What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Valence Photoemission

G. Margaritondo, in Encyclopedia of Condensed Matter Physics, 2005

Historical Background: The Photoelectric Effect

The photoelectric effect, discovered by Hertz in 1887, is the basis of some of the most powerful techniques capable of analyzing the electronic structure of condensed-matter systems – the techniques collectively labeled as “photoemission spectroscopy.” Even before the development of such techniques, the photoelectric effect had a fundamental (and often not correctly understood) impact on the history of modern science.

The main milestones after Hertz's work were Thompson's 1897 discovery of the electron, Einstein's 1905 hypothesis on the photon, and Millikan's 1918 reluctant experimental validation of Einstein's hypothesis. Einstein derived the photon hypothesis with a purely thermodynamic argument and used it to predict the frequency threshold in the photoelectric effect, which was not experimentally verified until the late 1910s.

The photoelectric effect provided, in this way, an experimental basis for quantum mechanics. This is undeniably a very significant contribution to the development of modern physics. The next scientific impact of the photoelectric effect – its practical use in spectroscopy – came almost 40 years later.

The reason for this long delay is the surface sensitivity of the photoelectric effect. In order to become a photoelectron, an electron in a solid must first absorb a photon to increase its energy above the photoelectric threshold. Since the threshold is of the order of several electronvolts, the photon must be in the X-ray or ultraviolet spectral range. Photons of this kind can penetrate quite deep in the solid before being absorbed. On the other hand, excited electrons can travel only over a very short distance before losing energy.

As a consequence, photoelectrons originate only from a thin slab near the surface of the sample. They carry information on the electronic structure of that slab, which typically consists of a very few atomic planes. If the surface is contaminated, the information is spurious. A suitably clean solid surface can be obtained by several different methods such as cleaving, scraping, and in situ growth. However, most clean surfaces do not stay clean unless under ultrahigh vacuum (pressure <10−9 Pa). This level of vacuum could not be routinely achieved before the 1960s, and this explains the long delay in the use of the photoelectric effect in spectroscopy.

After solving the vacuum and contamination problems, photoemission spectroscopy initiated a steady growth. This expansion was accelerated in the late 1960s by the advent of synchrotron radiation sources. With their help, photoemission became the premier tool for probing the electronic structure of solids and condensed systems in general.

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Nuclear Medicine and Molecular Imaging

K.P. Schäfers, ... T. Viel, in Comprehensive Biomedical Physics, 2014

1.10.2.1.3.1 Photoelectric effect

The photoelectric effect is the process of a photon being absorbed by an atom ejecting an electron. This process of interaction usually occurs between photons and electrons of inner shells. Consequently, more loosely bound outer shell electrons can fill the inner shell leading to the emission of x-rays. Alternatively, a second electron can be emitted removing the excess energy. This electron is then called Auger electron. The recoil energy of this process (typically some electron volts) is absorbed by the entire atom. The photoelectric effect dominates in lead for rather low photon energies (see Figure 10).

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 10. Mass attenuation coefficient in lead as a function of incident photon energy. For γ energy at the binding energy of the K-shell (88 keV), a maximum in photoelectric absorption occurs, whereas at higher incident photon energies, the Compton effect and pair production effects dominate.

Adapted from Grupen C (1993) Teilchendetektoren. Mannheim: BI-Wissenschaftsverlag.

The ejected electron will have an energy of

Ee=hv−EB

with Planck's constant h, the frequency of the photon v, and the binding energy of the electron EB. The binding energy depends on the shell (K-, L-, or M-shell) the electron is ejected from. For photon energies above the energy of the K-shell, K-shell electrons are usually involved. The probability for an interaction of two particles is usually expressed as the so-called cross section. The total cross section of the photoelectric effect in the K-shell can be described as

σphotoK= 32εγ712α4Z5σThe

The energy εγ represents the reduced photon energy Eγ 2 (mec2)− 1 and the fine-structure constant α can be approximated by 1/137. The Thomson cross section σThe for elastic scattering of photons with electrons is determined by

σThe=83πre2

with the classical electron radius re. The cross section σphoto(K) strongly depends on the atomic number, as evidenced by the factor Z5  in the previous equations. High-Z elements like lead have thus a comparatively high photoelectric cross section.

