MUON CATALYZED FUSION PROCESS - TDL
Transcript of MUON CATALYZED FUSION PROCESS - TDL
MUON CATALYZED FUSION PROCESS
by
MYEUNG HOI KWON, B.S.
A THESIS
IN
PHYSICS
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
Accepted
May, 1987
ACKNOWLEDGMENTS
I would like to thank Dr. Y.N. Kim for his guidance and
advice during this work. Also, I would like to thank Dr.
V.K. Agarwal for serving on my thesis committee. Finally,
I would like to thank my family for their support which made
this effort possible.
11
CONTENTS
ACKNOWLEDGMENTS ii
LIST OF TABLES iv
LIST OF FIGURES v
PREFACE vii
CHAPTER I. THE CONCEPTS OF MUON CATALYZED
NUCLEAR FUSION 1
CHAPTER II. PROCESSES OF MUON CATALYZED NUCLEAR FUSION 10
2.1. Formation of Muonic Atom 12 2.2. Formation of Muonic Molecule 13 2.3. Nuclear Fusion Reaction 21 2.4. Probability of Muon Loss 25
CHAPTER III. KINETICS OF MUON CATALYZED NUCLEAR FUSION IN A MIXTURE OF DEUTERIUM AND TRITIUM 28
CHAPTER IV. NUMERICAL RESULTS AND CONCLUSION 4 6
REFERENCES 58
1 1 1
LIST OF TABLES
Table 2.1. Binding Energy of Muonic Molecule —
Table 2.2. Muonic Molecular Formation Rates —
Table 2.3. Nuclear Fusion Rate at Each State
of Muonic Molecule D|J.T
Table 2.4. Nuclear Fusion Rates
Table 2.5. Probability of Initial Muon Sticking in ^He
Table 4.1. Assumed Molecule Formation Rates
19
21
23
24
26
48
IV
LIST OF FIGURES
Figure 1.1. Nuclear Fusion Number in D-T Mixture 9
Figure 2.1. General Scheme for Muon Catalysis in a Deuterium-Tritium-Protium Mixture 11
Figure 2.2. Transition Scheme of D|l Muonic Atom
in D-T Mixture 14
Figure 2.3. Energy Level of Muonic Molecule 18
Figure 3.1. Muon Catalyzed Nuclear Fusion Processes
in the D2 + T2 Mixture 29
Figure 3.2. Maximum Number of Nuclear Fusion (1) 33
Figure 3.3. Nuclear Fusion Number Depend on Density 34
Figure 3.4. Maximum Number of Nuclear Fusion (2) 35 Figure 3.5. Transition and Charge Exchange of
Muonic Atoms 37
Figure 3.6. Population of T|i Muonic Atom 39
Figure 3.7. The Effect of the Existence of D|ID Muonic Molecules in Nuclear Fusion Cycle 41
Figure 3.8. Effective Muonic Molecular Formation
Rate 42
Figure 3.9. Concept of Cycle-by-cycle Analysis 45
Figure 4.1. Muon Cycling Rate Depend on Temperature in Liquid Target 50
Figure 4.2. Number of Nuclear Fusion per Muon Depend on Temperature 51
Figure 4.3. Muon Cycling Rate Depend on Target Density at Various Temperature 52
V
Figure 4.4. Number of Nuclear Fusion per Muon Depend on Target Density at Low Temperature 53
Figure 4.5. Number of Nuclear Fusion per Muon Depend on Target Density at High Temperature 54
Figure 4.6. Number of Nuclear Fusion per Muon Depend on Concentration of Tritium at High Temperature 55
Figure 4.7. Cycling Rate Depend on Concentration of Tritium at Low Temperature 56
VI
PREFACE
Nuclear synthesis in hydrogen isotopes was predicted
theoretically by F.C. Frank [1] and A.D. Sakharov [2] in 1947
and confirmed experimentally by L.W. Alvarez, Q^ .ai. [3] in
1957. After this discovery, a large number of theoretical
and experimental works came out. Finally systematic surveys
of muon catatlyzed nuclear fusion were done by J.D. Jackson
[4] and Ya.B. Zeldovich, Q^ ai. [5]. Unfortunately, these
studies showed muon catalyzed nuclear fusion was not useful
as a new energy source because of small muonic molecular
formation and fusion rates.
Interest in muon catalyzed nuclear fusion was revived in
1977, upon the theoretical predictions of high muonic
molecular formation rates calculated by S.I. Vinitsky, Q^ ai.
[6], following E.A. Vesman's assumption [7] of resonant
muonic molecular formation. Since then, considerable
theoretical and experimental works related to this system
have been done by many groups demonstrating the possibility
as a new energy source. But, there exist several
discrepancies between theoretical predictions by S.S.
Gershtein, ^ ai- [8], L.I. Menshikov, ^ ai- [9] and
experimental results by S.E. Jones, ^ al- [10] because of
Vll
the complexity of kinetic processes of muon catalyzed nuclear
fusion.
This work will describe the current status of muon
catalyzed nuclear fusion first and then show my calculations
based on a cycle-by-cycle analysis of muon catalyzed nuclear
fusion process which leads to lesser discrepancies between
theory and experiment.
viii
CHAPTER I
THE CONCEPTS OF
MUON CATALYZED NUCLEAR FUSION
It is known that nuclear synthesis of hydrogen isotopes
occurs at high temperature in thermonuclear reactions,
releasing energy among the isotopes as follows.
P + P->D + e" + V+2.2 MeV.
P + D-> ^He+Y+5.4 MeV.
D + D ^ T + P+4 MeV.
- ^He + n + 3 . 3 MeV.
^He + 7+ 24 MeV,
D + T_> ^He+n + 17. 6MeV.
Here, P, D and T are hydrogen isotopes proton, deutron and
triton respectively. In order for these syntheses to occur,
the reacting nuclei must approach to within a distance on the
order of the radius of action of the nuclear forces. This in
turn requires that nuclei must gain sufficient kinetic energy
to overcome the coulomb barrier between the charged nuclei.
