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AUG 041988

7~LE THEORETICALSURVEYOF MUON CATALYZEDFUSION

LA-uR--88-18O2

DE88 014463AU7HOm8):M. LEON

8UDMnW070 Proceeding of Muon-Catalyzed Fueion WorkshopSanibel Ieland, Florida “by 2-6, 1988

DIXIAMER

Unltd Sum ~ a sayqasy tbaaf.

ILomhnmmLosb

Alamos NationalLaboratorAlamos,New Mexi 4!$

ifiikROISTRIBUTMINOF THISDWlllli ii I t’; Uf,lI:

THEORETICAL SURVEY OF p~F

M. LmnLos Alamos National Laboratory

INTRODUCTION

Since not everpne attending this Workshop is a pCF expert, I willtry to give a helpful introduction to the theory of this fascinacin subject.

fconcentrating on the d-t fusion cycle. As many of you know, t, e largest

E“ Uart of this body of theory has been developed by Leonid Ponomarev andISco eagues and collaborators over the last two decades. We are fortunate

in having Professor Ponomarev with us at this Workshop, and look forwardto hearing from him later this morning.

The main steps in the muon-catalyzed d-t fiwion cycle are shown inFi .1. Most of the stages are very fast,●and therefore do not contribute sig-

Ini cantly to the cycling time. Thus at h uid Ha densities (4=1 in the stan-‘lledard convention) the time for stopping t negative muon, its subsequent

capture and deexcitation to theF

d state isestirnated to be * 10-11 sec.’The muon spends essentially of lts time in either the (alp) ground state,

Jwaiting for transfer to a tp) ground state to occur, or in the (t p) groundstate. writ ing for rnolec ar formation to occur. Followi

Ythe format ion

kof this “mesomolectden actually a muonic molecular ion), eexcit ation andfusion are again fast. hen the muon is (usuall ) liberated to go around

1agtin. We will now discuss these ctepe in some etail.

Fig. 1. The muon-catalyzed d-t fusion cycle.

MOLECULAR FORMATION

The Auger mechanism, where the energy released in forming the me-somolecule is carried off by an ejected electron, has been calculated byPonomarev and l%ifman,2 using a twcAevel (or Born-Oppenheimer) ap-proximation. The rea~ting molecular fomnation rate is quite slow fordtp : A~tA ~ 3x10-4 s- at room temperature. (These rates are mnven-tiondly normalized to liquid H2 tar t density, 4.25 x 1022 atoms/cm3.)

rThis Auger rate is completely warfed by the rate for resonant molec-ular fomdio~ the mechanism first au

Ysteal by Vesman3 in 1967 to ex-

plain the stnk~ obeerved temperature ependence for ddp fomnation. Thismechanism reqwes the existence of a loosely bound state, with binding en-ergy ~ 4 eV, so that the energy releaid upon mesomolecular formation cango into the vibration and rotation of the resulting compound molede:

tp + (D2)0,0 + [(dtp)”dee]~,J (1)

This ondmdy final state means that the cross section for this recess con-?tains b-functions at the resonana energies; temperature depen ence of A~tK

from this mechanism cornea &om the overlap of these d-functions with theMaxwell distribution of tp + Da kinetic energy.

The hst requirement in prov~ the wdidity of this resonance ideais an accurate calculation of the binding energies of the ostulated bound

t?states. This task was embarked upon by Ponomarev an his collaboratorfollowing Vemnan’sau

O!rtion, and required their developing the adiabatic

representation (rneth of perturbed stationary statea) into a powerful toolfor calculating meeomolecular wave functions (solving the non-relativisticCoulomb threebody roblem). B 1977 Vinitsky aud Ponomarev4 eztab

rlished that for the ( ,v)=(l 1) ({s angular rnomm.tum, v ~ vibrationquantum number) etate d dcipand dtp, ell s -2.2ev and -1.lev re-@ectively. This &abled Gerzhtein and Ponomarev5 to point out that theresonant molecular formation mte A~~~s ~0%’1,0~ that s 100 fuzionz permuon in a denee D-T tar~t inpoazible. Tim prechctmn of a hundred fuziomper mum did much to trigger the -t blossoming of intmt in pCF.

