Soft x-ray absorption spectrocopy
Peter Krüger
Graduate School of Advanced Integration Science
Chiba University, Japan

2013.09.16

Peter Krüger (Chiba University)

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Outline
1

Introducion
NEXAFS-XANES

2Experimental
Measurement modes

3

TheoryLight-Matter interaction
XAS formula
Single-electron approximation
Selection rules
XAS and DOS
Linear and circular dichroism
XMCD and sum rules
Crystal field multiplet theory

4

Original research part
X-ray absorption and linear dichroism of individual titania
nanostructures - Theory and STXM experiments
Peter Krüger (Chiba University)

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Soft X-rays
100 < hν < 3000 eV↔

12 > λ > 0.4 nm

penetration depth in condensed matter < 1µm

Principle of Light absorption

incoming flux I0 , after distance x, I (x)dI ∼ −I (x)dx absorbed in (x, x + dx)µ ≡ −dI /Idx = absorption coefficient
I (x) = I0 exp(−µx) absorption law[Beer-Lambert]
Absorption spectra = µ(λ).
[Rehr Albers RMP 2000]
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Absorption edges

From I. Koprinarov, A. P. Hitchcock

1st row elements: 1s = K-edge, transition metals: 2p = L23-edges, etc
Peter Krüger (Chiba University)

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4 / 41NEXAFS Near-edge x-ray absorption fine structure /XANES X-ray absorption near-edge structure
Chemical analysis. Example: C K-edge of polymers

From I. Koprinarov, A. P. Hitchcock
Peter Krüger (Chiba University)

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Measurement modes
Transmission.
Most direct. Correct by definition. Good for hard x-rays.
Problem: needs homogeneous sample, no thickness variation.
For soft x-rays, short attenuation length (i.e. µ large), → thinsamples required, difficult to have no thickness variation.

FY. Fluorescence yield.
Decay of core-hole by radiation (fluorescence).
Number of fluorescence photons measured.
Bulk sensitive. Liquids, organic matter, high pressure ok.
But FY small for soft x-rays, where non-radiative (Auger) decay
dominatesPeter Krüger (Chiba University)

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Measurement modes
TEY. Total electron yield.
Most used for soft XAS from solids. Shine on sample.
Count all photoemitted electrons, most easily from drain current.
Probing depth limited by electron escape depth ∼ 3–10 nm.
Surface sensitive. Vacuum conditions required.

PEY. Partial electron yield.
Measure only a part of the emitted electrons, e.g. number of
secondary electron of a certain Auger decay.
Can yield improved energy resolution and insight into decay process,but PEY may not be exactly proportional to XAS.
Surface sensitive. High vacuum required.

Peter Krüger (Chiba University)

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Theory: Light-Matter interaction
Electronic hamiltonian. Electron operators ri , pi . Nuclei at Rj .
H({ri , pi }) =

X p2
i

i

2m

+ V ({ri }, {Rj })X-ray field expressed with vector potentialA(r, t) = A0 exp(ik · r − iωt) ,

E=−

∂A
,
∂t

B=∇×A.

Interaction with electron. Momentum increased by field term.
ep −→ p − A(r, t)
c
p2
p2
e
e
e2
−→
−
A(r, t) · p +[p, A] +
A2
2m
2m mc
2mc
2mc 2
Peter Krüger (Chiba University)

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p2
p2
e
e
e2
−→
−
A(r, t) · p +[p, A] +
A2
2m
2m mc
2mc
2mc 2
Coulomb gauge ∇ · A = 0 → [p, A] = 0.
A term → single photon processes = absorption, stimulated emission.
A2 term → two-photon processes (esp. scattering).
→ Interaction Hamiltonian for absorption
Hint (t) = Hint exp(−iωt) ,Hint = −e XA(ri ) · pi
mci

1st order perturbation → transition rate
Wif =“Fermi’s Golden Rule”

2π|hφf |Hint |φi i|2 δ(Ef − Ei − ~ω)~(H(t) ∼ exp(iωt) →Wif ∼ δ(Ef − Ei + ~ω) →Peter Krüger (Chiba University)

Soft x-ray absorption spectrocopy

stimulated emission)
Cheiron School 2015.09.16

9 / 41XAS formula
X-ray beam = plane wave with polarization e.