To produce this amount of kinetic energy, we must use an
accelerator or an atomic bomb.
In nature, this penetration of coulomb barrier occurs
occasionally at much lower energies due to quantum mechanical
tunneling effect through the coulomb barrier. The
penetration factor (B) of this coulomb barrier is calculated
from the equation
B= Exp 2 r^ I
V 2M ( U (x) - E ) dx h Jx
1
Here X-j and X2 are positions of nuclei, M is reduced mass of
nuclei, U(X) is coulomb potential energy and E is kinetic
energy of nuclei. The penetration factor B for the ordinary
hydrogen molecule is about 1 x 10~^. This means that the
probability of nuclear reaction in the hydrogen molecule is
about 10"^ reaction during one year in 1 m- hydrogen liquid.
Here, as is known in quantum mechanics, the tunneling
probability depends on the separation between the hydrogen
nuclei significantly. For example, if the separation between
the hydrogen nuclei is decreased by one half, the possibilty
of a nuclear reaction is increased by a factor of 10^^.
The muon catalyzed nuclear fusion is defined as the
reaction induced by the presence of negative muons in cold
hydrogen leading to a nuclear fusion from following
processes.
When the negative muon enters a mixture of hydrogen
isotopes, it replaces the lighter electron of hydrogen
isotope to form a muonic atom. This reaction is
energetically favorable and rapid, because the muon greatly
overweighs the electron. This process occurs in
approximately less than lO"-'- second.
The same holds true for the succeeding muonic molecular
formation. The produced muonic atoms collide with the
molecules and replace the initial atoms in target molecules
and form muonic molecules. This muonic molecule is 200 times
smaller than an ordinary hydrogen molecule because of the
heavy mass of muon. Thus the muon confines the hydrogen
isotopes to a very tiny volume. Although the temperature of
the target is near room temperature, conditions inside the
muonic hydrogen molecule are similar to those found inside a
dwarf star.
In these local-star-like conditions, nuclear fusion will
occur very rapidly, resulting in the release of a muon. This
released muon participates in the muon induced nuclear fusion
cycle again. These processes are due to the properties of
the muon, which are:
1. The muon possesses the same charge as an electron.
2. The muon is almost 207 times heavier than an
electron.
3. The muon has a relatively long lifetime ( TQ =
2.2 X 10"^ second ) compared to other processes of
muon catalyzed nuclear fusion.
The fundamental process which determines the life time of the
muon in hydrogen is its decay into an electron, neutrino and
antineutrino.
i -> e + v_ + v. '
which occurs with a decay rate XQ = 0.455 x 10^ second"-'-.
In his Nobel prize acceptance lecture, Luis Alvarez [11]
described the first observation of the phenomenon of muons
stopped in a liquid hydrogen isotopes.
We had a short but exhilarating experience when we thought we had solved all of the fuel problems of mankind for the rest of the time. A few hasty calculations indicated that in liquid HD a single negative muon would catalyze enough fusion reactions before it decayed to supply the energy to operate an accelerator to produce more muons, with energy left over after making the liquid HD from the sea water. While everybody else had been trying to solve this problem by heating hydrogen plasmas to millions of degree. We had apparently stumbled on the solution, involving very low temperatures instead.
Further theoretical calculations and experimental
observations decreased this dream, because the rates of
muonic molecular formation and nuclear fusion were too small
to be useful due to the short muon life time. For example,
in the mixture of liquid hydrogen and deuterium case, fusion
occurs as follows from [12], [13] and [14].
First, the injected muon will be captured by a hydrogen
or deuterium atom:
P2 + IX~-> PM'+P + e~,
Ap„ = 4 x 10 second ,
—13
Tp^ =2.5x10 second ,
D2 + |r -^DM.+ D+ e~. Xdu = 4 X 10^^ second ^,
~13
T^^= 2.5x10 second
Here' X „ and X „ are the formation rates of muonic atoms per
second and T „ and T „ are the formation times of muonic
atoms.
Following the muon captured process, these muonic atoms
collide with target molecules and form muonic molecules.
P|i+P2-^P|^P + p,
- 1 Xp^p= 2 . 2 x 1 0 s e c o n d ,
— 7
Tp„p= 3 . 8 x 1 0 s e c o n d
Pm-Do - PM-D + D,
A<_i,d= 5 . 9 X 10 second , -pUd
_7
Tp^^= 1.7x10 second.
D|I+D;i DM-D+D,
jj„< = 0 . 8 X 10 second ,
Tjj j= 1.25 X 10 second.
These muonic molecules have the characters of
local-star-like conditions and induce nuclear fusion as shown
below.
p)j.p_>D + e' + |I+2.2 MeV,
^pupf = 2 . 6 x 1 0 s e c o n d ,
_7
Tp„pf = 4 x 1 0 s e c o n d .
PM- D ^ ^He 4- |i + Y + 5 . 4 MeV,
^pLidf - 0 -26 X 10 s e c o n d ^
TpHdf = 3 . 8 x 1 0 s e c o n d .
D|iD - ^ T + P + | I + 4 MeV,
^He + n + | l + 3 . 3 MeV,
''He + Y + | I + 24 MeV,
7
d idf =7x10 second ^
•d ldf _7
= 1.4x10 second ,
Comparing the lifetime of muon (TQ = 2.2 X 10"^ second) with
the time required for one fusion cycle ( T „ + T „ + " pudf ~
10" second ) demonstrates that a muon can only induce one
fusion reaction during its whole life time.
The interest in the problem of muon catalyzed nuclear
fusion was revived in 1977 by L.I. Ponomarev [14], S.S.
Gershtein [15] and S.I. Vinitsky, ot. al. [6] upon the
theoretical prediction of a high rate of DjiT muonic molecule
formation, which was confirmed experimentally later by V.M.
