X-ray spectroscopy course of Advanced Spectroscopy of ... introduction.pdfX-ray spectroscopy Course...

X-ray spectroscopy Course of Advanced Spectroscopy of Nanomaterials

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X-ray spectroscopy course of Advanced Spectroscopy of Nanomaterials Lectures: 7 and 14 september, plus 2, 9, 16 and 23 november; all lectures in room N016. Lecture 1: Introduction & Understanding X-rayAbsorption Spectroscopy Lecture 2: Experimental aspects of XAS Lecture 3: Atomic Multiplet Theory & XAS Lecture 4: Crystal Field Theory & XAS Lecture 5: Charge Transfer Multiplet Theory & XAS + XPS Lecture 6: Applications of X-ray Spectroscopy in nanoscience Course Material:

- These notes - Selected pages from the book Core Level Spectroscopy of Solids. - Slides from the Powerpoint presentations.

Course website: http://www.anorg.chem.uu.nl/people/staff/FrankdeGroot/ASNX2009.htm The website contains links to the power point slides, the computer program and additional material.

Core Level Spectroscopy of Solids Chapter 1 & 2 Chapter 4 Chapter 5 till section 5.3. Chapter 6 till section 6.3.2


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Introduction X-rays are defined as the electromagnetic radiation with a wavelength between Ultraviolet radiation and gamma radiation. The border between Ultraviolet radiation and X-rays is not clearly defined but usually set around 10-8meter, i.e. 10 nm. The upper limit of x-rays is usually set at 10-12 meter, i.e. 1 pm, although radiation with an energy above 10-11 meter is also called gamma radiation, in particular if it is created with a radioactive element. X-rays are sub-divided into soft x-rays, between 10 nm and 100 pm and hard x-rays between 100 pm and 1 pm. Soft x-rays are called 'soft' because they do not penetrate air and as such are relatively safe to work with. We will discuss the penetration length of x-rays below. Because of historical reasons, x-rays are often given in Angstroms, where 1 Å equals 10-10 meter or 100 pm.

The bottom two axes in the figure give the frequency and the energy of the electromagnetic radiation, using the inverse proportionality between frequency and wavelength , i.e. =c/, where c is the speed of light 3·108 m/s. This yields x-ray frequencies between 1016 and 1021 Hertz. It is more common to indicate x-rays with their energies E, using E=h, with the energy given in electron volts (eV) or kilo electron volts (keV), where they range between 100 eV and 106 eV. The wavelength of x-rays are of the same size as molecules, which range from 100 pm for small molecule such as water to 10 nm for macromolecules such as enzymes. In particular the distance of 1 Å is important in matter, because it corresponds to a typical interatomic distance. This makes diffraction with x-rays of approximately 1 Å an important technique to determine the interatomic distances and, in general, the structure of matter.


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X-rays are traditionally produced using x-ray tubes. In an x-ray tube, electrons are accelerated between the anode and the cathode. When they hit the cathode with high energy, they are decelerated through inelastic collisions with atoms in the target material. If the energy of the electrons is high enough this produces a continuous spectrum of x-rays. In addition, the impinging high-energy electrons excite core electrons from the cathode material. Core holes decay, in part, via radiative processes in which a shallow core electron fills a deep core hole, thereby emitting x-rays. These x-rays are characteristic for the cathode material and are confined to only a few energies. This process will be discussed in detail below.

X-ray tubes are used in many x-ray applications; for example (almost) all medical x-ray images use x-ray tubes. In addition, laboratory x-ray sources used in x-ray photoemission and x-ray diffraction make use of x-ray tubes. In x-ray spectroscopy however the role of x-ray tubes has been declining over the last few decades. The reason is evident from the figure: X-ray tubes produce a maximum of 1010 'photons per second', whereas x-rays produced with synchrotrons reached up to 1019 photons per second. These photon counts are usually given with the 'brightness'. The brightness is normalised to an area of 1 mm2 and further normalised to the x-ray beam divergence and the energy (range). It is clear that with 109 times more photons per second, one can carry out much faster experiments. In addition, it allows a range of new experiments that have very low x-ray cross-sections. It should be noted that the x-rays produced with synchrotrons have very high intensity, but still they cannot be considered as x-ray lasers, because in general they have insufficient coherence.


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Synchrotron Radiation

Experimental: Generating x-rays with Synchrotrons The Intensity of x-ray sources

X-ray tubes:

Wilhelm Conrad Röntgen discovered x-rays in 1895 whilst working with cathode-ray tubes. Using the principle of fast electrons hitting a metallic target, a first substantial gain in brilliance was not obtained until the introduction of rotating anode sources (1960).

