Hyperfine Interactions Curso 2008 Clase 5- Página-1 Centro Brasileiro de Pesquisas Física - Rio de...

Hyperfine InteractionsCurso 2008Clase 5- Página-1

Centro Brasileiro de Pesquisas Física - Rio

de Janeiro -Brasil.Hyperfine Interactions

The electric charges present in the nucleus interact with the electrons that surround it. The electric currents ( or the magnetic moments) associated with the electrons and the nuclei also interact.

The main contributions to the interaction between the nucleus and its surrounding, involve the following nuclear moments:

Electric part nuclear electric monopole moment, Ze.

nuclear electric quadrupole moment, Q.

Magnetic part nuclear magnetic dipole moment,

The interaction of the nuclear electric monopole moment with the electric field of the electrons is de Coulomb interaction, and does not concern us here.




The other interactions between nuclei and electrons are called hyperfine interactions.

Experimentally, it is observed that the hyperfine interactions are much weaker that other electronic, atomic or ionic interactions.

For example, for rare earths:

ELS~ 104 k K Eexch~ 103 k K ECF ~ 102 k K Ehf~ 10-4k K

The nuclei are characterized by the atomic number Z and by the mass number A.

The angular momenta of the nucleons coupled in such way as to produce:

I = 0 when ( and only when) both Z and A are even.

I, the total nuclear angular momentum, measured in units of , can be integer or half- integer.

h




The nuclei having nonzero have an associated magnetic dipole moment given by: Iμ NIg

where gI is the nuclear g-factor and N is the nuclear magneton.

18362B

pN M

he B is the Bohr magneton.

The nuclear magnetic moment is also written: Iμ hHere, is the giromagnetic ratio.

Since N << B and the g-factors of nuclei and electrons are of the order of 1, it follows that the nuclear magnetic moments are much smaller than the ionic moments. For this reason, the nuclear magnetism of matter produces more subtle effects than the electronic magnetism. Every nucleus with I 0 has a magnetic dipolar moment. The nuclei that have I > 1/2 also possess a quadrupole moment Q, since their charge distributions lacks spherical symmetry.



de Janeiro -Brasil.Electrostatic Interactions

The nuclei located in a solid interact with the electric charges in their neighborhood. These charges can be the electrons bound to the same atom or to neighbor atoms, conduction electrons and other nuclei.

The interaction energy of a localized charge distribution with an electric potential produced by other charges is: dvrVrW )()(

The integration is made over the volume occupied by the nucleus.

The potencial V(r) can be expanded in a Taylor series around the origin:

...21)0()(

0

2

,0

ji

jji

iii

i xxVxx

xVxVrV

Summing and substrating the term:

0

2

22

0

2

,

2

61

61

iijiij

ji xVr

xxVr

where ij is the Kronecker delta, we obtain:




...)3(61

61)0()(

0

22

,0

2

22

0

ji

ijjji

iiiii

i xxVrxx

xVr

xVxVrV

Substituting in the integral:

...)()3(61

)(61)()()0(

2

0

2

,

2

0

2

2

0

dvrrxxxxV

dvrrxVdvrx

xVdvrVW

ijjijiji

iii

ii

dvrrxxQ

dvrrxVQ

xxVEpqVW

ijjiij

iiij

ji ji

)()3(

)(61

61)0()0(

2

2

0

2

2

, 0

2




dvrrxVQ

xxVEpqVW

iiij

ji ji

)(61

61)0()0( 2

0

2

2

, 0

2

Electrostatic energy of the nucleus taken as a point charge.

The electric dipole moment of the nucleus is cero, because the center of mass and the center of charge coincides.

This term only gives a displacement in the total energy. It does not depend on the nuclear orientation.

Interaction between the nuclear quadrupole moment and the Electric Field Gradient



de Janeiro -Brasil.

dvrrxxQ ijjiij )()3( 2

Electrostatic Interactions

We will obtain the expression of the electric quadrupole interaction in quantum mechanics. We susbtitute the charge density , by the operator:

k

krrer )()( ρ The sum extends over the Z protons, at positions rk, with coordinates xik.

The quadrupole moment tensor

becomes the tensorial operator:

dvrrrxxe kijjik

)()3( 2 ijQ

)3( 2 k

ijkjkik rxxe ijQ




The hamiltonian of the quadrupole interactions results: ij

ijijQ V QH

61

The matrix elements of this hamiltonian can be written in simple form using the Wigner -Eckart theorem. We apply this theorem to the matrix elements of the operators Qij:

C is a constant and represents other quantum numbers besides I and m.

