Jan Vogel

Present position: Directeur de Recherche at the CNRS (Institut Néel, Grenoble)

Research subjects: X-ray Magnetic Circular Dichroism, PhotoEmission Electron Microscopy (PEEM), Magnetization reversal dynamics

Post-doc: 1995-1997, Laboratoire Louis Néel + ESRF

Thesis: University of Nijmegen (Netherlands) 1994

“X-ray Linear and Circular Dichroism in thin Transition Metal and Rare Earth overlayers”

HERCULES 1991

- X-ray Absorption: general introduction- Multiplet Theory for XAS

- Some quantummechanics (orbitals, configurations and terms)

- Line strengths, cross-sections: selection rules

- Magnetic and Crystal Field Effects, polarization dependence of absorption spectra

L-edges of Transition Metals, M-edges ofRare Earths

vale

nce

band EF

core

leve

ls

2p

2s

EF

2p

2s

Ek=hν-EB

vale

nce

band

core

leve

ls

EF

2p

2s

Fluorescence (radiative decay)

Typical timescales 10-15 s Auger decay(non-radiative decay)

One electron model:Fermi ’s Golden Rule:

wabs = (2π/h) |< Φf |T | Φi >|2 ρf(Ehν - Ei )

|<Φf|T|Φi>|2 : matrix element of electromagnetic field operator T between initial state |Φi> and final state <Φf|

ρf(E) density of valence states at E ( > Efermi)

Ei core-level binding energy

One-photon transitions (absorption): T1 ≈ H1 = (e/mc) p⋅A

Plane wave: A = eq A0exp[i k⋅r]T1 = (eA0/mc) Σq[eq ⋅ p + i (eq ⋅ p)(k⋅r)]

q : light polarization

First term: dipole transitions (∆l = ± 1, ∆s = 0)Second term: quadrupolar transitions (∆l = 0, ± 2, ∆s = 0)

k⋅ r ≈ √hν (eV)/ 80Z

K-edges: C (hν = 284eV, Z = 6) → 0.03

Zn (hν = 9659eV, Z = 30) → 0.04Transition probability T2 → quadrupolar transitions about 1000 times smaller than dipolar ones

Dipole approximation:

Transition probability : wabs ∝ Σq |< Φf | eq ⋅ r | Φi >|2 ρf (Ehν - Ei )

K-edge: 1s → empty p-statesL1-edge: 2s → pL2,3-edges: 2p1/2, 3/2 → s,dM4,5 -edges: 3d3/2, 5/2 → p,fetc.

2p1/2

2p3/2

Spin-orbit coupling: l ≥ 1 Spin parallel/anti-parallel to orbit: j= l + s, l - s

p → 1/2, 3/2d → 3/2, 5/2

Branching ratios: -j ≤ mj ≤ jp1/2 → mj = -1/2, 1/2p3/2 → mj = -3/2, -1/2, 1/2, 3/2Intensity ratio p3/2 : p1/2 = 2 : 1

d5/2 : d3/2 = 3 : 2

valenceband

corelevels

2p1/2

2p3/2

EF

dsp

∆l = ±1∆s = 0

• Intuitively nice: Most valence electrons are in bands

• Direct comparison with band structure calculations?

• No!The core hole spoils it all!

Si K-edge absorption of NiSi2 compared to Si p-DOS.

Right: including energy dependence of matrix elements

Discrepancies:- Influence of core hole

- Dynamics of transition

Si* p : p-density of states in presence of core hole

P.J.W. P.J.W. WeijsWeijs et al., Phys. Rev. B. 41, 11899 (1991)et al., Phys. Rev. B. 41, 11899 (1991)

• Core Hole pulls down DOS• Final State Rule: Spectral shape of XAS looks like final state DOS • Initial State Rule: Intensity of XAS is given by the initial state

initialstate dn

finalstate

multipletpdn+1

J

J’

∆J=0,±1

• Accounts naturally for core hole

• ‘Easier’ exact theory• Good for RE M45• Extensions for less

localized final states:– Crystal fields– Impurity models

(Anderson)> mixed valence> hybridization

Weakly correlated limit (interaction with neigbouring atoms >> intra-atomic interactions):

- Near edge region (0-50 eV): multiple scattering (band structure) →Y. Joly

- Extended region ( > 50 eV): single scattering (EXAFS) →S.Pascarelli

Strongly correlated electron systems (intra-atomic interactions >> interaction with environment):

- Atomic calculations:

M4,5-edges of Rare Earths (3d → 4f transitions)

- Magnetic, crystal field: weak perturbation

L2,3-edges of Transition Metals (2p → 3d transitions)

- Crystal Field (environment) more important.

