Physics and Verification

Inclusive Cross-sections

All cross-section data are taken from the ENDF/B-VI[1] evaluated data library.

All inclusive cross-sections are treated as point-wise cross-sections for reasons of performance. For this purpose, the data from the evaluated data library have been processed, to explicitly include all neutron nuclear resonances in the form of point-like cross-sections rather than in the form of parametrisations. The resulting data have been transformed into a linearly interpolable format, such that the error due to linear interpolation between adjacent data points is smaller than a few percent.

The inclusive cross-sections comply with the cross-sections data set interface of the GEANT4 hadronic design. They are, when registered with the tool-kit at initialisation, used to select the basic process. In the case of fission and inelastic scattering, point-wise semi-inclusive cross-sections are also used in order to decide on the active channel for an individual interaction. As an example, in the case of fission this could be first, second, third, or forth chance fission.

Elastic Scattering

The final state of elastic scattering is described by sampling the differential scattering cross-sections ${{\rm d} \sigma \over {\rm d} \Omega}$. Two representations are supported for the normalised differential cross-section for elastic scattering. The first is a tabulation of the differential cross-section, as a function of the cosine of the scattering angle $\theta$ and the kinetic energy $E$ of the incoming neutron.

$${{\rm d} \sigma \over {\rm d} \Omega}~=~ {{\rm d} \sigma \over {\rm d} \Omega}\left(\cos{\theta,~E}\right)$$

The tabulations used are normalised by $\sigma/(2\pi)$ so the integral of the differential cross-sections over the scattering angle yields unity.

In the second representation, the normalised cross-section are represented as a series of legendre polynomials $P_l(\cos{\theta})$, and the legendre coefficients $a_l$ are tabulated as a function of the incoming energy of the neutron.

$${2\pi\over\sigma (E)}{{\rm d} \sigma \over {\rm d} \Omega}\le... ...right)~=~ \sum_{l=0}^{n_l} {2l+1\over 2}a_l(E)P_l(\cos{\theta})$$

Describing the details of the sampling procedures is outside the scope of this paper.

An example of the result we show in figure 33.1 for the elastic scattering of 15 MeV neutrons off Uranium a comparison of the simulated angular distribution of the scattered neutrons with evaluated data. The points are the evaluated data, the histogram is the Monte Carlo prediction.

In order to provide full test-coverage for the algorithms, similar tests have been performed for ${\rm ^{72}Ge}$, ${\rm ^{126}Sn}$, ${\rm ^{238}U}$, ${\rm ^{4}He}$, and ${\rm ^{27}Al}$ for a set of neutron kinetic energies. The agreement is very good for all values of scattering angle and neutron energy investigated.

Figure 33.1: Comparison of data and Monte Carlo for the angular distribution of 15 MeV neutrons scattered elastically off Uranium ($^{238}U$). The points are evaluated data, and the histogram is the Monte Carlo prediction. The lower plot excludes the forward peak, to better show the Frenel structure of the angular distribution of the scattered neutron.

Radiative Capture

The final state of radiative capture is described by either photon multiplicities, or photon production cross-sections, and the discrete and continuous contributions to the photon energy spectra, along with the angular distributions of the emitted photons.

For the description of the photon multiplicity there are two supported data representations. It can either be tabulated as a function of the energy of the incoming neutron for each discrete photon as well as the eventual continuum contribution, or the full transition probability array is known, and used to determine the photon yields. If photon production cross-sections are used, only a tabulated form is supported.

The photon energies $E_\gamma$ are associated to the multiplicities or the cross-sections for all discrete photon emissions. For the continuum contribution, the normalised emission probability $f$ is broken down into a weighted sum of normalised distributions $g$.

$$f\left(E\rightarrow E_\gamma\right)~=~ \sum_{i}p_i(E)g_i(E\rightarrow E_\gamma)$$

The weights $p_i$ are tabulated as a function of the energy $E$ of the incoming neutron. For each neutron energy, the distributions $g$ are tabulated as a function of the photon energy. As in the ENDF/B-VI data formats[1], several interpolation laws are used to minimise the amount of data, and optimise the descriptive power. All data are derived from evaluated data libraries.

The techniques used to describe and sample the angular distributions are identical to the case of elastic scattering, with the difference that there is either a tabulation or a set of legendre coefficients for each photon energy and continuum distribution.

As an example of the results is shown in figure33.2 the energy distribution of the emitted photons for the radiative capture of 15 MeV neutrons on Uranium ($^{238}$U). Similar comparisons for photon yields, energy and angular distributions have been performed for capture on ${\rm ^{238}U}$, ${\rm ^{235}U}$, ${\rm ^{23}Na}$, and ${\rm ^{14}N}$ for a set of incoming neutron energies. In all cases investigated the agreement between evaluated data and Monte Carlo is very good.

Figure 33.2: Comparison of data and Monte Carlo for photon energy distributions for radiative capture of 15 MeV neutrons on Uranium ($^{238}U$). The points are evaluated data, the histogram is the Monte Carlo prediction.

Fission

For neutron induced fission, we take first chance, second chance, third chance and forth chance fission into account.

Neutron yields are tabulated as a function of both the incoming and outgoing neutron energy. The neutron angular distributions are either tabulated, or represented in terms of an expansion in legendre polynomials, similar to the angular distributions for neutron elastic scattering. In case no data are available on the angular distribution, isotropic emission in the centre of mass system of the collision is assumed.

There are six different possibilities implemented to represent the neutron energy distributions. The energy distribution of the fission neutrons $f(E\rightarrow E')$ can be tabulated as a normalised function of the incoming and outgoing neutron energy, again using the ENDF/B-VI interpolation schemes to minimise data volume and maximise precision.