Synchrotron Radiation Facilities:

The progress of high energy physics, with the construction of powerful particle accelerators gave birth to what we now call First generation synchrotron sources (1970). Using the deflection of high energy electrons by a magnetic field for the production of x-rays proved so promising that a number of dedicated Second generation sources were built (1980). Relying on the combination of needle thin electron beams and Insertion Devices, Third generation synchrotron sources (1995) are now emitting synchrotron x-ray beams that are a trillion (1012) times more brilliant than those produced by x-ray tubes.

(Based on the ESRF www-page: http://www.esrf.fr/ )

Initially synchrotron radiation was seen as an annoying side product of charged particle accelerators. When charged particles are accelerated they start to produce radiation: the so-called synchrotron radiation. Only after the theory about EXAFS was established dedicated synchrotrons for radiation studies (like EXAFS) were built.

In a synchrotron charged particles are generated and accelerated to the desired energy before they are stored in a ring: the so-called storage ring. Without the presence of an external magnetic field the electrons will travel in a straight line but when a magnetic field is applied they will bend caused by the Lorentz force. This force causes an acceleration of the particles and therefore radiation is formed. Of course this radiation will cause the kinetic energy of the electrons to reduce and therefore the radius of the circular orbit will decrease. To prevent that after a short period the electrons will have lost their energy, the energy loss is compensated by a Radio Frequency cavity. It is important to mention here that not only x-rays are produced during bending of the electron but also a much wider spectrum of radiation is formed (depending on e.g. the magnetic field strength, the beam current, the energy of the beam etc). Synchrotron radiation emerges at the bending magnets where the orbit of the electrons is changed. Between the magnets the electrons follow a straight path without the formation of radiation. Insertion devices can be placed in the electron beam in order to produce radiation. Examples of such devices are wigglers and undulators. Basically these devices ‘wiggle’ the electrons in the straight path in order to produce radiation.


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Synchrotrons: Radiation from a fast moving particle source appears to the observer in the laboratory as being all emitted in the general direction of motion of the particle. Think of tomatoes being thrown in all directions from a fast moving truck. In synchrotron radiation sources (storage rings) highly relativistic electrons are stored to travel along a circular path for many hours. Radiation is caused by transverse acceleration due to magnetic forces in bending magnets (forming the circular path) or periodic acceleration in special insertion device magnets like undulators and wigglers.

Bending magnet: Radiation is emitted tangentially to the orbit well collimated in the nondeflecting, or mostly vertical plane. Because the geometry of the storage ring is determined by bending magnets, it is not possible to freely choose the field strength and the critical photon energy is therefore fixed.

Undulator: The electron beam is periodically deflected by weak magnetic fields. The spectral resolution of the radiation is proportional to the number of undulator periods and varying the magnetic field can shift its wavelength. Most radiation is emitted within a very small angle.

Wiggler: Increasing the magnetic field strength causes the pure sinusoidal transverse motion of electrons in an undulator to become distorted due to relativistic effects. For very strong fields many harmonics are generated which eventually merge into a continuous spectrum from IR to hard x-rays. The spectral intensity varies little over a broad wavelength range and drops off exponentially at photon energies higher than the critical photon energy. Compared to bending magnet radiation, wiggler radiation is enhanced by the number of magnet poles.

(Based on the SSRL www-page: http://www-ssrl.slac.stanford.edu/welcome.html)

A XAFS spectrum originates from the fact that the probability of an electron to be ejected from a core level is dependent on the energy of the incoming beam. For this reason the energy of the x-rays is varied during an experiment. Of course this requires a monochromatic beam. However the radiation generated by a synchrotron is polychromatic. Therefore the desired wavelength has to be filtered from the polychromatic beam. For this purpose a monochromator is installed in the beamline (see sheet: XAFS beamline). The incoming beam is ‘reflected’ on a crystal due to Bragg reflections. The wavelength of the reflected beam is given by sin..2. dn . By varying the angle between the x-ray propagation direction and the surface of the crystal, the energy of the x-ray beam can be varied. In principle one crystal is enough to obtain the desired wavelength of the beam. In practice a double crystal monochromator is used in order to obtain an outgoing beam that is parallel to the incoming beam (sheet XAFS beamline). After the beam has passed the monochromator, its intensity (I0) has to be measured. This can be done by an ionization chamber. In the next step, the x-ray interacts with the sample of interest. To measure the intensity after the sample a second ionization chamber can be used. This is the transmission mode of x-ray absorption.


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The Interaction of x-rays with matter If an x-ray hits a solid material, the material starts to emit electrons. This is essentially the photoelectric effect that also occurs with ultraviolet radiation. In the case when a metal surface is exposed to ultraviolet, the following experimental observations can be made:

a. No electrons are ejected unless the radiation has a frequency above a certain threshold value characteristic of the metal.

b. In case of x-rays, there is always some electron emission, but one observes intensity jumps in the emitted electrons at certain energies.

c. Electrons are ejected immediately, however low the intensity of the radiation.

d. The kinetic energy of the ejected electrons increases linearly with the frequency of the incident radiation.

