The deepest gratitude to my supervisor Professor George Panayiotakis for offering me the opportunity to make this PhD and for his continuous support and guidance during all these years

The above should not change or deteriorate with time and as a consequence of repeated exposure to x-rays. That is, x-ray fatigue and x-ray damage should be negligible.

The photoconductor should be easily coated onto the active matrix panel, for example, by conventional vacuum techniques. Special processes are generally more expensive.

The photoconductor should have uniform characteristics over its entire area.

For the time being, the large area coating requirements in mammography (typically over 30 × 30 cm2 or more), makes amorphous (a-) and polycrystalline (poly-) photoconductors to be more suitable for digital detectors as compared to crystalline materials which are difficult to grow in such large areas with current techniques. a-Se is one of the most highly developed photoconductor due to its commercial use as an electrophotographic photoreceptor. It can be easily coated as thick films (e.g. 100-500 ìm) onto suitable substrates by conventional vacuum deposition techniques and without the need to raise the substrate temperature beyond 60-70 oC. In addition, its amorphous state maintains uniform characteristics to very fine scales over large areas (Kasap and Rowlands 2000, 2002a, 2002b). Nevertheless, due to its high µ § compared to other materials there has been an active research to find x-ray photoconductors that could replace a-Se in flat panel image detectors (Kasap and Rowlands 2000, 2002a , 2002b).

A perfect elemental crystal consists of a regular spatial arrangement of atoms, with precisely defined distances (the interatomic spacing) separating adjacent atoms. Every atom has a strict number of bonds to its immediate neighbors (the coordination) with a well defined bond length and the bonds of each atom are also arranged at identical angular intervals (bond angle). This perfect ordering maintains a long range order and hence a periodic structure (Kabir 2005). A hypothetical two-dimensional crystal structure is shown in figure 4.1(a).

An amorphous solid exhibits no crystalline structure or long range order and it only possesses short range orders because the atoms of an amorphous solid must satisfy their individual valence bonding requirements, which leads to a little deviation in the bonding angle and length. Thus, the bonding geometry around each atom is not necessarily identical to that of other atoms, which leads to the loss of long-range order as illustrated in figure 4.1(b). As a consequence of the lack of long-range order, amorphous materials do not possess such crystalline imperfections as grain boundaries and dislocations, which is a distinct advantage in certain engineering applications (Kabir 2005).

Figure 4.1. Two dimensional representation of the structure of (a) a crystalline solid and (b) an amorphous solid.

Materials like a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2 satisfy some of the ideal characteristics mentioned earlier and hence are potential candidates as photoconductors in direct detectors for digital mammography. The most important ones are a-Se, CdTe, CdZnTe, PbO, PbI2 and HgI2 due their amorphous and polycrystalline structure and due to certain properties that will be discussed in detail. The information given is mainly obtained from Kabir (2005) and Nesdoly (1999).

Figure 4.2. (a) The grain structure of polycrystalline solids. (b) The grain boundaries have impurity atoms, voids, misplaced atoms, and broken and strained bonds (Kasap 2002).
4.4.1. Amorphous Selenium (a-Se)

Selenium is a member of the group VI column of the periodic table. The family name of the elements of this group is chalcogens. The atomic number (Z) of selenium is 34, and it has six valence electrons. Its electronic structure is [Ar]3d104s2p4. The density of a-Se is 4.3 g/cm3, relative permittivity år = 6.7, and energy gap Eg = 2.22 eV.

Amorphous selenium can be quickly and easily deposited as a uniform thick film (for example 100-1000 ìm) over large areas (for example 40 cm × 40 cm or larger) by conventional vacuum deposition techniques and without the need to raise the substrate temperature beyond 60-70 °C. Nevertheless, pure a-Se is thermally unstable and crystallizes over time. Alloying pure a-Se with As (0.2 ¨C 0.5% As) greatly improves the stability of the composite film and helps to prevent crystallization. However, it is found that arsenic addition has adverse effect on the hole lifetime because the arsenic introduces deep hole traps. If the alloy is doped with 10 ¨C 20 parts per million (ppm) of a halogen (such as Cl), the hole lifetime is restored to its initial value. Thus, a-Se film that has been alloyed with 0.2 ¨C 0.5% As (nominal 0.3% As) and doped with 10 ¨C 20 ppm Cl is called stabilized a-Se. Stabilized a-Se is currently the preferred photoconductor for clinical x-ray image sensors. Crystalline Se is unsuitable as an x-ray photoconductor because it has a much lower dark resistivity and hence orders of magnitude larger dark current than a-Se.

Stabilized a-Se has excellent transport properties, with typical hole and electron ranges (ìô products) being 30 x 10-6 cm2/V and 5 x 10-6 cm2/V respectively. At typical operating fields (>10 V/ìm) the hole mean free path length is µ §30 mm whereas that for electrons µ § 5 mm. Since as mentioned in the previous chapter most a-Se detectors are 200-500 ìm thick, these large mean free paths ensure that no free charges will be lost due to trapping. The dark resistivity of a-Se is ~ 1014 Ù-cm. The dark current in a-Se detectors is less than the acceptable level (1 nA/cm2) for an electric field as high as 20 V/ìm. The image lag in a-Se detectors is under 2% after 33 ms and less than 1% after 0.5 s in the fluoroscopic mode of operation (Choquette et al 2001). Therefore, image lag in a-Se detectors is considered as negligible. The pixel to pixel sensitivity variation is also negligible in a-Se detectors.

The charge transport properties of a-Se as compared to a-As2Se3 are better for both electrons and holes. In a-As2Se3 electrons are trapped and holes have much smaller mobility. In addition, the dark current of a-Se is much smaller than a-As2Se3 (Kasap and Rowlands 2000).

Where a-Se suffers in comparison to other materials is in two areas: x-ray absorption and x-ray sensitivity (µ §). Since the Z of Se is 34, a-Se is a rather poor absorber of x-rays and a thicker detector must be used to absorb the same amount of x-ray radiation with a detector composed of a material with higher atomic number (for example CdZnTe that has Zeff = 50). For the typical value of the electric field used in a-Se devices (10 V/ìm) the value of µ § is about 45 eV when for polycrystalline mercuric iodide (poly-HgI2) and polycrystalline Cadmium Zinc Telluride (poly-CdZnTe) the value of µ § is typically 5-6 eV.

