FISSIONING URANIUM PLASMA
FOR ROCKET PROPULSION
By
HENRY FOUST ATWATER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
To my Father and Mother,
for their continued, encouragement.
ACKNOWLEDGEMENTS
The author expresses his sincere appreciation to
faculty members and fellow students for their interest and
assistance during the preparation of this dissertation.
Special appreciation is extended to Dr. R. B. Perez, who
served as chairman of the author's graduate committee,
until leaving to accept another position. The author
appreciates the guidance of Dr. M. J. Ohanian, who assumed
the duties of chairman following Dr. Perez's departure.
The author is extremely grateful to Dr. H. E. Wilhelm for
many explanations and suggestions which were essential to
the entire study. The participation of Dr. W. H. Ellis is
appreciated. Several helpful suggestions from Dr. R. T.
Schneider are acknowledged. The author also thanks
Dr. R. G. Blake and Dr. R. A. Blue for their participation.
Special appreciation is extended to Mr. W. G.
Wolfer for patiently explaining the notation of molecular
and atomic spectroscopy. The author appreciates the
invaluable assistance so generously provided by Mrs. Jennie
Grossman of the University of Florida Computing Center.
The financial assistance received from two Atomic
Energy Commission Special Fellowships and from the National
iii
Aeronautics and Space Administration is gratefully ac
knowledged. This study was performed under NASA Grant
NGR 10005068.
It is a pleasure to thank my wife, Vann, for
typing the first draft.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ..................................... ii
LIST OF TABLES ........................................ vii
LIST OF FIGURES .................................... viii
ABSTRACT ................ ............................ xi
Chapter
I. INTRODUCTION .................................. 1
II. APPLICATION OF DIFFUSION THEORY TO
NEUTRONS IN A FISSIONING URANIUM
PLASMA ....................................... 11
III. NUCLEONIC ANALYSIS OF EXTERNALLY
MODERATED URANIUM PLASMA .................. 41
IV. PRESSURETEMPERATURE ANALYSIS OF
URANIUM PLASMA FUEL REGION ................ 50
V. PRESSURETEMPERATURE ANALYSIS OF
HYDROGEN PLASMA COOLANT REGION ............ 82
VI. PRESSURETEMPERATURE ANALYSIS OF
HYDROGENURANIUM MIXING REGION ........... 108
VII. SIMULATION OF HYDROGENURANIUM
MIXING IN STEADY FLOW ..................... 127
VIII. RADIANT ENERGY TRANSFER AND
TEMPERATURE DISTRIBUTION ......... ........ 134
IX. CONCLUSIONS AND RECOMMENDATIONS ............ 169
Appendices
A. UPLAZ2 PROGRAM ............................... 178
TABLE OF CONTENTSContinued
Appendices Page
B. HPLAZ PROGRAM ................ ................. 191
C. HUPLAZ2 PROGRAM .............................. 198
LIST OF REFERENCES .................................. 200
BIOGRAPHICAL SKETCH .................................. 206
~
LIST OF TABLES
Table Page
1. THEORETICAL SPECIFIC IMPULSE ................ 4
2. THEORETICAL IONIZATION POTENTIALS
OF URANIUM .................................. 179
3. DATA USED IN HYDROGEN PLASMA
COMPOSITION CALCULATIONS .................. 197
vii
LIST OF FIGURES
Figure Page
1. Vortex flow plasma reactor ................... 7
2. Axial flow plasma reactor with magnetic
containment ........................... ....... 7
3. Wheel flow plasma reactor .................... 8
4. Axial flow plasma reactor ................... 8
5. Neutron spectrum in graphite ................. 43
6. Neutron spectrum in graphite moderated
uranium plasma ............................... 46
7. Thermal neutron flux in graphite
reflected uranium plasma reactor ............. 48
8. Uranium plasma composition at pressure
of 100 atmospheres ........................... 72
9. Uranium plasma composition at pressure
of 500 atmospheres ........................... 73
10. Uranium plasma composition at pressure
of 1000 atmospheres .......................... 74
11. Effect of temperature on total uranium
density ...................................... 77
12. Pressure trajectories for constant
uranium density ........... ... ................ 80
13. Hydrogen plasma composition at pressure
of 100 atmospheres ........................... 105
14. Effect of temperature on total hydrogen
density ...................................... 106
15. Composition of hydrogenuranium plasma
at pressure of 500 atmospheres ............... 126
viii
LIST OF FIGURESContinued
Figure Page
16. Radial density profiles of hydrogen
and uranium ................................ 129
17. Rosseland opacity of uranium ................ 153
18. Rosseland opacity of hydrogen .............. 154
19. Effect of coolant seeding on graphite
temperature ................................. 159
20. Plasma temperature distribution at
100 atmospheres ................ ............ 160
21. Temperature distributions for centerline
temperatures from 20,0000K to 80,000K ...... 161
22. Pressure required forcritical plasma
reactor ...................................... 163
23. Net power and fission fragment power as
a function of centerline temperature ........ 167
24. Energy of hydrogen coolant nuclei due
to flow and thermal motion ................ 174
25. Uranium plasma composition at pressure
of 100 atmospheres .......................... 181
26. Uranium plasma composition at pressure
of 200 atmospheres .......................... 182
27. Uranium plasma composition at pressure
of 300 atmospheres .......................... 183
28. Uranium plasma composition at pressure
of 400 atmospheres .......................... 184
29. Uranium plasma composition at pressure
of 500 atmospheres .......................... 185
30. Uranium plasma composition at pressure
of 600 atmospheres .......................... 186
LIST OF FIGURESContinued
Figure Page
31. Uranium plasma composition at pressure
of 700 atmospheres .......................... 187
32. Uranium plasma composition at pressure
of 800 atmospheres .......................... 188
33. Uranium plasma composition at pressure
of 900 atmospheres ................. ........ 189
34. Uranium plasma composition at pressure
of 1000 atmospheres ......................... 190
35. Hydrogen plasma composition at pressure
of 100 atmospheres .......................... 192
36. Hydrogen plasma composition at pressure
of 200 atmospheres .......................... 193
37. Hydrogen plasma composition at pressure
of 300 atmospheres .......................... 194
38. Hydrogen plasma composition at pressure
of 400 atmospheres .......................... 195
39. Hydrogen plasma composition at pressure
of 500 atmospheres .......................... 196
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
FISSIONING URANIUM PLASMA
FOR ROCKET PROPULSION
By
Henry Foust Atwater
March, 1968
Chairman: Dr. M. J. Ohanian
Major Department: Nuclear Engineering Sciences
A fissioning uranium plasma for use as a high
temperature selfsustaining energy conversion device is
studied theoretically. The study is oriented with par
ticular emphasis on the use of the fissioning plasma as
a heat source for rocket propulsion systems, although the
ideas and methods which are developed can also be applied
to a closed cycle heating system which utilizes the
fissioning plasma principle.
The effects of coupling mechanisms between nucle
onic and plasma properties of the fissioning plasma are
identified and analyzed for a reference model. The model
is based on a central fuel region of fully enriched
uranium surrounded by a hydrogen coolant region, contained
within a cylindrical graphite chamber. The uranium region
serves as a heat source through liberation of radiant energy
from fissioning uranium nuclei. The radiant heat transfer
from the uranium plasma to the hydrogen coolant plasma is
accomplished by seeding the coolant with a material which
is sufficiently opaque to reduce the coolantwall interface
temperature below the melting point of graphite. The
radial mixing of flowing uranium and hydrogen is simulated
by use of analytic expressions. Numerical techniques are
developed for the engineering analysis of the principle
characteristics of the fissioning plasma reactor, under
conditions of local thermodynamic equilibrium.
The reference model study leads to several basic
requirements which must be satisfied in order to develop
an operable fissioning plasma reactor. The nuclear analy
sis of such problems as critical mass, reactivity balance,
and kinetic behavior must treat both the fuel and coolant
as plasma regions rather than ordinary gases. In order to
sustain a critical uranium mass at the desired high
temperatures, pressures as high as 1000 atmospheres will
be required. In order to heat the relatively transparent
hydrogen coolant and also to prevent melting of the
graphite containment, the coolant will have to be seeded.
Additional theoretical and experimental studies of plasma
instabilities associated with the coaxial flow of uranium
and hydrogen are needed to analyze such problems as system
xii
stability and fuel loss rate. Finally, a substantial high
pressure, high temperature experimental program is needed
to determine statistical weights and energy levels of the
various uranium ions.
xiii
CHAPTER I
INTRODUCTION
This study presents the identification and analysis
of some of the prominent physical effects which must be
examined in order to develop a coherent understanding of
a fissioning uranium plasma. For the benefit of those who
(like this writer) have had more experience in the area of
nuclear engineering than in plasma physics, it will be
helpful to have a basic definition of a fissioning uranium
plasma. In the most general sense, a fissioning uranium
plasma is a mass of ionized uranium gas which is undergoing
nuclear fission. From this definition, it is immediately
seen that the analysis of such a system must be built upon
a combination of two areas of physics which are ordinarily
treated separately. The first is nuclear physics, which
forms the basis for the disciplines of reactor physics and
nuclear engineering. The second is plasma physics, which
treats the static and dynamic behavior of high temperature
ionized gases.
In terms of reactor technology, a fissioning uranium
plasma can be considered as another member of the large
collection of reactor types that have been conceived over
the past two decades as successors to presentday operating
 1 
 2 
power reactors. Even so, the plasma core reactor must
certainly be designated as one of the most advanced
concepts in view of its position in the evolution from
solid to liquid to plasma fuel reactors. It is more
important however, to recognize that the concept is ad
vanced because of the magnitude of the complex physical
phenomena which must be understood and the many techno
logical problems associated with the high temperature,
high pressure plasma experiments which must be performed,
before such a system is built and operated.
A common characteristic of the solid, liquid and
plasma core reactors is the energy production mechanism.
In each case, energy is released when the fuel nuclei
undergo neutron induced fission. After this initial
energy production, the plasma reactor concept differs
greatly from either the solid or liquid core reactor. In
the solid and liquid core reactors, the energy from fission
heats some working fluid or coolant by conductive and
convective heat transfer. In the plasma core reactor, the
fuel plasma is heated by collisional energy transfer from
the fission fragments. The coolant plasma is then heated
by radiant energy transfer from the uranium plasma.