These observations follow naturally from the quantum nature of electromagnetic radiation; in fact the photoelectric effect was used by Einstein to (re)define electromagnetism. The same phenomenon is repeated in the X-ray range. At a certain X-ray energy extra electrons will appear from the sample.

The characteristics of the photoelectric effect can be explained in terms of photons. If the incident radiation has frequency , it consists of a stream of photons of energy h. These particles collide with the electrons in the metal. If the energy of a photon is less than the energy required removing an electron from the metal, then an electron will not be ejected, regardless of the intensity of the radiation. The energy required to remove an electron from the surface of a metal is called the work function of the metal and denoted . However, if the energy of the photon is greater than , then an electron is ejected

with a kinetic energy, E=½mv2, equal to the difference between the energy of the incoming photon and the work function. In case of x-rays, the explanation is analogous, with the addition that a core electron is excited instead of a valence electron. The binding energy of the core electron must be added and the kinetic energy is given as:

BEmv 221


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In fact, this equation defines the binding energy of a core level and x-ray photoemission experiments are used to measure core level binding energies. The binding energy EB is closely related to the ionisation energy that is defined as EB+. The photoelectric effect is not the only interaction of x-rays with matter. Another important interaction is the elastic scattering of x-rays due to electrons, which is the origin of x-ray diffraction experiments. Elastic scattering is also known as Thompson scattering. In a x-ray diffraction experiment one hits a sample with a x-ray and measures the diffracted x-rays. The diffraction angle can be correlated with the interatomic distances using Bragg's law. In this course we focus on spectroscopy and x-ray diffraction will not further be treated. With respect to spectroscopy, it is important to note that the amount of x-rays involved in x-ray diffraction is almost constant with respect to the photon energy. As such, it only gives an (almost constant) background upon which the photoelectric effect is superimposed.

This figure gives all contributions to the x-ray absorption process for a manganese atom (in a gas or in a solid). The intensity axis is given on a logarithmic scale and it can be seen that the photoelectric effect dominates up to approximately 50 keV. At higher energies the

100 1k 10k 100k 1M

1

10

100

Photonuclear

Electron-

positron

Photoelectric

Thompson

Compton

Inte

nsi

ty (

log

)

Energy (eV)


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photoelectric effect dies out and in addition to Thompson scattering, Compton scattering becomes important. Compton scattering refers to inelastic scattering. Around 1 MeV an additional x-ray process takes place, i.e. the photonuclear absorption, which refers to the absorption of a x-ray in a nuclear transition. This process is well known under the name of Mössbauer effect. At even higher energies the photon energy is high enough to cause the creation of electron-positron pairs, which is the inverse process of the electron-positron annihilation as used in positron emission tomography (PET) scans. In the following we will consider x-ray energies up to 100 KeV and assume that only Thompson scattering and the photoelectric effect take place.

X-ray absorption spectroscopy A straightforward experiment is to measure the transmission of an x-ray through a sample. A photon entering solid can be scattered or annihilated in the photo-electric effect. All other photons will be transmitted and will leave the sample. In x-ray absorption spectroscopy (XAS) the absorption of x-rays by a sample is measured. The intensity of the beam before the sample I0 and the intensity of the transmitted beam I are measured. Consider an infinitely thin layer, of thickness, dl, of the absorbing material. The intensity I0

of the incident beam is reduced by dI on passing through dl. dI is proportional to dl and dI is also proportional to the total intensity I, i.e.:

dlIdI is a proportionality constant called the linear absorption coefficient. It incorporates the combined effects of all photoelectric () and scattering () processes. One can rewrite the equation to:

dldII

1

Integration over a finite thickness from 0 to x yields:


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0ln

0

xl

I

II x

xII x 0lnln The linear absorption coefficient is dependent on the x-ray energy (indicated with its frequency ) and rewriting the equation yields:

xx eII )(

0 )()( alternatively, the linear absorption coefficient can be given as:

tx I

I 01 ln)(

The figure below indicates the intensity drop through sample of thickness x equal to 2/. This reduces the intensity after the sample to e-2·I0. The next task is to determine the absorption cross section as a function of the energy . We assume that the scattering is small and constant and limit ourselves to the photoelectric effect. An x-ray acts on charged particles such as electrons. As an x-ray passes an electron, its electric field pushes the electron first in one direction, then in the opposite direction, in other words the field oscillates in both direction and strength. In most descriptions of x-ray absorption one uses the vector field A to describe the electromagnetic wave. The vector field A is given in the form of a plane wave of electromagnetic radiation:

)(02

10 ˆ)cos(ˆ tkxi

qq eAetkxAeA The cosine plane wave function contains the wave vector k times the displacement x and the frequency times the time t. The cosine function is rewritten to an exponential function, for which only the absorbing e-it term has been retained. êq is a unit vector for a polarisation q. (These polarisation effects will not be treated explicitly below.) The Hamiltonian describing the interaction of x-rays with electrons can be approximated in perturbation theory and its first term can be written as:

ApH mce 1

The first term of the interaction Hamiltonian H1 describes the action of the vector field A on the momentum operator p of an electron. The electric field E is collinear to the vector field, in other words this term can also be understood as the electric field E acting on the electron moments. The proportionality factor contains the electron charge e, its mass m and the speed of light c. The Golden Rule states that the transition probability W between a system in its initial state i and final state f is given by:

if EEiffiW2

2

The initial and final state wave functions are built from an electron part and a photon part. The photon part of the wave function takes care of the annihilation of a photon in the x-ray


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absorption process. In the text below, it will not be included explicitly. The delta function takes care of the energy conservation and a transition takes place if the energy of the final state equals the energy of the initial state plus the x-ray energy. The squared matrix element gives the transition rate. The transition operator T1 describes one-photon transitions such as x-ray absorption. It is in first order equal to the first term of the interaction Hamiltonian, i.e. T1=H1. The transition rate W is found by calculating the matrix elements of the transition operator T1. Two-photon phenomena, for example resonant x-ray scattering, are described with the transition operator in second order. The interaction Hamiltonian is found by inserting the vector field into H1:

k

rikqkVm

e

q

epeaTk

k21 ˆ

This equation can be rewritten using a Taylor expansion of eikr=1+ikr+…. Limiting the equation to the first two terms 1+ikr, the transition operator is:

k

kqkqVme

kq

rkpeipeaTk

))(ˆ(ˆ21

T1 contains the electric dipole transition (êkq·p). The electric quadrupole transition originates from the second term. The value of kr can be calculated from the edge energy in eV and the atomic number Z.

ZeVrk edge 80/)()( In case of the 1s core levels (called K edges) from carbon (Z=6, edge = 284 eV) to zinc (Z=30, edge = 9659 eV), the value of kr lies between 0.03 and 0.04. The transition probability is equal to the matrix element squared, hence the electric quadrupole dipole transition is smaller by approximately 1.510-3 than the dipole transition and can be neglected, which defines the well-known dipole approximation. The transition operator in the dipole approximation is given by:

k

kqVme

kq

peaTk

ˆ21

Omitting the summation over k and using the commutation law between the position operator r and the atomic Hamiltonian (p=m/i[r,H] ) one obtains the familiar form of the x-ray absorption transition operator:

rereV

eT qq

qq

ˆˆ2

1


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Basics of XAS Spectroscopy

Including this operator into the Fermi golden rule gives:

if EEiqfq

fi reW2

ˆ

This equation will form the basis for the rest of the lectures. It is a rather theoretical and formal description. What happens in practice? If an assembly of atoms is exposed to x-rays it will absorb part of the incoming photons. At a certain energy (depending on the atoms present) a sharp rise in the absorption will be observed (Figure 1.2). This sharp rise in absorption is called the absorption edge.

Figure: The x-ray absorption cross sections of manganese and nickel. Visible are the L2,3 edges at respectively 680 and 830 eV and the K edges at respectively 6500 eV and 8500 eV.

The energy of the absorption edge is determined by the binding energy of a core level. Exactly at the edge the photon energy is equal to the binding energy, or more precisely the edge identifies transitions from the ground state to the lowest empty state. The figure shows the x-ray absorption spectra of manganese and nickel. The L2,3 edges relate to a 2p core level and the K edge relates to a 1s core level binding energy. Many complicating aspects do play a role here and we will come back to this issue later. In the case of solids, when other atoms surround the absorbing atom, a typical XAFS spectrum is shown below.

Figure: The L3 XAFS spectrum of platinum metal. The edge jump is seen at 11564 eV and above one observes a decaying background modulated by oscillations.

11000 11500 12000 12500 13000

Energy (eV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Intensity (a.u)


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Instead of a smooth background we see here changes in the intensity of the absorption. Oscillations as function of the energy of the incoming x-rays are visible. The question, which rises immediately, is what causes these oscillations? An answer can be found by assuming that the electron excitation process is a one electron process. This makes it possible to rewrite the initial state wave function as a core wave function and the final state wave function as a free electron wave function (). One implicitly assumes that all other electrons do not participate in the x-ray induced transition. We will come back to this approximation below.

