MATHEMATICAL PHYSICS

M. V. MASLENNIKOV

ON THE GENERAL PROBLEM OF THE THEORY OF NEUTRON SLOWING DOWN

(Presented by Academician M. V. Keldysh, 13 VIII 1957)

The state of a neutron in the theory of slowing down is determined by specifying the corresponding phase point \(Q=(\mathbf r,\vec\Omega,u)\), where \(\mathbf r\) is the spatial coordinate of the neutron; \(\mathbf r\in G\); \(G\) is a connected domain of three-dimensional space \(R_3\), in which the process under consideration takes place. The complement of \(G\) to the whole space is occupied by black bodies*. \(\vec\Omega=|\mathbf V|^{-1}\mathbf V\), where \(\mathbf V\) is the velocity of the neutron; \(\vec\Omega\in S\); \(S\) is the unit sphere of three-dimensional space. The lethargy
\[ u=-\ln\frac{|\mathbf V|^2}{2E_0};\qquad E_0=\sup\frac{|\mathbf V|^2}{2}, \]
where the supremum is taken over all neutrons; \(u\in U_1=[0,\infty)\). Thus, \(Q\in H=G\times S\times U_1\). Slowing down occurs on nuclei of various kinds, filling \(G\) in an arbitrary manner. Here, for simplicity, it is assumed that \(G\) decomposes into parts, each of which contains nuclei of one and the same mass: \(M=M(\mathbf r)\) (the mass of the neutron is taken as unity). If \(l_s(\mathbf r,u)\), \(l_c(\mathbf r,u)\) are the total mean free paths for scattering and absorption, then let
\[ \beta(\mathbf r,u)=l_s^{-1}(\mathbf r,u)+l_c^{-1}(\mathbf r,u) \]
and
\[ h(\mathbf r,u)=\beta^{-1}(\mathbf r,u)\,l_s^{-1}(\mathbf r,u). \]
Then the phase density of neutron collisions \(\psi(Q)\) satisfies the integral equation

\[ \psi(Q)=\hat A\psi(Q)+T(Q); \tag{1} \]

\[ \hat A F(Q)=\beta(\mathbf r,u)\int_0^{|\mathbf r-\vec\rho_0|} \exp\left[-\int_0^\rho \beta(\mathbf r-r'\vec\Omega,u)\,dr'\right]d\rho \times \]

\[ \times \int_S d\vec\Omega'\int_0^u h(\mathbf r-\rho\vec\Omega,u')\,g(\mathbf r-\rho\vec\Omega,\vec\Omega\vec\Omega',u',u)\, F(\mathbf r-\rho\vec\Omega,\vec\Omega',u')\,du'. \tag{2} \]

Here \(\vec\rho_0=\vec\rho_0(\mathbf r,\vec\Omega)=\mathbf r-t\vec\Omega\in \overline G\setminus G\); \(t\) is the distance from \(\mathbf r\) to the boundary \(\overline G\setminus G\) of the domain \(G\) in the direction \(-\vec\Omega\); \(t=\inf t'\), \(\mathbf r-t'\vec\Omega\in \overline G\); \(g(\mathbf r,\mu,u',u)\) is the scattering indicatrix through an angle \(\arccos\mu\) with a change of lethargy from the value \(u'\) to the value \(u\).

The free term of equation (1) is determined by the boundary conditions and the distribution of sources:

\[ T(Q)=\beta(\mathbf r,u)\int_0^{|\mathbf r-\vec\rho_0|} \exp\left[-\int_0^\rho \beta(\mathbf r-r'\vec\Omega,u)\,dr'\right] S(\mathbf r-\rho\vec\Omega,\vec\Omega,u)\,d\rho+ \]

\[ +|\mathbf V(u)|\,\beta(\mathbf r,u)\,B(\vec\rho_0,\vec\Omega,u) \exp\left[-\int_0^{|\mathbf r-\vec\rho_0|} \beta(\mathbf r-r'\vec\Omega,u)\,dr'\right]. \tag{3} \]

* If \(G\) is convex, the “black body” may be a vacuum.

