Stern-Gerlach experiment, but with three sets of magnets. Quantum Mechanicsfails to give the observed result for this situation, whereas the Reciprocal System does. My comment on this YouTubevideo is:

"Quantum Mechanics is not the only game in town. A competing theory called the Reciprocal System can explain thisgentleman's video, as follows. The neutral atoms (silver, hydrogen, whatever) in the experiment do not initially have amagnetic moment or charge at all. Upon passing through the first magnet, each atom is induced to have two charges (oneN and one S). Consequently, the beam splits. Upon leaving the first magnet, the induction ends, the charges revert tothermal energy, etc..." [YouTube does not provide sufficient comment space for a full explanation, so it's implied.]

Obviously, the same result occurs for the second set and third set--the beam becomes charged (with two poles) and splitsin two, each time! If there's no gap between the sets of magnets, then the charges do not revert back to thermal energy.Either way, the beam is charged (with two poles) and it must split because, by probability, half are oriented in onedirection, and half in the other direction! This demonstrates that the Reciprocal System is correct, and QuantumMechanics is not. However, the conventional theoreticians refuse to accept this verdict and invoke the HeisenbergUncertainty Principle, which is what they always do when they find themselves in a tight spot.

Oh, and by the way, silver is diamagnetic in the solid state!


7. Ferromagnets and Magnetic Hysteresis Curves

a. Elements

Quantum Mechanics has numerous theories attempting to explain ferromagnetism--but none of them work. Jiles (Ref.[30], pp. 291-321) reviews these attempts and concludes: "It seems therefore that although the Heisenberg model isa useful concept the interactions between electrons in real solids are probably not the simple direct Heisenbergexchange."....[Also] "The drawback of the intinerant electron theory is that it is extremely difficult to make fundamentalcalculations based on it." [Also] "Therefore although the whole approach of Heitler-London, Heisenberg and Bethestill provides a useful conceptual framework for discussing the magnetic interactions of electrons, the method seemsto be ultimately inadequate and we await a better description which can give more accurate values...from firstprinciples."

The waiting is now over. The Reciprocal System is a unified, general theory and so it can be applied to any aspect ofthe physical world, including ferromagnetism.

Quoting Larson (Ref. [1], pp. 215-216):

"For an explanation in terms of the theory of the universe of motion, we need to consider the nature of the atomicmotion. If a two-dimensional positive [time] rotational vibration is added to the three-dimensional combination ofmotions that constitutes the atom it modifies the magnitudes of those motions, and the product is not the same atomwith a magnetic charge, it is an atom of a different kind [an isotope]. A magnetic charge, as a distinct entity, can existonly where an atom is so constituted that there is a portion of the atomic structure that can vibrate two-dimensionallyindependently of the main body of the atom. The requirements are met, so far as the magnetic rotation is concerned,where this rotation is asymmetric, that is, there are n displacement units in one of the two magnetic dimensions [ts]and n+1 in the other [tp].

"On this basis, the symmetrical B group of elements, which have magnetic rotations 1-1, 2-2, 3-3, and 4-4 areexcluded. While the magnetic charge has no third dimension, the electric rotation [te] whith which it is associated inthe three-dimensional motion of the atom must be independent of that associated with the remainder of the atom. Theelectric rotational displacement must therefore exceed 7, so that one complete unit (7 displacement units plus aninitial unit level) can stay with the main body of the magnetic rotation, while the excess applies to the magnetic charge.Furthermore, the electric displacement must be positive, as the reference system cannot accomodate two different


negative displacements (motion in time, [space displacements]), in the same atomic structure. The electronegativedivisions (III and IV) are thus totally excluded. The effect of all these exclusions is to confine the magnetic charges toDivision II elements of Groups 3A and 4A."

The ferromagnetic elements are listed in Table VIII, together with the properties we will now calculate. Theseproperties include the saturation magnetic flux density at 0 K, Bs; the magnetic remanence, Br; the external coercivemagnetic flux density, Bext_c; and the Curie temperature, Tc.

1) saturation magnetic flux density at 0 K, Bs

The equations for the susceptibilities given previously for diamagnets and paramagnets do not apply toferromagnets, because the latter have hysteresis curves, which means that a given external flux density can result indifferent internal flux densities and hence permeabilities and susceptibilities. Importantly, the reduction factor, u_SI,does not applyHowever, we can adapt the numerator of Eq. (127a) to calculate the saturation magnetic flux densityat 0 K.

Bs_calc IM

nf floorZuc

2

nM Mu_weber edgeuc

Vuc_SI tesla (137a)

or

teslaBs_calc IM

nf floorZuc

2

nM Mu_weber s0

Vuc_SI (137b)

depending on whether the magnetic dipole is along an edge of the unit cell (typically edge_c, if the unit cell is not cubic)or between the atoms the shortest distance apart (s0). IM is the "magnetic transmission ratio"--this is similar to theinterregional ratio, IR, and the rotational ratio, Id0, as described in other Reciprocal System books and papers.


IR1

156.4444 IR 0.0064 (used in the atomic force repulsion equation, time region)

Id0 2.1475 109

(used in the denominator of the natural gravitationalforce equation, time-space region)

Magnetic flux density has the space-time dimensions

t2

s4

This can be considered to be the square of the space-time dimensions of one-dimensional force:

t

s2

2

So, in a sense, the magnetic flux density is a two-dimensional force or "field intensity." (This is another one of thereasons why we have no need to use the H-vector of conventional magnetic physics. Also, note that magnetomotiveforce has the dimensions t2/s3, which means that it's really a two-dimensional "potential.") Like the other forcesdiscussed above, there must be a "transmission ratio" or factor in the equation for ferromagnetic Bs. For onedimension, there are three possible orthogonal directions, so the starting ratio is 1/3. But the magnetic flux density istwo-dimensional, so this must be squared: (1/3)2 = 1/9. And: a magnetic dipole includes two charges, so we finally getfor the magnetic transmission ratio:

IM1

3

21

3

2

IM 0.01235 (138)

The detailed calculations follow for each ferromagnetic element. The crystal data come from the Reciprocal SystemData Base. The observed values are from the ferromagnetic sections of Ref. [12], Ref. [15], and Ref. [30].


Co 3-2-9 FCC nM_Co 3 Zuc 4 nf .9 Vuc_SI 42.637 1030

m3

s0_Co 2.4123 1010

edgec_Co Vuc_SI

1

3 edgec_Co 4.0496 10

10 m

Bs_Co_calc IM

nf floorZuc

2

nM_Co Mu_weber s0_Co

Vuc_SI Bs_Co_calc 1.8116

Bs_Co_obs 1.797 (tesla)

(The HCP version of Co gives similar results).(99 % pure)

Fe 3-2-8 BCC nM_Fe 3 Zuc 2 nf 1 Vuc_SI 23.258 1030

m3

edgec_Fe Vuc_SI

1

3 edgec_Fe 2.8545 10

10 m (all edges same, 0 K) s0_Fe 2.4825 10

10 m

Bs_Fe_calc IM

nf floorZuc

2

nM_Fe Mu_weber edgec_Fe

Vuc_SI Bs_Fe_calc 2.1832 Bs_Fe_obs 2.187

(tesla) (99.95 % pure)


(The HCP version of Ni gives similar results.)

(99% pure)

Bs_Ni_obs .653

teslaBs_Ni_calc 0.6829Bs_Ni_calc IM

nf floorZuc

2

nM_Ni Mu_weber s0_Ni

Vuc_SI

(1/2 face diagonal in unit cell, confirmsillus. in Ref. [30], p. 229))

s0_Ni 2.5061 1010

(all edges)edgec 3.5175 1010

Nickel's nominal rotational displacements are 3-3-(8); soprobably only half have the alternative equivalent positiverotational displacements 3-2-10 which can take aferromagnetic charge. The maximum value of nM is 2, ratherthan 3.

edgec Vuc_SI

1

3Vuc_SI 43.522 10

30nf .5Zuc 4nM_Ni 2FCC3-2-10Ni


(tesla)

Bs_Gd_obs 2.488Bs_Gd_calc 2.4341Bs_Gd_calc IM

nf floorZuc

2

nM_Gd Mu_weber edgec_Gd3

2

Vuc_SI

ms0 3.5708 1010

edgec_Gd 5.6255 1010

edgea 3.4195 1010

m3Vuc_SI 65.778 1030

nf .8Zuc 2nM_Gd 4HCP4-3-10Gd

All of the rare earth elements have four magnetic charges and are HCP, but the axial ratio (edge_c/edge_a) is considerablyless than 1.633, which means that the atoms in the same basal plane are usually a little farther apart than those immediatelyabove and below (Wychoff, Ref. [65], p. 10); so, approximately, 50% of the time, the dipole length is edge_c, and 50% of thetime the dipole length is 2 x edge_c, because of the alternate misalignment. The average dipole length is then 1.5 x edge_c.According to Martin (Ref. [23], p. 76), Tb, Dy, Ho, and Er have "helical anti-ferromagnetism" or "helical ferromagnetism"--thismeans that the angle of the magnetic moment changes by some angle with each unit cell basal plane. From the standpointof the Reciprocal System, this effect seems to be captured by the 1.5 factor for dipole length of these elements. In all cases,except for Tm, the value of nf = 1. Tm has a reduction to .9. On the basis of Larson's statement above, elements Sm, Eu, andYb should be ferromagnetic, but they are anti-ferromagnetic or paramagnetic, unless in compounds (like V, Cr, and Mn). Ceamd Nd are also anti-ferromagnetic, except in compounds. Sm is treated as a paramagnet in this paper; Eu is on the fence.

