Magnetic Properties Magnetism References Definitions Energy

• A. F. Orchard, “Magnetochemistry”. • Drago, “Physical Methods in Chemistry”, Chapter 11. • C. J. O’Connor, Prog. Inorg. Chem., 1982, 29, 203. • R. L. Carlin, “Magnetochemistry”, Springer-Verlag, Magnetic Properties Berlin, 1986. • F. E. Mabbs and D. L. Machin, “Magnetism and Transition Metal Complexes”, Chapman and Hall, Fundamentals and structural London, 1973. correlations • W. E. Hatfield, Magnetic Measurements, Chapter 4 in “Solid State Chemistry: Techniques”, A. K. Cheetham and P. Day, Eds., Clarendon Press, Oxford, 1987. • B. N. Figgis, M. A. Hitchman, “Ligand Field Theory and its Applications”, Chapter 9. 1 2

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Definitions Energy relationships HB E = (erg/cm3) B = H + 4πM • B is the flux density 8π • HB/8π is the magnetic energy B M H 1+ 4πχ H ⎛ ⎞ ‘stored’ in any medium. (Seems inside a sample, H is ( v ) 1 χv 2 = 1+ 4π = 1+ 4πχv E = = + H H H 8π ⎜ 8π 2 ⎟ weird, but if you integrate HB/8π the applied field, and M ⎝ ⎠ χv: volume susceptibility (dimensionless) over all space with no sample is the induced H 2 present, you obtain the energy χ : gram susceptibility (cm3 g–1): In a vacuum, E = (χ = 0) g magnetization. vac 8π v that was needed to magnetize χ v χv 2 the magnet!) χg = ∴ χ is a measure of the E − E = H ρ ← density vac 2 3 –1 sample’s ‘susceptibility’ to or, choosing E to be the zero of energy χ : molar susceptibility (cm mol ): vac m being magnetized, per unit χ = χ • M χv 2 m g E = H F is the force that a volume applied field. 2 z ∂M (Technically, χ = , which is important element of the sample ‘feels’ as v ∂H • Strictly, M doesn’t have to dE ⎛ dH ⎞ it moves in an inhomogeneous F = = χ H when dealing with magnetically-ordered be parallel to H, and then z dz v ⎜ dz ⎟ χ ⎝ ⎠ magnetic field (i.e., in or out of materials where M is not proportional to H.) is a matrix. the field of a magnet along the z-direction). 3 4

3 4 χ measurement; Faraday method χ measurement; Evans (solution) NMR Evans, J. Chem. Soc., 1959, 2003. to microbalance method Oatfeld & Cohen, J. Chem. Ed., 1972, 49, 829. F =mHdH z dz χg • Two, concentric, NMR tubes are used. z • Inner tube: internal standard, dissolved in solvent • Outer tube: internal standard, dissolved in solvent, plus sample is moved into paramagnetic solution species to be measured. }constant range N S The presence of paramagnetic ions in the outer tube shifts the field felt by the standard molecules in the outer tube. H dH dz shift of resonance field → ∆ H 2π = ∆ χv Zero Field: Fz = mg g - gravitationalal constant H 3

dE ⎡ ⎛ dH ⎞ ⎤ where ∆ χv is the difference in the volume susceptibilities of the liquids With Field: F = = m⎢g + H χ ⎥ z dz ⎜ dz ⎟ g ⎣ ⎝ ⎠ ⎦ gram susceptibility of paramagnetic substance: ⎛ ρ − ρ ⎞ 3 ∆ν ← frequency shift (Hz) 1 0 S • Pole pieces shaped to yield field gradient requirements (or χg = χ0 ⎜ + ⎟ + ⎝ m ⎠ 2πm ν ← resonance frequency (Hz) “shaping coils” mounted to faces of the magnet). !########" correction for grams of paramagnetic solute density changes m = • Advantages: good sensitivity; no ‘packing error’; Small !##########" ml solution sample allows good thermostatting of sample accounts for sol- ρ = solvent density vent displaced 0 • Disadvantage: magnetic anisotropies difficult to measure.5 ρS = solution density 6 χ = gram susceptibility of solvent 0 5 6

–6 χdia ~ 10 - present in all matter T independent The Basic Very weakly destabilizing in H Schemes –3 –2 χpara ~ 10 – 10 - unpaired spins 1/ T dependence Zero temperature weakly stabilizing in H ‘snapshots’ of magnetic moments in each paramagnetic (zero χanti ~ χdia at T = 0 field), ferromagnetic, if exactly stoiciometric antiferromagnetic, and slowly rises with T until TN ferrimagnetic materials.

6 χferro ~ up to 10

slowly decreases with T until near TC stabilizing in H

7 8

7 8 Diamagnetism – Pascal’s Constants Pascal’s Constants not perfect

calc. exptl. H All paramagnetic pyridine: 50.4 10–6 49 10–6 H H measurements should be − × − × corrected for H diamagnetism: H AsMe2 −147 × 10–6 −194 × 10–6 χobs = χpara + χdia H N H (+) (–) H AsMe2 Fortunately, it is usually the case that These can be estimated |χpara| >> |χdia | and errors like this can –6 –6 H 5 × C(ring) = 5 × (–6.24 × 10 ) = −31.2 × 10 using Pascal’s constants be tolerated. Nevertheless, if –6 –6 5 × H = 5 × (–2.93× 10 ) = −14.6 × 10 (from atomic/ionic diamagnetic complexes analogous to 1 × C(ring) = –4.6 × 10–6 susceptibilities) –6 3 –1 –1 paramagnetic species can be Total: − 50.4 × 10 cm mol (or emu mol ) measured, they yield better

