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Optical properties and Interaction of radiation with matter. S.Nannarone TASC INFM-CNR & University of Modena. Outline. Elements of Classical description of E.M. field propagation in absorbing/ polarizable media  Dielectric function

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Optical properties and interaction of radiation with matter l.jpg

Optical propertiesandInteraction of radiation with matter

S.Nannarone

TASC INFM-CNR & University of Modena


Outline l.jpg

Outline

  • Elements of Classical description of E.M. field propagation in absorbing/ polarizable media  Dielectric function

  • Quantum mechanics microscopic treatment of absorption and emission and connection with dielectric function

Physics related to a wide class of Photon-in Photon-out experiments including Absorption, Reflectivity, Diffuse scattering, Luminescence and Fluorescence or radiation-matter interaction

[some experimental arrangements and results, mainly in connection with the BEAR beamline at Elettra

http://new.tasc.infm.it/research/bear/]


Systems l.jpg

Systems

  • bulk materials the whole space is occupied by matter

  • Surfaces  matter occupies a semi-space, properties of the vacuum matter interface on top of a semi-infinite bulk

  • Interfaces  transition region between two different semi-infinite materials

Information [see mainly following lectures]

  • Electronic properties full and empty states, valence and core states, localized and delocalized states

  • Local atomic geometry /Morphology electronic states – atomic geometry different faces of the same coin

Energy range Visible, Vacuum Ultraviolet, Soft X-rays)


Slide4 l.jpg

Synchrotron and laboratory sources/LAB

Conceptually Shining light on a system, detecting the products and measuring effects of this interaction

This can be done by Laboratory sources

They cover in principle the whole energy range nowadays covered by synchrotrons (J.A.R.Samson Techniques of vacuum ultraviolet spectroscopy)

  • Incandescent sources

  • Gas discharge

  • X-ray e- bombardment line emission

  • Bremsstrahlung continuous emission sources

  • Higher harmonic source


Slide5 l.jpg

Synchrotron and laboratory sources / Synchrotron

  • Some well known features

  • Collimation

  • Intrinsic linear and circular polarization

  • Time structure (typically 01-1 ns length, 1 MHz-05GHz repetition rate)

  • Continuous spectrum, high energy  access to core levels

  • Reliable calculability of absolute intensity

  • Emission in clean vacuum, no gas or sputtered materials

  • High brilliance unprecedented energy resolution

  • High brilliance  small spot  Spectromicroscopy

“The one important complication of synchrotron source is, however, that while laboratory sources are small appendices to the monochromators, in a synchrotron radiation set-up the measuring devices becomes a small appendices to the light source. It is therefore recommendable to make use of synchrotron radiation only when its advantages are really needed.”

C.Kunz, In Optical properties of solids New developments, Ed.B.O.Seraphin, North Holland, 1976


Radiation matter interaction polarization and current induction in e m field l.jpg

Radiation-Matter Interaction  Polarization and current induction in E.M. field

Matter polarizes in presence of an electric field Result is the establishment in the medium of an electric field function of both external and polarization charges

Matter polarizes in presence of a magnetic field Result is the establishment in the medium of a magnetic field function of both external and polarization currents

The presence of fields induce currents

  • Mechanisms and peculiarities of polarization and currents induction in presence of an E.M. field

  • Scheme to calculate the E.M. field established and propagating in the material

  • Basis to understand how this knowledge can be exploited to get information on the microscopic properties of matter


Slide7 l.jpg

Basic expressions - Charge polarization

and induced currents

Polarization vectors


Charge and magnetic current polarization closer look l.jpg

-Ze-

+Ze

Charge and Magnetic/current polarization – closer look


Induced currents l.jpg

Induced currents

Ze+

e-

_

+

Motion of charge under the effect of the electric field of the E.M. field but in an environment where it is present an E.M. field


Slide10 l.jpg

Expansion of polarisation

Physical meaningElastic limit  the potential is not deformed by the field

Linear and isotropic media

Dielectric function

Permeability function


Linear versus non linear optics l.jpg

Linear versus non linear optics

Formally linear optics implies neglecting terms corresponding to powers of the electric field

Physicallyit meansE.M. forcesnegligible with respect to electron-nuclei coulomb attraction

Nuclear atomic potential is deformed  not harmonic (out of the elastic limit) response  distortion  higher harmonic generation


