Neutronics Computation with Angular Flux

Transport Equation

Than Yan Ren

supervised by

Prof. Lim Hock

Assoc. Prof. Chung Keng Yeow

An Honours Thesis submitted in partial fulfilment of the requirements for

the

Degree of Bachelor of Science

(Honours)

to the

Department of Physics

National University of Singapore

AY2015-2016

Abstract

Computer simulations are crucial in the analysis of nuclear reactors both

in the developmental phase and for accident scenarios due to the fact that they

provide a risk-free method for experimenting with the situation. Huge collections

of programs that can together simulate the entire meltdown process of a nuclear

power plant have been developed. The aim of this study is to investigate the

basics in building a simulation for core nuclear kinetics based on the solving the

time independent neutron transport equation. MATLAB codes were developed

from first principles based on methods from E. Lewis for various set ups in the

spherical geometry to investigate steady state behaviours. Special attention has

also been given to the choice of input parameters they have to fit approximations

taken in solving the transport equation. As initial exercises to test the code, the

criticality of pure uranium spheres were investigated. The results were found

to be in good agreement with known results like from diffusion approximation.

After we are satisfied with the workings of the code, we look at how it can be

applied to more complex situations. This will be illustrated with examples in

analysing the yield of a uranium bomb and the behaviour of the pebble bed

reactor in steady state conditions.

Acknowledgements

I would like express my sincere gratitude to Prof Lim Hock and Assoc Prof Chung Keng

Yeow for taking time out of their busy schedules for our regular meetings, and also for their

guidance and mentorship throughout the duration of this project.

I would also like to thank my family and my friends for their continued support and under-

standing throughout this demanding period.

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope of project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Theory 2

2.1 Neutron chain fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Cross section of interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Neutron transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Diffusion approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Three energy groups in moderated reactors . . . . . . . . . . . . . . . . . . . 9

3 Solving the time-independent neutron transport equation 10

3.1 Derivation of discretised equations . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Algorithm for solving equations . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Meaning behind the iteration process . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Determining input cross section values and energy groups . . . . . . . . . . . 24

3.4.1 Angular dependence of scattering cross sections . . . . . . . . . . . . 24

3.4.2 Group-to-group scattering . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Initial results 27

4.1 Critical mass of pure U-235 sphere . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Critical mass of enriched uranium sphere . . . . . . . . . . . . . . . . . . . . 29

4.3 Temperature profile of water-cooled uranium sphere . . . . . . . . . . . . . . 31

5 Possible Applications 35

5.1 Estimating efficiency of uranium bomb . . . . . . . . . . . . . . . . . . . . . 35

5.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Pebble bed reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusion 47

A Alternate derivations of discretised equations 50

B Obtaining temperature profile from given neutron flux 51

C Derivation of energy loss from scattering 53

D MATLAB codes 55

1 Introduction

1.1 Motivation

Nuclear power plants are power plants which obtain energy from fission of radioactive nuclei

(most commonly Uranium-235) in a nuclear reactor core. It provides a far more efficient

source of energy than the traditional fossil fuels and generates much less waste in compari-

son, about 1100GWh per ton of uranium[1] in a typical Pressurised Water Reactor (PWR)

plant to about only 2.7MWh per ton of coal[2], thus it is an attractive option that many

countries have begun to adopt. However the safety of such power plants has been a huge

concern due to the volatility of the radioactive materials, and indeed large scale accidents

have happened like the Chernobyl and Fukushima Daiichi disaster. On top of the immediate

loss of lives incurred, the radioactive material released could potentially spread hundreds

of miles away through being carried by wind or water and cause widespread health issues.

Thus many have been aiming to create safer reactor designs that are less prone to melting

down in the event of an accident. Crucial to the design process would be using computer

simulation is a safe way to analyse the dynamics of the hypothetical reactors.

Besides that, accident scenarios can also be analysed with computer simulation to deter-

mine the optimal response to minimise the damage done. Large scale and complex sets of

simulation codes like the Accident Source Term Evaluation Code (ASTEC) have been devel-

oped for such purposes and with the introduction of the upcoming Generation IV reactors,

even more simulations will have to be developed.

For anyone who wishes to work on or even develop such simulations, it will be very helpful

that he or she understands the basics working principles behind the simulations well in order

to identify and handle possible shortcomings in the code.

1.2 Scope of project

The aim of this project is then to conduct a study on these basic working principles of a

simulation. The type of simulation that will be investigated here is done through numerical

approximation of the neutron transport equation that describes the dynamics within fission-

able materials. The other major type of simulation is done through Monte-Carlo and this

is also being looked at in a seperate project. We will aim to understand the physical prin-

1

ciples behind how and why certain approximations are taken. The first part of the project

will consist of writing up a basic algorithm for solving time independent problems based on

material from [3] with proper understanding of its workings. After writing up the code, it

will be applied to simple examples in order to verify that it is working as intended. Once

we are satisfied with that, we will then go on to discuss possible ways of adapting the code

into the analysis of more complex situations. Initial results will also be shown for a more

complete illustration of the method.

2 Theory

2.1 Neutron chain fission

Fissionable materials release energy primarily through the neutron-induced fission, which

causes a heavy nucleus like Uranium-235 to fission into two smaller daughter nuclei and

possibly other small fragments like protons and neutrons. The energy released comes from

the decrease in total binding energy of the fission products in comparison to the original

nucleus and thus it is this extra binding energy that is released from the fission event.

Figure 1: Plot of binding energy versus nucleon number [4]

As one can see from figure (1), after the element Iron, the binding energy tends decrease

as the nuclei gets heavier. Due to the fact that there are often a few neutrons present in

the fission products, these second generation neutrons can go on to trigger even more fission

2

events. If the conditions are favourable, an exponentially increasing chain of fissions events

can occur, leading to the release of large amounts of energy. When this happens, the materi-

als is said to have reached criticality. This possibility is determined by whether the neutrons

produced from fission is able to compensate for those lost to leakage or other interactions.

The easiest way to reach criticality is simply to increase the amount of fissile material used,

since volume increases by r3 while the surface area only increases by r2, the fission rate

that occurs within the volume of the material will increase faster than the leakage that only

happens on the surface. Thus for fissionable materials, the critical mass or size will be of

great interest.

It is the possibility of creating this chain reaction that allows significant amounts of energy

to be extracted from fissile materials. This indicates that properties of fissionable materials

are best studied by looking at the dynamics of the neutrons within the material.

The neutrons in general can be described with the quantity N(~r, Ω, E, t), which is the total

number of neutrons in a dV volume about position ~r travelling in the direction within a dΩ

cone about directional vector Ω and having energy from E to E+dE at time t, as illustrated

in figure (2).

Figure 2: Illustration of quantity N(~r, Ω, E, t).

The next step is now to describe the interaction of these neutrons with fissionable materials

3

and other matter. This is done using the concept of the cross section of interaction.

2.2 Cross section of interaction

The cross section value describes the probability for the neutron to interact with the atomic

nuclei present, in the sense that these nuclei will appear to the neutrons to have a certain

cross sectional area as shown in figure (3). So should the neutron pass through that area,

Figure 3: Interpretation of the cross section of interaction. The neutron marked with a crossdoes not interact. Note that this cross section may not necessarily coincide with the physicalsize of the nucleus.

the neutron will interact with the nucleus.

Each individual type of interaction will have their own respective cross section and the com-

bined cross section of separate interactions will just be the sum of the individual cross section

values. For example, when an interaction happens, the neutron can either be absorbed or

scattered, so we have

σt = σs + σa (2.1)

where σt is the total cross section and σs and σa are the scattering and absorption cross

sections respectively. This is not the full picture yet however, as these cross sections do

change according to the incoming energy of the neutron and scattering cross sections can

also be split into the angle and energy that the incoming neutron is scattered into. So

4

equation (2.1) should instead read

σt(~r, E) =

∫dE ′

∫dΩ′ σs(~r, E

′ → E, Ω · Ω′) + σa(~r, E). (2.2)

Note that due to the symmetry of a collision event, σs will depend on just Ω · Ω′ instead of

both Ω and Ω′ separately. Equation (2.2) can also depend on time since materials can be

moving or expand due to heat and such, but in this case we will not be considering such

effects.

The most important interaction here is of course the fission, of which cross section is given

by σf (~r, E) and is considered a subset of the absorption reaction.

The reaction rate at each dV volume in space using this concept can then be given as

the total number of neutrons passing through the total cross section contributed by the

nucleons in the dV volume per unit time. This can be written more formally as

Rt(~r, t) =

∫dE

∫dΩ σt(~r, E)v(E)N(~r, E, Ω, t). (2.3)

The rate of each specific type of interaction can also be written is very similar form, which

will be used in describing the dynamics of the system. Thus it is useful to define the quantity

ψ(~r, Ω, E, t) = v(E)N(~r, E, Ω, t) (2.4)

which is also called the neutron flux.