9 estimates. 10

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Statistical Thermodynamics Magnetization of an Ideal Paramagnet If we call m the average magnetic moment of a paramagnetic molecule, Magnetic susceptibility measurements are “bulk then NAm is the molar magnetization the compound in question. property” measurements, and results are If we apply a magnetic field along the z-axis to a sample of noninteracting averages over the so-called ‘ensemble’ of molecules, each molecule will experience Zeeman splitting of its energy levels. m is given by substituting into the formula we have have just presented as follows molecules in the system. J J − EM kBT M g µ H k T A g e−εi kBT µ e J M g µ e J J B B ∑ i i ∑ M J ∑ J J B M =− J M =− J A = i ⇒ m = J = J Probability the an energy level with energy ε is occupied −εi kBT J J i g e − EM kBT M g H k T ∑ i e J e J J µB B −ε k T i ∑ ∑ g e i B M =− J M =− J p i where g is the degeneracy of the ith level. J J i = k T i What do all the symbols mean? g e−εi B ∑ i • J is the total angular momentum, but in many cases, J ! S. i • M is the z-component of J, so that in spin-only cases, M = M . The 'thermal average' of property A one will measure for a system in equilibrium: J J S • µ is the z-component of the magnetic moment for each Zeeman k T M J A g e−εi B ∑ i i level, so µ = M g µ . (We can ignore x and y components since M J J J B A = p A = i ; where A is the value of property A in the ith state ∑ i i −ε k T i they cancel each out when averaged over all molecules.) i g e i B ∑ i • E is the energy of each Zeeman level, so , E = −µ • H = − M g µ H i M J M J M J J J B 11 12

11 12 Pictorial View Magnetization and Susceptibility

J J M g H k T 1 at temperatures over a few K. Therefore, we can expand − E k T J J µB B ≪ MJ B M J gJ µB H kBT µ M e M J gJ µBe ∑ J ∑ M J gJ µBH kBT M =− J M =− J e 1+ M g µ H k T m J J ! J J B B = J = J − E k T M g H k T e MJ B e J J µB B J J ∑ ∑ M J gJ µBH kBT M J M J M g µ e M 1+ M g µ H k T J =− J =− ∑ J J B ∑ J ( J J B B ) M =− J M =− J –1 m = J ! g µ J µB H = 0.4574 cm for H = 1.0 T (10,000 G) relative J J B J M g µ H k T µ H populations e J J B B 1+ M g µ H k T B = 0.6581 K for H = 1.0 T (10,000 G) ∑ ∑ ( J J B B ) k at 298 K, M =− J M =− J B H = 1.0 T J J J 2 ⎛ x ⎞ E 2 2 M 2 2 math notes: e ! 1+ x for x ≪1 E = +2µ H 0.991 ∑ J g H ⎜ ⎟ B gJ µB H J J µB J J − m J(J 1) ⎜ x 0 ; (1) 2J 1 ⎟ = ⇒ = + ∑ J = ∑ J = + k T J 3k T ⎜ − − ⎟ B B ⎜ J 2 1 ⎟ 0.994 1 ⎜ x = J(J +1)(2J +1) ⎟ E = +µBH ∑ ( ) ⎝ ∑− J 3 ⎠ M J =− J 0.996 N 2 2 E = 0 The molar bulk magnetization, M, is then M = N m = A ⎡g J(J + 1)µ ⎤ H A ⎣ J B ⎦ 3kBT E = –µ H 0.998 B The molar magnetic susceptibility is then,

E = –2µBH 1.000 ∂M NA 2 Curie χm = = µeff ; and µeff = gJ J(J +1)µB 13 14 H ∂H 3kBT Law - derived

13 14

Classical Result Curie’s Law C Empirical form: χm = ⎛ µH ⎞ T x = cosθ ; dx = sinθdθ a = ( ) ⎜ k T ⎟ ⎝ B ⎠ N 2 C A µeff 2 π −1 χm = ; and C = ! 0.125gJ J(J + 1) µH cosθ k T ax N µ cosθe B sinθdθ xe dx T 3kB A ∫ ∫ 0 1 ⎧ 1 ⎫ N m = = N µ = N µ coth a − 2 A π A −1 A ⎨ ⎬ a 23 -1 ⎛ −21 erg⎞ µH cosθ kBT ax ⎩ ⎭ (6.023× 10 mol ) 9.2740 × 10 e sinθdθ e dx !##"##$ N 2 ⎜ G ⎟ ∫ ∫ A µB ⎝ ⎠ erg -1 -1 0 L(a) NOTE: = ! 0.125 K mol = 0.125 K mol 1 3k ⎛ erg⎞ G 2 Langevin B 3 1.3807 × 10−16 3 function ⎜ K ⎟ 1 a a ⎝ ⎠ 2 Normally , a ≪ 1 (µH ≪ kBT ) , ∴ use coth a − = − +" a ⎛ J ⎞ a 3 45 (6.023× 1023 mol-1) 9.2740 × 10−24 ? N µ2 ⎝⎜ T⎠⎟ J 2 3 BUT (!): A B = 1.25 K mol-1 - 1 µa µ H ! 2 = 0.125 K mol µ = N µL(a) # N = N L(a) 3k ⎛ 23 J ⎞ T A A 3 A 3k T B 3 1.3807 × 10− B ⎝⎜ K ⎠⎟ 2 2 Comparison of the Q.M. result with the classical result yields: Why? 1 G = 1 erg , BUT 1 T = 10 J

N Aµ 2 NA µB ⎛ −7 N ⎞ ⎛ J -1⎞ −6 3 -1 N Amclassical = H kBT ≫ µH µ = g J(J + 1)µ Using SI units throughout for C: µ ! 4π × 10 1.25 K mol = 1.57 × 10 m K mol 3k T eff J 15 B 0 ⎜ 2 ⎟ ⎜ 2 ⎟ 16 B 3kB ⎝ A ⎠ ⎝ T ⎠

