1

Hyperfine interaction and Knight shift

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: July 10, 2012)

The Knight shift is a shift in the nuclear magnetic resonance frequency of a paramagnetic

substance first published in 1949 by the American physicist Walter David Knight. The Knight

shift refers to the relative shift K in NMR frequency for atoms in a metal (e.g. sodium) compared

with the same atoms in a nonmetallic environment (e.g. sodium chloride). The observed shift

reflects the local magnetic field produced at the sodium nucleus by the magnetization of the

conduction electrons. The average local field in sodium augments the applied resonance field by

approximately one part per 1000. In nonmetallic sodium chloride the local field is negligible in

comparison.

The Knight shift is due to the conduction electrons in metals. They introduce an "extra"

effective field at the nuclear site, due to the spin orientations of the conduction electrons in the

presence of an external field. This is responsible for the shift observed in the nuclear magnetic

resonance. The shift comes from two sources, one is the Pauli paramagnetic spin susceptibility,

the other is the s-component wave functions at the nucleus. Depending on the electronic structure,

the Knight shift may be temperature dependent. However, in metals which normally have a

broad featureless electronic density of states, Knight shifts are temperature independent.

http://en.wikipedia.org/wiki/Knight_shift

1. Introduction

There is an interaction between the magnetic moment of a nucleus and the magnetic

moment of electron (orbital magnetic moment and spin magnetic moment). This

interaction is very important in the nuclear magnetic resonance (NMR). Through this

interaction, the information on the properties of electrons surrounding the nucleus can be

observed by NMR. The interaction consists of the dipole-dipole interaction (spin-dipolar

interaction), the hyperfine interaction (Fermi contact field), and the crystal field (related

to the orbital angular momentum).

2. The coupling Hamiltonian between electron and nucleus (Abraham)

2

The behavior of an electron [ q = -e (charge of electron); m (mass of electron)] in a

magnetic field B produced by a nucleus, is given by the Hamiltonian

BsAp )2()(2

1 2

Bc

q

mH .

where B is the Bohr magneton of electron

mc

eB

2

ℏ .

s = ℏ/S and S is the spin angular momentum (in the units of ℏ ). The second term arises

from the spin magnetic moment in the presence of magnetic field B. According to the

classical electromagnetic theory, the magnetic moment of nucleus )( Iμ ℏ produces at a

point removed from it by a vector r, a magnetic field B

AB ,

with

3r

rμA ,

where A is the magnetic vector potential.Noting that

3

3

1

rr

r .

Then we get

][]1

[)(3

rrr

μμ

rμAB ,

since

rrrrr

1111

μμμμμ

.

(b) The vector A satisfies the Coulomb gauge,

0 A

since

0)()(

)(

33

3

rr

r

rμμ

r

rμA

________________________________________________________________________

((Mathematica))

4

Near the origin, A has a singularity of order r-2 and B has a singularity of order r-3, so

some care must be exerted in the calculation of its interaction.

____________________________________________________________________

3. Magnetic field ditribution due to the magnetic moment of nucleus

Suppose that the magnetic moment of the nucleus is directed along the z axis at the

origin.

),0,0( 0μ , 3r

rμA

The magnetic field produced by the magnetic mpoment of nucleus is

222

5

0 2,, zyxyzzxr

AB

We make a plot of B in the y-z plane by using the StreamPlot (Mathyematica), where 0

= 1.

Clear@"Global`∗"D;

Needs@"VectorAnalysis`"D; SetCoordinates@Cartesian@x, y, zDD;

= 81, 2, 3<; r = x2 + y2 + z2 ; R = 8x, y, z<; R1 = R ë r3;

A1 = Cross@µ, R1D êê Simplify

9z µ2 − y µ3

Ix2 + y2 + z2M3ê2,

−z µ1 + x µ3

Ix2 + y2 + z2M3ê2,

y µ1 − x µ2

Ix2 + y2 + z2M3ê2=

Div@A1D êê FullSimplify

0

Curl@R1D êê FullSimplify

80, 0, 0<

B1 = Curl@A1D êê FullSimplify

92 x2 µ1 − Iy2 + z2M µ1 + 3 x Hy µ2 + z µ3L

Ix2 + y2 + z2M5ê2,

3 x y µ1 − x2 µ2 + 2 y2 µ2 − z2 µ2 + 3 y z µ3

Ix2 + y2 + z2M5ê2,3 z Hx µ1 + y µ2L − Ix2 + y2 − 2 z2M µ3

Ix2 + y2 + z2M5ê2=

5

((Mathematica))

