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).
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