15 16 Spin-Only Magnetism χ and χ–1 C N χ = = A µ2 ; and µ = 2 S(S + 1)µ = n(n + 2)µ m T 3k T eff eff B B N 2 B C A µeff 2 χ = ; and C = ! 0.125g J(J + 1) (assumes g = 2) n = # of unpaired electrons m T 3k J e B n S µeff 1 1/ 1.73 Note: χ –1 2 χ 2 1 2.83 The high-T slope of a χ–1 3 3/ 3.87 vs. 1/T plot is a more 2 reliable measure of µ 4 2 4.90 eff than the direct use of the 5 5 /2 5.92 Curie formula from a single measurement at a 0 6 3 6.93 0 T T fixed temperature. 7 17 7 /2 7.94 18

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Other coordination geometries: dn Spin-Only Trigonal Bipyramidal complexes Examples L

L n S µeff (µB) M L L 3 3 /2 3.87 5 5 /2 5.92 L

High-spin d5- and d3-complexes often behave as Spin states are determined nearly ideal spin-only systems. In these cases, µeff by varying ligand field obtained by assuming the Curie formula is correct strengths in axial and and plugging in the temperature, 300K, gives good equatorial ligands – see if to excellent agreement with experiment. (Why?)19 you can discern trends. 20

19 20 C Non-ideal Paramagnets: Curie-Weiss Law: χ = Non-ideal Paramagnets; Molecular Systems T − θ

θ < 0 ; paramagnetic behavior well above antiferromagnetic ordering temperature (T ) N C ⎛ T ⎞ Curie-Weiss Law: χ = ; χT = C χ –1 T − θ ⎝⎜ T − θ ⎠⎟ χT θ > 0 ; net ferromagnetic coupling yields χT > C

χT = C Curie Law: χT = C C 1 ⎛ 1 ⎞ ⎛ θ ⎞ Curie-Weiss Law: χ = ; χ − = T − T − θ ⎝⎜ C ⎠⎟ ⎝⎜ C ⎠⎟ TN θ > 0 ; paramagnetic behavior well above ferromagnetic 0 ; net antiferromagnetic coupling yields T C ordering temperature (T ). Applies only for T > θ. θ < χ < C TC θ C ⎛ 1 ⎞ Curie Law: χ = ; χ −1 = T T ⎜ C ⎟ T ⎝ ⎠ 21 22

21 22

H k T Paramagnet Saturation: µB > B Paramagnet Saturation: µB H > kBT

Orchard, p. 49 & section 3.10 J J − E k T M g H k T Orchard, section 3.10 µ e MJ B M g µ e J J µB B ∑ M J ∑ J J B M J M J Scaled saturation curves J =− J =− µ = J = J − E k T M g H k T ∑ e MJ B ∑ e J J µB B M J =− J M J =− J J 1.0 g µ H d ⎛ M x ⎞ Let x = J B ; then µ = g µ ln⎜ e J ⎟ k T J B dx ∑ B ⎝ M J =− J ⎠ S = 7/2

J 0 2J ⎛ ∞ M ∞ M ⎞ M x Jx M x Jx − M x Jx −x J −x J e J = e e J = e e J = e ⎜ e − e ⎟ ∑ ∑ ∑ ∑ ( ) ∑ ( ) M/Msat S = 5/2 M J =− J M J =−2J M J =0 ⎝ M J =0 M J =2J +1 ⎠

⎛ ∞ M 2J +1 ∞ M ⎞ Jx −x J −x −x J = e ⎜ ∑ (e ) − (e ) ∑ (e ) ⎟ = S = 3/2 ⎝ M J =0 M J =0 ⎠ ∞ Jx −( J +1)x M J ⎛ e − e ⎞ eJx 1− e−(2J +1)x e−x = ( ) ∑ ( ) ⎜ 1 e−x ⎟ M J =0 ⎝ − ⎠ (2J +1)x ⎛ (2J +1)x ⎞ (2J 1)x 2 (2J 1)x 2 sinh sinh e + − e− + d ⎜ ⎟ = = 2 ⇒ µ = g µ ln 2 x 2 −x 2 x J B dx ⎜ x ⎟ e − e sinh ⎜ sinh ⎟ 2 ⎝ 2 ⎠ g µ H µ = g µ JB ( y) ; y = J J B 0.00 10 20 30 40 J B J k T B H/T 2J + 1 2J + 1 1 y B ( y) = coth y − coth J 2J 2J 2J 2J 23 24 ex + e−x coth x 1 as x B ( y) → 1 as y → ∞ Brillouin Functions = x x → → ∞ J e − e− 23 24 Saturation Measurements to identify Spin States Origins of Curie Law Deviations The Brillouin functions have a characteristic shape for • The Weiss constant, θ, incorporates each J (or S for spin-only cases). the effects of intermolecular/interionic When magnetic ion magnetic coupling above the ordering concentrations or g-values are not known, the saturation temperatures of ferro-, antiferro-, and measurements can yield J (or S) for near-ideal Curie- ferrimagnets. Law systems. • There are several sources of non- Saturation measurements are only useful for getting the Jg µ H Curie Law behavior that have an µ = g µ JB ( y) ; y = J B low-T moment and only if the J B J k T intramolecular origin. system if nearly ideal (i.e., B 2J + 1 2J + 1 1 y molecular, noninteracting, BJ ( y) = coth y − coth moments) 2J 2J 2J 2J 25 26

25 26

Origins of Curie Law Deviations Magnetism of Ln3+ Complexes

• The simplest deviation: g ≠ ge. This doesn’t alter the temperature dependence, but it does alter the computed value of S. The correct value of g to use 2 1 2 2 2 is: g = /3(gx + gy + gz ) • Temperature-independent (Van Vleck) paramagnetism (TIP). To more carefully discuss the origin of this kind of case, we’ll need to use the Generalized Treatment of Susceptibility - see below! • Another kind of T-independent paramagnetism arises from the conduction electrons in metals (Pauli paramagnetism). • Zero Field Splitting and first-order orbital effects. • Temperature variations due to coupling within a

