PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

No. 251　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1970

Calculation of the Characteristics of Bounded Bremsstrahlung Beams Behind Thick Absorber Layers

Vorob’ev V. A., Gankin Yu. D., Tareev G. P.

Presented at the joint seminar of the sectors
DCM and MRD, Research Institute of Nuclear Physics

1. Bounded bremsstrahlung beams are used in many areas of applied physics. The quantitative characteristics of such beams when they pass through layers of material are, as a rule, effectively obtained by computation. We have developed a program for the M-20 suitable for the joint computation of any characteristics of a bremsstrahlung beam that are functionals of the spectral distribution of intensity on the beam axis behind a layer of absorber. The program makes it possible simultaneously to carry out calculations for 5 radii of the irradiating beam behind 5 thicknesses.

2. The computational algorithm used is as follows. The spectral distribution of the intensity of unscattered radiation is calculated by mono-lines by a numerical method; the spectral distribution of scattered radiation (for each thickness and radius) is estimated by the Monte Carlo method using effective modifications, which reduce to the following.

a) The method of collision density is used; the collision density is estimated by means of “importance sampling,” with both the initial (according to Schiff) distributions with the multiplier proposed by us \(q(E)\) and the transition probability density being changed by means of an exponential transformation [1, 3].

b) Analytical averaging of the absorption and production of annihilation quanta is applied.

c) The reciprocity principle is used; that is, the beginning of all trajectories of the “played quanta” is taken at the point of the anterior

layer boundaries on the beam axis, and the contributions are recorded behind the corresponding thicknesses \(H_l\), in circles \(F_{cj}\) of radius \(r_j\) of the irradiating beams.

The contribution to the component of the spectrum intensity of scattered radiation attributable to one tree of simulated trajectories has the form:

\[ \xi_{jk}^{i}=\sum_{j=1}^{N_k-1}\left[T_{jk}^{i}+2\sum_{l=0}^{N_j-1}T_{ljk}^{i}\right], \tag{1} \]

where summation over \(j\) takes into account the partial contributions from the \(j\)-th collision along the branch of the trajectory tree, while the sums over \(l\) take into account the contributions from annihilation quanta born in the \(j\)-th collision (for simplification, the trajectories of both born quanta were combined). Here

\[ T_{jk}^{i}=E_j Q_{jk}P_j^{i}\frac{1}{|\cos\theta_j|}, \tag{2} \]

\(Q_{jk}\) is the weight, \(P_j^{i}\) is the probability that a quantum from the collision point \(r_j\) will enter the element \((F_c * \Delta E_i)\) without collisions.

The initial distribution \(J_0(E)\) is biased by means of the function

\[ \varphi'(E)=C\cdot\exp[-\mu(E)H]\cdot E, \tag{3} \]

so that the weight multiplier for eliminating this bias has the form

\[ u_b\varphi^{n}=\frac{a}{E_k\exp[-\mu(E_k)H]}, \tag{4} \]

In this case the coefficient \(a\), with allowance for normalization to \(1\ \mathrm{MeV}/\mathrm{cm}^{2}\) of incident radiation, is calculated as

\[ a=\frac{z\int J_0(E)\exp[-\mu(E)H]\,dE}{\gamma\int J_0(E)\,dE}. \tag{5} \]

Here \(J_0(E)\) is the initial intensity distribution. The weight multipliers eliminating the bias of transition probabilities (for a multiplicative form of the exponential transformation) have the form

\[ \beta_l=\frac{C^{5}(E_{l-1})}{1-\gamma\cos\theta_{l-1}}\, \exp[-\mu(E_{l-1})\gamma\Delta Z_l], \tag{6} \]

Fig. 1.

Fig. 2a.

Fig. 3.

Fig. 2b.

where \(\gamma = 0.6 \div 0.8\), \(C^f\) is the probability of avoiding absorption in a collision. After the spectral distributions have been calculated, a block of processing subroutines operates to compute the required beam characteristics by quadrature formulas.

1. Figure 1 shows the change in the spectral composition of the intensity of scattered radiation with \(E_m = 30\) MeV behind a layer of Fe of thickness \(H = 30\) cm, as a parametric function of the radius of the “irradiation field,” calculated by the program described. As can be seen, as the beam radius increases the spectral distribution converges uniformly over all spectral components to the limiting distribution \((R=\infty)\), which indicates the establishment of equilibrium [2]; moreover, the scattered radiation is concentrated in the energy region \(4 \div 10\) MeV.

Figure 2 shows the behavior of the energy buildup factor on the beam axis as a function of the radius of the “irradiation field.” As can be seen, the value \(B_E\) for layers of large thickness (2b) converges more slowly to the limiting value \(B_E^{(\infty)}\) than for layers of smaller thickness (2a), and the convergence worsens as the maximum energy in the spectrum decreases.

Figure 3 gives the dependence of the mean energy in the complete spectrum of the transmitted radiation, for a broad bremsstrahlung beam, on the thickness of the Fe layer, for a primary distribution truncated below at the level 0.05 MeV. For initial distributions with \(E_m > 15 \div 20\) MeV and the material Fe, this quantity may reach a maximum at certain \(H\). This is explained by the fact that, for such spectra at small thicknesses, rapid attenuation of the components of the low-energy part of the spectrum occurs, together with still weak attenuation of the components of the high-energy part; moreover, the scattered radiation is concentrated in the middle of the energy range, which leads to an increase in \(E_{\mathrm{cp}}\). At large thicknesses the low-energy part of the spectrum has already been attenuated, and, owing to the weakening of the high-energy part, \(E_{\mathrm{cp}}\) decreases.

Literature

1. The Monte Carlo Method in the Problem of Radiation Transport. Collection edited by Corresponding Member of the USSR Academy of Sciences G. I. Marchuk, Atomizdat, Moscow, 1967.

2. U. Fano, L. Spencer, M. Berger. Gamma-Ray Transport. Atomizdat, Moscow, 1963.

3. G. A. Mikhailov. Izvestiya VUZov, Fizika 10, 1968, pp. 76–79.