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For this case they cite the scattering probability as given by

where

Then they plot the probability distribution for some incident neutron energies:

I've been trying to reproduce this plot but I just don't seem to understand how.

So, for example, if ##E_i = kT## and ##E_f/E_i = 1.0 ## then ##E_f = kT## and $$P(E_i \rightarrow E_f) = \frac {1} { kT} erf \sqrt { \frac {kT} {kT}} = \frac {0.84} {kT}$$.

I'm not sure what I'm supposed to do next, as the expected result in the plot is about 0.5. I know ##P(E_i \rightarrow E_f)## is a probability distribution so in order to get rid of the ##kT## I need to integrate but from where to where?

Can you help me figure it out?