\(B(\vec\rho_0,\vec\Omega,u)\) is the phase density of neutrons at a boundary point \(\vec\rho_0\in \bar G\setminus G\). The function \(B\) is defined for all \(u>0\), \(\vec\rho_0\in \bar G\setminus G\), and those \(\vec\Omega\) which correspond to the direction from \(\vec\rho_0\) into \(G\); \(S(Q)\) is the phase strength of the sources; \(Q\in H\).

Let \(H(\bar u)\) be the part of \(H\) on which \(u\in[0,\bar u)\), and let \(K(\gamma)\) be the set of all functions \(F(Q)\) measurable on \(H\) such that \(\exp(-\gamma|r|)F(Q)\) is bounded on every \(H(\bar u)\), \(\bar u<\infty\). Put
\[ \alpha(r)=\frac{(M(r)+1)^2}{4M(r)} \]
and
\[ \omega_+=\sup_{Q\in H}\omega(Q),\qquad \omega_-=\inf_{Q\in H}\omega(Q), \]
for any \(\omega(Q)\) defined on \(H\).

The study of equation (1) is carried out under the following assumptions:

1. \(\beta(r,u)\in K(0)\); \(h(r,u)\in K(0)\); \(M(r)\in K(0)\), \(M_+<\infty\).

2. \(0<\beta_-\leq \beta_+<\infty\).

3. The primary radiation has lethargy \(u\) not exceeding \(u_0\in(0,\infty]\).

4. \(S(Q)\in K(\delta)\), \(\delta\in[0,\beta_-)\), and
   \[ S^+=\sup_{Q\in H} S(Q)e^{-\delta|r|}<\infty . \]

5. \(B(\vec\rho_0,\vec\Omega,u)\leq B^+e^{\delta|\vec\rho_0|}\), \(B^+=\mathrm{const}\).

The assumption \(\beta_->0\) means that domains \(G\) not containing voids are being considered.

The indicatrix \(g(r,\mu,u',u)\) is uniquely determined by the law of a single scattering and can be written, for example, as follows:
\[ \begin{aligned} g(r,\mu,u',u) &=\frac{\alpha(r)}{\pi}e^{-(u-u')} \Bigg\{ \sigma_0(r,\mu,u',u)\,\delta\!\left(\mu-\mu_0(r,u-u')\right) \\ &\quad+\sum_{n=1}^{\infty} \frac{\sigma_n(r,\mu,u',u)} {\left(1-\frac{M(r)+1}{M(r)}\frac{E_n(r)}{E_0}e^{u'}\right)^{1/2}} \, \delta\!\left(\mu-\mu_0(r,u-u')-\frac{M(r)E_n(r)}{2E_0}e^{(u-u')/2}\right) \\ &\quad+\frac{2}{M(r)} \frac{W(r,\mu,u',u)} {\left(1-\frac{2\mu}{M(r)+1}e^{(u-u')/2} +\frac{1}{(M(r)+1)^2}e^{u-u'}\right)^{1/2}} \Bigg\}, \end{aligned} \tag{4} \]
\[ \mu_0(r,u)=\frac{M(r)+1}{2}e^{-u/2}-\frac{M(r)-1}{2}e^{u/2}. \]

The terms in the right-hand side of (4) that contain the \(\delta\)-function correspond to excitation of a nucleus of mass \(M(r)\) as a result of inelastic scattering of a neutron to the level \(E_n(r)\); the last term corresponds to the continuous spectrum of excitation of the nucleus. Possible anisotropy of scattering in the center-of-mass system is taken into account by the factors \(\sigma_k(r,\mu,u',u)\) and \(W(r,\mu,u',u)\).

