Intrinsic sticking in dt muon-catalyzed fusion: Interplay of atomic, molecular and nuclear phenomena

ANNALS

OF PHYSICS

192, 158-203

(1989)

Intrinsic Sticking in dt Muon-Catalyzed Interplay of Atomic, Molecular and Nuclear MICHAEL

National

Fusion: Phenomena*

DANOS

Center for Radiation Research, Bureau of Standards, Gaithersburg.

MD

20899

AND

ALFONS A. STAHLHOFEN Duke

AND

DEDICATED

TO HERMAN

C. BIEDENHARN

of Physics, Durham. NC 27706

Department University, Received

LAWRENCE

January

FESHBACH

9, 1989

IN HONOR

OF HIS 70TH

BIRTHDAY

A comprehensive reaction theory for the resonant muon catalyzed fusion of deuterium and tritium is formulated. Emphasis is put on non-perturbative, many body, treatment of the long range Coulomb force and its interference with the nuclear forces, with the aim of providing the theoretical framework for an accurate calculation of the branching ratio dtp + { (a~) + n )/{ a + n + n ] essential for muon catalyzed fusion. NJ 1989 Academic PEW. IX.

I. INTRODUCTION

The objective of this paper is to develop a theoretical framework adapted to accurate and reliable computations of the intrinsic sticking fraction w, in muon catalyzed deuterium-tritium (dt) fusion (&‘). Since an experiment capable of measuring o, precisely at high density and temperature is exceedingly difficult Cl] a reliable computation of this number is essential. The upper limit of the fusion yield is controlled by the reciprocal of W,vven if the muon lifetime were infinite, a sufficiently large sticking fraction w, would render ,& impractical. Recent experiments on pcf have shown that one muon can catalyze more than 175 fusions [2], a number which exceeds existing predictions. This fusion yield is already close to the break-even point of a “cold fusion” reactor, even when considering efficiencies in the energy to electricity conversion cycle [3]. Besides its practical aspects, the subject of this paper constitutes a challenging theoretical problem involving simultaneously several usually disparate fields of * Supported

in part

by the Department

of Energy

(Advanced

158 0003-4916/89

$7.50

Copyright c.r 1989 by Academic Press, Inc All rights of reproduction m any form reserved.

Energy

Projects)

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physics. The dt p molecule constitutes a rare instance of a system in which several very different energy regions are non-perturbatively interwoven [4]: (i) The nuclear subsystem is an excited state of ‘He, which has two open channels, cmand dt. This Snucleon system has a nuclear resonance at -55 keV above the dt threshold. (ii) The hydrogenic Coulomb binding energy of the muon is - 3 keV. (iii) The dt p molecule (in the absenceof nuclear interactions) has 5 bound states, with energies from 1 eV up to 300 eV. One major objective of this paper is to incorporate properly in a reaction theory the fact that the three-body Coulomb system at small dt distances is markedly influenced by the nuclear forces in a manner which cannot be treated perturbatively. Put differently, the presence of the muon transforms the two-channel nuclear S-matrix into an infinite-channel overall S-matrix. We now explain qualitatively the non-perturbative character of the interplay of the different subsystems[S]. Let us begin by focusing on the motion of the d and the t. Close together, but just outside the range of the nuclear forces, the dt wave function will be governed by long-range forces. The essential point is that this dt wave function, at least for some of its components (see Eqs. (2.13a, b), below), belongs to an open nuclear channel. (The nuclear interactions will affect only the J = 0 states significantly.) However the dt p molecule has, even in the presence of nuclear interactions, a finite perimeter. Including now the muon in the discussion, we see, that the effect of the muon is to generate reflections in the dt wave function so as to close this, otherwise open, channel at large distances. This closure results in molecular resonanceson an energy scale appropriate to molecular energies, that is, extremely narrow on a nuclear scale. We have constructed simple models which support this conclusion [6]. This effect is non-perturbative, since, for example, it is well known, that Coulomb bound states cannot be achieved perturbatively. We will focus our attention in this paper on the dt fusion alone and omit any discussion of the other hydrogen isotopes [4] in &since, from a practical point of view, only dt fusion seems feasible [3]. The particular characteristic of the dt system which necessitatesa different and more complex treatment is the dominance of the nuclear reaction rate by a unique channel, the 3/2+ state of the ‘He* compound system. The dt reaction threshold is within one line width of this resonance [4], which implies that the nuclear fusion reactions proceed exclusively through this reaction channel. In fusions of other hydrogen isotopes such a selectivity is absent and the reaction proceeds through the tails of the many distant resonancesof differing angular momentum and parity. Previous calculations of the sticking fraction were based on a heuristic incorporation of the nuclear interaction and were only partially non-perturbative [7]. The results indicated the need for a technically complete treatment of the sticking, taking fully into account the three-body dt p molecular wave functions, the nuclear physics, and the kinematics of the three-body ccpn system, as is done here. Our treatment is, however, non-relativistic and neglects hyperfine effects. We will also not address the problem of the formation of the muo-molecular complex [g].

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We begin with a short survey of the fusion reaction d+t+p+a+p+n+

17.6MeV,

using the standard view as developed by Ponomarev various groups [lo].

[9]

(1.1) and later extended by

1. A muon enters a liquid DT-mixture and is slowed down (within less than 10 ~ lo set). 2. The muon is captured by one of the hydrogen isotopes d or t, replacing an electron in the process. 3. (a) The muon cascades down to the 1s state of t, or (b) if captured in d, the muon is transferred to the heavier tritium isotope (dEr48 eV) either during the cascade in deuterium or only slowly after reaching the 1s orbit. (Even from the d,, state the transfer rate is 100 times faster than the competing dd p formation rate.) 4. A dt p molecule is formed in the field of a host D, molecule. (The resonant muo-molecular formation appears to be more than 1000 times faster than the muon lifetime.) 5. The muonic molecule de-excites (via the Auger effect) to the J=O muo-molecular state. 6. The nuclear fusion reaction d+ t --) c1+ n takes place, releasing 17.6 MeV energy. (The 3/2+ resonance of ‘He* causes the fusion reaction to be millions of times faster than the natural muon decay.) 7. (a) If the muon is set free it can repeat the cycle, otherwise (b) the muon becomes bound to the reaction product, the c( particle. These reactions define the intrinsic sticking fraction CO,.The captured muon can be stripped (reactivated) in a secondary reaction [lo]; this defines the effective sticking fraction which is appreciably ( z 20%) smaller than w, and is of practical importance for applications of jKf Following this sketch, we restrict our discussion to problems related to the interference between nuclear and molecular (muonic) phenomena. To carry out our non-perturbative program, we employ a time-independent description. We begin in Section II with a discussion of the dt p (fusing) system and introduce the methods of our approach. This section is followed by the main part of this paper: a detailed theoretical formulation for a calculation of the intrinsic sticking fraction w,. Some tedious derivations and additional information have been assembled in various appendices. II. MUON STICKING AS A BRANCHING RATIO The resonant production [4, 91 of the (muonic) molecule dt ~1 proceeds via a quasi-stationary state of the muo-molecular complex [(dt ,u)* d2e], where the

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excited state of the molecule (dr p)* is labelled by the quantum numbers (J, v)” = ( 1, 1) . (Here J(v) is the orbital angular momentum (vibrational) quantum number and rt the parity.) This “doorway state” then de-excites by an electromagnetic transition to the fusing states (0, l)+ or (0, O)+. The rate of the two-step transitions to the fusing states via the side-branch (2,O) + is comparatively small [ 111. Thus the fusing states are “prepared” by the Auger transition from the doorway state. We treat the electromagnetic transition in first order perturbation theory. (The perturbative approach for this part of the calculation is in fact an essential simplification for the complete calculation, as discussed in Appendix F.) Using a Hamiltonian W,,, for the interaction between the dt g molecule and the electronic cloud, the transition probability per unit time is given by the (formal) expression: (2.la) Here Iid) and Ii,-) denote the (electronic and nuclear) wave functions of the doorway and the fusing state, computed as stationary eigenstates of the (nuclear and Coulomb) Hamiltonian H,y of the six-body muon-nuclear system. The density of states, p,(E), is discussed in Appendices E and F. The complete Hamiltonian of the many-body problem of &is given by the sum of H, with the electromagnetic interaction Hamiltonian, Hint : H = H, + H,,,

(2.2)

The explicit form of Hint is determined by the fact that the transition from the ( 1, 1) to the (0, v) + states requires a LIP = 1~ angular momentum transfer. Thus only the dipole part of the multipole expansion of the Coulomb interaction (cf. Appendix A) has to be considered, which leads to HI”,=

-e’(r;+r;-r;,). =- -efj.($

(2.lb)

where d = e( y; + v; - y;,). The matrix element c3 in Eq. (2.lb), above, involves the wave functions electron of the H, molecule participating in the transition. The total width doorway state is given by a summation over all possible final states, i.e.,

w=c w,

(2.lc) of the of the

(2.ld)

where W,. has been defined in Eq. (2.la). Since the fusing states decay (only) via a fusion reaction, the Auger rate is identical to the fusion rate. The muon-nuclear

162

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components of the final states wave function Ii,), Eq. (2.la), are defined in the nap region (see below). Hence we can interpret the indexfas a set of quantum numbers for final states, distinguishing between sticking {fs} and non-sticking {fn} states. This allows a definition of the intrinsic sticking fraction o, as (2.le) where the denominator is, of course, the total width W defined in (2.ld). The main difticulty in determining o,~, Eq. (2.le), derives from the fact that the reaction proceeds via two intermediate resonances, as already indicated above. One resonance is associated with the molecular fusing states with J=O and the other with the nuclear 3/2+ state of ‘He *. We follow the conventional understanding of the reaction dynamics [9] and ignore direct fusion from J#O states, the influence of the host system on the dt p molecule, Coulomb excitation and polarization of ‘He by the muon, and (Coulomb) polarizability of the nuclei. (These effects can be incorporated perturbatively in the calculation, if a higher accuracy is needed.) The factorization of Hint into an electronic and a muon-nuclear part, Eq. (2.lb), allows one to define the state ID), generated by the electromagnetic transition operator d, as ID)dQ(d}.