Free ion wavefunctions: eigenfunctions of Hamiltonian operator:

H = - (h2/8π2m)∇2 - Zeff e2/r

H⏐ψ⟩ = E⏐ψ⟩

Orbitals (s, p, d, ..) can be written as linear combinations of hydrogen-likewave functions

ψn,l,ml = Rn,lYlml

n = principal quantum number (n ≥ 1)

l = orbital quantum number (0 ≤ l ≤ n-1)

ml = projection of l on z-direction (-l ≤ ml ≤ l)

Rn,l = Radial part of wavefunction

Ylml = Angular part of wavefunction

d0 d±1 d±2

p0p±1

s0

s-orbital: Y00 = 2-1/2(2π)-1/2

p-orbitals: Y11 = (3/4)1/2sinθ • (2π)-1/2eiϕ

Y10 = (3/2)1/2cosθ • (2π)-1/2

Y1-1 = (3/4)1/2sinθ • (2π)-1/2e-iϕ

d-orbitals: Y2±2 = (15/16)1/2 sin2θ • (2π)-1/2e±2 iϕ

Y2±1 = (15/4)1/2sinθcosθ • (2π)-1/2e± iϕ

Y20 = (5/8)1/2 (3cos2θ - 1) • (2π)-1/2

e± i mφ = cos (mφ) ± i sin (mφ) (De Moivre’s theorem)x = r sinθcosφ , y = r sinθsinφ , z = r cosθ

θ

ϕx

y

zr

Example:Rn,2• Y2

2 + Rn,2• Y-22 =

Rn,2 • (15/16)1/2 • (2π)-1/2 • sin2θ(cos 2φ + i sin2φ) +Rn,2 • (15/16)1/2 • (2π)-1/2 • sin2θ(cos 2φ - i sin2φ) =2Rn,2 • (15/16)1/2 • (2π)-1/2 • sin2θ cos 2φ =2Rn,2 • (15/16)1/2 • (2π)-1/2 • sin2θ(cos2φ - sin2φ) =2Rn,2 • (15/16)1/2 • (2π)-1/2 • (x2 - y2 )/ r2 ~ (x2 - y2 )

Normalization: d(x2 - y2) = 2-1/2 • Rn,2 (Y22 + Y-2

2)

dz2 = Rn,2 Y02

dyz = 2-1/2 • Rn,2 (Y12 - Y-1

2)dxz = 2-1/2 • Rn,2 (Y1

2 + Y-12)

dxy = 2-1/2 • Rn,2 (Y22 - Y-2

2)

pZ px py

http://en.wikipedia.org/wiki/Atomic_orbital

dz2 dxz

dyz dxy dx2-y2http://en.wikipedia.org/wiki/Atomic_orbital

Configuration: Number of electrons in a given (partly filled) orbital: 3d2, 3d94f8, etc.

Term: energy level of a system. Each configuration has several energy levels/terms.

Labelling terms: (2S + 1) X

L = 0 1 2 3 4 5 6

X = S P D F G H I

(2S + 1): Multiplicity

singlet doublet triplet quartet quintet sextet

S = 0 1/2 1 3/2 2 5/2

- L ≤ ML ≤ L - S ≤ MS ≤ S

Degeneracy of term: (2L + 1)(2S + 1)

Term 3P is 3x3 = 9-fold degenerate.

d1 (d9 ) :

ml = 2 1 0 -1 -2

ML = 2, MS = 1/2↑

d1, d9 → 2D

↑ ↑

d2 (d8 ) :

ml = 2 1 0 -1 -2

ML = 3, MS = 1

↑ ↑ ML = 1, MS = 1

↓ ↑ ML = -2, MS = 0

↓ ↓ or ↑ ↑: 2 x ( ) = 20 ↑ ↓: 5 x 5 = 25 Total: 45 possibilities52

Occurrencies for different ML and MS values:ML = ±4 (1 time each) MS = ±1 (10 times each)ML = ±3 (4) MS = 0 (25)

