PHYSICAL REVIE% A VOLUME 34, NUMBER 4 OCTOBER 1986

Theriiialization of the muonic tritium atom in deuterium-tritium mixtures

James S. CohenTheoretica/ BiUision, Los A/amos Nationa/ Laboratory, Los A/amos, Xeu Mexico 87545

(Received 6 May 1986)

Thermalization of hot tp atoms formed in D2-DT-T2 mixtures and relaxation of the triplet hyper-fine state of tp are treated by Monte Carlo simulation. The cross sections for collisions with molec-ular targets are derived from cross sections for colhsions with the component atoms. The statisticalhistograms for the time-dependent velocity distributions are fitted by sums of two Maxwellian func-tions, and the coefficients and temperatures are tabulated for tritium fractions of 10%, 50%, and90% and mixture temperatures of 30, 100, and 300 K. The results provide a basis for analyzingsome previously puzzling aspects of the observed cychng rate in recent muon-catalyzed-fusion ex-

periments. In particular, thermalization is found to be very incomplete for some low-temperature

targets. Also, the relative rates of kinetic thermalization and hyperfine-state relaxation depend

strongly on the target tritium fraction, and hyperfine effects may be observable. The velocity distri-bution of tp atoms at times of molecular dt's formation is shown to depend only rather weakly on

the initial velocity distribution, which is not well known.

I. INTRODUCTION

Muon-catalyzed-fusion experiments in recent yearshave provided incontrovertible evidence that formation ofthe mesomolecular ion dt's is very rapid compared to themuon lifetime. ' The fast molecular-formation rate isconsistent with the predicted resonant mechanism, where-

by the binding energy of dt's, formed in a collision of tpwith Di (or DT) is converted to rovibrational energy ofthe resulting compound molecule (in which the small dt'sacts as a nucleus). However, a distinctive target-temperature dependence characteristic of resonant pro-cesses has not been seen (at least for D2, which is the moreimportant target), and dt's formation appears still to befast at very low temperatures, below the lowest predictedresonance. Moreover, molecular formation has been ob-served to exhibit an even more rapid transient rate at veryearly times. ' In a recent Letter, it was shown that thistransient may come from epithermal molecular forma-tion. The tp atoms are formed with kinetic energy muchgreater than kT and pass through energies where the reac-tion rate significantly exceeds the thermal value. Undersome target conditions (in particular at low temperatures),the thermalization time turns out to be longer than themolecular-formation time. In such a case, epithermalmolecular formation must dominate, thereby obscuringthe energy dependence of the actual resonant behavior.

In the present work, the thermalization of tp atoms indeuterium-tritium mixtures is treated by Monte Carlosimulation of the time-dependent Boltzmann equation.Because the tp atom is small and neutral, the requiredcollision cross sections for the molecular targets (includ-ing rotational and vibrational excitation) can be obtainedfrom cross sections for collisions with the component nu-clei (d and t) Results for th. e actual molecular targetswill be compared with results obtained assuming atomictargets. As is well known, e.g., in low-energy neutronscattering, the thermalization time is decreased by the ef-fects of molecular binding. The greatest uncertainty in

the present calculations probably comes from the atomiccross sections themselves, which have been calculatedwith increasing accuracy over the years by Ponomarevand co-workers. For the present application, the bestavailable results were fitted by appropriate effective-rangeexpansions. Because the elastic cross sections for tp+dand tp+t are dramatically different from each other atlow energies, the thermalization process has an importantdependence on tritium fraction.

Another uncertainty in the present calculations derivesfrom the initial distribution of tp( ls) kinetic energies. Ifexcited states can be ignored, as at low densities where thecascade is mainly radiative, then this choice is relativelysimple; namely, the tp has energy —19 eV if a muon istransferred' from dp(ls) to t and —1 eV if a free muonis captured directly" on t However, . excited states maybe important. ' Muon transfer from excited states of dpyield different tp energies; furthermore, large elastic andsuperelastic (sometimes called Coulomb deexcitation)cross sections of excited tp, can decrease or increase thekinetic energy. Fortunately, as will be shown, the energydistributions at times of interest turn out to be rather in-sensitive to the initial condition as long as Eo»kTa,where Tx is the target temperature. In keeping with thecurrently limited knowledge of the cascade, the thermali-zation time is taken to be simply inversely proportional tothe density, and results are normalized to liquid-hydrogendensity (4.25 X 10 atoms/cm ).

At the time of its initial formation, the tp, atom is ex-pected to be statistically distributed between states withthe t and p spins coupled as singlet or triplet. During thesubsequent thermalization process, the tp atom tends torelax to its ground (singlet) state. An important questionis how complete this relaxation is when dt's formationoccurs. The rates for molecular formation from the sing-let and triplet states are quite different. In the systemconsisting of pure deuterium, ddp formation from bothspin states of dp is known to be important and gives riseto a distinctive transient effect. ' A similar transient ef-

1986 The American Physical Society

JAMES S. COHEN

feet in dtp formation was first interpreted in an analo-gous manner, but this explanation did not hold up. '"Nevertheless, dtp formation from triplet rp may still besignificant, and determination of the nonequihbrium dis-tribution of states is important. The elastic cross sectionsfor tp+t scattering likewise depend significantly on thetp spin state. For these reasons the spin state of tp isalso followed in the Monte Carlo simulations, and the rel-ative triplet population is given as a function of time.However, the distributions of kinetic energies of the atomsin different spin states do not differ greatly, and combin-ing them simplifies tabulation of the results.

