THEORY OF MULTIPLE SCATTERING OF GAMMA RAYS

V. S. Galitshev, V. I. Ogievetskii, A. N. Orlov

CONTENTS

I. Introduction . . . 161
§ 1. Basic processes of the interaction of gamma radiation with matter . . . 162
§ 2. Number of quanta, intensity, and build-up factor . . . 166

II. Equation of radiation transfer . . . 167
§ 3. The equation of radiation transfer and the expansion of the photon density in Legendre polynomials . . . 167
§ 4. Energy spectrum and angular distribution of photons in an unbounded homogeneous medium . . . 169

III. Method of polynomial expansions. Results and comparison with experiment . . . 172
§ 5. Foundations of the method of polynomial expansions . . . 172
§ 6. Plane isotropic source . . . 172
§ 7. Point (isotropic) source . . . 178
§ 8. Numerical results and comparison with experiment . . . 179
1. Plane source . . . 179
2. Point source . . . 185

IV. Small-angle approximation. Energy spectrum of scattered gamma radiation at large penetration depths . . . 191
§ 9. Introductory remarks . . . 191
§ 10. Equation of radiation transfer in the approximation of small scattering angles . . . 192
§ 11. Solution of the radiation-transfer equation in the small-angle approximation . . . 193
§ 12. Evolution of the angular distribution with increasing penetration depth . . . 195
§ 13. State of limited radiation equilibrium . . . 196
§ 14. Energy spectrum of gamma radiation at large penetration depths in the small-angle approximation . . . 196
§ 15. Allowance for angular deviations . . . 197
§ 16. Semi-asymptotic method of Spencer . . . 200

V. Other approximate methods for calculating multiple scattering . . . 202
§ 17. Monte Carlo method . . . 202
§ 18. Approximate direct methods . . . 207
§ 19. Method of successive passage through thin layers . . . 212

Conclusion . . . 214
References . . . 214

I. INTRODUCTION

In the last decade, powerful sources of gamma rays have come into wide use in science and technology. Propagating in various absorbing media (air, concrete, water, lead, etc.), gamma rays undergo absorption and scattering; moreover, for a large absorber thickness the principal

a fraction of the intensity is due not to primary quanta, but to multiply scattered quanta. It is not difficult to show (Gubanov ^44) that, for an initial quantum energy of several MeV, the number of scattering events before the quantum is absorbed by the photoelectric effect may reach 10–15 in light absorbers. The attenuation of the intensity \(I\) of gamma rays with penetration depth \(x\) in an absorber in the presence of multiple scattering does not obey the simple exponential law \(I = I_0 e^{-\mu x}\). The attenuation depends in a complicated manner both on the energy spectrum of the incident radiation and on the properties and geometry of the absorber. In order to clarify this dependence, many experimental investigations and theoretical calculations have been carried out. The reviews published so far ^40, ^42, ^57 on the interaction of gamma radiation with matter consider only individual works on multiple scattering of gamma rays.

The aim of the present review is a systematic exposition of the methods published in the literature for the theoretical study and calculation of multiple scattering of gamma rays. Experimental data are given only for comparison with the results of theory *).

We shall confine ourselves to consideration of gamma quanta with energies from 0.05 to 10 MeV. Below the lower limit of this interval, the probability of absorption of a quantum is considerably greater than the probability of scattering, so that multiple scattering is practically excluded; at energies above several MeV, cascade processes become important, the study of which is the subject of cascade shower theory (see, for example, ^39). Moreover, the energies of gamma quanta from natural and artificial radioactive sources lie below this limit.

§ 1. Basic processes of interaction of gamma radiation with matter

In the present section the elementary processes of interaction of gamma quanta with the material of an absorber are briefly considered, and formulas and notation necessary for the subsequent exposition are given.

Multiple scattering of gamma rays occurs by means of successive elementary acts of interaction of quanta with matter, more precisely, with various elementary particles or fields. At present one can indicate a number of types of such elementary interaction processes, which are classified according to the kind of fields (particles) with which the gamma quantum interacts, and according to the changes that the quantum undergoes in the interaction. Interaction of gamma quanta is possible with 1) electrons, 2) nucleons, 3) the electromagnetic field of electrons and nucleons, 4) the meson field of nucleons. As a result of each of these processes there may occur: A) complete absorption of the quantum, B) inelastic scattering, C) elastic scattering.

Thus, in all, \(4 \times 3 = 12\) different interaction processes are possible. Their probability depends on the energy of the quanta and, for some processes, is very small. It is very significant that scattering of gamma quanta by one another belongs to the low-probability processes. This permits the use of the principle of superposition of intensities of gamma radiation.

In the energy region of interest to us the following processes most often occur: (1A) photoelectric absorption (photoelectric effect), (1B) incoherent scattering by an electron (Compton effect), (3A) formation of an electron–positron pair. Processes (1A) and (3A) lead to complete absorption of the quantum. It is true that, in the bremsstrahlung of the electrons and positrons produced in these processes, secondary gamma quanta may appear, but because of the small

*) For reference purposes, in the list of literature experimental works are separated out and indicated, as are the measurements given in them. Also separated out are those theoretical works in which specific numerical results were obtained.

... the probabilities of such cascade processes we shall regard the Compton effect as the sole source of scattered quanta.

Passing to a quantitative description of the probabilities of absorption and scattering processes, we shall henceforth agree to express energy in units of the electron rest energy \(m_e c^2 = 0.5108 \simeq 0.5\) MeV and, in this case, to denote it by \(\alpha\). In discussing experimental data, and for convenience of comparison with the original papers, the energy will sometimes be expressed in MeV and then denoted by \(E\), so that

\[ \alpha = E/m_e c^2 \simeq 2E . \]

We shall express the wavelength \(\lambda\) of gamma rays in Compton units \(\lambda_k = h/mc = 0.02427 \cdot 10^{-8}\ \mathrm{cm}^{*}\).

We now give expressions for the probabilities of the three processes listed above, which we shall need in what follows. These probabilities are determined by effective cross sections, derivations and summaries of which are given in many monographs and reviews\(^{41,56}\). Below are formulas for the effective cross sections \(\sigma\), calculated for an atom with atomic number \(Z\).

In the photoelectric effect a gamma photon is absorbed by an atomic electron, which is detached from the atom. The energy of the quantum is partly spent in overcoming the binding of the electron to the atom, while the greater part is converted into the kinetic energy of the electron. The principal share of the total photoelectric-effect cross section falls on the electrons of the \(K\)-shell. In the nonrelativistic region \((\alpha < 1)\), in the Born approximation this share is equal to

\[ \sigma_\phi = \sigma_0 \frac{4\sqrt{2}}{(137)^4} Z^5 \alpha^{-7/2}, \tag{1,1} \]

where

\[ \sigma_0 = \frac{8\pi}{3}\left(\frac{e^2}{mc^2}\right)^2 = \frac{8\pi}{3} r_0^2 = 6.6537 \cdot 10^{-25}\ \mathrm{cm}^2 \tag{1,2} \]

is the Thomson scattering cross section. In the extreme relativistic region \((\alpha \gg 1)\)

\[ \sigma_\phi = \sigma_0 \frac{3}{2(137)^4} Z^5 \alpha^{-1}. \tag{1,3} \]

In the intermediate energy region \((\alpha \simeq 1)\) the theory gives more complicated expressions. To take into account the photoelectric effect on electrons of other shells, it is customary to multiply \(\sigma_\phi\) by \(5/4\) (more precisely, see, for example,\(^{66}\)).

The Compton effect may be regarded as the collision of a gamma quantum with an atomic electron\(^{47}\). Since the binding energy of an electron in an atom is considerably smaller than the photon energy, the electron may be treated as free. Using the relativistic conservation laws for energy and momentum in the collision, it is not difficult to obtain the relation between the energy \(\alpha_0\) of the incident photon, the energy \(\alpha\) of the scattered photon, and the scattering angle \(\theta\):

\[ \alpha = \frac{\alpha_0}{1+\alpha_0(1-\cos\theta)} . \tag{1,4} \]

The energy \(\alpha_0-\alpha\) is transferred to the recoil electron. Since numerically \(\lambda = \alpha^{-1}\), it follows from (1,4) that

\[ \lambda = \lambda_0 + 1 - \cos\theta . \tag{1,5} \]

\[ \rule{4cm}{0.4pt} \]

\(*\) Sometimes \(\lambda_k\) is understood to mean a quantity \(2\pi\) times smaller.

The effective cross section for Compton scattering is determined by the Klein—Nishina—Tamm formula

\[ d\sigma_{\mathrm{k}}=Z\frac{r_0^2}{2}\left(\frac{\alpha}{\alpha_0}\right)^2 \left(\frac{\alpha}{\alpha_0}+\frac{\alpha_0}{\alpha}-1+\cos^2\theta\right)d\Omega, \tag{1,6} \]

where \(d\Omega=\sin\theta\,d\theta\,d\varphi\) is the elementary solid angle about the direction of emission of the scattered quantum. The product \(d\sigma_{\mathrm{k}}\) by the number of atoms in \(1\ \mathrm{cm}^3\), \(n\), is numerically equal to the probability \(dW_{\theta,\varphi}\) of scattering when the quantum traverses a path of \(1\ \mathrm{cm}\), and may be represented in the form

\[ dW_{\theta,\varphi}=\overline{K}(\cos\theta)\sin\theta\,d\theta\,d\varphi, \tag{1,7} \]

where

\[ \overline{K}(\cos\theta)=nZ\frac{r_0^2}{2} \left(\frac{\alpha}{\alpha_0}\right)^2 \left(\frac{\alpha}{\alpha_0}+\frac{\alpha_0}{\alpha}-1+\cos^2\theta\right) \tag{1,8} \]

is the probability density. Integrating (1,7) with respect to the azimuth \(\varphi\), we obtain the probability of scattering independent of the azimuth,

\[ dW_\theta=K(\cos\theta)\sin\theta\,d\theta, \tag{1,9} \]

where

\[ K=2\pi\overline{K}. \tag{1,10} \]

In a number of cases an expression is required for the probability of scattering not in a specified direction \(\theta,\varphi\), but with a specified change of energy from \(\alpha_0\) to \(\alpha\). Replacing \(\cos\theta\) in (1,9) according to (1,4), we obtain

\[ dW_\alpha=K\left(\frac{1}{\alpha_0},\frac{1}{\alpha}\right)\frac{d\alpha}{\alpha^2} =K'(\alpha_0,\alpha)\,d\alpha. \tag{1,11} \]

Next it is easy to find the probability of scattering with a change of wavelength from \(\lambda_0\) to \(\lambda\). For this purpose, in (1,9) one should replace \(\cos\theta\) by \(\lambda\) according to (1,5). Then

\[ dW_\lambda=K(\lambda_0,\lambda)\,d\lambda, \tag{1,12} \]

where

\[ K(\lambda_0,\lambda)=nZ\pi r_0^2 \left(\frac{\lambda_0}{\lambda}\right)^2 \left[ \frac{\lambda_0}{\lambda}+\frac{\lambda}{\lambda_0} -2(\lambda-\lambda_0)+(\lambda-\lambda_0)^2 \right]. \tag{1,13} \]

Finally, integrating \(d\sigma_{\mathrm{k}}\) over all angles, we obtain the total effective cross section of the Compton effect for a gamma quantum with energy \(\alpha\):

\[ \sigma_{\mathrm{k}}=\sigma_0 Z\frac{3}{8\alpha} \left\{ \left[1-\frac{2(\alpha+1)}{\alpha^2}\right]\ln(2\alpha+1) +\frac{1}{2}+\frac{4}{\alpha}-\frac{1}{2(2\alpha+1)^2} \right\}. \tag{1,14} \]

At energies \(\alpha>2\) a process of type (3A) is possible—the absorption of a photon in the field of a nucleus (much more rarely in the field of an electron) with the formation of a pair: an electron and a positron. In this process the photon energy is transformed into the rest energy of the electron and positron and into the kinetic energy of these particles. Simple analytic expressions for the effective cross section for pair production \(\sigma_{\mathrm{p}}\) exist only for the extreme relativistic region, which is not considered in our review. For \(\alpha<2\) the cross section \(\sigma_{\mathrm{p}}=0\) and thereafter increases monotonically with \(\alpha\). Near \(\alpha=2\), \(\sigma_{\mathrm{p}}\sim Z^2\).

Adding the effective cross sections (1,1), (1,14), and \(\sigma_{\mathrm{p}}\), we obtain the total cross section of the interaction of a gamma quantum with an atom of atomic number \(Z\),

\[ \sigma=\sigma_{\mathrm{f}}+\sigma_{\mathrm{k}}+\sigma_{\mathrm{p}}. \tag{1,15} \]

Multiplication of \(\sigma\) by the number of atoms in \(1\ \mathrm{cm}^3\), \(n\), gives the absorption coefficient

\[ \mu=\sigma\cdot n, \tag{1,16} \]

determining the familiar law of attenuation of the intensity \(I\) of a narrow beam of gamma quanta with depth of penetration:

\[ I=I_0 e^{-\mu x}, \tag{1.17} \]

where \(I_0\) is the value of \(I\) at \(x=0\). The quantity \(\mu^{-1}=l\) is called the mean free path. Coordinates measured in mean free paths will be denoted by \(X=x/l\), etc.

In accordance with (1.15), \(\mu\) may be represented as the sum

\[ \mu=\mu_{\phi}+\mu_k+\mu_{\pi}. \tag{1.18} \]

The dependence of \(\mu\) on \(\alpha\) is shown for several elements in Fig. 1; moreover, for lead the separate terms of \(\mu\) are plotted with dashed lines. As is seen, for photons of low energy the photoeffect is of greatest importance; in the region

Fig. 1. Absorption coefficient \(\mu\) (\(\mathrm{cm}^{-1}\)) for gamma rays in lead, tin, copper, and aluminum as a function of energy.

of the minimum of the curve \(\mu(\alpha)\), attenuation of the beam occurs mainly at the expense of the Compton effect, while at still higher energies pair production predominates.

Theoretical values of \(\mu\) are in good agreement with experimental data. Especially accurate is the formula for Compton scattering (1.6), which plays the principal role in the theory of multiple scattering.

The relative numbers of quanta knocked out of the incident monochromatic beam as a result of the various interactions are respectively equal to \(\sigma_{\phi}/\sigma\), \(\sigma_k/\sigma\), and \(\sigma_{\pi}/\sigma\). These same ratios determine the relative fractions of absorbed energy. The energy of quanta absorbed by the photoeffect and by pair production is converted mainly into the kinetic energy of electrons and positrons. In the Compton effect, the energy of the incident quantum \(\alpha\) is transferred partly to the scattered quantum \((\alpha\sigma_p/\sigma)\), and partly to the recoil electron \((\alpha\sigma_s/\sigma)\). At the not very high energies considered by us, the electrons formed in all three processes rapidly lose the kinetic energy they have acquired, chiefly through ionization of the atoms of the absorber. The portion of the energy transferred to electrons is absorbed practically at the same place where the gamma quantum was absorbed. Unlike this energy, the energy transferred to the scattered photon is not absorbed near the scattering atom, but passes into the energy of the flux of scattered radiation.

To compute this energy, it is sufficient to multiply \(d\sigma_{\mathrm{k}}\) by \(a(\cos\theta)/a_0\) and integrate over all angles. Then we obtain

\[ \sigma_{\mathrm{p}}= Z\,\frac{3}{8}\sigma_0 \left[ \frac{1}{a^3}\ln(1+2a) + \frac{2(1+a)(2a^2-2a-1)}{a^2(1+2a)^2} + \frac{8a^2}{3(1+2a)^3} \right]. \tag{1,19} \]

Obviously,

\[ \sigma_{\mathrm{e}}=\sigma_{\mathrm{k}}-\sigma_{\mathrm{p}}. \tag{1,20} \]

Thus, the mean energy absorbed as a result of all interactions is equal to

\[ a\varepsilon_{\mathrm{p}} = \frac{a(\sigma_{\mathrm{f}}+\sigma_{\mathrm{e}}+\sigma_{\mathrm{p}})}{\sigma}, \tag{1,21} \]

and the scattered energy is

\[ a\varepsilon_{\mathrm{p}}=\frac{a\sigma_{\mathrm{p}}}{\sigma}. \tag{1,22} \]

§ 2. Number of quanta, intensity, and build-up factor

In the present section definitions are given for a number of quantities characterizing gamma radiation, and the relations between these quantities.

Let \(N(\mathbf r,\mathbf u,\lambda)\,du\,d\lambda\) denote the number of photons with wavelength in the interval from \(\lambda\) to \(\lambda+d\lambda\), moving in the direction of the unit vector \(\mathbf u\) in the solid-angle element \(du\), and crossing in 1 sec. \(1\ \mathrm{cm}^2\) of a surface perpendicular to the vector \(\mathbf u\), with center at the point \(\mathbf r\). The photon flux density, or, briefly, the photon density \(N(\mathbf r,\mathbf u,\lambda)\), will enter into all subsequent calculations. In comparing the theory with experiment it is usually necessary to know the energy flux density of photons \(I(\mathbf r,\mathbf u,E)\). This quantity can readily be expressed in terms of the function \(N(\mathbf r,\mathbf u,\lambda)\). For this it is sufficient to take into account that

\[ N(\mathbf r,\mathbf u,\lambda)\,du\,d\lambda = - N(\mathbf r,\mathbf u,E)\,du\,\frac{mc^2}{E^2}\,dE = -\bar N(\mathbf r,\mathbf u,E)\,du\,dE, \]

where \(\bar N(\mathbf r,\mathbf u,E)\) is the photon density per energy interval of \(1\ \mathrm{MeV}\). Then the energy flux density in MeV corresponding to this photon density \(\bar N\), by definition, is equal to

\[ I(\mathbf r,\mathbf u,E)=E\cdot \bar N(\mathbf r,\mathbf u,E). \tag{2,1} \]

This quantity is also called the intensity of gamma radiation*).

The argument \(E\) in (2,1) may be replaced by \(a\). Then

\[ I(\mathbf r,\mathbf u,a)=a\bar N(\mathbf r,\mathbf u,a). \tag{2,2} \]

In a number of problems the number of arguments of the functions \(N\), \(\bar N\), and \(I\) is reduced by considerations of symmetry. Thus, in the one-dimensional case (for example, an infinitely wide plane-parallel beam incident normally on a flat absorber perpendicular to the \(x\)-axis), \(I(\mathbf r,\mathbf u,E)=I(x,u_x,E)\). Integration of this quantity with respect to the angular variable gives \(I_0(x,E)\)—the spectral distribution of the intensity at a distance \(x\) from the source, independently of their angular distribution. Integration of \(I\) with respect to the variable \(E\) gives \(I_A(x,u_x)\)—the angular distribution of the intensity at a distance \(x\) from the source. Finally, integration of \(I\) simultaneously over angles and energies gives \(I_{\mathrm{t}}(x)\)—the total inten-

* The quantities \(N\) and \(I\) are sometimes also called the “differential spectra” of photons and of intensity, respectively.

sity of radiation. In the case of a point isotropic source one should set \(I(\mathbf r,\mathbf u,E)=I(r,u_r,E)\), where \(r=|\mathbf r|\), and \(u_r=\left(\mathbf u\,\dfrac{\mathbf r}{r}\right)\). Integration of the function \(I(r,u_r,E)\) with respect to \(u_r\), \(E\), or \(u_r\) and \(E\) simultaneously gives \(I_0(r,E)\), \(I_A(r,u_r)\), and \(I_\tau(r)\), respectively. In experimental work the task usually posed is that of determining one of the quantities \(I\), \(I_0\), \(I_A\), or \(I_\tau\). The results of measurements are often expressed in terms of the so-called build-up factor, defined by the relation

\[ B(x)=\frac{I_\tau}{I^{(0)}}=\frac{I^{(0)}+I'}{I^{(0)}}= \frac{\text{total energy flux (unscattered + scattered)}}{\text{unscattered energy flux}}, \tag{2,3} \]

which shows how, with increasing distance from the source, the fraction of scattered radiation in the total flux increases. Similarly to \(I_A\), \(I_0\), and \(I_\tau\), by integrating \(N(x,u_x,E)\) or \(N(r,u_r,E)\) one can determine the quantities \(N_A\), \(N_0\), \(N_\tau\), and also the build-up factor \(B_N\) in terms of the number of photons.

