X-ray absorption spectroscopy

USTV school on the characterization of glass structure,Grenoble, November 17-22, 2019

Institut Néel, CNRS/Université Grenoble Alpes

Yves Joly

2

References:

X-Ray Absorption and X-ray Emission Spectroscopy : Theory and ApplicationsEdited by J. A. van Bokhoven and C. Lamberti, Wiley and sons (2016).ISBN : 978-1-118-84423-6.

and more specifically, chapter 4 :"Theory of X-ray Absorption Near Edge Structure" Yves Joly and Stéphane Grenier.

About X-ray Raman spectroscopy:"Full potential simulation of x-ray Raman scattering spectroscopy“Y. Joly, C. Cavallari, S. A. Guda, C. J. SahleJ. Chem. Theory Comput. 13, 2172-2177 (2017).

Soon International Tables for Crystallography, Volume I on XAS

3

Outline

I – X-ray Matter interaction and absorption spectroscopy

II – Examples in XANES

III – X-ray Raman Scattering

A- Introduction

F – About ab initio simulation

B- Some properties

C – From multi-electronic to mono-electronic

D – Absorption cross section formula

E – Selection rules

G – EXAFS in Brief

4

ħ, I0

ħ, I

ħ’, I’

ħF, IF

Transmission

Fluorescence

Electrons

A-Introduction

Elastic scattering Resonant x-ray scattering

Inelastic scattering X-ray Raman

5

Monochromatic beam ħ, I

Absorption

Beer-Lambert law :

ln(I0/I) = D

n

1iiσV

1μ V : (cell) volume with n atoms

i = absorption cross section

For a solid :

cm2 or Mbarn 1 Mbarn = 10-18 cm2

I = I0exp(-D)

: total linear absorption coefficientD: sample thickness

homogeneousisotropicmaterial

I0

6J.-L. Hazemann, O. Proux et al.CRG FAME, ESRF

7

For any chemical element there is a set of absorption edges

K L2 M2

H 13.6

C 284.2

Fe 7112 720 52.7

Ag 25514 3524 604

U 115606 20948 5182

Some edges (eV)

Deeper is the edge

Shorter is the time life

Broader is the edge

n = 1

n = 2

n = 3

ℓ = 0

ℓ = 0

ℓ = 0ℓ = 1

ℓ = 2

ℓ = 1

Fermi

K

L1

L2

M2

L3

M1

M3M4

M5

spin-orbit spreads level between 2

Experiment Ch. Den Auwer et al.

8

X-ray absorption spectra are a signature of the probed material

B – Some properties of X-ray absorption

Dependence on the local geometry

9

Dependence on the polarization light

Pleochroism or dichroism is the change in color evident as the mineral is rotated under plane-polarized light.

Due to adsorption of particular wavelengths of light. transmitted light to appear colored.

Function of the thickness and the particular chemical and crystallographic nature of the mineral.

Also true in the X-ray energy range :

B. Poumellec et al., DCI-LURE

10

Dependence on the oxidation state

SH

SO

=

SO O

Pichon et al. IFP, LURE

K+2 (SO4)

2-

11

Circular polarization light and ferromagnetic materials

Measurements with circular polarization are sensitive to magnetic materials

XMCD = left - right

12

Atom shell 1

Atom shell 2

Final states are calculated by simple interference between the outgoing wave and the backscattered waves by the different shells

EXAFS gives information on

- The number of atoms per shell,

- The distance of the shells

EXAFS and XANES

0

0.5

1

1.5

8900 9000 9100 9200 9300 9400

Seuil K du Cuivre dans Cu et YCuO2.5

Energie (eV)

Cuivre, métal

YCuO2.5

XANES

EXAFS

XANES gives information on

- 3D arrangement

- Local symmetry,

- Electronic and magnetic structure

13

X-ray absorption spectroscopies are

- local spectroscopies

- Selective on the chemical specie

- Process involved are complex…

EF Absorption cross section

g

f

Core hole

Pertubated electronic structure

C – From multi-electronic to mono-electronic

14

Characteristic times

1 – Time of the process « absorption of the photon »t1 = 1/Wfi, Wfi absorption probabilty

t1 < 10-20 s

2 – Time life of the core holet2 = ħ /Ei, Ei width of the level

for 1s for Z = 20 up to 30, Ei ≈ 1 eVt2 ≈ 10-15 à 10-16 s

3 – Relaxation time of the electronEffect on all the electrons of the field created by the hole and the

