Application of the multigroup diffusion theory to the study of the fusion-fission hybrid reactor concept using MatLab.

1. 1
Application of the multigroup neutron diffusion
theory to the study
of the fusion-fission hybrid reactor concept
Facoltà di Ingegneria Civile e Industriale
Dipartimento di Ingegneria Astronautica, Elettrica ed Energetica
Corso di Laurea Magistrale in Ingegneria Energetica
Sabino Miani
Matricola 1346890
Relatore Correlatore
Prof. Renato Gatto Prof. Augusto Gandini
A.A. 2018 – 2019
Titolo della tesi titolo della tesi titolo della tesi
titolo della tesi
Facoltà di
Dipartimento di
Corso di laurea in
Mario Rossi
Matricola xxxxx
Relatore Correlatore
Nome e cognome Nome e cognome
A.A. 0000-0000

2. 2
to my family

3. 3
Acknowledgements
I would first and foremost like to thank my thesis advisor, Professor Gatto, for allowing
me the opportunity to work on such a current topic. His advices have not only been
helpful and inspiring, but also engaging and leading. Always an email away, Professor
Gatto has been such an accessible advisor. Secondly, I could not have done this with out
my family’s love, support, and guidance. I am forever grateful, thank you.

4. 4
Summary
1 Introduction.....................................................................................................................6
1.1 Overlook of the topic ..................................................................................................6
1.2 The multiplying system...............................................................................................7
1.3 The geometry .............................................................................................................8
1.4 The medium ...............................................................................................................9
1.5 Structure of the thesis ..............................................................................................10
2 The software..................................................................................................................11
2.1 Introduction to MatLab..............................................................................................11
2.2 The current folder.....................................................................................................12
2.3 The workspace.........................................................................................................13
2.4 The variables............................................................................................................15
2.5 The script .................................................................................................................15
3 The ABBN Library.........................................................................................................18
3.1 Changes in the library ..............................................................................................18
4 The materials.................................................................................................................23
4.1 The medium’s script .................................................................................................23
4.2 Production of tritium ................................................................................................25
4.3 Burning of americium ...............................................................................................26
4.4 Boundary materials ..................................................................................................26
5 The multigroup diffusion equation..............................................................................28
5.1 Multigroup approximation.........................................................................................28
5.2 The diffusion equation in its general form.................................................................29
5.3 Discretized diffusion equation ..................................................................................31
5.4 The system of equations ..........................................................................................35
5.5 Diffusion equation in presence of a cavity................................................................36
6 The iterative process....................................................................................................42
6.1 The terms of the diffusion equation ..........................................................................42
6.2 First iterative process ...............................................................................................44
6.3 Calculation of the neutron flux..................................................................................46
6.4 Computation of the multiplication factor ...................................................................51
6.5 Iterations on the multiplication factor........................................................................53
6.6 Second iterative process with the external source ...................................................55
6.7 Calculating the dimension of the system..................................................................56

5. 5
7 The Code .......................................................................................................................57
7.1 Structure of the code................................................................................................57
7.2 Medium composition ................................................................................................58
7.3 Computation of the initial multiplication factor ..........................................................59
7.4 Beginning of the first iterative process......................................................................64
7.5 Second iterative process..........................................................................................66
7.5.1 Computation of the initial multiplication factor .......................................................68
7.6 Insertion of the external source ................................................................................69
8 Conclusions ..................................................................................................................76
APPENDIX A.1..................................................................................................................87
APPENDIX A.2..................................................................................................................87
References .......................................................................................................................90

6. 6
Chapter 1
Introduction
1.1 Overlook of the topic
The work described in the present paper is included in a much larger context regarding
the production of energy using nuclear reactions, in particular with focus on the burning
of isotopes with relatively long half-life and the production of fuel for fusion uses. Both
these topics are still actual since a definitive solution has not yet been found. The
production of radioactive waste has always been a primary issue regarding not only the
energy production field but also the medical one, for example. For an instance, one might
think that since in Italy there are no functioning nuclear power plants, the processing of
nuclear waste would be of no concern, but as we have said this issue also regards fields
different from that one. Whenever a radioactive material is involved in an industrial
process, we would have to deal with a certain amount of waste. In some cases, it is
convenient to stock the spent material and wait for it to decay, when it is not possible to
burn it. The burning process we are talking about is here strictly related to the nuclear
field, by it we intend the disappearing of specific elements through particular nuclear
reactions. The reactions we are talking about involve neutrons, therefore we need a
neutron multiplying medium. While, in general, any reactions that implicate the
absorption of neutrons lead to the destruction of the colliding element, here we will only
consider fission reactions of a specific isotope. Regardless, the same considerations can be
done when a different nuclear reaction, such as neutron capture, is taken into account.
However, while on one side we will be burning radioactive isotopes, thus reducing the
actual spent fuel that will end up being stock, on the other we will be producing them. In
particular, we will consider the production of tritium from neutron capture of lithium. We
know in fact that energy production from fusion reactions has become quite appealing in
the past years and specific facilities able to produce particular types of fuels are needed,
especially when dealing with nuclear fuels. While there is actually a wide range of
possible fusion reactions, and therefore of elements, we shall here concentrate on tritium,
being (together with deuterium) one of the most suitable elements for fusion purposes.
This means that the system here studied has the two goals of burning radioactive isotopes
deriving from spent fuel and production of fuel destined for fusion reactors. Moreover,
since both processes need neutrons in order to happen, we will need a system capable of
producing neutrons and sustain a stable chain reaction. The production of neutrons will be
possible by choosing a set of multiplying materials, such as uranium and plutonium
compounds. The latter can also derive from spent fuel and therefore be destroyed in the
system, while still being able to produce neutrons to keep up the chain reaction. Other
than these, a coolant will be present, serving the only purpose of cooling down the core to
avoid overheating and possible meltdown of the core.

7. 7
The medium in general will be a mixture of these materials (compounds), where each one
of them will occupy a different fraction of the total volume. The geometry of our system
will be cylindrical, and a more detailed description will be given in the following chapters.
The calculations carried over in this paper are simulations of a simplified system where,
for example, a homogeneous mixture of materials has been considered instead of one
where the reactor is actually divided into two regions that serve different purposes. In fact,
while the central core holds the fissile material, the outer blanket is used for the
production of tritium.
The software used for the computations is MatLab and shall be described in a chapter
dedicated to it.
1.2 The multiplying system
While most of the systems that involve nuclear fission reactions are critical systems, an
increasing role is being played by reactors defined as subcritical. The main reason why the
interest towards such system has been growing throughout is years is due to their intrinsic
safety. How this is possible, can be understood after describing the difference, in terms of
neutrons population, between critical and subcritical systems. Let’s take the “critical
equation” (Ref. 1):
𝐾𝑒𝑓𝑓 =
𝐾∞ 𝑒−𝐵2 𝜏 𝑡ℎ
1 + 𝐿2 𝐵2
= 1 (1.1)
whose values are used to differentiate such systems. The term on the left hand side of the
equation is the multiplication factor, a similar symbol is also found on the right hand side,
but with an ‘infinite’ subscript. This term has the same meaning of the previous one, when
considering a system with infinite dimensions, thus without leakages. It is now easy to
understand how the rest of the terms take into account the leak of neutrons when the
reactor has finite dimensions. While each term of the critical equation will be described in
a different chapter, we wish here to focus on the general meaning of that equation. In
particular, the factor is defined as the number of neutrons produced per neutrons that
disappears, whether for absorption (capture or fission) or leakage. Based on the materials
compositions and the reactor’s dimensions, we can have three main situations:
• 𝐾 𝑒𝑓𝑓 = 1, which using the explanation given before means that for each neutron
that disappears from the system in a certain instant of time, another one is
produced, thus keeping constant their population. This is the situation that applies
to the majority of the reactors, where we always have a stable population of
neutrons. Such a system can be obtained by fixing the dimensions and modifying
the composition of the materials inside, or by modifying the dimensions with fixed
compositions.

8. 8
• 𝐾 𝑒𝑓𝑓 > 1 is a situation in which a reactor should neve be found. In fact, here the
number of neutrons produced grows with time and is always larger than the one of
the neutrons that disappear.
• 𝐾 𝑒𝑓𝑓 < 1 is the case of our system, where in each instant of time, the number of
neutrons produced by the chain reactions is lower than the number of neutrons that
disappear from our system.
Obviously, such a system (multiplication factor lower than unity) is not able to keep up a
stable chain reaction, since the number of neutrons produced diminishes with time. If the
system is left by itself it will, at one point, shut down. Thus, an external source of neutrons
is here used to keep the system critical and maintain a stable chain reaction. The reason
why a subcritical reactor is safer than a critical one, resides in what we have said in the
previous sentence. A critical system is a system capable of functioning without external
help, but in our case the reactor is considered critical only when an external source is used,
thus when the latter is removed or shutdown the whole system becomes again subcritical
and will eventually stop producing energy.
The system here described is a cylinder with finite radius and infinite height with a central
cavity, where the external source should be located. This neutron source is usually
obtained by the collision of a beam of protons and a set of target nuclei. The collision takes
place into the central cavity, this way the neutrons can enter our system from the central
axis and be considered as a positive current (positive in the radial direction). The medium
is an homogeneous mixture of fissile, fissionable and absorbing materials, where he latter
is mainly the coolant.
1.3 The geometry
As previously stated, the system that we will describe resembles a typical hybrid fusion –
fission subcritical system, with some main differences. The shape is that of a cylinder with
finite radius and infinite height. The radius here considered is the extrapolated, which is
not the physical boundary of the reactor but the distance to which the neutron flux falls to
zero. We know in fact that, right outside the border of our system the flux cannot be zero
due to neutrons exiting and eventually entering (if a shielding material is used) the core;
however, a condition that takes into account the annulment of the flux, together with
another set of conditions, is needed in order to have an exact solution of the diffusion
equation (this shall be more clear when explaining the diffusion theory). This means that
the system will be actually smaller (on both sides) than the one here described. Regarding
the shape, we know that the cylindrical one is widely used even though is not the most
efficient from the neutron economy point of view. However, when choosing a shape the
feasibility of such a system also needs to be considered. In fact, while on one side the
spherical shape reduces neutrons leakages (having the smallest external surface to internal
volume ratio), on the other it increases the difficulties from the construction point of view.

