soft x-ray absorption and resonant scatteringcheiron2007.spring8.or.jp/pdf/djhuang.pdf3. resonant...

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Soft X-ray Absorption and Resonant Scattering

Di-Jing HuangNat’l Synchrotron Radiation Research Center, TaiwanDept. of Physics, Nat’l Tsing Hua University, Taiwan

Chairon 2007, SPring8, JapanSept. 16

[email protected]

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Soft x-ray: 250 eV ~ a few keV

2

3

Interaction of photons with matter:

• lattice structure: arrangement of atoms • electronic states• magnetic order• excitations (electronic states or phonons)

Photoelectric effectPhotoabsorptionScattering/diffraction

SynchrotronRadiation

Transmission(absorption)

Inelastic Scattering

photoemissionReflected Photons

Fluorescence

Crystal

Bragg Diffraction

4


1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption

Experimental SetupApplications

- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn

3. Resonant Soft X-ray Scattering BasicsExamples

- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D

magnetic ordering of La0.5Sr1.5MnO4

3

5

X-ray absorption

( ) ( ) μ− =dI z I z dz

( ) ( )- dI z I zdz

μ= zIzI 0e)( μ−=

absorption coefficient μ

dz

0I )(zIΔΩΦ

=Ω

ΔΩ

0

Wddσ

00 A

I=Φ

( ) ( )aW I z dz I z dzρ σ μ= =# of absorption events

# of atoms/area aμ ρσ=

1 δ β≡ − +n i (α and β are real numbers)

kzkzinkzi βδ −−= eeEeE )1(0

)(0

The wave propagating in the medium is

2μ β=

absorption cross section

6

EM waves scattering by an electron

0 0e , ˆE E E //i tin xω−=

r'z

x

y

rR

X

ψ

( + )a E v B= − ×in inm e

The dipole approximation 'r r− ≈ r 20

14

(r, ) (r ', / ) r 'A Jπε

≈ −∫c r Vt t r c d

AE ∂= − Φ −

∂t∇

Far away from the electron:ctt /'rr' −−=2

0

14

(r ', ')(r, ) r 'r r '

JAπε

=−∫c V

tt dThe retarded vector potential

0(t) e cos(t)

i

in

rr

ψ⋅

= −k rE

E

Thomson scattering length(the classical electron radius)

The scattering involves a phase shift of π

2

0 20

5

4

2.82 10 Å

ermcπε

−

=

= ×2 2

0 cosd rd

σ ψ=Ω

2 2 20 0

8cos3T r d rπσ ψ= Ω =∫

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7

X-rays scattered by an atom

'q k k≡ −k 'k

r-k r⋅ 'k r⋅

'k k= = k

kq 'k

θ2θ

2 sinq k θ=

elastic scattering

A volume element at will contribute an amount to the

scattered field with a phase factor .

3rd re q r⋅i

30 ( ) rrρ−r d

Momentum transfer (in units of )

2 2

2 2 2

2

2 2

'

2 'cos 22 (1 cos 2 )4 sin

q k k

θ

θ

θ

= −

= − +

= −

=

q k kk kkk

e q r⋅i

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X-rays scattering

∫ ⋅≡ r)er()q( rq0 df iρ

Scattering amplitude: )q(re)r( 00

rq0 frd-r i −=∫ ⋅ρ

Atomic scattering factor

Z)q( ,0 0 =→ fq (the number of electrons in the atom)

All of the different volume elements scatter in phase; each electron contributes -r0 to the scattered field.

f 0 is independent on photon energy ; it is the scattering amplitude in units of –r0.

f 0 is the Fourier transform of the charge distribution.

ω

04 4'' Im( )r f fk kπ πσ = − = −

Optical theorem

absorption cross-section

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The forced charge oscillator: a classical model of an electron bound in an atom.

'em κ= − −− Γx x xEdamping force

02 E esi tx x x e

mωω −−+ Γ + = mms /' ,/ Γ≡Γ≡ κω

)(1 ,e 22

000 Γ−−

−== −

ωωωω

imeExxx

s

ti

How does the scattering amplitude f change if the photon energy approaches an absorption edge?

