Help needed -- Electron beam at low pressure

[Ash Khan]

Hey! I am trying to figure out this one problem. Some help would be appreciated.
How can i relate the maximum distance traveled by an electron at a given pressure?
So electron is colliding with air molecules. I wonder if there is a formula or derivation which relate maximum penetration and pressure of air?

 

[Topolfractal]

You are looking for a diffusion formula. Why are you only interested in the penetration of 1 particle and not many or even atoms?

 

[Ash Khan]

I am actually interested in a beam of electron

 

[Topolfractal]

Sorry just the way you phrased your question led me to wonder.

 

[Topolfractal]

Okay, because you are asking about a beam of electrons , I change my advice to that of looking up the resistance of air. The pressure is correlated with temperature and how fast the particles are moving. Although because you are only considering a beam of electrons and not regular atoms, the scattering is not what is considered a diffusion type scattering. Resistance depends on pressure though ( through the correlation with temperature. Find out the resistivity of air.

 

[Topolfractal]

The penetration length is calculated from the distance at which the current dissipates to zero from the resistance of the air.

 

[mfb]

I think you need the cross-section for collisions of electrons at your beam energy with the molecules in air.

Pressure will determine the density of molecules, and cross-section times density gives the scattering probability per length, which then leads to the fraction of unscattered electrons as function of distance.

Edit: at atmospheric density, radiation length in air at atmospheric pressure is about 300 meters if I remember correctly. This can be used for high-energetic electrons.

 

[berkeman]

I am actually interested in a beam of electron

What is the context of your question? Why will you have a beam of accelerated electrons in a vessel that has some air in it?

It seem like how far the beam makes it into the air portion of the vessel will also depend on the anode configuration. Is the beam aimed at an anode surface?

 

[Ash Khan]

This makes sense but i am just confused how i should correlate pressure directly with the maximum penetration distance of electron beam. I tried using deceleration, momentums and other techniques

 

[Ash Khan]

So, i am doing this experiment where i have a hot filament which is shooting a beam of electrons. Beam is accelerated by two plates with a constant potential. I have a vacuum pump to pump out air and decrease the pressure in the container (cylindrical chamber). I want to derive a formula where i can just enter the value of pressure and figure out how much distance the beam will travel before it stops.

 

[Topolfractal]

The penetration length is calculated from the distance at which the current dissipates to zero from the resistance of the air.

 

[ChrisVer]

You can use the formula for the number density $n= \frac{P_{vac}}{k_B T}$
Where $P_{vac}$ the pressure and $T$ is the temperature. Since you don't know what kind of temperatures you have in your device, using the room temperature is the only natural choice.

So the number of scatterings of the electrons over the gas is $N_{scat} = \sigma_{e/gas} \frac{P}{k_BT} L$ in a total path $L$.

Now the quantity that multiplies the distance L is giving you the inverse mean free path of your electrons within the gas: $\lambda_{mfp} = \frac{1}{\sigma n} = \frac{k_B T}{\sigma_{e/gas} P}$. After 1 $1 \lambda_{mfp}$ your electron beam intensity has dropped by $e^{-1}$.