27 28

27 28 Pauli Paramagnetism Generalized Treatment of Valence electron µ H χ = µ 2N(E ) Susceptibility (Van Vleck) DOS for Mo. B B F • Energy shifts are generally small compared to spacings between levels (except for degeneracies). E k T N e− i B ∑µi H M i = E k T (0) e− i B Ei Ei ∑ antibonding i (1) The Zeeman (0) (1) (2) 2 (0) ⎛ ⎞ − E k T − E k T −( E H + E H +!) k T − E k T Ei H Perturbation expansion: e i B = e i B e i i B " e i B ⎜1− ⎟ interaction introduces ⎝ kBT ⎠

(0) (1) (2) 2 a small difference in Ei = Ei + Ei H + Ei H +! (1) ⎛ E H ⎞ E(0) k T N E(1) + 2E(2) H 1− i e− i B up- and down-spin E ∑( i i )⎜ ⎟ ∂ i i ⎝ kBT ⎠ E = −µ • H ⇒ µ = − M ! ( i i ) i (1) (1) populations. (0) ∂H ⎛ E H ⎞ E k T bonding 1− i e− i B D. D. Koelling, F. M. Mueller, A. J. Arko, J. B. (1) (2) ∑⎜ k T ⎟ µi " Ei + 2Ei H i ⎝ B ⎠ Ketterson, Phys. Rev. B, 1974, 10, 4889 29 30 (/atom)

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Generalized Treatment of Generalized Treatment of Susceptibility (Van Vleck) Susceptibility (Van Vleck) If a material is not magnetically 2 ordered, it will have no permanent ⎛ (1) ⎞ E (0) ⎜ ( i ) (2) ⎟ − E k T magnetization. Therefore, MH=0 = 0. N − 2E e i B ∑⎜ k T i ⎟ E(0) k T i B N E(1)e− i B ⎜ ⎟ (2) ∑ i ⎝ ⎠ i χ = (0) M H 0 = (0) = 0 ; ∴ numerator = 0 = E k T − E k T ∑e− i B ∑e i B H i H i 2 (0) ⎛ (1) ⎞ (0) E E E (0) E E i i i E k T i i N ⎜ ( ) − 2E(2) ⎟ e− i B ∑⎜ i ⎟ i kBT ⎝⎜ ⎠⎟ ∂M Using this and expanding equation (1) to terms linear in H: M ! (0) H ; and finally, since χ = , − Ei kBT ∂H (1) (2) ∑e Ei and Ei are obtained from perturbation theory. i (0) (0) (0) (0) 2 Ψ H Ψ Ψ H Ψ ⎛ (1) ⎞ (2) i Zeeman n n Zeeman i E (0) i (2) − E k T E = N ⎜ ( ) − 2E ⎟ e i B i ∑ (0) (0) E (1) and E (2) are obtained ∑⎜ k T i ⎟ (1) (0) (0) n≠i E – E i i i ⎜ B ⎟ E = Ψ H Ψ = n i ⎝ ⎠ (2) i i Zeeman i 2 from perturbation theory. χ = (0) (0) (0) − E k T (0) (0) Ψ L + 2S Ψ e i B µ H Ψ L + 2S Ψ i z z n ∑ B i z z i i = ∑ (0) (0) 31 E – E 32 n≠i n i

31 32 Temperature Independent (Van Curie’s Law from the Van Vleck Formula Vleck) Paramagnetism (TIP) The Van Vleck formalism can be used to derive Curie’s Law; • TIP is significant when the ground state is diamagnetic just retain the leading term in the numerator of eqn. (2): (E (0) = 0), but there are many paramagnetic excited m 1 2 i S = (0) 2 2 states with energies substantially greater than k T (E ⎛ (1) ⎞ ⎛ (1) ⎞ B n Ei (0) Ei (0) ⎜ ( ) (2) ⎟ − Ei kBT ⎜ ( ) ⎟ − Ei kBT g µ H (0) N − 2E e N e e B – E0 >> kBT), but still not too high in energy. ∑⎜ k T i ⎟ ∑⎜ k T ⎟ i ⎜ B ⎟ i ⎜ B ⎟ ⎝ ⎠ ⎝ ⎠ m = −1 2 χ = ! H S • Only the second-order Zeeman mixing of excited states E(0) k T E(0) k T ∑e− i B ∑e− i B into the ground state (by the field) makes an i i S appreciable contribution. Use of eqn. (2) shows the T- (0) (1) (1) ⎛ (1) = 2S + 1 ⎞ Now, E = E + E H, where E = gM µ ⎜ ∑−S ⎟ i i i i S B S independence: ⎜ x2 1 S(S 1)(2S 1) ⎟ ⎝ ∑ S = 3 + + ⎠ − diamagnetic ground state: µ H Ψ(0) L + 2S Ψ(0) = 0 2 B i z z i ⎛ (1) ⎞ 2 E (0) ⎛ ⎞ i − E k T S (0) ⎜ ( ) ⎟ i B 2 2 MS − E k T 2 N e Ng ⎜ ( ) ⎟ e i B ⎛ (1) ⎞ ∑⎜ ⎟ µB E (0) i kBT ∑ ∑ k T i (2) − E k T ⎜ ⎟ i ⎜ M =−S B ⎟ 2 2 N ⎜ ( ) 2E ⎟ e i B ⎝ ⎠ ⎝ S ⎠ Ng µ S(S + 1) ∑ − i 2 B ⎜ k T ⎟ (0) (0) χ = (0) = = i ⎜ B ⎟ (2) Ψ H Ψ − Ei kBT ⎛ S ⎞ (0) 3k T ⎝ ⎠ NE0 (2) 0 Zeeman n e − E k T B χ = ~ ; E = ∑ 1 e i B (0) (0) 0 ∑ (0) (0) ⎜ ⎟ − Ei kBT − Ei kBT E – E i ∑⎜ ∑ ( )⎟ e e n≠i n 0 i ⎝ MS =−S ⎠ 33 ∑ ∑ 34 i i