9. 9
This makes the cylindrical geometry more suitable, even though it has a larger surface to
volume ratio. Moreover, among all the cylindrical geometries that can be taken, the shape
that reduces the loss of neutrons through its borders is the one where its height is
somewhat similar to its diameter. Of course, considering both dimensions would
complicate the calculations, in fact the height will also have to be taken into account in the
Laplacian term of the multigroup diffusion equation:
∇2
=
𝜕2
𝜕𝑟2
+
1
𝑟
𝜕
𝜕𝑟
+
𝜕
𝜕𝑧
(1.2)
which then obviously makes the neutron flux dependent from both dimensions, thus
complicating the solution of the diffusion equation.
1.4 The medium
The composition of the system here considered is also quite different from a real one,
being this too a simplified version of it. The layout is somehow similar to the one of a
critical system, with distinct fuel rods each containing the fissile and fissionable material.
In the case of hybrid reactors, the system is divided into two main regions where the inner
one contains the fuel, i.e. the fissile and fissionable material, while the outer one is the
blanket that contains the material used to produce fusion fuel. The first part, being the
region containing the fuel, is the one in which the fission reactions take place; these serve
not only to sustain the chain reaction but also to burn the eventual actinides or to produce
fissile elements whenever needed. The second part is basically an absorbing medium with
no fuel. In our case we will not consider such a layout, being too complicated to describe,
but instead our system will be filled with a homogeneous mixture of both fissile and non –
fissile materials. Therefore, we will consider a multiplying medium (fissile and fissionable
isotopes) in which we have a homogeneously distributed mixture of absorbing materials,
e.g. isotopes for the production of fusion fuel and coolant. The fissile and fissionable
isotopes will be a mixture of uranium (mostly U – 238, enriched in U – 235) and plutonium
(Pu – 238, Pu – 239, Pu – 240, Pu – 241 and Pu – 242, where the last three cover the largest
percentage) isotopes, these will be used both to maintain the chain reaction and generate
new fissile material. Another fissile element that will compose the medium is americium
(Am – 241), this is the one that will be burned in our system through fission reactions. The
absorbing elements are a lithium compound (both Li – 6 and Li – 7, enriched in the
former), where Li – 6 is used to produce tritium through neutron capture, and lead (in
particular Pb – 208) used as coolant.
The compositions of each compound (within a single element) can be modified at will, just
like the volumetric fraction of each material.

10. 10
1.5 Structure of the thesis
The main topic on which this paper is based is the derivation of the multigroup diffusion
equation for subcritical hybrid systems, where the title ‘hybrid’ is due to the fact that these
reactors will both be able to burn radioactive isotopes deriving for example from nuclear
power plants based on fission reactions, and to produce fuel for fusion purposes.
Regarding the former, in some cases it will be also possible to produce fuel for fission uses.
The set of equations derived will serve as a tool to build a code that will be able to
determine the composition of a subcritical system of given dimensions and fixed
multiplication factor. Such a system, with the addition of an external source of neutrons
will then be able to sustain a stable chain reaction and eventually be used to simulate a
hybrid system capable of both burning and producing specific isotopes. While the last part
is dedicated to the code itself, on the rest of the work we mainly focus on describing the
instruments that were needed to create the program. Below the list of the chapter and their
relative description:
• the first chapter is the introductory one, in which an overall description of the whole work
has been given;
• the second chapter is dedicated to the description of the software used to write code, with a
major focalization on the tasks considered useful for our purposes;
• the third chapter focuses on the data used to make the calculations and how it was possible
to import everything into the software;
• the fourth chapter is centered on the materials to which the data belongs to and that will be
used to carry on the simulations;
• following the previous chapter, in the fifth we will start out with the multigroup diffusion
equation and derive ones needed for our purposes;
• the sixth chapter will be dedicated to the iterative process followed to determine the
wanted physical quantities of our system;
• the seventh chapter is where we shall give an elucidation of the code used to simulate our
system;
• the eight chapter is the last one and will hold the conclusions and future developments of
the code.

11. 11
Chapter 2
The software
We will here give a brief illustration of the software used to write and consequently run
the simulations of the system previously described. The major tasks and tools of the
MatLab software will be shown and described, dwelling on the parts that most concern
our purposes. While the first paragraphs are centralized on the software itself, the last part
shows a part of the script holding the code, and some of the functions used to compute the
wanted values.
2.1 Introduction to MatLab
As anticipated in the introductory chapter, the software used to develop the code is called
MatLab. In order to understand how the code works, we first need to give an explanation
of the program used to create it. The name MatLab derives from the fusion of the two
words Matrix and Laboratory, in fact the main objects used in this software are matrices,
and this includes vectors (being only a more particular arrangement of numbers). The
program itself can be used as a normal calculator or as a programming tool, able to solve a
great deal of problems. We will go a little bit more into details now, on how it works and
what are its main functions.
When opening MatLab, a window, called desktop, becomes visible and it appears like this:
Fig. 2.1 Shows the MatLab desktop shown when the software is opened

12. 12
The central window (Command Window) is where calculations take place, the prompt
command (>>) on the top right is where the input expression has to be written when using
MatLab as a calculator. However, when running a script, the command window is where
the computations can eventually appear. Whether they do or they do not depends on how
we structured the code (a specific command allows us to show the results of a calculation
carried over by the software).
The top part of the interface is filled with commands that can be used to import data from
other files, to clean the command window or to write a script, in which case a new
window appears where the code can be written on and it will be shown in one of the
following paragraphs.
The windows on both sides of the Command Window are the Current Folder (on the left
hand side) and both the Workspace and the Command History (right hand side). The
former is the primarily folder used by MatLab, where the data is saved on or imported
from for example. The other two windows, that share the right hand side, are the places
where the created variables are shown. The next paragraphs will focus on these windows.
2.2 The current folder
The Current Folder, shown in the picture below, is basically the main folder used by
MatLab to permanently store data for example, and it is created as soon as we run the
software for the very first time.
Fig. 2.2 The current folder is represented, with a set of data saved on it
As we can see the horizontal window on the top part of the screen shows us where this
folder is located. The main folder is called ‘MATLAB’ and contains a series of folders
where everything regarding the software is stored.

13. 13
The MATLAB folder is created by the software itself when the software is opened;
furthermore, other folders are directly created by the program whenever needed.
Going back to the Current Folder now, we can see from the picture some of the files saved
on it. These can be written scripts or generic data. In our case for example, the scripts with
different geometries have been saved under different names. Also, the modified libraries
that hold the data that will be used are saved and ready to use through a specific
command.
To open up a script, involving a particular geometry composition, we just need to click on
it, and the script opens in a new window. The situation is a bit different when considering
the library scripts. In this case we can see how the same name shows up under different
files defined as ‘m – files’ and ‘mat – files’, the difference between these two types of files
will be here illustrated:
• the m – file is where the actual script has been saved, i.e. the data and the computations
regarding a particular set of materials. This means that the set of equations, for example
used to determine physical parameters (such as the diffusion coefficient), are located in this
file.
• If we wish to actually use that script and compute the data that can be determined by it, we
need to save it somewhere and in this case the data is stored as a mat – file. Once the data is
saved, it can be used whenever.
The type of files that can be saved into the Current Folder are not only m – files or MatLab
related files, in fact this works also when we are using different tools such as Excels
worksheets for example or when we need to use a graph. Regarding data saved on an
Excel file, if we were to use it, first we would need to save it onto the Current Folder. In
this case, the Excel files will be saved as ‘.xlsx.’ files, instead of ‘.m’ which states that the
file is a MatLab format file.
2.3 The workspace
The workspace is where all the variables created, within a script or in the Command
Window, are stored temporarily. It is thus possible to recall a variable and use it in
different ways. If, for example, we are making simple calculations then it is possible to
recall a variable through the prompt command into the Command Window. This can be
helpful when we wish to check only a certain part of a script, this way we can easily recall
one or more variables and use them right away.
However, when dealing with a script it is far more convenient to save any data directly
into the Current Folder, since this way it can always be available; in fact, when a variable
having the same name of one precedingly stored into the Workspace is saved, it
automatically takes the place of the first one. This can hence create confusion.

14. 14
Being usually large, the number of created variables, it is also convenient to name each one
of them with a name of common sense, representing for example its function. It is also
important to note that whenever needed, it is possible to erase the variables from the
workspace, to make room for new ones.
The other window that appears on the right hand side is the Command History, which
holds all the commands used in the Command Window or the ones used when we run a
script.
The figure below shows some of the variables created and saved when running the created
simulation code. The variables, in this case, belong to the workspace which is the one
highlighted in blue:
Fig. 2.3 The workspace with its data saved is illustrated in the figure
As we can see, the screen is vertically divided into two parts, the one on the right hand
side holds the name of the variables created while on the left hand side we have the saved
data. In particular, if the saved array is a scalar quantity then a single value will appear,
the number of iterations (‘iter’) is an example. However, if we are dealing with matrices,
what is shown is the dimensions of it, i.e. the number of values that have been saved.
For an instance, all the data imported from the excel file is saved in terms of vectors or
matrices, where each value belongs to a particular energy group. For example, the first
vector is called ‘csi’ and it holds the energy spectrum values, and on the right hand side it
says [1 x 26]. In fact, when running that script we were dealing with 26 energy groups,
which means that each physical parameter will hold 26 values. The diffusion coefficient
for example, is a function of the total cross sections which, regarding a certain material,
depends on the energy of the neutrons, i.e. on the energy group.

15. 15
Since the excel file holds the values, belonging to a single vector, in a column, we need to
transpose all the vectors in order to have row arrays , easier to use in this environment.
This is done into the library script, for each one of the variables imported.
2.4 The variables
It is important to note that, other than matrices, vectors and scalars we have another
particular set of values, whose dimensions are represented by three numbers. Creating
these particular arrays was necessary when dealing with the neutron flux in one
dimensional calculation. While for the rest of the variables we have assumed that they
remain constant throughout the whole system, and only change from one group to the
other one, for the neutron flux such an assumption would be too coarse.
In this case, the changes regard not only values belonging to different energy groups, but
also within the same group, on where inside the system the flux is determined. The object
used in this case is basically a set of row vectors, where each vector holds Ng (where ‘Ng’
is the total number of groups) values of the neutron flux; the number of these vectors
depends on the number of nodes in which the system has been divided into. This means
that it depends on the number of points in which we wish to determine the neutron flux.
In our case we have picked N = 40 and being Ng = 26 the set’s dimensions will be [1 x 26 x
40] if we allocate groups values into a single vector, thus ending up with 40 vectors;
otherwise we will have [1 x 40 x 26], if the numbers inside the single vector hold the values
that the neutron flux, belonging to a single energy group, has in each one of the nodes. We
will use one or the other depending on the operations we are dealing with.
The reason why the examples present 26 energy groups is that, when first deriving the
code, the data used in the computations had been determined using a splitting of the
energy range into 26 groups. Nonetheless, a coarser division is considered when
simulating our system, in fact the number of groups falls to 6.
2.5 The script
In this last part we will show what a typical script looks like and describe some of the
functions that can be used. As seen in the first paragraph, the software can be used as a
normal calculator, in which case everything can be done using the command window; or
as an actual programming software, in which case a new tool has to be used. This
instrument is a new window that opens, and it holds a blank page with a top bar, filled
with the main functions.
The blank page is where the script has to be written, together with titles and comments. In
this window it is also possible to run the code as a whole or just some parts of it; as well as
print the script on a file different from the MatLab ones, for example on Word.

16. 16
The left hand side presents a set of numbers in column, these are useful to identify errors
that occur when running the code. In the following page a typical window holding a script
is presented.
The figure below shows one of the scripts created to compute the critical dimensions of a
system with fixed composition, in this case the geometry was cylindrical with finite radius
and infinite height.
Fig. 2.4 The picture shows the edit window used to compute and run a script
The top bar, just like we saw on the MatLab desktop, is where the main commands are,
from there we can save or run the code for example. The ‘Publish’ bar on the top left is the
one that allows us to save the script into a different format.
Note that the blank page, where the script is written, is divided into two sections; of these
two, only the first one is highlighted, and this can be done by writing ‘%%’ in order to
separate different sections. Moreover, it is possible to run sections individually through
the ‘Run Section’ or ‘Run and Advance’ commands, in the latter case after running a
section the succeeding one is highlighted and ready to be computed.
Just like the figure shows, the top part is where we have the data that will be used
throughout the code, it’s mostly values that will be used at the beginning of the code or
scalars that don’t change (the number of groups or the number of nodes for example)
throughout the whole computation. All these values can be easily modified by just typing
the new values after the equal sign. The semicolon at the end of each value keeps the code
from publishing all the values, hence only the ones without it will appear on the
Command Window.
Before closing this chapter, we will linger on the first part of the code, right below the
highlighted section of the figure, since a previously described variable appears.

