-
Workshop in MagnetochemistryMolecular Magnetism (DFG-SPP
1137)
Kaiserslautern, 29.09. – 02.10.2003Foundations
Heiko Lueken, Institut für Anorganische Chemie, RWTH Aachen
Contents
1 Magnetic quantities 3
2 Magnetisation and magnetic susceptibility 4
3 Paramagnetism 53.1 Curie law . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 53.2 Free lanthanide ions and
Hund’s rules . . . . . . . . . . . . . . . . . . . . 73.3
Lanthanide ions in octahedral ligand fields . . . . . . . . . . . .
. . . . . . 83.4 3d ions in octahedral ligand fields . . . . . . .
. . . . . . . . . . . . . . . . 113.5 Curie-Weiss law . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 143.6 Exchange
interactions in polynuclear compounds . . . . . . . . . . . . . . .
16Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 16
4 Magnetic ordering 174.1 Ferromagnetism . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Substances and materials . . . . . . . . . . . . . . . . .
. . . . . . . 174.1.2 Hysteresis loop and magnetisation curve . . .
. . . . . . . . . . . . 18
4.2 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 194.3 Ferrimagnetism . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 20
4.3.1 Substances and materials . . . . . . . . . . . . . . . . .
. . . . . . . 204.3.2 Magnetic saturation moment . . . . . . . . .
. . . . . . . . . . . . . 204.3.3 Paramagnetism above the Curie
temperature . . . . . . . . . . . . 22
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 24
5 Theory of free ions 255.1 Foundations of quantum-mechanics . .
. . . . . . . . . . . . . . . . . . . . 255.2 Perturbation theory .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3
One-electron systems . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 295.4 Angular momentum . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 335.5 Spin . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 355.6
Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 36Problems . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 44
1
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6 Exchange interactions in dinuclear compounds 456.1
Parametrization of exchange interactions . . . . . . . . . . . . .
. . . . . . 456.2 Heisenberg operator . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 496.3 Exchange-coupled species in a
magnetic field . . . . . . . . . . . . . . . . . 506.4 Mechanisms
of cooperative magnetic effects in insulators . . . . . . . . . .
56Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 57
7 Exchange interactions in chain compounds 59
8 Exchange interactions in layers and 3 D networks 608.1
Molecular-field model . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 608.2 High-temperature series expansion . . . . . . .
. . . . . . . . . . . . . . . . 62
9 Magnetochemical analysis in practice 64
References 65
Appendix 66
2
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1 Magnetic quantities
• The legal SI units are not generally accepted. The CGS/emu
system is still widelyused in magnetochemistry. Therefore, use
magnetic quantities which are indepen-dent of the two systems, e.
g., µeff or µ
2eff instead of χmolT
• do not mix the systems, e. g., use G (gauss) instead of T
(tesla) in the CGS/emusystem
• use B0 = µ0H in graphical representations (conversion factor
10−4 T/Oe)
Table 1: Definitions, units and conversion factors [1, 2]
quantity SI CGS/emu
permeabilityµ0
of a vacuum4π/107 Vs/(Am) 1
magnetic flux T = Vs/m2 G = (erg/cm3)1/2B
density 1T =̂ 104 G
magnetic field A/m OeHa
strength 1A/m =̂ (4π/103)Oe
B = µ0(H + M) B = H + 4πM
M magnetisation A/m G
1A/m =̂ 10−3 G
m = MV m = MV
mmagnetic di-
Am2 Gcm3pole moment
1Am2 =̂ 103 Gcm3
eh̄/(2me) eh̄/(2me)µB Bohr magneton9.27402 × 10−24 Am2 =̂
9.27402 × 10−21 G cm3
magnetic di- σ = M/ρ b σ = M/ρ
σ pole moment Am2/kg G cm3/g
per unit massc 1Am2/kg =̂ 1G cm3/g
Mmol = MMr/ρd Mmol = MMr/ρ
Mmolmolar
Am2/mol G cm3/molmagnetisatione
1Am2/mol =̂ 103 Gcm3/mol
atomic magnetic µsm/µB = Msmol/(NAµB)
f µsm/µB = Msmol/(NAµB)µsm
saturation moment µB µB
aIf B0 = µ0H is used instead of H , e. g., in graphs, the
conversion factor is 10−4 T/Oe.
bρ specific density.cSpecific magnetisation; σs specific
saturation magnetisation.dMr molar mass.eMsmol molar saturation
magnetisation.fNA Avogadro constant.
3
-
Table 1: Definitions, units and conversion factors [1, 2]
(cont.)
quantity SI CGS/emu
M = χH M = χH
χmagnetic volume
1 1susceptibility
1 =̂ 1/(4π)
χg = χ/ρ χg = χ/ρ
χgmagnetic mass
m3/kg cm3/gsusceptibility
1m3/kg =̂ 103/(4π) cm3/g
χmol = χMr/ρ χmol = χMr/ρ
χmolmagnetic molar
m3/mol cm3/molsusceptibility
1m3/mol =̂ 4π/106 cm3/mol
[3kB/(µ0NAµ2B)]
1/2[χmolT ]1/2a [3kB/(NAµ
2B)]
1/2[χmolT ]1/2
µeffeffective Bohr mag-
1 1neton number [3]
1 =̂ 1
akB Boltzmann constant.
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 atomsvacuum 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 ofelectron orbits about the applied
magnetic field (shown by all substances);(ii) usually independent
of both T and B for purely diamagnetic materials (closed
shellsystems);(iii) allowed for as diamagnetic correction in the
evaluation of experimental susceptibilitydata of open shell
systems.
4
-
Paramagnetism
(i) χ > 0 and M > 0 owing to net spins and orbital angular
momentum polarized indirection 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 appliedfields, but inverse dependency at strong fields
(paramagnetic saturation, see Figure 1)where B(α) = Mmol/M
∞
mol with M∞
mol = NA gJ J µB;temperature independent paramagnetism: weak
forms of paramagnetism, called Pauliparamagnetism (of conduction
electrons observed in metals) and Van Vleck paramag-netism (TIP,
second order effect involving mixing with the first excited
multiplet by theapplied field; example: Eu3+).
1.0
0.8
0.4
0.2
1 2 3 4 5 60
B
B a( )
B
kBT
m=a
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: kBT � µB paramagnetic saturation: kBT � µB
3 Paramagnetism
3.1 Curie law
χmol =MmolH
=C
Twhere C = µ0
NAµ2
3kB(4)
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
0 Ta
C/c
mol
0 Tb
C/
cm
ol
T
1
0
mef
fef
f/
(0)
m
c
Curie behaviour (in reduced magneticquantities χmol/C and
µeff/µeff(0)(C and µeff(0) refer to the free ion)
Fig. 2 a: variation χmol vs. TFig. 2 b: variation χ−1mol vs.
TFig. 2 c: variation µeff vs. T
µ = g√S(S + 1)µB spin-only formula (5)
Gd2(SO4)3 · 8 H2O and (NH4)2Mn(SO4)2 · 6 H2O withGd3+ [4f 7]
(ground state 8S7/2, S = 7/2, µ = 7.94µB)Mn2+ [3d5] (ground state
6A1, S = 5/2, µ = 5.91µB)
6
-
3.2 Free lanthanide ions and Hund’s rules
µ = gJ√J(J + 1)µB except 4f
4, 4f 5, 4f6 systems (6)
with Landé factor
gJ = 1 +J(J + 1) + S(S + 1) − L(L+ 1)
2J(J + 1)(7)
S, L, J correspond to the total spin angular momentum, the total
orbital
angular momentum and the total angular momentum, respectively,
of theground state.
The 4f electrons of free Ln ions are influenced by
interelectronic repulsionHee (splitting energy 10
4 cm−1) and spin-orbit coupling HSO (103 cm−1),i. e., Hee >
HSO. To determine the free ion ground state use the followingscheme
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
ms +12
+12
+12
+12
+12
+12
+12
−12
−12
−12
−12
−12
−12
−12
ml +3 +2 +1 0 −1 −2 −3 +3 +2 +1 0 −1 −2 −3
1. The term with maximum S (or multiplicity 2S + 1) lies lowest
inenergy (
∑ims,i = MS → S).
2. For a given multiplicity, the term with the highest value of
L lieslowest in energy (
∑iml,i = ML → L).
3. For atoms with less than half-filled shells, the level with
the lowestvalue 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:
Pr3+ [4f 2]: S = 1, L = 5, J = 4; gJ =45, µ = 3.58µB (
3H4).Dy3+ [4f 9]: S = 5/2, L = 5, J = 15/2; gJ =
43, µ = 10.65µB (
6H15/2).
7
-
Table 2: Lanthanide ions: term symbol (ground state),
one-electronspin-orbit coupling parameter ζ4f [cm
−1], gJ , gJJ , gJ [J(J+1)]1/2 andµexpeff(295 K)
Ln3+ 4f N 2S+1LJ ζ4fa) gJ gJJ gJ [J(J + 1)]
1/2 µexpeff
La3+b) 4f 0 1S0 0
Ce3+ 4f 1 2F5/2 625 6/7 15/7 2.535 2.3–2.5
Pr3+ 4f 2 3H4 758 4/5 16/5 3.578 3.4–3.6
Nd3+ 4f 3 4I9/2 884 8/11 36/11 3.618 3.4–3.5
Pm3+ 4f 4 5I4 1000 3/5 12/5 2.683 2.9c)
Sm3+ 4f 5 6H5/2 1157 2/7 5/7 0.845 1.6
Eu3+ 4f 6 7F0 1326 0 0 0 3.5
Gd3+ 4f 7 8S7/2 1450 2 7 7.937 7.8–7.9
Tb3+ 4f 8 7F6 1709 3/2 9 9.721 9.7–9.8
Dy3+ 4f 9 6H15/2 1932 4/3 10 10.646 10.2–10.6
Ho3+ 4f 10 5I8 2141 5/4 10 10.607 10.3–10.5
Er3+ 4f 11 4I15/2 2369 6/5 9 9.581 9.4–9.5
Tm3+ 4f 12 3H6 2628 7/6 7 7.561 7.5
Yb3+ 4f 13 2F7/2 2870 8/7 4 4.536 4.5
Lu3+b) 4f 14 1S0 0
a) The relation between ζ4f and λLS of the Russell-Saunders
ground term is given byλLS = ±(ζ4f/2S), where (+) and (−) sign
correspond to N ≤ 2l+1 and N ≥ 2l+1 respectively.
b) diamagneticc) 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
4f electrons. The ligand field effect produces splittings of HLF
≈ 102 cm−1leading generally to temperature dependent µ and
described by the effective
8
-
Bohr magneton number µeff :
χmol = µ0NAµ
2effµ
2B
3kBTwhere µeff =
(3kBTχmolµ0NAµ2B
)1/2= 797.7(Tχmol)
1/2(8)
As a rule, µeff approaches the free Ln ion value for T above 200
K (seeTable 2, Figures 3 – 7).