2222

ˆˆˆ Mcrerecre qiqiiqf

Figure: The schematic density of states of an oxide. The 1s core electron at 530 eV binding energy is excited to an empty state: the oxygen p-projected density of states. The squared matrix element M2 is in many cases a number that is only little varying with energy and one can often assume that it is a

constant. The delta function of the Golden Rule implies that one effectively observes the density of empty states ().

2~ MI XAS The x-ray absorption selection rules determine that the dipole matrix element M is non-zero if the orbital quantum number of the final state differs by 1 from the one of the initial state (L = 1, i.e. s p, p s or d , etc.) and the spin is conserved (S = 0). The quadrupole transitions imply final states that differ by 2 from the initial state (L = 2, i.e. s d, p f or (L = 0, i.e. s s, p p). They are some hundred times weaker than the dipole transitions and can be neglected in most cases. It will be shown below that they are visible though as pre-edge structures in the K edges of 3d-metals and the L2,3 edges of the rare earths. In the dipole approximation, the shape of the absorption spectrum should look like the partial density of the (L = 1) empty states projected on the absorbing site, convoluted with a Lorentzian. This Lorentzian broadening is due to the finite lifetime of the core-hole, leading to an uncertainty in its energy according to Heisenberg's principle. A more accurate approximation can be obtained if the unperturbed density of states is replaced by the density of states in presence of the core-hole. This approximation gives a relatively adequate


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simulation of the XAS spectral shape, at least in those cases where the interaction between the electrons in the final state is relatively weak. This is often the case for 1s 4p transitions (K-edges) of the 3d metals.

XANES The single electron approximation as stated in section 1.3 gives an adequate simulation of the XANES spectral shape if the interactions between the electrons in the final state are relatively weak. This is the case for all K edges and to a very good degree also for all other edges with binding energies above 3 keV (with a few exceptions). We will assume this is correct and use the equivalence of IXAS and the density of states as a starting point.

Figure: Schematic band picture of (a) a TiO6- cluster and (b) a FeO6- cluster. Ti4+ in TiO2 has an empty 3d band and Fe3+ in Fe2O3 contains a half filled 3d band. A usual picture to visualise the electronic structure of a transition metal compound, such as oxides and halides, is to describe the chemical bonding mainly as a bonding between the metal 4sp states and the ligand p states, forming a bonding combination, the valence band


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and empty antibonding combinations. The 3d states also contribute to the chemical bonding with the valence band that causes them to be antibonding in nature. This bonding, taking place in a (distorted) cubic crystal-line surrounding, causes the 3d states to be split into the so-called t2g and eg manifolds. This situation is visualised for a TiIVO6 cluster in the figure above. For comparison the oxygen 1s X-ray absorption spectrum is given. On the right this picture is modified for a partly filled 3d band in a FeIIIO6 cluster. The 3d band is split by the crystal field splitting and a large exchange splitting. This qualitative description of the electronics structure of transition metal compounds can be worked out quantitatively within DFT.

The picture of the left is a DFT calculation of the density of states of TiO2. The total DOS is projected to oxygen (O) and titanium (Ti) respectively. The picture below shows the broadened oxygen DOS in comparison with the experimental spectrum. In line with the dipole selection rule only the oxygen p-projected DOS is used.

We assume that the density of states of complex systems is calculated using the DFT approximation. Programs to calculate the X-ray absorption spectral shape include FEFF, GNXAS and Wien2k.


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XANES spectral shape analysis We have seen that the oxygen K edge x-ray absorption spectrum of TiO2 can be simulated with the oxygen p-projected DOS as calculated by DFT. (Actually in case of the oxygen K edge of TiO2, the core hole effect is small and can be neglected). The similarity between x-ray absorption and the projected DOS allows a qualitative analysis of the spectral shapes without the actual inclusion of calculations.

Above we give the oxygen K edges of a series of transition metal oxides, ranging from Sc2O3 to CuO. Sc2O3 contains ScIII and is formally a 3d0 compound, i.e. it has all ten 3d states empty. CuO is a 3d10 compound and has all ten 3d-states occupied. One can distinguish two regions in the spectrum: the dashed region is related to the empty 3d-band, actually to the oxygen p-projected DOS inside this 3d-band. The structure at 540 eV is related to the empty 4s and 4p states of copper. The dashed region is split by the cubic crystal field splitting and, in first approximation, one observes two peaks that are respectively related to the t2g (xy, xz, yz) and eg (z2, x2-y2) orbitals.


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X-ray attenuation lengths Figure: The attenuation length of x-rays through carbon (solid), iron (dotted) and lead (dash-dotted).