CdTe has a moderate atomic number (Zeff ~ 50) and a low µ §~ 4.5 eV. As a consequence, it is highly efficient to absorb the incident x-ray radiation and convert it to charge carriers. As opposed to a-Se where both electrons and holes are mobile, in CdTe only electrons are mobile. The major disadvantages of CdTe is the relatively high dark current which is of the order of ~ 10 nA/cm2 and the high substrate and annealing temperatures (180-190 oC) required to deposit large area polycrystalline CdTe layers by vacuum deposition techniques. The latter can cause damage in the electronics which lie beneath the photoconductor’s layer.

CdZnTe (<10% Zn) polycrystalline film has been used as a photoconductor layer in x-ray AMFPI. Introduction of Zn into the CdTe lattice increases the bandgap, decreases conductivity and hence largely reduces dark current. Hole mobility in CdZnTe decreases with increasing Zn concentration whereas electron mobility remains nearly constant. Furthermore, addition of Zn into CdTe increases lattice defects and hence reduces carrier lifetimes. The poly-CdZnTe has a lower crystal density resulting in lower x-ray sensitivity than its single crystal counterpart. Furthermore, for a detector of given thickness, the x-ray sensitivity in CdZnTe detectors is lower than in CdTe detectors. Nevertheless, the CdZnTe detectors show a better signal to noise ratio and hence give better detective quantum efficiency (DQE). The measured sensitivities are higher than other direct conversion sensors (e.g. a-Se) and the results are encouraging.

Although CdZnTe can be deposited on large areas, direct conversion AMFPI of only 7.7×7.7 cm2 (512 × 512 pixels) from a polycrystalline CdZnTe has been demonstrated. The CdZnTe layer thickness varies from 200-500 ìm. Temporal lag and nonuniform response were noticeable in early CdZnTe sensors which are attributed to large and nonuniform grain sizes.
4.4.4. Polycrystalline Lead Oxide (poly-PbO)

Direct conversion flat panel X-ray imagers of 18 × 20 cm2 (1080 × 960 pixels) from a poly-PbO with film thickness of ~300 ìm have been demonstrated (Simon et al 2004). One advantage of PbO over other x-ray photoconductors is the absence of heavy element K-edges for the entire diagnostic energy range up to 88 keV, which suppresses additional noise and blurring due to the K-fluorescence. PbO has µ §~8 eV, density 4.8 g/cm3, energy gap Eg=1.9 eV and resistivity in the range 7-10 x 1012 Ù-cm (Simon et al 2004). PbO photoconductive polycrystalline layers are prepared by thermal evaporation in a vacuum chamber at a substrate temperature of ~100°C. The dark current in PbO sensors is ~40 pA/mm2 at an electric field of 3 V/ìm (Simon et al 2004). PbO reacts with the air and this leads to an increase in dark current and a decrease in x-ray sensitivity. Additionally, thick PbO layers degrade from prolonged x-ray exposure.

Polycrystalline HgI2 layers can be prepared by both physical vapor deposition (PVD) and screen printing (SP) from a slurry of HgI2 crystal using a wet particle-in-binder process (Street et al 2002). Direct conversion x-ray AMFPI of 20 × 25 cm2 (1536 × 1920 pixels) and 5 × 5 cm2 (512 × 512 pixels) size have been demonstrated using PVD and SP poly-HgI2 layer, respectively (Street et al 2002, Zentai et al 2004). HgI2 has µ §~ 5 eV, density 6.3 g/cm3, energy gap Eg= 2.1 eV and resistivity ~ 4 x 1013 Ù-cm.

HgI2 tends to chemically react with various metals and hence a thin blocking layer (typically ~1 ìm layer of insulating polymer) is used between the HgI2 layer and the pixel electrodes to prevent the reaction and also to reduce the dark current. The HgI2 layer thickness varies from 100-400 ìm.

The dark current of HgI2 imagers increases superlinearly with the applied bias voltage. The dark current of a PVD HgI2 detector strongly depends on the operating temperature (it increases by a factor of approximately two for each 6°C of temperature rise). It is reported (Zentai et al 2004) that the dark current varies from ~ 2 pA/mm2 at 10°C to ~ 180 pA/mm2 at the 35°C for an applied electric field of 0.95 V/ìm. On the other hand, the dark current in the SP sample is an order of magnitude smaller than in PVD sample and more stable against temperature variation. The only disadvantage of SP detectors is that they show ~2|4 times less sensitivity compared to PVD detectors.

Electrons have much longer ranges than holes in HgI2. Furthermore, it is reported that HgI2 image detectors with smaller grain sizes show good sensitivity and also an acceptable uniform response. As reported in the literature, poly-HgI2 imagers show excellent sensitivity, good resolution, and acceptable dark current, homogeneity and lag characteristics, which make this material a good candidate for diagnostic x-ray image detectors.
4.4.6. Polycrystalline Lead Iodide (poly-PbI2)

PbI2 photoconductive polycrystalline layers are prepared by PVD at a substrate temperature of 200 to 230°C. Direct conversion AMFPI of 20 × 25 cm2 size (1536 × 1920 pixels) have been demonstrated using PVD polycrystalline PbI2 layer (Zentai et al 2003). PbI2 coating thickness varies from 60-250 ìm. PbI2 has µ §~ 5 eV, energy gap Eg=2.3 eV and resistivity in the range 1011-1012 Ù-cm.

PbI2 detectors have a very long image lag decay time that depends on the exposure history. The dark current of PbI2 imagers increases sublinearly with the applied bias voltage and it is in the range 10-50 pA/mm2 at an electric field of 0.5 V/ìm. Furthermore, it is much higher than that of PVD HgI2 detectors, making it unsuitable for long exposure time applications. The resolution of PbI2 imagers is acceptable but slightly less than that of HgI2 imagers. Also, the x-ray sensitivity of PbI2 imagers is lower than that of HgI2 imagers. The pixel to pixel sensitivity variation in PbI2 imagers is substantially low. Its dark conductivity is larger than that for a-Se (Kasap and Rowlands 2000).
4.5. Table of material properties

In table 4.1 some of the material properties of a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2 are presented.

Chapter 5

Physics Of Image Formation

5.1. Introduction

The x-ray-matter interactions produce charges (i.e. electrons) that as they drift inside the photoconductor interact with the material giving birth to secondary charged particles. The final image is created from those charges that have not been recombined or trapped during their drifting towards the collecting electrodes. This chapter describes the basic physics of x-ray¨Cmatter interactions, of atomic deexcitation mechanisms and of electron interactions. Furthermore, it discusses some aspects of the charge carrier transport inside a-Se.
5.2. X-ray ¨C matter interactions

In the mammographic energy range (photon energies smaller than 40 keV) three are the types of x-ray-matter interaction:

Coherent (Rayleigh) scattering.