Radiant energy transfer is the characteristic which most
clearly distinguishes a plasma core reactor from a solid
or liquid core reactor. Since radiation has not been
 3 
utilized as the primary energy transfer mechanism in a
nuclear reactor, it is expected that radiant energy transfer
will also produce some of the most difficult but interesting
theoretical and engineering problems associated with plasma
core concept.
The usual impetus for studying a new reactor concept
is derived from the continuing efforts of the nuclear
industry to develop more efficient and economical re
actors (1,2,3). The plasma core reactor is an exception.
The initial interest in the gaseous or plasma reactor
concept (411) has resulted in research directed toward
its use as a high performance space propulsion device.
In order to compare the propulsion potential of the plasma
core reactor with other concepts, the specific impulse,*
Isp, is used as a relative measure of performance. Since
specific impulse is related to propellant mass (M) and
temperature (T) through the relation Isp /T79 (12), a
high temperature, low molecular weight propellant is desira
ble. The maximum propellant temperature is limited by the
maximum allowable temperature of the reactor fuel. The
temperature limit in a solid core reactor is the fuel
melting point, while the temperature of a liquid core
*Specific impulse is defined as engine thrust per
unit weight flow rate of propellant.
 4 
reactor is limited by the fuel boiling point. The plasma
core reactor can theoretically operate at arbitrarily high
temperatures. In practice the temperature limit will be
determined by the maximum fuel temperature for which the
plasma containment chamber can be cooled below its melting
point. The following are typical values of specific
impulse (13,14) indicating the high performance that can
be expected from a plasma reactor propulsion system.
TABLE 1
THEORETICAL SPECIFIC IMPULSE, I (seconds)
sp
Chemical Rocket 400
Solid Core Reactor 1000
Liquid Core Reactor 2500
Plasma Core Reactor 6500
Although the present interest in plasma core
reactors seems to be entirely concerned with rocket
propulsion applications, the concept should not be con
sidered as a single purpose device. Without specifying
its application, a fissioning plasma embedded in a coolant
plasma can be considered as a general energy conversion
device. The high temperature coolant might be used in
numerous open or closed cycle systems. Central station
power plants operating with gas heat exchangers or directly
driven gas turbines could utilize the plasma reactor as
the energy source. When considering the maximum cycle
5 
efficiency as given by the Rankine cycle, the possibility
of extremely high coolant temperatures in a plasma reactor
are particularly attractive.
A brief survey of recent plasma reactor* research
will be indicative of the wide range of problems which are
being studied. The basic plasma core concept is simple.
A heavy, low velocity gaseous fuel and a light, high
velocity gaseous propellant are injected into a reaction
chamber having walls composed of some suitable neutron
reflecting material. Fissions in the fuel heat the pro
pellant which is then exhausted through a nozzle to give
thrust. The gaseous state of the fuel gives the previously
mentioned advantage of high temperatures, but also causes
a serious problem of fuel separation and containment.
Some of the fuel will mix with the propellant and be
exhausted from the reactor. In order to maintain a
critical fuel mass, additional fuel must be continually
injected into the reaction chamber. This is an inefficient
use of expensive nuclear fuel which also results in a
reduction of specific impulse, due to the increase in the
mean mass of the propellant. Several approaches to the
fuel containment problem have been proposed. Four of the
concepts which have received attention are the vortex flow,
The plasma core reactor is also referred to as a
gaseous core or cavity core reactor.
axial flow with magnetic containment, wheel flow, and
coaxial flow (9,13,15,16). The vortex flow reactor illus
trated in Figure 1 is designed to achieve fuelpropellant
separation through use of centrifugal forces. Hydrogen
propellant diffuses radially inward through the gaseous
uranium vortex, with the centrifugal forces associated
with the heavier uranium counteracting the diffusion drag
of the hydrogen. The disadvantage of this concept is due
to the hydrogen drag, which is great enough to cause large
uranium losses except at very low hydrogen flow rates.
Figure 2 illustrates the magnetic containment scheme.
Magnetic fields of force confine the uranium plasma while
the propellant flows axially around the fuel region.
This concept involves the technical difficulties associ
ated with magnetic containment, in addition to propellant
drag problems similar to those of the vortex flow. Another
variation is the wheel flow concept shown in Figure 3.
Hydrogen propellant enters and leaves the reactor tan
gentially along the outer periphery after circulating
around the inner cylindrical uranium region. In this
concept the main difficulty still seems to be excessive
fuel loss due to hydrogenuranium mixing. The coaxial
flow reactor is shown in Figure 4. This concept is based
on the coaxial flow of a central cylindrical uranium mass
and a surrounding annulus of hydrogen. No vortex motion
 6 
Neutron
assure shell Propellant inlet
__"__[_ _..___ V e
Rotating plasma fuel
Figure 1. Vortex flow plasma reactor.
Neutron reflector
Pressure shell Propellant inlet
e
Currentcarrying Magnetic field
coils lines
Plasma fuel
Figure 2. Axial flow plasma reactor with
magnetic containment.
lector
 8 
Propellant
exhaust
Propellant/,'
inlet
Plasma fuel 
Propellant
inlet
Figure 3. Wheel flow plasma reactor.
Neutron reflector
Pressure shell
Fuel inlet
Plasma fuel
Figure 4. Axial flow plasma reactor.
 9 
is involved. It is hoped that the hydrogenuranium mixing
can be controlled by proper selection of initial velocity
profiles and upstream geometry.
Studies related to the plasma reactor concept can
be divided into three areas. These are nuclear reactor
physics, radiant energy transfer, and turbulent plasma
flow. A very comprehensive summary of NASA/Lewis supported
research in these areas has recently been published by
Ragsdale and Rom (17). The past studies have generally
approached the three problem areas by treating each area
separately, without including the coupling mechanisms by
which nuclear, plasma and radiation effects are related
physically. The present study will consider only the
coaxial flow system, which appears to be the simplest
concept. This system presents a sufficient number of
problems, without considering the added complexities of
vortex flow or magnetic containment. The purpose of the
present study is to identify specific problems in the
above three areas, develop methods for relating these
problems, and analyze them in a consistent manner. The
high temperature uranium is properly treated as a plasma
rather than an ordinary gas, and the effects of the plasma
characteristics on nuclear criticality are determined.
The effects of fuel and coolant inhomogeneities on radiant
energy transfer are included. Temperature distributions
1
 10 
determined by radiant energy transfer are then included
in the calculation of the critical uranium mass. The
emphasis is on the development and application of general
methods of analysis, rather than a conceptual design of
an entire plasma reactor system. In order that methods
of this study may be extended to actual design problems,
numerical techniques are developed for several of these
methods.
The effects of the high fuel and coolant tempera
tures on neutron reactions are treated in Chapters II and
III, within the framework of the diffusion approximation
to the Boltzmann equation. The composition of the
uranium plasma region and the hydrogen plasma region,
and the influence of plasma effects on reactor criticality
are analyzed in Chapters IV and V. The composition of
the hydrogenuranium mixing region is analyzed in
Chapter VI. In Chapter VII an analytic simulation of
hydrogen and uranium mixing is developed. In Chapter VIII
the calculated plasma compositions and the simulated flow
distributions are used to determine the temperature distri
butions resulting from radiant energy transfer. The
temperature distributions are then related to the critical
uranium mass. Recommended theoretical and experimental
extensions of this study are presented in Chapter IX.
CHAPTER II
APPLICATION OF DIFFUSION THEORY TO NEUTRONS
IN A FISSIONING URANIUM PLASMA
A General Boltzmann Equation for Neutrons
in a Fissioning Uranium Plasma
A fundamental macroscopic description of the
neutron population in a solid fuel reactor is given by
the Boltzmann neutron transport equation. This integro
differential equation specifies the neutron distribution
in space, velocity, time and direction of motion. Using
notation similar to that of Weinberg and Wigner (18), the
transport equation is
1 a 4 ,
v at(r,vQt) + V(r,v,,t) + t (r,v) (r,v, ,t)
= X(v) f dv' f d (v')Z (r,v')((rv' ',t)
V' t '
+ J dv' f dS 's C v' v ( 'rv','t) (2.1)
External neutron sources and delayed neutrons are not
included in (2.1).
The symbols are defined in the usual manner:
 11 
 12 
Singular neutron flux (number of neutrons
per unit area per unit time per unit
solid angle per unit speed in. the volume
4
element dr about r, traveling within the
solid angle d in the direction and
having a speed v in dv at time t)
Zt(r,v) = macroscopic collision cross section for
neutrons at r with speed v
X(v) = spectrum of fission neutrons
v(v) = mean numbers of neutrons resulting from
a fission caused by a neutron with
speed v
= macroscopic scattering cross section at
r for changing the speed and direction
v', S' into a speed and direction range
dv, dQ at v,
The individual terms of (2.1) describe the following
reactions:
1 9 = change per unit time in the net number
v t of neutrons of direction and speed v
in dr
V4 =
net loss per unit time from dr of neutrons
of direction in dn by leakage
St = number of neutrons of direction and
speed v which are removed per unit time
from dr by absorption and scattering
collisions
XJdv'fdQ'vZf =
jdv'fdri'Z E =
s
number of neutrons of direction 2 and
4
speed v gained per unit time in dr from
fissions due to neutrons with all speeds
v and directions
number of neutrons of direction 3 and
speed v gained per unit time in dr from
scattering collisions which scatter
neutrons from all speeds v' and directions
I
E (S ,v'+v,6'+ )
s
S(r,v,At)
 1.3 
For a steady state neutron population, the first
term of (2.1) is zero. This gives the steady state
Boltzmann equation for which there are many analytical
and numerical methods for obtaining solutions. Exact and
approximate solutions which have been obtained for solid
core reactors are numerous and will not be presented here.
A qualitative description of the usual treatment of the
steady state Boltzmann equation will demonstrate that
modifications are necessary, in order to properly describe
neutrons in a plasma core reactor. When applying the
Boltzmann equation to neutrons in a solid core reactor, the
usual procedure is to classify neutrons as either epithermal
or thermal, according to their energy. The high energy
epithermal group includes those neutrons which are emitted
at fission energies (average fission energy = 2 MeV) and
slowed down to some thermal energy boundary (usually taken
as 1 eV to 2 eV). Thermal neutrons are those which have
slowed down and are in approximate thermal equilibrium at
some characteristic temperature, T, of the medium. Thus
the thermal energy range extends from zero to a few eV.