(2.3)

Here Id) and ID) denote three-body muon-nuclear wave functions. The state ID) is not an eigenstate of the Hamiltonian H,, but can be expanded in a basis of (timeindependent) continuum eigenstates of H,. (Note that the state Id) exists only in a compact region of configuration space. Accordingly the state ID), generated by the action of the operator sz’ on Id), has again support only in a compact region, whereas the continuum eigenfunctions of the Hamiltonian H, extend over the complete (noncompact) configuration space.) Since the muon-nuclear wave functions are defined in the nap channel as well, these eigenstates involve both the sticking and the non-sticking sectors of the Hilbert space. (We defined in Eq. (2.le) the intrinsic sticking fraction as the branching ratio into these states.) Before this expansion can be described in detail, the incorporation of the nuclear interaction and the degeneracies (multiplicities) of the various systems involved in reaction (1.1) have to be discussed. The nuclear interaction is, of course, known only in approximate form. This, together with the large dimensionality of the configuration space, makes a complete treatment inaccessible at present. Fortunately, an accurate solution nonetheless can be achieved because the relevant length scales allow a separation of the problem. The distance scale of the nuclear forces is approximately 3-5 fm, while the dt p molecule has the nuclei separated by -500 fm (the muonic Bohr radius is -250 fm). Thus we may use the Wigner-Eisenbud matching-radius (r,) scheme to separate the nuclear and the molecular effects with I-~ N 5-10 fm being amply large. (The criterion is that rm be sufficiently large so that channel orthogonality is

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obtained, with no significant contribution from closed nuclear channels.) The accuracy of this approximation is comparable to the uncertainties in the experimental data of the nuclear system, i.e., 5He*. In the Wigner-Eisenbud treatment the data describing the nuclear interaction are replaced by the (energy dependent) R-matrix parameters at the boundary [12]. Accordingly the configuration space of the six-body system allows a splitting into three regions: (a)

the dt p region, where the Hamiltonian

effectively becomes:

H, + Hctrp=

(2.4)

Note that near the boundary of this region, i.e., near Ir;, 1= a( -5 fm), the dt nuclear channel for a molecular resonance is open, but the dt p channel itself is closed. (b)

the nap region, where the Hamiltonian (2.5)

applies. (c) the nuclear region, with Ir;, ( < a z 5 fm, Ir’,, I < h z 3 fm, which-because a, b< the muon Bohr radius-appears to the muon effectively as the point mass ‘He*, so that the muon is described by Coulomb wave functions with 2 = 2. Since the Hamiltonian separates in the different regions in different ways, different product expressions for the wave function in the three regions are obtained (see below). The wave functions for the various channels in the three regions of configuration space are coupled through the Hamiltonian H, of Eq. (2.2) or, equivalently, by matching conditions at \r;, I = II, Iv’,,,,I = h, respectively. Although the dt p channel is closed, the coupling to the my channei-which is open--shows that the relevant solutions to the Hamiltonian H,r lie in the continuum. To determine the degeneracy of the time-independent solutions at euclz energy E, we must consider the open channels. There are two (independent and orthogonal) sets of states (for the nap system) at each energy E: (i) a denumerably infinite set of bound CL~and recoiling neutron states; (ii) a continuously infinite set of states, each consisting of an al-1 scattering state and the corresponding recoiling neutron. For both cases the wave function for the nccp system (in the nctp region) \r’,,l > h (i.e., away from the channel radius) can be expanded in the form

for

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AND

BIEDENHARN

where the factorizing wave functions Yn,,iB1 can be written as (2.7)

The wave functions t,bi(i = pa, n) contain the orbital and the internal wave functions of the system i; the internal wave function of the alpha particle has been neglected. The set {/I} labels the functions (2.7) and denotes a complete set of quantum numbers, which are explicitly defined in Section III. (The Coulombic quantum numbers of the a~ system, the angular momenta of the neutron, and the total energy and the total angular momentum constitute the set { p}.) The functions (2.7) form a complete set for the asymptotic region, defined by p, -+ co. Thus the set { 8) specifies the “physical channels,” since these quantum numbers are (in principle) accessible to experimental determination. The coordinates used in Eq. (2.7) are not adapted to specifying the wave function of the channel radius defined by Ir’,,l = h. Consequently we will need to employ two different coordinate systems: one appropriate to the cmp system as required for I?“,[ z b, and another needed for the asymptotic region of the nap system, as used above in Eq. (2.7). We introduce the coordinates shown in Fig. 1, defining ii as the coordinates of

n FIG.

1.

Definition

of the coordinates

of the nap system

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the three particles, ~1,n, p, with respect to the center of mass;fi, are the coordinates of particle i with respect to particle j: fll, = J{,- r’, )

(2.8a)

v’n,E r; - lZl.

(2.8b)

The center of mass coordinates of the system (ij), d,,, are defined as (m,, + tn,) I?,,, = nz/,y;, +m.v’,,

(2.9a)

(m,, + m,) if,,, = In,?,, + m,Fz.

(2.9b)

Not shown in Fig. 1 are the coordinates p’,, which are the coordinates of particle i with respect to the center of massof the two other particles: #6,(= r; - I7n*

(2.10a)

p’, = r’, - R’,LZ

(2.10b)

Since the vectors on the right-hand sides of Eqs. (2.10) are collinear, the RHS can be expressed as d,L =

p’,, =

m, + m,, + m m,+m, m,+m,+m m,+m,,

’ i ,I ’

(2.1 la)

’ i

(2.11b)

”

The transformation between the two different sets of Jacobi coordinates is given by t,,, = d,, + m,+m,

r

“l’

ro‘” = r;,, - ,n, f m, + m,, “/’ p’n =

m.(m,+m,+m,,) (m, + m,)(m,

+ m,)

+ r,,

mp m, + m,,

_ Pp.

(2.12c)

The entire set of Equations (2.8) to (2.11) is easily rederived for the dt ,u three-body system by mere change of the indices cc+ t, n + d. Let us briefly discuss the coordinates of Eqs. (2.6) and (2.7). These coordinates are appropriate for the asymptotically large distances between the neutron and a~ systems. At the matching radius I?,,, I = b the wave functions of Eqs. (2.6) (2.7) have to be rewritten in terms of the coordinates r‘,,,, and b,,, which are the coordinates needed for the matching procedure at the boundary of the nuclear region. In this way we

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obtain a complete representation of the wave function of the ncrp region from the asymptotic region (p, + co) down to the matching radius r’,, = b. When written in the coordinates r’,, and pP, the wave function YIIoL,, can be matched between the nccp region and the nuclear (i.e., ‘He*-p) region of configuration space. These coordinates allow effectively a mapping of each component of the nccp wave function onto the corresponding components of the ‘He*-p system (at each given set of quantum numbers (E, Z, r~)~~~~,),since the wave function in the (compound) ‘He*-p region has the explicit form:

Here the set {p} = (fiZjf d enotes the quantum numbers of the 5He*-p hydrogenlike Coulomb system at the energy .stPi ; xHe is the internal wave function of the compound ‘He* nucleus at the energy qp,

=E-&{p).

(2.13b)

The summation in Eq. (2.13a) is over all possible muon states. The product form of (2.13a) is-within the set of general assumptions listed in Section I--exact. This shows that the description of the compound nucleus ‘He* (existing in the continuum at approximately 17 MeV above the cm threshold), is needed for all values of the R-matrix energies ErP). Since for each value of E * the corresponding are known, the information about the characteristics of the system can be transmitted between the nctp and the dt p regions. But owing to the energy-dependence of the R-matrix, the different components { p} of (2.13a) are mapped differently between the nap and dt p region. We now turn to the dt ,u channel. Away from the matching radius, i.e., for I?“, 1>a, the wave function exists for any energy above the an threshold and factorizes into the internal wave functions for the deuteron and triton, and a Coulomb wave function for the dt p three-body system, written as tidrPlal. The index {/I} indicates that the degeneracy of the dt p wave function reflects the degeneracy of rl/ ~np~Bl as we show below. The nuclear part of the ‘He*-p wave function, xHe(ETP)) in Eq. (2.13a), splits into components, labeled by {p}. Since the R-matrix elements are energy dependent, the different components xHe therefore require different R-matrix elements, i.e., R,= Rv(Erpl). Therefore the three-body wave function rjdrlr must be expanded in the form (2.13a) to allow the matching; each componentAefmed by { p})-must be treated separately. Having accomplished the expansion (cf. Section III), we introduce Wigner’s notation for the nuclear wave function tid,: (2.14)

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and introduce analogous definitions at the matching radius I?,,) = b. (Note that the R-matrix elements are a complete substitute for the explicit form of the nuclear wave functions $,, and IJ+~,.) Now we can match at the two channel radii and obtain: (2.15) The linear relation V= RD is valid for any number of channels. For two channels it can be written in the convenient form (2.16) (The explicit form of the matrix elements P, is given in Section III.) This relation, connecting the two matching radii, has to be evaluated separately for each given set of quantum numbers (1, ?I, E),,,. The transmission of the degeneracy from the my- to the dt p-region is now easy to identify: every component $nalliP; contains certain values of D,,(b) and I’,,(b), which determine, via the inverse of the transformation (2.16), the corresponding values of Drl,(a) and Vdr(a). Since this procedure determines the boundary conditions for rjdrP at the matching radius, the degeneracy and multiplicity of the fusing states is indeed determined by the corresponding degeneracy and multiplicity of the nap region. Whereas Eqs. (2.15) or (2.16) give the explicit matching conditions for the nuclear part of the wave functions, those for the muonic part are comparatively easy to formulate. According to our general set of assumptions (cf. Section I), the muon is a spectator of the nuclear reaction. (We assumedin particular, that there is no energy exchange between the muon and ‘He* in the nuclear region.) Therefore the state of the muon is unchanged by the nuclear reaction and the muonic wave function, i.e., ti5We..,, in Eq. (2.13a), remains the same when matching between the two matching radii. This apparently simple condition, however, implies the enormous intricacy of the problem, especially the necessity of determining the wave function simultaneously in the nctp and the dt p region while respecting the high degeneracy. A particular form of the wave function $d,,l can be constructed at any given energy E with arbitrary wave function I’(,, and D‘,, at jr;,) = a for each of the infinitely many { p ). = (yifj) components, yielding an infinite number of distinct, energy-degenerate solutions. Using the R-matrix, the phase shifts in the related, infinitely degenerate, ncrp components are determined. The problem, viewed in this way, is one of almost unlimited complexity. To resolve the problem, we clearly need a criterion for selecting from these infinitely many possible wave functions the correct linear combination representing the particular fusing state under consideration.

168

DANOS, STAHLHOFEN AND BIEDENHARN

An unambiguous criterion is provided by the physical process underlying Eq. (2.1). According to our assumption, the molecular doorway state Id), once populated, decays only via the electromagnetic transition. This implies that the discrete unperturbed doorway state exists only in a finite compact part of the dr p region. This property of the state Id) determines similar characteristics of the unique state ID), which is generated by the action of the operator n’ on the unique state Id) (cf. Eq. (2.3)). The state ID) can be expanded at each given energy in terms of the (infinitely degenerate) set of eigenstates of H,, that is, over the fusing states Ifi(E These states extend over the full configuration space and form a complete orthonormal set. Let 1D(E)) be defined as the component of 1D) at energy E; (D(E)), henceforth called a “dipole eigenstate,” is by definition an eigenstate of H,. It can be written as

ID(E)) =P(E) ID> =c C,(E) Ifi(E

(2.17)

where the operator P(E), defined by (2.18a)

P(E) = 1 Vi(E)> (f,(E)I, is the projection are given by

operator on states of good energy; the coefficients of the expansion C,(E) E =
d Ia>.

