monte carlo methods in reactor...
TRANSCRIPT
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.
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Contents
Introduction . . . . . . . . . 3
Nuclear power. . . . . . . . . 3
Introduction to Nuclear Reactor Technology . . . . 4
Reactor kinetic equations . . . . . . . 5
Principles of Monte Carlo . . . . . . . 7
- Neutron multiplication factor . . . . . 10
- Delayed neutrons . . . . . . . 12
- Prompt neutron lifetime . . . . . . 15
- Prompt jump. . . . . . . 15
Summary. . . . . . . . . 17
References . . . . . . . . . 17
1. Introduction
The Monte Carlo method provides approximate solutions to a variety of mathematical
problems by performing statistical sampling experiments on a computer. The method applies
to problems with no probabilistic content as well as to those with inherent probabilistic
structure. Among all numerical methods that rely on N-point evaluations in M-dimensional
space to produce an approximate solution, the Monte Carlo method has absolute error that
decreases as N-1/2 whereas, in the absence of exploitable special methods all others have
errors that decrease as N-1/M at best.
The method was born in 1930s and began to be studied in depth during Second World War in
Los Alamos National Laboratory. It is called after the city in the Monaco principality, famous
by its roulette, a simple random number generator. The name and the systematic development
of Monte Carlo methods dates from about 1944. At that time basic steps were made by
physics researchers Ulam, Fermi, Von Neumann and Metropolis while analysing neutron
behaviour. At that time, there was no typical realization because of poor computational
technology, but nowadays the method corresponds perfectly to digital processors, that are
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getting faster every day, and is widely used in all fields of science, especially in nuclear
physics.
2. Principles of Fission Chain Reaction Nuclear reactors use a chain reaction to induce a controlled rate of nuclear fission in fissile
material, releasing both energy and free neutrons. The splitting of the nucleus is possible if it
is energetically feasible and the potential barrier is passed. Since the probability for
spontaneous fission of naturally occuring elements (uranium) is very small and cannot be
controlled externally, nuclear reactors are driven by induced fission with neutrons. Neutrons
are used because every fission of a heavy nucleus apart from two smaller nuclei and a lot of
kinetic energy produces 2 or 3 additional neutrons.
The newly produced neutrons can either trigger new fission to produce new generation of
neutrons or be absorbed by other material. The other reason why neutrons are convenient is
that in addition to the kinetic energy they carry also binding energy when they enter nuclei.
Because of the spin-coupling effect the difference in energy released by neutron capture of
two nuclei of similar but different parity mass and atomic number is about 2MeV. Therefore
nuclei (around uranium) with even number of protons and odd number of neutrons (for
example 235U) can be split by thermal neutron, whereas odd-odd nuclei (for example 238U) can
be split only by neutron with kinetic energy over 1MeV.
Fissioned nucleus usually splits into two lighter nuclei and neutrons. The former are called the
fission products. They are mostly radioactive because of the overplus of neutrons. Due to the
''shell'' structure of the target nucleus, asymmetric decay is energetically more favourable that
the symmetric one [1]. Therefore, if the incident neutron energy is low, the probability for
asymmetric decay is much bigger compared to the symmetric decay. On the other hand, if
most of the neutrons causing fission have high kinetic energies, the symmetric decay
predominates [1].
The neutron population at any instant is a function of the rate of neutron production (due to
fission processes) and the rate of neutron losses (via non-fission absorption mechanisms and
leakage from the system). When a reactor’s neutron population remains steady from one
generation to the next, the fission chain reaction is self-sustaining and the reactor’s condition
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is referred to as "critical". In order to be steady state, the multiplication factor for the chain
reaction, defined as:
1i
i
nkn+= (1)
where ni is the number of fissions in i-th generation, has to be exactly 1. This means that the
power of the reactor is constant. However, because the time between successive reactions,
which is per definition equal to the average neutron lifetime in reactor, is of order 10-5 s, even
small change of k would cause explosion or shut down before the operator could interfere.
