UNIVERSITY OF LJUBLJANA Faculty of Mathematics and Physics Department of Physics Seminar on Monte Carlo Methods in Reactor Physics Author: Andrej Kavčič Mentor: prof. dr. Matjaž Ravnik Ljubljana, January 2008 Abstract The Monte Carlo method is being widely used to solve neutron transport problems in nuclear reactor cores along with the advancements in computer technology. The method is originally as old as neutron is, however, it was not until the last decade that it became popular due to growing computer capacities. Today almost all neutron parameters and their effects on reactor behaviour can be simulated. The next pages represent a step by step introduction, from the primary Monte Carlo idea to the calculation of three most important parametres in nuclear reactor physics: multiplication factor, delayed neutrons factor and prompt neutron lifetime. 1
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UNIVERSITY OF LJUBLJANA Faculty of Mathematics and Physics
Department of Physics
Seminar on
Monte Carlo Methods in Reactor Physics
Author: Andrej Kavčič
Mentor: prof. dr. Matjaž Ravnik
Ljubljana, January 2008
Abstract
The Monte Carlo method is being widely used to solve neutron transport problems in nuclear
reactor cores along with the advancements in computer technology. The method is originally
as old as neutron is, however, it was not until the last decade that it became popular due to
growing computer capacities. Today almost all neutron parameters and their effects on reactor
behaviour can be simulated. The next pages represent a step by step introduction, from the
primary Monte Carlo idea to the calculation of three most important parametres in nuclear
*Data for fast-neutron induced fission. Table 1. Total delayed neutron yields (number of delayed neutrons on 100 fission events) for thermal neutron induced fission of different isotopes [6].
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The protocol of calculating the effectiveness of delayed neutrons (example
for TRIGA type reactor)
i generation: 1000 At the begining there are
N fission neutrons (1000in the left example).
Fission (average n = 2.439)
Prompt neutrons Delayed neutrons
17 b0 = 0.69% 2422
0.5 1 1.5 2 2.5
EHMeV L
0.2
0.4
0.6
0.8
1
χHEL Energijski spekterzakasnelih nevtronov
1 2 3 4 5
EHMeV L
0.1
0.2
0.3
0.4
χHEL Energijski spektertakojš njih nevtronov
992 8
i+1 generation:
47% survival beff = 0.8%
41% survival
1000
Prompt and delayed neutronenergies are sampled fromdifferent sprectrums. After picking time, energy andangle parametres the journeystarts. Because of different startingenergies, delayed neutronsappear to be more effective(this statement is valid onlyfor small reactors - likeTRIGA) [7].
The results of kinetic parametres for the upper example would be:
1 1000 11000
i
i
nkn+= = = , 0
17 0.69%2439
β = = , 8 0.80%1000effβ = = (18)
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4.3. Prompt neutron lifetime The prompt neutron lifetime is the average time between the emission of neutrons and either
their absorption in the system or their escape from the system [8]. The term lifetime is used
because the emission of a neutron is often considered its ‘birth’, and the subsequent
absorption is considered its ‘death’. For thermal (slow-neutron) fission reactors, the typical
prompt neutron lifetime is on the order of 10-4 seconds, and for fast fission reactors, the
prompt neutron lifetime is on the order of 10-7 seconds [9].
In the beginning when the neutron was born its energy has been defined from the neutron
fission spectrum. Along its journey we have the information about scattering and a loos of
energy which gives us exact information about velocity. And together with the travelling
distance we receive the neutron lifetime.
The problem is that all the neutrons are not important for the reactor core. In other words,
neutrons that escape out of the reactor core can either return or not. Only the lifetime of the
returned neutrons is important but it is hard to separate between them. Because of that we will
rather choose another method, simulation of a neutron prompt jump.
4.3.1. Prompt jump (or drop):
The prompt jump factor is the factor by which power undergoes immediate (but not
instantaneous) change in response to a step change in reactivity. This power change occurs
over several hundred prompt neutron lifetimes, through alteration of the prompt neutron
population because of change in the source multiplication factor. The change is so rapid that
delayed neutrons can be neglected in the point kinetic equation (7), thus it follows:
( )dn n
dtρ β−
=Λ
(19)
Which solution is: 0( )eff t
n t n eρ β−Λ= (20)
In earlier chapters we have already calculated the multiplication factor and delayed neutrons
factor. What we need now is the change of the neutrons density during time and we can get
the mean generation time from the equation (20).
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The prompt jump beings in the order of 10-4 seconds and involves only a small fraction of one
second (exact time depends on kinetic parametres).
At t = 0 ms, the neutron source is put in the reactor and after 15ms we get the homogenous
neutron flux across the whole reactor (Fig. 8). After 35ms the source is put outside the core
and the flux decreases with tipical period which is defined from the kinetic parametres.
10 20 30 40 50tHmsL
0.02
0.04
0.06
0.08
0.1
0.12φHtL
Figure 8: the curve of neutron flux when we put the neutron source into the reactor for 35ms
In the last step the exponential function has to be fitted and the time we are looking for can be
easily calculated by following:
0( )eff t
n t n eρ β−Λ= or 0
0( )ttn t n e= , where 0
eff
tρ βΛ
=−
(21)
The final result of a prompt neutron lifetime for TRIGA type reactor is: 38 sμΛ =
10 20 30 40 50tHmsL
0.02
0.04
0.06
0.08
0.1
0.12φHtL
Prompt dropL= 38.8 ms
Prompt jumpL= 37.6ms
Figure 9: the promt jump and drop.
5. Summary
The professional Monte Carlo neutron codes are rapidly developing from year to year, for
instance, delayed neutrons were regulary implemented in the last two years. However, there
are still some problems with the prompt neutron lifetime calculations. In general, the method
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(especially General Monte Carlo N-particle Code [10]) is gaining a major role in kinetic
calculations and stands as highly perspective. Exact calculations of kinetic parametres are
cruical for the reactor safety and Monte Carlo method is highly respected when talking about
precise evaluations.
References:
[1] M. Rosina, Jedrska fizika, FMF (2005).
[2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering, Third Edition,
Prentice Hall, Inc. (2001).
[3] M. Ravnik, Reaktorska in radiacijska fizika, FMF (2005).
[4] Criticality is another expression for multiplication factor (especially usefull for
operators in nuclear power plants) and is defined as: 1kk
ρ −= .
[5] Available at Janis 2.2 Java-based Nuclear Data Display Program: