Question

expression for mean free path? full derivation

Answer 1

Mean free path is the distance traveled by a particle such as electron, in between two successive collisions.  It could be an atom, molecule in gases or liquids, or electrons in any medium or photons too.

l = 1 /  sigma * n
where sigma = effective cross sectional area offered by each particle, for collision
           n = number of particles per unit volume = concentration of particles

One can prove this by estimating probability of a particle hitting or being stopped by collision in a small volume dV of area A & thickness dx:

From a beam many particles are travelling parallelly through a vertical slab of area A and thickness dx. In this volume we have n A dx number of particles.  Each particle has an effective surface area for collision = sigma. It is because only one side may be available for collision etc.  Total surface area of these that is available for collision is n A dx sigma. A particle that entered into this vertical slab of area A can encounter a surface area  n A dx sigma as obstruction or wall. So the expression for probability :
P (particle stopped or collision in volume dV) = n A dx  sigma / A  = n sigma dx

Now the Intensity I of a beam of particles is decreased by an amount dI.  dI is proportional to Intensity and the probability P.

dI  =  - I n sigma dx        -ve because intensity decreases due to collisions
dI / I  = -  n sigma  dx        In I = - n sigma x    =>  I(x) = I0  exp (- n sigma x)
I0 is I at x=0
Find the probability of a particle being stopped from x to x+dx. This is
P1(x) = [ I(x) - I(x+dx) ] / I0  =  (n sigma)  exp(-n sigma x)  dx

Expected value of x  is  Integral from 0 to infinity  of   x  * P1(x)
 integral of x (n sigma) exp ( - n sigma x ) dx      =  1/n sigma 
We call  1 / n * sigma = mean free path length = l