A0 (r) = eA0 exp(ik · r).

hν < 1 keV ⇔ λ > 12 Å, much larger than core orbital.
⇒ for matrix element calculation, exp(ik · r) ≈ 1 “dipole approximation”
Instead of p, we can use r. [r, H] = i~m p and |φi’s are eigenstates of H.
Then, dropping all constants, the XAS intensity for a given initial state Φini
X
X
Iini (ω) =|hΦf |e ·ri |Φini i|2 δ(Ef − Eini − ω)ifAt low T , only ground state populated, so
I (ω) =

X
f

Peter Krüger (Chiba University)|hΦf |e ·X

ri |Φg i|2 δ(Ef − Eg − ω)iSoft x-ray absorption spectrocopy

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From many-electron to single-electron formula

I (ω) =

X|hΦf |e ·X

ri |Φg i|2 δ(Ef − Eg − ω)if

If |Φg i = |Φc0 i|φc i|Φf i = |Φ̃cj i|φk i
hφk |φc i = 0cccc
Sj ≡ hΦ̃j |Φ0 i Eg = E0 + c Ef = Ẽj + k ∆j ≡ Ẽjc − E0c
I (ω) =X
j=

X

|Sj |2X|hφk |e · r|φc i|2 δ(k − c − ω + ∆j )k

|hφk |e · r|φc i|2 δ(k − c − ω) ∗X

|Sj |2 δ(ω − ∆j )jkConvolution of one-electron XAS formula with N − 1 electron excitation
spectrum (∼ core-level XPS).

Peter Krüger (Chiba University)

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Single-electron approximation
Only j = 0 (“fully relaxed channel”) →I (ω) = |S0 |2

X|hφk |e · r|φc i|2 δ(k − ˜c − ω) ,˜c = c − ∆0k

XAS can be calculated approximately from single electron states φcand φk .
Intensity reduced |S0 | < 1. Rest 1 − |S0 | in many-body excited states(∼ photoemission shake-up satellites etc) which adds to background.
Core-level energy shifted by N − 1 electron relaxation energy tocore-hole, ∆0 . (In practice k − ˜c = Ef − Eg , from ∆SCF calc.)

Peter Krüger (Chiba University)

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Core hole — orbital relaxation — screening
ground state

final state
without relaxation
with relaxation
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What is the best potential for final state |ki ?
Final state rule
photoelectron wave function|ki computed in presence of
core-hole, and all electronic
states fully relaxed
Slater transition statecompute |ci and |ki with halfa core-holetransition: ground (0 holes)
to final (1 hole)in solids: impurity problem,
supercell calculation
Peter Krüger (Chiba University)

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Dipole selection rules
Matrix element hφk |e · r|φc i. Expand φk in atomic-like orbitals at sites Ri .
X
k
φk =
Bilm
φilm , φilm (r) = Rilk (ri )Ylm (Ωi ) , ri ≡ r − Ri ≡ (ri , Ωi )ilm

Ylm are spherical harmonics = complete, orthonormal set of angularfunctions (∼ s,p,d,f. . . orbitals). The core-orbital is localized at site ic , soonly orbitals at ic contribute to the matrix element.
Consider core-s pstate and linearly polarized light along z, e = z.
e · r = z = rY10 4π/3 spherical harmonic
Z
Z
Z
1
2
∗
3
∗
hφlm |z|φs i = dΩ dr r Rl Ylm z Rs Y00 = √dr r Rl Rs dΩYlm
Y10
3
Ylm orthonormal set ⇒ only (lm) = (10) gives non-zero integral.
⇒ selection rulehφlm |z|φs i = 0, except for (lm) = pz .

Peter Krüger (Chiba University)

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Light polarization q = 0 linear z, q = ±1 circular left/right.
Wigner-Eckart theorem
 0

0
0
l
1 l
hn0 l 0 m0 s 0 |rq |nlmsi = δs 0 s (−1)l −m
hn0 l 0 ||r ||nli−m0 q mWigner 3-j symbols (. . . ) (=angular integrals) non-zero only for:l0 = l ± 1

m0 = m + q

s0 = sdipole selection rules
circular pol., spherical harmonics
q
s
p0
p1
p−1
0
p0
s,d0
d1
d−1
1
p1
d1
d2
s, d0
−1 p−1 d−1 s, d0d−2Peter Krüger (Chiba University)

linear pol., cubic harmonics
q0
s
px
py
pz
x px s, de
dxy
dxz
y py
dxy
s, de
dyz
z pz
dxz
dyz
s, de
de = {dx 2 −y 2 , d3z 2 −r 2 }Soft x-ray absorption spectrocopy

