Milne problem for non-absorbing medium with extremely anisotropic scattering kernel in the case of specular and diffuse reflecting boundaries

Abstract

The Milne problem is studied in one speed neutron transport theory using the linearly anisotropic scattering kernel which combines forward and backward scatterings (extremely anisotropic scattering) for a non-absorbing medium with specular and diffuse reflection boundary conditions. In order to calculate the extrapolated endpoint for the Milne problem, Legendre polynomial approximation ($P_{N}$ method) is applied and numerical results are tabulated for selected cases as a function of different degrees of anisotropic scattering. Finally, some results are discussed and compared with the existing results in literature.

1 Introduction

The Milne problem is a well known problem for both the radiative transfer field and neutron transport theory. The problem is to obtain the angular density distribution function everywhere in the half-space $0\le x\le \infty$ with zero incident flux. Neutrons diffuse from a source at $x=+\infty$. There is a vacuum or reflecting medium in the region $x<0$. The extrapolation endpoint of neutrons leaving from the boundary at $x=0$ is determined with Milne problem (Case and Zweifel [1]). Placzek and Seidel [2] and Noble [3] used the Wigner-Hopf technique for solving the Milne problem with vacuum boundary condition. The extrapolated endpoint of the half space was obtained by LeCaine [4] and Marshak [5] with variational method. The formulations of the problem for the isotropic and the linearly anisotropic cases were considered by McCormick and Ku$\check{s}\check{c}$er [6], McCormick [7] and Case and Zweifel [1]. Shure and Natelson [8] used the Case singular eigenfunction method to calculate extrapolated endpoint for the special cases of absorbing and non-absorbing media. Using the Weigner-Hopf method, Williams [9] solved the problem analytically with the diffuse reflecting boundary condition. The problem with the reflecting boundary condition for non-absorbing medium was solved by Razi [10] and AbdelKrim and Degheidy [11] using variational approach. Tezcan [12], used the $P_{N}$ method, and calculated extrapolated endpoint in the case of extremely anisotropic scattering kernel for non-absorbing medium. Atalay [13, 14] solved the same problem including absorbing medium by using the singular eigenfunction method for linearly anisotropic scatterig. Loyalka and Naz [15] calculated the linear extrapolation distance for conservative case of isotropic scattering (c=1) using Gaussian quadratures method. Degheidy and El-Shahat [16] studied the problem by a technique based on constructing integral equations. Using the variational principle, Grzesik [17] solved the Milne problem with linear anisotropic scattering.

In this paper, we considered the Milne problem for non-absorbing medium under the specular and diffuse boundary condition. We obtained the extrapolation distance $z_{0}$ by using Legendre polynomial approximation in the case of linearly and extremely anisotropic scattering.

In the plane geometry, the neutron transport equation with one speed, time independent and source free is given by Case and Zweifel [1], as follows

$$\begin{aligned} \mu \frac{\partial \Psi (x ,\mu )}{\partial x}+\Psi (x,\mu )=2\pi c\int \limits _{-1}^{+1}f(\mu ,\mu ^{\prime })\Psi (x ,\mu ^{\prime })d\mu ^{\prime }, \end{aligned}$$

(1)

where $\Psi (x ,\mu )$ is the angular distribution of neutrons, c is the number of secondary neutrons per collisions, $\mu$ is the cosine of the angle between the direction of the neutron velocity and the positive x axis and $f(\mu ,\mu ^{\prime })$ is the scattering kernel which is assumed to be of the form of the combination of linearly anisotropic scattering and extremely anisotropic scattering (Siewert [18]), (Razi [19]), (de Azevedo et al. [20]), (Degheidy et al. [21]), (Degheidy et al. [22]). Explicitly, the function $f(\mu ,\mu ^{\prime })$ is given by,

$$\begin{aligned} f(\mu ,\mu ^{\prime })=\frac{1}{4\pi }a(1+3f_{1}\mu \mu ^{\prime })+\frac{b}{2\pi }\delta (\mu ^{\prime }-\mu )+\frac{d}{2\pi }\delta (\mu ^{\prime }+\mu ), \end{aligned}$$