In the third, fourth, and fifth terms of (2.1), the
neutron speed v (or v'), appearing in the macroscopic cross
sections is the relative speed between neutron and nucleus.
Since solid core reactors typically operate with coolant
temperatures in the 5000K to 600K range (corresponding to
an energy range kT = 0.04 eV to 0.05 eV), these scattering
nuclei are considered to be unbound and also stationary
with respect to epithermal neutrons. In this case, the
relative speed is the absolute speed of the neutron. As
long as neutron speeds are above a few electron volts,
nuclear motion does not affect the neutron slowing down
process. All scattering collisions between epithermal
neutrons and stationary nuclei will result in a decrease
in neutron energy. The absorption of epithermal neutrons
by nuclei having large resonance cross sections cannot be
treated by assuming that the absorbing nuclei are at rest.
At the maximum solid core fuel temperatures of about 30000K
(corresponding to an energy kT = 0.26 eV), the translational
motion of the nuclei is large enough to strongly influence
resonance absorption through the Doppler effect. Resonance
absorption will be discussed later in this chapter.
When neutrons are slowed down from the epithermal
energy range, into the thermal energy range, the neutron
speeds are comparable to nuclear speeds and the nuclear
motion cannot be ignored. The effects of nuclear motion
are important in two aspects of neutron interactions. The
first involves the treatment of thermal neutron scattering
reactions and the second deals with the specifications of
low energy absorption cross sections. The energy distri
butions of nuclei and thermal neutrons can be described by
 14 
 15 
modified MaxwellBoltzmann distributions having a character
istic energy kT. Because of the distribution of energies
of the scattering nuclei, a scattering collision does not
always result in a decrease in neutron energy. Collisions
in which the neutron gains energy become possible and the
probability for collisions with an energy loss become
smaller. Thermal neutron scattering is further complicated
by chemical binding effects associated with the lattice
structure of solids and the molecular structure of liquids.
The effects of nuclear motion and chemical binding on
scattering collisions are included in the theory of neutron
thermalization (19), and will not be treated here. A
concise description of neutron thermalization problems is
given by Beckurts and Wirtz (20).
Microscopic thermal neutron absorption cross
sections are functions of the relative neutron speed. In
general, this speed dependent cross section is not the
same as the experimentally measured average cross section.
In order to use measured absorption cross sections in
reactor calculations, the experimentally determined values
must be properly weighted to include the effects of relative
speed (21).
The preceding discussion of the Boltzmann transport
equation has considered only neutrons in a solid core
reactor. To summarize this discussion, the application
 IG 
of the Boltzmann equation to solid core reactors can be
characterized by two features: (1) the assumption that
epithermal neutrons are slowed down by collisions with
stationary nuclei and (2) the methods of thermalization
theory are used to treat the effects of nuclear motion
and chemical binding on thermal neutron collisions.
In order to properly apply the Boltzmann equation
to neutrons in plasma core reactor, several modifications
are necessary. The need for these modifications is due to
the extremely high temperatures of the plasma core reactor,
as compared to solid core reactor temperatures. Tempera
tures on the order of 100,0000K are anticipated in the
plasma core reactor. Scattering and absorbing nuclei at
this elevated temperature have an energy kT = 8.6 eV.
Recalling that scattering and absorbing nuclei in a solid
core reactor have a maximum energy of about 0.05 eV and
0.26 eV respectively, it is clear that nuclear motion
becomes a more important consideration in a plasma core
reactor. To formally account for the possible effects of
this increased nuclear motion, the usual form of the
Boltzmann equation (2.1) is modified to include the
dependence of collision processes on the relative speed.
The following symbols for the several velocities are
used:
 17 
v v = neutron velocity
n n n
S= VNN = nuclear velocity
v = relative velocity
The relative speed v, is then defined as
v =  = i NI, where v = V2E m (2.2)
in N n n
Collision processes are related to the relative speed
through the neutron flux < = n(r,v,,t)v, and the energy
dependent microscopic cross sections, a(E), where n(r,v,,t)
is the neutron density. For a neutron of speed vn and a
nucleus of speed VN, the collision processes of (2.1) must
now be integrated over all nuclear speeds VN and directions
N. The expressions for these collision processes become:
Neutrons removed by absorption and scattering =
SdVN dN( ,VN Nt)t(v)n(,vn" nt)
N N
Neutrons gained by fission =
X(v) f dVN N dv'n dv(v')N(r,V NN,t)f(v)
N N n n
x n(r,v'n ,t)v'n
 18 
Neutrons gained by "scattering in" =
SdN dN dv'n d 'nN( ,VN' Nt)o s ,v'v,'n
VN v n
x n(r,v' n t)v' n
The general Boltzmann equation for neutrons in a fissioning
plasma can now be written as
~(,vnr,,t) + vn Vn(,vn nt)
+ f dVN d NN(N,VN',N,t)at(v)n(r,vn nt)V'n
= X(v) dV d dv' dI (n )N(r,VN ,t)f(v)
x n(r,vnn tvn n
+ I dV N dN f dv' f di'nN(,VN'N,t)as(rv'v' n n)
x n(r,vnn,t)v'n (2.3)
Equation (2.3) shows the means by which the relative
velocity enters into the individual collision processes.
This form of the Boltzmann equation is still quite general.
As such, it does not immediately suggest obvious methods
of reduction to simpler forms which are more amenable to
 19 
analytical or numerical solutions. In order to retain the
significant effects that may result from the inclusion of
exaggerated nuclear motion, some characteristics of a
typical plasma core reactor will now be considered. The
introduction of a model of such a reactor allows physical
arguments to be used in making simplifying assumptions.
For purposes of describing the nucleonic characteristics,
the reactor is basically a central region of uranium plasma
surrounded by a cylindrical annulus of hydrogen coolant
plasma. To provide the desired neutron reflection while
operating at high temperatures, the reaction chamber is
assumed to be graphite. The determination of the neutron
energy spectrum for this model is of primary importance,
since all nuclear characteristics of the system will be
directly affected by the neutron spectrum. The simplest
case to consider is the zero power plasma core reactor
which consists of a critical mass of uranium plasma sur
rounded by graphite. No hydrogen coolant is included
since there is no significant heat production at zero
power. For an adequately reflected zero power reactor,
leakage is assumed to be negligible. For a high power
rocket reactor, leakage due to neutron streaming through
the nozzle would have to be included. The neutron spectrum
in the zero power reactor will be determined by scattering
collisions with graphite and uranium, and by absorption
 20 
collisions with uranium. The relative importance of
scattering collisions in graphite and uranium can be
determined by comparing the scattering mean free paths
of neutron in the two materials. Graphite has an atom
density of N = 0.08 x 1024/cc and an epithermal micro
scopic scattering cross section of ac = 4.8 barns. The
corresponding scattering mean free path is
Xs i = 1/N a s = 2.6 cm
graphite c c
Nuclear criticality calculations (which are discussed in
the following chapter) indicate that a typical critical
uranium235 mass can be achieved with an atom density of
approximately N = 10'9/cc. With an average epithermal
scattering cross section of 10 barns, the scattering mean
free path in uranium is
us = 1/Nu a = 104 cm
These mean free path values show that a neutron slowing
down from fission energy to thermal energy will have about
4000 scattering collisions in graphite for each scattering
collision in uranium. Thus the neutron spectrum will be
determined primarily by collisions in graphite. This
conclusion greatly simplifies the computation of effective
absorption cross sections in the fuel region since the
 21 
neutron spectrum in graphite can be used to obtain averaged
uranium cross sections, even though the uranium contains
no graphite. The neutron spectrum of graphite will
actually be slightly modified in the fuel region due to
epithermal and thermal absorptions in uranium. The
temperature of the graphite will essentially determine
the distribution of thermal neutrons. The influence of
uranium absorptions on the thermal neutron spectrum is
through second order effects, since the absorption mean
free path for thermal neutrons in the fuel region of a
zero power plasma reactor is about Xua = 1700 cm. For
this value, a 78 meter thickness of uranium plasma would
be required to attenuate the thermal neutron intensity
by one per cent. The epithermal neutron spectrum of
graphite will be modified at energies coinciding with large
uranium resonance cross sections. From the above argu
ments, the zero power plasma reactor is seen to be well
moderated, with an epithermal neutron spectrum determined
by slowing down collisions in graphite, and a thermal
spectrum determined by the graphite temperature.
Extensive one and two dimensional transport theory
calculations have been used to study the effects of
materials, temperature and dimensions on the criticality
of gaseous core reactors (22). For the purpose of
calculating detailed design parameters, it will probably
be necessary to use the more exact computational methods
 22 
such as these. However, since the reactor is very thermal,
it is reasonable to expect that diffusion theory methods
will suffice for survey and comparison calculations. In a
parametric study of gaseous core reactors using both
transport and diffusion theories, Plunkett (23) has demon
strated that multigroup diffusion theory is adequate for
calculations of this type. In the present study, the
effects of elevated temperatures and the resulting ex
aggerated nuclear motion will be treated within the
framework of multigroup diffusion theory. This approach
leads to the development of calculational methods which
are designed specifically for application to diffusion
theory codes. Such methods may be directly applicable to
transport theory codes. If not, it will be necessary to
include the effects of nuclear motion by reduction of
equation (2.3) to a form which can be solved numerically.
Diffusion Approximation to Boltzmann Equation
The previous discussion of the plasma core reactor
has considered the zero power reactor which contains no
hydrogen. A power producing plasma reactor will operate
with a high velocity hydrogen coolant plasma surrounding
the uranium plasma. In this case the exaggerated motions
of both absorbing fuel nuclei and scattering coolant nuclei
become important. The inclusion of these effects will
 23 
require special treatment of several terms in the multigroup
diffusion equation. Using notation similar to that of
Baller (24), the multigroup diffusion equation is
N
D.V2.() + i (r) = X.S(r) + E S .j (r)
Di 1i t. 1 1 s ,j
1 j=l
j i
(2.4)
where
N
t = i + DiB. j=l
N
S(r) = (vxf)i(Wr
i=l
for 1 < i < N, N = number of groups.
The symbols are defined as
.i(r) = neutron flux in ith group at position r
D. = diffusion coefficient for the ith group
1
E. = total removal from ith group
1
E = absorption cross section for the ith
a group
B2 = transverse buckling for the ith group
s,i+j = scattering from group i to group j
Xi = integral of the fission spectrum over the
energy range represented by group i
 24 
S(r) = fission source distribution
(vEf)i = average number of neutron produced by
fission in the ith group times the
fission cross section in the ith group
X = criticality eigenvalue
The effects of nuclear motion of absorbing nuclei and
scattering nuclei will be treated separately. The ab
sorption cross section, Eai, will include the effects of
1
the motion of uranium nuclei in the treatment of the
Doppler broadening of the uranium absorption resonances.