Bystritsky [16] . The high rates of Dp.D and D lT molecular
formation are due to the weakly bounded rotational and
vibrational state with quantum number J, vibrational number
V and rotational numbers K ( J = 1, V = 1, K ). For example,
the T|l muonic atom and deuterium molecule meet and form the
muonic molecule in the process,
*
T^+ D2-^ [ ( D^T) D2e ]vk.
From the calculations by Vinitsky, ^i ^ . [6], the molecular
formation rate of the D|IT molecule, its fusion rate and muon
sticking probability are
8
X,jj„ >10 second ,
dutf' lO second ,
W^^t^O.Ol.
Compared to the muon lifetime (TQ = 2.2^g ), these resonant
formation processes are so fast that one muon can induce
multiple nuclear fusions.
The processes of muon catalyzed nuclear fusion are too
complicated to understand exactly, since most of the
parameters in each process depend on the temperature,
concentrations of hydrogen isotopes and densities of target
materials. For these reasons, predicted number of induced
nuclear fusion reactions by one negative muon and the
observed values disagree substantially as seen in Figure 1.1
200
c o
Q.
o JO
E
c o "<0
(0
o 3
100 _
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Density
Figure 1.1.
Nuclear Fusion Number in D-T Mixture
(1 ) The average number of muon catalyzed nuclear fusion in D-T target observed by the BYU/INEL/LANL collaboration [10].
( 2 ) The maximum number of muon catalyzed nuclear fusion in D-T target on theoretical prediction by S.S. Gershtein, £i 3X- [8].
(3) The maximum number of muon catalyzed nuclear fusion in D-T target on theoretical prediction by L.I. Menshikov, QX. 3L1. [9] .
CHAPTER II
PROCESSES OF MUON CATALYZED NUCLEAR FUSION
The simple scheme of muon catalyzed nuclear fusion was
explained in Chapter I and it can be summarized like Figure
2.1. The notations for each process are as follows.
'0
' il
i|Ij
51^
w.
Muon decay rate.
Formation rate of muonic atom.
Rate of charge exchange between hydrogen isotopes i and j.
Formation rate of muonic molecule.
Nuclear fusion rate of each muonic molecule
Probability of muon sticking in He atom.
Concentration of hydrogen isotope.
These processes can be divided into 4 categories. The
first category details processes of formation of muonic atoms
including the transitions and charge exchanges of these
muonic atoms. The second category includes the processes of
formation of muonic molecules, the third category explains
the nuclear fusion processes of muonic molecules and the last
category is the steps of muon sticking to He atoms. These 4
categories will be explained step by step following the
procedures of muon catalyzed nuclear fusion.
10
11
1 He + Y
r e l e a s e d muon
F i g u r e 2 . 1 .
Genreal Scheme for Muon Catalysis in a Deuterium-Tritium-Protium Mixture
12
2.1. Formation of Muonic Atom
When a negative muon with several MeV kinetic energy is
injected in the mixture of hydrogen isotopes, it is captured
by hydrogen isotope instantly because of the mass difference
between muon and electron. The basic details involved in
this process are as below.
1. Negative muons are slowed down from the collisions
with target molecules,
2. Negative muons are captured by the hydrogen
isotopes in a very short time ( lO"-*- second ) ,
3. And transit to the ground states of muonic atoms
within 10"^ second.
The real schemes of these processes, however, are not simple
because of their dependence on the densities of target
materials and concentrations of the hydrogen isotopes of the
target.
When considering the density of the target, these
highly exited muonic atoms ( n > 14 ) reach exited muonic
atom states ( n = 5 to 7 ) in a time of 0.5 x 10"^^ x p"^
second and finally in a time of 1.4 x 10"^^ x p"^ second, they
reach the Is ground state of the muonic atoms [9]. Usually,
the concentrations of muonic atoms are treated the same
ratios as those of the isotopes without discussion [17], or
they are assumed that all of the lighter muonic atoms are
changed to the heavier muonic atoms by the charge exchange
between them in the condition of C^ > 0.1 [18] . Here C^
13
means the concentration of isotope which is heavier than
initial muonic atom.
These reactions of isotopic charge exchange are
irreversible because of the difference of binding energy
between them. The difference of binding energy between P|J,
and Djl muonic atoms which originates from the difference in
the reduced mass between them is about 135eV and different
energy between D|I and T|J, muonic atoms is about 4 5eV.
The transfer of the muon from a light isotope to a
heavy isotope is an elastic process in which the difference
in binding energies of the muonic atoms are converted to
kinetic energy of the relative motion of the charge-exchanged
nuclei.
L.I. Menshikov [9] organized the transition processes
of muonic atom using the diagram like Figure 2.2. For this
diagram, he used the calculated results of paper [19], [20],
[21] and [22].
2.2. Formation of Muonic Molecule
The ground state muonic atoms, which were slowed down
as results of elastic collisions with molecules of the
target, form the muonic molecules. In such a process, the
binding energy of the muonic molecule can, in general, be
given off either radiation or to the electron of the hydrogen
isotope molecule or to a neighboring nucleus to dissociate
the collided molecule.
14
n = 5
n = 4
n = 3
n = 2
i c ^O
----wiopq:
sop
n = 1
•--•B^spq
3/9 q3
1/9 q3 ?-' P t
0.18P
•--,_2. 5 P C ^
0.03 17pCt
> 4
^-—__3.10pCt
n = 5
n = 4
n = 3
n = 2
n = 1
Figure 2.2.
Transition Scheme of Dji Muonic Atom in D-T Mixture[9]
-11 (unit :10 second)
15
These processes are explained by the electric dipole El
transition of muonic molecules with conversion of the atomic
electron from the state J = 0 of the continuous spectrum of
the muonic atom and nuclear system into a bound state of the
muonic molecules with orbital angular momentum J = 1. This
case, one atom in collided molecule combines with muonic atom
for producing a muonic molecule. The released energy from
this formation of muonic molecule is carried away with the
Auger electron.