photoelectron. Many kinds of process, multielectronic.t3 ≈ 10-15 à 10-16 s

4 – Transit time of the photoelectron outward from the atomDepends on the photoelectron kinetic energy, for Ec = 1 à 100 eV

t4 ≈ 10-15 à 10-17 s

5 – Thermic vibrationt5 ≈ 10-13 à 10-14 s

multi-electronic process can be seen at low energy of the photoelectron

X-ray absorption takes a snap shot of the pertubated material

15

Non localized final statesLocalized final states

EF Absorption cross section

g

Absorption cross section

EF

g

- Interaction with the hole mono-electronic approach

Ground state theory: DFT

spatialy

In energy

signal back quickly to 0

- Several possible electronic states…

16

A. Scherz, PhD Thesis, Berlin

Non localized final statesLocalized final states

Ground-state theorymultiplet

O. Proux et al.FAME, ESRF

Intermediatesituation …

17

Improvements in progress:Bethe Salpeter Equation (Shirley…)Time-Dependent DFT (Schwitalla…)Multiplet ligand field theory using Wannier orbitals (Haverkort…)Multichannel multiple scattering theory (Krüger and Natoli)Dynamic mean field theory (Sipr…)Quantum chemistry techniques, Configuration interaction…

Multiplet ligand field theory:multi-electronic but mono-atomic L23 edges of 3d elements M45 edges of rare earth

DFT:Multi-atomic but ground state theory (mono-electronic) K, L1 edges L23 edges of heavy elements

18

Plane wave :

Interaction Hamiltonian:

𝑨 𝒓, 𝑡 𝐴 𝑎𝑒 𝒌.𝒓 𝜺 𝑎 𝑒 𝒌.𝒓 𝜺∗

𝑬 𝒓, 𝑡 𝑖𝜔𝐴 𝑎𝑒 𝒌.𝒓 𝜺 𝑎 𝑒 𝒌.𝒓 𝜺∗

𝑩 𝒓, 𝑡 𝑖𝐴 𝑎𝑒 𝒌.𝒓 𝒌 𝜺 𝑎 𝑒 𝒌.𝒓 𝒌 𝜺∗

𝐻𝑒𝑚 𝒑. 𝑨

𝒑 𝑖ℏmomentum

D – Absorption cross section formula

o

f

g

ℏ𝒌𝜺

Signal depends on :

- initial states g

- final states f

- a transition operator o

light polarisation light wave vector

19

𝜎 𝜔 4𝜋 𝛼ℏ𝜔 𝐹 𝑜 𝐼 𝛿 ℏ𝜔 𝐸 𝐸

E1El. dipole

E2El. quadrupole

o

f

g

𝑃2𝜋ℏ 𝑓 𝑜 𝑔 𝛿 ℏ𝜔 𝐸 𝐸

𝑓 𝑜 𝑔 𝑓 𝒓 𝑜 𝒓 𝑔 𝒓 𝑑𝒓

Golden Rule : Dirac (1927) called by Fermi in 1950 Golden Rule n° 2

𝐹 and 𝐼: multi-electronic final and initial states

o 𝜺 · 𝒓 𝜺 · 𝒓𝒌 · 𝒓

Multi-electronic system Transition from 𝐼 to 𝐹

Absorption cross section formula

20

Multielectronic:

Mono-electronic absorption cross section formula

𝜎 𝜔 4𝜋 𝛼ℏ𝜔𝑆 𝑓 𝑜 𝑔 𝛿 ℏ𝜔 𝐸 𝐸 ∆𝐸

𝜎 𝜔 4𝜋 𝛼ℏ𝜔 𝐹 𝑜 𝐼 𝛿 ℏ𝜔 𝐸 𝐸

Ground state (≈ mono-electronic) approximation:

Relaxation effect of the “other” electrons

o 𝜺 · 𝒓 𝜺 · 𝒓𝒌 · 𝒓

𝑓 is by default calculated in an excited state:- with a core-hole- an extra electron on the first non occupied level

