#### Foreword

This treatise, of less than 400 pages, manages to cover the entire field of
nuclear power production, from *E = mc*^{2} almost, but not quite, to the construction
of a thermonuclear weapon. (No need for that anyway, as Google
will provide many feasible designs.) It covers reactor physics in detail, including almost all popular (and many unpopular) calculational tools; experimental
techniques; engineering plans; and almost anything else relevant to
the design and operation of a nuclear power plant (including state-of-theart
descriptions of what’s going on around the world of nuclear power. It
even describes uranium enrichment, via gaseous diffusion and centrifuges,
but does stop short of explaining how tritium is produced, an essential ingredient
in boosting weapon yields.) Some mathematical and engineering
details are added in a series of five appendices. Throughout, the three authors
display their deep understanding of the physical, mathematical and
engineering processes underlying the nuclear power industry.

A theoretical physicist, uninterested in experiment or engineering details,
could still profit greatly from this book. As could an engineer or an experimental
physicist, or chemist, for that matter. She would only have to
skip the irrelevant portions of the text. But is this the right thing to do? In
this day of holistic information theory I think a reactor scientist, whatever
her academic background, would be well served to have at least a minimal
amount of contact with all aspects of the field. This has been emphasized
to me by papers I have listened to devotedly at the biannual ICTT^{2} meetings,
which in the past few years have begun to move from isolated topics
in mathematics and physics (primarily) to actual, detailed descriptions of
(all) the processes taking place in nuclear power plants. The senior author
of this book, a regular attendee at these meetings and himself the organizer
of one, is/was surely not unaware of this trend, and, indeed, it is tempting to
speculate that the schema of this book was inspired by the papers presented
at these meetings.

It is interesting to contrast what might be called this "modern" approach to
the way I learned about reactors. I had my first introduction to the art in the
summer of 1950. I was still a graduate student at Duke University, but my
thesis adviser, Eugene Greuling (co-inventor of the Greuling-Goertzel approximation
still used in slowing-down calculations, and described in this
book) arranged for me to spend the summer working with him at Oak Ridge,
on the design of a nuclear reactor for aircraft propulsion. It seems like an
idiotic idea now, but at that time when air-to-air refueling had not been invented
it was clear that a chemically-fueled bomber would never have the
range to fly from the U.S. to the U.S.S.R and return safely. Adding more fuel
simply increased the weight of the aircraft without extending the range. I
tried vainly to learn a little reactor physics, but the only text available was
a short monograph by Soodak and Campbell (Harry Soodak was a fellow
Dukie, by the way, but already had his degree). The book simply did not
go into enough detail to give me a grasp of the fundamentals. The aircraft
project sputtered along for several years before the Boeing Company figured
out the better solution mentioned above.

Later that year, as I was the only graduate student at Duke with a Q-clearance,
I was assigned to work on a secret contract Duke had received from the U.S.
government to help design something misnamed the "MTR" for "materials
testing reactor". However it was not a reactor at all, but a high-energy
proton accelerator designed to bombard depleted U-238 and change it into
Pu-239. This scheme, invented by E.O. Lawrence, was set up at a secret
laboratory built expressly for him, now known as the Lawrence Livermore
Laboratory. I never knew that the laboratory was supposed to be secret,
and I talked about it to many uncleared friends before its existence, as a
previously secret entity, appeared in the newspapers. (I was lucky not to be
prosecuted, as this happened at the height of Sen. McCarthy’s reign). I don’t
think I contributed much to the project; at least, when I won the Lawrence
medal in 1972, this project was not mentioned in my citation. One problem
might have been that a few moments after the accelerator was finally turned
on the entire uranium target melted due to heat deposition from the proton
beam, when the program itself its quietus made. Here is a prime example, I
aver, of why engineering should not be separated from physics. Another is
the aircraft reactor project, where a savvy aeronautical engineer/pilot might
have figured out a better and much cheaper way to threaten the U.S.S.R from
the very beginning.

After graduate school, in 1953, I went to work at the Knolls Atomic Power
Laboratory, designing reactors for submarines. Five years later I joined
the Nuclear Engineering Department at the University of Michigan. In the
interim, I taught a night course in reactor physics at Union College in Schenectady,
where the Knolls Lab was located. The only text available to me
(besides the previously cited Soodak and Campbell) was the reactor physics
book of Glasstone and Edlund, so I used that, as inadequate as it was (already
ten years out of date). I did learn a great deal of reactor physics, but
I never really learned how reactors worked. Things like fuel management,
heat transfer, nucleate boiling, accident prevention, etc., all topics discussed
in detail in this book, were a deep dark mystery to me. When it became apparent
that I would have to learn all that stuff if I wanted to remain a nuclear
engineer, I left the University of Michigan, joined the physics department
at Virginia Tech and resigned as a Fellow of the American Nuclear Society.
Kenneth Case, physics professor at the University of Michigan, had
already begun to transform me into a mathematical physicist, a transformation
exacerbated by my excellent relations with the mathematics department
at Virginia Tech. And yet, if I had stayed in reactor physics? Who knows
what might have been. In the early 1990’s, more than twenty years after
leaving Michigan for Virginia Tech, I was invited to participate in my
first American Nuclear Society meeting after many years away. I couldn’t
believe the adulation I drew. Everybody knew my name and my work in
reactor physics, even after so many years. I lost my chance to be famous
by forsaking nuclear engineering. Perhaps if this book had been available
then, my entire life would have been different!