Clear@"Gobal`"D

Needs@"VectorAnalysis`"D

SetCoordinates@Cartesian@x, y, zDD

Cartesian@x, y, zD

r = 8x, y, z<; m = 80, 0, m0<;

A =1

Hr.rL3ê2Cross@m, rD êê Simplify

9−m0 y

Ix2 + y2 + z2M3ê2,

m0 x

Ix2 + y2 + z2M3ê2, 0=

B = Curl@AD êê Simplify

93 m0 x z

Ix2 + y2 + z2M5ê2,

3 m0 y z

Ix2 + y2 + z2M5ê2, −

m0 Ix2 + y2 − 2 z2M

Ix2 + y2 + z2M5ê2=

rule1 = 8m0 → 1, x → 0<;

B1 = B ê. rule1

90,3 y z

Iy2 + z2M5ê2, −

y2 − 2 z2

Iy2 + z2M5ê2=

f1 = StreamPlot@8B1@@2DD, B1@@3DD<, 8y, −3, 3<, 8z, 3, −3<,StreamStyle → PurpleD;

f2 = Graphics@8Red, Thick, Arrow@880, −0.5<, 80, 0.5<<D,Black, Thin, Line@88−3, 0<, 83, 0<<D,Line@880, −3<, 80, 3<<D,Text@Style@"y", Black, 12D, 83.2, 0<D,Text@Style@"z", Black, 12D, 80, 3.2<D <D;

Show@f1, f2D

6

Fig. The distribution of the magnetic field B produced by the magnetic moment of

nucleus at the origin (along the z axis) in the y-z plane.

4. Orbital magnetic moment contribution

The Hamiltonian arising from the orbitral motion of the electron isgiven by

'

2)]([

2

1

)()(2

1

)(2

1

0

2

2

22

2

L

L

HH

mc

e

c

e

m

c

e

c

e

m

c

q

mH

ApAApp

ApAp

Ap

with

2

02

1p

mH ,

y

z

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

7

and

2

2

2

2)(

2' ApAAp

mc

e

mc

eH L .

In the first order (the order of A) perturbation , we have

)]()(2

' pAAppAAp ℏ

BL

mc

eH

.

Here we calculate

)()]()(' rpAApr

ℏ

BLH .

where )(r is an arbitrary wave function and p is the quantum mechanical operator

defined by

i

ℏp .

Using the formula (vector analysis),

ArrArA )()())(( ,

we get

)(2

)(2

)()(2

)()(2

)(2

)](2)([)('

33

3

3

rμlrμL

rμpr

rpr

μ

rA

rAArr

rr

r

r

i

iH

BB

B

B

B

BL

ℏ

ℏ

ℏ.

wehere we use 0 A and

8

3r

rμA

The orbital angular momentum L is defined as

prlL ℏ .

in the quantum mechanics. Then the Hamiltonian can be written as

32'

rH BL

μl

We note that the orbital magnetic moment L is

lLμ BB

L

ℏ

.

5. Spin magnetic moment contribution

The interaction between the spin magnetic moment of electron and is

])([2

][[2

)2('

2

rr

r

H

B

B

BS

μμs

μs

Bs

where the spin magnetic moment is given by

Ssμℏ

BBS

2)2( .

and S is the spin angular momentum (in the units of ℏ ). Here we use the formula

FFF 2)()(

where F is any vector. We note that

9

rrrr

111 μμμ

μ

where is independent of r. Then we have

rrrrr

1)()

1(

1)()

1()(

μμμμμ

rr

122 μμ

The interaction 'SH can be written as

)1

]()())([(2

)]1

()1

()[(2'

2

2

r

rrH

B

BS

μssμ

μμs

This can be rewritten as

)1

]()())([(2' 2

rH BS μssμ

Here we notice that for 0r

5

2))((3)(

)1

)()((r

r

r

rsrμsμsμ

_______________________________________________________________________

((Mathematica))

10

___________________________________________________________________

6. The expression of 'SH near the origin (White): hyperfine interaction

We start with the spin Hamilonian

)(2' rH BS As

The matrix element of the Hamiltonian for the wave function )(r

rrB

B

s

s

m

dd

d

HW

)()()()()()([2

)()]()[(2

'

**

*

rrAsrrrrAsrr

rrAsrr

where the radius defines a sphere which encloses the nucleus. Outside the sphere A(r) is

given by

3r

rμA

The second term in s

mW ( )2(m

sW ) gives the dipole-dipole interaction since r>. The first

term in s

mW is the additional interaction

Clear@"Global`∗"D; Needs@"VectorAnalysis`"D;SetCoordinates@Cartesian@x, y, zDD; s = 8s1, s2, s3<; = 81, 2, 3<;

r = x2 + y2 + z2 ;