0 100 200 300 4000
100
200
3 00
400
500
T/K
0.0
0.4
0.8
1.2
1.6
2.0
2.4
meff
cm
ol
105
-1m
ol
/m
-3
a
b
c
d
e
f
Fig. 3: Ce3+ in cubic ligand fields; χ−1mol–T - (b,d,f) and
µeff–T diagrams(a,c,e); ∆ = 605 cm−1 (e,f; oct.), ∆ = −605 cm−1
(c,d; tet); straight lines(a,b) refer to the free ion.
9
-
0
40
80
120
160
200
0 100 200 300 400T/K
U5+
Yb3+
3+Ce
c-1
mol
/10
6m
ol
m-3
()
Fig. 4: Variation χ−1mol vs. T of Ce3+, Yb3+,
and U5+ in octahedral coordination.
0
1
2
3
4
0100 200 300 400
T/K
Ce3+
Yb3+
U5+
meff
Fig. 5: Variation µeff vs. T of Ce3+, Yb3+,
and U5+ in octahedral coordination.
0
40
80
120
160
200
240
100 200 300 4000.0
1.0
2.0
3.0
4.0
-15
-3/10
mol m
mol
c
meff
/KT
0 0 100 200 300 400
0
50
100
150
200
0.00.40.81.21.62.02.42.83.23.6
meff
mol
c10
/-1
5-3
mm
ol
T/K
Fig. 6: χ−1mol–T and µeff–T diagrams for Pr3+ (left) and Nd3+
(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
-
0 100 200 300 4000.0
1.0
2.0
3.0
4.0
T/K
mef
f
dc
ba
0 100 200 300 4000
1
2
3
4
5
6
7
8
T/K
cm
ol
-1/10-
7m
mol
-3
a
b
c
d
Fig. 7: µeff–T (left) and χ−1mol–T diagrams (right) for Sm
3+ and Eu3+, cal-
culated with the spectroscopically determined data of Cs2NaYCl6
: Sm3+
(a) and Cs2NaEuCl6 (d), respectively; free Sm3+ ion (b), free
Eu3+ ion (c).
3.4 3d ions in octahedral ligand fields [4, 8, 6]
Tab. 3: Overview on the estimated magnetic behaviour of 3d
systems (octahedralligand field)
system energetic order N a ground state χmol(T )b
5(hs) 6A1 χmol = C/T
1, 2, 6(hs), 7(hs) T χmol = f(T )
3dNHee ≈ HLF > HSO
4(hs), 9 E χmol = C′/T + χ0
3, 8 A2 χmol = C′/T + χ0
HLF > Hee > HSO 6(ls)1A1 χmol = χ0
ahs and ls assign high-spin and low-spin configuration,
respectively.bχmol = f(T ) stands for complicated variation χmol
vs. T (see Figure 8); C is the Curie constant, C
′ is aconstant which deviates more or less from C caused by HLF
and HSO , and χ0 is a temperature independentparameter.
11
-
Tab. 4: Ions with 3dN high-spin configuration: term symbols
(ground state), one-electron spin-orbit coupling parameter
ζ3d[cm−1] [4], S, 2 [S(S + 1)]1/2 and µexpeff (295 K).
Ion 3dN 2S+1LJ ζ3d S 2[S(S + 1)]1/2 µexpeff
Ti3+ 3d1 2D3/2 154 1/2 1.73 1.65 – 1.79
V3+ 3d2 3F2 209 1 2,83 2.75 – 2.85
V2+ 3d3 4F3/2 167 3/2 3.87 3.80 – 3.90
Cr3+ 3d3 4F3/2 273 3/2 3.87 3.70 – 3.90
Cr2+ 3d4 5D0 230 2 4.90 4.75 – 4.90
Mn3+ 3d4 5D0 352 2 4.90 4.90 – 5.00
Mn2+ 3d5 6S5/2 347 5/2 5.92 5.65 – 6.10
Fe3+ 3d5 6S5/2 (460) 5/2 5.92 5.70 – 6.00
Fe2+ 3d6 5D4 410 2 4.90 5.10 – 5.70
Co3+ 3d6 5D4 (580) 2 4.90 5.30
Co2+ 3d7 4F9/2 533 3/2 3.87 4.30 – 5.20
Ni3+ 3d7 4F9/2 (715) 3/2 3.87
Ni2+ 3d8 3F4 649 1 2.83 2.80 – 3.50
Cu2+ 3d9 2D5/2 829 1/2 1.73 1.70 – 2.20
12
-
Tab. 5: Ions with 3dN low-spin configuration: no. of unpaired
elec-
trons N ′, S, 2 [S(S + 1)]1/2 and µexpeff (295 K).
Ion 3dN structure N ′ S 2[S(S + 1)]1/2 µexpeff
Cr2+ oct.(dist.)a) 3.20 – 3.30
Mn3+3d4
oct.(dist.)2 1 2.83
3.18
Mn2+ oct.(dist.) 1.80 – 2.10
Fe3+3d5
oct.(dist.)1 1/2 1.73
2.0 – 2.5
Fe2+ oct. 0
Co3+3d6
oct.0 0 0
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.
3d1: Example for temperature dependent µeff
µeff ∼√T for T ≤ 100 K
χ−1mol ≈ linear increase for T ≥ 150 K
Fig. 8: Variation of µeff vs. T and χ−1
mol vs. T plots of
a 3d1 ion in octahedral (——) and orthorhombic (– – –)
surrounding.
13
-
3.5 Curie-Weiss law
The Curie-Weiss law
χmol =C
T − Θpwith Θp =
2S(S + 1)
3kB
∑
i
ziJi (9)
• 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
(seeFigure 8) and f systems (except Gd3+, Eu2+)
– only applicable to pure spin paramagnetism with C
correspondingwith the permanent magnetic moment µ and Θp measuring
the
total exchange interactions of a magnetically active centre
withall its magnetic neighbours (nearest, next-nearest, etc.)
300200-100 0 100
T/K
25
75
50
d
c
a
b
mol
10
mol m
c-1
/(
)
-35
100
Fig. 9: χ−1 as a function of T for Curie law (a) and
Curie-Weissbehaviour (schematic); (b): EuO, Θp = 74.2K, TC = 69K;
(c):MnF2, Θp = −113K, TN = 74K; (d): Na2NiFeF7, Θp = −50K,TC =
88K)
14
-
0
0
=0
0
0p
0p
0p
Ta
C/c
mol
0
0p
0p
0p
0
0
=0
T
C/
b
cm
ol
0p
0p
0p 0
0
=0
T
1
0
mef
fef
f/
(0)
m
c
Curie behaviour (bold lines) and
Curie-Weiss behaviour in reducedmagnetic quantities χmol/C
and
µeff/µeff(0) (C and µeff(0) refer tothe free ion)
Fig. 10 a: variation χmol vs. TFig. 10 b: variation χ−1mol vs.
TFig. 10 c: variation µeff vs. T
15
-
3.6 Exchange interactions in polynuclear compounds [9]
Example: Dinuclear complex [Cu(CH3COO)2(H2O)]2
oxygen(water)
copper
acetate
S =1/21 S =1/22DE = -2J
S =0
S =1
1,0
0,8
0,6
0,4
0,2
0 100 200 300
T/K
cm
ol/
10
-8m
3m
ol-
1Fig. 11: [Cu(CH3COO)2(H2O)]2; left: molecular structure; right:
χmol versus T diagramwith J = −148 cm−1 (Ĥex = −2J Ŝ1 · Ŝ2), g =
2.16, χ0 = 76 × 10−11 m3 mol−1. Toexplain J = −148 cm−1 both the
direct exchange interactions between the dx2−y2 orbitalsand the
superexchange via the doubly occupied orbitals as well as
unoccupied orbitals ofthe bridging ligands have to be accounted
for.
χmol = µ0NAµ
2Bg
2
3kBT
[1 +
1
3exp
(−2JkBT
)]−1
+ χ0, see section 6.3, page 54
Problems
1. How are the quantities (i) m, (ii) µeff , (iii) µ, (iv) µsm
defined?
2. Calculate the spin contributions to the molar susceptibility
of hydrogen atoms at298K (SI units).µ0NAµ
2B/(3kB) = 1.57141 × 10−6m3 K mol−1
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 bea paramagnet obeying the Curie-Weiss law χmol =
C/(T − Θp), permitting areasonable interpretation C and Θp? Give
examples!
5. Write the ground state term symbols 2S+1LJ 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,(e) 4D? How many states (distinguished by
the quantum number MJ) belong toeach level?
16
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4 Magnetic ordering
4.1 Ferromagnetism ⇑⇑⇑⇑4.1.1 Substances and materials
Table 6: Ferromagnetic elements and compounds
substance structure TC/K Θp/K µsm/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 210−385CrBr3 BiI3 32.7 54 3.0
Tab. 7: Ferromagnetic materials: properties and application. TC
[K], µsm
[µB] per formula unit, (BH)max [kJm−3] (mr: magnetic data
recording;
sm: softmagnetic material; hm: hardmagnetic material)
material structure TC µsm (BH)max
a) 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
H
,J
Br
Hc
Hc
M
(H+M)=
m0
m0
S
B
M
p
B
B
Fig. 12: Hysteresis loop of a ferromagnet; B as afunction of H
(—) and µ0M as a function of H (· · ·),µ0M = Jp, Jp: magnetic
polarisation)
0
200
100
200100 300
T/K
s
s-
/(J
T1kg
)- 1
Fig. 13: σs as a function of T of gadolinium (σs =M s/ρ)
18
-
a b c
H H
Fig. 14: Schematic representation of domains in a ferromagnetic
monocrystal: (a) un-magnetised; (b) magnetisation through movement
of domain boundary walls; domainsoriented parallel to H grow at the
expense of antiparallel domains; (c) magnetisationby rotation of
the magnetisation vector of whole domains. The domains remain
orientedalong a preferred direction; stronger fields are required
to swing the magnetisation vectorstowards the applied field
[2].
