NA I L INSI. Ur Ot I

AlllDb 038136

NBSIR 87-3532

REFERENCE

Model for the Nuclear Effects of the

dt/i Fusing State

Michael Danos

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD

and

L.C. Biedenharn and Alfons A. Stahlhofen

Department of Physics

Duke University

Durham, NC 27706

April 1987

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

€

00

157

'87-3532

987

NBSIR 87-3532

MODEL FOR THE NUCLEAR EFFECTS OF THE

dt/A FUSING STATE

Michael Danes

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD

and

L.C. Biedenharn and Alfons A. Stahlhofen

Department of Physics

Duke University

Durham, NO 27706

April 1987

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Abstract

Simple models are constructed which are used to

illustrate the interplay between the strong nuclear

forces and the Coulomb molecular forces in the

description of the resonant dtp-fusion process.

Key words: deuterium; meso-mol ecul e; mu-meson; muon catalyzed fusion

muon molecule; t-d-fusion

1
1. Introduction

The six-body system of muon-catalyzed fusion [l] 1 (five nucleons plus the

muon) is so complicated and intricately interwoven that direct visualization

of the reaction becomes extremely difficult. In order to make the complicated

interplay of the molecular and nuclear physics accessible we will present here

models which focus schematically on the fusion dynamics alone, while stripping

away many of the details and their associated complications. These details

are, in fact, quite essential for an accurate calculation of the fusion rate

and the branching ratios into the various final states; for example, in an accurate calculation of the sticking fraction. The models, however, are

useful didactic tools, helpful in understanding the complexities of the

complete problem.

Let us detail how we arrive at our models in the following steps. (1) We

replace all potentials. Coulomb and nuclear, by square wells and barriers.

(2) We drop all angular functions, and retain only the remnant of the radial

parts, which we further truncate- to a one-dimensional problem, that is, to a single coordinate, x. The image of the dt-channel , denoted as channel d, is

taken as the region x > a; that of the na-channel , channel n, as x < -a, and

x = ±a represent the R-matrix channel radii of the respective channels.

(3) We ignore the presence of the muon in the na-channel, and we represent its

interaction with the system in the dt-channel as a weak attractive square well

located at c s x < c + t = i. In this way we arrive at our potential shown in

figure 1.

-* CHANNEL n CHANNEL d *

(n a ) ( d t )

Fig. 1, Potentials for the one dimensional model. See text for explanation.

lumbers in square brackets indicate the literature references at the end of

this paper.

2
The full line is the potential of the caricature of the complete

tdu-system. This potential can be adjusted to reproduce the spectrum of the

pnysical system, so that it yields a resonance at ED (to mimic the 55 keV resonance in 5 He), while the widths and heights of the barriers, t

x

, h

1 and

t

2

, h

2 can be adjusted to reproduce the partial widths. Finally, the

parameters of the "muonic potential well", t, h^ and c, can be adjusted to

yield a bound state resonance at E^ = -300 eV and the desired fusion width.

For simplicity we shall assume that there exists only a single "molecular

bound state," instead of the two J = 0 bound states of the real dty-system.

This model, although simple, is still much too complicated to allow for a transparent analytical solution. Let us therefore begin by replacing the

finite-range potentials by zero-range (i.e., 6-function) potentials. In this

way we will achieve a description which allows a completely explicit solution

and thus lends itself to full discussion of the mechanism of the interplay of

the different physical components. We do this in section 2. Of course, with

this doubly simplified pedagogical model we have moved far enough from

actuality that there is no possibility of any quantitative use. Hence we present the discussion of the somewhat more realistic fi nite-range .model in

section 3.

2. The Zero-Range Model

In order to solve easily for the nuclear effects we now construct a zerorange nuclear model. Note that on the scale of the y-molecular i 0 n (~ 500 fm)

the nuclear dimensions (~ 2-3 fm) are indeed small.

The zero-range nuclear model uses the approximation: k cot 6 = - 1/a,

where a is the Fermi scattering length associated with the nuclear cross

secti on

:

a = 4tt A

2 sin 2 6 . (2.1)

In order to generalize this to two channels, we must first work the in

terms of a Hamiltonian:

H ip(r) = (T + V eff ) 4>(r) = E r) . (2.2)

3
The effective potential V ^(r) is to be singular in such a way that

0 +

/

V P ff

= w

o ^ 0+ ) 0 - (2.3)

Integrating the Hamiltonian equation, (2.2) above, over the same infintesimal

interval we find (we use units such that ft = c = 1)

- ^ v(0+) + W

0 i|;(O+) = 0 . (2.2a)

In other words, the zero-range model is equivalent to imposing the boundary

condition

JkL

ip 0 + 2mW

Q

1a (2.4)

where ip( r) = A sin(kr + 6).