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16 / 41Density of states (DOS)Eigenstates ψk , eigenvalues k .
X
ρ() =
δ( − k ) total DOSkρ(, r) =X|ψk (r)|2 δ( − k ) local (point) DOSkρilm () =X|hφilm |ψk i|2 δ( − k )partial (ilm) DOSk

φilm = normalized basis function centered on site i, symmetry lm

Peter Krüger (Chiba University)

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17 / 41XAS and DOS
Iq (ω) =X|hψk |rq |φc i|2 δ(k − c − ω)kDevelop |ki in local basis |ilmi
hk|rq |ic lc mc i =X

hk|ilmihilm|rq |ic lc mc i

ilm

=

X
hk|ic , lc ± 1, m + qihic , lc ± 1, m + q|rq |ic lc mc i±

Localization of |ci and selection rules → only one or two terms survive,
e.g. c = s, q = z ⇒ h||i = 0 except for i = 0, (lm) = p0
XX
Iq (ω) =|hk|ic l± mihic l± m|rq |ci|2 δ(k − c − ω)±Peter Krüger (Chiba University)kSoft x-ray absorption spectrocopy

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Iq (ω) ≈X|hic l± m|rq |ci|2±

X|hk|ic l± mi|2 δ(k − c − ω)kIq (ω) ∼ A+ ρic l+ m (ω + c ) + A− ρic l− m (ω + c )K-edge, s → p, A− = 0. L23 edges often A+  A−So, in single electron approximation, the
XAS is approximately proportional to a partial DOS at absorber site
Example: K-edge, x-polarization.

XAS ∼ px -DOS→ element-resolved / local electronic structure (unoccupied states)Peter Krüger (Chiba University)

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Example: SrTiO3, O-K edge
Partial density of states fromDFT (Wien2k) code

XAS spectrum essentially broadened O-p projected DOS. Some
improvement with final state rule.
[G. Ratdke, G. Botton, Microscopy and Microanalysis 2010]
Peter Krüger (Chiba University)

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Multiple scattering
Continuum wave expanded in energy-dependent
spherical waves |ilmi located at sites i.
X k
|ki =
Bilm |ilmi
ilm
k
Bilm
=

X

τilm,jl 0 m0 Akjl 0 m0

jl 0 m0

Akjl 0 m0 = plane-wave coefficientsτij = scattering path operatorXτij = δij ti + ti Gij tj + ti
Gik tk Gkj tj + . . .
kti atomic scattering matrix (phase shifts)
Gij free electron propagatorPeter Krüger (Chiba University)

Z. Wu et al J. Synchrotron Rad. (2001)

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Dichroism= polarisation dependence of light absorption
Linear Dichroism (LD) is the difference in absorption of light linearlypolarized parallel and perpendicular to an orientation axis (e.g. molecularaxis, crystal axis, magnetisation direction)
LD = I (||) − I (⊥) = Iz − Ix

LD is sensitive to anisotropy of electronic density (and/or atomic
structure) around the absorber sitePeter Krüger (Chiba University)

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Linear dichroism

α-quartz single crystal
hexagonal, c6=a ⇒ LD6= 0
I (||) − I (⊥) ∼ ρpz − ρpx
Si K-edge XAS[Taillefumier et al. PRB 2002]

Peter Krüger (Chiba University)

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Circular Dichroismis the difference in absorption of leftand right-handed circularly polarized
light.

Chirality/Handedness
Peter Krüger (Chiba University)

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X-ray circular dichroism of chiral molecules

Serine. [Physica Scripta T115, 873]
∼ optical activity.
Peter Krüger (Chiba University)

But effect much weaker for x-rays than vis-UV light.
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X-ray Magnetic Circular Dichroism

Fe K-edge.

Peter Krüger (Chiba University)

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Origin of magnetic circular dichroism
circular light is chiral (= parity-odd) but
also time-reversal odd −→ dichroism
for time-reversal broken statesPeter Krüger (Chiba University)

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Circular light has angular momentum (helicity)
absorption → angular momentum transferred to orbital moment of electron
if states orbitally polarized → transition probabilites different for left/right→ circular dichroism detects orbital polarization of electronic states
Ex. K-edge. 1s → p.
XMCD ∼ DOS(p+ ) - DOS(p− )orbital magnetic polarization Lz () of p-projected conduction band

However, orbital polarization of conduction-p band usually small, because
spin-orbit coupling  hybridization
Peter Krüger (Chiba University)

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L23-edge (2p → 3d) XMCD of transition elements
Strong absorption “white lines”. Strong XMCD.