The scattering cross section, incle
s,1j, will include the
effects of the motion of hydrogen nuclei.
Effect of Thermal Motion of Absorbing Nuclei on Epithermal
Neutrons
The macroscopic absorption cross section for the
ith group is the flux weighted average cross section
which is defined as
J E
a. E
S JE i(r,E)dE
E
i (2.5)
where Eui and EL. are the upper and lower boundaries,
respectively, of the ith group. In the epithermal energy
respectively, of the ith group. In the epithermal energy
range, the uranium235 microscopic absorption cross
section, a (E), exhibits a large number of resonances
(25). The usual procedure in evaluating the integral in
the numerator of (2.5) is to assume that each well defined
resonance is accurately represented by the single level
BreitWigner formula (26). This formula expresses the
energy dependent absorption cross section as
r r
o0(E) = 2 gj n y
a (Er E0)2 + (r/2)2 (2.6)
The symbol E0 denotes the energy of the resonance and Er
is the kinetic energy associated with the relative velocity
between neutron and nucleus. In the case of a stationary
nuclei, Er is given by
E = 1 2~ (2.7)
r 2 n
where v is the neutron speed in the laboratory coordinate
system, and p is the reduced mass as defined by
mM (2.8)
m + M
with m and M representing the mass of the incident neutron
and the target nucleus, respectively. The symbols Fy and
r are the radiative capture and neutron widths (27) which
n
define a total width r as
F = F + F
"Y n
 25 
(2.9)
 26 
The reduced neutron wavelength is given by
1 h
27r 27T(2pE ) 1/2 (2.10)
The statistical spin factor g (28) is equal to (2J + 1)/
2(21 + 1), where J is the spin quantum number of the
compound nucleus formed by the target nucleus and the
neutron, and I is the spin quantum number of the target
nucleus. The peak value of the total (absorption plus
scattering) cross section, a0, occurs at E = EO and has
the value
r
S= 4Xt2gj n (2.11)
Using (2.11) and (2.6), the absorption cross section
becomes
r1
a (Er) = 0 o 2 (2.12)
E E
T/2
If the absorbing nuclei were stationary, Eai would
be evaluated from (2.5), with a (Er ) given by (2.12) and
Er given by (2.7). In reality, all absorbing nuclei have
a kinetic energy at least as great as that corresponding
to the temperature of the surrounding medium. The thermal
motion associated with the kinetic energy effectively
 27 
shifts the energy at which neutrons are absorbed to higher
or lower values, depending on the direction of the target
nuclei with respect to that of the incident neutron. This
energy shift is referred to as Doppler broadening (18) of
the resonance cross section, since the neutron "sees" a
cross section which is spread over an increasingly wider
energy range as the nuclear motion increases. Doppler
broadening strongly affects neutron absorption since the
neutron density is a rapidly varying function in the
neighborhood of a resonance. At the comparatively low
temperatures of solid core reactors, the Doppler effect
is very important in determining reactor criticality and
stability. The kinetic energy of an absorbing nuclei in
a plasma core reactor will correspond to temperatures in
the 50,0000K to 100,0000K range, as compared to a maximum
of about 30000K for a solid core reactor. At such high
fuel temperatures, the effects of Doppler broadening in a
plasma core reactor are expected to be much larger than
the corresponding effects in a solid core reactor. In
order to compare the Doppler broadening equations in the
plasma and solid core reactors, the approximations used
in treating the Doppler effect in a solid core reactor
will be examined briefly.
In a solid core reactor operating at any appreciable
power level, the fuel nuclei will have sufficient thermal
energy to require an accurate determination of the relative
energy appearing in the resonance cross section as given in
equation (2.12). The relative kinetic energy of the
neutronnucleus collision is
r I n VN
E = n. (2. 13)
where vn is the velocity of the incident neutron, VN the
velocity of the nucleus, and I the reduced mass as defined
by (2.8). Expanding (2.13), the relative energy can be
written as
pn P
E E + E n N (2.14)
r m n M N m + M14
where the following relations have been used:
E mv 2 = kinetic energy of the neutron
n 2 n
S1
En 2 MVN2 = kinetic energy of the nucleus
P = mn = momentum of the neutron
n n
N = MV = momentum of the nucleus
"N N
Note that, if the nuclei were stationary, the second and
third terms of (2.14) would be zero and the expression
reduces to (2.6). The kinetic energy of nuclei at the
maximum solid core fuel temperature of 30000K is 0.26 eV,
_IL~i
 28 
 29 
while a neutron at the lowest uranium235 resonance has an
energy of 1.0 eV. From these values, the second term of
(2.14) is seen to be much smaller than the first, since
the largest ratio of the second to the first term is
approximately 0.001. Therefore, the second term is con
sidered negligible in dealing with solid fuel resonance
absorption. The third term of (2.14) is proportional to
the component of momentum of the target nucleus parallel
to the direction of motion of the incident neutron. Due
to its large mass as compared to that of the neutron, the
nucleus has a large enough momentum to cause the third
term to have a significant effect on the relative energy.
Denoting by Vz the component of the velocity of the nucleus
which is parallel to that of the neutron, the third term
can be written as
Pn PN Nm)
S N n)v V (2.15)
m+M m+M nz
Using (2.15) and the expression for the relative kinetic
i 2 equation (2 14)
energy for stationary nuclei, E = 2v equation (2.14)
is simplified as
E = E v V = E 7E" V (2.16)
r m n nz z
Since only the nuclear motion parallel to the
neutron direction is considered to be significant, the
 30 
probability of a neutron collision with a nucleus with
velocity Vz is proportional to the number of nuclei with
this velocity. The number of such nuclei is given by the
one dimensional Maxwellian distribution
MN 1/2
P(Vz) = (2 eMNVz2/2kT (2.17)
Hence the effective cross section for a neutron with kinetic
energy E is
O(E) = ((Er)P(Vz)dVz (2.18)
Using equation (2.16), the effective resonance absorption
cross section is
E E dE
a (E) = o a(Er)P( r (2.19)
This cross section is evaluated by substituting (2.12) and
(2.17) into (2.19). Using the notation of Dresner (29),
this substitution yields
r
ca(E) = C0 Y Y(e,x) (2.20)
a 0 r
where
SrC. exp[ e2(x y)2]
Y(e,x) 2/ _ 1 + dy (2.21)
2 VT 2
L
 31 
and
x = 2(E EO)/r, y 2(Er EO)/, 0 = FA, A = (41pEkT/) 1/2
(2.22)
Doppler broadened resonance absorption cross section
in solid fuel reactors are computed using (2.20), with
Y(9,x) evaluated from tabulated values based on the reso
nance parameters r and E0, and the fuel temperature T.
Doppler broadening in a plasma core reactor is
expected to differ quantitatively from the solid core
reactor, due to the larger kinetic energy of the plasma
core fuel nuclei. The following treatment of the Doppler
effect in a plasma reactor is based on the assumption that
the single level BreitWigner formula (2.12) accurately
describes resonance absorption cross sections. The relative
kinetic energy of the neutronnucleus collision is again
given by equation (2.14). From the same arguments used in
the investigation of the Doppler effect in solid core
reactors, the third term of (2.14) will have a significant
effect on the relative energy in the plasma fuel reactor
also. For fuel nuclei at a temperature of 100,000K and
a neutron at the energy of the first uranium resonance,
the ratio of the second to the first term is 0.04, or forty
times larger than the corresponding ratio for the solid
core reactor. Thus, the second term of (2.14) can result in
 32 
a significant correction to the relative energy, and hence
to the amount of resonance absorption. At the extremely
high temperatures which are being considered in the plasma
core reactor, this correction may strongly influence re
actor criticality and stability.
Expanding equation (2.13) as before and retaining
all terms, the relative energy is
E E + V 2 /2iE V (2.23)
r 2 N z
where E = v2 again denotes the relative kinetic energy
of the neutron stationary nuclei. The relative energy is
now a function of all three velocity components (V, Vy, V ),
rather than the single component Vz as in the case of solid
fuel nuclei. Hence, the probability of a neutron collision
with a nucleus with velocity VN = (Vx, Vy Vz) is proportional
to the three dimensional Maxwellian distribution of nuclei
P(VN), where
P(V )dV = ( /2 e MVN2/2kT 4V2dV (2.24)
N N =27 kT e4VN N
where V 2 V + V 2 + V2. The effective resonance
N x y z
absorption cross section for a neutron with kinetic energy
E is
 33 
CO CO CO
a(E) a(Er)P(VNdV N = a(Er}P(VN) ddV dVz
N CO C 0co
(2.25)
where a (Er) is given by equation (2.12), Er by (2.23)
and P(VN) by (2.24).
Equation (2.25) is the plasma fuel equivalent of
the solid fuel cross section given by (2.18). An analytical
evaluation of (2.25) was investigated by expanding the
integrand in a threevariable Taylor's series about the
most probable velocities (Vxp, Vyp, Vzp ) = (kT, kT, kT).
The advantages of this approximate analytic approach do
not justify the difficulty involved in obtaining sufficient
accuracy. A more direct method of computing the effective
cross section is based on the use of the energy dependent
form of the Maxwellian distribution. Expressed in terms
of the kinetic energy, EN, of the nucleus, the distribution
is
P(EN)dEN 27w /EN eE/kT (2.26)
1N4) N (7kT)3/2 N
Since all directions of motion of the absorbing nuclei are
1 2 1
=E = E where E. 1 MV 1
equally probable, E = E z i = 2 MV = EN;
i = x, y, z. The velocity component of the nucleus parallel
to the direction of travel of the neutron is
 34 
V = /2E~7M = /2E 73M (2.27)
From (2.23) and (2.27), the relative energy is
Er E + EN 2/"EEN 3M (2.28.)
Using (2.12), (2.26) and (2.28), the computational form
of the effective resonance cross section is
r
oa(E) = a0 4 H(a, 8, y) (2.29)
with the Doppler broadening function H(a, 3, y,) defined as
H(a, 8, y) = de
/ 0 1 + ya / + E)
(2.30)
where
a(E, T) = (E EO)/kT
g(E, T) = 2/VifkTE/3M
y(r, T) = 4kT/r2 (2.31)
and the reduced energy variable is e = EN/kT.