Incoherent (Compton) scattering.

Photoelectric absorption.

5.2.1. Coherent (Rayleigh) scattering

Rayleigh scattering is a process in which the energy of the initial photon is not converted to kinetic energy of another particle but it is all scattered.

Using classical physics, one can derive the differential cross section of scattering of a photon from a free electron (Thomson scattering) as:

µ § (5.1)

where ro is the classical radius of the electron, è is the angle between the initial trajectory of the electromagnetic wave and the new one after the scattering from the electron, and dóï/dÙ is the differential cross section per electron of the classical scattering that gives the fraction of the incident energy that is scattered by the electron into the solid angle dÙ=1/r2 or the fractional number of photons scattered into unit solid angle at angle è.

Suppose now that the photon of energy E passes over an atom. Since a photon is an electromagnetic wave, its oscillating electric field sets the electrons of the atom in a momentary vibration. These oscillating electrons emit radiation of the same energy E as the incident radiation. The scattered waves from electrons combine with each other to form the scattered photon. The differential cross section for Rayleigh scattering is given as:

µ § (5.2)

where F(µ §, Z) is the so called ‘atomic form factor’, which is the probability that all the electrons take up recoil momentum without absorbing any energy, and µ §is related to the momentum transfer during the collision given by:

µ §(Å-1) (5.3)

where mc2 is the electron rest energy.
5.2.2. Incoherent (Compton) scattering

In Compton scattering a photon with energy E collides with an atomic electron, transfers some of its energy and momentum to the electron and is being deflected at an angle è, with respect to its initial direction, with energy µ § given by:

µ § (5.4)

The differential cross section of incoherent scattering including electron binding effects can be given as the product of the Klein-Nishina differential cross section dóKN/dÙ (for Compton collision between a photon and a free electron) and the incoherent scattering function of an atom S(µ §, Z). The latter represents the probability that an atom will be raised to any excited or ionized state when a photon imparts a recoil momentum to an atomic electron. Therefore the differential cross section of incoherent scattering is given by:

µ § (5.5)
The relationship between the electron scattering angle èe and the photon scattering angle è, derived from angle calculations in the scattering process, is given by:

µ § (5.6)

In the photoelectric effect the incident photon is completely absorbed by the atom that ejects a photoelectron. The more tightly bound electrons are the important ones in bringing about photoelectric absorption and the maximum absorption occurs when the photon has just enough energy to eject the bound electron. The photoelectron is ejected with energy Ee given by:

Ee=E-Bs (5.7)

where Bs is the binding energy of the electron in the ‘s’ shell. To a first approximation, at low energies the emission is entirely due to the electric vector of the incident wave acting on the electron. Neglecting relativity and spin corrections, the number of photoelectrons per solid angle is (Davisson and Evans 1952):

µ § (5.8)

where â=õ/c with õ being the electron velocity, and µ § are the polar and azimuthal angles of the photoelectron with respect to the direction of the incident photon.

The atom after having interacted with photoelectric effect deexcites releasing energy. During a deexcitation sequence, the vacancy initially created in an inner shell migrates to outer shells with radiative and non-radiative atomic transitions. The radiative transitions are the emission of fluorescent photons while the non-radiative transitions the ejection of Auger or Coster-Kronig (CK) electrons.

Figure 5.1. illustrates schematically the three types of atomic deexcitation. In a radiative transition (figure 5.1(a)), the vacancy in a shell is filled in with an electron that jumps from a higher shell or subshell with the concomitant emission of electromagnetic radiation (fluorescent photon). The inter-subshell fluorescent transitions (e.g. the emission of XiXj photon in figure 5.1(a)) have almost negligible probability of occurrence. The fluorescent photons are isotropically emitted with energy Efl = Bi-Bj, where Bi and Bj are the binding energies of the shells i and j involved in the transition.

isotropically ejected, while their energy is EAug/CK = Bi-Bj-Bk, where Bi, Bj and Bk are the binding energies of the shells i, j and k involved in the transition.

Figure 5.1. The three types of atomic deexcitation. (a) Fluorescent photon emission, (b) Auger electron ejection and (c) CK electron ejection.

5.4. Electron interactions

Five are the types of electron-matter interaction:

Emission of Bremsstrahlung radiation.

Phonon interactions.

Elastic scattering.

Inelastic scattering.

Collective oscillations (plasma waves).

The emission of Bremsstrahlung radiation takes place at high energies and high Z materials. Thus, in the mammographic energy range this effect can be ignored. The phonon interactions, alternatively called ‘inelastic collisions of low energy electrons with phonons’, are not well understood and take place for electron energies lower than 50 eV (Fourkal et al 2001). Consequently this kind of interaction is not discussed.

Elastic interactions are those in which the initial and final quantum states of the target atom are the same. During an elastic collision, there is a certain energy transfer from the projectile to the target, which causes the recoil of the latter. Because of the large mass of the target, the average energy lost by the particle is a very small fraction of its initial energy and is usually neglected. This is equivalent to assuming that the target has an infinite mass and does not recoil. For a wide energy range (a few hundred eV to ~1GeV), elastic collisions can be described as scattering of the projectile by the electrostatic field of the target.

