# One group modified and Two group theory

1. May 16, 2010

### koab1mjr

What is the big difference between the two reactor calculations. They seem to be virtually the same?

2. May 17, 2010

### Hologram0110

I would disagree. The two-group model is also often expanded into a multi-group model with more than two energy groups.

The two group model solves two-coupled differential equations for the neutron flux instead of one. The one group model also requires a lot of information which is more suited to a multi-group or monte-carlo model such as the non-leakage probabilities, resonance absorption, fast-fission factor ect. In most cases these are artificial values or calculated by one of the more advanced models. In the two-group model these variables are treated more naturally because you can account for some of the energy dependence of the cross-sections.

The multi-group model is also very intuitive. For each of the energy groups you have some probability that the neutrons in it will upscatter to one of the higher energies, get absorbed, down scatter (moderated), or diffuse to a different location. The magnitude of these probabilities are determined by the coefficients.

The effects of these differences become more clear when you start modeling non-homogenous assemblies. Since neutrons created locations in the fuel have different distances to the moderator they also have different non-absorption probabilities. In the one group modified this usually is not accounted for (unless you have a position dependent value which I have never seen done, but there is no reason you couldn't that I'm aware of). Similar things happen near control rods where the effect depends on the energy level. The boundary conditions on the flux can also be different for the multi-group model.

3. May 17, 2010

### koab1mjr

OK thanks a bunch. I was not satisfied with the answer my professor gave me. One follow on.

How is upscatter accomplished? I would think that the max would be no energy loss due to a glancing scatter. Or is the neutron being absorbed by an already excited nucleus and then is ejecting all of its excitation energy to the neutron to regain a ground state.

We never talked about that. I saw the multi group model thing in the book though, So really 2 group is jsut the simplest case of the multigroup model.

4. May 17, 2010

### Hologram0110

Upscatter is generally a small, but non-zero effect. It isn't usually discussed when being introduced to the multi-group model because it makes the equations look more complicated. I included it because I like the resulting symmetry it produces in the equations. To me it makes it obvious that the coefficients mean things are scattering out of the neutron energy range of that group.

The easiest way to explain up scatter is with the billiard ball model of neutrons. Think of the speed of atoms in thermal equilibrium. There a distribution of energies, some have above average speed, some have below average speed. If this is at equilibrium then there must be some phenomena that prevents them from all having the same energy. Now think of the geometry of the collisions.

If you have two neutrons separated with velocity vectors 90 degrees apart, with equal energy and on a collision course. If they hit each other with perfect timing and geometry they will both scatter and end up with the same energy. Now if instead, one neutron then hits the side or back of the other neutron it can transfer it's energy and momentum to the first. This results in one neutron with higher energy and one with lower.

While I described it with neutrons, the same process can take place with anything that undergoes collisions. Neutron-nucleus for example also works. You can end up with fast moving atomic nuclei in different ways, fission fragments for example can have significant energy for short periods of time. The total effect from up-scatter is usually small but it is still included especially when you have very narrow energy groups.