Luckily, aproximately 1% of the neutrons (called delayed neutrons) arise from fission
products with lifetimes of up to around 10s. Although their relative number is small, they
increase the average neutron lifetime to about 1s and the chain reaction becomes controlable.
Fissile isotopes are capable of sustaining fission chain reaction in collisions with low-energy
(or thermal) neutrons [2]. Examples are 235U, 239Pu and 233U but only 235U is present in nature,
so others have to be produced artificially.
In general fissile materials have large cross sections ( > 102 barn) for slow (E < 0.1eV)
neutrons. The probability for inducing fission is higher for slow neutrons. To reduce the
critical mass and required fissile fuel concentration (enrichment) most reactors use materials
to slow down neutrons, called moderators.
A good moderator has a high cross section for neutron (mostly elastic) scattering and on the
other hand the cross section of neutron absorption has to be as low as possible. The moderator
should be dense (to decrease the mean free path of the neutron) and, most importantly, contain
low mass nuclei in order to increase neutron energy loss per collision [3]. Examples of good
moderators are: hydrogen, deuterium, graphite or beryllium.
3. Reactor Kinetic Equations All the equations written down are based on independence of space and time distribution of
neutron flux, which for thermal neutrons can be explain as follows:
( , , ) ( , , )r E t vn r E tΦ = (2)
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where represents the space, energy and time distribution of neutron flux,( , , )r E tΦ ( , , )n r E t
is distribution of neutron density and v is velocity of thermal neutrons.
During a transient time distribution of neutron density mostly depends on multiplicator factor
k of fission assembly, prompt neutron lifetime and delayed neutron parameters. In one neutron
lifetime, l, the number of neutrons changes as
( 1)n kn n k nΔ = − = − (3) We can rewrite upper equation in the infinitesimal form (4), count out the delayed neutrons (1
- β ) and add the factor that actually describes them (λ C ) in order to get the reactor kinetic
equation:
1dn k ndt l
−= (4)
(1 ) 1dn k n Cdt l
β λ− −= + , C is calculated as dC C n
dt lβλ= − + (5)
where is multiplication factor, l - prompt neutron lifetime in fission assembly, k β is
delayed neutron fraction,C precursors concentration of delayed neutrons and λ decay
constant of precursors of delayed neutrons.
The prompt neuron lifetime is related to prompt neutron generation time l Λ through the
neutron reproduction number . Relation between bought prompt neutron times is: k
lk
Λ = (6)
Using term for reactivity r [4] and prompt neutron generation time the kinetic equations can
be edited by following:
( )dn n C
dtρ β λ−
= +Λ
(7)
It is clear that the the most important factors how the reactor behaves are: reactivity, effect of
delayed neutrons and neutron generation time. In the next chapter there will be shown how to
calculate these crucial parametres. There are some deterministic principles, but it is almost
impossible to get the final result without approximations in comparison to Monte Carlo
method.
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4. Principles of Monte Carlo Basic idea of Monte Carlo method is to calculate the travelling distance of each neutron. We
need to propagate the neutron trajectory in the reactor like it really behaves (the best way is 'to
think like a neutron').
The probability of a first collision for particle between l and l + dl along its line of flight is
given by:
( ) lp l dl e dl−Σ= Σ (8)
where S is the macroscopic total cross section of the medium and is interpreted as the
probability per unit length of a collision. Setting ξ the random number on [0,1) it follows
that
0
1l
s le ds eξ −Σ= Σ = −∫ −Σ (9)
and
1 (1 )l ln ξ= − −Σ
(10)
But, because 1 - ξ is distributed in the same manner as ξ, and hence may be replaced by ξ,
we obtain the well-known expression for the distance to collision:
1 ( )l ln ξ= −Σ
(11)
and after i collisions in one dimmension it can be written:
11 ( )i il l ln ξ+ = −Σ
(12)
In this chapter there will not be discussion about random numbers, let just say that they could
be generated with quite many modern mathematical algorhythms. Untill now there has not
been anybody who could proove unrandomness in these generators. In the beginning there
were actually some problems, but nowadays modern mathematical software corporations (like
Wolfram Research Inc.) have officially published invitation for those who will prove
unrandomness in their generators.