= ±2 (5)= ±1 (8)= 0 (9)

ML = +4 and -4 occur 1 time each → 1 G

Subtracting 1 for each ML value and 2L+1 = 9 times MS = 0 :ML = ±3 (3) MS = ±1 (10)

= ±2 (4) MS = 0 (16)= ±1 (7)= 0 (8)

ML = +3 and -3 occur 3 times each → 3 x 1F or 1 x 3F

3 x 1F gives 3 x (2L + 1) = 21 times MS = 0 → not possible

d2, d8 → 1G + 1D + 1S + 3F + 3P

Configuration Terms

d1, d9 2D

d2, d8 3F,3P,1G,1D,1S

d3, d7 4F,4P,2H,2G,2F,2x2D,2P (2D occurs twice)

d4, d6 5D,3H,3G,2x3F,3D,2x3P,1I,2x1G,1F,2x1D,2x1S

d5 6S,4G,4F,4D,4P,2I,2H,2x2G,2x2F,3x2D,2P,2S

Term with lowest energy: Hund’s rules

1) For a given configuration, the terms with the highest multiplicity (spin) lie lowest.

2) Of the terms with the highest spin, the one with the largest value of L lies lowest.

ml= 2 1 0 -1 -2 ML MS Ground term

d1 ↑ 2 1/2 2D

d2 ↑ ↑ 3 1 3F

d3 ↑ ↑ ↑ 3 3/2 4F

d4 ↑ ↑ ↑ ↑ 2 2 5D

d5 ↑ ↑ ↑ ↑ ↑ 0 5/2 6S

d6: 5D d7: 4F d8: 3F d9: 2D

∑∑∑∑ ⋅+++= −

Niii

pairsre

NrZe

Nm

p slrHiji

i )(222

2 ζEnergy of term:

∑N

mpi2

2

Kinetic energy of electrons

∑ −

NrZe

i

2Electrostatic interaction with the nucleus

∑pairs

re

ij

2(Hee) Electron-electron repulsion

∑ ⋅N

iii slr )(ζ (Hls) Spin-orbit coupling of each electron

First two parts are the same for all terms in the configuration → average energy of configuration Hav H = Hav + H′ee + Hls

∑∑ −=−=pairs

re

pairsre

eeeeee ijijHHH 22'

H = Hav + H′ee + Hls

Electron repulsion parameters: Coulomb, exchange.

∑∑ +=++

k

kk

k

kkJ

Sre

JS GgFfLL 1212 ||

12

2

Fk (fk) , Gk (gk) : radial (angular) part of direct Coulomb repulsion and exchange interaction

Examples :

2p2 : f0F0 + f2F2 2p3p : f0F0 + f2F2 + g0G0 + g2G2

3d2 : f0F0 + f2F2 + f4F4 2p3d : f0F0 + f2F2 + g1G1 + g3G3

Properties fk : f0 always present, kmax = 2 x lowest l - value

Properties gk : only present for e- in different shells

k is even if l1+l2 is even, odd if l1+l2 is odd

kmax = l1 + l2

Example 1s2s, term symbols 1S and 3S

)21()21(|| 001112

2 ssGssFSS re +=

)21()21(|| 003312

2 ssGssFSS re −=

1S ↔2G0(1s2s)

3S

3d2: 1S, 3P, 1D, 3F and 1G

F4 ≈ 0.62 F2

2 electrons: f0 = 1

RelativeEnergy

RelativeEnergy

1S F0 + 2/7 F2 + 2/7 F4 0.46F2 4.6 eV3P F0 + 3/21 F2 - 4/21 F4 0.02F2 0.2 eV1D F0 - 3/49 F2 + 4/49 F4 -0.01F2 -0.1 eV3F F0 - 8/49 F2 - 1/49 F4 -0.18F2 -1.8 eV1G F0 + 4/49 F2 + 1/441 F4 0.08F2 0.8 eV

Energy of terms of 3dn configurations:

Term: (2L + 1)(2S + 1)-fold degenerate.