2.3 Neutron transport equation

We would expect an equation describing neutron dynamics to have the form

∂

∂tN(~r, Ω, E, t) =Factors removing or introducing neutrons into

particular phase space considered

5

These factors should consist, in some form, of the interaction terms as described in the

previous section. The actual neutron transport equation is in fact in this form and reads

∂

∂tN(~r, Ω, E, t) = − Ω · ~∇ψ(~r, Ω, E, t)− σt(~r, E)ψ(~r, Ω, E, t) +

qex(~r, Ω, E, t) +

∫dE ′

∫dΩ′ σs(~r, E

′ → E, Ω · Ω′)ψ(~r, Ω′, E ′, t) +

χ(E)

∫dE ′νσf (~r, E

′)φ(~r, E ′, t)

(2.5)

where

φ(~r, E, t) =

∫dΩ ψ(~r, Ω, E, t). (2.6)

The first term on the right describes the loss of neutrons from the net number of neutrons

moving out of the volume dV around ~r while the third term accounts for external sources.

Neutrons that interact and considered to be removed first, which is accounted for by the

second term, and then reinserted as their appropriate internal source terms according to

type of interaction. So the fourth term describes the neutrons that have been scattered from

E ′ and Ω′ into E and Ω. And finally the last term describes the neutrons produced from

fission events with energy E travelling in direction Ω; ν here is a constant describing the

average number of neutrons released per fission event (which depends on the fissile material)

and χ(E) is the energy distribution of the fission neutrons. One reasonable approximation

of χ(E) is as follows[5],

χ(E) = 0.453 exp(−1.036E) sinh(√

2.29E) (2.7)

in units of MeV. This approximation is found to be in very good fit with the experimental

data[6] as shown in the following figure.

6

Figure 4: MATLAB plot of equation (2.7) against actual experimental data

Note that for the full neutron dynamics will also consist of delayed neutrons, which are

neutrons not emitted at the instant of the fission event but rather at a later time by the

daughter nuclei produced by the fissioning as they could be in an excited state. However,

this will not matter here as we will only be looking at time independent form of the equation,

which is∂N

∂t= 0. This is already very useful since only materials that are exactly critical

can have time independent solutions. So in these steady state situations, it does not actually

matter when the neutrons are released. So finally, the equation that we will be handling is

given as [Ω · ~∇ + σt(~r, E)

]ψ(~r, Ω, E) =

qex(~r, Ω, E) +

∫dE ′

∫dΩ′ σs(~r, E

′ → E, Ω′ · Ω)ψ(~r, Ω′, E ′) +

χ(E)

∫dE ′νσf (~r, E

′)φ(~r, E ′)

(2.8)

which is an integro-differential equation to be solved for ψ(~r, Ω, E, t).

7

2.4 Diffusion approximation

This section will describe a simple way to solve the time independent version of the neu-

tron transport equation using the diffusion approximation. To illustrate this, first integrate

equation (2.8) through all energies and then all angles, obtaining

σt(~r)φ(~r) + ~∇ · ~J(~r) = qex(~r) + σs(~r)φ(~r)

+ νσf (~r)φ(~r)(2.9)

noting that assumptions, which will be covered in detail in the next chapter, have been made

in order for the scattering and fission terms to be integrated into this form. Supposing that

there are sufficient neutrons moving in all directions, Fick’s first law of diffusion gives the

relation~J(~r) = − 1

3σt(~r)~∇φ(~r) (2.10)

thus resulting in the equation

∇2φ(~r) = 3σt(~r)(σa(~r)−ν

kσf (~r))φ(~r) + qex(~r). (2.11)

One simple result from this approximation comes from considering the situation where the

fissile material is homogeneous, source-free and spherically symmetric, this greatly simplifies

the transport equation into

∂2

∂r2rφ(r) = 3σt(σa −

ν

kσf )rφ(r). (2.12)

This is a simple ordinary differential equation which can be solved with the appropriate

boundary condition φ(R + 23σt

) = 0 to obtain

φ(r) =C

rsin(√

3σt(ν

kσf − σa) r

)(2.13)

as the only family of physically valid solutions under the condition that

3σt(νσf − σa) =( π

R + 23σt

)2

. (2.14)

8

The above condition also sets a restriction on the radius to be

R =

√π2

3σt(νσf − σa)− 2

3σt(2.15)

which we can identify as the critical radius, thus we can see that this gives a very quick way

to estimate the critical radius of any fissile material that roughly fits the assumptions taken.

This will be a very useful method to use as a quick check for the results obtained in this

project. Also note that for materials where νσf − σa < 0, criticality cannot be achieved as

square root term will become imaginary.

2.5 Three energy groups in moderated reactors

In analysing situations within the typical moderated reactor, it is common to just take three

energy groups labelled fast (F), intermediate (I) and thermal groups (T)[5]. This method is

chosen due to the properties of the fissile materials and the neutrons. Firstly, the fast range

covers all energies above 0.1MeV as this is the range where practically all fission neutrons

are released into. The intermediate range, which covers 1eV ≤ E ≤ 0.1MeV, is where the

scattering and absorption cross sections are dominated by closely packed resonance peaks

due to the atomic spectra of the nuclei. It is typically seen as an intermediate group that

fission neutrons go through to eventually reach the thermal range, where the fission cross

section increases dramatically (from 1 to 2) in the case of uranium-235, one of the commonly

used fuels in reactors. Finally the thermal range covers energies below 1eV as this is where

the neutron energies are comparable to the thermal energies of the nuclei. This also means

upscatter into the I and F ranges is impossible as they are beyond the reach of the thermal

energies.

The energy dependent fission cross section shown in figure (5) clearly illustrates the three

distinct regions and the tightly-packed resonance peaks can also be seen in the intermediate

region. The energy distributions of the neutrons within this context can also be roughly

described as

ψF (E) = χ(E) (2.16)

for the fission group,

ψI(E) ∝ 1

E(2.17)

9

Figure 5: Plot of σf (E) of U-235 obtained from International Atomic Energy Agency[7]

for the intermediate group, and finally

ψT (E) =E

(kBT )2exp(−EkBT

)(2.18)

for the thermal group, which unsurprisingly is the Maxwell-Botlzmann distribution for par-

ticles in thermal equilibrium.

3 Solving the time-independent neutron transport equa-

tion

3.1 Derivation of discretised equations

This section will describe the method adapted from [3] to discretise the neutron transport

equation such that it will be suitable for computational schemes to handle.

10

The equation to solve here is[Ω · ~∇ + σ(~r, E)

]ψ(~r, Ω, E) = χ(E)

∫dE ′

ν

kσf (~r, E

′)φ(~r, E ′)

+

∫dE ′

∫dΩ′ σs(~r, E

′ → E, Ω · Ω′)ψ(~r, Ω′, E ′).

(3.1)

This is then split in three energy groups fast, intermediate and thermal, which results in the

following,[Ω · ~∇ + σg(~r)

]ψg(~r, Ω) = χg

ν

k

∑g′

σfg′(~r)φg′(~r) +∑g′

∫dΩ′ σg′g(~r, Ω · Ω′)ψg′(~r, Ω′) (3.2)

where g = T, I, F is the index for the energy groups. This is done by assuming the energy

seperability of the angular flux

ψ(~r, Ω, E) ≈ fg(E)ψg(~r, Ω) Eg < E 6 Eg−1 (3.3)

where the function fg(E) is normalised with∫gdE fg(E) = 1. The cross sections can then

be seen to be weighted and averaged over this distribution for example

σg(~r) =

∫g

dE σ(~r, E)f(E). (3.4)

The energy seperability need not be assumed however, in which case to arrive at the same

result, we would require ∫gdE σ(~r, E)ψ(~r, Ω, E)∫

gdE ψ(~r, Ω, E)

?= σg(~r). (3.5)

We can see that mathematically, σg should have angular dependence. However, physically

speaking σg(~r) should be correct since even if the nuclei are not spherically symmetric, they

should be randomly oriented thus on average all incoming angles should be equal. Even if

this is not convincing, an alternative derivation shown in Appendix A will still yield the

same result.

Note that using the spherical symmetry of the problem, any neutron streaming through

the domain of the system can be broken down as shown in figure(6).

Thus it can seen that the vectors ~r and Ω can be replaced with scalars r and µ = cos Θ. So

11

Figure 6: Particle streaming through a spherical domain with u being a coordinate along Ωwith impact parameter b

equation (3.2) can be re-written as

[µ∂

∂r+

1− µ2

r

∂

∂µ+σg(r)

]ψg(r, µ) = χg

ν

k

∑g′

σfg′(r)φg′(r) +∑g′

∫dΩ σg′g(r, Ω · Ω′)ψg′(r, µ′)

(3.6)

with the quantity Ω · Ω′ being related to µ and µ′ as follows,

Ω · Ω′ = µµ′ +√

1− µ2√

1− µ′2 cos(ω − ω′) (3.7)

and ω is the angle shown in figure (7).

Then note that the scattering cross section is a function defined on −1 6 Ω · Ω 6 1, meaning

it can be expanded using the Legendre polynomials as follows,

σg′g(r, Ω · Ω′) =∞∑l=0

(2l + 1)σlg′g(r)Pl(Ω · Ω′). (3.8)

Here the Pl(Ω · Ω′) are the Legendre polynomials and

σlg′g(r) =

∫dΩ Pl(µr)σg′g(r, µr) =

∫dµr2

Pl(µr)σg′g(r, µr) (3.9)

where µr = Ω · Ω′. The term Pl(Ω · Ω′) from equation (3.8) can be further simplified using

12

Figure 7: Coordinate ω in relation to Ω and r

the Legendre addition theorem to be

Pl(Ω · Ω′) =1

2l + 1

l∑m=−l

Y m∗l (Ω)Y m

l (Ω′) (3.10)

where

Y ml (θ, ϕ) =

√(2l + 1)

(l −m)!

