CALCULATION OF THE CHARACTERISTICS OF POSITRON TRANSPORT THROUGH MATTER

A. A. Vorob’ev, O. B. Evdokimov, V. A. Kuz’minykh,
S. A. Vorob’ev, I. A. Tsekhanovskii

(Presented by the scientific seminar of the nuclear-reaction sector of the Scientific Research Institute of Nuclear Physics, Electronics, and Automation)

In this work, a calculation is performed of the distributions of stopped positrons and absorbed energy during the passage of positrons with energies \((0.23 \div 1.0)\) MeV through absorbers with different atomic numbers. A comparison is made of the spatial distributions obtained with analogous distributions for electrons.

Introduction

In the practical use of positron beams, it is necessary in some cases to know precisely the characteristics of the distribution of the flux of charged particles in an absorber. Often in such cases it is assumed, approximately, that positrons and electrons of equal energy are absorbed in matter in the same way. It is known, however, that the interaction parameters of positrons and electrons with matter have certain differences, due to the difference in the signs of their charges [1]. The influence of these differences has been investigated in many works, for example [2, 3], but no estimates have been made of the degree of difference in the absorption of energy and in the distribution of the positron flux with absorber depth from the analogous characteristics for electrons.

The aim of the present work was to calculate the stopped positrons and the energy lost by them in various absorbers. Unlike earlier works carried out by other authors, we used correct cross sections for the processes of interaction of positrons with matter. The calculation of the characteristics of the positron flux in the absorber was performed according to the theory of transport through matter of fast beta particles in the continuous-slowing-down approximation, without taking into account in-flight annihilation of positrons.

Method of calculation and basic relations

The equation for the transport of particles through matter in the one-dimensional case within the framework of the segment model has the form [4]:

\[ I(x, S_j, \bar{\varepsilon}_{j+1}) = \int_{4\pi} I(x - \lambda_j \bar{s}_j, S_{j-1}, \bar{\varepsilon}_j)\, q(x,\lambda_j,S_j,\omega_j)\,d\Omega, \tag{1} \]

here \(q(x,\lambda_j,s_j,\omega_j)\) is the angular distribution on the \((j+1)\)-st segment relative to the direction on the \(j\)-th segment of the trajectory,

\(x\) is the spatial coordinate,

\(\lambda_j\) is the length of the trajectory segment,

\(s_j\) is the residual range of the particle in the material,

\(\varepsilon_j \equiv \cos\theta_j\), where \(\theta_j=\vec n_j \vec n_x\), \(\omega_j=\vec n_j \vec n_{j+1}\), and \(\vec n_j\) indicates the direction of motion of the particles on the \(j\)-th segment of the trajectory.

We define the kinetic energy of the particles at the beginning of the \(j\)-th segment as

\[ T_j=T_{j-1}-\left(\frac{dT}{ds}\right)\lambda_j, \]

where the ionization energy losses of positrons, with allowance for the density effect, are calculated from known relations [5, 6]. The angular distribution of positrons is calculated using the differential scattering cross section in the second Born approximation, taking into account the screening effect of the Coulomb field of the nucleus [7].

Equation (1) is reduced in this case to a recurrence relation for the angular moments of the harmonic coefficients of the differential flux \(I(x,s_j,\varepsilon_j)\) [8, 9]:

\[ J_{lk}^{j}=\sum_{k'=-\infty}^{\infty}\psi_{kk'}^{lj}\sum_{l'=0}^{\infty}J_{l'k'}^{j-1}(2l+1)E_{ll'k'}^{*j}, \tag{2} \]

where

\[ E_{ll'k'}^{*j}=\frac{1}{2}\int_{-1}^{1}P_l(\varepsilon)P_{l'}(\varepsilon)e^{-i\mu'\lambda_j\varepsilon}\,d\varepsilon, \tag{3} \]

\[ \mu=\frac{2\pi k}{L}, \]

\(L\) is the period of the function describing the differential flux. The expression for \(\psi_{kk'}^{lj}\), depending on the geometry of the problem, takes the form

a) for an infinite medium

b) for a semi-infinite medium \(\psi_{kk'}^{lj}=q_l^j\delta_{kk'}\);

\[ \psi_{kk'}^{lj}\frac{1+q_l^j}{2}\delta_{kk'}+ i\frac{1-q_l^j}{2\pi}\, \frac{\left[(-1)^{k'-k}-1\right]}{k'-k}(1-\delta_{kk'}), \tag{4} \]

where

\[ q_l^j=2\pi\int_{-1}^{1}q^j(x,\lambda,s,\omega)P_l(\cos\omega)\,d\cos\omega. \tag{5} \]

The differential flux of particles in the absorber is determined by the expression:

\[ I(x,s_j,\varepsilon_j)=\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}P_l(\varepsilon) \left[\sum_{k=0}^{\infty}a_{lk}^{j}\cos\mu x+b_{lk}^{j}\sin\mu x\right]. \tag{6} \]

The coefficients \(a_{lk}^{j}\) and \(b_{lk}^{j}\) entering into (6) are calculated from the following relations derived by us:

\[ a_{lk}^{j}=\frac{1}{2l+1} \begin{cases} (-1)^{l/2}\chi_{lk}^{j}, & l=2n,\\ (-1)^{l+1/2}\chi_{lk}^{j}, & l=2n+1; \end{cases} \]

\[ b_{lk}^{j}=\frac{1}{2l+1} \begin{cases} (-1)^{l/2}\chi_{lk}^{j}, & l=2n,\\ (-1)^{l-1/2}\chi_{lk}^{j}, & l=2n+1, \end{cases} \tag{7} \]

where \(n=0,1,2,3,\ldots\)

