22  Physics Models Used in the Particle Matter Interaction Model

22.1 Command Overview

The core attributes are:

The resulting interaction model is then attached to beamline elements that represent material boundaries or residual-gas regions.

22.2 The Energy Loss

The mean ionization energy loss is modeled with the Bethe-Bloch equation,

\[ -\frac{dE}{dx} = \frac{K z^2 Z}{A \beta^2} \left[ \frac{1}{2}\ln\left(\frac{2 m_e c^2 \beta^2 \gamma^2 T_{\max}}{I^2}\right) - \beta^2 \right], \tag{22.1}\]

where Z, A, I, beta, gamma, and Tmax have their usual stopping power meaning and

\[ T_{\max} = \frac{2 m_e c^2 \beta^2 \gamma^2} {1 + 2 \gamma m_e/M + (m_e/M)^2}. \tag{22.2}\]

The manual states that this form is used for incident PROTON, DEUTERON, MUON, HMINUS, and H2P beams over the documented energy ranges, and also for ALPHA particles in their supported range.

At low energy the stopping power is switched to Andersen-Ziegler-style semi-empirical fits. The implementation also models energy straggling with a Gaussian width

\[ \sigma_0^2 = 4 \pi N_A r_e^2 (m_e c^2)^2 \rho \frac{Z}{A} \Delta s. \tag{22.3}\]

Particles whose remaining energy drops below LOWENERGYTHR are removed and written to the corresponding loss file.

22.3 The Coulomb Scattering

The Coulomb-scattering model is split into:

• multiple Coulomb scattering
• large-angle Rutherford scattering

The distributions are written in the legacy manual as

\[ P_M(\alpha)\, d\alpha = \frac{1}{\sqrt{\pi}} e^{-\alpha^2}\, d\alpha \tag{22.4}\]

and

\[ P_S(\alpha)\, d\alpha = \frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\, d\alpha, \tag{22.5}\]

with transition scale

\[ \theta_0 = \frac{13.6\,\mathrm{MeV}}{\beta c p}\, z \sqrt{\Delta s / X_0}\, [1 + 0.038 \ln(\Delta s / X_0)]. \tag{22.6}\]

22.3.2 Large Angle Rutherford Scattering

The rare large-angle tail is handled separately as a Rutherford-scattering process. The legacy implementation samples this branch probabilistically and then draws the corresponding scattering angle from the cumulative distribution.

22.4 Beam Stripping Physics

Beam stripping covers:

• interaction with residual gas
• electromagnetic or Lorentz stripping

The common stochastic model assumes a mean free path lambda and interaction probability

\[ P(x) = 1 - e^{-x/\lambda}. \tag{22.7}\]

22.5 The ScatteringPhysics Substeps

The implementation uses internal substeps when the main tracking step is too large for accurate material-interaction physics. In the legacy code path, ScatteringPhysics.cpp subdivides the step so that the material-interaction substep stays below the documented threshold. Particles already inside the element and particles entering the element are then advanced consistently over the same physical time interval.

22.6 Available Materials in OPAL

The material database includes the standard beamline materials listed in the legacy manual, among them:

• Air
• Aluminum
• AluminaAl2O3
• Beryllium
• BoronCarbide
• Copper
• Gold
• Graphite
• GraphiteR6710
• Kapton
• Molybdenum
• Mylar
• Titanium
• Water

For each material the original manual provides:

• atomic number Z
• atomic weight A
• mass density rho
• radiation length X0
• mean excitation energy I
• Andersen-Ziegler low-energy fit coefficients

22.7 Example of an Input File

The original manual points to particlematterinteraction.in as the reference input deck. It combines material elements with collimator-style aperture settings and a particle-matter interaction definition.

22.8 A Simple Test

The documented benchmark uses a cold Gaussian beam passing through a copper slit or elliptic collimator and compares both absorbed and scattered populations as well as the downstream energy and angular spectra.