4.2 Antiferromagnetism ⇑⇓⇑⇓
Tab. 8: Antiferromagnetic compounds
substance structure TN/K Θp/K µsm/µB
MnO NaCl 118 −610 5.0FeO NaCl 185 −570 3.3CoO 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
(rutiletype); A, B: Mn2+ sublattices
19
-
4.3 Ferrimagnetism ⇑↓⇑↓4.3.1 Substances and materials
Tab. 9: Ferrimagnetic compounds and materials
substance/material structure TC/K Θp/K µsm/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
µsm = MsMr/(ρNA) = M
smol/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 onthe two sublattices; (c)
two equal moments and one unequal [2].
Example Fe3O4 =̂ FeIII[FeIIFeIII]O4 (inverse spinel):
µsm = gSµB with g = 2 (pure spin magnetism) of the iron ions
occupyingtetrahedral (A) and octahedral holes (B):
Fe3+(A): µsm = 5µBFe3+(B): µsm = 5µBFe2+(B): µsm = 4µB
20
-
Resultant moment µsm(Fe3O4):
µsm(B) − µsm(A) = (5 + 4)µB − 5µB = 4µB per formula unit(exp.:
4.1µB)
Néel’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 thespins within each sublattice parallel, despite their
mutual antiferromagnetic
interaction. This is supported by the fact, that ZnFe2O4, which
has thenormal structure, has no net saturation moment.
Examples: Mixed ferrites made of MII[FeIII]O4 and
FeIII[MIIFeIII]O4:
Mixing two ferrites, three cases can be distinguished: 1.) Both
ferritesare 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 inverseMIIi (amount x, µ
sm(Mi)) and M
IIj (amount 1 − x, µsm(Mj)) on site B:
µsm = xµsm(Mi) + (1 − x)µsm(Mj)
Example: Fe[NixMn1−xFe]O4:Fe[NiFe]O4: µ
sm ≈ 2.3µB, TC ≈ 900 K
Fe[MnFe]O4: µsm ≈ 4.7µB, TC ≈ 580 K
linear variation µsm vs. x (and also TC vs. x)
2. Normal and inverse base ferrite
Provided MIIi is diamagnetic (µsm(Mi) = 0), M
IIi [Fe
III2 ]O4. Substituting
MIIj in FeIII[MIIj Fe
III]O4 by MIIi , the latter occupies site A. In return
for it, FeIII changes from A to B. The net moment of the mixed
ferriteamounts to
µsm = (1 − x)µsm(Mj) + 2xµsm(FeIII)
Since always µsm(Mj) < µsm(Fe
III), the net moment increases with fur-
ther incorporation of MIIi . For x = 1, however, µsm must be
zero, be-
cause MiFe2O4 is a normal ferrite, which is antiferromagnetic.
Thus,
µsm passes through a maximum, reflecting the increasing
antiferromag-netic coupling B–B and the decreasing A–B interaction
according tothe increasing Mi amount.
21
-
4.3.3 Paramagnetism above the Curie temperature TC [5]
In the ferrite MFe2O4 the MII ions are considered as diamagnetic
and the
magnetically active Fe3+ ions are distributed over the A and B
sites. M smol,Aand M smol,B assign the magnetisation per mole of
the Fe
3+ ions, if they
occupy the A and B sites, respectively. In general, M smol,A and
Msmol,B
differ on account of the different ligand field effects acting
on the iron ions.The net magnetisation (T < TC) of the ferrite
is
M smol = xMsmol,A + yM
smol,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
χmol=
T
C+
1
χ0− σT − Θ with (11)
χ−10 = n(2xy − x2α − y2β),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,
whereasnα and nβ refer to the interaction in the sublattices A and
B, respectively.
The variation χ−1mol vs. T (Gl. (11)) is not linear in contrast
to the Curie-Weiss law. Instead, a hyperbola with asymptotes
χ−1mol = T/C + χ−10 (12) and T = Θ (13)
is obtained (see Figure 17).
T T0p C
1 1c
molc
mol c= T
C+ 1
0
Q
0
1c
T Q=
Fig. 17: χ−1mol vs. T of a simple ferrimagnet
22
-
Remarks to eq. (11):
• The characteristic feature of a ferrimagnetic material is the
hyperbolicvariation χ−1mol vs. T (in contrast to ferromagnetic
materials where themolecular 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
Θp =−C/χ0 is called asymptotic Curie point. It corresponds with
theparamagnetic Curie temperature of an antiferromagnet in the
caseof x = y = 1/2 and α = β.
• The intersection of the hyperbola eq. (11) with the T axis is
of greatimportance:
TC =nC
2
{xα+ yβ + [(xα− yβ)2 + 4xy]1/2
}
TC > 0 means, that at T = TC the susceptibility diverges (χ →
∞),i. e., for T < TC spontaneous magnetisation exists. In the
case of TC <
0, the substance behaves paramagnetically in the whole
temperaturerange.
• When α > 0 and β > 0 ferrimagnetism exists. If the
absolute valuesof 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 arequenched, 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
magneticstructure 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 <40 K. Neutron diffraction yields the magnetic
saturation moment µsm =
3µB for the magnetically active centres. What are the
magneticallyactive 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,so long as pure spin contributions are considered? What is
the µsmvalue per formula unit determined, e. g., by SQUID
magnetometry?
5. EuO (NaCl type) is paramagnetic at T > 70 K and obeying
theCurie-Weiss law. How large is the expected paramagnetic
moment
µ, deduced from the slope of the Curie-Weiss straight line?
Whattype 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-spincompounds above the magnetic ordering temperature TC
(Curie tem-perature) and TN (Néel temperature), respectively? What
informa-
tion can be gained from a corresponding measurement with regard
tothe 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 satisfythe commutation relation
q̂p̂q − p̂qq̂ = [q̂, p̂q] = ih̄ (q = x, y, z; i =√−1) (14)
Example 5.1 Application of the commutation relation
x̂ = x · and p̂x =h̄
i
d
dx
x̂p̂xΨ = xh̄
i
dΨ
dx
p̂xx̂Ψ =h̄
i
d(xΨ)
dx=h̄
i
(Ψ + x
dΨ
dx
)
(x̂p̂x − p̂xx̂)Ψ = −h̄
iΨ = ih̄Ψ
5.2 Perturbation theory ([5]83–91)
1. Non-degenerate states
Ĥ(0)Ψ(0)n = E(0)n Ψ
(0)n unperturbed system (15)
Hamilton operator of the perturbed system:
Ĥ = Ĥ(0) + λĤ(1)
Schrödinger equation of the perturbed system:
ĤΨn = EnΨn; find En,Ψn (16)
Series expansion of Ψn und En:
25
-
Tab. 10: Classical and quantum-mechanical forms of Ekin and
Epot
QuantityDa) classical quantum-mechanical
Ekin 1mev
2x
2=
p2x2me
p̂2x2me
=1
2me
(h̄
i
d
dx
)2= − h̄
2
2me
d2
dx2
3 p2
2me− h̄
2
2me
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)= − h̄
2
2me∇2 b)
Epotc) 1 −eV (x) −eV̂ (x) = −eV (x)·
3 −eV (r) −eV̂ (r) = −eV (r)·a) Dimension.b) ∇ is the Nabla
operator.c) Valid for one electron with charge −e in the potential
V .
Ψn = Ψ(0)n + λΨ
(1)n + λ
2Ψ(2)n + . . . (17)
En = E(0)n + λE
(1)n + λ
2E(2)n + . . . . (18)
Insert the series into eq. (16):
(Ĥ(0) + λĤ(1))(Ψ(0)n + λΨ(1)n + λ
2Ψ(2)n + . . .) =
(E(0)n + λE(1)n + λ
2E(2)n + . . .)(Ψ(0)n + λΨ
(1)n + λ
2Ψ(2)n + . . .),
Ordering of the terms with regard to powers of λ:
Ĥ(0)Ψ(0)n + λ(Ĥ(1)Ψ(0)n + Ĥ
(0)Ψ(1)n)
+ λ2(Ĥ(1)Ψ(1)n + Ĥ
(0)Ψ(2)n)
+ . . . =
E(0)n Ψ(0)n + λ
(E(1)n Ψ
(0)n + E
(0)n Ψ
(1)n
)+
λ2(E(2)n Ψ
(0)n + E
(1)n Ψ
(1)n +E
(0)n Ψ
(2)n
)+ . . . .
λ0
λ1
λ2
...
Ĥ(0)Ψ(0)n = E(0)n Ψ
(0)n (19)
(Ĥ(0) − E(0)n )Ψ(1)n = (E(1)n − Ĥ(1))Ψ(0)n (20)(Ĥ(0) − E(0)n
)Ψ(2)n = E(2)n Ψ(0)n + (E(1)n − Ĥ(1))Ψ(1)n (21)
...