Dielectric function and response l.jpg

Dielectric function and response

In very general way

External stimulus

Note is defined as a real quantity


Slide13 l.jpg

Summary material properties within linear approximation

And

Conduction under a scalar potential – Usual ohmic conduction

Conduction in an e.m. field


Maxwell equations in matter for the linear case l.jpg

Maxwell equations in matter for the linear case

Corresponding equations for vacuum case


Slide15 l.jpg

Wave equation - Vacuum

Vacuum supports the propagation of plane E.M. waves with dispersion / wave vector energy dependence

Wave equation - Matter

Matter supports the propagation of E.M. waves with this dispersion

Formally q is a complex wavevector

Wave vector eigenvalue/dispersion depends on the properties of matter

through    (all real quantities)


Complex refraction index l.jpg

Complex refraction index

Absorption

Phase velocity

Real and imaginary parts not independent

Absorption coefficient

Lambert’s law


Slide17 l.jpg

Complex dielectric constant – Complex wave vector


Slide18 l.jpg

Supported/propagating E.M. modesdepend on the properties of matter through   

The study of modes of the e.m. field supported/propagating in a medium and the related spectroscopical information is the essence of the optical properties of matter

Relation between (r,t), (r,t) (r,t) or (q,) (q, ) (r ) and the properties of matter

1st part  Classical scheme / macroscopic picture

2nd part  Quantum mechanics / microscopic picture


Spatial dispersion l.jpg

Spatial dispersion

extension on which the average is made

Note  0 wavevector does not mean lost of dependence on direction  anisotropic materials excited close to origin


Slide20 l.jpg

Unknowns and equations

   (real quantities) are the unknowns related with the material properties

(r,t) is close to unity at optical frequencies  magnetic effects are small

(not to be confused with magneto-optic effects: i.e. optics in presence of an external magnetic field)

Generally a single spectrum – f.i. absorption – is available from experiment

(An ellipsometric measurement provides real and imaginary parts at the same time.

It is based on the use of polarizers not easily available in an extended energy range)

Real and imaginary parts are related through Kramers – Kronig relations

Sum rules


Slide21 l.jpg

Kramers – Kronig dispersion relations

Under very general hypothesis including causality and linearity


Slide22 l.jpg

Models for the dielectric constant / Lorentz oscillator

Mechanical dumped oscillator forced by a local e.m. field Neglecting the magnetic term

e-

Induced dipole

Out of phase – complex/dissipation – polarizability (Lorentzian line shape)


Complex dielectric function l.jpg

Complex dielectric function

From


Lorentz oscillator dielectric function l.jpg

Lorentz oscillator Dielectric function


Lorentz oscillator refraction index l.jpg

Lorentz oscillator Refraction index


Lorentz oscillator absorption reflectivity loss function l.jpg

Lorentz oscillator Absorption Reflectivity Loss function

Physics

Difference between transverse and longitudinal excitation

EEL spectroscopy

Optical spectroscopy


Slide27 l.jpg

Non linear Lorentz oscillator

Anarmonic potential

  • induced dipole at frequency  and 2 

  • the system is excited by a frequency  but oscillates also at frequency 2 

  • re-emitting both  and 2 


Slide28 l.jpg

Lorentz oscillator in a magnetic field 1/2

x and y motions are coupled

Solving for x and y

Larmor frequency


Slide29 l.jpg

Lorentz oscillator in a magnetic field 2/2


Lorentz oscillator in a magnetic field 1 3 l.jpg

Lorentz oscillator in a magnetic field 1/3

The dielectric function is a tensor

[ Physically lost of symmetry for time reversal ]


Propagation in a magnetised medium 1 2 l.jpg

Wave equation

Eigenvalue equation

Propagation in a magnetised medium 1/2

Note≠ 0 in anisotropic media

with


Propagation in a magnetised medium 2 2 l.jpg

Propagation in a magnetised medium 2/2

Considering the medium with B||z


Magneto optics effects l.jpg

Elliptically polarized

 Rotation according to n+-n-

Linear polarized

Longitudinal geometry

N+ Right circular polarized wave

N-  Left circular polarized wave

Dichroism

Magneto-optics effects

Two waves propagating with two different velocities and different absorption

Magneto-optic effects e.g. Faraday and Kerr effects/geometries


Dielectric tensor l.jpg

Dielectric tensor

are in general tensorial quantities


Dielectric tensor35 l.jpg

Dielectric tensor

Scalar medium

Magnetized medium


Slide36 l.jpg

Longitudinal and transverse dielectric constant 1/2

Any vector field F can be decomposed into two vector fields one of which is irrotational and the other divergenceless

If a field is expanded in plane waves FT is perpendicular to the direction of propagation.


