introduction
One issue in application with neutrons or other particles is the damage induced in the materials, indeed, high energy neutrons can collide with atoms creating “collisional cascades” meaning an increase of vacancy and interstitial populations. In this page you will find a short explanation about a simple way to simulate neutron and interstitial interaction with matter through Monte Carlo method.
two body elastic collision
Let’s consider the collision between two particles, one of them (called target) is assumed at rest in the Laboratory reference frame. we will consider lowercase letters for the projectile and uppercase letters for the target. In figure A you can see the rapresentation of the collision in lab and center of mass reference frames:
By definition \vec V_L = \vec v_{cm} + \vec V_C but since \vec V_L = 0 we have that \vec v_{cm} = - \vec V_C . Since in center of mass frame the modulus of the velocities is conserved we can state that v_{cm} = V'_C . From this we can easly notice from figure B that \gamma = \frac{\pi - \theta_c}{2} .
The scattering angle \theta_c can be extracted uniformly between zero and 180°.
Thanks to some trigonometric derivations we can easly find that the angle in Lab frame is:
cos\theta_L = \frac{1+Acos\theta_c}{\sqrt{A^2 + 2Acos\theta_c +1}}
Where A is the fraction M/m.
Finally the energy of the projectile after the collision is:
E' = E\frac{(1+\alpha) + (1-\alpha)cos\theta_c}{2}
Where E is the energy before the collision and \alpha is defined as
\alpha = \left ( \frac{A-1}{A+1} \right )^2
.
All these formulas can be found in: Lamarsh Introduction to nuclear reactor theory, or in this other page.
When we simulate the collision between neutrons and heavy nuclei we can consider just the center of mass scattering angle, but when two particles of same mass collide we must consider the lab angle and also the direction before the collision (with respect our coordinate system). For example in this figure, the dashed line rapresent the arriving direction of the projectile.
Then the positions of the two particles after the impact will be:
x_{projectile} = x_0 + dsin(\beta + \theta_L)sin(\phi)
y_{projectile} = y_0 + dsin(\beta + \theta_L)cos(\phi)
z_{projectile} = z_0 + dcos(\beta + \theta_L)
x_{target} = x_0 + dsin(\beta - \gamma)sin(\phi)
y_{target} = y_0 + dsin(\beta - \gamma)cos(\phi)
z_{target} = z_0 + dcos(\beta - \gamma)
cross section
For Neutron – Atom interactions just absorption and elastic scattering cross sections have been considered. Data can be obtained by databases like this, then the file can be converted in a vector function of energy.
For Atom – Atom collision an Hard sphere cross section has been adopted. this mean \sigma = \pi (2R)^2 where R is the interaction radius.
algorithm
