Mean free path neutrons

OpenMC can also be used find the mean free path of a neutron in a specific material.

This python notebook allows users find the mean free path of some common fusion materials.

This follows on from the previous notebook where material (Macroscopic) cross sections were plotted.

These plots show the Macroscopic cross section which is the the effective target area of all of the nuclei contained in the volume of the material.

Macroscopic cross section (Σ) is related to Microscopic cross section (σ) with the following equation.

The units of (Σ) Macroscopic cross section are \(\mathrm{m}^{-1}\) and therefore the mean free path is just 1/Σ at the energy of the neutron

First import OpenMC and configure the nuclear data path

import openmc
from pathlib import Path
import matplotlib.pyplot as plt
# Setting the cross section path to the correct location in the docker image.
# If you are running this outside the docker image you will have to change this path to your local cross section path.
openmc.config['cross_sections'] = Path.home() / 'nuclear_data' / 'cross_sections.xml'

We will now make some materials and get the mean free path at 14MeV (14e6eV) which is the rough energy DT neutrons are emitted with.

We will also get the mean free path at thermal neutron energy, this is the energy neutrons have when they have been slowed down (moderated) to be in equilibrium with the material.

We can start with a minimal steel.

mat1 = openmc.Material(name = 'Steel')
mat1.add_element('Fe', 0.975)
mat1.add_element('C', 0.025)
mat1.set_density('g/cm3', 7.7)

mean_free_path_14 = mat1.mean_free_path(energy=14e6)
mean_free_path_thermal = mat1.mean_free_path(energy=0.025)

print(f'Mean free path of a neutron in {mat1.name} at 14e6 eV = {mean_free_path_14:.4f} cm')
print(f'Mean free path of a neutron in {mat1.name} at 0.025 eV = {mean_free_path_thermal:.4f} cm')

Mean free path of a neutron in Steel at 14e6 eV = 4.6263 cm
Mean free path of a neutron in Steel at 0.025 eV = 0.8608 cm

Next material is tungsten which is often used as first wall armour.

mat2 = openmc.Material(name = 'Tungsten')
mat2.add_element('W', 1)
mat2.set_density('g/cm3', 19.3)

mean_free_path_14 = mat2.mean_free_path(energy=14e6)
mean_free_path_thermal = mat2.mean_free_path(energy=0.025)

print(f'Mean free path of a neutron in {mat2.name} at 14e6 eV = {mean_free_path_14:.4f} cm')
print(f'Mean free path of a neutron in {mat2.name} at 0.025 eV = {mean_free_path_thermal:.4f} cm')

Mean free path of a neutron in Tungsten at 14e6 eV = 2.9505 cm
Mean free path of a neutron in Tungsten at 0.025 eV = 0.6684 cm

Next up is a lithium ceramic, Li4SiO4 is a candidate breeder material

mat3 = openmc.Material(name = 'Li4SiO4')
mat3.add_element('Li', 4.0, percent_type='ao')
mat3.add_element('Si', 1.0, percent_type='ao')
mat3.add_element('O', 4.0, percent_type='ao')
mat3.set_density('g/cm3', 2.32)

mean_free_path_14 = mat3.mean_free_path(energy=14e6)
mean_free_path_thermal = mat3.mean_free_path(energy=0.025)

print(f'Mean free path of a neutron in {mat3.name} at 14e6 eV = {mean_free_path_14:.4f} cm')
print(f'Mean free path of a neutron in {mat3.name} at 0.025 eV = {mean_free_path_thermal:.4f} cm')

Mean free path of a neutron in Li4SiO4 at 14e6 eV = 6.1760 cm
Mean free path of a neutron in Li4SiO4 at 0.025 eV = 0.2780 cm

Finally concrete which is often used for bioshields, this is regular concrete and one might also consider borated or other concretes

mat4 = openmc.Material(name='Concrete')
mat4.add_element("H",0.168759)
mat4.add_element("C",0.001416)
mat4.add_element("O",0.562524)
mat4.add_element("Na",0.011838)
mat4.add_element("Mg",0.0014)
mat4.add_element("Al",0.021354)
mat4.add_element("Si",0.204115)
mat4.add_element("K",0.005656)
mat4.add_element("Ca",0.018674)
mat4.add_element("Fe",0.00426)
mat4.set_density("g/cm3", 2.3)

mean_free_path_14 = mat4.mean_free_path(energy=14e6)
mean_free_path_thermal = mat4.mean_free_path(energy=0.025)

print(f'Mean free path of a neutron in {mat4.name} at 14e6 eV = {mean_free_path_14:.4f} cm')
print(f'Mean free path of a neutron in {mat4.name} at 0.025 eV = {mean_free_path_thermal:.4f} cm')

Mean free path of a neutron in Concrete at 14e6 eV = 8.2203 cm
Mean free path of a neutron in Concrete at 0.025 eV = 1.5279 cm

This code block then plots the mean free path of the materials for comparison

all_materials = [mat1, mat2, mat3, mat4]

labels = [m.name for m in all_materials]
mfp_14 = [m.mean_free_path(energy=14e6) for m in all_materials]
mfp_thermal = [m.mean_free_path(energy=0.025) for m in all_materials]

bar_width = 0.35
y = range(len(labels))

fig, ax = plt.subplots(figsize=(8, 4))
ax.barh([i + bar_width/2 for i in y], mfp_14, bar_width, label='14e6 eV')
ax.barh([i - bar_width/2 for i in y], mfp_thermal, bar_width, label='Thermal (0.025 eV)')

ax.set_yticks(y)
ax.set_yticklabels(labels)
ax.set_xlabel('Mean Free Path (cm)')
ax.set_title('Mean Free Path of Neutrons in Different Materials')
ax.legend()
plt.tight_layout()
plt.show()

Learning Outcomes:

• OpenMC can be used to find the mean free path of a neutron in the material

• The mean free path varies with neutron energy

• The mean free path varies with material (composition and density)

• The mean free path is shorter at thermal energies compared to the birth energy of a DT fusion neutron (14MeV)

Additional information

The mean free path information is useful when setting mesh sizes for weight window generation, more on that in the variance reduction tasks.