Rare Earth Elements


edgea 3.3654 1010

edgec_Dy 5.56 1010

s0 3.5089 1010

m

Bs_Dy_calc

IM nf floorZuc

2

nM_Dy Mu_weber edgec_Dy3

2

Vuc_SI Bs_Dy_calc 3.1412 Bs_Dy_obs 3.768

Ho 4-3-13 HCP nM_Ho 4 Zuc 2 nf 1 Vuc_SI 60.043 1030

m3

edgea 3.3589 1010

edgec_Ho 5.3247 1010

s0 3.4570 1010

m

Bs_Ho_calc IM

nf floorZuc

2

nM_Ho Mu_weber edgec_Ho3

2

Vuc_SI Bs_Ho_calc 3.155 Bs_Ho_obs 3.909

(tesla)

Tb 4-3-11 HCP nM_Tb 4 Zuc 2 nf 1 Vuc_SI 62.736 1030

m3

edgea 3.4001 1010

edgec_Tb 5.4268 1010

s0 3.5028 1010

m

Bs_Tb_calc IM

nf floorZuc

2

nM_Tb Mu_weber edgec_Tb3

2

Vuc_SI Bs_Tb_calc 3.0775 Bs_Tb_obs 3.407

(tesla)

Dy 4-3-12 HCP nM_Dy 4 Zuc 2 nf 1 Vuc_SI 62.972 1030

m3


(tesla)

Bs_Tm_obs 2.884Bs_Tm_calc 2.8789Bs_Tm_calc IM

nf floorZuc

2

nM_Tm Mu_weber edgec_Tm3

2

Vuc_SI

ms0 3.4080 1010

edgec_Tm 5.2294 1010

edgea 3.3350 1010

m3Vuc_SI 58.161 1030

nf .9Zuc 2nM_Tm 4HCP4-3-15Tm

(tesla)

Bs_Er_obs 3.439Bs_Er_calc 3.1875Bs_Er_calc IM

nf floorZuc

2

nM_Er Mu_weber edgec_Er3

2

Vuc_SI

ms0 3.4628 1010

edgec_Er 5.4516 1010

edgea 3.3409 1010

m3Vuc_SI 60.848 1030

nf 1Zuc 2nM_Er 4HCP4-3-14Er


The average ratio of calculated to observed for all of the ferromagnetic elements is

Bs_Fe_calc

Bs_Fe_obs

Bs_Co_calc

Bs_Co_obs

Bs_Ni_calc

Bs_Ni_obs

Bs_Gd_calc

Bs_Gd_obs

Bs_Tb_calc

Bs_Tb_obs

Bs_Dy_calc

Bs_Dy_obs

Bs_Ho_calc

Bs_Ho_obs

Bs_Er_calc

Bs_Er_obs

Bs_Tm_calc

Bs_Tm_obs

90.9444

Note that our calculations are based on 100% pure crystal unit cells without imperfections. Actual experimentalspecimens undoubtedly have some impurities and crystal defects, and so that would account for the fairly smallaverage difference between calculated and observed values.


Co FCC nM_r_Co 1 Zuc 4 nf .9 Vuc_SI 43.238 1030

m3

s0 2.4235 1010

edgec Vuc_SI

1

3 edgec 3.5098 10

10 m

Br_Co_calc IM

nf floorZuc

2

nM_r_Co Mu_weber s0

Vuc_SI Br_Co_calc 0.5982 Br_Co_obs .5

(tesla) (99 % pure)

(The HCP version of Co gives similar results).

2) magnetic remanence

When the external magnetic flux density goes to zero, the ferromagnet loses some of its magnetism--one or twoferromagnetic charges revert back to thermal energy, leaving the atom with just one or two charges. Also the dipolelengths often shorten to the smallest interatomic distance, s0. Temperature seems to have little effect on remanence, andso all calculations here are for room temperature parameters, using Eq. (137a) or (137b).

Fe BCC nM_r_Fe 2 Zuc 2 nf 1 Vuc_SI 23.556 1030

m3

edgec Vuc_SI

1

3 edgec 2.8666 10

10 s0 2.4886 10

10 m

Br_Fe_calc IM

nf floorZuc

2

nM_r_Fe Mu_weber s0

Vuc_SI Br_Fe_calc 1.2529 Br_Fe_obs 1.30

(tesla) (99.8% pure)


Data for the remanence of rare earth elements are not available. Prediction: their values for nM_r would be 1 or 2,or possibly 3.

Br_Fe_calc

Br_Fe_obs

Br_Co_calc

Br_Co_obs

Br_Ni_calc

Br_Ni_obs

31.0086

Certainly within theexperimental accuracy.

(tesla)

Br_Ni_obs .4Br_Ni_calc 0.3462Br_Ni_calc IM

nf floorZuc

2

nM_r_Ni Mu_weber s0

Vuc_SI

ms0 2.5178 1010

edgec 3.5067 1010

edgec Vuc_SI

1

3

m3Vuc_SI 43.122 1030

nf .5Zuc 4nM_r_Ni 1FCCNi


3) external magnetic coercive flux density, Bext_c

The external magnetic coercive flux density (that required to reduce the induction of a magnetized material to zero)depends on the impurities or "foreign" elements present, domain walls, magnetostriction, strain, and heat treatment.It is a secondary property and is determined by the form of the hysteresis curves, which themselves are based onBs, Br, and r_avg. Other ferromagnetic theories cannot make use of this, because they do not have the correctcharacterization of the curves.

4) Curie temperature and magnetic saturation as a function of Curie temperature

At the Curie temperature, Tc, a ferromagnet converts to a paramagnet. Tc is the key variable for the curve ofsaturation magnetic flux density vs. temperature, and as with other such Reciprocal System equations, it musttherefore be in the denominator. Whenever the key variable is in the denominator for a time region function, it mustbe squared. This means that the temperature, T, must be in the numerator, and must also be squared for dimensionalbalance. At T = 0, the magnetic flux density is at it's maximum value, Bs and so the factor T2/Tc2 must be subtractedfrom 1. Thus we obtain

Bs_T

Bs1

T2

Tc2

n

=

Now the question is: what should be the value of the exponent n? Most likely, n could be 1, 1/2, 1/3, or 1/4. Let x = T/Tc.The graphs follow.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 x( )2

1

1 x( )2

1

2

1 x( )2

1

3

1 x( )2

1

4

x

Figure 26. Variation of Magnetic Saturation, Bs_T/Bs with temperature, T/Tc

Based on the experimental data (Ref. [30], p. 231, p. 308), it's clear that n = 1/4. This is the same value neuristicallyderived by Larson, Ref. [1], p. 251. Now we turn to the calculation of Tc.


The energy of a magnetic dipole is simply

EM_dipole

nM Mu_weber nM Mu_weber

4 0_SI r_0 edgeucconvjoulestoev eV (139a)

or

EM_dipole


4 0_SI r_0 s0convjoulestoev eV (139b)

depending on whether the dipole is along the edge of a unit cell or the shortest length between two atoms.

The Reciprocal System Data Base has very precise calculations of enthalpy for solid matter as a function of temperature.But a good approximation for the thermal energy of a solid atom is simply

Eth 3 kB_ev T(141)

When Eth_atom = EM_dipole, the ferromagnet converts to a paramagnet. Therefore,

Tc


4 0_SI r_0 edgeucconvjoulestoev

3 kB_ev (142a)


KTc_Co_obs 1403.15Tc_Co_calc 1315.9505Tc_Co_calc

EM_Co

3 kB_ev

eVEM_Co 0.3402EM_Co

nM_Co Mu_weber nM_Co Mu_weber

4 0_SI r_0 s0_Coconvjoulestoev

Co

KTc_Fe_obs 1043.15Tc_Fe_calc 1112.1074Tc_Fe_calc

EM_Fe

3 kB_ev

eVEM_Fe 0.2875EM_Fe

nM_Fe Mu_weber nM_Fe Mu_weber

4 0_SI r_0 edgec_Feconvjoulestoev

Fe

Of course, there is a probability distribution of temperatures in solids, and so some atoms will convert before others; that'swhy the transition from ferromagnetism to paramagnetism takes place over a large zone of temperatures centered on Tc.

(142b)Tc


4 0_SI r_0 s0convjoulestoev

3 kB_ev

or


Ni

EM_Ni

nM_Ni Mu_weber nM_Ni Mu_weber

4 0_SI r_0 s0_Niconvjoulestoev EM_Ni 0.1455 eV

Tc_Ni_calc

EM_Ni

3 kB_ev Tc_Ni_calc 562.9761 K

Tc_Ni_obs 631.15 K

There is considerable uncertainty in the observed values of Tc for the rare earth elements; nonetheless, we'llmake the attempt at a comparison of theory to experiment. Also, note that these elements have the 3/2 factorapplied to their unit cell edge lengths.