G. Conforto, ^t aJ.. [12], E.J. Bleser, ^t ai.. [13], V.B.
Belyaev, ^t ^ . [23] and V.P. Dzhelepov, Q^ ^ . [14]
calcualted the formation rates of muonic molecules following
this scheme and got the values as follows:
X „ = 2 . 2 X 10^ s e c o n d " ^ . p|ip
X „ , --= 5 . 9 X 10^ s e c o n d " ^ .
X, ..^ = 0 . 4 X 10^ s e c o n d " ^ .
X,.., = 0 . 5 3 X 10^ s e c o n d " ^ d|ld
X,„^ = 2 X 10^ s e c o n d " ^ . d|lt
X = 0 . 7 X 10^ s e c o n d " ^ .
In 1968, Vesman [8] suggested the existence of weakly
binding energy states of muonic molecule. The concept of the
resonant formation mechanism for muonic molecule is based on
16
the idea that the energy released from the formation of the
DjiD molecule transfers to the excited vibration of the
molecule which was produced when the muonic atom collided
with the molecule.
In the mixture of D2 and T2 case, the formation schemes
of muonic molecules are
DM- + D,
+
[ ( D|ID ) D2e ] j ^
• D
• D
® v_y
D D
D
T|i + D2 [ ( DIIT ) D2e ]vk ,
(V) • D
• D
^
\ ^
D|IT
D
This mechanism is acceptable only if at least one of the
binding energy levels of the DfID molecule has an energy lower
than the electron ionization energy ( electron ionization
energy of deutron is about 15 eV ) and dissociation energy of
molecule ( dissociation energy of deuterium is about 4.5 eV).
Also the kinetic energy of muonic atom has to satisfy the
resonance condition.
E = AE - E B
17
Where AE is the excitation energy of a muonic molecule and Eg
is the binding energy.
For a long time, the theoretical calculations did not
confirm the existence of such a weakly bounded system. S.I.
Vinitsky, QX^ ^ . [6] showed the existence of this small
binding energy level of D|ID muonic molecule from the
effective scheme of the adiabatic represention of the three
body problem. According to their calculation, they showed
the existence of 5 binding energy levels Ej^ in D|ID molecule
as shown in Figure 2.3. The calculated binding energies Eg
of these states of muonic molecules are given in Table 2.1.
Here J is orbital angular momenta and V is vibrational
numbers.
The rates of the resonant formation of the muonic
molecules are calculated by S.I. Vinitsky, ^t ^ . [6] from
the equation
X = No J 27Ch . I Tj J . 5(Ef - Ei) . 7(e, ET) de.
where E^ and E ^ are the energies of the final and initial
states of the system, y (e, e,j, ) is the Maxwellian
distribution of the incident muonic atom with the energy at a
given temperature T, corresponding to the average energy of
the thermal motion.
18
V
> i
u 0) G w H (0
-rH +J q 0) •p 0
200eV-
400eV-
600eV-
R
Distance
F i g u r e 2 . 3 .
E n e r g y L e v e l of Muonic M o l e c u l e
E n e r g y l e v e l of muon ic m o l e c u l e DflD i n s t a t e w i t h d i f f e r e n t o r b i t a l a n g u l a r
momenta J and v i b r a t i o n a l number s V [ 6 ] .
19
Csl
0)
*^
P O 0) .H O
o -H G O P S
o > i tn U Q)
W
tn C
-H
-o •H pq
>
-P -H C P
'>;
t
o
CNJ
r^ ^
tH
""
O »
TH
"^""^
iH ^
O
^
^~. o
o
•o o 4J 0)
s
o cr> •
CTi
cr>
r-•^ • o 1
o TH
o CO CN
iH CNJ .
CN ro
"^ o .
r-iH CO
a Q
.H <U
lev
o ^ H
<^ in .
CO 00
^ «X) . o
00
o "^ CNl CN
•^ iH •
CO CO
cr» KO •
CN CN CO
Q
C O -H -P (d
ox
irr
j-i 0. P. (d
*J3 iH .
00 o
00 VO .
o
TH *^
CN 00 CM
O
r-•
•^ 00
cr> o •
en iH 00
a Q
G O -H 4-) (d
.urb
>
-p M (U (i
"r 00 .
LO 00
00 00 «
x-i
^
r yo CN CN
KO VD •
in 00
cr> cr» .
^ CN 00
a Q
O (U ^ •P
1 1 1
1 1 1
1 1 1
in
en •
CN 00
r~ o .
CO iH 00
a Q
.H (d G O
,ati
<
-H M (d >
1 1 1
1 1 1
in in
(X) CN CN
*X)
r~ •
CN 00
r-CN •
' T CN 00
a Q
G O •H
ula
t
O •H fd O
20
3 &P = —X kT. ^ 2
Here k is boltzmann constant and Y ( e, e,j, ) is
, , , 27 e V 1 Y(e, ET) = -— X — . — exp ' ^ ' 27C er J ET ^ 26^
Matrix element of the transition is
I Tf i I ' = X X I dR^^^dP V'^*(Y. R)P ' *(P) H int mk m j
p(%)V'\r,R)M.
Where y^^'^) ( r, R ) and p<^'^^ (p) are the wave functions of
the initial (i) and final (f) states of the muonic molecule
and the ordinary molecule, respectively.
The resonant formation rates of muonic molecules of D|ID
and D|J,T depend on the temperature strongly. From theoretical
calculation by S.I. Vinitsky, ot. ai [6], muonic molecular
formation rates X^^j^^ is about 0.8 x 10° s" and iit -^ bigger
than lO s"-*- and less than lO s"- . Muonic molecular formation
rates are shown in Table 2.2 from [6], [24], [25] , [26] and
[27] .
21
Table 2.2.
Muonic Molecular Formation Rates
Theory
Experiment
Maximum
Minimum
T « 540
T ;= 100
^dMd
= 10^
<10^
> 10^
<10^
^d(it-T
.10^
<10^
8 = 3 x 1 0
= 0.12x10^
^dHt-D
< IC?