The adiabatic ~ntation has continued to develop,e but ma over-taken (at least as fkr az claimed pmcizion in the binding en- is concerned)about two years

6%%’variational cdculationz. The rment when ClI =

-1.975 eV (-O. e ) for ddp(dtp) ehould be accurate to a hction ofan rneV, m that it is the %elativiztlcn correction to these energies thatare of greatezt current interemt. Rx ddp, the vacuum polarization term is

t, providing + 10 meV out d a total correction to ei 1compktdy dominanof +11 meV.7 In contrast, for dtp the nuclear charge distribution is domi-nant, contributing 15 meV of a total + 23 meV correction. Higher precisioncdculationz are now under way, and we can hope to hear about. them zoom

There are two channels of molecular formation that are eupecidly sen-sitive to theze energiez, because the rewnance energies turn out to be omdl.Fimt, in the ddp reaction

(dP)F + (D2)0,0 + [(d4)fidee]7,1 (2)

~ and ddp total spin S= ~, the resonance energywith dp hyperfine state F= ~according to Zmeskal et al.8 is only about 4 meV; this gives rise to thespectacular (and unexpected) hyperfine effect observed at SIN.9 Since the

‘* should eventually allow the determinationtemperature’dependence of Aj~Pof this resonance energy to within about 0.1 meV, this case will provide anextremely stringent test of the calculated ddp J= 1, v= 1 binding energy,including all the corrections.

The other sensitive reaction is in dtp formation:

(l’)

with F=O, S= 1. Here the stron est resonances (those with orbital angulartmomentum L=O,l ) actually lie elow threshold! This circumstance is be-

lieved to lead to the three-body contribution to J~tP discovered at LAMPFIO(see below),

The pioneering calculation of the rates for resonant molecular for-mation (the Vesman mechanism) was made by Vinitsky et al.11 in 1977;the importance of L,J and hyperfine effects,12 and electron screening13 waspointed out somewhat later. This left the computed molecular formationrates significantly smaller than the experiment al value. 14’15’6 The situat imiwaa remedied when Menshikov and Faifmanle pointed out the importance ofusing undistorted dp + d and tp + d wave functions in calculating the tran-

‘ 17for the ddp case ulled allsition matrix element. Finally Menshikov et ti.the many strands together - rincluding “back decay” (the reverse o reaction2)1a and transitions of the [(ddp)dee]* complex - in a beautiful ah initiocalculation, which fits the experiment al results8’14IIS very well indeed.

The extension of this calculation to Ad: is straight forward for allchannels exce t the crucially important F=O, ~2 reaction, with its strong,

7below-thresho d resonances ( resumably J ~ J’: O ~ 1, 0 ~ 2, 1 - 2,retc. ). The precise location o these resonances will soon be known, when

the precise calculations of the relativistic corrections are completed. At lowenough density, treatment of the F=O, Da channel is also straightforward:the Auger deexcitation of the [(dtpl 1&e]* complex broadens the resonance&function9 into Lorentzians, lg with width ra s 0.8 meV.20 As a result thecontributions of the below-threshold reaonancea are dominant, thr higher- Lrewmances being ouppreseed by a centrifugal barrier factor. 12 This assumes

\that the tp stoma are therrrdized at the target temperature.At higher densities (4 > few %), three-body contributions to molecular

formation become signification. 21 The impact approximation, so importantfor the theory of collhional line broadening in spectmecopy, haa been appliedextensively to this problem, 19,**-*5 Thie ent~ln inclu~ng in the total widthof the Lorentzians a cohiond contrition, expressing the effect of collisionswith neighboring molecules:

r=r, +drc (3)

withr. =no4 vu>, (4)

But what collision cross section u should be used here? Petrovlg andot hers22\23’25 have used only the inelastic cross section on the final state,while in optics the initial state inelastic scattering plus the integral of thesquare of the difference of the elastic scattering arnplitudes also enters: 27