2

0 2 2

e( , )( )

ikr

ss

ca inrrr t

iω

ω ω ω−

− + Γ=E E

Resonant scattering amplitude in units of -r0:

2

2 2( )ss

fi

ωω ω ω

=− + Γ

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0( , ) ( ) '( ) ''( )q qω ω ω= + +sf f f i f

Resonant scattering

As the photon energy approaches the binding energy of one of the core-level electrons,

ω

dispersion corrections

2222

222

)()()('Γ+−

−=

ωωωωωω

s

ssf

2222

2

)()(''

Γ+−Γ

−=ωωω

ωω

s

sf

Absorptionω

'f

''f

' ''f f i f= +

4 Im( )fkπσ = −

6

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12

core level

Continuum

Photon energy

Abs

orpt

ion

Atom

core level

Continuum

Solid

EF

EV

Photon energy

Abs

orpt

ion

Band resonance + atomic multiplet

atomicbackground

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13

( )2

f if i

d E E hd

σ δ ν∝ ⋅ ⋅ − −Ω ∑ Ψ ΨPAPhoto-absorption

2 3p d→

1s np→ K-edge XAS can be accurately described with single-particle methods.

L-edge XAS: the single-particle approximation breaks down and the pre-edge structure is affected by the core hole wave function. The multiplet effect exists.

Dipole approximation: 1 1ie i⋅ ≈ + ⋅ ≈k r k r( )i t i te eω ω⋅ − −≈k rA ε ε=

( ) ( )2 2f i f i

d f i E E h f i E E hd

σ δ ν δ ν∝ ⋅ ⋅ − − ∝ ⋅ − −Ω

⋅∑ ∑ ε rε P

polarization of x-ray

ε ε εˆ[ , ] ( )f iim imf i f H i E E f i im f iω⋅ ≈ ⋅ = − ⋅ = ⋅ε P r r r∵

ˆ[ , ] , if [ ( ), ] 0H V ri m

= =Pr r

If ⋅k r 1,

1s

2s

3s

K

L1

M2,3

L2

L32p1/2

2p3/2

3p1/2, 3/2M1

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ˆijM f i∝ ⋅ε r)(2 2

ifij EEMW +−= ωδπ

2p

3d

Dipole transition

1f il l lΔ ≡ − = ±

Δ ≡ −( )l f im m m

( )θφθφθ cos,sinsin,cossinr̂ =

),(cos 0,134 φθθ π Y= ),(sin 1,13

8 φθθ πφ±

± =⋅ Ye i ∓

41,1 1, 1 1,2 03 2

r̂ ( )xx y yiz

i Y Y Yεε επ ε ε− +−

+⋅ = + +ε

r̂ sin cos sin sin cosx y ze e e⋅ = + +ε θ φ θ φ θ

Absorption probability:

L. circularly polarized

Y1,1: Δ ml = +1R. circularly polarized

Y1,-1: Δ ml = - 1

Δ ml = 0

5 1 62 3 2 3n nijM p d p d+∝ ⋅ε rFor 2p 3d:

Selection rule:

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•Two absorption “peaks” from 2p 3d:

3 22 /p

L3

d

1 22 /p

L2

•Absorption cross section is proportional to numbers of d holes and density of states in the core level.

L3

L2

Fe

L-edge XAS provides information on the chemical state, orbital symmetry, and spin state of materials.

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2 64 3s d

2 74 3s d2 84 3s d 1 104 3s d

Each element has specific absorption energies. “finger print” element specific spectroscopy

Fe

CoNi

Cu

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Experimental Setup: how to measure XAS spectra?Applications





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Incident light

transmitted light

Measurement of Soft X-ray absorption

electrometer

mesh

detector

I0

10

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Photoelectric absorption

2p1/2

2p3/2

3p1/2

3p3/2

3d

1s

2s

3s

K

L1

M

L2

L3

3d

K

L1

M

L2

L3

Fluorescent x-ray emission

3d

K

L1

M

L2

L3

Auger electron emission

Photoexcitation and Relaxation

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Photoelectrons or Auger electrons

hν

M1

L2

L3

Photoabsorption

2p

d

3s

Auger electrons

secondary electrons

Kinetic energy

Inte

nsity

(log

)

secondary electrons

PhotoelectronsAuger electrons

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Auger electrons(20 Å)

Fluorescence(1000 Å)

Incident light

transmitted light

electrometer

total electrons(40 ~ 50 Å)

Measurement of Soft X-ray absorption

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soft x-ray absorption & scattering

TM: 2p 3dO: 1s 2p

direct, element-specific probing of electronic structure of TMO

TM 2p

In transition-metal oxides

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•Coulomb interactions of 3d•Interaction between core hole and valence states

53d

5 62 3p d

cdU

43d

5 52 3p d'cdU

+2Mn3Mn +

2 54 3Mns d

hole doping:LaMnO3 La1-xSrxMnO3

La0.7Ce0.3MnO3 electron doping?