KTc_Tb_obs 220Tc_Tb_calc 173.3218Tc_Tb_calc

EM_Tb

3 kB_ev

eVEM_Tb 0.0448EM_Tb

nM_Tb Mu_weber nM_Tb Mu_weber

4 0_SI r_0 edgec_Tb3

2

convjoulestoev

apparently Tb loses two units of magnetic charge prior to converting to a paramagnetnM_Tb 2

Tb

Tc_Gd_calc 376.1997KTc_Gd_obs 293Tc_Gd_calc

EM_Gd

3 kB_ev

eVEM_Gd 0.0973EM_Gd

nM_Gd Mu_weber nM_Gd Mu_weber

4 0_SI r_0 edgec_Gd3

2

convjoulestoev

apparently Gd loses a unit of magnetic charge prior to converting to a paramagnetnM_Gd 3

Gd


eVEM_Er 0.0112EM_Er

nM_Er Mu_weber nM_Er Mu_weber

4 0_SI r_0 edgec_Er3

2

convjoulestoev

apparently Er loses three units of magnetic charge prior to converting to a paramagnetnM_Er 1

Er

KTc_Ho_obs 20Tc_Ho_calc 44.1613Tc_Ho_calc

EM_Ho

3 kB_ev

eVEM_Ho 0.0114EM_Ho

nM_Ho Mu_weber nM_Ho Mu_weber

4 0_SI r_0 edgec_Ho3

2

convjoulestoev

apparently Ho loses three units of magnetic charge prior to converting to a paramagnetnM_Ho 1

Ho

KTc_Dy_obs 87Tc_Dy_calc 42.2924Tc_Dy_calc

EM_Dy

3 kB_ev

eVEM_Dy 0.0109EM_Dy

nM_Dy Mu_weber nM_Dy Mu_weber

4 0_SI r_0 edgec_Dy3

2

convjoulestoev

apparently Dy loses three units of magnetic charge prior to converting to a paramagnetnM_Dy 1

Dy


For all the ferromagnetic elements:

KTc_Dy_calc Tc_Ho_calc Tc_Er_calc Tc_Tm_calc

443.6383

For these elements, the average calculated value is:

KTc_Dy_obs Tc_Ho_obs Tc_Er_obs Tc_Tm_obs

442.75

Note that for the last four elements the average observed value is:

KTc_Tm_obs 32Tc_Tm_calc 44.9661Tc_Tm_calc

EM_Tm

3 kB_ev

eVEM_Tm 0.0116EM_Tm

nM_Tm Mu_weber nM_Tm Mu_weber

4 0_SI r_0 edgec_Tm3

2

convjoulestoev

apparently Tm loses three units of magnetic charge prior to converting to a paramagnetnM_Tm 1

Tm

KTc_Er_obs 32Tc_Er_calc 43.1333Tc_Er_calc

EM_Er

3 kB_ev


Tc_Fe_calc

Tc_Fe_obs

Tc_Co_calc

Tc_Co_obs

Tc_Ni_calc

Tc_Ni_obs

Tc_Gd_calc

Tc_Gd_obs

Tc_Tb_calc

Tc_Tb_obs

Tc_Dy_calc

Tc_Dy_obs

Tc_Ho_calc

Tc_Ho_obs

Tc_Er_calc

Tc_Er_obs

Tc_Tm_calc

Tc_Tm_obs

101.0415

So 10% off, on average. Of course actual solid atoms have energy a bit different from 3 x kB x T. Also, thereare impurities and crystal defects. Nonetheless, our calculations seem to be close enough to confirm thetheory, particularly as there is a great amount of experimental uncertainty regarding the rare earth elements.And keep in mind that the experimenters may be measuring the temperature at which essentially all atomshave converted--by the probability distribution of temperatures in solids, this bulk temperature could beconsiderably higher than Tc.


(Ref. [15], p. 766)Bs_Tb_100_obs 3.17

Bs_Tb_100 2.7813Bs_Tb_100 Bs_Tb_calc 1100

Tc_Tb_calc

2

1

4

(Ref. [15], p. 766)Bs_Gd_160_obs 2.194

Bs_Gd_160 2.3157Bs_Gd_160 Bs_Gd_calc 1160

Tc_Gd_calc

2

1

4

(Ref. [30], p. 165)Bs_Ni_300_obs .603

Bs_Ni_300 0.6282Bs_Ni_300 Bs_Ni_calc 1300

Tc_Ni_calc

2

1

4

(Ref. [30], p. 165)Bs_Co_300_obs 1.759

Bs_Co_300 1.7876Bs_Co_300 Bs_Co_calc 1300

Tc_Co_calc

2

1

4

(Ref. [30], p. 165)Bs_Fe_300_obs 2.15

Bs_Fe_300 2.1423Bs_Fe_300 Bs_Fe_calc 1300

Tc_Fe_calc

2

1

4

Now we will calculate the value of the magnetic saturation flux density (in tesla) at room temperature (300 K here) for theferromagneic elements other than the rare earth elements--for these an appropriate temperature between 0 and Tc will beused.


Note: the values from Ref. [15] are in graph form expressed in units of A x m2 /kg = emu/g. They have been

converted to tesla by multiplying the value taken from the graph by the density in g/cm3 and by 4 x 10-4.

(Ref. [15], p. 767)Bs_Tm_20_obs 2.787

Bs_Tm_20 2.7246Bs_Tm_20 Bs_Tm_calc 120

Tc_Tm_calc

2

1

4

(Ref. [15], p. 767)Bs_Er_18_obs 2.981

Bs_Er_18 3.0386Bs_Er_18 Bs_Er_calc 118

Tc_Er_calc

2

1

4

(Ref. [15], p. 767)Bs_Ho_10_obs 3.739

Bs_Ho_10 3.1138Bs_Ho_10 Bs_Ho_calc 110

Tc_Ho_calc

2

1

4

Ref. [15], p. 766)Bs_Dy_30_obs 3.498

Bs_Dy_30 2.9486Bs_Dy_30 Bs_Dy_calc 120

Tc_Dy_calc

2

1

4


Bs_Fe_300

Bs_Fe_300_obs

Bs_Ni_300

Bs_Ni_300_obs

Bs_Co_300

Bs_Co_300_obs

Bs_Gd_160

Bs_Gd_160_obs

Bs_Tb_100

Bs_Tb_100_obs

Bs_Dy_30

Bs_Dy_30_obs

Bs_Ho_10

Bs_Ho_10_obs

Bs_Er_18

Bs_Er_18_obs

Bs_Tm_20

Bs_Tm_20_obs

90.9622

Therefore the calculations are within 5% of the observed values, which is quite probably within the experimentaluncertainy, especially for the rare earth elements. The expression used in conventional physics works quite well for Fe,Co, and Ni, but does not work very well for the rare earth elements. See the figure on p. 99, Vol. 3, of Ref. [7]. In asense, this confirms both the calculation of Tc and the use of the 1/4 exponent.


5) magnetic hysteresis curves

According to Jiles, Ref. [30], p. 148:

"When a magnetic field is applied to a demagnetized ferromagnetic material the changes in magnetic induction Bwhen traced on the B, H plane generate the initial magnetization curve. At low fields the first domain process occurswhich is a growth of domains which are aligned favorably with respect to the field according to a minimization of thefield energy ... and a consequent reduction in size of domains which are aligned in directions opposing the field....

"At moderate field strengths a second mechanism becomes significant; this is domain rotation, in which the atomicmagnetic moments within an unfavorably aligned domain overcome the anisotropy energy and suddenly rotate fromtheir original direction of magnetization into one of the crystallographic 'easy' axes which is nearest to the fielddirection.

"The final domain process which occurs at high fields is coherent rotation. In this process the magnetic moments,which are all aligned along the preferred magnetic crystallographic easy axes lying close to the the field direction, aregradually rotated into the field direction as the magnitude of the field is increased. This results in a single-domainsample."

For solid state properties, conventional theoretical physicists use rather complicated statistical functions like theFermi-Dirac distribution to describe the motion of their charged electrons--which they mistakenly use to explainvalence, atomic-molecular bonding, spectroscopy, electric currents, and magnetism. This is in contrast with theReciprocal System, where we commonly use the Gaussian or Normal distribution to describe the solid stateproperties. It's rather obvious, from inspection, that the ferromagnetic hysteresis curves are cumulative probabilitydistributions representing the fraction of unit cells fully magnetized in the direction of the external field (and referencesystem), and so the error function, erf, applies. (This function is somewhat misnamed--it's simply an integration of theGaussian function.) The external flux density, Bext, must be divided by its maximum value, Bext_max, to obtain an inputvariable ranging between -1 and 1:

Bext_ratio

Bext

Bext_max (143)


Then, with two statistical parameters, and , the argument for the error function will be

Bext_ratio

With this, the internal magnetic flux density can be expressed as

(144)Bint erf

Bext_ratio

Bs_T

where Bs_T = the saturation internal magnetic flux density at the temperature T. For the hysteresis plots which follow, theabscissa (Bext_ratio) will range from -1 to +1. For proper scaling (and to visually show r), the ordinate must be scaled byBext_max, as well.

