Workshop in Magnetochemistry Molecular Magnetism (DFG-SPP 1137) Kaiserslautern, 29.09. – 02.10.2003 Foundations

Heiko Lueken, Institut fur¨ Anorganische Chemie, RWTH Aachen

Contents

1 Magnetic quantities 3

2 Magnetisation and magnetic susceptibility 4

3 Paramagnetism 5 3.1 Curie law ...... 5 3.2 Free lanthanide ions and Hund’s rules ...... 7 3.3 Lanthanide ions in octahedral ligand fields ...... 8 3.4 3d ions in octahedral ligand fields ...... 11 3.5 Curie-Weiss law ...... 14 3.6 Exchange interactions in polynuclear compounds ...... 16 Problems ...... 16

4 Magnetic ordering 17 4.1 Ferromagnetism ...... 17 4.1.1 Substances and materials ...... 17 4.1.2 Hysteresis loop and magnetisation curve ...... 18 4.2 Antiferromagnetism ...... 19 4.3 Ferrimagnetism ...... 20 4.3.1 Substances and materials ...... 20 4.3.2 Magnetic saturation moment ...... 20 4.3.3 Paramagnetism above the Curie temperature ...... 22 Problems ...... 24

5 Theory of free ions 25 5.1 Foundations of quantum-mechanics ...... 25 5.2 Perturbation theory ...... 25 5.3 One-electron systems ...... 29 5.4 Angular momentum ...... 33 5.5 Spin ...... 35 5.6 Spin-orbit coupling ...... 36 Problems ...... 44

1 6 Exchange interactions in dinuclear compounds 45 6.1 Parametrization of exchange interactions ...... 45 6.2 Heisenberg operator ...... 49 6.3 Exchange-coupled species in a magnetic field ...... 50 6.4 Mechanisms of cooperative magnetic effects in insulators ...... 56 Problems ...... 57

7 Exchange interactions in chain compounds 59

8 Exchange interactions in layers and 3 D networks 60 8.1 Molecular-field model ...... 60 8.2 High-temperature series expansion ...... 62

9 Magnetochemical analysis in practice 64

References 65

Appendix 66

2 1 Magnetic quantities

The legal SI units are not generally accepted. The CGS/emu system is still widely • used in magnetochemistry. Therefore, use magnetic quantities which are indepen- 2 dent of the two systems, e. g., µeff or µeff instead of χmolT do not mix the systems, e. g., use G (gauss) instead of T (tesla) in the CGS/emu • system

−4 use B0 = µ0H in graphical representations (conversion factor 10 T/Oe) •

Table 1: Definitions, units and conversion factors [1, 2]

quantity SI CGS/emu

permeability 7 µ0 4π/10 Vs/(Am) 1 of a vacuum 2 3 1/2 B magnetic flux T = Vs/m G = (erg/cm ) density 1 T = 104 G Ha magnetic field A/m Oe strength 1 A/m =b (4π/103) Oe

B = µ0(H + M) B = H + 4πM M magnetisation A/m b G 1 A/m = 10−3 G m = MV m = MV magnetic di- m Am2 b G cm3 pole moment 1 Am2 = 103 G cm3

eh¯/(2me) eh/¯ (2me) µB Bohr magneton − b − 9.27402 10 24 Am2 = 9.27402 10 21 G cm3 × × magnetic di- σ = M/ρ b σ = M/ρ b σ pole moment Am2/kg G cm3/g per unit massc 1 Am2/kg = 1 G cm3/g

d M = MMr/ρ M = MMr/ρ molar mol b mol M Am2/mol G cm3/mol mol magnetisatione 1 Am2/mol = 103 G cm3/mol s s f s s s atomic magnetic µm/µB = Mmol/(NAµB) µm/µB = Mmol/(NAµB) µm saturation moment µB b µB

a −4 If B0 = µ0H is used instead of H, e. g., in graphs, the conversion factor is 10 T/Oe. bρ specific density. cSpecific magnetisation; σs specific saturation magnetisation. d Mr molar mass. e s Mmol molar saturation magnetisation. f NA Avogadro constant.

3 Table 1: Definitions, units and conversion factors [1, 2] (cont.)

quantity SI CGS/emu M = χH M = χH magnetic volume χ 1 1 susceptibility 1 = 1/(4π)

χg = χ/ρ χg = χ/ρ magnetic mass b χ m3/kg cm3/g g susceptibility 1 m3/kg = 103/(4π) cm3/g

χmol = χMr/ρ χmol = χMr/ρ magnetic molar b χ m3/mol cm3/mol mol susceptibility 1 m3/mol = 4π/106 cm3/mol [3k /(µ N µ2 )]1/2[χ T ]1/2a [3k /(N µ2 )]1/2[χ T ]1/2 effective Bohr mag- B 0 A B mol B A B mol µ 1 b 1 eff neton number [3] 1 = 1

a kB Boltzmann constant. b 2 Magnetisation and magnetic susceptibility magnetic flux density in vacuo in matter

B = µ0H B = µ0(H + M) (1) magnetisation M = χH (2) magnetic susceptibility magnetic volume susceptibility χ

magnetic mass susceptibility χg = χ/ρ

magnetic molar susceptibility χmol = χgMr (3)

substance class range diamagnets χ < 0 10−4 . . . 10−6 closed shell atoms − − vacuum χ = 0 paramagnets χ > 0 10−2 . . . 10−5 open shell atoms

Diamagnetism (i) χ < 0 and M < 0 owing to small additional currents attributable to the precession of electron orbits about the applied magnetic field (shown by all substances); (ii) usually independent of both T and B for purely diamagnetic materials (closed shell systems); (iii) allowed for as diamagnetic correction in the evaluation of experimental susceptibility data of open shell systems.

4 Paramagnetism (i) χ > 0 and M > 0 owing to net spins and orbital angular momentum polarized in direction of the applied field; (ii) observed in various forms, differing in magnitude and dependency on T and B: Curie paramagnetism: inverse dependency on T , independence on B at weak applied fields, but inverse dependency at strong fields (paramagnetic saturation, see Figure 1) ∞ ∞ where B(α) = Mmol/Mmol with Mmol = NA gJ J µB; temperature independent paramagnetism: weak forms of paramagnetism, called Pauli paramagnetism (of conduction electrons observed in metals) and Van Vleck paramag- netism (TIP, second order effect involving mixing with the first excited multiplet by the applied field; example: Eu3+).

B(a) 1.0

0.8

0.4

B 0.2 mB a = kBT

0 1 2 3 4 5 6 Fig. 1: Brillouin function B(α); inserts: magnetic dipoles (left: randomly distributed, B = 0; middle: weakly polarised, weak B, high T ; right: completely aligned, strong B, low T )

Curie limit: k T µB paramagnetic saturation: k T µB B  B  3 Paramagnetism

3.1 Curie law 2 Mmol C NAµ χmol = = where C = µ0 (4) H T 3kB

C: Curie constant; µ: permanent magnetic moment of an atom

Preconditions for Curie behaviour, examples magnetically isolated centres •

5 thermally isolated ground state temperature independent C • −→

l

o C

m /

l c

o /

m

C c

0 T 0 T a b

(0)

f

f 1

e

m

/ Curie f behaviour (in reduced magnetic

f e quantities χmol/C and µeff /µeff (0)

m (C and µeff (0) refer to the free ion)

0 Fig. 2 a: variation χmol vs. T T −1 c Fig. 2 b: variation χmol vs. T Fig. 2 c: variation µeff vs. T

µ = g S(S + 1) µB spin-only formula (5) p Gd2(SO4)3 8 H2O and (NH4)2Mn(SO4)2 6 H2O with 3+ 7 · 8 · Gd [4f ] (ground state S7/2, S = 7/2, µ = 7.94 µB) 2+ 5 6 Mn [3d ] (ground state A1, S = 5/2, µ = 5.91 µB)

6 3.2 Free lanthanide ions and Hund’s rules 4 5 6 µ = gJ J(J + 1) µB except 4f , 4f , 4f systems (6) with Landp e´ factor J(J + 1) + S(S + 1) L(L + 1) g = 1 + − (7) J 2J(J + 1) S, L, J correspond to the total spin angular momentum, the total orbital angular momentum and the total angular momentum, respectively, of the ground state. The 4f electrons of free Ln ions are influenced by interelectronic repulsion 4 1 3 1 Hee (splitting energy 10 cm− ) and spin-orbit coupling HSO (10 cm− ), i. e., Hee > HSO. To determine the free ion ground state use the following scheme and apply Hund’s rules:

Ln3+ Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

N 1 2 3 4 5 6 7 8 9 10 11 12 13 14

m +1 +1 +1 +1 +1 +1 +1 1 1 1 1 1 1 1 s 2 2 2 2 2 2 2 −2 −2 −2 −2 −2 −2 −2 m +3 +2 +1 0 1 2 3 +3 +2 +1 0 1 2 3 l − − − − − − 1. The term with maximum S (or multiplicity 2S + 1) lies lowest in energy ( m = M S). i s,i S → 2. For a givPen multiplicity, the term with the highest value of L lies lowest in energy ( m = M L). i l,i L → 3. For atoms with lessP than half-filled shells, the level with the lowest value of J lies lowest (J = L S ), while the opposite rule applies | − | (highest J lies lowest) when a subshell is more than half full (J = L + S).

Examples: 3+ 2 4 3 Pr [4f ]: S = 1, L = 5, J = 4; gJ = 5, µ = 3.58 µB ( H4). 3+ 9 4 6 Dy [4f ]: S = 5/2, L = 5, J = 15/2; gJ = 3, µ = 10.65 µB ( H15/2).

7 Table 2: Lanthanide ions: term symbol (ground state), one-electron 1 1/2 spin-orbit coupling parameter ζ4f [cm− ], gJ , gJ J, gJ [J(J +1)] and exp µeff (295 K)

3+ N 2S+1 a) 1/2 exp Ln 4f LJ ζ4f gJ gJ J gJ [J(J + 1)] µeff 3+b) 0 1 La 4f S0 0 3+ 1 2 Ce 4f F5/2 625 6/7 15/7 2.535 2.3–2.5 3+ 2 3 Pr 4f H4 758 4/5 16/5 3.578 3.4–3.6 3+ 3 4 Nd 4f I9/2 884 8/11 36/11 3.618 3.4–3.5 3+ 4 5 c) Pm 4f I4 1000 3/5 12/5 2.683 2.9 3+ 5 6 Sm 4f H5/2 1157 2/7 5/7 0.845 1.6 3+ 6 7 Eu 4f F0 1326 0 0 0 3.5 3+ 7 8 Gd 4f S7/2 1450 2 7 7.937 7.8–7.9 3+ 8 7 Tb 4f F6 1709 3/2 9 9.721 9.7–9.8 3+ 9 6 Dy 4f H15/2 1932 4/3 10 10.646 10.2–10.6 3+ 10 5 Ho 4f I8 2141 5/4 10 10.607 10.3–10.5 3+ 11 4 Er 4f I15/2 2369 6/5 9 9.581 9.4–9.5 3+ 12 3 Tm 4f H6 2628 7/6 7 7.561 7.5 3+ 13 2 Yb 4f F7/2 2870 8/7 4 4.536 4.5 3+b) 14 1 Lu 4f S0 0

a) The relation between ζ4f and λLS of the Russell-Saunders ground term is given by λLS = (ζ f /2S), where (+) and ( ) sign correspond to N 2l+1 and N 2l+1 respectively.  4 − ≤ ≥ b) diamagnetic c) observed for Nd2+ compounds.

