Physics

I. N. Minin

On the Theory of Radiation Diffusion in a Semi-Infinite Medium

(Presented by Academician V. A. Ambartsumian on February 6, 1958)

V. V. Sobolev has shown \((^1)\) that the solution of any problem on the luminosity of a plane layer of infinitely large optical thickness with a spherical scattering indicatrix, in the presence of radiation sources whose power depends only on the optical depth \(\tau\), is reduced to finding the function \(\Phi(\tau)\), defined by the equation

\[ \Phi(\tau)=K(\tau)+\int_{0}^{\tau}K(\tau-\tau')\Phi(\tau')\,d\tau', \tag{1} \]

where

\[ K(\tau)=\frac{\lambda}{2}\int_{0}^{1} e^{-\tau/\eta}\varphi(\eta)\frac{d\eta}{\eta}; \tag{2} \]

\(\lambda\) is the ratio of the scattering coefficient to the attenuation coefficient; \(\varphi(\eta)\) is a function introduced by V. A. Ambartsumian \((^2)\) and defined by the equation

\[ \varphi(\eta)=1+\frac{\lambda}{2}\eta\varphi(\eta)\int_{0}^{1}\frac{\varphi(\zeta)}{\eta+\zeta}\,d\zeta. \tag{3} \]

In the present note exact analytic solutions of equation (1) are found, and some consequences from them are obtained.

Applying the one-sided Laplace transform to equation (1), we obtain

\[ \overline{\Phi}(p)=\frac{\overline{K}(p)}{1-\overline{K}(p)}, \tag{4} \]

where

\[ \overline{\Phi}(p)=\int_{0}^{\infty}\Phi(\tau)e^{-p\tau}\,d\tau,\qquad \overline{K}(p)=\frac{\lambda}{2}\int_{0}^{1}\frac{\varphi(\eta)}{1+p\eta}\,d\eta. \tag{5} \]

Using the well-known Mellin—Riemann inversion formula

\[ \Phi(\tau)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\overline{\Phi}(p)e^{p\tau}\,dp \tag{6} \]

and carrying out the integration along the imaginary axis \((c=0)\), we find

\[ \Phi(\tau)=\frac{1}{\pi}\int_{0}^{\infty}[A(x)\cos\tau x+B(x)\sin\tau x]\,dx, \tag{7} \]

where

\[ A(x)=\frac{\lambda \dfrac{\arc\operatorname{tg} x}{x}-a(x)} {1-\lambda \dfrac{\arc\operatorname{tg} x}{x}}, \qquad B(x)=-\frac{x b(x)} {1-\lambda \dfrac{\arc\operatorname{tg} x}{x}}, \]

\[ a(x)=\frac{\lambda}{2}\int_0^1 \frac{\varphi(\eta)}{1+\eta^2x^2}\,d\eta, \qquad b(x)=\frac{\lambda}{2}\int_0^1 \frac{\eta\varphi(\eta)}{1+\eta^2x^2}\,d\eta . \]

We note that \(a(x)\) and \(b(x)\) are connected by the relation

\[ x^2 b^2(x)+a^2(x)-2a(x)+\lambda\frac{\arc\operatorname{tg} x}{x}=0. \]

However, the use of formula (7) for computations is inconvenient.

Another way of obtaining the transform \(\overline{\Phi}(p)\) can be indicated. The point is that \(\overline{\Phi}(p)\) has at the point \(p=-k\) a pole of the first order, and the quantity \(k\) is connected with \(\lambda\) by the equation

\[ \frac{\lambda}{2k}\lg\frac{1+k}{1-k}=1. \]

Moreover, the point \(p=-1\) is a branch point for \(\overline{\Phi}(p)\). Taking into account the presence of such singular points and using the method of contour integration, we obtain

\[ \Phi(\tau)= \frac{e^{-k\tau}} {\dfrac{\lambda}{2}\int_0^1 \dfrac{\varphi(\eta)\eta}{(1-k\eta)^2}\,d\eta} + 2\lambda\int_1^\infty \frac{x e^{-\tau x}} {\pi^2\lambda^2+\left(2x+\lambda\lg\frac{x-1}{x+1}\right)^2} \frac{dx}{\varphi\left(\dfrac{1}{x}\right)} . \tag{8} \]

