11. Diamagnetism and Paramagnetism
• 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
Ref: D.Wagner, “Introduction to the Theory of Magnetism”, Pergamon Press (72)
Bohr-van Leeuwen TheoremM = 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
diamagnetism
Electronic structure Moment
H: 1s M S
He: 1s2 M = 0
unfilled shell M 0
All filled shells M = 0
Magnetic subsceptibility per unit volumeM
H
Magnetization M magnetic moment per unit volume
χ M = molar subsceptibilityσ = specific subsceptibility nuclear moments ~ 10−3 electronic moments
In vacuum, H = B.
Larmor Precession
Magnetic (dipole) moment: 31
2d x
c m x J x
3r
m xA
31d x
c
J x
Ax x
For a current loop:
1
2I d
c m x l
IArea
c
For a charge moving in a loop:
3d x I dJ l
qq J x v x x ( charge at xq )
31
2 qd x qc
m x v x x2 q
q
c x v
2
q
m c L L
Classical gyromagnetic ratio 2
q
m c
Torque on m in magnetic field:d
d t
LΓ m B L B
→ L precesses about B with the Larmor frequency2L
q BB
m c
Lorentz force:d q
mdt c
vv B → cyclotron frequency c
q B
m c 2 L
2B
e
m c
Caution: we’ll set L to L in the quantum version
Langevin Diamagnetism Equation
Diamagnetism ~ Lenz’s law: induced current opposes flux changes.
Larmor theorem: weak B on e in atom → precession with freq 2L
e B
m c 1
2 C
Larmor precession of Z e’s:
2
2 4L Z e B
I Zem c
21I
c
22
24
Z e B
m c
2 2 2x y 2 2 2 2r x y z 2 23
2r →
For N atoms per unit volume:2
226
N N Z er
B m c
Langevin diamagnetism same as QM result
experiment
Good for inert gasesand dielectric solids
Failure: conduction electrons (Landau diamagnetism & dHvA effect)
χ < 0
Quantum Theory of Diamagnetism of Mononuclear Systems
Quantum version of Langevin diamagnetism
Perturbation Hamiltonian [see App (G18) ]: 2
222 2
i e e
m c m c A A A
H
Uniform ˆBB z → 1, , 0
2B y x A
2 2
2 224 8
i e B e Bx y x y
m c y x m c
H
0 A→1
2B y x
x y
A
2 2
2 222 8z
e B e BL x y
m c m c
The Lz term gives rise to paramagnetism.
1st order contribution from 2nd term:2 2
228
e BE
m c
2 2
26
e rEB
B m c
same as classical result
2 22
212
e Br
m c
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
Magnetic moment of free atom or ion: μ J Bg J J L S
γ = gyromagnetic ratio.g = g factor. Bg
For electrons g = 2.0023
For free atoms,
1 1 11
2 1
J J S S L Lg
J J
μB = Bohr magneton.
2B
e
m c
~ spin magnetic moment of free electron
U μ BJ Bm g B , 1, , 1,Jm J J J J
For a free electron, L = 0, S = ½ , g = 2, → mJ = ½ , U = μB B.
B
B B
N e
N e e
B
B B
N e
N e e
Anomalous Zeeman effect
Caution: J here is dimensionless.
x
x x
N e
N e e
x
x x
N e
N e e
B
Bx
k T
M N N x x
x x
e eN
e e
tanhN x
High T ( x << 1 ):2
B
N BM N x
k T
B JM N g J B x B
B
g J Bx
k T
Curie-Brillouin law:
Brillouin function:
2 12 1 1
2 2 2 2J
J xJ xB x ctnh ctnh
J J J J
2 12 1 1
2 2 2 2J
J xJ xB x ctnh ctnh
J J J J
High T ( x << 1 ):31
3 45
x xctnh x
x
2 21
3B
B
N J J gM
B k T
2 2
3B
B
N p
k T
C
T
Curie law
1p g N J J = effective number of Bohr magnetons
B JM N g J B x B
B
g J Bx
k T
Gd (C2H3SO4) 9H2O
Rare Earth Ions
ri = 1.11A
ri = 0.94A
Lanthanide contraction
4f radius ~ 0.3APerturbation from higher states significant because splitting between L-S multiplets ~ kB T
Hund’s Rules
Hund’s rule ( L-S coupling scheme ):Outer shell electrons of an atom in its ground state should assume1.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 LS < 0.
For filled shells, spin orbit couplings do not change order of levels.
Mn2+: 3d 5 (1) → S = 5/2 exclusion principle → L = 2+1+0−1−2 = 0
Ce3+: 4 f 1 L = 3, S = ½ (3) → J = | 3− ½ | = 5/2
Pr3+: 4 f 2
(1) → S = 1 (2) → L = 3+2 = 5 (3) → J = | 5− 1 | = 4
25/2F
34H
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.
2B μ L S L is quenched → μ is quenched
L = 1 electron in crystal field of orthorhombic symmetry ( α = β = γ = 90, a b c ):
2 2 2e A x B y C z 2 0 0A B C
Consider wave functions: j jU x f r 2 1 2j j jU L L U U L
For i j, the integral 3 *
i jd r U U is odd in xi & xj , and hence vanishes.
i.e., i j i j i iU e U U e U
23 4 2 2 2 2x xU e U d r f r A x B x y A B x z 1 2A I I
where 23 41 jI d r f r x 23 2 2
2 i jI d r f r x x 1 2y yU e U B I I Similarly 1 2z zU e U A B I I
→
2 2 2e Ax By A B z →
Uj are eigenstates for the atom in crystal field.
Orbital moments are zero since 0j z jU L U Quenching
→ 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.
2 2 2e A x y z
For lattice with cubic symmetry,
2 0 0A
there’s no quadratic terms in e φ .
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 LS turns ψ0 into
1 22 2x y zU i U U
where1 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.
1zL
Thus
2z B z zL S 1
1 B
BE g H Energy difference between Ux α and Ux β in field B : 1
2 1 BH
→1
2 1g
Van Vleck Temperature-Independent Paramagnetism
Consider atomic or molecular system with no magnetic moment in the ground state , i.e.,
0 0 0z zs s
In a weak field μz B << Δ = εs – ε0 ,
0 0 0 0z
Bs s
20 0 0 0 2 0z z z
Bs
2
2 0z z z
Bs s s s s
a) Δ << kB T
220
2zB
B NM s
k T
2 10z
B
N sk T
b) Δ >> kB T
220z
BM s N
22
0z
Ns
0 2sB
N N Nk T
0 sN N N
Curie’s law
van Vleck paramagnetism
0 0 z
Bs s s s
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 < 0 as H increases.
i.e., H aligns μ and makes system more ordered.
→ Removing H isentropically (Δ = 0) lowers T.
Lattice entropy can seeps in during demagnetization
Magnetic cooling is not cyclic.
Isothermal magnetization
Isoentropic demagnetization
Spin entropy if all states are accessible: ln 2 1N
Bk S S ln 2 1BN k S
T2
2 1
BT T
B BΔ = internal random field
Population of magnetic sublevels is function of μB/kBT, or B/T.
Δ = 0 →1 2
BB
T T or
is lowered in B field since lower energy states are more accessible.
Nuclear Demagnetization
5.58~
2 1836p e
→ T2 of nuclear paramagnetic cooling~ 10–2 that of electronic paramagnetic cooling.
B = 50 kG, T1 = 0.01K, →1
0.5p
B
B
k T
Δ on magnetization is over 10% max. → phonon Δ negligble.
Cu: T1 = 0.012K
T2 = T1 ( 3.1 / B )
BΔ =3.1 G
2 1
BT T
B
~ 1836p em m
→~ 5.58pg
~ 3.83ng
3~ 1.52 10 B
56.72 10 /B
B
G Kk
27
2
10 1000.01 2 10
50
GT K K
kG
Paramagnetic Susceptibility of Conduction Electrons
Classical free electrons:2B
B
MN B
k T
~ 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.
2 2B B
B F B F
N B N BT
k T T k TM
Pauli paramagnetism at T = 0 K
1
2
F
B
N d D B
1 1
2 2
F
F
B
d D B D
T = 0
1
2
F
B
N d D B
1 1
2 2
F
F
B
d D B D
PauliM N N 2FB D
23
2 B F
NB
k T
Landau diamagnetism:2
2LandaB F
u
N
TM B
k
→
2
2Pauli LandB F
au
NBM M M
k T
χ is higher in transition metals due to higher DOS.
parallel moment
anti-parallel moment
χ > 0 , Pauli paramagnetism
Prob. 5 &6