Elementary Processes of the Transformation of Radiation Energy in Matter

Chapter 8

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter Outline 8.1 Dissipation of Energy of the Primary Particle in Matter 8.1.1 Production of Charge Carriers in Gases 8.1.2 Formation of Charge Carriers in Solids 8.2 Formation and Structure of a Track

161 161 164 165

8.3 Fluctuations of Ionization: The Fano Factor 8.4 Electron Thermalization References

169 170 173

8.1 DISSIPATION OF ENERGY OF THE PRIMARY PARTICLE IN MATTER 8.1.1 Production of Charge Carriers in Gases A primary charged particle spends its energy on ionization and excitation. The delta electrons formed during ionization ionize and excite the atoms of matter and form, in particular, secondary delta electrons. Secondary delta electrons, in turn, create tertiary and then those quaternary electrons, etc. This process is called the ionization cascade. Because the energy, initially concentrated in one particle, is exchanged into small portions and transferred to a large number of low-energy electrons, it is said that degradation or dissipation of energy occurs. In the process of exchange, the energy of all delta electrons becomes lower than the ionization energy, then new delta electrons cease to appear, and the available ones can lose energy in inelastic collisions of excitation, until their energy min becomes less than Uex . The further transformation of these subthreshold delta electrons is discussed in Section 8.4. Let us figure out the balance of energy lost by the primary particle with energy E0 in matter. By creation of each electron-ion pair, the energy Ui is consumed, while in the ionization cascade, Ni pairs are produced by the primary particle and delta electrons of all generations. Because of the random nature of the interaction of the charged particle with the atoms of matter, the value of Ni fluctuates. For energy balance, we use the mean value Ni . In addition to single ionization, when the particle moves, multiple ionization events can occur, mainly because of the ionization of the inner shells. Multiply charged ions colliding with neutral atoms can be recharged, forming new singly charged ions. However, doubly charged ions in noble gases can no longer form two single ions, as the energy of double ionization is lower than the doubled energy of a single ionization. Thus, in argon, Ui ¼ 15.8 eV and UiII ¼ 27.6 eV and in xenon Ui ¼ 12.1 eV and UiII ¼ 21.2 eV. Therefore, excited atoms appear in collisions of the second kind and the excess energy passes into the kinetic energy of the atoms: Arþþ þ Ar/Arþ þ Ar .

(8.1)

As a result of such process, some of the energy that has gone into ionization is not included into the whole balance. To consider it, R.L. Platzman [1] has proposed to introduce a small coefficient ki. Thus, the total energy spent for ionization is equal to ki Ni Ui .

Radiation. https://doi.org/10.1016/B978-0-444-63979-0.00008-2 Copyright © 2019 Elsevier B.V. All rights reserved.

161

162 PART | I Fundamentals

The total energy spent for excitation can be obtained by summing up all the transitions in accordance with the spectrum of excitation levels. Introducing the average excitation energy Uex , we have X Uex ¼ Nex Uex . (8.2) The remaining energy of the primary particle is contained in the kinetic energy of the subthreshold electrons distributed in the spectrum. Introducing the average energy of the subthreshold electrons Ed , we obtain X Ed ¼ Nd Ed . (8.3) As a result, the energy balance has the form E0 ¼ ki Ni Ui þ Nex Uex þ Nd Ed .

(8.4)

Because delta electrons are produced only in ionization processes, it can be assumed that Ni ¼ Nd. To quantify the formation of ions, the energy of a particle is divided by the number of ion pairs. The quotient is called the average energy of the formation of a pair of ions: E0 w ¼ ¼ ki Ui þ Uex Nex þ Ed . (8.5) Ni To calculate all the quantities in the energy balance, Eq. (8.5) is rather complicated because the corresponding cross sections are poorly known. R.L. Platzman made an estimate, according to which approximately equal relations are valid for all noble gases [1]: ki ¼ 1:06;

Ed Uex Nex ¼ 0:31; ¼ 0:85; ¼ 0:4: Ui Ui Ni

(8.6)

Hence, for the average energy of formation of a pair of ions, we obtain w=Ui ¼ 1:71:

(8.7)

Subsequently, more detailed estimates have shown that the ratios (8.7) in different noble gases are somewhat different, but the errors associated with the approximate nature of the calculations in most cases do not exceed 10%. A characteristic feature of the ionization and excitation of noble gases is the rather narrow energy range of the excitation levels, which leads to a small fraction of the transitions to the region of the discrete spectrum associated with the excitation of atoms, with respect to transitions to the continuum associated with the ionization of atoms. Thus, in min argon, the ionization energy Ui ¼ 15.76 eV and the lowest resonant excitation level Uex ¼ 11.61 eV. Because of this,  the ratio Uex Ui is high, and w/Ui is relatively low compared with other gases. In molecular gases, the calculated value of the ratio (8.7) is much higher. Therefore, in hydrogen, w/Ui ¼ 2.50.  It was shown earlier that in inert gases Nex Ni ¼ 0; 4. In molecular gases, this ratio is approximately equal to unity. The average energy of ions pair formation is a very important parameter characterizing the effect of radiation on matter. A special issue of the report of the International Commission on Radiation Units and Measurements (ICRU) is devoted to the discussion on the parameter available at the time of writingdreport No. 31, 1979 [2]. An investigation of dependence of this parameter magnitude on the type and energy of the particles continues up to the present [3e5]. The dependence of w on energy and on particle type has been long studied in detail. It is established that wa/wb z 1 for noble gases and hydrogen and w1.05e1.07 for other molecular gases. Apparently, this contributed elastic energy losses at the ends of the ranges of alpha particles. It is well known that w for slow heavy ion is noticeably higher than wa and wb. This circumstance is usually represented as the presence of an “ionization defect,” Edef: E0 w ¼ wb . (8.8) ðE0  Edef Þ This representation assumes that the energy (E0  Edef) goes to the formation of electrons with an efficiency of 1/wb, and the energy Edef is transmitted in elastic collisions with a very low probability of ionization. It is assumed that w grows in the region of low particle energies. Assuming that at a low particle energy the number of charge carriers it generates is proportional not to the energy itself but to the difference between its kinetic energy and the ionization energy, one can obtain a simple expression for w: wN ; (8.9) w ¼ ð1  Ui =EÞ where wN is the average energy of formation of a pair of ions at high particle energy.