(l +m)!Pml (cos(θ))eimϕ (3.11)

are the spherical harmonics. The scattering term now becomes

∑g′

∫dΩ′ σg′g(r, Ω · Ω′)ψg′(r, µ′) =

∑g′

∞∑l=0

σlg′g(r)l∑

m=−l

Y m∗l (Ω)

∫dΩ′ Y m

l (Ω′)ψ(r, µ′).

(3.12)

In this problem, the angular flux ψ is not dependent in the ω coordinate referenced in figure

(7). Thus it can be shown that for m 6= 0,∫dΩ′ Y m

l (Ω′)ψ(r, µ′) =

∫dΩ′ Y m

l (Θ′, ω′)ψ(r, µ′) = 0 (3.13)

13

since eimω′

is integrated over its entire period. This leads to

∑g′

∞∑l=0

σlg′g(r)l∑

m=−l

Y m∗l (Ω)

∫dΩ′ Y m

l (Ω′)ψ(r, µ′) =∑g′

∞∑l=0

(2l + 1)Pl(µ)σlg′g(r)φlg′(r)

(3.14)

where

φlg(r) =

∫dµ′

2Pl(µ

′)ψg(r, µ′) (3.15)

since

Y 0l (Ω) =

√2l + 1 Pl(µ). (3.16)

The next step is to discretise equation (3.6) in terms of angle, or more specifically the

µ coordinate. The method[8] of using a Gaussian quadrature here will be optimal since

integrals over µ will have to be approximated. Note that since the integral∫dµf(µ) is

unweighted and defined over the interval [−1, 1], it can be approximated using the Gauss-

Legendre quadrature as∑

nwnf(µn) for some weightage wn > 0 normalised to∑

nwn = 2.

The quadrature rule allows the approximated integral to be exact for polynomials degree

(2N −1) or less since there are 2N parameters wn and µn. According to the quadrature

rule, the associated orthogonal polynomials are the Legendre polynomials, thus the µnwill satisfy

PN(µn) = 0. (3.17)

For the remaining N weightages, it is sufficient to demand that polynomials up to (N − 1)

be integrated correctly. A convenient choice of polynomials to check this condition can be

the Legendre polynomials themselves. This leads to the relation

1

2

∫dµ Pl(µ) =

1

2

∑n

wnPl(µn) = δlo for 0 6 l 6 N − 1 (3.18)

which can be used to determine the weightage set wn. It is also important to note that

due to the symmetry of the situation, the wn and µn have to satisfy

µN+1−n = −µn for 1 6 n 6N

2(3.19)

and

wN+1−n = wn for 1 6 n 6N

2(3.20)

14

which fortunately is indeed satisfied by the Gauss-Legendre quadrature.

Before taking approximation of the differential term, it is important to re-write the dif-

ferential operators in conservation form such that approximations will still retain particle

conservation. The conservation form is one that when integrated over all streaming angles,

satisfies the following neutron balance equation

~∇· ~Jg(~r)+σg(~r)φg(~r) = χgν

k

∑g′

σfg′(~r)φg′(~r)+∑g′

∫dΩ

∫dΩ′ σg′g(~r, Ω·Ω′)ψg′(~r, Ω′) (3.21)

which itself is equation (3.2) integrated over all angles. This means that the differential

operator has to be derived from its divergence form ~∇ · Ωψ. For this case, the conservation

form is

µ∂

∂r+

1− µ2

r

∂

∂µ=

µ

r2

∂

∂rr2 +

∂

∂µ

1− µ2

r. (3.22)

A simple way to approximate the angular differential term is then

∂

∂µ

1− µ2

rψg(r, µn) '

(1− µ2n+1/2)ψg(r, µn+1/2)− (1− µ2

n−1/2)ψg(r, µn−1/2)

(µn+1/2 − µn+1/2)r(3.23)

where half-integer indexes are introduced to aid the computation with the relation

ψg(r, µn) =1

2

[ψg(r, µn+1/2) + ψg(r, µn−1/2)

](3.24)

with µ1/2 = −1 and µN+1/2 = 1. However, when one considers the trivial solution in infinite

medium limit ψ = constant, which results in

Ω · ~∇ψg(r, µ) = 0 =⇒ µnr2

2r +µ2n−1/2 − µ2

n+1/2

(µn+1/2 − µn+1/2)r= 0 (3.25)

which implies condition µn =1

2(µn+1/2 +µn−1/2). However, this happens to be incompatible

with the µn determined from the Gauss-Legendre quadrature in equation (3.17). To rectify

this, angular differencing coefficients αn±1/2[8] is used to replace the µn±1/2 terms and the

weightages wn can be introduced into equation (3.23) such that

∂

∂µ

1− µ2

rψg(r, µn) ≈

2αn+1/2ψg(r, µn+1/2)− 2αn−1/2ψg(r, µn−1/2)

rwn. (3.26)

15

Comparing this with the case in equation (3.23), we expect that

2αn+1/2 ' 1− µ2n+1/2 (3.27)

which will be a consistency check for the replacement used. Then looking at the case of

infinite medium again, the condition from demanding Ω · ~∇ψ = 0 is now

µnr2

2r + 2αn+1/2 − αn−1/2

rwn= 0 =⇒ αn+1/2 = αn−1/2 − µnwn. (3.28)

For the resulting equation to satisfy the neutron balance equation, a further condition can

be obtained by just looking at the left hand side of equation (3.21)

~∇ · ~Jg(~r) + σg(~r)φg(~r) =1

r2

∂

∂rr2(∫

dΩ (r · Ω)ψg(~r))

+ σg(~r)(∫

dΩ ψg(~r))

=1

r2

∂

∂rr2(1

2

∫dµ µ ψg(r, µ)

)+ σg(r)

(1

2

∫dµ ψg(r, µ)

)≈ 1

r2

∂

∂rr2(1

2

∑n

wnµnψg(r, µn))

+ σg(r)(1

2

∑n

wnψg(r, µn)).

(3.29)

Another way to obtain the above is to discretise the left hand side of equation (3.6) and

integrate over all streaming angles as shown,∫dΩ[µ∂

∂r+

1− µ2

r

∂

∂µ+ σg(r)

]ψg(r, µ)

=1

2

∫dµ[ µr2

∂

∂rr2 +

∂

∂µ

1− µ2

r+ σg(r)

]ψg(r, µ)

≈ 1

2

∑n

wn

([µnr2

∂

∂rr2 +

∂

∂µ

1− µ2n

r+ σg(r)

]ψg(r, µn)

)≈ 1

r2

∂

∂rr2(1

2

∑n

wnµnψg(r, µn))

+∑n

(αn+1/2ψg(r, µn+1/2)− αn−1/2ψg(r, µn−1/2)

r

)+ σg(r)

(1

2

∑n

wnψg(r, µn)).

(3.30)

16

Comparing the above two equations give the condition∑n

αn+1/2ψg(r, µn+1/2)− αn−1/2ψg(r, µn−1/2) = 0

=⇒ α1/2ψg(r, µ1/2) = αN+1/2ψg(r, µN+1/2)

(3.31)

and simply setting α1/2 = αN+1/2 = 0 will ensure the above is satisfied for any solution

ψg(r, µn) thus also allowing the µn+1/2s to be arbitrarily determined. This claim can be

made due to the symmetry of the µn and wn from equations (3.19) and (3.20) which

gives

αN+1/2 = α1/2 −∑n

µnwn = α1/2 −N/2∑n=1

µnwn −N∑

n=N/2+1

µnwn = α1/2. (3.32)

The resulting k-eigenvalue equation is as follows,

µnr2

∂

∂rr2ψgn(r) +

2αn+1/2ψg,n+1/2(r)− 2αn−1/2ψg,n−1/2(r)

rwn+ σg(r)ψgn(r, n) =

χgν

2k

∑g′

σfg′(r)∑n′

wn′ψg′n′(r) +1

2

∑g′

N−1∑l=0

(2l + 1)Pl(µn)σlg′g(r)∑n′

wn′Pl(µn′)ψg′n′(r)

(3.33)

where ψgn(r) represents the computational solution for ψg(r, µn). Substituting equation

(3.24) into (3.33) to remove the n+ 1/2 indexes results in

µnr2

∂

∂rr2ψgn(r) +

4αn+1/2ψgn(r)− 2(αn+1/2 + αn−1/2)ψg,n−1/2(r)

rwn+ σg(r)ψgn(r, n) =

χgν

2k

∑g′

σfg′(r)∑n′

wn′ψgn′(r) +1

2

∑g′

N−1∑l=0

(2l + 1)Pl(µn)σlg′g(r)∑n′

wn′Pl(µn′)ψg′n′(r)

(3.34)

Since the k-eigenvalue equation does not contain any integrals over space, the spatial dis-

cretisation can be done simply by integrating equation (3.34) over an incremental spherical

shell

Vi = 4π

∫ ri+1/2

ri−1/2

dr r2. (3.35)

17

The discretised angular flux is then taken as

ψgni =4π

Vi

∫ ri+1/2

ri−1/2

dr r2ψgn(r) (3.36)

which is simply the average over the corresponding shell. Another approximation

4π

∫ ri+1/2

ri−1/2

dr rψgn(r) ≈ 4π

∫ ri+1/2

ri−1/2

dr rψgni =1

2(Ai+1/2 − Ai+1/2)ψgni (3.37)

where Ai = 4πr2i is well justified if ri+1/2 − ri−1/2 is small and finally the integration of the

spatial derivative term will be

4π

∫ ri+1/2

ri−1/2

dr r2µnr2

∂

∂rr2ψgn(r) = 4π

∫ ri+1/2

ri−1/2

dr(µnr

2 ∂

∂rψgn(r) + 2µnrψgn(r)

)= 4πµnr

2ψgn(r)∣∣∣ri+1/2

ri−1/2

+ 4π

∫ ri+1/2

ri−1/2

dr(

2µnrψgn(r)− 2µnrψgn(r))

= µn(Ai+1/2ψgn,i+1/2 − Ai−1/2ψgn,i−1/2).

