11. Diamagnetism and Paramagnetism
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Ref:
Langevin Diamagnetism Equation
Quantum Theory of Diamagnetism of Mononuclear Systems
Paramagnetism
Quantum Theory of Paramagnetism
Rare Earth Ions
Hund Rules
Iron Group Ions
Crystal Field Splitting
Quenching of The Orbital Angular Momentum
Spectroscopic Splitting Factor
Van Vieck Temperature-Independent Paramagnetism
Cooling by Isentropic Demagnetization
Nuclear Demagnetization
Paramagnetic Susceptibility of Conduction Electrons
D.Wagner, “Introduction to the Theory of Magnetism”, Pergamon Press (72)
Bohr-van Leeuwen Theorem
M = γ L = 0 according to classical statistics.
→ magnetism obeys quantum statistics.
Main contribution for free atoms:
• spins of electrons
• orbital angular momenta of electrons
• Induced orbital moments
paramagnetism
Electronic structure
Moment
H: 1s
MS
He: 1s2
M=0
unfilled shell
M0
All filled shells
M=0
diamagnetism
Magnetization M  magnetic moment per unit volume
Magnetic subsceptibility per unit volume  
χ M = molar subsceptibility
σ = specific subsceptibility
M
H
In vacuum, H = B.
nuclear moments ~ 10−3 electronic moments
Larmor Precession
1 3 J  x 
A   d x
c
x  x
Magnetic (dipole) moment:
For a current loop:
1
d 3 x x  J  x 

2c
m
J d 3x  I d l
For a charge moving in a loop:
m
m
1
2c
A
 x I dl 
J  x   q v   x  xq 
q
1
q
3



d
x
x

q
v

x

x

x

v

L


q
q

2c
2c
2mc
Classical gyromagnetic ratio
Torque on m in magnetic field:

q
2mc
Γ
Lorentz force:
dv q
 vB
dt c
I
Area
c
( charge at xq )
Caution: we’ll set L to  L
in the quantum version
 L
B 
e
2mc
dL
 mB   LB
dt
→ L precesses about B with the Larmor frequency
m
mx
r3
→ cyclotron frequency
qB
2mc
qB
c 
 2L
mc
L   B 
Langevin Diamagnetism Equation
Diamagnetism ~ Lenz’s law: induced current opposes flux changes.
L 
Larmor theorem: weak B on e in atom → precession with freq
Larmor precession
of Z e’s:
L
Z e2 B
I    Ze 

2
4 m c
 2  x2  y 2
For N atoms per unit volume:
1
 I
c
r 2  x2  y 2  z 2
1
eB
 C
2
2mc

r2 
→
N
N Z e2 2


r
B
6 m c2
χ<0
2
Z e2 B

4 m c2
2
3
2
2
Langevin diamagnetism
same as QM result
Good for inert gases
and dielectric solids
experiment
Failure: conduction electrons (Landau diamagnetism & dHvA effect)
Quantum Theory of Diamagnetism of Mononuclear Systems
Quantum version of Langevin diamagnetism
ie
e2
H 
A2
   A  A   
2
2mc
2mc
Perturbation Hamiltonian [see App (G18) ]:
Uniform B  B zˆ
→ A
1
B   y , x, 0
2

→  A  0
A  
1 

 
By
x

2  x
y 
i eB  
  e2 B 2 2
2
eB
e2 B 2 2
2
H 
x

y

x

y


Lz 
x

y


2 


4 m c  y
x  8 m c
2mc
8 m c2
The Lz term gives rise to paramagnetism.
1st order contribution from 2nd term:
e2 r 2
 E


B
B
6 m c2
e2 B 2
2
E 

8 m c2
e2 B 2
2

r
12 m c 2
same as classical result
Paramagnetism
Paramagnetism: χ > 0
Occurrence of electronic paramagnetism:
• Atoms, molecules, & lattice defects with odd number of electrons ( S  0 ).
E.g., Free sodium atoms, gaseous NO, F centers in alkali halides,
organic free radicals such as C(C6H5)3.
• Free atoms & ions with partly filled inner shell (free or in solid),
E.g., Transition elements, ions isoelectronic with transition elements,
rare earth & actinide elements such as Mn2+, Gd3+, U4+.
• A few compounds with even number of electrons.
E.g., O2, organic biradicals.
• Metals
Quantum Theory of Paramagnetism
μ 
Magnetic moment of free atom or ion:
γ = gyromagnetic ratio.
g = g factor.
g  B  
For electrons g = 2.0023
For free atoms,
g  1
U  μ  B  mJ g B B
N
eB
 B
N e
 e   B
μB = Bohr magneton.
B 
J  LS
Caution: J here is
dimensionless.
e
~ spin magnetic moment of free electron
2mc
J  J  1  S  S  1  L  L  1
2 J  J  1
mJ   J ,  J  1,
For a free electron, L = 0, S = ½ , g = 2,
→
mJ =  ½ , U =  μB B.
N
e   B

