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The steady-state continuous energy diffusion equation

In this text we make no effort in making a derivation^1.1 of the steady-state diffusion equation^1.2--we merely postulate that the transport of neutrons in a media is controlled by

$$\begin{eqnarray*} \lefteqn {- \nabla \cdot \left[ D(\hspace{0.2ex}\underline{r}... ...},E') \Phi(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex},E') dE' \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

where

>

denotes the spatial dependence (i.e. a position vector).

E

denotes the energy [eV].

>

is the neutron flux density^1.3 [neutrons/ $(\mbox{cm}^2\cdot\mbox{s}\cdot\mbox{eV})$].

D

is the diffusion coefficient of the media [cm].

>

is the macroscopic absorption cross section of the media [ $\mbox{cm}^{-1}$].

>

is the macroscopic scattering cross section per energy at position r for scattering of neutrons with initial energy E' resulting in neutrons with energy E [ $\mbox{cm}^{-1} \mbox{eV}^{-1}$].

>

is the average number of neutrons released by a fission with incident neutron energy E [--].

>

is the fraction of neutrons released by fission with the energy E [--], $\int_0^\infty \chi (E) dE = 1$.

>

is the macroscopic fission cross section [ $\mbox{cm}^{-1}$].

>

is the fundamental (i.e. the largest) eigenvalue of the system [--].

We note that (1.1) is an eigenvalue problem and that the only physically acceptable solution is the eigensolution $\{\lambda_0,\Phi\}$ which corresponds to the eigenvalue with the largest modulus, $\lambda_0$. The (real positive) constant $\lambda_0$ is also called the multiplication constant. The multiplication constant shows whether the system is super-critical ($\lambda_0 > 1$), critical ( $\lambda_0 \equiv 1$) or sub-critical ($\lambda_0 < 1$). We see directly from (1.1) that if $\{\lambda_0,\Phi\}$ is a solution so is $\{\lambda_0, C\Phi\}$, where $C \in \Re_+$. In order to determine $\Phi$ uniquely we could for instance require a critical reactor, ie demand that $\lambda_0 = 1$. This is accomplished by altering either the material composition of the reactor, the geometry of the reactor or a combination of both.

Strictly speaking, the only physically acceptable solution $\{\lambda_0 , C\Phi\}$ satisfies $\lambda_0 = 1$, since for $\lambda_0 \ne 1$ no steady-state solution exists. None the less as a first approximation one often assumes [14] that the required reactor modification in order to achieve criticality is so little that the true (physical) flux for $\lambda_0 = 1$ is not significantly different in shape compared to the one for $\lambda_0 \ne 1$.

It should be noted that (1.1) is a mathematical formulation - in any realistic situation the integrations are only performed on a finite energy interval, ie from 0 up to some maximum energy E_max. In calculations involving thermal nuclear reactors it is allowable to put $E_{max} = 15 \:\mbox{Mev}$. Furthermore in the case of a critical system $\lambda_0 = 1$ we may note that (1.1) in fact states the neutron balance^1.4 in the steady-state case. The physical importance of the different terms in (1.1) is given below (from left to right):

.

Total outflow of neutrons with energy E due to diffusion.

.

Neutrons with energy E lost by absorption.

.

Neutrons scattered "out of energy E".

.

Neutrons scattered "into energy E".

.

Total number of neutrons produced by fission which have an energy of E (divided by the fundamental eigenvalue).

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