3.3 Lanthanide ions in octahedral ligand fields [5] The chemical environment of the Ln ions has only a minor effect on the 2 1 4f electrons. The ligand field effect produces splittings of H 10 cm− LF ≈ leading generally to temperature dependent µ and described by the effective

8 Bohr magneton number µeff :

2 2 1/2 NAµ µ 3k T χ χ = µ eff B where µ = B mol = 797.7(T χ )1/2(8) mol 0 3k T eff µ N µ2 mol B  0 A B  As a rule, µeff approaches the free Ln ion value for T above 200 K (see Table 2, Figures 3 – 7). a 500 c 2.4 e 2.0

400

3 -

m f 1.6

300 f

d f e

mol 1.2 m 5 200

10 0.8 /

l b

o

1 m - 100

c 0.4

0 0.0 0 100 200 300 400 T/K

3+ 1 Fig. 3: Ce in cubic ligand fields; χmol− –T - (b,d,f) and µeff –T diagrams 1 1 (a,c,e); ∆ = 605 cm− (e,f; oct.), ∆ = 605 cm− (c,d; tet); straight lines − (a,b) refer to the free ion.

9 200 3+ ) 5+ 4

3 U Yb

- 160 m 3

mol 120 6 f f

e 3+ 10 2

m Ce

( 80 /

l 3+ o

1 Ce m - 1

c 40 5+ U Yb3+ 0 0 0 100 200 300 400 0 100 200 300 400 T/K T/K −1 3+ 3+ 3+ 3+ Fig. 4: Variation χmol vs. T of Ce , Yb , Fig. 5: Variation µeff vs. T of Ce , Yb , and U5+ in octahedral coordination. and U5+ in octahedral coordination.

4.0 240 3.6 200 200 3.2 3.0 2.8 160 150

3 2.4

- 3

- 2.0 m 2.0 f

120 f f

f e

100 1.6 e m

m

molm

80 mol 1.2 5

5 1.0 50 0.8 10

10

40 /

/

l l 0.4

o

o

1 1

m

m

-

0 0.0 - 0 0.0 c c 0 100 200 300 400 0 100 200 300 400 T/K T/K

1 3+ 3+ Fig. 6: χmol− –T and µeff –T diagrams for Pr (left) and Nd (right), calculated with the spectroscopically determined data of Cs2NaPrCl6 and Cs2NaNdCl6, respectively (solid lines: full basis; dottet lines: ground mul- tiplet; dashed lines: free ions)

10 4.0 8 d 7 c a 3.0 6

3 - 5

m 2.0

4 b

f f

mol e

7

-

m b 3 c 1.0 a /10 l 2

o

1 d

m

-

c 1 0.0 0 0 100 200 300 400 0 100 200 300 400 T/K T/K

1 3+ 3+ Fig. 7: µeff –T (left) and χmol− –T diagrams (right) for Sm and Eu , cal- 3+ culated with the spectroscopically determined data of Cs2NaYCl6 : Sm 3+ 3+ (a) and Cs2NaEuCl6 (d), respectively; free Sm ion (b), free Eu ion (c).

3.4 3d ions in octahedral ligand fields [4, 8, 6]

Tab. 3: Overview on the estimated magnetic behaviour of 3d systems (octahedral ligand field)

a b system energetic order N ground state χmol(T ) 6 5(hs) A1 χmol = C/T 1, 2, 6(hs), 7(hs) T χ = f(T ) H H > H mol N ee LF SO 3d ≈ 4(hs), 9 E χmol = C0/T + χ0 3, 8 A2 χmol = C0/T + χ0 1 HLF > Hee > HSO 6(ls) A1 χmol = χ0

ahs and ls assign high-spin and low-spin configuration, respectively. b 0 χmol = f(T ) stands for complicated variation χmol vs. T (see Figure 8); C is the Curie constant, C is a constant which deviates more or less from C caused by HLF and HSO, and χ0 is a temperature independent parameter.

11 Tab. 4: Ions with 3dN high-spin configuration: term symbols (ground state), one-electron spin-orbit coupling parameter ζ3d 1 1/2 exp [cm− ] [4], S, 2 [S(S + 1)] and µeff (295 K). N 2S+1 1/2 exp Ion 3d LJ ζ3d S 2[S(S + 1)] µeff 3+ 1 2 Ti 3d D3/2 154 1/2 1.73 1.65 – 1.79 3+ 2 3 V 3d F2 209 1 2,83 2.75 – 2.85 2+ 3 4 V 3d F3/2 167 3/2 3.87 3.80 – 3.90 3+ 3 4 Cr 3d F3/2 273 3/2 3.87 3.70 – 3.90 2+ 4 5 Cr 3d D0 230 2 4.90 4.75 – 4.90 3+ 4 5 Mn 3d D0 352 2 4.90 4.90 – 5.00 2+ 5 6 Mn 3d S5/2 347 5/2 5.92 5.65 – 6.10 3+ 5 6 Fe 3d S5/2 (460) 5/2 5.92 5.70 – 6.00 2+ 6 5 Fe 3d D4 410 2 4.90 5.10 – 5.70 3+ 6 5 Co 3d D4 (580) 2 4.90 5.30 2+ 7 4 Co 3d F9/2 533 3/2 3.87 4.30 – 5.20 3+ 7 4 Ni 3d F9/2 (715) 3/2 3.87 2+ 8 3 Ni 3d F4 649 1 2.83 2.80 – 3.50 2+ 9 2 Cu 3d D5/2 829 1/2 1.73 1.70 – 2.20

12 Tab. 5: Ions with 3dN low-spin configuration: no. of unpaired elec- 1/2 exp trons N 0, S, 2 [S(S + 1)] and µeff (295 K).

N 1/2 exp Ion 3d structure N 0 S 2[S(S + 1)] µeff Cr2+ oct.(dist.)a) 3.20 – 3.30 3d4 2 1 2.83 Mn3+ oct.(dist.) 3.18 Mn2+ oct.(dist.) 1.80 – 2.10 3d5 1 1/2 1.73 Fe3+ oct.(dist.) 2.0 – 2.5 Fe2+ oct. 0 3d6 0 0 0 Co3+ oct. TIPb) Co2+ 3d7 oct.(dist.) 1 1/2 1.73 1.8 Ni2+ 3d8 square planar 0 0 0 0

a) Distorted due to Jahn-Teller effect. b) Temperature independent paramagnetism. 1 3d : Example for temperature dependent µeff

µeff √T for T 100 K 1 ∼ ≤ χ− linear increase for T 150 K mol ≈ ≥

−1 Fig. 8: Variation of µeff vs. T and χmol vs. T plots of a 3d1 ion in octahedral (——) and orthorhombic (– – –) surrounding.

13 3.5 Curie-Weiss law The Curie-Weiss law C 2S(S + 1) χ = with Θ = z (9) mol T Θ p 3k iJi p B i − X is the most overworked formula in paramagnetism [7] • has only a limited applicability: • – not applicable to magnetic diluted systems – not applicable to systems like Ti3+ in octahedral surrounding (see Figure 8) and f systems (except Gd3+, Eu2+) – only applicable to pure spin paramagnetism with C corresponding with the permanent magnetic moment µ and Θp measuring the total exchange interactions of a magnetically active centre with all its magnetic neighbours (nearest, next-nearest, etc.)

d 3

- 100

75 c

molm 5

10 a /() 50

1

mol

-

c 25 b

-100 0 100 200 300 T/K

Fig. 9: χ−1 as a function of T for Curie law (a) and Curie-Weiss

behaviour (schematic); (b): EuO, Θp = 74.2 K, TC = 69 K; (c): MnF2, Θ = 113 K, T = 74 K; (d): Na2NiFeF7, Θ = 50 K, p − N p − TC = 88 K)

14 0 0 p 0p 0

l 0 =0

C p

0p=0 o /

m

l

o c

m /

c 0p 0 C

0p 0

0 T 0 T a b

0p 0 Curie behaviour (bold lines) and 0p =0 (0) Curie-Weiss

f behaviour in reduced f 1 e magnetic quantities χ /C and

m mol

/ 0 0 f p µeff /µeff (0) (C and µeff (0) refer to

f e the free ion)

m

Fig. 10 a: variation χmol vs. T 0 T Fig. 10 b: variation χ 1 vs. T c mol− Fig. 10 c: variation µeff vs. T

15 3.6 Exchange interactions in polynuclear compounds [9]

Example: Dinuclear complex [Cu(CH3COO)2(H2O)]2

1 1,0 -

0,8

mol 3 0,6 S=1

m S1=1/2 S2=1/2 8 DE =-2J copper - 0,4

S=0 /10

l 0,2

acetate o m oxygen c 0 100 200 300 (water) T/K

Fig. 11: [Cu(CH3COO)2(H2O)]2; left: molecular structure; right: χmol versus T diagram −1 −11 3 −1 with = 148 cm (Hˆex = 2 Sˆ1 Sˆ2), g = 2.16, χ0 = 76 10 m mol . To J − −1 − J · × explain = 148 cm both the direct exchange interactions between the dx2−y2 orbitals and theJsuperexc− hange via the doubly occupied orbitals as well as unoccupied orbitals of the bridging ligands have to be accounted for.

2 2 −1 NAµBg 1 2 χ = µ0 1 + exp − J + χ0, see section 6.3, page 54 mol 3k T 3 k T B   B  Problems

s 1. How are the quantities (i) m, (ii) µeff , (iii) µ, (iv) µm defined? 2. Calculate the spin contributions to the molar susceptibility of hydrogen atoms at 298 K (SI units). µ N µ2 /(3k ) = 1.57141 10−6m3 K mol−1 0 A B B × 3. What are the electronic and structural conditions for a compound to be a param- agnet obeying the Curie law? Give examples!

4. What are the electronic, structural, and thermal conditions for a compound to be a paramagnet obeying the Curie-Weiss law χ = C/(T Θ ), permitting a mol − p reasonable interpretation C and Θp? Give examples!

2S+1 5. Write the ground state term symbols LJ of free ions with electronic configuration [Ar] 3dN (N = 0, 1, 2, . . . , 10).

6. What levels (multiplets J) may arise from the terms (a) 1S, (b) 2P , (c) 3P , (d) 3D, 4 (e) D? How many states (distinguished by the quantum number MJ ) belong to each level?

16 4 Magnetic ordering

4.1 Ferromagnetism ⇑⇑⇑⇑ 4.1.1 Substances and materials Table 6: Ferromagnetic elements and compounds

s substance structure TC/K Θp/K µm/µB α-Fe bcc 1044(2) 1100 2.216 β-Co hcp 1388(2) 1415 1.715 Ni ccp 627.4(3) 649 0.616 Gd hcp 293.4 317 7.63 EuO NaCl 69 74.2 6.94

La1 xSrxMnO3 perovskite 210 385 − − CrBr3 BiI3 32.7 54 3.0

s Tab. 7: Ferromagnetic materials: properties and application. TC [K], µm −3 [µB] per formula unit, (BH)max [kJ m ] (mr: magnetic data recording; sm: softmagnetic material; hm: hardmagnetic material)

s a) material structure TC µm (BH)max appl.

Fe40Co40B20 amorphous > 800 1.43 sm supermalloyb) ccp 673 sm Alnicoc) d) > 800 25 hm

SmCo5 CaCu5 1000 160 hm

Sm2Co17 Th2Zn17 225 hm

Nd2Fe14B Nd2Fe14B 585 37.6 360 hm

CrO2 Rutil 392 2.00 mr

a) (BH)max is the performance of a hardmagnetic material. b) Typical composition (mass %): Ni (79), Fe (15.5), Mo (5.0), Mn (0.5). c) Alnico 2 (mass %): Ni (18–21), Al (8–10), Co (17–20), Cu (2–4), Nb (0–1), Fe (rest). d) Heterogeneous system: ferromagnetic segregations (Fe/Co) in a (Ni/Al) matrix.

17 4.1.2 Hysteresis loop and magnetisation curve

B = m0(H+M)

B,Jp

S m0 M

Br

BHc H

M Hc

Fig. 12: Hysteresis loop of a ferromagnet; B as a function of H (—) and µ0M as a function of H ( ), · · ·

µ0M = Jp, Jp: magnetic polarisation) )

1 200

-

kg

1 -

100

/(JT

s s

0 100 200 300 T/K Fig. 13: σs as a function of T of gadolinium (σs = M s/ρ)

18 H H

a b c

Fig. 14: Schematic representation of domains in a ferromagnetic monocrystal: (a) un- magnetised; (b) magnetisation through movement of domain boundary walls; domains oriented parallel to H grow at the expense of antiparallel domains; (c) magnetisation by rotation of the magnetisation vector of whole domains. The domains remain oriented along a preferred direction; stronger fields are required to swing the magnetisation vectors towards the applied field [2].