(2)

where $f_{1}$ represents linearly anisotropic parameter ($-1$ $\le$ $3f_{1}$ $\le$ $+1$). Setting $a=1-\alpha$, $b=\alpha$ and $d=0$ $f(\mu ,\mu ^{\prime })$ represents the forward scattering plus linearly anisotropic scattering. Also setting $a=1-\alpha$, $b=0$ and $d=\alpha$ it represents the backward scattering plus linearly anisotropic scattering. By normalizing the scattering kernel, one obtain that $\alpha$ is a real constant ($0\le \alpha \le 1$) and

$$\begin{aligned} a+b+d=1, \end{aligned}$$

(3)

The boundary conditions are given by the following expressions for the specular and diffuse reflecting boundary

$$\begin{aligned} \Psi (0,\mu )=\, & {} \rho ^{s}\Psi (0,-\mu )+2\rho ^{d}\int \limits _{0}^{+1}\mu ^{\prime }\Psi (0,\mu ^{\prime })d\mu ^{\prime }, \qquad \mu >0 \end{aligned}$$

(4)

$$\begin{aligned} \lim _{x \rightarrow \infty }\Psi (x,\mu )=0 \end{aligned}$$

(5)

where $\rho ^{s}$ ($0 \le \rho ^{s} \le 1$) and $\rho ^{d}$ ($0 \le \rho ^{d} \le 1$) are specular and diffuse reflectivities of the boundary, respectively.

2 $P_{N}$ method and calculations

Inserting Eq. (2) into (1), the time independent source free one speed transport equation for the forward, backward and linearly anisotropic scattering may be rewritten as

$$\begin{aligned} \mu \frac{\partial \Psi (x ,\mu )}{\partial x}+\Psi (x,\mu )=\frac{ca}{2}\int \limits _{-1}^{+1}(1+3f_{1}\mu \mu ^{\prime })\Psi (x,\mu ^{\prime })d\mu ^{\prime } \nonumber \\ +\,b\int \limits _{-1}^{+1}\Psi (x,\mu ^{\prime })\delta (\mu ^{\prime }-\mu )d\mu ^{\prime }\nonumber \\ +\,d\int \limits _{-1}^{+1}\Psi (x,\mu ^{\prime })\delta (\mu ^{\prime }+\mu )d\mu ^{\prime }, \end{aligned}$$

(6)

For the solution of Eq. (6), the angular distribution and Dirac Delta function can be expanded in terms of Legendre polynomials as follows:

$$\begin{aligned} \Psi (x,\mu )= & {} \sum\nolimits _{n=0}^{\infty }\frac{2n+1}{2}\psi _{n}(x)P_{n}(\mu ), \end{aligned}$$

(7a)

$$\begin{aligned} \delta (\mu -\mu ^{\prime })= & {} \sum _{n=0}^{\infty }a_{n}P_{n}(\mu ),\end{aligned}$$

(7b)

$$\begin{aligned} \delta (\mu +\mu ^{\prime })= & {} \sum _{n=0}^{\infty }b_{n}P_{n}(\mu ) \end{aligned}$$

(7c)

where

$$\begin{aligned} \psi _{n}(x)= & {} \int \limits _{-1}^{+1}P_{n}(\mu )\Psi (x,\mu )d\mu ,\qquad n=1,3,\ldots ,\infty ,\nonumber \\ a_{n}= & {} \frac{2n+1}{2}P_{n}(\mu ^{\prime })\nonumber \\ b_{n}= & {} (-1)^{n}a_{n}. \end{aligned}$$

(8)

Multiplying Eq. (6) by $(2m+1)P_{m}(\mu )$ and integrated both side of Eq. (6) over $\mu$ on the interval [−1, +1], and using the following orthogonality and recurrence relations for $P_{m}(\mu )$

$$\begin{aligned} \int \limits _{-1}^{+1}P_{n}(\mu )P_{m}(\mu )d\mu= & {} \frac{2}{2m+1}\delta _{nm},\nonumber \\ \mu P_{m}(\mu )= & {} \frac{1}{2m+1}[(m+1)P_{m+1}(\mu )+mP_{m-1}(\mu )], \end{aligned}$$

(9)

the following moment equation for linearly plus extremely anisotropic scattering kernel which consist of the infite sets of coupled differantial equations with $n=0,1,\ldots$ is obtained as

$$\begin{aligned} n\frac{d\psi _{n-1}(x)}{dx}+(n+1)\frac{d\psi _{n+1}(x)}{dx}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\ +(2n+1)(1-c(b+(-1)^{n}d))\left[ 1-\frac{ac}{1-c(b+d)}(\delta _{n0}+\frac{\gamma a}{3(1-cb+cd)}\delta _{n1})\right] \psi _{n}(x)=0, \end{aligned}$$