The Doppler broadened resonance absorption cross
section in a plasma core reactor is given by (2.29), with
H(a, 6, y) evaluated from tabulated values based on the
resonance parameters F and E0, and the fuel temperature T.
 35 
Effect of Thermal Motion of Hydrogen Nuclei
In the previously discussed zero power plasma
reactor with a graphite reflector, the thermal neutron
spectrum was described by a Maxwellian distribution
corresponding to the graphite temperature. When the
reactor operates at significantly high power levels, the
hydrogen coolant required for heat removal will exist at
temperatures which result in a large increase in the thermal
motion of these scattering nuclei. When neutrons in thermal
equilibrium with graphite have a scattering collision with
hydrogen nuclei at a higher temperature, the neutrons will
gain kinetic energy. If the collision rate with hydrogen
is comparable to that with graphite, the characteristic
temperature of the neutron spectrum will be shifted to a
higher value. This shift, or "hardening," of the spectrum
will influence reactor criticality by reducing the neutron
density at energies where the uranium absorption cross
section is large.
Based on two dimensional criticality calculations
of gaseous core reactors containing hydrogen, Hyland et
al. (30) have concluded that neutron scattering collisions
with hydrogen have a negligible effect on the critical mass.
This conclusion does not appear to be justified since the
"upscattering" effects of hydrogen are not included. The
calculations are based on the assumption that the thermal
 36 
neutron spectrum is determined by the graphite temperature,
and do not include the spectral hardening due to hydrogen
at a higher temperature. While it may be true that hydrogen
scattering collisions are negligible in certain restricted
cases, this is not expected to be true in most cases of
interest. The extent to which hydrogen influences criti
cality will be determined by the mass of hydrogen in the
core, and by the spatial distribution of the hydrogen
density and temperature. Thus, in order to properly
evaluate the effects of hydrogen, the heat transfer and
fluid flow characteristics of hydrogen must be considered
simultaneously with the neutron scattering problem.
A more rigorous treatment of hydrogen scattering
has been used by Herwig and Latham (31) in their multigroup
diffusion theory calculations of spherical gaseous core
reactors containing hot hydrogen. By defining an effective
macroscopic hydrogen scattering cross section, the effects
of high temperature hydrogen on critical mass are investi
gated. The effective cross section is defined by con
sidering the scattering collision rate of a Maxwellian
distribution of neutrons at temperature Tn with a
Maxwellian distribution of hydrogen atoms at T For a
neutron density n and a hydrogen density NH, the collision
rate is
 37 
NH a nV = E nV = Z' nV (2.32)
Hs r s r s n
where the effective macroscopic scattering cross section
for hydrogen is given by
ss Vr/ n (2.33)
The symbols are
Vr = average relative velocity between hydrogen
and neutrons
V = average neutron speed = (8kT /nm ) 1/2
H = average hydrogen speed = (8kTH/TmH) 1/2
o = microscopic scattering cross section of
hydrogen
The effective multigroup hydrogen cross sections
are calculated by the following relation:
NHo V V)
N s a H 1
= = NCs.C.ij = zs.1i
1 V 13 1
(2.34)
where
VH V. = average relative speed between hydrogen
at a given temperature and neutrons in
group i
as. = microscopic scattering cross section for
I group i
 38 
V = average neutron velocity in group i
pij= neutron energy transfer coefficient from
group i to group j
The GAM1 (32) and SOPHIST1 (33) codes are used to calcu
late the transfer coefficients. Using the effective
hydrogen scattering cross section with fuel and reflector
cross sections computed in the usual manner, a one
dimensional multigroup diffusion theory was used to make
a parametric study based on various hydrogen temperatures
and pressures, fuel nuclei, reflector materials, and fuel
distributions. The details of these calculations and the
assumptions involved are presented in the original article
and will not be discussed here. However, it is worthwhile
to discuss the results of this study, which are significant
for two reasons. First, the groupaveraged cross sections
include the spectral hardening effects of high temperature
hydrogen and second, the fact that high temperature hydrogen
strongly influences the critical mass through a complex
interaction of multiple processes is clearly demonstrated.
Within the framework of multigroup diffusion theory, the
technique of using an effective hydrogen scattering cross
section is a practical approach to the treatment of thermal
neutron scattering by high temperature hydrogen. Before
adopting this technique as a standard method of calculating
 39 
thermal group cross sections, a more rigorous and possibly
more accurate method should be considered. The fundamental
approach to the treatment of thermal neutron scattering is
through application of appropriate scattering kernels
within the framework of thermalization theory. In the
hydrogen cooled plasma core reactor, the thermal group
averaged cross section of hydrogen, Es, is determined by
a mixed neutron spectrum. The spectrum results from
scattering collisions with both the high temperature
hydrogen and the relatively low temperature reflector
material. The scattering cross section Es in (2.34) was
obtained using the TEMPEST code (34). This code averages
microscopic cross sections over a thermal neutron spectrum
based upon one of the following approximations:
1. The WignerWilkins light moderator equation
2. The Wilkins heavy moderator equation
3. The Maxwellian distribution
Since each of these approximations is characterized by a
single moderator temperature, it appears that Es in (2.34)
is determined from a spectrum based upon a temperature and
scattering kernel corresponding to either hydrogen or the
reflector, rather than from a mixed spectrum. A more
rigorous calculation of Es would require the determination
of the mixed spectrum. Such a spectrum could be computed
 40 
from an appropriate weighting of a hydrogen spectrum based
upon a free gas scattering kernel and a reflector spectrum
based upon the proper scattering kernel (such as the Parks
kernel for graphite) at the reflector temperature. The
spectral weighting would be determined by the spatial
distribution of the hydrogen density and temperature. The
decision to use either the mixed spectrum method or the
effective hydrogen cross section method should be based on
the particular reactor being studied. The effective cross
section method may be adequate for most parameter studies,
while the mixed spectrum method might be required for
detailed core design studies. If a very elaborate mixed
spectrum calculation is necessary to obtain multigroup
cross sections, then the accuracy of the multigroup dif
fusion approximation may be less than that involved in the
spectrum calculation. In this case, the accuracy of the
cross sections would provide no additional accuracy in the
multigroup results. If additional accuracy is required,
one and two dimensional transport theory codes should be
used.
CHAPTER III
NUCLEONIC ANALYSIS OF EXTERNALLY MODERATED
URANIUM PL:F.'.'
The multigroup diffusion approximation to the
Boltzmann equation is used to make a parametric investi
gation of some of the essential nuclear characteristics
of an externally moderated uranium plasma. These calcu
lations provide a quantitative check on the assumption
that the neutron energy spectrum in the uranium plasma is
primarily determined by the moderating characteristics of
the surrounding reflector. The results of these calcu
lations are used to define a reference model of the plasma
core reactor. This model is used to study the coupling
between the plasma and nucleonic characteristics of a
fissioning uranium plasma.
The neutron energy spectrum in graphite is calcu
lated using the GAM1 code (32) for the epithermal portion
of the spectrum and the TEMPEST code (34) for the thermal
portion. A value of 1.125 eV is used for the thermal
epithermal energy boundary (maximum graphite temperature
corresponds to 0.35 eV), and the U235 fission spectrum
is used as the neutron source. The calculated spectrum
represents the energy distribution of neutrons which
originate from fissions in a central U235 plasma and are
 41 
 42 
slowed down by collisions in a surrounding graphite re
flector. Since the reflector contains no uranium, the
spectrum is determined entirely by scattering and ab
sorption in graphite. The calculated neutron spectra
corresponding to several reflector temperatures are shown
in Figure 5. Changing the graphite temperature affects
the thermal portion of the spectrum by shifting the most
probable energy, E = kT, at which the thermal neutron
flux is a maximum.* Once the graphite temperature is
specified, the calculation of the neutron energy spectrum
and multigroup cross sections in the reflector is
straightforward.
The neutron spectrum and cross sections in the
central uranium plasma are calculated using the following
argument. Consider a homogeneous mixture of graphite and
a small amount of U235. Most of the epithermal neutron
scattering will be due to graphite, while thermal neutron
absorption will be almost entirely due to uranium. For a
sufficiently small amount of U235, the epithermal
scattering effects of uranium and the thermal absorption
effects of graphite are negligible. In this case, the
total neutron spectrum can be considered to be the
The thermal energy flux c(E) has a maximum at
E = kT, while the thermal lethargy flux (u) has a maximum
at a lethargy corresponding to E = 2kT.
bKi
Ii
10 /
6 
Graphite Temperature
r  K2000K 1
1 J 320000K I
30000K
Sj 40000K _
2
I \ j
Lethargy 20 15 I 1 5
Energy .02eV ley 10eV lkeV IkeV i0keV 1MeV MeV
Figure 5. Neutron spectrum in graphite.
 44 
superposition of an epithermal graphite spectrum and a
thermal uranium spectrum. Now consider the heterogeneous
system consisting of a central U235 plasma surrounded by
a graphite reflector. Due to the low uranium density, the
epithermal macroscopic scattering cross section of uranium
is negligible compared to that of graphite and the uranium
plasma is essentially transparent to high energy neutrons.
Hence, the higher energy epithermal neutrons in the core
region have a spectrum determined by the reflector and
experience a negligible attenuation in the uranium plasma.
The low energy portion of the epithermal neutron spectrum
will be determined by graphite scattering plus uranium
absorption due to the U235 resonances in the 1 eV to 100
eV range. Epithermal neutrons which are slowed down by
scattering collisions in the reflector become the source
of thermal neutrons in the core region. The thermal
neutron spectrum emerging from the reflector is then
modified by absorptions in the uranium plasma. The flux
weighted thermal and epithermal multigroup cross sections
of the uranium plasma region are calculated using the
neutron spectrum of graphite containing a dilute quantity
of homogeneously mixed U235. This method of flux weighting
includes the spectral effects of both graphite scattering
and uranium absorption and thereby provides a more accurate
 45 
description of the physical system than does the method of
weighting uranium cross sections by the unmodified neutron
spectrum in graphite.