4.2 Antiferromagnetism ⇑⇓⇑⇓ Tab. 8: Antiferromagnetic compounds s substance structure TN /K Θp/K µm/µB MnO NaCl 118 610 5.0 − FeO NaCl 185 570 3.3 − CoO NaCl 292 3.8 NiO NaCl 523 2.0 Co3O4 spinel 40 3.02 ZnFe2O4 spinel 10.6 0 4.0 MnF2 rutile 74 113 4.98 −

F Mn A Mn B

Fig. 15: Spin structure of the antiferromagnet MnF2 (rutile type); A, B: Mn2+ sublattices

19 4.3 Ferrimagnetism ⇑↓⇑↓ 4.3.1 Substances and materials

Tab. 9: Ferrimagnetic compounds and materials

s substance/material structure TC /K Θp/K µm/µB

Mn3O4 spinel 42 1.85

Fe3O4 spinel(inverse) 858 4.1

NiFe2O4 spinel(inverse) 858 2.3

Na2NiFeF7 weberite 88 -50 2.2 (Ni) 5.0 (Fe)

4.3.2 Magnetic saturation moment s s s µm = M Mr/(ρNA) = Mmol/NA Arrangements of magnetic dipoles in ferrimagnets

+ = + = + =

a b c

Fig. 16: Three possible arrangements of magnetic dipoles in ferrimagnetic materials: (a) unequal numbers of identical moments on the two sublattices; (b) unequal moments on the two sublattices; (c) two equal moments and one unequal [2].

III II III Example Fe3O4 = Fe [Fe Fe ]O4 (inverse spinel):

µs = gSµ with g = 2 (pure spin magnetism) of the iron ions occupying m B b tetrahedral (A) and octahedral holes (B): 3+ s Fe (A): µm = 5 µB 3+ s Fe (B): µm = 5 µB 2+ s Fe (B): µm = 4 µB

20 s Resultant moment µm(Fe3O4): µs (B) µs (A) = (5 + 4) µ 5 µ = 4 µ per formula unit m − m B − B B (exp.: 4.1 µB) Neel´ ’s suggestion: all interactions in the ferrites are antiferromagnetic, but the A–B interaction is considerably stronger than A–A or B–B. Thus, in the inverse spinel structure the dominating A–B interaction makes the spins within each sublattice parallel, despite their mutual antiferromagnetic interaction. This is supported by the fact, that ZnFe2O4, which has the normal structure, has no net saturation moment. II III III II III Examples: Mixed ferrites made of M [Fe ]O4 and Fe [M Fe ]O4:

Mixing two ferrites, three cases can be distinguished: 1.) Both ferrites are inverse, 2.) one is normal, the other inverse, 3.) both are normal. While 3.) is practically of no relevance, 1.) and 2.) are of interest. 1. Both base ferrites inverse MII (amount x, µs (M )) and MII (amount 1 x, µs (M )) on site B: i m i j − m j µs = x µs (M ) + (1 x) µs (M ) m m i − m j Example: Fe[NixMn1 xFe]O4: s − Fe[NiFe]O4: µm 2.3 µB, TC 900 K s ≈ ≈ Fe[MnFe]O4: µm 4.7 µB, TC 580 K ≈s ≈ linear variation µm vs. x (and also TC vs. x) 2. Normal and inverse base ferrite II s II III Provided Mi is diamagnetic (µm(Mi) = 0), Mi [Fe2 ]O4. Substituting II III II III II Mj in Fe [Mj Fe ]O4 by Mi , the latter occupies site A. In return for it, FeIII changes from A to B. The net moment of the mixed ferrite amounts to

µs = (1 x) µs (M ) + 2x µs (FeIII) m − m j m s s III Since always µm(Mj) < µm(Fe ), the net moment increases with fur- II s ther incorporation of Mi . For x = 1, however, µm must be zero, be- cause MiFe2O4 is a normal ferrite, which is antiferromagnetic. Thus, s µm passes through a maximum, reflecting the increasing antiferromag- netic coupling B–B and the decreasing A–B interaction according to the increasing Mi amount.

21 4.3.3 Paramagnetism above the Curie temperature TC [5] II In the ferrite MFe2O4 the M ions are considered as diamagnetic and the 3+ s magnetically active Fe ions are distributed over the A and B sites. Mmol,A s 3+ and Mmol,B assign the magnetisation per mole of the Fe ions, if they s s occupy the A and B sites, respectively. In general, Mmol,A and Mmol,B differ on account of the different ligand field effects acting on the iron ions. The net magnetisation (T < TC) of the ferrite is s s s M mol = xM mol,A + yM mol,B with x + y = 1, (10) where x and y give the amount of Fe3+ ions on A and B sites, respectively. For T > TC, molecular field theory leads to (see ref. [5]): 1 T 1 σ = + with (11) χmol C χ0 − T Θ 1 2 − 2 χ− = n(2xy x α y β), 0 − − σ = n2Cxy [x(1 + α) y(1 + β)]2 , − Θ = nCxy[2 + α + β]. The positive molecular field parameter n stands for the strength of the antiferromagnetic interaction between the sublattices A and B, whereas nα and nβ refer to the interaction in the sublattices A and B, respectively. 1 Curie- The variation χmol− vs. T (Gl. (11)) is not linear in contrast to the Weiss law. Instead, a hyperbola with asymptotes

1 1 χmol− = T/C + χ0− (12) and T = Θ (13) is obtained (see Figure 17).

1 1 = T+ 1 cmol cmol C c0

1 c0

T=Q

Qp 0 TC T

−1 Fig. 17: χmol vs. T of a simple ferrimagnet

22 Remarks to eq. (11):

The characteristic feature of a ferrimagnetic material is the hyperbolic • 1 variation χmol− vs. T (in contrast to ferromagnetic materials where the molecular field approximation gives a straight line). A linear dependence is obtained for σ = 0 and x(1 + α) = y(1 + β), • fulfilled for x = y = 1/2 and α = β, which corresponds to antiferro- magnetism. The intersection of the asymptote eq. (12) with the T axis at Θ = • p C/χ is called asymptotic Curie point. It corresponds with the − 0 paramagnetic Curie temperature of an antiferromagnet in the case of x = y = 1/2 and α = β. The intersection of the hyperbola eq. (11) with the T axis is of great • importance: nC T = xα + yβ + [(xα yβ)2 + 4xy]1/2 C 2 − n o T > 0 means, that at T = T the susceptibility diverges (χ ), C C → ∞ i. e., for T < TC spontaneous magnetisation exists. In the case of TC < 0, the substance behaves paramagnetically in the whole temperature range. When α > 0 and β > 0 ferrimagnetism exists. If the absolute values • of α and β exceed those values, which fulfill the condition αβ = 1, the moment ordering on the sublattices is antiferromagnetic or nonmag- netic. If the absolute values are smaller than the values which fulfill αβ = 1, the antiferromagnetic interactions within the sublattices are quenched, so that spontaneous magnetisation results.

23 Problems 1. What is a hard magnetic material? Give examples!

2. Fe3O4 adopts the inverse spinel structure type. What type of magnetic structure is expected? How large is the magnetic saturation moment per formula unit so long as pure spin magnetism is assumed?

3. Co3O4 is a spinel with an antiferromagnetic magnetic structure at T < s 40 K. Neutron diffraction yields the magnetic saturation moment µm = 3 µB for the magnetically active centres. What are the magnetically active centres? Is the spinel normal or inverse? What are the spin configurations and total spin quantum numbers of the centres?

4. NiFe2O4 is an inverse spinel. What type of magnetic ordering is ex- pected? How large is the magnetic saturation moment of each centre, s so long as pure spin contributions are considered? What is the µm value per formula unit determined, e. g., by SQUID magnetometry? 5. EuO (NaCl type) is paramagnetic at T > 70 K and obeying the Curie-Weiss law. How large is the expected paramagnetic moment µ, deduced from the slope of the Curie-Weiss straight line? What type of magnetic structure exists at T < 70 K, if neutron diffraction yields only an increase of reflection intensities, but no extra lines com- pared to those at T > 70 K? Consequently, what sign is expected for Θp? 6. What law is expected to describe the temperature dependence of the magnetic susceptibility of magnetically concentrated Mn(II) high-spin compounds above the magnetic ordering temperature TC (Curie tem- ´ perature) and TN (Neel temperature), respectively? What informa- tion can be gained from a corresponding measurement with regard to the magnetic centres and the existing interactions?

24 5 Theory of free ions

5.1 Foundations of quantum-mechanics ([10]9–12) Postulate 1. The state of a system is fully described by the wavefunction Ψ(r1, r2, . . . , t). Postulate 2. Observables are represented by operators chosen to satisfy the commutation relation qˆpˆ pˆ qˆ = [qˆ, pˆ ] = ih¯ (q = x, y, z; i = √ 1) (14) q − q q − Example 5.1 Application of the commutation relation h¯ d xˆ = x and pˆ = · x i dx

h¯ dΨ xˆpˆ Ψ = x x i dx h¯ d(xΨ) h¯ dΨ pˆ xˆΨ = = Ψ + x x i dx i dx   h¯ (xˆpˆ pˆ xˆ)Ψ = Ψ = ih¯Ψ x − x − i

5.2 Perturbation theory ([5]83–91) 1. Non-degenerate states

ˆ (0) (0) (0) (0) H Ψn = En Ψn unperturbed system (15) Hamilton operator of the perturbed system:

Hˆ = Hˆ (0) + λHˆ (1)

Schrodinger¨ equation of the perturbed system:

Hˆ Ψn = EnΨn; find En, Ψn (16)

Series expansion of Ψn und En:

25 Tab. 10: Classical and quantum-mechanical forms of Ekin and Epot

Quantity Da) classical quantum-mechanical

2 2 2 2 2 2 mevx px pˆx 1 h¯ d h¯ d Ekin 1 = = = 2 2m 2m 2m i dx −2m dx2 e e e   e 2 2 2 2 2 p2 h¯ ∂ ∂ ∂ h¯ b 3 + + = 2 ) 2m −2m ∂x2 ∂y2 ∂z2 −2m ∇ e e   e E c) 1 eV (x) eVˆ (x) = eV (x)· pot − − − 3 eV (r) eVˆ (r) = eV (r)· − − − a) Dimension. b) is the Nabla operator. ∇ c) Valid for one electron with charge e in the potential V . −

(0) (1) 2 (2) Ψn = Ψn + λΨn + λ Ψn + . . . (17) (0) (1) 2 (2) En = En + λEn + λ En + . . . . (18) Insert the series into eq. (16): ˆ (0) ˆ (1) (0) (1) 2 (2) (H + λH )(Ψn + λΨn + λ Ψn + . . .) = (0) (1) 2 (2) (0) (1) 2 (2) (En + λEn + λ En + . . .)(Ψn + λΨn + λ Ψn + . . .), Ordering of the terms with regard to powers of λ: ˆ (0) (0) ˆ (1) (0) ˆ (0) (1) 2 ˆ (1) (1) ˆ (0) (2) H Ψn + λ H Ψn + H Ψn + λ H Ψn + H Ψn + . . . = (0) (0) (1) (0) (0) (1) En Ψn + λ En Ψn + En Ψ
n +
2 (2) (0) (1) (1) (0) (2) λ En Ψn + En Ψn + En Ψn
+ . . . .

0
λ ˆ (0) (0) (0) (0) H Ψn = En Ψn (19) 1 λ (Hˆ (0) E(0))Ψ(1) = (E(1) Hˆ (1))Ψ(0) (20) − n n n − n 2 ˆ (0) (0) (2) (2) (0) (1) ˆ (1) (1) λ (H En )Ψn = En Ψn + (En H )Ψn (21) . − . − . .