Im')(23ImIm')3(Im 22

ijijjik

ijkjkik IIIIICrxxe

ijijji

ijijQ IIIIIV

IIeQ 2)(

23

)12(6HThe hamiltonian remains:

where Q is a number, called electric quadrupole moment, defined as:

IIrxxeIIeQk

ijkjkik )3( 2




The Electric Field Gradient tensor is a symmetrical tensor

0

2

jiij xx

VV

Therefore, can be put in diagonal form. Using the Laplace equation (∇2V = 0), it is posible to define the EFG tensor with only two values:

zz

yyxxzz V

VVandeqV

The axes are chosen in such way that the EFG components satisfy: xxyyzz VVV

Taking the axes coincident with the principal axes of the EFG, the quadrupole hamiltonian becomes:

)(3)12(4

22222

yxzQ IIIIIIqQe

H

10




Examples

I = 3/2 21

2222

311()1(3

)12(4

IIm

IIhqQeEQ

I = 5/2




Now, we will examinate the last term in W:

dvrr

xVWii

)(61' 2

0

2

2 This term involves the laplacian of V.

Since some electrons have nonzero density at the nucleus, the potential satisfy the Poisson´s equation:

0

2

eV

Then,

222

0

2

0

)0(61)0(

61' rZerZeW e




For a free ion of total angular momentum J, it can be shown that the interaction between the nuclear quadrupole moment and the EFG produced by the electrons is (Bleaney 1967):

)12()12(2

)1()1()(3 232

IIJJIIJJ

BIJIJ

H

where )12(32 JJJJrqQeB

JJ is a number tabulated for each ion (for rare earths see Elliot and Stevens (1953))



de Janeiro -Brasil.Magnetic Dipolar Interactions

The magnetic dipolar hyperfine interaction may be written as the interaction of the nuclear magnetic moment with the magnetic field produced by extranuclear sources.

hfhf BμH

)()( rArB

'')'(

4)( 0 dv

rrrJrA

v

3

'1'

1rrr

rrr



de Janeiro -Brasil.

0')'( dvrJv

')'('

21')'(' dvrJrrdvrJrr

vv

')'('21 dvrJrm

v

3

0

4)(

rrmrA

5

20 )(3

4)(

rrmrrmrB

Magnetic Dipolar Interactions



de Janeiro -Brasil.

Contribution of the Electronic Spin to the Magnetic Hyperfine Field

rkikk erur

)()(2

),()( ruri

iiB

iiiiB rsgrrsgrM )()()()(

vrB

rrMrrMw s

IIs)()(3

4 5

20

μμ

is

iiiBs rsrrsgB 30 ˆ)ˆ(3

4

dv

rr

rv

i

is

3

3 )(



de Janeiro -Brasil.Fermi contact term

MB o

38

4

2)0()0()0( sgsgM BB

)0(3

84

sgB Bo

c

sI I

gW IB

oc

)0(3

84

n

nsnstot 22

),0(),0()0(



de Janeiro -Brasil.Core polarization field

)0(3

84

's

ocp MB

JgBcp )1(6

The incomplete shells (and the conduction electrons) may also give to another contribution to the hyperfine field, through the modification to the hyperfine field, through the modification of the radial distribution of the closed shells, thus producing a noncompensated spin density at the origin.

This leads to an s magnetization equal to M's(0) at the nucleus, and this term of the hyperfine field, called the core polatization field , acts through the contact term and is written

This term is dominant in the hyperfine field of the S -state of rare earth ions, such as Gd+3 (-21T) , and in the ions of d transition metals, such as Fe ( -27.5 T in metallic Fe).

In the series of tripositive rare-earths ions, Bcp es proporcional to the spin component of the of the total angular momentun J, given in Tesla, approximately by



de Janeiro -Brasil.

Orbital contribution to the magnetic hyperfine Field

30

4)(

rr

rA I

μ

dV

rrJr

dVr

rrJdVrArJW

V

eI

V

Ie

Ve

3

03

0 )(4

)(4

)()(

μμ

dqvrddtdqdVrJ e

)( dq

rvrW

VI

3

0

4

μ

hlvmr

3

3

lq

rerdq



de Janeiro -Brasil.

Orbital contribution to the magnetic hyperfine Field

il

N

iiIB

o

il

N

iiI

oL rlr

mhelW 33 2

44

μμ

33 24

24

lI

Bo

lIBo

L rIμ

rW LILμ

LIL BW μ 32

4 lB

oL rB L



de Janeiro -Brasil.Free atom

SLJ JIH Ahf

In the more general case A is the hyperfine tensor.

The description of the inetraction in terms of the hyperfine field Bhf in fact applies when A has uniaxial symmetry (Az= A >>Ay , Ay)

hfIhf BA μJIH J

NIhf g

AB

JIF F is the hyperfine quantum number



de Janeiro -Brasil.

Contributions to Bhf in the free ion

cpdiporbhf BBBB



de Janeiro -Brasil.

Hyperfine Interactions Curso 2008 Clase 5- Página-1 Centro Brasileiro de Pesquisas Física - Rio de...

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