>10^
7 X 10^
= 4 xlO^
* T menas Temperature, unit is k
2.3. Nuclear Fusion Reaction
The muonic molecule is formed in an excited state which
will be deexcited rapidly. Then, the two nuclei in their
vibrational motions can penetrate the classical forbidden
Coulomb barrier and come within a nuclear interaction
distance of each other. In this case, a compound nuclear
system is formed, which subsequently deexcites or takes part
22
i n n u c l e a r f u s i o n c h a i n .
D^T->( ^ H e ) * ^ ^He + n + |i
-^ ( He ) ,
DIID-> ( ^He )* -> ^He + ^
( H e ) ,
The rate of nuclear reaction is described by L.N
Bogdanova [28] from the equation.
X = Ap jd^r |z1 R\Kj (r,R) pR=0,
Here Vj/j (r, R) is the value at R = 0 of the wave function
describing the relative motion of the nuclei in the muonic
molecule. In the muonic atom D|I case, it is represented by
the equation.
A„= lim(vap / 9k^Ci) ^ v = 0
Here penetration coefficient C^ is
9
ZiZ2e
23
and V, k are the relative velocity and momentum of the
nuclei.
The rates of deexcitation from the excited compound
nuclei are assumed to follow an Auger transition rate ^^n' '
The nuclear fusion rates at each state of muonic molecule are
shown in Table 2.3 [28].
Table 2.3.
Nuclear Fusion Rate at Each State of Muonic Molecule DjXT [28]
Tl
5
4
3
2
1
( J V )
(11)
(01)
(20)
(10)
(00)
^ n n -
n ' = 4
n' = 3
n ' = 1
n ' = 2
n ' = 2
n ' = 1
11 - 1 ,X 10 s e c
, 1 1 . 4
, 1 . 3
, 0 . 0 2
, 0 . 4 4
, 0 . 5 6
, 0 . 4 2
1
3 . 9 X 10^
12 1 .0 X 10
5 1 .0 X 10
8 1 .0 X 10
12 1 .0 X 10
The probabilities of the nuclear reactions from the
various rotational-vibrational states ( J V ) are calculated
from the comparison of X^^x and ^j^ . The nuclear fusions of
24
compound nuclei in J = 0 states are dominant among the
possible states.
Nuclear fusion rates of muonic molecules are summarized
in Table 2.4 from the results of [5], [28] and [29].
Table 2.4.
Nuclear Fusion Rates
nuclear reaction reference
D|J-D He+n+| l L.N.Bogdanova [28] 1 . 0 X 1 0 '
DM-T
TM-T
He+n+| I
He + 2n+I l
Ya.B.Zeldovich [5]
L.N.Bogdanova [28]
C.Y.Hu [2 9]
Ya.B.Zeldovich [5]
12 1.0 X 10
1.1 X 10 12
0.7 X 10 12
1.0 X 10 11
2.4. Probability of Muon Loss
When a nuclear reaction occurs in a muonic molecule,
there exist the processes which lead to muon loss after
nuc lear fusions. In this process, the muon sticks to a ^He
25
atom following from the nuclear fusion. This sticking
coefficient W is calculated by S.S. Gershtein [33] from the
equation,
= X ^ (n) W
where W(n)= \p^^)\
|Jdry^(r)Xl/i(r) \\
Here W(n) is the probability of a muon being captured in an
emitted muonic atom with state n = (nlm) described by a wave
function \|/j (r) . Since, the fusion reaction takes place at
very small distance R = 0 between the nuclei, the initial
state wave function Y''"( ) of ^^^ system practically coincides
with the wave function of the muonic atom (jlA)''", where A is
the intermediate compound nucleus. From these concepts, muon
sticking probabilities W ^ are found to be Table 2.5.
The produced muonic Helium atoms have an initial energy about
4 MeV, which may lose muons in the reaction of ionization and
charge exchange from the collisions with targets. Here, the
ionization process is
(iHe)"*" + D2 ( or T2) -> He"^ + D2 (or T2) + |i~,
and the charge exchange process is
(llHe)"*" + D2 ( or T2) -^ He^ + D^.
Table 2.5.
Probability of Initial Muon Sticking in He
26
Reaction Reference w
D|iD — ^ M- He + n L.N. Bogdanova, et al. [32] 0.133
D.Caf f rey , e t a l . [30] 0.895x10
D j i T — • ^ He + n L.N. Bogdanova, et al. [32] - 2
0 . 8 4 8 x 1 0
C.Y. Hu [29] - 2
0 . 9 x 1 0
T |JT —>• p. He + n S .S . G e r s h t e i n [33] 0.1
L.I. Menshikov, si ^ . [34] used the concepts of the
transition processes of muonic atoms to calculate the
stripping factor of muonic Helium as a function of the target
density.
' 't = ( 1 - f ^ ) =<P (-0 • 26 - 0 • 069 )'
27
and
Ydud = ( 1 - Q^3^^p ) exp (-0 . 05 - . 039 ),
Here Y M and Ydud ^^^ ^^® stripping constants of D|IT and D}ID
muonic molecules respectively. So, the effective sticking
coefficients of muonic molecules are
Wd^it = ( 1 - T d n t ) Wd^t ^
and
^ d u d = ( 1 - Yd^id ) ^d^id
Here W^ and W are initial and effective muon sticking
coefficients, respectively.
CHAPTER III
KINETICS OF MUON CATALYZED NUCLEAR FUSION
IN A MIXTURE OF DEUTERIUM AND TRITIUM
Among the several processes of muon catalyzed nuclear
fusion in mixture of hydrogen isotopes, the deuterium and
tritium ( D2 + T2 ) target is the most interesting because of
the possibility of enhancing the multiple number of fusion
cycles per muon.
The general scheme of the muon catalyzed nuclear fusion
for the D2 + T2 mixture is shown in Figure 3.1.