26 (1) The first-order correction to the energy En (0) (premultiply both sides of eq. (20) with Ψn ∗ and integrate)

(0) (0) (1) (0) (0) (1) (1) (0) (1) (0) Ψ ∗Hˆ Ψ dτ E Ψ ∗Ψ dτ = E Ψ ∗Hˆ Ψ dτ n n − n n n n − n n Z Z Z 0

|E(1) = n Hˆ (1) n {z } (22) n | |

The first-order correction to the wavefunction:

m Hˆ (1) n Ψ(1) = | | Ψ(0) (23) n − (0) (0) m m=n Em En X6 −

The second-order correction to the energy:

2 m Hˆ (1) n E(2) = | | (see Fig. 18) (24) n − (0) (0) m=n Em En X6 − E

Y (0) 2

E (0) 2 H Y (0) 12 Y (0) 2 E (0) E (0) 1 1 2 H H E (0) H 21 12 2 22 E (0) E (0) E (0) H 1 2 2 22 Y (0) 1

E (0) 1 H Y (0) 21 Y (0) 1 E (0) E (0) 2 2 1

E (0) H 1 11 H H E (0) H 12 21 1 11 E (0) E (0) 2 1

(a) (b) (c) Fig. 18: Illustration of the possible effects of a perturbation on two non-degenerate levels; (a) 0th,(b) 1st,(c) 2nd order

27 2. Degenerate states Eq. (22) – (24) apply also in this case; in addition, the correct zeroth-order wavefunctions have to be determined (see Fig. 19): Example: doubly degenerate pair of states ˆ (0) (0) (0) (0) H Ψn,i = En Ψn,i (i = 1, 2) (25)

(0) (0) (0) Ψn = u1Ψn,1 + u2Ψn,2 (26) The ’correct’ linear combinations are those which correspond to the per- turbed functions for λ 0. → Determination of u1 and u2: Substituting eq. (26) in eq. (17); eq. (20) now reads:

Hˆ (0) E(0) Ψ(1) = E(1) Hˆ (1) u Ψ(0) + u Ψ(0) (27) − n n n − 1 n,1 2 n,2
(0) (0)

Multiply with Ψn,1∗ and Ψn,2∗, respectively and integrate:

(1) u1 H11 En + u2H12 = 0 (0) ˆ (1) (0) − where Hij = Ψn,i∗ H Ψn,j dτ u H + u H E(1) = 0 1 21 2
22 − n Z Under the condition that
the determinant of the coefficients of u1 and u2 disappears the non-trivial solutions of this pair of equations are obtained:

(1) H11 En H12 − (1) = 0 (28) H21 H22 En −

(1) 2 2 E = H11 + H22 /2 (H11 H22) /4 + H12 n(1,2)  − | | p (1)
u1(1,2) H12 u1(1,2)(H11 En(1,2)) + u2(1,2)H12 = 0; x(1,2) = = (1) − u2(1,2) −H E 11 − n(1,2) Normalisation: 1 u2(1,2) = 2  x(1,2) + 1 x2 u2 + u2 = 1 (1,2) 2(1,2) 2(1,2) ⇒  x(1,2)  u1(1,2) = xq(1,2)u2(1,2) =  2 x(1,2) + 1   q  28 (1) Correct zeroth-order wavefunction for the energy En(1,2):

(0) (0) (0) Ψn(1,2) = u1(1,2)Ψn,1 + u2(1,2)Ψn,2

E

Y (0) Y (0) 2 2 H H H H E (0) E (0) E (0) H 21(1) 1(1)2 21(2) 1(2)2 2 2 2 22 E (0) E (0) 1 2 E (0) H 2 22

Y (0) u Y (0) u Y (0) 1(2) = 1(2) 1,1 2(2) 1,2 Y (0) Y (0) Y (0) u Y (0) u Y (0) E (0) E (1) 1,1 , 1,2 1(1)= 1(1) 1,1 2(1) 1,2 1 1(2) H H E (0) E (0) E (0) E (1) 21(2) 1(2)2 1 1 1 1(2) E (0) E (0) 2 1

E (0) E (1) 1 1(1) H H E (0) E (1) 21(1) 1(1)2 1 1(1) E (0) E (0) 2 1

(a) (a´) (b) (c) Fig. 19: Illustration of the possible effects of a perturbation on a doubly degenerate ground state and a non-degenerate excited state; (a) 0th, (a’) correct 0th, (b) 1st,(c) 2nd order

5.3 One-electron systems ([5]91–96) Schrodinger¨ equation (spin ignored): h¯2 2 eVˆ (r) ψ(r) = Eψ(r). (29) − 2me ∇ − h i For convenience the eigenfunctions (atomic orbitals) are given in spherical polar coordinates:

l l 1 imlφ ψ(r) = ψn,l,ml (r, θ, φ) = Rn,l(r) Yml (θ, φ) = Rn,l(r) Θml(θ) e r2π radial f. angular f.

l | {z } | {z } The functions Yml (θ, φ) are the (in general, complex) spherical harmon- ics specified by the quantum numbers l and ml (see Table 11). They play a predominant role in magnetism.

29 x = r sin θ cos φ · · z y = r sin θ sin φ · · z = r cos θ (30) · P(x,y,z) r.cosq 2 2 2 2 r r = x + y + z q y cos θ = z/r r.sin q r.sincosq. f f tan φ = y/x (31) r.sinsinq. f x Fig. 20: Relation between carte- sian coordinates and spherical po- lar coordinates

Real functions (see Table 12) are gained by linear combinations

1 1 l [ ψn,l,ml + ψn,l, ml ] = Rn,l(r)Θ m (θ) cos mlφ √2 − √π l 1 − 1 l− (32) √ [ ψn,l,ml ψn,l, ml ] = √π Rn,l(r)Θ ml (θ) sin mlφ i 2 − − − − 1 1 l [ψn,l,ml + ψn,l, ml] = Rn,l(r)Θm (θ) cos mlφ √2 − √π l 1 1 l (33) [ψn,l,ml ψn,l, ml ] = Rn,l(r)Θm (θ) sin mlφ. i√2 − − √π l For completely describing the wave function, the spin has to be taken into consideration. If spin-orbit coupling is ignored, the total function (spin orbital) is 1 ψ(r, θ, φ; σ) = ψ(r, θ, φ) ψ(σ) where σ = . (34) 2

30 Table 11: Spherical harmonics for l = 0, 1, 2, 3

l a) l l ml Yml (θ, φ) Yml (x, y, z) 1 1/2 1 1/2 0 0 4π 4π  3 1/2  3 1/2 z 0 cos θ 4π 4π r 1   1/2   1/2 3 iφ 3 x iy 1 sin θ e   ∓ 8π ∓ 8π r 5 1/2 5 1/2 3z2 r2 0 (3 cos2θ 1) − 16π − 16π r2   1/2   1/2 15 iφ 15 z(x iy) 2 1 cos θ sin θ e   ∓ 8π ∓ 8π r2  1/2  1/2 2 15 2 i2φ 15 (x iy) 2 sin θ e   32π 32π r2  7 1/2  7 1/2 z(5z2 3r2) 0 (5 cos3θ 3 cos θ) − 16π − 16π r3   1/2   1/2 2 2 21 2 iφ 21 (5z r ) 1 sin θ(5 cos θ 1)e (x iy) −  ∓ 64π − ∓ 64π  r3 3  1/2  1/2 2 105 2 i2φ 105 z(x iy) 2 cos θ sin θ e   32π 32π r3   1/2   1/2 3 35 3 i3φ 35 (x iy) 3 sin θ e   ∓ 64π ∓ 64π r3     a) Phase factor corresponding to the Condon-Shortley convention, i. e., 1 for odd positive ml and +1 otherwise. −

31 Table 12: Real orthonormal linear combinations of the spherical harmonics l Yml (θ, φ) for l = 1, 2, 3.

l function designation

3 1/2 3 1/2 z cos θ = p 4π 4π r z  3 1/2  3 1/2 x 1 sin θ cos φ = p 4π 4π r x  3 1/2  3 1/2 y sin θ sin φ = p 4π 4π r y   1/2   1/2 2 2 5 2 5 3z r (3 cos θ 1) = − d 2 16π − 16π r2 z  15 1/2  15 1/2 xz cos θ sin θ cos φ = d 4π 4π r2 xz  15 1/2  151/2 yz 2 cos θ sin θ sin φ = d 4π 4π r2 yz   1/2   1/2 2 2 15 2 15 x y sin θ cos 2φ = − dx2 y2 16π 16π r2 −  15 1/2  15 1/2 xy sin2θ sin 2φ = d 16π 4π r2 xy  1/2   1/2 2 2 7 3 7 z(5z 3r ) (5 cos θ 3 cos θ) = − f 3 16π − 16π r3 z  1/2  1/2 2 2 21 2 21 x(5z r ) sin θ(5 cos θ 1) cos φ = − f 2 32π − 32π r3 xz  1/2  1/2 2 2 21 2 21 y(5z r ) sin θ(5 cos θ 1) sin φ = − f 2 32π − 32π r3 yz  1051/2 105 1/2 xyz 3 cos θ sin2θ sin 2φ = f 16π 4π r3 xyz  1/2  1/2 2 2 105 2 105 z(x y ) cos θ sin θ cos 2φ = − fz(x2 y2) 16π 16π r3 −  1/2  1/2 2 2 35 3 35 x(x 3y ) sin θ cos 3φ = − fx(x2 3y2) 32π 32π r3 −  1/2  1/2 2 2 35 3 35 y(3x y ) sin θ sin 3φ = − fy(3x2 y2) 32π 32π r3 −    

32 5.4 Angular momentum ([5]97–103) Classical definition of angular momentum l: l = r p. (35) ×

l

r p

Fig. 21: Definition of the angular momentum

l = lxi + lyj + lzk (36) = (yp zp )i + (zp xp )j + (xp yp )k z − y x − z y − x Length of the angular momentum vector l 2 = l2 + l2 + l2. (37) | | x y z

Derive quantum mechanical operators ˆlx, ˆly, ˆlz by substituting the position operator and the linear momentum operator for the corresponding classical quantity, i. e. h¯ ∂ q qˆ = q p pˆ = (q = x, y, z; i = √ 1) → · q → q i ∂q −

ˆl = yˆpˆ zˆpˆ ; ˆl = zˆpˆ xˆpˆ ; ˆl = xˆpˆ yˆpˆ (38) x z − y y x − z z y − x h¯ ∂ ∂ h¯ ∂ ∂ h¯ ∂ ∂ ˆl = yˆ zˆ ; ˆl = zˆ xˆ ; lˆ = xˆ yˆ (39) x i ∂z − ∂y y i ∂x − ∂z z i ∂y − ∂x       Commutation relations

[ˆlx, ˆly] = ih¯lˆz; [ˆly, ˆlz] = ih¯ˆlx; [ˆlz, ˆlx] = ih¯ˆly , (40)

Operator ˆlz in spherical polar coordinates: h¯ ∂ ˆl = (41) z i ∂φ

33 ˆlz acts on the φ depending part of the atomic orbitals (Table 11): ˆ Dirac lz l ml = lz l ml = mlh¯ l ml notation ˆ l ml lz l ml = ml h¯ l ml l ml

1

| {z } generally Hˆ Ψ = E Ψ, (Ψ normalised eigenfunction of Hˆ )

Ψ∗Hˆ Ψ dτ = E Ψ∗ Ψ dτ = E Z Z 1

Ψ∗Hˆ Ψ dτ Ψ|Hˆ Ψ{z matrix} element (Dirac notation) ≡ h | | i Z Application of lˆz: h¯ ∂Y 2(θ, φ) ˆl 2 2 = R (r) 2 z n,2 i ∂φ h¯ ∂ 15 1/2 = R (r) sin2θ ei2φ n,2 i ∂φ 32π "  # h¯ 15 1/2 ∂ei2φ = R (r) sin2θ n,2 i 32π ∂φ   h¯ 15 1/2 = R (r)i2 sin2θ ei2φ = 2¯h 2 2 n,2 i 32π   2 2 ˆl l ml = l(l + 1)¯h l ml (42)

