Neutron density, flux and reaction rates

In reactor physics it is customary[32] to define the number of neutrons per unit volume, per velocity bin and solid angle unit $n(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)$ such that the number of neutrons in volume d³r, at a position $\overrightarrow{r} ,$ with a velocity between v and v+dv pointing in direction $\overrightarrow {\Omega}$(unit vector) within solid angle $d^{2}\Omega$ is $n(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)d^{3}rdvd^{2}\Omega.$ The flux of neutrons is defined as

$$\phi(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)=vn(\overrightarrow {r} ,v,\overrightarrow{\Omega} ,t)$$ (4.8)

The number of neutrons per time unit with velocity v and direction $\overrightarrow {\Omega}$ which cross a planar unit surface at position $\overrightarrow{r}$ and unit normal vector $\overrightarrow{u}$ is: $\phi$ $(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)\overrightarrow{\Omega } .\overrightarrow{u} .$ The case where $\phi$ is isotropic is particularly interesting, since it is closely realised in reactors. Then, if one measures the angles with respect to the normal vector $\overrightarrow{u} ,$ we obtain the total number of neutrons crossing the surface, irrespective of their direction by:

$$4\pi\phi(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)\in... ...ta=2\pi\phi(\overrightarrow{r} ,v,\overrightarrow {\Omega} ,t)$$

We, now, consider a thin slab with unit surface, atomic thickness of n_s identical nuclei per unit surface. The nuclei have reaction cross-section $\sigma.$ The number of reactions in the slab per unit time reads:

$$n_{reac}=4\pi\phi(\overrightarrow{r} ,v,\overrightarrow{\Omeg... ...i(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)n_{s}\sigma$$

The total flux is the directional flux integrated over angle, thus:

$$\varphi(\overrightarrow{r} ,v,t)=4\pi\phi(\overrightarrow{r} ,v,\overrightarrow {\Omega} ,t)$$

This quantity is, usually, named neutron flux. In term of this quantity, the number of reactions per time unit is, thus:

$$n_{reac}=n_{s}\sigma\varphi(\overrightarrow{r} ,v,t)$$ (4.9)

while the number of neutrons crossing a unit surface plane per time unit is $\frac{\varphi(\overrightarrow{r} ,v,t)}{2}$.^4.1

Instead of computing the number of reactions per unit time, it is instructive to compute the total length travelled by all neutrons traversing a thin, unit surface, slab, which we assume to have thickness l, that is very small compared to the transverse dimensions of the slab. Then the total length travelled by the neutrons is

$$L=4\pi\phi(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)l... ...^{\pi /2}\sin(\theta)d\theta=\varphi(\overrightarrow{r} ,v,t)l$$

where the volume V of the slab is simply l, since it has unit surface. Thus we obtain an expression for the flux:

$$\varphi(\overrightarrow{r} ,v,t)=\frac{L}{V}$$ (4.10)

The formula was demonstrated for an infinitely thin slab. However, for isotropic neutron fluxes, it can be generalized to any arbitrary volume. Indeed, any volume can be subdivided, at will, into n small, thin, parallelograms. For each of the elementary volumes we have the flux value $\varphi_{i}=\frac{L_{i}}{V_{i}}$. The average flux over the volume is the average of the $\varphi_{i}$ weighed by the volume:

$$<\varphi>=\frac{\sum V_{i}\varphi_{i}}{\sum V_{i}}=\frac{\sum L_{i}}{V} =\frac{L}{V}$$ (4.11)

since the total length travelled by the neutrons is clearly the sum of the elementary lengths. This is an important formula since it is the one used in all Monte Carlo simulations. It can be shown that this formula also holds for anisotropic neutron fluxes.

Note that the definition of the neutron flux is different from the usual definition of flux in other fields of physics. For example, in thermodynamics, a finite heat flux through a surface requires a temperature gradient across this surface. The analogous of the heat flux in neutronics is the neutron current defined as:

$$\overrightarrow{J} (\overrightarrow{r} ,v,t)=\int_{(4\pi)}\ov... ...i(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)d^{2}\Omega$$ (4.12)

This current is only different from 0 for anisotropic neutron fluxes. One sided currents are, also, frequently used and defined by $\overrightarrow {J^{+}} (\overrightarrow{r} ,v,t)=\int_{\overrightarrow{\Omega}... ...arrow{\Omega}\phi(\overrightarrow{r} ,v,\overrightarrow{\Omega } ,t)d^{2}\Omega$ and $\overrightarrow{J^{-}} (\overrightarrow{r} ,v,t)=\int _{\overrightarrow{\Omega}... ...tarrow{\Omega}\phi(\overrightarrow{r} ,v,\overrightarrow{\Omega} ,t)d^{2}\Omega$ where $\overrightarrow{N}$ is the direction in which the flux is measured.

The number of interactions of type $(\alpha)$ per cm³ is $\Sigma ^{(\alpha)}\varphi$ where $\varphi$ is the neutron flux expressed in n/cm²/s. The neutron mean free path for reaction $(\alpha)$ is $\Lambda^{(\alpha)}(cm)\ =\ \frac{1}{\Sigma^{(\alpha)}}$. The most important macroscopic cross-sections are the scattering cross-section $\Sigma_{S}$, the absorption cross-section $\Sigma_{a}$ and the fission cross-section $\Sigma_{f}=\ \Sigma_{a}P_{f}$ where P_f is the fission probability.