Second perturbation: spin-orbit coupling.

s.o.coupling << electron repulsion: Russel-Saunders scheme. Splitting in states with

| L - S | ≤ J ≤ | L + S | (2J + 1)-fold degenerate: -J ≤ MJ ≤ J

single-electron spin-orbit coupling parameter ζ: interaction strength between spin and orbital angular momenta of a single e- in a configuration.

ζl • s with ζ = (Zeff e2/2m2c2) r-3

For a term: λL•S with λ = ± ζ/2S

Number of states: smallest of (2S + 1) and (2L + 1), ∆EJ,J+1= λ (J + 1)

Shell less than half full: λ positive → |L - S| lowest in energy

more than half full: λ negative → |L + S|Hund ’s third rule

Full symbol for state: (2S+1)LJ

“normal” multiplet (d2) “inverted” multiplet (d8)

electron repulsion >> s.o.coupling Russel-Saunders

s.o.coupling >> e- repulsion j-j coupling

l + s = j for each e-, splitting determined by coupling between total angular momenta j of each e-.

Russel-Saunders ⇒ jj-coupling for d2

For 3d-metals Coulomb, exchange ≈ 1000 x s.o.

4f-metals ≈ 10 x ⇒intermediate coupling

4f- metals (Lanthanides or Rare Earths)

Ce 4f 1 2F7/2 Tb 4f8 7F9

Pr 4f 2 3H4 Dy 4f9 6H15/2

Nd 4f 3 4I9/2 Ho 4f10 5I8

Pm 4f 4 5I4 Er 4f11 4I15/2

Sm 4f5 6H5/2 Tm 4f12 3H6

Eu 4f6 7F0 Yb 4f13 2F7/2

Gd 4f7 8S7/2 Lu 4f14

Splitting GS ⇔ first excited state:

200 - 300 meV for Pr and Nd

50 - 100 meV for Sm and Eu

2 eV for Gd

300 → 800 meV for Tb → Yb

X-ray absorption: two open shells → multiplication of (symmetry of) terms of both shells.

Shell A with LA and SA, shell B with LB and SB.

|LA - LB| ≤ L ≤ LA + LB |SA - SB| ≤ S ≤ SA + SB

Simple case 2p6 3d 0 → 2p5 3d1

p5: 2P (L = 1, S = 1/2) d1: 2D (L = 2, S = 1/2)2P⊗ 2D = 1P1 + 1D2 + 1F3 + 3P0,1,2 + 3D1,2,3 + 3F2,3,4

Coupling: p5 strong spin-orbit coupling → jj

d1 weak s.o.coupling → LS

p5 d1 intermediate coupling

Fermi ’s Golden Rule: wabs∝Σq |<Φf |eq ⋅ r|Φi >|2 ρf (Ehν- Ei )

Atomic wavefunctions: <Φf |eq ⋅ r|Φi > ⇒ <φ (J′M′)|eq ⋅ r |φ (JM)>

(q = polarization of light)

Wigner-Eckhart theorem:

<φ (J′M′)|Pq |φ (JM)> = (-1)J-M ( ) <φ (J′)||Pq ||φ(J)>

3J symbol ≠ 0 if: ∆ J = (J′- J) = -1, 0, +1 and J′+ J ≥ 1

∆M = (M′- M) = q

J 1 J′- M q M′

|<φ (J′)||Pq ||φ (J)>|2 : Line strength of transition

Hartree-Fock theory with relativistic corrections

Slater electronic (Fk) and exchange (Gk) integrals.

dd: F0, F2, F4

ff : F0, F2, F4, F6

pd : F0, F2, G1, G3

df : F0, F2, F4, F6, G1, G3, G5

2p6 3d 0 → 2p5 3d1 : 1P1 (LS coupling), J = 1 (jj coupling)

Intermediate coupling: also 3P1 and 3D1

Ti4+, atomicmultiplet.