Shift operators:

ˆl+ = ˆlx + iˆly; ˆl = ˆlx iˆly. (43) − − Reverse operations: 1 1 ˆlx = (ˆl+ + ˆl ); lˆy = (ˆl+ ˆl ). (44) 2 − 2i − −

ˆlz l ml = ml h¯ l ml ˆ2 2 l l ml = l(l + 1) h¯ l ml (45)

ˆ l l ml = l(l + 1) ml( ml 1) h¯ l ml 1 .  −   p

34 z

ml =+2 +1

0

1

2

Fig. 22: Specified orientation of l (l = 2) with regard to the component lz while lx and ly are unspecified

5.5 Spin ([5]103–105) Spin orbital of a one-electron system:

l ψ(r) ψ(σ) = ψn,l,ml (r, θ, φ) ψ(σ) = Rn,l(r) Yml(θ, φ) ψ(σ) atomic orbital | 1{z } ms = 2 : α Spin function: ψ(σ) s ms 1 ≡ | i ms = : β  −2

2 2 1 sˆ s ms = s(s + 1) h¯ s ms with s = 2 1 sˆz s ms = msh¯ s ms with ms = 2 (46)  sˆ s ms = s(s + 1) ms(ms 1) h¯ s ms 1  −   p 2 2 2 2 where sˆ = sˆx + sˆy + sˆz; sˆ+ = sˆx + isˆy; sˆ = sˆx isˆy − 1 1 − sˆx = (sˆ+ + sˆ ); sˆy = (sˆ+ sˆ ) 2 − 2i − −

35 5.6 Spin-orbit coupling ([5]105–116) Spin-orbit coupling results on account of the interaction of the electron’s magnetic spin moment µs e µs = gs = γegs (g = 2.002 319 314), γe magnetogyric ratio −2me with the orbital magnetic moment at the centre of the orbit e µl = l = γe l −2me caused by the circulating charged particle. Only certain orientations be- tween l and s are allowed given by the vector sum j = l + s, l + s 1, . . . , l s j = l + s, with − | − | (47) as s = 1/2 j = l 1/2  →  mj = ml + ms

2 2 2 2 ˆ = ˆx + ˆy + ˆz ˆ+ = ˆx + iˆy ˆ = ˆx iˆy (48) − −

2 2 ˆ j mj = j(j + 1) h¯ j mj

ˆz j mj = mjh¯ j mj (49)

ˆ j mj = j(j + 1) mj(mj 1) h¯ j mj 1 .  −   p 3 ∆E = E(2P ) E(2P ) = ζ; generally: ∆E(J, J 1) = J ζ 3/2 − 1/2 2 − Operator of spin-orbit coupling e 1 ∂V (r) ˆ ˆl·s Hso = ξ(r) ˆ where ξ(r) = 2 2 . (50) −2mec r ∂r Example 5.2 Spin-orbit coupling of the p1 system

1 1 1 3 p system: l = 1, s = 2, j = 2 and 2 (see Figs. 23 and 24, Table 13)

36 E -1 n/cm 2P 3/2

16973 1 (4) 17cm 1 z 16956 - 2P 2 1/2 (0) E (6)

1 2

2S (2) 1/2 -z 0 n ohne mit Spin-Bahn-Wechselwirkung l= 589.8nm 589.2nm Fig. 23: Splitting of the p1 Fig. 24: Term scheme of levels by spin-orbit interac- the sodium atom tion (ζ: one-electron spin- orbit coupling constant) Unperturbed sixfold degenerate states: ˆ (0) (0) (0) (0) H ψi = E ψi (i = 1, 2, . . . , 6). Eq. (27) reads in this case: (Hˆ (0) E(0))ψ(1) = (E(1) Hˆ (1)) u ψ(0) + . . . + u ψ(0)). (51) − − 1 1 6 6 (0) Multiplication with ψ1 ∗ from the left and integration gives: (0) (0) (0) (1) (0) (1) (1) (0) (0) ψ ∗ Hˆ E ψ dτ = ψ ∗ E Hˆ u ψ + . . . + u ψ dτ 1 − 1 − 1 1 6 6 Z Z 0


| {z (1) (0)} (0) (1) (0) (0) 0 = u1E ψ1 ∗ ψ1 dτ + . . . + u6E ψ1 ∗ ψ6 dτ Z Z (52) (0) (1) (0) (0) (1) (0) u ψ ∗ Hˆ ψ dτ . . . u ψ ∗ Hˆ ψ dτ. − 1 1 1 − − 6 1 6 Z Z (0) Proceeding similarly with ψi ∗ (i = 2, . . . , 6) and using the abbreviation (0) (0) ψ ∗Hˆ (1)ψ dτ H we obtain a system of six equations: i j ≡ ij R0 = u H E(1) + u H + + u H 1 11 − 2 12 · · · 6 16 (1) 0 = u1H21 + u2 H
22 E + + u6H26 − · · · (53) . . . .
0 = u H + u H + + u H E(1) 1 61 2 62 · · · 6 66 − 37
Non-trivial solutions for the coefficients of u1, u2, . . . , u6: Calculation of the integrals Hij

2 ∗ ˆ 0 0 ψn,l,ml (r, θ, φ) ψms(σ) HSB ψn,l,ml (r, θ, φ) ψms(σ) r dr sin θ dθ dφ dσ. (54) Z
∞ R (r) ξ(r) R (r) r2 dr (55) n,l n,l × Z0 π 2π 1/2 0 m m l ∗ ˆ· l 0 Yl (θ, φ) ψms (σ) l sˆ Yl (θ, φ) ψms(σ) sin θ dθ dφ dσ. 0 0 1/2 Z Z Z−

2 ∞ 2 hc ζn,l = h¯ Rn,l(r) ξ(r) Rn,l(r) r dr. (56) Z0 ζn,l: one-electron spin-orbit coupling constant basis functions in Dirac notation: ml ms 1 1 0 1 1 1 1 1 0 1 1 1 . (57) 2 2 − 2 − 2 − 2 − − 2 The integral eq. (55) has the short form

hcζn,l ml ms ˆl·sˆ m0 m0 . (58) h¯2 l s

Determination of 36 matrix elements of the spin-orbit coupling operator ˆl·sˆ = lˆxsˆx + ˆlysˆy + ˆlzsˆz. Replace the x and y components by the step operators (eq. (44,46)):

1 1 1 1 ˆl·sˆ = lˆzsˆz + (ˆl+ + ˆl ) (sˆ+ + sˆ ) + (ˆl+ ˆl ) (sˆ+ sˆ ) 2 − 2 − 2i − − 2i − − 1 = lˆzsˆz + (ˆl+sˆ+ + ˆl sˆ+ + ˆl+sˆ + lˆ sˆ 4 − − − − ˆl+sˆ+ + ˆl sˆ+ + ˆl+sˆ lˆ sˆ ) − − − − 1− − = lˆzsˆz + (ˆl+sˆ + ˆl sˆ+) (59) 2 − − The general matrix element (58) is ˆ· ˆ 1 ˆ ˆ ml ms l sˆ ml0 ms0 = ml ms lzsˆz + (l+sˆ + l sˆ+) ml0 ms0 2 − − ˆ = ml ms lzsˆz ml0 ms0 1 ˆ + 2 ml ms l +sˆ m l0 ms0 − 1 ˆ +2 ml ms l sˆ+ ml0 ms0 . (60) − ˆ (1) General hints to the evaluation of matrix elements m H n :

38 (i) Evaluate Hˆ (1) n . This will result in a constant a multiplied by a wave- function which may or may not be the same as the original. For the present

let us assume Hˆ (1) n = a n . (ii) The result of (i) is then premultiplied by m giving m an .

(iii) Since a is a constan t we have m an = a m n and we are thus left

with the task of evaluating m n . Provided m and n are orthonor-

malised, m n = 1 when m = n but is zero otherwise.

On account of orthonormalised states

0 0 ml ms ml0 ms0 = δml,ml δms,ms , (61) the integral is not zero when m = m and m = m . The wavefunctions l l0 s s0 are eigenfunctions of ˆlz und sˆz, so that the application of the operator products in eq. (60) on the wavefunction on its right-hand side yields:

2 ˆlzsˆz ml ms = ml ms h¯ ml ms

ˆl+sˆ ml ms = − 2 l( l + 1) ml(ml + 1) s(s + 1) ms(ms 1) h¯ ml + 1 ms 1 − − − − ˆl sˆ+ ml ms = − p p 2 l( l + 1) ml(ml 1) s(s + 1) ms(ms + 1) h¯ ml 1 ms + 1 − − − − p 1 p ˆ where s = 2. For diagonal elements only lzsˆz may contribute, whereas for non-diagonal elements only the step operators may account:

ml ms ˆl+sˆ ml 1 ms + 1 ml ms ˆl sˆ+ ml + 1 ms 1 − − − − Matrix elemen ts (58) which may contribute are restricted to the condition

ml + ms = ml0 + ms0 (62) The non-zero matrix elements are:

1 1 1 1 1 ˆl+sˆ 0 0 ˆl+sˆ 1 − 2 − 2 − 2 − − 2 1 ˆ 1 1 ˆ 1 0 2 l sˆ+ 1 2 1 2 l sˆ+ 0 2 . − − − − −

39 (63) m m 1 1 1 1 0 1 0 1 1 1 1 1 l s 2 − 2 2 − 2 − 2 − − 2 1 1 1 ζ 2 2 1 1 1 ζ 1 ζ − 2 −2 2 0 1 1 ζ q0 2 2 0 1 q 0 1 ζ − 2 2 1 1 1 ζ q1 ζ − 2 2 −2 1 1 q 1 ζ − − 2 2

ζ 1 1 1 1 1 1 H11 = 1 ˆlzsˆz 1 = ζ 1 1 1 = ζ h¯2 2 2 · · 2 2 2 2 ζ 1 1 1 1 H22 = 1 ˆlz sˆz 1 = ζ 1 ( ) = ζ h¯2 − 2 − 2 · · −2 −2 ζ 1 1 1 H33 = 0 ˆlz sˆz 0 = ζ 0 = 0 h¯2 2 2 · · 2 ζ 1 1 1 H44 = 0 ˆlzsˆ z 0 = ζ 0 ( ) = 0 h¯2 − 2 − 2 · · −2 ζ 1 1 1 1 H55 = 1 ˆlzsˆz 1 = ζ ( 1) = ζ h¯2 − 2 − 2 · − · 2 −2 ζ 1 1 1 1 H66 = 1 ˆlz sˆz 1 = ζ ( 1) ( ) = ζ h¯2 − − 2 − − 2 · − · −2 2 ζ 1 1 1 1 1 1 1 H23 = 1 lˆ +sˆ 0 = ζ √2 1 1 1 = ζ = H32 h¯2 − 2 2 − 2 2 · · − 2 − 2 2 q ζ 1 1 1 1 1 H45 = 0 lˆ+sˆ 1 = ζ √2 1 = ζ = H54 h¯2 − 2 2 − − 2 2 · · 2 q Diagonalisation of the 2 2 blocks of the H matrix: × 1 ζ E(1) 1 ζ −2 − 2 = ( 1ζ E(1))( E(1)) 1ζ2 = 0 1 q (1) −2 − − − 2 2 ζ E − q (1) 1 (1) E = ζ; E = ζ. (1) 2 (2) − (1) 1 Evaluation of the zeroth-order functions for E(1) = 2 ζ:

0 = 1 ζ 1 ζ u + 1 ζ u −2 − 2 2(1) 2 3(1) q
40 u2(1) 1 1 2 x(1) = = 2; u2(1) = 3; u3(1) = 3 u3(1) q q q ψ = 1 1 1 + 2 0 1 . (64) 2 3 − 2 3 2 (1) q q For E = ζ, the result is: (2) − 0 = 1 ζ + ζ u + 1 ζ u −2 2(2) 2 3(2) u 2(2) √
q 2 1 x(2) = = 2; u2(2) = 3; u3(2) = 3 u3(2) − − q q ψ = 2 1 1 + 1 0 1 . (65) 3 − 3 − 2 3 2 q q Evaluating the second 2 2 blo ck the resulting states are × E(1) = 1 ζ : ψ = 1 1 1 + 2 0 1 (66) (1) 2 4 3 − 2 3 − 2 (1) q q E = ζ : ψ = 2 1 1 1 0 1 . (67) (2) − 5 3 − 2 − 3 − 2 q q The functions are not only eigenfunctions of the operators ˆl·sˆ and sˆ·ˆl but also of ˆl 2 + lˆ·sˆ + sˆ·ˆl + sˆ2 = ˆl + sˆ 2 = ˆ2. (68)

2 If ˆ acts on a quartet state
function ψQ (ψ1, ψ2, ψ4, ψ6) and on a doublet state function ψD (ψ3, ψ5), respectively the result is 2 2 2 ˆ ψQ = ˆl + 2 ˆl·sˆ + sˆ ψQ = h¯2 2 + 2 1 + 3 ψ = h¯2 (15) ψ = h¯2 (3)(5) ψ · 2 4
Q 4 Q 2 2 Q = h¯2 j (j + 1) ψ where j = 3 (69) 1 1 Q
1 2 2 2 2 ˆ ψD = ˆl + 2 ˆl·sˆ + sˆ ψD = h¯2 2 2 1 + 3 ψ = h¯2 (3) ψ = h¯2 (1)(3) ψ − · 4
D 4 D 2 2 D = h¯2 j (j + 1) ψ where j = 1. (70) 2 2 D
2 2

41 Table 13: Functions and energies of the p1 system

(1) ψ ml ms j mj mj = ml + ms j E

1 3 3 3 ψ1 1 2 2 2 2 1 1 2 1 3 1 1 ψ2 1 + 0 3 − 2 3 2 2 2 2 3 1 2 2 ζ ψ q1 1 1 + q2 0 1 3 1 1 4 3 − 2 3 − 2 2 − 2 −2 ψ q 1 1 q 3 3 3 6 − − 2 2 − 2 −2 2 1 1 1 1 1 1 ψ3 1 + 0 − 3 − 2 3 2 2 2 2 1 2 ζ ψ q2 1 1 q1 0 1 1 1 1 − 5 3 − 2 − 3 − 2 2 − 2 −2 q q Example: Energy eigen values and eigenfunctions of 4f 1 (Ce3+) and 4f 13 (Yb3+) On account of j = l s = 3 1 (eqn. (47)) for one-electron and one-hole   2 f systems we have

Ln3+[4f N ] ground multiplet E excited multiplet E Ce3+[4f 1] 2F (j = 5/2) 2ζ 2F (j = 7/2) +3ζ 5/2 − Ce 7/2 2 Ce Yb3+[4f 13] 2F (j = 7/2) 3ζ 2F (j = 5/2) +2ζ 7/2 −2 Yb 5/2 Yb Eigenfunctions are obtained with the help of so-called vector coupling co- efficients (Clebsch-Gordan coefficients) (see Table 14). Table 14: Vector coupling coefficients for systems with j2 = 1/2

j = m = 1 m = 1 2 2 2 −2 1 1 1 j1 + m + 2 j1 m + 2 j1 + 2 − s 2j1 + 1 s 2j1 + 1 1 1 1 j1 m + 2 j1 + m + 2 j1 2 − − −s 2j1 + 1 s 2j1 + 1

42 Table 15: Spin-orbit coupled eigenfunctions of Ce3+ and Yb3+ free ions

3+ 3+ a) (Ce ) (Yb ) j mj ml ms EJ EJ φ ; φ 5 5 = 6 3 1 1 2 1 1 6 2  2 ∓ 7  ∓ 2  7   2 φ ; φ 5 3 = q5 2 1 q2 1 1 2ζ 2ζ 2 5 2  2 ∓ 7  ∓ 2  7   2 − φ ; φ 5 1 = q4 1 1 q3 0 1 3 4 2  2 ∓ 7  ∓ 2  7  2 7 7 q 1 q φ0 ; φ0 = 3 1 8 2  2   2 φ ; φ 7 5 = 1 3 1 + 6 2 1 20 70 2 2 7 2 7 2   ∓   3ζ 3ζ 7 3 q2 1 q5 1 2 2 φ0 ; φ0 = 2 + 1 − 3 6 2  2 7  ∓ 2 7   2 7 1 q3 1 q4 1 φ0 ; φ0 = 1 + 0 4 5 2  2 7  ∓ 2 7  2 a) Short form of the functions:q the first sym bol refersq to the upp er sign, the second to the lower one.

5 5 The calculation of the coefficients is demonstrated for 2 2 (first line in Table 15).

Assignments: j = 5/2, m = m = 5/2, j = l = 3, and m = m = 1/2 j 1 2 s  (j2 = s = 1/2) The roots of the lower row of Table 14 become

1 5 1 1 j1 + m + 2 3 + 2 2 6 m2 = : = − = −2 s 2j1 + 1 s 7 r7 1 5 1 1 j1 m + 2 3 2 + 2 1 m2 = : − = − = 2 −s 2j1 + 1 s 7 −r7 Since the Condon-Shortley standard assignment is j s and j l 1 → 2 → the sign of the coefficients has to be changed according to the phase relation j +j j j j jm = ( 1) a b− j j jm . Finally, we obtain | b a i − | a b i 5 5 = 6 3 1 + 1 2 + 1 . 2 2 − 7 − 2 7 2 q q

43 Problems ˆ 1. Calculate the matrix elements l, ml lq l, ml0 (where q stands for z, +, ) ˆ ˆ h | | ˆi2 ˆ ˆ − (a) 0, 0 lz 0, 0 , (b) 2, 2 l+ 2, 1 , (c) 2, 2 l+ 2, 0 , (d) 2, 0 l+l 2, 0 . h | | i h | | i h | | i h | −| i 2. The 14 microstates m m of an f 1 system (l = 3, s = 1/2) yield under | l si the influence of the spin-orbit coupling operator 14 eigenstates jm | ji which are linear combinations of the microstates. Use Table 14 to eval- uate the vector coupling coefficients for the coupled states 5/2 1/2 , | i 5/2 1/2 , 7/2 3/2 , and 7/2 3/2 . Control your results with | − i | i | − i the entries of Table 15.

44 6 Exchange interactions in dinuclear compounds

6.1 Parametrization of exchange interactions Heitler-London model of H • 2

2 r12 1 ra 2 rb1 rb2 ra1 a ra b b

Fig. 25: H2 model; a and b assign the nuclei, 1 and 2 the electrons Valence bond ansatz: construction of products with orbital configuration φaφb using the four spin orbitals

φaα φaβ φbα φbβ

Product states in consideration of the Pauli principle: D = φ (1)α(1) φ (2)β(2) φ (2)α(2) φ (1)β(1) (71) 1 a b − a b D = φ (1)β(1) φ (2)α(2) φ (2)β(2) φ (1)α(1) (72) 2 a b − a b D = φ (1)α(1) φ (2)α(2) φ (2)α(2) φ (1)α(1) (73) 3 a b − a b D = φ (1)β(1) φ (2)β(2) φ (2)β(2) φ (1)β(1) (74) 4 a b − a b

45 Construction of eigenfunctions of the total spin •

Sˆ0 = Sˆ1 + Sˆ2 (S1 = S2 = 1/2)

2 2 D and D are eigenfunctions S0 M 0 of Sˆ0 = (Sˆ + Sˆ ) and Sˆ0 = 3 4 | S i 1 2 z Sˆ + Sˆ with S0 = 1 and M 0 = 1 and 1 (Spin triplet functions 1 1 z1 z2 S − | i and 1 1 ), respectively. | − i D + D = 1 0 (spin triplet function) 1 2 ⇒ | i D D = 0 0 (spin singlet function) 1 − 2 ⇒ | i Φ = D D Φ = D Φ = D + D Φ = D 1 1 − 2 2 3 3 1 2 4 4 0 0 1 1 1 0 1 1 | i | i | i | − i

φa and φb are normalised:

φa(1)∗φa(1)dτ1 = φb∗(2)φb(2)dτ2 = 1, Z Z but not orthogonal: overlap integral S = φ (1)∗φ (1)dτ = φ (2)∗φ (2)dτ = 0 ab a b 1 a b 2 6 Z Z Normalised functions of the dinuclear unit: Φ = N [φ (1)φ (2) + φ (2)φ (1)] 1 [α(1)β(2) α(2)β(1)] 1 g a b a b 2 − sym q anti | {z } Φ2 | α(1)α(2){z } 1 Φ  = Nu [φa(1)φb(2) φa(2)φb(1)]  [α(1)β(2) + α(2)β(1)] 3 − 2   Φ4  anti  qβ(1)β(2)  | {z }  sym   2 1 2 1 where N = [2 + 2S ]− 2 and N = [2 |2S ]− 2 . {z } g ab u − ab

46 Symmetry of the functions with regard to exchange of electrons • total function: anti Φ (1, 2) = Φ (2, 1) i − i singlet function: orbital sym (g), spin function anti triplet functions: orbital anti (u), spin function sym

Symmetry of the orbital forces a distinct multiplicity of the spin func- tion⇒ on account of the Pauli principle

Evaluation of the energy E(S) and E(T ) of the singlet and triplet • states h¯2 e2 e2 h¯2 e2 e2 e2 Hˆ = 2(1) 2(2) + −2me ∇ − ra1 − rb1 − 2me ∇ − ra2 − rb2 r12 hˆ(1) hˆ(2) | {z } | {z }

1 g ˆ 1 g 2(h + habSab) + Jab + Kab E(S) = Φ1 H Φ1 = 2 (75) h | | i 1 + Sab 2(h h S ) + J K E(T ) = 3Φu Hˆ 3Φu = − ab ab ab − ab (76) h 2| | 2i 1 S2 − ab where

h = φa(i) hˆ(i) φa(i) (one-centre D E = φb(i) hˆ(i) φb(i) one-electron integral)

D E hab = φa(i) hˆ(i) φb(i) (transfer or hopping integral)

D 2 E Coulomb Jab = φa(1)φ b(2) e /r12 φa(1)φb(2) ( integral) 2 Kab = φa(1)φb(2) e /r12 φa(2)φb(1) (Exchange integral).

47 1 g ˆ 1 g Example: Evaluation of E(S) = Φ1 H Φ1 D E 1. Integration over the spin:

1 α(1)β(2) β(1)α(2) α(1)β(2) β(1)α(2) = 2 h − | − i 1 α(1)β(2) α(1)β(2) + α(2)β(1) α(2)β(1) 2 h | i h | i −  1 1 α(2)β(1) α(1)β(2) α(1)β(2) α(2)β(1) = 1. h | | {z i −}h | | {z i } 0 0  2. Integration| over the{zspace: } | {z } E(S) = (77) 2 ˆ ˆ 2 Ng φa(1)φb(2) + φb(1)φa(2) h(1) + h(2) + e /r12 φa(1)φb(2)+

D 2 φb(1)φa(2) = 2N [2(h + habSab) + Jab + Kab]. g E Singlet-triplet splitting:

∆E(T, S) = E(T ) E(S) − 2K 4h S + 2S2 (2h + J ) ≈ − ab − ab ab ab ab

Application of the Heitler-London model to dinuclear complexes • 1 having S1 = S2 = 2 centres

Example: Ln0 Cua– L– CubLn0

As distinguished from the strong covalent bond in H2 the interactions be- tween both magnetically active electrons is weak. small ∆E(T, S). The⇒highest singly occupied antibonding orbitals φa and φb of the fragments Ln0 CuaL and LCubLn0 , respectively take over the role of the 1s orbitals of the H atoms. φa and φb have mainly d character. They are centered at the metal ions and partially delocalised in the direction of the ligands.