17. 17
The for loop (with its keywords in blue) is there used to create the neutron flux vectors
and assign them a unitary value, to initialize the computations. In this case, each vector
will hold N values, just like the total number of nodes, of unitary values each:
phi = ones(1,N)
the values have been assigned using the MatLab function ‘ones’, between parenthesis we
have the dimensions of the array (in this case a row vector). Then, as we have seen
already, a physical condition is necessary, so the flux falls to zero on the boundary of our
system. Furthermore, while each vector is created it has to be stored somewhere in order
for it to be easily traceable; at each step of the for loop (Ng in total, just like the groups) a
new vector is stored into an array defined this way:
phi_n0( :, :, n) = phi
where the fist index refers to the number of rows of each vector, by using ‘:’ we are
considering all of them instead of a particular one. The second index refers to the number
of nodes while the last one to the number of groups. If we wish to recall the value of the
neutron flux in a particular node and belonging to a certain group we just need to replace
the indices with the proper numbers.
This is only of the many functions that have been used in the code and that will be
presented in an appropriate chapter.

18. 18
Chapter 3
The ABBN library
In this third chapter we will give an illustration of the work carried over in order to
implement and adjust the data (physical parameters saved as an Excel file) in order to be
able to use it in the software. Furthermore, a description of the parameters will be given,
with reference to the scripts used in the code. In this case, we will use the data related to a
different compound (uranium), in order to show a distinct point of view (the number of
isotopes composing the material change, just like the physical quantities).
3.1 Changes in the library
The Library’s data is presented as an Excel file made of 22 worksheets, each sheet
containing the parameters of a particular element. The first 12 sheets refer to a 26 energy
group’s division; while the remaining 10, to a 6 group parting of the energy of the
neutrons. In every sheet we find the nuclear parameters of that precise element. Some of
these parameters were already there (e.g. microscopic cross sections) while some others
(e.g. transport microscopic cross sections and diffusion coefficients) have been calculated
using simple Excel functions. The inelastic scattering cross sections (from one group to the
other ones) are allocated into a square matrix whose dimensions depend on the number of
energy groups considered. In our case we have a 6x6 square matrix, since the calculations
in this paper are carried over considering 6 energy groups; being group 1 the one at
highest energies (approximately 10 MeV) and the last one the thermal one. However, the
code has been written in such a way that calculations can be done also considering the 26
energy group’s library.
Now, for this data to be used properly in the diffusion equation, we need to evaluate the
macroscopic cross sections. These are linked to the atomic densities of the elements
composing our nuclear system. Considering a system characterized by a homogeneous
mixture of materials, in which each one of them occupies a precise volume fraction and
knowing the physical characteristics of the components, it is possible to determine the
atomic densities.
The calculation of these densities was at first carried out by hand, while the macroscopic
cross sections were determined using Excel functions. The problem was that, when
changes were needed, the atomic densities had to be determined all over again and the
macroscopic cross sections updated, this way simple calculations became complicated.
A way to solve this involves the utilization of the MatLab software.

19. 19
A script is here used to calculate directly the atomic densities and the macroscopic cross
sections. The advantage of such a script is that it can be used for every kind of element or
material, in fact each parameter can be modified to have, for example, the wanted density
or enrichment. However, in order to write the script a derivation of the equations was still
necessary; this has been carried out in Appendix A.1.
The basic nuclear parameters are taken directly from the Excel file in which the library has
been saved and a MatLab function is used to take the values of the considered parameter
and allocate them into a column vector, properly transposed. The script is divided into
two parts, where the first one operates the atomic densities calculations whereas the
second one withdraws the data from the Excel file. Using simple operations, the
macroscopic cross sections can then be computed and saved into the so called ‘Current
Folder’, previously described. This is the folder in which all the data is saved into and
taken from when needed. The Library too must be saved into this folder for it to be used
from the software.
The scripts have been written both for the single material and for a mixture, in case more
than one material is found into the system.
An example of such a script is presented below, in this case the material considered is a
fuel material (Uranium Dioxide with the 100% enrichment in 𝑈92
235
). The enrichment has
been indicated with the letter x, while N_av stands for Avogadro’s number and PM for
molecular weight (the reason why we have chosen PM over MW as the molecular weight’s
initials is that the latter can be easily confused with the already known unit of measure
‘MegaWatt’). Moreover, knowing that the effective density is usually smaller than the
theoretical one (due to a certain void fraction related to how the fuel was made), we have
indicated the former with ‘ro’ and the latter with ‘ro_th’ imagining a void fraction of the
0.05%. Below the set of initial values is shown.
x = 1;
N_av = 6.022*10^23;
ro_th = 10.838;
fraz_ro = 0.955;
PM_A = 235;
PM_B = 238;
PM_C = 16;
The atomic density of the molecular complex (UO2) is first calculated, computing the
molecular weight of the molecule and converting it into grams (instead of atomic mass
units), obtaining the generic value here called N. To determine a specific atomic density,
i.e. related to a particular isotope, we simply have to multiply it by its atomic abundance.
ro = (ro_th*fraz_ro);
PM1 = (x*PM_A) + (1 - x)*PM_B + 2*(PM_C);
PM = (1.66054*10^(-24))*(6.022*10^23)*PM1;
N = (N_av*ro)/(PM)*(10^(-24));

20. 20
The following section becomes necessary when we have a mixture of components, such as
fuel and coolant for example or fuel and a structural material; in fact, in this case each one
of them will occupy a fraction of the total volume, a quantity that must be specified. This
parameter can also be easily modified to obtain the wanted mixture.
F = 0.8;
The macroscopic cross sections can now be computed. The following section is the one
responsible for it.
With the first equation the atomic density of a particular element can be calculated while
the other commands are used to take the data from the Excel file and save it into the
MatLab folder. The equation we are talking about is actually a command ‘N_1 = x*N’ and
it represents the product between the generic atomic density and the enrichment of the
isotope of which we wish to compute the atomic density.
The values of a single parameter (fission microscopic cross section for example) are
collected into a column vector (since the values are displaced into columns in the Excel
file), transposed into a row vector and then multiplied by the atomic density of that
element and the volumetric fraction of that material, if the material is actually available in
the system.
The previously described processes have to be followed when the data belongs to a
microscopic cross section, otherwise it is just saved directly into the Current Folder (the
vector ‘csi’ defining the neutron spectrum for example). The function that withdraws the
data from the Excel file is:
vector = xlsread(‘Filename’, ‘Worksheet’, ‘Range’)
All the needed parameters are presented in the box below. The neutron spectrum,
associated with each group, is presented as ‘csi’, this considers the fraction of neutrons,
deriving from fissions, characterized by a certain energy range (the one indicated by the
group to which the spectrum belongs).
With ‘fiss’ we have indicated the product between the fission microscopic cross section
and the average number of neutrons that result from every fission reaction, for every
energy group.
The diffusion coefficients are labeled as ‘diff’ and the matrix containing the inelastic
scattering cross sections is called ‘sigma’ (the only one that hasn’t been transposed, being a
matrix). The elastic scattering cross section, from a preceding group to the present one, is
the one indicated with ‘sigma_rem2’ while with ‘sigma_rem1’ we have defined the
removal cross section (equal to the sum of the capture, fission, inelastic ed elastic cross
sections). Each one of these quantities is calculated for each one of the isotopes, in relation
to a certain atomic density and volumetric fraction.

21. 21
The box below holds what has been said in the previous paragraph, the data here
presented is the one related to the isotope U – 235.
N_1 = x*N;
csi1 = ((xlsread('LibUranio.xlsx',1,'P3:P28'))');
fiss_macro1 = F*N_1*((xlsread('LibUranio.xlsx',1,'E3:E28'))');
diff_macro1 = F*(1/N_1)*((xlsread('LibUranio.xlsx',1,'Q3:Q28'))');
sigma_rem1_macro1 = F*N_1*((xlsread('LibUranio.xlsx',1,'O3:O28'))');
sigma_rem2_macro1 = F*N_1*((xlsread('LibUranio.xlsx',1,'M3:M28'))');
sigma_macro1 = F*N_1*((xlsread('LibUranio.xlsx',1,'U3:AT28')));
We can see from the data between brackets in the ‘xlsread’ function (in the above and
below boxes) that except for the worksheet number (related to a particular element)
everything else stays the same. In fact, as described earlier, the library has been modified
so that the data could be extracted in a simpler way. If we wish to use a certain material
over another we only have to check the worksheet where its data has been located and
change the number on the script. Here the data of the fissionable element, U – 238, is
presented, in fact its atomic density is computed by multiplying the variable ‘N’ by (1 – x)
being ‘x’ the enrichment in U – 235.
N_2 = (1 - x)*N;
csi2 = ((xlsread('LibUranio.xlsx',2,'P3:P28'))');
fiss_macro2 = F*N_2*((xlsread('LibUranio.xlsx',2,'E3:E28'))');
diff_macro2 = F*(N_2)*((xlsread('LibUranio.xlsx',2,'Q3:Q28'))');
sigma_rem1_macro2 = F*N_2*((xlsread('LibUranio.xlsx',2,'O3:O28'))');
sigma_rem2_macro2 = F*N_2*((xlsread('LibUranio.xlsx',2,'M3:M28'))');
sigma_macro2 = F*N_2*((xlsread('LibUranio.xlsx',2,'U3:AT28')));
Since we have considered uranium dioxide, we also need to calculate the data related to
the oxygen found in the compound, this is presented beneath.
N_3 = 2*N;
csi3 = ((xlsread('LibUranio.xlsx',12,'P3:P28'))');
fiss_macro3 = F*N_3*((xlsread('LibUranio.xlsx',12,'E3:E28'))');
diff_macro3 = F*(1/N_3)*((xlsread('LibUranio.xlsx',12,'Q3:Q28'))');
sigma_rem1_macro3 = F*N_3*((xlsread('LibUranio.xlsx',12,'O3:O28'))');
sigma_rem2_macro3 = F*N_3*((xlsread('LibUranio.xlsx',12,'M3:M28'))');
sigma_macro3 = F*N_3*((xlsread('LibUranio.xlsx',12,'U3:AT28')));
The data presented above relates to the single elements found inside the compound, each
with different parameters. Since we are dealing with a composite element, we need to put
together the single parameters and determine the final one. This, has to be done for each
one of the quantities characterizing the medium.

22. 22
The data belonging to the uranium dioxide molecule is presented below. It is though
important to note that while we always speak in terms of singular quantities, we
obviously intend for them to be plural.
csiA = (csi1 + csi2 + csi3);
fissA = (fiss_macro1 + fiss_macro2 + fiss_macro3);
diffA = (diff_macro1 + diff_macro2 + diff_macro3);
sigma_rem1A = (sigma_rem1_macro1 + sigma_rem1_macro2 + sigma_rem1_macro3);
sigma_rem2A = (sigma_rem2_macro1 + sigma_rem2_macro2 + sigma_rem2_macro3);
sigmaA = (sigma_macro1 + sigma_macro2 + sigma_macro3);
Published with MATLAB® R2018a
We can now easily understand the reason why the previous calculations have been made
using a MatLab script instead of carrying them by hand, with the help of Excel. This way
there is no actual need to make any calculations and the parameters can be modified by
just changing the values.