We generalize the singular potential to two channels to obtain:

and

(2.5)

( 2 . 6 )

Of course, V, = V ..

* dn nd

Integrating over the interval at the origin, as before, we obtain (we now take into account the different reduced masses in the two channels)

*j(0+)

^(0+)

+ W

A^

d (0+)

B^

n (0+),

= 0 , (2.7)

with W a symmetric, constant, 2x2 matrix.

We now explicty limit our attention to the case where the nuclear system

*

( 5 He ) has an energy such that the dt-channel d is closed and the na-channel n

is open.

4
Introducing in the effective nuclear interaction an energy shift between

the n and d channels to account for the difference in the channel thresholds

one has

E nucl 'shi ft ( 2 . 8 )

Thus at the energy under consideration, < is the wave number in the open

n channel and k is the wave number in the closed d channel, both correspond!’ ng

* to the same energy for the 5 He system. Thus the wave function for the system

has the form:

Introducing this into

-kr

y = A e

B sin(<r

n + 5 n

)

2.7) we find that:

(2.6a)

M (2.7a)

The requirement that a non-trivial solution exist, is the condition that

Det M = 0 (2.9a)

where M is the matrix above. This condition is found to be:

(k + 2m, W,,l f- < cot 6 + 2m W 1 = 4m

^ Q dd J ^ n n nn J (Wv ,)2

nd ' (2.9b)

Since according to eq (2.8) the d-channel wave numbers k is determined by

the n-channel wave number < , and since all the other parameters are constant

we see that eq (2.9) expresses the phase shift 6 p as a function of the

n-channel energy. Recall that this is the region where the d-channel is

closed, the n-channel open.

Now let us introduce the muon into the problem.

5
There are very few solvable models for the hydrogen molecular ion — the

prototype for the (dtp) + ion — so we must be very schematic in order to

achieve a solvable model. Physically the muon binds the dt-system since it

has a significant probability of being between the two nuclei developing

thereby an effective attraction. A reasonable facsimile of this situation

would be the solvable model of a weak, long-range, attraction between the d

and t nuclei as done in the finite-range model, figure 1. An even simpler

model replaces this by a 5-function attraction at a distance b between the

nucl ei

.

Thus we add to the system the attractive interaction

V dt

= -Y2 S(r

d

- b) . (2.10)

The effect of this interaction is to replace the (dt) wave function in

eq (2.6a) by

-kr +kr

^

d = A(e + a e

d

)

, 0 < r d< b . (2.11)

Integrating across the singularity at r d = b yields

-kb kb

- it: P

+ v^---\b 1 - 0 • ( 2 - 12a > d e + a e or,

2kb

k

[ 1 + “ %kb

~

1

] * 2m y

2 r . ( 2 . 12 b) a e +1

Thus

:

a -2kb ,2k

(2.13)

Let us interpret this seemingly complicated result.

The significant point is that the "muon interaction" has introduced a qualitative change in the (dt) wave function, correspond!

- ng to a reflection

(the term a) from the muonic interaction potential. The interaction (given by

r) is of order of 10 2 eV, the molecular binding energy. The "reflection

coefficient" a is almost everywhere exponentially small (owing to the

6
— 2 k b "penetration factor" e with b large, b * 500 fm) except when the (dt) wave number is itself very small (~ r, i.e., small on a nuclear scale). The

reflection is therefore a very sensitive function of the energy in the

vicinity of the molecular binding energy.

The introduction of the muon interaction also changes the coupled system

eq (2.7a) at nuclear distances; eq (2.9a) for Det M = 0 now becomes

Det

(1 + cO

X

X

< cot 6 - A

n n

= 0 . (2.14)

[Here we have re-named the parameters for simplicity of writing. In effect

” A

d

X the short-range nuclear interaction is parametrized by W + ( ^ ^ ).]

Let us discuss the meaning of these results.

First: The reflection parameter a is a function of the energy, k, in the

channel d, eq (2.13), and thus a function of the total energy E

t eq (2.8).

