# E' vs. E_2 Neutron Scattering and Logarithmic Energy Loss

#### PlasMav

Hello,

I just had a little debate with my professor after taking my final exam. He had given us an additional formula sheet at the last second (hand written on the projector) which confused me.

The question was a 7 MeV neutron collides with several U-238 atoms before reaching 2 MeV. How many collisions did it take to get there and what was the average loss.

So one of the equations he gave us was:

n = ln(E/E')/zeta

This equation confused me enough to screw up most of the problem. Afterward I looked it up and the correct equation is:

n = ln(E_1/E_2)/zeta

Which makes more sense to me but he argued they are the same thing. I am familiar with E' being elastic scattering with needs angles to solve (based on the formula sheet given to us) which is what messed me up.

Does my argument have a foundation?

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#### PeterDonis

Mentor
Does my argument have a foundation?
It doesn't look like it to me. It looks like you are just quibbling over notation. The point is that you take the logarithm of the ratio of energies before and after. Whether you call those energies $E$ and $E'$ or $E_1$ and $E_2$ is a matter of notation and has nothing to do with the physics. Which seems to be what your professor was saying.

#### PlasMav

In our text which is what he references E' is something different: the elastic scattering energy

Reference:

This was mostly the same as our question:

Question more involving E':

I was using the E' formula for the first one because the formula the professor provided was:

n = ln(E_1/E_2)/zeta

n = ln(E/E')/zeta

E_2 is not the same as E' here.

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#### Astronuc

Staff Emeritus
In our text which is what he references E' is something different: the elastic scattering energy

I was using the E' formula for the first one because the formula the professor provided was:

n = ln(E_1/E_2)/zeta

n = ln(E/E')/zeta

E_2 is not the same as E' here.
I suppose it is confusing for one to use E and E' in one case, usually the energy before collision and energy after collision, respectively, for a single collision, then for successive multiple collisions. In the context of the exercise, one can assume that one collision, or successive collisions are all elastic. For one collision, there is a defined relationship between E' and E in terms of the mass of the target nuclei and the scattering angle, and one could determine an average scattering energy by integrating over all angles. See equation (26) in the following link.

See some notes here: http://mragheb.com/NPRE 402 ME 405 Nuclear Power Engineering/Neutron Collision Theory.pdf

E and E' are variables, whereas E1 and E2 are particular values, and one could simply say, solve the problem when E = E1 and E' = E2, to which PeterDonis alluded. See equations (31) and (37) in the above link (and note E' and E'' are duplicated in the text before eq (37)).

It would be useful to work through the derivations and become comfortable with the theory.

Other notes - https://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_6.pdf - warning: they use bright yellow highlight.

"E' vs. E_2 Neutron Scattering and Logarithmic Energy Loss"

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