Document Overstraeten improvement

Sentaurus TCAD user manual as:

```math

\gamma (T) = \tanh \left(\frac{0.063 \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063 \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1}

\gamma (T) = \tanh \left(\frac{\hbar \omega_{op}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{\hbar \omega_{op}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1}

```

The value of the reference temperature $`T_0`$ is not entirely clear as it is never stated explicitly, a value of

$`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in Allpix Squared are taken from the abstract

with $`\hbar \omega_{op} = 0.063 \,\text{eV}`$. The value of the reference temperature $`T_0`$ is not entirely clear as it is never

stated explicitly, a value of $`T_0 = 300 \,\text{K}`$ is assumed.

### Original publication

The other model parameter values implemented in Allpix Squared are taken from the abstract

of \[[@overstraeten]\] as:

```math

This model can be selected in the configuration file via the parameter `multiplication_model = "overstraeten"`.

### Optimized parameters

An optimized parametrization of the Van Overstraeten-De Man model based on measurements with an infrared lased TCT setup is implemented in Allpix Squared, based on Table 3 of \[[@rd50ionization]\] with the following parameter values:

```math

\begin{aligned}

    a_{\infty, e} &= 1.149\times 10^{6} \,\text{/cm}\\

    b_{e} &= 1.325\times 10^{6} \,\text{V/cm}\\

    \\

    a_{\infty, h} &= 2.519\times 10^{6} \,\text{/cm}\\

    b_{h} &= 2.428\times 10^{6} \,\text{V/cm}\\

    \\

    \hbar \omega_{op} = 0.0758 \,\text{eV}

\end{aligned}

```

In contrast to the original model, this publication uses a parametrization without differentiating between low and high field regions.

This model can be selected in the configuration file via the parameter `multiplication_model = "overstraeten_optimized"`.

## Okuto-Crowell Model

The Okuto-Crowell model \[[@okuto]\] defines the impact ionization coefficient similarly to the above models but in addition