# Cross Section Models – pyne.xs.models¶

This module provides functions to compute cross sections, and related quantities, from fundamental physical models. From an empirical perspective, physical models are not as valuable as experimental data. However, these functions exist to be called upon when experimental data is not available.

All functionality may be found in the models module:

from pyne.xs import models


Many of these functions may be called with either scalar or vector arguments.

The following terminology applies for this module:

• G: The number of low-resolution energy groups, indexed by g.
• N: The number of high-resolution energy groups, indexed by n.
• E_g: The low-resolution group structure [MeV]. Must be monotonic.
• E_n: The high-resolution group structure [MeV]. Must be monotonic in the same direction as E_g.
• phi: Neutron flux [n/cm^2/s].
• E: The incident neutron energy [MeV].
• E’: The exiting neutron energy [MeV].
• theta: The scattering angle [radians].
• M_A: The atomic mass [amu] of the target material.
• T: The temperature [kelvin] of the target material.

## Models API¶

This module provides physical cross-section models and helper functions.

pyne.xs.models.E_prime_min()

The minimum possible exiting enegy of a neuron after a scattering collision. This is based on the incident energy and the mass of the target. For a proof, use the conservation of energy and momentum.

$\mbox{min}(E^\prime) = \left(\frac{M_A - m_n}{M_A + m_n}\right)^2 E$
Parameters: E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. M_A : float (or array), optional Atomic mass of the target nucleus [amu]. min_E_prime : float (or array) Minimum exiting energy.
pyne.xs.models.alpha()

Scattering kernel alpha value.

$\alpha = \frac{E^\prime + E - 2\sqrt{E^\prime E}\cos\theta}{\frac{M_A}{m_n}kT}$
Parameters: E_prime : float (or array) The exiting energy of the neutron after scattering event [MeV]. E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. theta : float (or array) Scattering angle in [radians]. M_A : float (or array), optional Atomic mass of the target nucleus [amu]. T : float (or array), optional Tempurature of the target material [kelvin]. a : float (or array) alpha value
pyne.xs.models.alpha_at_theta_0()

Scattering kernel alpha value at the lower bound of the scattering angle.

$\alpha_{\theta=0} = \frac{E^\prime + E - 2\sqrt{E^\prime E}}{\frac{M_A}{m_n}kT}$
Parameters: E_prime : float (or array) The exiting energy of the neutron after scattering event [MeV]. E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. M_A : float (or array), optional Atomic mass of the target nucleus [amu]. T : float (or array), optional Tempurature of the target material [kelvin]. a : float (or array) alpha value with theta = 0.
pyne.xs.models.alpha_at_theta_pi()

Scattering kernel alpha value at the upper bound of the scattering angle.

$\alpha_{\theta=\pi} = \frac{E^\prime + E + 2\sqrt{E^\prime E}}{\frac{M_A}{m_n}kT}$
Parameters: E_prime : float (or array) The exiting energy of the neutron after scattering event [MeV]. E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. M_A : float (or array), optional Atomic mass of the target nucleus [amu]. T : float (or array), optional Tempurature of the target material [kelvin]. a : float (or array) alpha value with theta = pi.
pyne.xs.models.beta()

Scattering kernel beta value.

$\beta = \frac{E^\prime - E}{kT}$
Parameters: E_prime : float (or array) The exiting energy of the neutron after scattering event [MeV]. E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. T : float (or array), optional Tempurature of the target material [kelvin]. b : float beta value.
pyne.xs.models.chi()

Calculates the fission neutron spectrum (frequency) at energy E. E may be either a float or an array of floats. This is based off of the values for U-235, which are representative for other isotopes. See Lamarsh or ‘Comparison of prompt-fission neutron multiplicities and energy spectra for intermediate energy proton-and neutron-induced fission’ –Oleg Batenkov, Georgy Boikov, Vilen Eismont, Mikhail Majorov, Sergey Soloviev, Jan Blomgren, and Walter Loveland.

pyne.xs.models.fastspect()

This function produces a rough estimate fast neutron spectrum. Parameters ———- E : array

Array representing the energy groups of interest.
T : float
The temperature of the reactor.
lower: float
The point at which the shape of the flux switches from high energy to low energy.
Returns: phi : array flux values for the reactor
pyne.xs.models.group_collapse()

Calculates the group cross-sections for a nuclide for a new, lower resolution group structure using a higher fidelity flux. Note that g indexes G, n indexes N, and G < N.

This function has two optional ways of being called. If the group boundaries E_g and E_n are provided, this will collapse the flux automatically. However, if a partial energy matrix and flux collapse has already been performed you can shortcut their recalculation by calling this function with the phi_g and partial_energies keyword arguments.