(3.38)

Then as with the angular case,

ψgni =1

2(ψgn,i+1/2 + ψgn,i−1/2) (3.39)

will relate the integer indexes to the half integer indexes. Note that similar to the angular

case r1/2 = 0 and rI+1/2 = R is taken, where R is the radius of the U-235 ball. Taking the

integral using the relations defined above and then dividing by Vi results in the equation

µnVi

(Ai+1/2ψgn,i+1/2 − Ai−1/2ψgn,i−1/2) +(Ai+1/2 − Ai+1/2)

Viwn

(2αn+1/2ψgni − (αn+1/2 + αn−1/2)ψg,n−1/2,i

)+σgiψgni

j+1

=χg

ν

2k

∑g′

σfg′i∑n′

wn′ψg′n′i +1

2

∑g′

N−1∑l=0

(2l + 1)Pl(µn)σlg′gi∑n′

wn′Pl(µn′)ψg′n′i

j(3.40)

where j is the iteration index. This method is known as iteration over source since only

the source terms are from the previous iteration. Now that the set of discretised equations

are defined, the boundary conditions for the problem can be introduced. First is due to the

18

symmetry at the center of the ball, which is most frequently approximated by

ψg,N+1−n,1/2 = ψgn,1/2 for 1 6 n 6N

2. (3.41)

Also since the U-235 ball is taken to be in vacuum, another boundary condition is

ψgn,I+1/2 = 0 for 1 6 n 6N

2. (3.42)

Now from equations (3.24), (3.39) and (3.40) and the boundary conditions, there are a total

of 3NI + N equations within a group g while there are a total of 3NI + I + N unknowns

since all ψg,n+1/2,i+1/2 are not solved for. To account for the lack of I equations, one method

is to extend the boundary condition (3.42),

ψg,1/2,I+1/2 = 0 (3.43)

since equation (3.40) solves from µn−1/2 to µn. Then another equation needs to be introduced

to link ψg,1/2,i+1/2 to ψg,1/2,i. This is done by attempting to extend equation (3.40) to µ1/2.

Conveniently, at µ1/2 = −1,∂

∂µ

1− µ2

rψ =

2µ

rψ, thus no approximation of the angular

derivative is required. Taking integral then gives

4π

∫ ri+1/2

ri−1/2

dr r2 2µ1/2

rψg,1/2(r) = −(Ai+1/2ψg,1/2,i+1/2 − Ai−1/2ψg,1/2,i−1/2) (3.44)

thus resulting in−2

Vi(Ai+1/2ψg,1/2,i+1/2 − Ai−1/2ψg,1/2,i−1/2) + σgiψg,1/2,i

j+1

= Sjgni (3.45)

where Sjgni is simply the right hand side of equation (3.40). Further substituting in equation

(3.39) gives 2

Vi

(2Ai−1/2ψg,1/2,i − (Ai+1/2 + Ai−1/2)ψg,1/2,i+1/2

)+ σgiψg,1/2,i

j+1

= Sjgni. (3.46)

While (3.46) provides I equations as required, they are not useful unless the rest of ψg,1/2,i+1/2

are known, therefore giving an additional I equations to be solved. Fortunately, since

19

ψg,1/2,I+1/2 is known, a simple extension of equation (3.39),

ψg,1/2,i =1

2(ψg,1/2,i+1/2 + ψg,1/2,i−1/2) (3.47)

will ensure that all ψg,1/2,i and ψg,1/2,i+1/2 can be solved. Overall, this process created 2I

equations and also an additional I unknowns, not counting equation (3.43), thus exactly

balancing the total equations and unknowns to be solved for.

3.2 Algorithm for solving equations

The main idea in solving for all the ψgni is to start from ψg,1/2,I+1/2 and ψgn,I+1/2 for n 6N

2and then solve for increasing µ and decreasing r in half integer steps. This requires the

substitution of equation (3.39) into (3.40) which gives−µnVi

(2Ai−1/2ψgni − (Ai+1/2 + Ai−1/2)ψgn,i+1/2)

+(Ai+1/2 − Ai+1/2)

Viwn

(2αn+1/2ψgni − (αn+1/2 + αn−1/2)ψg,n−1/2,i

)+ σgiψgni

j+1

= Sjgni

(3.48)

and then solving for ψj+1gni gives

ψj+1gni =

1

−2µnAi−1/2 +2(Ai+1/2−Ai−1/2)

wnαn+1/2 + Viσgi

(1

wn(Ai+1/2 − Ai−1/2)(αn+1/2 + αn−1/2)ψj+1

g,n−1/2,i

− µn(Ai+1/2 + Ai−1/2)ψj+1gn,i+1/2 + ViS

jgni

).

(3.49)

The same is done for its µ1/2 case yielding

ψj+1g,1/2,i =

1

4Ai−1/2 + Viσgi

(2(Ai+1/2 + Ai−1/2)ψj+1

g,1/2,i+1/2 + ViSjgni

). (3.50)

Figure (8) shows the direction in which the solutions can be obtained in the discretised

grid, the bold arrows indicate equations (3.49) and (3.50) while the dotted arrows indicate

equations (3.39) and (3.33), which are also known as the diamond difference relations. Note

that some of the points have two bold arrows leading to it, this is because equation (3.49)

20

Figure 8: Illustration of the direction in which each of the discretised points of µ and r are

solved for n 6N

2in the case of I = 3 and N = 4

requires two input points from the grid. Now for the case of n >N

2, boundary condition

(3.41) can be used to obtain ψgn,1/2, which can then be solved in increasing µ and increasing

r for the rest of the ψ values. So now the system will be solved from ri−1/2 to ri thus the

diamond difference relation (3.39) have to be subsituted again into equation (3.40) to obtain

ψj+1gni =

1

2µnAi+1/2 +2(Ai+1/2−Ai−1/2)

wnαn+1/2 + Viσgi

(1

wn(Ai+1/2 − Ai−1/2)(αn+1/2 + αn−1/2)ψj+1

g,n−1/2,i

+ µn(Ai+1/2 + Ai−1/2)ψj+1gn,i−1/2 + ViS

jgni

).

(3.51)

The way to solve the upper half of the grid is illustrated in figure (9). This process can be

done seperately for each of the energy groups to obtain the full set of solutions.

21

Figure 9: Illustration of the direction for solving the remaining points of the solution grid inthe same case as figure (8)

3.3 Meaning behind the iteration process

This entire iterative process can be seen as repeatedly applying a matrix M into a state

vector Ψ where

Ψ =

ψT,1,1...

ψT,N,I......

ψF,N,I

(3.52)

and M can be seen as the discretised form of the operator

(Ω · ~∇+ σt)−1(F + S) (3.53)

where the operators F and S are

Fψ(~r, Ω, E, t) = χ(E)

∫dE ′ νσf (~r, E)

∫dΩ′ ψ(~r, Ω′, E ′, t) (3.54)

and

Sψ(~r, Ω, E, t) =

∫dE ′

∫dΩ′ σs(~r, E

′ → E, Ω · Ω′)ψ(~r, Ω′, E ′, t). (3.55)

22

Putting the iteration index in we get

Ψj+1 = MΨj. (3.56)

To illustrate what happens after many iterations, we first break down the initial condition

Ψ0 into normalised eigenvectors ψk of the matrix M as

Ψ0 =∑k

Ckψk (3.57)

which leads to

Ψj =∑k

(λk)jCkψk. (3.58)

Then suppose that the eigenvector index k is ordered such that λn > λm if n < m. So if the

iteration index j is large enough, we have (λ1)jC1 (λ2)jC2 which gives

Ψj ≈ (λ1)jC1ψ1. (3.59)

Should this λ1 be simply equal to 1, it would mean that

(Ω · ~∇+ σt)ψ1 = (F + S)ψ1 (3.60)

thus ψ1 will be the solution to the time independent neutron transport equation that we

have been seeking and Ψj would naturally converge to it. So as we can see, as long as Ψj

is chosen such that it has a component along ψ1, the algorithm will naturally output the

correct solution.