As discussed in Chapter II, the injected negative muon
is captured to form a muonic atom DjJ. or T}I. This process is
simple because it depends only on the concentrations of
deuterium and tritium in the target.
Those muonic atoms can form muonic molecules or transfer
muons to form heavier muonic atoms.
DM^D )
D|-tT ^
DJJT \
T M-T )
28
29
00
0) u p
-H
0) M P -P X
-H
B^
Q
(I) ^ -P
C -H
CO Q) CO (0 Q) O O U
C O
-H CO P
0) .H O
p
<u N
> 1
(d 4-> (d u c o p S
30
These processes are very complicated because each
process depends on the density, concentration and the
temperature of the target and the energy of injected muon,
etc. If the muonic molecules are formed, nuclear fusion
occurs immediately because of the characteristics of muonic
molecules.
To describe this system, S.S. Gershtein, QX. . (1980)
[8], S.G. Lie, £lL . (1982) [35] and L.I. Menshikov and L.I
Ponomarev (1984) [9] used the following set of equations and
tried to solve the kinetics of this system.
First, they described the set of relevant muonic and
nonmuonic reactions, and then formulated the equations of
associated reaction rates. The relevant relations are
represented by the following sets:
U ~ - > e " + Ve+v;, .
T + |I~ -> T^ .
D + | l" ^ D^ .
T|I+ D2 ( DT )-> n + a + p. .
n + a|I .
T|I+ DT ( T2 ) -> n + a + |I .
n + a | I .
D|I+T-^D+T^
D |I+ D He + n + |I
-» |i He + V .
The rate equations describing the above reactions are
31
dN ^ = ( >.o + a ) N^- >-f ( 1 - W3 ) N^^t - >-fd[ 1 - I (Wd+ w; )
- Xf t ( 1 - Wt ) Nt ,t
N d|Id
dN d l
dt - ( 0 + dt Ct + d ld Cd ) ^d\l ~ ^a^dN^ /
dN t^ dt
= ( 0 + V t Cd+ t^t Ct ) N - Xdt Ct Ndji- >.a Ct N^
T ^fd^d^d^id /
dN d|It
dt = ( 0 + \ ) Nd^t - d it Ct Nt^ ,
dN tp.t _
dt = (XQ + Xft ) Nt^t ~ t it Ct Nt^
dN — — = (XQ + Xf^ ) N(j„d ~ dud Cd Ndu , dt
Here N- are the time dependent densities of muons, muonic
atoms and muonic molecules.
In this treatment, they did not consider the effects of
32
concentrations of target materials, density of liquid and
temperature adequately. Figure 3.2, Figure 3.3 and Figure
3.4 show the predicted and observed nuclear fusion numbers in
D2 and T2 target.
For the full consideration of these several conditions,
a cycle by cycle analysis is required to describe the entire
fusion processes of one muon during its lifetime.
The concept of cycle by cycle analysis is that the
processes of muon catalyzed nuclear fusion are described by
the time subsequent behavior of one muon. Typically, the
density of injected muons is around 10- /cm and the density
of liquid target is 4 x 10 /cm- . This means injected muons
can behave independently to each other. So, the cycle by
cycle method is valid for a real system. Also, this analysis
method is more beneficial than other methods because of its
accuracy and possibility to calculate several important
values of X^, X^ and the average muon loss per cycle. Here X
is the possible number of muon induced nuclear fusions per
muon and X^ is the cycling rate of muon per unit time.
The analytical steps of cycle by cycle analysis are
summarized as follows:
a ) Formation of muonic atom. From the Chapter 2.1,
muon is captured to form an highly excited muonic atoms.
33
c o
0> Q.
o .o E 3
C O U) 3
u 3
150
100
0.75 1.00
Density
Figure 3.2.
Maximum Number of Nuclear Fusion (1)
( 1 ) The average number of muon catalyzed D-T fusion cycles observed by the BYU/INEL/LANL collaboration as a function of target density for cold ( Tcr lOOK) temperature with C^ = 0.5 [35].
(2 ) The predicted maximum number of muon catalyzed D-T fusion with the assumption XH = 1x10^
dut s-1 [8]
(3) The predicted maximum number of muon catalyzed D-T fusion with the assumption X = 1x10 s " [8]
34
100
c o 3
<1> CL k. 4) Xi E 3
C (O
(0
o 3 z
Tritium Concentration
Figure 3.3
Nuclear Fusion Number Depend on Density
The predicted number of muon catalyzed nuclear fusion for various values of 5 and Ct with assumptions of X,dut-D = 7x10^ S , Xdut-D= 3x10^ s~ .
35
Density
Figure 3.4.
Maximum Number of Nuclear Fusion (2 )
( 1 )Observed number of muon-catalyzed nuclear fusion by Jones, et al. [36].
(2) Redrawinging graph using the results of L.I. Menshikov, et al. [9].
36
D + l -> ( D l )^
T + l -> ( T l )^
The initial distributions of muonic atoms depend on the
concentrations of hydrogen isotopes in the target materials.
Then, highly excited muonic atoms lose their kinetic energy
from the collisions with target molecules. During these
processes, excited muonic atoms transit to the lower state of
excited atoms or transfer their charge to the heavy atoms.
(D^l)^ • (D|l)„'+D(or T)
dt.
^0
( T|I )^ 4- D ( or T )
D + e + Vg + V^
or
( TH \ K TH )^ • + D ( or T )
0 •T + e + Vg + V i
Here X'- is the transition rate, X^^ is the charge exchange
rate and XQ is the muon decay rate. The entire processes of
these steps are shown in Figure 3.5. From these concepts.
37
-4AAA-(^
- *AA/^^
-•AA/MT
-*AA/M^
-•-wM^
F i g u r e 3 . 5 .
Transition and Charge Exchange of Muonic Atoms
38
ground state muonic atoms D|l and T|I are formed finally like
Figure 3.6.
