11. Diamagnetism and Paramagnetism

1. 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)

2. 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 diamagnetism MagnetizationM magnetic moment per unit volume Magnetic subsceptibility per unit volume In vacuum, H = B. χM = molar subsceptibility σ = specific subsceptibility nuclear moments ~ 10−3 electronic moments

3. Larmor Precession Magnetic (dipole) moment: For a current loop: For a charge moving in a loop: ( charge at xq) Caution: we’ll set L to  L in the quantum version Classical gyromagnetic ratio Torque on m in magnetic field: → L precesses about B with the Larmor frequency Lorentz force: → cyclotron frequency

4. Langevin Diamagnetism Equation Diamagnetism ~ Lenz’s law: induced current opposes flux changes. χ < 0 Larmor theorem: weak B on e in atom → precession with freq Larmor precession of Z e’s: → Langevin diamagnetism same as QM result For N atoms per unit volume: Good for inert gases and dielectric solids experiment Failure: conduction electrons (Landau diamagnetism & dHvA effect)

5. Quantum Theory of Diamagnetism of Mononuclear Systems Quantum version of Langevin diamagnetism Perturbation Hamiltonian [see App (G18) ]: Uniform → → The Lz term gives rise to paramagnetism. 1st order contribution from 2nd term: same as classical result

6. 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

7. Quantum Theory of Paramagnetism Magnetic moment of free atom or ion: Caution: J here is dimensionless. μB = Bohr magneton. γ = gyromagnetic ratio. g = g factor. ~ spin magnetic moment of free electron For electrons g = 2.0023 For free atoms, For a free electron, L= 0, S = ½ , g = 2, → mJ=  ½ , U =  μB B. Anomalous Zeeman effect

8. High T ( x << 1 ): Curie-Brillouin law: Brillouin function:

9. High T ( x << 1 ): Curie law = effective number of Bohr magnetons Gd (C2H3SO4)  9H2O

10. Rare Earth Ions Lanthanide contraction ri= 1.11A ri= 0.94A Perturbation from higher states significant because splitting between L-S multiplets ~ kB T 4f radius ~ 0.3A

11. 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 Maximum value of S allowed by exclusion principle. Maximum value of L compatible with (1). J = | L−S | for less than half-filled shells. J = L + S for more than half-filled shells. Causes: Parallel spins have lower Coulomb energy. e’s meet less frequently if orbiting in same direction (parallel Ls). Spin orbit coupling lowers energy for LS < 0. 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

12. Iron Group Ions L = 0

13. 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.

14. Quenching of the Orbital Angular Momentum Atom in non-radial potential → Lz not conserved. If  Lz  = 0, L is quenched.  L is quenched → μ is quenched L = 1 electron in crystal field of orthorhombic symmetry ( α = β = γ = 90, a  b  c ): → Consider wave functions: → For i j, the integral is odd in xi& xj, and hence vanishes. i.e., where Similarly

15.  Uj are eigenstates for the atom in crystal field. Orbital moments are zero since Quenching For lattice with cubic symmetry,  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.

16. 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 λLS turns ψ0 into where • α | β = 0 → term Uzβ ~ O(λ2) in any expectation values. •  It can be dropped in any 1st approx. Thus Energy difference between Ux α and Uxβ in field B : →

17. Van Vleck Temperature-Independent Paramagnetism Consider atomic or molecular system with no magnetic moment in the ground state , i.e., In a weak field μz B << Δ = εs – ε0 , a) Δ << kB T b) Δ >> kB T van Vleck paramagnetism Curie’s law

18. 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, ΔS < 0 as H increases. i.e., H aligns μ and makes system more ordered. → Removing H isentropically (ΔS = 0) lowers T. Lattice entropy can seeps in during demagnetization Magnetic cooling is not cyclic.

19. Isothermal magnetization Isoentropic demagnetization T2 Spin entropy if all states are accessible: 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 → or BΔ = internal random field

20. Nuclear Demagnetization → T2 = T1 ( 3.1 / B ) → T2 of nuclear paramagnetic cooling ~ 10–2 that of electronic paramagnetic cooling. B = 50 kG, T1 = 0.01K, → ΔS on magnetization is over 10% Smax. → phonon Δ S negligble. Cu: T1 = 0.012K BΔ =3.1 G

21. Paramagnetic Susceptibility of Conduction Electrons Classical free electrons: ~ 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. 

22. Pauli paramagnetism at T = 0 K T = 0 parallel moment anti-parallel moment χ > 0 , Pauli paramagnetism Landau diamagnetism: → χ is higher in transition metals due to higher DOS. Prob. 5 &6