Abstract Generated abstract
The paper proposes a method for calculating the integral absorbed power and energy efficiency of gamma radiation in macrosystems of arbitrary configuration. Starting from energy balance relations for primary, scattered, and incident radiation, it derives expressions for a centrally irradiated absorbing sphere using absorbed-energy buildup factors, then introduces an equivalent-sphere construction based on the mean attenuation of primary radiation through the actual system. This reduces the calculation for complex geometries to determining the relative primary transmitted power, from which an effective radius, scattered component, and efficiency are obtained. Comparisons with Monte Carlo calculations for point gamma sources indicate good agreement, supporting the proposed approach within the considered energy range and configurations.
Full Text
Physical Chemistry
B. I. Vainshtein
A METHOD FOR CALCULATING THE INTEGRAL ABSORBED POWER OF GAMMA RADIATION IN MACROSYSTEMS
(Presented by Academician S. S. Medvedev, 9 IX 1963)
The calculation of the integral absorbed power of $\gamma$-radiation in macrosystems (I.A.P.) for a number of cases has been considered in works ($^{1-5}$). In works ($^{1-3}$), only an approximate calculation of the I.A.P. was carried out (without rigorous allowance for multiple scattering) for some source—irradiated-object systems. In works ($^{4,5}$), a semiempirical method was proposed for calculating the I.A.P. in a cylindrical absorbing object from a point or linear source.
In the present work a method is proposed for calculating the I.A.P. ($w_a$) in arbitrary macrosystems.
Let us denote $w_a/w_0=\eta$; $w_{\mathrm{p}}/w_0=j_{\mathrm{p}}$; $w_{\mathrm{r}}/w_0=j_{\mathrm{r}}$; $w_{\mathrm{in}}/w_0=j_{\mathrm{in}}$; $w_{\mathrm{op}}/w_0=j_{\mathrm{op}}$, where $w_{\mathrm{p}}$, $w_{\mathrm{r}}$ are, respectively, the energy-flux powers of the primary attenuated and scattered radiation emerging from the surface of the irradiated object; $w_{\mathrm{in}}$ is the energy-flux power of $\gamma$-radiation entering through the surface of the irradiated object; $w_0$ is the power of the $\gamma$-radiation source; $w_{\mathrm{op}}$ is the power of the primary $\gamma$-radiation scattered without interaction with the macrosystem; $\eta$ is the energy efficiency of the macrosystem with respect to $\gamma$-radiation (e.c.p.); $j_{\mathrm{p}}$ and $j_{\mathrm{r}}$ are, respectively, the relative energy-flux powers of the primary attenuated and scattered $\gamma$-radiation emerging from the surface of the irradiated object; $j_{\mathrm{in}}$ is the fraction of the power of the $\gamma$-radiation source reaching the macrosystem; $j_{\mathrm{op}}$ is the relative power of the primary $\gamma$-radiation scattered without interaction with the macrosystem.
For a given power of the $\gamma$-radiation source, the value of the I.A.P. is uniquely determined by the e.c.p. of the macrosystem ($^{4,5}$). We note that
$$ \eta = 1 - (j_{\mathrm{op}} + j_{\mathrm{r}}), \tag{1} $$
where
$$ j_{\mathrm{op}} = 1 - j_{\mathrm{in}} + j_{\mathrm{p}}. \tag{2} $$
Thus, the calculation of the e.c.p. reduces to determining $j_{\mathrm{p}}$, $j_{\mathrm{r}}$, and $j_{\mathrm{in}}$. We note that when an isotropic radiation source is located inside the macrosystem, $j_{\mathrm{in}}=1$ and $j_{\mathrm{op}}=j_{\mathrm{p}}$.
Calculation of the e.c.p. of a sphere. Consider the case when a point isotropic source of $\gamma$-radiation lies at the center of an absorbing sphere of radius $R$ (in centimeters). Obviously, in this case $j_{\mathrm{p}}=e^{-\mu R}$, where $\mu$ is the linear attenuation coefficient of $\gamma$-radiation in the substance filling the sphere (in inverse centimeters). To calculate the value of $\eta$, we integrate the energy absorbed in an elementary spherical layer of thickness $d\rho$, located at a distance $\rho$ from the center of the sphere, over the entire volume of the sphere:
$$ \eta = \int_0^R B(E,\mu\rho)\, e^{-\mu\rho}\mu_a\,d\rho, \tag{3} $$
where $B(E,\mu\rho)$ is the buildup factor of absorbed energy, $\mu_a$ is the linear
the absorption coefficient of $\gamma$-radiation in the substance filling the sphere (in inverse centimeters).