3P1

3D11P1

[ ] [ ] [ ] 23,11

10

1 || PDFPSI XAS ∝

3P03P1

3D11P1

3P23D2

3F21D2

3D33F3

1F33F4

-3.281 1.0-2.954 -0.94 0.30 0.080.213 -0.19 -0.77 0.605.594 0.24 0.55 0.79

-2.381 0.81 -0.46 0.01 0.34-1.597 -0.03 -0.50 0.56 -0.653.451 0.04 -0.30 -0.82 -0.473.643 -0.57 -0.65 -0.06 0.48

-2.198 -0.21 0.77 0.59-1.369 0.81 -0.19 0.543.777 -0.53 -0.60 0.59

-2.481 1.0

dLSpLSELECTROeff HHHH 32 −− ++=1515 32||32

12

2 dpdpH re

ELECTRO =

pslpH ppppLS 2||22 ⋅=− ς

dsldH ddddLS 3||33 ⋅=− ς

000000000

MatrixEnergy

100010001

rsEigenvecto

Energy Levels Intensities0.00 3P 0.000.00 3D 0.000.00 1P 1.00

H = 0 : only 1P1 line present (inital state 1S0, ∆J = 0, ±1 ∆S = 0 ∆L = ±1 )

Eigenvectors:

1st line: 3P 2nd: 3D 3rd: 1P

Intensity: square of 1P-character in state

Energy: diagonalize energy matrix

591.3000671.0000345.1−

MatrixEnergy

100010001

rsEigenvecto

Energy Levels Intensities-1.345 3P 0.00+0.671 3D 0.00+3.591 1P 1.00

HELECTRO ≠ 0, no spin-orbit coupling :

still pure L-S coupling, no mixing of different terms

still only 1P1 line present

000.0335.1312.2335.1944.0635.1312.2635.1944.0

−

MatrixEnergy

577.0816.00.0408.0288.0866.0707.05.05.0

−−−−−−

rsEigenvecto

Energy Levels Intensities-1.888 3P 0.00-1.888 3D 0.666+3.776 1P 0.333

HELECTRO = 0, Hls-2p = 3.776 eV :

Mixing of LS terms:

‘3P’ = 0.5 3P - 0.866 3D

‘3D’ = -0.5 3P - 0.288 3D + 0.816 1P

‘1P’ = -0.707 3P - 0.408 3D - 0.577 1P

Intensities:

‘3P’ : 0 ‘3D’ : (0.816)2 = 0.666 ‘1P’ : (-0.577)2 = 0.333

591.3335.1312.2335.1289.2635.1312.2635.1615.1

−

MatrixEnergy

792.0603.0089.0248.0185.0951.0557.0776.0297.0

−−−

rsEigenvecto

Energy Levels Intensities-2.925 3P 0.008+0.207 3D 0.364+5.634 1P 0.628

HELECTRO : F2 = 5.042 eV, G1 = 3.702 eV, G3 = 2.106 eV

Hls-2p = 3.776 eV

-4 -3 -2 -1 0 1 2 3 4 5 6 70

2

4

6

(d)

(c)

(b)

(a)

Inte

nsity

Energy (eV)

3P 3D 1P

a) H = 0

b) Hls-2p

c) Hel.

d) Hel. + Hls-2p

• TiIV L2,3 edge: 3d0→2p53d1

• TiIV M2,3 edge: 3d0→3p53d1

• LaIII M4,5 edge: 4f0→3d94f1• LaIII N4,5 edge: 4f0→4d94f1

Edge Ti 2p Ti 3p La 3d La 4d

Average Energy (eV) 464.00 37.00 841.00 103.00

Core spin-orbit (eV) 3.78 0.43 6.80 1.12F2 Slater-Condon(eV) 5.04 8.91 5.65 10.45

Intensities:

Pre-peak 0.01 10-4 0.01 10-3

p3/2 or d5/2 0.72 10-3 0.80 0.01

p1/2 or d3/2 1.26 1.99 1.19 1.99

100 105 110 115 120 125 835 840 845 850 8551E-4

1E-3

0.01

0.1

1

10

100

1000

1E-4

1E-3

0.01

0.1

1

10

100

1000

La 3dLa 4d

Inte

nsity

(lo

g sc

ale)

Energy (eV)

35 40 45 50 55 460 465 470 475 480

Ti 3p Ti 2p

Energy (eV)