Usually the populations of Tjl and Dp. muonic atoms in
target were assumed that
1) they are the same as the ratios of concentrations of
D2 and T2 [17] ,,
or
2) if C . > 0.1, all of the D|I muonic atoms transfer to
T|I atoms [18] .
The above two assumptions did not consider the effects of the
density of target and the concentrations of hydrogen isotopes
adequately.
b) Formation of muonic molecule. The ground state
muonic atoms collide with target molecules and form muonic
molecules following resonant molecular formation processes:
1) Dp + D2-^[(DpD)D2e]*^ .
2) Tp + D^ ->[(DpT)D2eL vk
3) Tp + DT->[(D|j.T)T2e]^ *
k -
4) Tp + T^ -4[(T|iT)T2e]* ,
And the concentration ratio of D2 : DT : T2 in mixture of
hydrogen isotopes D2 and T2 is C^^ : 2C^C^ : C^^, because of
the charge exchange between D2 and T2 molecules.
39
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Tritium Concentration
Figure 3.6.
Population of Tjj, Muonic Atom.
The population of the ground state T|j, atom as a function of the tritium concentration for various densities P of the D2 + T2 mixture
40
Usually, total formation rate of muonic molecule DfiT is
treated as
dm a dm-D dm-
Here ^^ut-D ^^ ^^^ muonic molecular formation rate of process
2) and ^^ut-T -^ muonic molecular formation rate of process
3). Also, processes 1) and 4) are normally ignored because of
their small molecular formation rates relative to the
formulation rate of DjlT molecule. But, in the calculation of
the total number of nuclear fusions per muon, even a small
ratio can affect significantly because of the characteristics
of the cycle process.
Figure 3.7 shows the effect of the existence of D|J.D
molecules in the calculation of nuclear fusion number. And
Figure 3.8 shows molecular formation rate <„*- by the
concentration ratio of D2 : DT : T2.
c) Cycling rate X,_ of muon. Cycling rate means the
inverse of the average consumed time for one induced nuclear
fusion. It usually treats with the equation
1 Qis^^d , 1
Xc ^dtQ ^d^l(^d
or
4 1
1.00
c o 3
O
o
0.75 _
« 0 .50 _ LU
O
"o o
0.25 _
0.00
0 20 40 60 80 100 120
Cycling Number
F i g u r e 3 . 7 .
The Effect of the Existence of D}iD Muonic Molecules in Nuclear Fusion Cycle
• Line (1)
• Line (2)
• Line (3)
• Line (4)
Density ^^^^^ is 0% case
Density P ^ ^ is 1% case
Density P ,. is 5% case
Density P d^id
d^ld is 10% case
42
8 0>
CO
6 . (0
E o LL
k . (0 3 O 0)
,_
(0 00
o T —
. • ^ c 3
4
c o 3
2 -
Trltium Concentration
F i g u r e 3 . 8 .
Effective Muonic Molecular Formation Rate
( 1 ) Effective molecular formation rate^dM^t by the effect of concentration ratios of D2 / DT and T2 .
(2) Effective molecular formation rate X,d}xt by the effect of concetration ratios of D 2 and T2
^d|it-D= 7 X 10^ s^ ,X d It- T= 3 X 10^ s ^
43
1 QisxC
A.dtCt + 4*
1
^10 + d^tCd
4 4Ulo + d tCd.
^d^lt^^d
The product 0^3 x C^ is the probability that the muon reaches
the D|I ground state, X-^Q is the effective triplet quenching
rate which has contribution from both D and T collisions.
This cycling rate becomes a very important parameter in the
nuclear fusion cycle since it restricts the fusion cycle.
In a cycle-by-cycle analysis method, it is easily
calculated from the summation of consuming time of a muon at
each step as ,.
r = Xpi^Ti.
Where P ^ is the possibility of the muon passing by that stage
and T^ is the consuming time for that stage. So, total
consuming time per nuclear fusion is denoted as the equation,
T = C t X
X y ^ d U t i
T. H ^ ^djLlt
( Td i i t i "^ T^jUti) + It j t T, + " X, Dt
j d ' j d -d^ld
44
Here i, j denote each state of muonic atoms and molecules, f
means the fusion time at each muonic molecule, and T^ is the
sum of muon captured time by hydrogen isotopes and transition
time of muonic atom from n > 14 to n = 5 state.
The calculation processes of muon catalyzed nuclear
fusion are shown in Figure 3.9,
45
Muon
Muon l o s s
Muon r e c y c l i n g
F i g u r e 3 . 9
Concept of Cycle-by-Cycle Analysis
CHAPTER IV
NUMERICAL RESULTS AND CONCLUSION
As explained in Chapter III, the method of cycle-by-
cycle analysis can be used to study muon catalyzed nuclear
fusion system. For the numerical estimates of this system,
the values below were used for each process of muon catalyzed
nuclear fusion. And the muon cycling rate ( X_ ) and number
of nuclear fusions per muon were calculated. These results
came out with the dependency of density of the target,
concentrations of the hydrogen isotopes and temperature of
the target.
The main numerical values are as follows.
a) Formulation of muonic atom. This process includes
the slowing down of the muon in a mixture of hydrogen
isotopes, its capture into high levels of the muonic atom,
the de-excitation to the ground state and the transition of
the muon to the heaviest hydrogen isotope.
The rate of slowing down of the muon in hydrogen
isotopes is
X , ^ = 10^ to 10^° sec"l [19], slow down
46
47
After slowing down, muons with energy around 10 keV are
captured in highly excited states of muonic atoms with a
rate,
^ capture ' ^^^^ sec"! [38].
These highly excited muonic atoms cascade to an excited
state ( n ~ 5 ) with a rate
^ de-excitation ^ ^^^^ sec"! [39],
Then, the excited muonic atoms deexcite to the ground
muonic atoms, or transfer the muons to heavier atoms. For
these processes, the schemes of L.I. Menshikov, ^t aJL- [9]
were followed.
b) Formation of the muonic molecule. In collisions of
the muonic atoms with the molecules D2 and DT, the muonic
molecules DjlD, DjlT and T|IT are formed. For the rates of
muonic molecular formation, the results of S.I. Vinitsky, £JL
al. [6], W.H. Breunlich, st al- [25] and S.E. Jones, ^t. aJ..