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

163

An obvious confirmation of the increase in w with a decrease in particle energy was obtained for the energy range up to 1.5 keV [6]. When a substance is irradiated with X-ray quanta, the change in w correlates with the ionization of the L and M shells. In Table 8.1 the average energy of a pair of ions formation in some gases is presented. The average energy of a pair of ions formation has been measured many times by different authors in different laboratories and by different methods. Accordingly, the results of the measurements are somewhat different. In other reviews and manuals, the reader can find other values of the parameter w, although the spread is currently low. However, w ¼ 34.0 eV for air, in view of the importance of this parameter, is recommended by the International Commission on Radiological Units. The experimental determination of w is quite simple, but it is very difficult to obtain an exact value [1]. There is a large number of factors that can affect the result. Let us analyze the main ones. Usually, in the experiment, a quantity proportional to Ni is determined by measuring either the amplitude of the ionization pulse or the average current. The number of charges extracted from the track of the particle and participating in signal formation is highly dependent on recombination on the track (Chapter 11). Incorrect consideration of the role of recombination can lead to errors in the determination of Ni. In pulse measurements, an important role can play the capture of electrons by molecules of electronegative impurities (Chapter 9). True is the fact that, in the case of current measurements, this circumstance is not of much importance because as electrons are captured in gases, negative ions are formed, and they also contribute to the current. In current measurements, the contribution to the current of electrons and negative ions, despite their significantly different mobility, is equivalent. Significant errors can arise when the particle energy lost in the substance is incorrectly determined. The purity of the test substance plays an important role in the accuracy of measuring w. In some cases, small additives can significantly change the energy of formation of a pair of ions. Thus, in mixtures of certain gases, the energy of the lower excited level of the ground gas is higher than the ionization energy of the impurity. In this case, inelastic collisions can occur, leading to additional ionization. For example, in a mixture of Ne þ Ar the following reaction can occur: Ne þ Ar/Ne þ Arþ þ e.

(8.10)

The probability of such reactions is especially great because the lower excited level of noble gases is metastable and, therefore, its lifetime is quite long. The additional ionization that occurs in such reactions appreciably changes the measured value of w. For some mixtures, this change is given in Table 8.2. This effect of small admixture additives on the yield of ionization is sometimes called the Jesse effect [7]. It is used to reduce the average energy of formation of a pair of ions, mainly to reduce ionization fluctuations (Section 8.3). When exposed to radiation, some substances emit a portion of the absorbed energy in the form of lightdthey luminesce (scintillate) (Chapter 12). The luminescence intensity and the conversion efficiency of the radiation energy turning into light can be estimated by the number of photons emitted by the substance. To quantify the yield of photons by analogy with ionization, it is convenient to introduce the average energy expended on the formation of a photon. There are two sources for producing photons: first, when the excited atoms turn over to the ground state, and second, when ions are recombined.

TABLE 8.1 Average Energy of a Pair of Ions Formation in Some Gases [1] Substance

He

Ne

Ar

Kr

Xe

H2

N2

O2

CO2

CH4

Air

w, eV

42.3

36.6

26.4

24.1

21.9

36.3

34.9

30.8

32.8

27.3

34.0

w/Ui

1.72

1.70

1.68

1.72

1.80

2.35

2.24

2.55

2.38

2.10

e

TABLE 8.2 Average Energy of Formation of a Pair of Ions in Some Gas Mixtures Gas

He

He D 0.13% Ar

Ne

Ne D 0.12% Ar

Ar

Ar D 0.2% C2H2

w, eV

41.3

29.7

36.3

26.1

26.4

21

164 PART | I Fundamentals

It should be noted that the emission of photons occurs after rather complicated transformations, which undergo excited atoms, electrons, ions, and the products of their recombination. In the course of these transformations, nonradiative transitions are possible, which reduces the yield of photons. The average energy per photon is estimated neglecting the nonradiative losses. Thus, the quantities characterizing the limiting yield of photons are obtained. If photons are produced only from transitions of excited atoms, then Nph ¼ Nex in the limit. If recombination also matters and then assuming that each recombination act gives a photon, we obtain Nph ¼ Nex þ Ni. A calculation, using Eq. (8.7) for noble gases, offers the following: in the case of excitation process only, wph ¼ 4.1 Ui; in the case when both processes, excitation and recombination, produce photons, wph ¼ 1.2 Ui. In scintillation method of radiation detection, the concept of conversion efficiency u is, predominantly, used to characterize the transformation of particle energy into light: u ¼ Nph hnph =E0 ;

(8.11)

where hnph is the average energy of the emitted photons. It is easy to see that there is a simple connection between the average energy going to produce a photon and the conversion efficiency: u ¼ hnph =wph .

