General expression for the density effect for the ionization loss of charged particles

The equations for the density-effect correction $\delta \mathrm{}\left(X\right)\mathrm{}$ for the ionization loss as given in a previous paper have been extended to the case where $\delta \left(X_0\right)\equiv {\delta }_0>0\mathrm{}$. Here $X_0$ is the lower limit of the range $X_0<X<\mathrm{}{X}_1$, where a particularly simple expression for $\delta \mathrm{}\left(X\right)\mathrm{}$ applies; $X={log}_{10}\left(\frac{p}{m_{0}c}\right)$, where $p$ is the momentum and $m_0$ is the mass of the incident particle. The case ${\delta }_0>0\mathrm{}$ applies directly to the calculation of $\delta \mathrm{}\left(X\right)\mathrm{}$ for conductors (metals).