Second: the parameters A

d

» A

n

, X are constants defined by the nuclear

interaction. Both k (d-channel energy) and < (n-channel energy) are given by

E tot* Thus eq (2.14) simply determines the (not) phase shift 6^ of the channel

n as a function of the total energy E

t t

. To interpret these relations further let us disconnect the channels (that

is, put X = 0) . Then we see that we get (from the Hamiltonian) two separate

conditions

and

k

1 - Ct 1 + a

< COt6 = A

n n

“ 2m

d

W

dd * (2.15a)

2m W

n nn (2.15b)

Condition (2.15b) is obvious as it corresponds to the Fermi scattering

length, but condition (2.15a) is interesting. Without the muon (that is,

putting a = 0) we get k = A

d — the wave number in the closed channel d is

accordingly of nuclear energy scale. Now turn on the muon (a * 0). Solving

eq (2.15a), using eq (2.13) for a, we find:

k .j (1 + erb + 0 [(f )

2 ]} (2.16)

L A

d

a

= -1+2 ~ + ••• . (2.17)

A

d

7
Thus we find that k is of the order of r/2 — a wave-number of molecular energy

scale — and the reflection parameter a

= -1 + small. Accordingly we see that

there is a very strong reflection induced by the muon interaction. This is

required in order to change dramatically the order of magnitude of the

channel d energy. In other words, only if a = -1 + small, can the energy of

the dt-system acquire a molecular size, i.e., k << W^.

Having identified the molecular energy region let us now reconnect the

two channels and hence examine eq (2.14) for the phase shift of channel n with

W

n(j * 0. Choosing the molecular energy region for k (see eq (2.16)), means

that eq (2.15a) is satisfied. Accordingly, from eq (2.14), we see that

k cot6 - a must become larger. (Recall that W , * 0) . This implies that

n n rh nd ' r

5 n + 0 as k +-r/2 (1 + e"

rD

).

Now let us ask: at what neighboring energy do we get 6 n - tt/2 ? This is

a tedious calculation, but the answer is remarkable simple: the difference in

wave-numbers between 6=0 and 6 = tt/2 is given by

n n

Ak = r e~

rb << r . (2.18)

In other words, the width of the molecular resonance is exponentially

small compared with the molecular energy itself.

3. The Finite Range Model

Having seen the essential aspects of the interplay between the nuclear

and molecular interactions in generating sharp molecular resonances we revert

to the finite range model which is somewhat closer to the actual dtu-system.

Within that model we 'a! so will be in the position to discuss the diverse

possible simplifications which have been employed in the treatment of the

dtu-system, and to illustrate the resulting changes in the wave function from

the wave function of the "exact" treatment.

We begin with the pure nuclear case, i.e., in the absence of the muon.

In our model this is mimicked by el iminati ng -the molecular potential, i.e., by

putting h^ = 0. This is indicated in figure 1 by the dashed line.

We first consider an energy above, but close to, the channel d threshold,

i.e., E > 0. Here one has two independent solutions at the same energy E. One way of generating these solutions is to begin the integration of the

8
Schrddinger equation at an arbitrary point x = x

Q using the starting

conditions (in this section we will put in addition m

n = md

= 1)

= 1 and

ij/ 2) (x

0

) = 0

d_

dx «

(1)

(* 0

) = 0 (3.1a)

d__

dx * (2)

(x 0

) = 1 (3.1b)

The resulting two solutions are sketched in figure 2. By adjusting the

overall normalization two independent linear combinations of these solutions

can be written, for |x| >|a|, as

4^(x) = cos(kx + 6^) x > a (3.2a)

^

X) (x) = cos(<x - 6^) x < -a (3.2b)

and

4^

2 ^(x) = cos(kx + 5^) x > a (3.3a)

^

2) (x) = f/

2

^ cos(<x - 6^) x < -a . (3.3b)

Fig. 2. The two orthogonal solutions above the channel d thresholds.

9
The relative normalization constants and the phase shifts 6^

depend on the choice of x

Q

, but are well-defined. The starting point x

Q may

be chosen to be different for the two solutions. By judicious choice of x

0

( 1 ) (11 (21 ( 2 1 one can achieve that 6\ ' = 6

V

' and s\ ' = 6

V . The resulting solutions are

d n d n

3 called the eiyenchannel solutions, they diagonalize the S-matrix of the

system.