21

𝜎 𝜔 4𝜋 𝛼ℏ𝜔 𝑓 𝑜 𝑔 𝛿 ℏ𝜔 𝐸 𝐸

Core-hole and photoelectron time life effects

Lorentzian convolution

𝛤 𝛤 𝛤 𝐸

𝛤 : core-hole widthClassical experiment: known tabulated values

M. O. Krause, J. H. Oliver, J. Phys. Chem. Ref. Data 8, 329 (1979)

Experiment using High resolution fluorescence mode:Reduced value

𝛤 : photoelectron state widthDue to all possible inelastic processIncrease with energy

12𝜋

𝛤

𝐸 𝐸

22

E – Selection rules

Inside the absorbing atom (non magnetic case) :Spherical harmonic

Solution of the radial Schrödinger equationAmplitudes. Contains the main

dependence on the energy. Contains theinformation on the density of state

𝑓 𝒓 𝑎ℓ 𝐸 𝑏ℓ 𝑟 𝑌ℓ 𝑟ℓ

K edge :

LII edge :

∣ , ⟩ 𝑔 𝑟 0𝑌 ∣ , ⟩ 𝑔 𝑟 𝑌

0

∣ , ⟩ 𝑔 𝑟𝑌

𝑌∣ , ⟩ 𝑔 𝑟

𝑌

𝑌

ℓ 0

ℓ 1

Core states g

Final states f

𝑓 𝑜 𝑔

The expansion of 𝜺 · 𝒓 and 𝒌 · 𝒓 in real spherical harmonics gives :

For example, polarization along z, wave vector along x :

The transition matrix is then:

Radial integralGaunt coefficient

(tabulated constant related to the Clebch-Gordon coefficient)

non zero, only for some ℓ and m gives the

selection rules

Slowly varying with E

Strong dependence with ℓo

1o mo= 0

2o mo= 1

Transition operator o

𝜺 · 𝒓4𝜋3 r𝑌

𝜺 · 𝒓 𝑧 𝑟 cos 𝜃 𝑐 r𝑌

𝜺 · 𝒓 𝒌 · 𝒓 𝑘𝑧𝑥 𝑘𝑟 sin 𝜃 cos 𝜃 cos 𝜑 𝑐 𝑘𝑟 𝑌

𝑓 𝑜 𝑔 𝑐ℓ 𝑘ℓ 𝑎ℓ 𝐸ℓ

𝑏ℓ 𝑟, 𝐸 𝑔ℓ 𝑟 𝑟 ℓ 𝑑𝑟 𝑌ℓ∗𝑌ℓ 𝑌ℓ 𝑑 𝑟

23

24

Angular integral non zero only for :

ℓ : same parity than ℓg + ℓo

|ℓg - ℓo| ≤ ℓ ≤ ℓg + ℓo

with complex spherical harmonics :

m = mo + mg

Dipole: ℓ = ± 1Quadrupole : ℓ = 0, ± 2

Dipole probed state

Quadrupole probed state

K, LI, MI, NI, OI p d

LII, LIII, MII, MIII, NII, NIII, OII, OIII

s - d p - f

MIV, MV, NIV, NV, OIV, OV

p - f s - d - g

25

K edge case :

dipole component and polarization along z :

one probes the pz states projected onto the absorbing atom

quadrupole component, polarization along z, wave vector along x :

one probes the dxz states projected onto the absorbing atom

kB

e

z

x

XANES is very sensitive to the 3D environment

26

EF

g

f

Fluorescence

Transmission

Electrons

Whatever is the detection mode,

- one measures the transition probability between an initial state g and a final state f

- Thus one measures the density of state

- The density of state depends on the electronic and geometric surrounding of the absorbing atom

𝜎 𝜔 4𝜋 𝛼ℏ𝜔 𝑓 𝑜 𝑔 𝛿 ℏ𝜔 𝐸 𝐸 ∝ ℛ𝒜 𝑎ℓℓ

𝜌 𝐸 𝑓 𝑓 4𝜋𝑟 𝑏ℓ 𝑑𝑟 𝑎ℓℓ

27

F- About ab initio simulation

About the potential

As in most electronic structure calculations the choice of the potential is important

One body calculation = local density approximation (LSDA)

Potential = Coulomb potential + exchange-correlation potential

Depends just on the electron

density

Different theories

X

Hedin and Lundqvist

Perdew…..

