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 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
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
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)
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
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: 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
High T ( x << 1 ): Curie-Brillouin law: Brillouin function:
High T ( x << 1 ): Curie law = effective number of Bohr magnetons Gd (C2H3SO4) 9H2O
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
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 LS < 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
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. 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
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.
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 λLS 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 : →
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
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.
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
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
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.
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