5.6.1 Reactivity calculation

As already mentioned, in a Monte Carlo calculation, a particle is followed from its birth to its death. The code ends with the end of the last history. But in an (over-)critical system, neutron chains are infinite and such a calculation will not end. A special technique has been developed to handle this case. It allows us to obtain keff and tallies for any criticality. Starting with Ns source neutrons, the calculation develops over cycles (or neutron generations); within a cycle, a neutron is followed from its birth to its death but, this time, fission is considered as a cycle termination (like a capture). Within a cycle, at each collision point, the number of fission neutrons stored n is randomly sampled in order to have a mean value

-f W

keff

(W is the neutron weight, of-is/t is the microscopic fission/total cross-section and keff is estimated from the previous cycle, or is a user-given value if this is the first cycle). For the next cycle, M =^2 n particles (M ~ Ns) are emitted on the corresponding collision site. The effective multiplication factor can be obtained as

where Nifis and Niabs are respectively the number of fissions and of absorp­tions in generation i. In MCNP, three methods (based on cross-section calculations) are used to obtain the effective multiplication factor keff and the final value is a weighted average of the three factors.

To perform a critical calculation, one has to use the KCODE card (in the Data block card); a ‘starting’ source has to be defined to initiate the first cycle. The syntax is

KCODE Ns kexpt N NT

where Ns is the number of ‘source’ neutrons per cycle, kexpt is the expected value of the effective multiplication factor, NT is the total number of cycles (~100 is usually sufficient) and NI is the number of initial cycles that are
excluded from the keff (and tallies) calculation, in order to give enough time for the fission source to be established.

Going back to the example of the uranium sphere in a light water cylinder, the input file could be

First simple geometry c

c Cell cards c

1 1 -18.75 -1 $ the inner sphere

2 2 -1.0 -2 3 -4 1 $ the cylinder without the sphere

3 0 #2 #1 $ exterior

c

c Surface cards c

1 SO 5 $ centred sphere with R=5 cm

2 CZ 20 $ infinite cylinder with R=20cm

3 PZ -20 $ bottom plane intersecting the cylinder.

4 PZ 20 $ top plane intersecting the cylinder.

c Materials

M1 92235.60c 1 $ 235U

M2 1001.60c 2 8016.60c 1 $ H2O

c Source : kcode

sdef pos 0 0 0 erg 2.5

kcode 1000 1 10 80

totnu

Suppose that this input file is named lstgeo. MCNP can be run by enter­ing the command

mcnp n=1stgeo

At the end of the run, the keff value is displayed on the screen and is stored in the lstgeo file (after the short table summarizing the number of source neutrons, captures, escapes,…). The result is keff = 0.840 ± 0.003 (with more cycles, the precision is improved). If the sphere radius is changed from 5 cm to 6.312 cm, keff = 0.999 ± 0.003 which corresponds to the critical case.

Then, using the S(a,@) treatment (by inserting an MT2 lwtr.07 card after material M2), we see that keff is reduced to keff = 0.974 ± 0.003; this demonstrates the importance of this treatment for thermal neutrons.