(2.18b)

The states Ifi( and the dipole state ID(E)), being eigenstates of 1JS at the energy E, extend over the full configuration space, i.e., they have support in all three regions. (In (2.18) the symbol C stands for summation over the discrete and integration over the continuum (scattering) eigenstates Ifi( of the Hamiltonian H,. The problem of normalising the Ifi(E)) is discussed in Appendices E and F.) Note, that as a consequence of this expansion any linear combination of the Ih( E) ) orthogonal to 1D(E) ) has a vanishing dipole moment and can not be reached by the Auger transition described by Eq. (2.1). In other words, at the energy E the fusion proceeds exclusively through the state ID(E)), constructed in Eq. (2.17a). Thus the dipole eigenstates are the states selected by the Auger transition, i.e., they represent that special linear combinations out of the infinite set of energy eigenstates which describe the fusing process at each energy E. This completes the survey of the various problems and their resolution. We can now address the (technical) details of the calculation. III.

DETAILS OF THE CALCULATION

The formal construction of the dipole state, i.e., Eqs. (2.17) and (2.18), in principle still implies full knowledge of the highly degenerate wave function. A

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variational calculation of the dipole state provides a practical realization of this schemeas follows: (1)

choose a sufficiently large and flexible set of basis states in the dt p region,

(2) maximize the overlap of the variational ID> =fi Id), and

wave function

I$)

with

(3) simzdtaneousl~~ minimize the variational functional ($1 F(E) I$ ) in the dt p region, i.e., for I?<,,13 u.

Here I$) denotes an arbitrary linear combination over the functions of the basis set; E is a chosen energy (that is, the energy is not determined variationally); and (IF(E)1 ) is a functional appropriate for continuum states [13]. This procedure yields in the dt ~1 region the continuum eigenfunction Yy,,, at the energy E, approaching optimally the dipole state ID(E)) (with an undetermined normalization) and the correct boundary values at lr;, / = a which have to be used in the matching between the dt p and nap regions (cf. Eqs. (2.16) and (2.17)). To match the variationally determined wave function Yd,, between the dt p and ‘He*-p regions, $d,,, has first to be analyzed in terms of the complete set of muonic Coulomb states $$~..,,(p’,,) (cf. Eq. (2.13)). This expansion can be written in simplified notation (i.e., omitting the angular components connected to FJ,) as a projection at the matching radius jr; ( = a:

(3.lb)

The amplitudes 16:’ have been introduced in Eq. (2.13a), with the set of quantum numbers (11) = (50) and are appropriate to the ‘He*+ region. With these amplitudes the wave function in the dt p region, at IV;, 1= a, can be written as

and

ay,,, ‘d,=“=;; (&?‘I’(E*)4&?!*.,,(~LJ drd,

(3.2b)

(The energy E* has been defined in Eqs. (2.13).) The boundary conditions for the nccpwave function at the matching radiusi?,,, I = b are determined by the P-matrix relation, Eq. (2.17): (‘$‘)=(

P,,(E*) Px(E*)

P,,(E*) P,,(E*)

(3.3)

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The matrix elements P, read in terms of the R-matrix p II =-II R R,25 Pzl = -

P,, +, 12

R,,&,-R,,R,, R12

P,, = f ’

(3.4) 12

The relation (3.3) allows the construction of the complete wave function of the na,u region (i.e., including the muon) at the matching radius IF,,, 1= b in the form

and

where the angular momentum of the an system is not explicitly written. The index anp indicates, that the wave function (3.5), whose components are labeled by the muonic quantum numbers {cl) = (Co), is valid only at Ir’,, 1= b. Since such a restriction does not exist for the muonic coordinate P; the wave function (3.5) contains already, though still implicitly, all information about sticking. This information is extracted in the last step when we have to match at the boundary between the ‘He*-p and nap regions, i.e., the wave functions Y,,O and !P nap. Because of the linearity of the problem the matching conditions separate in {p> = (E/j). Therefore the matching conditions have the implicit form

where Yi$ has been defined in Eq. (3.5). The amplitudes 21;; are related to the amplitudes A i @), introduced in Eqs. (2.6), by a summation over { p}, i.e. A{,, F 1 iq;;]. {Pi

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This summation can be postponed to the final expression for the sticking fraction, given below. In order to determine now the amplitudes A’!“,1 in Eqs. (3.6) we use the facts that (i) the quantities x, x’, which are contamed in Eqs. (3.6), are (numerically) known at the matching radii (cf. Eqs. (3.3) and (3.5)); and (ii) that the muonic wave functions $&& form a complete orthonormal set. Therefore we obtain the amplitudes A, -i$! , by multiplying Eqs. (3.6) with the wave functions Y,12,,j,j I and ii Y,,,,I ,j j/dr,,x 1r,, = h= Y,‘,,, IBj , respectively, and forming the overlap:

(3.7b)

Note that the radial integrations of the overlaps (3.7) are restricted to p,, since rnc is fixed at h. In order to evaluate the formal matrix elements (3.7) we must write the wave functions explicitly. We begin by constructing the form of Y,,Pl s1 near the boundary, i.e., for I?,,,1 Z h, using the following observations. At the matching radius the nuclear particles by definition do not interact. Thus the relative wave function of the an system is a spherical Bessel(and Neumann) function including a phase shift (written as 6 II’ I ). The relative motion of the cm system (with angular momentum L = 2) represents a negligible perturbation for the Coulombic wave function of the muon, which is taken with respect to the ctn center-of-mass. (The non-coincidence of the center-of-mass and the center-of-charge could be corrected for by a perturbation expansion if required for higher accuracy.) The Coulomb interaction between the alpha particle and the muon prevents the continuation of the wave function YF1z,,P to the region I?,,, ( > h since in this region the ctn system is no longer adequately described by a free wave; it requires the switch to the coordinates (Y;,, , p’,). Incorporating all angular momenta, one component Y&l of Y,,,(cf. Eq. (3.5)) can be written at the matching radius as (cf. Appendix A for the notation)

n,(kh)sir16~,,~)x [ [19,51:21 $,“‘I

CJ1 [pl

P; dp,i C,,(P,,) j/b, q”]

1111

[‘I.

P,,) > (3.8a)

(A similar form for (Y,$) is obtained by using Eq. (3.11 ).) The representation of the muonic Coulomb function as a Fourier transform to

172

DANOS,

STAHLHOFEN

AND

momentum space is convenient. The amplitudes (3.1), are related to XL,“} via (cf. Eq. (3.5)): ~(~)(r,,, m

= b) = Dir)

{j,(kb)

BIEDENHARN

Dtp), resulting from the projection

cos iSi,) -nL(kb)

sin 6,,,},

(3.9a)

and similarly

The relative momentum

in the cm system, k, is defined by k2 -=E-Ejp)+Q, 2m,,

where rn,’ = m; 1 + m;’ is the reduced mass of the an system, and Q = 17.6 MeV for the d + I -+ c1+ n fusion reaction. The energy E is measured in the dt ,U channel while sip), Eq. (2.13b), is the Coulombic energy of the muon. The phase shift 6 ( ~), Eq. (3.8a), can be determined using the R-matrix relation in the form of Eq. (3.3). The identity $Z,(kr)

= Z;(kr)

;t

Z,(kr)

- kZ,+ 1(kr)

(3.11)

is needed for this purpose, where Z, stands for either j, or n,. (To avoid confusion with the logarithmic derivative in the dt p channel, the nuclear orbital angular momentum L (Eqs. (3.8a), (3.9)) is in Eqs. (3.12) denoted by 1.) The (straightforward) calculation gives tan bIPI =

where k is given by (3.10)

h,j,(kb)+j,+lW) h,n,W)

+ n/+ 1WI’

(3.12a)

and h, is defined by h ~ Lk’ I

- (l/b)

k

.

(3.12b)

The logarithmic derivative of the an wave function

is connected to the logarithmic derivative L, of the dt ~1channel LJ,? = (xm’/x~~J,

(3.12d)

INTRINSIC

via the relation

STICKING

IN

dt

MUON-CATALYZED

173

FUSION

cf. Eq. (3.3): (3.12e)

The matrix elements P, depend via the energy of the nuclear subsystem (defined as E* in Eq. (2.13a)) from the energy of the muon (cf. Eq. (3.3)); i.e., they are taken at different channel energies for each muon state. Therefore the logarithmic derivatives Li,‘:i (and lu$;i itself) depend on the energy of the ‘He*+ system. (This can also be seen in the definition of k, Eq. (3.10), and its appearance in Eq. (3.12b).) Let us summarize: starting with the variationally determined rlt p wave function one obtains the logarithmic derivatives Lir” at the channel radius. This determines the factor h,, (Eq. (3.12b)). The last step is the determination of the phase shift 6 IPI Considering the intimate coupling between the dt p and the cmp region, it becomes clear that the phase shift 6 I,,; (and similarly the phase shift r introduced below) shows the typical character of narrow resonanceswhen the energy sweeps over one of the dt p molecular resonance (i.e., the fusing states). Let us now consider the nap region away from the matching radius Ir’,, 1= b. The Hamiltonian IY,,,{~, Eq. (2.5), shows that the energy splitting between the 3 particles is determined by the Coulomb states of the ccc1system. Using the form (2.7) of the nap wave function (in the asymptotic coordinates Y;,, and 6,) gives after a Fourier transformation of the muonic wave function to momentum space, for one component labeled by ( fi) = j&k-, I,,k, I}:

(3.8b) The momentum P in the relative (n - CL~)motion is given by (cf. Eq. (3.10))

$E+Q-E,;.; r

(3.13a)

where Q is defined as in (3.10), a,,. is the muon energy, and the reduced mass is 1 1 -=-++ m, nl,

1 m, + m,’

(3.13b)

In order to match the wave functions !P$J (Eq. (3.8a) and Yy,,,,O, (Eq. (3.8b)) according to the (formal) conditions (3.6). we have to rewrite Y,g,i,: pi in Eq. (3.8b) in terms of the coordinates of (3.8a), i.e. r’,, and PP. The required expressions, given in Eqs. (2.12a), (2.12c), must be inserted into Eq. (3.8b). We employ the addition 5’)5.‘192.1-I?

