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Plotting the Bethe-Bloch Formula

Physics Asked on August 7, 2021

We had in our scriptum the following formula for the Bethe-Bloch formula:

$$-frac{dE}{dx} = Kfrac{rho Z}{A} frac{z^2}{beta^2} left[ lnleft( frac{2mgamma^2beta^2}{I} – beta^2 right) right], $$

where I set $c = 1$ for convenience, $K$ is a constant, $I$ is the average ionisation potential, and $rho$ denotes the density of the material.

My question is: I have often seen plots which show on the $x$-axis $betagamma$, but I am not sure how one would plot $-frac{dE}{dx}$ as a function of $betagamma$, given that in front of the $ln$, we have the factor $frac{1}{beta^2}$ (and not $frac{1}{beta^2gamma^2}$), and also in the $ln$, we have $-beta^2$ and not $-left( betagammaright)^2$.

I would love hear some opinions on this!

One Answer

If you know $beta$ you can compute $gamma = (1 - beta^2)^{-1/2}$, so it is just a matter of calculating $beta gamma$ for each value of $beta$ you are interested in. And if you know $betagamma$, you can extract $beta$ from it, e. g. by plotting $betagamma$ versus $beta$ (again, $c = 1$):

enter image description here

When you know $beta$, you can put it back into the Bethe-Bloch Formula.

Correct answer by Jacopo Tissino on August 7, 2021

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