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Interaction of Gamma Quanta With Matter

Ilya Obodovskiy, in Radiation, 2019

6.2 Photoeffect

A photoelectric effect is the absorption of a gamma quantum by an electron, in which the gamma quantum gives up almost its entire energy to the electron. Absorption by a free electron is impossible, as the law of momentum conservation is not satisfied.

Therefore, the photoelectric effect occurs on bound atomic electrons, and the higher the probability of the process, the stronger is the electron bound. The photoelectric effect on the K-shell nearest to the nucleus has the greatest probability, the smaller probability on the L-shell, and even smaller probability on the M-shell. The ratio of the cross sections on the K-, L-, and M-shells is

(6.1) σaK/σaL ∼5,σaL/σaM∼4.

Strict theoretical formulas for calculating the cross sections of the photoelectric effect of any energy quanta on atoms with any Z do not currently exist. The main theoretical calculations have an approximate character and a limited field of application. Therefore, while the expressions for the cross sections in the literature are not available, we only note the character of the cross section dependence on the atomic number of the substance and on the energy of the quanta:

(6.2)σaϕ∼Z5/hvathv≫IK;σ aϕ∼Z5/hv3,5athv>IK.

The dependence of the photoabsorption cross section on the energy of gamma quanta has the form of a decreasing stepped curve; each step is associated with the contribution of one of the inner shells of the atom. In intervals between steps, the cross section varies monotonically, and in double logarithmic coordinates, it is practically linear. The dependence of the photoelectric absorption coefficient on the gamma quantum energy is shown in Fig. 6.2.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 6.2. Schematic dependence of the photoeffect absorption coefficient in Xe on quantum energy.

It is seen that, starting at high energies, the cross section increases with decreasing energy, until the quantum energy reaches the binding energy of the electron on the K-shell. With further decrease of energy, the photoelectric effect on the K-shell becomes energetically impossible, and it is omitted from the process. A sharp vertical jump is observed on the curve, called the K-edge of absorption. At gamma quantum energy, which is smaller than the energy of the K-edge, only L-, M-, and the following shells participate in the photoelectric effect. As is known, these shells have subshells in accordance with different values of the orbital and total moments. The L-shell has three subshells, and the M-shell has five subshells; the absorption edge is divided into several ones.

As the energy of the gamma quantum decreases below the K-edge, the cross section again grows until an LI-edge appears, followed by LII- and LIII-edges, and so on. As the L-shell is switched off by parts, the magnitude of each jump is noticeably smaller than the jump corresponding to the K-edge of the absorption.

In Table 6.1, the energy values of the K-, L-, and M-edges of absorption of some elements are shown.

Table 6.1. Values of the Energy of the K-, L-, and M-Edges of Absorption of Certain Elements (keV)

AlSiFeCuGeAgIXeCsPb
K 1.56 1.84 7.11 8.98 11.1 25.5 33.5 34.7 36.0 88.0
LI 1.1 1.41 3.8 5.2 5.5 5.7 15.9
1.25 3.5 4.9 5.1 5.4 15.2
1.22 3.4 4.6 4.8 5.0 13.0
MI 1.2 3.9

The angular distribution of photoelectrons is shown in Fig. 6.3. At low energies (hν << mc2), photoelectrons are emitted predominantly in the direction of the electric vector of the incident electromagnetic wave, i.e., at the right angles to the direction of propagation of radiation. With increasing energy, the angular distribution extends forward.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 6.3. Angular distribution of photoelectrons.

As a result of photoeffect, a photoelectron appears with the energy

(6.3)E=hv−UK,L,M…,

where UK, L, M… is the binding energy of electrons on K-, L-, or M-shells.

After a photoelectron emerges out of the corresponding shell of an atom, with the greatest probability out of K-shell, a vacancy forms, followed by the cascade of transitions described in Section 4.6.