In the independent atoms approximation it is assumed that the target atoms are independent, neutral and at rest. To account for the effect of the finite size of the nucleus on the elastic differential cross section (DCS) (which is appreciable only for projectiles with energies larger than a few MeV) the nucleus can be represented as a uniformly charged sphere of radius:

Rnuc=1.05 x 10-15 Aw1\3 (5.9)

where Aw is the atomic mass in g/mol. The electric field that the projectile encounters is that of the nucleus and of the electron cloud. The electrostatic potential of the target atom is (Salvat et al 2003):

µ § (5.10a)

where ñ is the electron cloud density and önuc is the potential of the nucleus given by:

µ § (5.10b)

In the static-field approximation (Mott and Massey 1965, Walker 1971) the DCS for elastic scattering is obtained by solving the partial wave expanded Dirac equation for the motion of the projectile in the field of the atom. The interaction energy is:

V(r)= -eö(r)+Vex(r) (5.11)

with Vex(r) being the local approximation to the exchange interaction between the incident electron and the atomic electrons. It is assumed that the electron has randomly oriented spin. Therefore, the effect of elastic interactions can be described as a deflection of the projectile trajectory, characterized by the polar è and azimuthal ö angles. For a central field, the angular distribution of singly scattered electrons is axially symmetric about the direction of incidence (independent of ö). The DCS (per unit solid angle) for elastic scattering of a projectile with kinetic energy E, into the solid angle element dÙ about the direction (è,ö) is given by (Walker 1971):

µ § (5.12a)

where f(è) is the direct scattering amplitude given by:

µ § (5.12b)

with äl+ and äl- being the phase shifts and Pl(cosè) the associated Legendre polynomials, and g(è) is the spin flip scattering amplitude given by:

µ § (5.12c)

with Pl1(cosè) being the associated Legendre functions. In the above equations k is the wave number of the projectile given by:

µ § (5.13)

For calculation simplicity, the Rutherford elastic scattering differential cross section is usually used and it is given by (Shimizu and Ze-Jun 1992):

µ § (5.14)

where æ is the screening parameter. Nevertheless, the above DCS is not a realistic approximation.
5.4.2. Inelastic scattering

Inelastic collisions are the dominant energy loss mechanism for electrons with intermediate and low energies. They are interactions that produce electronic excitations and ionizations in the medium.

The theory of energy loss of fast charged particles (meaning that their kinetic energy is much bigger than the kinetic energy of the atomic electrons) caused by their inelastic collisions with atoms was established by Niels Bohr (1913) through a semiclassical treatment. In this approach, collisions are classified according to their impact parameter b, which is, roughly, the distance of closest approach of the incident particle to the center of the atom. The theory has been developed for the case of separate atoms (gases).

Bethe (1930, 1932) formulated the theoretical expression for inelastic scattering stopping power of electrons in gases (free atoms and molecules) at a quantum mechanical base. He classified the collisions according to their momentum transfer q, which is observable in contrast to b. The momentum transfer q is a function of the energy transfer W of the incident particle to the atom and of the deflection angle è experienced by the incident particle. The theory is based on the so called First Order (Plane Wave) Born Approximation:

First Order (Plane Wave) Born Approximation: The scattering field is considered to be a perturbation (to first order). That is, the interaction factor V (a Coulomb potential/field) between the particle and the atom, is calculated in the lowest order.

µ § (5.15)

where p, E, and u are the initial momentum, kinetic energy and velocity of the incident particle in respect, µ §, µ §and µ §the corresponding values after the collision and En the energy of the final stationary state of the atom (whose initial energy was E0=0). The ä function imposes energy conservation. Equation (5.15) can be rewritten as:

µ § (5.16)

Bethe’s calculations on the stopping power are based on the continuous slowing down approximation (CSDA). He included the contribution of all possible atomic excitation processes to the energy loss in a factor called the mean ionization energy J. Thus, the average energy loss ÄE of a penetrating electron for a given path length S is given by:

µ § (5.17a)

with µ § (5.17b)

where E is the energy of the incident electron, NA is the Avogadro number, ñ is the density and A is the atomic weight. Berger and Setzer (1964) developed an empirical formula for calculating J:

µ § (5.18)

Nevertheless, the Bethe stopping power (5.17b) has two basic problems:

It is not valid for energies lower than J because the logarithmic term becomes negative for energies < J/1.66

The CSDA does not allow the secondary cascade generation and simple excitation processes to be described.

Figure 5.2 presents a simplified schematic of an inelastic collision.

µ § (relativistic) (5.19a)

Q=µ § (non-relativistic) (5.19b)

Suppose now that the electron interacts with the electrons in an atom and that it transfers momentum q to the atom as a whole. Due to the fact that the atomic electrons, which are responsible for internal atomic excitations, are bound and move around the nucleus, in this case there is not a unique correspondence between q and W, è but W and è can acquire different values depending on the atomic excitation. That is, in the case of an inelastic collision of an electron with an atom, both the energy loss W and the scattering angle è of the projectile are stochastic quantities that are defined through certain probability density functions. For a better handling of calculations, it is more convenient to adopt the recoil energy Q instead of è, using equation (5.19) and:

µ § (5.20)
5.4.2.3. Bethe’s theory revisited

Inokuti (1971) has rewritten equation (5.16) for the non-relativistic case as:

µ § (5.21)

In this equation:

K is related to the momentum transfer q=µ §K=µ §(k-k’)=p-p’.

µ § is the recoil energy.

The factor µ § is the Rutherford cross section for the scattering of a particle with charge z0e (z0=1 for an electron) by a free and initially stationary e, which upon the collision receives recoil energy between Q and Q+dQ.

ån(K) is a matrix element related to q. The factor µ § accounts for the fact that if the incident particle transfers momentum to an atom as a whole, there is not a unique correspondence between momentum transfer q and energy transfer W. It gives the conditional probability that the atom makes a transition to a particular excited state n upon receiving a momentum transfer q. It is called the inelastic scattering form factor.

The concept of oscillator strength stems from the late 19th century model of the electrical and optical behavior of matter. Electrons were supposed to lie at equilibrium positions within atoms and to react elastically to weak disturbances. Thus, they would perform forced oscillations when exposed to electromagnetic radiation. The amplitude and phase of these oscillations would depend on the characteristic (angular) frequency of free oscillation of each electron ùs, on its weak damping constant ãs, and on the radiation frequency ù. In particular, if the disturbance is an electromagnetic wave with an oscillating electric field F = Fo exp(-iùt) and it is directed along z, an atomic electron, which is also subject to an elastic force ¨Cmùs2z and to a frictional force ¨Cmãsdz/dt, experiences a displacement z from its equilibrium position given by:

µ § (5.22)

Therefore the disturbed electron and the atomic nucleus form an oscillating electric dipole which has a dipole moment ez. The polarizability of this dipole a(ù), that is the dipole moment ez per unit field strength, is given as:

µ § (5.23)

The complex character of á(ù) serves to represent by a single number the phase lag and the magnitude of the displacement z with respect to the field oscillation. If we now consider the interaction of incident electromagnetic wave with all the atomic electrons, then if ns is the number of electrons at each state ‘s’, we have fs (=ns) number of oscillators at each state ‘s’ that take part in the interaction. The number fs is called the generalized oscillator strength (GOS), and is the number of dipole oscillators of natural frequency ùs, or else the number of electrons at state ‘s’.