(10)

where $\Psi _{-1}(x)=0$ and $\delta _{kl}$ is the kronecker delta function. Substituting the followings

$$\begin{aligned} c^{\prime }= & {} \frac{ac}{1-c(b+d)}, \end{aligned}$$

(11a)

$$\begin{aligned} \gamma ^{\prime }= & {} \frac{\gamma a}{3(1-cb+cd)}, \end{aligned}$$

(11b)

$$\begin{aligned} \gamma=\, & {} 3f_{1}, \end{aligned}$$

(11c)

$$\begin{aligned} z=\, & {} ((1-bc)^{2}-c^{2}d^{2})^{1/2}x, \end{aligned}$$

(11d)

$$\begin{aligned} \phi _{n}(z)=\, & {} (1-bc-(-1)^{n}cd)^{1/2}\psi _{n}(x), \end{aligned}$$

(11e)

into Eq. (10) one obtaines

$$\begin{aligned} n\frac{\phi _{n-1}(z)}{dz}+(n+1)\frac{d\phi _{n+1}(z)}{dz}+(2n+1)(1-c^{\prime }(\delta _{n0}+\gamma ^{\prime }\delta _{n1}))\phi _{n}(z)=0. \end{aligned}$$

(12)

Equation (12) is similar to the equation for the linearly anisotropic scattering. Consequently, Eq. (10) for linearly plus extremely anisotropic scattering kernels is transformed to Eq. (12) for linearly anisotropic scattering, where $\Psi _{m}$, x and c are replaced by $\Phi _{m}$, z, $c^{\prime }$, respectively. In the $P_{N}$ method, in order to terminate and to solve the infinite set of coupled equations for $\phi _{n}$ in Eq. (12) it is sufficient to get $d\phi _{n+1}(z)/dz=0$ and $n=0,1,\ldots ,N$. As a result of this, $N+1$ coupled equations with $N+1$ unknowns can be found. To obtain the solutions to the these $N+1$ coupled equations, the boundary condition given in Eq. (4) should be rearranged by using Marshak boundary condition (Marshak [23]):

$$\begin{aligned} \int \limits _{0}^{1}\mu ^{m}\Psi (0,\mu )d\mu =0,\qquad m=1,3,\ldots ,N \end{aligned}$$

(13)

Using Eq. (4) in (13), the boundary condition for specular and diffuse reflecting in $P_{N}$ method can be written as

$$\begin{aligned} \int \limits _{0}^{1}\mu ^{m}\Psi (0,\mu )d\mu -\rho ^{s}\int \limits _{0}^{1}\mu ^{m}\Psi (0,-\mu )d\mu -\frac{2\rho ^{d}}{m+1}\int \limits _{0}^{1}\mu ^{\prime }\Psi (0,\mu ^{\prime })d\mu ^{\prime }=0,\quad m=1,3,\ldots ,N. \end{aligned}$$

(14)

$P_{1}$ Approximation: In $P_{1}$ approximation, taking $N=1$, i.e. $n=0$ and $n=1$, two coupled moment equations $\phi _{0}$ and $\phi _{1}$ can be obtained using Eq. (12) as

$$\begin{aligned}&\frac{d\phi _{1}(z)}{dz}+(1-c^{\prime })\phi _{0}(z)=0, \end{aligned}$$

(15a)

$$\begin{aligned}&\frac{d\phi _{0}(z)}{dz}+3\,(1-\gamma ^{\prime })\phi _{1}(z)=0. \end{aligned}$$

(15b)

If we set $c=1$ for non-absorbing medium, we obtain $c^{\prime }=1$ from Eq. (11a), therefore $\phi _{1}(z)$ can be found as a constant

$$\begin{aligned} \phi _{1}(z)=-(1-b+d)^{1/2}. \end{aligned}$$

(16)

Substituting Eq. (16) into (15b) we obtain

$$\begin{aligned} \phi _{0}(z)=3(1-\gamma ^{\prime })(1-b+d)^{1/2}z+A_{0}. \end{aligned}$$

(17)

Using Eqs. (16, 17) in (11d, 11e), the moment solutions for linearly anisotropic scattering are

$$\begin{aligned}&\psi _{1}(x)=-1, \end{aligned}$$

(18)

$$\begin{aligned}&\psi _{0}(x)=(3(1-b+d)-\gamma a)x+\frac{A_{0}}{(1-b-d)^{1/2}}. \end{aligned}$$