**Paul Zweifel**

Virginia Tech University

#### Preface

The subject of the present work is the set of computational tools applied
in the description, operation, and control of man-made neutron multiplying
systems. Those systems may be used as neutron source, as various types of
research reactors, to study phenomena taking place in large-scale industrial
devices as with zero power reactors, or to produce energy in a power reactor.
The book aims at making the reader understand the main features of neutron
multiplying systems. Understanding is provided by a limited number of
fundamental physical models. In the models, a key point is the interaction
of neutrons with various nuclei, a question discussed in nuclear physics.
Once we are interested in the neutron population, those nuclear reactions
are classified from the point of view of neutron balance and a set of crosssections
1 is provided for every nucleus. The characterization of the material
requires more than specifying the XSs. The number of reaction per unit
time, the so called reaction rate, depends on the density of the nucleus and
the relative speed which depends upon the thermal motion of the nucleus.
A further essential component of a neutron multiplying system is the geometry.
Cross-sections, geometry, material composition and temperature fully
Once the geometry and material composition are known, we can write down
equations which govern the neutron distribution in space, time, and neutron
energy. Step-by-step we build up a sequence of models intended to master
the technique of designing, measuring and controlling neutron multiplying
systems. Simple models are utilized to introduce basic terms and concepts;
more sophisticated models are needed to take account of practical situations.
As the reader will see, none of these are describable simply. The XSs
depend on the neutron energy and exhibit a large variety of energy dependence.
Most multiplying systems are heterogeneous, thus the geometry is far from being simple. This is the reason for the extended use of numerical
models in the solution of the equations.

The text uses an extended amount of mathematics and physics. From the
former, the reader may need expertise in linear algebra, numerical methods,
and statistics. From the latter, some nuclear physics, mechanics, and statistical
physics are the prerequisites. In a limited book it is impossible to
touch upon the underlying mathematical and physical details. To fill that
gap, textbooks are recommended to collect deeper knowledge on each specific
subject.

The structure of the book is the following. Chapter 1 is a survey of the
problems discussed here in connection with reactor calculations. The six
subjects are treated only briefly, the emphasis being placed on the relations
among the subjects. Each subject will be unfolded in a separate subsequent
chapter. The first subject, in Section 2 discusses the components of the reactor
models and their relationships. Chapter 2 lays down the theoretical
background of the reactor calculation. Chapter 3 deals with problems that
can be solved exactly. Chapter 4 is a survey of approximate solution methods.
The approximate methods in Chapter 5 are organized into a calculational
model to solve specific problems of reactor physics. The next Chapter
6, is devoted to a generally utilized approximation, diffusion theory. Some
of the approximation relies on the spectral properties of the neutron distribution
which is the subject of Chapter 7. The result of the approximation
methods is a system of equations that we solve by the numerical methods
discussed in Chapter 8. In reactor physics, the temporal behavior of the reactor
is of primary importance, and this is the topic of Chapter 9. Chapters
10-11 deal with specific applications of the afore-mentioned reactor physical
models. Chapter 12 gives a brief overview of nuclear reactor types
applied for electricity generation. Some special problems are summarized
in the Appendices.

The models we present have been selected to correspond to the practical
tasks involved with neutron multiplying systems: design, measurement design
and processing (instrumentation and control), and operation.

The authors are grateful to Paul Zweifel and Yuri Orechwa for their advice
and editing the text. If any error has remained, it is the fault of the authors.

Collaboration with Dr. Ferenc Wettl in LaTeX problems is gratefully acknowledged.

#### List of Contributors

##### Author(s):

**Mihály Makai **

**Dániel Péter Kis **

**János Végh **

#### Reviews

##### Review 1

**Endorsement:**

*"Proposes a broader discussion of reactor technology… Provides an excellent discussion of modern analytical techniques and computational tools applied in the design, analysis, operation and control of neutron multiplying systems.” – Journal of Nuclear Engineering & Radiation Science, Volume 2, Issue 3, July 2016 – The American Society of Mechanical Engineers (ASME).*

https://nuclearengineering.asmedigitalcollection.asme.org/article.aspx?articleID=2546702