In the quantum mechanical treatment, the analogue of the classical oscillator strength fs is a function g(k, n) which is proportional to the probability of an electron passing from state |k> to |n>. Therefore for a one electron atom the sum of oscillator strengths over all the possible n states is 1. Thus for an atom of Z electrons the same sum is equal to Z. This is an initial description of the so called ‘Bethe sum rule’ later discussed in more detail.

Coming back to equation (5.21) one can define the GOS as (Inokuti 1971):

µ § (5.24a)

or
µ § (5.24b)

with µ §. The optical oscillator strength (OOS) is defined as:

µ § (5.25)

When one deals with excitation to continua (i.e. with ionization) the excitation energy En is not a discrete variable, but it is a continuous variable µ § that takes all real values greater than the first ionization threshold. Then the inelastic cross section for excitation to continuum states between µ § and µ §+dµ § is µ §. Thus µ § is the differential inelastic cross section perµ §. In this case the GOS is defined as (Inokuti 1971):

µ § (5.26)

The final continuum state is specified by µ § and a set Ù of all the other quantum numbers (i.e. angular momentum or direction of atomic electron ejection). The GOS describes the atom. It is difficult to be evaluated theoretically, because a sufficiently accurate eigenfunction of an atomic or molecular system in its ground state and especially in its excited states is seldom available. Atomic Hydrogen and the free electron gas are the only systems for which the GOS is known for every transition.

The effect of individual inelastic collisions on the projectile is completely specified by giving the energy loss W and the polar è and azimuthal ö scattering angles (W, è, ö) or (W, Q, ö). For media with randomly oriented atoms the DCS for inelastic collisions is independent of the azimuthal angle ö. Thus the important parameters are the recoil energy Q and the energy loss W. Therefore, it is more convenient to work in the so called (Q, W) representation.

In (Q, W) representation the GOS and OOS are defines as:

µ § (5.27)

The OOS is closely related to the photoelectric cross section for photons with energy W. That is, as long as the wavelength of the incident particle is sufficiently large compared to the atomic size (dipole approximation), the OOS is proportional to the cross section for photoelectric absorption of a photon with energy W by the atom. The knowledge of GOS does not suffice to describe the energy spectrum and angular distribution of secondary knock on electrons (ä-rays).
5.4.2.5. Bethe surface- Bethe sum rule

The plot of GOS on (Q, W) plane is called the Bethe surface. The physics of inelastic collisions is largely determined by a few global features of the Bethe surface:

Limit Q-> 0: In this limit the GOS reduces to OOS.

Limit of very large Q: In this limit, the binding and momentum distribution of the target electrons have a small effect on the interaction. Thus, in the large Q region, the target electrons behave as if they were free and at rest and consequently the GOS reduces to a ridge along the line W=Q which was named the Bethe Ridge.

For the discrete case the Bethe sum rule is (Inokuti 1971):

µ § (5.28)

whereas for the continuum case and in the (Q, W) representation (Salvat et al 2003):

µ § (5.29)

The Bethe sum rule roughly says that the average of the energy transfer to the atom over all modes of internal excitation for a given Q should be the same as the energy transfer to Z free electrons.
5.4.2.6. The differential inelastic scattering cross section

Fano (1963) using equations (5.21) and (5.24) calculated the differential cross section for inelastic scattering of a charged particle from an isolated atom (independent’s atom approximation). This cross section can be calculated adequately to lowest order in the particle-atom interaction, or else in the low-Q approximation. The obtained differential cross section can be written for the continuum case as (Salvat et al 2003):

µ § (5.30)

where â=u/c with u being the velocity of the incident electron and èr is the angle between the initial momentum of the projectile and the momentum transfer which is given by:

µ § (5.31)

In (5.30) the first term in the curled bracket deals with the longitudinal interactions. These are related to the Coulomb interaction between the incident electron and the atom, an interaction that is parallel (longitudinal) to the momentum transfer q. The second term in the curled bracket is related to the transverse interactions. The mechanism of transverse excitations by a fast particle is electromagnetic and as such becomes important only when the particle approaches the light velocity or when the energy taken up by an electron in the medium is itself relativistic. This interaction is also known as interaction through virtual photons and is called transverse because the photon fields are perpendicular to q.

The differential inelastic scattering cross section for dense media (solids) can be obtained from a semiclassical treatment in which the medium is considered as a dielectric, characterized by a complex dielectric function µ §, which depends on the wave number k and the frequency ù. In the classical picture, the electric field of the projectile polarizes the medium producing an induced electric field that causes the slowing down of the projectile. The dielectric function relates the Fourier components of the total (projectile’s and induced) electric field and the external electric potentials. The momentum transfer can be defined as µ § and the energy transfer as W=µ §and therefore µ §. The DCS obtained from the dielectric and quantum treatments are consistent (the former reduces to the latter for a low-density material) if one assumes the identity:

µ § (5.32)

where Ùp is the plasma energy of a free-electron gas given as:

µ § (5.33)

where N is the electron density of the medium. The differential inelastic scattering cross section for dense media is given by (Salvat et al 2003):

µ §(5.34)

The factor D(Q, W) accounts for the so-called density effect correction (Sternheimer 1952). The origin of this term is the polarizability of the medium, which screens the distant transverse interactions causing a net reduction of their contribution to the stopping power.
5.4.2.7. Secondary electron emission (ä-rays)

After the collision of the primary electron with an electron of an atom, the energy W lost by the primary electron is transferred to the secondary electron and if this electron is ejected from an inner shell it acquires energy Es=W-Bi, where Bi is the binding energy of the particular inner shell, or if it is ejected from an outer shell its energy is Es=W. If it is assumed that the atomic electron is initially at rest, then it is ejected at the direction of momentum transfer q and therefore its polar ejection angle is given by (5.31). Since the momentum transfer lies on the plane formed by the initial and final momenta of the primary electron (scattering plane), the azimuthal emission angle ös of the secondary electron is ös=ö+ð, where ö is the azimuthal angle of the scattered projectile.

5.4.3. Collective description of electron interactions

When an electron transverses a medium, it is likely the energy transferred to the medium to induce collective oscillations of electrons (plasma waves). This kind of interaction is possible in materials like a-Se (Fourkal et al 2001). A plasma wave is possible to decay into many electron-holes pairs.

