The rate of photoionization between heights z and is equal to


where for solar zenith angle ,

is the absorption cross-section, and n(z) is the number density of the absorbing gas. Assume an isothermal atmosphere so

Plugging (3) into (2) yields


Plugging (4) into (1) gives


But is monotonic, so f(x) and have maxima and minima at the same values of x. Therefore, there are extrema at


This is a maximum since

Physically, there are many absorbers but few photons near the surface, and many photons but few absorbers at the top of the atmosphere. The product of the two must therefore have a maximum where the two are balanced. Solving (8), the maximum is at

and the photoionization rate at this level is

The integrated rate over the entire atmosphere is then