Transition Ground Transitions Term Symbols3d0→2p53d1 1S0 3 123d1→2p53d2 2D3/2 29 453d2→2p53d3 3F2 68 1103d3→2p53d4 4F3/2 95 1803d4→2p53d5 5D0 32 2053d5→2p53d6 6S5/2 110 1803d6→2p53d7 5D2 68 1103d7→2p53d8 4F9/2 16 453d8→2p53d9 3F4 4 123d9→2p53d10 2D5/2 1 2

All transitions with |J-1| ≤ J′ ≤ J+1 (∆J = 0, ±1)

M4,5-edges: 3d104fn → 3d94fn+1

4f electrons screened by 5s, 5p , 5d and 6s electrons

Calculations: spherical,unperturbedwavefunctions

magnetic and crystal fields small perturbation.

Simplest case Yb (4f13):3d104f13 → 3d94f14 2F7/2 → 2D5/2 M5 (3d5/2)

2F7/2 → 2D3/2 M4 (3d3/2)

- 4f spin-orbit coupling ≈ 1.3 eV (energy difference 2F7/2 and 2F5/2 )

- Crystal field can mix some 2F5/2-character into GS

4f12 → 3d9 4f13 or equivalently f2 → df

1S3P

1D3F

1G3H

1I ΣGround

StateDeg. 1 9 5 21 9 33 13

s2 1 1 1S0p2 1 9 5 15 3P0d2 1 9 5 21 9 45 3F2f2 1 9 5 21 9 33 13 91 3H4

J-values0 0

12

2 234

4 456

6

For 4f12 Hund’s rule GS is 3H6

Intermediate coupling: 4f spin-orbit mixes some 1I6 character into ground-state (≈ 1%) (4f s.o. = 0.33eV)

F2 = 13.175 eV, F4 = 8.264 eV andF6 = 5.945 eV

025.2408.0408.0201.2

−−−MatrixEnergy

995.0095.0095.0995.0

61

63 −

IH

rsEigenvecto

Energy Levels-2.240 I ~3H62.064 II ~1I6

3d9 4f13: 2D⊗2F S = 0, 1 L = 1, 2, 3, 4, 5

Five singlets: 1P1, 1D2, 1F3, 1G4 and 1H5

Fifteen triplets: 3P0,1,2, 3D1,2,3, 3F2,3,4, 3G3,4,5 and 3H4,5,6

GS J = 6 → J = 5, 6, 7 1H5, 3G5, 3H5 and 3H6

Strong 3d spin-orbit coupling (18eV) → J = 5 states mix

965.1486088.10163.19088.10995.1469607.10163.19607.10706.1484

−−−−

MatrixEnergy

680.0733.0116.0341.0302.0890.0649.0609.0455.0

51

53

53

−−−−−−

rsEigenvecto

HGH

Energy Levels Intensities1464.44 I 4.111466.89 II 0.521510.33 III 0.231462.38 3H6 1.16

Tm M5-edge. Peaks from left to right:3H6

I = 0.455 | 3H5> - 0.89 | 3G5> + 0.116 | 1H5>

II = -0.609 | 3H5> - 0.302 | 3G5> + 0.733 | 1H5>

From top to bottom: increasing values of Slater-Condon parameters (0 → 80% of H.F. values).

M. Pompa et al., Phys. Rev. B 56, 2267 (1997)

Pr4+ : 4f1

Pr3+ (4f2) : alltransitions

Pr3+ (4f2) : dipoleallowed transitions (200)

Experiment

Ce Dy

Configuration of most RE in metallic state: 4fn(5d6s)3

Calculations for RE3+-ions (without 5d and 6s electrons).

B.T. Thole et al., Phys. Rev. B 32, 5107 (1985)

- Experimental broadening (Gaussian): energyresolution of monochromator (0.25 eV → 0.4 eV for La → Yb).

- Lifetime broadening (Lorentzian): ∆E ≈ h/τ(uncertainty)

(τ = lifetime)

In general: the larger E, the smaller τΓ3/2 from 0.2 eV for La to 0.3 eV for YbΓ5/2 from 0.4 eV for La to 0.6 eV for Tm.

4f

3d5/2

3d3/2

Decay channels:

M4: Coster-Kronig decay possible, M5 not → M4 larger.

A) Param. Thole et al.