(19)

Here, $A_{0}$ is constant. It can be obtained from Eq. (7a) for $x=0$ and $n=0,1$,

$$\begin{aligned} \psi (0,\mu )=\frac{1}{2}\psi _{0}(0)P_{0}(\mu )+\frac{3}{2}\psi _{1}(0)P_{1}(\mu ), \end{aligned}$$

(20)

and using boundary condition given by Eq. (14), $A_{0}$ is therefore found as

$$\begin{aligned} A_{0}=2\frac{(1-b-d)^{1/2}(1+\rho ^{s}+\rho ^{d})}{1-\rho ^{s}-\rho ^{d}}. \end{aligned}$$

(21)

The extrapolated endpoint $z_{0}$ is defined as a distance from a vacuum boundary to where the asymptotic intensity vanishes, and it can be written mathematically as $\Psi _{0}(x)\big |_{x=-z_{0}}=0$ in which exponential terms are eliminated. From the definitions one can get

$$\begin{aligned} z_{0}=\frac{A_{0}}{(3(1-b+d)-\gamma a)(1-b-d)^{1/2}}, \end{aligned}$$

(22)

and substituting Eq. (21) in (22), the extrapolated endpoint for the specular and diffuse reflectivities can be obtained as

$$\begin{aligned} z_{0}=\frac{2(1+\rho ^{s}+\rho ^{d})}{(3(1-b+d)-\gamma a)(1-\rho ^{s}-\rho ^{d})}. \end{aligned}$$

(23)

$P_{3}$ Approximation: In $P_{3}$ approximation, i.e. $n=0,1,2,3$, we get $c=1$, the moment equations from Eq. (12) can be written as

$$\begin{aligned}&\frac{d\phi _{1}(z)}{dz}=0, \end{aligned}$$

(24a)

$$\begin{aligned}&2\frac{d\phi _{2}(z)}{dz}+\frac{d\phi _{0}(z)}{dz}+3(1-\gamma ^{\prime })\phi _{1}(z)=0, \end{aligned}$$

(24b)

$$\begin{aligned}&3\frac{d\phi _{3}(z)}{dz}+2\frac{d\phi _{1}(z)}{dz}+5\phi _{2}(z)=0, \end{aligned}$$

(24c)

$$\begin{aligned}&3\frac{d\phi _{2}(z)}{dz}+7\phi _{3}(z)=0. \end{aligned}$$

(24d)

From Eqs. (24c) and (24d), a second order differential equation is obtained for $\phi _{2}(z)$

$$\begin{aligned} -\frac{9}{7}\frac{d^{2}\phi _{2}(z)}{dz^{2}}+5\phi _{2}(z)=0. \end{aligned}$$

(25)

The solution of Eq. (25) is given by

$$\begin{aligned} \phi _{2}(z)=A_{1}exp(-\alpha _{1}z); \qquad \alpha _{1}=(35/9)^{1/2} \end{aligned}$$

(26)

where $A_{1}$ is constant. Inserting Eq. (26) into (24d), $\phi _{3}(z)$ is found as

$$\begin{aligned} \phi _{3}(z)=\frac{3}{7}A_{1}\alpha _{1}exp(-\alpha _{1}z). \end{aligned}$$

(27)

Using Eqs.(26, 27) and substituting $\phi _{1}(z)=-(1-b+d)^{1/2}$ in Eq. (24b), we obtain

$$\begin{aligned} \phi _{0}(z)=3(1-\gamma ^{\prime })(1-b+d)^{1/2}z+A_{0}-2A_{1}exp(-\alpha _{1}z). \end{aligned}$$

(28)

Using Eqs. (11d) and (11e), we obtain

$$\begin{aligned} \psi _{3}(x)= & {} \frac{3}{7}\frac{A_{1}\alpha _{1}e^{-\alpha _{1}((1-b)^{2}-d^{2})^{1/2}x}}{(1-b+d)^{1/2}}, \end{aligned}$$

(29a)

$$\begin{aligned} \psi _{2}(x)= & {} \frac{A_{1}e^{-\alpha _{1}((1-b)^{2}-d^{2})^{1/2}x}}{(1-b-d)^{1/2}},\end{aligned}$$