(5)

In contrast, Menshikov 24 claims that, because of the momentum carried bythe tp, the appropriate cross section is the final state total cross section:

u ff= ‘~1 + ‘i~el. (6)

However, in spite of its seductiveness, it is evident that the basic condi-tions for the impact approximation are not met for this problem of molecularformation.26 There are two conditions: First, the ‘detuning” (displacementfrom the imperturbed energy) AE must be related to the collision durationTc by27

(7)

For a realistic inter-molecular interaction and the temperature range of in-terest, the RHS of this relation is X1 meV; thus the impact approximation isconfined to a completely uninteresting energy range. This limitation arisesbecause, for AE ~ %/TC, information about the scattering wave function isneeded for times t < rC, not just the asymptotic properties of this wave func-tion (phase shifts, etc.). A second condition is the requirement of purelybinary collisions; this leads to26

0.14 @300K (8)“{0.02 030K ‘

which by itself removes most of the experimental data points!It appears that what is needed is the generalization (to the massive

incoming tp as opposed to the photon) of the (maxny-body) quusiddic ap-

proximation of collisional line-broadening theory2e (which may be related tothe “Quasi-reeonant molecular formation” of Menshikov and Ponomarev21 ).In my opinion, the theory of three-body molecular formation barely existsat present.

A possibility mentioned several timm2gf30 for solid and liquid targetsis that one or more phonow c

7off enough energy to maLe the strongest

resonance (O + 1, at * - 12 meV accessible to the physical region. So far,only a calculation for a metallic, rather than molecular, hydrogen target hasappeared.31

DEEXCITATION AND FUSION

Once the [ dtp);ldee]o complex is formed, Auger transitions will carTyit quickly to a ~=0 state of dtp where fusion is very rapid. The chain oftransit ions has been studied by Bogdanova et al.20

The proper method of calculating the fusion rate ~f (and the sticking,discussed below) has generated some controversy. What we shall refer toas the orthodoz view, expounded by Bogdanova,32 Markushin33 and others,holds that a good approximation is provided by the simple formulae

Af=lc”p (9)

(10)

K=Iim-“4, [VC7C-2], (11)

where C is the Garnow factor for the d-t system, o is the reaction cross

[section d + t ~4He + n), and V “v (r,R) is the three-body wave functionfor the dtp) JVmesomolecule R being the d-t separation and r the muon

I/iposition relative to the d-t C ). This approximation could fail (in the or-thodox view) only if significant rearrangement of the dtp spectrum by the

strong interact ion occurs. This would happen if the 5He*( ~+ ) resonance,which dominates the fusion reaction, were to be closely degenerate with adtp bound state. However, both the resonance energy and its width serveto prevent this rearrangemeut.

This point of view is supported., e.g., by the recent R-matrix calcula-tion of Struensee et al., 34 which finds At values close to the orthodox results.Obviously, the dissenters don’t agree, and I am sure we will be told whyduring this Workshop.

STICKING AND R~ACTIVATION

Sticking of the negative muon to the dau ter He nucleus limits the#number of fusions per muon that can be attain . Two factors are involved:

initial stickin U“(n, t) in the (n,t) state of the?

y) a system, and the prob-ability [R( n, ; q$fifor reaciiration of the muon uring the slowing-down ofthe (pa)+:

Z - R(n, 4; d)]~~(n, ~).US(4) =n,t [1 (12)

The simplest approximation for the initial sticking uses the adiabatic(Born-Oppenheimer) ap roximation for the muon wave function as the d

[and t approach one anot er, and takes the overlap (sudden approximation)with the final state wavefunction of a muon traveling in a bound state aroundthe retreating a-particle:

d~(n,l!) =1< !PI,(p -sHe) Ie’Mv”rWnto(p -

Calculated in this way the total initial sticking comes

~ ~~(n,e) = 1.16%.

a) >y . (13)

out to be35

(14)

The largest correction to this ap roximation comes from the fact thatEthe true wave function does not adia atically adjust sa the d-t separation

goes to zero, but rather “lags-behind”. Thus instead of W1,(P -5 He ) in 13q.