As follows from (4), practically without restricting the generality of the physical formulation of the problem, one may assume that for all admissible values of the independent variables:
\[ 6.\quad \sigma_k(r,\mu,u',u)\leq \sigma_k^{0}<\infty,\quad k=0,1,2,\ldots;\qquad \sum_{k=1}^{\infty}\sigma_k=\sigma<\infty . \]
\[ 7.\quad W(r,\mu,u',u)\leq \tau<\infty . \]

The singularities appearing in (4) introduce no fundamental difficulties, since, using (4) and (2), one can give a definition of \(\hat A\) that contains no singularities.

The study of the properties of \(\hat A\) is greatly facilitated by the circumstance that this operator is an operator of Volterra type with respect to the variable \(u\).* It is not difficult to show that \(\hat A\) maps a measurable function into

* In particular, \(\hat A\) has no eigenfunctions in the classes \(K(\gamma)\), \(\gamma<\beta_-\) (cf. the proof of assertion 8). The eigenvalue problem acquires meaning when the structure of \(\hat A\) is changed, for example, when multiplication and acceleration of neutrons are included in the consideration.

measurable. Let \(\gamma<\beta_-\), \(p>0\). Then, by direct calculation, the inequality can be established

\[ \hat A e^{\gamma|\mathbf r|+(p-1)u}\leq \frac{\Phi(p)}{\beta_- -\gamma}\,e^{\gamma|\mathbf r|+(p-1)u}, \tag{5} \]

\[ \Phi(p)=2\alpha_+\beta_+h_+ \left[ \frac{4\tau+\sigma_0(1-e^{-pq_+})}{p} +\sigma B\left(p,\frac12\right) \right]. \tag{6} \]

Here \(B\) is Euler’s integral of the first kind, \(q_+=\ln\left(\dfrac{M_+ +1}{M_- -1}\right)^2\).

From (5) we obtain

\[ \hat A[K(\gamma)]\subset K(\gamma),\qquad \gamma<\beta_-. \]

1. If \(F(Q)\in K(\gamma)\), \(\gamma<\beta_-\), then, uniformly with respect to all \(Q\in H(\bar u)\), \(\bar u<\infty\),

\[ \lim_{\nu=\infty} e^{-\gamma|\mathbf r|}\hat A^\nu |F(Q)|=0. \]

Indeed, for \(p\geq 1\) and a suitable constant \(F_{\bar u}\),

\[ |F(Q)|\leq F_{\bar u}e^{\gamma|\mathbf r|+(p-1)u},\qquad Q\in H(\bar u). \]

According to (6), \(p\) can be chosen so large that \((\beta_- -\gamma)^{-1}\Phi(p)\leq 1/2\), so that for \(Q\in H(\bar u)\)

\[ e^{-\gamma|\mathbf r|}\hat A^\nu |F(Q)|\leq 2^{-\nu}F_{\bar u}e^{(p-1)\bar u}, \]

as was required.

Let \(p>0\), \(k\in[0,\beta_- -\delta)\), \(L>0\), and let \(R(\mathbf r)\) denote the distance from the point \(\mathbf r\) to the set of boundary points of \(\bar G\) at each of which either the source density \(S\) or the flux through the boundary \(B\) is nonzero. Put

\[ \psi_1(Q)=Le^{-kR(\mathbf r)+\delta|\mathbf r|+(p-1)u}. \tag{7} \]

Then

\[ \hat A\psi_1(Q)\leq \frac{\Phi(p)}{\beta_- -(\delta+k)} \left(1-e^{-R_0(\beta_- -(\delta+k))}\right)\psi_1(Q), \tag{8} \]

where \(R_0\) is the diameter of \(G\); \(R_0\in(0,\infty]\). As follows from 3–5 and formula (3), \(T(Q)\in K(\delta)\) and

\[ T(Q)\leq L_1 e^{\delta|\mathbf r|-(\beta_- -\delta)R(\mathbf r)}Y(u_0-u). \tag{9} \]