174

DANOS,

STAHLHOFEN

AND

BIEDENHARN

theorems of Appendix A to rewrite (3.8b) in these new coordinates. To shorten the notation, we introduce the abbreviations

TL The introduction

=47~(--)“*(~~+~~+‘)

[[II, (1,-J; T;,,-,2=(-)‘2

of “scaled” momenta,

(3.14)

cf. Eq. (2.12), is convenient:

m, Cm,+ m, + mJ xl=p(m,+mn)(m~+mp)~

L, = P--!q m,+m,

(3.15a)

=ppa;

(3.15b)

L2

we need in addition

T;,,,.

the definitions

With these definitions

A’= max(K,~,,, L, P,),

(3.16a)

Y= min(K,r,,,

(3.16b)

a component

L, p,).

‘FernPiP) reads now

k,k2;.,i.Z -

nk2

X

tx)

jk,

( y,

P$

&pa

sin

z { p} 1

Cnj.(Ppa)

T~,,j.,j,,(K2r,,)j;.,(L,

(i

P,)

>

This form is suitable for implementing the matching conditions at Ir’,,l = 6, and we can perform the overlaps (3.7). Using the definitions (3.9) for XL,“}, we obtain for the left-hand side of (3.7a) ( !PnariPI 1!PL,$ > = ~&jBi$;;

x j- P: &p f ~;a dp,, C,,(P,,) xjk,(Klb)[jk2(h

Pp)

cos

j- P: dp, Ca(p,J

~{8}-nkl(L~

Pp)

sin

‘{fill

Xj,,(K*b)j,,(L*P,)j,(P,P,)

x ( [ [,$/21[,X+1~C$1] x [~~1/21[~~~7.21]

CM] [i-l]

PI]

Ckl C’l

1 [[q,

c1/2l~,c~l]c~l[~~l/zIp~l]c~l]c’1)~

(3.18)

INTRINSIC

The right-hand

STICKING

IN dt

MUON-CATALYZED

175

FUSION

side of (3.7a) yields a more complicated

xik,(KIh)[jk,(L,p,,)coszI/,;-nk,(L,

expression:

PhL)sinTi/311

xik;(Klh)[jk~(L,p,)cos~i81-‘nki(L,p~)sint:,,:] X ji,

(K2b)

jiz(L

PO)

j;.;

(R*b)

.jj.;(

[L,p,,)

x ( [[~~l~rl[~~~llj~l]C~nl]r~l x ~‘1~~/27~j~~~l~~-~1]C~.l]C~I]C~1 x C’1)Y21CLn

“Cj;l;;;.~l]

1 ~~qF,~:21[j~~~lj~~~l]r~~~~l]C~‘1 Cj.‘l]C~‘l]

111 ),

(3.19)

The overlaps can be done in two steps. We begin with the angular overlaps, which are actually identical for Eqs. (3.7a) and (3.7b). (Note that the functions i$“ll, ?.Lk2] in Eqs. (3.17) to (3.19) are well defined: since pG$ b, they are given by r”,= r^,,, i,y = i, ) The angular overlap in Eq. (3.18) is defined by ( [[)l~“21[~~~ll~F,k’l]C’nl]r~l[~,‘

~1121 CTnz -r;,ij~.2i~C~.I~C~1~C13j

x [~[L~21i~I]c’1]c’1)=~U(LJ, n

lj,

Zlk,k21,,k,

[[‘l;~:zii,s;~~c-u A,j.2k,Z)

and is explicitly evaluated in Appendix B, which also contains the angular momentum matrix element of Eq. (3.19):

(3.20a) the explicit form of

( [[tlC1!*l[jlk~ljC~?l]C’~l]I~l ” x ~llt”‘~~~~“i:“‘~][‘ll’^]l”l

1 [[~~~/*l[~~~~l~~~l]C~~l]1~‘l

x [qb1121[r,;I -r~‘l-c~‘i1C~.‘l]C~‘11r~l~ rp.l

= f V(k, kzl,k,

2, &Ati, I) k; k;l;k’,

i; iii’ti’,

I)

(3.20b)

The overlaps associated with the matching Eqs. (3.7b) differ from Eqs. (3.18) and (3.19) only in the radial parts, which can be calculated employing the relation (3.11). Denoting the resulting overlaps by x:,“’ Big/(rB) and HI,iI,.i(r,, T,,.), etc., we see that Eqs. (3.7) are equivalent to the two sets of linear equations

176

DANOS,STAHLHOFENANDBlEDENHARN

Equations (3.21) have to be solved for the (numerical) determination of the amplitudes 2:;;. (Actually both equations are needed, since the phase shifts sB are not yet determined.) In the set (3.21) the expressions xi;‘, (xi,“})’ represent known numbers (defined in Eqs. (3.9)), the quantities Rip), (P1 Bj$j are the results of the overlap (3.18) and Ht,,,,.i, Ht,l,,P, are defined in Eqs. (3.19). It is important to note that the information concerning the molecular resonances in the dr .LI channel is contained in Eqs. (3.21) via the resonant phase shift 6{,] (cf. the definition in Eqs. (3.9) and the discussion in Section IV, below). The amplitudes ,? s/, connected to the amplitudes A (8t of the final states by a summation over { ~1i (cf. the remark below Eqs. (3.6)), determine the sticking fraction as an inspection of Eqs. (2.1) shows: the intrinsic sticking fraction into a given sticking state (labeled by { /?) 3 {IRK, I,, k, I}) is obtained by summing the amplitudes A”I# coherently over all quantum numbers except those designating the particular state and multiplying this expression by the corresponding ratio of densities of states. However as shown in Appendix E, in the present context this ratio equals unity. This then gives the result [14]

,iBl=

(3.22)

{ p}nkl!f~i*i*

The complete sticking fraction requires a summation of Eq. (3.22) over all possible sticking states, In Eq. (3.22) the total transition probability into a final state (cf. Eq. (2.le)) has disappeared in view of the observation that the sum (over the complete set (p}) of the squares of the amplitudes is equal to unity.

IV. DISCUSSION AND SUMMARY The reaction theory of dt p fusion, as formulated here, allows already some qualitative observations even before the actual numerical calculations are completed [15]. Let us first discuss the mechanism of the detuning effect [16], which cannot be explained in (earlier) perturbative models. The explanation is based on the consequences of energy conservation and our basic assumption that the muon does not exchange energy with the compound ‘He* nucleus (cf. Section I). Then the energy dependence of the R-matrix becomes crucial, since the coupling between the dt p and the nap regions decreases as the distance between the actual energy of the compound nucleus and the 3/2+ resonance increases. The translational energy of a muon in a sticking state ( -20 to 50 KeV) is determined by the recoiling c1 particle. A free muon, however, left behind after the fusion, has a comparatively low energy. Since the energy of the nuclear system depends on the energy of the muon (cf. Eqs. (2.13), (3.9)), the nuclear system has accordingly less energy available in fusions leading to sticking states as compared to non-sticking fusion reactions. As a first consequence of this fact let us consider the energy-momentum matching of the

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

FUSION

177

muon between the dt p and nap regions, concentrating on sticking states. Then the muon must have at the matching radii a “large” energy and momentum. It meansin terms of Eqs. (3.1) and (3.9) that iI must belong to the continuum. Since the dt p wave function is a rather low energy (and low momentum) system the overlap (3.1) will have a small value, i.e., ~2: will be rather “small.” Then, when employing the P transformation (3.3), the “smallness” of ~2: will be aggravated by the “smallness” of the P transformation, since E * is in this casefurther away from the 3/2 + resonance than for the case of non-sticking states. The related momentum matching is visible from the overlap integrals (3.18) and (3.19), taken together with the triangularity conditions (C.2a) on the momenta. Since the momenta in bound Coulomb states are “small,” the “large” momentum associated with the “boost” of Eq. (3.13a), which enters (Eqs. 3.18) and (3.19)) in the guise of L,,, demands “large” values of relative momenta, i.e., depends on the high-momentum tails of the Coulomb functions, which again have “small” amplitude. The next consequenceconcerns the zero-crossing of the sticking amplitude A”!;:! at some place within the dt p resonance line which, again, is not contained in the results of the earlier perturbative treatments. This can be seen as follows. A formal inversion of Eqs. (3.20), together with the definitions (3.9), shows an important dependence of the sticking amplitudes on 6 :~I) i.e., (4.1) where L is the orbital angular momentum of the ctn system contained in xi,!‘, Eq. (3.9). The function F is presumably a slow function of the energy and depends on the quantum numbers of the sticking state. (The coupling of the phase shift tiPi, implicitly contained in the function Fip,, to the dt ~1 molecular resonances is “washed out” compared to the coupling of the phaseshift 6 ( Pj (cf. Eqs. (3.12)). Therefore the main energy dependence of Eq. (4.1) is in the second factor.) In the second factor, 6 is the phase shift which changes by 71from 6, to 6, + rr as the ncrp energy sweepsover the width of the molecular resonance of the dt p system, i.e., as the energy goes over the (0,O) (or the (0, 1)) state. From this it follows that at some place the amplitude ac.[Bi Pi goes through zero within this “line width.” The actual place of this zero crossing depends on the “background” phase 6,, which can be different for the (0,O) and (0, 1) resonances, and for the different (nix) sticking states. On the other hand, the Auger transition strength from the (1, 1) state will essentially vanish outside of the line widths. A very important observation is that the dt p system has several essential characteristics not shared by the system of the other hydrogen isotopes. They are: the 312+ ‘He* system; the 3/2+ resonance close to, but above the dt p threshold; a unique entrance channel. These characteristics collaborate to achieve a single fusion channel, and a very small number of (with a single dominant) sticking states. Thus the detuning mechanism exists, and, the zero crossing does not get washed out by the presenceof many fusion channels, where each would have the zero crossing at a different energy, although still within the line width of the fusing resonance.

178

DANOS,

STAHLHOFEN

AND

APPENDIX

BIEDENHARN

A

We employ throughout the contra-standard (Biedenharn) time-reversal-covariant phase convention. Thus the usual spherical harmonics (Y,,,,(t)) are replaced by YE](f) E (-i)’

Y,(i)

s icl.

64.1)

This definition allows a compact notation for angular momentum re-coupling calculations [17]. The basic recoupling transformation of four tensors, needed in Appendix B, may serve as a typical example: [[&4~CW]C~l

[~C~~C~]C/I]LRI

= 1

hi

The square symbol

[ 1 a

b

d

e

f

c

h

i

g

[[@“l&4]

Chl [@bl#C’l]

is related to the ordinary

[iI]

9jcoupling

Cnl.

(A.2a)

coefficient by (using

P= (2c+ 1)“2 and analogous definitions)

(A2.b)

We now derive the addition functions. For the geometry

theorems

for spherical

Bessel (and Neumann) (A.3a)

R=?,+i’* the identity e

it5

.R

=eiL.il

erl;

(A.3b)

.i2

gives (via the plane wave expansion)

= (47c)* c ir1+r2 f,f,j,,(kr,) /I12 Vllf[/lI]

x Ck

co1 [/$Chljyl]

j,z(kr2) WI.