(8.12)

8.1.2 Formation of Charge Carriers in Solids In solids, a charged particle spends energy transferring electrons from the valence band to the conduction band, i.e., on the formation of an electron-hole pair, on the formation of excitons, and also on the excitation of crystal lattice vibrations, i.e., on the formation of phonons. The energy that goes into the formation of an electron-hole pair in solids must be compared with the width of the forbidden band. In many types of insulators, i.e., substances with a wide forbidden band, the valence band, as a rule, is rather narrow, and the conduction band is relatively wide. The minimum energy going to ionization is practically equal to the width of the forbidden band. In such substances, the energy of formation of a pair of ions is related to the width of the forbidden band Ug by the ratio w z 1.5 Ug. In typical semiconductors, the valence band and the conduction band have a significant width in comparison with the forbidden band. Electrons and holes can be formed with equal probability at any level of the corresponding zones. This means that electrons and holes formed in the ionization events have excess energy. It is possible to find the average carrier energy after ionization. By the definition of the average, R Ei Ef ðEÞgðEÞdE . (8.13) E ¼ R0 Ei f ðEÞgðEÞdE 0 where f(E) is the filling function of states, the FermieDirac distribution f ðEÞ ¼

1 ; 1 þ exp½ðE  EF Þ=kT

where g(E) is the density of states,

 gðEÞ ¼

21=2 p2



 m3=2 1=2 E ; Z3

(8.14)

(8.15)

EF is the Fermi level, and Ei is the energy required for ionization. In the corresponding zones f(E) ¼ const, and the bands can be considered parabolic; in this case g(E) w E1/2. Then, by integrating, we obtain E ¼ 0:6Ei .

(8.16)

In gases and insulators, practically all the energy of a particle can be transferred to electrons, as heavy ions receive very little energy. In semiconductors, because of the interzone transition two charge carriers are formed, an electron and a hole, with comparable masses. Therefore, we need to consider two relations: (1) the energy relationship E1 ¼ Ug þ E2 þ Ee þ Eh

(8.17)

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

165

and (2) the additional condition that follows from the law of momentum conservation p1 ¼ p2 þ pe þ ph ;

(8.18)

where E1 and p1 are the energy and the momentum of the incident particle prior to ionization and E2 and p2 are the same after ionization. Ee, Eh, pe, phdthe energy and the momentum of the formed electron and the hole. The simultaneous solution of Eqs. (8.17) and (8.18), if Ei has a minimal value, gives   me Ei ¼ Ug 1 þ ; (8.19) me þ mh where me and mh are masses of an electron and a hole. It is also necessary to consider the energy expended for exciting the phonons. Because the corresponding cross sections are not well known, in the radiation physics of semiconductors it is customary to use the parameter r, the average number of phonons per ionization. The values of the parameter are determined by comparing the calculated models with the experiment, and they are given in Table 8.3. Now we can figure out the energy needed for the formation of a pair of carriers in semiconductors by a fast charged particle. For each act of ionization, the energy Ei is required. Besides, for each act of ionization, the energy r$Zu is transformed into phonon energy. The formation of excitons can be neglected because in germanium and silicon the binding energy of excitons is very small and at all real temperatures they autoionize. In the balance of energy expended by the particle for ionization, it is also necessary to take into account the residual kinetic energy of the subthreshold electrons and holes Ee and Eh . As a result, we obtain w ¼ Ei þ rZu þ Ee þ Eh .

(8.20)

Assuming that Ee ¼ Eh ¼ 0:6Ei and me ¼ mh, we finally get w ¼ 3:3Ug þ rZu.

(8.21)

The values of the mean energy for the formation of a pair of charge carriers in certain semiconductors are given in Table 8.4. The relationship between the formation energy of a pair of charge carriers and the width of the forbidden band is shown in Fig. 8.1. Calculation by formula (8.21) using the parameters of Table 8.3 gives somewhat higher values.

8.2 FORMATION AND STRUCTURE OF A TRACK A charged particle, passing through matter, forms elementary excitationsdelectron-ion pairs and excited molecules. The area around the trajectory of the particle (and delta electrons), in which these elementary excitations are concentrated, is called the particle track. All subsequent transformations, which can occur with elementary excitations and which are discussed in this and subsequent chapters, essentially depend on the spatial distribution of elementary excitations, i.e., from the structure of the track.

TABLE 8.3 Phonon Energies and the Values of the Parameter r in Si and Ge Si

Ge

Zu, eV

0.063

0.037

r

17.5

57

TABLE 8.4 Values of the Mean Energy for the Formation of a Pair of Charge Carriers in Certain Semiconductors Si (300K)

Ge (77K)

CdTe (300K)

HgI2 (300K)

GaAs

PbI2

GaSe

AlSb

DUg, eV

1.12

0.74

1.47

2.13

1.42

2.6

2.03

1.62

w, eV

3.61

2.98

4.43

4.22

4.2

7.68

6.03

5.05

166 PART | I Fundamentals

FIGURE 8.1 Relationship between the formation energy of a pair of charge carriers w and the width of the forbidden band Ug. According to C.A. Klein, Bandgap dependence and related features of radiation ionization energies in semiconductors, J. Appl. Phys. 39 (4) (1968) 2029e2038.