Several calculated neutron spectra in a graphite
moderated uranium plasma are shown in Figure 6. These
spectra were calculated by the methods described in the
preceding paragraph. When the uranium density is less
than N235 = 101/cc, the epithermal spectrum in the uranium
plasma is equal to the spectrum in graphite, since reso
nance absorptions in uranium are negligible for such
dilute quantities of U235. As the uranium density is
increased to N235 = 101/cc, resonance absorptions begin
to slightly depress the epithermal spectrum in the 1 eV
to 100 eV range. At a density of N235 = 1020/cc, the
neutron spectrum is significantly modified by thermal and
epithermal absorptions in the uranium plasma.
The spectra shown in Figures 5 and 6 were used to
calculate multigroup cross sections for the reflector and
uranium regions, respectively. Using the AIM6 multigroup
diffusion theory code (35), a parameter study was made to
determine the range of values of uranium density, core
size, flux shapes, and reflector thickness for which a
critical U235 plasma reactor can be realized. The purpose
of these calculations is to determine a reasonable
reference model without attempting to optimize all the
i I i I I I I I I I i I i 1 1 i  .
1o I
i /
f \ Graphite Temp = 40000K / \\
I
i_ /U235 Plasma Density _
_I I_ 
1
\2 /
S1019 i
/ 1019/cc
S10 2o /cc 
Lethargy 2 15 5 1
Energy .02eV leV LeV IOkeV lMeV i MsH
Figure 6. Neutron spectrum in graphite moderated uranium plasma.
 47 
nuclear characteristics of the system. Typical thermal
flux profiles of critical reactor configurations are shown
in Figure 7. The thermal flux attenuation is seen to be
very insensitive to changes in the uranium density profile,
in the case of low fuel density. An average uranium
density of N235 = 101/cc was used in these two calcu
lations.
Based on the four group diffusion theory calcu
lations, the uranium plasma reactor reference model is
defined as follows:
Core radius 4 ft
Core height 8 ft
Graphite reflector thickness 2 ft
Graphite temperature at core
interface 40000K
Uranium composition 100% U235
The critical mass of U235 will depend on the uranium
density distribution, the plasma temperature distribution,
and pressure, and the hydrogen coolant density and
temperature distribution.
These parameters will have to be specified in
order to compute the critical mass. The diffusion theory
calculations indicate that average U235 densities on the
order of 0.5 x 1019/cc to 1.6 x 1019/cc will be required
to achieve a critical plasma reactor at the desired
w4W
10
oer
x 8
6
r1 4
2
Fuel Graphite reflector
Fuel Graphite reflctorI
0 40 80 120 160 190
Radius, cm
Figure 7. Thermal neutron flux in graphite reflected uranium plasma reactor.
MW
 49 
high temperatures. These densities correspond to U235
masses of about 25 kg to 75 kg. The choice of U235 as
the fuel for the reference model is based on the availa
bility of both plasma and nuclear data for uranium. Either
Pu239 or U233 appears to be a better plasma reactor fuel
when only the nucleonic criticality effects are considered
(23,31). However, since the plasma characteristics of the
fuel also have an important effect on the critical mass,
the per cent ionization, as a function of pressure and
temperature, must be considered when comparing the relative
merits of the three fuels.
CHAPTER IV
PRESSURETEMPERATURE ANALYSIS OF URANIUM
PLASMA FUEL REGION
Introduction
In order for the uranium fuel region to function
efficiently as a source of energy in the plasma core
reactor, the uranium should exist at temperatures in the
50,0000K to 100,0000K range. Under these conditions, heat
transfer is accomplished primarily by radiation. At these
temperatures, the uranium fuel is highly ionized and must
therefore be treated as a plasma rather than an ordinary
gas. Since uranium becomes appreciably ionized at tempera
tures above a few thousand degrees Kelvin, the plasma
effects are important at all temperatures of practical
interest. In performing a nuclear analysis or radiative
heat transfer analysis of an externally moderated high
temperature reactor, these plasma effects must be considered.
In order to account for these effects, it is necessary to
determine the composition of the uranium plasma fuel region
as a function of plasma temperature and pressure.
Basic Equations for Arbitrary Temperatures
An uranium plasma in local thermodynamic equilibrium
is characterized by the following set of equilibrium ion
ization and recombination reactions. The uranium atom and
 50 
 51 
all ions through twelvetimes ionized uranium are included.
The more highly ionized species are negligible for the
temperatures and pressures being considered in this study.
U O Ui+ + e
U1+ : U2+ + e
U2+ + U3+ + e
i+ (i+l)+
U +U + e
U11+ t U2+ + e
(4.1)
where
U = neutral uranium
i+
U = itimes ionized uranium
e = electron
The relation between the number of ions and
electrons in an equilibrium ionizationrecombination
reaction is described by the following Saha equation (36):
 52 
N N u. 2im k 3/2
Ni+le = 2 1+1 e T3/2eEi/kT L4.2)
i i h
where
Ni1 .= density of (i+1)times ionized uranium
Ni = density of itimes ionized uranium
Ne = density of electrons
m = electron mass
e
k = Boltzmann's constant
h = Planck's constant
T = plasma temperature
ui+ = partition function of (i+l)times ionized
uranium
u. = partition function of itimes ionized
uranium
E. = ionization potential for the reaction
Ui + U(i+l)+ + e
Due to the electrical neutrality of the plasma,
the net charge is zero. This balance is expressed as:
Total negative charge of electrons = Total positive charge
of ions
 53 
12
N = I Z. N.
e iO=
where
Z. = charge of ith uranium ion
1
The temperature, pressure and total particle density
are related by the equation of state for a perfect gas.
The total pressure is the sum of the partial pressures of
the individual species as given by
P = Pi = NOT kT
1
N kT =
TOT
12
. N.kT = NekT + N.kT
1 i=O
12
N TOT N + N.
NTOT Ne i=0 Ni
i=0
(4.4)
where
NTOT = total particle density
The right side of the Saha equation (4.2) is
dependent on the temperature as given by
u 2nm k 3/2
u. 3/2 T3/2eEi/kT
Ki(T) E 2 i+l ( T3/2 e/k
u1 i h
The ith Saha equation is then written as
(4.5)
(4.3)
 54 
Ni+ N
N. Ki(T) (4.6)
Combining equations (4.3), (4.4), and (4.6) gives the
complete set of equations describing the plasma composition.
N, N
e K, (4.71)
1
N N
Se = K, (4.72)
2
N. N
+ e K. (4.7i)
N. 1
1
N 13 N
N 1 2 K12
N12
13
N = Z. N
e i= 1
(4.712)
(4.713)
(4.714)
13
N = N + + i
TOT e 1
i=l
 55 
Note that each uranium index (i) has been increased by
one, so that within the formulation of the numerical
expressions used in the computer program, the following
definitions are used:
N = density of U in particles/cc
N = density of U+ in particles/cc
N13 = density of U12+ in particles/cc
Since the electrons and various uranium species will have
densities with numerical values in the approximate range
of 0 to 1020 particles/cc, it is convenient to scale
equations (4.71) through (4.714) so that all particle
densities can be expressed as dimensionless concentrations
Ci, where 0 < C. < 1. Dividing each term in equations
(4.71) through (4.714) by NTOT gives the following set
of scaled equations.
C, Ce *
c K (4.81)
C 1
1
C3 C
2 e K2* (4.82)
2
 56 
C C
i+1 Ce
C. K
1
C C
13 e
12
13
C =e
i=1
= K *
12
Zi Ci
1 1
(4.8i)
(4.812)
(4.813)
(4.814)
1 = C + c.
i=l
Ci = Ni/NTT Ce = Ne/N Ki* = Ki/NTT
(4.9)
From equations (4.81)through (4.812),
C2 =  K*
e
C2 C1 C1
C3 K2* K* K2* K2
e C C2
e e
where
 57 
C3 C C
K K K
C, = K3 1 K2* K3 K'3
e C C
e e
In general,
C.
C K =
Ci C ii
e
C
11 K K2. Ki1
C
C
K
e1 i1
e
i
K = H K.
1 j=l 3
From equations (4.813) and (4.10)
]3
C = Z iC
i=l
]3
= 1 Zi
i=l
C
S ii
C
e
13
i=l
Z ,
K.
C
e
(4.12)
From equations (4.814) and (4.10)
13 ]3
1 = C + C. = C + C1 + C.
i=l1 i=2
]3 K'
1 = C + Cl + i1
e i=2 C
(4.13)
Solving for C, from equations (4.12) and (4.13) and equating
the results,
where
(4.10)
(4.11)
 58 
C 1 C
13 K C 13 K' (4.14)
11 i1
Zil i 1 +
i=1 C i=2 C
e e
The set of equations (4.81) through (4.814) has thus
been reduced to a fundamental expression relating the
electron concentration to the pressure and temperature.
Equation (4.14) can be written as
C C 1
F(P,T,C ) e + e 3 = 0 (4.15)
13 K' 13 K1
il i1
i 11 1 + 11
i=2 C i=2 C
e e
For a given temperature and pressure, F(P,T,Ce) = F(Ce)
becomes a polynomial function of Ce, and (4.15) can be
expressed as
G(C )
e
F(Ce) = (4.16)
where
13 K' 13 K'
G(C) = C(l + ) + (Ce 1) Z. i (4.17)
i=2 C i=2 C
e e
13 K'. 13 K'.
H(Ce) = (1 + I  )( Z ii) (4.18)
i=2 C i=2 C
e e
Since all C K'., and Z. are positive, H(C ) is always
'* ~ J L
 59 
positive and the zeros of F(C ) are given by the zeros of
G(C ). Thus, the electron fraction, Ce, is given by the
solution of
G(C ) = 0 (4.19)
Simplifying equation (4.17), the polynomial equation (4.19)
is
13
G(C) = Ce + 1 + Zi)Ce Z. C
i=2 e
x K' i = 0 (4.20)
ii
Equation (4.20) is a 13th degree polynomial in C with
one real positive root. From physical intuition there
should be only one real root, since for a given tempera
ture and pressure, the number of electrons in a plasma is
uniquely determined. This can be shown rigorously by
application of Descartes' rule of signs (37) to equation
(4.20). Solutions to equation (4.20) are obtained using
the iterative NewtonRhapson method (38). The Newton
Rhapson recurrence formula for the solution to (4.20) is
G(Ce(n))
Ce(n + 1) = C (n) G (4.21)
e e G7~(n) )
 60 
where
Ce (n+l) = value of Ce after n + 1 iterations
C (n) = value of C after n iterations
e e
C (1) = initial estimate of C
e e
dG
G' (C (n)) = d
e ,e Ce = C (n)
G(Ce(n)) is obtained directly from (4.20). G' (Ce(n)) is
obtained by differentiation of (4.20).