[26] were followed, and the following molecular formation
rates were assumed as shown in Table 4.1.
48
Table 4.1.
Assumed Molecule Formation Rates
Low Temp.
Middle Temp
High Temp.
Vd
,0.8 X 10^
Vt
4lt-D
8 X 10^
6 X 10^
8 4 X 10
8 3 X 10
2 X 10^
1 X 10^
Vt
6 3 X 10
0 *High temperature ~ 1000 K, Middle temperature = 550 K, Low temperature = 100 K.
c) Nuclear reaction in muonic molecule. From the
characteristics of muonic molecules, the nuclear fusion
occurs rapidly. The following nuclear fusion rates of muonic
molecules from [28], [4] and [30] were used here.
-d|Jci xt,..^ « 1.0 X lo^
1^ 1.0x10^^.
xl Mt 1 . 0 X 10^^,
49
d) Muon sticking and stripping process. Muonic helium
atoms are produced from the reaction of nuclear fusion. These
muonic helium atoms are no longer neutral and the muons
remain in He atom for the duration of their life times. We
use the rates of these processes following results from [31],
[32] and [33].
^dud =0.133
d|it = 0.00848.
V t = 0-1.
These muonic helium atoms collide with hydrogen isotopes
and slow down from initial kinetic energy to the thermal
energy. During these processes, muons are stripped from the
helium atoms and these stripped muons can again take part in
the fusion chains. I used the concept of muon stripping
followed L.I. Menshikov, QX. al- [34] .
At this time, it is difficult to check the accuracy of
this cycle-by-cycle analysis method, because there is little
experimental data available. In this chapter, the numerical
results of the cycle-by-cycle analysis will be compared with
experimental data from [10], [26] and [31].
The muon cycling rates ( X^ ) and the number of nuclear
fusions per muon are shown in Figure 4.1, 4.2, 4,3, 4.4,
4.5, 4.6 and 4.7.
50
250
0 100K_,_ 400 K° 600 K° 800 K° 1000K°
Temperature
Figure 4.1
Muon Cycling Rate Depend on Temperature in Liquid Target
5^5,5 • Experimental data [24] : Calculated results.
51
150
c o 3
Q) Q .
E 3
C O w 3 u.
50
u 3
100 ¥P 300 K° 500 K° 700 K° 900 K° >
Temperature
F i g u r e 4 . 2 .
Number of Nuclear Fusion per Muon Depend on Temperature
Experimental data [24] Experimental data [8] . Calculated results. Experimental data [31].
52
125
(O
o o (0
100 _
0)
(0
D) c 73 >» O c o 3
75 _
50
25 _
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Density
Figure 4 .3
Muon Cycling Rate Depend on Target Density at Various Temperature
2,iE,S Experimental data [24] Calculated results.
53
c o a
^
^
0) M
-H
O
-o G <U Ou 9 (D
a
o
p -p (0 u <u g 0) H (U
o *^ - H CO P ^ >
(d
0) CO C (U Q o
0) M-l tJ^ O ^
(d
0)
P S
lO
o
00
I O CO Csl
-H ._.__, ro -p O
-H TJ
fd fd -p -P fd fd -O TJ
fd -P fd
rH fd fd (0 ^ fd -P -p 4J
CO O C G a . -H 0) <U (U
o -P 6 S g 0) -H -H -H
I, Jq M 5-1 M " O 0) 0) Q)
(u a a a P ^ X X X
o H w w w
0 KHl4^f*
o i n
o o o in
uoisnj JB8| onN lo jaquinN
54
150
c o 3
O Q .
O
E 3
C O
3 U .
CO
u
120 _
Density
F i g u r e 4 . 5 .
Number of Nuclear Fusion per Muon Depend on Target Density at High Temperature
55
150
120
c o
Q .
0>
n E 3 C _o "5) 3 U. k . (O
u 3
90
60
30
0.0 0.2 0.4 0.6 0.8 1.0
Tritium Concentration
F i g u r e 4 . 6 .
Number of Nuclear Fusion per Muon Depend on Concentration of Tritium at High Temperature
56
120 _
o
o
c
o < ^
QC
C
75 > . O c o
90
60 _
30 _
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Tritium Concentration
F i g u r e 4 . 7 .
Cycling Rate Depend on Concentration of Tritium at Low Temperature
Calculated results. Experimental data [31]. Best fitting line by group [31]
57
Figure 4.1 shows the temperature dependency of muon
cycling rates. In this graph, O, A and # represents
experimental data from S.E. Jones, eial. [24]. Numerical
results ( X^ ) from cycle-by-cycle analysis are in fairly
close agreement with the experimental results. Figure 4.2
shows the temperature dependency of the number of nuclear
fusions per muon. O and # represent the experimental data
from [24]. Figure 4.3 shows the density dependency of muon
cycling rates. In this graph, O, A and D represent
experimental data from [24]. The calculated results are in
close agreement except for the cases of low densities ( p <
0.3 ) . Figure 4.4 and 4.5 indicate the number of nuclear
fusions per muon depends on density. In figure 4.4, dot
line, O A • and • are based on calculated data by L.I.
Ponomarev, et a_l. [17] and experimental data by S.E. Jones,
jet, .ai. [10], [24]. The results of cycle-by-cycle analysis
are reasonably close to the published data. Figure 4.6 and
4.7 show the tritium concentration dependency of muon cycling
rates and the number of nuclear fusions. In figure 4.7, #
and solid line represent the experimental results from [31]
and calculated results.
The cycle-by-cycle analysis of muon catalyzed nuclear
fusion in hydrogen isotopes is well agree with the
experimental results except for the extreme cases of low
density of target. More accurate values for each process
would improve the accuracy of this model.
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