The two solutions, eq (3.2) and (3.3), (whether eigenchannel solutions or

not) can be used to compute the R-Matrix. To that end we define (V denotes

value and D denotes derivative, as in Wigner's original discussion):

^ := ^U) = cos (ka + 6^) 3.4a'

,(i) .. Hi

(i)

dx

(x=a)

= -k si n (ka + 6^ )

:= ^^(-a) = N (l) cos (<a + 6^

) )

,«) ..

n

* d(-x)

(x=-a)

= N^< si n a + 6^ ^

)

(3.4b)

(3.5a)

(3.5b)

The minus sign in the definition of arises since in the R-matrix

formalism it is defined by the derivative going away from the scattering

center. Also, we should account for the fact that the reduced masses in

channels d and n are different, which, however, for simplicity we shall

i gnore

.

With the definitions (3.4), (3.5) we have:

In detail, (3.6) consists of four homogeneous linear relations between

the R , and . These redundant relations can be used to derive the

cc c c four R-matrix elements, even though the R-matrix has only three independent

parameters. The fact that R dn = R nd can be used to check the accuracy of the

solutions. As one can see immediately from the definitions (3.4), (3.5), the

R-matrix depends on the choice of the "matching radius," here x = ± a.

10
As is well known, the R-matrix elements can be written as meromorphic

functions involving energy independent parameters, which can however be

approximated by a few almost energy independent parameters. It has been

shown, by fitting to experimental data, that the nuclear R-matrix of the

5 He-system in the neighborhood of the dt-threshold can be well represented by

a background term plus a single resonance term. 2 Our potential model has the

same characteristics. We can thus confidently use this same form to compute

the R-matrix in the vicinity of E = 0, say within aE = ± a few times E^. (In our model we could check the accuracy of this representation by computing the

solutions at a selection of energies).

We now are ready to consider the (purely nuclear) case E < 0 in more

detail. Then the channel d is closed and the wave function for x > a is of

the form,

,

s -Kx

( X ) ~ G (3.7)

where we have introduced

K := /2 | E | . (3.7a)

With this as "starting function" one can integrate the Schrcrdinger equation

towards negative values of x, i.e., into the channel n, which we assume to be

open at this energy. In channel n the wavefunction is given by

i|^(x) = N

n cos(<x - 6 n

) , x < -a , (3.8)

where again < is the wave number in channel n correspond! ng to the energy E.

The resulting function is sketched in figure 3. We can compute the following

quanti ti es

and

VE) = %

D

d

(E)

d ^

dx

-Ka

= e (x=a)

(3.9a)

-Ka

= -K e

(x = a)

(3.9b)

11
« CHANNEL n CHANNEL d

( n a )

( d t n )

Fig. 3. Solutions for the different potentials of Fig. 1 below the channel d

threshold. The little circles indicate the crossings of the channel

radii x = -a and x = a.

V

n (E) = (x=-a)

= N cos(<a + 6 ) n

v n'

d^r

D (E) = ~r

n

v

' d(-x)

(x=-a)

= -< N sin(<a + 6 ) n

' n

7

(3.10a)

(3.10b)

As before, the minus sign in (3.10b) arises since the definition of D requires

differentiation outward. These quantities are again connected by the nuclear

R-matrix

v

d

R ddR dn °d

V

n

R nd R nnD

n

(3.11)

We now restore the "presence of the muon," that is, we return to the

potential indicated by the full line in Fig. 1. In channel d, x > a, we have

three regions:

(1) x < c (3.12a)

(2) C < X < £ (3.12b)

(3) X > £ . (3.12c)

12
The wave function in these regions. for an energy -h^ < E < 0, has the form

(proceeding from the outside inward)

4*3 ( x ) = e“

Kx (3.13a)

^ ) — ^2 sin(k

d

x + n) (3.13b)

( x ) = A

1 -Kx _ +Kx

e + B

1 e (3.13c)

while in channel n the form (3.8) is maintained. The procedure by which one

can compute the full wave function can now be given as follows:

(i) compute A

2

, n from the matching conditions at x = ji, (ii) compute and B

x from the matching conditions at x = c, (iii) use (13c) to compute V

d

and D

d

at x = a,

( i v

)

use the R-matrix to compute V and D , (v) compute (the non-normal i zed amplitude) from V

p and D . (vi) divide all amplitudes by N/N

n where N is the desired normalization

constant of the continuum wave, eq (3.8).