The figure gives the attenuation lengths of x-rays from 300 eV to 10 keV through C, Fe and Pb. X-rays with energies less than 1 keV have an attenuation length of less than ~1 m, implying that for soft x-rays the sample has to be thin. The figure below shows the transmission through 1m and 10m iron and it can be seen that soft x-rays (<1 keV) have little to no transmission through 1m iron. At the iron K edge at 8 KeV the transmission through 10m iron drops from ~0.65% to less than 0.10%. The extra absorption of the iron 1s core level takes away most of the x-rays in the first 10m. In addition soft x-rays have a large absorption cross section with air, hence the experiments have to be performed in vacuum.

Figure: Transmission through air and bulk iron. Given are respectively 10 cm air (solid), 10 m air (dashed), 1m iron (dash-dotted) and 10m iron (dotted).


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Detection techniques

Transmission detection

After the beam has passed the monochromator, its intensity (I0) is measured in an ionisation chamber, or using a thin metallic foil or grid absorbing part of the photons. The x-ray interacts with the sample of interest and the intensity after the sample is measured with a second ionisation chamber or a photodiode. This is the transmission mode of x-ray absorption. Transmission experiments are standard for hard x-rays. An important limitation of transmission detection arises from the requirement for a homogeneous sample. Variations in the thickness or pinholes are reasons for the so-called thickness effect that can significantly affect the spectral shape by introducing non-linear responses. This is important in particular for the EXAFS analysis. The combination of the short attenuation length with the thickness effect makes transmission experiments not suitable for x-ray absorption below 1 keV. For the soft x-ray range other detection modes must be used. A special technique to measure transmission experiments is to use an energy dispersive monochromator. The monochromator crystal is bent and a range of energies is sent through the sample simultaneously. Each energy has its particular angle of incidence and after the transmission through the sample the complete x-ray absorption spectrum can be measured momentary with an array detector. Energy dispersive experiments suffer from similar limitations as normal transmission experiments with respect to attenuation length and thickness effect. A major advantage of energy dispersive x-ray absorption is that the complete XAS spectrum is obtained momentarily. It is clear that this yields interesting options for time resolved measurements. Real time (non repeated) experiments can be probed with typically 1 s time resolution. A typical energy dispersive XAS beamline, is beamline ID24 at ERSF in Grenoble. See their website: http://www.esrf.fr/UsersAndScience/Experiments/XASMS/ID24/

Fluorescence Yield

The decay of the core hole gives rise to an avalanche of electrons, photons and ions escaping from the surface of the substrate. By measuring any of these decay products, it is possible to measure samples of arbitrary thickness. In this section the different methods are discussed, specifically with regard to the conditions under which a specific yield measurement


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represents the x-ray absorption cross section and the related question of the probing depth of the specific yield method. An important prerequisite for the use of decay channels is that the channels are linearly proportional to the absorption cross section. In general this linear proportionality holds, but there are cases where for example the ratio between radiative a non-radiative decay varies significantly over a relatively short energy range. The reason behind this phenomenon is a varying fluorescence decay depending on the symmetry of the final state in the x-ray absorption process. It turns out that in cases where multiplet effects are important (discussed in chapter 3), for example the 2p x-ray absorption of 3d metals, the fluorescence decay varies by almost a factor of two over an energy range of approximately 15 eV, i.e. from the first to the last peak in the edge region. Because for soft x-rays Auger decay dominates over fluorescence, this effect is only visible in fluorescence yield detection. The fluorescent decay of the core hole can be used as the basis for the absorption measurement. The amount of fluorescent decay increases with energy and a comparison with the amount of Auger decay shows that Auger decay dominates for all core levels below 1 keV. In case of the 3d-metals, the K edges show strong fluorescence and all other edges mainly Auger decay. The photon created in the fluorescent decay has a mean free path of the same order of magnitude as the incoming x-ray, which excludes any surface effect. On the other hand it means that there will be saturation effects if the sample is not dilute. For dilute materials the background absorption B dominates the absorption of the specific edge and the measured intensity is proportional to the absorption coefficient. For less dilute materials the spectral shape is modified and the highest peaks will appear compressed with respect to the lower peaks, an effect known as self-absorption or saturation.

Figure: The effect of saturation due to self-absorption on the platinum L2 spectrum measured in fluorescence yield assuming a relative background absorption of 1.5. The transmission spectrum is given with the solid line. The saturated (and re-normalised) fluorescence spectrum with the dashed line. The figure shows the


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main effect of saturation due to self-absorption: a reduction of the peak heights and depths and as such a blurring of the spectrum and as such of the spectral information. Using the inverse formula one can in principle reconstruct the original absorption spectrum from the saturated one. An uncertainty in this procedure is the exact value of B and in addition this data treatment increases the noise. Partial Fluorescence Yield Recently it became possible to use fluorescence detectors with approximately 1 eV resolution to tune to a particular fluorescence channel. This could be denoted as partial fluorescence yield. The technique is also known as selective x-ray absorption because one can select for example a particular valence and measure the x-ray absorption spectrum of that valence only. Other possibilities are the selectivity to the spin orientation and the nature of the neighbouring atoms. Partial fluorescence yield effectively removes the lifetime broadening of the intermediate state. This effect can be used to measure x-ray absorption spectra with unprecedented spectral resolution.