N e   B  e   B
J   g B J
, J  1, J
Anomalous
Zeeman effect
x
B
k BT
N
e x
 x
N e  e x
N
ex
 x
N e  e x
e x  e x
M   N  N    N  x
 N  tanh x
e  e x
High T ( x << 1 ):
N 2 B
M  N x 
kB T
Curie-Brillouin law:
M  N g J B BJ  x 
x
g J B B
kB T
Brillouin function:
BJ  x  
  2 J  1 x  1
2J 1
 x 
ctnh 

ctnh



2J
2
J
2
J
2
J




M  N g J B BJ  x 
BJ  x  
x
g J B B
kB T
  2 J  1 x  1
2J 1
 x 
ctnh 
ctnh 


2J
2J
 2J 

 2J
High T ( x << 1 ):
1 x x3
ctnh x    
x 3 45
2 2
N p 2  B2 C
M N J  J  1 g  B




3
k
T
B
3 k BT
T
B
pg
N J  J  1
Curie
law
= effective number
of Bohr magnetons
Gd (C2H3SO4)  9H2O
Rare Earth Ions
Lanthanide
contraction
ri = 1.11A
ri = 0.94A
4f radius ~ 0.3A
Perturbation from higher states
significant because splitting
between L-S multiplets ~ kB T
Hund’s Rules
For filled shells, spin orbit couplings do not change order of levels.
Hund’s rule ( L-S coupling scheme ):
Outer shell electrons of an atom in its ground state should assume
1.Maximum value of S allowed by exclusion principle.
2.Maximum value of L compatible with (1).
3.J = | L−S | for less than half-filled shells.
J = L + S for more than half-filled shells.
Causes:
1. Parallel spins have lower Coulomb energy.
2. e’s meet less frequently if orbiting in same direction (parallel Ls).
3. Spin orbit coupling lowers energy for LS < 0.
Mn2+:
3d 5
(1) → S = 5/2
Ce3+:
4f1
L = 3, S = ½
Pr3+:
4f2
(1) → S = 1
exclusion principle → L = 2+1+0−1−2 = 0
(3) → J = | 3− ½ | = 5/2
(2) → L = 3+2 = 5
2
F5/2
(3) → J = | 5− 1 | = 4
3
H4
Iron Group Ions
L=0
Crystal Field Splitting
Rare earth group: 4f shell lies within 5s & 5p shells
→ behaves like in free atom.
Iron group: 3d shell is outer shell
→ subject to crystal field (E from neighbors).
→ L-S coupling broken-up; J not good quantum number.
Degenerate 2L+1 levels splitted ; their contribution to moment diminished.
Quenching of the Orbital Angular Momentum
Atom in non-radial potential → Lz not conserved.
If  Lz  = 0, L is quenched.
μ  B L  2S
 L is quenched → μ is quenched
L = 1 electron in crystal field of orthorhombic symmetry ( α = β = γ = 90, a  b  c ):
e   A x2  B y 2  C z 2
2  0
U j  xj f r 
Consider wave functions:
For i  j, the integral
i.e.,
Ui e  U j
d rU
3
Similarly
*
i
Uj
→
e   Ax2  By 2   A  B  z 2
L 2 U j  L  L  1 U j  2 U j
→
is odd in xi & xj , and hence vanishes.
 i j Ui e  Ui
U x e  U x   d 3r f  r 
where
 A B C  0
I1   d 3r f  r 
Uy e Uy
2
2
 A x 4  B x 2 y 2   A  B  x 2 z 2   A  I1  I 2 
x 4j
 B  I1  I 2 
I 2   d 3r f  r 
2
xi2 x 2j
U z e  U z    A  B  I1  I 2 
 Uj are eigenstates for the atom in crystal field.
Orbital moments are zero since
U j Lz U j
0
Quenching
For lattice with cubic symmetry,
e   A  x2  y 2  z 2 
2  0
 A0
 there’s no quadratic terms in e φ .
→ Ground state remains triply degenerate.
Jahn-Teller effect: energy of ion is lowered by spontaneous lattice distortion.
E.g., Mn3+ & Cu2+ or holes in alkali & siver halides.
Spectroscopic Splitting Factor
λ = 0 or H = 0 → Uj degenerate wrt Sz.
In which case, let A, B be such that ψ0 = x f(r) α is the ground state, where
α (spin up) and β (spin down) are Pauli spinors.
1st order perturbation due to λ LS turns ψ0 into