The cross sections of the photoelectric effect in a wide energy range for the air and lead are shown at the end of this chapter in Fig. 6.8.

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NEUTRON CAPTURE RADIATION SPECTROSCOPY

H. MOTZ, G. BÄCKSTRÖM, in Alpha-, Beta- and Gamma-Ray Spectroscopy, 1968

3.4 PHOTOELECTRON CONVERSION PROCESS

The photoelectric effect has been utilized in only one experiment in which both external and internal targets were used at the Brookhaven Graphite Reactor18. A thin lens spectrometer was employed for these measurements which for the external target extended only up to about 700 keV. A lead converter of 30 mg/cm2 was used for both boron and cadmium targets which essentially totally absorbed the thermal neutrons in the beam. Although the ease of target handling and the convenient comparison of the B10(n, α) Li7 γ-ray of 478 keV allowed intensity calibration, the method was abandoned due to the high background encountered. It is interesting to note in this work that a measurement of the background for a simple low-energy spectrum such as from a boron target is not a good indication of what one might expect from a complex high energy spectrum such as cadmium.

The internal target arrangement permitted a significant improvement in resolution and the cadmium and sodium spectra were observed up to ≈ 3 MeV with ≈ 3% resolution. Source strengths were much greater because the volume of the source was so large (≈ 6000 cm3). Gamma-ray beam collimation permitted the detection of the K-X-ray from the uranium converter in coincidence with the K-photoelectron and a large improvement in signal-to-background as well as improved spectral response resulted. The thin lens instrument is well-suited to the focussing of the photoelectrons present in a cone, but the mean cone angle corresponding to the maximum photoelectric cross-section changes with energy so that it is difficult to optimize such a system over a large energy range. In addition, the main γ-ray beam must pass into the spectrometer and an understandably large background results. This is definitely inferior to central-ray spectrometer designs in which the conversion electrons are deflected away from the main beam so that an angular separation results. This central-ray design is not suited to the photoelectric conversion process because of the large angles involved, although it is applicable to the Compton process. In summary, the photoelectric effect can be used with moderate resolution into the MeV region, but the crystal diffraction spectrometers are superior in both sensitivity and resolution.

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SURFACE AND INTERFACE ANALYSIS AND PROPERTIES

Ana Maria Botelho do Rego, Luis Filipe Vieira Ferreira, in Handbook of Surfaces and Interfaces of Materials, 2001

3.1.1 Basic Principles

The photoelectric effect is the basis of this technique. When a photon having an energy Ephoton impinges on a surface, as schematically displayed in Figure 8, an electron bound to the nucleus with a binding energy Eb is ejected with a kinetic energy Ek related by

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Fig. 8. Schematic representation of an XPS experiment.

(18)Ephoton=hv=Eb+Ek+ϕ

where ϕ is the work function of the spectrometer and needs to be adjusted every time the equipment is vented [20].

Equation (18) can be rewritten in the following ways:

()Eb=hv−Ek−ϕandEk=hv−Eb−ϕ

Eb and Ek vary, then, in a symmetrical way. In a photoelectron spectrum, the intensity of ejected electrons as a function of their kinetic energy is registered. With X-ray radiation, hν is high enough to eject inner shell electrons. These electrons, contrarily to the valence ones, have their binding energy to the nucleus almost unchanged by the atomic environment. They are, then, a fingerprint of the element where they originate. Since the inner-shell electron binding energy is the variable allowing for the identification of a given element, spectra are usually displayed in the form of photoelectron intensity as a function of the decreasing binding energy rather than the increasing kinetic energy.

In a first approximation, we can consider the electron structure as frozen under the photoelectron emission process and identify Eb with the Hartree–Fock energy eigenvalues of the orbitals (Koopman's theorem). A schematic representation of an expected photoelectron spectrum could then be the one in Figure 9.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Fig. 9. Schematic representation of an “expected” XPS spectrum.