The value of the statistical parameter can be expressed in terms of Bext_max, as follows:

At Bext = Bext_c (the external flux density at the coercivity point, where the demagnetizing curve crosses from positive tonegative), Bint = 0, and therefore we can solve for .

Bext_ratio

Bext_c

Bext_max at point of coercivity

0 erf

Bext_c

Bext_max

Bs_T=


Bext_c

Bext_max (145)

Data for ferromagnetic materials usually include the values of Bs_T, Br (the remanent induction--the internal flux densitywhen Bext = 0), and r_avg (the average permeability in the second quadrant between Bext = 0 and Bc). Note: some ofthe tables give r_max, for the first quadrant; but because the magnetizing and demagnetizing curves are mostly parallel,one can use the value of r_max, at least as a starting point for iteration, for r_avg.

Now we will compute the hysteresis plots for the three main ferromagnetic materials, Fe, Co, Ni. Note: we will use theobserved values of Bs_T, Br, and r_avg, as input because our purpose here is to show the validity of the use of the erfto describe the hysteresis curves. The previously calculated values of Bs_T, Br could, of course, be used but wouldhardly change the results. Keep in mind, as well, that even a small percentage of impurities have a big impact onhysteresis. With r_avg given, it's easy to see that

Bext_c

Br

r_avg (146)

(Or, if Bext_c is more certain, then we can get r_avg from this equation.) In what follows, for economy, we will simplyuse Bs to represent Bs_T. It's understood that we're computing the properties at room temperature.

To find the hysteresis loss--the area enclosed by the two curves--we subtract the area below the magnetizing curvefrom the area below the demagnetizing curve (dividing by 0_SI to get the correct units):

WH

1

1

Bext_ratioerfBext_ratio

BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

(147)


1

1

Bext_ratioerf

Bext_ratio

Bext_c

Bext_max

BsBext_max

0_SI erf

Bext_ratio

Bext_c

Bext_max

BsBext_max

0_SI

d WH_Fe_obs_99.95=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001

.5teslaBext_max .2 104

Starting Values:

There are two equations in two unknowns, and Bext_max, plus we would like to minimize the error in computing WH..

(reference values range from 30 to 60)J/m3WH_Fe_obs_99.95 40

(external magnetic flux density at coercivity point)Bext_c 5.9091 106

teslaBext_c

Br

r_avg

(avg. permeability, 2nd quad.--the reference max. values range from 180000 to 350000)r_avg 220000

(internal remanent magnetic flux density)teslaBr 1.30

(internal magnetic flux density saturation)teslaBs 2.15

Date from Ref. [30], p. 113 and Ref. [12], p. 4-142 and Ref. [16], p. 12-108, averaged99.95% Pure Iron


Bext_max

Find Bext_max Bext_max 1.9583 105

Bext_c

Bext_max 0.3017

0.5021

r_calc

erf

Bs 0

0 Bext_c r_calc 2.2 10

5 Br_calc erf

Bs Br_calc 1.3 tesla

r_calc

r_avg1 Br_calc

Br1

WH_Fe_99.95

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_Fe_99.95 40 J/m3


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 11.1 10

5

8.78 104

6.59 104

4.39 104

2.2 104

0

2.2 104

4.39 104

6.59 104

8.78 104

1.1 105

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 27. Hysteresis Curves for 99.95 % Iron

Br_Fe Br Bext_c_Fe Bext_c (for use in the section on compounds and alloys)


1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d WH_Fe_99.8=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001

.5teslaBext_max 1 103

Starting Values:

(Ref. [30], p. 113)J/m3WH_Fe_99.8 500


teslaBext_c

Br

r_avg

(avg. permeability for 2nd quadrant, adjusted to get Bext_c correct)r_avg 9000

(internal remanent magnetic flux density)teslaBr .77


Date from Ref. [30], p. 113, Ref. [12], p. 4-142, Ref. [69], p. 73999.8 % Pure Iron


Bext_max

Find Bext_max Bext_max 0.0003 0.9183 Bext_c

Bext_max 0.3035

r_calc

erf

Bs 0

0 Bext_c r_calc 9000 Br_calc erf


r_calc

r_avg1

Br_calc

Br1

WH_Fe_99.8

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_Fe_99.8 502.8717 J/m3


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 17592.14

6073.71

4555.28

3036.86

1518.43

0

1518.43

3036.86

4555.28

6073.71

7592.14

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 28. Hysteresis Curves for 99.8 % Pure Iron


0.0809Bext_c

Bext_max 0.3142Bext_max 0.0247

Bext_max

Find Bext_max

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


Starting Values:

WH_Co_obsThe references do not have a value for

(external magnetic flux density at coercivity point)Bext_c 0.002teslaBext_c

Br

r_avg

(avg. permeability in 2nd quadrant)r_avg 250



Date from Ref. [12], p. 4-142 and Ref. [30], p. 16599% Pure Cobalt


r_calc

erf

Bs 0



r_calc

r_avg1

Br_calc

Br1

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 171.14

56.92

42.69

28.46

14.23

0

14.23

28.46

42.69

56.92

71.14

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 29. Hysteresis Curves for 99% Pure Cobalt


WH_Co

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_Co 11198.0267 J/m3 (a prediction)

Br_Co Br Bext_c_Co Bext_c (for use in the section on compounds and alloys)


0.2657Bext_c

Bext_max 0.3911Bext_max 0.0025

Bext_max

Find Bext_max

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


Starting Values:

WH_Ni_obsThe references do not have a value for


Br

r_avg

(avg. permeability, considered to be approximately the same as the max. given in the data)r_avg 600


(internal magnetic flux density saturation)teslaBs .603

Date from Ref. [12], p. 4-142 and Ref. [30], p. 16599% Pure Nickel


r_calc

erf

Bs 0



r_calc

r_avg1

Br_calc

Br1

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1240.31

192.25

144.19

96.13

48.06

0

48.06

96.13

144.19

192.25

240.31

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 30. Hysteresis Curves for 99% Pure Nickel


WH_Ni

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_Ni 1277.9748 J/m3 (a prediction)

Br_Ni Br Bext_c_Ni Bext_c (for use in the section on compounds and alloys)

The curves calculated above look very similar to the experimental hysteresis curves given in the physics literature, andtherefore we can consider the erf function to be confirmed for this property. Unfortunately there are no readily availablecurves for the rare earth elements, so there wouldn't be much point to computing the theoretical curves for theseelements. It's also rather disconcerting that there are no values for the hysteresis loss of Co and Ni ferromagnets--soour calculations for these elements are predictions.


Note: Initial Magnetization Curve

A simple sigmoid probability curve, like the following, can represent, approximately, the initial magnetization curve:

Bint

Bs

1

1 e

Bext

Bext_max i

i

(148)

where i and i are the relevant statistical parameters. Leti .3 i .08

0 0.2 0.4 0.6 0.80

0.5

1

1

1 e

Bext_ratio i

i

Bext_ratio

Figure 31. Initial Magnetization Curve (Generic)

This curve agrees, qualitatively, with the curves in the literature (like Ref. [18], p. 215, Fig. 29.3) but there isn't sufficientempirical data to set up a detailed comparison of theory with experiment.


We'll use the observed, rather than the calculated value, of Tc for the following graphs (because the empiricalcurves are based on it).

(assumed approximately constant)Fe 7.481.0395 Fe 4

T Tc_Fe

According to Stoner (Ref. [28], p. 379), the (empirical) Curie-Weiss equation for iron above the Curie point is

w 55.8470Tc_Fe_obs 1043.15te 8ts 2tp 3

Vuc_SI 24.866 1030

edgec 2.9196 1010

nM 3Zuc 2nf 1

nf =1 for Fe when ferromagnetic or paramagnetic.

BCCFe

u_SI

nf

Zuc

2 nM Mu_weber edgec

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

T Tc

1

Vuc_SI

st_u 102

3

(149)

The paramagnetic equation for ferromagnetic elements above the Curie point is slightly modified from the previousequation for paramagnets, Eq. (131): the Curie temperature must be subtracted from the applied temperature.

6) paramagnetic behavior above the Curie temperature


1050 1100 11500.02

0.04

0.06

0.08

.0395 Fe 4

TT Tc_Fe_obs

u_SI

nf

Zuc

2 nM Mu_weber edgec

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

TT Tc_Fe_obs

1

Vuc_SI

st_u 102

3

TT

Figure 32. Iron Susceptibility Above Tc

The theoretical curve and the empirical curve have the same shape.