R = 8x, y, z<;

L1@a_D := Ha@@1DD D@� , xD + a@@2DD D@�, yD + a@@3DD D@�, zDL &;

f1 = L1@µD@L1@sD@1êrDD êê FullSimplify

1

Ix2 + y2 + z2M5ê2I3 s3 z Hx µ1 + y µ2L −

s3 Ix2 + y2 − 2 z2M µ3 + s2 I3 x y µ1 − x2 µ2 + 2 y2 µ2 − z2 µ2 + 3 y z µ3M +

s1 I2 x2 µ1 − Iy2 + z2M µ1 + 3 x Hy µ2 + z µ3LMM

f2 =1

r5I3 Hµ.RL Hs.RL − Hµ.sL r2M êê FullSimplify

−Ix2 + y2 + z2M Hs1 µ1 + s2 µ2 + s3 µ3L + 3 Hs1 x + s2 y + s3 zL Hx µ1 + y µ2 + z µ3L

Ix2 + y2 + z2M5ê2

f1 − f2 êê Simplify

0

11

r

B

m

s dW )]()([)(2 *)1( rrAsrr

Using the vector analysis

srrArrAssrrA )()()]()([])()([

we get

)]()([])()([ rrAssrrA

where we use the assumption that s is independent of r; 0 s . Then we get

r

B

m

s dW srArr )()(22)1(

since

srAr

srArsrArrsrA

)()(

)()())(()()]()([

Note that )(r is independent of r for r<; )0()( r . Then we have

rB

rB

rB

m

s

d

d

dW

)()0(2

)()0(2

)()0(2

2

2

2)1(

rAas

srAa

srAr

We also use the Gauss’s theorem. Because of the sphere of integrals has been chosen to

lie outside the nucleus,

3)(

r

rμrA

22

3

2)1( )0(3

16

3

8)0(2)()0(2

μsμs

rμas BB

rB

m

sr

dW

Note that

12

μ

μμ

μμ

μrrμ

μrrμ

rμ

rrμa

3

8

3

424

sin24

])(

[

)]([1

)()(

0

3

2

2

2

3

2

3

d

rd

rr

d

rrdr

rd

r

where da is the surface element normal to the surface of sphere. Note that

rdrdad r

rea 2

with

ddd sin

d is the solid angle. For simplicity we assume that is directed along the z axis.

)sin,sincossin,coscossin(

)cos,sincossin,coscos(sin),0,0(

)cos,sinsin,cos(sincos)(

2

2

2

2

2

r

re

rz

μrrμ

13

This is the contact hyperfine interaction

6. Magnetic hyperfine field

The resulting Hamiltonian is obtained as

35

2

1 2)()(3

16))((3)(2

rr

rH BBB

μlrμs

rsrμsμ

The first term is a dipole-dipole interaction and the second term is a contact term of the

hyperfine interaction. Since

locH Hμ 1

The local field Hloc seen by a nucleus is given by

14

)](3

8)(3[2

533rs

rsrslH

rrrBloc

7. Knight shift in metals (White)

The s-like conductuion electrons produce a contact hyperfine field at the nuclei of a

metal. Thus, if the conduction electrons are polarized by an external field, at a fixed

frequency the resonance of a nuclear spin is observed at a slightly different magnetic field

in a metal than in a diamagnetic solid. This effect is known as the Knight shift or metallic

shift.

0H

HK

z

loc .

The local field z

locH is given by

2* )0(3

16)()()(

3

16

zBzB

z

loc SdSH rrrr

The average conduction electron spin zS is related to the Pauli spin susceptibility s of

the conduction electrons,

0HSNgM szBz .

The Knight shift is rewritten as

22)0()0(

3

16

NNgK s

B

sB ,

where

2)0(

1

N,

is the ratio of the conduction electron concentration at the nucleus to the average

conduction electron. The spin susceptibility can be determined by very careful

conduction electron spin resonance experiment.

8. Pauli paramagnetism and density of states in metal

15

The susceptibility due to the Pauli paramagnetism of conduction electrons is obtained as.

)(2)(22

FBFBs ND . (41)

For metals which is well described by free electron model, the Knight shift is proportional to the

density of states at the Fermi level (per spin).

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REFERENCES

W.D. Knight, Solid State Phys. 2, 93 (1956).

A. Abragam, The Principle of Nuclear Magnetism (Oxford at the Clarendon Press)

R.M. White, Quantum Theory of Magnetism (Springer, Berlin, 2007)

H.J. Zeiger and G.W. Pratt, Magnetiuc interactions in solids (Clarendon Press, Oxford

1973).