(29b)

$$\begin{aligned} \psi _{1}(x)= & {} -1,\end{aligned}$$

(29c)

$$\begin{aligned} \psi _{0}(x)=\, & {} (3(1-b+d)-\gamma a)x+\frac{2A_{1}e^{-\alpha _{1}((1-b)^{2}-d^{2})^{1/2}x}}{(1-b-d)^{1/2}}+\frac{A_{0}}{(1-b-d)^{1/2}}, \end{aligned}$$

(29d)

where, $A_{0}$ and $A_{1}$ are constants. By using the lowest order of Marshak boundary condition given in Eq. (14) the constants $A_{0}$, $A_{1}$ can be found as follows

$$\begin{aligned} A_{0}= & {} \frac{3}{4}A_{1}+2\left( \frac{1+\rho ^{s}-\rho ^{d}}{1-\rho ^{s}-\rho ^{d}}\right) (1-b-d)^{1/2}, \end{aligned}$$

(30a)

$$\begin{aligned} A_{1}= & {} \frac{56(1-b-d)^{1/2}(2+(-3+\rho ^{d})\rho ^{d}-11\rho ^{d}\rho ^{s}-2(\rho ^{s})^{2})}{(-1+\rho ^{d}+\rho ^{s})\{175(-2+\rho ^{d}+2\rho ^{s})+96\alpha _{1}[\rho ^{d}-2(1+\rho ^{s})]\}}. \end{aligned}$$

(30b)

Finally, the extrapolated endpoint $z_{0}$ is calculated from Eq. (22) using $A_{0}$ in Eq. (30a) for $P_{3}$ approximation.

$P_{N}$ Approximation: In $P_{N}$ approximation for $c=1$, the moment equations are obtained from Eq. (12) as

$$\begin{gathered} \frac{{d\phi _{1} (z)}}{{dz}} = 0, \hfill \\ 2\frac{{d\phi _{2} (z)}}{{dz}} + \frac{{d\phi _{0} (z)}}{{dz}} + 3(1 - \gamma ^{\prime } )\phi _{1} (z) = 0, \hfill \\ 3\frac{{d\phi _{3} (z)}}{{dz}} + 2\frac{{d\phi _{1} (z)}}{{dz}} + 5\phi _{2} (z) = 0, \hfill \\ \begin{array}{*{20}c} \cdot \\ \cdot \\ \cdot \\ \end{array} \hfill \\ N\frac{{d\phi _{N} (z)}}{{dz}} + (N - 1)\frac{{d\phi _{{N - 2}} (z)}}{{dz}} + (2N - 1)\phi _{{N - 1}} (z) = 0, \hfill \\ N\frac{{d\phi _{{N - 1}} (z)}}{{dz}} + (2N + 1)\phi _{N} (z) = 0. \hfill \\ \end{gathered}$$

(31)

After some algebric manipulations, (N−1)th order differential equation is obtained from Eq. (31) and the solution of this equation is

$$\begin{aligned} \phi _{N-1}(z)=\sum \limits _{k=1}^{\frac{N-1}{2}}A_{k}exp(-\alpha _{k}z). \end{aligned}$$

(32)

The other solutions of moment equations can be found as

$$\begin{aligned} \phi _{N}(z)= & {} -\frac{N}{2N+1}\frac{d\phi _{N-1}(z)}{dz}\nonumber \\ \phi _{k-2}(z)= & {} -\frac{2k-1}{k-1}\int \phi _{k-1}(z)dz-\frac{k}{k-1}\phi _{k}(z), \qquad k=4,5,\ldots ,N\nonumber \\ \phi _{1}(z)= & {} -(1-b+d)^{1/2}\nonumber \\ \phi _{0}(z)= & {} -3(1-\gamma ^{\prime })(1-b+d)^{1/2}z-\frac{2}{3}\phi _{2}(z)+A_{0}. \end{aligned}$$

(33)

After these moment solutions $\phi _{n}(z)$ are transformed into $\psi _{n}(x)$ for $n=0,1,2,\ldots ,N$ using Eqs. (11d, 11e), the constants $A_{0}$ and $A_{k}$ can be computed by using Marshak boundary condition given in Eq. (14). Finally, the extrapolated endpoint $z_{0}$ can be calculated from $\Psi _{0}(x)\big |_{x=-z_{0}}=0$.

