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