(A.4)

We multiply Eq. (A.4) with &~c’lAc’l]col where A:] is an arbitrary rank I tensor. Recoupling the angular momenta and integrating over the direction of I? leads to

INTRINSIC

STICKING

IN dt

x [l I 1, I L] [A [“ipp]

MUON-CATALYZED

CO’.

This expression can be simplified using the arbitrariness for the 9jcoefticients. This gives the familiar addition theorem: j,(kR)

179

FUSION

(A.51 of ALi] and inserting values

it,511 ““‘I- +i2-- I) [l II, /l,] j,, (h-r,) j,,(kr2)[jF’l’jS’:‘]~1’.

=;4”‘-)

(A.61

Note that the phase is that of the plane wave expansion: the replacement i’l + ‘z~ I--t ( - )1’2(‘1+ ‘? ” is allowed since I, + I, - 1 is even, owing to the presence of the invariant matrix element: i7,(^2 1 I, o o A

[1~1,1L]=(-)‘:““+‘~+“-&

12 o 30. >

The derivation of the addition theorem for spherical Neumann involved [ 1S] and we give only the result:

(A.7) functions

is more

rz,(kR) i!’ =4nC(-) Ilk

‘12”1+‘2-‘1 [l II, /l,].i,,(kr,)n,,(kr,)

x [;yl~c,‘d]~fl

(A.8)

where r (. (r , ) is the smaller (larger) of r, The geometry R = i, - ?I enforces via yrhif WI -i2)

, rl.

= ( - j/2 yrq~,) m -

(A.9)

a change of the phase in (A.6) and (A.8) i.e., (- )“2(11+‘zm ‘) -+ (- )‘/2”1 m’2P”. Further, in this geometry the expression (A.8) yields the multipole expansion of the Coulomb potential (when defined as l/R) as the special case I= 0 upon multiplication with k in the limit k + 0.

180

DANOS, STAHLHOFEN AND BIEDENHARN APPENDIX

B

The evaluation of the angular momentum matrix elements (overlaps) U( -.-I -..) and V( ...I . ..) can be done, e.g., by using graphical recoupling methods [17]. In figure 2 (where I, has been denoted as 1,) two equivalent recoupling schemes are shown for U(LJ, lj, II k, k,l,k, =;

lv,l12hc, I)

[[[4~~/~l[y^~~ll~~21]C~~1]C~l

x

[‘1p21[~~~Il~~~21]

VI]

CKI]

Cl1

1 [ [yp’l~~~l]

CJl

[‘]y21r*;‘l]

CA]C’I]~

(B.1)

The top scheme requires two intermediate summations, i.e., over S and K, whereas the bottom scheme involves only one (i.e., K). The question of computational

FIG. 2. Two equivalent recoupling graphs for the evaluation of the overlap Eq. (B.l).

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

181

FUSION

efficiency depends on the structure of available codes, since both schemes have approximately the same number of terms: the double summation involves S = 0 and 1, thus restricting K to the values K = I and K = I- 1, Z, I+ 1, respectively; the single sum is restricted to the values I= 0 and I= 2.

The corresponding

expression

for the bottom graph reads

;

. A

K

f 1 j OKK (B.2b) The choice between these two forms numerical calculation.

is decided by the subroutines

available

for

DANOS,

FIG.

3.

Two

equivalent

STAHLHOFEN

recoupling

Figure 3 shows two equivalent V(k,k*l,k, +

A,&hc,

graphs

AND

BIEDENHARN

for the evaluation

of the overlap

Eq. (B.3).

schemes for the angular overlap matrix element

Zlk;k;l;kr,

~;&L’K’,

I)

[[[Yl~‘121[~,C,kll~~21]C~nl]C~l

x [~~1/21[~~~11~~21]1~1]r~1]~~1~

x [IIyl[~~~;lr^~;l]

The top graph yields

V’l]CK’l]

[ [r151/21[~,C~;l~~;l]C~.1]C~‘l

[‘I].

(B.3)

INTRINSIC

STlCKING

IN df MUON-CATALYZED

FUSION

183

Here the notation

has been introduced. The recoupling scheme of the bottom graph leads to the equivalent

expression:

(B.4b)

184

DANOS.

STAHLHOFEN

AND

BIEDENHARN

As previously, the choice of the form to be used must be made in consideration the details of the subroutines.

of

C

APPENDIX

The radial integrations of Eqs. (3.18) and (3.19) contain products of spherical Bessel and Neuman functions. The integral over three spherical Bessel functions can be evaluated in closed form using the Jackson-Maximon formula [19]. Correcting misprints in Ref. 17 (page 174), the formula can be written as

I r*drj,,(plr)j,~(p2r)j,3(p3r) =7L

* 6(p, p2p3) i-(‘l+‘2+h)

p1p2p3 cl, , l2 , 131CFv’iv21d5’311Co’

= 1~‘lh’3 PI P2P3’ Here the discontinuous

(C.1)

function

6(p, p2p3) =

1 if pl, p2, p3 form a non-degenerate triangle 4 if p,, p2, p3 form a degenerate triangle 0 if p, , p2, p3 do not form a triangle

(C.2a)

has been introduced. Furthermore, fi,, fi2, g3 are the directions in the triangle defined by PI +b2 +p3 = 0. The invariant product can be evaluated by chasing a coordinate system where bj is on the z-axis and j, and j2 are in the x-y plane. Then

=

11

12

m

-m

13

0>

yye,,

n)

m

rye,,m

0)

Y,‘3’(0,0)

(C.2b)

where cos8 ’

=p-p:-p: 2

PIP3

’

2

JP:-P:-Pp: 2 P2P3

’

(C.2c)

and coso

(C.2d)

INTRINSIC

STICKING

IN

dr

MUON-CATALYZED

FUSION

185

The integral involving a Neumann function x

,-h.h/, =

sh

PI. P2P3 -

r2

(C.3)

drn,,(p,r)j,z(p2r)j,,(p,r)

must be evaluated numerically. Note that in view of the triangularity condition between I,, I,, 1, the integration could be extended down to h = 0. We are not aware of a closed form for this integral. The integrals over four Besselfunctions, i.e., 0 (i/2/14

PI P? P3 P4 =

s

r2

(C.4)

drj,,(p,r)j12(p2r)j13(p3r)j4(p4r)

can (formally) be evaluated in terms of the above three-field integral by introducing the delta function 6(r-r’) z=- r

2 71s P’ dpjj.(pr)jj.(pr’).

(C.5

This gives the expression (-5) Here i. must be chosen such that for both sets, (AI, 12) and (I/,1,), the triangular rules are fulfilled in order not to obtain indeterminate values for the functions A. We see that the discontinuous character of 0 is again such that fi, + t2 + jjX + i)4 = 0 must be fulfilled. If this formal evaluation is useful for practical applications must (again) be decided by numerical tests. The integrations involving one or two spherical Neumann functions, i.e ~~~~~p3I~~=jr2dr~,,(~,r)j:?(~2r).i,~(~~r)n,,(~,r),

(C.7)

fi ;:s2 1‘d:“p4 = j r2 drj,,(PIr)j12(Prr)

(C.8 1

n,,hr)

n,(p,r),

can be evaluated similarly: 0 ;;$;,, 1z4= 1 j p2 dp A;$

jj;:A?p4

(C.7a) (C.Sa)

Inserting these relations in the radial integrations, Eqs. (3.18) and (3.19), allows their evaluation.

186

DANOS,

STAHLHOFEN

AND

BIEDENHARN

D

APPENDIX

The amplitudes A”{:; of the final (sticking) states are obtained by an inversion of Eqs. (3.21). An equivalent procedure, which might be easier to handle numerically, can be summarized as follows: We start again with Eq. (3.6) and use the explicit forms of the wave function $A$ (cf. Eq. Wa)) and $nsp(bl (cf. Eq. (3.17)). With these definitions Eq. (3.6a) can be written in the form

x s P:, dp,mCn,(p,d

X [[~~~/21[~~~~l~~]]C~~l]C~l x C?,

C~I~1[~~~1l~C~21~C~l~ P

Ch-IICU,

P.1)

A similar equation can be written down for Eq. (3.6b), using again Eq. (3.11). The first step is now (cf. Section III) to calculate the angular momentum overlap, i.e., to calculate the matrix element U( ...I ...) of Appendix B. The subsequent multiplication of Eq. (D.l) with a (muonic) Coulomb function C,.( p,) and integration over pP lead to: Dfpc) {j,(kb)cos

c?(,~ -nL(kb)

= c A”$ (PI ’

Tkk,,

sin 6,,,}

U(LJ, rj, I( k,k,l,,k, -k2

Ti,.

12 s

l,l,hi,

I)

P:, dp,, C,, (~,a)

x j,, (K,b) j,, (K,b)(cos ttBj ftij$”

-sin rip) Fk;$‘).

CD.21

Here the symbols r, F abbreviate the integrations

PidP,j/q(L, Pp)jiz(Lz P,) C,‘,(P~) P,) c,dP,); s

(D.3a)

and &$.fi’=

P;

dp,nk,CL,

Pp)ji2(L2

again a similar equation is valid for the derivatives by replacing Z,(kb) according to Eq. (3.11).

(D.3b)

with Zj(kb)

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

FUSION

187

Equation (D.2) is obviously equivalent to Eq. (3.2la); the strategy to be used in an actual numerical calculation has to be chosen according to the results of numerical tests. E

APPENDIX

In the transition probability the normalization and the density of states are interrelated and must be evaluated together. The electromagnetic interaction is treated in first order perturbation, yielding the golden rule formula (2.1). For its application the initial and final states entering the matrix element must be stationary states of the “strong” Hamiltonian H,. In one case, the initial state, Ii,, ), is the electro-molecular state H,e, in some of its rotation-vibration bound states, and where one of the “H” indices is the rfrp doorway state Id). The final state I&E,)) describes a (Hze)+e continuum state and the ID(E)) state of the ‘He*p system. In our treatment we consider the recoiling part of the H,e, molecule which does not participate in the process to be a spectator, which has been factorized and yields an overlap matrix element of value unity. This way, the final state for the sticking transitions is a three-body continuum state: e, n, per,while that for the nonsticking transitions is a four-body continuum state: e, n, p, g. The available total kinetic energy E, thus is different in the different asymptotic channels of the final state l&E,)): E, = 17.6 MeV - JEblnd (e)l - JEbind (dt ,u)j (-~6”“).

(E.1)

Here Eblnd are the binding energies of the Auger electron and of the dt p system in

the (1, 1) state. We shall use Jacobi coordinates in the hyper-spherical form (hyperJacobi coordinates) [20]. To that end recall the re-writing of the kinetic energy of an unequal-mass two-body system into CM and relative coordinates: M,,=m,

(E.2a)

+m,

(E.2b) (E.3a) (E.3b) ( E.4a ) *

*

*

Pl2 -=---

PI

P2

h2

ml

m2

T=p:+p:=

2m,

(E.4b) - p:‘,

2m2

2M,,

+ - Pf2

2p,?’