In equation (12) there is random walk of neutron in one dimension. The next logical decision
is to write it down in three dimensions. What we need are two more additional random
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numbers to specify the direction of flight of the neutron. One way is to write them with the
azimuthal and polar angle, but because of practical reasons the more common expression is to
write as j and Cos q.
For the homogenous distribution it can be written:
1( ) 4
dP dPd d d Cosϕ ϑ π
=Ω
= , 12
dPdϕ π
= , 1( )
dPd Cosϑ 2
= (13)
If we define another two random numbers xi, xi+1 , any direction of the neutron flight could be
defined as:
2i iϕ πξ= and 12i iCos 1ϑ ξ += − (14)
Figure 1: randomly sampled 500 directions from the same starting point (Eq. 13). They are
actually distributed on the surface of a sphere.
In the next step the length has to be included in each directon. For the coordinates we write
x = l sin q cosj, y = l sin q sin j and z = l cos q, where l is random lenght of neutron flight,
and a propagation of elastic scattering looks like:
21 1
1 ( ) 1 (2 1) (2 )j j i i ix x ln Cos 3ξ ξ π+ += − − −Σ
ξ +
21 1
1 ( ) 1 (2 1) (2 )j j i i iy y ln Sin 3ξ ξ+ += − − −Σ
πξ + (15)
1 31 ( )(2 1)j j i iz z ln ξ ξ+ += − −Σ
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Figure 2: random neutron walk in three dimensions.
The next logical step is to correct our model for the in-elastic scattering. When a neutron
collides with the hydrogen nucleus (proton) it looses for about half of its energy. That is also
the reason why the water is so good moderator. At this point we need exact information about
matherials (nuclear fuel, control rods, moderator...) for defining the loss of energy for each
collision.
Here it is usefull to include all the neutron interactions with matter, especially capture and
fission. When a particle collides with a nucleus, the following sequence should occur:
- the collision nuclide is identified,
- either elastic or inelastic scattering is selected,
- check probability for fission and determine the new energy and direction of the
outgoing neutrons,
- check probability for capture and kill the neutron.
All the probability informations are given in the evaluated nuclear data files [5] containing
evaluated (recommended) cross sections, spectra, angular distributions, fission product yields
and other data, with emphasis on neutron-induced reactions.
Figure 3: Fission cross sections s for interaction with neutrons for 235U as a function of
incident neutron kinetic energy E in log-log scale [7].
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At this point we have achieved the first reliable model of neutron transport. On the bottom
picture (Figure 4) there is the starting neutron (black line from the orange square) that scatters
and absorbs in the uranium. After fission we get two new neutrons, the green one and the blue
one with their own trajectories. The green one is captured in the absorber (for example B, In,
Cd, Ag…) and the blue one produces another fission with two new neutrons (light blue and
red one).
Figure 4: model of neutron fission (black or blue) and capture (green one).
The basic idea of such transport via Monte Carlo method is actually as old as neutron is.
Enrico Fermi used Monte Carlo (at that time the method had no special name) in the
calculation of neutron diffusion in 1930s but until now it was almost impossible to make an
exact calculation of a neutron transport. As soon as we have Monte Carlo neutron transport
code, we can start calculating major reactor paramethers.
4.1. Neutron multiplication factor
The neutron multiplication factor k, is the average number of neutrons from one fission that
cause another fission. The remaining neutrons either are absorbed in non-fission reactions or
leave the system without being absorbed. The value of k determines how a nuclear chain
reaction proceeds:
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- k < 1 (sub-critical mass): The system cannot sustain a critical reaction, and any
beginning of a chain reaction dies out over time. For every fission that is induced in
the system, an average total of 1/(1 − k) fissions occur.
- k = 1 (critical mass): Every fission causes an average of one more fission, leading to a
fission (and power) level that is constant. Nuclear power plants operate with k = 1.
- k > 1 (super-critical mass): For every fission in the material, it is likely that there will
be k fissions after the next mean generation time. The result is that the number of
fission reactions increases exponentially, according to the equation (7).
