MATTER

A Coulomb-scattering and energy-absorbing element simulating material in the beam path.

Parameter Name	Units	Type	Default	Description
L	$M$	double	0.0	length
XO	$M$	double	0.0	radiation length
ELASTIC		long	0	elastic scattering? If zero, then particles will lose energy due to material.
ENERGY_STRAGGLE		long	0	Use simple-minded energy straggling model? Ignored for ELASTIC scattering.
Z		long	0	Atomic number
A	$AMU$	double	0.0	Atomic mass
RHO	$KG/M^3$	double	0.0	Density
PLIMIT		double	0.05	Probability cutoff for each slice

This element is based on section 3.3.1 of the Handbook of Accelerator Physics and Engineering, specifically, the subsections Single Coulomb scattering of spin- ${\rm\frac{1}{2}}$ particles, Multiple Coulomb scattering through small angles, and Radiation length. There are two aspects to this element: scattering and energy loss.

Scattering. The multiple Coulomb scattering formula is used whenever the thickness of the material is greater than $0.001 X_o$, where $X_o$ is the radiation length. (Note that this is inaccurate for materials thicker than $100 X_o$.) For this regime, the user need only specify the material thickness (L) and the radiation length (XO).

For materials thinner than $0.001 X_o$, the user must specify additional parameters, namely, the atomic number (Z), atomic mass (A), and mass density (RHO) of the material. Note that the density is given in units of $kg/m^3$. (Multiply by $10^3$ to convert $g/cm^3$ to $kg/m^3$.) In addition, the simulation parameter PLIMIT may be modified.

To understand this parameter, one must understand how elegant simulates the thin materials. First, it computes the expected number of scattering events per particle, $E = \sigma_T n L = \frac{K_1 \pi^3 n L}{K_2^2 + K_2*\pi^2}$, where $n$ is the number density of the material, L is the thickness of the material, $K_1 = (\frac{2 Z r_e}{\beta^2 \gamma})^2$, and $K_2 = \frac{\alpha^2 Z^\frac{2}{3}}{\beta\gamma}$, with $r_e$ the classical electron radius and $\alpha$ the fine structure constant. The material is then broken into $N$ slices, where $N = E/P_{limit}$. For each slice, each simulation particle has a probability $E/N$ of scattering. If scattering occurs, the location within the slice is computed using a uniform distribution over the slice thickness.

For each scatter that occurs, the scattering angle, $\theta$ is computed using the cumulative probability distribution $F(\theta>\theta_o) = \frac{K_2 (\pi^2 - \theta_o^2)}{\pi^2 (K_2 + \theta_o^2)}$. This can be solved for $\theta_o$, giving $\theta_o = \sqrt{\frac{(1-F)K_2\pi^2}{K_2 + F \pi^2}}$. For each scatter, $F$ is chosen from a uniform random distribution on $[0,1]$.

Energy loss. Energy loss simulation is very simple. The energy loss per unit distance traveled, $x$, is $\frac{dE}{dx} = -E/X_o$. Hence, in traveling through a material of thickness $L$, the energy of each particle is transformed from $E$ to $E e^{-L/X_o}$.

Energy straggling. This refers to variation in the energy lost by particles. The model used by elegant is very, very crude. It assumes that the standard deviation of the energy loss is equal to half the mean energy loss. This is an overestimate, we think, and is provided to give an upper bound on the effects of energy straggling until a real model can be developed. Note one obvious problem with this: if you split a MATTER element of length L into two pieces of length L/2, the total energy loss will not not change, but the induced energy spread will be about 30% lower, due to addition in quadrature.