48 6.2 Heisenberg operator Phenomenological description of the interaction between the unpaired elec- trons of the centres by an apparent spin-spin coupling, whose magnitude and sign are given by the spin-spin coupling parameter (exchange param- eter) : J Hˆ = 2 Sˆ ·Sˆ where 2 = ∆E(T, S) (78) ex − J 1 2 − J

Hˆex is an effective operator, describing but not explaining the phenomenon.

ˆ 1 g 3 u Application of Hex to Φ1 (S0 = 0) and Φi (S0 = 1, i = 2, 3, 4): 2 Sˆ ·Sˆ 1Φg = 3 1Φg, 2 Sˆ ·Sˆ 3Φu = 1 3Φu − J 1 2 1 2 J 1 − J 1 2 i − 2 J i E(
S) E(T
) ∆E(T, S) =|E{z(T}) E(S) = 2 | {z } (79) ⇒ − − J < 0: singlet ground state (intramolecular antiferromagnetic interaction) J > 0: triplet ground state (intramolecular ferromagnetic interaction) J Hints to the evaluation of E(T ) and E(S):

2 ˆ 2 ˆ ˆ ˆ2 ˆ2 ˆ · ˆ S0 = S1 + S2 = S1 + S2 + 2S1 S2 ˆ · ˆ ˆ 2 ˆ2  ˆ2 2S1 S2 = S0 S1 S2 2 − − = h¯ S0(S0 + 1) S (S + 1) S (S + 1) − 1 1 − 2 2 0 or 2 3 3  4 4  | {z } | {z } | {z } Heisenberg operator for more than two centres:

Hˆ = 2 Sˆ ·Sˆ (80) Hˆ = 2 Sˆ ·Sˆ (81) ex − Jij i j ex − J i j i

(∂En/∂B) exp( En/kBT ) µ¯n exp( En/kBT ) n − n − Mmol = NA = NA − P exp( En/kBT ) P exp( En/kBT ) n − n − P P 49 (82)

Van Vleck equation

ˆ ˆ (0) ˆ (1) Operator: H = H + BzH

(0) (1) 2 (2) En = Wn + BzWn + Bz Wn + . . . .

(1) (2) Wn , Wn : First- and second-orderZeeman coefficients

µ¯ = ∂E /∂B = W (1) 2BW (2) . . . n − n − n − n − (1) 2 (2) (0) [(Wn ) /k T 2Wn ] exp( Wn /k T ) B − − B χ = µ N n (83) mol 0 A P (0) exp( Wn /kBT ) n − P Eq. (83) is valid for applied magnetic fields B 0 →

6.3 Exchange-coupled species in a magnetic field

Hˆ = 2 Sˆ ·Sˆ = 2 Sˆ Sˆ + Sˆ Sˆ + Sˆ Sˆ ex − J 1 2 − J z1 z2 x1 x2 y1 y2 1   = 2 Sˆz1Sˆz2 + Sˆ+1Sˆ 2 + Sˆ 1Sˆ+2 (84) − J 2 − − h  i Basis: spin functions in the form M M where the first M refers to | S S i S electron 1 and the second to electron 2 H-Matrix: (85) M M 1 1 1 1 1 1 1 1 S1 S2 2 2 − 2 2 2 − 2 − 2 − 2 1 1 / 2 2 2 −J 1 1 /2 − 2 2 J −J 1 1 /2 2 − 2 −J J 1 1 /2 − 2 − 2 −J

Evaluation of the diagonal element H11: 2 1 1 Sˆ Sˆ 1 1 = 2 1 1 = /2 − J 2 2 z1 z2 2 2 − J 2 2 −J

50 Evaluation of the off-diagonal element H23:

1 1 1 1 1 2 Sˆ 1Sˆ+2 = (1)(1) = − J − 2 2 2 − 2 − 2 −J −J

Result: Tab. 16: Spin functions and exchange 1 energies of the S1 = S2 = 2 system

Spin function MS0 S0 E

1 1 1 1 1 0 0 3 √2 2 − 2 − − 2 2 2J 1 1
2 2 1 1 1 1 + 1 1 0 1 1 √2 2 − 2 − 2 2 −2J 1 1
1 − 2 − 2 −

51 E 2S +1 S 11 30J 5

9 20J 4

7 12J 3

5 6J 2

3 2J 1 1 0 0

Fig. 26: Relative energies and multiplicities of the spin states 3+ 5 2+ 1 of a dinuclear Fe complex (S = 2); for Cu (S = 2) only the 3+ 7 two lowest levels are relevant, while for Gd (S = 2) the two levels with S0 = 6 (E = 42 ) and S0 = 7 (E = 56 ) have | J | | J | to be added.

52 Magnetic susceptibility of a spin-spin-coupled system with S = S = 1 • 1 2 2 10 10

1 c - MS cm 8 ) 8

1

- E/ 1 b 6

mol

3 6 a

0 m

1 4 7

S= - 4 D 10 2 E(T,S) (

/

-1 l

o

S=0 0 m 2 0 c -2 0 0 1 2 3 4 5 0 5 10 15 20 25 T B0/T /K

Fig. 27: Correlation diagram of a Fig. 28: χmol versus T diagram of 1 1 S1 = S2 = 2 system under the in- a S1 = S2 = 2 exchange-coupled 1 fluence of isotropic intramolecular system with = 2 cm− at ap- 1 J − spin-spin coupling ( = 2 cm− ) plied fields of B = 0.01 T (a), J − 0 and applied field 3.5 T (b), and 5 T (c) Application of the Van Vleck equation (83) to a dinuclear system with 1 S1 = S2 = 2 Zeeman-Operator:

Hˆ = γ g(Sˆ + Sˆ ) B = γ g Sˆ0 B Mz − e z1 z2 z − e z z Hˆ (1) | {z } (86) S0M 0 1 1 1 0 1 1 0 0 S −

1 1 g µBBz 1 0 0

1 1 gµ B − − B z 0 0 0

Matrix elemen ts: ˆ (1) 11 HMz 11 = gµBBz W 11 = gµB h | | i −→ | i ˆ (1) 1 1 HMz 1 1 = gµBBz W 1 1 = gµB h − | | − i − −→ | − i − 53 Zeeman (1) The remaining matrix elements ( coefficients) are zero. W 11 , (1) (0) (0) Van Vleck| i W 1 1 , WS = E(S), and WT = E(T ) are substituted into the | − i equation. After dividing by 2, the Bleaney-Bowers expression (χmol per centre) is obtained, here extended by χ0

2 2 1 N µ g 1 2 − χ = µ A B 1 + exp − J +χ , only applicable to a (Fig. 28)(87) mol 0 3k T 3 k T 0 B   B 

1.0 10 e a 9 b c c )

)

3 8 d 1 0.8

- - e

d m 7

mol 0.6 6

3

mol

m 5

7

7 - 0.4 b 4

10

10 (

( 3

/

/

l

l

o

o 0.2 a 2

1

-

m

m

c c 1 0.0 0 0 100 200 300 400 0 100 200 300 400 a T/K b T/K

Model calculations concerning 1 the system S1 = S2 = 2 with positive and negative ; 2.5 J

e 1 2.0 values [cm− ]: d J c curve a: 50 1.5 − b curve b: 25

f

f − e 1.0 curve c: 0

m curve d: +25 0.5 a curve e: +50 0.0 Fig. 29 a: χmol–T diagram 0 100 200 300 400 1 Fig. 29 b: χ− –T diagram c T/K mol Fig. 29 c: µeff –T diagram

54 Polynuclear unit of n equivalent centres:

E(S0) 2 2 S0 S0(S0 + 1)(2S0 + 1)Ω(S0) exp µ0 NAµBg − kBT χmol = 0 (88) n 3kBT E(S )  P 0 (2S0 + 1)Ω(S0) exp S − kBT P   Evaluation of S0, E(S0) and Ω(S0):

S0 of the coupled states: 1 S0 = nS, nS 1, . . . , 0 (nS integer) or (nS half integer) − 2 Relative energies E(S0): z E(S0) = J [S0(S0 + 1) nS(S + 1)] −n 1 − − z: number of nearest neighbours of a centre n: number of interacting centres

Frequency Ω(S0) of the states S0:

Ω(S0) = ω(S0) ω(S0 + 1) − S0 ω(S0) is the coefficient of X in the expansion

S S 1 S n (X + X − + . . . + X− )

Scope of validity of the Heisenberg model • 1. Localised magnetic moments (no band magnetism) 2. good quantum number S of the centres 3. orbital singlet as ground term 3d5-High-Spin 6A −→ 1 3d3(O ), 3d7(T ) 4A h d −→ 2 3d8(O ), 3d2(T ) 3A h d −→ 2 (3d9(O ) 2E) h −→

55 6.4 Mechanisms of cooperative magnetic effects in insulators

N O N

Cu Cu a

N O N

N O N

Cu Cu b

N O N

Fig. 30: Magnetic orbitals a and b, localised in the left and right 2+ fragment of [L20 Cu2(µ-OH)2] (Kahn,1993)

M

M

X M X M

M X M X a b

Fig. 31: CrCl3, CrBr3: Magnetic orbitals; top: direct exchange dxy–dxy0 ; bottom: 90◦ superexchange

56 Problems ˆ2 ˆ2 ˆ ˆ ˆ 1. Verify the equation S = Sz h¯Sz + S+S . − − 2. Show, that the function eq. (74)

Ψ = D = φ (1)β(1) φ (2)β(2) φ (2)β(2) φ (1)β(1) 4 4 a b − a b ˆ 2 ˆ ˆ 2 ˆ ˆ ˆ is an eigenfunction of S0 = (S1 + S2) and Sz0 = Sz1 + Sz2! How large are S0 and MS0 ? 3. Corresponding to E(S), eq. (77), evaluate the energy E(T ) of the triplet state and verify the result given in eq. (76). 4. Reconstruct the steps eq. (86) eq. (87). −→ 5. The Bleaney-Bowers formula, eq. (87), approaches for high tem- perature the Curie-Weiss law, eq. (9). Give the relation between J and Θp. Is the result in agreement with the right formula in eq. (9)? 6. What magnetic behaviour is obtained, if in the Bleaney-Bowers- Formel, eq. (87), is set to 0? J 7. Determine on the basis of matrix (85) and perturbation theory the correct zeroth-order wavefunctions and verify the entries in Table 16. 8. Write the Heisenberg spin operator for a trinuclear unit of equiva- lent centres (equilateral triangle). What spin operator applies for an isosceles triangle and what operator for a three-membered chain? 9. What general formula evaluates the magnetic susceptibility of an equi- lateral triangle? The derivation of the susceptibility expression for an isosceles triangle needs more expense. What steps have to be consid- ered if perturbation theory is consequently applied? Give the basis 1 functions (spin functions) for a S1 = S2 = S3 = 2 system. What are the resulting S0 states? What operator represents the perturbation by the magnetic field?

1 10. Give a rough drawing of the χmol− –T diagram of a homotrinuclear 1 cluster with equivalent antiferromagnetically coupled S = 2 centres. (Hint: Think what magnetic behaviour is expected for high and low temperature, k T and k T , respectively.) B  |J | B  |J |

57 1 11. Magnetochemical results are often presented as χmol–T , χmol− –T or Curie µeff –T diagrams. What type of diagram is suited for (a) paramagnetism, (b) intramolecular ferromagnetic interactions, (c) in- tramolekular antiferromagnetic interactions, (d) diamagnetic behaviour, (e) TUP behaviour? 12. To reliably characterise magnetic properties measurements at different field strength are essential. What is the reason? 13. A frequent mistake in magnetochemical investigations is the applica- tion of too strong magnetic fields. Why may this be unfavourable? 14. To evaluate the paramagnetic part of the susceptibility of a compound with macrocyclic ligands, the problem may occur that the incremental method for the diamagnetic correction is not as precise as necessary. What is to be done? 15. For a polynuclear complex one observes at high temperature Curie- Weiss behaviour with Θp > 0 and in the low-temperature region field-dependent susceptibilities. What magnetic collective effects can be expected?