Longitudinal and transverse dielectric constant 2 2 l.jpg

Longitudinal and transverse dielectric constant 2/2

Optics  EELS/e- scattering

The description in terms of longitudinal and transverse dielectric function is equivalent to the description in terms of the usual (longitudinal) dielectric function and magnetic permeability. They are both/all real quantities together with conductivity. They combine together to forming the complex dielectric constant defined here.


Transverse and longitudinal modes 1 3 l.jpg

Transverse and longitudinal modes 1/3

Propagating waves and excitation modes of matter are two different

manifestation of the same physical situation

Plasmon is a charge oscillation at a frequency defined by the normal modes oscillation produces a field  only a field of this kind is able to excite this mode

_

+

  • Modes can be transverse or longitudinal in the same meaning of transverse and longitudinal E.M. field

  • searching for transverse waves is equivalent to searching for

    transverse modes


Transverse and longitudinal modes 2 3 l.jpg

Transverse and longitudinal modes 2/3

Searching for modes  eigenvectors of

Transverse modes  Polaritons

The quantum particles are coupled modes of radiation field and of the elementary excitations of the system, called Polaritons including transverse (opical) phonons, excitons,….

Longitudinal modes

the quantum particles are coupled modes of radiation field and of the elementary excitations of the system: Plasmons, longitudinal opical phonons, longitudinal excitons,….


Transverse and longitudinal modes 3 3 l.jpg

Transverse and longitudinal modes 3/3

Polarization waves


Slide41 l.jpg

Sum rules for the dielectric constant

Examples of sum rules

Of use in experimental spectra interpretation


Quantum theory of the optical constants l.jpg

Quantum theory of the optical constants

Macroscopic optical response

Microscopic structure

Transition probability

Ground state HRADIATION + HMATTER perturbed by radiation-matter interaction

  • Two approaches

  • fully quantum mechanics

  • semi classical

  • Three processes

  • Absorption

  • Stimulated emission

  • Spontaneous emission


Microscopic description of the absorption and emission process l.jpg

O ° O

O° O °°O

Term neglected for non relativistic particles

Microscopic description of the absorption and emission process

System

Radiation

mi,ei

mj,ej

  • Interaction Hamiltonian HI

  • Effect of the interaction on the states of the unperturbed HR + HI

Matter


Hamiltonian of a charged particle in e m field l.jpg

Hamiltonian of a charged particle in E.M. field


Particle radiation interaction l.jpg

Particle radiation interaction

Matter Hamiltonian + perturbation Hamiltonian

Problem to be solved

Eigenstate and eigenvector of the matter radiation system in interaction


Important notes l.jpg

Important notes

The solution is found by a perturbative method

  • it is assumed here – formally - that the problem in absence of interactions has been solved.

  • In practice this can be done with more or less severe approximations.

  • The calculation of the electronic properties of the ground state is a special and important topic of the physics of matter

Many particles state

Generally obtained by approximate methods


Transition between states of ground state due to the perturbation term l.jpg

Transition between states of ground state due to the perturbation term

The effect of perturbation HI on the eigenstates of H0

Obtained by time dependent perturbation theory


Matrix elements 1 3 l.jpg

Matrix elements 1/3

The evolution of the state m is obtained calculating the matrix element

System states under perturbation due to

Changes of photon occupation and matter (f.i. electronic) state


Matrix elements 2 3 l.jpg

Matrix elements 2/3

It is found that for photon mode k, only

contribute linear terms to matrix elements

+1  photon emission -1 photon absorption

Probability of transition of the system from state


Matrix element 3 3 l.jpg

Matrix element 3/3


Transition probabilities l.jpg

Spontaneous emission

Stimulated emission

Spontaneous emission present only in quantum mechanics treatment

Transition probabilities

Integrating in time from 0 to infinity for the transition probabilities

per unit time


Slide52 l.jpg

for

probability of finding the state in a state n, at thermodynamic equilibrium

Dielectric function and microscopic properties

Dissipated

power


Microscopic expression of the dielectric function l.jpg

Microscopic expression of the dielectric function

Physical meaning  Sum of all the absorbing channels at that photon energy

Note dissipation originates from non radiative de-excitation channels


Intuitive meaning of the expression for absorption coefficient l.jpg

Intuitive meaning of the expression for absorption coefficient

En’