However, in an accurate calculation of a binding energy, the relaxation energy of the remaining electronic structure to a new hole state has to be included [21]. Besides, many other factors contribute to render the schematic picture displayed in Figure 9 very different in real situations and great care needs to be taken in the interpretation of features appearing in a XPS spectrum. Some of the examples of features other than the main photoelectron peak, which can appear in a XPS spectrum, are shake-up and shake-off peaks, X-ray source satellites (for nonmonochromatic X-rays), “cross-talk” peaks, and Auger peaks [22]. Moreover, the ejection of an electron from the inner shell of a given element does not usually give rise to a single peak. Reasons for this are chemical shift, orbit-spin coupling, and spin coupling (when nonpaired electrons exist in the element). Finally, photoelectrons suffering single or multiple inelastic collisions in the medium lose energy and leave the surface with a lower kinetic energy. This implies that every photoelectron peak has a background at lower kinetic energies (higher binding energies) larger than the background at higher kinetic energies (lower binding energies). In Figure 10, we can see a real survey spectrum of a sample containing a single element from the second period of the periodic table—highly oriented pyrolitic graphite—and exhibiting several of the above mentioned features. The spectrum was obtained with a freshly pealed sample with a Kratos XSAM800 spectrometer operating in a fixed analyzer transmission mode (see Section 3.1.4 under a pressure of the order of 10−9 mbar and using a pass energy of 10 eV. The magnesium nonmonochromatic radiation was used (main component at hv = 1253.6 eV).

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Fig. 10. Survey XPS spectrum of a sample of highly oriented pyrolithic graphite showing several features, other than the photoelectron peak.

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Optical Detectors

Robert H. Kingston, in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I.C Electronic Detectors

The internal photoelectric effect, just as infrared radiation, was also first observed in the 19th century, when certain minerals such as selenium or lead sulfide were found to increase their electrical conductivity in the presence of light. These photoconductors depend upon the photoexcitation of bound electrons and/or holes into the conduction and/or valence bands of the material. Then, at the turn of the century, external photoemission was discovered in vacuum diodes. As first explained by Einstein, the photoelectric effect was found to have a threshold wavelength determined by the relation hν = hc/λ ≥ E, where E is the energy required for the electron to exit the material. In the case of a semiconductor, the excitation energy, E, is that of the gap between the valence and conduction bands or the ionization energy of an impurity in the material. The electronic detector family has two main branches, the first being the vacuum photodiode and its more useful adaptation, the vacuum photomultiplier. Following these are the semiconductor devices, the photoconductor and the photodiode. The former behaves as a light-controlled variable resistance, while the semiconductor photodiode, almost identical to the vacuum photodiode in characterization, is a high impedance current source with current magnitude proportional to the incident radiation.

Electronic detectors offer the ultimate in frequency response, as high as tens of gigahertz, and especially in the visible, approach photon-counting or quantum-limited performance. As such, they offer magnitudes of improvement in sensitivity over thermal devices. In the limit of photon-counting performance, the signal measurement fluctuation or noise is produced by the random production of photo electrons. In many cases, electrical noise in the postdetection amplifier, rather than “photon” noise, limits the sensitivity.

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Origins of Quantum Theory

J.E. House, in Fundamentals of Quantum Mechanics (Third Edition), 2018

1.5 The Photoelectric Effect

In 1887, H. R. Hertz observed that the gap between metal electrodes became a better conductor when ultraviolet light was shined on the apparatus. Soon after, W. Hallwachs observed that a negatively charged zinc surface lost its negative charge when ultraviolet light was shined on it. The negative charges that were lost were identified as electrons from their behavior in a magnetic field. The phenomenon of an electric current flowing when light was involved came to be known as the photoelectric effect.

The study of the photoelectric effect is made possible with an apparatus like that shown schematically in Fig. 1.9. An evacuated tube is arranged so that the highly polished metal that is to be illuminated, such as sodium, potassium, or zinc, is made the cathode. When light shines on the metal plate, electrons flow to the collecting plate (anode), and the ammeter placed in the circuit indicates the amount of current flowing. Several observations can be made as the frequency and intensity of the light varies:

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Fig. 1.9. An illustration of the photoelectric effect.