Depends also on the electron kinetic energy

0

0,05

0,1

0,15

0,2

0,25

0 50 100 150 200

Energy (eV)

Example of calculation

Xanes spectra of the copper K-edge in copper fcc

X

Energy dependent Hedin and Lundqvist

28

And about the shape of the potential

The muffin-tin approximation the MT of the LMTO program

(almost) always used in the multiple scattering theory

Constant between atoms

Spherical symmetry inside the atoms

Before approximation After approximation

Overlap

Empty sphere

With the muffin-tin, there are always 2 parameters : overlap and interstitial constant

29

The multiple scattering theory

Two ways to explain it :

the Green function approach

the scattering wave approach

Just one atom :

We build a complete basis in the surrounding vacuum (Bessel and Hankel functions)

We look how the atom scatters all the Bessel functions (phase shift theory)

Atomic scattering amplitude

Bessel Outgoing Hankel

Photoelectron wave vectorAmplitude

Solution of the radial Schrödinger

equation

𝜑 𝒓 𝑎ℓ𝑏ℓ 𝑟 𝑌ℓ 𝑟 𝑗ℓ 𝑘𝑟 𝑖𝑡ℓ ℎℓ 𝑘𝑟 𝑌ℓ 𝑟

30

Several atoms ( cluster )

Each atom receives not only the central Bessel function but also all the back scattered waves from all the other atoms

The problem is not anymore spherical

We have to fill a big matrix with the scattering atomic amplitudes of each atom and the propagation function from one atom to another

Matrix containing the atomic scattering amplitudesMatrix containing the

geometrical terms corresponding to the scattering from any site

“a” of the harmonic L=(ℓ,m) towards any site “b” with the

harmonic L’

𝜏1

1 𝑇𝐻 𝑇

31

one gets for the absorption cross section:

Multiple scattering amplitude

Green’s function

From the optical theorem :

𝜎 𝜔 4𝜋 𝛼ℏ𝜔 ℐ 𝑔 𝑜∗ 𝑏ℓ𝑌ℓ 𝜏ℓℓ 𝑏ℓ 𝑌ℓ 𝑜 𝑔

ℓ ℓ

𝑎ℓ 𝑎ℓ∗ ℐ 𝜏ℓ

ℓ

𝜑 𝒓 𝑎ℓ 𝐸 𝑏ℓ 𝑟, 𝐸 𝑌ℓ 𝑟 χℓ

Wave function in the atom:

When no spin-orbit:

Then one gets the scattering amplitude of the central atom in the presence of its neighboring atoms.

32

When considering only one scattering process : EXAFS

When considering a limited expansion of scattering processes : path expansion XANES without the first eV

When considering all the scattering processes : XANES including the edge

𝜏1

1 𝑇𝐻 𝑇

𝜏1

1 𝑇𝐻 𝑇 ≅ 𝑇 𝑇𝐻𝑇

𝜏1

1 𝑇𝐻 𝑇 ≅ 𝑇 𝑇𝐻𝑇 𝑇𝐻 𝑇 ⋯ 𝑇𝐻 𝑇

Different codes:- FeffFit- GNXAS

33

Absorbing atom

The finite difference method

22

2 2h

xhxhxx

x ffff

Discretization of the Schrödinger equation on a grid of points

016,2,2

jjfifi h

EVh

𝜑 𝒓 𝑎ℓ 𝐸 𝑏ℓ 𝑟, 𝐸 𝑌ℓ 𝑟 χℓ

𝜑 𝒓 𝑗ℓ 𝑘𝑟 𝑌ℓ 𝑟 𝑖 𝑠ℓ 𝐸 ℎℓ 𝑘𝑟 𝑌ℓ 𝑟ℓ

Interest : free potential shapeDrawback : time consuming Use of MUMPS library (sparse matrix solver)

40 times faster low symmetry possibleS. Guda, et al. J. Chem. Theory Comput. 11, 4512 (2015)

+ continuity at area bordersBig matrix, unknowns: if ,

34

Code for XANES using the mono-electronic approach (not complete)

C. R. Natoli(INFN, Frascati, Italy, 1980)

Cluster approach -Multiple scattering theory

Now with a fit by M. Benfatto

CONTINUUMThe first !