13, the true three-body wave function, with R+ o, is needed. The resultswit h the wave function calculated by several different methods (quantumMonte Carlo, 3g adiabatic represent ation, 35 variationa137) agree quite welland reduce the sticking b!y about 2570:38

Ew: = 0.85%. (15)

Next, we consider strong interaction effects on w:, in particular the

effect of the energy dependence of the $+ sHe* t-matrix element. Accord-.ing to the orthodox view, this aifects only the non-adiabatic corrections toU“9? and merely increases X! w: by < 3%.32*33The dissenting view is quited~fferent; e.g., Rafelski et al.3g claim the strong interaction effects reduceW: by a factor of two! We will certainly hear from both sides during theWorkshop.

The reactivation of muons in stripping or transfer collisions is veryimportant, especially for dt p beci~u= here the recoil velocity of the CKis solarge (u s 6 au.). The reactivation depends on the competition amenall the excitation, deexcitation, Stark mixing and transfer processes, an3the slowi

?of the (ap)+. Since the R.(n,t;#) obviously depends drastically

on the (n, ) values, there is an important dependence on the initial popu-lations u~(n,t). The most complete calculations are those of Cohen40 andMarkushi-n,41which in fact agree very well for R(d) and the x-ra intensitiesKu(d), Kp(#), . . . . fThe x-ray intensities (ddp as well as dtp are in fair

42 The theoretical R(d) variation with density #agreement with SIN results.is too small to account for the variation in w,(4) reported from the LAMPF. .neutron data. 10

SCAVENGING BY HELIUM

3He from tritium decay appears in any d-t experiment, so He scaveng-ing will always be present. There are obviously six possible reactions:

xp +A He -+A Hep + x (16)

!rx i!= p,d,t; A = 3,4.he reactions are elieved to proceed via the (Auger) formation of the

(xHep) mesomolecular state 2p0, folkowed by radiative dissociation:43

zp + He + (zHep)2PU + e-

The rates J,fic aa functions of temperature have been calc~ated bYFomic.hev et ak.” The predictions that (1) AXfi~ (T) increasea as T de-cnmaea, and (2) ~t~~ (T) is an order of magnitude larger than ~&, (T),appear to be experimentally verifkcL4s’45

In addition to this ~und-state transfer, for d-t targets (with theirlarqe cycling rates) the imtial capture by, and excited-state transfer to, Heis significant; the total scavenging rate will have both cent ribut ions:

The quantity WH. is analogous to (I-ql,) (see below). So far, there is nocalculation of this quantity.

ELASTIC SCATTERING

Elastic scattering of muonic atoms from target mokcules plays an ex-tremely important role in the catalysis c cle, since it determines the rate of

Jthermalization of the dp and tp atoms. he early calculations of Matveenkoand Ponomarev46 used a simple two-level (Bom-Oppenheimer) approxima-

ition. More recent work Melezhik and collaborators47 *4e)uses the adiabaticrepresentation (PSS met od) for calculating the nuclear scattering (dp + d,etc. ). To this is added the efkct of electron screening (in Born a proxima-

aftion), and of molecular structure (using the Fermi pseudo-potenti method).An example of the resulting cross sections, for dp (F=*) scattering fkom d,D,and D2 is shown in Fig. 2.

30

(20s am (MS 0.20

COLLISION f NERGY [ W ]

-\

, L=------- . .4.---”--”-”-”-”-”--~~”; .~”~

.0”/“H \

/I /“ k

/“I .0”-.\ ,./”’.-.

Fig. 2. Elastic scattering of F=} dp atomn frmn d (crm), D atoms (ua~) andDa mokules (a.OI) (from ref. 48).