33 34

Examples: T-independent effects Empirical Corrections

Diamagnetic susceptibilities † χm C compound Curie-Weiss plus T-independent corrections: * –6 3 –1 χ = χ0 + χm (10 cm mol ) T − θ ion (10–6 cm3 Li2[CrO4] +20.7 • In practice, a fit of the χ data including a T- mol–1) Na [CrO ] +15.4 2 4 independent term, χ , is often included to account Li+ – 0.6 K [CrO ] –3.6 0 2 4 for diamagnetism and TIP empirically (χ = χ + Na+ – 5 0 dia Cs2[CrO4] –51.7 χTIP). K+ – 13 K[MnO ] +27.8 4 WARNING: empirical corrections like this can be Cs+ – 31 Cs[MnO4] +3.5 dangerous when used indiscriminately! The magnitude Notes: of the correction has to be sensible (and small!). † The values given in the table at right are from Orchard Furthermore, since neither χdia nor χTIP is dependent on (p. 81), divided by 4π × 10–6 to convert them to cgs units. field, the H-dependence of the data should be checked! See p. 6 for more. *From Selwood. 35 36

35 36 ZFS: Effect on Susceptibility ZFS: Effect on Susceptibility E E = E(0) + E(1) H + E(2) H 2 +! i i i i E 2 MS = +1 8µB MS = +1 3k gµBH Bearing in mind the definitions of B the terms that go into Van gµ H M = –1 B D S (2) Vleck’s formula (Ei = 0) : M = –1 E (0) = 0 ; E (1) = 0 D S MS = 0 0 0 2 χT 4µB E (0) = D M = 0 D H ±1 S 3kB = 5K ⎛ S(S + 1)⎞ E (1) = ± gM µ k H = µ H • g • Sˆ + D Sˆ2 − ±1 S B H B eff B ⎝⎜ z 3 ⎠⎟

2 (1) 2 2 E (0) i E k T gµ –gµ ( ) − i B ( B ) − D kBT ( B ) − D kBT 0 N e 0 + e + e 2 D k T 0 50 100 150 200 250 300 ∑ 2 − D kBT − B i k T k T k T 8µ e 8 χ = B N B B N B µB e T (0) = D k T = − E k T − B − D kBT χ = N e i B 1+ 2e kBT 1+ 2e ∑ ( ) k T − D kBT i B 1 2e ( + ) (1) (1) (2) ( E H E H 2 ) k T E H CAUTION: e− i + i +! B " 1− i actually only applies when k T ≫ E(1) H 37 38 k T B i B

37 38

E 3+ M = + 3/ Cr ZFS: S = 3/2 ZFS: S = 3/2 S 2 E (0) (1) (2) 2 M = + 3/ E = E + E H + E H +! S 2 i i i i ± 3/ 3gµ H 5µ 2 2 B Bearing in mind the definitions of B 1 3 kB 2D MS = + /2 the terms that go into Van ± /2 3gµBH Vleck’s formula (E (2) = 0) : 1 1 gµ H i 2D MS = + /2 ± /2 B M = – 3/ S 2 (0) (1) 1 E±1/2 = 0 ; E±1/2 = ± /2gMSµB 1 gµ H 1 ± /2 B M = – 3/ MS = – /2 (0) S 2 E = 2D T D ±3/2 1 χ 4K (1) MS = – /2 = H E = ± 3/ gM µ k ±3/2 2 S B B 2 E(0) k T N E(1) e− i B H ∑ i 2 ( ) µB χ = i E(0) k T k k T e− i B B B ∑ i 2 −2D k T B 0 µB 1+ 9e 2 2 2 2 2D k T 0 50 100 150 200 250 300 1 1 ⎡ 3 3 ⎤ − B χ = N gµ + – gµ + gµ + – gµ e 2 −2D k T T ( 2 B ) ( 2 B ) ⎢( 2 B ) ( 2 B ) ⎥ µ 1+ 9e B k T −2D kBT χ = N ⎣ ⎦ = N B B 1+ e 2D k T 2D k T − B k T − B ( ) kBT 2 + 2e B 1+ e ( ) 39( ) 40

39 40 Orbital Angular Some Room-temperature Effects momentum • In tetrahedral • In some cases, orbital complexes with some contributions to tetragonal distortion magnetism can be quite (due to an asymmetric large. environment) ZFS can significant affect the • This occurs orbitally susceptibility. degenerate states so competition with J-T • For Co(II), ζ is distortions makes the significant (ca. 530 cm– actual observations 1 ) and ∆t isn’t very variable. large; mixing of the • Well-known plots 2 excited states ~ ζ /∆t, including orbital so anisotropy gives contributions to relatively large41 ZFS magnetism are42 found in