26
-
The first-order correction to the energy E(1)n
(premultiply both sides of eq. (20) with Ψ(0)∗n and
integrate)
∫Ψ(0)∗n Ĥ
(0)Ψ(1)n dτ −E(0)n∫
Ψ(0)∗n Ψ(1)n dτ
︸ ︷︷ ︸0
= E(1)n −∫
Ψ(0)∗n Ĥ(1)Ψ(0)n dτ
E(1)n =〈n|Ĥ(1)|n
〉(22)
The first-order correction to the wavefunction:
Ψ(1)n = −∑
m6=n
〈m|Ĥ(1)|n
〉
E(0)m −E(0)n
Ψ(0)m (23)
The second-order correction to the energy:
E(2)n = −∑
m6=n
∣∣〈m|Ĥ(1)|n〉∣∣2
E(0)m −E(0)n
(see Fig. 18) (24)
Y2(0)
Y1(0)
2(0)
E
1(0)
E
1(0)
E 11H
Y1(0)
Y2
(0)
1(0)
E2(0)
E
21H
(a) (b) (c)
E
1(0)
E 11H12H
1(0)
E2(0)
E
21H
Y2(0)
Y1
(0)
2(0)
E1(0)
E
12H
2(0)
E 22H
12H21H
2(0)
E 22H
2(0)
E1(0)
E
Fig. 18: Illustration of the possible effects of a perturbation
ontwo 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
Ĥ(0)Ψ(0)n,i = E
(0)n Ψ
(0)n,i (i = 1, 2) (25)
Ψ(0)n = u1Ψ(0)n,1 + u2Ψ
(0)n,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:
(Ĥ(0) −E(0)n
)Ψ(1)n =
(E(1)n − Ĥ(1)
)(u1Ψ
(0)n,1 + u2Ψ
(0)n,2
)(27)
Multiply with Ψ(0)∗n,1 and Ψ
(0)∗n,2 , respectively and integrate:
u1(H11 − E(1)n
)+ u2H12 = 0
u1H21 + u2(H22 −E(1)n
)= 0
where Hij =
∫Ψ
(0)∗n,i Ĥ
(1) Ψ(0)n,j dτ
Under the condition that the determinant of the coefficients of
u1 and u2disappears the non-trivial solutions of this pair of
equations are obtained:∣∣∣∣∣H11 − E(1)n H12
H21 H22 − E(1)n
∣∣∣∣∣ = 0 (28)
E(1)n(1,2) =
(H11 +H22
)/2 ±
√(H11 −H22)2/4 + |H12|2
u1(1,2)(H11 − E(1)n(1,2)) + u2(1,2)H12 = 0; x(1,2)
=u1(1,2)u2(1,2)
= − H12H11 − E(1)n(1,2)
Normalisation:
x2(1,2)u22(1,2) + u
22(1,2) = 1 ⇒
u2(1,2) =1√
x2(1,2) + 1
u1(1,2) = x(1,2)u2(1,2) =x(1,2)√x2(1,2) + 1
28
-
Correct zeroth-order wavefunction for the energy E(1)n(1,2):
Ψ(0)n(1,2) = u1(1,2)Ψ
(0)n,1 + u2(1,2)Ψ
(0)n,2
Y2
2
(0)
(0)E
Y2
2
(0)
(0)E
1(0)
E
E
1(0)
1(2)(1)
EE
2(0)
22HE
1(0)
1(1)(1)
EE
(a) (a )´ (b) (c)
Y(0)
1(2)= Y(0)
Y(0)1,1u1(2) 1,2u2(2)
Y(0)
1(1)= Y(0)1,2Y
(0)1,1u1(1) u2(1)Y
(0)1,2Y
(0)1,1 ,
1(0)
E
2(0)
22HE1(1)2 1(2)2H H21(1) 21(2)H H
2(0)
E1(0)
E
1(0)
1(2)(1)
EE1(2)2H21(2)H
1(0)
E2(0)
E
1(0)
1(1)(1)
EE1(1)2H21(1)H
1(0)
E2(0)
E
Fig. 19: Illustration of the possible effects of a perturbation
on
a doubly degenerate ground state and a non-degenerate
excitedstate; (a) 0th, (a’) correct 0th, (b) 1st,(c) 2nd order
5.3 One-electron systems ([5]91–96)
Schrödinger equation (spin ignored):
[− h̄
2
2me∇2 − eV̂ (r)
]ψ(r) = Eψ(r). (29)
For convenience the eigenfunctions (atomic orbitals) are given
in spherical
polar coordinates:
ψ(r) = ψn,l,ml(r, θ, φ) = Rn,l(r)︸ ︷︷ ︸radial f.
Y lml(θ, φ)︸ ︷︷ ︸angular f.
= Rn,l(r) Θlml(θ)
√1
2πeimlφ
The functions Y lml(θ, φ) are the (in general, complex)
spherical harmon-ics specified by the quantum numbers l and ml (see
Table 11). They playa predominant role in magnetism.
29
-
z
y
x
P( )x,y,z
r cos
r sin
r sin sin
r sin cos
r
q
q
q
q
q
f
ff
.
.
.
..
.
x = r · sin θ · cosφy = r · sin θ · sinφz = r · cos θ (30)
r2 = x2 + y2 + z2
cos θ = z/r
tanφ = y/x (31)
Fig. 20: Relation between carte-
sian coordinates and spherical po-lar coordinates
Real functions (see Table 12) are gained by linear
combinations
1√2[−ψn,l,ml + ψn,l,−ml] = 1√πRn,l(r)Θl−ml(θ) cosmlφ
1i√
2[−ψn,l,ml − ψn,l,−ml] = 1√πRn,l(r)Θl−ml(θ) sinmlφ
(32)
1√2[ψn,l,ml + ψn,l,−ml] =
1√πRn,l(r)Θ
lml
(θ) cosmlφ1
i√
2[ψn,l,ml − ψn,l,−ml] = 1√πRn,l(r)Θlml(θ) sinmlφ.
(33)
For completely describing the wave function, the spin has to be
taken
into consideration. If spin-orbit coupling is ignored, the total
function (spinorbital) is
ψ(r, θ, φ;σ) = ψ(r, θ, φ)ψ(σ) where σ = ±12. (34)
30
-
Table 11: Spherical harmonics for l = 0, 1, 2, 3
l ml Ylml
(θ, φ) a) Y lml(x, y, z)
0 0( 1
4π
)1/2 ( 14π
)1/2
0( 3
4π
)1/2cos θ
( 34π
)1/2zr
1
±1 ∓( 3
8π
)1/2sin θ e±iφ ∓
( 38π
)1/2x± iyr
0( 5
16π
)1/2(3 cos2θ − 1)
( 516π
)1/23z2 − r2r2
2 ±1 ∓(15
8π
)1/2cos θ sin θ e±iφ ∓
(158π
)1/2z(x± iy)r2
±2( 15
32π
)1/2sin2θ e±i2φ
( 1532π
)1/2(x± iy)2r2
0( 7
16π
)1/2(5 cos3θ − 3 cos θ)
( 716π
)1/2z(5z2 − 3r2)r3
±1 ∓( 21
64π
)1/2sin θ(5 cos2θ − 1)e±iφ ∓
( 2164π
)1/2(x± iy)(5z
2 − r2)r3
3
±2(105
32π
)1/2cos θ sin2θ e±i2φ
(10532π
)1/2z(x± iy)2r3
±3 ∓( 35
64π
)1/2sin3θ e±i3φ ∓
( 3564π
)1/2(x± iy)3r3
a) Phase factor corresponding to the Condon-Shortley convention,
i. e., −1 for odd positive mland +1 otherwise.
31
-
Table 12: Real orthonormal linear combinations of the spherical
harmonicsY lml(θ, φ) for l = 1, 2, 3.
l function designation
( 34π
)1/2cos θ =
( 34π
)1/2 zr
pz
1( 3
4π
)1/2sin θ cosφ =
( 34π
)1/2 xr
px
( 34π
)1/2sin θ sinφ =
( 34π
)1/2 yr
py
( 516π
)1/2(3 cos2θ − 1) =
( 516π
)1/2 3z2 − r2r2
dz2
(154π
)1/2cos θ sin θ cosφ =
(154π
)1/2 xzr2
dxz
2(15
4π
)1/2cos θ sin θ sinφ =
(154π
)1/2 yzr2
dyz
( 1516π
)1/2sin2θ cos 2φ =
( 1516π
)1/2 x2 − y2r2
dx2−y2
( 1516π
)1/2sin2θ sin 2φ =
(154π
)1/2 xyr2
dxy
( 716π
)1/2(5 cos3θ − 3 cos θ) =
( 716π
)1/2 z(5z2 − 3r2)r3
fz3
( 2132π
)1/2sin θ(5 cos2θ − 1) cosφ =
( 2132π
)1/2 x(5z2 − r2)r3
fxz2
( 2132π
)1/2sin θ(5 cos2θ − 1) sinφ =
( 2132π
)1/2 y(5z2 − r2)r3
fyz2
3(105
16π
)1/2cos θ sin2θ sin 2φ =
(1054π
)1/2 xyzr3
fxyz
(10516π
)1/2cos θ sin2θ cos 2φ =
(10516π
)1/2 z(x2 − y2)r3
fz(x2−y2)( 35
32π
)1/2sin3θ cos 3φ =
( 3532π
)1/2 x(x2 − 3y2)r3
fx(x2−3y2)( 35
32π
)1/2sin3θ sin 3φ =
( 3532π
)1/2 y(3x2 − y2)r3
fy(3x2−y2)
32
-
5.4 Angular momentum ([5]97–103)
Classical definition of angular momentum l:
l = r × p. (35)
l
rp
Fig. 21: Definition of the angular momentum
l = lxi + lyj + lzk (36)
= (ypz − zpy)i + (zpx − xpz)j + (xpy − ypx)k
Length of the angular momentum vector
|l|2 = l2x + l2y + l2z. (37)
Derive quantum mechanical operators l̂x, l̂y, l̂z by
substituting the position
operator and the linear momentum operator for the corresponding
classicalquantity, i. e.
q → q̂ = q · pq → p̂q =h̄
i
∂
∂q(q = x, y, z; i =
√−1)
l̂x = ŷp̂z − ẑp̂y; l̂y = ẑp̂x − x̂p̂z; l̂z = x̂p̂y − ŷp̂x
(38)
l̂x =h̄
i
(ŷ∂
∂z− ẑ ∂
∂y
); l̂y =
h̄
i
(ẑ∂
∂x− x̂ ∂
∂z
); l̂z =
h̄
i
(x̂∂
∂y− ŷ ∂
∂x
)(39)
Commutation relations
[l̂x, l̂y] = ih̄l̂z; [l̂y, l̂z] = ih̄l̂x; [l̂z, l̂x] = ih̄l̂y ,
(40)
Operator l̂z in spherical polar coordinates:
l̂z =h̄
i
∂
∂φ(41)
33
-
l̂z acts on the φ depending part of the atomic orbitals (Table
11):
l̂z∣∣ l ml
〉= lz
∣∣ l ml〉
= mlh̄∣∣ l ml
〉Dirac notation
〈l ml
∣∣l̂z∣∣ l ml
〉= ml h̄
〈l ml
∣∣ l ml〉
︸ ︷︷ ︸1
generally Ĥ Ψ = EΨ, (Ψ normalised eigenfunction of Ĥ)∫Ψ∗Ĥ Ψ
dτ = E
∫Ψ∗ Ψ dτ
︸ ︷︷ ︸1
= E
∫Ψ∗Ĥ Ψ dτ ≡ 〈Ψ|Ĥ |Ψ〉 matrix element (Dirac notation)
Application of l̂z:
l̂z∣∣ 2 2
〉= Rn,2(r)
h̄
i
∂Y 22 (θ, φ)
∂φ
= Rn,2(r)h̄
i
∂
∂φ
[(15
32π
)1/2sin2θ ei2φ
]
= Rn,2(r)h̄
i
(15
32π
)1/2sin2θ
∂ei2φ
∂φ
= Rn,2(r)i2h̄
i
( 1532π
)1/2sin2θ ei2φ = 2h̄
∣∣ 2 2〉
l̂2∣∣ l ml
〉= l(l + 1)h̄2
∣∣ l ml〉
(42)
Shift operators:
l̂+ = l̂x + il̂y; l̂− = l̂x − il̂y. (43)
Reverse operations:
l̂x =1
2(l̂+ + l̂−); l̂y =
1
2i(l̂+ − l̂−). (44)
l̂z∣∣ l ml
〉= ml h̄
∣∣ l ml〉
l̂2∣∣ l ml
〉= l(l + 1) h̄2
∣∣ l ml〉
l̂±∣∣ l ml
〉=
√l(l + 1) −ml(ml ± 1) h̄
∣∣ l ml ± 1〉.