HYPERFINE TRANSITIONS

The hyper&te transitions

(d@ F-f + d z @p)F=* + d

(tp)J’=’ + t + (qp’ + t

(19)

(20)

come about because of ezchange scattering. Matveenko and Ponomarev’scalculated these rates using the two level approximateion. Recent calculationsof Melezhik and collaborators47 use the adiabatic represent ation. Resultsfor the rates for dp + d are shown in Fig. 3, along with the low tempera-ture SIN point .g While this looks like good agreement, the theoretical valuewill be significantly raised when the contribution born resonant hyperjkequenching,17*49, i.e., the sequence

(alp)* + D2 - [(ddp)”dee]” ~ (c@ + D2, (21)

is included, thus leaving a discrepancy.For quenchi

Yof the triplet tp state the older two-level calculations’s

giving 3: = 9 x 10 S-l agree quite well with adiabatic representation result9.1 x 106s-1 of Melezhik.A7 Kammel et als” at SIN looked for the ‘build-up’ in time of the neutron signal at very low Ct (= tritium fhction) butsaw only a very much faster build-up; this may inchcate an experimental AtsignMcant ly larger than the predicted value. More experimental information

Fi .3.f47.

Hyperlhe

m -

as “

04 “

m

Q2“

a

o aoammT~TURE (K]

transition rates for dp atoms colliding with d’s (from ref.

d --+ t TRANSFER

The twdevel calculation of Matveenko and Ponomarev4e gave 1.9 x10Os-l fm the (ground date) d ~ t transfer, while the recent calculationsof Melezhik81 and Kobayashi et al.sz “ve 2.7 and 2.6 x 10*s-l res

T rtively.

In discussing the measured values of ~t, we must keep in mind t at these

depend on what is assumed about the triplet quenching rate At. The experi-mental numbers from several groups, =2.8 x 10ss-l ,14’15~53154correspond to

neglect of the tp triplet state, i.e., At = co; smaller values of A~ give linger

A~t values.

KINETICS OF THE d-t CYCLE

In general, in order to calculate the rate of production of fusion neu-trons as a function of time in terms of the underlying physical rates, it isnecessary to solve the kinetic equations describing the system. 55 However,for the steady-state cycling rate J= (i.e., after any transients have becomenegligible), it is sticiently simply to add the times the muon spends in eachstate, to get the cycle time12:

A:l = Td + Tt, (22)

(23)

Tt=T; +T; (triplet and singlet tp ground state); (24)

here,

(25)

and the branching ratio x is given by

(27)

These times and rates are normalized to liquid-hydrogen density, and theddp and ttp channels are neglected. If the high temperature ratio of D2 andDT molecules holds, we can write for the molecular formation rates

(2!3)

The factor ql, #,Ct ) is the probability of a muon, initially capturedAinto a hi hly excite dp atom, not transferring to a t during the (mainly

3collision ) deexcitation cascade. While some calculations of ql * have beencarried out ,s8’s7 the experiments seem to favor less drastic ~ and C~depen-dence than predicted. It seems likel that the deexcitation cascade is much

Jmore complex than the present mo els allow.

CONCLUSIONS

While much has been accomplished toward a complete quantitativedescription of the d-t catalysis cycle, the job is by no means finished. Inparticular, a really quantitative theory of three-body (many-body) molecu-lar formation is sorely lacking. A more detailed treatment of the deexcitationcascade, from the initial atomic capture of the p- to the ground state dp ortp, is needed for calculating w He and mare realistic ql ,(#,Ct ) values. Andaccording to the dissentin

#view mentioned above, more careful trest ment

of the strong-interaction e ects on At and u.; is called for.More precise calculations of the relativistic corrections to the dtp and

ddp binding energies will soon become available, so that the two-body molec-ular formation rates ~ddp and ~~~~can be computed more precisely. pre-sumably, more complete results for elastic scat tering, hyperfine quenchingand d ~ t transfer will also soon be completed, allowing more definitiveconclusions about thermalization, kinetits, etc. Then comparison with ex-periment will show where problems remain.

ACKNOWLEDGEMENT

This survey has benefit ted from many helpful and instructive conver-sations with various pCF colleagues, most especially with James S. Cohen.

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