FIGURE 8.2 Track of a weakly ionizing particle.

Three circumstances make the structure of the track very complicated: first, the statistical nature of the interaction of charged particles with molecules, second, the presence of delta electrons, and, third, multiple scattering. In Fig. 8.2 the track of a weakly ionizing particle, i.e., a relativistic particle with an energy corresponding to the minimum specific energy loss (Section 5.2.6), is shown. Acts of excitation in the figure are not marked. A particle in some cases forms isolated pairs and, predominantly, relatively slow electrons, but with still enough energy for ionization. The trajectory of slow electrons due to multiple scattering is noticeably curved, and the ionization density is much higher than in the track of the primary particle. It can be seen from the figure that ionization is quite unevenly distributed over the track and concentrated in isolated cells (spurs). In radiation physics, the structure of the track is detailed very carefully [9e12]. Besides the cells (spurs), blobs are identified, i.e., cells with a large number of ion pairs in them, corresponding to a local energy release of 100e500 eV and short tracks that are cylindrical regions of overlapping spurs with a high ionization density, formed by delta electrons with energy in the range 500e5000 eV. The ratio of spurs, blobs, and short tracks depends on the energy of the primary particle. This dependence is shown in Fig. 8.3. The spurs or ionization cells are distributed in space statistically. The average distance between cells is determined by the density of ionization, which in turn depends on the density of matter, charge, and energy of the incoming particles. The average distance between the ionization cells [av can be found as the reciprocal value of the primary ionization n0 (Section 5.4): [av ¼ 1=n0 (8.22)

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

167

1.0

Spurs

Part of energy

0.8 0.6 0.4

Short tracks

0.2 0.0

Blobs 1

10

100

1000

Energy of primary electron, keV FIGURE 8.3 Ratio of spurs, blobs, and short tracks in the dependence on energy of primary particle. Figure is built on the basis of data from Ref. S.M. Pimblott, J.A. LaVerne, A. Mozumder, N.J.B. Green, Structure of electron tracks in water. 1. Distribution of energy deposition events, J. Phys. Chem. 94 (1990) 488e495.

As the relativistic particle loses dE per unit path dx and produces n0 primary interactions, the average energy release in the cell is DE ¼ The average number of pairs in a cell is

dE=dx . n0

(8.23)

 N ¼ DE w.

(8.24)

[av ntot ¼ ðdE=dxÞ=w

(8.25)

We also give the obvious relation For Ar at the normal conditions [av ¼ 0.34 mm, DE ¼ 83 eV; Nz3O4. The average diameter of a cell can be set equal to the practical range of a delta electron with an energy corresponding to the average energy loss. For argon, under normal conditions, it is approximately 5 mm. As the gas pressure rises, the average distance between the cells begins to decrease, but as the cell size also decreases because of the decrease in the range of the delta electrons, then ionization in the cells is typical for the action of weakly ionizing particles on condensed bodies, e.g., liquids. Thus, in the water for an electron with an energy of 1 MeV, the average cell spacing is 300 nm, the average cell diameter is 2 nm, the energy per cell is 60 eV, and the average number of pairs of ions and excited molecules per cell is w4. The distance between the cells decreases with increasing dE/dx at a constant gas pressure. At dE/dx w 170 keV/cm, the average distance between the cells becomes equal to their mean diameter, and the cells merge into a solid cylindrical track. This is exactly what the track of an alpha particle with an energy of 5e10 MeV in gas looks like. At the initial part of its range, the alpha particle has a specific loss of w1 MeV/cm, which grows as the particle slows in matter, reaches a maximum at an alpha-particle energy of w900 keV, and then falls off. The total range of an alpha particle with an energy of 5 MeV in argon under normal conditions is 3.67 cm; on this path it forms an average of 1.9$105 ion pairs. At the initial stage of the range, the real distance between pairs is less than calculated and on the final stage distance is greater. The track of the strongly ionizing particle is shown schematically in Fig. 8.4. The track has a central part with a high charge density, the so-called “core,” and the outside part, that is called “coat,” formed by d-electrons emerging from the core. In a condensed matter, the density of ionization in tracks of heavy ions and, especially, of fission fragments is so great that a cylindrical cloud of electrons and ions can be considered a plasma (the Debye screening radius is less than the radius of the ionization column). The volume of the track is an extremely important parameter for all questions concerning the effect of ionizing radiation on any type of substance. It is in the region of the track that the main processes of converting radiation energy into structure defects, into chemical reactions, into biological effects, and other manifestations of impact occur. And if the length of the track is a more or less definite quantity associated with the range of the particle, then the track radius depends strongly on what specific task it is meant to. Accordingly, the radius of the track is determined differently depending on the specific task. We here indicate two possible options for determining the radius of the track. The first option is the average distance from the track axis on which ionization is still carried out. The second option is the average distance to which electrons reach during the thermalization process. This option is considered in Section 8.4; here we analyze the first one.

168 PART | I Fundamentals

FIGURE 8.4 The track of the strongly ionizing particle. The central part, called “core,” and the outside part, called “coat,” are shown.

FIGURE 8.5 The definition of the track radius. The arrow shows the direction of the primary particle movement.