1.

The light must have some minimum or threshold frequency, ν0, in order for the current to flow.

2.

Different metals have different threshold frequencies.

3.

If the light striking the metal surface has a frequency greater than ν0, the electrons are ejected with a kinetic energy that increases with the frequency of the light.

4.

The number of electrons ejected depends on the intensity of the light, but their kinetic energy depends only on the frequency of the light.

An electron traveling toward the collector can be stopped if a negative voltage is applied to the collector. The voltage required to stop the motion of the electrons, which is known as the stopping potential, V, and causes the current to cease, depends on the frequency of the light that caused the electrons to be ejected. In fact, it is the electrostatic energy of the repulsion between an electron and the collector that exactly equals the kinetic energy of the electron. Therefore, the two energies can be equated to yield the relationship

(1.31)Ve=12mv2

In 1905, Albert Einstein explained the major aspects of the photoelectric effect. Einstein based his analysis on the relationship between the energy of light and its frequency that Planck established in 1900. It was assumed that the light behaved as a collection of particles (called photons) and the energy of a particle of light was totally absorbed by its collision with an electron on the metal surface. Electrons are bound to the surface of a metal with an energy called the work function, w, which is different for each type of metal. When the electron is ejected from the surface of the metal, it will have a kinetic energy that represents the difference between the energy of the incident photon and the work function of the metal. Therefore, the energies are related by the equation

(1.32)12mv2=hv−w

It can be seen that this is the equation of a straight line when the kinetic energy of the electron is plotted against the frequency of the light. By varying the frequency of the light and determining the kinetic energy of the electrons (from the stopping potential), a graph such as that shown in Fig. 1.10 can be prepared to show this relationship.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Fig. 1.10. The relationship between the kinetic energy of electrons ejected in the photoelectric effect and frequency of the light.

The intercept is ν0, which is the threshold frequency, and the slope is Planck's constant, h. One of the significant points in the interpretation of the photoelectric effect is that light is considered to be particulate in nature. In other experiments, such as the diffraction experiment of T. Young, it was necessary to assume that light behaved like a wave. Many photovoltaic devices in common use today (e.g., light meters, optical counters, etc.) are based on the photoelectric effect.

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Radiation Therapy Physics and Treatment Optimization

B. Nilsson, A. Brahme, in Comprehensive Biomedical Physics, 2014

9.01.3.2.1 Total cross section

The probability for a photoelectric effect is based approximately on calculations for a K-electron using an unscreened Coulomb potential, for energies not too close to the binding energies. Correction factors are then applied to take into consideration the effect of the K-edge and the contributions from L-, M-,… electrons and the screened potential. For low photon energies, Heitler (1954) obtained the relation

[73]τK=4 21/2α4σ0Z5mec27/2hν7/2

where α is the fine structure constant 1/137 and σ0 is the Thomson scattering cross section:

[74]σ0=83πre2

At high energies, relativistic corrections must be included and an expression obtained by Sauter is

[75]τK=σ032α4Z5 mec2hν

As is obvious from the equations for the cross sections, there is a high dependence on both the atomic number, Z, and the photon energy, hν. At low energies where the photoelectric effect is dominant, the variation with energy is close to (hν)− 3. At high photon energies, the variation with energy has decreased to (hν)− 1. The variation with atomic number is, after corrections, changing with energy between Z4 at low energies and Z5 at very high energies.

The aforementioned cross sections hold for the photoelectric effect in the K-shell. The interactions with outer shells must be added, and if the photon energy is lower than the binding energy of a specific shell, then the effect can only be applied to outer shells. Figure 27 shows the relation τshell/τ for the K-, L1-, and M1-shells as a function of atomic number for photon energies close to the binding energy. An approximate expression for the relation τ/τK is given by Hubbell (1969):

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 27. Ratio of the cross section for photoelectric effect in different subshells to the total cross section.

