REFLECTION OF GAMMA RADIATION FROM TWO-LAYER ABSORBERS

A. M. Kolchuzhkin, S. I. Durinov

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

In the practice of radiation defectoscopy, methods based on measuring reflected gamma radiation are widely used. In [1] a general formula was obtained for the flux of gamma quanta reflected from a two-layer barrier. If the first layer has thickness \(h\), and the second layer is semi-infinite, then

\[ \Phi^{(-)}(\Omega,E)=\Phi_0^{(-)}(\Omega,E)- \]

\[ -\,n_2 \int_{0}^{\frac{n_1}{n_2}h} dt' \int_{4\pi} d\Omega' \int_{0}^{E_0} dE' \left( \frac{\Sigma_1(E')}{n_1} - \frac{\Sigma_2(E')}{n_2} \right) \Phi^{+}(t',\Omega',E')\, \Phi_0(t',\Omega',E'), \tag{1} \]

Here \(\Phi^{(-)}(\Omega,E)\) is the differential flux of radiation reflected from a two-layer barrier;

\(\Phi_0(t',\Omega',E')\) is the differential flux in a homogeneous semi-infinite absorber made of material 2;

\(\Phi_0^{(-)}(\Omega,E)\) is the flux reflected from this homogeneous absorber;

\(\Phi_0^{(+)}(t',\Omega',E')\) is the adjoint function for the two-layer barrier;

\(\Sigma_1, \Sigma_2\) are the linear attenuation coefficients of layers 1 and 2;

\(n_1, n_2\) are the electron densities.

Calculations by formula (1) were carried out under the assumption that the flux \(\Phi_0\) and the adjoint function \(\Phi^{+}\) are constant in the first layer and equal to their values at the boundary [2]. A more accurate expression can be obtained by assuming an exponential character for the behavior of the flux and the adjoint function in the first layer:

\[ \Phi_0(t',\Omega,E')= \begin{cases} \dfrac{1}{\cos\vartheta_0}\, e^{-\Sigma_2(E_0)\frac{t'}{\cos\vartheta_0}}\, \delta(\Omega'-\Omega_0)\,\delta(E'-E_0), & \cos\vartheta' > 0, \\[1.2em] \Phi_0^{(-)}(\Omega',E')\, e^{\Sigma_2(E')\frac{t'}{|\cos\vartheta'|}}, & \cos\vartheta' < 0. \end{cases} \tag{2} \]

\[ \Phi^{+}(t',\Omega',E') = \begin{cases} e^{-\Sigma_1(E')\left(h_1-\dfrac{n_2}{n_1}t'\right)\dfrac{1}{\cos\vartheta'}} \Phi_0^{(-)}(\Omega',E'\to\Omega,E)\times \\[6pt] \qquad {}\times e^{-\Sigma_1(E)\dfrac{h_1}{|\cos\vartheta|}}, & \cos\vartheta'>0,\\[10pt] \dfrac{1}{|\cos\vartheta|}\, e^{-\Sigma_1(E)\dfrac{n_2}{n_1}\dfrac{t'}{|\cos\vartheta|}}\, \delta(\Omega-\Omega')\,\delta(E-E'), & \cos\vartheta'<0. \end{cases} \]

(the source is assumed to be plane, \(t=0\), unidirectional \(|\Omega_0|\), monoenergetic \((E_0)\). By \(\Phi_0^{(-)}(\Omega',E'\to\Omega,E)\) is denoted the flux of quanta with parameters \((\Omega,E)\), reflected from the second layer, if the radiation incident on it is characterized by the parameters \((\Omega',E')\). Substituting (2) into (1), we obtain

Table 1

\[ \Phi^{(-)}(\Omega,E)= \Phi_0^{(-)}(\Omega,E)\, e^{-\Sigma_1(E)\dfrac{h}{|\cos\vartheta|}} \left\{ \left[ e^{-\Sigma_1(E_0)\dfrac{h}{\cos\vartheta_0}} - e^{-\Sigma_2(E_0)\dfrac{n_1}{n_2}\dfrac{h}{\cos\vartheta_0}} \right] - \left[ e^{+\Sigma_1(E)\dfrac{h}{|\cos\vartheta|}} - e^{+\Sigma_2(E)\dfrac{n_1}{n_2}\dfrac{h}{|\cos\vartheta|}} \right] \right\}. \tag{3} \]

In this formula the terms depending on \((E_0,\cos\vartheta_0)\) and \((E,\cos\vartheta)\) describe the perturbation of the incoming and outgoing flux, respectively.

Fig. 1. Dependence of the intensity of reflected radiation on the thickness of the iron layer

If the initial energy \(E_0\) is such that the main process of interac-

action is Compton scattering, the perturbation of the incident flux will be equal to zero and (3) becomes

\[ \Phi^{(-)}(\Omega,E)=\Phi_0^{(-)}(\Omega,E) \left[ 1-e^{-\frac{n_1 h}{|\cos\vartheta|} \left(\frac{\Sigma_1(E)}{n_1}-\frac{\Sigma_2(E)}{n_2}\right)} \right], \tag{4} \]

i.e., to calculate the flux of gamma radiation reflected from a two-layer barrier, it is necessary to know the flux of quanta reflected from a homogeneous semi-infinite absorber made of material \([2]\). A large quantity of such data is available in \([3]\).

Formula (4) was used to calculate the angular distribution of the intensity of gamma radiation reflected from a two-layer Fe + Al barrier at \(E_0 = 0.66\) MeV. The results of the calculations are given in Table 1. The values obtained for this combination by the Monte Carlo method (4000 histories) are also given there.

Figure 1 shows the dependence of the intensity of the reflected radiation on the thickness of the iron layer. The results presented show that formula (4) makes it possible to calculate the flux of radiation reflected from a two-layer absorber over a wide range of thicknesses of the first layer.

REFERENCES

1. A. M. Kolchuzhkin, V. V. Uchaikin. Atomic Energy, 25, 442, 1968.
2. A. M. Kolchuzhkin, V. V. Uchaikin. “Izv. vuzov,” Physics, 12, 127, 1968.
3. B. P. Bulatov, V. A. Efimenko, V. G. Zolotukhin, V. A. Klimanov, V. P. Mashkovich. Albedo of Gamma Radiation. Atomizdat, Moscow, 1968.