N(E) density of states

(Number of states/eV)

Joint Density Of States N(E) N(E’)

En


Dipole approximation l.jpg

Dipole approximation

Matrix elements of position operator


Semi classical approach l.jpg

Semi classical approach

Note that the same result can be obtained by considering the transition probability between quantized states of the matter system under the effect of classical external perturbation of the E.M. field with given by the same expression of

This semi classical approach gives identical results for absorption and stimulated emission probabilities, but does not account for spontaneous emission


Selection rules for hydrogen atom l.jpg

Selection rules for Hydrogen atom

Generic light polarization


Selection rules 2 l.jpg

Selection rules/2

For radiation polarized along z

[linear polarized light ]

Expressions valid in any central field


Hydrogen selection rules circular polarization l.jpg

Hydrogen - Selection rules Circular polarization


Calculation of matrix elements optical properties of matter l.jpg

Calculation of matrix elements - Optical properties of matter

The basic step in calculation involves

many particles wavefunctions

Born - Oppenheimer approximation

Nuclear motions separated from electronic motions

One electron description

One electron WF Solution of motion in an average potential generated by all other electrons


Dielectric function in one electron approximation l.jpg

Dielectric function in one electron approximation

Case of crystals

K reduced vector within the Brillouin zone

Crystal states  E(k)

Joint Density of States - JDOS


Phenomenology of absorption l.jpg

Phenomenology of absorption

  • Interband transitions

  • direct/indirect

  • Intraband absorption

  • Phonon contribution

  • Core/localized (e.g. molecular) level absorption

Local field effects - Local (Lorentz) field corrections


Decay and relaxation of excited states l.jpg

Decay and relaxation of excited states

Probability of relaxation/decay of excited state as integral on all the spontaneous emission channels of field and matter states

As a consequence the dependence of Im () has to be modified

  • Lorentzian broadening

  •  function substituted for by Lorentzian curve

(e.g. see Lorentz oscillator)


Lorentzian broadening l.jpg

Lorentzian broadening


Exploitation of emission radiative decay l.jpg

Exploitation of emission / radiative decay

Total/Partial yield measurement of absorption through electron (Secondary, Auger,..) and photon (fluorescence, luminescence,…) yields

  • De-excitation spectroscopies

  • Fluorescence

  • Luminescence  XEOL

  • Auger electron and photon induced – Selection rules and surface sensitivity

matrix element ~ constant I  Density of states and  3


Boundaries reflectivity l.jpg

Boundaries reflectivity

From material filling the whole space to material with boundaries and matter-vacuum interfaces

Reflectivity - Measure of the reflected intensity as a function of incident intensity

Fresnels relations based on boundary conditions of fields link

reflected intensity with dielectric function


Reflectivity from a semi infinite homogeneous material l.jpg

Reflectivity from a semi-infinite homogeneous material

Surface plane

Normal to surface

Modellisation of surfaces and interfaces

Multiple boundaries


S and p reflectivity l.jpg

S and p reflectivity


Diffuse scattering 1 2 l.jpg

Diffuse scattering 1/2

Small and/or rough objects

Scattered wave

scattering medium

Incident field

Term neglected if   dimensions

Inhomogeneous filling of space

Scalar theory of scattering(single Cartesian component)

Defining:Scattering potential


Diffuse scattering 2 2 l.jpg

Diffuse scattering 2/2

Born approximation

The scattering amplitude is the Fourier transform of the scattering potential

Inverting  F(r)  n(r)


Conclusions l.jpg

Conclusions

Classical scheme Introduction of the dielectric function

Microscopic (quantum mechanics) treatment of emission and absorption

Relation between macroscopic dielectric function (measured quantity) and microscopic properties

http://www.gfms.unimore.it/


Calculation of ij elements l.jpg

Calculation of ij elements


Source l.jpg

Source

3

5

x 1012

T

320eV

r = 1 m 0.1% BW

4.5

2

4

3.5

//

1

3

Eoy N/C

2.5

0

2

x 1012

1.5

-1

1

Eoz N/C

0.5

-2

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y (mrad)