[76] τ/τK=1+0.01481lnZ2−0.00079lnZ3

Corrections to the aforementioned equations have also to be applied due to the screening effect, which reduces the cross section with a percentage that is rather independent of the photon energy and around 2% for the K-shell but up to 30% for the L-shells.

Close to the energy of the electron binding energies, there is a large discontinuous change in the cross section because if the energy is just below, for example, the binding energy of the K-shell, then no K-electrons can be expelled, but with an energy just above the binding energy, this is possible. Tables of photon cross sections always have two lines corresponding to the binding energy. One is the value of the cross section just below the binding energy, and one is the value of the cross section just above the binding energy. Figure 28 illustrates the photoelectric cross section for Pb and water for energies between 0.01 and 100 MeV. The discontinuities at the absorption edges for the K- and the L-shells for Pb are clearly expressed. For water, the binding energy is lower than 10 keV.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 28. Cross section for photoelectric effect in water and lead as a function of photon energy.

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COLLISION KINEMATICS

Esam M.A. Hussein, in Radiation Mechanics, 2007

Photoelectric absorption

Kinematic analysis of the photoelectric effect is not quite straightforward because it involves a bound electron as a target, but the atom as a whole recoils. The photoelectric effect cannot take place in free space, in the absence of the atom. On the other hand, one cannot precisely define the target of the interaction, and has to rely on the probabilistic arguments of quantum mechanics (see Section 3.5.5). Nevertheless, one can represent the photoelectric effect by the two-body interaction 2(1,3)4, with M1 = 0, the product particle is an electron, M3 = Me, and M4 = M2 – Me + Be, i.e. the target loses an electron in the process and gains some excitation energy, Be. Then the threshold energy for the interaction, according to Eq. (2.88) is:

(2.159)Ef=(M2+Be)−M222M2= Be+Be22M2≈Be

The approximation is made possible by the fact that Be, the electron's binding energy, is at most in the keV range, while the mass of an atom is at least in the GeV range. It is also reasonable to assume that M2 and M4 are much larger than both Me and E1. Given this, it is also reasonable to assume that Cr coincides with L. Since photoelectric absorption takes place only in the field of the atom, it is not unreasonable to reduce the Cr energy by the mass of the target atom, M4, which includes the atomic field potential, Be. The remaining energy is then assumed by the electron. Therefore,

(1.160)E3≈E′3= s−M4≈M2+E1(M2−Me+ Be)=E1−Be+Me

where s=M22+2M2E1 and s≈M2(1+E1M2) Keeping in mind that E3 = M3 + T3 = Mc + M3, the above is the well-known Einstein's (1905) photoelectric equation4, in which Be is called the work function and is equal to the binding energy of the atomic electron liberated in the interaction (typically a K-shell electron). Note that under this approximation, E4 = M4 + Be, i.e. the momentum given to the residual atom is Be. The momentum of the electron is then:

(2.161)P3 2=(E1−Be+Me)2−Me 2=E12+2MeE1+Be2−2Be(Me+E1)

Given that Be < Me, since the maximum value of Be is for the K shell is at most on the order of 100 keV (see Section 1.2.3) Me = 511 keV, then P3 > P1(=E1). When E1 > Be, then P32=E12+2MeE1, and the momentum given to the recoil atom can be neglected. The momentum balance (P3Sin ϑ3 = P4 sin ϑϑϑ4)shows that as P4 → 0, sin ϑϑ3 → 0 and the electron will tend to emerge in the same direction as the incident photon, i.e. with a small scattering angle. Then the residual atom will recoil backwards, since ϑ3 + ϑ4= 0 given that Cr and L almost coincide. When E1, on the other hand, is close in value to Be, the momentum of the electron becomes almost zero, and ϑ4 → 0. Then, the electron and the residual atom travel in opposite directions, i.e. ϑ3 → π at low energies. As Section 3.5.5, Eq. (3.166) shows, at low photon energy electron emission tends to be in a direction normal to that of the incident photon.

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Scintillation detectors and photodetectors

Syed Naeem Ahmed, in Physics and Engineering of Radiation Detection (Second Edition), 2015

A.1 Photocathode

We learned about the photoelectric effect in Chapter 2 and saw that some particular materials emit electrons when they absorb photons of energy above a certain threshold. This threshold energy is called the work function and is characteristic of the material. The material itself is called a photocathode. If we assume that no energy is lost during this process, then the energy of the emitted photon would simply be equal to the difference between the photon energy and the work function of the material, i.e.,

(6.5.1) Ee=hcλ−Φ,

where λ is the wavelength of the incident light and Φ is the energy threshold or work function of the material. For a given material, this expression can be used to determine the maximum detectable wavelength (see example below).

Example:

A material having a work function of 2 eV is used to convert photons into electrons, which are then detected by a PMT. Compute the maximum wavelength of the photons it can convert into electrons.

Solution:

The maximum wavelength can be obtained by setting Ee=0 in Eq. (6.5.1), which simply means that all of the incident energy has been used in the conversion process and the electron has not taken away any of the photon’s energy. Hence we have

0=hcλmax−Φ ⇒λmax=hcΦ=(6.63×10−34)(2.99×108)2×1.602×10−19=6.187×10−7m=618.7nm.

Looking at Eq. (6.5.1) it is apparent that choosing a material with low work function is advantageous in terms of delivering energy to the outgoing electrons. However, this is not the only criterion for selecting a photocathode for use in a PMT. For example, if the PMT is to be used to detect scintillation photons, then the most important factor is the conversion efficiency of the material at the most probable wavelength of the scintillation light. If the material does not have high enough efficiency at that wavelength, then even a very low work function would not matter much. Since the realization of this problem, a number of extensive studies have been carried out to find the optimum photocathode materials at different scintillation wavelengths. Most of these efforts have gone into understanding the spectral response of the materials. By spectral response we mean the efficiency of the production of photoelectrons as a function of light wavelength. This efficiency is generally referred to as photocathode quantum efficiency and is defined as the ratio of the number of emitted photoelectrons to the number of incident photons, i.e.,

(6.5.2)QE=Ne Nγ,

where Ne is the number of electrons emitted and Nγ is the number of incident photons. Quantum efficiency can also be expressed in terms of more convenient quantities, such as incident power and photoelectric current. To do this, we first note that the incident power can be calculated from

(6.5.3)Pγ=nγhv,

where nγ represents the number of photons of frequency ν incident on the detector per unit time. The expression for quantum efficiency can then be written as

(6.5.4)QE=nePγ/hv=IpehvePγ.

Here ne is the number of photoelectrons ejected per unit time and Ipe=ene is the photoelectric current. This equation can be used to determine the photoelectric current for a particular value of the incident power (see example below).

Example:

A photocathode produces a current of 20 nA when exposed to light of wavelength 510 nm. Determine the photoelectric current if the wavelength of light is changed to 475 nm such that the incident power remains the same. Assume that the material has the same quantum efficiency at the two wavelengths.

Solution:

Equation (6.5.4) for photoelectric current can be written as

Ipe=(QE)ePγhcλ=Kλ,

where λ is the wavelength of the incident light and we have lumped together all the constant terms into one parameter K. This constant can be eliminated by writing the above equation for the two wavelengths and then dividing one by the other. Hence the required current is

(6.5.5)Ipe,2=Ipe,1λ2λ1 =20475510=18.6nA.

Figure 6.5.2 shows a typical spectral response curve of a photocathode. Here the quantum efficiency of the photocathode is plotted against the wavelength of incident light. The interesting thing to note here is that the curve has a plateau where the variation in efficiency is not very large. As soon as one goes beyond this plateau on either side, efficiency decreases rapidly. Therefore, to build a good scintillation detector with a PMT, one should ensure that the spectrum of scintillation light has a peak somewhere in the middle of this plateau. There are some materials that have very short plateaus as well, and using those in PMTs is generally not a good idea unless the scintillation spectrum is also narrow and has a clearly defined peak that occurs at or near the peak of the spectral response curve.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 6.5.2. Quantum efficiency of a typical photocathode as a function of wavelength of incident photons.