  U x  i



U y  
Uz 
21
2 2


where
1   y   x
2   z   x
α | β  = 0 → term Uz β ~ O(λ2) in any expectation values.
 It can be dropped in any 1st approx.
 Lz   
Thus

1
z  B  Lz  2Sz 
 

    1  B
 1 
Energy difference between Ux α and Ux β in field B :
→


g  2 1  
 1 


E  g  B H  2 1   B H
 1 
Van Vleck Temperature-Independent Paramagnetism
Consider atomic or molecular system with no magnetic moment in the ground state , i.e.,
0 z
0  s z
s 0
In a weak field μz B << Δ = εs – ε0 ,
0

0  0 
B
s

0 z
0
s z
s
s z 0


0  z
s  z
a) Δ << kB T
N0  N s 
M
2B

N
0  2
B

s z
0
s  2
B

s z
0
B
0

0 z s
2
2
b) Δ >> kB T

N
2 k BT
N0  N s  N
2
s z 0
s z 0
s  s 

s
2
N
2 kB T
1
kB T
2B

s z 0
2
2N

s z 0
2
M
Curie’s law

N
van Vleck
paramagnetism
Cooling by Isentropic Demagnetization
Was 1st method used to achieve T < 1K.
Lowest limit ~ 10–3 K .
Mechanism: for a paramagnetic system at fixed T, Δ S < 0 as H increases.
i.e., H aligns μ and makes system more ordered.
→ Removing H isentropically (Δ S = 0) lowers T.
Lattice entropy can seeps in
during demagnetization
Magnetic cooling is not cyclic.
Isothermal magnetization
Isoentropic demagnetization
T2
Spin entropy if all states are accessible:
S  k B ln  2S  1
N
 N kB ln  2S  1
S is lowered in B field since lower energy states are more accessible.
Population of magnetic sublevels is function of μB/kBT, or B/T.
 ΔS=0→
B B

T1 T2
or
T2  T1
B
B
BΔ = internal random field
Nuclear Demagnetization
T2  T1
m p ~ 1836 me
g p ~ 5.58
B
B
5.58
3
→  p ~ 2  1836 e ~ 1.52 10  B
gn ~ 3.83
T2 = T1 ( 3.1 / B )
→ T2 of nuclear paramagnetic cooling
~ 10–2 that of electronic paramagnetic cooling.
B = 50 kG, T1 = 0.01K, →
B
kB
p B
k B T1
 0.5
 6.72 105 G / K
ΔS on magnetization is over 10% Smax.
→ phonon Δ S negligble.
Cu: T1 = 0.012K
BΔ =3.1 G
102 100G
T2  0.01K
 2 107 K
50kG
Paramagnetic Susceptibility of Conduction Electrons
Classical free electrons:
N  B2 B
M
k BT
~ Curie paramagnetism
Experiments on normal non-ferromagnetic metals : M independent of T
Pauli’s resolution:
Electrons in Fermi sea cannot flip over due to exclusion principle.
Only fraction T/TF near Fermi level can flip.

N  B2 B T
N  B2 B
M
 
kBT TF
k B TF
Pauli paramagnetism at T = 0 K
T=0


1 F
1 F
1
N   d  D     B    d  D      B D  F 
2  B
2  B
2
parallel moment


1 F
1 F
1
N   d  D     B    d  D      B D  F 
2  B
2  B
2
M Pauli    N  N    B D  F 
2
Landau diamagnetism: M Landau
3N  2

B
2k BTF
N2

B
2k BTF
→
χ is higher in transition metals due to higher DOS.
anti-parallel moment
χ > 0 , Pauli paramagnetism
M  M Pauli  M Landau
N2

B
2k BTF
Prob. 5 &6