What is lambda R in Richardson's Law?

Question

I've got to calculate the thermionic emission through a diode, so I need to use Richardson's Law. However, one thing's got me confused - according to the Wikipedia page:

$$J = A_GT^2e^\frac{-W}{kt}$$

I could live with that, but for $A_G$. Apparently, I'm not the only one; Wikipedia's a bit cryptic about what $A_G$ is, mentioning that physicists have struggled with it for decades, "but there is agreement that $A_G$ must be written in the form:"

$$A_G = \lambda_RA_0$$

"Where $\lambda_R$ is a material-specific correction factor that is typically of order 0.5, and $A_0$ is a universal constant."

That's the last real mention of $\lambda_R$, and the only reference for it is in French.

So, what is $\lambda_R$, really? How can I figure out its numerical value so that I can actually use Richardson's Law?

Answer 1

The quantity $\lambda_R$ is the dimensionless extracted tunneling/nucleation amplitude for electrons to get out of the metal. It is not simple to compute because it is an average over the thermal motion of the electrons of the probability of the electron getting far enough away from the metal in order to escape to infinity.

There are many models in which you can calculate $\lambda_R$, but it's value is best extracted from experiment--- it will depend on the type of metal and the roughness of the surface geometry, and on surface impurities and dirt. It is hopeless to calculate except in idealized situations.