Determination of the Effective Energy of Secondary Electrons in Alkali-Halide Salt Crystals under Irradiation by Betatron Bremsstrahlung

V. A. Vorob’ev, V. N. Rudenko

The transfer of the energy of betatron bremsstrahlung to the absorber material occurs mainly through the absorption of secondary electrons in the substance [1]. Secondary electrons in a material under the action of betatron bremsstrahlung are formed as a result of the photoelectric effect, the Compton effect, and the pair-production process [2]. The contribution of each of these principal processes of interaction of electromagnetic radiation with matter to the total number of secondary electrons depends on the energy of the radiation quanta. The secondary electrons produced by bremsstrahlung have different energies and form a continuous spectrum, which can be characterized by the effective energy of the spectrum. By effective energy is meant the energy of those electrons that have the same absorption coefficient in the material under study as does the flux of the electrons under consideration.

Knowledge of the effective energy of secondary electrons makes it possible to determine the distribution of the absorbed dose in the irradiated object, which is an important problem in solid-state physics, radiation chemistry, medicine, and radiation defectoscopy.

In the present work, the effective energy was determined for secondary electrons formed in the alkali-halide crystals KCl, KBr, and KJ under the action of betatron bremsstrahlung in the energy range 10–30 MeV, on the basis of an experimental study of the extrapolated ranges of monoenergetic electrons and of the penetration depths of secondary electrons into the crystals under investigation.

Studies of the dose distribution over the depth of crystals when they were irradiated with electrons of different energies, for the purpose of determining the extrapolated ranges of electrons, were carried out by measuring the ionization current in miniature ionization chambers made from the crystals under investigation and having a sensitive volume of \(0.2 \div 0.4\ \text{cm}^3\) [3].

The rear front of the dose-distribution curve over the thickness of the crystal characterizes the depth of penetration of electrons into the crystals under investigation.

crystals, and by extending its most straight portion to its intersection with the abscissa axis one obtains the value of the extrapolated range, which depends little on the experimental conditions and therefore is most often used to determine the depth of penetration of electrons into a substance [4].

The value of the extrapolated range of electrons is determined by the electron energy and depends on the chemical composition of the crystals.

On the other hand, the depth of penetration of secondary electrons formed in crystals by bremsstrahlung quanta was determined from the position of the maximum of the absorption curves of betatron bremsstrahlung in crystals, measured with a thin-walled ionization chamber [5]. The thickness of the crystal layer in front of the chamber, corresponding to the maximum of the absorption curve and ensuring the effect of electronic equilibrium, determines the penetration depth of electrons of energy equal to the effective energy.

Fig. 1. Dependence of the effective energy of secondary electrons on the effective atomic number of alkali-halide crystals at different maximum bremsstrahlung energies.

The electron energy is related to the penetration depth in alkali-halide crystals of the potassium series by the simple empirical relation 3, obtained on the basis of a study of extrapolated ranges, which is valid for electron energies of 5–11 MeV.

\[ T = \frac{R + B}{A}, \tag{1} \]

where \(T\) is the kinetic energy of the electrons in MeV, \(R\) is the extrapolated range in \(\mathrm{g/cm^2}\), and \(A, B\) are constants, whose values are given in Table 1.

Table 1

The values of the effective energy of the secondary electrons, obtained from formula 1, are given in the form of curves in Figs. 1–4 for the bremsstrahlung-energy range 10–30 MeV.

As is seen from the graphs of Figs. 1–4, there is an increase in the effective energy of the secondary electrons on going to salts with a larger atomic number of the halogen and a relative increase in the ratio of the maximum betatron bremsstrahlung energy to the effective energy of the secondary electrons as the maximum bremsstrahlung energy increases. For the chosen experimental geometry, with the aid of an ioniza-

…only those electrons can be registered by the ionization chamber whose direction of motion is close to the axis of the bremsstrahlung beam.

Fig. 2. Dependence of the ratio of the maximum energy of the betatron bremsstrahlung to the effective energy of secondary electrons on the effective atomic number of the alkali-halide compound.