13
G'(C ) = 1 + I (2i) (+Z )C (li)Z. C K'
i=2 1
(4.22)
Using equations (4.20) and (4.22) with (4.21), the electron
fraction is then given by
C (n+l) = C (n)
Ce(n) + [(l+Zi)C (n) Zi 1(n) K'
i=2
1 + I. (2i)(l+zi)C 1i(n) (i)z. C (n Ki'.
i=2 e e 1
(4.23)
Beginning with an assumed average electron concentration
of C (1) = 0.5 as the initial estimate, expression (4.23)
 61 
is evaluated repeatedly until the desired degree of accu
racy is obtained. The electron density is then given by
N e C N
e =e TOT
From equation (4.12),
C
C 13e and N = C NTOT
S 13 K'. 1 1 TOT
11
zi i1
i=2 C
From equation (4.7i), N.i+ = K. N /N for i = 1, 2, .,
12, gives the particle densities N2, N3, ., N13. The
particle densities Ne N1, N2, ., N13 then completely
describe the uranium plasma composition.
Low Temperature Equations
For temperatures less than about 10,0000K, the
iterative method for solving equation (4.14) does not
readily converge. Although uranium is not highly ionized
at temperatures below 10,0000K, a knowledge of the low
temperature uranium plasma composition is necessary in
order to describe initial or startup conditions in a
gaseouscore reactor. A different mathematical formulation
is used to calculate the plasma composition at low tempera
tures.
 62 
In the temperature range from 10000K to 10,0000K,
the only species that are present in significant quantities
are U, U and electrons. A single reaction accounts for
the equilibrium ionization and recombination of these three
species.
U t U1+ + e (4.24)
The corresponding Saha equation is
N+ e 2r7m kT 3/2 u
N 2 ( e  e kT (4.25)
No h2 o
The total particle density is
P
NOT = T = No + N + e (4.26)
The charge neutrality is given by
N = e (4.27)
Using N, = No, N2 = N+, Ne = e, and
27rm kT 3/2 u /
K = 2 e) eEo/kT
h2 uo
equations (4.25), (4.26) and (4.27) are simplified as
N2 N
N2e K (4.28)
N,
 63 
NTOT = N + N + Ne
TOT 2 e
(4.29)
(4.30)
N2 = N
Dividing each term in equations (4.28), (4.29), and(4.30)
by NTOT and combining these equations gives
C 2 + 2K*C K* = 0
e e
where
(4.31)
K* = K
TOT
The solutions of equation (4.31) are
Ce = K*( 1 /i + T7K)
Since K* > 0, /i + 1/K* > 1, and the positive radical must
be used to give a positive value of the electron concentration.
The low temperature plasma composition is then given by
N = NTO K* (1 + V1 + 1/K*)
N = NTOT
 2Ne
N2 = N
2 e
(4.32)
(4.33)
(4.34)
Uranium Partition Functions
The partition functions appearing in the general
Saha equation (4.2) are functions of the plasma temperature
64 
and the electron configuration of the particular uranium
atom or ion. The electronic partition function of itimes
ionized uranium is given by (39)
C Xi j/kT
ui = gij eij/kT
Si0 + i,l eX/kT + i,2 eX/kT + (4.35)
where
gi = statistical weight of the jth term of
'J itimes ionized uranium
X. = exitation energy level of the jth term
of itimes ionized uranium
The jth statistical weight for the itimes ionized uranium
is (40)
9i,j = (2Lj + 1)2Si + 1) (4.36)
where
L. = orbital angular momentum
S. = spin angular momentum
The statistical weights of the uranium atom are
calculated from the spectroscopic data of Steinhaus, Blaise,
and Diringer (41). Since experimental uranium ion data
from which the necessary Lij and Sij values can be
determined are not presently available, the isoelectronic
 65 
approximation (42) is used to calculate the statistical
weights and excitation energies of the uranium ions. This
approximation is based on the observation that atoms or
ions having the same number of orbital electrons have
similar electron configurations. The gij values are
calculated from (4.36), using the Lij and Sij values of
the neutral atom having i orbital electrons. The Xij are
obtained from spectroscopic measurements (43) of energy
levels of the neutral atoms.
Lowering of Ionization Potentials
The ionization potential Ei in equation (4.2)
represents the energy required for the ionization of an
isolated atom or ion. At high pressures the plasma
density increases until the atoms and ions can no longer
be considered as isolated. Each particle then moves in
an electric field associated with adjacent charged particles.
The ionization reactions are influenced by plasma micro
fields associated with electrostatic polarization (Debye
effect). The resulting effect is a lowering of the energy
required for ionization of an atom or ion, the effect
increasing with increasing charge density. To account for
the lowering of the ionization potentials, an effective
eff
ionization potential, E. is defined as
E.eff = E. AE. (4.37)
1 1 1
 66 
for the reaction Ui U(i+l)+ + e
where
E. = uncorrected ionization potential
AE. = lowering of ionization potential
Griem (44) has used the DebyeHuckel theory (36) for
ionized gases to obtain an analytical expression for the
amount of potential lowering. For an uranium plasma with
a maximum of twelve degrees of ionization, the potential
lowering is given by
13
AEi = 2(Zi + l)e3(r/kT)/2(Ne + Zi2Ni)1/2
i=2
(4.38)
where
Z. = charge of ith ion
e = electrostatic charge of the electron
N = electron density
N. = ith ion density
Cutoff Quantum Number as a Function of Effective Ionization
Potential
The electronic partition function of uranium was
given by
u = eXi,j/kT (4.35)
j=0
 67 
The index j is the principal quantum number of the jth
energy level of itimes ionized uranium. The partition
function of a hypothetical isolated particle cannot be
evaluated exactly since (4.35) involves an infinite number
of terms. When the summation extends to j = m, the
partition function diverges since Xij approaches a
constant value (the ionization energy), and gij is not
monotonically decreasing as j  However, due to the
effect of the Debye field, the summation can be terminated
at a finite cutoff quantum number, n corresponding to
the highest energy level at which the electron is still
bound. The calculational form of the partition function
based on bound electron states is
n (i)
c Xi,j/kT (4.39)
S j=0
The cutoff quantum number is a function of the effective
ionization potential since nc (i) is the largest integer
eff
value for j for which X. < E. For the ith uranium
1,3 1
ion, the cutoff number is expressed as
nc(i) = 3 Xij < E ef (4.40)
The relation between principal quantum number and
energy level for a hydrogenlike atom is (45)
 68 
pZ2 e4
eff
E = (4.41)
n 2h2n2
where p is the reduced mass, Z the nuclear charge, e the
electrostatic electron charge, h the reduced Planck's
constant, and n the principal quantum number. For the
multielectron atoms and ions in the uranium plasma, the
energy levels and principal quantum numbers are not
explicitly related by expressions such as (4.41). However,
it can be argued that a highly excited electron in a
multielectron atom is nearly hydrogenic, since the atom
can be regarded as a nucleus having an electron moving
about it in a large orbit. Based on this reasoning,
Drellishak et al. (46) have empirically treated the
lowering of the ionization potential by assuming that the
amount of potential lowering should vary inversely with
the square of the cutoff quantum number, according to
E.
AE = (4.42)
n 2(i)
c
where the symbols are defined as before. This relation
will be used in the following section to obtain initial
estimates of the cutoff quantum numbers of uranium atoms
and ions.
Computation of Uranium Plasma Composition
The computation of the uranium plasma composition
requires the determination of a consistent relation
between (1) the electron fraction (4.23) (or (4.31) for
low temperatures), (2) the lowering of the ionization
potential (4.38), and (3) the cutoff quantum number (4.40).
These expressions are seen to be completely coupled since
the plasma composition is dependent on the partition
functions, the partition functions are dependent on the
effective ionization potentials, and effective ionization
potentials are dependent on the plasma composition. The
uranium plasma composition is calculated using an iterative
technique which searches for the consistent solution of
equations (4.23), (4.38), and (4.40). A program called
UPLAZ2* was written to perform these iterative calcu
lations.
For a given pressure and temperature, the UPLAZ2
iteration is carried out as follows:
Step 1.Starting with equation (4.23), the electron,
atom, and ion densities are computed using only the first
term (ground state) of (4.39) in the partition functions.
These particle densities are used to compute the ionization
*Details of UPLAZ2 and other programs written for
this study are given in the Appendices.
 69 
 70 
potential lowering, AEi, according to (4.38). From
equation (4.42), the initial estimate of the quantum
cutoff numbers are calculated as
N ()(i) = /E~.E i = 2, 3, ., 13
c 1 1
(4.43)
Step 2.Returning to equation (4.23), the particle
densities are again calculated, this time including
n () (i) terms in the partition function series (4.39).
Using the new number densities, the next value of the
ionization potential lowering is calculated from (4.38).
The second estimate of the cutoff quantum numbers is
then determined from
c (i) = [nc(i)] (4.44)
where [nc(i)] is the largest integer value not exceeding
j, for which
X < E. AE. (4.45)
1, 1 1
where
E. = uncorrected ionization potential
AEi = current value of ionization potential lowering
 71 
Step 3.If n (2) (i) n1) (i) > 1 for any i,
Step 2 is repeated r times until n(r) (i) nc (i) < 1
for all i. When this condition is satisfied, the plasma.
composition is determined consistent with the ionization
potential lowering and cutoff quantum numbers.
Results of Uranium Plasma Composition Calculations
The UPLAZ2 program was used to calculate the
uranium plasma composition for pressures from 1 atm to
1000 atm and temperatures from 50000K to 120,0000K.
Typical results are shown in Figures 8, 9, and 10, for
pressures of 100, 500, and 1000 atmospheres. The electron
density and the densities of individual uranium species
are given in per cent of total particle density. Ad
ditional calculated uranium plasma compositions for
pressures from 100 atm to 1000 atm are shown in Figures 25
through 34, Appendix A.
The fact that the fuel region of the "gaseous"
core reactor is a plasma, rather than a ordinary gas, is
clearly demonstrated in Figures 8, 9, and 10. At a
temperature of 20,0000K, the total number of particles in
the fuel region consists of about 50% uranium and 50%
electrons. Above 40,0000K, the fuel region contains
approximately 80% electrons and only 20% uranium. In
 72 
100 71
U5+ U6+
10
H
w 7+
ClU
z8+
O 9+
010
0
0 40,000 80,000 120,000
Temperature, oK
Figure 8. Uranium plasma composition at pressure of 100 atmospheres.