This chain of steps can be characteri zed as a description of how the

Hamiltonian of the system carries the consequences of the condition ^(x) -» 0

for x + 0° through the system into the matching point to the open channel n,

x = -a, to finally determine the scattering phase shift 5. This procedure is

valid for all energies E above the threshold E^. of the na-channel n, as long

as one has available the R-matrix at those energies. In particular, one

could, in principle, find the explicit behavior of 6 (the resonance associated

with the quasi-bound "molecular state") by computing 6 "point-by-point," at

different energies, and fitting a suitable (arctan) curve to the points.

This, of course, is a very unsatisfactory procedure. One would like to have

to compute only the resonance position, the width of the resonance, and the

background phase shift, since as we have seen in section 2 the resonance is

extremely narrow, with a width r << |E M

|, where E^ is the "molecular" energy.

To illustrate this let us consider the case where the presence of the

nucleus, i.e., the influence of the region x < a, is negligible. In that case

one may use the potential indicated in figure 1 by the dash-dotted line. This

potential is infinitely repulsive for x = 0 and constant (V = 0) except within

the "muon well." Such a potential mimics the actual dtu-system when neglecting the nuclear forces entirely, and implies the new boundary condition

13
ip(x = 0) = 0 . (3.14)

With this boundary condition (3.14) and the boundary condition ^( + ®) = o one

has a well-defined eigenvalue problem, with a real eigenenergy. The wave function in the region x < c is given by (3.13c) with E = and

A<°> = - B<°> (3.15)

owing to (3.14). Let us remark that the requirement given by (3.14)

effectively disconnects the two channels and hence the energy in the channel d

becomes discrete. For this case the R-matrix becomes diagonal; the channel

coupling matrix elements vanish. This discrete state as we have seen in

section 2, will actually be "embedded in the continuum," here the open

net-channel n, by the fact that, in our case, the off-diagonal R-matrix

elements do not vanish. We will return to this complete case below.

We now compare the solutions for our three potentials at the energy E^,

at the channel radius x = a. We consider the logarithmic derivative.

L

= d 1 og d

dx

(x = a)

(3.16)

for our solution. The "pure nuclear" solution yields (K^ := /2 1 E^ | )

Lm = - K m < 0 , N M

the "pure molecular" solution yields

LM

K„ [1 + e2 aK

H]

[1 - e'

2aK

M]

> 0

while the full solution yields

L = K m [-A

1 e"

aK

M + B

x e aK

M]

l\

x e”

aK

M + B

l e aK

M

(3.17a)

(3.17b)

(3.17c)

where neither Aj nor vanishes.

14
As to be expected, and as can be inferred from figure 3, the full

solution lies in between the "pure nuclear" and the "pure molecular"

solutions. Depending on the dynamics, and also on the choice of the matching

radius a, L in (3.17c) can be positive or negative.

We now will discuss the response of the system for the case where the

"molecular resonance" is narrow in the sense that all other parameters

character! zing the system can be considered to be constant over the width of

the resonance.

It will turn out that the introduction of a propagation matrix will be

very useful in the discussion to follow. We thus introduce the definition

which we will use to propagate the logarithmic derivative in the dt-channel

from x = c to x = a. We construct this matrix using the form (3.13c) of the wave function which is valid in the region a < x < c. Introducing the

abbreviation

(3.18)

5 = K(c - a) (3.19)

we obtain

P n = cosh e, =: C

P 12 = - K sinh e, = -KS , P 22 = cosh e, = C . (3.20)

Note that Det P = 1.

Using these results we find

1 - xtanh e,

*

X - tanh e, (3.21)

where we have introduced the loyarithmic derivative (divided by K) at x = c:
We obtain x from the wave function of the "molecular potential well,"

that is, the wave function (3.13b). To simplify the expressions we assume a stronyly repulsive potential at x > £, so that we can replace (3.13a) by the

condition

^

d

( £) = 0

Then we have.

