The mean free path is the average distance between 2 collisions. We simply copy the things in Bohr and Mottelson, Bertulani and Banielewicz, and John Lilley in here.

Since a nucleus has finite size, the mean free path can be view as transparency of nucleon scattering. Since the total cross section is smaller for higher energy, the mean free path is proportional to energy.

The wave number K under a complex optical potential V+iW is

K=\sqrt{\frac{2m}{\hbar^2}(E-V-iW)}=k_r+ik_i.

There are two solutions, one has imaginary k_r,

k_r=\frac{\sqrt{m}}{\hbar}\sqrt{(E-V)+\sqrt{(E-V)^2+W^2}}

k_i=\frac{m}{\hbar^2}\frac{W}{k_r}=\frac{1}{2\lambda}

where m is the nucleon mass, E is the incident energy, v is the velocity inside the nucleus, and \lambda is the mean free path,

\lambda=\frac{\hbar}{2W\sqrt{m}}\sqrt{(E-V)+\sqrt{(E-V)^2+W^2}}\approx\frac{\hbar}{W\sqrt{2m}}\sqrt{(E-V)}

Since the imaginary potential W\approx-15-0.07E, V\approx-39+0.11E from 150 MeV to 400 MeV. The calculated mean free path for proton is shown in here. (Take \hbar = 197.33 [MeV\cdot fm/c], proton mass m=938.272 [MeV/c], the \lambda is in fm.)

Screenshot from 2016-01-18 18:57:43.png

The blue line on the plot is exact calculation, the purple line is approximation W << E-V.

Screenshot from 2016-01-18 18:37:48

This plot is taken from S.S.M. Wong, showing the radial shapes of the volume term (simialr to central term) of proton-nucleus optical potential.

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in J. Killey, the k_r is taken as,

k_r = \frac{\sqrt{2m(E-V)}}{\hbar},

which is the wave number without W. The result only for weak W << E-V.

 

 

 

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