3 4Mn Mn+ +→

+2Mn

+2Mn 3Mn +

MnO2

MnO

4Mn +

LaMnO33Mn +

Mitra et al., PRB 67, 92404 (2003)

3 2Mn Mn+ +→

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CuO

La/Sr

High Temperature Superconducting Cu oxides

Sr doping: La2CuO4 La2-xSrxCuO4

x2-y2

3z2-r2

eg

t2g

3d crystal field

dxy

dyz, dzx

hole doping: 3d9 3d9 + O 2p hole

La2CuO4(Cu2+ : 3d9)

28Chen et al. PRL 68, 2543 (1992)

Orbital Characters of Doped Holes in La2-xSrxCuO4

3z2-r2

eg

t2g

3d

dxy

dyz, dzx

L2

L3

ˆ ˆσ σ⊥c E cE

x2-y2

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O 1s X-Ray Absorption on TMO

O 1s

O 2p

⋅⋅⋅++=Φ + Ldd nnG

1βαIn a cluster approach

O 1s XAS

1+→ nn dd

⋅⋅⋅+=Φ +1' ndα

(neglecting 1s core hole)

A: d 9L d 9

spectral weight arising from doped holes ( α 2 << β 2)

B: d 9 d 10

upper Hubbard band

EF

Chen et al.,PRL (1991)

Photon Energy (eV)

A B

La2-xSrxCuO4

d 9 d 10L

d 10

Spectral weight transfer:a fingerprint of strong electron correlations

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ferromagnetism (FM) ferrimagnetismanti-ferromagnetism (AFM)

Magnetic materials:In some solids, individual ions have non vanishing average vector moments below a critical temperature. Such solids are called magnetically ordered.

If a solid exhibits a spontaneous magnetization, its ordered state is described as ferromagnetic or ferrimagnetic. A solid with magnetic ordering has no spontaneous called antiferromagnetic.

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Fe Cu

Density of states Density of states

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Polarization of Synchrotron Radiation

Linearly polarized Left-handed circularly polarized

Right-handed circularly polarized

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Soft X-Ray Magnetic Circular Dichroism in Absorption

MCD is defined as the difference in absorption intensity of magnetic systems excited by left and right circularly polarized light.

−+ −≡ σσMCD

Right-handedcircularly polarized light preferentially excites spin-downelectrons.

Left-handed circularly polarized light preferentially excites spin-upelectrons.

For 2p3/2 3d

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MCD sum rules

Experimental confirmation:Chen et al., PRL 75, 152 (1995)

L3

L23

( )L

A μ μ+ −≡ −∫

2

( )L

B μ μ+ −≡ −∫

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Soft X-Ray Magnetic Circular Dichroism in Absorption

Soft X-ray MCD in absorption providesa unique means to probe:

element-specific magnetic hysteresisorbital and spin momentsmagnetic coupling.

There are two ways to obtain a MCD spectrum:1) Fixing M, measure XAS with left and right

circular lights.2) Fixing the helicity of light, measure XAS with

two opposite directions of M.

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ps →MCD in K-edge or L1-edge XAS is not sensitive to spin magnetic moments. Instead, the MCD measures orbital moments.(H. W.)

2sL1

2p

L1

L2

L3

lm10

-1

1=Δ lm

0=−σ

0≠+σ

d

2s

dp →

MCD in L-edge XAS measures both spin and orbital magnetic moments because of the spin-orbit interaction in the core levels.

p

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Chen et al., PRB 48, 642 (1993)

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Nature 416, 301 (2001)

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42

T

R B = 0

B = 15 T

T

R

• Variety of fascinating macroscopic phenomena• Tunable electronic and magnetic properties

T

R

MITHTSC CMR

Correlated-Electron Materials

High Temperature SuperConductivity

Metal-to-Insulator Transition

Colossal Magneto-Resistance

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spin

charge

orbital

lattice

charge ordering: spatial localization of the charge carriers on certain sites

2+

3+

spin ordering: long range ordering of local magnetic moments

orbital ordering:periodic arrangement of specific electron orbitals

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Charge-orbital ordering in correlated electron systems

Sternlieb et al. (1996)

La0.5Sr1.5MnO4

La1.6-xNd0.4SrxCuO4

J. Tranquda et al. (1995)

Moritomo et al. (1995)

Murakami et al.(1998)

C. H. Chen et al. (1998)

La1/3Ca2/3MnO3

Small valencedisproportionation !

