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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