-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y (mrad)

Source: 3.3 m of arc, 3.1 mx 3.3 m vertical x horizontal

two fields – vertical and horizontal – out of phase of ±/2according to the sign of take off angle  (J.Schwinger PR 75(1949)1912)

Electric fields

≈ 103 photons/bunch - bunch duration ≈ 20 ps

320eV

r = 1 m 0.1% BW


Polarimetry 100 ev ellipse l.jpg

Polarimetry 100 eV ellipse

Selector fully open:

Zc= 45 mm, Zg = 1 mm

S1=-0.9S2=0.011 S3=-0.068

Ey=0.95Ez=0.04 =-1.4 

Ellipticity,  =0.04

Polarization selector position:

Zc = 34 mm, Zg = 41 mm

(aperture 4 mm)

S1=-0.97 S2=0.011 S3=0.082

Ey=0.98Ez=0.04 =-1.44 

Ellipticity,  =0.33

Polarization selector position: Zc = 31 mm, Zg = 31 mm (aperture 14 mm)

S1=-0.77 S2=0.08 S3=-0.57Ey=0.93 Ez=0.31 =1.43 


Transport and conditioning optics l.jpg

P2

Helicity selector

BPM

GAS CELL

P1

EXIT SLITS

MONO

Transport and conditioning optics

Source  4 m HxV

Mirrors in sagittal focusing  reduction of slope errors effects in the dispersion plane

Intensity monitor

Light spot

Energy range 3- 1600 eV

Energy resolution E/E ≈ 3000 (peak 5000) at vertical slit (typically 30 μm) x 400 μm (variable)Variable

divergence (maximum, variable) 20 m vert x hor

ellipticity variable horizontal/vertical (typically in the range 1.5 – 3.5, Stokes parameters (normalized to the beam intensity) S1 0.5 - 0.6, S2 0 - 0.1, S3 0.75 -0.85 )

helicity variable (typical value for rate of circular polarization P or S3 0.75 – 0.95)

plane-grating-plane mirror monochromator based on the Naletto-Tondello configuration


Slide78 l.jpg

Examples and experimental arrangements at BEAR (Bending magnet for Absorption Emission and Reflectivity)

Bulk materials

Surfaces

Interfaces

Absorption

Reflectivity

Fluorescence

Luminescence – XEOL

Diffuse scattering

Experimental arrangements

BEAR (Bending magnet for Emission Absorption Reflectivity) beamline at Elettra


Experimental scattering chamber l.jpg

Experimental/scattering chamber

Detection

e- analyser /photodiodes

(2 solid angle)

VIS Luminescencemonochromator

Goniometers

M,A 0.001°

A 0.01°

S 0.05°

C 0.1°

(Positive -Differentially pumped joints)

Sample manipulator

6 degree of freedom

Rotation around beam axis  any position of E in the sample frame


Optical constants of rare hearths l.jpg

Optical constants of rare hearths

See f.i. Mónica Fernández-Perea, Juan I. Larruquert, José A. Aznárez1, José A. Méndez Luca Poletto, Denis Garoli, A. Marco Malvezzi, Angelo Giglia, Stefano Nannarone, JOSA to be published


Interfaces surface physics in periodic structures multilayer optics l.jpg

  • ultra-thin deposited films

  • buried interface spectroscopy

  • Devices of use in spectroscopy

Z

Interfaces & surface physics in periodic structures(multilayer optics)

See also posterP III 26

ML : Artificial periodic stack of materials

(Optical technology  Band pass mirrors)

BRAGG

At Bragg Z dependent standing e.m. field establishes both inside the structure and at the vacuum-surface interface modulated in amplitude and position


Standing waves excitation l.jpg

Scanning through

Bragg peak

In energy or angle

Spectroscopy of interfaces

Physics of mirror/Reflection

Standing waves & excitation

Si

Mo

Si

Local modulation of excitation

Photoemission, Auger, fluorescence, luminescence etc..