Δq/qtotal << 1

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• With ω =E3d − E2p, fres enhanced dramatically and ordering of Ψ3d focused.

0, k ε , 'k ε2 pΨ

3dΨ

'q k k≡ −momentum transfer

modulation vector

0

3 3 2

32 23'

~( )

dik r ik r

resd d

p pd

p

re ref

E E iε ε

ω

⋅ ⋅⋅ ⋅ΨΨ Ψ

−

Ψ

− − Γ∑

Advantage of Resonant Soft X-ray Scattering

TM

46

Elastic x-ray scattering

2

qfdd

∝Ωσ

scattering form factor

k 'k

q k' k= −momentum transfer

q

θλπθ sin4sin2 == kq

θ2

A volume element at will contribute an amount to the scattering field with a phase factor .

r3d rreiq⋅

jrj(r )eiq

qj

f ρ ⋅≡ ∑ Fourier transform of charge distribution.

Bragg condition:q = modulation vector of

charge, spin , or orbital order

rrk ⋅ r'k ⋅

' kk =elastic scattering

Fourier transform of spin distribution.

jrj(r )eiq

qj

S S ⋅≡ ∑

2

qSdd

∝Ωσ

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47

2 2

2 2 22

2 22 )int (( ) ( )e e

j jmc me e

j jmc mcj

cj j j

AA r A P AH At

s s+ ⋅ − ⋅∇×=∂

− ⋅ ×∂∑ ∑∑ ∑

2

222

int

2

2 2

2

2 2 ' 'j jiq

j

rq r

j

ijf ec e

V mi f

f H i

mcce s ii ε ε

πσ

π π ε εωω

⎛ ⎞= ⎜ ⎟

⎝ ⎠− ⋅ ×⋅∑ ∑ ii

kinetic energy m.B SO

Non-resonant

mag 32

charge

~ ~ 10f

f mcω − 6

e

mag 10~ −

σσ

X-ray magnetic scattering

Resonant

int i2

nt2 ( )n i n

i f

f H n n HE

iE

E Eπσ δ

ω −−

+∑

Blume, J. Appl. Phys. (1985)

for ~ 600 eVω

48

Resonant X-ray magnetic scattering

1±=Δ lm

electric dipole transitions

1=Δ lm

0ε ε

F1,1 F1,-1

'k

kq σ=0ε

π=ε

[ ]001

[ ]100

[ ]010

σπ ×

scattering amplitudes )(ˆ)εε(83

1,11,10*

−−⋅×−= FFzif resmag π

λHannon et al., PRL(1988)

As a result of spin-orbit and exchange interactions,magnetic ordering manifests itself in resonant scattering.

1−=Δ lm

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Resonant Soft X-ray Scattering

Charge-orbital ordering in manganite and magnetite• Wilkins et al., PRL (2003)• Thomas et al., PRL (2004)• Dhesi et al., PRL (2004)• Huang et al., PRL (2006)

…….

CDW or charge stripes in cuprate and nickelate• Abbamonte et al., Nature (2004)

• Abbamonte et al., Nature Physics (2005)• Schüßler-Langeheine et al., PRL (2005)

……

Antiferromagnetic ordering of manganite• Wilkins et al., PRL (2003)

• Okamoto et al., PRL(2007)

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magnetic ordering AFM of La0.5Sr1.5MnO4

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Inverted spinel structure (cubic)1/3: tetrahedral (A-site) Fe3+

2/3: octahedral (B-site) Fe3+, Fe2+

The Verwey transition of magnetite (Fe3O4)

oxygen

Fe3O4 is believed to be a classic example of charge ordering.

1st order

A-siteFe

B-siteFe

Verwey model:

charge order-disorder transition of B-site Fe3+ & Fe2+ with TV ~ 120 K

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• Charge ordering deduced from the Fe-O distance.

• Valence of B-site Fe: 2.4 & 2.6 (rather than 2 & 3)

• (0 0 1)c and (0 0 1/2)ccharge modulation along the c-axis

Refinement of x-ray and neutron diffraction

Wright, Attfield, and Radaelli, PRL (2001), PRB (2002)

2a

Fe2+ Fe3+B-site

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53

LDA+U calculations: charge-orbital orderingJeng, Guo, and Huang, PRL (2004)

z = 0

a2

2/a

2/a

OFe3+A Fe3+B Fe3+

B Fe2+ O

electron-rich Fe4O4 cube(3 B-Fe2+ and 1 B-Fe3+)

electron-poor Fe4O4 cube(3 B-Fe3+ and 1 B-Fe2+)

Fe2+

Fe3+ Fe3+

Fe3+

Fe3+

Breakdown of Anderson’s criterion can be explained by the existence of orbital ordering.