Obtaining λ1 6= 1 is also useful as that would mean

(Ω · ~∇+ σt)λ1ψ1 = (F + S)ψ1 (3.61)

which translates to∂

∂tψ1 = (λ1 − 1)(Ω · ~∇+ σt)ψ1. (3.62)

Given that σt and ψ1 are positive definite, and that Ω · ~∇ is a neutron conserving term, the

overall sign of∂

∂tψ1 should be decided by the (λ1−1) factor. Thus we can see that the value

of λ1 will also tell us if the system is subcritical or supercritical, allowing us to adjust the

23

parameters accordingly.

3.4 Determining input cross section values and energy groups

Determining the cross section values could prove to be a difficult task as there may not be

well-established theories on the energy and angular dependence of the cross sections. Instead,

these values are determined experimentally and a collection of such data is made publicly

available by the International Atomic Energy Agency (IAEA)[7]. Each set of data provides

roughly 10000 data points across 13 orders of magnitudes of energy in terms of eV. Assigning

an energy group to each data point could prove to be rather computational exhausting,

thus one will typically have to construct broader energy groups based on observations or

assumptions and then construct the group cross section based on the assumed in-group

energy spectrum if required based on equation (3.4). This work will use rough values for the

cross sections as the aim of this project is to study the possible methods in which calculations

regarding fissionable materials can be done. Thus being able to obtain a decent estimate

will good enough as further improvements to the results can always be come by taking finer

discretisations.

3.4.1 Angular dependence of scattering cross sections

For simplicity’s sake the angular dependence of σg′g(r, µr) will be determined using classical

hard sphere scattering throughout this work. It is safe to assume that the impact point of

the neutron is uniformly distributed over the scattering cross section as shown in figure (10).

Hard sphere scattering with the approximation mM gives the relation

db = −1

2Rn sin

(Θs

2

)dΘs (3.63)

where Θs = cos−1(µr) and Rn is the radius of the nucleus. The distribution of incoming neu-

trons on the impact parameter is2πb

πR2n

db, giving the corresponding distribution to be just

1

2dµr. So assuming that σg′g(r) is known, σg′g(r, µr) can be obtained by distributing evenly

over the cosine of the scattering angle µr. The fact that this distribution is uniform also has

implications on the σlg′g(r) as the identity from equation (3.18), together with the definition

of σlg′g(r) in equation (3.9) would mean that σlg′g(r) = δl0 σg′g(r), thus greatly simplifying

the expression of Sgni. So in situations where there is only heavy nuclei (like uranium), it

is sufficient to use this simple expression for the cross section, thus making calculations faster.

24

Figure 10: Visualisation of the scattering cross section

For situations with light nuclei, a more accurate expression for σs(µ) dµ will need to be

found without considering the infinite mass limit. This can be done by looking at the fol-

lowing two identities[9]

σs(θ) = σs(θCM)×

(mM

cos θ +√

1− m2

M2 sin2 θ)2

√1− m2

M2 sin2 θ(3.64)

and

θCM = sin−1(mM

sin θ)

+ θ (3.65)

where the cross section in the CM frame is just the cross section from the infinite mass limit.

So the scattering cross section in lab frame turns out to be

σs(µ) dµ = σs(θ)dθ

dµdµ

= sin(

sin−1(mM

√1− µ2

)+ cos−1 µ

)×

(mMµ+

√1− m2

M2 (1− µ2))2

√1− m2

M2 (1− µ2)

× 1

−√

1− µ2dµ

(3.66)

25

up to a constant factor such that

∫dµ

2σs(µ) = σs. This can then be incorporated in the

discrete ordinate scheme by obtaining the scattering moments

σlg′g(r) =

∫dµ

2Pl(µ)σg′g(r, µ). (3.67)

3.4.2 Group-to-group scattering

To even begin determining group-to-group scattering cross sections, we must look at the

energy dependence of the scattering, again using the hard-sphere model for the target nu-

cleus. The details of this calculation will be left in Appendix C but the final result is that

for(M −m)2

(M +m)2E 6 E ′ 6 E, we have

p(E → E ′) =(M +m)2

4mMEπR2

n. (3.68)

Thus we can see that the cross section will simply have1

Edependence. So after normalising,

we get

σs(E → E ′) dE ′ =σs(E)

(1− α)EdE ′ for αE 6 E ′ 6 E (3.69)

where α =(A− 1

A+ 1

)2

for collision with nucleus of nucleon number A. Handling group-to-

group scattering can be tricky, as the respective cross sections also depends on the deflection

angle of the scattering event. This is more so when the energy groups are broad, as will be

the case when the three-group approximation is used. The reason is that the algorithm will

be insensitive to energy losses from within-group scattering. A more rigid approach would

be to use Monte-Carlo methods to simulate the accurate down-scatter probabilities given

that fission neutrons are generated with a specific distribution. For the case of this project,

an energy distribution of the neutrons will need to be assumed in order for there to be a

quick way of determining the approximate values for these cross sections. Also, to further

simplify matters, the down-scatter probability will not be coupled with the angular part of

the cross section. To put in simply, we have

σg′g = σsg′

∫g′dE

φg′(E)

v(E)

Eg − αE(1− α)E

for Eg 6 E 6Egα

(3.70)

where φg′(E) is assumed.

26

4 Initial results

4.1 Critical mass of pure U-235 sphere

In this example, only the fast energy group will be used, since that is where fission neutrons

are generated. Furthermore, neutrons will only lose up to 1.7% of their energy per collision,

which together with the fact thatσaσt≈ 0.17 in this range, means neutrons are unlikely to

lose much energy before being either absorbed or simply leaking out of the material. This

justifies using only the fast energy group since φ(E) will then not deviate very far from χ(E).

Using parameters from [5], the critical radius obtained for the pure U-235 sphere is 8.09 ±0.01cm, compared to 8.70cm obtained from diffusion approximation. In terms of mass this

will be 42.4kg compared to 52.7kg from diffusion theory. In contrast, the critical mass by the

European Nuclear Society[10] (ENS) reads 46.7kg. Not much conclusion can be drawn from

the ENS value as it is not stated how their value was obtained. However, what can be said

is that the three values are in good agreement with the highest percentage difference being

21.6%. What is more convincing is that the solution for φ(r) in both cases agree very well

with each other as shown in figure (11). Note that neutron flux plots throughout this work

obtained in time-independent settings like this have arbitrary units, since the total number

of neutrons will necessarily be the same as in the initial input flux. Figure (12) shows the

convergence of the solution through the iteration process as ψ1 starts to dominate over all

other eigenvectors. This plot is important also as it shows that the algorithm did indeed

converge in the desired manner. The algorithm is also capable of giving us plots of the an-

gular neutron flux like the ones shown in figure (14) and figure (13) for this example. This

is something that the simple diffusion approximation will not be capable of. In spherically

symmetric situations, this is most useful in comparing the outward and inward fluxes. In

this situation the boundary condition can be clearly seen in the 3D plot while the greater

outward flux compared to the inward flux is more clearly seen in the contour plot.

27

Figure 11: Plot of φ(r) solution obtained (lower curve) in comparison with solution fromdiffusion approximation (upper curve).

Figure 12: Plot of ratio of total neutrons in current iteration in comparison to previousgeneration, showing convergence of solution.

28

Figure 13: Contour plot of angular flux solution.

Figure 14: 3D plot of angular flux solution.

4.2 Critical mass of enriched uranium sphere

The critical mass of a homogeneous sphere of uranium enriched to arbitrary levels is obtained.

In this case, it is appropriate to model it with at least two distinct energy groups, due to

29

the existence of a threshold fission energy in U-238. It is typically agreed that this threshold

energy lies at around 1MeV and this can also be clearly seen in figure (15). As follows

Figure 15: Plot of σf (E) for U-238 in log scale, showing the threshold energy at around1MeV[7].

from the case of pure U-235, only the fast energy group needs to be considered, but the

difference here is that it will have to be split into two at 1MeV due to the threshold property

of the cross section. The two isotopes have similar neutron yield per fission, 2.536 for U-235

and 2.819 for U238, and similar fission neutron spectrum as shown in figure (16). In fact,

given the two-energy groups taken here, 70.0% of neutrons generated for U-235 have energy

greater than 1MeV while for U-238 it is 69.5%. This means that we can simply use the same

parameters as the pure U-235 case for ν and χ(E). The results obtained can be summarised

in the figure (17). Note that as the enrichment approaches 0, the critical radius is diverging.

This is the expected result as pure U-238 is known to be unable to reach criticality. In fact,

natural uranium (0.7% enrichment) is also known to be unable to reach criticality and it has

been verified using this method up to a sphere of 10m in radius, equivalent to 80000 tonnes

of uranium!

30

Figure 16: Plot of fission neutron spectrum for U-238 in comparison with U-235. The uppercurve is for U-238[7].

Figure 17: Plot of critical radius versus enrichment of uranium

4.3 Temperature profile of water-cooled uranium sphere

In this section, we look at the case of a tampered fissile material. The configuration chosen

here is the pure U-235 immmersed in flowing, pressurised water to draw some similarities

31

with the water-cooled fuel in Pressurised Water Reactors (PWR) that are common nowadays.

