Diamagnetism and paramagnetism • Langevin diamagnetism • paramagnetism • Hund’s rules atom • Lande g-factor • Brillouin function • crystal field splitting • quench of orbital angular momentum • nuclear demagnetization • Pauli paramagnetism and free electron Landau diamagnetism Dept of Phys gas M.C. Chang B=(1+χ)H Curie’s law χ=C/T important Basics • System energy E(H) (EFETST→=− if ≠0) 1 ∂E • magnetization density MH()=− VH∂ • susceptibility ∂M 1 ∂2 E χ ≡=− ∂H VH∂ 2 Atomic susceptibility 2 2 ⎛⎞pi ee2 HVLgSHA=+++⋅+∑∑⎜⎟iBμμ() iB, = ii⎝⎠222mmcmc =+ΔHH0 Order of magnitude • μμωBBc(LgSH+⋅≈≈) H ≈=10−4 eV when H 1T H • Ayx=−(,,0) iii2 2 eeH2 ⎛⎞ 2 Ama22≈ , a ≡ ∑ i ⎜⎟ 0 0 2 2mci ⎝⎠ mc me 2 ()ωc −5 ≈ 2 ≈ 10 of the linear term atH =1 T e / a0 important • Perturbation energy (to 2nd order) 2 nHnΔ ' Δ=En nHnΔ +∑ nn'≠ EEnn− ' 2 2 e nLμB ()+ gSHn⋅ ' nL gSn H n A2 n ' = μBi+⋅+2 ∑ + ∑ 2mc i n' Enn− E ' • Filled atomic shell (applies to noble gas, NaCl-like ions…etc) Ground state |0〉: LS000== e2 2 EH220 r 0 (for spherical charge dist) ∴ Δ= 2 ∑ i 83mc i For a collection of N ions, NE∂Δ22 eN χ = −=−000r 2 < 22∑ i VH∂ 6 mcV i Larmor (or Langevin) diamagnetism An atom with many electrons • Without SO coupling • single electron ground states ⎛⎞p2 1 i Degeneracy D=d HVV=++∑⎜⎟iij∑ iij⎝⎠22m … • Maximally mutually commuting set 22 HL,,,, S Lzz S • N-electron ground states • Eigenstates (including ground states) • Without e-e interaction α,,,LSmLS , m Degeneracy D=Cd • With SO coupling (weak) N 2 ⎛⎞pi 1 … HVV=+++∑⎜⎟iij∑ ∑λiiS ⋅ L i iij22m i ⎝⎠ • With e-e interaction • Maximally mutually commuting set 222 HL,,, S J , Jz … many-electron • Eigenstates (including ground states) levels Ground states α,,,,LSJmJ w/o SO: labeled by L,S D=(2L+1)(2S+1) w/ SO: labeled by L,S,J D=(2J+1) important Ground state of an atom with unfilled shell (no H field yet!): • Atomic quantum numbers nlm,,ls , m • Energy of an electron depends on nl, ( no mls, m ) • Degeneracy of electron level ε nl , : 2(2l+1) • If an atom has N (non-interacting) valence electrons, then the 2(2l+ 1) degeneracy of the “atomic” ground state (with unfilled ε nl , shell) is C non-interacting N e-e interaction will lift this degeneracy partially, and then • the atom energy is labeled by the conserved quantities L and S, each is (2L+1)(2S+1)-fold degenerate • SO coupling would split these states further, which are labeled by J (later). interacting What’s the values of S, L, and J for the atomic ground state? To reduce Coulomb repulsion, Use the Hund’s rules (1925), electron spins like to be parallel, electron orbital motion likes to be in 1. Choose the max value of S that is consistent high ml state. Both help disperse the with the exclusion principle charge distribution. 2. Choose the max value of L that is consistent with the exclusion principle and the 1st rule Example: 2 e’s in the p-shell (l1 = l2 =1, s1 =s2 =1/2) (a) (1,1/2) (b) (0,1/2) (c) (-1,1/2) (a’) (1,-1/2) (b’) (0,-1/2) (c’) (-1,-1/2) 6 C 2 ways to put these 2 electrons in 6 slots • Spectroscopic notation: 21S + Energy levels of Carbon atom XXSPDJ (,,...)= 13 1 313 SP00,1,22, , D are o.k.; SPD 01, , 3 are not. (It's complicated. See Eisberg and Resnick App. K for more details) S=1 L=1 ml = 1 0 -1 3 • Ground state is P 0,1,2 , physics.nist.gov/PhysRefData/Handbook/Tables/carbontable5.htm (2L+1)x(2S+1)=9-fold degenerate • There is also the 3rd Hund’s rule related to SO coupling (details below) important v Review of SO coupling • An electron moving in a static v BE=× E E field feels an effective B field eff c • This B field couples with the electron spin HBSO=−μ ⋅ eff ⎛⎞⎛⎞qv dφ e =−⎜⎟⎜⎟SE ⋅ ×, Er =−ˆ for central force, φ=+ ⎝⎠⎝⎠mc c dr r ⎛⎞qdφ (x 1/2 for Thomas precession, 1927) =⎜⎟22 SL⋅ ⎝⎠mc rdr λ> 0 for less than half-filled (electron-like) ≡⋅λSL λ< 0 for more than half-filled (hole-like) λ 22 2 = ()JLS−− Quantum states are now labeled by L, S, J 2 (2L+1)x(2S+1) degeneracy is further lifted to become (2J+1)-fold degeneracy Hund’s 3rd rule: 3 • if less than half-filled, then J=|L-S| has the lowest energy P0 is the ground state in • if more than half-filled, then J=L+S has the lowest energy