The quantity $B(E,\mu\rho)$, as is known $^{(6)}$, can be represented in the form:
\[ B(E,\mu\rho)=A_1 e^{-\alpha_1\mu\rho}+A_2 e^{-\alpha_2\mu\rho}, \tag{4} \]
where $A_1$, $A_2$, $\alpha_1$, $\alpha_2$ are coefficients depending on $E$. Substitution of (4) into equation (3) gives
\[ \eta=\frac{\mu_a}{\mu}\left\{\frac{A_1}{1+\alpha_1}\left[1-e^{-(1+\alpha_1)\mu R}\right] +\frac{A_2}{1+\alpha_2}\left[1-e^{-(1+\alpha_2)\mu R}\right]\right\} \tag{5} \]
Fig. 1. Dependence of $B_j$ on $b$ and $E$
It is obvious that for $R\to\infty$, $\eta\to1$, and, consequently, the equality must hold
\[ \eta_{R\to\infty}=\frac{\mu_a}{\mu}\left[\frac{A_1}{1+\alpha_1}+\frac{A_2}{1+\alpha_2}\right]=1. \tag{5'} \]
Calculations show that when the coefficients $A_1$, $A_2$, $\alpha_1$, and $\alpha_2$ given in $^{(6)}$ are used, equation (5′) is not satisfied. In this connection, when determining the efficiency of the sphere by (5), it is necessary to use coefficients corrected according to equation (5′). Thus, in what follows, the efficiency of the sphere is calculated from equation (5), taking into account correction of the coefficients by (5′).
As shown in works $^{(4,5)}$, it is convenient to calculate the efficiency by the formula
\[ \eta=1-B_j j_{\mathrm{p}}, \tag{6} \]
where $B_j$ is the factor introduced in $^{(4,5)}$ for the build-up of the integral energy current of $\gamma$-radiation emerging from the surface of a macrosystem. By definition $^{(4,5)}$:
\[ B_j=1+j_{\mathrm{p}}/j_{\mathrm{п}}. \tag{7} \]
Let us consider the relation between $j_{\mathrm{п}}$ and $j_{\mathrm{p}}$, having calculated $B_j$ from equation (6) by substituting into (6) the values of the sphere efficiency obtained from equation (5), taking into account the correction by (5′). This calculation showed that in the energy interval $E=0.7$–$5.0$ MeV, for $b=\mu R=0\div4.0$, the values of $B_j$ (see Fig. 1) are described with an accuracy of approximately 5% by the equation
\[ B_j=1+k(E)b, \tag{8} \]
where it turned out that $k(E)=\mu_s/\mu$, where $\mu=\mu-\mu_a$. From (7) and (8) it follows that
\[ j_{\mathrm{p}}=\frac{\mu_s}{\mu}\,b j_{\mathrm{п}}, \tag{9} \]
i.e., the relative energy current of scattered radiation emerging from the surface of the sphere can be expressed as the product of three dimensionless criteria: $\mu_s/\mu$, $b$, and $j_{\mathrm{п}}$.
It follows from (6) and (8) that for a sphere
\[ \eta = 1-\left[1+\frac{\mu_s}{\mu}b\right]j_{\mathrm{p}} . \tag{10} \]
General course of calculating the integral absorbed dose in a macrosystem of arbitrary configuration. Suppose that for a macrosystem of specified shape and size one can find an equivalent sphere of radius \(R_{\mathrm{eff}}\), in which the quantities \(\eta\), \(j_{\mathrm{p}}\), and \(j_{\mathrm{r}}\) are equal to the corresponding quantities of the given macrosystem. Consequently, the calculation of the integral absorbed power and \(\eta\) in the irradiated macrosystem is reduced to the corresponding calculation of these quantities in a sphere, as described above.
The attenuation of the primary \(\gamma\)-radiation in an element of solid angle of the macrosystem \(d\Omega\) of thickness \(\rho_\Omega\) is equal to \(e^{-\mu \rho_\Omega}\), and over the whole macrosystem the following attenuation \(\bar{\xi}\) takes place:
\[ \bar{\xi}=\frac{1}{\Omega_k}\int_{\Omega_k} e^{-\mu \rho_\Omega}\,d\Omega, \tag{11} \]
where \(\Omega_k\) is the solid angle under which the surface of the irradiated system is seen from the location of the radiation source.