Pure uranium is used here even though uranium dioxide is what is usually used as fuel so as

to make comparisons with the previously obtained results. Since this is a moderated system,

using the threee energy groups mentioned at the end of Chapter 2 will be appropriate. Figure

Figure 18: Plot critical radius of uranium vs thickness of water tamper.

(18) shows the expected trend of decreasing critical radius as more tamper is used. The next

two figures show the neutron flux and the temperature profile taken for the case of 5cm of

tamper. Note the presence of a slight bump in the flux near the interface, this is likely due

to the better downscattering effect of the tamper layer, which feeds thermal neutrons back

into the core, giving rise more fission in the outer layer of the core. Indeed figure (21) shows

the incoming thermal neutrons generated in the moderation layer. This effect is interesting

to note as we could also expect it to happen for other geometries given its mechanism. This

shows that one may not even need to use the appropriate geometry in initial studies of

certain situations as the effects may already be present in the spherical geometry.

32

Figure 19: Plot of scalar flux versus radius in water moderated U-235 sphere.

Figure 20: Close up of bump in neutron flux near core-moderator interface.

33

Figure 21: Plot of angular flux of thermal neutrons in water moderated uranium sphere.Core radius here is 6.98cm.

Figure 22: Temperature profile of water moderate uranium sphere.

34

Figure (22) shows the temperature profile taken for only the uranium core. This is done

by solving the time independent heat equation with the energy generated from the fission

events as the source term. The flowing water surrounding the core is taken to be at 300C[11],

typical in PWR reactors, which gives the boundary condition needed. The exact method for

this is included in Appendix B. The specific value of the temperature here is not important

since it can be adjusted by changing the total number of neutrons in the system. The

important part is that we can see that the temperature profile is very similar in shape to

the neutron flux, which is not surprising and can be useful in making quick evaluations of

situations. Being able to take the temperature profile like this can also be very useful for

full simulations of reactors.

5 Possible Applications

From its ability to reproduce the correct results in a few simple configurations, we should

now be confident that the algorithm is working as we intended it to. With that, we can start

to use it to analyse interesting situations like for example the yield of nuclear weapons or

designs of new and upcoming Generation IV nuclear reactors. The purpose of this section

is then to illustrate how the algorithm developed in Chapter 3 can be applied in the study

of such situations. Initial results will also be shown as demonstration but further analysis

will not be made here as it is not within the scope of the project to study these individual

situations in detail.

5.1 Estimating efficiency of uranium bomb

5.1.1 Theory

The purpose of this section is to illustrate how the computational solution to the time-

independent neutron transport equation can also be used to tackle a time dependent problem

like the detonation of a uranium bomb. This is a faster alternative to solving the full time

dependent equation, which will involve( 1

v∆t+ Ω · ~∇+ σt + F + S

)ψ(ti + ∆t) =

1

v∆tψ(t). (5.1)

35

We can see here that we will need to derive the discretised form of the inverse operator( 1

v∆t+ Ω · ~∇+ σt + F + S

)−1

(5.2)

which could involve significant amount of work.

In this case, the efficiency of a spherical uranium bomb with tamper is estimated. First

begin by looking at the overall picture of the detonation process. The bomb is driven by

energy from the fissioning of the uranium nuclei, which is to be sustained as a chain reac-

tion mediated by neutrons. This bomb would start as a supercritical assembly of fissionable

material which would have positive neutron multiplicity, allowing the neutron flux and thus

the fission rate to build up with time. While this is happening, the energy released from

the fission events will also cause the bomb to expand. This effectively decreases the fission

cross sections by lowering the density of the material, causing the neutron multiplicity to

decrease. Eventually, the bomb will expand to the point where it is no longer critical and

the fission would then die down. Preliminary calculations have determined that the bomb

in general will not be able to fission all of its material before it becomes subcritical, thus

it is of interest to estimate the actual efficiency of uranium bombs in different configurations.

With the concept of neutron multiplicity, it is natural to assume the time dependence of

the neutron population in the material to be

N(t) = N0eαt (5.3)

for some effective multiplicity factor α. In terms of the neutron flux, the assumption would

be

ψ(~r, Ω, E, t) = ψ0(~r, Ω, E)eαt. (5.4)

Now it is important to note that in actual fact, this α factor would change with time due to

the expansion of the material causing the neutron multiplicity to decrease, thus using α(t)

would be more appropriate. However, if we are to only consider a very small time interval,

this α can be taken as a constant and the whole calculation would then consist of evaluating

the situation at each of these small time intervals.

36

Inserting the approximation back into the neutron transport equation would yield

1

v

∂

∂tψ(~r, Ω, E, t) =

α

vψ(~r, Ω, E, t) =

(− Ω · ~∇− σt(~r, E) + F + S

)ψ(~r, Ω, E, t). (5.5)

In order to proceed, we now bring in the computational algorithm, which is able to solve for(− Ω · ~∇− σt(~r, E) + F + S

)ψ(~r, Ω, E, t) = 0. (5.6)

Simply introduce an additional factor κ which can be seen as an adjustment to the value of

σt(~r, E) such that the equation to solve is(− Ω · ~∇− σt(~r, E)− κ+ F + S

)ψ(~r, Ω, E, t) = 0. (5.7)

Then should a solution be found to this equation, it will satisfy

1

v

∂

∂tψ(~r, Ω, E, t) =

α

vψ(~r, Ω, E, t) = κψ(~r, Ω, E, t) (5.8)

and the expression

ψ(~r, Ω, E, t) = ψ0(~r, Ω, E)eκvt (5.9)

will be valid for the small time interval ∆t as taken in the approximation. We then have at

each time step ti,

ψ(~r, Ω, E, ti + ∆t) = ψ(~r, Ω, E, ti)exp(κiv∆t) (5.10)

and the total number of fissions from ti to ti + ∆t is given by

fi(~r, ti) =

∫dE

∫ ∆t

0

dt′∫dΩ σf (~r, E)ψ(~r, Ω, E, ti)exp(κivt

′)

=1

κivσf (~r)φ(~r, ti)

(exp(κiv∆t)− 1

) (5.11)

and the total energy released up to this time will be

U(~r, ti) =i∑

j=0

ξffi(~r, tj) (5.12)

where ξf is simply the average energy released per fission event. Now to determine how

this energy that is released contributes to the expansion of the material, we look at the

37

following picture[12]. Consider the fact that per kilogram of weapons-grade uranium (> 80%

enrichment), the potential yield is about 3.28× 1011J while the energy required to bring the

same amount of uranium to boiling point is only of the order 106J. Thus it is appropriate

to model the uranium sphere as a gas and to simplify calculation, we can assume that

all uranium gas particles are expanding outwards at the same speed u(t). This is an okay

assumption since for spherical geometry we expect most of the fission to happen in the center,

thus most of the pressure pushing the material outward will originate near the center. This

means that we can assume the entire sphere is being pushed outward by pressure from the

center point, thus all layers of the sphere would indeed expand at the same rate. The rate of

change of this expansion is then determined by the total pressure exerted by the fission energy

released, which for the case of non-relativistic gas pressure, should be P (ti) =2

3

∫d3~r U(~r, ti).

But for per particle energies greater than around 2keV, radiation pressure arising from

photons emitted by the deceleration of fission fragments should dominate, as should be the

case here with around 200MeV released per fission event. So the pressure should instead be

P (~r, ti) =1

3

∫d3~r U(~r, ti). (5.13)

To show that radiation pressure dominates one simply need to express the respective pres-

sures in the form

Pcl =NU

VkBT (5.14)

and

Prad =( 8π5k4

B

45c3h3

)T 4 (5.15)

and then find the respective per-particle energy E =3

2kBT such that Prad > Pcl.

The work-energy theorem then gives the following relation of the total pressure to the out-

ward expansion speed of the material

P (t)dV

dt=dKE

dt= Mu

du

dt=

4

3πR2dR

dt

∫d3~r U(~r, t). (5.16)

38

Noting thatdR

dt= u(t) and taking the time discretised form of the above equation yields

u(ti + ∆t) = u(ti) +(4πR2P (ti)

M

)∆t (5.17)

and the radius of the expanding sphere at each time step is then

R(ti + ∆t) = R(ti) + u(ti + ∆t)∆t. (5.18)

This updated radius will then be subsequently used to update the cross sections which will

decrease by a factor of( R(ti)

R(ti + ∆t)

)3

and with this, calculations can begin for the growth

factor κv for the next time step.

At some point in time, system will expand to the point that it is no longer critical, with

κv < 0. The fission will then start dying down but not completely stop at this point, thus

it will be safer to continue running the process till the fission rate drops below a reasonable

threshold (say 0.1% of the peak fission rate). At the end the total number of fissions is then

compared to the amount of uranium available at the beginning to obtain the efficiency of

the bomb.

A final point to discuss will be the initial condition, that is the flux at t = 0, to be inserted

into the process. In fact, this calculation should be insensitive to the initial conditions.

Consider two different cases with N0 = 1 in comparison to N0 = 1000, the first case will

reach the same number of neutrons as the second case after aboutlog(1000)

log ν≈ 8 generations

with the release of roughly1000

ν

σtσt − σa

× 200MeV ≈ 1.5× 10−8J of energy, which will have

negligible contribution to the expansion. So it turns out that the first case will evolve exactly

into the second case, thus one should expect that both cases yield the same efficiency only

that the first case will take ∼ 8 generations longer in time to detonate.