previous example Paramagnetism of an atom with unfilled shell 1) Ground state is nondegenerate (J=0) 2 2 e 0 μB (Lg+⋅S ) H n ΔE =+μ 00LgSH++⋅ 00A2 ' Bi2 ∑ ∑ 2mc i n E0 − En (A+M, Prob 31.4) 2) Ground state is degenerate (J≠0) Van Vleck PM Then the 1st order term almost always >> the 2nd order terms. mLSJS=−μμBB( +2 ) =−( + ) L • Heuristic argument: J is fixed, L and S rotate around J, maintaining the triangle. So the magnetic moment is given by the component of L+2S parallel to J, H J JS⋅ J 22 2 SJJLS// ==22() −+ S JJ2 J =+−+++[]JJ(1)(1)(1) LL SS 2(JJ+ 1) • ΔE()mJJBJ= g μ mH, so χ= 0? ∴meff= −gJ μ B J No! these 2J+1 levels are closely JJ(1)(1)(1)+ −+++ LL SS packed (< kT), so F(H) is Lande g-factor gJ =+1 (1921) 2(J J + 1) nonlinear (next page). Langevin paramagnetism J −Em()/JB kT Ze=Δ==∑ ,() EmgmHKHJJBJμ ()∼ 1 at 1 T mJJ =− FETSkTZ=− =−B ln NF∂ N ⎛⎞gJHJBμ MgJ=− = JBμ BJ ⎜⎟ VH∂ V ⎝⎠kTB 21JJ++⎛⎞ 21 1 ⎛⎞ x where BxJ ( )≡− coth⎜⎟ x coth ⎜⎟ 22JJJJ⎝⎠22⎝ ⎠ Brillouin • kTBJB<<>> gμ JH(1) x function N MgJ= JBμ , χ = 0 V J +1 • kT>> gμ JH(1) x << B ()xx∼ BJB J 3J N 2 JJ(1)+ M = ()gJBμ H VkT3 B χ()T • at room T, χ(para)～500χ(dia) calculated earlier • Curie’s law χ=C/T (note: not good for J=0) 2 N ()μ p effective Bohr C = B , where pg= J(1)J + V 3k J magneton number f-shell (Lanthanides) In general (but not always), energy from low to high: 鑭系元素 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d … Due to low-lying J-multiplets (see A+M, p.657) • Before ionization, La: 5p6 6s2 5d1; Ce: 5p6 6s2 4f2 … 3d-shell (transition metal ions) ? • Curie’s law is still good, but p is mostly wrong • Much better improvement if we let J=S Crystal field splitting In a crystal, crystal field may be more important than the LS coupling • Different symmetries would have different splitting patterns. 淬滅 Quench of orbital angular momentum • Due to crystal field, energy levels are now labeled by L (not J) • Orbital degeneracy not lifted by crystal field may be lifted by Spontaneous 1) LS coupling, or 2) Jahn-Teller effect, or both. lattice distortion • The stationary state ψ of a non-degenerate level can be chosen as real when tt→−, ψ →ψ * ()=ψ if nondegenera t e • ψψLr= ψψ×∇ is purely imaginary i but ψ L ψ has to be real also ∴ ψ L ψ = 0 2 ()ψ L ψ can still be non-zero •for 3d ions, crystal field > SO interaction •for 4f ions, SO interaction > crystal field (because 4f is hidden inside 5p and 6s shells) •for 4d and 5d ions that have stronger SO interaction, the 2 energies maybe comparable and it’s more complicated. • Langevin diamagnetism • paramagnetism • Hund’s rules • Lande g-factor • Brillouin function • crystal field splitting • quench of orbital angular momentum • nuclear demagnetization • Pauli paramagnetism and Landau diamagnetism Adiabatic demagnetization (proposed by Debye, 1926) • The first method to reach below 1K • Without residual field J −Em()/J kT Z = ∑ eEmH, assume (J ) ∝ mJJ =− 增 ⎛⎞H 加 FkTZ=− ln ⎜⎟ 磁 ⎝⎠kT 場 ∂FH⎛⎞ 絕熱去磁 SS=− = ⎜⎟ ∂TkT⎝⎠ H f • If S=constant, then kT～H TTfi= Hi ∴ We can reduce H to reduce T Freezing is effective • With residual field only if spin (due to spin-spin int, specific heat crystal field… etc) is dominant (usually need T<<TD) Can reach 10-6 K (dilution refrig only 10-3 K) analogy 增 加 磁 場 → 環境 絕 熱 去 磁 ← 樣品 wikipedia Pauli paramagnetism for free electron gas (1925) • Orbital response to H neglected, consider only spin response • One of the earliest application of the exclusion principle NN=+↑↓ N 1 MNN=−()μB V ↑↓ For TT<< F , nn↑↓−≅ g()εμFB Hggg , =+ ↑ ↓ 2 ∴Mg= ()εμFB H 2 26⎛⎞α 1 − m 2 ⇒=χεμPauligka() F B =⎜⎟() F 0 ∼ 10 g()ε = 2mε a = ⎝⎠2π π 23 0 me2 e e2 μB = α = 2mc c • unlike the PM of magnetic ions, here the magnitude ～ DM’s (supressed by Pauli exclusion principle) Landau diamagnetism for free electron gas (1930) • The orbital response neglected earlier gives slight DM • The calculation is not trivial. For free electron gas, ek2 χ =− F Landau 12π 22mc 1 =− χ 3 Pauli • So far we have learned PM and DM for a free electron gas. How do we separate these contributions in experiment? X-ray magnetic circular dichroism (XMCD) Hoddeson, Our of the crystal maze, p.126.