Integration of \(N(\mathbf r,\mathbf u,\lambda)\) over all space and over all angles gives the so-called equilibrium spectrum

\[ \mathfrak N(\lambda)=\int N(\mathbf r,\mathbf u,\lambda)\,d\mathbf r\,d\mathbf u. \tag{2,4} \]

In some works the results of measurements are described by means of the transmission, which is the product of the build-up factor by \(\exp(-x/\cos\psi_0)\) (\(\psi_0\) is the angle between the direction of the incident radiation and the normal to the plane of the absorber).

To estimate the effect exerted by gamma rays on an absorber, it is important to know not the amount of energy that has passed through it, but the amount of absorbed energy. The amount of gamma-ray energy absorbed in \(1\ \text{cm}^3\) is called the dose, and the dose received by the absorber in 1 sec. is called the dose rate. According to (1,21) and (2,2), it is equal to

\[ M=aN\bar\varepsilon_{\mathrm n}ns=I(\sigma_{\mathrm n}+\sigma_\phi+\sigma_s)n. \tag{2,5} \]

II. THE RADIATION TRANSPORT EQUATION

§ 3. The radiation transport equation and expansion of the photon density in Legendre polynomials

A rigorous theory of the passage of gamma quanta through matter is based on the study of the integro-differential transport equation. This equation describes the change in the photon density \(N(\mathbf r,\mathbf u,\lambda)\) in passing from one point to another along the direction of propagation of the quanta and, in the case of a homogeneous medium, has the form

\[ \mathbf u\,\operatorname{grad} N(\mathbf r,\mathbf u,\lambda) = -\mu(\lambda)N(\mathbf r,\mathbf u,\lambda) + \]

\[ + \int_{\lambda-2}^{\lambda} d\lambda' K(\lambda',\lambda) \int d\mathbf u'\,\frac{1}{2\pi}\, \delta(1-\mathbf u\mathbf u'-\lambda+\lambda')\, N(\mathbf r,\mathbf u',\lambda') + S(\mathbf r,\mathbf u,\lambda). \tag{3,1} \]

On the left stands the change of \(N(\mathbf r,\mathbf u,\lambda)\); the terms on the right-hand side represent: a) a decrease in photon density as a result of absorption and scattering, b) an increase in photon density due to Compton scattering of photons with wavelength \(\lambda'\) \((\lambda-2\leq \lambda'\leq \lambda)\), c) an increase in photon density due to

source of radiation. The delta function ensures fulfillment of the scattering law (1.5), the factor \((2\pi)^{-1}\) is introduced for normalization reasons, \(K(\lambda',\lambda)\) is determined by formula (1.13), and the form of the function \(S(\mathbf r,\mathbf u,\lambda)\) depends on the geometrical dimensions, the spectral composition, and the angular distribution of the source. In those cases when the source emits quanta with wavelength \(\lambda_0 > \lambda - 2\), the lower limit of the integral should be replaced by \(\lambda_0\).

Let us begin the consideration of the transfer equation with the important special case of a source located on the infinite plane \(x=0\) and emitting quanta of wavelength \(\lambda=\lambda_0\). The distribution of the intensity of gamma rays from such a source depends only on one coordinate \(x\).

The function \(S(\mathbf r,\mathbf u,\lambda)\) in this case has the form

\[ S(\mathbf r,\mathbf u,\lambda)=\delta(x)f(u_x)\delta(\lambda-\lambda_0), \tag{3.2} \]

where \(f(u_x)\) describes the angular distribution of the emitted quanta. In particular, for an isotropic source \(f=\mathrm{const}\), and for a directed one, i.e., one giving radiation in the form of an infinitely wide parallel beam with a specified angle \(\psi_0=\arccos u_{x0}\) to the normal to the plane \(x=0\), \(f(u_x)=\mathrm{const}\cdot\delta(u_x-u_{x0})\). The function \(f(u_x)\) is assumed normalized so that

\[ \iint d\mathbf r\,d\mathbf u\,S(\mathbf r,\mathbf u,\lambda)=\delta(\lambda-\lambda_0). \tag{3.3} \]

Therefore, for a plane directed oblique source

\[ S(\mathbf r,\mathbf u,\lambda)=\frac{1}{2\pi}\delta(u_x-u_{x0})\delta(x)\delta(\lambda-\lambda_0), \tag{3.4} \]

and for a plane isotropic source

\[ S(\mathbf r,\mathbf u,\lambda)=\frac{1}{4\pi}\delta(x)\delta(\lambda-\lambda_0). \tag{3.5} \]

The radiation transfer equation (3.1) for a plane source (3.2) is now written as follows:

\[ u_x\frac{\partial N(x,u_x,\lambda)}{\partial x} = -\mu(\lambda)N(x,u_x,\lambda)+ \]

\[ +\int_{\lambda-2}^{\lambda}d\lambda'K(\lambda',\lambda) \int d\mathbf u'\,(2\pi)^{-1} \delta(1-\mathbf u\mathbf u'-\lambda+\lambda') N(x,u_x',\lambda') + \]

\[ +\delta(x)\delta(\lambda-\lambda_0)f(u_x). \tag{3.6} \]

For further applications it is expedient to transform this equation to another form. To this end we separate the dependence of \(N(x,u_x,\lambda)\) on \(u_x\) by expanding in Legendre polynomials

\[ N(x,u_x,\lambda)=\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}N_l(x,\lambda)P_l(u_x), \tag{3.7} \]

and replace the delta function according to the formula

\[ \delta(1-\mathbf u\mathbf u'-\lambda+\lambda') = \sum_{l=0}^{\infty}\frac{2l+1}{2} P_l(1-\lambda+\lambda')P_l(\mathbf u\mathbf u'). \tag{3.8} \]

Substitute (3.7) and (3.8) into equation (3.6), multiply it by \(P_l(u_x)\), and integrate with respect to \(\mathbf u\). In doing so one takes into account: 1) in transforming the left-hand side, the recurrence formula for Legendre polynomials

\[ u_x P_l(u_x)=\frac{1}{2l+1}\bigl[(l+1)P_{l+1}(u_x)+lP_{l-1}(u_x)\bigr], \tag{3.9} \]

2) in transforming the second term on the right, the addition theorem for Legendre polynomials

\[ P_l(\mathbf u\mathbf u')=P_l(u_x)P_l(u'_x)+2\sum_{n=1}^{l}\frac{(l-n)!}{(l+n)!}\, P_l^n(u_x)P_l^n(u'_x)\cos n(\varphi-\varphi'). \tag{3.10} \]

The final result of the integration has the form

\[ \frac{1}{2l+1}\left[(l+1)\frac{\partial N_{l+1}}{\partial x} +l\frac{\partial N_{l-1}}{\partial x}\right] = -\mu(\lambda)N_l(x,\lambda)+ \]

\[ +\int_{\lambda-2}^{\lambda} d\lambda'\,K(\lambda',\lambda)P_l(1-\lambda+\lambda')N_l(x,\lambda') +\delta(x)\delta(\lambda-\lambda_0)\Lambda_l \tag{3.11} \]

\[ (l=0,1,2,\ldots), \]

where

\[ \Lambda_l=\int d\mathbf u\,P_l(u_x)f(u_x). \tag{3.12} \]

In particular, if \(f(u_x)=\frac{1}{2\pi}\delta(1-u_x)\) (cf. (3.4)), then \(\Lambda_l=1\); and if \(f(u_x)=(4\pi)^{-1}\) (see (3.5)), then \(\Lambda_l=\delta_{l0}\).

Thus, solving equation (3.6) is equivalent to solving the infinite system of equations (3.11) for the coefficients of the expansion of the photon density \(N\) in Legendre polynomials.

§ 4. Energy spectrum and angular distribution of photons in an unbounded homogeneous medium

In the case where the sources are distributed uniformly in an unbounded homogeneous medium, the function \(N\) clearly does not depend on \(\mathbf r\) and \(\mathbf u\). Then the equilibrium spectrum \(\mathfrak N(\lambda)\), defined by (2.4), coincides with \(N\) up to a factor.

The equation determining the equilibrium spectrum is obtained from the transport equation (3.1) if the integrations indicated in (2.4) are carried out in it and the delta function is separated from the solution (cf. the transformation of equation (35) on p. 194 in \({}^{65}\)). In this case, for \(\lambda_0>\lambda-2\) we shall have

\[ \mu(\lambda)\mathfrak N(\lambda) = \int_{\lambda_0}^{\lambda} d\lambda'\,K(\lambda',\lambda)\mathfrak N(\lambda') +K(\lambda_0,\lambda). \tag{4.1} \]

In equation (4.1) one can pass from wavelengths to energy, if one uses the relations (1.11) and

\[ \mathfrak N(\lambda)d\lambda = -\frac{\mathfrak N(\alpha)\,d\alpha}{\alpha^2} = -\overline{\mathfrak N}(\alpha)d\alpha. \tag{4.2} \]

Then, instead of (4.1), we obtain

\[ \mu(\alpha)\,\overline{\mathfrak N}(\alpha) = \int_{\alpha}^{\alpha_0} d\alpha' K'(\alpha',\alpha)\,\overline{\mathfrak N}(\alpha') + K'(\alpha_0,\alpha). \tag{4.3} \]

The solution of equation (4.3) was obtained by Karp and Lamkin\(^{64}\) by numerical integration. Their results are presented in Fig. 2 for the equilibrium photon density \(\overline{\mathfrak N}\) (calculated per \(1\) MeV and expressed in units of the source density) for aluminum, copper, and lead at the initial energy \(\alpha_0=10\). The increase of \(\overline{\mathfrak N}\) with decreasing energy is due to the increase in the number of quanta as a result of multiple scattering. With a further decrease in energy, the sharp increase in the photon absorption coefficient due to the photoelectric effect begins to make itself felt, which causes \(\overline{\mathfrak N}\) to decrease.

Fig. 2. Equilibrium spectrum of photons with \(E_0=10\,mc^2\) as a function of energy for aluminum, copper, and lead.

The curves in Fig. 2 have discontinuities at the energy equal to the minimum energy of a singly scattered quantum,

\[ \alpha=\alpha_1=\frac{\alpha_0}{2\alpha_0+1}=\frac{10}{21}\;(0.24\ \text{MeV}). \]

This is connected with the fact that the second term on the right-hand side of equation (4.3), which determines single scattering of source quanta, is different from zero only for \(\alpha\) varying from \(\alpha_0\) to \(\alpha_1\), and is equal to zero for \(\alpha<\alpha_1\). As the atomic number of the element decreases, the maximum of the function \(\overline{\mathfrak N}\) shifts into the region of lower energies.

In order, in addition to the spectral distribution of the scattered quanta, to determine also their angular distribution, the initial equation (3.1) should be integrated only over the spatial coordinates. Then, instead of (4.3), for the distribution function \(\overline{\mathfrak N}(u_x,\alpha)\) we obtain the equation\(^{67}\)

\[ \mu(\alpha)\,\overline{\mathfrak N}(u_x,\alpha) = \int_{\alpha}^{\alpha_0} d\alpha' K'(\alpha',\alpha) \int du'\,\frac{1}{2\pi}\, \delta\!\left(\mathbf{u}\mathbf{u}'-1+\frac{1}{\alpha}-\frac{1}{\alpha'}\right) \overline{\mathfrak N}(u'_x,\alpha') + \]

\[ {}+ K'(\alpha_0,\alpha)\, \delta\!\left(u_x-1+\frac{1}{\alpha}-\frac{1}{\alpha_0}\right), \tag{4.4} \]

where \(K'\) is defined by (11.1).

Expand \(\overline{\mathfrak N}(u_x,\alpha)\) and the delta functions in Legendre polynomials (cf. § 3). Then multiplying equation (4.4) by \(P_l(u_x)\) and integrating over \(\mathbf u\), it is easy to show that,

that the expansion coefficients \(\overline{\mathfrak N}_l(\alpha)\) of the function \(\overline{\mathfrak N}(u_x,\alpha)\) in Legendre polynomials satisfy the equation

\[ \mu(\alpha)\overline{\mathfrak N}_l(\alpha)= \int_{\alpha}^{\alpha_0} K'(\alpha',\alpha)P_l \left(1-\frac{1}{\alpha}+\frac{1}{\alpha'}\right) \overline{\mathfrak N}_l(\alpha')\,d\alpha' + K'(\alpha_0,\alpha)P_l\left(1-\frac{1}{\alpha}+\frac{1}{\alpha_0}\right). \tag{4.5} \]

In particular, for \(l=0\) equation (4.5) coincides with equation (4.3).

Equation (4.5) was solved \({}^{67}\) by the method of successive approximations for aluminum and an initial energy \(\alpha_0=10\). The density of singly scattered quanta was taken as the zeroth approximation, the density of doubly scattered quanta as the first approximation, and so on. The results are presented in Figs. 3 and 4. In these graphs there are discontinuities of the delta-function type (shown by vertical solid lines) and of the “step” type.

Fig. 3. Equilibrium angular distribution of photons in aluminum. Energy of the scattered photons \(>0.243\) MeV.

Discontinuities of the first kind are associated with singly scattered quanta and are due to the fact that, for given values of the initial energy \(\alpha_0\) and the energy after scattering \(\alpha_1\), all singly scattered quanta are deflected by one and the same angle \(\theta_1\), and, consequently, the distribution function of these quanta \(N^{(0)}(u_x,\alpha)\), which is part of \(N(u_x,\alpha)\), becomes infinite at the point \((\alpha_1,\theta_1)\). The case \(\alpha=\alpha_{180^\circ}=0.476\) corresponds to single scattering through an angle of \(180^\circ\). For \(\alpha<\alpha_{180^\circ}\) single scattering is altogether absent (Fig. 4).

Discontinuities of the second kind are associated with doubly scattered quanta and appear when the angle \(\theta\) is equal to the maximum value of the photon deflection angle as a result of double scattering, \(\theta_m\). It is not difficult to see that

\[ \cos\theta_m = 1-2\left(\frac{1}{\alpha}-\frac{1}{\alpha_0}\right) + \frac{1}{2}\left(\frac{1}{\alpha}-\frac{1}{\alpha_0}\right)^2 . \]

Fig. 4. Equilibrium angular distribution of photons in aluminum. Energy of the scattered photons \(\le 0.243\) MeV.

Putting \(\alpha_0=10\), and taking \(\alpha\) successively equal to the energy values indicated on the curves in Figs. 3 and 4, it is easy to note that, as the energy of the scattered quanta decreases down to \(\alpha=0.476\), the value of the angle \(\theta_m\) increases (Fig. 3), while with a further lowering of the energy it decreases (Fig. 4).

It is also seen from Figs. 3 and 4 that, as the energy decreases, the number of photons scattered through an angle greater than the maximum angle \(\theta_m\) in double scattering increases. In other words, the role of higher orders of scattering increases as the energy decreases.

III. METHOD OF POLYNOMIAL EXPANSIONS. RESULTS AND COMPARISON WITH EXPERIMENT

§ 5. Fundamentals of the method of polynomial expansions

The solution of the integro-differential equation (3.1) in most even simple problems is associated with serious mathematical difficulties. At the same time, the form of the solution can often be predicted in general outline (exponential decrease with penetration depth \(x\)). We shall denote it by \(w(x)\). In such cases it is advantageous to use the method of polynomial expansions, in which \(w(x)\) is taken as the weight function. Let us explain the basic principles of the method of polynomial expansions using as an example a distribution function \(N(x)\) depending on a single variable \(x\). Suppose that \(x\) varies from \(-\infty\) to \(+\infty\), and that the density \(N(x)\) vanishes at both ends of this region. To obtain a polynomial expansion of the density, a weight function \(w(x)\) is chosen such that \(w(\infty)=w(-\infty)=0\). Then a sequence of polynomials \(p_0(x), p_1(x), \ldots\) is chosen, which has an associated sequence of polynomials \(p_0^{+}(x), p_1^{+}(x), \ldots\), satisfying the condition

\[ \int_{-\infty}^{\infty} w(x)p_{n'}^{+}(x)p_n(x)\,dx=\delta_{nn'} . \tag{5.1} \]

Then

\[ N(x)=w(x)\sum_{n=0}^{\infty} a_n p_n(x), \tag{5.2} \]

and the coefficients \(a_n\), on the basis of (5.1) and (5.2), are given by the formula

\[ a_n=\int_{-\infty}^{\infty} p_n^{+}(x)N(x)\,dx . \tag{5.3} \]

In practice, the determination of the coefficients \(a_n\) reduces to the solution of a system of integral equations that are simpler than (3.1).

When this method is applied to finding the density of gamma quanta, it is necessary to take into account the geometrical features of the particular problem and, in accordance with them, to choose the weight function so that it describes as well as possible the assumed form \(N(\mathbf r,\mathbf u,\lambda)\). This will ensure rapid convergence of the series (5.2).

§ 6. Plane isotropic source

In this case the radiation source is the infinite plane \(x=0\), all points of which radiate isotropically.

On the basis of (3.11), the radiation-transport equation reduces to the system

\[ \frac{1}{2l+1}\left[(l+1)\frac{\partial N_{l+1}}{\partial x} +l\frac{\partial N_{l-1}}{\partial x}\right] = -\mu(\lambda)N_l(x,\lambda)+ \]

\[ +\int_{\lambda-2}^{\lambda} d\lambda'\,K(\lambda',\lambda)P_l(1-\lambda+\lambda')N_l(x,\lambda') +\delta(x)\delta(\lambda-\lambda_0)\delta_{l0} \tag{6.1} \]

\[ (l=0,1,2,\ldots). \]

Proceeding to solve equations (6.1), let us assume that the source is surrounded on both sides by an infinitely extended medium. Then the density \(N\) is symmetric with respect to the plane of the source:

\[ N(x,u_x,\lambda)=N(-x,-u_x,\lambda). \tag{6.2} \]

The coefficients of the expansions in Legendre polynomials of the left- and right-hand sides of expression (6.2) are related to one another by

\[ N_l(x,\lambda)=(-1)^l N_l(-x,\lambda). \tag{6.3} \]

From this, in particular, there follows an important consequence for the moments of the photon density, defined by the formula

\[ b_{ln}(\lambda)=\int_{-\infty}^{\infty} y^n N_l(y,\lambda)\,dy. \tag{6.4} \]

If \(l\) is even, then, according to (6.3), \(N_l(x,\lambda)\) will be an even function of \(x\), and hence all odd moments will be equal to zero, i.e.

\[ b_{l1}=b_{l3}=\ldots=0 \quad (l \text{ even}). \tag{6.5} \]

Similarly, for odd \(l\),

\[ b_{l0}=b_{l2}=\ldots=0 \quad (l \text{ odd}). \tag{6.6} \]

Consequently, in the case of a plane isotropic source, every other moment vanishes and cannot be used for determining the density \(N\).