Leonov et al., PRL (2004)

54

Fe 3d O 2p

O 1s

≈ 1

R is the factor accounting for the enhancement of the form factor due to the O K-edge resonance.

The change in the structure factor with q=(0 0 ½)c around O K-edgeis negligible:

(0 0 L)

Γ

c

Correlation length ξ≡1/Γ = 900 Å

(0 0 ½)c resonant scattering at O K-edge of Fe3O4

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(0 0 ½)c resonant scattering at O K-edge of Fe3O4

ξ ~ 900 Å

ρ

(0 0 ½)

Huang et al., PRL (2004)Evidence for the ordering of electronic states associated with the Verwey transition.

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thermallydisordered(renormalized classical)

quantum disordered

quantum critical

Low-dimensional quantum magnetism

QCP

non-thermal parameter g(P, B, doping, …)

T

ordered at T=0

Chakravarty et al., PRL 60, 1057 (1988)

2D quantum Heisenbergantiferromagnet on a square lattice with

2 /( ) e S Bk TT πρξ ∝spin correlation

12s =

Mermin-Wagner (1966):Pure 2D magnetic order only exists at T=0. renormalized

classical

ordered

classical critical,

1JJ⊥

1C

TT

νξ

−−∼

quasi 2D

58Greven et al., PRL(1994)

Cu2+, S=1/2, TN=256 KSr2CuO2Cl2(1/2, 1/2, L/2)

TN

410JJ

−⊥ ∼

2 /( ) e S Bk TT πρξ ∝

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59

MnO

La/Sr

LaSrMnO4 La0.5Sr1.5MnO4

Half-doped single-layered manganitedoping Sr

Mn3+ (d4)

AFM(1/4, 1/4, 1/2)

TN ~ 110 K

C.O.(1/4, 1/4, 0)TCO ~ 220 K

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2cosI φ∝

(¼, ¼, 0)orbital ordering

0φ =

Mn L-edge (2p 3d)

(¼, ¼, ½)AFM ordering

q 001

(r. l

. u.)

q110 (r. l. u.)

Resonant soft x-ray scattering on La0.5Sr1.5MnO4

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61

La0.5Sr1.5MnO4(1/4, 1/4, 1/2 )

116 K

113 K

( 1) 0.75 0.04νξ ν−∝ − = ±, N

TT

T=123 KT=114 K

62

La0.5Sr1.5MnO4

116 K

2 /( ) e S Bk TT πρξ ∝semi- classical

(1/4, 1/4, 1/2 )113 K

Extension of zero-temperature AFM to T > TN

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63

Appendix: Basic of Magnetic Circular Dichroism in X-ray Absorption

Considering L-edge 2p3/2 3d absorption, and ignoring the spin-orbit interaction in the 3d bands,

2p3/2

3d

1/3 1/3 1/6

m j = 3/2 1/2 -1/2 -3/2L3

m l = 2 1 0 -1 -2

σ+=5/6

22

1,1( )2

,lm jx yi

R r Y Y j mε ε

σ +

−∝ ⋅

2ˆ, r ,l jl m j mσ ∝ ⋅ε

1 1/3 1/18

m j = 3/2 1/2 -1/2 -3/2

L3

m l = 2 1 0 -1 -2

Left-handed circularly polarized light preferentially excites spin-up electrons.

For left-handed circular light

radial part1/6 1/3 1/3

1/18 1/3 1

σ−=25/18

22

1, 1( )2

,x ylm jR r Y Y j m

iε εσ −−

+∝ ⋅

For light-handed circular light

Right-handed circularly polarized light preferentially excites spin-down electrons.