In Fig. 8.5 a diagram explaining the calculation of a track radius is shown. For the radius of the track, a transverse projection of the practical range of the delta electron is rtrack ¼ Rd sin4.

(8.26)

The range of a delta electron is related to its energy by a known relation, Eq. (5.97) Rd ¼ aEdb .

(8.27)

And the energy of the d-electron depends on the emission angle, Eq. (5.13a) Ed ¼ Edmax cos2 4.

(8.28)

b  rtrack ¼ a Edmax cos2b 4 sin4.

(8.29)

Thus,

As the angle increases, the sine values increase and the cosine values decrease, so for some value of the angle the value of the track radius rtrack passes through a maximum. The usual methods of finding the extremum of a function show that the maximum in Eq. (8.29) is attained for tg2 4 ¼ 1 2b. If we take R ¼ 0.012$E1.35 mg/cm2, where E is in keV, for the  max  practical range of an electron in an NaI crystal, then we find 4 rtrack ¼ 31 . The maximum energy of delta electron in the nonrelativistic case is determined by the velocity of an incident particle v: Edmax ¼ 2me v2 ;

(8.30)

From Eq. (8.29) it is clear that the radius of the track depends on the energy of primary particle and it decreases as particle decelerates. Finally we find   2:7 cm; (8.31) rtrack ¼ 1:32$106 v 109 where v is in cm/s. Calculation by Eq. (8.31) gives for NaI the value of the track radius of the order of 100e700 nm.

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

169

8.3 FLUCTUATIONS OF IONIZATION: THE FANO FACTOR Because a charged particle spends its energy through several channels, namely, through ionization, excitation, and kinetic energy of electrons, and the specific energy loss depends on the collision parameter, which is random, the number of pairs of ions and excited atoms fluctuates. As the spread in the number of the formed ion pairs determines the spread in the magnitude of the ionization signal, which, in turn, determines the accuracy of the energy measurement, it is very important to know the laws and quantitative characteristics of these fluctuations. The fluctuations of ionization were analyzed in detail by U. Fano [13]. He showed that the dispersion of the number of ions pairs is proportional to their average number DðNi Þ ¼ FNi .

(8.32)

The coefficient of proportionality F was called the Fano factor. If the ionization fluctuations occur according to the Poisson law, then F ¼ 1. Both experimental studies and Fano’s theoretical analysis showed that F < 1. It is easy to show that the ionization fluctuations do not really obey the Poisson law. According to Poisson law, a random variable can take any values, including arbitrarily large ones. Even if there is a limit in the value of a random variable, then, for the validity of the Poisson law, this limit must be much larger than the mean value. A charged particle forms on the average Ni ¼ E0 w ion pairs, and, according to the law of conservation of  energy, the limiting value, which could result if all the particle energy went to ionization only, is Nimax ¼ E0 Ui . The values of Ni and Nimax are quite close and they differ only by a factor of 2e2.5, so the fluctuations in the number of formed ion pairs cannot be described by the Poisson law. As is known, the presence of a limiting number of events is characteristic of the binomial law of probability distribution (it describes, e.g., radioactive decay). The expression for the variance in the binomial law has the form   N Dbinom ¼ N 1  . (8.33) Nmax Comparing this expression with the expression for the variance of the number of ion pairs, Eq. (8.32), and using the expressions for Ni and Nimax written above, we find that in this case the Fano factor is F ¼ 1

Ui . w

(8.34)

For example, for argon we get F w 0.4 from Eq. (8.34), which is close to the first estimates made by U. Fano. However, the ionization fluctuations are described not by a binomial law but by a much more complicated one. This is because of the fact that the primary fluctuating quantity is the energy loss, and the numbers of the produced pairs of ions and excited atoms are only derivatives of losses. It is obvious that the Fano factor depends on the relationship between the probabilities of ionization and excitation. With a decisive contribution of ionization, the Fano factor is small, and for a decisive contribution of excitation, the Fano factor tends to unity. Theoretical calculations of the Fano factor offer the values given in Table 8.5. Let us point out that the experimental value of the Fano factor for Ar is F ¼ 0.19. The Fano factor is significantly affected by the Jesse effect described in Section 8.1.1. Actually, as a result of this effect, part of the energy that went into excitation is pumped to the ionization channel and then the number of ion pairs increases, which naturally reduces the fluctuation in their number. If all the excited atoms could turn into a pair of ions because of this reaction (8.10), then the Fano factor would decrease to 0.05. If all the particle’s energy would turn exceptionally to ionization so that Nmax pairs of ions could be formed, then F w 0. In Table 8.6, the values of the Fano factor calculated for some gas mixtures at an impurity concentration of 0.5%e1% are shown. Fluctuations in the number of photons formed by a particle occur because of the same laws as the fluctuations in the number of ion pairs, but the Fano factor in this case appears to be close to unity. The fact is that the process of photons formation can be represented as consisting of, at least, two stages. At the first stage, excited states are formed, but at the