Co / stepped
Pt(111)
wire, ML, bulk
strong variation of
Co orbital magnetic
moment [µB ]
wire
0.68

ML
0.37

bulk
0.31

obtained with“XMCD sum rules”
P. Gambardella et al. Nature 416, 301 (2001)
Peter Krüger (Chiba University)

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L23-edge XMCD
Iq (m) =X|hdm|rq |mc i|2

mc
2
= Rdp
Aq (m)

XAS =

m
-2
-1
0
1
2

A−
6
3
1

A0
3
4
3

A+

1
3
6

XMCD
A+ − A−
-6
-3
0
3
6

XAS
P
q Aq
6
6
6
6
6

2
mq Iq (m)hn̄m i = 6 Rdp hnh iPhn̄m i = # holes in orbital (dm), hnh i = # holes in 3d shell.
X
X
2
2
XMCD =(I+ (m) − I− (m))hn̄m i = Rdp
3m hn̄m i = − 3 Rdp
hlz i
m

XMCD
hLz i
=−
XAS
2hnh iPeter Krüger (Chiba University)morbital sum rule

[B.T.Thole et al. PRL 1992]

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Spin-orbit coupled p-states: |jµi =2 A (m, σ, j, µ)|hdmσ|rq |jµi|2 = Rdp
q
Aq
−2 ↓
−1 ↓
0↓
1↓
2↓

( 32 , − 32 )
18−
90
3+

( 32 , − 21 )
6−
80
6+

( 32 , 12 )
1−
30
6+

P

1
mc σc (1 2 mc σc |jµ) × |mc σc i

( 32 , 32 )( 12 , − 12 )
3−
40
3+

( 12 , 12 )
2−
60
12+

Rel. transition probabilities to d ↓I+ I−diff sum
2p3/2 15 25 −1040
2p1/2 155
10
20
Applies directly to |g i = |d 5 ,6 Si
d ↑ shell full, d ↓ shell empty.
E.g. high spin Mn2+ , Fe3 .

Peter Krüger (Chiba University)

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XMCD sum rules (for L23-edges)[B. T. Thole et al. PRL 68, 1943 (1992). P. Carra et al. PRL 70, 694 (1993)]

If final 3d-orbital sufficiently localized then the (3d-) orbital moment of
the atom can be found from integrated spectra as
R
XMCD
hLz i
L3+L2
R
=−
2hnh i
L3+L2 XASXMCD = I+ − I− , XAS = I+ + I− + I0
If 2p3/2 → 3d and 2p1/2 → 3d transitions do not mix, then also
RR
− 2 L2 XMCD
2hSz i + 7hTz i
L3 XMCDR=−
3hnh i
L3+L2 XAS
Tz = magnetic dipole term.
Measures anisotropy of spin-density. Often negligible.
Sum rules have many limitations, but are easy to apply and very popular.
Peter Krüger (Chiba University)

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Correlation effects – many-electron theory
For transition metal (TM) L23-edges and rare-earths (RE) M45-edge thesingle electron formula for XAS yields very poor results.
Reason: spin and orbital degrees of freedom of core-hole strongly couple to
the rather localized electrons in open final shell (3d in TM, 4f in RE)
This gives rise to strong configuration mixing between several Slater
determinants, and thus the assumption |Φf i = |Φ̃cj i|φk i is very bad.
Back to multielectron formula:XX|hΦf |e ·I (ω) =
ri |Φg i|2 δ(Ef − Eg − ω)fPeter Krüger (Chiba University)iSoft x-ray absorption spectrocopy

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Multiplets
Multiplet is a term from atomic physics. For the same electronic
configuration, i.e. C (1s2 2s2 2p2 ) the energy levels of the many electronstates split into multiplets (singlets, doublets, triplets,. . . ).
The spltting is due to the electron-electron Coulomb (and exchange)interaction.
For free atoms, total angular momentum J = L + S is a good quantum
numberwhen spin-orbit interaction is neglected (ok for light elements) then L and
S are also individually good quantum numbers.
Ex. (1s2 2s2 2p2 ) has three LS-terms:
L = 2, S = 0 (1 D), L = 1, S = 1 (3 P), L = 0, S = 0 (1 D).
The states of each term are degenerate, but there is energy splitting
between the three terms.
Peter Krüger (Chiba University)

