The average value of the time elapsed after the last collision of all the electrons in a perturbed system is known as the relaxation time. It is represented by τ. τ=ne2σm
Mean free path
The mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties. l=(σn)(−1) Where l is the mean free path, n is the number of target particles per unit volume, and σ is the effective cross-sectional area of collision.
Relation of drift velocity with electric field
An electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Hence, N1∑i=1Nvi=0
Now, if an electric field is present. Electrons will be accelerated due to this field by a=m−eE where −e is the charge and m is the mass of electron.
Consider the ith electron at a given time t. This electron would have had its last collision some time before t, and let ti be the time elapsed after its last collision. If vi was its velocity immediately after the last collision, then its velocity Vi at time t is Vi=vi+(m−eEti).
The average velocity of the electrons at time t is the average of all the Vis.