For \(\lambda=1\), from (8) we find

\[ \Phi(\tau)=\sqrt{3} + 2\int_1^\infty \frac{x e^{-\tau x}} {\pi^2+\left(2x+\lg\frac{x-1}{x+1}\right)^2} \frac{dx}{\varphi\left(\dfrac{1}{x}\right)} . \tag{9} \]

As shown in (1), the solution of some problems reduces to determining the function

\[ \Psi(\tau)=1+\int_0^\tau \Phi(\tau)\,d\tau . \]

For \(\Psi(\tau)\) it is easy to find

\[ \Psi(\tau)= \frac{1}{\sqrt{1-\lambda}} - \frac{e^{-k\tau}} {k\,\dfrac{\lambda}{2}\int_0^1 \dfrac{\varphi(\eta)\eta}{(1-k\eta)^2}\,d\eta} - \]

\[ -\,2\lambda\int_1^\infty \frac{e^{-\tau x}} {\pi^2\lambda^2+\left(2x+\lambda\lg\frac{x-1}{x+1}\right)^2} \frac{dx}{\varphi\left(\dfrac{1}{x}\right)} , \tag{10} \]

and for \(\lambda=1\) we have

\[ \Psi(\tau)=1+\tau\sqrt{3} + 2\int_1^\infty \frac{1-e^{-\tau x}} {\pi^2+\left(2x+\lg\frac{x-1}{x+1}\right)^2} \frac{dx}{\varphi\left(\dfrac{1}{x}\right)} . \tag{11} \]

We note the relation

\[ \frac{1}{\dfrac{\lambda}{2}\displaystyle\int_0^1 \frac{\varphi(\eta)\eta}{(1-k\eta)^2}\,d\eta} = \frac{k}{\dfrac{\lambda}{1-k^2}-1} \left( 1-\frac{\lambda}{2}\int_0^1 \frac{\varphi(\eta)}{1+k\eta}\,d\eta \right), \]

which may prove useful in carrying out computations by formulas (8) and (10).

It follows from formula (8) that for \(\tau \gg 1\) the integral term may be neglected and, consequently, the nonintegral term is an exact asymptotic expression for \(\Phi(\tau)\) for large \(\tau\). For \(\Psi(\tau)\), from (10), for \(\tau \gg 1\) we find the value \(\dfrac{1}{\sqrt{1-\lambda}}\).

If pure scattering of radiation occurs in the medium \((\lambda=1)\) and the sources of radiation are located at infinitely great depth, then the solution of the problem is usually represented in terms of the function \(q(\tau)\), which, as found by V. V. Sobolev \({}^{1}\), is related to \(\Psi(\tau)\) by

\[ q(\tau)=\frac{\Psi(\tau)}{\sqrt{3}}-\tau . \tag{12} \]

Using (11) and (12), for \(q(\tau)\) we have

\[ q(\tau)=\frac{1}{\sqrt{3}} \left[ 1+2\int_1^\infty \frac{1-e^{-\tau x}} {\pi^2+\left(2x+\lg\frac{x-1}{x+1}\right)^2} \,\frac{dx}{\varphi\left(\dfrac{1}{x}\right)} \right]. \tag{13} \]

The function \(q(\tau)\) in the form (13) was obtained by another method by Mark \({}^{3}\). Here we have found (13) from the general solution (10), putting \(\lambda=1\) in it.

Table 1

In conclusion we give Table 1 of the functions \(\Phi(\tau)\), \(\Psi(\tau)\), and \(q(\tau)\) for \(\lambda=1\), calculated by formulas (9), (11), and (13).

Received
1 II 1958

References Cited

\({}^{1}\) V. V. Sobolev, DAN, 116, No. 1 (1957).
\({}^{2}\) V. A. Ambartsumian, DAN, 38, No. 8 (1943).
\({}^{3}\) C. Mark, Phys. Rev., 72, No. 7 (1947).