Electron yield

With the total electron yield method one detects all electrons that emerge from the sample surface, independent of their energy. One can use a number of detection devices: (i) one can detect the current that flows to the sample with a pico-ampere meter, and (ii) one can use a channeltron to amplify the emitted electrons to a detectable signal. In addition it can be shown that if the measurements are carried out in a gaseous atmosphere with a pressure of approximately 10 mbar, the detected ionised gas molecules yield a signal that is under certain conditions proportional to the absorption cross section. The interaction of electrons with solids is much larger than the interaction of x-rays which implies that the electrons that escape from the surface must originate close to the surface. The probing depth of total electron yield lies in the range of approximately 3 to 10 nm, depending also on the material studied. A quantitative study on the oxygen K edge of Ta2O5 determined an electron escape depth of 4 nm. Studies of rare earth overlayers on Ni revealed very short escape depths of the order of 1 nm. The extremely large x-ray absorption cross sections of the rare earth M4,5 edges (related to 3d to 4f transitions) makes that the x-ray penetration depth falls in the same range as the electron escape depth. This implies that both the electron escape depth e and the (angular dependent) x-ray penetration depth p()sin affect the measured x-ray absorption cross section (Ie), according to:

sin)(

)(p

eeI


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Angular dependent measurements on Dy/Ni(110) revealed an x-ray penetration depth of only 12 nm at the M5 resonance of Dy. In most other cases, the x-ray penetration depth will be significantly longer in line with the x-ray transmission plots as given above. Instead of just counting the escaped electrons, one can detect their respective energies and as such measure the partial electron yield. The figure shows a schematic spectrum of the electron decay energies of the nickel K edge.

Figure: Schematic electron emission spectrum for a Ni K edge experiment. The peak at low energy goes up to 80.0 on the scale of the figure. The primary Auger channel is the 1s2p2p channel at 6.5 keV. The main secondary channel is the 2p3d3d Auger channel.

This figure shows the primary 1s2p2p Auger electrons with kinetic energies of approximately 6500 eV. These primary electrons can loose part of their energy in inelastic scattering phenomena after which they have energies anywhere between 6500 eV and zero. In addition secondary Auger channels are visible as separate peaks, for example the 2p3d3d channel at approximately 850 eV. One can view this process as follows: The 1s x-ray absorption process has created a 1s core hole at ~8200 eV. The main Auger channel is the 1s2p2p channel that leaves two core holes in the 2p level, each with a binding energy of 850 eV. This implies that the kinetic energy of the Auger electron is 8200 - 850 - 850 = 6500 eV (neglecting correlation effects). However the system is now stuck with two core holes in the 2p core level. Both these holes can decay again via 'cascade' Auger processes, for example 2p3d3d. The 2p3d3d Auger electron has an energy of 850 - 0 -0 = 850 eV and the system is left with only valence holes that can decay usually via phonons and recombination processes Each such secondary peak brings with it its inelastic tail. The majority of the emitted electrons from a sample are true secondary electrons with emission energies of less than 100


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eV. These electrons have lost most of their energy (due to inelastic scattering processes) in their route to the surface of the sample. The figure shows an actual Auger decay spectrum for NiO. This figure shows that apart from the main channels such as 2p3d3d, essentially all Auger decay channels do contribute. The figure also includes the relatively weak 2p3s3s, 2p3s3p and 2p3p3p Auger decay channels.

Figure: Auger decay spectrum for NiO excited with x-rays of 853.4 eV. The structures visible are respectively related to the 2p3p3p, 2p3s3p and 2p3s3s Auger decay channels. The dark areas at the bottom relate to the multiplet calculations of these channels (see chapter 3).

If one detects an Auger decay channel with high resolution one can perform similar experiments as with high-resolution fluorescence detection and do selective x-ray absorption experiments.