23. 23
Chapter 4
The materials
The present chapter is dedicated to the materials and their purpose in the system
described. As a matter of fact, the system we are trying to simulate is not a usual critical
system used for energy production, thus every element composing the medium will serve
a particular task. This will be main point of the paragraphs composing this chapter, except
for the first one where a general view on the script, holding the materials information, will
be given.
4.1 The medium’s script
A major role in this work is played by the materials that will compose our system. Since
the main goal is to both burn/produce certain elements, we will have a set of fixed
elements and a certain amount of boundary materials. The calculations of the parameters
for each material will be carried out by a MatLab script, a part of which is presented
below. The complete script will contain all the needed elements with the possibility of
choosing one over another. Quantities such as enrichment or volumetric fraction can be
easily modified to obtain the wanted amounts. The section presented in the following page
is the one holding the data regarding the plutonium compound.
The first part is where the main data is entered, both fixed (molecular weight for example)
and non-fixed quantities (enrichment of the different isotopes, void fraction and
theoretical density). With these parameters, the generic atomic density is then determined,
this quantity will give the actual density of each element when multiplied by the right
enrichment. Afterwards, the macroscopic cross section is determined for each isotope and
then the parameters are put together to calculate the cross sections of the compound.
The example shown only related to the Pu – 238 isotope, but the computations are equally
carried over for each one of them.
Even though the table presents isotopes that go from 238 to 242 (molecular weight), it is
possible for example to remove one or more elements by adjusting their atomic percentage
(enrichment) and setting it to zero. The volumetric fraction, this time related to the
compound, can also be modified to increase or reduce (up to zero) the space occupied by
that particular element inside the system. The equations that appear in the box will be
described in detail in one of the following chapters, being this one strictly related to the
materials.

24. 24
Below, a part of the script used to import and determine some of the material’s parameters
is shown:
%Data (Plutonium)
x238 = 0.20;
x239= 0.51;
x240 = 0.17;
x241 = 0.11;
x242 = 0.01;
N_av = 6.022*10^23;
ro_th = 11.478;
fraz_ro = 0.955;
PM_A = 238;
PM_B = 239;
PM_C = 240;
PM_D = 241;
PM_E = 242;
%Atomic Density Calculation
ro = (ro_th*fraz_ro);
PM1 = (x238*PM_A) + (x240*PM_C) + (x241*PM_D) + (x242*PM_E) + (1 - (x238 + x239 + x241 +
x242))*PM_B;
PM = (1.66054*10^(-24))*(6.022*10^23)*PM1;
N = (N_av*ro)/(PM)*(10^(-24));
%Volumetric Fraction
F = 0.4;
%Macroscopic Cross Section Calculation (Pu - 238)
N_0 = x238*N;
csi0 = ((xlsread('LibUranio.xlsx',16,'P3:P8'))');
fiss_macro0 = F*N_0*((xlsread('LibUranio.xlsx',16,'E3:E8'))');
diff_macro0 = F*(1/N_0)*((xlsread('LibUranio.xlsx',16,'Q3:Q8'))');
sigma_rem1_macro0 = F*N_0*((xlsread('LibUranio.xlsx',16,'O3:O8'))');
sigma_rem2_macro0 = F*N_0*((xlsread('LibUranio.xlsx',16,'M3:M8'))');
sigma_macro0 = F*N_0*((xlsread('LibUranio.xlsx',16,'U3:Z8')));
%Homogeneous Medium (Pu - 239, Pu - 240, Pu - 241, Pu - 242)
csiA = (csi0 + csi1 + csi2 + csi3 + csi4);
fissA = (fiss_macro0 + fiss_macro1 + fiss_macro2 + fiss_macro3 + fiss_macro4);
diffA = (diff_macro0 + diff_macro1 + diff_macro2 + diff_macro3 + diff_macro4);
sigma_rem1A = (sigma_rem1_macro0 + sigma_rem1_macro1 + sigma_rem1_macro2 +
sigma_rem1_macro3 + sigma_rem1_macro4);
sigma_rem2A = (sigma_rem2_macro0 + sigma_rem2_macro1 + sigma_rem2_macro2 +
sigma_rem2_macro3 + sigma_rem2_macro4);
sigmaA = (sigma_macro0 + sigma_macro1 + sigma_macro2 + sigma_macro3 + sigma_macro4);

25. 25
The next two paragraphs will be dedicated to the ‘fixed’ elements, where we will give an
overview of the main parameters that characterize them and describe the main reactions
that these elements will undergo in the system. The isotopes that appear in these two
paragraphs are the ones responsible for the production of tritium (through neutron
absorption) and the burning of americium (through fission reactions).
In the last paragraph we will give an overview of the ‘boundary’ materials, i.e. the fuel,
used to sustain the chain reaction, and the coolant.
4.2 Production of tritium
From the mass – energy equivalence formula we know that reactions in which the mass is
reduced, a certain amount of energy (related to the lost mass) is consequently released.
There are two main families of reactions that undergo this exoenergetic process, fission
and fusion. In both cases the final mass is lower than the one that the particles had before
undergoing the reaction, hence the release of energy. In this paragraph we shall
concentrate only on the fusion reactions, whereas the rest of the paper mainly focuses on
fission phenomena.
To describe the probability of a certain fusion reaction to take place, we can use the same
parameter utilized in the fission field, the so called ‘cross section’. Since the phenomena
involved take place one after another, the total probability is equal to the product of the
single ones:
𝜎𝑓𝑢𝑠𝑖𝑜𝑛 = (𝜋𝜆2) (
tunnel
effect
) (
reaction
factor
) (4.1)
The first term on the right hand side is a sort of geometric factor. The reaction will take
place if the two nuclei are relatively close to each other, so this term holds the cross section
of the interacting particles. Since the distances involved are too small, the De Broglie
wavelength is here used instead of the radius. Now that the nuclei are moving towards
each other, they need to overcome the coulomb barrier and react, the probability of this is
determined by the second term on the right hand side.
Once the particles overcome the barrier and get together, they need to form a new nucleus,
and this depends on the type of reaction taking place, hence the name ‘reaction factor’. For
instance, interactions built on the strong nuclear force are usually faster and have a bigger
probability to end up with the formation of a nucleus, than the ones linked to the weak
nuclear force.
This leads to the conclusion that whether one reaction is more likely to take place at a
certain energy over another, depends on the combination of these factors. Among all
possible interactions, the ones that interest us the most are those involving hydrogen
isotopes, i.e. deuterium and tritium.

26. 26
While deuterium isotopes can be extracted from water at relatively low costs, tritium can
be produced using lithium. The reaction involving lithium is the following:
𝐿𝑖3
6
+ 𝑛 → 𝐻𝑒2
4
+ 𝑇 & 𝐿𝑖3
7
+ 𝑛 → 𝐻𝑒2
4
+1
3
𝑇1
3
+ 𝑛
But there is a reason why only the one involving Li – 6 is used, in fact this reaction can
take place with neutrons of any energy, while the second one only happens if the neutrons
have energies above a certain level making the former more suitable for tritium
production. Since natural lithium contains no more than 7.5% of Li – 6, being the rest Li –
7, we will need enriched lithium in our system. As seen in the previous chapters, this
quantity can be easily modified to obtain the wanted amount.
4.3 Burning of americium
The other important element composing our system is americium (Am – 241), which
derives for example from nuclear spent fuel. In fact, we know how some of the elements
characterized by relatively long half-life (e.g. 432 years for Am – 241) are also extremely
radiotoxic, hence the need to destroy them. When we talk about destroying elements, we
actually mean to convert them into non-radioactive isotopes, or still radioactive but with a
shorter half-life.
The conversion happens when the number of protons or neutrons in the nucleus changes,
this process is called transmutation and it can happen naturally (through radioactive
decay) or artificially (by nuclear reaction), in our case we will consider transmutation
through nuclear reactions. The ones that can take place when we have to deal with Am –
241 are neutron capture and fission, we will here consider only the latter. Even though in
spent nuclear fuel we usually find Am – 241, Am – 242, Am – 242m and Am – 243 we shal
focus only on the first isotope of that chain, since it’s usually way more abundant than the
others.
4.4 Boundary materials
We have seen the reason why elements such as lithium and americium have been
introduced into the system, but in order for the wanted reactions to take place these
cannot be the only elements that will be used.
The reactions involved take place only if, other than the specified elements, neutrons are
involved. In the present, neutrons come from fission reactions other than an external
neutron source, which will be introduced in the chapter regarding the derived equations.

27. 27
The fission reactions take place if fissile (and fissionable) material is introduced, hence the
need of uranium and plutonium isotopes; while the external source is used to maintain a
stable chain reaction.
As previously anticipated, the enrichment of each isotope can be chosen at will, but the
fuel will be mainly composed by U – 238, Pu – 240, Pu – 241 and Pu – 242.
Together with the fission process comes the production of heat, only to name one of the
processes involved. In fact, the particles resulting from the splitting of the fissile atom are
left with a certain amount of energy under the form of kinetic energy. This energy is
released throughout the medium by collisions with the surrounding atoms, producing a
temperature increase. To remove the heat produced a coolant is needed, in this case we
will use lead (in particular the Pb – 208 isotope).

28. 28
Chapter 5
The multigroup diffusion equation
In this chapter we will derive all the equation that will be used, after a proper
transformation, into the software. We will start out with the most generic form of the
multigroup diffusion equation and obtain its compact and simplified form, in order for it
to be used in the one dimensional calculation. the geometries considered will be both
cylindrical and spherical with a central cavity.
Once the main equations have been derived we shall apply some conditions and derive
the equations when a current of neutrons is introduced into our system.
5.1 Multigroup approximation
The equation from which we will derive all the other relations is the multigroup diffusion
equation, which is obtained through a series of approximations of the transport equation.
In fact, the latter can be solved with exact solutions only in a restricted number of
situations, thus the need to apply particular conditions in order to obtain simplified
equations.
One of the most common approximations (Ref. 1) of the transport equation is the so called
‘Pn approximation’, which will lead to a set (‘n’ in number) of coupled equations instead of
a single one, i.e. the initial transport equation. This approximation applies to the one -
dimensional transport equation, this means that the neutrons can only move along one
coordinate (z – axis for example). The direction of its motion can form a certain angle with
the axis, as long as the neutron moves along that fixed axis. Moreover, we will also
consider stationary conditions, so that the time variable will disappear.
It may look like an unusual approximation, since we go from dealing with a single
equation to a set of coupled ones; however, by doing this we get rid of the angular
variable. Furthermore, we actually do not need the complete set of equations derived, in
fact the first two form a system that encloses all the characteristics typical of the diffusion
theory with the transport corrections.
With the multigroup approximation, we take one of these equations and we integrate each
term on a certain energy interval. This interval derives from the splitting of the energy
range in which a neutron can be found. We know that from when neutrons are born to
when they thermalize (in a thermal system) their energy, regardless of the type of
collision, will always fall in this range. We can thus divided into intervals and assume that
the physical quantities found in the equation, can change from an interval to the other, but
remain constant within the same interval.