Red one produces another fission
(k=1)
After fission 2 neutrons are created (blue and green line)
Two neutrons (green and red line) produce
another fission (k=2)
After fission 3 neutrons are created (red, brue
and green line)
Figure 5: production of fission neutrons and calculation of multiplication factor k.
After a lot of fission neutrons during each generation the effective multiplication factor can be
defined (or the reactivity). In practice reactivity is written to five decimal places and the unit
is pcm (''percent milli'').
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4.2. Delayed neutrons:
As it was already mentioned nearly all of the fission neutrons – more than 99% of them
– are emitted just after the fission process, without considerable delay. The fission
fragments that usually have much higher excitation energy than the neutron separation
energy emit these prompt neutrons. The half-life of neutron-emission of these highly
excited states is in the order of 10-15 s or even smaller. However, not all fission
fragments emit neutrons, some of them decay by b- emission.
After the emission of the prompt neutrons there is usually no more neutron emissions,
the fission products undergo several successive b- decays to reduce their neutron excess.
However, in some cases a daughter nucleus is formed after a b- decay, where the
excitation energy is higher than the neutron separation energy. This nucleus will emit
again a neutron, nearly promptly after its formation. These are the delayed neutrons. For
example the decay chain for 87Br is summarised as follows:
87 87 * 86Br Kr Krβ −
n⎯⎯→ → + (16)
The 87Br nucleus is called delayed-neutron precursor, the 87Kr nucleus is called
delayed-neutron emitter. Obviously, for these neutrons the “delay time” is determined
by the half-life of the precursor nucleus, which can be quite large, since the b– decay is
governed by the weak interaction. Another interesting consequence of this decay chain
is that the excitation energy of the delayed-neutron emitter nucleus is usually much
lower than the excitation energies of the direct fission fragments. Therefore, the average
energy of the delayed neutrons is also smaller (300 - 600 keV) than that of the prompt
neutrons (2 MeV in average).
Delayed neutron
is born.
Radioactivedecay
fission
146La
87Br 87Kr
235U
Figure 6: birth of delayed neutrons.
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The total yield of the fission neutrons (average number of neutrons, ν) is the sum of the
yield of the prompt and the delayed neutrons (νp, and νd):
dp vvv += (16)
And delayed neutron fraction is defined as:
vvd=β
. (17)
Figure 7. shows the dependence of the delayed-neutron yield for 235U and 239Pu on the energy of
the fission-inducing neutron.
0,1 0,3 1,0 3,0 10,0 (MeV)
Energy of the neutron, which induced the fission
239Pu
235U
0,015
0,01
0,005
Delayed neutron yield (delayed neutrons / fission)
Figure 7: Delayed neutron yield in function of the energy of the fission inducing neutron[6].
The curves show that the delayed neutron yield is practically independent of the neutron energy,
at least in the energy interval 0 < E < 4 MeV. However, the total delayed neutron yields are
strongly dependent on the composition of the fissile nucleus. The values are shown in Table 1.
Fissile nucleus νd (neutron/100 fissions) 233U 235U 238U*
0,667+0,0029 1,621+0,05 4,39+0,10
239Pu 240Pu* 241Pu 242Pu*
0,628+0,038 0,95+0,08 1,52+0,11 2,21+0,26
*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:
http://www.nea.fr/janis . Accessed: December, 2007.
[6] G. R. Keepin: Physics of Nuclear Reactors, Addison-Wesley Publishing Co.,
Massachusetts (1965).
[7] Nuclear Power Fundamentals, Effective Delayed Neutron Fraction, 10.01.2008,
http://www.tpub.com/content/doe/h1019v2/css/h1019v2_105.htm .
[8] Lamarsh, John; Baratta, Anthony. Introduction to Nuclear Engineering. Prentice Hall
(2001).
[9] Duderstadt, James; Hamilton, Louis. Nuclear Reactor Analysis. John Wiley & Sons,
Inc. (1976).
[10] MCNP5 - A General Monte Carlo N-Particle Transport Code, Version 5 – Los Alamos
Diagnostics Applications Group (2007).
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