TABLE 8.5 Theoretical Values of the Fano Factor in Gases Substance

H

He

Ne

Ar

H2

F

0.39

0.21

0.13

0.16

0.34

170 PART | I Fundamentals

TABLE 8.6 The Fano Factor in Some Gas Mixtures Substance

He D Ar

He D Xe

He D CH4

Ne D Ar

Ar D C2H2

F

0.054

0.057

0.089

0.063

0.096

second stage, the excited states emit photons, and the transition of the molecule from the excited state to the ground state without the formation of a photon competes with the photon emission. Thus, the photon emission is a two-stage process, and therefore, to calculate the distribution parameters, methods for calculating fluctuations in cascade processes can be used. Generally, fluctuations in cascade processes can give a variance greater than Poisson’s, but in the case of photon fluctuations it is almost equal to the Poisson one, which means that the Fano factor is an order of unity. In semiconductors, the fluctuations in the number of charge carriers formed by the particle are determined by the random distribution of the particle energy between the ionization processes and phonon excitation. The result of these fluctuations is manifested in the following values of the Fano factor: F (Si) ¼ 0.155  0.002 for X-ray quanta with an energy of 2e3.7 keV at T ¼ 170K [14] and F (Ge) ¼ 0.112  0.001 at T ¼ 77K [15]. In view of the important role played by the Fano factor in the operation of spectrometric nuclear radiation detectors, theoretical analysis and experimental measurements of this parameter continue [16e19].

8.4 ELECTRON THERMALIZATION Rutherford was one of the first to draw attention to the fundamental fact that an electron separated from an atom in the process of ionization stays in the field of the parent ion. If this field is large enough, the electron is attracted to the ion as a result of electrostatic interaction and then recombines with it. Thus, the charges formed in the interaction of particles with matter can very quickly disappear. It is clear that this circumstance is of fundamental importance for all kinds of radiation science. An electron formed as a result of ionization has some excess kinetic energy. Such electrons are called hot. Moving in matter, a hot electron loses energy in acts of interaction with molecules of the medium until its energy becomes of the order of thermal energy, i.e., it becomes thermalized. The thermalization process takes place in the electric field of the parent ion, and at any stage of energy loss, the electron can recombine with the ion. To consider the interaction of an electron with an ion, at the continuous change in the energy of the electron and change of the relative location of recombining partners, is very difficult. Therefore, for an approximate analysis of the problem, it is possible to assume, for simplicity, that there is no interaction, while the electron is hot, and when it becomes thermal, the interaction is switched on jumpwise, and the cross section becomes equal to the interaction cross section for a thermal electron. Then the energy of interaction between a thermal electron and an ion turns out to depend on their mutual distance. We use the concept of the capture sphere, i.e., an area in which the potential energy of the interaction of the parent ion and electron is greater than the thermal energy. Then the radius of the sphere is determined from relation e2 ¼ kT. 4pεε0 rcr

(8.35)

e2 . 4pεε0 kT

(8.36)

Hence rcr ¼

In gas (ε ¼ 1) at room temperature, rcr ¼ 57 nm. If an electron in the process of thermalization moves away from the parent ion and goes out of the capture sphere, then the external electric field is capable of picking it up and then the electron can participate in the formation of the ionization signal or in radiation chemical reactions. If, however, an electron, in the process of thermalization, remains inside the capture sphere, then the electric field of the ion turns out to be many times stronger than the technically reasonable external field, and the charges recombine. It should be noted that recombination may appear radiative. Then the energy directed to ionization is highlighted in the form of a photon, the fate of which has to be discussed separately. In radiation detectors particularly, photons arising from recombination contribute to the scintillation signal. Therefore, the possibility of recombination in a pair of electron ion is determined by the process of electron thermalization.

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

171

The process of thermalization depends essentially on the substance in which it occurs. In an atomic gas, electrons can lose energy only through elastic collisions, and the loss in each collision is extremely small. In any molecular matter, electrons can lose energy by excitation of vibrational and rotational states of molecules, i.e., by inelastic collisions. Let us consider the thermalization of an electron in an atomic gas. The formalism of this problem is close to the formalism of the problem of slowing down neutrons. It is known from the conservation laws for elastic collision (Section 3.2.2) that the electron loses part of its energy DE, determined by the relation (3.54), at each collision with an atom. In the notation of this section it has the form DE ¼ E0

4mMcos2 4 ðM þ mÞ

2

;

(8.37)

where m is an electron mass, M is an atom mass, and 4 is the angle of electron emitting. Averaging over angles with the condition M [ m results in DE ¼ E0

2m . M

(8.38)

As can be seen, the absolute mean loss decreases with decreasing of electron energy, but the relative loss of DE/E0 remains constant. Thus, a logarithmic energy scale is introduced in describing the deceleration process. Hence, the electron energy after the n-th collision is   2m (8.39) En ¼ E0 exp  n . M The demanded number of collisions for thermalization is n ¼ M

lnðE0 =EÞ . 2m

(8.40)

If we assume for argon E0 ¼ Ed ¼ 0.31, Ui z 5 eV, and M z 1840$40 m and En ¼ 0.025 eV, we obtain n ¼ 2$105. It can be seen that when thermalization is due to elastic collisions, the required number of collisions is really large. The total path that an electron passes during the thermalization process consists of its individual paths from collision to collision, i.e., mean free paths. In describing the thermalization process, it is possible to operate with an average free run L ¼ 1=Nsmax .

(8.41)

The dependence of the cross section for the elastic collision of electrons with atoms of noble gases on the energy of electrons is given in Fig. 5.25. The nonmonotonic nature of this dependence is called the Ramsauer effect. Thus, the mean free path is also a function of the electron energy. Taking this dependence into account, the total path of an electron before thermalization is L ¼

n X

Li .