Equation (E.5) suggeststhe introduction of scaled coordinates

(E.5)

188

DANOS,

STAHLHOFEN

AND

BIEDENHARN

(E.6a (E.6b and

PI2= j-12JPl2

(E.7a)

h, = 212 J;M,,.

(E.7b)

Thus kp =pr. Returning to our system we obtain from the above relations for the non-sticking states: 1 1 1 -=-+ m,+mP+m, h me

(E.8a)

1 1 -=-+p2 m,

(E.8b)

1

1

ii=m+m.P

1 m,+m, 1

(E.8c)

0:

The momenta conjugate to the coordinates j3, are defined as: (E.9a) (E.9b) (E.9c) With these definitions the scaled hyper-Jacobi coordinates are p,=pcoso,(sin9,coscp,,sin9,sincp,,cos$,),

(E.lOa)

fi2 = p sin 0, cos o,(sin 9, cos q2, sin 9, sin (p2, cos Q2),

(E.lOb)

fi3 = p sin w, sin o,(sin 9, cos cp3,sin 9, sin cp3,cos Q3),

(E.lOc)

while similarly kL=kcos~,(sin9,coscp,,sin9,sincp,,cos9,),

(E.9a’)

1, = k sin g1 cos g2 (sin 3, cos (p2, sin g2 sin cp2,cos &),

(E.9b’)

L3 = k sin 0, sin c2 (sin g3 cos (p3, sin S3 sin q3, cos g3);

(E.9c’)

(of course, ki and fii are not parallel).

INTRINSIC

STICKING

Then, in the center-of-mass

IN

dt

MUON-CATALYZED

189

FUSION

system, the kinetic energy is (E.1 la)

for the non-sticking

channels and kf

k;

k’

(E.llb)

T=T+T=T

for the sticking channels. (For the sticking channels put m2= 0 in (E.lO) and CJ? =0 in (E.9’)). We also have for the volume element d3R rf dr, rz dr? r-i dr, = (p, pL2 p3)312 d’R p8 dp do, dw,

(E.12a)

for the non-sticking channels and (E.12b)

d3R rf dr, rz drz = (p, /A)~‘* d3R p’d pd w

for the sticking states. We also introduce the notation

for the non-sticking and

P.,= (PI P*)‘!2

(E.12d)

for the sticking states. Equations (E.11) show the advantage of the hyper-Jacobi coordinates: the kinetic energy is contained in the single radial component; the energy division between the different sub-systems at p -+ cc’ is described by the hyperangies. Thus, there we have for the Schrijdinger equation for the non-sticking states ; v’gcP;,+k’g(b,)=~(p”ia+Rp’i,)(6+k’B+~~=O

b

(E.13a)

where the second form has been obtained by splitting the Laplace operator of the wave equation in angular and radial parts. Note that in (E.13a) the eigenvalue k’ is scaled k’=

2T.

(E.13b)

Introducing (E.14)

190

DANOS,

STAHLHOFEN

AND

BIEDENHARN

we find that for large p the solution is x 7

cp= 0

w P-PJe(kp+6.)F(,,1({52})

(E.15b) with p = 712 for the non-sticking and p = 412 for the sticking states. The index v of the Bessel function is a function of the quantum numbers y: (E.16a)

v=vc{Y)L {Y> = {iliz,

f,m,, 12m2, 4m3),

(E.16b)

where [,, c2 are the discrete quantum numbers specifying the functions of the hyperangles w, , w2. The various constants and phase shifts entering the final wave function in Region II are contained in the argument of the Bessel function in the form of the phase shift: 6, = ~(IJ, z, A,, A/,) where A,, A, are the Coulomb phase shifts of the electron and the muon. We assume, that we have solved the problem and constructed a normalized Fiyj; i.e., the angular part of the final wave function fulfills the normalization condition:

I

df2 IFi7112= 1,

(E.17)

where we have written dQ for the surface element of the 9- (or 6-) dimensional unit sphere. In the following we shall use the procedure: perform the calculations in a normalization box of radius R, and at the end let R -+ co. This procedure is simpler, but in the end fully equivalent to, the Weyl eigendifferential method. At any rate, both the normalization of the continuum wave function, and the density of states, then are functions of R, and we have (in Eq. (2.la)) (E.lSa) (E.18b) (E.18~) where we have defined tpRE N,‘Cp

(E.19a)

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

191

FUSION

with (E.19b) where the indices symbolize region:

the region of integration.

Also we have in the dt p (E.20)

(PC/,,,= @ELI,,, and hence its normalization Section III:

is that of the (numerical)

where, for example, nrlrlc could be equal unity. Herewith W= I(D(E)ldlli)j’

(w)’

calculation

in

we then have

lim N,‘p,(E). R- I

As in Eq. (2.1 b), (u)’ is the square of the matrix the Auger transition. Our task now is to compute

described

(E.15d)

element of the electron part of

(E.22) In equation

(E.22) the first term, given by (E.21a), and the third term Shea,,

d’,

=

‘*5He-,,

(E.21b)

is independent of R. We thus concentrate on the ncxp region. There the wave function acquires the asymptotic form, Eq. (E.15b) for, say p 2 R,,. We thus break up the normalization integral into two parts and begin with considering a single asymptotic channel ( b) : 7 i‘!,1,1d-R[ ,j / = nm,, I p; +j;,e,,;

(E.23a)

with

The constant A / ,I; represents the matching of the wavefunction between the dtp and ncrp regions, and has been introduced in (2.6). In view of (E.17) the second integral in (E.23) becomes (the normalization must be computed in terms of the

192

DANOS,

STAHLHOFEN

AND

unscaled variable r; m = n, s for the non-sticking after combining all contributions of Eq. (E.15) R

BIEDENHARN

and sticking states, respectively)

r,dr(SWp+d))2

sRO

kp8

=T

[R - R, + sin(2kR + 24) - sin(2kR, + 2A)] (E.24b)

Thus, combining

the R-independent

terms from (E.24a) and (E.24b): (E.23b)

Herewith we obtain in view of (E.19b) for the normalization

Next we consider the density of states, again one asymptotic channel at a time, and again we consider the two box radii R O and R. Choose R, such that at the considered energy it is between two nodes of the wave function. Call the number of nodes in the interval between R, and R, NR, and that for r < R,, N,,. Then N =k(R-R,)+A,,l(P)-A,,:(po) R n

71

We must keep the second term in view of the presence of the Coulomb A N log kp. Then we have for the density of states

=;;

(E.26) phase

R-R,+$(AiBi(p)-Alar(po) I

=$

{R-R,+.fi~l(k))

= L fl - const. Tck

(E.27)

INTRINSIC

STICKING

IN df

MUON-CATALYZED

193

FUSION

Hence we find ( l/71)( R/k) - const

lim N,‘p,(E)= R- T

(14,’ ‘Pk’)

R n, (E.28)

We now regard the normalization of I/I,,~, 81, Eq. (3.8b) and consider a single component .( 0, Pt. We write it in the asymptotic domain, augmented by the wave function of the Auger electron. We have for the non-sticking states p: dp t-i, drLra rz dr e

4R

sin(p,P,)sin(p,,rp,,)sin(p,r,)’ P,, PI1

d”Q p: dp r& dr,,

1

1 2 grc I’n I’n +I’!Jnr!Ar +Pt,‘c) I

Plls rpz rf dr,.

R3

Pert.

2i pn P, pPI rILx pcrc

+ cc

=N;

( E.29 )

=impf.

Performing the same calculation for the sticking states we find a difference by a factor R from (E.26). The origin of these differences is that the correlations in the wave function associated with the constraint (E.l) are missing in (3.8b). We now must resolve these differences. To prepare for it, consider that for r, -+ cc;, the Hamiltonian becomes (take the 3-body case, i.e., 2 Jacobi coordinates) H=Vf$V;+k’

(E.30)

since only the total energy is fixed. On the other hand, (3.8b) seemsto imply the use of the Hamiltonian H’=V;+V;+k:+k;.

(E.31)

We note that owing to the fact that the Hamiltonian (E.30) is separable, any general solution of (E.30) can be expanded in terms of the solutions of (E.31) as (E.32)

This is, in fact, the form of our solution, since, e.g., in Eq. (2.6), a linear superposition of the kind (E.32) is implied. Thus, without further ado, our solutions are already implicitly the hyperspherical solutions, i.e., the solutions of (E.30). We only must recognize them, i.e., re-write them ,formally as hyperspherical solutions.

194

DANOS,

STAHLHOFEN

AND

BIEDENHARN

To this end we write kr = k,r,

cos o cos a + k,r,

sin w sin (T

(E.31a)

or kr=k,r,cosw,cosa,+k~r2sinw,sina,sina,cosa, + k,r,

sin w, sin o2 sin O, sin a2

(E.31b)

for the sticking and non-sticking channels, respectively. We now write the radial part of the asymptotic form of (2.6) in the two ways, say, for the sticking channels

By evaluating the right-hand side of (E.32) for a selection of values of the k,ri, such as to keep kr constant one can determine the function f(w, a). To obtain A one now repeats this for a few values of kr, such that one traces out, say, one wavelength of sin(kr + A); the variation of the logarithmic Coulomb phase then can be safely neglected. The functions f(o, a), of course, must be the same for the different choices of kr. This, in fact, is an exceedingly strong overall test for the numerical accuracy of the obtained solution. We now consider the case of a sequential decay in which the emission of the first particle leads to a well-defined narrow intermediate state of system 2, i.e., the system reached after the emission of the first particle. We consider the limiting case where the width of that state is sufficiently small so that all other variables stay constant whithin that width. Then the probability of the reaction as a function of a, will have the form m

W-(E2-E,)2+r2,4f(a,,...)

(E.33)

where E, is the energy of the long-lived intermediate state, and E,=E-E,

= E[ 1 - (cos or)‘] = E(sin aI)*

(E.34)

is the energy delivered to the system 2. Consider now the integrated probability Jda

W=jda

(E(sin a,)2yER)2

=gg!E.L),

+ I-2/4f(a) (E.35)

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

FUSION

19.5

where (E.36)

sincr.=mE, and the averaging is over the width states

of the resonance. Now

consider the density of

(E.37) where V, is the phase space of system 1, and I’/2 the phase space associated with system 2. In our case system 1 is the two-body system consisting of the Auger electron and the recoiling dt k system in one of the rather long-lived fusion states. The essential observation now is that I’, depends only on E,, but is independent of E, i.e., of the total energy, and therefore is a constant. Hence here

P(E)= V2-$ v,, i.e., the density of states to be used in (2.1) is given by the phase space of the Auger process. Thus it is the same for the sticking and the non-sticking transitions. as used in Eq. (3.22).