Figure 33. Cobalt Susceptibility Above Tc

1500 16000

0.02

0.04

0.06

0.08.0217 Co 4

TT Tc_Co_obs

u_SI

nf

Zuc

2 nM Mu_weber s0

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

TT Tc_Co_obs

1

Vuc_SI

st_u 102

3

TT

Co 8.116Tc_Co_obs 1403.15

w 58.9332te 9ts 2tp 3

Vuc_SI 47.4104 1030

s0 2.4991 1010

edgec 3.6193 1010

Zuc 4nM 3nf .5

nf = .5 for Co when paramagnetic

FCCCo


Figure 34. Nickel Susceptibility Above Tc

Again, the curves are very close.

700 800 900 10000

0.02

0.04

0.06

0.08.00555 Ni 4

TT Tc_Ni_obs

u_SI

nf

Zuc

2 nM Mu_weber s0

Vuc_SI

MP_nat_t

tp

1

ts

1

te

1

u_SI nf Zucw

1

Tt_u

TT Tc_Ni_obs

1

Vuc_SI

st_u 102

3

TT

Ni 7.357Tc_Ni_obs 631.15

w 58.6900te 10ts 2tp 3

Vuc_SI 43.5232 1030

s0 2.5382 1010

edgec 3.4904 1010

nM 2Zuc 2nf .25

nf = .5 for Ni when ferromagnetic, nf = .25 when paramagnetic.

FCCNi


Element At. No. uc tp ts te n_f Z_uc n_M edge_a edge_c s0 Vuc Bs_calc Bs_obsFe 26 BCC 3 2 8 1.0 2 3 2.8545 2.8545 2.4825 23.258 2.1832 2.1870

Co 27 FCC 3 2 9 0.9 4 3 4.0496 4.0496 2.4123 42.637 1.8116 1.7970

Ni 28 FCC 3 2 10 0.5 4 2 3.5175 3.5175 2.5061 43.522 0.6829 0.6530

Gd 64 HCP 4 3 10 0.8 2 4 3.4195 5.6255 3.5708 65.778 2.4341 2.4880

Tb 65 HCP 4 3 11 1.0 2 4 3.4001 5.4268 3.5028 62.736 3.0775 3.4070Dy 66 HCP 4 3 12 1.0 2 4 3.3654 5.5600 3.5089 62.972 3.1412 3.7680

Ho 67 HCP 4 3 13 1.0 2 4 3.3589 5.3247 3.4570 60.043 3.1550 3.9090

Er 68 HCP 4 3 14 1.0 2 4 3.3409 5.4516 3.4628 60.848 3.1875 3.4390

Tm 69 HCP 4 3 15 0.9 2 4 3.3500 5.2294 3.4080 58.161 2.8789 2.8840

Element At. No. uc tp ts teFe 26 BCC 3 2 8

Co 27 FCC 3 2 9

Ni 28 FCC 3 2 10

Gd 64 HCP 4 3 10

Tb 65 HCP 4 3 11Dy 66 HCP 4 3 12

Ho 67 HCP 4 3 13

Er 68 HCP 4 3 14

Tm 69 HCP 4 3 15

n_M_r Br_calc Br_obs Dipole len. n_M_c Tc_calc Tc_obs T Bs_T_calc Bs_T_obs2 1.2529 1.3000 edge_c 3 1112.11 1043.15 300 2.1423 2.150

1 0.5982 0.5000 s0 3 1315.95 1403.15 300 1.7876 1.759

1 0.3462 0.4000 s0 2 562.98 631.15 300 0.6282 0.603

3/2 edge_c 3 376.20 293.00 160 2.3157 2.194

3/2 edge_c 2 173.32 220.00 100 2.7813 3.1703/2 edge_c 1 42.29 87.00 30 2.9486 3.498

3/2 edge_c 1 42.16 20.00 10 3.1138 3.739

3/2 edge_c 1 43.13 32.00 18 3.0386 2.981

3/2 edge_c 1 44.97 32.00 20 2.7246 2.787

Table VIII. Summary of Properties of Ferromagnetic Elements

Note: edge_a, edge_c, s0 in units of 10-10 m; Vuc in units of 10-30 m3; Bs_calc, Bs_obs, Br_calc, Br_obs, Bs_T_calc,

Bs_T_obs, all in units of tesla (webers/m2)


b. Compounds and Alloys

Von Hippel explains (Ref. [20], p. v.) that "...molecular engineering [is] the building of materials to order. We now begin todesign materials with properties prescribed for the purpose in hand." By adding various non-ferromagnetic elements to aferromagnetic material we can obtain compounds and alloys with desired values of external magnetic flux density coercivityand internal magnetic flux density and remanence. Generally, adding non-ferromagnetic elements reduces the magneticremanence (Br) and magnetic saturation (Bs) of the pure material, while increasing the coercivity (Bext_c). Essentially, thenon-ferromagnetic elements block, to some degree, the magnetic dipole moments of the ferromagnetic atoms, andincrease the energy required to demagnetize the pure material.

If, after the addition of non-ferromagnetic atoms and after various kinds of heat and cold treatment of the resulting alloy, wecould know the exact placement of the atoms in a crystal unit cell, we could calculate--using the previously-given equations--the new values of Br and Bs. But this is usually not the case--we don't know precisely how many of the "foreign" atomsare in the way of the magnetic dipoles, and how many are not. So, what we will do is define two new constants, nr and kc.

nr factor_of_remanence_due_to_foreign_atoms (usually < 1, but not always)

kc factor_of_coercivity_due_to_foreign_atoms (usually > 1, but not always)

nr and kc will be given for each compound/alloy which follows. Of course, the base values for Br and Bext_c will be thoseof the predominant ferromagnetic material. Using the erf equations previously given we will calculate the hysteresis lossand BHmax values (commonly tabulated by commercial vendors) and compare with such empirical values as areavailable.


1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d WH_78Permalloy_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


Starting Values:

(Ref. [30], p. 113)WH_78Permalloy_obs 20


teslaBext_c

Br

r_avg

(avg. permeability, 2nd quadrant)r_avg 77000



Date from Ref. [12], p. 4-142 and Ref. [30], p. 13078 Permalloy


Bext_max


Bext_c

Bext_max 0.2256

0.6549

r_calc

erf

Bs 0



r_calc

r_avg1

Br_calc

Br1

WH_78Permalloy

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_78Permalloy 17.0285 J/m3


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 14.65 10

4

3.72 104

2.79 104

1.86 104

9292.88

0

9292.88

1.86 104

2.79 104

3.72 104

4.65 104

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 35. Hysteresis Curves for 78 Permalloy

78 Permalloy is 78% Ni, 22% Fe.

nr_78Permalloy

Br

Br_Ni nr_78Permalloy 1 (no reduction because, of course, Fe is ferromagnetic!)


kc_78Permalloy

Bext_c

Bext_c_Ni kc_78Permalloy 0.0078

Now we'll compute |BHmax|, the maximum energy product. The maximum value of "B x H" occurs at .5 x Bext_c.

BHmax erf

.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI J/m3

(150)

But Bext_c

Bext_max so

BHmax erf

.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI J/m3

BHmax 0.4256 J/m3 BHmax_78permalloy BHmax

Unfortunately, none of the references given at the end of this paper provide an experimental value.


1

1

Bext_ratioerf

Bext_ratio

Bext_c

Bext_max

BsBext_max

0_SI erf

Bext_ratio

Bext_c

Bext_max

BsBext_max

0_SI

d WH_SiFe_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


Starting Values:

WH_SiFe_obs 140


teslaBext_c

Br

r_avg

(avg. permeability, considered to be approximately the same as the max. given in the data)r_avg 40000



Date from Ref. [16]. p. 12-108--there are many other sources, all indisagreement

Silicon Iron (3% Si) Oriented


Bext_max


Bext_c

Bext_max 0.4071

0.8586

r_calc

erf

Bs 0


Bs Br_calc 1 tesla

r_calc

r_avg1

Br_calc

Br1

WH_SiFe

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

WH_SiFe 140 J/m3


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 13.27 10

4

2.62 104

1.96 104

1.31 104

6546.88

0

6546.88

1.31 104

1.96 104

2.62 104

3.27 104

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 36. Hysteresis Curves for Oriented Silicon Iron


Silicon iron is 96% Fe, 4% Si.

nr_SiFe

Br

Br_Fe nr_SiFe 0.7692

kc_SiFe

Bext_c

Bext_c_Fe kc_SiFe 4.2308

BHmax_SiFe erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI BHmax_SiFe 4.5301 J/m3

The references do not provide an experimental value.

Ref. [68] provides data for commercial magnets, including the maximum energy product, so we'll select a few ofthese for comparison with the theory; unfortunately Ref. [68] does not have values for Bs. Ref. [13], [30], [31],and [32] also have tables of data--but usually one or more necessary values are missing, which means thatapproximations have to be made.


erf .5Bext_c

Bext_max

Bs.5 Bext_c

0_SI BHmax_Alnico5_obs=

r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=Br erf

Bext_c

Bext_max

Bs=

Given

CTOL .000001TOL .000001

.5teslaBext_max 3Starting Values:


Br

r_avg

(Ref. [30], p. 377)BHmax_Alnico5_obs 40000

(average permeability, approximated from Ref. [31], pp. 156-158)r_avg 4


(approx. from Ref. [31], pp. 156-158)(internal magnetic flux density saturation)teslaBs 1.5

Alnico 5 Anisotropic Cast


The references do not provide an experimental value--but clearly one would not use this alloy in a transformer!