For the case of the uranium sphere covered with a tamper material, the calculation will

have a small difference, since heat is not directly generated in the tamper, it is not reason-

able to expect the tamper material to also undergo thermal expansion but instead simply

be pushed outwards by the expanding core. For ease of calculation, we assume that the

tamper is at least in a melted state. which will be smeared out on the surface of the core as

39

it expands. So given that the tamper itself does not expand, its thickness at each step can

be obtained after the new core radius is calculated as per above, which gives

Rtamp = 3

√R3init −R3

c,init +R3c −Rc. (5.19)

5.1.2 Results

Using the aforementioned method, we aim to give an estimate of the yield of the ’Little

Boy’ weapon used during World War II while identifying any interesting dynamics of the

detonation process. Given the weapon design [13][14], it is reasonable to approximate in this

case as a sphere containing 64kg of uranium enriched to 80% encased with a 9cm layer of

Tungsten Carbide as tamper. Only one energy group will be used here, similar pure U-235

case. This is because there is only 20% U-238 in the core, thus it the bomb will be mainly

running on U-235 fission, making it unnecessary to split the energy group for U-238. Cross

sections from only the fission energy group will be used since the time-scale of the detonation

is too short for neutrons to be slowed significantly. The time step used here will have to be

less than the growth parameter κv for accuracy, which is of order 10−7 in this case. For this

case, 10−8s is used as further decreasing it 10−9s does not significantly change the results

but would render the calculation much slower.

Given these parameters, the yield of the bomb is calculated to be 2.06% which is equiv-

alent to 23kt of TNT. This is actually 53% higher than the actual reported yield of 15kt[15]

but nevertheless as just an estimate, it is a decent result. The yield obtained here also means

that assumptions taken regarding the physical state of the uranium core and the dominant

form of pressure in the theory are justified, which is important.

The dynamics of the detonation is also interesting to look at. As seen in figure (23), most

of the fission events and thus the energy release comes from just the final 0.2µs of the det-

onation. The rest of the time was just used to build the fission rate to the point where it

becomes significant. Thus we can see here that it is very important for the tamper of the

bomb to hold on to the core for as long as possible since every bit of time towards the end

will contribute greatly to the yield. This effect can also be seen from figure (24) where the

expansion of the core is tracked, again emphasizing the significance of the last 0.2µs. With

this, it will be interesting to then investigate the actual effect varying the tamper mass on the

bomb yield. This will not be done in this project however, as it is not our aim to investigate

40

Figure 23: Plot of fission rate versus time

Figure 24: Plot uranium core radius versus time

41

the fission bomb in detail.

5.2 Pebble bed reactor

5.2.1 Theory

With the call for safer and more efficient reactor designs, the pebble reactor is one of the

newer Generation IV designs that is being developed. The reactor core consists of a gas-

cooled bed laden with fuel pebbles as shown in the figure below.

Figure 25: Schematic for a pebble bed reactor developed by MIT[16].

As can be seen from figure (26), the pebbles are simply thrown into the core without any

42

Figure 26: Image of the core of an actual pebble bed reactor.

specific configuration in mind. The basic idea is that each pebble alone will not be enough

to go critical, but will be when large numbers of them are assembled together as shown.

This also allows for easy maintenance as individual pebbles can easily be swapped out dur-

ing operation. Each pebble also in turn consists of more than 10000 microspheres, which

contain the uranium-dioxide fuel, embedded in graphite, which acts as moderation for the

fuel. Each microsphere has a layered structure as shown in figure (27), which is perfect for

analysis using our code that is developed for spherical geometries.

We will analyse the reactor by looking at just a single microsphere. This microsphere will be

surrounded by many other identical microspheres and the ensemble itself is also surrounded

by many other identical fuel pebbles. Thus we can see the situation as each single micro-

sphere being put in a constant bath of neutron flux, which is what allows the fuel inside to

be able to go critical despite not being of critical mass. So the boundary condition in this

case will be set as

ψ(R, µ) = const for µ < 0. (5.20)

Since there is an external source in this situation, a time-independent solution to the neutron

transport equation is guaranteed. The computational algorithm can then be used to find

43

Figure 27: Cross sectional diagram of each fuel pebble and the microspheres embeddedwithin.

this time-independent solution from which the leakage can be obtained as

ψout =

∫ 1

0

dµ ψ(R, µ) (5.21)

and then subsequently compared to the incoming flux

ψin =

∫ 0

−1

dµ ψ(R, µ) (5.22)

which is specified at the beginning of the problem to determine if the dimensions of the

microsphere are sufficient to sustain the constant bath imposed.

Another benefit of the design is that due to how spread out the fuel pieces are, the flux

and thus the temperature throughout the reactor is expected to be very even. This is ideal

as with the lack of hot-spots, the design will be less prone to melting down in an emergency

situation.

44

5.2.2 Results

Since this is a reactor situation with moderator included to slow the neutrons down to ther-

mal range, the usual three energy groups and their respective averaged cross sections will

be used. Ideally the fast group should also be split into two here to account for the U-238

given the low enrichment of the fuel, but since the main purpose of this is to illustrate the

method, using only three groups should be good enough. The microspheres are taken to be

TRISO particles [17]

Figure (28) shows the flux within each micropebble. As expected, the flux is higher at

the center as that is where the fission is concentrated but also evens out towards the outside

due to the presence of incoming neutrons. Also, the overall deviation between the flux is

Figure 28: Plot of scalar flux against radius for individual micropebble

also very small, within 5%, which should produce a more even temperature profile due to

the direct relationship between temperature and flux we saw in the water moderated sphere

example. This will indeed achieve the goal of the design for a smoother temperature profile.

The plot for the neutron flux is obtained by assuming an equal amount of incoming flux

in each of the energy groups. Each micropebble will in this case generate about 0.13% of

45

the incoming flux, with the outgoing flux having the proportions

φout,Fφin,F

= 1.0141

φout,Iφin,I

= 0.9964

φout,Tφin,T

= 0.9934.

(5.23)

The next step could be then to determine if the moderation surrounding each micropeb-

ble to slow the neutrons down to its original energy distribution. Also if the overall ratio

φF : φI : φT in the fuel pebble could be found, then the incoming flux is already fixed and

further analysis should be straightforward.

An interesting observation made while changing around the incoming flux ratios is thatφout,Tφin,T

do not seem to be very sensitive to adjustments to the ratio of φin,T with respect to

the other two energy groups. For example, setting φin,F : φin,I : φin,T = 1 : 1 : 10 gives

φout,Fφin,F

= 1.1353

φout,Iφin,I

= 0.9964

φout,Tφin,T

= 0.9932

(5.24)

which is only about 0.02% change inφout,Tφin,T

. What changes significantly, however, is the flux

generated by the sphere, which now increases to 0.533%.

Increasing the incoming thermal flux yet again to φin,F : φin,I : φin,T = 1 : 1 : 100 gives

similar effect on outgoing flux ratios but only increased the generated flux to 0.65%. Using

φin,F : φin,I : φin,T = 1 : 1 : 1000 again gives similar results. This could be indication that

neutron moderation should be designed to effective up till a certain threshold, after which

the returns would be minimal.

46

6 Conclusion

We have developed a working algorithm for solving the time independent form of the neu-

tron transport equation, which allows us to investigate steady state situations in fissionable

materials. The equation in principle, can be solved as accurately as we want it to since the

approximations taken are based on taking discretisations. So it is up to ones skill to be able

to determine how finely to discretise in order to obtain results of desired accuracy without

sacrificing too much computation time. Throughout this project, the discretisation used was

mostly rough, especially in the energy variable since the aim of the project is mainly illus-

tration. After writing out the code on MATLAB, it is tested on several toy model situations

like the pure uranium sphere in Chapter 4. The results obtained were satisfactory and in

line with known estimates like the diffusion approximation. In Chapter 5, possible studies

that can be done with this code are suggested with some initial results presented to illustrate

how the study can proceed. These initial results may not have the desired accuracy for a

proper study in each case but the method has be presented and it is up to future work to

improve on them if desired.

47

References

[1] U.S. Energy Information Administration. Table 3. annual spent fuel discharges and bur-

nup, 1968 - 2002. http://www.eia.gov/nuclear/spent_fuel/ussnftab3.htm, Octo-

ber 2004.

[2] Juliya Fisher. Energy density of coal. http://hypertextbook.com/facts/2003/

JuliyaFisher.shtml, 2003.

[3] E. E. Lewis and Jr. W. F. Miller. Computational Methods of Neutron Transport. John

Wiley & Sons, Inc., 1984.

[4] R Nave. Nuclear binding energy. http://hyperphysics.phy-astr.gsu.edu/hbase/

nucene/nucbin.html.

[5] Elmer E. Lewis. Fundamentals of Nuclear Reactor Physics. Academic Press, 2008.

[6] N. Kornilov, F. J. Hambsch, I. Fabry, S. Oberstedt, T. Belgya, Z. Kis, L. Szentmiklosi,

and S. Simakov. The 235u(n, f) prompt fission neutron spectrum at 100k input neu-

tron energy. Nuclear Science and Engineering: The Journal of The American Nuclear

Society, August 2010.