3 Numerical results

Numerical values of the extrapolated endpoint are given for isotropic scattering for diffuse reflection in Table 1 and for specular reflection in Table 2. They are compared with the exact values given in Williams’s paper [9]. Table 3 shows the extrapolated endpoints for different values of $\rho ^s$ and $\rho ^d$ for isotropic scattering and they are compared with Degheidy [16]. It can be seen from these tables that the distance $z_{0}$ at which the flux drops off the zero, increases with increasing the specular and diffuse components of reflectivity. In Tables 4 and 5, the numerical values of $z_{0}$ are given for different values of $\rho ^s$ and $\rho ^d$ for the linearly anisotropic scattering and compared with Atalay [14]. In these tables, the extrapolated endpoint values increase with increasing linearly anisotropic scattering coefficient $f_{1}$. Table 6 shows the emergent angular distribution $\Psi (0,-\mu )$ for various values of $\rho ^s$ and $\rho ^d$ for isotropic scattering $(f_{1}=0)$. The results are in good agreement with the literature for higher order $P_{N}$ approximation. In Tables 7, 8, the extrapolated endpoint values are given for the extremely anisotropic scattering kernel. Figures 1, 2, 3 and 4 show the behaviors of the extrapolated endpoint as a function of b and d for different values of $\rho ^{s}$ and $\rho ^{d}$. From these figures, it is seen that when forward scattering coefficent b increases, $z_{0}$ also increases. When backward scattering coefficient d increases, $z_{0}$ also decreases. We have carried out all our numerical computations in Mathematica programming.

Table 1 The extrapolated endpoint for the diffuse reflection $(\rho ^s=0)$ and isotropic scattering $(f_{1}=0)$

Table 2 The extrapolated endpoint for the specular reflection $(\rho ^d=0)$ and isotropic scattering $(f_{1}=0)$

Table 3 The extrapolated endpoint for the different values of $\rho ^s$, $\rho ^d$ and isotropic scattering case $(f_{1}=0)$ in $P_{29}$ approximation

Table 4 The extrapolated endpoint for the different specular and diffuse reflection coefficients in linearly anisotropic scattering for $P_{29}$ approximation

Table 5 The extrapolated endpoint different values of $\rho ^{s}$ and $\rho ^{d}$ in linearly anisotropic scattering ($f_{1}\ne 0$) for $P_{29}$ approximation

Table 6 The emergent angular distribution $\Psi (0,-\mu )$ for the selected values of $\rho ^s$ and $\rho ^d$ in isotropic scattering $(f_{1}=0)$

Table 7 The extrapolated endpoint for the different values of b and using $P_{29}$ approximation in the case of $f_{1}=0$, $d=0$

Table 8 The extrapolated endpoint for the different values of d and using $P_{29}$ approximation in the case of $f_{1}=0$, $b=0$

4 Conclusions

The Milne problem with the specular and diffuse reflecting boundary conditions is solved using Legendre polynomial approximation for a non-absorbing half space medium. In this work, the combination of the linearly and extremely anisotropic scattering functions are considered. The moment equations, infinite set of coupled differential equations which are obtained from the neutron transport equation for linearly plus extremely anisotropic scattering kernel are transformed into the equations written for linearly anisotropic scattering kernel. After that, applying N th order approximation in the $P_{N}$ method, the set of N+1 coupled differential equations are obtained and reduced to one (N−1) th order differential equation for $\Psi _{N-1}(x)$. The remaining N moments for $\Psi _{N}(x)$ are derived after finding the solution of $\Psi _{N-1}(x)$. The constants of these solutions are calculated using Marshak boundary condition with specular and diffuse reflectivities of the boundary. And then the distance $z_{0}$ is obtained using $A_{0}$ constant for each N th order approximation. Finally some results for the distance $z_{0}$ and the emergent angular distributions are tabulated in tables and compared with the literature (Williams [9]), (Degheidy and El-Shahat [16]), (Atalay [14]). The comparisons with the present numerical results in given tables show that the $P_{N}$ method agrees with the available results reported in the literatures. Furthermore, the numerical values of extrapolated endpoint $z_{0}$ are calculated using forward and backward anisotropic scattering kernel for the non-absorbing medium for the first time in the literature.

Fig. 1

The behaviors of the extrapolated endpoint as a function of b for different values of $\rho ^{s}$

Fig. 2

The behaviors of the extrapolated endpoint as a function of d for different values of $\rho ^{s}$

Fig. 3

The behaviors of the extrapolated endpoint as a function of b for different values of $\rho ^{d}$

Fig. 4

The behaviors of the extrapolated endpoint as a function of d for different values of $\rho ^{d}$