For expanding the function \(N_l(x,\lambda)\) in this case, Spencer and Fano\(^{18}\) proposed introducing new polynomials. The weight function is chosen with allowance for the symmetry conditions of the distribution function (6.3). For even \(l\), the weight function is taken to be \(\frac12 e^{-\beta|x|}\); for odd \(l\), \(\frac12 \beta x e^{-\beta|x|}\). Knowing the weight functions, we can write (cf. (5.2))

\[ N_l(x,\lambda)=\frac12 e^{-\beta|x|} \sum_{n=0}^{\infty} a_{ln}(\lambda)U_n(\beta|x|) \quad (l \text{ even}), \tag{6.7} \]

\[ N_l(x,\lambda)=\frac12 \beta x e^{-\beta|x|} \sum_{n=0}^{\infty} a_{ln}(\lambda)V_n(\beta|x|) \quad (l \text{ odd}), \tag{6.8} \]

where \(U_n\) and \(V_n\) are polynomials of degree \(n\), to whose definition we now pass.

It is desirable to choose \(U_n\) and \(V_n\) so that a small number of the first terms of the expansions (6.7) and (6.8) already gives a sufficiently good approximation for \(N_l(x,\lambda)\). In other words, it is necessary to require that, beginning with some \(n'\), the moments of the function \(N_l(x,\lambda)\) of order \(n>n'\) be equal to zero, i.e., for even \(l\),

\[ \int_{-\infty}^{\infty} x^{2n'} e^{-\beta|x|}U_n(\beta|x|)\,dx=0 \quad (n>n'). \tag{6.9} \]

a: for odd \(l\).

\[ \int_{-\infty}^{\infty} x^{2n' + 1}\beta x e^{-\beta |x|} W_n(\beta |x|)\,dx = 0 \qquad (n>n'). \tag{6,10} \]

For brevity of notation it is convenient to introduce into relations (6,9) and (6,10) the variable \(y=\beta x\) and, in addition, to transform them to the form

\[ \int_0^\infty e^{-y} y^{2n'} U_n(y)\,dy = 0 \qquad (n>n') \tag{6,11} \]

and

\[ \int_0^\infty y e^{-y} y^{2n'} V_n(y)\,dy = 0 \qquad (n>n'). \tag{6,12} \]

Hence the polynomials \(U_n\) and \(V_n\) are determined up to arbitrary constants if they are sought in the form

\[ U_n(y)=\sum_{l=0}^{n} a_l y^l, \]

\[ V(y)=\sum_{l=0}^{n} b_l y^l. \]

This is easy to verify by putting successively \(n=0, 1, 2, \ldots\). Then

\[ U_0(y)=a_0,\quad U_1(y)=-a_1(1-y),\quad U_2(y)=a_2(3-5y+y^2),\ldots, \]

\[ V_0(y)=b_0,\quad V_1(y)=-b_1(3-y),\quad V_2(y)=b_2(15-9y+y^2),\ldots . \]

Let us consider the associated polynomials \(U_{n'}^{+}\) and \(V_{n'}^{+}\). According to (5,1), they are defined as follows:

\[ \int_0^\infty e^{-y} U_{n'}^{+}(y) U_n(y)\,dy = 0 \qquad (n<n'), \tag{6,13} \]

\[ \int_0^\infty y e^{-y} V_{n'}^{+}(y) V_n(y)\,dy = 0 \qquad (n<n'). \tag{6,14} \]

In order not to fall into contradiction with the preceding exposition when replacing \(n'\leftrightarrows n\), we must regard the polynomials \(U_{n'}^{+}\) as composed only of even powers of \(y\), and the polynomials \(V_{n'}^{+}\) of odd powers of \(y\), that is,

\[ U_{n'}^{+}(y)=\sum_{l=0}^{n'} c_l y^{2l},\qquad V_{n'}^{+}(y)=\sum_{l=0}^{n'} d_l y^{2l+1}. \]

In that case, with the aid of relations (6,13) and (6,14), one can obtain explicit expressions for the associated polynomials, but again only up to arbitrary ...

arbitrary constants. Indeed, setting \(n'=0, 1, 2, \ldots\), we find

\[ U_0^+(y)=c_0,\qquad U_1^+(y)=c_0\left(1-\frac{1}{2!}y^2\right),\qquad U_2^+(y)=c_0\left(1-y^2+\frac{1}{4!}y^4\right),\ldots, \]

\[ V_0^+(y)=e_0y,\qquad V_1^+(y)=e_1\left(2y-\frac{1}{3!}y^3\right), \]

\[ V_2^+(y)=e_2\left(3y-\frac{1}{2}y^3+\frac{1}{5!}y^5\right),\ldots, \]

where

\[ e_0=d_0,\qquad e_1=\frac{1}{2}d_0,\qquad e_2=\frac{1}{3}d_0,\ldots \]

It remains to determine the arbitrary constants \(a_n\), \(b_n\), \(c_0\), and \(d_0\). For this purpose we use the normalization conditions

\[ \left. \begin{aligned} \int_0^\infty e^{-y}U_n^+U_n\,dy&=1,\\ \int_0^\infty ye^{-y}V_n^+V_n\,dy&=1. \end{aligned} \right\} \tag{6,15} \]

Since they contain polynomials pairwise with their adjoints, we may take \(c_0\) and \(e_n\) arbitrary in the adjoint polynomials, and determine \(a_n\) and \(b_n\) from (6,15). Setting \(c_0=1\), we obtain

\[ U_0=1,\qquad U_1=\frac{1}{2}(1-y),\qquad U_2=\frac{1}{8}(3-5y+y^2),\ldots; \]

\[ U_0^+=1,\qquad U_1^+=1-\frac{1}{2}y^2,\qquad U_2^+=1-y^2+\frac{1}{24}y^4,\ldots \]

Next, setting \(e_n=1\), we find

\[ V_0=\frac{1}{2},\qquad V_1=\frac{1}{8}(3-y),\qquad V_2=\frac{1}{48}(15-9y+y^2),\ldots, \]

\[ V_0^+=y,\qquad V_1^+=2y-\frac{1}{3!}y^3,\qquad V_2^+=3y-\frac{1}{2}y^3+\frac{1}{5!}y^5,\ldots \]

In the general case

\[ U_n(y)=\frac{(-1)^n}{2^n}\left(\frac{\partial}{\partial y}-1\right)^{2n} \sum_{j=0}^{n} C_{n+j}^{j}\,\frac{2^{-j}y^{\,n-j}}{(n-j)!}, \tag{6,16} \]

\[ V_n(y)=\frac{1}{2(n+1)}\left(\frac{\partial}{\partial y}-1\right)^2 U_n(y), \tag{6,17} \]

\[ U_{n'}^+(y)=\sum_{j=0}^{n'} C_{n'}^{j}\frac{(-y^2)^j}{(2j)!}, \tag{6,18} \]

\[ V_{n'}^+(y)=-\sum_{j=1}^{n'} C_{n'+1}^{j}\frac{(-y^2)^j}{(2j-1)!\,y}. \tag{6,19} \]

Finally, one should indicate a simpler, but completely equivalent, notation for formulas (6.11)—(6.15) (cf. (5.1)):

\[ \int_0^\infty e^{-y} U_n^+ U_{n'}\,dy=\delta_{nn'}, \tag{6.20} \]

\[ \int_0^\infty y e^{-y} V_n^+ V_{n'}\,dy=\delta_{nn'}, \tag{6.21} \]

from which it follows (cf. (5.3)):

\[ a_{ln}(\lambda)=\int_{-\infty}^{\infty} U_n^+(\beta x)N_l(x,\lambda)\beta\,dx= \]

\[ =\sum_{j=0}^{n} C_n^j\frac{(-1)^j}{(2j)!}\,b_{l,2j}(\lambda) \qquad (l\ \text{even}), \tag{6.22} \]

\[ a_{ln}(\lambda)=\int_{-\infty}^{\infty} V_n^+(\beta x)N_l(x,\lambda)\beta\,dx= \]

\[ =\sum_{j=1}^{n+1} C_{n+1}^j\frac{(-1)^{j-1}}{(2j-1)!}\,b_{l,2j-1}(\lambda) \qquad (l\ \text{odd}), \tag{6.23} \]

where \(b_{ln'}\) is the \(n'\)-th spatial moment of \(N_l(x,\lambda)\) (see formula (6.4)).

It remains for us to find equations for the coefficients \(a_{ln}\) in (6.7) and (6.8). To do this, we multiply equation (6.1) by \(\beta U_n^+(\beta x)\), if \(l\) is even, or by \(\beta V_n^+(\beta x)\), if \(l\) is odd, and integrate with respect to \(x\) from \(-\infty\) to \(+\infty\). Then on the left there will appear integrals which are evaluated by integration by parts; they will give a term vanishing upon substitution of the limits, and an integral containing either the derivative \(\dfrac{dU_n^+}{dx}\) (for even \(l\)) or \(\dfrac{dV_n^+}{dx}\) (for odd \(l\)). These derivatives are replaced with the aid of the formulas *):

\[ \frac{dU_n^+}{dy}=-V_{n-1}^+,\qquad \frac{dV_n^+}{dy}=\sum_{l=0}^{n} U_l^+ . \tag{6.24} \]

Thus, the coefficients \(a_{ln}\) will satisfy the equations **)

\[ \frac{\beta}{2l+1}\big[(l+1)a_{l+1,n-1}+l a_{l-1,n-1}\big] =-\mu(\lambda)a_{ln}(\lambda)+ \]

\[ +\int_{\lambda-2}^{\lambda} d\lambda' K(\lambda',\lambda) P_l(1-\lambda+\lambda')a_{ln}(\lambda') +\beta\delta(\lambda-\lambda_0)\delta_{l0} \qquad (l\ \text{even}) \tag{6.25} \]

\[ \underline{\phantom{xxxxxxxxxxxx}} \]

*) The first of these relations follows from (6.18) and (6.19); in proving the second one should, in addition, use the formula

\[ C_n^k=\sum_{i=1}^{n-k+1} C_{n-i}^{k-1}. \]

**) In the original paper \({}^{18}\), in formula (20), for odd \(l\), a minus sign should stand on the left.

\[ -\frac{\beta}{2l+1}\sum_{n'=0}^{n}\left[(l+1)a_{l+1,n'}+l a_{l-1,n'}\right] = -\mu(\lambda)a_{ln}(\lambda)+ \]

\[ +\int_{\lambda-2}^{\lambda} d\lambda'\,K(\lambda',\lambda)P_l(1-\lambda+\lambda')a_{ln}(\lambda') \qquad (l\ \text{odd}). \tag{6.26} \]

Equations (6.25) and (6.26) can be integrated numerically if the delta functions are separated from the solutions of these equations (cf. § 3). It is essential that \(a_{ln}=0\) for \(n<\dfrac{l}{2}-1\), since the term with the source is present only in the equation with \(l=0\). Taking this circumstance into account, let us write out the system of equations that must be solved in order to determine the first four coefficients \(a_{0n}\) in the expansion of the function \(N_0(x,\lambda)\)—the density of photons at a given point, irrespective of their direction. This system consists of ten equations:

\[ \left. \begin{aligned} F(a_{00})+\beta\delta(\lambda-\lambda_0)&=0,\\ F(a_{10})&=-\frac{1}{3}\beta a_{00},\\ F(a_{01})+\beta\delta(\lambda-\lambda_0)&=\beta a_{10},\\ F(a_{21})&=-\frac{2}{5}\beta a_{10},\\ F(a_{11})&=-\frac{1}{3}\beta(a_{00}+2a_{21}+a_{01}),\\ F(a_{02})+\beta\delta(\lambda-\lambda_0)&=\beta a_{11},\\ F(a_{31})&=-\frac{3}{7}\beta a_{21},\\ F(a_{22})&=\frac{1}{5}\beta[3a_{31}+2a_{11}],\\ F(a_{12})&=-\frac{1}{3}\beta[a_{00}+2a_{21}+a_{01}+2a_{22}+a_{02}],\\ F(a_{03})+\beta\delta(\lambda-\lambda_0)&=\beta a_{12}, \end{aligned} \right\} \tag{6.27} \]

where

\[ F[a_{ln}(\lambda)]= \]

\[ =-\mu(\lambda)a_{ln}(\lambda)+ \int_{\lambda-2}^{\lambda} d\lambda'\,K(\lambda',\lambda)P_l(1-\lambda+\lambda')a_{ln}(\lambda'). \tag{6.28} \]

Let us note that in the case of a plane isotropic source all calculations can also be carried out with the moments \(b_{ln}\) instead of directly determining the coefficients \(a_{ln}\). This is sometimes more convenient.

Such a detailed exposition of the solution of the transport equation for a plane isotropic source is explained both by the practical importance of this problem and by the fact that the calculations of the present paragraph are easily carried over to the case of a plane directed source emitting photons of arbitrary energy.

along the normal to the plane\(^{18}\)*) and are taken as the basis for investigations of a plane directed inclined source\(^{3}\). For lack of space we cannot dwell on these questions.

§ 7. Point (isotropic) source

The problem of a point isotropic source is reduced to that considered in the preceding paragraph**). To this end, let us first note that instead of expansion (6.7) one should now write

\[ N_l^*(R,\lambda)=\frac{e^{-|R|}}{4\pi R^2} \sum_{n=0}^{\infty} a_{ln}^*(\lambda)U_n(|R|), \tag{7.1} \]

where \(R=\beta r\), \(r=|\mathbf r|\) is the distance to the source; the asterisk indicates that the corresponding quantity belongs to a point isotropic source.

Hence (cf. (6.22))

\[ a_{ln}^*(\lambda)=\int_{0}^{\infty} U_n^{+}(R)N_l^*(R,\lambda)\,4\pi R^2\,dR = \sum_{\nu=0}^{n} C_n^\nu \frac{(-1)^\nu}{(2\nu)!}\,b_{l,2\nu}^*(\lambda). \tag{7.2} \]

In particular, if one is interested in the density of photons at a given point irrespective of their direction, that is, if \(l=0\), then it is easy to establish a relation between the coefficients \(a_{0n}^*\) and those for a plane isotropic source. For this it is necessary to take into account that (the proof is given in Appendix A to Fano’s paper\(^{7}\)):

\[ N_0^*(R,\lambda)=-\frac{1}{4\pi R}\,\frac{\partial N_0(R,\lambda)}{\partial R}, \tag{7.3} \]

whence, in turn, the corresponding relation for the moments follows:

\[ b_{0,2\nu}^*=(2\nu+1)b_{0,2\nu}. \tag{7.4} \]

Finally, with the aid of formulas (7.2), (7.4), and (6.22) it is easy to show that

\[ a_{0n}^*(\lambda)=(2n+1)a_{0n}(\lambda)-2n a_{0,n-1}(\lambda). \tag{7.5} \]

Thus, the coefficients \(a_{0n}^*\) for a point isotropic source are known as soon as the coefficients \(a_{0n}\) for a plane isotropic source are known.

More complicated is the problem of a point directed source. Such a source may be regarded as elementary. Any other source may be considered as a collection of point directed sources. In view of the cumbersome nature of the presentation, we do not dwell on this case, referring interested readers to the original work\(^{18}\).

\[ \text{――――――――――} \]

) For photons of high energy the calculations are considerably simplified (see \(^{18}\), p. 448).
*) It can also be solved independently\(^{18}\).

§ 8. Numerical results and comparison with experiment

1. Plane source

In a number of experimental works the passage of gamma rays through plane absorbers was investigated under conditions in which the source (usually a radioactive preparation) is located so far from the absorber that the incident rays form a broad, practically parallel beam. If scattering in air is neglected, such a source is equivalent to a plane directed source at the boundary of the absorber.

As indicated in § 6, such a problem was solved for the case of high energy and normal incidence by the method of polynomial expansions in work \(^{18}\) (§ IV, 1). Numerical results were obtained for absorption in lead of quanta with initial energy \(10.2\) MeV. The spectrum of the scattered radiation was calculated down to an energy of \(1\) MeV. At lower energy the simplifying assumption of the method \(^{18}\) is violated.

Fig. 5 illustrates the dependence on energy of the quantity \(e^{\mu_m x} I_0(\mu_m x, E)\) for \(\mu_m=\mu(3.2\ \mathrm{MeV})=0.469\ \mathrm{cm}^{-1}\) (the minimum value of the attenuation coefficient) for five penetration depths. The position of the source on the energy scale is indicated by a solid line, and its intensity by the area of the dashed rectangle. The ordinate scale is assumed to be normalized to unit source intensity at \(x=0\). The dashed curve \(\mu_m x=0\) deviates slightly from the abscissa axis, and this deviation characterizes the accuracy of the calculations. Typical of all the curves is the presence of a maximum at an energy of about \(3\) MeV, corresponding to the minimum of the attenuation coefficient (cf. Fig. 2).

Fig. 5. Differential spectrum \(I_0(E,\mu_m x)\) of gamma-ray intensity at various distances from a plane directed source of \(10\) MeV in lead.

Calculations for a plane source emitting photons of arbitrary energy along the normal to the surface were carried out by Spencer and Stinson \(^{19}\). As an example they considered a plane source of energy \(1.33\) MeV in water. In order to write down in this case the polynomial expansion for the photon energy density \(I_0(\mu_0 x,E)\), it is sufficient to represent the density \(N_0\) as the sum of symmetric and antisymmetric components

\[ N_0(x,\lambda)=N_0^{(s)}(x,\lambda)+N_0^{(a)}(x,\lambda), \]

where \(N_0^{(s)}\) must be expanded in the polynomials \(U_n\), and \(N_0^{(a)}\) in the polynomials \(V_n\).

V. S. GALITSKY, V. I. OGIEVETSKY, A. N. ORLOV

Passing from the photon density \(N_0(x,\lambda)\) to the photon energy density \(I_0(\mu_0 x,E)\), we shall have

\[ I_0(\mu_0 x,E)=e^{-\mu_0 |x|} \left\{ \sum_{n=0}^{\infty} I_{0n}^{(s)}(E)U_n\left(|\mu_0 x|\right) +\mu_0 x \sum_{n=0}^{\infty} I_{0n}^{(a)}(E)V_n\left(|\mu_0 x|\right) \right\}. \]

The authors \(^{19}\) calculated the first three terms in each of the sums entering here for \(\mu_0=0.0612\ \mathrm{cm}^{-1}\). From them the curves in Fig. 6 were constructed. The curve for \(\mu_0 x=-1\) corresponds to backscattering.

Fig. 6. Differential spectrum of the intensity of gamma rays at various distances from a plane directed source of \(1.33\ \mathrm{MeV}\) in water.

The angular distribution of the radiation from a plane isotropic source was investigated by Berger \(^{1}\). Fig. 7 illustrates typical results of his calculations for a plane isotropic \(1\ \mathrm{MeV}\) source in water at \(\mu_0 x=1\) and represents the dependence of \(e^{\mu_0 x} I(x,u_x,E)\) on \(u_x\). The maxima of the curves in Fig. 7 are mainly due to singly scattered radiation. As \(\mu_0 x\) increases, they shift toward \(u_x=1\).

Berger also carried out calculations for a plane directed inclined source. His results for a \(0.66\ \mathrm{MeV}\) source in water are given in Figs. 8 and 9. Fig. 8 \((a,b,c,d)\) shows the energy spectrum \(I_0^{\psi}(x,E)\) of the scattered radiation (in units of \(\dfrac{\mathrm{MeV}}{\mathrm{cm}^2\,\mathrm{sec}}\cdot\dfrac{1}{\mathrm{MeV}}\)) at various distances \(x\) from the plane of the source for inclination angles \(\psi=0^\circ, 30^\circ, 60^\circ\), and \(90^\circ\)). Fig. 9 \((a,b)\) presents the dependence of the total intensity \(I_T^{\psi}(x)\) on the angle of inclination of the source and on the distance from the plane of the source).