 73 
100
10
Z U7+
E1
U
H
H
0.1
H
z
U
0.1
0 40,000 80,000 120,000
Temperature, oK
Figure 9. Uranium plasma composition at pressure of 500 atmospheres.
 74 
100
H UO' U'
10
H
U U
H
0+
S1
U
0.1
0 40,000 80,000 120,000
Temperature, oK
Figure 10. Uranium plasma composition at pressure of 1000 atmospheres.
 75 
order to determine the radiative emission and absorption
characteristics of the uranium plasma fuel region, the
particle densities of the various uranium species must
be considered, because each ion has different radiative
properties. In order to determine the nuclear criticality
characteristics of the reactor, only the total uranium
density is considered, since neutron induced fission is
a nuclear effect and is not influenced by ionization of
uranium atoms. This is not meant to imply that plasma
effects have no influence on nuclear criticality, since
the plasma characteristics of uranium actually determine
the critical uranium mass. This static coupling between
plasma and nuclear effects is shown by the equation of
state (4.3), which can be written as
12
P = P + Y P. (4.46)
i=0
where
P = N kT, partial pressure of electrons
e e
P. = N.kT, partial pressure of the uranium species
1 1
Ui+
This expression shows that in thermodynamic equilibrium,
one free electron creates the same partial pressure as one
uranium atom or ion. Thus, for a given constant reactor
 76 
pressure P, the formation of each free electron by ioni
zation is balanced by the removal of one uranium atom or
ion. From Figures 8, 9, and 10, the electron fraction is
seen to increase monotonically as the temperature increases,
which means that any temperature increase at constant
pressure requires a reduction in the amount of uranium
in the fuel region. The quantitative effects of temperature
on the uranium density are shown in Figure 11. The slope
dN
of the curve (i) gives the rate of uranium reduction
P
with increasing temperature at constant pressure. The
uranium reduction rate decreases at higher temperatures,
since the electron fraction changes very slowly for tempera
tures greater than 40,0000K. From Figure 10, a basic
characteristic of the plasma core reactor is seen to be
strong dependence on the uranium density on the plasma
temperature. For a given constant pressure, there is
only one temperature which will sustain the critical
uranium mass. Suppose that the reactor is to be operated
at a constant pressure of 400 atm and that the critical
mass corresponds to a total uranium density of l019/cc.
From Figure 11, it is seen that a temperature of 59,0000K
would be required to sustain the desired uranium mass. An
increase in the temperature would cause the reactor to
become subcritical and similarly, a temperature decrease
 77 
C.)
1021
S1028 II
0 40,000
Temperature, 'K
H
H
z
I 1 "
0 40,000
Temperature, 0K
Pressure, atmospheres
1000
Effect of temperature on total uranium density.
Figure 11.
 78 
would result in a supercritical reactor. This dependence
of critical mass on plasma temperature indicates that a
control system which automatically compensates for tempera
ture variations would be a necessary component of the
plasma reactor system. In an operating rocket reactor, a
change in the thrust requirement, accomplished by varying
the power output, will involve temperature changes in the
uranium plasma. Since a large temperature change at
constant pressure can significantly affect the criticality
and hence the reactor stability, the pressure must be
allowed to vary in order to maintain a constant uranium
mass as the temperature changes. Thus, the criticality
problem becomes a question of finding a pressuretempera
ture relation which sustains the desired uranium mass.
This relation can be expressed mathematically as a problem
of finding temperature dependent pressure trajectories
along which the uranium mass has the critical value. The
general pressure trajectory is expressed as
Pc = f(MU,T) (4.47)
where
P = pressure trajectory
MU = critical uranium mass
T = plasma temperature
 79 
f = functional relation between Pc, MU, and T
The functional relations which define pressure
trajectories are determined numerically using the UPLAZ2
program. For the uranium density corresponding to a given
critical mass and a given temperature, the UPLAZ2 equations
are solved for the pressure P. By solving the equations
for temperatures from 50000K to 120,0000K, the resulting
pressures define trajectories along which the uranium mass
is constant. The nuclear calculations discussed in
Chapter III indicate that a critical reactor would require
uranium densities in the neighborhood of 1019/cc. Calcu
lated pressure trajectories for several uranium densities
are shown in Figure 12. The most important trend shown
by these curves is that as the critical uranium density
increases, extremely high pressures are required in order
to operate the reactor at high temperatures. Therefore,
every effort should be made to reduce the critical mass
by optimizing the reactor geometry, the reflector geometry
and materials, and the fluid flow characteristics. From
Figure 11, it is seen that if the critical uranium density
can be reduced from 1.0 x 1019/cc to 0.5 x 101'/cc, the
pressure required to operate the reactor at 100,0000K is
reduced from 950 atm to 480 atm.
 80 
1000
800
0.5xl0
600
400
300
200 U235 density(/cc)
m O.1xlO10 19
100
10
0 40,000 80,000 120,000
Temperature, K
Figure 12. Pressure trajectories for constant uranium density.
 81 
The pressure trajectories for an actual hydrogen
cooled plasma core reactor will differ quantitatively
from those in Figure 12. This is to be expected since
the critical mass will not remain constant over wide
temperature variations, due to the temperatures dependence
of hydrogen scattering and uranium resonance absorption
as discussed in Chapter II. However, the application of
pressure trajectories in determining the critical mass
will still be valid when these two effects are included,
and the general form of the trajectories in Figure 12 can
be expected to be similar to those in the hydrogen cooled
uranium plasma.
CHAPTER V
PRESSURETEMPERATURE ANALYSIS OF HYDROGEN
PLASMA COOLANT REGION
Introduction
The high temperature plasma core reactor will be
cooled by hydrogen which is injected concentrically around
the central uranium region. Cooling will be accomplished
by radiant energy transfer from the high temperature
uranium plasma to the lower temperature hydrogen. When
injected at low temperatures, the hydrogen will be in a
diatomic molecular gaseous state. As the hydrogen coolant
flows coaxially around the uranium fuel, the hydrogen
temperature will increase due to absorption of radiant
energy from the fissioning uranium plasma. At high
temperatures hydrogen becomes dissociated and ionized so
that the exhausted coolant will consist of numerous hydrogen
species resulting from the various dissociation and ioni
zation reactions. In order to determine the neutron
scattering effects and radiant energy absorption properties
of the hydrogen coolant, it is necessary to calculate the
composition of a hydrogen plasma as a function of plasma
temperature and pressure.
 82 
 83 
Equilibrium Hydrogen Reactions
For the range of pressures and temperatures
expected in the plasma core reactor, the hydrogen coolant
may undergo any of the following equilibrium reactions:
1. Molecular dissociation
2. Atomic ionization
3. Atomic electron
attachment
4. Molecular ionization
5. Dissociative recombination
6. Dissociation of molecular
ions
7., 8., 9. Twobody
combination
10., 11. Threebody
recombination
H2 Z H + H
H + e < H
H +
+ +
H2 +e H + H
+ H+
H2 + + H
H + H2 H,
H2 + H+ H3
H + H + H2 + H
H + H + H H2 + H
From an examination of the reaction energies, absorption
properties, and expected coolant temperatures, the hydrogen
coolant is considered to be adequately described by the
first four reactions. Reaction 9 can indirectly enhance
the absorption properties of hydrogen at higher pressures
(47), but will not be included in the present analysis.
 84 
Hydrogen Reaction Equations
1. Molecular dissociation H2 f H + H.The
equilibrium relation between molecular and atomic hydrogen
involved in molecular dissociation at absolute temperature
T is (39)
N NH 27mHmk 3/2 UH H E'H /kT
H H _____ T3/2 e 2 (5.1)
NH mH h2 H,2
where
mH = atomic weight of H
mH = atomic weight of H
uH = partition function of H
UH2 = partition function of H
E'H_ = dissociation energy of H2
H2
k = Boltzmann's constant
h = Planck's constant
The partition function of atomic hydrogen is
uH = gj Xj/kT (5.2)
where
gj = statistical weight of jth term
X = excitation energy of jth term
 85 
The nth state of atomic hydrogen is 2n2 times degenerate
X0
and has an energy X, = X0 , where n is the principal
n
quantum number and XO is the ground state energy level.
On this basis, the partition function may be written
explicitly as
nc (1 XO/kT
UH = 2n2 e n (5.3)
n=l
where nc is the principal quantum number of the last bound
state. Defining a reduced energy variable a t X0/kT,
(5.3) is simplified as
n
H = 2e n n2 en2 (5.4)
n=l
For a sufficiently large quantum number, nL, the energy
levels approach a continuum. For n > nL the integral
approximation of IvanovKholodnyi et at. (48) can be
applied to the partition function. Replacing the near
continuum portion of (5.4) by an integral,
nL n/(5
UH = 2ea n2 + n e dn (5.5)
n=l nL
For
a/n 2 < 1
 86
n n
uH 2e n2 e + ndn
n= nL
ij
UH = 2e
x e + 4e/4 + 9e/9 + n eC/nL2 + 1(nc n3
(5.6)
The number of terms to be retained in the series (5.6) is
determined by the requirement that n 3 3 e /nL
This requirement also provides that n3 >> nL 3 so that
the partition function may now be written in a form which
is convenient for computation:
nL
u = 2ea n2 e + 1 n (5.7)
n=1
The quantum number, n at which the electron
becomes free due to preionization of hydrogen, is a
function of the charge density of the plasma. Ivanov
Kholodnyi et aZ. have derived a relation between nc and
the electron density Ne as follows:
log Ne = 21.65 6 log (nc + 1)
e C
(5.8)
 87 
This expression is consistent with the integral approxi
mation used in (5.5) and will be used to calculate the
quantum number appearing in the partition function (5.7).
The usual form of the partition function of
molecular hydrogen is
uH E uR u (5.9)
where
uE = electronic partition function
uR = rotational partition function
u = vibrational partition function
The electronic partition function is
u = g ej/kT (5.10)
The statistical weight is given by
g. = 2J(j) + 1 (5.11)
j H2
where J is the total angular momentum of the jth state
H2
of H,.
When the diatomic hydrogen molecule is pictured as
a rigid dumbbell with moment of inertia I, rotating in
three dimensions, the energy and degeneracy of the Jth