(prediction)J/m3WH_Alnico5 1.5251 106

WH_Alnico5

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

J/m3BHmax_Alnico5 40000BHmax_Alnico5 erf0.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1

Br_calc 1.28Br_calc erf

Bsr_calc 4r_calc

erf

Bs 0

0 Bext_c

0.366

0.3756Bext_c

Bext_maxBext_max 0.852

Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 11.76

1.41

1.06

0.7

0.35

0

0.35

0.7

1.06

1.41

1.76

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 37. Hysteresis Curves for Alnico 5


Alnico 5 is 51% Fe, 8% Al, 14% Ni, 24% Co, and 3% Cu.

nr_Alnico5

Br

Br_Fe nr_Alnico5 0.9846 (clearly with Ni and Co, there's not much reduction)

kc_Alnico5

Bext_c

Bext_c_Fe kc_Alnico5 54153.8462 (wow)


erf .5Bext_c

Bext_max

Bs .5Bext_c

0_SI BHmax_PtCo_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


(Ref. [30], p. 382, but adjusted down)BHmax_PtCo_obs 45000


Br

r_avg

(average permeability, from Ref. [13], p. 5-165)r_avg 1.2


(assumed to be the same as Co)(internal magnetic flux density saturation)teslaBs 1.759

Platinum Cobalt



J/m3WH_PtCo 2.9174 106

WH_PtCo

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

J/m3BHmax_PtCo 45000BHmax_PtCo erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1


Bsr_calc 1.2r_calc

erf

Bs 0

0 Bext_c

0.6309

0.2128Bext_c


Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.7

0.56

0.42

0.28

0.14

0

0.14

0.28

0.42

0.56

0.7

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 38. Hysteresis Curves for Platinum Cobalt


Platinum cobalt is 77% Pt and 23% Co.

nr_PtCo

Br

Br_Co

nr_PtCo 1.29 (Pt increases the remanence in this case.)

kc_PtCo

Bext_c

Bext_c_Co kc_PtCo 268.75


erf .5Bext_c

Bext_max

Bs .5Bext_c

0_SI BHmax_NdFeB_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


(some values in the literature go as high as 320000, but that seemsimprobable)

BHmax_NdFeB_obs 250000


Br

r_avg

(average permeability, based on but rounded up from Bext_c, given in Ref. [30], p. 377)r_avg 1.0345


(Ref. [30], p. 385)teslaBs 1.6

Nd2-Fe14-B



J/m3WH_NdFeB 5.9948 106

WH_NdFeB

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

J/m3BHmax_NdFeB 2.5 105

BHmax_NdFeB erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1


Bsr_calc 1.0345r_calc

erf

Bs 0

0 Bext_c

0.6104

0.5689Bext_c


Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.72

0.58

0.43

0.29

0.14

0

0.14

0.29

0.43

0.58

0.72

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 39. Hysteresis Curves for Nd2-Fe14-B

nr_NdFeB

Br

Br_Fe

nr_NdFeB 1

kc_NdFeB

Bext_c

Bext_c_Fe kc_NdFeB 2.1266 10

5 (This is an amazing increase--no wonder this

material is so popular.)


erf .5Bext_c

Bext_max

Bs .5Bext_c

0_SI BHmax_Cu2MnAl_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001


(converted from MGO value given in

above Ref. to J/m3)

BHmax_Cu2MnAl_obs 27852.115BHmax_Cu2MnAl_obs 3.5 7957.74715


Br

r_avg

(average permeability, based on Bext_c, given in Ref. [32], p. 858 for "Manganese Aluminum" )r_avg .2877

(see calc. below)teslaBr .4028

(Ref. [28], p. 525--stated to be the same as Ni--but see calc. below)teslaBs .6042

Cu2MnAl--a Heusler Alloy



J/m3WH_Cu2MnAl 2.6927 106

WH_Cu2MnAl

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

J/m3BHmax_Cu2MnAl 27852.115BHmax_Cu2MnAl erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1



erf

Bs 0

0 Bext_c

0.2148

0.1469Bext_c


Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.063

0.051

0.038

0.025

0.013

0

0.013

0.025

0.038

0.051

0.063

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 40. Hysteresis Curves for Cu2MnAl

nr_Cu2MnAl

Br

Br_Ni nr_Cu2MnAl 1.007

kc_Cu2MnAl

Bext_c

Bext_c_Ni kc_Cu2MnAl 2100.1043


(which compares with 630.15 K, the supposedobservation)

KTc_Cu2MnAl 533.5239Tc_Cu2MnAl

EM_Cu2MnAl

3 kB_ev

eVEM_Cu2MnAl 0.1379

EM_Cu2MnAl

nM_Cu2MnAl Mu_weber nM_Cu2MnAl Mu_weber

4 0_SI r_0 edgecconvjoulestoev

teslaBs_Ni_obs 0.653which compares withBs_Cu2MnAl 1.0049

Bs_Cu2MnAl

IM nf floorZuc

2

nM_Cu2MnAl Mu_weber edgec

Vuc_SI

m3Vuc_SI 2.1064 1028

Vuc_SI edgec3

(assumed to be approx. constant from 0to 300 K)

(all edges same)medgec 5.95 1010

nf 1Zuc 4nM_Cu2MnAl 3

Note: The Heusler alloys are unique in that they are constructed of non-ferromagnetic elements but, which whencombined, are ferromagnetic. Ref. [70] states that crystal volume unit cell is FCC, with the edge being 5.95 A. The Mnatoms are ferromagnetically-charged here, not Cu or Al. Larson specifically says, Ref. [1], p. 216, "Under some specialconditions, the [electric] displacements of chromium (6) and manganese (7) are increased to 8 and 9 respectively byreorientation relative to a new zero point, as explained in Chapter 18 of Volume 1 [Ref. 2, 2nd ed.]). These elements arethen also able to accept magnetic charges." The Heusler alloys are often compared to nickel.

The two magnetic dipoles are between the two pairs of Mn atoms at the edges of the unit cell.


Bs_Cu2MnAl_300 Bs_Cu2MnAl 1300

Tc_Cu2MnAl

2

1

4

Bs_Cu2MnAl_300 0.9138

nM_r_Cu2MnAl 2

Br_Cu2MnAl2

3Bs_Cu2MnAl_300 Br_Cu2MnAl 0.6092

Everything checks out.


erf .5Bext_c

Bext_max

Bs .5Bext_c

0_SI BHmax_BaFe12O19_obs=

Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001

.5teslaBext_max .4Starting Values:

(adjusted down from Ref. [31], p. 182; Ref. [68] says it's 8350--prob. aniso.)BHmax_BaFe12O19_obs 5000


Br

r_avg

(average permeability, based on Bext_c, given in Ref. [31], p. 182)r_avg 1.1958


(see calc. below)teslaBs .3043

Barium Ferrite BaO-6Fe2O3 = BaFe12O19 isotropic form



J/m3WH_BaFe12O19 1.555 105

WH_BaFe12O19

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d

J/m3BHmax_BaFe12O19 5000BHmax_BaFe12O19 erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1



erf

Bs 0

0 Bext_c

0.6413

0.4385Bext_c


Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.79

0.63

0.47

0.31

0.16

0

0.16

0.31

0.47

0.63

0.79

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 41. Hysteresis Curves for Barium Ferrite

nr_BaFe12O19

Br

Br_Fe nr_BaFe12O19 0.156

kc_BaFe12O19

Bext_c

Bext_c_Fe kc_BaFe12O19 28700.4516


(using the interatomic distance of the dominant dipole)eVEM_BaFe12O19 0.282

EM_BaFe12O19

nM_BaFe12O19 Mu_weber nM_BaFe12O19 Mu_weber

4 0_SI r_0 s0_1convjoulestoev

teslaBs_BaFe12O19 0.3103

Bs_BaFe12O19

IM nf 8 nM_BaFe12O19 Mu_weber s0_1

Vuc_SI

IM nf 4 nM_BaFe12O19 Mu_weber s0_2

Vuc_SI


m3

Vuc_SI 6.9756 1028

Vuc_SI edgea2

edgec sin 60 deg( )

ms0_2 2.778 1010

s0_1 2.910 1010

medgec 23.194 1010

edgea 5.893 1010

nf 1Zuc 2nM_BaFe12O19 3

Note: Ref. [71] describes in great detail the crystal structure of barium ferrite, BaFe12O19. The unit cell is, overall,hexagonal, with a = 5.893 A and c = 23.194 A. There are two formula units per cell, so there are 24 Fe atoms, whichmeans there are 12 magnetic dipoles--8 of these point in the direction of the magnetic field, and 4 point in the directionopposite to the direction of the magnetic field--substances like this are called "ferrites" and they exhibit "ferrimagnetism."The iron atoms are in the interstices between the O atoms--not at the corners. The average interatomic distance for the 8dipoles is 2.910 A (Fe(4)-Fe(4) in the above Ref.); the average distance for the 4 dipoles is 2.778 A (Fe(5)-Fe(5) in theabove Ref.).