(45)
34
-
z
ml=+2
+1
0
1
2
Fig. 22: Specified orientation of l (l = 2) with regardto the
component lz while lx and ly are unspecified
5.5 Spin ([5]103–105)
Spin orbital of a one-electron system:
ψ(r)ψ(σ) = ψn,l,ml(r, θ, φ)ψ(σ) = Rn,l(r)Ylml
(θ, φ)︸ ︷︷ ︸atomic orbital
ψ(σ)
Spin function: ψ(σ) ≡ | sms 〉{ms =
12 : α
ms = −12 : β
ŝ2∣∣ sms
〉= s(s+ 1) h̄2
∣∣ sms〉
with s = 12
ŝz∣∣ sms
〉= msh̄
∣∣ sms〉
with ms = ±12ŝ±
∣∣ sms〉
=√s(s+ 1) −ms(ms ± 1) h̄
∣∣ sms ± 1〉
(46)
where ŝ2 = ŝ2x + ŝ2y + ŝ
2z; ŝ+ = ŝx + iŝy; ŝ− = ŝx − iŝy
ŝx =12(ŝ+ + ŝ−); ŝy =
12i(ŝ+ − ŝ−)
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
µs = −e
2megs = γegs (g = 2.002 319 314), γe magnetogyric ratio
with the orbital magnetic moment at the centre of the orbit
µl = −e
2mel = γe l
caused by the circulating charged particle. Only certain
orientations be-
tween l and s are allowed given by the vector sum
j = l + s, with
{j = l + s, l + s− 1, . . . , |l − s|as s = 1/2 → j = l ± 1/2
(47)
mj = ml +ms
̂2 = ̂2x + ̂2y + ̂
2z ̂+ = ̂x + îy ̂− = ̂x − îy (48)
̂2∣∣ j mj
〉= j(j + 1) h̄2
∣∣ j mj〉
̂z∣∣ j mj
〉= mjh̄
∣∣ j mj〉
̂±∣∣ j mj
〉=
√j(j + 1) −mj(mj ± 1) h̄
∣∣ j mj ± 1〉.
(49)
∆E = E(2P3/2) −E(2P1/2) =3
2ζ; generally: ∆E(J, J − 1) = J ζ
Operator of spin-orbit coupling
Ĥso = ξ(r) l̂·ŝ where ξ(r) = −e
2m2ec2
1
r
∂V (r)
∂r. (50)
Example 5.2 Spin-orbit coupling of the p1 system
p1 system: l = 1, s = 12, j =12 and
32 (see Figs. 23 and 24, Table 13)
36
-
E
E (6)
(4)
(2)
1-2
ohne mit
Spin-Bahn-Wechselwirkung
z
z-
(0)
Fig. 23: Splitting of the p1
levels by spin-orbit interac-tion (ζ: one-electron spin-
orbit coupling constant)
/cm-1
16973
16956
0S
1 2
2P1/
1/
2
2
2P3/2
589.2nml
n
n
17cm1
589.8nm=
2
Fig. 24: Term scheme ofthe sodium atom
Unperturbed sixfold degenerate states:
Ĥ(0)ψ(0)i = E
(0)ψ(0)i (i = 1, 2, . . . , 6).
Eq. (27) reads in this case:
(Ĥ(0) − E(0))ψ(1) = (E(1) − Ĥ(1))(u1ψ
(0)1 + . . .+ u6ψ
(0)6 ). (51)
Multiplication with ψ(0)∗1 from the left and integration
gives:∫
ψ(0)∗1
(Ĥ(0) − E(0)
)ψ(1)dτ
︸ ︷︷ ︸0
=
∫ψ
(0)∗1
(E(1) − Ĥ(1)
)(u1ψ
(0)1 + . . .+ u6ψ
(0)6
)dτ
0 = u1E(1)
∫ψ
(0)∗1 ψ
(0)1 dτ + . . .+ u6E
(1)
∫ψ
(0)∗1 ψ
(0)6 dτ
−u1∫ψ
(0)∗1 Ĥ
(1) ψ(0)1 dτ − . . .− u6
∫ψ
(0)∗1 Ĥ
(1) ψ(0)6 dτ.
(52)
Proceeding similarly with ψ(0)∗i (i = 2, . . . , 6) and using
the abbreviation∫
ψ(0)∗i Ĥ
(1)ψ(0)j dτ ≡ Hij we obtain a system of six equations:
0 = u1(H11 −E(1)
)+ u2H12 + · · · + u6H16
0 = u1H21 + u2(H22 −E(1)
)+ · · · + u6H26
......
0 = u1H61 + u2H62 + · · · + u6(H66 −E(1)
)
(53)
37
-
Non-trivial solutions for the coefficients of u1, u2, . . . ,
u6:Calculation of the integrals Hij
∫ (ψn,l,ml(r, θ, φ)ψms(σ)
)∗ĤSB ψn,l,m′l(r, θ, φ)ψm′s(σ) r
2 dr sin θ dθ dφ dσ. (54)
∫ ∞
0
Rn,l(r) ξ(r)Rn,l(r) r2 dr × (55)
∫ π
0
∫ 2π
0
∫ 1/2
−1/2
(Y mll (θ, φ)ψms(σ)
)∗l̂·ŝ
(Y
m′ll (θ, φ)ψm′s(σ)
)sin θ dθ dφ dσ.
hc ζn,l = h̄2
∫ ∞
0
Rn,l(r) ξ(r)Rn,l(r) r2 dr. (56)
ζn,l: one-electron spin-orbit coupling constantbasis functions
in Dirac notation:
∣∣ml ms〉
∣∣ 1 12〉 ∣∣ 0 12
〉 ∣∣ −1 12〉 ∣∣ 1 − 12
〉 ∣∣ 0 − 12〉 ∣∣ −1 − 12
〉. (57)
The integral eq. (55) has the short form
hcζn,l
h̄2〈ml ms
∣∣l̂·ŝ∣∣m′l m′s
〉. (58)
Determination of 36 matrix elements of the spin-orbit coupling
operator
l̂·ŝ = l̂xŝx + l̂yŝy + l̂zŝz.Replace the x and y components
by the step operators (eq. (44,46)):
l̂·ŝ = l̂zŝz +12(l̂+ + l̂−)
12(ŝ+ + ŝ−) +
12i
(l̂+ − l̂−) 12i(ŝ+ − ŝ−)= l̂zŝz +
14(l̂+ŝ+ + l̂−ŝ+ + l̂+ŝ− + l̂−ŝ−
−l̂+ŝ+ + l̂−ŝ+ + l̂+ŝ− − l̂−ŝ−)= l̂zŝz +
12(l̂+ŝ− + l̂−ŝ+) (59)
The general matrix element (58) is〈ml ms
∣∣l̂·ŝ∣∣m′l m′s
〉=
〈ml ms
∣∣l̂zŝz + 12(l̂+ŝ− + l̂−ŝ+)∣∣m′lm′s
〉
=〈ml ms
∣∣l̂zŝz∣∣m′l m′s
〉
+12〈mlms
∣∣l̂+ŝ−∣∣m′l m′s
〉
+12〈mlms
∣∣l̂−ŝ+∣∣m′l m′s
〉. (60)
General hints to the evaluation of matrix elements〈m
∣∣Ĥ(1)∣∣n
〉:
38
-
(i) Evaluate Ĥ(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 presentlet us assume Ĥ(1)
∣∣n〉
= a∣∣n
〉.
(ii) The result of (i) is then premultiplied by〈m
∣∣ giving〈m
∣∣an〉.
(iii) Since a is a constant we have〈m
∣∣an〉
= a〈m
∣∣n〉
and we are thus leftwith 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〈ml ms
∣∣m′lm′s〉
= δml,m′l δms,m′s, (61)
the integral is not zero when ml = m′l and ms = m
′s. The wavefunctions
are eigenfunctions of l̂z und ŝz, so that the application of
the operatorproducts in eq. (60) on the wavefunction on its
right-hand side yields:
l̂zŝz∣∣ml ms
〉= ml ms h̄
2∣∣ml ms
〉
l̂+ŝ−∣∣ml ms
〉=√
l(l + 1) −ml(ml + 1)√s(s+ 1) −ms(ms − 1) h̄2
∣∣ml + 1ms − 1〉
l̂−ŝ+∣∣ml ms
〉=√
l(l + 1) −ml(ml − 1)√s(s+ 1) −ms(ms + 1) h̄2
∣∣ml − 1ms + 1〉
where s = 12 . For diagonal elements only l̂zŝz may contribute,
whereas fornon-diagonal elements only the step operators may
account:〈ml ms
∣∣l̂+ŝ−∣∣ml − 1ms + 1
〉 〈mlms
∣∣l̂−ŝ+∣∣ml + 1ms − 1
〉
Matrix elements (58) which may contribute are restricted to the
condition
ml +ms = m′l +m
′s (62)
The non-zero matrix elements are:〈1 − 12
∣∣l̂+ŝ−∣∣ 0 12
〉 〈0 − 12
∣∣l̂+ŝ−∣∣ −1 12
〉〈0 12
∣∣l̂−ŝ+∣∣ 1 − 12
〉 〈−1 12
∣∣l̂−ŝ+∣∣ 0 − 12
〉.