The energy distribution can also be represented as a general evaporation spectrum,

$$f(E\rightarrow E')~=~f\left(E'/\Theta(E)\right).$$

Here $E$ is the energy of the incoming neutron, $E'$ is the energy of a fission neutron, and $\Theta(E)$ is effective temperature used to characterise the secondary neutron energy distribution. Both the effective temperature and the functional behaviour of the energy distribution are taken from tabulations.

Alternatively energy distribution can be represented as a Maxwell spectrum,

$$f(E\rightarrow E')~\propto~\sqrt{E'}{\rm e}^{E'/\Theta(E)},$$

or a evaporation spectrum

$$f(E\rightarrow E')~\propto~E'{\rm e}^{E'/\Theta(E)}.$$

In both these cases, the temperature is tabulated as a function of the incoming neutron energy.

The last two options are the energy dependent Watt spectrum, and the Madland Nix spectrum. For the energy dependent Watt spectrum, the energy distribution is represented as

$$f(E\rightarrow E')~\propto~{\rm e}^{-E'/a(E)}\sinh{\sqrt{b(E)E'}}.$$

Here both the parameters a, and b are used from tabulation as function of the incoming neutron energy. In the case of the Madland Nix spectrum, the energy distribution is described as

$$f(E\rightarrow E')~=~{1\over 2}\left[g(E',<K_l>)~+~g(E',<K_h>)\right].$$

Here

$$g(E',<K>)~=~ {1\over 3\sqrt{<K>\Theta}}\left[u_2^{3/2}E_1(u_2)-u_1^{3/2}E_1(u_1) +\gamma(3/2, u_2) - \gamma(3/2, u_1)\right],$$

$$u_1(E',<K>) = {(\sqrt{E'}-\sqrt{<K>})^2 \over \Theta},~{\rm and}$$

$$u_2(E',<K>) = {(\sqrt{E'}+\sqrt{<K>})^2 \over \Theta}.$$

Here $K_l$ is the kinetic energy of light fragments and $K_h$ the kinetic energy of heavy fragments, $E_1(x)$ is the exponential integral, and $\gamma(x)$ is the incomplete gamma function. The mean kinetic energies for light and heavy fragments are assumed to be energy independent. The temperature $\Theta$ is tabulated as a function of the kinetic energy of the incoming neutron.

Fission photons are describes in analogy to capture photons, where evaluated data are available. The measured nuclear excitation levels and transition probabilities are used otherwise, if available.

As an example of the results is shown in figure33.3 the energy distribution of the fission neutrons in third chance fission of 15 MeV neutrons on Uranium ($^{238}$U). This distribution contains two evaporation spectra and one Watt spectrum. Similar comparisons for neutron yields, energy and angular distributions, and well as fission photon yields, energy and angular distributions have been performed for ${\rm ^{238}U}$, ${\rm ^{235}U}$, ${\rm ^{234}U}$, and ${\rm ^{241}Am}$ for a set of incoming neutron energies. In all cases the agreement between evaluated data and Monte Carlo is very good.

Figure 33.3: Comparison of data and Monte Carlo for fission neutron energy distributions for induced fission by 15 MeV neutrons on Uranium ($^{238}U$). The curve represents evaluated data and the histogram is the Monte Carlo prediction.

Inelastic Scattering

For inelastic scattering, the currently supported final states are (nA$\rightarrow$) n$\gamma$s (discrete and continuum), np, nd, nt, n$^3$He, n$\alpha$, nd2$\alpha$, nt2$\alpha$, n2p, n2$\alpha$, np$\alpha$, n3$\alpha$, 2n, 2np, 2nd, 2n$\alpha$, 2n2$\alpha$, nX, 3n, 3np, 3n$\alpha$, 4n, p, pd, p$\alpha$, 2p d, d$\alpha$, d2$\alpha$, dt, t, t2$\alpha$, $^3$He, $\alpha$, 2$\alpha$, and 3$\alpha$.

The photon distributions are again described as in the case of radiative capture.

The possibility to describe the angular and energy distributions of the final state particles as in the case of fission is maintained, except that normally only the arbitrary tabulation of secondary energies is applicable.

In addition, we support the possibility to describe the energy angular correlations explicitly, in analogy with the ENDF/B-VI data formats. In this case, the production cross-section for reaction product n can be written as

$$\sigma_n(E, E', \cos(\theta))~=~\sigma(E)Y_n(E)p(E, E', \cos(\theta)).$$

Here $Y_n(E)$ is the product multiplicity, $\sigma(E)$ is the inelastic cross-section, and $p(E, E', \cos(\theta))$ is the distribution probability. Azimuthal symmetry is assumed.

The representations for the distribution probability supported are isotropic emission, discrete two-body kinematics, N-body phase-space distribution, continuum energy-angle distributions, and continuum angle-energy distributions in the laboratory system.

The description of isotropic emission and discrete two-body kinematics is possible without further information. In the case of N-body phase-space distribution, tabulated values for the number of particles being treated by the law, and the total mass of these particles are used. For the continuum energy-angle distributions, several options for representing the angular dependence are available. Apart from the already introduced methods of expansion in terms of legendre polynomials, and tabulation (here in both the incoming neutron energy, and the secondary energy), the Kalbach-Mann systematic is available. In the case of the continuum angle-energy distributions in the laboratory system, only the tabulated form in incoming neutron energy, product energy, and product angle is implemented.

First comparisons for product yields, energy and angular distributions have been performed for a set of incoming neutron energies, but full test coverage is still to be achieved. In all cases currently investigated, the agreement between evaluated data and Monte Carlo is very good.