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Toy model for multiplet effects in XAS
1s → 2p absorption of a (1s 2 ) or (1s 2 2s 2 ) ground state
single-electron states( = orbitals )Peter Krüger (Chiba University)

many-electron states

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with spin-orbit coupling

many-electron states

single-electron states

final levels and absorption intensities
Peter Krüger (Chiba University)

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L23- and M45-edge spectra: Crystal Field Multiplet model
Most popular method for calculating transition metal L23- and rare-earth
M45-edge spectra based on:
R. Cowan: Atomic structure and spectra (1980) book + program
B.T. Thole adds crystal field and magnetism (∼ 1985)
F.M.F. de Groot (Utrecht, NL) maintains code “CTM4XAS”
Choose element and valency, e.g. Mn4+
Ground state electronic conf.: (1s 2 2s 2 2p 6 3s 2 3p 6 3d 3 ) = [Ar] 3d 3Possible states: 3 electrons in 10 spin-orbs,
↑

↑ ↑

↑↓ ↑

Peter Krüger (Chiba University)

103= 120 states

S=3/2, L=2+1+0=3

term 4 F

S=1/2, L=2+2+1=5

term 2 H

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Form all possible basis states (Slater determinants) |ni
Compute Hamiltonian matrix: hn|H|mi
H = kinetic energy + V(electron-nuclei) + V(el.-el.) + spin-orbit
Matrix elements computed by Cowan’s program for free ion
Diagonalize matrix

hn|H|mi→ energy levels (eigenvalues) and wave functions (eigenstates)
Eigenstates = linear combinations of Slater determinants (= basis states):
configuration interaction
Compute and diagonalize hn|H|miFinal state energy levels Ef and wavefunctions |f i.
XAS ∼ |hf |e · r|g i|2 δ(Ef − Eg − ω)Peter Krüger (Chiba University)

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Rare-earth M45-edge (3d) absorption spectra.
Experiment and multiplet calculations (free ions. Cowan’s program.)[T. Kroll et al. New Journal Phys. 11 (2009) 025019.]

Peter Krüger (Chiba University)

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Peter Krüger (Chiba University)

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diagonalize H = H(atomic) + V(crystal field)
Multiplet interactions (coulomb, exchange integrals, spin-orbit) and crystal
field comparable→ lots of lines, states all mixed up (L,S, spin-orbit, C.F.) simple
assignment generally impossible
Crystal field approach very successful for ionic compounds in cubic
symmetry (only one empirical CF parameter)
Covalent or metallic bonding, low symmetry →many CF parameters/reduction factors, CFM model questionable
Recently, several ab initio multiplet approaches.
quantum chemistry “CAS-SCF” [H Ikeno et al PRB 83, 155107]
Wannier-orbital approach [M. Haverkort et al PRB 85, 165113]
particle-hole theories:
Time-dependent DFT [J. Schwitalla et al PRL 80, 4586]
Bethe-Salpeter equation [R Laskowski et al PRB 82, 205104]
I Multichannel multiplet scattering [P. Krüger et al PRB 70, 245120]
I
I

Peter Krüger (Chiba University)

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X-ray absorption of individual
titania nanostructures - Theory
and STXM experiments
Peter Krüger

千葉

大学院融合科学 ナノ物性

Xiaohui Zhu
Adam Hitchcock
McMasters, Hamilton, Canada

Carla Bittencourt
University of Mons, Belgium

Why ?
●

Titanium oxide has wide range of applications:catalysis, solar cells, paint, tooth paste …

●

nanostructures enhance surface/bulk ratio → higher efficiency●

1-D like shapes (nano-tubes, ribbons) promising for devices●

prope and understand electronic structure of individual nanoparticles●

our tool: x-ray absorption spectroscopy + microscopy (STXM)

Scanning Transmission X-ray Microscopy
EPU

Canadian Light Source
SpectroMicroscopy Beamline

Elliptical Polarized Undulator
all 4 quadrants movable
Linear polarization any angle
30 nm spot

Circular polarization

3

Na,H-Titanate nanotubes / “scrolls”
TiO2 (anatase) + NaOH(aq) → NaxH2-xTi3O7hydrothermal synthesis T~ 95 - 135 C[P. Umek et al., J. Nanosci. Nanotechnol. 7, 3502 (2007)]

TEM[ C. Bittencourt, P. Krüger et al, B. J. Nanotechnol.3, 789 (2012) ]Structure of nanotubesa1000 nmc

10 nm

Bulk H2Ti3O7

b

a

c

Structural model taken from S. Zhang et al, PRB 71, 014104 (2005)

