# Bethe Formula for Stopping Power Motivation: filling in holes from [[Stopping Power]] Using relativistic quantum mechanics, the Bethe Formula for stopping power can be derived. Also known as Bethe-Bloch Equation. $ -\frac{dE}{dx} = \frac{4\pi k_0^2z^2e^4n}{mc^2\beta^2}\left[\text{ln}\left( \frac{2mc^2 \beta^2}{I(1-\beta^2)}\right) - \beta^2 \right] $ There are a lot of terms in this equation: - $k_0 = 8.99 \times 10^9 \, Nm^2C^{-2}$ - $z$ is the atomic number of the heavy particle - $e$ is the magnitude of the electron charge - $n$ number of electrons per unit volume in the medium - $m$ is the electron rest mass - $c$ is speed of light in a vacuum - $\beta$ is the speed of the particle relative to the speed of light, or $\frac{v}{c}$. - $I$ is the [[Mean Excitation Energy]] of the medium In theory, you could also separate each of value into a specific term with a physical meaning, but it offers minimal insight in my opinion. ## for electrons in particular for electrons, we must consider collisional and radiative losses, which expands this equation further. From [[@knollRadiationDetectionMeasurement2010]], $ \frac{dE}{dx} = \left(\frac{dE}{dx} \right)_{col} + \left(\frac{dE}{dx} \right)_{rad} $ where $ \left(\frac{dE}{dx} \right)_{col} = \frac{2\pi e^4 NZ}{m_0v^2}\left[ ln \left(\frac{m_0v^2 E}{2I^2(1-\beta^2)} \right) - ln(2)\left(2 \sqrt{1 - \beta^2} - 1+\beta^2\right) + \frac{1}{8}\left(1 - \sqrt{1-\beta^2} \right)^2 \right] $ and $ \left(\frac{dE}{dx} \right)_{rad} = \frac{NEZ(Z+1)e^4}{127m_0^2c^4}\left(4 \, \cdot ln\left(\frac{2E}{m_0c^2} - \frac{4}{3}\right) \right) $ where the ratio between these two loss terms is roughly $ \frac{col}{rad} \approx \frac{EZ}{700} $