MXAN

J. Rehr,A. Ankudinov et al.(Washington. U., USA, 1994)

Cluster approach -Multiple scattering theory - path expansionfit – self consistency

FEFF feff.phys.washington.edu/feff/

T. Huhne, H. Ebert(München U., Germany)

Band structure approach – Full potential

SPRKKR olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR/

P. Blaha et al.(Wien, Austria)

Band structure, FLAPW Wien-2k susi.theochem.tuwien.ac.at

Y. Joly, O. Bunau(CNRS, Grenoble)

Cluster approach, MST and FDM

FDMNES www.neel.cnrs.fr/fdmnes

K. Hermann, L. Pettersson(Berlin, Stockholm)

LCAO STOBE w3.rz-berlin.mpg.de/~hermann/StoBe/

D. Cabaret et al.(LMPC, Paris)

Band structure, Pseudo potential

Xspectra /Quantum-espresso

www-ext.impmc.jussieu.fr/~cabaret/xanes.html

35

G. EXAFS in brief

A first order approximation

Absorbing atom part Backscattering by the neighboring atom

𝜎 𝐸 𝐾 𝑓 𝜺 · 𝒓 𝑔

𝑓 𝑓 𝛿𝑓

𝜎 𝐸 𝐾 𝑓 𝜺 · 𝒓 𝑔 2𝐾ℛ𝑒 𝛿𝑓 𝜺 · 𝒓 𝑔 𝑔 𝜺 · 𝒓 𝑓

𝛿𝑓

𝑓

𝜎 𝐸

36

Inelastic mean free path

Thermal disorder

and Fourier transform fit of R, N and

Theory of phase shift:

𝜒 𝐸𝜎 𝜎

𝜎2ℛ𝑒 𝛿𝑓 𝜺 · 𝒓 𝑔 𝑔 𝜺 · 𝒓 𝑓

𝑓 𝜺 · 𝒓 𝑔

𝛿𝑓 𝑓 𝜋1

2𝑖𝑘 1 ℓ 2ℓ 1 𝑒 ℓ sin 𝛿ℓℓ

→Electron scattering amplitude

𝜒 𝑘𝑁

𝑘𝑅𝑒 𝑒 𝑓 𝜋 sin 2𝑘𝑅 2𝛿 𝛿

𝛿 𝛿

𝑘 ℏ 𝐸 𝐸

𝑘𝑅

𝑅

phase shifts

37

Fourier transform

38

Examples in XANES

39

Linear dichroism in rutile TiO2

Rutile TiO2

0 10 20 30 40 50

A1A

2 A3

Full line : Calculation // z

perp. z

e

e

Dotted experiment

ke

(b)

k

e

(c)

z

xy

ke

(a)

dipolequadrupole

Important linear dichroism

Influence of the core-hole

Shift of the 3d

Experiment by Poumellec et al.

40

3 4 5 6 7 8 9 10 11 12Energy (eV)

A1

A3A

2

d

dq

q

dz2

pz

By the dipole component which probes the pstates, we also observe the projection of the

d states of the neighboring Ti

With a precise analysis of the XANES features, we get a detailed description of the electronic

structure

Quantitative analysis of the pre-edge

41

Organic molecule on surface : acrylonitrile

For the light element

- Long hole life time

- Good energy resolution

- Study of the first non occupied molecular orbitals

Normal

grazing

The molecules are deposited on a surfaceThe experiment is performed along 2 directions

Normal incidence, x-ray probe px and py orbitals, projections of the antibonding molecular orbitals y* and sGrazing incidence, x-ray probe pz orbitals, projections of the antibonding molecular orbitals z*

y

xN C

H

Scheme of acrylonitrile

py

y*

px

42

Tourillon, Parent, Laffont (LURE)

XANES lets to determine how are arranged the molecules

Normal = ½cos(x+y)+sinz

z

43

Iron in solution

Fe2+ Fe-O = 2.16 AFe3+ Fe-O = 2.06 A

No need of mixing Fe2+ - Fe3+

With Wang and Vaknin, Ames Laboratory

shoulder

Shift

Effect of second shell

44

Pt13 cluster on -Al2O3 under H2

High resolution XANES+

DFT-Molecular dynamics(VASP)

+XANES simulation

Many parameters13 Pt positionsH numberH positions2 difference facesSome size dispersionSeveral site absorption…

A. Gorczyca et al., coll. IFPEN, Solaize, FranceJ.-L. Hazemann, O. Proux…

Exp: FAME / ESRF

11560 11570 11580 115900,0

0,5

1,0

1,5

2,0

2,5

Nor

m. A

bs.