39
-
mlms∣∣ 1 1
2
〉 ∣∣ 1 − 12
〉 ∣∣ 0 12
〉 ∣∣ 0 − 12
〉 ∣∣ −1 12
〉 ∣∣ −1 − 12
〉
〈1 12
∣∣ 12 ζ〈
1 − 12∣∣ −12 ζ
√12 ζ
〈0 12
∣∣√
12 ζ 0
〈0 − 12
∣∣ 0√
12 ζ
〈−1 12
∣∣√
12 ζ −12 ζ
〈−1 − 12
∣∣ 12 ζ
(63)
H11 =ζ
h̄2〈1 12
∣∣l̂zŝz∣∣ 1 12
〉= ζ · 1 · 12
〈1 12
∣∣ 1 12〉
= 12 ζ
H22 =ζ
h̄2〈1 − 1
2
∣∣l̂zŝz∣∣ 1 − 1
2
〉= ζ · 1 · (−1
2) = −1
2ζ
H33 =ζ
h̄2〈0 12
∣∣l̂zŝz∣∣ 0 12
〉= ζ · 0 · 12 = 0
H44 =ζ
h̄2〈0 − 12
∣∣l̂zŝz∣∣ 0 − 12
〉= ζ · 0 · (−12) = 0
H55 =ζ
h̄2〈−1 12
∣∣l̂zŝz∣∣ −1 12
〉= ζ · (−1) · 12 = −12 ζ
H66 =ζ
h̄2〈−1 − 12
∣∣l̂zŝz∣∣ −1 − 12
〉= ζ · (−1) · (−12) = 12 ζ
H23 =ζ
h̄2〈1 − 12
∣∣12 l̂+ŝ−
∣∣ 0 12〉
= 12ζ ·√
2 · 1〈1 − 12
∣∣ 1 − 12〉
=√
12 ζ = H32
H45 =ζ
h̄2〈0 − 12
∣∣12 l̂+ŝ−
∣∣ −1 12〉
= 12ζ ·√
2 · 1 =√
12 ζ = H54
Diagonalisation of the 2 × 2 blocks of the H matrix:∣∣∣∣∣∣−12 ζ
− E(1)
√12 ζ√
12 ζ −E(1)
∣∣∣∣∣∣= (−12ζ −E(1))(−E(1)) − 12ζ2 = 0
E(1)(1) =
12 ζ; E
(1)(2) = −ζ.
Evaluation of the zeroth-order functions for E(1)(1) =
12 ζ:
0 =(−12 ζ − 12 ζ
)u2(1) +
√12 ζ u3(1)
40
-
x(1) =u2(1)u3(1)
=√
12; u2(1) =
√13; u3(1) =
√23
ψ2 =√
13
∣∣ 1 − 12〉
+√
23
∣∣ 0 12〉. (64)
For E(1)(2) = −ζ, the result is:
0 =(−12 ζ + ζ
)u2(2) +
√12 ζ u3(2)
x(2) =u2(2)
u3(2)= −
√2; u2(2) = −
√23; u3(2) =
√13
ψ3 = −√
23
∣∣ 1 − 12
〉+
√13
∣∣ 0 12
〉. (65)
Evaluating the second 2 × 2 block the resulting states are
E(1)(1) =
12 ζ : ψ4 =
√13
∣∣ −1 12〉
+√
23
∣∣ 0 − 12〉
(66)
E(1)(2) = −ζ : ψ5 =
√23
∣∣ −1 12
〉−
√13
∣∣ 0 − 12
〉. (67)
The functions are not only eigenfunctions of the operators l̂·ŝ
and ŝ·l̂ but
also of
l̂ 2 + l̂·ŝ + ŝ·l̂ + ŝ2 =(l̂ + ŝ
)2 = ̂2. (68)
If ̂2 acts on a quartet state function ψQ (ψ1, ψ2, ψ4, ψ6) and
on a doubletstate function ψD (ψ3, ψ5), respectively the result
is
̂2 ψQ =(l̂ 2 + 2 l̂·ŝ + ŝ2
)ψQ
= h̄2(2 + 2 · 12 + 34
)ψQ = h̄
2 (154 )ψQ = h̄2 (32)(
52)ψQ
= h̄2 j1(j1 + 1)ψQ where j1 =32 (69)
̂2 ψD =(l̂ 2 + 2 l̂·ŝ + ŝ2
)ψD
= h̄2(2 − 2 · 1 + 34
)ψD = h̄
2 (34)ψD = h̄2 (12)(
32)ψD
= h̄2 j2(j2 + 1)ψD where j2 =12 . (70)
41
-
Table 13: Functions and energies of the p1 system
ψ∣∣ml ms
〉 ∣∣ j mj〉
mj = ml +ms j E(1)
ψ1∣∣ 1 12
〉 ∣∣ 32
32
〉32
ψ2
√13
∣∣ 1 − 12〉
+√
23
∣∣ 0 12〉 ∣∣ 3
212
〉12
ψ4
√13
∣∣ −1 12〉
+√
23
∣∣ 0 − 12〉 ∣∣ 3
2 − 12〉
−1232
12 ζ
ψ6∣∣ −1 − 12
〉 ∣∣ 32 − 32
〉−32
ψ3 −√
23
∣∣ 1 − 12〉
+√
13
∣∣ 0 12〉 ∣∣ 1
212
〉12
ψ5
√23
∣∣ −1 12〉−
√13
∣∣ 0 − 12〉 ∣∣ 1
2 − 12〉
−1212 −ζ
Example: Energy eigenvalues and eigenfunctions of 4f 1
(Ce3+)
and 4f 13 (Yb3+)On account of j = l ± s = 3 ± 12 (eqn. (47)) for
one-electron and one-holef systems we have
Ln3+[4fN
] ground multiplet E excited multiplet E
Ce3+[4f 1] 2F5/2 (j = 5/2) −2ζCe 2F7/2 (j = 7/2) +32ζCeYb3+[4f
13] 2F7/2 (j = 7/2) −32ζYb 2F5/2 (j = 5/2) +2ζYb
Eigenfunctions are obtained with the help of so-called vector
coupling co-
efficients (Clebsch-Gordan coefficients) (see Table 14).
Table 14: Vector coupling coefficientsfor systems with j2 =
1/2
j = m2 =12
m2 = −12
j1 +12
√j1 +m+
12
2j1 + 1
√j1 −m+ 12
2j1 + 1
j1 − 12 −
√j1 −m+ 12
2j1 + 1
√j1 +m+
12
2j1 + 1
42
-
Table 15: Spin-orbit coupled eigenfunctions of Ce3+ and Yb3+
free ions
a)∣∣ j mj
〉 ∣∣ml ms〉
E(Ce3+)J E
(Yb3+)J
φ1;φ6∣∣ 5
2 ± 52〉
= ∓√
67
∣∣ ±3 ∓ 12〉±
√17
∣∣ ±2 ± 12〉
φ2;φ5∣∣ 5
2 ± 32〉
= ∓√
57
∣∣ ±2 ∓ 12〉±
√27
∣∣ ±1 ± 12〉
−2ζ 2ζ
φ3;φ4∣∣ 5
2 ± 12〉
= ∓√
47
∣∣ ±1 ∓ 12〉±
√37
∣∣0 ± 12〉
φ′1;φ′8
∣∣ 72 ± 72
〉=
∣∣ ±3 ± 12〉
φ′2;φ′7
∣∣ 72± 5
2
〉=
√17
∣∣ ±3 ∓ 12
〉+
√67
∣∣ ±2 ± 12
〉
φ′3;φ′6
∣∣ 72 ± 32
〉=
√27
∣∣ ±2 ∓ 12〉
+√
57
∣∣ ±1 ± 12〉 32ζ −32ζ
φ′4;φ′5
∣∣ 72 ± 12
〉=
√37
∣∣ ±1 ∓ 12〉
+√
47
∣∣0 ± 12〉
a) Short form of the functions: the first symbol refers to the
upper sign, the second to the lowerone.
The calculation of the coefficients is demonstrated for∣∣ 5
252
〉(first line in
Table 15).Assignments: j = 5/2, m = mj = 5/2, j1 = l = 3, and m2
= ms = ±1/2(j2 = s = 1/2)
The roots of the lower row of Table 14 become
m2 = −1
2:
√j1 +m+
12
2j1 + 1=
√3 + 52 − 12
7=
√6
7
m2 =1
2: −
√j1 −m+ 12
2j1 + 1=
√3 − 52 + 12
7= −
√1
7
Since the Condon-Shortley standard assignment is j1 → s and j2 →
lthe sign of the coefficients has to be changed according to the
phase relation
|jbjajm〉 = (−1)ja+jb−j|jajbjm〉. Finally, we obtain∣∣ 52
52
〉= −
√67
∣∣ 3 − 12〉
+√
17
∣∣ 2 + 12〉.
43
-
Problems
1. Calculate the matrix elements 〈l,ml|l̂q|l,m′l〉 (where q
stands for z,+,−)(a) 〈0, 0|l̂z|0, 0〉, (b) 〈2, 2|l̂+|2, 1〉, (c) 〈2,
2|l̂2+|2, 0〉, (d) 〈2, 0|l̂+l̂−|2, 0〉.
2. The 14 microstates |mlms〉 of an f 1 system (l = 3, s = 1/2)
yield underthe influence of the spin-orbit coupling operator 14
eigenstates |jmj〉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〉,|5/2 − 1/2〉, |7/2 3/2〉, and |7/2 − 3/2〉.
Control your results withthe entries of Table 15.
44
-
6 Exchange interactions in dinuclear compounds
6.1 Parametrization of exchange interactions
• Heitler-London model of H2
1
2
a br
r
r
r
a
a
b
b
2
2
1
1 rb2
ra1
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:
D1 = φa(1)α(1)φb(2)β(2)− φa(2)α(2)φb(1)β(1) (71)D2 =
φa(1)β(1)φb(2)α(2)− φa(2)β(2)φb(1)α(1) (72)D3 = φa(1)α(1)φb(2)α(2)−
φa(2)α(2)φb(1)α(1) (73)D4 = φa(1)β(1)φb(2)β(2)− φa(2)β(2)φb(1)β(1)
(74)
45
-
• Construction of eigenfunctions of the total spin
Ŝ′ = Ŝ1 + Ŝ2 (S1 = S2 = 1/2)
D3 and D4 are eigenfunctions |S ′M ′S 〉 of Ŝ′2 = (Ŝ1 + Ŝ2)2
and Ŝ ′z =Ŝz1 + Ŝz2 with S
′ = 1 and MS′ = 1 and −1 (Spin triplet functions | 1 1 〉and | 1
−1 〉), respectively.D1 +D2 =⇒ | 1 0 〉 (spin triplet function)D1 −D2
=⇒ | 0 0 〉 (spin singlet function)
Φ1 = D1 −D2 Φ2 = D3 Φ3 = D1 +D2 Φ4 = D4| 0 0 〉 | 1 1 〉 | 1 0 〉 |
1 −1 〉
φa and φb are normalised:∫φa(1)
∗φa(1)dτ1 =
∫φ∗b(2)φb(2)dτ2 = 1,
but not orthogonal:
overlap integral Sab =
∫φa(1)
∗φb(1)dτ1 =
∫φa(2)
∗φb(2)dτ2 6= 0
Normalised functions of the dinuclear unit:
Φ1 = Ng [φa(1)φb(2) + φa(2)φb(1)]︸ ︷︷ ︸sym
√12[α(1)β(2)− α(2)β(1)]
︸ ︷︷ ︸anti
Φ2
Φ3
Φ4
= Nu [φa(1)φb(2) − φa(2)φb(1)]︸ ︷︷ ︸anti
α(1)α(2)√12[α(1)β(2) + α(2)β(1)]
β(1)β(2)︸ ︷︷ ︸
sym
where Ng = [2 + 2S2ab]
− 12 and Nu = [2 − 2S2ab]−12 .