41 42

First-Order Spin-Orbit First-Order Spin-Orbit Coupling in T- Γ 7 Coupling in T-states, states, TiIII example; set up III Ti example; symmetry No JT- 1 λ Γ For the t configuration, there are 6 spin-orbitals: 2 Distortion 7 2g T2 ( ) aspects assumed Ψ d α 1 ⎡ 2α − −2α ⎤ 1 xy 2 ⎣ ⎦ 2 λ Ψ −1β λ T2 2 / No JT- 1 2 Ψ d β ⎡ 2β − −2β ⎤ 3 xy 2 ⎣ ⎦ Distortion sin(J + 1 2)α O χ(C ) = λ Ψ 1α h Γ8 assumed α /2 4 sinα 2 Ψ5 1β Γ 6(3C 3RC ) 8 Ψ −1α 2 + 2 6 O′ E R 8C3 8RC3 2 2 6C4 6RC4 12(6C2′ + 6RC2′ ) (= C4 + RC4 ) To evaluate the matrix elements involving 2 2 2 A1 1 1 1 1 1 1 1 1 x + y + z these basis functions, we need: A 1 1 1 1 1 1 1 1 2 − − − λ λL • SΨ = − Ψ + d 2 2 α E 2 2 −1 −1 2 0 0 0 (2z2 – x2 − y2 ,x2 − y2 ) 1 2 2 x − y The spin-orbit perturbation is evaluated T 3 3 0 0 1 1 1 1 (R , R , R ); (x, y, z) 1 1 − − x y z λL • SΨ = λ ⎡ −2α + Ψ ⎤ in the basis of these six functions 2 ⎣ 2 2 ⎦ T2 3 3 0 0 −1 −1 −1 1 (xy,xz, yz) λ λL • SΨ = Ψ + d 2 2 α Γ6 2 −2 1 −1 0 2 − 2 0 = Γ1/2 3 2 4 x − y 1 Γ7 2 −2 1 −1 0 − 2 2 0 ⎡ ⎤ λL • SΨ4 = λ 2β + Ψ4 Γ 4 −4 −1 1 0 0 0 0 = Γ ⎣ 2 ⎦ 8 3/2 λL • SΨ = − λ Ψ 43 5 2 5 44 T ⊗ Γ = Γ ⊕ Γ λL • SΨ = − λ Ψ 2 1/2 7 8 6 2 6

43 44 First-Order Spin-Orbit Coupling in T- 1st-Order S-O Coupling in T-states, TiIII states, TiIII example; secular equation example; solutions, magnetic moments

No JT- 1 No JT- Γ7 For the t configuration, there are 6 spin-orbitals: Γ7 Eigenvalues and eigenfunctions, magnetic moments Distortion ( 2g ) Distortion assumed 1 assumed 1 Ψ1 dxyα ⎡ 2α − −2α ⎤ E = +λ Φ = ⎡Ψ − 2Ψ ⎤ −µ 2 ⎣ ⎦ 1 3 ⎣ 1 2 ⎦ B λ λ 2 1 2 1 T2 Ψ2 − β T2 2-fold degenerate Φ = ⎡Ψ + 2Ψ ⎤ µ 2 3 ⎣ 3 4 ⎦ B 1 Ψ3 dxyβ ⎡ 2β − −2β ⎤ 1 ⎡ ⎤ 2 ⎣ ⎦ Φ3 = 2Ψ1 + Ψ2 0 λ λ 3 ⎣ ⎦ /2 Ψ4 1α /2 1 ⎡ ⎤ E = − λ 2 Φ4 = 2Ψ3 − Ψ4 0 1 3 ⎣ ⎦ Ψ5 β 4-fold degenerate Φ = Ψ = 1β 0 Γ8 Γ8 5 5 Ψ6 −1α Φ = Ψ = −1α 0 6 6 Secular Equation: Example calculations of Magnetic Moments: M z = µB (Lz + 2Sz ) Ψ1 Ψ2 Ψ3 Ψ4 Ψ5 Ψ6 Φ M Φ = µ 1β L + 2S 1β = µ ⎡1+ 2 • −1 ⎤ = 0 5 z 5 B z z B ⎣ 2 ⎦ 0 − E − λ The spin-orbit Ψ1 2 2µ λ λ B Ψ − − E perturbation is Φ1 M z Φ1 = 2µB Φ1 Lz + 2Sz Φ1 = Ψ1 − 2Ψ2 Lz + 2Sz Ψ1 − 2Ψ2 2 2 2 0 3 λ evaluated in the basis µ 2µ Ψ 0 E B B 3 − 2 L 2S L 2S = 0 of these six functions = 3 Ψ1 z + z Ψ1 + 3 Ψ2 z + z Ψ2 = Ψ λ λ E 4 2 2 − µB µB 2µB Ψ5 λ = 2α L + 2S 2α + −2α L + 2S −2α + −1β L + 2S −1β = − 2 − E 0 6 z z 6 z z 3 z z 2 2 2 Ψ 0 λ µB 1 1 µB 1 µB µB 6 0 − − E 45 ⎡2 + 2 • − 2 + 2 • ⎤ + ⎡−1+ 2 • − ⎤ = + (−2) =46−µ 2 6 ⎣ 2 2 ⎦ 3 ⎣ 2 ⎦ 6 3 B

45 46

st III 1 -Order S-O Coupling in T-states, Ti 3+ Γ7 Ti : S.O. coupling example; the big picture 2µBH (0) (1) (2) 2 Eigenvalues and eigenfunctions, magnetic moments E = E + E H + E H +! 2 λ i i i i T2 E = +λ Φ = 1 ⎡Ψ − 2Ψ ⎤ −µ Bearing in mind the definitions of the 1 3 ⎣ 1 2 ⎦ B terms that go into Van Vleck’s 2-fold degenerate Φ = 1 ⎡Ψ + 2Ψ ⎤ µ λ/ 2 3 3 4 B 2 (2) ⎣ ⎦ formula (assume Ei = 0) : Φ = 1 ⎡ 2Ψ + Ψ ⎤ 0 3 3 ⎣ 1 2 ⎦ Oh Γ8 E = − λ 2 Φ = 1 ⎡ 2Ψ − Ψ ⎤ 0 (0) (1) 4 3 ⎣ 3 4 ⎦ 4-fold degenerate Φ = Ψ = 1β 0 H 5 5 E–λ/2 = 0 ; E–λ/2 = 0 Φ = Ψ = −1α 0 6 6 Γ7 2µBH 2 E(0) k T N E(1) e− i B E(0) = 3λ/ ; E (1) = ± µ ∑( i ) λ 2 λ B 2 λ 2 2 i T 2µBH χ = 2 E(0) k T k T e− i B 3λ/ B ∑ 2 i λ/ 2 2 2 2 ⎡ ⎤ −3λ 2kBT 4 0 – e 2 3 2k T ( ) + (µB ) + ( µB ) − λ B ⎣⎢ ⎦⎥ µB e No JT- O Γ χ = N = N h 8 −3λ 2kBT k T −3λ 2kBT Distortion k T 4 + 2e B 2 + e B ( ) ( ) assumed H 47 48

47 48 T-terms: Orbital Effects (Max.) T-terms: Orbital Effects (Max.)

