Information on a fission reaction

[ptabor]

I'm trying to write a stochastic simulation of a chain fission reaction in an enriched uranium sample (I plan to vary the concentration of U235 - from what I've read 4%U235 vs 96%U238 is what is used in commercial reactors).

from my basic understanding of the process, a neutron is fired into the sample, where it is captured by a U235 atom which undergoes fission into two lighter elements and some stray neutrons. These stray neutrons go on to propogate the reaction. If not enough are released then the reaction will fizzle, if too many are produced you get a runaway reaction (as in the A-bomb).

My question is: where can I find the laws that govern
a) how many neutrons are emitted
b) how much kinetic energy will they have when they are ejected.

part b is relevant to my simulation because if the neutrons are too energetic, they won't be captured by a U235 atom and thus won't serve to propogate the reaction.

 

[Grogs]

My question is: where can I find the laws that govern
a) how many neutrons are emitted
b) how much kinetic energy will they have when they are ejected.

i'm pretty sure these can only be determined emperically. Here's what I've got from my reactor theory book:

a) $\nu(E)$ = 2.432 + 0.066E, for 0<E<=1MeV
$\nu(E)$ = 2.349 + 0.15E, for E > 1 MeV.

$\nu$ is defined as the number of neutrons released per fission, and notice also that $\nu$ is explicitly a function of energy, and this formula is only valid for U-235.

b) The spectrum is:

$$\chi (E) = 0.453 e^{-1.036E} \sinh(\sqrt{2.29E})$$

Again, this is valid only for U-235. E for both formulas is in MeV.

EDIT: I guess I should add that, from what you've described of your model, I suspect you'll find sustaining a chain reaction is impossible. This is because the average energy of a fission neutron is ~2-MeV. If you're assuming only low-energy neutrons can cause a fission, I suspect you'll get << 1 neutron per fission that goes on to produce another fission. In a real-world thermal reactor, you've got a moderator (usually water) that slows the neutrons down to thermal energies before they are absorbed.

 

[ptabor]

Grogs, can you please cite the textbook? perhaps I'll venture over to the library and check it out.

is the Energy in these equations the energy of the impact neutron?

 

[Grogs]

Grogs, can you please cite the textbook? perhaps I'll venture over to the library and check it out.

Sure. It's 'Nuclear Reactor Analysis' by Duderstadt and Hamilton. I don't know your level of expertise, but it's a pretty intimidating text (and I say that as a person working on an MS in Nuclear Engineering.) You might also check out 'Introduction to Nuclear Engineering' by Lamarsh. It's a bit more user-friendly. Finally, if you're not particularly proficient in physics/engineering, there's 'Nuclear Energy: Principles, Practices, and Prospects' by Bodansky. It's geared more towards the curious novice than the NE student, so it's easy to understand, but pretty skimpy on details.

Most likely, from the sound of what you're modelling, look for the 'four-factor' or 'six-factor' formula in the index. That's a pretty easy qualitative to compare reactivity to enrichment.

is the Energy in these equations the energy of the impact neutron?

The first part (#neutrons per fission) is. For thermal reactors where most of the fissions are caused by low-energy neutrons, we usually just use 2.43 for $\nu$.

The second equation is giving the energy of the neutrons *produced* by a fission, or more correctly, it's giving you the probability that a neutron will be produced at a particular energy. The energy of the incoming neutron would have very little impact on this equation.

 

[ptabor]

Ahh, thanks for the clarification.

What is the threshold energy for an emitted neutron to be captured? this is to say, how energetic is too energetic to propogate the reaction?

Also, in the probability density, what is a reasonable range for E?

Ima head over to the library and check out Lamarsh's book, as I have no background in Nuclear physics, other than the standard undergraduate modern physics class.
thanks for all your help.

 

[Astronuc]

Neutrons can be absorbed at any energy, with varying probability (micrscopic cross-section). Also, whether or not they will cause fission, also varies with energy.

http://wwwndc.tokai.jaeri.go.jp/cgi-bin/Tab80WWW.cgi?/data2/JENDL/JENDL-3.3prc/intern/U235.intern

http://wwwndc.tokai.jaeri.go.jp/cgi-bin/nuclinfo2004?92,235

See - Cross Sections (taken from JENDL-3.3) - need special software to process.

Table of U-235 .
Figures of U-235 : type-1: type-2: type-3.

Basically one needs a program to process the microscopic cross-sections into macroscopic cross-sections in order to calculate the reaction rates.

One also needs temperature and density of the U-235. The question - will the sample be moderated/cooled?

Commercial reactors use UO₂ with U-235 up to 5% of U. They are also moderated, i.e. neutron energies are slowed to ~0.025 in thermal equilibrium with the core materials. Fission neutrons have high energies on the order of a few MeV.

Basically one has to determine the reaction probability per unit energy then integrate over the energy spectrum to determine the overall criticality or reaction rate.

 

[ptabor]

For the sake of simplicity (this is an open ended assignment), perhaps I should stick to modeling the process that occurs in the a-bomb. Then I wouldn't have to worry about the moderator.

astro, if you have the time, can you perhaps provide a little explanation of these tables?

 

[Grogs]

For the sake of simplicity (this is an open ended assignment), perhaps I should stick to modeling the process that occurs in the a-bomb. Then I wouldn't have to worry about the moderator.

astro, if you have the time, can you perhaps provide a little explanation of these tables?

What you'll need to do is make some simplifying assumptions. The key component of the 4- and 6- factor formulas are the cross-sections, which represent the probability the neutrons will interact with a given nucleus. Unfortunately, these cross-sections are strongly energy dependent. Take a look at http://wwwndc.tokai.jaeri.go.jp/j33fig/jpeg/u235_f1.jpg graph from Astronuc's post for an example. What you'll have to do is find a value of the cross-sections that's pretty representative of the neutrons present in the Uranium. With those cross-sections, if you can get somewhat decent guesses at the value of 4 of the 6 terms of the 6-factor formula, you can come up with an algebraic expression relating the enrichment on 235-U to the multiplication factor, k. To make a bomb, k must be >1, so that's your cutoff point.

Since the cross-sections are really at the heart of that problem, take a look in the LaMarsh book and get familiar with what microscopic and macroscopic cross-sections if you haven't already encountered them in your Modern Physics class.

 

[ptabor]

Does anyone know where I can find a semi analytic expression for the cross sections, as a function of energy/temperature?