Tc_BaFe12O19

EM_BaFe12O19

3 kB_ev K

Tc_BaFe12O19 1090.8822 K (which compares with 723.15 K, the supposedobservation)

Bs_BaFe12O19_300 Bs_BaFe12O19 1300

Tc_BaFe12O19

2

1

4

Bs_BaFe12O19_300 0.3043

nM_r_BaFe12O19l 2

Br_BaFe12O192

3Bs_BaFe12O19_300 Br_BaFe12O19 0.2028 obs. = .20, Ref. [31], p. 182, for

"isotropic barium ferrite"

Everything checks out--there is no need for the nonsense of the Quantum Mechanics "superexchange" concept.Anisotropic forms of barium ferrite have higher coercivities and BHmax values--see Ref. [7], Vol. 4, p. 462.


Br erf

Bext_c

Bext_max

Bs=r_avg

erf

Bext_c

Bext_max

Bs 0

0 Bext_c=

Given

CTOL .000001TOL .000001

.5teslaBext_max 102

Starting Values:


Br

r_avg

(average permeability, based on Bext_c, given in Ref. [31], p. 478)r_avg 83.3


(see calc. below)teslaBs .6247

Magnetite Fe3O4



J/m3WH_Fe3O4 9677.3941

WH_Fe3O4

1

1


BsBext_max

0_SI erf

Bext_ratio

BsBext_max

0_SI

d


J/m3BHmax_Fe3O4 265.7254BHmax_Fe3O4 erf.5 Bext_c

Bext_max

Bs.5 Bext_c

0_SI

Br_calc

Br1

r_calc

r_avg1


Bsr_calc 83.3r_calc

erf

Bs 0

0 Bext_c

0.5607

0.3836Bext_c


Bext_max

Find Bext_max


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 147.93

38.34

28.76

19.17

9.59

0

9.59

19.17

28.76

38.34

47.93

erfBext_ratio

Bs

Bext_max

erfBext_ratio

Bs

Bext_max

Bext_ratio

Figure 42. Hysteresis Curves for Magnetitie

nr_Fe3O4

Br

Br_Fe nr_Fe3O4 0.3204

kc_Fe3O4

Bext_c

Bext_c_Fe kc_Fe3O4 846.1538


(using the interatomic distance of the dominant dipole)eVEM_Fe3O4 0.1452

EM_Fe3O4

nM_Fe3O4 Mu_weber nM_Fe3O4 Mu_weber

4 0_SI r_0 s0_1convjoulestoev

teslaBs_Fe3O4 0.6794

Bs_Fe3O4

IM nf 8 nM_Fe3O4 Mu_weber s0_1

Vuc_SI

IM nf 4 nM_Fe3O4 Mu_weber s0_2

Vuc_SI

m3

Vuc_SI 591.921 1030


ms0_2 5.6519 1010

s0_1 5.6519 1010

(all edges same)medgec 8.3963 1010

nf 1Zuc 8nM_Fe3O4 3

Note: Magnetite is "lodestone"--the original natural magnetic material; Lucretius wrote about its wonders prior to 55B.C.E. Ref. [65], Vol. 2, p.83 describes the crystal structure of magnetite, Fe3O4. The unit cell is, overall, FCC, with eachedge =8.3963 A. There are eight formula units per cell, so there are 24 Fe atoms, which means there are 12 magneticdipoles--8 of these point in the direction of the magnetic field, and 4 point in the direction opposite to the direction of themagnetic field. The iron atoms are in the interstices between the O atoms--not at the corners. From the ReciprocalSystem Data Base calculations, the interatomic distance for the 8 dipoles is 5.6519 A; the interatomic distance for the 4dipoles is also 5.6519 A. (Other interatomic distances between the iron atoms are 4.1982, 3.6357, and 3.7842 A, butthese are not used for the dipoles--only the longest distance is used.)


Tc_Fe3O4

EM_Fe3O4

3 kB_ev K

Tc_Fe3O4 561.6637 K (which compares with 848.15 K, the supposedobservation, Ref. [31], p. 477)

Bs_Fe3O4_300 Bs_Fe3O4 1300

Tc_Fe3O4

2

1

4

Bs_Fe3O4_300 0.6247 tesla

nM_r_Fe3O4 2

Br_Fe3O42

3Bs_Fe3O4_300 Br_Fe3O4 0.4165 tesla obs. = .44, Ref. [31], p. 478

Everything checks out--there is no need for the nonsense of the Quantum Mechanics "superexchange" concept.

The sample calculations cover all of the major classes of ferromagnetic materials and verify the conclusions of theReciprocal System: each ferromagnetic atom has two magnetic charges, one N and one S, orthogonal to oneanother. In a solid, magnetic dipoles are formed, and all of the observed properties are thereby explained. Therefore,James Alfred Ewing, the great physicist who theorized that atoms were tiny magnets, and who was the first todiscover magnetic hysteresis, has been vindicated.


Conclusion

The Reciprocal System represents a new paradigm for theoretical physics. It is a unified, general theory,applicable to all of the categories of physical phenomena, including the electric and magnetic susceptibilities ofmatter. This paper applies the theory to the calculation of all of the important properties of dielectrics, diamagnets,paramagnets, and ferromagnets; included are 150 equations, 42 figures, and 8 tables covering the calculated andobserved values of these properties. The Reciprocal System calculations are usually in agreement with theobservations, and where there are non-negligible differences, theoretical explanations are readily available.Unlike conventional physics, the Reciprocal System does not use charged electrons to explain these properties;massless, chargeless electrons are the physical entities involved in ordinary electric circuits, and the additionalspace and time of the atoms of a dielectric account for the value of the index of refraction and dielectric constant.Magnetic charges do exist, contrary to the assertions of conventional physics, and account for the properties ofdiamagnets, paramagnets, and ferromagnets.


Acknowledgements

Funding for this work came from my company, Transpower Corporation. Of course, great thanks go to Dewey B.Larson, who served as my theoretical physics mentor from 1965 until his death in 1990. He was, by far, the mostintelligent and most logical of any individual I've ever known.


References

[1] D. Larson, Basic Properties of Matter (Salt Lake City, UT: International Society of Unified Science, 1988).

[2] D. Larson, The Structure of the Physical Universe (Portland, OR: North Pacific Publishers, 1959); Nothing ButMotion (Portland, OR, North Pacific Publishers, 1979. The latter is Volume I of the 2nd ed. of The Structure of thePhysical Universe.

[3] D. Larson, New Light on Space and Time (Portland, OR: North Pacific Publishers, 1965).

[4] D. Larson, The Case Against the Nuclear Atom (Portland OR: North Pacific Publishers, 1963).

[5] D. Larson, The Neglected Facts of Science (Portland, OR: North Pacific Publishers, 1982).

[6] H. Young, R. Freedman, Sears and Zemansky's University Physics, 11th ed. (San Francisco, CA: Pearson,Addison Wesley, 2004).

[7] J. Thewlis, Encyclopaedic Dictionary of Physics (New York, NY: The Macmillan Company, 1962).

[8] R. Rose, L. Shepard, J. Wulff, The Structure and Properties of Materials, Vol. IV--Electronic Properties (New York,John Wiley & Sons, Inc., 1966).

[9] M. Podesta, Understanding the Properties of Matter, 2nd ed. (London and New York: Taylor & Francis, 2002).

[10] J. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford: ClarendonPress, 1985).

[11] R. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure (New York, NY: Oxford University Press,2005).

[12] E. Condon, H. Odishaw, ed., Handbook of Physics, 2nd ed. (New York, NY: McGraw-Hill Book Company, 1967).

[13] D. Gray, ed., American Institute of Physics Handbook, 3rd ed. (New York, NY: McGraw-Hill Book Company, 1972).


[14] W. Martienssen, H. Warlimont, ed., Springer Handbook of Condensed Matter and Materials Data (Berlin,Germany: Springer, 2005).

[15] I. Grigoriev, E. Meilikhov, ed. Handbook of Physical Quantities (Boca Raton, FL: CRC Press, 1997).

[16] D. Lide, ed., CRC Handbook of Chemistry and Physics, 88th ed. (Boca Raton, FL: CRC Press, 2008).

[17] J. Speight, ed., Lange's Handbook of Chemistry, 16th ed. (New York, NY: McGraw-Hill Book Company, 2005).

[18] A. Von Hippel, Dielectrics and Waves (New York, NY: John Wiley & Sons, 1954).

[19] A. Von Hippel, ed., Dielectric Materials and Applications (Cambridge, MA: The M.I.T. Press, 1954).

[20] A. Von Hippel, ed. Molecular Science and Molecular Engineering (Cambridge, MA and New York, NY: The M.I.T.Press and John Wiley & Sons, Inc., 1959).

[21] J. Edminister, Electric Circuits (New York, NY: McGraw-Hill Book Company, 1965).

[22] J. Van Vleck, The Theory of Electric and Magnetic Susceptibilities (London: Oxford University Press, 1932).

[23] D. Martin, Magnetism in Solids (Cambridge, MA: The M.I.T. Press, 1967).