29. 29
Obviously, the bigger the number of intervals in which the range is divided, the more
precise the approximation. By following this path we obtain the multigroup diffusion
equation, which will be explained in the next paragraph.
5.2 The diffusion equation in its general form
The multigroup diffusion equation in its compact form is the one from which we will
derive all the others (Ref. 2):
𝐷𝑖∇2
𝜙𝑖(𝑟⃗) − Σ 𝑡,𝑖 𝜙𝑖(𝑟⃗) + ∑(Σ 𝑠,𝑗→𝑖)𝜙𝑗(𝑟⃗) + 𝜒𝑖 ∑(𝜈Σ 𝑓) 𝑗 𝜙𝑗(𝑟⃗) = 0
𝑁𝑔
𝑗 = 1
𝑁𝑔
𝑗 = 1
(5.1)
For our purpose, equation (5.1) has to be written in a more explicit form, so it will be easier
to use it inside the code, this can be done with the utilization of some approximations and
by rewriting the scattering terms into a more proper form.
The approximations considered are presented below (Ref. 2):
• ‘directly coupled’ for the elastic scattering:
∑ (Σ 𝑒𝑙,𝑖→𝑗) → (Σ 𝑒𝑙,𝑖→(𝑖 + 1))
𝑁𝑔
𝑗 = 1; 𝑗 ≠ 𝑖
∑ (Σ 𝑒𝑙,𝑗→𝑖)𝜙𝑗 → (Σ 𝑒𝑙,(𝑖 − 1)→𝑖 )𝜙(𝑖 − 1)
𝑁𝑔
𝑗 = 1; 𝑗 ≠ 𝑖
where we have taken into account only groups that either precede or follow the one
considered. This means that when the neutrons collide with a certain isotope, they can
either get out of a group (ith
group for example) to end up in the succeeding one or end up
in a certain group (ith group) from the following one. When talking about neutrons exiting
or entering a group we always refer to the energy groups, hence when losing energy the
neutrons may exit a certain group and eventually enter another one.
• ‘no up – scattering’ for the inelastic scattering:
∑ (Σ𝑖𝑛𝑒𝑙,𝑖→𝑗) →
𝑁𝑔
𝑗 = 1; 𝑗 ≠ 𝑖
∑ (Σ𝑖𝑛𝑒𝑙,𝑖→𝑗)
𝑁𝑔
𝑗 = 𝑖 + 1
∑ (Σ𝑖𝑛𝑒𝑙,𝑗→𝑖)𝜙𝑗 →
𝑁𝑔
𝑗 = 1; 𝑗 ≠ 𝑖
∑ (Σ𝑖𝑛𝑒𝑙,𝑗→𝑖)𝜙𝑗
(𝑖 − 1)
𝑗 = 1

30. 30
Which implies that neutrons that undergo inelastic scattering can only lose their energy
when colliding. This is an important condition to specify, in fact we know that neutrons in
the thermal group for example, don’t have all the same energy, which has a certain
distribution. This means that some neutrons will have more energy, and some will have
less energy than the average one. Furthermore, when a neutron having an energy value
below the average one collides, it may eventually gain energy in the process. In our case
this phenomenon is forbidden, thus simplifying the equation.
The conditions just depicted lead to the following equation (Ref. 2):
−𝐷𝑖∇2
𝜙𝑖 + (Σ 𝑐,𝑖 + Σ 𝑓,𝑖)𝜙𝑖 + (Σ 𝑒𝑙,𝑖 →(𝑖+1))𝜙𝑖 + ∑ (Σ𝑖𝑛𝑒𝑙,𝑖 →𝑗 𝜙𝑖)
𝑁𝑔
𝑗=𝑖+1
=
= (Σ 𝑒𝑙,(𝑖−1)→𝑖 𝜙(𝑖−1)) + ∑(Σ𝑖𝑛𝑒𝑙,𝑗 →𝑖 𝜙𝑗)
𝑖−1
𝑗= 1
+ 𝜒𝑖 ∑(𝜈Σ 𝑓,𝑗 𝜙𝑗)
𝑁𝑔
𝑗=1
(5.2)
that can be represented in a more compact form (Ref. 2):
𝐷𝑖∇2
𝜙𝑖 + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.3)
Here the second term is the removal macroscopic cross section, this parameter holds all
the macroscopic cross sections written on the left hand side of the equation (5.2). These
cross sections are the ones that consider the loss of neutrons, hence the name ‘removal’.
The so called source term has been identified by ′Si′ and it contains the first two terms on
the right hand side of equation (5.2). Since neutrons can be also gained through elastic and
inelastic scattering, parameters that consider these phenomena are needed, thus the source
term. The “gain” of neutrons in this case means that particles that are characterized by a
certain energy (i.e. belong to a specific energy group), end up in the ith
group through
scattering, thus increasing the neutron flux. The last term is the one identified by the letter
‘F’ and it holds the neutrons produced by fission reactions.
The diffusion equations presented above can serve several purposes based on what
approximations are used.
Consider now the multigroup diffusion equation, applied to a system big enough so that
the neutron’s diffusion length can be considered small. Here the spatial and the energy
dependence of the flux can be separated, and the Helmholtz equation used (for the spatial
dependence of the flux):
∇2
𝜙(𝑟⃗) + 𝐵2
𝜙(𝑟⃗) = 0 (5.4)
This helps get rid of the Laplacian term, when introducing the buckling inside the
diffusion equation.

31. 31
The diffusion equation thus becomes:
𝐷𝑖B2
𝜙𝑖 + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.5)
in its compact form. An iterative process can be followed in order to determine the
neutron flux for each one of the groups, with an initial guess of the buckling and the
fission term.
This is the so called zero dimensional equation or half dimensional, to be more precise,
since the spatial dependence of the neutron flux becomes implicit with the use of the wave
equation (Helmholtz equation).
However, if we decide to keep the Laplacian, a discretization process must be done, and
this will be the topic of the next paragraph.
5.3 Discretized diffusion equation
The discretization of the derivatives , in the diffusion equation, is carried over to facilitate
its usage in some common problems. The finite difference method can be used for this
purpose. We will derive the discretized diffusion equation for two main geometries,
spherical and cylindrical (with infinite height). The general equation remains the same
whereas the Laplacian term changes based on the shape of our system. Note that, unless
specified, the subscript ‘i’ will from now on refer to a generic position inside our system.
Considering a sphere of radius ‘R’ surrounded by the vacuum, the Laplacian can be
written as below:
∇2
𝜙 =
𝑑2 𝜙
𝑑𝑟2
+
2
𝑟
𝑑𝜙
𝑑𝑟
(5.6)
where the two derivatives can be discretized using the finite difference method, thus
obtaining (Ref. 3):
(
𝑑2
𝜙
𝑑𝑟2
)
𝑖
≅
𝜙𝑖 − 1 − 2𝜙𝑖 + 𝜙𝑖 + 1
∆2
(5.7)
(
𝑑𝜙
𝑑𝑟
)
𝑖
≅
𝜙𝑖 + 1 − 𝜙𝑖 − 1
2Δ
(5.8)
The complete methodology used to discretize first and a second order derivatives is
presented in Appendix A.2.

32. 32
The discretized multigroup diffusion equation becomes then (Ref. 3):
− 𝐷𝑖 [
𝜙𝑖 − 1 − 2𝜙𝑖 + 𝜙𝑖 + 1
∆2
+
2
𝑖Δ
𝜙𝑖 + 1 − 𝜙𝑖 − 1
2Δ
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.9)
where i = 0, 1, … , N is the total number of nodes in which the system has been divided
into, while the delta term identifies the distance between two contiguous nodes. The
neutron flux is now calculated not only for each one of the energy groups, but also in each
node. We will see, on the chapter that describes the code, how to convert the neutron flux
into a function. In fact, in this case for each energy group we will have a set of components
corresponding to the values that the neutron flux has along the radius. Explicating the two
terms in the parenthesis and grouping the neutron flux values for each node we get:
−
𝐷
Δ2
(1 −
1
𝑖
) 𝜙𝑖 − 1 + (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖) 𝜙𝑖 −
𝐷
Δ2
(1 +
1
𝑖
) 𝜙𝑖 + 1 = S𝑖 + 𝜒𝑖 𝐹 (5.10)
where we can define three coefficients (Ref. 3):
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2
(1 −
1
𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
𝑖
)
(5.11)
These coefficients apply for all the nodes except for the one in the origin, in order to
calculate the coefficients within this node, we have to follow a different approach. If we
look at equation (5.9) we can see how for 𝑟 → 0 the term ‘2/(i∆)’ tends to infinity. To go
around this we can use the first order Taylor’s series expansion of the generic function
‘f(x)’:
𝑓(𝑥) ≈ 𝑓(0) + 𝑥 ∗ 𝑓 ′(0) + 𝑂(𝑥2
) (5.12)
Deriving now this equation with respect to ‘x’, we obtain:
𝑑𝑓(𝑥)
𝑑𝑥
≈
𝑑𝑓(0)
𝑑𝑥
+ 𝑓 ′(0) + 𝑥 (
𝑑𝑓 ′(0)
𝑑𝑥
) (5.13)
Furthermore, by analyzing each term of equation (5.13):
•
𝑑𝑓(0)
𝑑𝑥
= 0, since f(0) is a scalar its derivative is zero;
• 𝑓 ′(0) = 0, in fact here the function is actually the neutron flux and can either be at
its maximum value (without a central cavity) or at its minimum (with a central
cavity), being in the latter case zero, when the radius is zero. Its first order
derivative has to be consequently null.

33. 33
We then obtain:
𝑑𝑓(𝑥)
𝑑𝑥
≈ 𝑥 (
𝑑𝑓 ′(0)
𝑑𝑥
) = 𝑥(𝑓 ′′(0)) (5.14)
that considering a spherical geometry becomes in which the function represents the
neutron flux (Ref. 4):
𝑑𝜙
𝑑𝑟
≈ 𝑟 (
𝑑2
𝜙
𝑑𝑟2
)
0
(5.15)
That can be also written in the following way:
2
1
𝑟
𝑑𝜙
𝑑𝑟
≈ 2 (
𝑑2
𝜙
𝑑𝑟2
)
0
(5.16)
more suitable for our purposes. This equality can be used to simplify the Laplacian term
when deriving the diffusion equation in the first node. In fact, we can write:
∇2
𝜙 =
𝑑2
𝜙
𝑑𝑟2
+
2
𝑟
𝑑𝜙
𝑑𝑟
= (
𝑑2
𝜙
𝑑𝑟2
)
0
+
2
𝑟
(
𝑑𝜙
𝑑𝑟
)
0
≈ 3 (
𝑑2
𝜙
𝑑𝑟2
)
0
(5.17)
The diffusion equation becomes for the generic node ‘i’ is:
− 𝐷𝑖 [(
𝑑2
𝜙
𝑑𝑟2
)
𝑖
+
2
𝑟𝑖
(
𝑑𝜙
𝑑𝑟
)
𝑖
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.18)
and by applying the simplification from above (where the node is the one located in the
coordinate r = 0) we have (Ref. 3):
− 𝐷𝑖 [3 (
𝑑2
𝜙
𝑑𝑟2
)
𝑖
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.19)
which represents our simplified equation in the center of the system.The same equation is
presented below, where we have highlighted the node in which it’s calculated:
− 3𝐷𝑜 (
𝑑2
𝜙
𝑑𝑟2
)
𝑜
+ Σ 𝑟𝑒𝑚,𝑜 𝜙 𝑜 − S0 = 𝜒 𝑜 𝐹 (5.20)
Proceeding with the discretization of the second order derivative, we obtain:
− 3𝐷𝑜
𝜙− 1 − 2𝜙0 + 𝜙 1
∆2
+ Σ 𝑟𝑒𝑚,𝑜 𝜙 𝑜 − S0 = 𝜒 𝑜 𝐹 (5.21)

