# Zero k-eff uncertainty in infinite homogeneous medium 2-group problem

Code version
Version | 0.13.0
Git SHA1 | cff247e35785e7236d67ccf64a3401f0fc50a469

Background
I was working out a simple infinite box problem to learn about multigroup cross section generation and running in MG mode. I came across an odd result. In continuous energy mode, all three criticality estimators returned an uncertainty, per below:
k-effective (Collision) = 1.00091 +/- 0.00102
k-effective (Track-length) = 1.00083 +/- 0.00104
k-effective (Absorption) = 0.99895 +/- 0.00112
Combined k-effective = 1.00006 +/- 0.00075

However, when running the problem in 2-group multigroup mode, the absorption estimator returns a 0-valued uncertainty, as:

k-effective (Collision) = 1.00084 +/- 0.00099
k-effective (Track-length) = 1.00084 +/- 0.00101
k-effective (Absorption) = 0.99970 +/- 0.00000
Combined k-effective = 0.99970 +/- 0.00000

I have compared the fast/thermal fluxes in both problem variants and they are in excellent agreement.

Problem statement
The geometry is a simple cube with reflective boundaries. There is a single material, composed of a mix of H-1 and U-235 in the ratio of 2058:1. The model temperature is 600K. The cross section library is the NNDC, and the WMP method is used to handle temperatures.

Two models are used. The box_U.py runs the continuous energy problem, with tallies setup to create a simple two-group library for the fuel “gas”, The resulting mgxs.h5 file is read by the box_U2g.py problem, which is set to run in multigroup.
mgxs.h5 (13.1 KB)
box_U2g.py (955 Bytes)
box_U.py (4.3 KB)

@AlexandreTrottier Interesting question. The absorption estimate of keff works by scoring w\Sigma_f/\Sigma_a each time a neutron is absorbed (where w is the particle weight). The fundamental reason that the uncertainty is so small (it’s actually not zero but when rounded to 5 digits appears as 0) is that a vast majority of the neutrons are absorbed in a single energy group. Thus, most of the keff estimates are exactly the same. There is a small probability of absorption in the other group, which means there is some non-zero variance in the estimate. If that probability were zero or if you were running a one-group problem, you would indeed get an uncertainty of exactly zero.