APPENDIX

F

The purpose of this Appendix is to discuss, from a conceptual viewpoint, some theoretical difficulties inherent in an accurate calculation of dt pcf, in particular the “density of states” problem. The essential problem in &is the quantum mechanical calculation of the transition from a well-defined initial state of the muo-molecular complex ((dt p)*dee)* to a multiparticle continuum final state consisting of an Auger electron, a neutron, an alpha particle, and a muon-all in the continuum-plus a deuterium atom (de), which under our assumptions, we regard as a “spectator”. Viewed in this way, the problem appears dauntingly complicated: quantum mechanically the transition necessarily involves all possible reaction paths, including all possible sequences of the four particle emissions which can lead from the initial state to the final configuration. All such paths will, moreover, interfere coherently. A reaction theory of such complexity would be out of the question except for one fortunate circumstance: the initial step in the transition is electromagnetic so that a perturbative approach, for this part of the transition, is fully justified. Were it necessary to treat the entire transition non-perturbatively an accurate calculation would be infeasible. The fact that the Auger transition is electromagnetic, and hence perturbative, does not, of itself, resolve the question of coherently interfering reaction paths. Let us consider the possibility that the two reaction paths: (a) Auger emission followed

196

DANOS,

STAHLHOFEN

AND

BIEDENHARN

by fusion versus (b) fusion allowed by Auger emission, must be taken into account. Numerical estimates [ 1l] show that the direct fusion process (which is hindered by a parity mis-match) is disfavored by a factor of order 105. It follows that any coherent interference effects are of order 1 in 300, and can thus be safely ignored. This is a very major simplification in the calculation, and has already been incorporated in our discussion of the fusion process in the main text of this paper. A second major simplification occurs for the sticking states where the emergent alpha particle and muon form a bound state. For this situation, the final state continuum has only three particles, and not four, as would be the case otherwise. In this three body continuum consisting of an Auger electron, a neutron and an (a~)atom, energy can be shared among the particles, even if the emission sequence is well-defined. However, the Auger emission process is smooth, without structure in the energy distribution of the electron. By contrast-as discussed in the main text on the influence of the closed dt p channel-the neutron plus (UP) continuum is characterized by sharp resonances of milli-ev width. Thus the spectrum of the n + (alp) continuum states shows that these states are many orders of magnitude longer-lived than the Auger emission process. Correspondingly it is fully justified to consider the overall process as the sequence: Auger emission (with a well-defined energy) followed by fusion to a two body, (n + (UP)), sharply resonant continuum final state. It follows from these considerations, that the density of states for the three-body decay factors into a smoothly varying (Auger) electron density of states multiplied by the appropriate (strongly varying, resonant) two-body density of states, as we now discuss. Consider a perturbative transition from a system A to a two-body system which can decay in a second reaction, that is, A+B*+e

\

(F.1) B+b

We seek to discuss the influence of the decay of the excited system B* on the spectral distribution of emergent electron states. If the intermediate state B* were to be a discrete (bound) state, then the electron probability per unit time would simply be given by the Fermi golden rule:

where pp, is the (electron) density of states:

per= (vl(2~2fi3).

(F.3)

If, however, the state B* lies in the continuum then to use the Fermi golden rule requires that one solve two connected problems: (a) the continuum wave function

INTRINSIC

STICKING

IN dt

MUON-CATALYZED

FUSION

197

for B* must be effectively normalized, (b) this normalization must be used to construct the required generalization of the density of states function, which enters the calculation of the transition probability. An approximate calculation, adapted to a single resonance, is given in Ref. 21 based on the eigendifferential method [22]. We wish, however, to present a general treatment with possibly overlapping resonances. To begin the development, let us consider first the general problem of a nuclear reaction involving a two-body initial (asymptotic) state, proceeding through a many level intermediate compound state, into two-body (reaction channels plus entrance channel) asymptotic states: A + B + C* J D + E. To discuss this process effectively. we will need a precise notation: we will define an alternative (denoted a) to be a pair of separated nuclei, A and B say, with relative angular momentum 1, channel spin S =j, +j, (where j,, (j,) is th e intrinsic spin of system A (resp. B), and having total angular momentum I= I+ S. The spin-angle component of the wave function for the alternative denoted (as/) will be defined to be: $,Z,s,oj., where G is the projection of the channel spin s and h is the projection of the orbital angular momentum 1. We will designate the system given by alternative a, channel spin s, angular momentum 1, projections cr and i, to be a channel, denoted by c = (crslo%). The S-matrix, which by definition relates the in-going to the out-going components of the reaction, is described invariantly by the matrix elements: (F.4) Expressing this same S-matrix channels. we have:

in covariant

S,!,, = c (S’I’O’IZ’IJM) JM

form as matrix

elements labelled by

S~“.,.,,,,(slaE,)JM),

(F.5)

where the (. 1. ) are vector coupling coefficients. The most general form for a wavefunction at total energy lk in the external region (outside the region of strong nuclear interaction, but not necessarily outside the range of Coulomb and centrifugal interactions) is given by:

(slog 1JM)(s’l’d%’

1JM) Aa.r.,.o,j..

(F.6)

where the sum is over: c = aslol, c’ = (ds’l’d1~‘) and JM. The radial functions I,,(O,,) are in (out) spherical waves (angular momentum 1, alternative X) normalized to unit flux at infinity; v, is the velocity for alternative LX. The result in Eq. (F.6) is completely general, but formal in the sense that the amplitudes A (. are unspecified. If we specify that the reaction is induced by a plane wave of particles (CL’S’) then the amplitudes A,.. are determined to be: A a,\,I’a’;,, = i(7rv,.)‘!2 &!(21’ + 1)“2 sy.

(F.7)

198

DANOS,

STAHLHOFEN

AND

BIEDENHARN

Introducing this value for A,., in Eq. (F.6), and adding and subtracting amount of out-going waves, we find *E-ICI planeWaVe(a’s’a’)

+

an equal

+reaction3

where

(u,‘/u,)1’2(sIaA

1JM)(s’l’a’O

1JM)(21’

+ 1)“2.

(F.8)

The reaction amplitude

is obtained from Eq. (F.8) as (a,/~,,)~‘~ times the coefficient the amplitude of the spherical outgoing waves of *,,,Od in *reaction--i.% corresponding to alternative 01,channel spin S, with projection c (normalized so as to eliminate the ratio of velocities that enters in the definition of the cross section). Thus

. (s’Z’a’0) JM)(G,,I;,y,~

- S~s,;r’s’,‘).

(F.9)

The differential cross-sections now follow as da .‘S’d:aSu/dQ=Iq

(F.lO)

1s0,11’s’0’ uwl2.

For unpolarized processes where a’, a are not measured we sum over the states a’, and average over a, so that da,.,.;,,/dQ

= (2s + l)-’

(F.ll)

c )qasb;rcsfor12. 60’

Angular momentum techniques simplify this last equation. Application techniques leads to a Legendre series for the differential cross section: da,~,~,,,/dQ = (2s + 1 )-I 1: c B, (a’s’; as) P, (cos e),

of these (F.12)

where the coefficients BL are defined by B,(a’s’;

as) = l/4( -)“-“‘c

Z(l,J,Z2J2;

sL) Z(I;J,l;J,;

s’L)

x ReC(d,,,;- S2?;+,,)(~,,,; - S~s~~;ms,,)*l. (Re signifies taking the real part of the expression.)

(F.13)

INTRINSIC

The sum in Eq. (F.13) defined by Zr

STICKING

IN dt

MUON-CATALYZED

199

FUSION

rt,, rc2, I,, I;, I,, 1;. The Z coefficient

is over J,,J?,

is

I(21, + 1)(2/, + 1)(25, + 1)(2Jz + l)]“? x(I,1,001LO)

(F.14)

W(I,J,I,J,;sL),

where W here is a Racah coefficient. So far we have merely repeated standard nuclear reaction theory. In order to see how to relate these results to the fusion reaction, we must put these results into spectral form. Using the fundamental property that the S-matrix is unitary (S+ = S ‘, from flux conservation) and symmetric (time-reversal symmetry) we can put SJn in the canonical form: sJ”

=

(u’nj

-1

(F.15)

e2~~J”(~‘“),

The elements of the where: A’” is real and diagonal, and U Jn is real and orthogonal. diagonal matrix A’” are called the eigen-phaseshifts. The matrices U’” may be written in terms of their eigen-vectors Up, where k = 1, 2, .... N with N being the number of open channels at energy E. Using these eigen-vectors and the eigen-phaseshiftsallows one to put the S-matrix in spectrulform: S’” = i (exp(2idp)) k=l

U;d”0 Up.

(F.16)

Introducing these results into Eqs. (F.12) and (F.13) allows one to give a spectral of the cross-sections. The coefficient, B,, now has the form

parametrizarion

x sinAJ,o,sin~~~~~~~ cos(AJ,+, -- AJTnZk2),(F.17) where

(F.18)

1 Z(l;J,l~J,;.v’L)(U/:~“~),.,,,;(~~~””),.,~,;

I;/;

>

These results, which may appear rather formal, have an important physical interpretation. The spectral result for B, shows that the most general reaction is a sum of factors which relate two coherently interfering eigensystems: (Jnk), and (Jnk),. The amplitude for this general interference term is given by TCJnk,l, CJnA-)2.This amplitude is real and factors into an entrance channel part multiplied by an exit

200

DANOS,

STAHLHOFEN

AND

BIEDENHARN

channel part. Were the reaction to go through a single eigen-phaseshift resonance then the reaction itself factors (just as T does) into a part relating to the mode of formation and a part relating to the mode of decay. HOW does this general result for reaction theory relate to the fusion problem at hand? The crucial observation is once again that the initial step (the Auger process) is perturbative. If the general nuclear reaction treated above were to have the entrance channel treated perturbatively, then the eigenvectors (Up) would have for their components ( UkJ”)lSl values proportional to the reduced matrix elements of the perturbative interaction. In other words, the entrance channel parameters for the reaction would be completely determined perturbatively. The eigen-phaseshifts themselves, and the properties of the exit channel, would still be non-perturbative. The result is, however, still a nuclear reaction and not a cascade (sequence) as required for the Auger plus fusion reaction. It is the singular advantage of a perturbative transition that an absorption can be formally replaced by emission with only kinematic modifications. Thus we can replace the perturbative entrance channel matrix element for absorption leading to the eigensystem (Jzk), by a perturbative emission matrix element (the Auger transition) leading to the same eigensystem (Jnk).

