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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