A comparison of the curves presented, in particular Figs. 5, 7, and 9, makes it possible to trace the change in the spectrum of the scattered radiation with the thickness of the absorber for different values of the initial energy \(E_0\). For small thicknesses the number of scattered quanta is small, and the form of the spectrum is determined mainly by the spectrum of the source. At medium and large thicknesses, multiple scattering is substantial and the form of the spectrum approaches the spectrum in an infinite medium (see Fig. 2). In light elements there is a maximum in the soft region of the spectrum. In Figs. 6 and 8 it lies beyond the energy limits for which the calculations were carried out. In heavy elements (see, for example, Fig. 5) this maximum is absent, but instead there is a maximum near the energy corresponding to the minimum of the absorption coefficient.

*) The calculations were carried to \(E=0.133\ \mathrm{MeV}\). The breaks in the curves near \(0.2\ \mathrm{MeV}\) correspond to the minimum energy of singly scattered quanta.

Fig. 7. Angular distribution of the energy flux \(I(\mu_0 x, u_x, E)\) of several spectral components at a distance \(\mu_0 x = 1\) from a plane isotropic source of \(1\ \text{MeV}\) in water.

Fig. 8. Differential spectrum \(I^{(\psi)}(X,E)\) of the flux of scattered gamma rays at various distances \(X = \mu_0 x\) from a plane directed inclined source of \(0.66\ \text{MeV}\) in water.

Experimental investigations of the spectral and angular distribution of scattered radiation in the case of a “plane source” are not available in the literature accessible to us. Of the theoretical results cited for plane

Fig. 9. Total flux of scattered gamma rays from a plane directed inclined source of 0.66 MeV in water.

sources, only the total radiation intensity has been subjected to experimental verification.

Let us consider the experimental works relevant here. Beach, Theus, and Faust \(^{20}\) studied the passage of gamma rays of Cs\(^{137}\) (\(0.66\) MeV), Na\(^{24}\) (\(1.38\) and

Fig. 10. Comparison of theoretical and experimental data for transparency as a function of the distance to the source plane.

\(2.76\) MeV), and Co\(^{60}\) (\(1.17\) and \(1.33\) MeV) through iron. In the setup they used, the source was placed at a distance of \(4.5\) m from the absorber (stacks of iron plates \(120 \times 120 \times 0.6\ \text{cm}^3\)) to ensure the same beam intensity over the entire surface of the absorber. Gamma radiation was recorded

by a counter placed behind the absorber. The experimental results and the theoretical transparency curves of Spencer and Fano are shown in Fig. 10. The experimental points for \(Na^{24}\) and \(Co^{60}\) fall well on the theoretical curves. For \(Cs^{137}\) there are deviations which apparently are explained mainly by the large scattering of the radiation in air, as a result of which, in addition to the primary quanta, scattered quanta also fall on the absorber; this was not taken into account in the calculation.

In the work of Kern, Kennedy, and Wikov \(^{30}\), the passage of gamma radiation from a flat inclined source of \(Cs^{137}\) through concrete slabs was investigated. In Fig. 11 their results for buildup factors are compared with Berger’s theoretical curves \(^{3}\) for a 0.66 MeV source in water and for inclination angles \(\psi = 0^\circ\) and \(\psi = 60^\circ\).

Fig. 11. Theoretical and experimental buildup factors for radiation from a flat directed inclined source of 0.66 MeV as a function of distance in the direction of the incident radiation. The theoretical curves correspond to an infinite mass of water; the experimental curves—to a plane-parallel concrete slab.

In both cases the experimental curves lie below the theoretical ones. This is explained by the fact that the buildup factors for concrete are lower than for water. In addition, the theoretical calculations refer to an infinite medium and, consequently, take into account the entry into the detector of radiation scattered by more deeply situated layers, whereas in the experiments concrete slabs of finite thickness were always used and such backscattering was excluded; moreover, the theoretical and experimental curves are almost parallel to one another for

\[ \frac{x}{\cos \psi} > 3, \]

which is in agreement with the results \(^{2}\) obtained by the Monte Carlo method (see § 17).

Royce, Shure, and Taylor \(^{33}\) investigated the passage of a broad beam of gamma rays of \(N^{16}\) (6 MeV) through water. In the work, the relative dose of gamma radiation was measured at various distances from the source by means of an anthracene scintillation counter. As can be seen from Fig. 12, up to distances of 160 cm the experimental data (crosses) agree well with those calculated according to the theory of Spencer and Fano \(^{61}\).

Fig. 12. Dose attenuation of gamma radiation \(N^{16}\) (\(6\ \mathrm{MeV}\)) in water. The solid curve was calculated by the Spencer and Fano method; the crosses on the curve are experimental data.

Fig. 13. Differential spectrum \(I_0(E,\mu_0 r)\) of the intensity of gamma rays at various distances \(r\) from a point isotropic source \(Co^{60}\) in water.

Fig. 14. Spectrum of Fig. 13 on a logarithmic scale.

2. Point Source

Numerical calculations18 for the case of a point isotropic source of Co60 in water were carried out by the method of § 7. The first four expansion coefficients \(N_0(r,\lambda)\) in the polynomials \(U_n\) were found. The results of the calculations18 are presented in Figs. 13—15. Figure 13 gives the radiation intensity \(I_0(\mu r,E)\), multiplied by \(4\pi r^2 e^{\mu_0 r}\), at various distances \(r\) from the source. The solid vertical lines indicate the positions of the two components of the Co60 source. The areas of the dotted rectangles correspond to the intensities of these lines. The ordinate scale in this case is normalized to unity for the intensity of one of the source components (1.33 MeV). (The narrow-beam attenuation coefficient of this component is \(\mu_0=\mu(1.33\ \text{MeV})=0.0612\ \text{cm}^{-1}\).)

In Fig. 14 the same curves as in Fig. 13 are shown on a semilogarithmic scale. It can be seen from them that at low energies the spectral distribution does not change with increasing penetration depth, since all the curves are parallel to one another. This indicates that the numbers of quanta produced by scattering and absorbed through the photoelectric effect in this spectral region are in “equilibrium.”

Finally, Fig. 15 makes it possible to compare the theoretically computed build-up factor (solid curve) with the experimental data of White35. The theoretical curve was obtained with the aid of the spectral distribution of Fig. 13. The agreement between the experimental and theoretical values is very good. It is seen from Fig. 15 that the build-up factor increases with distance somewhat faster than linearly.

Fig. 15. Comparison of the calculated curve for the build-up factor with experimental data35.

The second experimental work with a point source of Co60 immersed in water was carried out by Hayward27. He investigated the spectrum of secondary electrons in water at various distances from the source with the aid of a scintillation spectrometer.

Figure 16 shows the differential spectrum of electrons produced by quanta from a Co60 source for six different distances from it. Along the ordinate axis is plotted the number of electrons falling within an energy interval of 1 MeV per quantum emitted by the source and per gram of anthracene crystal used as the phosphor in the scintillation counter. The solid curves in Fig. 16 were obtained theoretically by Spencer. The position of the experimental points relative to the theoretical curves indicates that the general character of the change of the spectrum with distance is correctly predicted by the theory. At a small distance (10 cm) the hard end of the spectrum is clearly revealed, and two peaks are outlined corresponding to the two components of the Co60 source. At distances of about a meter and more, the changes in the shape of the spectrum on the high-energy side are too slight and were not detected because of the low resolving power of the spectrometer.

Fig. 16. Differential spectrum of electrons produced by quanta from a \( \mathrm{Co}^{60} \) source in water for six different distances.

Fig. 17. Comparison of experimental data for the soft part of the spectrum of a \( \mathrm{Co}^{60} \) source with calculation results. \(A\)—theoretical spectrum; \(B\)—experimental spectrum.

Among the works under discussion are also the measurements carried out by Weiss and Bernstein\(^ {34}\) of the energy spectrum of gamma rays from a point source of Co\(^ {60}\) in water for energies below \(0.1\) MeV, using a scintillation spectrometer. The maximum distance from the source was \(190\) cm. A typical spectral curve\(^ {34}\) for the distance \(r = 160\) cm is shown in Fig. 17 (solid curve). The dashed curve corresponds to the calculations of Spencer and Fano. As can be seen, the two curves are close to one another. When \(r\) was varied from \(70\) to \(190\) cm, no change was observed either in the shape of the spectrum or in the position of the maximum. Some broadening of the spectral curve and a displacement of the maximum toward higher energies was observed only for \(r < 30\) cm. Weiss and Bernstein also measured the attenuation of the soft part of the spectrum of Co\(^ {60}\) gamma rays with increasing distance from the source.

Fig. 18. Attenuation of the soft branch of the spectrum of a Co\(^ {60}\) source as a function of distance. \(A\)—theoretical curve; \(B\)—experimental points.

Fig. 19. Differential distribution \(I(E,\mu_2,\mu_0 r)\) of the intensity of gamma rays for several spectral components at a distance from a point isotropic Co\(^ {60}\) source in water.

Fig. 18 shows that in this case as well the theoretical predictions are excellently confirmed by the experimental data. Similar results were obtained with an Hg\(^ {203}\) source.

In the work of Spencer and Stinson\(^ {19}\) cited above, along with a plane directed source, point isotropic and point directed sources were also considered. For a point isotropic Co\(^ {60}\) source in water, the angular distribution of photons was calculated for several energy values (Fig. 19); \(\theta_r\) is the angle of deviation of a photon from the line drawn from the source to the photon, and for a point directed source of energy \(1.33\) MeV in water, owing to the complexity of the calculations, the problem was solved under strongly simplifying assumptions, and the dependence of the mean-square radial displacement of the radiation on energy at various depths was calculated.

Also of interest is the work of Elliott et al.\(^ {23}\), carried out with the aim of testing the applicability of the Spencer–Fano theory for heavy elements and a point isotropic Co\(^ {60}\) source. As a detector of the scattered

of the radiation a special photographic film was used. The experimental value of the build-up factor was determined as the ratio of the total

![Figure 20 graph]

Fig. 20. Build-up factor for a point isotropic source of \(Co^{60}\) in lead. The solid curve was calculated\(^{23}\) by the Spencer and Fano method.

blackening of the film to the blackening calculated for the case of the action only of unscattered radiation. As is seen from Fig. 20, the agreement between theory (solid curve) and the experimental data is good.

![Figure 21 graph]

Fig. 21. Theoretical and experimental build-up factors for a point isotropic source of \(Co^{60}\) in iron as a function of distance from the source. — \(A\)—theory; \(B\)—experiment based on theoretical absorption coefficients equal to \(0.0549\ \text{cm}^2/\text{g}\) and \(0.0516\ \text{cm}^2/\text{g}\); \(C\)—experiment based on absorption coefficients 0.4% larger, \(\mu_1 = 0.0551\ \text{cm}^2/\text{g}\) and \(0.0518\ \text{cm}^2/\text{g}\).

A detailed study of the build-up factors for a point source of \(Co^{60}\) in iron and lead is the subject of the work of Garrett and White\(^{26}\). The experiments con-

were carried out for an iron thickness up to 14 mean free paths and a lead thickness of somewhat more than 16 mean free paths. The ionization was measured with air-equivalent condenser chambers. The results of the measurements and of calculations according to the theory of Spencer and Fano are presented in Figs. 21 and 22.

Fig. 22. Theoretical and experimental build-up factors for a point isotropic source of Co\(^{60}\) in lead as a function of distance from the source. \(A\)—theory; \(B\)—experiment based on theoretical absorption coefficients equal to 0.0585 cm\(^2\)/g and 0.0537 cm\(^2\)/g; \(C\)—experiment based on absorption coefficients smaller by 0.2%, and correspondingly equal to 0.0584 cm\(^2\)/g and 0.0536 cm\(^2\)/g.

Wyatt\(^{36}\) carried out a study to determine the values of the intensities \(I_0(r,E)\) (Fig. 23), \(I_A(r,u_r)\) (Fig. 24), and \(I(r,u_r,E)\) (Fig. 25) for a Co\(^{60}\) source placed in concrete. The dotted and solid curves in Fig. 23 are, respectively, the theoretical spectral curves for water and iron, calculated by Goldstein and Wilkins\(^{62}\). The agreement between theory and experiment, as the author notes, is within the limits of measurement accuracy. Figure 24 presents the mean angular distribution of the intensity of scattered radiation. The series of curves in Fig. 25 gives the gamma-ray intensities as a function of energy for various values of the angles at \(\mu_0 r = 6.52\).

In a recently published work by Tsypin, Kuktevich, and Kazanskii\(^{38}\), the dose attenuation of gamma rays was measured (the ratio of the ionization measured by a chamber in the substance to the ionization measured by a chamber at the same point without the substance under investigation) for point sources of Na\(^{24}\) in water and in lead, and of Au\(^{198}\) in iron. As the authors note, their experimental data on dose attenuation for Na\(^{24}\) coincide with the calculated data obtained on the basis of the tables of dose build-up factors and absorption coefficients given by Fano\(^{57}\).

There are reports in the literature\(^{61}\) that at the National Bureau of Standards (USA) extensive work\(^{62}\) has been carried out on calculating gamma-ray distributions for 206 cases. The calculations were performed for most elements of the periodic system, including substances with pure Compton scattering, for point isotropic and plane directed sources. The calculations

...were carried out for distances from 1 to 20 mean free paths from the source, with an initial gamma-quantum energy from 0.5 to 10 MeV.

Fig. 23. Differential spectrum \(I_0(E,\mu_r)\) of the intensity of gamma rays at various distances \(r\) from a point isotropic \(^{60}\mathrm{Co}\) source in concrete.

As can be seen from the data presented in this section, the spatial, angular, and spectral distributions of the radiation of a point source do not reveal any qualitatively distinct features in comparison with a plane source.

Fig. 24. Average angular distribution. The distance from the origin corresponds to the relative intensity of quanta of all energies per unit solid angle for a given \(\theta\).

Agreement between the theoretical data obtained by the transport-equation method and the experimental results is good; however, the computations become

Fig. 25. Differential energy and angular distribution of the intensity of gamma radiation at \(\mu_0 r = 6.52\).

very cumbersome if the penetration depth exceeds 15–20 mean free paths.

IV. THE SMALL-ANGLE APPROXIMATION. ENERGY SPECTRUM OF SCATTERED GAMMA RADIATION AT LARGE PENETRATION DEPTHS

§ 9. Introductory remarks

The method of polynomial expansions set forth in the preceding chapter is a very constructive method for solving the problem of the propagation of gamma radiation in matter, but it is suitable for relatively small penetration depths and is numerical. For a qualitative analytical approach to the problem one may use the fact that, in the Compton effect for gamma radiation with energy of the order of several \(M e\), scattering through small angles is much more probable than scattering through large angles.

The present chapter consists of two parts. In the first part, in the small-angle approximation, the analytical form of the energy spectrum and angular distribution of scattered gamma radiation is considered \({}^{8,48,49,55,58,59,60}\). The second part is devoted to the extremely important work of Spencer and Fano \({}^{7,17,55,58}\) on the study of asymptotic formulas for the energy spectrum of gamma rays at large penetration depths, taking angular deviations into account.

It should be noted that the small-angle approximation \((\cos \theta \simeq 1,\ \sin \theta \simeq \theta;\) “straight-ahead” approximation) is rather crude and can lay claim, basically, only to a qualitative description. Indeed \({}^{7}\), in the case of a plane directed source, for example, for a photon flying at an angle \(\theta\) to the normal to the plane of the sources, the effective absorption coefficient is equal to \(\dfrac{\mu(\lambda)}{\cos \theta}\). If a photon with wavelength \(\lambda_0\) and direction

\(\theta_0=0\), as a result of the Compton effect it is deflected through an angle \(\theta=\arccos(1-\delta\lambda)\), then the effective absorption coefficient changes from \(\dfrac{\mu(\lambda_0)}{\cos\theta_0}=\mu(\lambda_0)\) to

\[ \frac{\mu(\lambda_0+\delta\lambda)}{1-\delta\lambda} \sim \mu(\lambda_0)+\left(\frac{d\mu(\lambda)}{d\lambda}\right)_{\lambda=\lambda_0}\delta\lambda+\mu(\lambda_0)\delta\lambda . \]

The first correction term, due to the change in \(\mu(\lambda)\), for most materials has the same order of magnitude as the second correction term, due to angular deflections. Therefore, taking angular deflections into account is at least as important as taking into account the energy dependence of the absorption coefficient. Note, however, that in light elements the absorption coefficient over a wide energy interval depends only very weakly on energy, and in this case it is reasonable, in parallel with the small-angle approximation, to neglect the dependence of the absorption coefficient on energy. The small-angle approximation has also proved interesting in that, as Fano\(^7\) showed, allowance for angular deflections does not change the analytical form of the energy spectrum of gamma rays at large penetration depths, but leads only to a change in certain numerical constants.

§ 10. The radiation transport equation in the small-angle scattering approximation

In the case of a plane directed source, in the small-angle approximation the angular distribution of the scattered gamma radiation has a sharply pronounced forward directionality. Therefore, in the expansion (3,7) in Legendre polynomials, the principal role is played by the distant terms with large \(l\). Using the known relation

\[ \lim_{l\to\infty} P_l\left(\cos\frac{\theta}{l}\right)=I_0(\theta), \tag{10,1} \]

one can pass from an expansion in Legendre polynomials to an expansion in Bessel functions of zero order (the Hankel transform):

\[ N(x,\theta,\lambda)=\frac{1}{2\pi}\int_0^\infty N_l(x,\lambda) I_0(l\theta)\,l\,dl \tag{10,2} \]

and

\[ N_l(x,\lambda)=2\pi\int_0^\infty N(x,\theta,\lambda) I_0(l\theta)\,\theta\,d\theta \tag{10,3} \]

(such an operation was used in the study of electron scattering by Molière\(^ {46}\)).

The energy spectrum of gamma rays is evidently described by the function

\[ N_0(x,\lambda)=2\pi\int_0^\infty N(x,\theta,\lambda)\,\theta\,d\theta . \]

Taking into account also that, for large \(l\),

\[ P_l(1-\lambda+\lambda')\sim I_0\left(l\sqrt{2(\lambda-\lambda')}\right) \tag{10,4} \]

and

\[ \frac{l+1}{2l+1}\frac{\partial N_{l+1}}{\partial x} + \frac{l}{2l+1}\frac{\partial N_{l-1}}{\partial x} \sim \frac{\partial N_l}{\partial x}, \tag{10,5} \]

one can reduce the coupled infinite system of equations (3,11) to a од-

to the equation[^48]

\[ \frac{\partial N_l(x,\lambda)}{\partial x} = -\mu(\lambda)N_l(x,\lambda) + \int_{\lambda_0}^{\lambda} d\lambda' \, K(\lambda',\lambda) I_0\!\left(l\sqrt{2(\lambda-\lambda')}\right) \times \]

\[ \times N_l(x,\lambda')+\delta(x)\delta(\lambda-\lambda_0). \tag{10,6} \]

The reduction carried out corresponds to the small-angle approximation \(\cos\theta\sim 1\), \(\sin\theta\sim\theta\).

For the approximation in which \(\cos\theta\sim 1-\dfrac{\theta^2}{2}\), one should (see Wick’s paper[^69]) add to the right-hand side of (10,5) \(\dfrac{1}{2}\dfrac{\partial^3 N_l(x,\lambda)}{\partial l^2\partial x}\). The differential scattering coefficient \(K(\lambda',\lambda)\) (1,13) is approximated in a number of works[^8],[^48],[^49],[^58]–[^60] by the expression

\[ K(\lambda',\lambda)\sim \left(\frac{\lambda'}{\lambda}\right)^n C = \left(\frac{\lambda'}{\lambda}\right)^n \frac{3}{4}\mu_T, \tag{10,7} \]

where \(\mu_T=\sigma_0 n Z\) is the Thomson scattering coefficient (see (1,2)). The best value of the exponent in the least-squares sense[^48] is \(n=1.8\). In the works of Fano’s group \(n\) is chosen equal to 0. In Foldy’s works[^8],[^59] \(n=1\). As is not difficult to see[^48],[^60], the solution of equation (10,6) in the case of nonzero \(n\) differs from the solution for \(n=0\) only by the factor \(\left(\dfrac{\lambda_0}{\lambda}\right)^n\). Since, moreover, we are interested, in the small-angle approximation, in a narrow spectral interval, this factor will be omitted below.

