# Why is the mean free time of electrons independent of applied electric field?

In the drude model we can easily get that the mean free time $$\tau$$ is $$\tau =\frac{m_e}{\rho n e^2}$$

So from above, we can see that the mean free time depends on properties of the material alone (i.e. the resistivity $$\rho$$ etc) and not on the applied field. We further see this independence from $$\vec{E}$$ when we use the Drude theory to calculate that $$\frac{d\vec{p}(t)}{dt}=-\frac{\vec{p}(t)}{\tau}+\vec{f}(t)$$ where $$\vec{f}(t)$$ is the electric force per electron. That is, $$\vec{f}(t)$$ gives a measure of $$\vec{E}$$ and from above, we can freely adjust $$\vec{E}$$ and hence increase or decrease the drift velocity all the while maintaining a constant $$\tau$$. The fact that $$\tau$$ remains constant even if we increase $$\vec{E}$$ does not sit well with me. If we apply a larger field, then the acceleration felt by each particle will increase and hence they should cross the mean free distance in a shorter amount of time. I have 3 ways of potentially fixing this issue although I'm not sure whether they are correct or not and would love some help on this issue.

My first solution would simply be to assume that $$\tau$$ is in fact dependent on $$\vec{E}$$ however it is extremely weakly dependent on $$\vec{E}$$ because at standard temperatures, the average velocity of the electrons is very high ($$\pm 500 m/s$$). Thus the additional speed gained due to the presence of a field of normal strength is very small in comparison to the speed without the field present and so we can approximate this as a constant $$\tau$$ provided the field is not absurdly strong. I'm not very confident in this explanation at all though.

The second solution is to observe that when an electron collides with an ion, the directionality of its velocity is random. Those electrons who attain velocities in the direction of the acceleration caused by the $$\vec{E}$$ field (the direction of electron flow in the field) will have their mean free times reduced as they speed up due to the field. However, for each electron that attains a forward velocity, there is on average one that attains a backward velocity (opposite the direction of electron flow in the field) and these electrons decelerate/ slow down due to the field and hence the time taken to travel their mean free paths increases thus increasing $$\tau$$ for them. Overall then, we get a constant $$\tau$$ on average. I am more confident in this solution but still not totally confident.

The final solution I have is to assume that the mean free distance changes in the presence of an electric field. A stronger $$\vec{E}$$ means faster electrons which could potentially mean that the collision cross section of the ions decreases. Thus the electrons accelerate in the presence of the field and hence travel faster but now they have to travel a larger mean free distance on average since they are less likely to collide. I am also not very confident in this.

Are any of these explanations correct or is it some combination of the above? Any help on this problem would be greatly appreciated as this issue is giving me nightmares!

## 1 Answer

For a second if we let that $$\tau$$ depends on another parameter than that of the material itself, then $$\frac{d\mathbf{p}}{dt}=\mathbf{F}-\frac{\mathbf{p}}{\tau}$$ Consider only the electric field, We get

$$\frac{d\mathbf{p}}{dt}=-e\mathbf{E}-\frac{\mathbf{p}}{\tau}$$ $$\frac{d\mathbf{p}}{dt}=0=-e\mathbf{E}-\frac{\mathbf{p}}{\tau}\ \ \text{(in steady state)}$$ $$e\mathbf{E}=-\frac{m\mathbf{v}}{\tau}$$ Further for conductors $$\mathbf{J}=\sigma \mathbf{E}=-n e\mathbf{v}\Rightarrow e\mathbf{E}=-\frac{ne^2}{m\sigma}m\mathbf{v}$$ Comparing the two equation we get $$\boxed{\tau=\frac{m\sigma}{ne^2}}$$

• Thanks for the response! Okay so I know how to derive that $\tau =\frac{m_e}{\rho n e^2}$. My issue is that I can't figure out why $\tau$ is independent of $\vec{E}$ on a molecular/atomic level. Regarding your analogy with the cars on the road colliding into each other, the Drude model posits that the electrons collide into the stationary ions, not other moving electrons and so I don't see how your analogy works. Mar 11, 2021 at 14:00