The theoretical formulation of the collective oscillations of electrons has been formulated by Pines and Bohm (1952). The theory developed deals with the physical picture of the behavior of the electrons in a dense electron gas. In a dense electron gas, the particles interact strongly because of the long range of the Coulomb force. In fact, each particle interacts simultaneously with all the other particles.

Suppose a particle that moves inside a dense electron gas with velocity u0. If its velocity is smaller than the mean thermal speed of electrons in the gas, the electrons respond in such a way that when a steady state is established, the field of the particle is screened out within a distance ëD which is the Debye length given by:

µ § (5.35)

where kB is the Boltzmann’s constant, n the electron density and T the temperature. The picture is as follows: suppose an electron at position xi having velocity smaller than the thermal speed. The particle is surrounded by a comoving cloud in which the electron density is reduced below the average. The cloud is elliptical, being shortened in the direction of particle motion by a specific ratio. The comoving cloud represents a region from which electrons have been displaced by the repulsive Coulomb potential of the ith electron. Most of the electron cloud is located within a distance ~ ëD. The electric field of the ith electron for r>ëD is negligible (screening).

If the particle has u0 bigger than the mean thermal speed, the field of the particle continues to be screened, but also a new phenomenon appears: the excitation of a wake trailing behind the particle consisting of collective oscillations that carry energy away from the particle. The energy loss to the collective oscillations is of the same order of magnitude as the loss caused by short-range Coulomb collisions with the individual particles. The energy lost per unit distance to the collective oscillations is given by:

µ § (5.36)

where Av is the average value of the mean thermal velocity of electrons in solid and E0=mu02/2. The mean free path an electron travels for the emission of an energy quantum to collective oscillations is:

µ § (5.37)

where µ § and ùp is the plasma frequency. The induction of collective oscillations from a passing electron does no produce angular deflection to the electron.
5.5. Charge carrier transport inside a-Se

As it was mentioned, the x-ray matter interactions produce electrons (primary electrons). These electrons as they travel inside the solid cause ionizations along their tracks and hence the creation of electron-hole pairs. For many semiconductors the energy µ § required to create an electron-hole pair has been shown to depend on the energy bandgap Eg via the Klein rule (Klein 1968):

µ § (5.38)

The phonon term Ephonon accounts for energy losses (like phonon production) and is typically very small (~0.5 eV). Thus, practically µ §. Que and Rowlands (1995b) have shown that for the case of amorphous semiconductors Klein’s rule is written as:

µ § (5.39)

For the case of a-Se (Eg~2.2. eV) one would expect µ §~5 eV. Nevertheless, experimental measurements have found that µ §>>5 eV. In addition, experiments show that µ § decreases with increasing the electric field (Rowlands et al 1992) and that it has an energy dependence (Mah et al 1998). These observations imply that the charge carriers inside a-Se are subject to recombination and trapping. When an electron-hole pair is created inside a-Se it may:

Recombine with the other half of the same pair before they are separated (geminate recombination).

Separate to recombine with other electrons or holes in the same electron track. This is the so-called columnar recombination, so named because a column of ionization forms around the electron track.

Separate, escape from the track to recombine in the bulk of the a-Se with electrons or holes from other tracks (bulk recombination).

Separate, escape from the track to become trapped in the photoconductor layer (bulk trapping).

Separate, escape the track, avoid trapping and reach one of the surfaces of the a-Se layer.

Haugen et al (1999) showed that bulk recombination inside a-Se is negligible. Furthermore, Kasap et al (2004) showed that only drifting holes recombine with trapped electrons, following Langevin recombination with coefficient:

µ § (5.40)

where ìh is the hole drift mobility, åoår is the permittivity of a-Se, whereas the recombination of drifting electrons with trapped holes is negligible.

5.5.1. Geminate (Onsager) recombination

The theory for geminate (initial) recombination of ions was formulated by Onsager (1938). Later on, Pai and Enck (1975) used Onsager theory to investigate the photogeneration process inside a-Se in the optical regime.

Each absorbed optical photon creates one electron-hole pair in a-Se. The excess kinetic energy Ek carried by the electron or hole is not sufficient to generate secondary electrons and holes, and is presumed to be dissipated by exciting phonons. The process by which the electron-hole pair loses excess energy and reaches an equilibrium state is called the ‘thermalization process’. After the electron-hole pair is thermalized, the electron and hole are separated by a distance r and at an angle è with the applied field F. According to Onsager theory, such a thermalized pair can either recombine (geminate recombination) or escape their mutual Coulomb attraction and separate into a free electron and a free hole. The probability of escaping geminate recombination is:

µ § (5.41)

where µ § and å is the relative dielectric constant, kB the Boltzann’s constant and T the temperature. The probability p(r, è, F) increases as r, è, F increase. The distribution of r in the electron-hole pairs is given by (Pai and Enck 1975):

µ § (5.42)

where ro is characteristic thermalization length determined experimentally. If the photogeneration efficiency n is defined as the fraction of electron-hole pairs which do not recombine relative to all electron-hole pairs created, then:

µ §µ § µ § (5.43)

where µ § and Il are the modified Bessel functions. The photogeneration efficiency increases as å, ro and T increase, whereas n and µ §depend on the energy of incident photon Eph , and F, T. Knights and Davis (1975) assumed that during the thermalization process the motion of the carriers is diffusive and the rate of energy dissipation to phonons is híp2. Thus, they have calculated the thermalization distance ro as:

µ § (5.44)

where t is the thermalization time, D is the diffusion constant and hí is the incident photon’s energy.

Que and Rowlands (1995b) have extended the Onsager theory into the x-ray regime. At high electric fields (i.e. 10 V/ìm) g(r,è)=g(r) and the distribution g(r) is given as:

µ § (5.45)

where µ § is the distribution of the kinetic energies Ek of the electron-hole pairs and Ec=2.067 eV (for a-Se). The photogeneration efficiency is now given as:

µ § (5.46)

where µ §. To a first order with respect to F (5.46) is written as:

µ § (5.47)

The thermalization distance r is obtained from:

µ § (5.48)

where µ § and for a-Se: ì = ìh = 0.14 cm2/Vs and híp2=15 meV. The n andµ § do not depend on Eph but still depend on F, T.

Jaffe (1913) formulated the theory for columnar recombination. As mentioned earlier, in columnar recombination the electron-hole pairs separate and the released charges recombine with other electrons or holes within the same electron track. Jaffe’s assumptions are:

Only one electron track is taken into consideration. The time needed for the creation of the track is much smaller than the recombination time of the carriers.