Labels in the figure: \(E_{\max}^{x}/E_{\mathrm{eff}}^{e}\); \(Z_{\mathrm{eff}}\); effective atomic number of the absorber material; \(10\), \(15\), \(20\), \(25\), \(30\) MeV; \(o\)—KCl, \(x\)—KBr, \(\Delta\)—KI.

The energy of secondary electrons obtained through the Compton effect is described by the formula [6]

\[ E^{e}=\frac{E^{x}}{1+\frac{1}{2E^{x}}(1-\cos\Theta)}, \tag{2} \]

where \(\Theta\) is the scattering angle of the quantum,
\(E^{x}\) is the energy of the quantum before scattering.

Fig. 3. Dependence of the effective energy of secondary electrons on the maximum energy of the betatron bremsstrahlung.

Labels in the figure: effective energy of secondary electrons; maximum energy of the betatron bremsstrahlung; \(10\), \(15\), \(20\), \(25\), \(30\) MeV; \(o\)—KCl, \(x\)—KBr, \(\Delta\)—KI.

For \(\Theta=\pi\) (the electron has acquired a forward direction of motion), the energy of secondary Compton electrons is close to the energy of the incident quantum.

The mean energy of the spectrum of secondary electrons formed as a result of the pair-production process, owing to the distribution of the quantum energy between the positron and the electron, is

\[ E^{e}=\frac{E^{x}}{2}-m_{0}c^{2}, \tag{3} \]

where \(m_{0}c^{2}\) is the rest energy of the electron.

From formulas 2 and 3 it is evident that the secondary electrons produced by Compton scattering have a higher energy.

In connection with the above, the increase in the ratio of the maximum energy of bremsstrahlung to the effective energy of secondary electrons with increasing maximum energy of the bremsstrahlung is explained by the relative increase in the number of secondary electrons,

Fig. 4. Dependence of the ratio of the maximum energy of betatron bremsstrahlung to the effective energy of secondary electrons on the maximum energy of betatron bremsstrahlung.

formed as a result of the pair-production process, since with increasing quantum energy the cross section of the pair-production process increases in comparison with the cross section of the Compton effect.

The increase in the effective energy of secondary electrons with increasing effective atomic number of an alkali-halide compound can be explained as follows. The position of the maximum of the absorption curve is determined to a considerable extent by the dependence of the intensity of the beam of transmitted electrons on the crystal thickness and is determined by the magnitude of the ionization losses, which play the principal role in the energy range of the electrons under consideration. Since the ionization losses depend only weakly on the effective atomic number, the position of the maximum depends only weakly on \(Z_{\mathrm{eff}}\), but depends substantially on the density.

With increasing \(Z\), the total cross section for the production of secondary electrons increases both owing to the Compton effect and owing to the pair-production process, with the cross section of the pair-production process increasing to a greater degree.

The increase in the effective energy of secondary electrons at the exit from the crystal is explained by the absorption of low-energy electrons in the crystal, which leads to a relative increase in the high-energy component in the spectrum of secondary electrons.

LITERATURE

1. U. Fano, L. Spencer, M. Berger. Gamma-Ray Transport, Gosatomizdat, 1963.

2. Experimental Nuclear Physics. Ed. by E. Segrè, vol. 1, Foreign Literature Publishing House, Moscow, 1955.

3. B. A. Kononov, V. N. Rudenko. “The effect of radiation on the properties of materials,” Proceedings of the Conference, Novosibirsk, issue 2, p. 36, 1963.

4. V. N. Rudenko. Atomic Energy, No. 2, 150, 1964.

5. V. A. Vorob’ev, V. N. Rudenko. “Attenuation of a narrow beam of betatron bremsstrahlung in thick layers of alkali-halide crystals.” Present collection.

6. O. I. Leipunskii, B. V. Novozhilov, V. N. Sakharov. Propagation of Gamma Quanta in Matter. State Publishing House of Physico-Mathematical Literature, Moscow, 1960.