B. N. Figgis, M. A. Hitchman, 1.0 < A < 1.5 “Ligand Field Theory and its (strong field limit) (weak field limit) Applications”, Chapter 9.

1.0 < A < 1.5 (strong field limit) (weak field limit) B. N. Figgis, M. A. Hitchman, “Ligand Field Theory and its Applications”, Chapter 9. 49 50

49 50

Interactions between Magnetic Ions Energy Levels from HHDVV • We’ve been mostly concerned with the way magnetic • Using manipulations similar to those used to treat spin- moments can be altered by the electronic structure orbit coupling, we can rewrite HHDVV: ˆ ˆ within the ion. H HDVV = −2JS A •SB ˆ ˆ ˆ ˆ2 ˆ ˆ ˆ2 ˆ2 ˆ ˆ • We begin to consider interactions between ions by ST = S A + SB ; ST = ST •ST = S A + SB + 2S A •SB considering dinuclear systems and the 2Sˆ •Sˆ = Sˆ2 − Sˆ2 − Sˆ2 These are operators! phenomenological Heisenberg Dirac-Van Vleck A B T A B (HDVV) Hamiltonian for coupling: • The eigenvalues (energies) of HHDVV are then: These are J 0 ferromagnetic coupling E = −J ⎡S (S + 1) − S (S + 1) − S (S + 1)⎤ ˆ ˆ > n ⎣ T T A A B B ⎦ scalars! H = −2JS •S J < 0 antiferromagnetic coupling HDVV A B • In general, the coupling strength (~J) is much smaller than the splittings between the spin-states of the • While this reminds us of a form expected for dipole- individual ions - which are in turn much greater than dipole coupling, remember that a phenomenological kBT. ∴ We can view SA and SB as constants and Hamiltonian is just another name for an effective absorb the two last terms into the zero of energy: Hamiltonian - we’ll worry about where J comes from 51 E = −JS (S + 1) ; energy spacing: E (S ) − E (S − 152) = −2JS n T T n T n T T

51 52 E = E(0) + E(1) H + E(2) H 2 +! i i i i Example: Cr3+ 3+ 2+ Example: Cr coupled to Ni Bearing in mind the definitions of the 2+ terms that go into Van Vleck’s formula coupled to Ni (2) • Possible spin states of the coupled system are (Ei = 0) : E 1 MS = + /2 determined as a vector sum (formally like we saw 1 (1) S = /2 gµBH (0) 1 for L-S coupling and S.O. coupling) 1 3 5 MS = – /2 E5/2 = 0; E5/2 = ± /2gµB, ± /2gµB, ± /2gµB 3J (0) (1) • Van Vleck expression can be used to calculate 3 1 3 S = /2 3gµ H E3/2 = 5J; E3/2 = ± /2gµB, ± /2gµB B expected susceptibility. (0) (1) E 1 1 E1/2 = 8J; E1/2 = ± /2gµB 5J MS = + /2 2 (0) S = 1/ gµ H (1) − E k T 3 2 B N E e i B S ; S 1 1 ∑ i 5 Cr3+ = 2 Ni2+ = MS = – /2 ( ) S = / 5gµBH 3J i 2 χ = (0) S = S 3+ + S 2+ − E k T T Cr Ni 3 k T e i B 5 3 1 S = /2 3gµBH B ∑ possible values of S : , , i T 2 2 2 2 2 2 2 2 2 H 1 3 5 ⎡ 1 3 ⎤ −5J kBT ⎡ 1 ⎤ −8J kBT E = −JS (S + 1) 5J 2 gµ + 2 gµ + 2 gµ + 2 gµ + 2 gµ e + 2 gµ e n T T ( 2 B ) ( 2 B ) ( 2 B ) ⎣⎢ ( 2 B ) ( 2 B ) ⎦⎥ ⎣⎢ ( 2 B ) ⎦⎥ −35J χ = N 5 5 −5J kBT −5J kBT E5 2 = −J ( + 1) = kBT 6 + 4e + 2e 2 2 4 5 5gµ H ( ) S = /2 B 15J 3J −5J k T −8J k T − − 2 35 5e B 1 e B E3 2 = ; E1 2 = µB 2 2 + + 2 4 4 χ = N g k T −5J kBT −5J kBT H B 6 + 4e + 2e (J > 0 case shown)53 ( ) 54

53 54

A General Formula for Two Coupled Ions Ni2(abpy)(CH3CN)2(NO3)2] Using Van Vleck’s formula and the previous example as guidance, it isn’t hard to write a general formula for any two coupled centers: E 2 2 − E(0) k T 2 − E(0) k T n B n B 1 Ng µB ane g ane M = + / ∑ 3 ∑ S 2 χ = n ! n S = 1/ gµ H m − E(0) k T − E(0) k T 2 B k T b e n B 8 T b e n B 1 B ∑ n ∑ n MS = – /2 n n 3J 2 NµB 3 3 Where we've used the fact that ! 0.375 = . S = /2 3gµBH kB 8 1 Also, a = S (S + 1)(2S + 1) ; b = (2S + 1) 5J n 3 T T T n T Note that the sum over n refers to all the values that − E(0) k T 5 2 n B ST takes, from S A + SB down to S A – SB S = / 5gµ H g a e 2 2J T 8J T 2 B 3 ∑ n 3 g 0 + 2e + 10e Fitted line has added χ = n = ( ) m − E(0) k T 2J T 8J T We'll express the coupling constant J in kelvin: J(K) = J k 8 T b e n B 8 T 1+ 3e + 5e term (C/T) with C = B ∑ n ( ) n 0.08, J = –11K, g = 2.03 H (J < 0) (J > 0 case shown)

55 56

55 56 Dipolar Interaction: Weak! What controls coupling? • The energy of the magnetic dipole-dipole interaction falls off as the cube of the distance between dipoles and • How does the sign and 3– Cl Cl is too weak to explain magnetic coupling. magnitude of J change Cl with angle (θ)? Cu Cu Cl θ Cl • delocalized and localized Cl Cl

r picture (Hay, Thiebault, rˆ = r and Hoffmann, J. Am.