σ+=25/18σ−=5/6

64

↑= 1,123

23 , Y

↓+↑= 1,131

0,132

21

23 , YY

↓+↑= −−

0,132

1,131

21

23 , YY

↓= −−

1,123

23 , Y

22

1, 1( ) ,2

x ylm j

iR r Y Y j m

ε εσ ± ±∝ ⋅

∓

2

3 1 12,2 1,1 2 2 3,

2x y

j

iY Y j m

ε εσ +

−∝ = = =

2ˆ, r ,l jl m j m∝ ⋅εσ

( )θφθφθ cos,sinsin,cossinr̂ =

),(cos 0,134 φθθ π Y= ),(sin 1,13

8 φθθ πφ±

± =⋅ Ye i ∓

41,1 1, 1 1,03 2 2

r̂ ( )x y x yi izY Y Yε ε ε επ ε− + +

−⋅ = + +ε

r̂ sin cos sin sin cosx y ze e e⋅ = + +ε θ φ θ φ θ

31

1,11,12,231

21

23

1,12,2 , ==== YYYmjYY j∵

∑+

−=

=21

21

2211)|( 2211

ll

llllmmlml YlmmlmlYY

2211

21

),|( 2211 mlmlmm

lm YYmlmlmlY ∑=

Clesbsch-Gordan coefficients

1/3 1/3 1/6

1/18 1/3 1

m j = 3/2 1/2 -1/2 -3/2

j=3/2

L3

m l = 2 1 0 -1 -2

2

3 1 12,0 1, 1 2 2 18,

2x y

j

iY Y j m

ε εσ − −

+∝ = = =

.....0,261

1,11,1 +=− YYY181

1,11,10,231

21

23

1,10,2 , ==== −− YYYmjYY j∵ 2,21,11,1 YYY =

33

65

↑= 1,123

23 , Y

↓+↑= 1,131

0,132

21

23 , YY

↓+↑= −−

0,132

1,131

21

23 , YY

↓= −−

1,123

23 , Y

22

1, 1( ) ,2

x ylm j

iR r Y Y j m

ε εσ ± ±∝ ⋅

∓

2

3 1 12,1 1,1 2 2 3,

2x y

j

iY Y j m −

+

−∝ = = =

ε εσ

2ˆ, r ,l jl m j mσ ∝ ⋅ε

41,1 1, 1 1,03 2 2

r̂ ( )x y x yi izY Y Yε ε ε επ ε− + +

−⋅ = + +ε

3 1 2 12,1 1,1 2,1 1,1 1,02 2 3 3, −= = = =∵ jY Y j m Y Y Y

∑+

−=

=21

21

2211)|( 2211

ll

llllmmlml YlmmlmlYY

2211

21

),|( 2211 mlmlmm

lm YYmlmlmlY ∑=


1/3 1/3 1/6

1/18 1/3 1

m j = 3/2 1/2 -1/2 -3/2

j=3/2

L3

m l = 2 1 0 -1 -2

2

3 1 12, 1 1, 1 2 2 3,

2x y

j

iY Y j m

ε εσ −

− − −

+∝ = = =

.....1,221

0,11,1 += YYY

3 1 2 12, 1 1, 1 2, 1 1, 1 1,02 2 3 3, −

− − − −= = = =∵ jY Y j m Y Y Y

.....1,221

0,11,1 += −− YYY

66

↑= 1,123

23 , Y

↓+↑= 1,131

0,132

21

23 , YY

↓+↑= −−

0,132

1,131

21

23 , YY

↓= −−

1,123

23 , Y

22

1,1( ) ,2

x ylm j

iR r Y Y j m

ε εσ +

−∝ ⋅

2

3 1 12,1 1,1 2 2 3,

2x y

j

iY Y j m

ε εσ +

−∝ = = =

3 1 2 12,1 1,1 2,1 1,1 1,02 2 3 3, jY Y j m Y Y Y= = = =∵

∑+

−=

=21

21

2211)|( 2211

ll

llllmmlml YlmmlmlYY

2211

21

),|( 2211 mlmlmm

lm YYmlmlmlY ∑=


1 1/3 1/18

m j = 3/2 1/2 -1/2 -3/2

j=3/2

L3

m l = 2 1 0 -1 -2

.....0,261

1,11,1 +=− YYY3 1 1 1

2,0 1,1 2,0 1,1 1, 12 2 3 18, jY Y j m Y Y Y −= = = =∵2,21,11,1 YYY =

1/3 1/3 1/6

m j = 3/2 1/2 -1/2 -3/2

j=3/2

L3

m l = 2 1 0 -1 -2

.....1,221

0,11,1 += YYY2

3 1 12,0 1,1 2 2 18,

2x y

j

iY Y j m

ε εσ −

+

−∝ = = =

11,1 1, 1 2,06 .....Y Y Y− = +

2

3 32,2 1,1 2 2, 1

2x y

j

iY Y j m

ε εσ +

−∝ = = =

3 32,2 1,1 2,1 1,1 1,12 2, 1jY Y j m Y Y Y= = = =∵

soft x-ray absorption and resonant scatteringcheiron2007.spring8.or.jp/pdf/djhuang.pdf3. resonant...

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