B) Further reduction of Slater integrals (70-80%)

C) ∆J dependent broadening

Absorption line strengths and 3d3d4f εd-decay

lifetime broadening state dependent

Polarization dependence of X-ray Absorption Spectra (Dichroism): q = -1 (right circularly polarized light)

q = 0 (linearly, // to quantization.-axis)

q = +1 (left circularly polarized light)

X-ray Magnetic Circular Dichroism (XMCD): difference in absorption for left- and right circularly pol.light.

X-ray Linear Dichroism: diff. in absorption for linearly polarized light ⊥ and // to quantization axis (q = ± 1 and q = 0 ).

Difference between q = +1 and q = -1 transitions (circular dichroism): proportional to M

Difference between q = 0 and q = ±1 transitions (linear dichroism): proportional to M2

Crystal Field: no circular dichroism, only linear.

Magnetic Field: circular and linear dichroism.

Magnetic field → (2J + 1)-fold degeneracy lifted (Zeeman-splitting)

J= 7/2

MJ = 7/2

MJ = 5/2

MJ = 3/2

MJ = 1/2

MJ = -1/2

MJ = -3/2

MJ = -5/2

MJ = -7/2

Energy of MJ - levels: EM = -gαJµBHM

Occupation of Mj-levels: Boltzmann-distribution nj (T) = e- (E/kT)/ Σ e- (E/kT)

T = 0K: only lowest lying level (Mj = -J) occupied

M5 M4

Total

q = ±1q = 0

q = +1q = -1

XLDXMCD

Axial crystal field (symmetry O20 )(Yb)

CF: no splitting of +MJ and -MJ → no circular dichroism

Ce 4f 1: Hund ’s rule G.S. 2F7/2

Absorption + dichroism: contribution of J = 5/2M.Finazzi et al, Phys.Rev.Lett. 75, 4654 (1995).

- 4f levels well screened → free ion wavefunctions can be used.

- magnetic + crystal fields small perturbation with respect to Coulomb + S.O. → Hund’s rule ground state.

- dichroism: polarization dependence of XAS: direct information on Magnetic (CD + LD) and CF effects.

3d-electrons: CEF more important (10-100 x 4f)

spin-orbit 0.1-0.01 x 4f

Octahedral symmetry (Oh): dZ2 and dx2-y2 pointing towards ligands

Tetrahedral symmetry (Th): dxy, dxz and dyz

Group theory, cubic symmetry:

eg : dz2, dx2 - y2 t2g: dxy, dxz, dyz

Octahedral symmetry Tetrahedral

Calculations Ti4+ (d0) as a function of 10DqF.M.F de Groot et al, Phys.Rev.B 41, 928 (1990).

Splitting related, but not equal, to 10 Dq

Splitting of atomic terms by crystal field (Oh): Tanabe-Sugano diagrams

Tanabe-Sugano diagram for d2

10Dq = 2.5 eV

H. Ikeno, F.M.F de Groot, E. Stavitski and I. Tanaka, J. Phys. : Cond. Matter 21, 104208 (2009).

Ti4+ 2p63d0 → 2p53d1 transition for several CF symmetries

Calculations for other TM in Oh-symmetry more complicated, number of lines very large: example Mn2+

F.M.F. de Groot et al., Phys.Rev.B 42, 5459 (1990)

Oh-symmetry, 10Dq = 0.9 eVAtomic multiplet

Mn in different ionic compounds → different valencies

Several bio-organic compoundsS.P.Cramer et al., J.Am.Chem.Soc. (1991)

Effect of crystal field on magnetic state of ion: high-spin to low spin transition. Depending on relative weight of Crystal Field and Exchange interaction

Exchange > CEF (10Dq) : high spin

Example: Co3+

C. Pinta et al, Phys.Rev.B 78, 174402 (2008).

Statistical branching ratio L3 : L2 = 2 : 1 or L3/(L2 + L3) = 2/3

Valid if 2p spin-orbit coupling >> pd Slater integrals (pure jj-coupling) and ζ3d = 0

Influence of ζ2p and pd Slater integrals on 3d8 → 2p53d9.(G. v.d. Laan and B.T.Thole, Phys.Rev.Lett.60, 1977 (1988)). Branching ratios for 3d

TM-ions (multiplet model)

Spin-orbit splitting in d-levels

d5/2

d

d3/2

- 2p3/2 → 4d3/2, 5/2

- 2p1/2 → 4d3/2, 5/2

Intensity shift from L2 to L3 edge → L3 : L2 ≥ 2 : 1

Absorption of Ni in NiO + theory.

a) ζ3d = 0 b) ζ3d = 0.1 eV(G. v.d. Laan and B.T.Thole, Phys.Rev.Lett.60, 1977 (1988)).