§ 11. Solution of the radiation-transport equation in the small-angle approximation

In the case of an absorption coefficient linearly dependent on the wavelength,

\[ \mu(\lambda)=\mu_0+\dot{\mu}_0(\lambda-\lambda_0), \]

applying the Laplace transform with respect to \(\lambda\) to \(N_l(x,\lambda)\),

\[ F_l(x,s)=\int_{\lambda_0}^{\infty} d\lambda\, \exp[-(\lambda-\lambda_0)s]\,N_l(x,\lambda), \tag{11,1} \]

we have

\[ N_l(x,\lambda) = \frac{1}{2\pi i} \int_{\delta-i\infty}^{\delta+i\infty} ds\,\exp[(\lambda-\lambda_0)s]\,F_l(x,s), \]

where for \(F_l(x,s)\) we obtain[^48],[^49] the expression

\[ F_l(x,s) = \exp[-\mu_0 x]\, \exp\left\{ C\int_{0}^{x} dy\, \frac{ \exp\!\left[-\dfrac{l^2}{2(s-\dot{\mu}_0 y)}\right] }{ s-\dot{\mu}_0 y } \right\}. \tag{11,2} \]

If the dependence of the absorption coefficient on energy is neglected \((\dot{\mu}_0=0)\), then the expression for \(F_l(x,s)\) simplifies:

\[ F_l(x,s) = \exp[-\mu_0 x]\, \exp\left\{ \frac{a x}{s}\exp\left(-\frac{l^2}{2s}\right) \right\}. \tag{11,3} \]

The inversion of \(F_l(x,s)\) by Hankel and Laplace transforms can be carried out by expanding \(\exp(\mu_0 x)F_l(x,s)\) in powers of the penetration depth \(x\).

We present the result in the case of a constant absorption coefficient \({}^{8,48,59}\)

\[ \begin{aligned} N(x,\theta,\lambda)=&\frac{1}{2\pi}e^{-\mu_0 x}\Biggl\{ \delta(\lambda-\lambda_0)\delta\!\left(\frac{\theta^2}{2}\right) +\frac{\rho^2}{4(\lambda-\lambda_0)} \delta\!\left(\lambda-\lambda_0-\frac{\theta^2}{2}\right) \\ &\quad+\frac{1}{(\lambda-\lambda_0)^2} \sum_{n=2}^{\infty} \frac{n-1}{(n!)^2} \left(\frac{\rho}{2}\right)^{2n} \left(1-\frac{\theta^2}{2n(\lambda-\lambda_0)}\right)^{n-2} u\!\left(\lambda-\lambda_0-\frac{\theta^2}{2n}\right) \Biggr\}, \end{aligned} \tag{11,4} \]

where

\[ \rho=2\sqrt{Cx(\lambda-\lambda_0)}, \tag{11,5} \]

\(u(x)\) is the unit step

\[ u(x)= \begin{cases} 1, & x>0,\\ 0, & x<0. \end{cases} \tag{11,6} \]

The solution obtained has the form of a power series with respect to the depth of penetration of gamma radiation into the substance, multiplied by the exponential \(\exp(-\mu_0 x)\), which describes the absorption of unscattered radiation. Each term of this series has a simple physical meaning. The first term describes radiation that has not undergone scattering. The second term represents singly scattered radiation. The following terms, as is seen from the correlation between angle and wavelength, represent the fractions of gamma radiation that have undergone Compton scattering \(2,3,\ldots\) times.

As was to be expected, the maximum angle of deflection for twice-scattered radiation is
\(\theta_{\max}^{(2)}=\sqrt{4(\lambda-\lambda_0)}\), and for \(n\)-fold scattered radiation
\(\theta_{\max}^{(n)}=\sqrt{2n(\lambda-\lambda_0)}\).

The intensity of unscattered radiation is described by the product of two \(\delta\)-functions, that of singly scattered radiation by one \(\delta\)-function; the intensity of twice-scattered radiation has a discontinuity at \(\theta=\theta_{\max}^{(2)}\), the intensity of thrice-scattered radiation is continuous but has a discontinuity in the first derivative, and the intensity of \(n\)-fold scattered radiation contains a discontinuity in the \((n-2)\)-th derivative. The curve representing the intensity of \(n\)-fold scattered radiation becomes smoother as \(n\) increases, which is natural. The angular distribution of gamma radiation scattered two or more times therefore has a characteristic step-like form.

These considerations concerning the continuity of the intensities of multiply scattered radiations are also valid for an arbitrary absorption coefficient.

As the depth of penetration increases, the higher orders of scattering begin to play the principal role; the angular distribution becomes smoothed and tends to the Gaussian
\(\exp[-\theta^2/2(\lambda-\lambda_0)]\).

An expression of the type (11,4), only more cumbersome, can be obtained also for an absorption coefficient depending linearly and quadratically on wavelength \({}^{48}\). For the energy spectrum \(N_0(x,\lambda)\), closed expressions can be found from formulas (11,2) and (11,3):

for the case \(\mu(\lambda)=\mu_0\)

\[ N_0(x,\lambda)=e^{-\mu_0 x} \left\{ \delta(\lambda-\lambda_0) +\frac{\rho}{2(\lambda-\lambda_0)}J_1(\rho) \right\}, \tag{11,7} \]

for the case \(\mu(\lambda)=\mu_0+\mu'_0(\lambda-\lambda_0)\)

\[ N_0(x,\lambda)=e^{-\mu_0 x} \left\{ \delta(\lambda-\lambda_0) +Cx\,{}_1F_1\!\left(1-\frac{C}{\mu'_0};\,2;\,-\mu'_0(\lambda-\lambda_0)x\right) \right\}, \tag{11,8} \]

where \(J_1(\rho)\) is a Bessel function of imaginary argument of the first order, and \({}_1F_1(\alpha;\beta;x)\) is the confluent hypergeometric function. The energy

the spectrum (11.8) was first found in closed form in the work of Bethe, Fano, and Karr[^55]. In these cases, from (11.2) and (11.3) one can also obtain the angular moments

\[ \overline{\theta^{2n}}=2\pi\int\limits_{0}^{\infty}\theta^{2n}N(x,\theta,\lambda)\,\theta\,d\theta, \]

which, in the case of a constant absorption coefficient, are expressed in terms of Bessel functions of imaginary argument, and, in the case of a linear absorption coefficient, in terms of degenerate hypergeometric functions[^49].

§ 12. Evolution of the Angular Distribution with Increasing Penetration Depth

It was noted above that, as the penetration depth increases, the angular distribution of scattered gamma radiation tends to a Gaussian. Knowing the moments of this distribution[^49], one can, by means of the Fano and Spencer method[^18] of polynomial expansions with weight

\[ \exp\left[-\frac{\theta^{2}}{2(\lambda-\lambda_{0})}\right], \]

find the angular distribution of gamma rays at large depths. Namely,[^49] for the case of a constant absorption coefficient,

\[ \begin{aligned} N(x,\theta,\lambda) &=\frac{4C^{2}x^{2}}{\pi}\exp(-\mu_{0}x) \exp\left[-\frac{\theta^{2}}{2(\lambda-\lambda_{0})}\right] \Biggl\{ \frac{1}{\rho^{3}}J_{1}(\rho) \\ &\quad -\frac{2}{\rho^{4}}\left[J_{0}(\rho)-\frac{2}{\rho}J_{1}(\rho)\right] L_{2}\left[\frac{\theta^{2}}{2(\lambda-\lambda_{0})}\right] \\ &\quad -\frac{16}{\rho^{5}}\left[J_{1}(\rho)-\frac{4}{\rho}J_{0}(\rho)+\frac{8}{\rho^{2}}J_{1}(\rho)\right] L_{3}\left[\frac{\theta^{2}}{2(\lambda-\lambda_{0})}\right]+\ldots \Biggr\}, \end{aligned} \tag{12.1} \]

where \(L_n(x)\) are the Chebyshev–Laguerre polynomials.

Figure 26 shows the evolution of the angular distribution of multiply scattered gamma radiation. The curves for \(\rho=1,4\), and \(6\) were computed from formula

Fig. 26. Evolution of the angular distribution of gamma rays with increasing penetration depth. The normalized angular distribution of radiation with energy \(9\) MeV, scattered two or more times, is shown. \(E_{0}=12.5\) MeV; 1) \(\rho=1\), 2) \(\rho=4\), 3) \(\rho=6\), 4) \(\rho=16\), 5) \(\exp[-\theta^{2}/2(\lambda-\lambda_{0})]\).

(11.4), the curve for \(\rho=16\) by the method of polynomial expansions (12.1). For comparison, the Gaussian distribution is also shown.

§ 13. The state of restricted radiation equilibrium

In this and the following sections of the chapter we shall briefly present the results of the work of the Fano and Spencer group7, 17, 55, 58, devoted to the study of gamma radiation at great penetration depths in matter. Let us begin with a qualitative description. As the penetration depth in matter increases, the fraction of secondary, scattered radiation increases. It may be expected that at sufficiently great depths there will be a tendency toward a distribution which, in a certain sense, may be called a state of radiation equilibrium.

By the true state of radiation equilibrium is meant a state characterized by the following properties: 1) the ratio of the intensity of the secondary radiation to the intensity of the primary radiation tends, as the penetration depth increases, to a certain maximum value; 2) the quality of the secondary radiation ceases to depend on the penetration depth; 3) the change in the intensity of all penetrating radiation is determined by the attenuation of the intensity of the primary radiation, that is, follows an exponential law. The existence of such a state considerably simplifies the study of the problem of radiation propagation.

Unfortunately, in the propagation of gamma radiation the conditions of true radiation equilibrium are not satisfied. Indeed, in Compton scattering some gamma quanta will have almost the same energy, almost the same direction, and almost the same attenuation coefficient as before scattering. Owing to this, properties 1) and 3) are absent: the ratio of the intensity of the secondary gamma rays to the intensity of the primary rays increases without bound. At the same time property 2), concerning the quality of the secondary radiation, which has an absorption coefficient markedly different from the maximum absorption coefficient, is preserved. Fano calls such a state the state of restricted equilibrium. In the state of true radiation equilibrium the radiation flux at great depths is described as the product of an exponential function of the penetration depth and a function of the energy and direction of the secondary radiation. In the state of restricted equilibrium of interest to us, the function of depth loses its exponential character; however, gamma radiation at great depths, with an energy differing markedly from the primary one, can be described by a formula of the form

\[ N(x,\lambda,\theta)\sim N_x(x)\cdot N_{\lambda\theta}(\lambda,\theta). \tag{13.1} \]

Fano calls the ratio of \(N_x\) to the exponential \(\exp(-\mu_m x)\), which describes the penetration of the hardest radiation, the “build-up factor”; its determination is the aim of a series of works7, 17, 55, 58.

§ 14. Energy spectrum of gamma radiation at great penetration depths in the small-angle approximation

Let us first consider, in the small-angle approximation, the case of a plane directed monochromatic source. If the primary radiation is the most penetrating, then the absorption coefficient may be approximated by the expression \(\mu(\lambda)=\mu_0+\mu'_0(\lambda-\lambda_0)\); the energy spectrum for this case was obtained in § 12. Using the asymptotic formula for the degenerate hypergeometric function appearing in (11.8) at large values of the argument, we find an expression for the build-up factor \(B(x)\)

in the form[^55]

\[ B(x)=x^K;\qquad K=\frac{C}{\mu_0}, \tag{14.1} \]

\[ N_x(x)=x^K\exp(-\mu_0 x). \tag{14.2} \]

The investigation is somewhat more complicated if the initial photon energy is higher than the energy corresponding to the minimum of the absorption coefficient. In this case a parabolic approximation of the absorption coefficient of the form
\(\mu(\lambda)=\mu_m+\ddot{\mu}_m(\lambda-\lambda_0)^2/2\) may be adopted. The equation for the energy spectrum may be obtained from the general transport equation (10.6) by setting \(l\) equal to zero:

\[ \frac{\partial N_0(x,\lambda)}{\partial x} = -\mu(\lambda)N_0(x,\lambda) + \int_{\lambda_0}^{\lambda} K(\lambda',\lambda)N_0(x,\lambda')\,d\lambda' + \beta(x)\delta(\lambda-\lambda_0). \tag{14.3} \]

Putting \(K(\lambda',\lambda)=C\) and applying the two-sided Laplace transform

\[ \left. \begin{aligned} y_0(p,\lambda)&=\int_{-\infty}^{\infty} e^{px}N_0(x,\lambda)\,dx,\\ N_0(x,\lambda)&=\frac{1}{2\pi i}\int_{-i\infty}^{+i\infty} e^{-px}y_0(p,\lambda)\,dp \end{aligned} \right\} \tag{14.4} \]

for \(y_0(p,\lambda)\), Fano found the expression[^7][^58]

\[ y_0(p,\lambda) = C(\mu(\lambda)-p)^{-1} \exp\left\{\int^{\lambda}\frac{C\,d\lambda'}{\mu(\lambda')-p}\right\} = \]

\[ = C[\mu(\lambda)-p]^{-1} \exp\left\{2\pi C/[2\ddot{\mu}_m(\mu_m-p)]^{1/2}\right\}. \tag{14.5} \]

Inverting expression (14.5) by Laplace’s method of steepest descents leads to the growth factor

\[ B(x)\sim x^{-5/6}\exp\left(bx^{1/3}\right), \tag{14.6} \]

where

\[ b=\frac{3\pi^2 C^2}{2\ddot{\mu}_m}. \]

§ 15. Allowance for angular deviations

A new treatment of the question of the asymptotic laws of the distribution of radiation intensity at great depths is given in Fano’s extensive study[^7]. In the present section we dwell above all on this work, closely connected with Spencer’s work[^17], in which the method now actually used for calculating radiation density at great depths is developed. The basic ideas of Spencer’s method and the results obtained with its aid are presented in the next section.

Our exposition will be limited to the case of a plane source, situated in the plane \(x=0\), with wavelength \(\lambda=\lambda_0\). For the investigation

of the transport equation (3.6) Fano applies the Laplace transform

\[ y(p,u_x,\lambda)=\int_{-\infty}^{\infty} e^{px}N(x,u_x,\lambda)\,dx, \tag{15,1} \]

\[ N(x,u_x,\lambda)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} e^{-px}y(p,u_x,\lambda)\,dp. \tag{15,2} \]

The equation determining the angular and spatial distributions for each value of \(p\) is derived from equation (3.6) with the aid of the transform (15,2). As a result one obtains the equation

\[ [\mu(\lambda)-pu_x]\,y(p,u_x,\lambda) = \int_{\lambda_0}^{\lambda} d\lambda' K(\lambda',\lambda) \int_{4\pi}du'\,(2\pi)^{-1}\delta(1-\mathbf{u}\mathbf{u}'-\lambda+\lambda') \times \]

\[ \times\,y(p,u'_x,\lambda')+\delta(\lambda-\lambda_0)\delta(\hat{x}). \tag{15,3} \]

Because of the presence of the factor \(\mu(\lambda)-pu_x\) on the left-hand side of equation (15,3), \(y(p,u_x,\lambda)\) becomes infinite when \(\mu(\lambda)=pu_x\). This occurs only for real positive values of \(p\) greater than \(\mu(\lambda)\), or for real negative quantities less than \(-\mu(\lambda)\). Therefore the singularity of equation (15,3) is associated with values of the complex variable \(p\) which are bounded by segments of the real axis to the right of \(\mu_s\) and to the left of \(-\mu_s\), where \(\mu_s\) is the smallest value of \(\mu(\lambda)\) in the wavelength region from \(\lambda_0\) to \(\lambda\).

Further, examination of the integrand in (15,3) shows that, on the one hand, it has a minimum when approaching the point \(p=\mu_s\) along the real axis and, on the other hand, a maximum when approaching it along the imaginary axis; that is, the point \(p=\mu_s\) is a saddle point. In the neighborhood of the saddle point the integral (15,2) can be evaluated by the saddle-point method. In view of the circumstance noted, below we shall be interested only in the behavior of \(y(p,u_x,\lambda)\) near the point \(p=\mu_s\), which corresponds to finding approximate analytic solutions of equation (15,3) valid for \(\lambda\sim\lambda_0\).

Expanding \(y(p,u_x,\lambda)\) in Legendre polynomials, we reduce equation (15,3) to the system of equations

\[ \mu(\lambda)y_l(p,\lambda) - p(2l+1)^{-1}\bigl[(l+1)y_{l+1}(p,\lambda)+ly_{l-1}(p,\lambda)\bigr] = \]

\[ = \int_{\lambda_0}^{\lambda} K(\lambda',\lambda)P_l(1-\lambda+\lambda')y_l(p,\lambda')\,d\lambda' + I_l\delta(\lambda-\lambda_0). \tag{15,4} \]

The first equation of this system has the same form as in the small-angle approximation, except that on the left-hand side \(y_1\) appears instead of \(y_0\). It is convenient, following Spencer\(^{17}\), to use the parameter

\[ g_1(p,\lambda)= \int_{4\pi}du\,(1-u_x)y(p,u_x,\lambda) /\int_{4\pi}y(p,u_x,\lambda)\,du, \tag{15,5} \]

which represents the first moment of the angular distribution. If in (15,4) we put \(y_1=(1-g_1)y_0\), then the first equation is written as

\[ [\mu(\lambda)-p+pg_1(p,\lambda)]y_0(p,\lambda) = \int_{\lambda_0}^{\lambda}K(\lambda',\lambda)y_0(p,\lambda')\,d\lambda' + I_0\delta(\lambda-\lambda_0). \tag{15,6} \]

In the wavelength region \(\lambda \sim \lambda_s\), the solution of equation (15.6) can be represented in the form

\[ y_0(p,\lambda)= \frac{C[\mu(\lambda)-p]^{-1}}{1+\dfrac{p g_1(p,\lambda)}{\mu(\lambda)-p}} \exp\left\{ \int^\lambda \frac{C}{1+\dfrac{p g_1(\lambda,p)}{\mu(\lambda)-p}} \frac{d\lambda'}{[\mu(\lambda')-p]} \right\}. \tag{15.7} \]

This expression differs from that obtained in the small-angle approximation (14.5) only by replacing \(C\) by

\[ C\left[1+\frac{p g_1(p,\lambda)}{\mu(\lambda)-p}\right]^{-1}. \tag{15.8} \]

In his work\(^{18}\), having improved Wick’s method\(^{69}\) and relying to a considerable extent on Spencer’s work\(^{17}\), Fano showed that the factor appearing in the denominator tends to a constant when \(p\) approaches the singular point \(p=\mu_s\). Fano thus proved a very important result: allowing for angular deviations does not change the form of the formulas for the growth factors, but merely refines the constants entering these formulas.

Table I

Quantities \(\dot{\mu}_s/\mu_s\) and \(C/\mu_s\), for various substances and energies

Source energy (MeV)	H\(_2\)O \(\dot{\mu}_s/\mu_s\)	H\(_2\)O \(C/\mu_s\)	Al \(\dot{\mu}_s/\mu_s\)	Al \(C/\mu_s\)	Fe \(\dot{\mu}_s/\mu_s\)	Fe \(C/\mu_s\)	Pb \(\dot{\mu}_s/\mu_s\)	Pb \(C/\mu_s\)
10	7.5	7.54	4.1	6.21	—	—	—	—
8	6.6	6.87	4.0	5.91	0.48	4.64	—	—
6	5.5	5.96	3.8	5.43	1.2	4.56	—	—
4	4.0	4.91	3.3	4.68	2.0	4.25	—	—
3	3.1	4.22	2.7	4.10	2.1	3.89	0.33	2.88
2							1.7	2.65
1	0.96	2.37	0.96	2.37	0.89	2.36	1.7	1.75
0.8	0.73	2.12	0.73	2.12	0.72	2.11	1.6	1.43
0.6	0.52	1.86	0.52	1.86	0.54	1.84	1.5	1.033
0.4	0.32	1.58	0.32	1.57	0.37	1.52	1.35	0.579
0.3	0.22	1.41	0.22	1.40	0.32	1.32	1.2	0.333
0.2	0.13	1.23	0.155	1.21	0.31	1.01	0.95	0.133
0.15	0.086	1.12	0.148	1.09	0.33	0.767	0.77	0.0647

Thus, for the growth factors one may use, also in the general case, the expressions

\[ B(x)\sim x^K, \tag{15.9} \]

if the initial photon energy \(E_0\) is below the energy \(E_m\) corresponding to the minimum of the absorption coefficient, and

\[ B(x)\sim x^{-\frac{5}{6}}\exp\left[H(\mu_m x)\right]^{\frac{1}{3}}, \tag{15.10} \]

if \(E_0>E_m\). In formulas (15.9) and (15.10)

\[ K=\frac{\bar C}{\dot{\mu}_s},\qquad H=3\cdot 2^{-\frac{2}{3}}D^{\frac{1}{3}} =3\left(\frac{\pi^2\bar C^2}{2\mu_m^2\ddot{\mu}_m}\right)^{\frac{1}{3}}. \tag{15.11} \]

The values of these constants for various materials and energies are given in Tables I–III, taken from Appendix (C) to Fano’s paper\(^{7}\). White’s results\(^{35}\) \((\mu_m x \leqslant 15)\) agree with Fano’s theory\(^{7}\).

Table II

Values of \(K=\dfrac{\overline{C}}{\mu_s}\) for various combinations of the parameters \(\dfrac{\mu_s}{\mu_s}\) and \(\dfrac{C}{\mu_s}\)

\(\dfrac{\mu_s}{\mu_s}\)	\(\dfrac{C}{\mu_s}=0.02\)	0.05	0.1	0.25	0.5	1	2	4	6	8
0.02	0.0200	0.0516	0.1095	0.374	2.63	13.00	42.3	113.0	189.5	272
0.05	0.01939	0.0494	0.1052	0.332	1.328	5.49	17.21	45.5	76.5	108.9
0.1	0.01846	0.0473	0.0988	0.293	0.871	2.98	8.84	23.0	38.5	54.7
0.25	0.01617	0.0411	0.0844	0.231	0.547	1.439	3.81	9.48	15.69	22.2
0.5	0.01341	0.0338	0.0687	0.1798	0.388	0.887	2.11	4.96	8.16	11.32
1	0.01003	0.0252	0.0506	0.1289	0.265	0.559	1.211	2.67	4.24	5.87
2	—	—	0.0335	0.0841	0.1696	0.345	0.709	1.477	2.28	3.11
4	—	—	—	—	—	—	—	0.821	1.173	1.674
6	—	—	—	—	—	—	—	0.577	0.870	1.165
8	—	—	—	—	—	—	—	0.447	0.671	0.897

Table III

Values of \(\mu_m\), \(H\), \(D=\dfrac{4H^3}{27}\), and \(\overline{C}\) for various substances

Substance	\(\mu_m\), cm\(^{2}\)/g	\(H\)	\(D\)	\(\overline{C}\) (cm\(^{2}\)/g at Compton wavelength units)
H\(_2\)O	0.0167	2.0	1.3	0.118
Al	0.0216	2.1	1.4	0.0942
Fe	0.0300	2.8	3.4	0.0826
Sn	0.0351	2.6	2.7	0.0689
W	0.0391	2.5	2.4	0.0619
Pb	0.0410	2.3	1.8	0.0594
U	0.0425	2.1	1.4	0.0565

§ 16. Spencer’s Semiasymptotic Method

We now turn to the second principal work on great depths (Spencer\(^{17}\)), the only work that gives a numerical method for calculating radiation at great depths.

Spencer notes that at great penetration depths the angular distribution must have a sharply pronounced forward directionality. Such a distribution requires, for its description, a very large number of spherical harmonics. Therefore, to characterize the angular distribution of gamma rays at great penetration depths, Spencer proposes using “angular moments”

\[ y_n(p,\lambda)=\frac{1}{2\pi}\int_{4\pi} du\,(1-u_x)^n\,y(p,u_x,\lambda). \tag{16,1} \]

The equation for the angular moments in the case of a source with plane symmetry is obtained from equation (15,3) by multiplying it by \(\dfrac{1}{2\pi}(1-u_x)^n\) and integrat-

by integration over all directions

\[ [\mu(\lambda)-p]y_n(p,\lambda)+p y_{n+1}(p,\lambda)= \]

\[ =\int_{\lambda_0}^{\lambda} d\lambda'\, K(\lambda',\lambda) \sum_{n'=0}^{n}(\lambda-\lambda')^n S_n^{\,n-n'}(\lambda-\lambda')\,y_n(p,\lambda') +Q_n\delta(\lambda-\lambda_0), \tag{16,2} \]

where

\[ (-1)^{n'}[2^n-n'!\,(n-n')!]S_n^{\,n'}(z)= \frac{d^m}{dz^m}\{z^{\,n-n'}(2-z)^n\}, \]

\[ Q_n=\frac{1}{2\pi}\int_{4\pi}du\,(1-u_x)^n f(u_x). \]

In contrast to spherical harmonics, a sharply forward-directed angular distribution can be characterized by a small number of moments \(y_n(p,\lambda)\). Spencer concluded that two, at most four, of the first equations of the system (16,2) give sufficient accuracy. To solve these equations Spencer developed a semiasymptotic method; for details we refer to his work\(^{17}\).

Fig. 27. Differential spectrum of the radiation intensity of a plane directed source. \(E_0=10.22\) MeV, lead.

Fig. 28. Differential spectrum of the radiation intensity of a plane directed source. \(E_0=5.11\) MeV, iron.

work\(^{17}\). We shall point out only that the integrals entering the right-hand side of (16,2) are replaced, following Chandrasekhar\(^{52}\), by quadrature formulas. Some results of Spencer’s work\(^{17}\) are shown in Figs. 27 and 28. Fig. 27 refers to a plane directed source of energy \(10.22\) MeV in lead and gives the differential spectrum of the radiation intensity for penetration to a depth \(\mu_m x=160\). Along the ordinate is plotted the quantity \(I_0(\mu_m x,E)e^{\mu_m x}\), in units of

\[ \frac{\mathrm{MeV}}{\mathrm{cm}^2\,\mathrm{sec}}\frac{1}{\mathrm{MeV}} =\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}. \]

The intensity of the source is taken to be \(10.22\ \mathrm{MeV}/\mathrm{cm}^2\,\mathrm{sec}\), and the attenuation coefficient of the most penetrating component is \(0.466\ \mathrm{cm}^{-1}\). For comparison with the polynomial method \(^{18}\), the same figure, for \(\mu_m x = 10\), gives a dashed curve calculated by this method.

Fig. 28 corresponds to analogous calculations for a plane directed source of energy \(5.11\ \mathrm{MeV}\) in iron. The source intensity in this case is \(5.11\ \mathrm{MeV}/\mathrm{cm}^2\,\mathrm{sec}\), and the attenuation coefficient of iron corresponding to an energy of \(5.11\ \mathrm{MeV}\) is \(0.246\ \mathrm{cm}^{-1}\). Along the ordinate axis in Fig. 28 the quantity \(I_0(\mu_0 x, E)e^{\mu_0 x}\) is plotted in the same units as in Fig. 27. The calculations in this case were carried only to the value \(\mu_0 x = 50\).

Such are the principal results of theoretical work on the study of radiation penetration to great depths.

V. OTHER APPROXIMATE METHODS FOR CALCULATING MULTIPLE SCATTERING

§ 17. The Monte Carlo Method

In recent years this name has been applied widely to a direct stochastic method for calculating the number of scattered quanta. It is based on the direct computation of a large number of quantum trajectories, in which successive acts of scattering are regarded as a sequence of random processes (a Markov chain), the probability of each of which is determined by the known formulas of § 1. The trajectories of quanta resemble, in appearance, the trajectories of Brownian particles. The desired radiation intensity is measured by the number of trajectories passing through a given point of space and representing quanta of the given energy and direction*).

To illustrate the method, let us consider the simplest computational scheme for the passage of gamma rays through a plane absorbing layer of thickness \(d\), situated perpendicular to the \(x\)-axis. The source is located on the surface \(x=0\). The calculation is carried out in the following stages:

1) From the prescribed distribution of the quanta emitted by the source over the angles \(\psi_0\) and (in the case of a nonmonochromatic source) over the energy \(\alpha_0\), a random quantum \((\alpha_0,\psi_0)\) is selected. (The selection of a random quantity from a prescribed distribution is performed by a computing machine according to a separate program; see, for example, \(^{2}\).)

2) The point of the first interaction of the quantum with an atom of the absorber \(x_1\) is selected randomly in accordance with the distribution

\[ h(x_1) \simeq \frac{\mu(\alpha_0)}{\cos \psi_0}\exp\left(-\frac{\mu(\alpha_0)x_1}{\cos \psi_0}\right). \]

*) Multiple scattering of gamma rays is only one of many problems for which the Monte Carlo method is used. A large number of calculations on neutron scattering have been performed by this method \(^{63}\). The use of stochastic methods in the study of the scattering of charged particles is discussed in detail by Chavchanidze \(^{51}\). A monograph by Chandrasekhar \(^{53}\) is devoted to general questions of the study of stochastic processes. The computational scheme in the case of gamma-ray scattering was given in Berger’s work \(^{2}\). Referring the reader to these sources, we note only that, with the present wide dissemination of electronic computing machines, the Monte Carlo method is very promising, especially in solving problems with complicated boundary conditions. The concrete form of the computational scheme depends to a considerable degree on the features of the available computing machine, and the construction of special machines for carrying out calculations by the Monte Carlo method is desirable \(^{54}\).

3) In the case of an absorber of complex chemical composition, the atomic number of the atom with which the quantum interacts is chosen at random (in accordance with the relative concentration of the elements).

4) In accordance with their relative probabilities, the type of interaction is determined (absorption or scattering). In the case of absorption, the trajectory is terminated at this point, and the calculation of the next trajectory is resumed from operation 1.

5) In the case of scattering, the scattering angle \(\theta_1\) is determined (the probability of the different values of \(\theta_1\) is determined by formula (1.6)) and the azimuth \(\varphi_1\). From \(\theta_1\) and \(\varphi_1\) the energy \(\alpha_1\) of the scattered quantum is determined (by formula (1.4)) and the angle with the normal \(\psi_1\) (by the well-known formula of spherical trigonometry (18.2)). Operations 2–5 are repeated until one of the following events occurs: a) absorption of the quantum, b) decrease of the energy below the prescribed limit \(\alpha_{\min}\), beyond which, under the conditions of the problem, the form of the spectrum is of no interest, c) exit of the quantum through the plane \(x=d\) (transmission), d) exit of the quantum through the plane \(x=0\) (backscattering). The ratio of the number \(N\) of trajectories ending by method c) to the total number \(N_0\) of calculated trajectories gives the probability of transmission of the quantum

\[ p=\frac{N}{N_0}. \]

Since the number \(N\) of transmitted quanta decreases rapidly with the thickness of the absorber \(d\), to compute \(P\) for large \(d\) it is necessary to calculate a very large number \(N_0\) of trajectories, of which only a small part \(N\) reaches the plane \(x=d\).

The probable absolute error in determining the transmission probability \(P\) is equal to

\[ r=0.6745\sqrt{\frac{P(1-P)}{N_0}}. \]

Since attenuations down to \(P=10^{-6}\) are often of practical interest, to obtain an accuracy of the order of 10% it is necessary to compute \(N_0\sim 10^8\) trajectories. This problem is too cumbersome even with electronic computers.

For this reason the Monte Carlo method is more suitable for problems in which the total attenuation of the radiation is not very large, but which, because of complicated boundary conditions or for other reasons, cannot be solved by other methods.

We shall now give some results obtained by the Monte Carlo method.

Heyward and Hubbell\(^9\) considered a point directed source of quanta with energy \(1\ \mathrm{MeV}\) and constructed, from several tens of trajectories calculated with the aid of desk calculating machines, the albedo of a plane wall as a function of atomic number and angle of incidence \(\psi_0\). The calculation of each trajectory required one man-day. The albedo in energy is shown in Fig. 29. The results obtained agree with the authors’ experimental data\(^ {28}\) on reflection

Fig. 29. Energy albedo as a function of \(z\) for different angles of incidence.

gamma rays of Co\(^{60}\) from walls made of wood and pressed coils of steel wire of equal density\(*\).

To determine the energy spectrum of the reflected quanta it is necessary to compute a considerably larger number of trajectories. Some results of such calculations by Perkins\(^{14}\), carried out on an electronic computer, are shown in Fig. 30. The maximum in the soft region of the spectrum near \(\alpha = 0.4\) is characteristic not only of the reflected radiation, but also of the scattered radiation that has passed through an absorber (cf., for example, Fig. 13), while the second maximum in the hard part is apparently associated with singly scattered

Fig. 30. Energy spectrum of reflected photons at various angles of incidence of primary photons from a Co\(^{60}\) source.

quanta. This maximum was observed experimentally on various materials\(^{28,29}\).

Berger\(^{2}\) carried out calculations of the energy spectrum of scattered photons for the incidence of a plane-parallel beam of quanta with energy \(0.66\) MeV at angles \(\psi_0 = 0^\circ\) and \(60^\circ\) onto the surface of water. Cases were calculated in which the absorber thickness along the path of the primary rays was

\[ D'=\frac{\mu_0 d}{\cos \psi_0}=1,\,2,\,4 \text{ and } \infty \]

and graphs were given of the spectra of the reflected and transmitted radiation. For \(\alpha > 0.4\) the spectra of the reflected radiation for different \(D'\) differ little, whereas for \(\alpha < 0.4\) a semi-infinite absorber gives 2–3 times more soft quanta than a layer of thickness \(D' = 2\). In the angular distribution of the reflected radiation for \(D'=\infty\), near \(\cos \psi_0 = -1\), the number of quanta is twice as large as for \(D'=2\).

In the case of appreciable attenuation of the radiation, it is possible, without increasing the error \(r\), to reduce the number \(N_0\) by various artificial devices, for example by introducing weight factors that single out families of such trajectories which, with high probability, reach the point where the intensity is to be calculated, or by combining the Monte Carlo method with analytical methods.

One such device was used\(^{15}\) in calculating the intensity of radiation transmitted through layers of H\(_2\)O, Al, Fe, Sn, and Pb of thickness up to \(\mu_0 x = 12\), for normal and oblique incidence of a plane-parallel beam of gamma—

\(*\) Grazhdankina and Fakidov\(^{43}\) determined, by a photographic method, the dependence of the intensities \(I'\) of reflected radiation on the density \(\rho\) of the material of the scattering wall and found that \(I'\) has a maximum near \(\rho = 2\).

THEORY OF MULTIPLE SCATTERING OF GAMMA RAYS

rays of Cs\(^{137}\) and Co\(^{60}\). For certain values of the parameters, the energy spectrum and angular distribution of the transmitted radiation were calculated. Comparison with the data of Goldstein and Wilkins\(^{62}\) for an infinite half-space, calculated by the method of polynomial expansions\(^{18}\), shows that the build-up factor \(B\) in the case of an absorber of finite thickness increases more slowly with \(\mu_0 x\) than for a semi-infinite one. Curves are given for the angular distribution of radiation transmitted through aluminum. As the thickness \(x\) increases, this distribution approaches the isotropic one.

Berger and Doggett\(^{4}\) obtained a large number of results by combining the Monte Carlo method with analytical calculations. In this work, all trajectories with the same sequence of values of the energy \(E_n\) and angle \(\theta_n\) (\(n\) is the number of the scattering event) are combined into \(E,\theta\)-families. The probability of the occurrence of various \(E,\theta\)-trajectories is found by multiplying the probabilities (1,6) of elementary scattering events. For each set \(E,\theta\), a transport equation is solved for the probability \(P\) of passage of a quantum through the plane \(x=t\) after the \(n\)-th scattering.

The results of these calculations are expressed in terms of the build-up factors \(B_{tt}\), \(B_{t\infty}\), and \(B_{0t}\). The first refers to radiation that has passed through a layer of finite thickness \(t\), the second to radiation at depth \(t\) in a semi-infinite absorber. The third is defined as the ratio of the flux density of quanta (or, correspondingly, of energy) near the surface of the scattering object facing the source to the flux density at the same point if the scatterer is removed.

Let the mean value of \(B\) over all \(i\) calculated trajectories be denoted by \(\overline{B}\). The accuracy of the calculation for a given \(i\) is determined by the variance

\[ \sigma=\left[\frac{1}{i}\sum_{j=1}^{i}\left(B_j-\overline{B}\right)^2\right]^{\frac{1}{2}}. \]

Consider the ratio

\[ q=\frac{\left(\overline{B}_{tt}-1\right)}{\left(\overline{B}_{t\infty}-1\right)} \]

of the number of scattered quanta at the exit from a layer of thickness \(t\) to the number of scattered quanta at the same depth \(t\) in a semi-infinite absorber. It characterizes the influence of the more deeply situated layers of the absorber on the number of quanta at the given depth. It turns out that, owing to the strong correlation of \(\overline{B}_{tt}\) and \(\overline{B}_{t\infty}\), the variance of \(q\), denoted by \(\sigma_q\), is very small and, for thicknesses \(\mu_0 t \ge 4\), practically does not depend on \(t\). Since for medium and large \(t\) the quantity \(\overline{B}_{t\infty}\) is practically equal to \(\overline{B}_{\infty\infty}\), which is calculated exactly by analytical methods (see Chapter III), the smallness of \(\sigma_q\) makes it possible, from the known \(\overline{B}_{\infty\infty}\), to find \(\overline{B}_{tt}\) with considerably less labor (with a smaller number \(i\)) than by direct calculation.