[24] R. Fraser, Molecular Rays (London: Cambridge University Press, 1931).

[25] N. Ramsey, Molecular Beams (Oxford: Clarendon Press, 1956).

[26] S. Bhatnagar, K. Mathur, Physical Principles and Applications of Magnetochemistry (London: Macmillan andCompany, Ltd., 1935).

[27] P. Selwood, Magnetochemistry, 2nd. ed. (New York, NY: Interscience Publishers, Inc., 1956).

[28] E. Stoner, Magnetism and Matter (London: Methuen & Company, Ltd., 1934).

[29] E. Stoner, Magnetism, 2nd ed. (London, Methuen & Company, Ltd., 1936).


[30] D. Jiles, Introduction to Magnetism and Magnetic Materials, 2nd. ed. (Boca Raton, FL: CRC Press, 1998).

[31] D. Hadfield, ed., Permanent Magnets and Magnetism: Theory, Materials, Design, Manufacture, and Applications(London: ILIFFE Books, Ltd., 1962).

[32] L. Moskowitz, Permanent Magnet Design and Application Handbook, 2nd ed. (Malabar, FL: Krieger PublishingCompany, 1995).

[33] R. McQuistan, Scalar and Vector Fields: A Physical Interpretation (New York, NY: John Wiley & Sons, 1965).

[34] A. Baden Fuller, Engineering Field Theory (Oxford: Pergamon Press, 1973).

[35] A. Baden Fuller, Worked Examples in Engineering Field Theory (Oxford: Pergamon Press, 1976).

[36] J. Artley, Fields and Configurations (New York: Holt, Rinehart and Winston, Inc., 1965).

[37] L. Setian, Engineering Field Theory with Applications (Cambridge, England: The University Press, 1992).

[38] M. Bradshaw, W. Byatt, Introductory Engineering Field Theory (Englewood Cliffs, NJ: Prentice-Hall, Inc., 1967).

[39] P. Moon, D. Spencer, Field Theory for Engineers (Princeton, NJ: D. Van Nostrand Company, Inc., 1961).

[40] D. Vitkovitch, ed., Field Analysis: Experimental and Computational Methods (London: D. Van NostrandCompany, Ltd., 1966).

[41] FlexPDE 6 Software Package (Spokane Valley, WA: PDE Solutions, Inc., 2008).

[42] G. Backstrom, Simple Fields of Physics by Finite Element Analysis: Applications in 2D and 3D toElectromagnetism, Heat, Waves, and Fluid Dynamics (Uppsala University, Sweden: GB Publishing, 2006).

[43] D. Goodstein, The Mechanical Universe and Beyond (California Institute of Technology: Annenberg Media,1985-1986).

[44] P. Falstad, www.falstad.com.


[45] R. Satz, The Unmysterious Universe (Troy, NY: The New Science Advocates, 1971).

[46] R. Satz, "The Gravitational Formula at High Velocities," Reciprocity, Vol. IV, No. 2, July 1974.

[47] R. Satz, "Further Mathematics of the Reciprocal System," Reciprocity, Vol. X, No. 3, Autumn 1980.

[48] R. Satz, "Photoionization and Photomagnetization," Reciprocity, Vol. XII, No. 1, Winter 1981-1982.

[49] R. Satz, "Theory of Electrons and Currents," Reciprocity, Vol. XIII, No. 1, Autumn 1983.

[50] R. Satz, "Permittivity, Permeability, and the Speed of Light in the Reciprocal System," Reciprocity, Vol. XVII, No. 2,Autumn 1988.

[51] R. Satz, "The Unit of Magnetic Charge," Reciprocity, Vol. XVIII, No. 1, Winter 1988-1989.

[52] R. Satz, "Theory of the Capacitor," Reciprocity Forum, July 4, 2007--latest version is athttp://transpower.wordpress.com.

[53] R. Satz, "Theory of Electrical Resistivity," Nov. 24, 2009, http://transpower.wordpress.com

[54] R. Satz, "Theory of Thermoelectricity, Thermomagnetism, and Thermal Resistivity," February 21, 2010,http://transpower.wordpress.com

[55] J. Ewing, Magnetic Induction in Iron and Other Metals (London: D. Van Nostrand Company, 1900). GoogleBooks has this work online athttp://books.google.com/books?id=noBCAAAAIAAJ&pg=PA13&dq=Magnetic+Induction+of+Iron+and+Other+Metals&hl-=en&ei=1l9XTOeyCoKC8gaHx7H8BA&sa=X&oi=book_result&ct=book-preview-link&resnum=1&ved=0CC0QuwUwAA-#v=onepage&q&f=false.

[56] W. Hughes, E. Gaylord, Basic Equations of Engineering Science (New York, NY: Schaum Publishing Company,1964).

[57] R. Siegel, J. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Washington, D.C.: Hemisphere PublishingCorporation, 1992).


[58] J. Donnay, H. Ondik, Crystal Data Determinative Tabels, 3rd. ed. (Washington, D.C.: National Bureau ofStandards, 1973).

[59] J. Tuma, Handbook of Physical Calculations (New York, NY: McGraw-Hill Book Company, 1976).

[60] R. Satz, "Calculation of the Gravitational Limits and the Hubble Constant for the Local Group," May 28, 2008,http://transpower.wordpress.com.

[61] R. Bronson, Modern Introductory Differential Equations (New York, NY: McGraw-Hill Book Company, 1973).

[62] J. Markus, Guidebook of Electronic Circuits (New York, NY: McGraw-Hill Book Company, 1974).

[63] S. Parker, ed., McGraw-Hill Concise Encyclopedia of Science and Technology (New York, NY: McGraw-Hill, Inc.,1984).

[64] C. Metz, 2000 Solved Problems in Physical Chemistry (New York, NY: McGraw-Hill, Inc., 1990).

[65] R. Wyckoff, Crystal Structures, six volumes (New York: John Wiley & Sons, 1963).

[66] R. Gautreau, W. Savin, Theory and Problems of Modern Physics (New York, NY: McGraw-Hill Book Company,1978).

[67] denito9474, "Philosophy of Quantum Physics, No BS | 1. Stern-Gerlach Exp.", [A YouTube video on theStern-Gerlach Experiment], http://www.youtube.com/watch?v=waK4eKNXB4A

[68] Magnetic Materials Producers Association, "Standard Specifications for Permanent Magnet Materials--No.0100-00" (Chicago, IL: MMPA); no date given, circa 2000. www.smma.org/PDFs/0100-00.pdf.

[69] D. Askeland, P. Phule, The Science and Engineering of Materials, 5th Ed. (Florence, KY: Cengage Learning,2006).

[70] http://en.wikipedia.org/wiki/Heusler_alloy

[71] W. Townes, J. Fang, A. Perrotta, "The Crystal Structure and Refinement of Ferrimagnetic Barium Ferrite,BaFe12O19", Zeiteschrift fur Kristallographic, Bd. 125, S. 437-449, 1967.


0 0.005 0.01 0.015

40

20

0

20

40

sin k6t

tu

cos k6t

tu

k6

sin k6t

tu

10 cos k6t

tu

k6

t


4 cos ti Vmax C convCV R 4 cos tf 2 Vmax2

4 cos tf Vmax C convCV R C2

convCV2

R2

2 16 R C conv

tf ti

max2

8 cos ti Vmax2

cos tf 4 cos ti Vmax C convCV R 4 cos tf 2 Vmax2

4 cos tf Vmax C convCV R C2

tf ti


0.002 0.0025 0.003


t

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

t L

R

d


texpR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

d


0 5 104

0.001 0.0015 0.002 0.0025 0.003

0

50

100

150

200

atan L

R

Vmax

R2

2L

2

sin t ac atan L

R

t


t L

R

d

texpR

Lt

Vmax

R2

2

L2

sin ac atan L

R

Vmax

R2

2

L2

sin t ac atan L

R

d


EC 2

R T 2 EC 2

L

1

2


2C

4 Vmax


2 ieff

T

convCV 2

R T 2 C4 Vmax


2 ieff

T

convCV


8 Vmax t C convCV L 8 Vmax2

t2

cos t 4 Vmax t2

C convCV R 4 C2

convCV2

2L

2 8 C convCV L cos t Vmax


chi_sat

2.1304E-048.8298E-04

5.4804E-03

3.0900E-04

1.1329E-03

2.7918E-03

4.6741E-031.3803E-02

1.7143E-02

1.7768E-02

4.2436E-04

1.4301E-03

3.4467E-03


K


H_Fe_obs_99.95


H_SiFe_obs


convCV Vmax cos tf cos ti 16 R C convCV Vmax sin tf sin ti 16 R C convCV Vmax

1

2

2convCV

2R

2

2 16 R C convCV Vmax cos tf cos ti 16 R C convCV Vmax sin tf sin ti 16 R C convCV V


CV 2

L

1

2


max t 4 C2

convCV2

2L R t 4 cos t 2 Vmax

2t2

4 cos t Vmax t2

C convCV R C2

convCV2

R2

2t2

16 t R t C conv


max

1

2


convCV Vmax cos t 16 t R t C convCV Vmax

1

2

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