34. 34
Where we can remove the subscript from the medium related parameters (i.e. diffusion
coefficient, removal macroscopic cross section and the neutron’s energy spectrum) if we
assume that they only change with the energy of the neutrons, remaining constant
throughout the medium (for a specific group). Furthermore, being the radial neutron flux
symmetrical with respect to the z – axis (height of the system) we have 𝜙− 1 = 𝜙 1, thus
the equation becomes:
6𝐷
𝜙0 − 𝜙1
∆2
+ Σ 𝑟𝑒𝑚 𝜙 𝑜 − S0 = 𝜒𝐹 (5.22)
That can be written in a way such as to explicitly show the coefficients:
(
6𝐷
∆2
+ Σ 𝑟𝑒𝑚) 𝜙 𝑜 −
6𝐷
∆2
𝜙1 = S0 + 𝜒𝐹 (5.23)
We then have, for the generic node ‘i’, the same coefficients derived earlier (Ref. 3):
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2
(1 −
1
𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
𝑖
)
(5.24)
and for the node located at the center of our system the coefficients are (Ref. 3):
{
𝑎0,0 = (
6𝐷
∆2
+ Σ 𝑟𝑒𝑚)
𝑎0,1 = −
6𝐷
∆2
(5.25)
We can proceed in the same way in order to derive the coefficients related to a cylindrical
geometry. Starting from the Laplacian term, that will now be different:
∇2
𝜙 =
𝑑2
𝜙
𝑑𝑟2
+
1
𝑟
𝑑𝜙
𝑑𝑟
(5.26)
adopting the approximation of finite radius and infinite height, we can neglect the neutron
flux variation with respect to the z – axis (height) and thus write the above equation.

35. 35
The coefficients are in this case (Ref. 3):
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2
(1 −
1
2𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
2𝑖
)
(5.27)
for a generic node ‘i’, while and for the one at the center of our system (Ref. 3):
{
𝑎0,0 = (
4𝐷
∆2
+ Σ 𝑟𝑒𝑚)
𝑎0,1 = −
4𝐷
∆2
(5.27)
Note that the coefficients remain the same except for multiplicative constants.
5.4 The system of equations
All the coefficients computed in the previous paragraph, when allocated into an array,
compose a tridiagonal matrix. This matrix is unique for each energy group, since the
coefficients depend on parameters that change with the group (removal macroscopic cross
section and diffusion coefficient), just like the neutron flux and the source term since we
are dealing with the multigroup diffusion equation. Writing everything in a more compact
form (Ref. 4), we will have for the generic group ‘g’:
𝐴 𝑔 𝜙 𝑔 = 𝑆′ 𝑔 (5.29)
where ‘A’ is the tridiagonal matrix belonging to the gth group and the term on the right
hand side of the equation is the source term, given by the sum of the scattering and the
fission terms.
We can easily see, from what we have written so far, how the only unknown quantity in
the above equation is the neutron flux. In fact, once the composition of the medium has
been fixed, both the coefficients of the matrix and the “source term” can be determined.
We only need to rewrite the equation so that the neutron flux is the only quantity on the
left hand side (Ref. 5):
𝜙 𝑔 = 𝐴 𝑔
−1
𝑆′ 𝑔 (5.30)
Where the resulting neutron flux is not a scalar but a vector whose components are the
values that the flux, belonging to the generic g - th group, has in all the nodes. The
computation just described has to be carried on Ng times (just like the number of groups),
so at the end we will end up with Ng vectors each with N elements.

36. 36
The iterative process that we will follow, in order to calculate the neutron flux and reach
stable conditions in our system, will be described in detail in the following chapter. Here
we will concentrate on deriving the equation that were used to write the code. The
equations derived so far apply to a full sphere/cylinder with a homogeneous medium on
the inside and surrounded by vacuum. We wish now to derive those equation when
considering the same exact geometries but with a central cavity, these equations will be
obtained in the succeeding paragraph.
5.5 Diffusion equation in presence of a cavity
The cavity here considered has no medium in it, so just like the physical boundary of our
system (in correspondence with the radius) we can assume that the net current across the
surface that separates our system from the cavity is zero. this mean that either the current
exiting the system is zero, together with the one entering it, or that they are equal. Using
this condition and knowing that in this case the first node belong to the boundary surface,
where the neutron flux is not null, we can derive the diffusion equation on the boundary
surface. This leads to the calculation of the coefficients in that node, the rest will remain
the same since the system doesn’t change. We will determine the coefficients both for
spherical and cylindrical geometry, starting with the former.
Being the net current zero means that:
𝐽 = (𝐽+) − (𝐽−) = 0 (5.31)
where the positive current (of neutrons entering our system) and the negative one
(neutrons exiting the system and entering the cavity) can be written using Fick’s law (Ref.
1) considering that our system is now a sphere:
𝐽+
=
𝜙𝑖
4
−
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
𝑖
& 𝐽−
=
𝜙𝑖
4
+
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
𝑖
(5.32)
It is important to note that when deriving Fick’s law, we have assumed stationary
conditions for the neutron flux, that will thus remain the same with time. However, if it
were to change with it, then due to the time necessary for the neutron to travel from the
scattering volume through the surface (through which we are calculating the net current)
then the flux would be known only in the time interval preceding the instant in which the
neutrons cross the surface, which is where we wish to calculate the net current.
This is due to the fact that the time necessary for the neutron to reach the surface is ‘t = r/v’
where ‘r’ is the distance separating the two points and ‘v’ is an average velocity. Now, if
the flux changes with time, it will change right before reaching the surface since the we
will be over ‘t = r/v’.

37. 37
Another thing worth noting is when we substitute the above equations in the one of the
net current, obtaining:
𝐽 = (𝐽+) − (𝐽−) = [
𝜙𝑖
4
−
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
𝑖
] − [
𝜙𝑖
4
+
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
𝑖
] = − 𝐷 (
𝑑𝜙
𝑑𝑟
)
𝑖
(5.33)
which tells us that the gradient of the flux is opposite in direction to the net current. This is
obvious if we think that the gradient is positive in the direction where the function to
which it’s applied grows. In our case the neutron flux grows in the positive direction
radial direction, when going from the cavity towards the system, since the cavity holds
low or no neutrons at all. Now, since the diffusion of particles (in this case) occurs in the
direction where we have less concentration, then the net current will be in this direction
thus opposite to the gradient. Going back to our equation now, we can write it considering
that the node ‘i’ is the one on the boundary (i = 1) and discretizing the first order equation
(Ref. 8) :
𝐽 = − 𝐷 (
𝑑𝜙
𝑑𝑟
)
1
= −𝐷 (
𝜙𝑖 + 1 − 𝜙𝑖 − 1
2Δ
)
1
= −𝐷
𝜙2 − 𝜙0
2Δ
= 0 (5.34)
Thus, leading to the following equality 𝜙2 = 𝜙0. Writing this equation for the node on the
boundary, applying the condition on the Laplacian derived earlier we have:
− 𝐷𝑖 [3 (
𝑑2
𝜙
𝑑𝑟2
)
𝑖
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.35)
Making the same assumption made earlier (on the parameters that don’t change
throughout the system radial z direction) and substituting the second derivative with its
discretized value we have:
− 3𝐷
𝜙𝑖 − 1 − 2𝜙𝑖 + 𝜙 𝑖+1
∆2
+ Σ 𝑟𝑒𝑚 𝜙𝑖 − S𝑖 = 𝜒𝐹 (5.36)
That for the node on the boundary (i = 1) becomes:
− 3𝐷
𝜙 𝑜 − 2𝜙1 + 𝜙 2
∆2
+ Σ 𝑟𝑒𝑚 𝜙1 − S1 = 𝜒𝐹 (5.37)
and with the condition on the flux (𝜙2 = 𝜙0) we final get to the multigroup diffusion
equation on the boundary surface between the cavity and the medium:
− 3𝐷
2𝜙2 − 2𝜙1
∆2
+ Σ 𝑟𝑒𝑚 𝜙1 − S1 = 𝜒𝐹 (5.38)
(
6𝐷
∆2
+ Σ 𝑟𝑒𝑚) 𝜙1 −
6𝐷
∆2
𝜙2 − S1 = 𝜒𝐹 (5.39)

38. 38
The situation changes if the cavity, instead of being empty, holds something that injects a
current of neutrons into our system. We will now have a positive current of neutrons
different from zero. Since we are still near the boundary the condition on the Laplacian
stays, but we have to modify the one on the current, that now becomes:
𝐽+
= [
𝜙𝑖
4
−
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
𝑖
] = 𝐽𝑖 ≠ 0 (5.40)
where ‘𝐽𝑖’ is the numerical value of the current, value that in the code can be chosen at will.
Referring once again to ‘i = 1’ we can write:
𝐽+
= [
𝜙1
4
−
𝐷
2
(
𝑑𝜙
𝑑𝑟
)
1
] =
𝜙1
4
−
𝐷
2
(
𝜙𝑖 + 1 − 𝜙𝑖 − 1
2Δ
)
1
(5.41)
𝐽+
=
𝜙1
4
−
𝐷
2
𝜙2 − 𝜙0
2Δ
= 𝐽1 (5.42)
Rewriting the above equation in order to have ‘𝜙0’ on the left hand side:
𝜙0 = −
4Δ
𝐷
(
𝜙1
4
− 𝐽1 −
𝐷
4∆
𝜙2) (5.43)
and substituting once again in the multigroup diffusion equation evaluated on the
boundary of our system:
− 3𝐷
𝜙 𝑜 − 2𝜙1 + 𝜙 2
∆2
+ Σ 𝑟𝑒𝑚 𝜙1 − S1 = 𝜒𝐹 (5.44)
We finally get:
(
6𝐷
∆2
+
3
∆
+ Σ 𝑟𝑒𝑚) 𝜙1 −
6𝐷
∆2
𝜙2 −
12
∆
𝐽1 = S1 + 𝜒𝐹 (5.45)
which is the multigroup diffusion equation in spherical geometry with a central cavity,
with a current of neutrons entering the system and exiting the cavity. The equation has
been derived on the boundary between the cavity and the system, so the only coefficients
that need modification are then ones belonging to that node; the rest of them stays the
same.
The same can be done when considering a cylinder with finite radius and infinite height.