[7] International Atomic Energy Agency. Evaluated nuclear data files endf. https://

www-nds.iaea.org/exfor/endf.htm.

[8] B. G. Carlson and K. D. Lathrop. Transport theory - the method of discrete ordinates.

In H. Greenspan, C. N. Kelber, and D. Okrent, editors, Computing Methods in Reactor

Physics. Gordon and Breach, New York, 1968.

[9] Stephen T. Thorton and Jerry B. Marion. Classical Dynamics of Particles and Systems.

Brooks/Cole Cengage Learning, 2008.

[10] European Nuclear Society. Critical mass. https://www.euronuclear.org/info/

encyclopedia/criticalmass.htm.

[11] Jacopo Buongiorno. Pwr description. Engineering of Nuclear Systems Lecture, 2010.

[12] B. Cameroon Reed. Arthur compton’s 1941 report on explosive fission of u-235: A look

at the physics. American Journal of Physics, 2007.

48

[13] Bruce Cameron Reed. The Physics of the Manhattan Project. Springer-Verlag Berlin

Heidelberg, 2011.

[14] John Coster-Mullen. Atom Bombs: The Top Secret Inside Story of Little Boy and Fat

Man. John Coster-Mullen, 2002.

[15] John Malik. The yields of the hiroshima and nagasaki nuclear explosions, 1985.

[16] Andrew C. Kadak. A future for nuclear energy: pebble bed reactors. International

Journal of Critical Infrastructures, 2005.

[17] R. N. Morris, D. A. Pett, D. A. Powe, and B. E. Boyack. Triso-coated particle fuel

phenomenon identification and ranking tables (pirts) for fission product transport due to

manufacturing, operations, and accidents. NUREG/CR-6844, Volume 1: Main Report,

2004.

49

A Alternate derivations of discretised equations

This section is motivated by the fact that the assumption made in equation (3.3) may not

be valid and thus a different way of discretising the equations may be needed. The desired

form is still ∫g

dE σ(~r, E)ψ(~r, Ω, E)?= σg(~r)ψg(~r, Ω). (A.1)

However, it turns out that simply taking the integral like this would result in∫g

dE σ(~r, E)ψ(~r, Ω, E) = σg(~r, Ω)ψg(~r, Ω) (A.2)

since ∫gdE σ(~r, E)ψ(~r, Ω, E)∫

gdE ψ(~r, Ω, E)

= σg(~r, Ω). (A.3)

This process results in an unwanted angular dependence in the resulting energy-averaged

total cross section. One way to circumvent this is to expand the angular dependence of the

angular flux in terms of Legendre polynomials, giving

ψ(~r, Ω, E) = ψ(r, µ, E) =∞∑l=0

(2l + 1)Pl(µ)φl(r, E) (A.4)

where

φl(r, E) =

∫dµ

2Pl(µ)ψ(r, µ, E). (A.5)

The integral over energy group can then be taken, resulting in∫g

dE σ(r, E)ψ(r, µ, E) =∞∑l=0

(2l + 1)Pl(µ)σlg(r)φlg(r) (A.6)

where

σlg(r) =

∫gdE σ(r, E)φl(r, E)

φlg(r)(A.7)

and φlg(r) is exactly as defined in equation (3.15). Now note that the scattering source is

also expanded similarly in terms of the Legendre polynomials, thus taking∫gdE on equation

50

(3.1) gives

[ µr2

∂

∂rr2+

∂

∂µ

1− µ2

r

]ψg(r, µ) =

∑g′

∞∑l=0

(2l+1)Pl(µ)(σlg′g(r)−σlg(r)δgg′

)φlg′(r)+χgν

∑g′

σfg′(r)φg′(r).

(A.8)

If one wishes to retain a form similar to the main derivation, the following term

σg(r)ψg(r, µ) = σg(r)∞∑l=0

(2l + 1)Pl(µ)φl(r, E) (A.9)

can be added back into the equation, resulting in

[ µr2

∂

∂rr2+

∂

∂µ

1− µ2

r+σg(r)

]ψg(r, µ) =

∑g′

∞∑l=0

(2l+1)Pl(µ)σlg′g(r)φlg′(r)+χgν∑g′

σfg′(r)φg′(r)

(A.10)

which has the exact same form as the main derivation with the exception that the cross

sections indicated with a tilde indicate different quantities with

σg(r) = σ0g(r) =

∫gdE σ(r, E)φ(r, E)

φg(r)(A.11)

and

σlg′g(r) = σlg′g(r) +(σg(r)− σlg(r)

)δgg′ . (A.12)

B Obtaining temperature profile from given neutron

flux

The heat equation for stationary state is

α∇2T (~r) = −S(~r) (B.1)

where α is the thermal diffusivity. Due to the rotational symmetry of the problem, the heat

equation can be simplified into

1

r2

d

drr2 d

drT (r) = −S(r)

α. (B.2)

51

Setting U(r) = T ′(r) then gives

U ′(r) = −2

rU(r)− S(r)

α(B.3)

which is to be solved together with the boundary conditions

U(0) = 0 (B.4)

and

−αU(R) = C(T (R)− Tc) + σ(T (R)4 − T 4c ). (B.5)

The second boundary condition given here comes from the fact that heat loss on the surface,

which is the left-hand side term, is given by the convective heat loss for a cooling fluid in

contact with the surface and the radiation heat loss, which give the first and second terms

of the right respectively. The exact form of this boundary condition is given by the physics

of the situation and can change accordingly.

The1

rfactor in equation (B.3) means an implicit scheme will have to be used when solving

numerically such that the singularity is avoided.The step size h is chosen to be equal to each

half-integer step (i.e. ri+1/2 − ri). Using a fully-implicit scheme, the differential equations

are

U(r0 + h) = U(r0)− h( 2

r0 + hU(r0 + h)− S(r0 + h)

α

)=⇒ U(r0 + h) =

1

1 + 2hr0+h

(U(r0)− hS(r0 + h)

α

) (B.6)

and

T (r0 + h) = T (r0) + hU(r0 + h). (B.7)

To determine the source term, first note the relation

E =3

2NkBT (B.8)

where N is the number of neutrons. Taking this equation on a per unit volume basis leads

to

QfφF (r)σf =3

2N(r)kBT (B.9)

52

with Qf = 200 MeV being the energy released per fission. So the source term can now be

written as

S(r) =QfφF (r)σf

32N(r)kB

(B.10)

where

N(r) =∑g

φg(r)

vg. (B.11)

To solve this set of equations, shooting method is used, taking an initial guess G0 of the

value of T (0) and then adjusting it in order to best fit the condition given in (B.5). This in

turn is done by defining from equation (B.5),

residue(G0) = αU(R) + C(T (R)− Tc) + σ(T (R)4 − T 4c ) (B.12)

noting that U(R) and T (R) are now considered functions of G0. So finding the correct G0 is

equivalent to finding the root of residue(G0), which can be easily done by bisection or other

root-finding algorithms.

C Derivation of energy loss from scattering

The can be analysed in the center of mass frame picture as seen in figure (29). Consider a

Figure 29: Analysis of collision in center of mass frame with impact parameter b.

neutron coming in with mass m and initial speed u colliding with a nucleus of mass M . The

53

center of frame would then move with speed

VCM =mu

m+M(C.1)

relative to the lab frame, which means that the initial speed of the nucleus in CM frame is

given by

U ′ = VCM (C.2)

with primed quantities indicate they are taken in the CM frame. Since the collision consid-

ered here is elastic, the CM speeds of both the nucleus and neutron would have to be equal

before and after the collision, giving

u′ = v′ and U ′ = V ′ (C.3)

where v and V are the final speeds of the neutron and nucleus respectively. Solving for the

final velocities then give

V ′ =mu

m+M(C.4)

and

v′ = u′ = u− VCM =Mu

m+M. (C.5)

In the CM frame, since the total momentum will be 0, the neutron and nucleus will always

travel in opposite directions even after collision. Furthermore, according to figure (29), the

neutron would be deflected by an angle π − 2α. The final speed of the neutron in lab frame

is then obtained by vectorially adding ~VCM to ~v′, which gives

v =√v′2 + V 2

CM − 2v′VCM cos(2α). (C.6)

The initial and final energies are E =1

2mu2 and E ′ =

1

2mv2 respectively, which gives

E ′ = E

(M2 +m2 − 2mM cos(2α)

(M +m)2

). (C.7)

Now note that sin(α) =b

Rn

and thus cos(2α) = 1− 2b2

R2n

, which leads to

E ′ =E

(M +m)2

((M −m)2 + 4mM

b2

R2n

). (C.8)

54

So for a fixed E,

σs(E → E ′) dE ′ = 2πb db = π db2. (C.9)

Here, it is important to first note that from equation (C.7) that σs(E → E ′) is only non-zero

at(M −m)2

(M +m)2E 6 E ′ 6 E and so finally,

σs(E → E ′) =(M +m)2

4mMEπR2

n (C.10)

within this region, which can be seen is independent of scattering angle.

D MATLAB codes

This section will include some important parts of the MATLAB codes that were written and

used for the project. They will be attached to the back of the report.

55