Energy (eV)

500°C, P(H2) = 10-5 bar

25 °C, P(H2) = 10-5 bar

500 °C, P(H2) = 1 bar

25 °C, P(H2) = 1 bar

P(H2) ↗ or T ↘:• ↘ intensity of white line• WL wider• Energy shift• ↗ post white line features

Experimental observations

45

46

Pt13 / – alumina case

Area of calculationcontaining the Pt (H) clusterand substrate (distorted) atoms

L3 simulation in the ground state

1 calculation gives the 13 atomsabsorption spectra

MST

Set of atomistic modelsfrom DFT (VASP)

11560 11580 11600 116200,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

Nor

m. A

bs.

Energy (eV)

18 H

Experimental spectrum25°C, P(H2) = 1bar

Spectra sensitivity on models

(100) face only

47

48

11560 11580 11600 116200,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

Nor

m. A

bs.

Energy (eV)

48

0 H

18 H

Experimental spectrum25°C, P(H2) = 1bar

Spectra sensitivity on models

(100) face only

49

49

11560 11580 11600 116200,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

Nor

m. A

bs.

Energy (eV)

18H

Experimental spectrum25°C, P(H2) = 1bar

20H

Sensitive tool for the quantificationof hydrogen coverage and morphology

(100) face only

Spectra sensitivity on models

11560 11570 11580 115900

1

2

3

4

5

Nor

mal

ized

Abs

orpt

ion

(A. U

.)

Energy (eV) Pt13H18

25°C, P(H2) = 1bar

500°C, P(H2) = 1bar

25°C, P(H2) = 10-5 bar

500°C, P(H2) = 10-5 bar

Pt13H16

Pt13H4

Pt13H10

Pt13H20

Pt13H14

Pt13H8

Pt13H2

(100)(110)

Sim

Exp

Simulations vs experiments : Best fits

Identification of hydrogen coverage / morphology on eachsurface and for each experimental condition

Pt Al

O H

A. Gorczyca et al. Angew. Chem. Int. Ed., 53, 12426 (2014)50

51

X-ray Raman Scattering

52

X-ray Raman Scattering (XRS)

Inelastic scattering technique energy loss ≈ absorption edge energy

or Non Resonant X-ray Inelastic Scattering (or EELS on Trans. Elec. Micr.)

EF

g

f

𝒒 𝒌 𝒌2

𝜔 , 𝒌𝜔 , 𝒌ℏ𝜔

ℏ𝜔

ℏ𝜔 ℏ𝜔 ℏ𝜔 𝐸 𝐸

First experiments by Suzuki et al. (end of 60th)

Main interest access to low energy edges using hard X-rayin situ, operando, extreme conditions…

Drawback low signal

But new synchrotron generation, new spectrometers new XRS beamlines

53

First approximation :

Same than (dipole) XANES, with q

𝑆 𝒒, 𝜔 𝑓 𝑒 𝒒.𝒓 𝑔 𝛿 ħ𝜔 𝐸 𝐸,

𝑑 𝜎𝑑𝛺 𝑑ℏ𝜔 𝑟

𝜔𝜔 𝜀 . 𝜀 𝑆 𝒒, 𝜔

𝑓 𝑒 𝒒.𝒓 𝑔 ≅ 𝑓 1 𝑖𝒒. 𝒓 𝑔 ≅ 𝑓 𝒒. 𝒓 𝑔

Dynamic structure factor:

Cross section:

Exact expansion:

Bessel function

𝑒 𝒒.𝒓 4𝜋 𝑖 ℓ𝑗ℓ 𝑞𝑟ℓ

𝑌ℓ∗ 𝑟 𝑌ℓ 𝑞

The formula

𝑞𝒒𝑞

54

Monopoleℓ = 0

Dipoleℓ = ±1

Dependence on q (scat. angle)

Probe of the different ℓ

sin 𝑞𝑟𝑞𝑟

cos 𝑞𝑟𝑞𝑟

sin 𝑞𝑟𝑞𝑟

1

𝑆 𝒒, 𝜔 𝑓 𝑗 𝑞𝑟 4𝜋𝑖𝑗 𝑞𝑟 ∑ 𝑌 ∗ 𝑟 𝑌 𝑞 ⋯ 𝑔 𝛿 ħ𝜔 𝐸 𝐸,

𝑌ℓ∗ 𝑟 𝑌ℓ

∗ 𝑟 𝑌ℓ 𝑟 𝑑𝑟 0

selection rule from:

Disordered material case (powder, glasses)

𝑆 𝑞, 𝜔 𝑓 𝑗 𝑞𝑟 𝑔 4𝜋 𝑓 𝑗 𝑞𝑟 𝑌 ∗ 𝑟 𝑔 ⋯,

𝛿 ħ𝜔 𝐸 𝐸

55

ExpID20, ESRF

CubicF𝑚3𝑚

Pt. group𝑚3𝑚

SCFR = 8 Å

ExamplesF edge in LiF

56

Sp. groupP63/mmc

Pt. group6m2

SCFR = 8 Å (B)R = 10 Å (N)

h-BN

57Y. Joly, C. Cavallari, S. A. Guda, and C. J. Sahle, J. Chem. Theory Comput. 13, 2172-2177 (2017).DOI: 10.1021/acs.jctc.7b00203

𝑝 𝑠𝑝 𝑝 𝑠𝑝

3-fold axis along c 𝑠𝑝 in basal plane

m plane no 𝑝 𝑠 hybridization

3.33 Å

1.45 Å

Anti-bondingB𝑠𝑝 N𝑠𝑝

B𝑝 N𝑝

58

Thèse Emmanuelle de Clermont Gallerande (IMPMC)

Etude de la structure locale d’Alcalin par XRS

Effect of non-bridging Oxygen

59

Comparison glass-Crystal

xNa2O-(1-x)B2O3

60

Tutorial on FDMNES

61

The FDMNES code1995: ESRF at Grenoble + Denis Raoux + Rino Natoli

Starting of the XANES theoretical study

1996: first version of FDMNESXANES calculation beyond the muffin-tin approximationXAFS IX, Grenoble, 26-30 Août 1996

1999: Resonant diffraction

2000–2009: Multiple Scattering TheoryMagnetism - Spin-orbitSpace group symmetry analysisTensor analysisFit procedureSelf-consistency

2010-2018: LDA + U, TD-DFTXES (valence to core)X-ray RamanSurface resonant X-ray Diffraction

COOP

fdmfile.txt VO6_inp.txt

fdmnes

VO6_out.txt

VO6_out_conv.txt

VO6_out_bav.txt

VO6_out_sd0.txt

fdmnes_error.txt

Same directory

VO6_out_scan.txt VO6_out_atom1.txt

VO6_out_sph_atom1.txt VO6_out_sph_xtal.txt

Input and output files

62

FiloutSim/VO6

Range-2. 0.1 0. 0.5 60.

Radius 2.5

Quadrupole

Polarization

Molecule2.16 2.16 2.16 90. 90. 90. 23 0.0 0.0 0.0 8 1.0 0.0 0.0 8 -1.0 0.0 0.0 8 0.0 1.0 0.0 8 0.0 -1.0 0.0 8 0.0 0.0 1.0 8 0.0 0.0 -1.0

Convolution

End

63

Examples of FDMNES indata file FiloutSim/Fe3O4

Range-2. 0.1 -2. 0.5 20. 1. 100.

Radius5.0

GreenQuadrupole

DAFS0 0 2 1 1 45.0 0 6 1 1 45.4 4 4 1 1 0.

SpgroupFd-3m:1

Crystal8.3940 8.3940 8.3940 90 90 90

26 0.6250 0.6250 0.6250 ! Fe 16d 26 0.0000 0.0000 0.0000 ! Fe 8a8 0.3800 0.3800 0.3800 ! O 32e

Convolution

End

FiloutSim/VO6

Range-2. 0.1 0. 0.5 60.

Radius 2.5

Quadrupole

Polarization

Molecule2.16 2.16 2.16 90. 90. 90. 23 0.0 0.0 0.0 8 1.0 0.0 0.0 8 -1.0 0.0 0.0 8 0.0 1.0 0.0 8 0.0 -1.0 0.0 8 0.0 0.0 1.0 8 0.0 0.0 -1.0

Convolution

End

64

Examples of FDMNES indata file FiloutSim/Fe3O4

Range-2. 0.1 -2. 0.5 20. 1. 100.

Radius5.0

GreenQuadrupole

DAFS0 0 2 1 1 45.0 0 6 1 1 45.4 4 4 1 1 0.

Cif_fileSim/in/Fe3O4.cif

Convolution

End