Cr sc cr oxide interface as received l.jpg

Cr2O3

(6 Å)

Cr

(15 Å)

Sc

(25 Å)

Cr/Sc Cr-Oxide interface (As received )

573 eV

X 60

Qualitative analysis

-Opposite behavior of Cr and Sc

  • Different chemical states of the buried Sc

  • Two signalsfrom oxygen: one bound to Cr at the surface, the second coming from the interface

    - Carbon segregation at the interface


Ru clean si buried interface l.jpg

Mo

(39.6 Å)

Ru

(15 Å)

Si

(41.2 Å)

Ru (Clean) -Si buried interface

Angular scan through the Bragg peak

at 838 eV

X 40

silicide


Model ru si interface l.jpg

Model – Ru-Si interface

  • Interface morphology

  • Calculation of e.m. field inside Ml

  • Photoemission was calculated, (Ek= h - EB)

  •  Minimum position and lineshape depend critically on the morphology profile

Ru

Ru-Si


Mo si ml i f roughness l.jpg

Mo/Si

61.2 %

58.4 %

Wavelength [nm]

Mo-Si ML & i.f. roughness

Motivation role of ion kinetic energy and flux during ML growth

ML (P  8 nm,   0.44) Performance- R (10°)

Ion assistance

Ions EK: 5 eV (1st nm), 74 eV > 1nm Controlled activation of surface mobility


Performance diffuse scattering l.jpg

Performance & diffuse scattering

Performance differences are to be related to interface quality

Diffuse scattering around the specular beam was measured

KS=Ki + qZ + q//


Diffuse scattering mls l.jpg

 f.i. Stearns jAP 84,1003

diffuse scattering - MLs

I.f.roughnes produces diffuse scattering around the specular beam

I.f.roughnes can/can not be coherently correleted through the ML

Description on a statistical base, ….fractal properties

Single interface – Autocorrelation function

In plane Fourier transform on q//of potential

KS=Ki + qZ + q//

MLs : S(q)  two terms

  • incoherent scattering by single interfaces

  • correlated/coherent scattering among i.f.

  • (interlayer replica of roughness)

See f.i. Stearns JAP, 84, 1003, 1998


Diffuse scattering l.jpg

Diffuse scattering

Detector

qdetector  0.003 nm-1

At 0.48 nm (13.1 eV)

 - scan

 Rocking scan

Incident beam

Divergence

qdiv  0.0005 nm-1/m

At 0.48 nm (13.1 eV)


Mo si ml diffuse scattering l.jpg

ξ=400 Å

ξ=200 Å

ξ= 300 Å

ξ=120 Å

Correlation function

In plane correlation function - absence of interface correlation

  • ξ , correlation length  h, fractal dimension/jaggedness

Mo-Si ML diffuse scattering

See also poster Borgatti et al. P III 17

Mo/Si

Ion assistance


Pentacene on ag 111 l.jpg

Pentacene on Ag(111)

See also poster Pedio et al. P II 33

Chemisorption morphology - tilt angle & electronic structure

( Concentrating on 1 Mono layer )

  • Premise about C22H14/substrates

  • He scattering on pentacene deposited by hyperthermalbeams 1ML planar

  • Morphology and electronic properties ( delocalization of the p electrons)  transport properties highly anisotropic;

  • on Metals: nearly planar orientation  a condition hindering the formation of an ordered overlayer;

  • on semiconductors/oxides: SiO2 standing

  • GeS lying.


Surface cell l.jpg

Surface Cell

( By He scattering )

Danışman et al. Phys. Rev. B 72, 085404 (2005)

C3 symmetry

Oblique cell Periodicity (6 x 3) ,

1 monolayer


Slide93 l.jpg

XAS

(At magic angle 54.7°)

Gas phase XAS

(Alagia et al.JChemPhys 122(05)124305)

Redistribution of the oscillator strength in the C1s – LUMO excitation region (1-3 of gas phase)

Tilt angle


Vb photoemission l.jpg

VB photoemission

LDA calculationsC22H14/Al(100)

Simeoni et al. S.Science 562,43 (2004)

EF

Redistribution of  states upon chemisorption

 HOMO-LUMO gap increasing


Xas 1 ml precession scan l.jpg

XAS 1 ML - precession scan

Dichroism/Bond directionality & Tilt angle of the molecule

ε= EV/EH = 0.29

i= 10°


Xas 1 ml deconvolution l.jpg

XAS – 1 ML - deconvolution


Precession scan formulae l.jpg

Fit parameter: θ (polar angle of dinamic dipole )

Precession scan - Formulae

Fit function for Single domain

P


Tilt angle fit l.jpg

Tilt angle - Fit


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