Neutron quantity normalization in an eigenvalue computation

[froztiz]

TL;DR Summary

Should we normalize scores by the eigenvalue in a critical calculation?

Dear Community,
I am having a question. I have developed a simple code to perform iteration power algorithm and find the keff value of a system. However, it is not still totally clear in my mind if I have to normalize all my scores by the eigenvalue, i.e. multiply by the keff (fluxes, power maps, etc...).
Does anybody knows what should be done, and could you explain why?
Thanks a lot.
Best.

 

[Astronuc]

What kind of system?

Is this a time-dependent (k_eff < 1, or k_eff > 1) or steady-state system (k_eff = 1)?

What is the relationship of k_eff to the eigenvalue?

Note that for a critical system, k_eff = 1, so multiplying or dividing by unity will have no effect on the values/functions being multiplied.

The power is related to the fission rate, which is related to the product of the fissile (fertile) atom densities, their fissile cross-sections and flux(es).

Note that in a conventional thermal system (e.g., LWR), the system actually 'mixed' spectrum with fission from both thermal neutrons (in ²³⁵U, or ²³³U, and with burnup and conversion of ²³⁸U, in ²³⁹Pu and ²⁴¹Pu) and fast neutrons (²³⁸U, ²⁴⁰Pu and ²⁴²Pu).

 

[froztiz]

What kind of system?

Is this a time-dependent (k_eff < 1, or k_eff > 1) or steady-state system (k_eff = 1)?

What is the relationship of k_eff to the eigenvalue?

Note that for a critical system, k_eff = 1, so multiplying or dividing by unity will have no effect on the values/functions being multiplied.

The power is related to the fission rate, which is related to the product of the fissile (fertile) atom densities, their fissile cross-sections and flux(es).

Note that in a conventional thermal system (e.g., LWR), the system actually 'mixed' spectrum with fission from both thermal neutrons (in ²³⁵U, or ²³³U, and with burnup and conversion of ²³⁸U, in ²³⁹Pu and ²⁴¹Pu) and fast neutrons (²³⁸U, ²⁴⁰Pu and ²⁴²Pu).

Thank you Astronuc for your reply and sorry for the time I took to answer.
Using the power iteration algorithm, I compute the k-eigenvalue. It s a mathematical method that permits to compute the eingenvalue and the associated eigenvector (the flux) by iterations, just like it is done within MCNP, for example. So my keff is the eigenvalue. It can be depending on the geometry, below, above or equal to 1. I agree that for a keff=1 dividing by the keff won't have any effect. My question is very naive and I was just wondering if, for a flux calculation for instance, solving the eigenvalue problem, I should divide my flux by the eigenvalue. I have no idea how this is done within MCNP

 

[Alex A]

It sounds like you are doing an eigenvalue calculation by iteration on some sort of theoretical system (a matrix?). This isn't what MCNP does. MCNP runs a transport simulation with 1/k and evolves the guess for k until the system balances. This is the method that defines k_eff. I don't fully understand why this is different to calculating increase of n per generation, but it is.

Until you explain what you are doing I don't see how anyone can help.

If you don't want to show anyone your problem/code then simulate your system and compare with the output of your program. OpenMC is open source and early versions e.g. 0.7 I found really easy to use.

 

[froztiz]

According to my knowledge, MCNP is using (as most of the MC calculation codes) the power iteration method, as explained in the following document: https://mcnp.lanl.gov/pdf_files/la-ur-06-7094.pdf
In particular, this method is detailed after page 14.
As a consequence, my question is well defined and I am just asking if, in case of an eigenvalue problem, we need to divide all quantity scored by the eigenvalue. The legitimacy of my question is based on the mathematical aspect of an eigenvalue problem. In linear algebra, we can define a norm as being the highest eigenvalue. The power iteration method provides the highest eigenvalue. Hence, I would like to know if I have to normalize my quantities with the eigenvalue.

To answer your question, calculating the increase of neutron per generation will not provide a Monte-Carlo estimator of your eigenvalue. I believe the estimator you construct will be biased, in particular your accuracy will strongly depends on the number of neutron per cycle and not anymore of the number of collisions, as it is done for instance, in the collision estimator or in the tracklength estimator of the k-eigenvalue.

 

[rpp]

You are correct that it is an eigenvalue problem and the eigenvalue is k-eff. If you write the equation in matrix form ($Ax=\lambda x$) you can see that the magnitude of the flux ($x$) is arbitrary. You can multiply the flux by any non-zero value, and it will still be a valid solution to the eigenvalue problem.

So how do set the value of the flux? There are several different conventions. If you know the total power of the system, you can normalize the flux to the power. This will give you the correct physical flux level for that power. Another convention is to normalize the flux to one fission particle. Still another convention would be to normalize the flux so that the average is 1. There is no "right" answer, it depends on what you are trying to do and what the computer code does.

But the simple answer is, no, you do not need to divide the flux by the eigenvalue, but you should normalize your results to some reasonable convention.

 

[froztiz]

Thank you very much RPP