TiO2 anatase: O-K edge XAS
Calculation: Density function theory, plane wave code (VASP)O-K edge XAS ~ O-p projected Density of states ○ broadening function

A -peakT2g = pi-bondB -peak
Eg = sigma-bondNanotubes: O-K edge XAS
Nanotubes
each O-site
has individual
spectralsignature

Nanotubes

O-K edge XAS of nanotubes
A B
B/A intensity increases with O-Ti coordination→ B/A = local measure of connectivity of octahedra

2

O-Ti coordination = number of O-Ti sigma bonds

2

sigma bond → Ti-3d Eg orbital → B-peak

2
1
3
4
4

C. Bittencourt, PK et al, B.J.Nanotechnol.3, 789 (2012)

OH

OH

L23-edge spectra: TiO2
Final state multiplets →all 1-electron schemes (DFT) failInstead:multichannel multiple scattering=
electron-hole multiplet coupling(ab initio)+
multiple scattering on big cluster(300 atoms)

TiO2 anatase

TiO2 rutileCalc→ short and long range effects→ spectra well reproduced forthe 1st time
[ P. Krüger, PRB 81, 125121 (2010) ]Exp

L23-edge XAS of TiO2 nanotubes
Calculation

Experiment

SrTiO3← nanotubes →TiO2anataseC. Bittencourt, PK et al, BJNano2012

D-E splitting
experiment

D-E splitting:TiO2 yes

SrTiO3 noconnectivity of TiO6 octahedra
Ti

O-Ti coordination
TiO2

Ti

3

O
Ti

SrTiO32

H2Ti3O7

2.6 (average)

Ti

O

Ti

D-E splitting= rough mesure of O-Ti coordination= number of edge shared octahedra

Nano-ribbons
2 μm

Hydrothermal synthesis
T > 155 C`

200 nm1.02 nm[A. Hitchcock et al, unpublished]Structure of nanoribbons
H2Ti3O7

Ribbon

10000

50 nm

b

100

c

NaTi3O6(OH)x2H20.

I. Andrusenko et al. Acta Cryst. B 67 (2011) 218

NaTi3O6(OH)x2H20 Optimization

DFT-LDA optimized structure. Variance with exp: a=0.9%, b=0.1%, c=3.3%.
Hydrogen position agree with chemical insight (hydrogen bonds)

Nanoribbons: STXM

2 um

OD 465 eV
5 um[X. Liu, A. Hitchcock et al, unpublished]

500 nmNanoribbons: O-K edge dichroism
linear dichroism
between along (y)and across (x,z)
polarization
Polarization

500 nm

H2Ti3O7

Experiment

NaTi3O6(OH)x2H20

Calculation

good agreement

Origin of O-K edge dichroism
view along ribbons long axis (b)cainner oxygens:
OTi4 ~ tetrahedrally coordinated→ x,y,z equivalent → no dichroismAπouter oxygen:
OTi2, all Ti-O bonds in (010) plane
Ti-O-Ti ~ right angle

Ti

Ti
O

B
σ

Origin of O-K dichroism
right angle Ti-O-Ti bond in xz plane
z

y

O-py
O-pz

O

` Ti

Ti
z

x

Ti

y
x

pz or px-orbital →py-orbital →one π + one σ bondtwo π- bonds

Origin of O-K dichroism
Three inner O atoms:→ no dichroism

O
Four outer O atoms:
in plane (ac)-plane polarisation→½σ+½π↔ A peak ~B peakz

out-of-plane (b) polarisation→ π only↔ A peak only
→ strong linear dichroism due to
directional bonding of outer O atoms

Ti
x

Ti
A

π

B

σ

Nanoribbons: Ti-L23 edge dichroism

A. Hitchcock et al. (unpublished)Nanoribbons: Ti-L23 edge dichroism
Exp

Calc

Calc

Calculation in multichannel theory
Ti 1,2,3 spectra aligned with experiment
Dichroism:
Eg-peak shift E(along) < E(across)
Calculation

E(y) < E(x,z)

agrees!

Conclusions
●

●

●

●

Nanotubes:
O-K A/B ratio = local measure of octahedra connectivity
Nanoribbons:O-K dichroism: along/across rod ↔ π/σ bonds
Ti-L23 edge well reproduced in multichannel theory,
numerically light, nanostructures no problem
First observation and first principles calculation of
L23 edge dichroism in individual nanoparticles