Parameters: sigma_n : array-like of floats) A high-fidelity cross-section. phi_n : array-like of floats The high-fidelity flux [n/cm^2/s] to collapse the fission cross-section over (length N). phi_g : array-like of floats, optional The low-fidelity flux [n/cm^2/s] to collapse the fission cross-section down to (length G). If present, partial_energies is needed as well. partial_energies : 2D array-like of floats, optional A partial energy matrix as provided by a previous call to the function partial_energy_matrix(). If present, phi_g is needed as well. E_g : array-like of floats, optional Lower resolution energy group structure [MeV] that is of length G+1. If present, E_n is needed as well. E_n : array-like of floats, optional Higher resolution energy group structure [MeV] that is of length N+1. If present, E_g is needed as well. sigma_g : ndarray An array of the collapsed fission cross-section.
pyne.xs.models.one_over_gamma_squared()

The inverse of the Lorentz factor sqared. Sometimes used as a realitivistic correction factor for the bound scattering length.

$\frac{1}{\gamma^2} = \left( 1 - \frac{2E}{931.46 \cdot m_n} \right)$
Parameters: E : float (or array) The incident energy of the neutron prior to scattering event [MeV]. inv_g2 : float (or array) Inverse of gamma squared.
pyne.xs.models.partial_energy_matrix()

Gerenates a matrix of fractional values that may be used to converts a high-resolution flux array with group structure E_n to a low-resolution flux array with group-structure E_g. The group structures must have the same monotonicity. This is useful for performing group collapses.

Parameters: E_g : sequence of floats Lower resolution energy group structure [MeV] that is of length G+1. E_n : sequence of floats Higher resolution energy group structure [MeV] that is of length N+1. pem : 2d numpy float array of fractions This is a GxN sized matrix that when dotted with a high-resolution flux (or cross section) produces a low-resolution flux (or cross section).
pyne.xs.models.partial_energy_matrix_mono()

Generates a matrix of fractional values that may be used to converts a high-resolution flux array with group structure E_n to a low-resolution flux array with group-structure E_g. Here, both of the energy arrays must be monotonic. This is useful for performing group collapses.

Parameters: E_g : 1d numpy float array Lower resolution energy group structure [MeV] that is of length G+1. Ordered based on slope. E_n : 1d numpy float array Higher resolution energy group structure [MeV] that is of length N+1. Ordered based on slope. slope : int, optional Gives the monotonicity of E_g and E_n. If positive, then they are monotonicly increasing (lowest-to-highest). If negative, they are monotonicly decreasing (highest-to-lowest). pem : 2d numpy float array of fractions This is a GxN sized matrix that when dotted with a high-resolution flux (or cross section) produces a low-resolution flux (or cross section).
pyne.xs.models.phi_g()

Calculates a lower resolution flux, phi_g, from a lower resolution group stucture E_g, a higher resolution groups E_n, and a higher resolution flux phi_n.

Parameters: E_g : sequence of floats Lower resolution energy group structure [MeV] that is of length G+1. E_n : sequence of floats Higher resolution energy group structure [MeV] that is of length N+1. phi_n : sequence of floats The high-fidelity flux [n/cm^2/s] to collapse the fission cross-section over (length N). phi_g : numpy array of floats The flux collapsed to G energy groups.
pyne.xs.models.sigma_s()

Computes the scattering cross section from an analytic model. The model accounts for both one-over-v dependence and relativistic effects and the bound scattering length provided. This model does not include resonances. This function works on both float and array values for the energy.

$\sigma_s(E) = 4 \pi b^2 \cdot \left( 1 - \frac{2E}{931.46 \cdot m_n} \right) \cdot \left( 1 + \frac{m_n}{M_A} \frac{kT}{E} \cdot e^{-\frac{M_A}{m_n}\frac{E}{kT}} \right) \cdot \left( 1 - \mbox{Exp}\left[-\sqrt{\frac{0.1}{E}}\right] \right)$
Parameters: E : float or array-like The incident energy of the neutron prior to the scattering event [MeV]. b : float, optional The bound scattering length of the target nucleus [cm]. M_A : float, optional Atomic mass of the target nucleus [amu]. T : float, optional Tempurature of the target material [kelvin]. sig_s : float or ndarray The scattering cross section evaluated at the given energy.

See also

pyne.data.b
scattering length data.
pyne.data.atomic_mass
Atomic mass data.
pyne.xs.models.sigma_s_const()

Computes the constant scattering cross-section based on the scattering length.

$\sigma_s = 4 \pi b^2$
Parameters: b : float (or array) The bound scattering length [cm] of the target nucleus. sig_s : float (or array) The micorscopic scattering cross-section [barns].

See also

pyne.data.b
scattering length data.
pyne.xs.models.thermspect()

This function produces a rough estimate thermal spectrum for an average thermal reactor. Parameters ———- E : array

Array representing the energy groups of interest.
T : float
The temperature of the reactor.
lower: float
The point at which the shape of the flux switches from high energy to low energy.
Returns: phi : array flux values for the reactor