It is important to note that this formal transformation (from reaction to sequential decay)-although indeed allowed by perturbation theory-would still not be fully correct were it not for the fact that (in the Auger plus fusion process) we have verified ab initio that the alternative (coherently interfering) path reversing the sequence does not occur appreciably. With this important detail accounted for, we can assert that the formal transformation is wholly correct for the fusion problem at hand. Let us now give the details as to how this transformation of nuclear reaction theory is to be carried out so as to lead to a calculation of sticking. Recall that the variational calculation determined, to within a overall constant, an Auger-weighted (dt p) state in the continuum (at each energy E) which was continued (via the R-matrix) into the (ncrp) system and tinally matched to the asymptotically valid (a~) + n wavefunction, 1+4~.(Note that tiE is fully determined up to an overall constant factor.) To accord with our discussion of reaction theory, let us note that the channel label c = (crslc~ll) now is to denote: tl= [(alpha + muon) atom in state N ( =principal quantum number) element A of the pair, with the neutron being the second element B]. with j, = angular s = [channel spin where 3 =j, +j,, + l/z) and j, = l/2 (spin of neutron)] (1, =Lo” I= [orbital angular momentum of the pair] g = [projection

of sJ,

1= [projection

of I).

momentum

as the first of atom

INTRINSIC

The wave function

STICKING

IN

dr

MUON-CATALYZED

$E can thus be written

201

FUSION

in the form: (F.19)

where the sum is over: ctsl, a’s’l’, Jnk, M. This result is the same as Eq. (F.6), except that we have introduced the spectral form for the S-matrix, and the general amplitudes A.,,,,S,,,, in Eq. (F.6) have been combined with the reduced matrix elements of the Auger transition (the ( UJnk)a,,Y.,,) to form the calculated coefficients (AJ”k.M )a,s.,. of the Auger-weighted state. Instead of the cross-section, for a specific nuclear reaction, in sequential decay problems the formalism, transformed as discussed above, leads to the probability that the Auger initiated transition (~‘$1’) leads to a definite final channel (crsl). Calling this probability P(a’s’l + ctsl) we have: P( u’s’l’ +asl)=(2s+

1))’

1

(2J+

l)sind,,,,,,

Jk,ka x sin x

A (Jnk ,z co@(J,k,,

[(up”“)3(.1(

-

Awrk,z)

U~~‘Z’)a,,][(A(J~k)‘)il..s.,‘(A’Jnk’?)Z.,s,,,]

psj.

(F.20)

This result is complicated and requires discussion. The Auger process is described by pe, (a phase space density of states) and the product (~4’~“~“) x (A’““k)z) which represents the product of the reduced matrix elements for forming eigen-channel states (Jnk), (respectively (Jnk),). Fortunately cl’s’l’ is but a single Auger channel, and the notational complexity of the general formalism is unnecessary. The sticking states are denoted by ctsl. It can be seen from the general result above that we have a complicated interference involving the (numerically)determined eigen-phaseshifts and eigen-vector components. If the interference effects can be neglected we see that the probability for obtaining a given sticking state (ctsl) takes the form: Probability

=

&I

Jnk

GJ+l) (F.21)

This (diagonal) form, if specialized to a single resonance (Jnk), agrees with the elementary calculation given in Ref 21. Note that these probabilities are defined at each energy E and can be expected to depend critically on this energy. The result given in Eq. (F.21) applies to sticking states since only for these are the final continuum states two-body states pius an Auger electron. For the nonsticking states one can either use completeness, or use the fact that the Coulomb (c+) states are smooth and permit a smooth continuation into the unbound (I.Y+ 11) states. Both procedures have practical difficulties (using completeness requires an absolute normalization for the fusing state, which is only numerically defined).

202

DANOS,

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AND

BIEDENHARN

To resolve these practical difficulties, we use yet another fortunate special circumstance of the @-fusion process. The nuclear fusion reaction takes place at distances very small compared to the muon scale ( < 5 fm vs N 250 fm). Thus there is a region where the nuclei are moving outside the influence of the nuclear interaction essentially independently of the muon (more precisely the total nuclear plus muon wavefunction can be split into components defined by sharp Coulomb eigenstates for the muon). (The nuclear reaction may be considered as taking place in a “punctured

three-space’, R3 - (0).) If we apply (molecular) reaction theory to this system, then the S-matrix is characterized by (J, rc, E) with components (channels) denoted by the muon quantum numbers (n, 1,j). (The energy of the nuclear sub-system is given by E* = E- E(muon).) We have used this description in the text, (cf. Eqs. (3.5)ff.). The essential points for the present discussion are: (i) Because the energy scale for the (dt p) molecular wave function is GZ100 eV, the wave function in this “reaction region” is dominated by the lowest muon bound states, (ii) hence the wave function in this reaction region has essentially only a twobody continuum structure (justifying the analysis leading to Eq. (F.21) if applied to this reaction region). The great advantage of considering this reaction region is that many fewer channels are involved, and for these only bound muon channels occur appreciably. An equally great advantage is that the wave function in the reaction-region is very likely dominated by a single resonant molecular eigen-channel, that is, only a single phase-shift will enter in Eq. (F.21) if applied to the reaction-region. (This can be checked without great difftculty for the numerically defined fusing state wave function in the reaction-region,) This analysis allows us now to resolve a basic problem inherent in our procedure for calculating the sticking fraction. Consider the (formal) analysis of the dipole-state 10) in terms of the dipole eigenstates ID(E)), Eqs. (2.17)-(2.19). This analysis required that the energy eigenstates of H,, the set {IL.)}, are all properly normalized in the continuum. If we consider the final system to consist of an Auger electron, a muon, an alpha and a neutron, then a very large number of eigenstates can be expected to enter, and the determination of the proper relative normalizations numerically is out of the question. (The Up vectors which appear in Eq. (F.21), and earlier, are required to be normed to unity). By contrast the energy eigenstates in the reaction-region (of R3 - (0)) can be expected to be resonant and (probably) unique at each energy. Determining the relative normalization of two overlapping resonant energy eigenstates in the reaction-region is actually feasible numerically, but-with luck!-should be unnecessary.

INTRINSIC

STICKING

IN

dt

MUON-CATALYZED

FUSION

203

REFERENCES I. S. JONES. Nature 321 (1986), 127; W. BREUNLICH et al.. Muon Catalyzed Fusion 1 (1987). 67. 2. S. JONES, Survey of Experimental Results in Muon Catalyzed Fusion, in “Proc. of MuCF Workshop Florida (1988),” AIP 181 (H. Monkhorst. S. E. Jones, and J. Rafelski, Eds.), New York (1989) 2. 3. J. RAFELSKI AND S. JONES. SC;. Amer. 255 (1987), 84; T. TAJIMA. A New Concept for a Muon Catalysed Fusion Reactor, in “Proc. of MuCF Workshop Florida (1988).” AIP 181 (H. Monkhorst. S. E. Jones, and J. Rafelski, Eds.), New York. 4. M. LEON, Theoretical Survey of M&F, to appear in “Proc. of MuCF Workshop Florida (1988):’ AIP 181 (H. Monkhorst, S. E. Jones, and J. Rafelski, Eds.). New York (1989). 423. 5. M. DANOS. L. C. BIELENHARN, ANI) A. A. STAHLHOFEN, Comprehensive Theory of Nuclear Effects on the Intrinsic Sticking Probability II, to appear in “Proc. of MuCF Workshop Florida (1988),” AIP 181 (H. Monkhorst, S. E. Jones. and J. Rafelski, Eds.), New York (1989). 308. 6. M. DANOS, L. C. BIEDENHARN, AND A. A. STAHLHOFEN, NBS Publication (1987). NBSIR 8773532. 7. L. N. BO~;~ANOVA CI al., Nucl. Phys. A 454 (1986) 653; G. M. HALE, EI al., Boundary-value approach to Nuclear Effects in Muon Catalyzed d-r Fusion, in “Proc. of MuCF Workshop Florida (1988),” AIP 181 (H. Monkhorst, S. E. Jones. and J. Rafelski, Eds.), New York (1989). 344; M. DANOS, B. MUELLER. ANU J. RAFELSKI, Muon Culalyzed Fusion 3 ( 1988). 443. 8. .I. RAFELSKI. The Challenges of Muon Catalyzed Fusion, in “Proc. of MuCF Workshop Florida (1988);’ AIP 181 (H. Monkhorst. S. E. Jones and J. Rafelski. Eds.), New York. 9. L. T. PONOMAREV, Muon Cara!uxd Fusion 3 (1988). 629. IO. H. E. RAFELSKI L’I ul., Muon Reactivation in Muon-Catalyzed D-T fusion, Univ. of Arizona preprint AZPH-TH 188812 ( 1988). to appear in “Particle and Nuclear Physics.” 11. Numerical estimates of the transition rates are listed in References 4 and 9. 12. M. C. STRUENSEE. <“I ui.. Plt,r. Rec. A 37 (1988). 340; Ref. 5; G. HALE. ef al. (Ref. 7). 13. R. K. NESBITT, “Variational Methods in Electron-Atom Scattering Theory,” Plenum Press, New York, London (1980); C. W. MCCURDY. T. N. RESCIGNO, AND B. L. SCHNEIDER, Phys. Ret,. A 36 (1987) 2061; M. DANOS. to be published. 14. One obtains Eq. (3.22) by inserting Eq. (2.1~) in the definition of the sticking fraction (Eq. (2.le)) and factorizing the resulting expression (cf. Ref. 5). The density of states can be calculated using either hyperspherical harmonics (Appendix E) or Weyl’s Eigendifferential method (Appendix F), (cf. Refs. 18-21). 15. The extenstve numerical calculations are performed by H. MONKHORST et al.; K. SZALEWICS el al. 16. M. DANOS, B. MUELLER. AND J. RAFELSKI, P/~Y.F. Rep. A 34 (1986), 3642. 17. The notation used here is summarized in M. DANOS, V GILLET ANO M. CAIJVIN. “Methods in Relativistic Nuclear Physics,” North-Holland, Amsterdam (1984), Chapter 4. A survey of frequently used notations is contained in L. C. Biedenharn and J. Louck, “Angular Momentum in Quantum Physics,” Encycl. of Marh. and i!.r Appl. 8, Cambridge University Press (1984). 18. M. DANOS AND L. C. MAXIMON. J. Math. Phys. 6 (1965). 766. 19. A. D. JACKSON AND L. C. MAXIMON, SIAM J. Math. Anal. 3 (1972) 446. 20. L. M. DELVES, Nucl. Ph~~.v. 20 (1960). 275: M. DANOS AND W. GREINER, Z./Y physjk 202 (1967), 125. 21. G. C. PmLLtps. T. A. GRIFFY, ANU L. C. BIEDENHARN, (i) Nucl. Phys. 21 (1960), 327; (ii) zpjt .fi Physik 205 (1967), 420. 22. A. SOMMERFELD, Partial Differential Equations in Physics. Academic Press (1949).