The holes and electrons are ejected from the central axis of the column in opposite directions.

The mobilities of electrons and holes are equal.

Diffusion, external field and recombination are taken into account. The electric field of electrons and holes is neglected.

Figure 5.3. The geometry that Jaffe used in his calculations for columnar recombination.
N particles (charges) are created homogeneously per unit length of the track, whereas the initial distribution around the center of the column is Gaussian:

µ § (5.49)

with b being the column’s initial radius.

Jaffe calculated, as a function of time, the spatial distribution of charges in the column n(r, t) as well as the fraction of electrons that escape columnar recombination: (i) in the absence of any applied field F, (ii) when F is parallel with the track, (iii) when F is perpendicular to the track and (iv) when F makes an angle ö with the track. The geometry used is shown in figure 5.3.

The fraction of electrons that escape recombination are assumed to be those that travel a distance R from the central axis of the column. Hence:

µ § (5.50)

with µ § and No the initial number of charges, á the recombination coefficient, D the diffusion coefficient and b the column’s initial radius. In the absence of an applied field the number of electrons that escape recombination is negligible.

If one assumes that initially No charges were created in length d, then the fraction of charges that escape columnar recombination is:

µ § (5.51)

where ì is the charge mobility and µ §and µ §

In this case the fraction of electrons that escape recombination is:

µ § (5.52)

with

5.5.2.4. Field at an angle ö with the track

The fraction of electrons that escape recombination is:

µ § (5.53)

with µ §. It is seen that for small F, µ §, whereas for intermediate and large F, µ §.

Chapter 6

Monte Carlo Simulation

6.1. Introduction

The Monte Carlo method has long been recognized as a powerful technique for performing certain calculations, generally those too complicated for a more classical approach. Since the use of high-speed computers became widespread in the 1950s, a great deal of theoretical investigation has been undertaken and practical experience has been gained in the Monte Carlo approach.

Historically, a primitive Monte Carlo method was first used by Captain Fox to determine ð in 1873. During World War II, Von Neumann and Ulam introduced the term Monte Carlo as a code word for the secret work at Los Alamos. It was suggested by the gambling casinos at the city of Monte Carlo in Monaco. The Monte Carlo method was then applied to problems related to the atomic bomb. After 1944, Fermi and Ulam used the method to study the Schrödinger equation in quantum mechanics and Goldberger to study nuclear fusion (1948). During the period 1948-1952, Wilson R R, Berger M and McCracken, used Monte Carlo techniques to conduct research in x-rays and a-rays showers. Soon after the initiation of computers in Monte Carlo calculations, the method was used to evaluate complex multidimensional integrals and to solve certain integral equations, occurring in physics, which were not amenable to analytic solution.

The application of Monte Carlo methods is a procedure which can be considered as a two input-one output problem as shown in figure 6.1. The two inputs are a large source of high quality random numbers and a probability distribution which describes the considered problem, whereas the output is the result of the random sampling of the probability distribution.
Figure 6.1. General block diagram of a Monte Carlo procedure.

In general, the primary components of the Monte Carlo method are the following:

Probability Distribution Functions (PDFs): the physical (or mathematical) system must be described by a set of PDFs.

Random Number Generator: a source of random numbers uniformly distributed on the unit interval must be available.

Sampling rule: a prescription for sampling from the specified PDFs, assuming the availability of random numbers on the unit interval, must be given.

Scoring (or tallying): the outcomes must be accumulated into overall tallies or scores for the quantities of interest.

Error estimation: an estimate of the statistical error (variance) as a function of the number of trials and other quantities must be determined.

Variance Reduction Techniques: methods for reducing the variance in the estimated solution to reduce the computational time for the Monte Carlo simulation.

Parallelization and vectorization: algorithms to allow Monte Carlo methods to be implemented efficiently on advanced computer architectures.

In the next sections some aspects of the mathematical foundation for Monte Carlo calculations as well as the Monte Carlo methods will be described. The information is obtained from Morin et al (1988).

The necessity to use random numbers emerged basically for three reasons:

Due to the need to study physical phenomena that their nature was random (e.g. the thermionic emissions of electrons from a metal).

Due to the fact that some physical problems were very expensive or very dangerous to be studied from an experiment or there were no experimental data.

Due to the fact that some phenomena because of their complex nature (e.g. the random Brownian motion), were better to be studied using random numbers than actually be studied analytically.

A random number is a particular value of a continuous variable uniformly distributed on the unit interval, which together with others of its kind, meets certain conditions. A high quality random number sequence is a long stream of numbers with the characteristic that the occurrence of each number in the sequence is unpredictable and that the stream of digits of the sequence passes certain tests which are designed to detect departures from randomness. The quality of a supposedly random sequence of numbers can be established only after a careful analysis aimed at discovering pattern in the sequence. The larger the number of tests applied, and the higher the level of sophisticated of these tests, the higher is the quality of a sequence which passes the test.

There are many random number sources, which are usually classified based on their method of production into three categories: tables, physical sources and algorithms. The latter mentioned, may appear to be a contradiction because an algorithm is a detailed set of rules to obtain a specific output from a specific input. Such an algorithm is termed a Random Number Generator (RNG), and its output is formally called a pseudorandom number, reflecting its deterministic production.

One important disadvantage of the use of algorithmic random number generators, is the fact that after a certain number of distinct elements have been produced, the sequence begins to repeat. If the period of the sequence, that is the number of distinct digits, is large, then the periodic behavior of a particular algorithmic random number generator is of no practical importance. So, it is important to establish the fact that the period is sufficiently large for the intended purpose of the generator. Consequently, the important characteristics of a RNG is the long period and the uniformity, in the sense that equal fractions of random numbers should fall into equal ‘areas’ in space.

Lehmer’s method is still the most commonly used for RNG. It is called the multiplicative-linear-congruential method. Given a modulus M, a multiplier A, and a starting value îo (the seed), random numbers îi are generated according to:

îi=(Aîi-1+B)moduloM (6.1)

where B is a constant (Andreo 1991).