Here \(Y(x)\) is the Heaviside function:

\[ Y(x)=0\ \text{for }x\leq 0,\qquad Y(x)=1\ \text{for }x>0;\qquad L_1=B^+\beta_+\sqrt{2E_0}+\frac{S^+\beta_+}{\beta_- -\delta}; \]

\[ \hat A\psi_1(Q)+T(Q)\leq \psi_1(Q) \left\{ \frac{L_1}{L}e^{-(\beta_- -(k+\delta))R(\mathbf r)-(p-1)u}Y(u_0-u) + \frac{\Phi(p)}{\beta_- -(k+\delta)} \left(1-e^{-R_0(\beta_- -(\delta+k))}\right) \right\}. \tag{10} \]

Obviously, there exists exactly one \(p=p(k)\) such that

\[ (\beta_- -(k+\delta))^{-1}\Phi(p(k)) \left(1-\exp[-R_0(\beta_- -(\delta+k))]\right)=1. \]

Let \(p>\max\{p(k),0\}\). If \(u_0=\infty\), require in addition \(p\geq 1\). Then there exists \(L\) such that the expression in braces in (10) does not exceed unity for all values \(Q\in H\). Fix these \(p\) and \(L\):

\[ \hat A\psi_1(Q)+T(Q)\leq \psi_1(Q),\qquad Q\in H, \tag{11} \]

whence, for any nonnegative integers \(n\) and \(m\),

\[ \sum_{\nu=n}^{n+m}\hat A^\nu T(Q)\leq \hat A^n\psi_1(Q). \tag{12} \]

(12) means that the series

\[ \sum_{\nu=0}^{\infty} \hat A^{\nu} T(Q)=\psi(Q) \]

converges for all \(Q\in H\), and

\[ \psi(Q)\leqslant \psi_{1}(Q),\qquad Q\in H. \tag{13} \]

Hence, \(\psi(Q)\in K(\delta)\). Moreover, from 8 and (12) it follows that, uniformly with respect to all \(Q\in H(\bar u)\), \(\bar u<\infty\),

\[ \lim_{n=\infty} e^{-\delta |r|}\sum_{\nu=0}^{n}\hat A^{\nu}T(Q)=e^{-\delta |r|}\psi(Q). \tag{14} \]

Using 8 and (14), it is not difficult to show that \(\psi(Q)\) is a solution of equation (1). However, this also follows immediately from (12) and Lebesgue’s theorem on dominated convergence.

Thus we obtain:

1. Under assumptions 1—7, equation (1) has exactly one solution \(\psi(Q)\) in the class of functions \(\displaystyle \bigcup_{\gamma<\beta_-} K(\gamma)\). This solution belongs to \(K(\delta)\), is represented by the Neumann series, and admits the estimate (13) for a suitable choice of the constants \(L\) and \(p\) depending on \(\delta\) and \(k\).

The uniqueness of the solution is an immediate consequence of 8. According to 9, in \(\displaystyle \bigcup_{\gamma<\beta_-} K(\gamma)\) the homogeneous equation (1) has only the trivial solution.

In the particular case of elastic scattering one may drop the requirement \(p>0\). For example, if \(G\) is the whole space \(R_{3}\), filled with a homogeneous moderator scattering neutrons elastically and isotropically in the center-of-mass system, and the sources have a plane-parallel structure and are concentrated in a layer of finite thickness orthogonal to the Cartesian axis \(Oz\), then the estimate (13) of the solution \(\psi(Q)\equiv\psi(z,\vec\Omega,u)\) takes the form

\[ \psi(z,\vec\Omega,u)\leqslant Le^{-k|z|+(p-1)u}, \tag{15} \]

\(k\in[0,1]\), \(p>p^{*}(k)\), where

\[ \frac{\alpha h}{1-k}\cdot \frac{1}{p(k)}\left(1-e^{-qp(k)}\right)=1,\qquad \alpha=\frac{(M+1)^{2}}{4M}, \]

\[ q=\ln\left(\frac{M+1}{M-1}\right)^{2}. \]

(15) is valid under the assumption that the characteristics of the medium do not depend on the neutron energy. The mean free path is taken to be unity.

I express my gratitude to E. S. Kuznetsov and V. A. Chuyanov for discussing the results of the work.

Department of Applied Mathematics
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
8 VIII 1957