Multiplets and hybridization (charge transfer model)

3dN: configuration in ionized state (Co2+, Ni2+ in CoO, NiO)

Partly covalent bonding: Ground State bonding combination of 3dN and 3dN+1L

L : hole on ligand

∆ = Charge transfer energy

U = Coulomb interaction

Ground state: Φi = sin α ∗ [3dN] + cos α ∗ [3dN+1L]

Final state: Φf1 = sinβ ∗ [2p53dN+1] + cos β ∗ [2p53dN+2L]

Φf1 = -cosβ ∗ [2p53dN+1] + sin β ∗ [2p53dN+2L]

Intensity main peak: cos2(β - α) Satellite: sin2(β - α)

Experimental (a) and simulated (b,c) CrL2,3 spectra for CsINiII(CrIII[CN]6).2H2O.

(c ) Without charge transfer (d3)

(b) With 20% |3d2L>

M.A.Arrio et al., J. de Physique7, C2-409 (1997)

X-ray Absorption Spectra: screening by extra e- → hybridization similar in GS and FS → small satellites

⇒ information on ground state symmetry

X-ray Photoemission Spectra: e- out → different hybridization in GS and FS

⇒ information on ground state configuration

Interaction between multiplet effects and Charge Transfer

∆ > δ/2: compression of multiplet structure (≈ reduction of Slater integrals)

Ab-initio Configuration Interaction calculations

Charge Transfer Multiplet calculations

Configuration Composition

H. Ikeno, F.M.F de Groot, E. Stavitski and I. Tanaka,

J. Phys. : Cond. Matter 21, 104208 (2009).

Conclusion:

- Crystal (ligand) fields important → crystal field multipletcalculations including symmetry and strength of CF

- Comparison XAS and calculations:

- valency

- symmetry

- spin-state

Golden rule (again): wabs = (2π / h) |< f |T |i >|2 ρf (Ehν - Ei )

T1 = CΣq[eq ⋅ p + i (eq ⋅ p)(k⋅r)]

Replacing p by r (r =(ih/m). p)

wabs ∝ |< f |e.r |i >|2 + (1/4) |< f |e.r k.r |i >|2

|< f |e.r k.r |i >|2 ≈ 0.01 |< f |e.r |i >|2

Selection rules for quadrupolar transitions:

∆ l = 0, ± 2 ∆ s = 0 ∆ J = 0, ±1, ±2

Transitions: 1s → 3d in 3d Transition Metals

2p → 4f in Rare Earths

Quadrupolar (1s → 3d) transitions vs. p – d hybridization

E. Gaudry, D. Cabaret, C. Brouder, I. Letard, A. Rogalev, F. Wilhelm, N. Jaouen, and P. Sainctavit, Phys. Rev. B 76, 964110 (2007).

Er L3-edge (top) and dichroism (bottom)

Er2Fe14B

QP

J.C.Lang et al., Phys.Rev.B46, 5298 (1992)

- R.D. Cowan, ‘’The Theory of Atomic Structure and Spectra’’, Berkeley: University of California Press (1981).

- B.N. Figgis, ‘’Introduction to Ligand Fields’’, Interscience Publishers, John Wiley & Sons, New York (1966).

- P.H. Butler, ‘’Point Group Symmetry Applications: Methods andTables’’, Plenum Press, New York (1981).

- http://www.anorg.chem.uu.nl/people/staff/FrankdeGroot/multiplet1.htm

- F.M.F. de Groot and J. Vogel, ‘‘Fundamentals of X-ray absorption anddichroism: the multiplet approach’’, Neutron and X-ray Spectroscopies, HERCULES Vol. V.

- Core Level Spectroscopy of Solids (book), Frank de Groot and AkioKotani (Taylor & Francis, 2008)