(3.23)

4

>d(x) = M sin k(x - t) , (3.24a)

i|/j(x) = Mk cos k(x - i) , (3.24b)

where M is the amplitude and

k = /2(h

M

+ E) . (3.25a)

For the case at hand the kinetic energy T is

0 < T = h M + E < h MMM . (3.25b) v

This way we obtain

X = — cot kt

K (3.22b)

The last step is to re-write the R-matrix in the form of a propagation

matrix, propagating the wave function from the channel d to the channel n (we

write di, r for the channel d and d for the channel n)

d

Y

n

(3.26)

16
which has the elements

Herewith

-R

~R

_n

12

R

2 1 R 12

" R

11 2 2

12

‘22

12

*"n 4*

R 11 L * R

1

2

n l(x=-a) R

2 1 L + R 22

( 3.27

)

(3.28)

We can now discuss the influence of the "molecular resonance" on the open

channel n. This we do in two steps. First we rewrite eq (3.21) as

K_ r , coth £

- tanh £ tanh £

L “ coth £

- x

•>

K +K 1 tanh £ (sinh £)

2 coth £

- \

(3.21a)

This shows that for small penetrability, where cothg = 1 + e with e << 1 and

sinh g >> 1, L is essentially constant with value = -K/tanh £ except in a very narrow region around the pole x = coth £. The width of this region is

given by a * (sinh £)- 2 = e << 1; in this region L sweeps out the values to

+«>. Continuing now to the na region, x < -a, one sees from (3.28) that again

only over the small region around the pole of (3.21a) will the value of L vary from its steady value , the "background" value associated with L^.

Evidently if A << h M then one may forget about the energy dependence of all

slowly variable parameters, including L^° . Next we want to determine the amplitude M of the "molecular state" in

terms of the amplitude of the channel n scattering state wave function, N,

eq (3.10). This we can now do since we know the phase shift 6:

L

6 = arc tan ( — 1 - <a , (3.29)

n ^ <

17
or

s sin(<a + 6)

(< 2 + L

2 )'* (3.29a)

c cos(<a + 6) = - -----

(< 2 + l

2 )'2 (3.29b)

and we can retrace the chain from channel n through to the "molecular" region.

~ -

1

In terms of the inverse transformation Cf = F of (3.26) we have at x - a

(3.30)

where, in terms of the R-matrix elements we have the explicit form of Q

-R

~R

22

21

R 12 R

21

' R

11 22

‘21

12

22

‘21

11

‘21 (3.31)

Proceeding now to x = c by means of the transformation Q

= p- 1

= c

1 1 u 9 Q 1 2 = SK ,

21

= S/K 9 Q 2 2 = C ’ (3.32)

we have

Vc

= Mk cos k t = Q n ^(a) + Q

<P C

= -M sin k t = Q 21 ^(a) + Q

which, together with (3.29), (3.30) and using ^

=

Mk cos kt = [Q u (Q n L

n + Q 12 ) + Q 12 (Q 21 L

n

- Mk sin kt = [Q 21 (Q u

L

p + Q 12 ) + 0 22 (Q 21 L

n

12

22 ^d

(a)

L

n ^n ^ ie1ds

+ Q 2 2 )

N <

/I_2 + <

2

+

k N <

/L

^ + <

^

(3.33a)

(3.33b)

(3.34a)

(3.34b)

18
Introducing the matrix D as

D := Q Q

we obtain squaring and summing (3.34a) and (3.34b)

(Mk) 2 = [(D n L

n + D

12 )2 + k2(D 21 L

n + D

22 )2] jjf

*^ 2 (3.35)

This equation yields our desired result, i.e., the dependence of the molecular

state amplitude M on the energy, which is reflected in the logarithmic

derivative L at x = a, in view of eq (3.28).

To obtain the extrema of M 2 as a function of L we re-write (3.35) in the

form

(Mk) 2 = <2 N 2 F(L2 + F, L

n + F

0

)

L

n

2 <2 (3.36)

where

2 2 2 F := D n + k D

21

2(D uD

12 + k

2 D,,D„)

F

1

:: 212 2

2 2 2 ;= (D 12 + k D„)/F '22 (3.37)

and, evaluating

d( Mk) 2 d( Mk) 2 dL

R dL

dE dL dL dE

n

we find the condition

L2 - n

2< 2 - 2F,

L - k

2 = 0

n (3.38a)

which yields

L

n = [,2 . F

0 ± - k

2

( 2 F

Q + f\) + Fq] (3.38b)

19
As the last point of this analysis we consider the behavior of the system

far from resonance. To that end we retreat to (3.33), insert the explicit

values for Q^, from (3.32), square and sum, to arrive at

M2 = N

2 (P 21 + L) 2 [(c + fM

2 - (3.39a)

22

Far from the resonance we have (from eq (3.21a)) that L -*• tJ Inserting this value for L into (3.39a) we see that M

2 simplifies to:

c

2 M

2 = N2

( P

2 1 + P 22

s

2

s

*

(3.39b)

Since far from the resonance- we also have coth £ = 1 + e, wi th e << 1 , we see

that

L_ = ...A—- . £ « i S

2 (sinhc) 2 £ 1 *

These results suffice to show that: far from resonance the amplitude M

(in the muonic region) is very small compared with the amplitude N in the

na-channel , that is, M

2 = e N

2 << 1. This completes the calculation.