The values of the energy build-up factor \(B_E\) calculated by the authors are given in Table IV, and the values of \(q\) in Table V. As can be seen, the larger \(Z\) and \(E\), the smaller the depth from which \(q\) ceases to increase. In lead, already at \(\mu_0 t=1\), practically \(B_{tt}=B_{t\infty}\). The accuracy of the experimental data is usually such that insignificant deviations of \(q\) from 1 cannot be observed, and therefore, when the estimates are not very precise, one may take \(B_{tt}=B_{t\infty}\) for \(\mu_0 t>2\). The influence of the lower-lying layers of the absorber on the form of the spectrum at depths \(\mu_0 t=4\) and 8 in water for \(E_0=0.66\) MeV

Table IV

Growth factor \(B_E\)

Material	Energy	\(\mu_0 t = 0,5\)	\(\mu_0 t = 1,0\)	\(\mu_0 t = 2,0\)	\(\mu_0 t = 4,0\)	\(\mu_0 t = 8,0\)	\(\mu_0 t = 16,0\)
Water	0,66	1,49	1,96	3,10	5,99	13,3	39,4
Water	1	1,40	1,80	2,72	5,01	10,5	25,7
Water	4	1,22	1,42	1,83	2,60	4,21	7,20
Iron	1	1,40	1,72	2,43	4,07	7,80	17,8
Iron	4	1,20	1,36	1,72	2,50	4,17	7,45
Iron	10	1,07	1,16	1,35	1,75	2,80	5,85
Tin	1	1,29	1,56	2,10	3,15	5,31	10,2
Tin	4	1,16	1,31	1,63	2,35	4,12	9,41
Tin	10	1,06	1,12	1,26	1,59	2,75	8,22
Lead	1	1,20	1,35	1,63	2,09	2,87	4,24
Lead	4	1,11	1,23	1,44	1,98	3,28	7,46
Lead	10	1,03	1,08	1,17	1,40	2,17	6,47

Table V

Correlation coefficient \(q\)

Material	Energy	\(\mu_0 t = 0,5\)	\(\mu_0 t = 1,0\)	\(\mu_0 t = 2,0\)	\(\mu_0 t = 4,0\)	\(\mu_0 t = 8,0\)	\(\mu_0 t = 16,0\)
Water	0,66	0,601	0,663	0,713	0,783	0,785	0,784
Water	1	0,661	0,720	0,754	0,821	0,828	0,830
Water	4	0,849	0,885	0,912	0,920	0,926	0,933
Iron	1	0,790	0,798	0,851	0,890	0,895	0,894
Iron	4	0,890	0,910	0,923	0,936	0,932	0,949
Iron	10	0,941	0,959	0,972	0,974	0,978	0,977
Tin	1	0,889	0,911	0,924	0,935	0,938	0,946
Tin	4	0,941	0,926	0,955	0,967	0,974	0,978
Tin	10	0,951	0,960	0,962	0,973	0,971	0,969
Lead	1	0,939	0,951	0,969	0,975	0,979	0,982
Lead	4	0,941	0,977	0,982	0,990	0,992	0,994
Lead	10	0,986	0,990	0,995	0,992	0,994	0,995
Estimated accuracy in %	—	±5,0	±2,0	±1,5	±1,5	±2,0	±2,5

is presented in Fig. 31. The unshaded area represents the spectrum at the exit from a layer of thickness \(t\), while the shaded area represents the soft radiation backscattered by the deeper layers in the semi-infinite absorber. The corresponding plots of the angular dependence of the energy flux at depths \(\mu_0 t = 4\) and \(8\) show that, at \(t = \infty\), the radiation propagating at angles

\[ \theta > \frac{\pi}{2} \]

constitutes about 20% of the total energy flux at depths \(\mu x = 4\) and \(8\), independently of the value of \(\mu_0 t\), while the increase of the flux in the direction \(\theta < \frac{\pi}{2}\) due to an increase of \(d\) is very small.

The spectral and angular distribution of the radiation reflected from a semi-infinite water absorber was also calculated \((E_0 = 0,66\ \text{MeV})\).

In both distributions there are two maxima. This confirms Perkins’ data\(^{14}\), as well as Bopp’s results\(^{15}\), obtained with allowance only for twice-scattered radiation. The first maximum, at an energy close to

\[ \alpha=\frac{\alpha_0}{1+\alpha_0}, \]

corresponds to the first scattering through an angle close to \(90^\circ\), with subsequent deflection of the quanta through angles close to \(0^\circ\). The corresponding

Fig. 31. Spectrum of scattered radiation that has passed through a layer of water of thickness \(X=4\) and \(X=8\) for normal incidence of primary quanta with energy \(0.66\) MeV. The smooth curves were calculated by the method of moments for an infinite absorber.

maximum in the angular distribution lies near \(\theta=90^\circ\). The second maximum, at

\[ \alpha=\frac{\alpha_0}{2+\alpha_0}, \]

is due to a single deflection through an angle close to \(180^\circ\), while in the remaining collisions the direction of the quanta changes little. In the angular distribution this gives a maximum near \(\theta=180^\circ\).

§ 18. Approximate Direct Methods

In most approximate methods for calculating scattered radiation, the total density of quanta \(N\) (or intensity \(I\)) is represented as the sum of the densities \(N_i\) of quanta scattered \(i\) times,

\[ N=\sum_{i=0}^{\infty} N_i(\mathbf{r}, \mathbf{u}, \lambda). \tag{18,1} \]

The density of the primary quanta \(N_0\) and of the once-scattered quanta \(N_1\) is found without difficulty. The computation of \(N_2\) is usually also not excessively cumbersome. To estimate \(N_i\) for \(i>2\), various approximate methods are used, which are considered below.

However, a number of features of scattered radiation already appear when considering only the terms \(N_1\) and \(N_2\), which we shall carry out for the example of a plane problem.

Consider a homogeneous layer of material containing \(n\) electrons in \(1\ \mathrm{cm}^3\), of thickness

\[ d=\frac{X}{\mu_0} \]

in the \(x\)-direction and infinite in the \(y\)- and \(z\)-directions.

... directions (Fig. 32). Let a gamma quantum with energy \(\alpha_0\) enter in a direction making an angle \(\psi_0\) with the normal to the surface of the layer, travel a distance \(s_0\) inside the layer before the first collision, be scattered in a direction making an angle \(\theta_1\) with the initial direction of the ray and an angle \(\psi_1\) with the normal to the surface of the layer, travel a distance \(s_1\) before the second collision, be scattered in a direction characterized by the angles \(\theta_2\) and \(\psi_2\), and so on. In addition to the angles \(\theta_k\) and \(\psi_k\), the direction of the ray is also characterized by the azimuthal angle \(\varphi_k\); these angles are related by

Fig. 32. Scheme of multiple scattering in a plane absorber.

\[ \cos\psi_{k+1}=\cos\psi_k\cos\theta_{k+1}+ +\sin\psi_k\sin\theta_{k+1}\cos\varphi_{k+1}. \tag{18,2} \]

The energy of the gamma quantum between the \(k\)-th and \((k+1)\)-st collisions is equal to \(\alpha_k\); the total absorption coefficient of the material of the layer for this energy shall be denoted by \(\mu_k=\mu(\alpha_k)\).

The probability that a photon will pass through the layer along any possible path, having undergone \(i\) collisions, is equal to the number \(N_i\) of photons that have passed through, referred to one incident photon, and is expressed by the integral \({}^{13}\)

\[ N_i=\iint\ldots\int e^{-\mu_i s_i}\prod_{k=0}^{i-1}e^{-\mu_k s_k}\overline{K}_{k+1}\sin\theta_{k+1}d\theta_{k+1}ds_kd\varphi_{k+1}, \tag{18,3} \]

where \(\overline{K}_{k+1}\) is determined by formula (1,8) with the substitution \(\theta=\theta_{k+1}\). The limits of integration in (18,3) are determined by the relations

\[ 0\leq \varphi_k\leq 2\pi,\qquad 0\leq \theta_k\leq \pi, \tag{18,4} \]

\[ 0\leq s_{k-1}\gamma_{k-1}\leq a-\sum_{i=0}^{k-2}s_i\gamma_i,\qquad \text{if}\quad 0\leq \psi_{k-1}\leq \frac{\pi}{2}, \tag{18,5} \]

\[ 0\geq s_{k-1}\gamma_{k-1}\geq -\sum_{i=0}^{k-2}s_i\gamma_i,\qquad \text{if}\quad \frac{\pi}{2}<\psi_{k-1}\leq \pi, \tag{18,6} \]

where \(\gamma_i=\cos\psi_i\). If, after the last collision before leaving the layer, \(\gamma_i>0\), then the quantum will pass through the layer. If \(\gamma_i<0\), it will leave from the same side from which it entered, that is, it will be reflected from the layer. For \(k=1\) the expression \(\sum_{i=0}^{k-2}s_i\gamma_i\) is replaced by zero.

In expression (18,3) the integration with respect to \(s_k\) is easily carried out, whereas the integration with respect to the angular variables can be performed only numerically. Introducing the notation \(\tau_i=\dfrac{\mu_i d}{\gamma_i}\) and \(d\Omega_i=\sin\theta_i d\theta_i d\varphi_i\), we have, for \(i=1\) and \(i=2\), after

integration over $S_0$ and $S_1^*$):

\[ N_1^{+}=ae^{-v_0}\int \frac{1-e^{-(v_1-v_0)}}{(v_1-v_0)\gamma_0}\,\overline K_1\,d\Omega_1, \tag{18,7} \]

\[ N_1^{-}=a\int \frac{1-e^{-(v_0-v_1)}}{(v_0-v_1)\gamma_0}\,\overline K_1\,d\Omega_1, \tag{18,8} \]

\[ N_2^{++}=a^2e^{-v_0}\int \frac{ \dfrac{1-e^{-(v_2-v_0)}}{(v_2-v_0)\gamma_0} - \dfrac{1-e^{-(v_1-v_0)}}{(v_1-v_0)\gamma_0} }{ (v_1-v_2)\gamma_1 }\, \overline K_1\overline K_2\,d\Omega_1d\Omega_2, \tag{18,9} \]

\[ N_2^{-+}=a^2e^{-v_0}\int \frac{ \dfrac{1-e^{-(v_2-v_1)}}{(v_2-v_1)\gamma_0} - \dfrac{1-e^{-(v_2-v_0)}}{(v_2-v_0)\gamma_0} }{ (v_0-v_1)\gamma_1 }\, \overline K_1\overline K_2\,d\Omega_1d\Omega_2, \tag{18,10} \]

\[ N_2^{+-}=a^2\int \frac{ \dfrac{1-e^{-(v_1-v_2)}}{(v_1-v_2)\gamma_0} - \dfrac{1-e^{-(v_0-v_2)}}{(v_0-v_2)\gamma_0} }{ (v_0-v_1)\gamma_1 }\, \overline K_1\overline K_2\,d\Omega_1d\Omega_2, \tag{18,11} \]

\[ N_2^{--}=a^2\int \frac{ \dfrac{1-e^{-(v_0-v_2)}}{(v_0-v_2)\gamma_0} - \dfrac{1-e^{-(v_0-v_1)}}{(v_0-v_1)\gamma_0} }{ (v_1-v_2)\gamma_1 }\, \overline K_1\overline K_2\,d\Omega_1d\Omega_2. \tag{18,12} \]

Here the indices $+$ and $-$ denote, respectively, scattering at angles $0<\psi_k\le \dfrac{\pi}{2}$ and $\dfrac{\pi}{2}<\psi_k\le\pi$ (Fig. 33). The first sign in $N_2$ refers to $\gamma_1$, the second to $\gamma_2$.

Fig. 33. Trajectories of quanta in single and double scattering in a plane absorber.

To find the energy carried by photons that have undergone $i$ collisions, it is necessary to multiply the integrand in (18,3) by

\[ \alpha_i= \frac{\alpha_0}{ \displaystyle \prod_{k=0}^{i-1}\left[1-\alpha_0(1-\cos\theta_{k+1})\right] }. \]

Table VI gives numerical values of $N_0$, $N_1$, $N_2$ for gamma rays with energy $\alpha_0=0.2\div 5$, incident normally on a layer of material of thickness $X$. It is assumed that absorption occurs only by the Compton effect, so that the results are valid for any absorber.

\[ \text{*) In these formulas misprints were allowed in the original paper.} \]

V. S. GALISHCHEV, V. I. OGIEVETSKII, A. N. ORLOV

Table V

\(a_0\)	\(X\)	\(N_0\)	\(N_1^+\)	\(N_1^-\)	\(N_2^{++}\)	\(N_2^{-+}\)	\(N_2^{--}\)	\(N_2^{+-}\)
5	0,5	0,6065	0,167	0,0264	0,0407	0,0054	0,011	0,0093
5	1	0,3679	0,174	0,0292	0,0672	0,0034	0,015	0,012
5	2	0,1353	0,103	0,0294	0,0613	0,00085	0,016	0,012
5	4	0,0183	0,0210	0,0294	0,0167	0,00012	0,016	0,012
2,5	0,5	0,6065	0,169	0,0405	0,0385	0,0078	0,015	0,013
2,5	1	0,3679	0,180	0,0458	0,0659	0,0066	0,020	0,017
2,5	2	0,1353	0,110	0,0462	0,0616	0,0018	0,022	0,018
2,5	4	0,0183	0,0229	0,0462	0,0179	0,00022	0,022	0,018
1,0	0,5	0,6065	0,160	0,0606	0,0374	0,010	0,018	0,017
1,0	1	0,3679	0,172	0,0720	0,0645	0,012	0,028	0,024
1,0	2	0,1353	0,104	0,0744	0,0609	0,0052	0,032	0,027
1,0	4	0,0133	0,0215	0,0745	0,0181	0,00055	0,032	0,027
0,2	0,5	0,6065	0,133	0,0996	0,0278	0,013	0,021	0,023
0,2	1	0,3679	0,144	0,119	0,0531	0,020	0,033	0,036
0,2	2	0,1353	0,0886	0,128	0,0516	0,013	0,039	0,044
0,2	4	0,0183	0,0179	0,130	0,0153	0,0022	0,040	0,045

Analysis of Table VI leads to the following conclusions:

1) The number of transmitted photons \(N_1^+ + N_2^{++} + N_2^{-+}\) increases as the energy \(a_0\) increases and the thickness \(X\) decreases. The fraction of photons scattered backward, \(N_2^{-+}\), is much smaller than each of the other two terms, especially at large \(a_0\). The fraction of twice-scattered photons \(N_2^{++} + N_2^{-+}\) increases with \(X\) up to \(X = 1\) and depends only weakly on \(a_0\).

2) The number of reflected photons \(N_1^- + N_2^{+-} + N_2^{--}\), with increasing thickness, tends to a certain limiting value \(N_\infty^-\), which is reached in the case \(a_0 = 5\) at \(X = 1\), and in the case \(a_0 = 0,2\) at about \(X = 4\). As \(a_0\) increases, both the absolute value of the number of reflected photons and the ratio of the number of reflected photons to the number of transmitted photons decrease. As the energy increases, the fraction of twice-scattered photons \(N_2^{+-} + N_2^{--}\) in the total sum of reflected photons \(N_1^- + N_2^{+-} + N_2^{--}\) increases.

Comparison of the various terms of the transmitted and reflected energy \(E\) leads essentially to the same conclusions as in the case of \(N\). It should only be noted that at high energies (\(a_0 = 5\) and \(2,5\)), among all the terms describing double scattering, the largest is \(E^{++}\). The others are considerably smaller.

From these conclusions it follows, in particular, that in computing the number of transmitted quanta \(N^+ + N^{++} + N^{-+}\), the quanta scattered backward, \(N^{-+}\), may be neglected. The same applies, to an even greater extent, to the fraction of the energy \(E^{-+}\), which is much smaller than \(E^+ + E^{++}\).

However, in the case of small \(X\) (\(X \sim 1\)) and at low energies (\(a_0 < 1\)), backscattering \(N_2^{--} + N_2^{+-}\) makes a noticeable contribution to the total energy of the scattered photons. Therefore, in computing the probability of passage of gamma rays of low energy through a thin layer of material, it is more important to take backscattering into account than the photoelectric effect.

The calculation of the energy \(E_i\) and the probability of passage \(N_i\) for large \(i\) is very difficult. Meanwhile, as indicated above, a quantum with energy of several \(Mc^2\) may undergo up to 10–15 collisions before it is absorbed by the photoelectric effect*).

*) This gives grounds for considering the propagation of gamma quanta as a kind of diffusion process.⁴⁵

Usually, already after the first 2–3 scattering events, the quantum energy falls to several tenths of an MeV. (The exception is scattering through small angles, but in this case the formulas are simplified and it is possible to compute scattering of multiplicity greater than 2.) It is therefore important to determine the fraction of quanta scattered 0, 1, 2, ... times in the energy of the transmitted or reflected radiation. A comparison\(^{11}\) of the values \(N_1\) and \(N_2\) obtained from formulas (18.7)—(18.12) with \(N_i\) \((i>2)\), estimated by extrapolation, shows that, for example, in the case of lead at energy \(\alpha_0 = 5\) and normal incidence \((\gamma_0 = 1)\), multiply scattered quanta do not make a substantial contribution to the radiation emerging from the layer. For brevity, let us call the order of a photon \(k\) the number of collisions it has undergone before leaving the layer. Then one may say that, as the thickness of the layer decreases, the order \(k_{\max}\) of those photons that still make a noticeable contribution to the energy of the transmitted radiation decreases: for \(X = 20\), \(k_{\max} = 8\); for \(X = 4\), \(k_{\max} = 5\). It is significant here that, for a layer thickness \(X = 1\), it is sufficient to take into account only the first two scatterings (single and double).

Thus, to compute the total probability of transmission of gamma rays through a thick layer of heavy material \((x<20)\), it is necessary to find the transmission probabilities of photons whose order is no higher than 5–6. In this case, as the estimates given show, the error in the computation will not exceed 20%. For lighter absorbers, the order of photons whose transmission must be taken into account will be higher.

Extensive material on the results of calculations carried out by the direct method described, for various absorbers and geometrical conditions, is given in works\(^{11}\) and \(^{12}\).

Formulas (18.7)—(18.12) are not difficult to generalize to the case of a plane absorber consisting of several layers of unlike materials. The problem of the passage of gamma rays through a two-layer absorber, taking into account only singly scattered radiation, was considered by Orlov and Fedorov\(^{32}\) in order to determine which arrangement of the absorbers gives the greatest attenuation of intensity. It turned out that if the energy of the incident quanta is less than the energy corresponding to the minimum on the curve \(\mu(\alpha)\), then the intensity is attenuated more when the heavy absorber is placed behind the light one. If, however, a large fraction of the spectrum of the incident radiation falls in the region where pair production is substantial, then the attenuation is greater when the heavy absorber is placed in front. These conclusions are qualitatively confirmed by the experiments of Tsypin, Kukhtovich, and Kazanskii\(^{38}\) with Co\({}^{60}\) gamma rays (the case of low energies) and by those of Orlov and Fedorov\(^{32}\) with betatron bremsstrahlung (the case of high energies).

Let us now briefly consider some approximate methods for computing the terms (18.1) with \(i>2\). One of them\(^{10}\) is based on the assumption that the energy \(\alpha_i\) of all \(i\)-fold scattered quanta is the same. Consequently, their scattering angle \(\theta_i\) is also the same. Under this assumption it is possible, for all terms of the sum

\[ I = \sum_{(i)} I_i \]

to write down and successively solve the transport equation. In the case of the plane problem\(^{10}\), the coefficients of these equations include the mean values \(\cos \psi_i\) (\(\psi_i\) is the angle with the normal of the \(i\)-fold scattered quanta), which are found approximately, in a manner the more accurate the smaller the scattering angles \(\theta_i\). By this method a number of results have been obtained for absorption in lead, iron, water, and concrete for absorber thicknesses \(D = 1, 2, 5, 10\), and 20 and \(\alpha_0 = 2, 6, 10\). Some of these results are given in Gorshkov’s monograph\(^{42}\) and in the book by Rumyantsev and Grigorovich\(^{50}\).

The indicated method is also applicable in the cases of continuously distributed\(^{25}\) and point isotropic\(^{24,31}\) sources.

Another technique for estimating the terms of (18.1) with large \(i\) was used in a series of papers\(^6\), in which the dose rate inside or behind an absorber is calculated directly. The technique amounts to calculating the dose rate \(M\) in two ways, with a deficit (\(M^{-}\)) and with an excess (\(M^{+}\)). If, analogously to (18.1), one writes

\[ M=\sum_i M_i, \]

then \(M_0+M_1+M_2\) may be taken as \(M^{-}\).

To calculate \(M^{+}\), various approximations are used. In particular, such mean values of the energy \(\alpha'\) and of the cosine \(\gamma'\) of the angle of incidence of the scattered quanta that have passed through the absorber are chosen for which \(M\) is maximal. The true value of \(M\) lies between \(M^{-}\) and \(M^{+}\). This method has low accuracy, but is quite suitable for estimating the dose rate from scattered radiation in industrial radiography at energies up to \(2\)—\(3\) MeV, in shielding calculations, etc. As a rule, it is not applicable to problems with more complicated geometry, in particular to multilayer absorbers, and also in those cases where, in addition to the dose rate, it is necessary to determine the spectrum of the scattered radiation.

§ 19. Method of successive passage through thin layers\(^ {13}\)

The essence of this method is that the passage of gamma rays through a thick layer of material is regarded as successive passage through a number of thin elementary layers, of thickness not greater than the photon mean free path. In calculating the transmission and reflection of gamma rays from such a layer, it is sufficient in the sum (18.1) to restrict oneself to the terms with \(i=0,1,2\). This reduces the computations and decreases the error in the calculation. Let \(\psi\) denote the angle with the normal to the layer, \(\cos\psi=\gamma\). If the photon density \(N\) at the left boundary of the elementary layer with number \(i\) is in question, we shall denote the density by \(l_i(\lambda,\gamma)\), and for the right boundary by \(r_i(\lambda,\gamma)\) (Fig. 34). The functions \(l\) and \(r\) represent the distributions of photons in energy and angles and, for brevity, will be called distributions.

Fig. 34.

Let \(\hat O\) be an operator that transforms the distribution of gamma photons incident on a layer into the distribution of photons that have passed through it, and let \(\hat S\) be an operator that transforms the incident distribution into the distribution reflected from the layer. Obviously, for any distributions \(l\) and \(r\) the following expressions must hold:

\[ \hat O(l+r)=\hat Ol+\hat Or,\qquad \hat Ol+\hat Sl=(\hat O+\hat S)l,\qquad \hat O(\hat Sl)=\hat O\hat Sl. \]

Consider two adjoining layers (Fig. 34), and let \(l_1\) be the distribution entering the left plane of the first layer, and \(l'_1\) the distribution reflected from both layers. Let, further, \(l_2\) and \(r_2\) be the distributions entering the second layer, and \(l'_2\) and \(r'_2\) the distributions emerging from this layer. Finally, let \(\hat O_1,\hat S_1\) and \(\hat O_2,\hat S_2\) be the transmission and reflection operators, respectively, for the first and second layers. The distributions \(l\) and \(r\), as is easy to see, are related to one another

by the following relations:

\[ \begin{aligned} l_2&=\hat O_1 l_1+\hat S_1 l'_2, & l'_1&=\hat O_1 l'_2+\hat S_1 l_1,\\ l'_2&=\hat O_2 r_2+\hat S_2 l_2, & r_2&=\hat O_2 l_2+\hat S_2 r_2 . \end{aligned} \tag{19,1} \]

Substituting \(l'_2\) from the second equation into the first, we have

\[ l_2=\hat O_1 l_1+\hat S_1\hat O_2 r_2+\hat S_1\hat S_2 l_2. \]

Applying this relation repeatedly to \(l_2\), we obtain

\[ l_2=(\hat O_1+\hat S_1\hat S_2\hat O_1)l_1+ (\hat S_1\hat O_2+\hat S_1\hat S_2\hat S_1\hat O_2)r_2+ (\hat S_1\hat S_2)^2l_2 \]

and, repeating this process \(k\) times, we shall have

\[ \begin{aligned} l_2={}&[\hat O_1+\hat S_1\hat S_2\hat O_1+(\hat S_1\hat S_2)^2\hat O_1+\ldots+(\hat S_1\hat S_2)^k\hat O_1]\,l_1+\\ &+[\hat S_1\hat O_2+\hat S_1\hat S_2\hat S_1\hat O_2+(\hat S_1\hat S_2)^2\hat S_1\hat O_2+\ldots+(\hat S_1\hat S_2)^k\hat S_1\hat O_2]\,r_2+\\ &+(\hat S_1\hat S_2)^{k+1}l_2 . \end{aligned} \tag{19,2} \]

Substituting \(l_2\) from the first equation (19,1) into the second and carrying out the same calculations, we find

\[ \begin{aligned} l'_2={}&[\hat S_2\hat O_1+\hat S_2\hat S_1\hat S_2\hat O_1+(\hat S_2\hat S_1)^2\hat S_2\hat O_1+\ldots+(\hat S_2\hat S_1)^k\hat S_2\hat O_1]\,l' +\\ &+[\hat O_2+\hat S_2\hat S_1\hat O_2+\ldots+(\hat S_2\hat S_1)^k\hat O_2]\,r_2+(\hat S_2\hat S_1)^{k+1}l'_2 . \end{aligned} \tag{19,3} \]

The quantities \((\hat S_1\hat S_2)^{k+1}l_2\) and \((\hat S_2\hat S_1)^{k+1}l'_2\) are the distributions of those photons which are reflected repeatedly between the layers \(k+1\) times. For sufficiently large \(k\), these terms may be neglected, since the probability of such scattering is negligibly small.

Substituting expressions (19,2) and (19,3) into the third and fourth equations (19,1), we obtain:

\[ \begin{aligned} l'_1={}&[\hat S_1+\hat O_1\hat S_2\hat O_1+\hat O_1\hat S_2\hat S_1\hat S_2\hat O_1+\hat O_1(\hat S_2\hat S_1)^2\hat S_2\hat O_1+\ldots]\,l_1+\\ &+[\hat O_1\hat O_2+\hat O_1\hat S_2\hat S_1\hat O_2+\hat O_1(\hat S_2\hat S_1)^2\hat O_2+\ldots]\,r_2, \end{aligned} \tag{19,4} \]

\[ \begin{aligned} r'_2={}&[\hat O_2\hat O_1+\hat O_2\hat S_1\hat S_2\hat O_1+\hat O_2(\hat S_1\hat S_2)^2\hat O_1+\ldots]\,l_1+\\ &+[\hat S_2+\hat O_2\hat S_1\hat O_2+\hat O_2\hat S_1\hat S_2\hat S_1\hat O_2+\hat O_2(\hat S_1\hat S_2)^2\hat S_1\hat O_2+\ldots]\,r_2 . \end{aligned} \tag{19,5} \]

Expressions (19,4) and (19,5) give the distributions of photons emerging from the outer planes of two adjoining layers, expressed in terms of the distributions incident on these layers. The coefficients of \(l_1\) and \(r_2\) give expressions for the operators \(\hat O\) and \(\hat S\) for the double layer. Depending on the required accuracy of the calculation, one must take one or another number of terms in the series \(1+\hat S_1\hat S_2+(\hat S_1\hat S_2)^2+\ldots\). Each of the terms in the expression for \(\hat S\) and \(\hat O\) of the double layer has a simple physical meaning. Thus, for example, the term \(\hat O_2\hat O_1\) is the operator for direct passage through two layers; \(\hat O_2\hat S_1\hat S_2\hat O_1\) is an operator,

describing passage through the first layer, reflection from the second layer, reflection from the first layer and, finally, passage through the second layer. The coefficients at \(l_1\) and \(r_2\) represent all possible successive passages and reflections inside the double layer, which end respectively in passage or reflection. It is clear that the process considered above for finding the operators of a double layer can be applied to finding the operators \(\hat S\) and \(\hat O\) for a triple layer, a fourth layer, and so on.

The operators \(\hat O\) and \(\hat S\) for an elementary layer, in turn, are expressed in terms of \(e^{-\nu_0}\) and integrals of the form (18.7)—(18.12).

In papers \(^{11,12}\) results are given of calculations of the passage of gamma radiation through iron and lead, carried out by this method. To shorten the calculations, back scattering was not taken into account. The results show that the method can be applied for gamma-quantum energies in the interval \(1\)—\(20\) MeV and for depths not exceeding \(X=20\). The error in the number of quanta \(N\), associated with neglect of triple, quadruple, etc., scattering, in the case of iron may reach \(30\%\). The error in the energy of the transmitted gamma quanta will not exceed several percent. This is due to the fact that quanta scattered more than twice have, on average, a small energy. If the initial energy of the gamma quanta is less than \(1\) MeV, scattering of multiplicity \(i>2\) can no longer be neglected. In this case, to compute the energy of the transmitted radiation it is better to use the direct methods set forth in the preceding paragraph, if it is undesirable to reduce the thickness of the elementary layer.

The direct methods and the method of successive passages through thin layers give convergent results in that region of energies and depths where the method of successive passages is applicable. The method of successive passages evidently admits a generalization to the case of multilayer absorbers.

CONCLUSION

The calculation methods considered in this review make it possible to elucidate the general regularities of the propagation of gamma rays in thick absorbers and to compute, with a sufficient degree of accuracy, the spatial distribution and the energy spectrum of scattered gamma rays in all practically encountered problems. For the study of the most general features of the spectrum at large distances from the source, the analytical methods set forth in Chapter IV are suitable. For accurate numerical calculations in the case of simple geometrical conditions and not very large penetration depths (corresponding to an attenuation of the intensity by up to \(10^6\)—\(10^8\) times), one should use the method of polynomial expansions (Chapter III). In most practical problems (calculation of shielding devices, estimation of scattered radiation in gamma-defectoscopy, etc.) it is more advantageous to apply one of the approximate methods considered in Chapter V.

CITED LITERATURE

I. Theoretical works containing numerical data in the form of tables or graphs. (The quantum energy is indicated in MeV and the geometry of the source, the material and thickness of the absorber; p. — point source, pl. — plane.)

1. M. J. Berger, J. Appl. Phys. 26, 1504 (1955), \(2\) MeV, p., isotropic, water, \(\mu x\) up to 7.
2. M. J. Berger, J. Res. Nat. Bur. Stand. 55, 343 (1955), Cs\(^{137}\), pl., inclined, water, \(\mu x\) up to 4.
3. M. J. Berger, J. Res. Nat. Bur. Stand. 56, 111 (1956), Cs\(^{137}\), pl., inclined, water, \(\mu x\) up to 8.
4. M. J. Berger, J. Dogget, J. Res. Nat. Bur. Stand. 56, 89 (1956); \(0, 66; 1, 4, 10\) MeV, pl., inclined, water, Fe, Sn, Pb, \(\mu x\) up to 16.
5. F. Bopp, Ann. der Phys. 30, 35 (1937), \(2.6\) MeV, p., Pb \(0.2\) cm, Al \(0.4\) cm.

1. L. Cave, J. Corner, R. H. A. Liston, Proc. Roy. Soc. A 204, 223 (1950), J. Corner, R. H. A. Liston, ibid. 204, 323 (1950), J. Corner, F. A. G. Day, ibid. 204, 329 (1950), up to 5 MeV, t., p., any atomic number, \(\mu x\) up to 16.
2. U. Fano, J. Res. Nat. Bur. Stand. 51, 95 (1953), Co\(^{60}\), p., water; 10 MeV, p., Fe, \(\mu x=10\)—50.
3. L. L. Foldy, R. K. Osborn, Phys. Rev. 81, 400 (1951), 17 MeV, p., water up to 450 cm.
4. E. Hayward, J. Hubbell, Phys. Rev. 93, 955 (1954), 1 MeV, t., directed, Al, Cu, Sn, Pb, water—albedo.
5. J. O. Hirschfelder, J. L. Magee, M. H. Hull, Phys. Rev. 73, 852 (1948), J. O. Hirschfelder, E. N. Adams, ibid. 73, 863 (1948), 2, 3, 5 MeV, p., Fe, Pb, concrete, \(\mu x=1, 2, 5, 10, 20\).
6. G. H. Peebles, J. Appl. Phys. 24, 1272 (1953), 0.5—10 MeV, t., Fe, Pb, air, etc., \(\mu x\) up to 20.
7. G. H. Peebles, J. Appl. Phys. 24, 1437 (1953), 0.5—10 MeV, t., Fe, Pb, air, etc., \(\mu x\) up to 20.
8. G. N. Peebles, M. S. Plesset, Phys. Rev. 81, 430 (1951), 5 MeV, p., Pb, U, \(\mu x\) up to 35.
9. J. F. Perkins, J. Appl. Phys. 26, 655 (1955), up to 6 MeV, p., inclined, Al, concrete, \(\mu x\) up to 4.
10. J. F. Perkins, J. Appl. Phys. 26, 1372 (1955), Co\(^{60}\), Cs\(^{137}\), p., inclined, Al, Fe, Sn, Pb, \(\mu x\) up to 12.
11. M. S. Plesset, S. T. Cohen, J. Appl. Phys. 22, 350 (1951), 0.5—5 MeV, t., air 10—140 m.
12. L. V. Spencer, Phys. Rev. 88, 793 (1952), —10 MeV, p., isotropic, Pb \(\mu x\) up to 160.
13. L. V. Spencer, U. Fano, J. Res. Nat. Bur. Stand. 46, 446 (1951), summary in Phys. Rev. 81, 464 (1951), up to 10 MeV, p. t., Pb \(\mu x\) up to 8.
14. L. V. Spencer, F. A. Stinson, Phys. Rev. 85, 662 (1952), Co\(^{60}\) p., t., water, \(\mu x\) up to 9.

II. Experimental works (an asterisk marks works that also contain theoretical results).

1. L. A. Beach, R. B. Theus, W. H. Faust, Phys. Rev. 92, 355 (1953), Na\(^{24}\), Co\(^{60}\), Cs\(^{137}\), p., Fe 13 cm.
2. J. Clay, C. Wansdronk, T. J. Dekker, Physica 18, 582 (1952), Na\(^{24}\), RaC, p., Pb up to 30 cm.
3. W. R. Dixon, Phys. Rev. 85, 498 (1952), Co\(^{60}\), p., Pb, concrete, \(\mu x\) up to 15.
   23. J. O. Elliot, R. T. Farrak, R. D. Myers, C. F. Ravilious, Phys. Rev. 85, 1048 (1952), Co\(^{60}\), t., Pb \(\mu x\) up to 17.
   24. W. R. Faust, Phys. Rev. 77, 227 (1950), Co\(^{60}\), t., water up to 90 cm.
   25. W. R. Faust, M. H. Johnson, Phys Rev. 75*, 467 (1949), Co\(^{60}\), continuously distributed in water.
4. G. Garrett, G. N. Whyte, Phys. Rev. 95, 889 (1954), Co\(^{60}\), t., Pb, Fe \(\mu x\) up to 16.
5. E. Hayward, Phys Rev. 86, 493 (1952), Amer. J. Roentgenology 71, 333 (1954), Co\(^{60}\), t., water up to 160 cm.
6. E. Hayward, J. H. Hubbell, J. Appl. Phys. 25, 506 (1954), Co\(^{60}\), t., wood, Fe—albedo.
7. G. J. Hine, R. C. McCall, Nucleonics 12, No. 4, 27 (1954), Co\(^{60}\), Hg\(^{203}\), Cs\(^{137}\), t., Pb, Fe, Al, wood, water, spectrum of reflected radiation.
8. F. S. Kirn, R. J. Kennedy, H. O. Wyckoff, Radiology 63, 94 (1954), Co\(^{60}\), Cs\(^{137}\), Au\(^{198}\), t., obliquely directed, concrete 90 cm, Pb 12 cm.
   31. P. Maignan, Ann. de phys. 8, 202 (1953), RdTh t., water 50—200 cm.
   32. A. N. Orlov, G. V. Fedorov, ZhTF 26, 1991 (1956), up to 20 MeV, p., 66 cm of water + 11 cm Fe.
9. R. A. Roys, K. Shure, J. J. Taylor, Phys. Rev. 95, 911 (1954), N\(^{16}\), p., water up to 190 cm.
10. M. H. Weiss, M. Bernstein, Phys. Rev. 92, 1264 (1953), Co\(^{60}\), t., water up to 200 cm.
11. G. H. White, Phys. Rev. 80, 154 (1950), Co\(^{60}\), t., water 30—250 cm.
12. G. N. Whyte, Canad. J. Phys. 33, 96 (1955), Co\(^{60}\), t. inside concrete, \(\mu x\) up to 10.
13. H. O. Wyckoff, R. J. Kennedy, W. R. Bradford, J. Res. Nat. Bur. Stand. 41, 223 (1948), up to 1.5 MeV, p., concrete 60 cm, Pb 9 cm.
14. S. G. Tsypin, V. I. Kukhtevich, Yu. A. Kazanskii, Atomic Energy No. 2, 71 (1956), Na\(^{24}\), Au\(^{198}\), t., water 190 cm, Pb, Fe 15 cm.

III. Other Works Cited

1. S. Z. Belen’kii, Cascade Processes in Cosmic Rays, Gostekhizdat (1948).
2. G. A. Bethe, Yu. Ashkin, article in the book Experimental Nuclear Physics, vol. 1, ed. E. Segrè, IIL (1955).
3. W. Heitler, The Quantum Theory of Radiation, Gostekhizdat (1940).
4. G. V. Gorshkov, Gamma Radiation of Radioactive Bodies, Leningrad University Press (1956).
5. N. P. Grazhdankina, I. G. Fakidov, in: Gamma Defectoscopy of Metals, Publishing House of the Academy of Sciences of the USSR, p. 61 (1955).
6. A. I. Gubanov, ZhETF 15, 336 (1945).
7. I. G. Dyad’kin, Izv. AN SSSR, Geophysical Series, No. 4, 323 (1955).
8. A. S. Kompaneets, ZhETF 15, 235 (1945).
9. L. V. Kurnosova, UFN 52, 603 (1954).
10. V. I. Ogievetskii, ZhETF 29, 454 (1955).
11. V. I. Ogievetskii, ZhETF 29, 464 (1955).
12. S. V. Rumyantsev, Yu. A. Grigorovich, Quality Control of Metals by Gamma Rays, Metallurgizdat (1954).
13. V. V. Chavchanidze, Proceedings of the Institute of Physics, Academy of Sciences of the Georgian SSR 2, 119 (1954).
14. S. Chandrasekhar, Radiative Transfer, IIL (1953).
15. S. Chandrasekhar, Stochastic Processes in Physics and Astronomy, IIL (1947).
16. Yu. A. Shreider, Conference “Paths of Development of Soviet Mathematical Machine Building and Instrument Making,” abstracts of reports, p. 56, report 44, Moscow (1956).
17. H. A. Bethe, U. Fano, P. R. Karr, Phys. Rev. 76, 538 (1949).
18. C. M. Davisson, E. D. Evans, Rev. Mod. Phys. 24, 79 (1952).
19. U. Fano, Nucleonics 11, No. 8, 8 (1953); 11, No. 9, 55 (1953).
20. U. Fano, Phys. Rev. 76, 739 (1949).
21. L. L. Foldy, Phys. Rev. 81, 395 (1951).
22. L. L. Foldy, Phys. Rev. 82, 927 (1951).
23. H. Goldstein, J. E. Wilkins, L. V. Spencer, Phys. Rev. 89, 1150 (1953).
24. H. Goldstein, J. E. Wilkins, Calculations of the Penetration of Gamma-Rays. NDA 15c—41. Nuclear Development Associates, Inc., White Plains, New York (1954).
25. H. Kahn, Nucleonics 6, No. 5, 27; No. 6, 61 (1950).
26. P. R. Karr, J. C. Lamkin, Phys. Rev. 76, 1843 (1949).
27. R. E. Marshak, Rev. Mod. Phys. 19, 185 (1947).
28. N. Marty, J. Phys. Rad. 13, 401 (1952).
29. L. V. Spencer, F. Jenkins, Phys. Rev. 76, 1885 (1949).
30. M. Verde, G. C. Wick, Phys. Rev. 71, 852 (1947).
31. G. C. Wick, Phys. Rev. 75, 738 (1949).
32. Monte Carlo Method, Nat. Bur. Stand. Applied Mathematic Series 12, June 11 (1951).
33. Symposium on Monte Carlo Method, editor H. A. Meyer, New York, John Wiley and Sons, Inc., Copyright (1956).