Ion Yield

If the absorption process takes place in the bulk and the core hole decays via an Auger process a positively charged ion is formed, but due to further decay, inelastic electron scattering and screening processes the original charge neutral situation will be restored after some time, typically in the femtosecond range. If however the absorption process takes place at the surface, the possibility exists that the atom which absorbs the x-ray is ionised by Auger decay and escapes from the surface before relaxation processes can bring it back to a bound state. If the escaping ions are analysed as a function of x-ray energy the signal is in general proportional to the absorption cross section. Because only atoms from the top-layer are able to escape, ion yield is extremely surface sensitive. The mere possibility of obtaining a measurable signal from ion yield means that the surface is irreversibly distorted, but as there are of the order of 1015 surface atoms per cm2 this does not necessarily mean that a statistically relevant proportion of the surface has to be affected for a measurable signal of, say, 107 ions.

X-ray spectroscopy Course of Advanced Spectroscopy of Nanomaterials Exercises

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Exercise 1: a. Calculate the energy in electron volt for an x-ray with 1 Å wavelength. (use c=3·108

m/s, h=6.6·10-34 and 1 eV = 1.6·10-19 J) b. Check if the figure on page 2 is correct for this wavelength. c. Calculate the wave length of IR radiation of 0.1 eV.

Exercise 2: X-ray Photoemission In an x-ray photoemission (XPS) experiment, NiO is excited with Al K radiation with an energy of 1487 eV. One observes peaks in the XPS spectrum at respectively 607 eV and 637 eV, both related to the 2p core level of nickel. Assume that the work function of NiO is equal to 2 eV.

a. Calculate the binding energies of the core levels associated with the observed peaks. b. Why are there two Ni 2p peaks?

Exercise 3: The binding energies of the core levels of manganese are respectively 6539 eV (1s), 769 eV (2s), 645 eV (2p), 82 eV (3s) and 47 eV (3p). The 2p spin-orbit splitting is 12.0 eV, the 3p spin-orbit splitting is 0.6 eV. The oxygen core levels are found at 543 eV (1s) and 42 eV (2s). A MnO sample is excited with Al K radiation with an energy of 1487 eV. Its work function is 2.0 eV.

a. At which kinetic energies does one observe XPS peaks? Exercise 4:

a. Calculate the ionisation energy of a rubidium atom, given that radiation of wavelength 58.4 nm produces electrons with a speed of 2450 km/s. The mass of an electron is 9.1 10-31kg.

Exercise 5:

Calculate the screening constant for the Argon 1s XAS spectrum that has its edge at 3205 eV. Exercise 6: Simulating an X-ray absorption spectrum. You can download the file NIOOK.DAT. This contains an oxygen K edge X-ray absorption spectrum of NiO. Also download the file NIODOS1.DAT, which contains the (total) density of states of NiO, as taken from the database used I exercise 6. The zero energy in this file is the Fermi energy. a. Compare the experimental NiO spectrum to the total DOS, using a plotting program.

(take care to use only that part of the DOS that is visible in an XAS experiment) b. Should an oxygen K edge spectrum look like the total DOS?

X-ray spectroscopy Course of Advanced Spectroscopy of Nanomaterials Exercises

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Exercise 7: The thickness effect An experimental artefact in transmission X-ray absorption spectroscopy is known as the thickness effect or the pinhole effect. Assume that there is a hole in the sample you are measuring with an X-ray in transmission mode. Assume that 20% of the illuminated sample is missing and in this 20% no absorption takes place. a: How does this affect the transmission? Assume that you measure an X-ray absorption spectrum, where the peak has a transmission of 10%. b: What transmission does one measure? c: How does the pinhole effect affect the XAS spectral shape? You can download the file XASfile1.dat. This file contains a theoretical x-ray absorption cross section μ∙x as a function of the energy in eV. The cross section is given for a thickness of 1 μm. d: Plot the file with EXCEL or ORIGIN (or any other plotting program) e: Calculate and plot the expected transmission Ix/Io in the program. Assume that 20% of the sample is missing and the X-rays are transmitted in this 20% without

absorption. f: Calculate and plot the observed transmission. h: Calculate and plot the observed cross section. Is there a large effect? Change the thickness of the sample to 10 μm and answer the 4 questions (d to h) again. Exercise 8: a. Why is lead the best element to be used to protect against the transmission of x-rays? b. A synchrotron that yields x-rays up to 2 KeV is not protected with lead shielding. Why is

this not necessary? Exercise 9: Go to the webpage of the Center for X-ray Optics (CXRO) in Berkeley (http://www-cxro.lbl.gov/) and follow the link to X-ray tools. a. Check the attenuation lengths of copper and lead between 100 eV and the maximum

energy possible. Modify each plot also to a logarithmic scale. b. Check the transmission of x-rays for Be and quartz (SiO2) between 100 eV and the

maximum energy possible. c. In Ultra-High-Vacuum equipment one uses beryllium windows. Why Be? d. To attenuate the synchrotron beams at energies above 3 KeV (for example for radiation

sensitive experiments) one uses aluminium foils. Why is Al used and not for example Cu or Pb?

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