A random variable is a variable that can take on more than one value (generally a continuous range of values) and for which any particular value that will be taken cannot de predicted in advance. Even though the value of the variable is unpredictable, the distribution of the variable may well be known. The formal definition of a random variable is given as:

When modeling a physical system or a physical phenomenon using Monte Carlo techniques, the first step that must be taken is the interpretation of this system or phenomenon with a set of mathematical expressions. These expressions must include all the physics describing the system (phenomenon) and can be derived either directly from theory or using experimental results. From these expressions, a PDF or a set of PDFs is made. Obviously, depending on the mathematical expressions used, PDFs can either have their origins in experimental data or in a theoretical model. Modeling the physical process by one or more PDFs, one can sample an ‘outcome’ from them (for example sample the azimuthal angular distribution of photoelectrons in the photoelectric process). Thus, the actual physical process is simulated. The PDF can be defined as:

f(x)µ §0 for all xµ §R

µ §

P(a

where P(a

In some cases it is more convenient to use the cumulative distribution function (CDF) defined as:

F(x) = P(Xµ §x) =µ § (6.2)

As can be seen, F(x) is a monotonically non-decreasing function taking on values from zero to one.
6.4. Sampling techniques

Once the PDF describing the physical system is known, the next step in a Monte Carlo simulation is the sampling process. The sampling process is a procedure in which the PDF is randomly sampled basically by two methods:

The Inversion Method.

The Rejection Method.

The inversion method is described by the fundamental inversion theorem:

Fundamental Inversion Theorem: Let X be a random variable with PDF f(x), cumulative distribution function F(x), and let r* denote a uniformly distributed number drawn from the unit interval. Then the probability of choosing x* as defined by

r*=F(x*)=µ § (6.3)

is f(x*). The theorem is graphically illustrated in figure 6.2 where the relationships among the PDF, its associated CDF, the uniformly distributed number r* and the value x* of the random variable X are shown.

Suppose that the function f(x) is a PDF and x µ §[a,b]. The algorithm of the inversion technique can be described in the next steps:

Check if f(x) is normalized. That is check if µ §.

Calculate the CDF: F(x) =µ §.

Generate random numbers R.

Let F(x) =R and solve for x as x=F-1(R).

Figure 6.2. Graphical illustration of the relationships among the PDF, its associated CDF, the uniformly distributed number r* and the value x* of the random variable X.

6.4.2. The Rejection Method

The rejection method is an alternative when the inversion method cannot be implemented (for example when the equation F(x)=R is difficult to solve). An important limitation though of the rejection method is the fact that depending on the shape of the modeled PDF it may be time consuming. If f(x) is a PDF and x µ §[a, b] then the algorithm of the rejection method is illustrated in figure 6.3 and it is described as the follows:

Find the maximum of f(x) and name it fo.

Set g: g(x) = f(x)/fo.

Generate a random number R1µ §[0,1).

Generate a random value of xµ §[a,b), say x*=a+(b-a)R1.

Generate a random number R2µ §[0,1).

Compare R2 with g(x*):

If g(x*)µ §R2 then: reject x* and return to step 3

else: accept x* and return to step 3.

Figure 6.3. Sampling a PDF f(x) by the rejection technique. The random pairs (x*, R2) are assumed to be uniformly distributed over the circumscribing rectangle. Only those bounded by f(x) are accepted.

Chapter 7

Primary Electron Generation Model

7.1. Introduction

This chapter deals with the Monte Carlo modeling of primary electron generation inside the photoconductor’s bulk. Through experimental research is not feasible to isolate and study only the primary electrons produced inside a photoconductor. Thus, a complete validation of the model cannot be done. Nevertheless, an indirect index of the reliability of the method rises from the fact that the developed model is an extension of a recently presented model to simulate the primary electron production inside a-Se (Sakellaris et al 2005), which is based on a validated model developed by Spyrou et al (1998) that simulates the x-ray energy spectrum sampling as well as the x-ray photon interactions. This section focuses on the simulation of primary electron production from x-ray-matter interactions (incoherent scattering, photoelectric absorption) as well as due to atomic deexcitation (fluorescent photon production, Auger and CK electron emission) inside a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2.

Figure 7.1 presents the flowchart of interaction processes taken into account in the simulation. The photon interaction cross sections for photoelectric absorption, coherent and incoherent scattering are extracted from XCOM (Berger et al 2005), the atomic form factors and incoherent scattering functions from Hubbel et al (1975), while the shell and subshell binding energies from Bearden and Burr (1967).

The energy of the recoil electron (Ee) is given by the following equation:

µ § (7.1)

where Ep is the energy of the initial photon, mc2 is the electron rest energy and èp is the polar angle of the scattered photon. Thus, using the sampled values of Ep, èp (Spyrou et al 1998), random samples of Ee are taken.

PDF(öe)=µ § (7.2)

The direction of the recoil electron is calculated by sampling both the azimuthal öe and the polar angle èe of the electron. The azimuthal angle öe is uniformly distributed in the interval [0,2ð). Thus, the inversion method is used to sample the azimuthal PDF given by:

(7.2)

Figure 7.1. A flowchart of the interaction processes taken into account in the simulation model.

µ § (7.3)

which is derived from angle calculations in the scattering process.
7.3. Photoelectric absorption
7.3.1. Photoelectric absorption from a molecule

For the case of compound materials, the molecular photoelectric cross section is evaluated as the weighted sum of the photoelectric cross sections of the atomic constituents (additivity approximation). Therefore if AxBy is the compound, then the molecular photoelectric cross section of AxBy at energy E, µ §, is defined as:

µ § (7.4)

where µ § are the photoelectric cross sections (in cm2/g) and wA, wB the fractions by weight of elements A and B, respectively. If the probability of a photon to interact with atom A is defined as:

µ § (7.5)

then µ §. Thus, the atom which photoelectrically absorbs the photon is determined by a Monte Carlo decision, based on the probabilities µ § and µ §.

Similar to the case of a-Se (Sakellaris et al 2005), it has been assumed that photons with energies híµ §1.434 keV, which is the binding energy of Se LIII subshell, are not taken into account in the simulation process.

For the case of Se the differences between the binding energies of L subshells as well as the binding energies of M subshells are smaller or equal to 1 keV and were assumed to be negligible (Sakellaris et al 2005). Since this is also the case in the rest of the elements except for the heavy ones (Hg, Tl, Pb), in order to determine the shell (or subshell) from which the photoelectron is ejected, the formulation followed for a-Se (Sakellaris et al 2005) can be adopted. Therefore:

If hí > BK, where BK is the binding energy of the K shell, the photon is absorbed by the K shell, ejecting a photoelectron with energy Ee= hí-BK.

If BLIII