46
-
• Symmetry of the functions with regard to exchange of
electrons
total function: anti Φi(1, 2) = −Φi(2, 1)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
tripletstates
Ĥ = − h̄2
2me∇2(1) − e
2
ra1− e
2
rb1︸ ︷︷ ︸ĥ(1)
− h̄2
2me∇2(2) − e
2
ra2− e
2
rb2︸ ︷︷ ︸ĥ(2)
+e2
r12
E(S) = 〈1Φg1|Ĥ|1Φg1〉 =2(h+ habSab) + Jab +Kab
1 + S2ab(75)
E(T ) = 〈3Φu2|Ĥ|3Φu2〉 =2(h− habSab) + Jab −Kab
1 − S2ab(76)
where
h =〈φa(i)
∣∣∣ĥ(i)∣∣∣φa(i)
〉(one-centre
=〈φb(i)
∣∣∣ĥ(i)∣∣∣φb(i)
〉one-electron integral)
hab =〈φa(i)
∣∣∣ĥ(i)∣∣∣φb(i)
〉(transfer or hopping integral)
Jab =〈φa(1)φb(2)
∣∣e2/r12∣∣φa(1)φb(2)
〉(Coulomb integral)
Kab =〈φa(1)φb(2)
∣∣e2/r12∣∣φa(2)φb(1)
〉(Exchange integral).
47
-
Example: Evaluation of E(S) =〈
1Φg1
∣∣∣Ĥ∣∣∣ 1Φg1
〉
1. Integration over the spin:
12〈α(1)β(2)− β(1)α(2)|α(1)β(2)− β(1)α(2)〉 =
12
[〈α(1)β(2)|α(1)β(2)〉︸ ︷︷ ︸
1
+ 〈α(2)β(1)|α(2)β(1)〉︸ ︷︷ ︸1
−
〈α(2)β(1)|α(1)β(2)〉︸ ︷︷ ︸0
−〈α(1)β(2)|α(2)β(1)〉︸ ︷︷ ︸0
]= 1.
2. Integration over the space:
E(S) = (77)
N2g
〈φa(1)φb(2) + φb(1)φa(2)
∣∣∣ĥ(1) + ĥ(2) + e2/r12∣∣∣φa(1)φb(2)+
φb(1)φa(2)〉
= 2N2g [2(h+ habSab) + Jab +Kab].
Singlet-triplet splitting:
∆E(T, S) = E(T ) − E(S)≈ −2Kab − 4habSab + 2S2ab(2h+ Jab)
• Application of the Heitler-London model to dinuclear
complexeshaving S1 = S2 =
12 centres
Example: L′nCua– L– CubL′n
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 fragmentsL′nCuaL and LCubL
′n, respectively take over the role of the 1s orbitals of
the H atoms. φa and φb have mainly d character. They are
centered at themetal 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
magnitudeand sign are given by the spin-spin coupling parameter
(exchange param-
eter) J :
Ĥex = −2J Ŝ1·Ŝ2 where − 2J = ∆E(T, S) (78)
Ĥex is an effective operator, describing but not explaining the
phenomenon.
Application of Ĥex to1Φg1 (S
′ = 0) and 3Φui (S′ = 1, i = 2, 3, 4):
−2J Ŝ1·Ŝ2 1Φg1 =(
32
)J︸ ︷︷ ︸
E(S)
1Φg1, −2J Ŝ1·Ŝ2 3Φui = −(
12
)J︸ ︷︷ ︸
E(T )
3Φui
⇒ ∆E(T, S) = E(T ) − E(S) = −2J (79)J < 0: singlet ground
state (intramolecular antiferromagnetic interaction)J > 0:
triplet ground state (intramolecular ferromagnetic interaction)
Hints to the evaluation of E(T ) and E(S):
Ŝ′2 =(Ŝ1 + Ŝ2
)2= Ŝ21 + Ŝ
22 + 2Ŝ1·Ŝ2
2Ŝ1·Ŝ2 = Ŝ′2 − Ŝ21 − Ŝ22
= h̄2[S ′(S ′ + 1)︸ ︷︷ ︸0 or 2
−S1(S1 + 1)︸ ︷︷ ︸34
−S2(S2 + 1)︸ ︷︷ ︸34
]
Heisenberg operator for more than two centres:
Ĥex = −2∑
i
-
(82)
Van Vleck equation
Operator: Ĥ = Ĥ(0) + BzĤ(1)
En = W(0)n +BzW
(1)n + B
2zW
(2)n + . . . .
W(1)n , W
(2)n : First- and second-orderZeeman coefficients
µ̄n = −∂En/∂B = −W (1)n − 2BW (2)n − . . .
χmol = µ0NA
∑n
[(W(1)n )2/kBT − 2W (2)n ] exp(−W (0)n /kBT )
∑n
exp(−W (0)n /kBT )(83)
Eq. (83) is valid for applied magnetic fields B → 0
6.3 Exchange-coupled species in a magnetic field
Ĥex = −2J Ŝ1·Ŝ2 = −2J(Ŝz1Ŝz2 + Ŝx1Ŝx2 + Ŝy1Ŝy2
)
= −2J[Ŝz1Ŝz2 +
12
(Ŝ+1Ŝ−2 + Ŝ−1Ŝ+2
)](84)
Basis: spin functions in the form |MS MS 〉 where the first MS
refers toelectron 1 and the second to electron 2
H-Matrix:
MS1MS2∣∣ 1
212
〉 ∣∣ − 12 12〉 ∣∣ 1
2 − 12〉 ∣∣ − 12 − 12
〉
〈12
12
∣∣ −J /2〈− 12 12
∣∣ J /2 −J〈
12 − 12
∣∣ −J J /2〈− 12 − 12
∣∣ −J /2
(85)
Evaluation of the diagonal element H11:
−2J〈
12
12
∣∣ Ŝz1Ŝz2∣∣ 1
212
〉= −2J
(12
) (12
)= −J /2
50
-
Evaluation of the off-diagonal element H23:
−2J〈− 12 12
∣∣ 12Ŝ−1Ŝ+2
∣∣ 12 − 12
〉= −J (1)(1) = −J
Result:
Tab. 16: Spin functions and exchangeenergies of the S1 = S2
=
12
system
Spin function MS′ S′ E
1√2
(∣∣ 12− 1
2
〉−
∣∣ − 12
12
〉)0 0 3
2J
∣∣ 12
12
〉1
1√2
(∣∣ 12 − 12
〉+
∣∣ − 12 12〉)
0 1 −12J∣∣ − 12 − 12
〉−1
51
-
E
00
1
2
3
4
530J
20J
12J
6J
2J
SS2 1+
1
3
5
7
9
11
Fig. 26: Relative energies and multiplicities of the spin
states
of a dinuclear Fe3+ complex (S = 52); for Cu2+ (S = 12) only
the
two lowest levels are relevant, while for Gd3+ (S = 72) the
twolevels with S ′ = 6 (E = |42J |) and S ′ = 7 (E = |56J |) haveto
be added.
52
-
• Magnetic susceptibility of a spin-spin-coupled system with S1
= S2 = 12
0
0
1 2 3 4 5
10
8
6
4
2
0
-2
E/
-1cm
/TB0
M
1
0
-1=S
=S
S
0
1
E(T,S)D
Fig. 27: Correlation diagram of aS1 = S2 =
12 system under the in-
fluence of isotropic intramolecularspin-spin coupling (J = −2
cm−1)and applied field
T/K0 5 10 15 20 25
0
2
4
6
8
10
a
b
c
cm
ol
10
-7m
3m
ol-1
/(
)Fig. 28: χmol versus T diagram ofa S1 = S2 =
12 exchange-coupled
system with J = −2 cm−1 at ap-plied fields of B0 = 0.01 T
(a),3.5 T (b), and 5 T (c)
Application of the Van Vleck equation (83) to a dinuclear system
with
S1 = S2 =12
Zeeman-Operator:
ĤMz = −γeg(Ŝz1 + Ŝz2)︸ ︷︷ ︸Ĥ(1)
Bz = −γe g Ŝ ′z Bz
S ′MS′∣∣ 1 1
〉 ∣∣ 1 0〉 ∣∣ 1 −1
〉 ∣∣ 0 0〉
〈1 1
∣∣ gµBBz〈1 0
∣∣ 0〈1 −1
∣∣ −gµBBz〈0 0
∣∣ 0
(86)
Matrix elements:
〈11|ĤMz |11〉 = gµBBz −→W(1)|11〉 = gµB
〈1 −1|ĤMz|1 −1〉 = −gµBBz −→W(1)|1−1〉 = −gµB
53
-
The remaining matrix elements (Zeeman coefficients) are zero.