(8.42)

1

If we neglect the dependence L ¼ f ðEÞ for a rough estimate, we obtain L ¼ nL z 6 cm at atmospheric pressure. The duration of the thermalization process is n P

Dt ¼

Li ;

1

vn

(8.43)

where vn is the velocity of an electron with the energy En. If we assume an average velocity of w107 cm/s, we could obtain at atmospheric pressure Dt ¼ 6$107 s. As the mean free path is inversely proportional to the pressure, then Dt w 1/p. In the process of thermalization, the electron describes a rather complex broken line of the Brownian line type. Schematically, the thermalization process is shown in Fig. 8.6. During the movement of the electron, the parent ion moves from the place of formation to a negligible distance. Therefore, if we denote the distance, at which the electron reaches the thermal energy kT, by R, then for its square we can determine R2 ¼ ðr1 þ r2 þ . þ Þ2 ¼

n X 1

ri2 þ

n X 1

ri rk cos qik .

(8.44)

172 PART | I Fundamentals

FIGURE 8.6 Scheme of the process of electron thermalization. (A) The change of energy and (B) the spatial pattern of thermalization.

2

If between the two collisions the electron made the same run L, then the first sum turns out to be nL . If we assume that the scattering is spherically symmetric for each collision, then the second sum disappears during averaging because the average value of the cosine for isotropic scattering is zero. As a result we have 2

R2 ¼ nL .

(8.45)

Thus, from the simplest considerations, we obtain the first important conclusion: the average square of the electron removal during the thermalization process is proportional to the first power of the number of collisions. But in reality, between collisions an electron passes each time a different distance r, that is distributed according to the   law exp  r L). In this case, the mean square of the distance traveled between two collisions is not equal to the square of the mean free path. Therefore, instead of Eq. (8.45), we can derive R2 ¼ nr 2 ; and r 2 can be derived by the usual method of the calculation of an average:    RN 2 r exp r L dr 2 0 2    r ¼ RN ¼ 2L . exp r L dr 0

(8.46)

(8.47)

Thus, 2

R2 ¼ 2nL .

(8.48)

Calculation considering the dependence L ¼ f ðEÞ according to the Ramsauer effect for the atmospheric pressure in pffiffiffiffiffi argon gives R2 ¼ 2$102 sm. With increasing pressure, this quantity varies pressure inversely. It should be noted that the carried out calculations are of an evaluation nature. They are meant only to illustrate the physical picture of the phenomenon. They do not take into account, e.g., the asymmetry of scattering and many other important details of the process. But the order of the quantities turns out to be correct. In the case where the main mechanism of electron energy loss is by elastic collision with atoms, the root-mean-square removal during the thermalization process is very significant. Experimental studies of thermalization carried out to date show that for argon at atmospheric pressure the thermalization time after excitation with gamma rays with an energy of 600 keV is 1.7$106 s, and the mean-square removal is w101 cm (our estimates are, respectively, 6$107 s and 2$102 cm). For the “electron-parent ion” interaction, not only the mean square distance between them but also how the distances between separate pairs are distributed is significant. If we continue the analogy with neutron physics, we can then use the

Elementary Processes of the Transformation of Radiation Energy in Matter Chapter | 8

173

TABLE 8.7 Calculated Values of the Thermalization Length in Liquid Noble Gases Liquid

LAr

LKr

LXe

Rtherm, nm

1568

3600

4600

References

[20]

[21]

[22]

solution of the Fermi “age” equation for neutrons. The distribution of thermal neutrons over spherical layers relative to a point source of fast neutrons is given by    4r 2 f ðrÞ ¼ pffiffiffi 3 exp r 2 b2 . pb

(8.49)

Here the parameter b is related to the previously determined root-mean-square removal by the relation b2 ¼

2 2 R. 3

(8.50)

Experimental studies make it possible to find the distribution parameters for some liquids: for liquid xenon b ¼ 76 nm, for neopentane b ¼ 22 nm, and for water b ¼ 1.8 nm. In molecular gases, in the process of thermalization, an electron requires substantially fewer collisions, it traverses smaller paths and is removed to a shorter distance. Thus, in hydrogen the energy of the vibrational quantum is 0.6 eV, and the rotational quantum is 0.007 eV. However, as the electron is an easy particle, the transfer of energy to vibrations and rotation of heavy atoms is ineffective. In hydrogen at the energy range 0.6e10 eV, the inelastic vibrational collision cross section is approximately 20 times smaller than the elastic cross section. Approximately the same relation is valid for rotational collisions, but at a lower energy. Therefore, for the thermalization in molecular hydrogen, an electron requires several hundred collisions instead of several hundred thousands in atomic argon. Similar considerations are fully applicable to liquids. Electron moves in the process of thermalization at noticeably shorter distances, and the process of thermalization takes less time. This process is especially rapid in polar substances. The fact is that the collision of electrons with neutral molecules is determined by a rather short-range polarization interaction. If a molecule already has a dipole moment, then the interaction extends over a larger distance and, therefore, the probability of such interaction is greater. The calculated values of the thermalization length in liquid noble gases are given in Table 8.7. In water, the mean square thermalization path is several nm [9]. In solids, the thermalization of electrons essentially depends on the type of a solid. In ionic crystals, the thermalizing electrons lose energy in scattering by optical phonons, i.e., by relatively large portions. Therefore, electrons cannot depart far from the parent ion. In atomic crystals, such as typical semiconductors, e.g., germanium and silicon, or in such insulators as diamond, electrons lose energy when interacting only with acoustic phonons, which is kinetically similar to elastic scattering on atoms in gases. Therefore, in these substances electrons in the thermalization process can be removed from the parent ion over rather considerable distances and can be used to form the ionization signal. The situation with the problem of thermalization can be illustrated by the following example. Of the enormous number of different types of substances in nature, ionization detectors, in which recombination should be avoided, can be realized only on the basis of noble gases in gaseous, liquid, and solid state and on the basis of some solids as germanium, silicon, and a small number of other substances. On the contrary, scintillation detectors, where recombination is essential, can be constructed on the basis of the widest range of substances, i.e., gaseous, liquid, and solid; crystalline, amorphous, and polymeric; inorganic and organic.