16. With decreasing temperature the µeff data of a homodinuclear com- pound increase weakly and then, after passing a maximum at low temperature, steeply drop. What is the reason for this behaviour? What model (susceptibility expression) should be tried to simulate the behaviour? 17. You notice that the paramagnetic properties of a dinuclear centrosym- metric molecular compound is not satisfactorily described with the corresponding eq. (88). What extensions of the model are in principle possible?

III 5 18. The homodinuclear complex [Cp∗RuCl2] (Ru [4d ], low spin, S = 1/2) contains in the unit cell two isomeric forms (ratio 1:1) with dis- tinctly different Ru–Ru separations of 2.93 and 3.75 A.˚ Give a rough

drawing of µeff as a function of T assuming that the Ru centres in 1 the former are antiferromagnetically coupled with 400 cm− J ≤ − while in the latter the centres are coupled ferromagnetically with 1 = 12 cm− . J

58 7 Exchange interactions in chain compounds

Uniform chain

J M J M J M J −−−− i −−−− i+1 −−−− i+2 −−−− N 1 N − Hˆ = 2 Sˆ ·Sˆ γ gB Sˆ , (89) − J i i+1 − e z i,z i=1 i=1 X X

1.0 2.0 0.1 0.1 11 3 10/9 7 5 0.08

9

11/10

0.06 0.06 2

B

l m

o

2 6

m

c 8 4

A 10

Ng J

0 m 0.04

0.02

kBT J S(S+1) 0 0 0 0.5 1.0 1.5 2.0 2.5 Fig. 32: Temperature dependent magnetic susceptibil- ity (reduced units) of an S = 1/2 antiferromagnetically coupled 1 D system (Heisenberg model)

Simulation of the magnetic susceptibility for infinite chains of Cu(II): 2 2 2 NAg µB 0.25 + 0.074975x + 0.075235x χmol = µ0 2 3 (90) kBT 1.0 + 0.9931x + 0.172135x + 0.757825x with x = 2 /(k T ). | J | B Susceptibility equation for classical spins, i. e., spins without space quan- tization N µ2 g2S(S + 1) 1 + u 2 S(S + 1) k T χ = µ A B with u = coth J B .(91) mol 0 3k T 1 u k T −2 S(S + 1) B −  B  J

59 Tab. 17: Compounds with 1 D arrangements of magnetic ex- change-coupled centres

1 1 a) compound S bridge [cm− ] 0[cm− ] J J 1 1 2 Cu(C O ) H O C O − 146 2 4 · 3 2 2 2 4 − 1 2 (C H NH )CuCl (µ-Cl) 50 0.05, 10− 6 11 3 3 2 2 − 1 2 CuGeO O − 63 6; 0.6 3 2 − − b) CsNiCl 1 Cl− 9 3 − a) : ‘intrachain’ parameter; 0: ‘interchain’ parameter. J 0 − J b) / = 7 10 3. |J J | ×

Alternating chain α α J M J M J M J −−−− i −−−− i+1 −−−− i+2 −−−− The Heisenberg operator reads in this case N

Hˆex = 2 Sˆ 2i·Sˆ 2i 1 + αSˆ 2i·Sˆ 2i+1 . (92) − J − i=1 X   For α = 0 the model reduces to homodinuclear systems. For antiferro- magnetic couplings with α = 0 (except α = 1) the magnetic susceptibility 6 decreases to zero for T 0. → Chains with chain links –Cu2+–L–Mn2+– of antiferromagnetically cou- pled Cu(II)–Mn(II) are important as intermediate stage towards sponta- neously magnetised molecular 3 D systems.

8 Exchange interactions in layers and 3 D networks

8.1 Molecular-field model ([5] 330-350) The simplest and most na¨ıve effective field approximation consists in con- sidering only one magnetic atom and replacing its interaction with the remainder of the crystal by an effective field. It is assumed that the ef- fective field, the so-called molecular field, is proportional to the average magnetisation of the compound: 2 n z H = λ M with λ = i iJi . (93) MF MF mol MF µ N µ2 g2 0PA B J 60 λ is positive for ferromagnetic and negative for antiferromagnetic interac- tions. In the presence of an applied field H, the total field acting on the centre is

Heff = H + HMF . (94)

Owing to the molecular field, the paramagnetic behaviour (T > TC(TN )) is modified:

Mmol = χmol0 (H + HMF ) = χmol0 (H + λMmol)

Mmol/H = χmol = χmol0 (1 + λχmol) 1 1 χ− = (χ0 )− λ. (95) mol mol − 1 λ produces a parallel shift of the (χmol0 )− –T curve. If the isolated centre obeys the Curie law, the Curie-Weiss law is obtained:

1 T C χ− = λ = χ = where Θ = λC. (96) mol C − ⇒ mol T Θ p − p λ and Θp have the same sign. Positive and negative Θp values refer to predominating ferro- and antiferromagnetic interactions, respectively. The layer-type compound FeCl2 may serve as an example that is magnetically characterised by dominating ferromagnetic intralayer and weaker antifer- romagnetic interlayer interactions. This does lead to Θp > 0, but an anti- ferromagnetic spin structure is observed below TN . For pure spin systems Θ and the spin-spin coupling parameters are p Ji related by 2S(S + 1) n Θ = z , (97) p 3k iJi B i X th where zi is the number of i nearest neighbours of a given magnetic centre, stands for the exchange interaction between the ith neighbours and n is Ji the number of sets of neighbours for which = 0. Ji 6 The molecular field approximation is applicable to ferromagnetic, an- tiferromagnetic, and ferrimagnetic materials below and above the critical temperature TC(TN ) [5].

61 8.2 High-temperature series expansion (HTSE); ([5] 386 – 415)

-1 c Qp<0

Qp=0

Qp>0

TC Qp T

Fig. 33: Schematic representation (shaded area) of the deviation from Curie-Weiss behaviour

T r Sˆ exp Hˆ β ∂ i zi − χmol = µ0NAgγe , β 1/(kBT ) ∂B  hP Z  i ≡  

Hˆ = 2 Sˆ ·Sˆ gγ B Sˆ  − J i j − e z zi i

(βHˆ )2 ∞ 1 exp( βHˆ ) = 1 (βHˆ ) + . . . = ( βHˆ )k − − 2! − k! − Xk=0 k (2β )k T r Sˆ Sˆ ∞ J Sˆ ·Sˆ  zi zj k! i j  N g2γ2 i j k=0 " igroup R 3), TC = 32.7 K

1.n 2.n 3.n Hˆ = 2 Sˆ ·Sˆ 2 α Sˆ ·Sˆ 2 β Sˆ ·Sˆ ex − J1 i j − J1 k l − J1 m n X 2 X 3 X J J |{z} |{z} t C ∞ 1 r s χmol = 1 + at J with at = at0 r s,r,sα β T k T − − " t=1  B  # r+s t X X≤ Results

10 20

8 1 3 2 15 c 1 - - l 1 m o

6 o l m / 3 10 1 0 m 6 6 80 - 4 m 0 1 - l o 1 60 o l / m l m 3 o 40

m 5 m - 6 3 -

c 2 0 20 1 / l o

m 0 0 50 100 c T / K 0 0 0 100 200 300 400 T / K

3+ −1 Fig. 34: Cr (S = 3/2) partial structure in CrBr3 and χmol and χmol as a function of temperature (applied field: B0 = 0.1 T); the solid lines −1 refer to the HTSE model with g = 1.952, J1 = 6.900 cm , J2 = 0 −1 −1 cm (fixed), J3 = 0.080 cm (quality of the fit: SQ = 0.7 %; SQ = − (F Q/n)1/2 100%, with F Q = n [χ (i) χ (i)]/χ (i) 1/2) × i=1 { obs − cal obs } 1 1 1 Parameter g P [cm− ] [cm− ] [cm− ] J1 J2 J3 d(Cr–Cr) [pm] 364 612 631 susceptibility 1.952 6.90 0 0.60 0.08 0.19 → − → − spin wave analysis 6.85 0 0.115 ≈ −

63 9 Magnetochemical analysis in practice

1. use g as parameter (in addition to ) J 2. if necessary use χ0 (TIP) as parameter:

χmol = χmol0 + χ0 (98)

χmol0 : polynuclear unit 3. Intermolecular cooperative interactions, weak compared to the in- tramolecular interaction: 1 1 = λ (99) χmol χmol0 − λ: molecular field parameter 4. Intermolecular cooperative interactions, competing with the intramolec- ular exchange interaction: application of higher-dimensional models in the framework of high- temperature series expansions (CrBr3) 5. Mononuclear impurity: C χ = (1 ρ)χ0 + ρ (100) mol − mol T Θ − p Use ρ, C, Θ as parameters (in addition to g und ) p J 6. Spinfrustration: Observed in the case of antiferromagnetic interac- tions, if the topology does not allow a satisfactory antiparallel spin- spin coupling for all centres (example: tetranuclear rhombic unit with antiferromagnetic interactions in the direction of the edges as well as in the direction of the short diagonal) 7. Publication of magnetochemical results: indication of the spin operator (definition of ), dia J χmol, ∆T , B0, number of magnetic centres

64 References

[1] T. I. Quickenden, R. C. Marshall, J. Chem. Educ. 1972, 49, 114. [2] B. I. Bleaney, B. Bleaney, Electricity and Magnetism, 3rd ed., Oxford University Press, Oxford, 1994. [3] J. H. Van Vleck, Electric and Magnetic Susceptibilities, Oxford Uni- versity Press, Oxford, 1932. [4] J. S. Griffith, The Theory of Transition-Metal Ions, Cambridge Uni- versity Press, Cambridge, 1971. [5] H. Lueken, Magnetochemie, Teubner, Stuttgart, Leipzig 1999. [6] F. E. Mabbs, D. J. Machin, Magnetism and Transition Metal Com- plexes, Chapman and Hall, London 1973. [7] J. H. Van Vleck, Physica 1973, 69, 177-192. [8] H. L. Schl¨afer, G. Gliemann, Einfuhrung¨ in die Ligandenfeldtheorie, Akad. Verlagsges., Wiesbaden 1980; Engl. Ausgabe: Basic Principles of Ligand Field Theory, Wiley-Interscience, New York 1969. [9] O. Kahn, Molecular Magnetism, Wiley-VCH, New York 1993. [10] P. W. Atkins, R. S. Friedman, Molecular Quantum Mechanics, Oxford University Press, Oxford 1997.

65 Appendix

Tab. 18: Energy conversion factors

J eV s−1 cm−1 K T kJ/mol 1 J 1 6,24151 1,50919 5,03411 7,24292 1,07828 6,02214 1018 1033 1022 1022 1023 1020 × × × × × × 1 eV 1,60218 1 2,41799 8,06554 1,16045 1,72760 9,64853 10−19 1014 103 104 104 101 × × × × × × 1 s−1 6,62607 4,13567 1 3,33564 4,79922 7,14477 3,99031 10−34 10−15 10−11 10−11 10−11 10−13 × × × × × × 1 cm−1 1,98645 1,23984 2,99792 1 1,43877 2,14195 1,19626 10−23 10−4 1010 10−2 × × × × 1 K 1,38066 8,61739 2,08367 6,95039 1 1,48874 8,31451 10−23 10−5 1010 10−1 10−3 × × × × × 1 T 9,27402 5,78839 1,39963 4,66864 6,71710 1 5,58494 10−24 10−5 1010 10−1 10−1 10−3 × × × × × × 1 kJ/mol 1,66054 1,03642 2,50607 8,35933 1,20272 1,79053 1 10−21 10−2 1012 101 102 102 × × × × × ×

66