It can readily be shown that numerically \(\bar{\xi}=j_{\mathrm{p}}\). On the other hand, under our assumption concerning the properties of the equivalent sphere,
\[ \bar{\xi}=e^{-\mu R_{\mathrm{eff}}}. \]
Consequently,
\[ R_{\mathrm{eff}}=\frac{1}{\mu}\ln\frac{1}{j_{\mathrm{p}}}. \tag{12} \]
![Figure 2 graph]
Fig. 2. Dependence of \(\eta\) and \(j_{\mathrm{r}}\) on \(j_{\mathrm{p}}\) (or \(\mu R_{\mathrm{eff}}\)) for \(\gamma\)-radiation energies \(E=0.7;\ 1.25;\ 2.5\) MeV
Let us determine the quantity \(j_{\mathrm{r}}\). Since the propagation of \(\gamma\)-radiation in the case of a spherical object and a central position of an isotropic \(\gamma\)-source is completely symmetric with respect to the source, one may assume that the energy-current buildup factor \(B'_j\), determined for an elementary surface of the sphere in the elementary solid angle \(d\Omega\), is equal to \(B_j\). Consequently, by analogy with (9), one may assume that in the elementary solid angle of the macrosystem under consideration
\[ dj_{\mathrm{r}}=\frac{\mu_s}{\mu}(\mu\rho_\Omega)\,dj_{\mathrm{p}} \]
and over the whole macrosystem
\[ j_{\mathrm{r}}=\frac{\mu_s}{\mu}\frac{1}{\Omega_k}\int_{\Omega_k} e^{-\mu\rho_\Omega}(\mu\rho_\Omega)\,d\Omega. \tag{13} \]
Calculation of the quantity \(j_{\mathrm{r}}\) by formula (13) for a number of special cases shows that \(j_{\mathrm{r}}\), determined in this way, is equal to \(j_{\mathrm{r}}\) for a sphere with \(R_{\mathrm{eff}}\) determined by formula (12). Consequently, having computed for each specific case \(j_{\mathrm{p}}=\bar{\xi}\) by (14) and \(\mu R_{\mathrm{eff}}\) by (12), we find \(j_{\mathrm{r}}\) in the form
\[ j_{\mathrm{r}}=\frac{\mu_s}{\mu}(\mu R_{\mathrm{eff}})j_{\mathrm{p}}, \tag{14} \]
and equation (10) becomes
\[ \eta = 1-\left[1+\frac{\mu_s}{\mu}\left(\mu R_{\mathrm{eff}}\right)\right]j_p . \tag{15} \]
The expressions obtained, (12), (14), (15), make it possible to determine the efficiency and \(j_p\) as functions of the energy of the \(\gamma\)-radiation, of the absorbing-scattering properties of the medium, and of \(j_p\). The dependence of \(\eta\) and \(j_p\) on the quantity \(j_p\) (or \(\mu R_{\mathrm{eff}}\)) for various energies of the primary \(\gamma\)-radiation (\(E = 0.7;\ 1.25;\ 2.5\) MeV), calculated from formulas (12), (14), (15), is shown in Fig. 2. Consequently, in order to determine the efficiency of a macrosystem by the proposed method, it is necessary to find the value of \(j_p\).
Fig. 3. Comparison of the results of calculating the efficiency by the method proposed in the present work (curve) and by the Monte Carlo method (points).
\(E = 1.25\) MeV
The results of calculating the efficiency for various cases of macrosystems with a point source of \(\gamma\)-radiation were compared with the results of calculating the same quantities by the method of statistical trials (Monte Carlo method)*, which is essentially a numerical experiment \((^{7,8})\). Processing the results of the efficiency calculation by the Monte Carlo method as a function of \(j_p\) showed that these results agree well with the values of \(\eta\) obtained from equation (18) (see Fig. 3). The agreement confirms the correctness of the method proposed in the present work for calculating the i.a.p. and the efficiency for \(\gamma\)-radiation in macrosystems.
The author takes this opportunity to express gratitude to A. Kh. Breger, N. P. Syrkus, V. A. El’tekov, and B. M. Terent’ev for fruitful discussion of the present work, and to S. I. Berestetskaya for assistance in carrying out the numerical calculations and in preparing the paper.
Physico-Chemical Institute
named after L. Ya. Karpov
Received
9 IX 1963
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* The Monte Carlo calculations were carried out by B. M. Terent’ev and V. A. El’tekov.