REFERENCES [1] [2] [3] [4]

R.L. Platzman, Total ionization in gases by high-energy particles: an appraisal of our understanding, Int. J. Appl. Radiat. Isot 10 (1961) 116e127. Average Energy Required to Produce an Ion Pair, ICRU Report 31, 1979. S. Sasaki, W-values for Heavy Ions in Gases. in: Nuclear Science Symposium Conference Record (NSS/MIC), IEEE, 2009, pp. 702e705. C.S. Wedlund, G. Gronoff, J. Lilensten, H. Ménager, M. Barthélemy, Comprehensive calculation of the energy per ion pair or W values for five major planetary upper atmospheres, Ann. Geophys. 29 (2011) 187e195.

174 PART | I Fundamentals

[5] S. Sasaki, T. Sanami, K. Saito, T. Murakami, Average energy to produce an ion pair in gases for high energy heavy ions, in: Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC, Report N35-3), IEEE, 2011, pp. 1873e1876. [6] A. Pansky, A. Breskin, R. Chechik. A New Technique for Studying the FanoFactor and the Mean Energy per Ion Pair in Counting Gases, ICFA Instrumentation Bulletin. http://www.slac.stanford.edu/pubs/icfa/spring96/paper6/paper6.pdf. [7] W.P. Jesse, J. Sadauskis, Ionization by alpha particles in mixtures of gases, Phys. Rev. 100 (1955) 1755e1762. [8] C.A. Klein, Bandgap dependence and related features of radiation ionization energies in semiconductors, J. Appl. Phys. 39 (4) (1968) 2029e2038. [9] S.M. Pimblott, J.A. LaVerne, A. Mozumder, N.J.B. Green, Structure of electron tracks in water. 1. Distribution of energy deposition events, J. Phys. Chem. 94 (1990) 488e495. [10] A.M. Miterev. Theoretical aspects of the formation and evolution of charged particle tracks 45 (2002) 1019e1050. [11] L.H. Toburen, Ionization and charge-transfer: basic data for track structure calculations, Radiat. Environ. Biophys. 37 (4) (1998) 221e233. [12] D. Emfietzoglou, G. Papamichael, K. Kostarelos, M. Moscovitch, A Monte Carlo track structure code for electrons (approximately 10 eV-10 keV) and protons (approximately 0.3-10 MeV) in water: partitioning of energy and collision events, Phys. Med. Biol. 45 (11) (2000) 3171e3194. [13] U. Fano, Ionization yield of radiations. II. The fluctuations of the number of ions, Phys. Rev. 72 (1947) 26e29. [14] A. Owens, G.W. Fraser, K.J. McCarthy, On the experimental determination of the Fano factor in Si at soft X-ray wavelengths, Nucl. Instrum. Methods Phys. Res. A 491 (2002) 437e443. [15] S. Croft, D.S. Bond, A determination of the Fano factor for germanium at 77.4 K from measurements of the energy resolution of a 113 cm3HPGe gamma-ray spectrometer taken over the energy range from 14 to 6129 keV. International Journal of Radiation Applications and Instrumentation. Part A, Appl. Radiat. Isot. 42 (11) (1991) 1009e1014. [16] R. Devanathan, L.R. Corrales, F. Gao, W.J. Weber, Signal variance in gamma-ray detectors. A review, Nucl. Instrum. Methods Phys. Res. A 565 (2) (2006) 637e649. [17] A. Subashiev, S. Luryi, Correlation effects in sequential energy branching: an exactly solvable model of the Fano statistics, Phys. Rev. E 81 (2010) 021123 [pp. 1e10]; see also arXiv:0911.1532, 2009. [18] M.J. Harrison, D.S. McGregor, F.P. Doty, Fano factor and nonuniformities affecting charge transport in semiconductors, Phys. Rev. B 77 (2008) 195207. [19] D.V. Jordan, A.S. Reinolds, J.E. Jaffe, K.K. Anderson, L.R. Corralesa, A.J. Peurrung, Simple classical model for Fano statistics in radiation detectors, Nucl. Instrum. Methods Phys. Res. A 385 (2008) 146. [20] A. Mozumder, Free-ion yield in liquid argon at low-LET, Chem. Phys. Lett 238 (1e3) (1995) 143e148. [21] A. Mozumder, Free-ion yield and electron-ion recombination rate in liquefied rare gases: the case of liquid krypton, Chem. Phys. Lett 253 (5e6) (1996) 438e442. [22] A. Mozumder, Free-ion yield and electron-ion recombination rate in liquid xenon, Chem. Phys. Lett. 245 (4e5) (1995) 359e363.