As noted above, the particular requirements of the system drive the choice of a photocathode for a PMT. Since there are a number of scintillation materials available having their own unique scintillation spectra, efficient detection of photons with a PMT requires the use of photocathode materials with matching spectral response characteristics. There are also applications where the photons to be detected are not coming from a scintillator. For example, the use of PMTs for detecting Cherenkov photons has recently gotten a lot of attention in highly sensitive large-scale neutrino detectors. In essence, a PMT is a versatile photodetector that can be used to detect photons in virtually any environment provided it is equipped with a photocathode having the appropriate spectral response.

Photocathodes can be used in essentially two different modes: transmission and reflection. In transmission mode the photocathode is semitransparent to allow transmission of photoelectrons (see Figure 6.5.3(a)). Such a photocathode is constructed by depositing a very thin layer of the material on the inside of the photon entrance window. Since most of the photoelectrons are emitted in the direction of travel of the incident photons, it is called a transmission photocathode. Most PMTs are constructed with this type of photocathode. There are also some photocathode materials that have high quantum efficiencies but very poor transmission properties. These opaque materials are used to construct the so-called reflection photocathodes, as shown in Figure 6.5.3(b). A reflection photocathode is made by depositing a thin layer of the material on a metal electrode inside the PMT.

What kind of interaction occurs between a photon and an electron in the photoelectric effect?

Figure 6.5.3. (a) Semitransparent photocathode used as a transmission photoemission device in a head-on type PMT. (b) Reflection type photocathode. Such photocathodes are generally used in circular type PMTs.

A host of photocathode materials have been identified with varying characteristics and spectral responses, some of which are listed below.

AgOCs: This is one of the most widely used photocathode materials. It has a photoemission threshold of 1100 nm and a peak quantum efficiency at around 800 nm. The working range of this material is in the near infrared range of the photon spectrum. The main disadvantage of AgOCs is its very low quantum efficiency, which has a peak of less than 1%. It is mainly used as a transmission photocathode.

GaAs(Cs): This material has a spectral response that ranges from UV to 930 nm. This broadband response makes it suitable for use with a wide range of scintillators without the need for a wavelength shifter. In most instances it is used in the transmission mode. Since the quantum efficiency of GaAs(Cs) is temperature dependent with a peak at very low temperatures, it is sometimes operated at very low temperatures.

InGaAs(Cs): This material has greater sensitivity in the infrared range and a higher signal-to-noise ratio in the 900–1000 nm range than GaAs(Cs).

SbCs3: This was one of the earliest photocathode materials. It is still very popular among manufacturers as its spectral response ranges from UV to the visible region, with a peak quantum efficiency of around 20%. The photoemission threshold of SbCs3 lies at around 700 nm and it has a peak at approximately 400 nm. Since it has very poor transmission capabilities, it is generally used as a reflection photocathode.

Bialkali materials: Bialkali materials such as SbRbCs and SbKCs are the most widely used of all photocathode materials due to their high sensitivity to blue light generated by NaI scintillators. The reader might recall that NaI is the most popular scintillator for radiation detection. Their sensitivity to blue light is not the only reason for their popularity, though. These materials also have high quantum efficiencies, with peaks of just under 30%. Another advantage is their good stability at elevated temperatures. Some bialkali materials can be used at a temperature as high as 175°C. A common bialkali material, SbKCs has a photoemission threshold of about 700 nm and maximum quantum efficiency of 28% at around 400 nm.

Multialkali materials: These materials have very wide spectral response, ranging from UV to near infrared, making them highly suitable for use with a number of different scintillators. Their main disadvantage is high thermionic emission of electrons even at room temperature, and therefore external cooling is generally required. NaKSbCs is a common multialkali.

CsTe, CsI: These materials are sensitive to photons in the UV region only and are therefore not very widely used.

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URL: https://www.sciencedirect.com/science/article/pii/B9780128013632000061

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