Chem. Soc. 1975, 97, 4884). 2 2 ε2 − ε1 ε2 − ε1 2 2 3 2 2 3 2 2 3 1 ( ) ( ) g ⎛ 2 ⎞ ⎛ 2 ⎞ ET − ES ≡ −2J = J12 − 2 (Jaa + Jab ) + = −2Kab + ( µB ) ⎛ e! ⎞ ⎛ 1⎞ ⎛ e! ⎞ me 1 ⎛ a0 ⎞ ⎛ e! ⎞ ⎛ me⎞ ⎛ a0 ⎞ e E ≈ ≈ = ≈ 2K12 Jaa − Jab dipolar 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ r ⎝ mc⎠ ⎝ r ⎠ ⎝ mc⎠ ⎝ ! ⎠ a0 ⎝ r ⎠ ⎝ mc⎠ ⎝ ! ⎠ ⎝ r ⎠ ⎝ a0 ⎠ (ET ≡ ES=1 ; ES ≡ ES=0 ) φ1,φ2,ε1,ε2 : delocalized orbitals & energies ; 2 3 3 3 1 1 ⎛ e2 ⎞ ⎛ a ⎞ ⎛ a ⎞ ⎛ a ⎞ ⎛ 27.2 ⎞ φ = (φ +φ ) , φ = (φ −φ ) : localized orbitals = 0 27.2 eV = α 2 0 27.2 eV 0 eV a 2 1 2 b 2 1 2 ⎜ ⎟ ⎜ ⎟ ( ) ⎜ ⎟ ( ) " ⎜ ⎟ ⎜ 2 ⎟ 57 58 ⎝ !c⎠ ⎝ r ⎠ ⎝ r ⎠ ⎝ r ⎠ ⎝ 137 ⎠

57 58

L L L Orthogonal Ni 180˚ π orbital superexchange L O O Orbitals or Not

O L O O O L 3+ 8 3 Cr Ni [(L4Ni)3Cr] : (d )3d 4+ –1 J ~ –200 cm O O O L NH3 NH3 3+ 9 5 O L [(L4Cu)3Mn] : (d )3d NH3 NH3 O

L O L′ H3N Cr O Cr NH3 Ni L′ L′ H N L L Cu 3 H3N L, L´ = Schiff base L O L′ NH3 NH3 O (imine) ligands 2 L′ O O O O (ε1 − ε2 ) L′ 1 Cu Mn ET − ES = −2J = J12 − 2 (Jaa + Jab ) + O 2K12 L′ O O 2 L′ O (ε1 − ε2 ) O ET − ES = −2J = −2Kab + O L′ Pei, Journoux, & Kahn, J − J Cu aa ab Inorg. Chem. 1989, 28, 100. L′ L′ (ET ≡ ES =1 ; ET ≡ ES =0 ) 4– 59 L′ 60 Compare [Ru2(µ-O)Cl10]

59 60 Magnetic + II III – A Polymeric NBu4 [Cu Cr (ox)3 ]: Prussian Blues d8d3 Analog

HO HO O O O O Cr Cu Cr HO O O O HO O Cu O O O O O O Cr Cu Cr O O O O H Okawa, S Kida,et al., O O Chem. Lett. 1990, 87 E Ruiz, A Rodrguez-Fortea,S O Alvarez, M Verdaguer, Chem. O Eur. J. 2005, 11, 2135. 61 62

61 62

Magnetic Prussian Blues Careful Study of Structure Can Be Revealing

Ferromagnets S (µ/µB) TC(K) Cr2+ 2 5.8 57 Rb2CrCl4 * (d4 - HS) Cu2+ 1/2 1.8 6 K2CuF4 * (d4 - HS) Mn3+ (d4 HS), 2 & 3.7 350 La0.7Sr0.3MnO3 † Mn4+ (d3 HS) 3/2

† perovskite ; * K2NiF4-type

E Ruiz, A Rodrguez-Fortea,S P A Cox, “The Electronic Structure and Alvarez, M Verdaguer, Chem. Chemistry of Solids” Oxford, 1987.

Eur. J. 2005, 11, 2135.63 64

63 64 Spinels and Inverse Spinels Double-Exchange in Magnetite (Fe3O4)

• The ‘transfer’ of an up- spin electron would leave the ion from

FeII(hs) FeIII(hs) which the electron is transferred (the left- hand ion) in a higher “hop” energy state. ⇓ • e– transferred must have opposite spin to FeIII(hs) FeII(hs) those of the ‘receiving’ ion. (Pauli principle.) ∴ double-exchange gives 2 4 ZnFe65 O ferromagnetic coupling.66

65 66

3/2 M – M0 ~ T 1 Ferromagnetic

M Magnetization

3/2 M ~ (TC – T ) M is much larger for a ferromagnet than a 1 paramagnet. T/TC

In reduced coordinates, T/TC and M/M0, the behavior of ferromagnets is fairly universal, as long as the dimensionality of the coupled spins is the same (i.e., the coupling between moments extends in 3-dimensions). 67