39. 39
We have already derived the matrix coefficients when considering a full cylinder:
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2
(1 −
1
2𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
2𝑖
)
(5.46)
for a generic node ‘i’, while for the one in the center of our system:
{
𝑎0,0 = (
4𝐷
∆2
+ Σ 𝑟𝑒𝑚)
𝑎0,1 = −
4𝐷
∆2
(5.47)
The condition regarding the Laplacian, when the radius is approaching zero, doesn’t
change when considering a different geometry; in fact, no assumptions related to the
shape of the system have been made. However, being the Laplacian term different based
on the shape of our system, the diffusion equation will also be different. The equation,
with the Laplacian term explicitly written is:
− 𝐷𝑖 [(
𝑑2
𝜙
𝑑𝑟2
)
𝑖
+
1
𝑟𝑖
(
𝑑𝜙
𝑑𝑟
)
𝑖
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.48)
While the condition regarding the middle of our system is:
𝑑𝜙
𝑑𝑟
≈ 𝑟 (
𝑑2
𝜙
𝑑𝑟2
)
0
(5.49)
This leads to the following equation (Ref. 3):
− 𝐷𝑖 [2 (
𝑑2
𝜙
𝑑𝑟2
)
𝑖
] + Σ 𝑟𝑒𝑚,𝑖 𝜙𝑖 − S𝑖 = 𝜒𝑖 𝐹 (5.50)
where basically the only difference stands in the numerical coefficient of the second order
derivative.
Discretizing the derivative and remembering that the parameters related to the
composition of the medium do not change throughout our system, being different only
when considering different energy groups. The equation then becomes:
− 2𝐷
𝜙 𝑖− 1 − 2𝜙𝑖 + 𝜙𝑖+ 1
∆2
+ Σ 𝑟𝑒𝑚 𝜙𝑖 − S𝑖 = 𝜒𝐹 (5.51)

40. 40
Now, when dealing with a central cavity we need to introduce a boundary condition that
will help us rewrite the flux inside the cavity (𝜙0), as a function of the ones inside our
system. In fact, when considering the node on the boundary surface (i = 1), the above
equation becomes:
− 2𝐷
𝜙 0 − 2𝜙1 + 𝜙2
∆2
+ Σ 𝑟𝑒𝑚 𝜙1 − S1 = 𝜒𝐹 (5.52)
When the central cavity has no medium in it, the condition is that the net current on the
boundary surface is zero; this leads to the following equality on the neutron flux:
𝜙2 = 𝜙0 (5.53)
and the discretized diffusion equation then becomes (Ref. 6):
− 2𝐷
2𝜙2 − 2𝜙1
∆2
+ Σ 𝑟𝑒𝑚 𝜙1 − S1 = 𝜒𝐹 (5.54)
(
4𝐷
∆2
+ Σ 𝑟𝑒𝑚) 𝜙1 −
4𝐷
∆2
𝜙2 − S1 = 𝜒𝐹 (5.55)
However, if an external source of neutrons is introduced into the cavity, a positive non
zero current will be entering our system. Once again, this boundary condition helps us
rewrite the neutron flux inside the cavity:
𝜙0 = −
4Δ
𝐷
(
𝜙1
4
− 𝐽1 −
𝐷
4∆
𝜙2) (5.56)
This equation has to be introduced in the diffusion equation; with some simplifications
and adjusting it we get:
(
4𝐷
∆2
+
2
∆
+ Σ 𝑟𝑒𝑚) 𝜙1 −
4𝐷
∆2
𝜙2 −
8
∆
𝐽1 = S1 + 𝜒𝐹 (5.57)
Which is the diffusion equation derived on the boundary surface between our system and
the cavity. This equation, just like the previous ones regarding the first node, apply to all
the energy groups and change for each one of them since we have quantities that are
function of the energy of the neutrons.
These parameters are the diffusion coefficient, the removal macroscopic cross section, the
neutron energy spectrum and the source term; the fission term changes too but in this case
it represent the sum over all the groups, so it is the same in every equation.
It is now useful to rewrite the coefficients for the two geometries and in each one of the
three cases underlined throughout this chapter. First we will present the coefficients of all
the nodes except for the first one, since the latter changes when modifications are applied
in that position.

41. 41
Regardless of the composition of the cavity, the coefficients for the spherical and
cylindrical geometries are respectively:
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2 (1 −
1
𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
𝑖
)
(5.58𝑎)
{
𝑎𝑖,𝑖 − 1 = −
𝐷
Δ2 (1 −
1
2𝑖
)
𝑎𝑖,𝑖 = (
2𝐷
Δ2
+ Σ 𝑟𝑒𝑚,𝑖)
𝑎𝑖,𝑖+1 = −
𝐷
Δ2
(1 +
1
2𝑖
)
(5.58𝑏)
When the sphere and the cylinder have a central cavity with no medium in it, the
coefficients of the first node are respectively:
{
𝑎0,0 = (
6𝐷
∆2
+ Σ 𝑟𝑒𝑚)
𝑎0,1 = −
6𝐷
∆2
(5.59𝑎) {
𝑎0,0 = (
4𝐷
∆2
+ Σ 𝑟𝑒𝑚)
𝑎0,1 = −
4𝐷
∆2
(5.59𝑏)
However, when an external source of neutrons is introduced into the cavity, the
coefficients become:
{
𝑎1,1 = (
6𝐷
∆2
+
3
∆
+ Σ 𝑟𝑒𝑚)
𝑎1,2 = −
6𝐷
∆2
(5.60𝑎) {
𝑎1,1 = (
4𝐷
∆2
+
2
∆
+ Σ 𝑟𝑒𝑚)
𝑎1,2 = −
4𝐷
∆2
(5.60𝑏)
In this case, the right hand side of the equation changes too. In fact, since neutrons are
introduced also from an external source, the source terms changes and become:
{
𝑆1
′
= S1 + 𝜒𝐹 +
12
∆
𝐽1
𝑆1
′
= S1 + 𝜒𝐹 +
8
∆
𝐽1
(5.61)
For spherical and cylindrical geometry respectively.

42. 42
Chapter 6
The iterative process
This chapter will be dedicated to the description of the iterative process followed to
determine the neutron flux in our system. We will basically derive all the equations that
will be later used in the code, after going through a transformation. We have in fact said
how the software used utilizes arrays for any type of computation, this means that each
equation that will there be used, needs to be converted into an array sort of form. This will
be more clear in the eight chapter.
After giving a brief recall of the terms that form the multigroup diffusion equation, we
will see how to determine the subcritical neutron flux in the one dimensional calculation
with and without the introduction of an external source.
6.1 The terms of the diffusion equation
Now that we have derived the necessary equations, we can describe the iterative process
that will lead to the calculation of the subcritical neutron flux. In the past chapter we have
derived a set of equations for both spherical and cylindrical geometry, from now on we
will only concentrate on the latter. In particular, our system will be a cylinder with a finite
radius and infinite height, with a central cavity where we will eventually insert an external
source. The multigroup diffusion equation has to be discretized, since we are dealing with
a one dimensional problem, just like we have seen in the previous chapter. It’s important
to note how the generic equation now will refer not only to a certain energy group, but
also to a particular node. We will write this explicitly by using an index for the nodes (i =
1, 2, …,N) and a subscript for the groups (g = 1, 2, …, Ng). This was not necessary in the
chapter where we have derived all the equations, since we were only concentrating on the
nodes and each equation would refer to a single energy group. Below we have the
complete multigroup diffusion equation (Ref. 2):
−𝐷𝑔∇2
𝜙 𝑔
𝑖
+ (Σ 𝑐,𝑔 + Σ 𝑓,𝑔)𝜙 𝑔
𝑖
+ (Σ 𝑒𝑙,𝑔 →(𝑔+1))𝜙 𝑔
𝑖
+ ∑ (Σ𝑖𝑛𝑒𝑙,𝑔→𝑔′ 𝜙 𝑔
𝑖
)
𝑁𝑔
𝑔′=𝑔+1
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+ 𝜒 𝑔 ∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
(6.1)

43. 43
Equation (6.1) can still be represented in a more compact form where we have grouped
some of the terms:
• Σ 𝑟𝑒𝑚,𝑔 𝜙 𝑔
𝑖
= (Σ 𝑐,𝑔 + Σ 𝑓,𝑔)𝜙 𝑔
𝑖
+ (Σ 𝑒𝑙,𝑔 →(𝑔+1))𝜙 𝑔
𝑖
+ ∑ (Σ𝑖𝑛𝑒𝑙,𝑔 →𝑔′ 𝜙 𝑔
𝑖
)
𝑁𝑔
𝑔′=𝑔+1
• 𝑆 𝑔
𝑖
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′→𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′=1
• 𝜒 𝑔 𝐹 = 𝜒 𝑔 ∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
We will now recall the meaning of each one of them and see it in terms of the iterative
process. The first parameter is the removal macroscopic cross section and it represents the
loss of neutrons from a certain energy group due to capture, fission and scattering.
Regarding the elastic scattering we only consider coupled groups and down scattering, so
when neutrons belonging to a certain group elastically collide end up in the subsequent
group (g + 1). When considering the inelastic scattering, the only approximation that we
take is the down scattering of the neutrons when a collision occurs.
Having no restrictions on the groups the neutrons can scatter to any of them, reason why
the sum goes from the successive one (g + 1) to the last one (Ng). Of course, whether a
neutron can actually end up in only one of the successive groups or in each one of them
depends on the inelastic scattering cross section, i.e. on the element considered. As we can
see, the removal term is characterized by the fact that the neutron flux is the one of the g-
th group, so the one that the diffusion equation refers to. Hence, we can highlight it. This
for example can’t be done with the flux in the first term, with the Laplacian, since the
operator directly applies to it. In this case the discretization will become helpful.
The second term is the so called source term, and as the name recalls it takes into account
the neutrons that end up in the group we are considering. From this term we exclude the
neutrons produced by fission reactions (since they will appear in another term), so we
only have scattering neutrons. Just like in the removal term, we can have neutrons that go
through elastic scattering but only from coupled groups, this means that we will only
consider neutrons coming from the previous energy group (g – 1). When considering the
inelastic scattering, the coupling disappears but we can only have down scattering. Taking
a generic group ‘g’, the neutrons that can end up in it are the ones that go from the most
energetic group (Ng = 1) to the one shortly before the present (g – 1). In this case the
situation, regarding the neutron flux, is a bit different because the flux refers to the groups
that precede the one considered so we cannot write that term as a product of the flux and a
macroscopic cross section. Here the neutron flux is implicitly contained into the source
term. The last term is the fission term, that contains the sum over all the energy groups.
Here in fact we will have the flux belonging to all the groups. This is also a source term
since of all the neutrons coming out of the fission reactions (over all groups), we only take
the only related to the ‘g’ group, and we do this by multiplying the sum by the energy
spectrum of that group.

44. 44
Before proceeding with the description of the iterative process, it is useful to rewrite the
diffusion equation by explicating the Laplacian term (referring to a cylinder with finite
radius and infinite height):
−𝐷𝑔 [
𝑑2
𝑑𝑟2
+
1
𝑟
𝑑
𝑑𝑟
] 𝜙 𝑔
𝑖
+
+(Σ 𝑐,𝑔 + Σ 𝑓,𝑔)𝜙 𝑔
𝑖
+ (Σ 𝑒𝑙,𝑔 →(𝑔+1))𝜙 𝑔
𝑖
+ ∑ (Σ𝑖𝑛𝑒𝑙,𝑔 →𝑔′ 𝜙 𝑔
𝑖
)
𝑁𝑔
𝑔′=𝑔+1
=
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+ 𝜒 𝑔 ∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
(6.2)
6.2 First iterative process
We wish to compute the neutron flux for a subcritical system, this will then lead to the
calculation of the rate of production and burning of respectively tritium and americium.
Since we need a sub critical system, first we need to find the right composition. In general,
we talk about the critical composition, but in this case we need a medium that keeps our
system sub critical, given fixed dimensions. The first iterative process will serve to reach a
sub critical system (we get to pick the effective multiplication factor of the system) with
fixed dimension, this quantity too gets chosen at the beginning. The equation that we will
consider in the iterative process is somehow different from the previous one, having the
multiplication factor on the right hand side since that’s the quantity we are iterating on:
−𝐷𝑔 [
𝑑2
𝑑𝑟2
+
1
𝑟
𝑑
𝑑𝑟
] 𝜙 𝑔
𝑖
+
+(Σ 𝑐,𝑔 + Σ 𝑓,𝑔)𝜙 𝑔
𝑖
+ (Σ 𝑒𝑙,𝑔 →(𝑔+1))𝜙 𝑔
𝑖
+ ∑ (Σ𝑖𝑛𝑒𝑙,𝑔 →𝑔′ 𝜙 𝑔
𝑖
)
𝑁𝑔
𝑔′=𝑔+1 =
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+
𝜒 𝑔
𝑘
∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
(6.3)
Or in its compact form:
𝐷𝑔∇2
𝜙 𝑔
𝑖
+ Σ 𝑟𝑒𝑚,𝑔 𝜙 𝑔
𝑖
− S 𝑔 =
𝜒 𝑔
𝑘
𝐹 (6.4)
Now that the equation is set, we need to check that we have all the parameters we need,
and this can be done through the script that holds all the material related quantities.