4. Application to dtp Calculation

Having derived the diverse relations fulfilled by the model systems one

still needs a prescri pti on , an algorithm, on how to perform an actual

calculation. Furthermore, this cal cul ational procedure should not be model

dependent, i.e., it should be applicable to the actual case of dty-fusion.

Hence, we now re-write our expressions in a model -i ndependent form. We

restrict ourselves hereby describing a method which would be practicable in

the case of a single (or very few) channels. The general case will be treated

in Ref. 3.

Recall that the essential model lies to the right of x = a, Fig. 1. To

wit., the expressions for the region to the left of x = a are already written

in terms of the R-matrix parameters, which are known from experiment. Thus we only have to discuss the dynamics of the "molecular system" to determine which

features are model -dependent and which features are model -i ndependent

.

Here the essential point is that the "molecular system" exhibits a very narrow resonance in the sence that the width r << A, where a is the distance

to the next feature (beyond a smooth background) in that system; it could be

20
another resonance, or a threshold. (In fact, the ratio r/A < 10~ 5 in the

actual case.) In that case the response of the system is to an extreme

accuracy that of the "standard isolated resonance," and can be specified by

the energy dependence of the logarithmic derivative at the matching radius.

This "standard" logarithmic derivative then is in the vicinity of the resonance

Lr = cot (bE - c) - d (4.1)

where b, c, d are constants. Equation (4.1) can be written as

1/b

L

S

~ d +

E c/b (4.2)

VJe now can equate the standard form (4.1) with the model form (3.21a),

re-written as function of the energy. To that end we expand x around the

pol e

with

and obtain

X(E) = X(E

0

) - (E - E

0

) Xj

X

1

d_A

dE

E=E

o

> 0

KC. K 1 S A

X S

2 E - E

0

(4.3a)

(4.3b)

(3.21b)

which now has the standard form (4.2).

At this point one sees how one can provide for a model -i ndependent

treatment. Namely, one solves the dtp three-body problem with three choices

of L to obtain the three solutions E

l = E(

L

x

) , E

2 = E(L2 ), and E3 = (L

3 ).

With these values one can determine from eq (4.1a) or (4.1b) the three

model -i ndependent parameters b, c, d. Now one can use (3.28) and (3.29) to

compute the continuation of the fusing state into the channel n. Finally, the

normalization of the molecular state is determined by (3.30) by demanding

*

d (a) N(Q

21 L

n + q 22> —-—

r

(< 2 + L

n

) 2 (4.4)

where, as in (3.10), N is defined by the standard standing-wave continuum

normalization condition.

21
REFERENCES

1. S. Jones, Nature 321, 127 (1986), and references therein.

2 e G. Hale (private communication); see aslo R. E. Brown, et al„. Fusion —

energy reaction 3 H(d,n) at low energies, Los Alamos preprint LA-UR87-462

(1987) (unpublished).

3. M. Danos, L. C. Biedenharn, B. Mailer, J. Rafelski, and A. Stahl hofen, to

be published.

22
NBS-114A (rev. 2 -aci U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instruction s) NBSIR 87-3532 APRIL 1987

4. TITLE AND SUBTITLE

MODEL FOR THE NUCLEAR EFFECTS ON THE t-d-y FUSING STATE

5. AUTHOR(S)

Michael Danos , L. C. Biedenharn and Alfons A. Stahlhofen

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

10. SUPPLEMENTARY NOTES

:__2 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A simple model is constructed which is used to

illustrate the interplay between the strong nuclear

forces and the Coulomb molecular forces in the

description of the resonant d-t-u fusion process.

12. KEY WORDS (Six to twelve entries; alphabet! cal order; capitalize only proper names; and separate key words by semi colon s) deuterium; meso-mol ecul e ; mu-meson; muon catalyzed fusion; muon molecule;

t-d-fusion

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

Q(j Unlimited

| | For Official Distribution. Do Not Release to NTIS 25

|

1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. 15. Price

(~Xl Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $9.95

USCOMM-OC 6043* P 80
.