W(1)|11〉,
W(1)|1−1〉,W
(0)S = E(S), andW
(0)T = E(T ) are substituted into the Van Vleck
equation. After dividing by 2, the Bleaney-Bowers expression
(χmol percentre) is obtained, here extended by χ0
χmol = µ0NAµ
2Bg
2
3kBT
[1 +
1
3exp
(−2JkBT
)]−1+χ0, only applicable to a (Fig. 28)(87)
1.0
0.8
0.6
0.4
0.2
0.00 100 200 300 400
T/Ka
a
b
d
c
e
cm
ol/
10
mm
ol
3-7
-1(
)
0 100 200 300 4000123456789
10
T/K
ab
b
cde
c10
7m
olm
-3
mol
-1(
)/
meff
1.5
0.5
0 100 200 300 4000.0
1.0
2.0
2.5
T/K
a
b
c
c
de
Model calculations concerning
the system S1 = S2 =12 with
positive and negative J ;
J values [cm−1]:curve a: −50curve b: −25curve c: 0curve d:
+25curve e: +50
Fig. 29 a: χmol–T diagram
Fig. 29 b: χ−1mol–T diagramFig. 29 c: µeff–T diagram
54
-
Polynuclear unit of n equivalent centres:
χmol =µ0n
NAµ2Bg
2
3kBT
∑S′ S
′(S ′ + 1)(2S ′ + 1)Ω(S ′) exp(−E(S′)kBT
)
∑S′(2S
′ + 1)Ω(S ′) exp(−E(S′)
kBT
) (88)
Evaluation of S ′, E(S ′) and Ω(S ′):
S ′ of the coupled states:S ′ = nS, nS − 1, . . . , 0 (nS
integer) or 12 (nS half integer)Relative energies E(S ′):
E(S ′) = − zJn− 1 [S
′(S ′ + 1) − nS(S + 1)]
z: number of nearest neighbours of a centre
n: number of interacting centres
Frequency Ω(S ′) of the states S ′:
Ω(S ′) = ω(S ′) − ω(S ′ + 1)
ω(S ′) is the coefficient of XS′
in the expansion
(XS +XS−1 + . . .+X−S)n
• 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 term3d5-High-Spin −→ 6A13d3(Oh),
3d
7(Td) −→ 4A23d8(Oh), 3d
2(Td) −→ 3A2(3d9(Oh) −→ 2E)
55
-
6.4 Mechanisms of cooperative magnetic effects in insulators
N
NN
N
CuCu
O
O
N
N N
N
CuCu
O
O
a
b
Fig. 30: Magnetic orbitals a and b, localised in the left and
rightfragment of [L′2Cu2(µ-OH)2]
2+ (Kahn,1993)
M
M
M
M
MX
X
X
a b
XM
Fig. 31: CrCl3, CrBr3: Magnetic orbitals; top: direct
exchangedxy–d
′xy; bottom: 90
◦ superexchange
56
-
Problems
1. Verify the equation Ŝ2 = Ŝ2z − h̄Ŝz + Ŝ+Ŝ−.
2. Show, that the function eq. (74)
Ψ4 = D4 = φa(1)β(1)φb(2)β(2)− φa(2)β(2)φb(1)β(1)
is an eigenfunction of Ŝ′2 = (Ŝ1 + Ŝ2)2 and Ŝ ′z = Ŝz1 +
Ŝz2! How largeare S ′ and MS′?
3. Corresponding to E(S), eq. (77), evaluate the energy E(T ) of
thetriplet 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 Jand 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), J is set to 0?
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 anisosceles triangle needs more expense. What steps
have to be consid-
ered if perturbation theory is consequently applied? Give the
basisfunctions (spin functions) for a S1 = S2 = S3 =
12 system. What are
the resulting S ′ states? What operator represents the
perturbation bythe magnetic field?
10. Give a rough drawing of the χ−1mol–T diagram of a
homotrinuclearcluster with equivalent antiferromagnetically coupled
S = 12 centres.(Hint: Think what magnetic behaviour is expected for
high and low
temperature, kBT � |J | and kBT � |J |, respectively.)
57
-
11. Magnetochemical results are often presented as χmol–T ,
χ−1mol–T or
µeff–T diagrams. What type of diagram is suited for (a)
Curieparamagnetism, (b) intramolecular ferromagnetic interactions,
(c) in-
tramolekular antiferromagnetic interactions, (d) diamagnetic
behaviour,(e) TUP behaviour?
12. To reliably characterise magnetic properties measurements at
differentfield 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
compoundwith macrocyclic ligands, the problem may occur that the
incrementalmethod 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
regionfield-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
lowtemperature, 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
thecorresponding eq. (88). What extensions of the model are in
principle
possible?
18. The homodinuclear complex [Cp∗RuCl2] (RuIII[4d5], 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 Å.
Give a roughdrawing of µeff as a function of T assuming that the Ru
centres in
the former are antiferromagnetically coupled with J ≤ −400
cm−1while in the latter the centres are coupled ferromagnetically
with
J = 12 cm−1.
58
-
7 Exchange interactions in chain compounds
Uniform chainJ
−−−− MiJ
−−−− Mi+1J
−−−− Mi+2J
−−−−
Ĥ = −2JN−1∑
i=1
Ŝi·Ŝi+1 − γegBzN∑
i=1
Ŝi,z, (89)
0 0.5 1.0 1.5 2.0 2.50
0.06
0.12.01.0
0
0.02
0.04
0.06
0.08
0.1
3
6810
57
9
11
4
10/9
11/10
cm
ol
J
m0N
g
BA
22
m
J
k TB
S(S+1)
Fig. 32: Temperature dependent magnetic susceptibil-ity (reduced
units) of an S = 1/2 antiferromagneticallycoupled 1D system
(Heisenberg model)
Simulation of the magnetic susceptibility for infinite chains of
Cu(II):
χmol = µ0NAg
2µ2BkBT
0.25 + 0.074975x+ 0.075235x2
1.0 + 0.9931x+ 0.172135x2 + 0.757825x3(90)
with x = |2J |/(kBT ).Susceptibility equation for classical
spins, i. e., spins without space quan-
tization
χmol = µ0NAµ
2Bg
2S(S + 1)
3kBT
1 + u
1 − u with u = coth[2JS(S + 1)
kBT
]− kBT
2J S(S + 1).(91)
59
-
Tab. 17: Compounds with 1D arrangements of magnetic
ex-change-coupled centres
compound S bridge J [cm−1] J ′[cm−1]a)
Cu(C2O4) · 13 H2O 12 C2O2−4 −146
(C6H11NH3)CuCl312
(µ-Cl)2 50 0.05,−10−2
CuGeO312
O2− −63 −6; 0.6
CsNiCl3 1 Cl− −9 b)
a) J : ‘intrachain’ parameter; J ′: ‘interchain’ parameter.b) |J
′/J | = 7 × 10−3.
Alternating chain
J−−−− Mi
αJ−−−− Mi+1
J−−−− Mi+2
αJ−−−−
The Heisenberg operator reads in this case
Ĥex = −2JN∑
i=1
(Ŝ2i·Ŝ2i−1 + αŜ2i·Ŝ2i+1
). (92)
For α = 0 the model reduces to homodinuclear systems. For
antiferro-magnetic couplings with α 6= 0 (except α = 1) the
magnetic susceptibilitydecreases 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 3D networks
8.1 Molecular-field model ([5] 330-350)
The simplest and most näıve effective field approximation
consists in con-
sidering only one magnetic atom and replacing its interaction
with theremainder 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:
HMF = λMFMmol with λMF =2∑n
i ziJiµ0NAµ2Bg
2J
. (93)
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 thecentre is
Heff = H +HMF . (94)
Owing to the molecular field, the paramagnetic behaviour (T >
TC(TN))
is modified:
Mmol = χ′mol(H +HMF ) = χ
′mol(H + λMmol)
Mmol/H = χmol = χ′mol(1 + λχmol)
χ−1mol = (χ′mol)
−1 − λ. (95)
λ produces a parallel shift of the (χ′mol)−1–T curve. If the
isolated centre
obeys the Curie law, the Curie-Weiss law is obtained:
χ−1mol =T
C− λ =⇒ χmol =
C
T − Θpwhere Θp = λC. (96)
λ and Θp have the same sign. Positive and negative Θp values
refer to
predominating ferro- and antiferromagnetic interactions,
respectively. Thelayer-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 Θp and the spin-spin coupling parameters Ji are
related by
Θp =2S(S + 1)
3kB
n∑
i
ziJi, (97)
where zi is the number of ith nearest neighbours of a given
magnetic centre,
Ji stands for the exchange interaction between the ith
neighbours and n isthe number of sets of neighbours for which Ji 6=
0.
The molecular field approximation is applicable to
ferromagnetic, an-
tiferromagnetic, and ferrimagnetic materials below and above the
criticaltemperature TC(TN) [5].
61
-
8.2 High-temperature series expansion (HTSE); ([5] 386 –
415)
TpQ
c-1
TC
Qp0
Fig. 33: Schematic representation (shaded area) of the
deviationfrom Curie-Weiss behaviour
χmol = µ0NAgγe∂
∂B
Tr
[∑i Ŝzi exp
(−Ĥβ
)]
Z
, β ≡ 1/(kBT )
Ĥ = −2J∑
i
-
Example: CrBr3 (space group R 3), TC = 32.7K
Ĥex = −2J11.n∑
Ŝi·Ŝj − 2 αJ1︸︷︷︸J2
2.n∑
Ŝk·Ŝl − 2 βJ1︸︷︷︸J3
3.n∑
Ŝm·Ŝn
χmol =C
T
[1 +
∞∑
t=1
at
( J1kBT
)t ]with at =
∑
r+s≤ta′t−r−s,r,sα
rβs
Results
1 3
2
0 100 200 300 4000
2
4
6
8
10
T / K
cm
ol/10
-6m
3m
ol-
1
0
5
10
15
20
c-1
mol /
10
6m
olm
-3
0 50 1000
20
40
60
80
cm
ol/10
-6m
3m
ol-
1
T / K
Fig. 34: Cr3+ (S = 3/2) partial structure in CrBr3 and χmol and
χ−1
mol
as a function of temperature (applied field: B0 = 0.1T); the
solid lines
refer to the HTSE model with g = 1.952, J1 = 6.900 cm−1, J2 =
0
cm−1 (fixed), J3 = −0.080 cm−1 (quality of the fit: SQ = 0.7 %;
SQ =(FQ/n)1/2 × 100%, with FQ =
∑ni=1 {[χobs(i) − χcal(i)]/χobs(i)}
1/2)
Parameter g J1 [cm−1] J2 [cm−1] J3 [cm−1]d(Cr–Cr) [pm] 364 612
631
susceptibility 1.952 6.90 0 → 0.60 −0.08 → −0.19spin 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 = χ′mol + χ0 (98)
χ′mol: polynuclear unit
3. Intermolecular cooperative interactions, weak compared to the
in-
tramolecular interaction:
1
χmol=
1
χ′mol− λ (99)
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:
χmol = (1 − ρ)χ′mol + ρC
T − Θp(100)
Use ρ, C, Θp as parameters (in addition to g und 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 withantiferromagnetic 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 J ),χdiamol, ∆T ,
B0, number of magnetic centres
64
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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., OxfordUniversity 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. Schläfer, G. Gliemann, Einführung in die
Ligandenfeldtheorie,
Akad. Verlagsges., Wiesbaden 1980; Engl. Ausgabe: Basic
Principlesof 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
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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
1K 1,38066 8,61739 2,08367 6,95039 1 1,48874 8,31451
×10−23 ×10−5 ×1010 ×10−1 ×10−3
1T 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