45. 45
This will include the diffusion coefficient for each group, the macroscopic cross sections,
the neutron energy spectrum, etc. Then we need to choose a set of initial values like the
multiplication factor and the dimensions of the system; of these, the former changes with
each iteration while the latter stays the same throughout the whole process. Furthermore,
we need to pick the composition, in fact the multiplication factor changes because so does
the composition of our system. In this situation we chose to modify the composition by
changing the enrichment, in U235, of the uranium compound; we can obviously choose
another quantity, for example the enrichment, in any of his isotopes, of plutonium. The
reason why we haven’t considered the volumetric fraction, is that by changing this
quantity the code slows down too much, but in general any quantity can be picked.
Once all these quantities have been settled, we can proceed with the first iterative process
that leads to the calculation of the multiplication factor. The convergence condition will be
on two successive values of the ‘k’ (Ref. 4):
|𝑘 𝑡+1
− 𝑘 𝑡|
𝑘 𝑡+1
< 𝜖 𝑘 (6.5)
Where the index stands for the number of iterations while on the right hand side we have
the error, whose value we can pick at will. Together with this test we should actually
include the one on the neutron flux, but as we have said in the previous chapters, it slows
down the running time too much with little or no improvement in the results. The
condition is the following one (Ref. 4):
𝑚𝑎𝑥
|𝜙𝑖
(𝑡+1)𝑔
− 𝜙𝑖
(𝑡)𝑔
|
𝜙𝑖
(𝑡+1)𝑔
< 𝜖 𝜑 (6.6)
where the maximum has to be evaluated in each node and for each one of the groups, in
every single iteration. In a system where we have N = 40 nodes and Ng = 6 groups, the
condition has to be verified 240 times per iteration and when considering 26 energy
groups that number goes up to 1040.
In general, until these two conditions have been verified (we will use a while loop to do
this) the iterative process goes on. We will see the single steps in the next paragraph.

46. 46
6.3 Calculation of the neutron flux
In order to determine the neutron flux, first we need to discretize the Laplacian term and
the equation becomes (Ref. 3):
−𝐷𝑔 [
𝜙 𝑔
𝑖−1
− 2𝜙 𝑔
𝑖
+ 𝜙 𝑔
𝑖+1
∆2
+
1
𝑖Δ
𝜙 𝑔
𝑖+1
− 𝜙 𝑔
𝑖−1
2Δ
] +
+(Σ 𝑐,𝑔 + Σ 𝑓,𝑔)𝜙 𝑔
𝑖
+ (Σ 𝑒𝑙,𝑔 →(𝑔+1))𝜙 𝑔
𝑖
+ ∑ (Σ𝑖𝑛𝑒𝑙,𝑔 →𝑔′ 𝜙 𝑔
𝑖
)
𝑁𝑔
𝑔′=𝑔+1 =
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+
𝜒 𝑔
𝑘
∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
) (6.7)
𝑁𝑔
𝑔′=1
This equation can be written in a different form, by grouping the terms that refer to the
neutron flux in different nodes (see the index). Proceeding this way we obtain:
−
𝐷𝑔
Δ2
(1 −
1
2𝑖
) 𝜙 𝑔
𝑖−1
−
𝐷𝑔
Δ2
(1 −
1
2𝑖
) 𝜙 𝑔
𝑖+1
+ (
2𝐷𝑔
Δ2
+ (Σ 𝑐,𝑔 + Σ 𝑓,𝑔) + (Σ 𝑒𝑙,𝑔 →(𝑔+1)) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔→𝑔′)
𝑁𝑔
𝑔′=𝑔+1
) 𝜙 𝑔
𝑖
=
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+
𝜒 𝑔
𝑘
∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
) (6.8)
𝑁𝑔
𝑔′=1
Defining now the removal macroscopic cross section as we have already done:
Σ 𝑟𝑒𝑚,𝑔 = (Σ 𝑐,𝑔 + Σ 𝑓,𝑔) + (Σ 𝑒𝑙,𝑔 →(𝑔+1)) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔→𝑔′)
𝑁𝑔
𝑔′=𝑔+1
(6.9)
We can rewrite the equation as follows:
−
𝐷𝑔
Δ2
(1 −
1
2𝑖
) 𝜙 𝑔
𝑖−1
−
𝐷𝑔
Δ2
(1 −
1
2𝑖
) 𝜙 𝑔
𝑖+1
+ (
2𝐷𝑔
Δ2
+ Σ 𝑟𝑒𝑚,𝑔) 𝜙 𝑔
𝑖
=
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+
𝜒 𝑔
𝑘
∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
(6.10)

47. 47
Equation (6.10) can be written in a more compact form, when defining the coefficients
respectively as 𝑎 𝑔
𝑖,𝑖−1
, 𝑎 𝑔
𝑖,𝑖+1
𝑎𝑛𝑑 𝑎 𝑔
𝑖,𝑖
:
𝑎 𝑔
𝑖,𝑖−1
𝜙 𝑔
𝑖−1
+ 𝑎 𝑔
𝑖,𝑖+1
𝜙 𝑔
𝑖+1
+ 𝑎 𝑔
𝑖,𝑖
𝜙 𝑔
𝑖
=
= (Σ 𝑒𝑙,(𝑔−1)→𝑔 𝜙(𝑔−1)
𝑖
) + ∑ (Σ𝑖𝑛𝑒𝑙,𝑔′ →𝑔 𝜙 𝑔′
𝑖
)
𝑔−1
𝑔′= 1
+
𝜒 𝑔
𝑘
∑ (𝜈Σ 𝑓,𝑔′ 𝜙 𝑔′
𝑖
)
𝑁𝑔
𝑔′=1
(6.11)
The equation above refers to a particular node (even though we also have the neutron flux,
belonging to the same energy group, of the previous and subsequent nodes) and a
particular energy group, this means that when calculating the neutron flux for a set of
nodes (linked to the same group) we will have to deal with a matrix. We will then have a
set of equations, written in matrix form, that will refer to each one of the groups. As an
example, we can write the set of equations for each node and then represent it into matrix
form. The above equation can be written into a more compact form, using the symbol
(previously described) for the source term and for the fission term:
𝑎 𝑔
𝑖,𝑖−1
𝜙 𝑔
𝑖−1
+ 𝑎 𝑔
𝑖,𝑖+1
𝜙 𝑔
𝑖+1
+ 𝑎 𝑔
𝑖,𝑖
𝜙 𝑔
𝑖
= 𝑆 𝑔
𝑖
+
𝜒 𝑔
𝑘
𝐹 (6.12)
Now, just for the example purpose, we can imagine the energy range divided into 3
groups and our system into 5 nodes (i = 0, 1, …, 4). We will then have three set of
equations (just like the number of groups), and each set will hold a certain number of
diffusion equations (just like the nodes, plus some corrections). Since the total number of
equations can be quite big when considering a finer division of the energy groups and a
larger number of nodes, we have picked only a restrained set of the latter and a coarse
division of the energy range (first group for fast neutrons, second group for epithermal
and last group for thermal).
For each one of these sets of equations, we should have 5 equations each just like the total
number of nodes. However, since we are considering a cylindrical geometry with a central
cavity the node i = 0 will not be taken into consideration (the neutron flux there is zero,
since the cavity doesn’t hold any mediums). Another, more mathematical than physical,
condition is that the flux falls to zero on the boundary of our system (node 4); this of
course is not completely true since we will always have neutrons exiting the system (and
eventually entering it if a shielding material is located outside the core).

48. 48
Since the neutron flux in these two nodes is zero, we don’t need to write the set of
equations related to them, this allows us to have three set of equations (Ref. 4) with three
equations each:
{
𝑎1
1,1
𝜙1
1
+ 𝑎1
1,2
𝜙1
2
= 𝑆1
1
+
𝜒1
𝑘
𝐹
𝑎1
2,1
𝜙1
1
+ 𝑎1
2,3
𝜙1
3
+ 𝑎1
2,2
𝜙1
2
= 𝑆1
2
+
𝜒1
𝑘
𝐹
𝑎1
3,2
𝜙1
2
+ 𝑎1
3,3
𝜙1
3
= 𝑆1
3
+
𝜒1
𝑘
𝐹
(6.13𝑎)
{
𝑎2
1,1
𝜙2
1
+ 𝑎2
1,2
𝜙2
2
= 𝑆2
1
+
𝜒2
𝑘
𝐹
𝑎2
2,1
𝜙2
1
+ 𝑎2
2,3
𝜙2
3
+ 𝑎2
2,2
𝜙2
2
= 𝑆2
2
+
𝜒2
𝑘
𝐹
𝑎2
3,2
𝜙2
2
+ 𝑎2
3,3
𝜙2
3
= 𝑆2
3
+
𝜒2
𝑘
𝐹
(6.13𝑏)
{
𝑎3
1,1
𝜙3
1
+ 𝑎3
1,2
𝜙3
2
= 𝑆3
1
+
𝜒3
𝑘
𝐹
𝑎3
2,1
𝜙3
1
+ 𝑎3
2,3
𝜙3
3
+ 𝑎3
2,2
𝜙3
2
= 𝑆3
2
+
𝜒3
𝑘
𝐹
𝑎3
3,2
𝜙3
2
+ 𝑎3
3,3
𝜙3
3
= 𝑆3
3
+
𝜒3
𝑘
𝐹
(6.13𝑐)
Where first of all we can see how the multiplication factor stays the same, since it is
representative of the whole system. Moreover, each set of equations holds a different value
of the neuron’s energy spectrum (𝜒 𝑔), in fact the term ‘F’ expresses the neutrons coming
out of every fission reaction in our system and of these neutrons we will only consider the
ones that belong to the group the equation refers to. So, when considering the first group
we will only consider the neutrons that come out of fission reactions with an energy in
that range, and so on. The source term has the same meaning, only in this case we consider
scattering reactions. The neutrons are the ones that, from other groups, end in ours so the
‘S’ term will refer to the gth
group. Note that the terms holding the neutron flux in the first
(i = 0) and last (i = 4) node are zero, since the neutron flux there is null.
Writing the system down into matrix form we have:
𝜙 = [𝜙1
1
, 𝜙1
2
, 𝜙1
3
, 𝜙2
1
, 𝜙2
2
, 𝜙2
3
, 𝜙3
1
, 𝜙3
2
, 𝜙3
3] 𝑇
(6.14)
Where the subscript refers to the energy group and the index to the node the flux refers to.