What is the formula/method to calculate the range and energy loss by electron passing through an medium?

I want to know how it is calculated that an electron having certain energy will pass how much distance inside an material (solid).

Actually I want to shoot an electron from vacuum to air and for not losing vacuum I need a solid material through which high energy electrons can pass through. I can accelerate the electron to maximum $$30\,\mathrm{keV}$$ in vacuum and I need an electron of just $$10$$ to $$13\,\mathrm{eV}$$ in air side.

The purpose for this is to perform a selective ionization in air. Selective ionization has been done but the method was photoionization and I am tying to achieve it by electron impact although $$10-13\,\mathrm{eV}$$ electrons can not penetrate deep in air but if we increase the surface area of electron and air collision then we can notice it happening.

So I need a formula or process to calculate the energy loss of an electron inside a material and the second question is how much material thickness is required to pass an high energy electron for the electron to come out with the required energy?

Obviously, you need to read the PDB:

https://pdg.lbl.gov/2019/reviews/rpp2018-rev-passage-particles-matter.pdf

The famous Bethe-Bloch formula:

$$-\Big{\langle}\frac{dE}{dx}\Big{\rangle} =\frac{4\pi}{_ec^2}\cdot\frac{nz^2}{\beta^2}\cdot\Big(\frac{e^2}{4\pi\epsilon_0}\Big)^2\cdot \big[\ln\big(\frac{2m_ec^2\beta^2}{I\cdot(1-\beta^2)}\big)-\beta^2\big]$$

is the basic starting point. It doesn't exactly apply to electron b/c of their low mass, and quantum exchange concerns when interacting with atomic electrons.

Another problem is the Landau distribution (https://en.wikipedia.org/wiki/Landau_distribution), describing energy loss:

$$p(x) = \frac 1{\pi c}\int_0^{\infty}e^{-t}\cos\Big( t\big(\frac{x-\mu}c\big)+\frac{2t}{\pi}\log\big(\frac t c\big)\Big)dt$$

which has long tails:

It makes getting narrow spectrum through energy loss impossible.

As said in the other thread, you will always have a energy distribution after the material. So why do you not jaust use a discharge tube, first with low pressure later with higher pressure, you can calculate the mean distance of the air molecules or even see it through the light in bands in the tube. yo can then measure the voltage of the electrons which ionize or excite the air atoms.

• Would you please elaborate or attach any relevant study about ionization in discharge tube. I think pressure inside discharge tune must be very low for electron to gain sufficient energy to cause ionization. Commented Aug 5, 2021 at 6:26
• Could you explain how we can use discharge tube for pressure of 2 to 3 atm air? Commented Aug 5, 2021 at 8:34
• I don't think you can! but if you want to study ionization in air, why so high pressure? the mean free path is extremely short. from experiment in lower pressure you can calculate what happens with shorter free path Commented Aug 5, 2021 at 12:43
• i want to create ions at large concentration. not to study them but rather see it's scalability Commented Aug 5, 2021 at 13:12

Check out the Stopping and Range of Electrons in Matter website. They provide simple-to-run code that you can download to calculate exactly what you want using a detailed Monte Carlo simulation. It'll give you accurate results and you can easily take experimental aspects like a finite beam size or a finite window size etc. into account.