The values \(\varkappa_{lk}^{j}\) and \(\chi_{lk}^{j}\) for the geometry of the problem used are calculated from recurrence relations, which, for the case of an infinite medium, have the form

\[ \varkappa_{lk}^{j}=(2l+1)q_l^j\sum_{l'}\varkappa_{l'k}^{j-1}\dot E_{ll'k}, \]

\[ \chi_{lk}^{j}=(2l+1)q_l^j\sum_{l'}\chi_{l'k}^{j-1}\dot E_{ll'k}, \tag{8} \]

where

\[ \dot E_{ll'k}=(-i)^{l'-l}E_{ll'k}^{*}. \]

From expression (6), upon integrating it over \(d\varepsilon\) and for \(j=j_{\max}\), one obtains an expression for the distribution function of stopped particles in the absorber:

\[ I(x)=\sum_{k=0}^{\infty}a_{0k}^{j_{\max}}\cos\mu x+b_{0k}^{j_{\max}}\sin\mu x. \tag{9} \]

The expression for the distribution of lost energy using (6) in this case takes the form

\[ I_T(x)=\sum_{k=0}^{\infty}\left[ \cos\mu x\int \frac{dT}{ds}a_{0k}(s)\,ds +\sin\mu x\int \frac{dI}{ds}b_{0k}(s)\,ds \right]. \tag{10} \]

In an analogous manner, expressions are also obtained for the angular distribution of particles at various depths of the absorber. The initial conditions of the problem take into account the position of the particle source relative to the origin of the coordinate system \(x_0\) and the initial direction—the angle \(\Theta_0\).

\[ \varkappa_{lk}^{0}=\frac{2}{L}(2l+1)P_l(\xi_0)\cdot \begin{cases} (-1)^{l/2}\cos\mu x_0, & l=2n,\\ (-1)^{l-1/2}\sin\mu x_0, & l=2n+1; \end{cases} \]

\[ \chi_{lk}^{0}=\frac{2}{L}(2l+1)P_l(\xi_0)\cdot \begin{cases} (-1)^{l/2}\sin\mu x_0, & l=2n,\\ (-1)^{l+1/2}\cos\mu x_0, & l=2n+1, \end{cases} \tag{11} \]

where \(n=0,1,2,3\ldots\)

Results of the calculation and their analysis

The calculation of the distributions of stopped positrons and electrons was carried out by us in the infinite-medium representation on the BESM-4 electronic computer for the case of deep penetration of particles with energy \((0.23 \div 1.0)\) MeV in absorbers with different atomic numbers.

The total range of beta particles [6], calculated for the selected energy and substance, was divided in our case into 50 intervals \((j_{\max}=50)\).

The number of terms in the expansion of the differential particle-flux function was taken to be 10, i.e., \(l_{\max}=k_{\max}=10\). As the investigation of the calculation model carried out by us showed, the chosen number of harmonics is quite sufficient for describing, in the interval \((-1 \div 1)\), the distribution function of stopped beta particles, but is not sufficient for describing the absorbed-energy distribution, which has a nonsmooth character.

Fig. 1. Distribution of stopped positrons with initial energy \(0.23\) MeV and \(1.0\) MeV in aluminum.

Figure 1 shows the calculated curves obtained by us for the distribution of stopped positrons in an aluminum absorber \((Z=13)\) at initial energies \(0.23\) MeV \((R_0=0.0697\ \mathrm{g/cm^2})\) and \(1.0\) MeV \((R_0=0.56\ \mathrm{g/cm^2})\). The curves obtained proved to be asymmetric in shape, with a considerable “tail” in the region of small depths, which is explained by the energy dependence of the scattering cross sections of beta particles

Fig. 2. Distribution of stopped electrons and positrons with initial energy \(1.0\) MeV in graphite, aluminum, and copper. 1 — electrons in Cu, 2 — positrons in Cu, 3 — electrons in Al, 4 — positrons in Al, 5 — electrons and positrons in C.

in the absorber. It is also seen that the fraction of positrons backscattered from the target decreases with increasing particle energy, and the minimum of the distribution shifts toward greater thicknesses in agreement with the known concepts regarding the interaction of beta particles with matter. On

Fig. 2 shows calculated curves for the distributions of stopped positrons and electrons with energy 1.0 MeV in absorbers made of graphite \((Z=6)\), aluminum \((Z=13)\), and copper \((Z=29)\). It is evident that the curves for the distributions of stopped electrons and positrons are fairly similar in shape, but the known differences in the interaction of electrons and positrons with matter are reflected in the position of the maxima of the calculated curves. The positions of the maxima of the distributions of stopped positrons and electrons for graphite practically coincide; in aluminum the maximum of the distribution of stopped positrons lies 6.3% deeper than for electrons, while in copper the difference reaches approximately 8.5%. The results we obtained are a consequence of the weaker scattering of positrons in comparison with electrons in matter. As the atomic number of the absorber increases, the difference in the form of the calculated curves for the distributions of stopped beta particles increases, which agrees with the known dependence for the difference in elastic scattering of electrons and positrons.

We also obtained calculated distributions of absorbed energy over the depth of the absorber for positrons and electrons; moreover, the nature of the difference in the curves for absorbed energy proved analogous to the difference described above in the curves for the distributions of stopped beta particles.

REFERENCES

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