II. CROSS SECTIONS FOR tp COI.I.ISIGNS

A. Atomic targets

Elastic, as well as rotationally and vibrationally inelas-tic, cross sections for collisions of tp with Dz, Tz, and DTcan be approximated using available theoretical calcula-tions of elastic cross sections for tp+d and tp+t (no ex-perimental data are available). This treatment is alsoapplicable to hyperfine quenching. Because tp is verysmall compared to the electron orbit in the 0 or T atom„theoretical treatments generally assume that the electroncan be ignored. ' The present work requires cross sectionsonly at rather low energies, so effo:tive-range expansionsare expected to be useful. At relatively large distances(i.e., r &&a„but r «ao, where a& and ao are the muonicand electronic Bohr radii, respectively), the interaction isnormally dominated by the long-range (r ) polarizationpotential, and the usual effective-range expansion must bemodified. ' The tp+d p-wave cross section turns out tobe a partial exception to this rule, but notwithstanding,low-energy expansions were found that were deemed ade-quate for E & 10 eV in all cases.

The most accurate calculations available for tp scatter-ing are those of Melezhik, Ponomarev, and Faifman, '

who expanded the wave functions in a large adiabaticbasis. That work gave cross sections for tp+d scatteringat E & 10 eV and for tp+i scattering below the thresholdfor excitation of the triplet hyperfine level of tp atBE=0.241 eV For tp+t . p-wave scattering and fortp+t s-wave scattering at higher energies, we resort tothe earlier calculations of Matveenko and co-workers, ' '9whose values were modified slightly to make them con-sistent with the later work; some extrapolation was stillrequired. The hyperfine quenching cross sections, whichare strongly dominated by the tp+ t muon-exchange reac-tion, were calculated by Matveenko and Ponomarev. '

Elastic scattering of the triplet hyperfine level is compli-cated by the fact that it can interact with another triton ineither a doublet or quartet state. The results ofMatveenko and Ponomarev ' for tp+t scattering in even(g) and odd (u) states were useful for estimating the quar-tet scattering. In general, the cross sections for tp+t areprobably not as accurate as those for tp+d. This failingis mitigated, except for targets consisting of mostly triti-um, by the fact that the tp+d cross sections are muchlarger than the tp+t cross sections at near-thermal ener-

gies. The fits utilized for all the required cross sectionsare given in the Appendixes. The integrated (over scatter-ing angle) cross sections are shown in Fig. 1.

effm~ ——

foal g +Plymc1+ sin P

where m~ and m~ are the actual atomic masses in mole-cule BC, the extramolecular interaction occurring with 8,and P is the average angle between the internuclear axis ofBC and the velocity of the projectile A during the col-lision.

{jo

E

R

CO

)0

o oa o.~ o.e o.sE {SV}

FIG. l. Integrated cross sections for elastic tp+d collisions(solid curve), elastic tp(g J, )+t collisions (chained long-dashedcurve), elastic tp( g & )+t (chained short-dashed curve), hyperfineexcitation (long-dashed curve) curve, and hyperfine deexcitation(short-dashed curve).

B. Molecular targets

Because tp is small and neutral, methods developed forscattering of slow neutrons by molecular gases are ap-propriate. State-to-state cross sections for tp+Dz, etc. ,coHisions could be calculated from the atomic cross sec-tions of Sec. IIA using the impulse approximation. z

However, since we only care about the projectile-energyloss, a simpler method is more efficient. The Sachs-Tellermass-tensor method is suggested by the fact that themost relevant collisions occur at energies large comparedto the target rotational quanta (-0.003 eV) but smallcompared to the vibrational quanta (-0.4 eV).z4 Sachsand Teller showed that under these conditions therotational-vibrational motion can be treated classicallywith the actual molecular target replaced by a hypotheti-cal mass point. This point moves with the velocity of thetarget atcm, but has a tensor mass that depends on thestructure of the molecule. This method is therefore ourchoice. For a diatomic molecule target BC, the effectivetarget mass is simply

34 THERMALIZATIGN QF THE MUGNIC TRITIUM ATOM IN. . .

The cross section for the A;+BC~AJ+BC collision[the subscript on A designates the hyperfine-spin state, 1

for the singlet (ground) state and 2 for the triplet state] isthen given by

drT; (k,8)

dO

P

d~', (kV Aa/t Aa 8)

PAa dQ

pAa =mAma/(mA +ma ),luAa =mAma /(mA +ma ),eff eff eff

(3)

k =p'Aa V„i/fi .

The magnitude of the relative velocity is given by

Vrei =(UA +Ua —2UAuacosQ)2 2 1/2

uA —(2EA/m—A)'~ (7)

ua (2Eaclm——a )'i

in terms of the translational energies E„and Eac of atomA and molecule BC, and u is the angle between vA andvg.

III. MONTE CARLO SIMULATION

The velocity and hyperfine-spin-state distributions ofhot tp, atoms as a function of time in a thermal (Maxwel-lian) reservoir of molecules (Dz, T2, DT) are described bythe time-dependent Boltzmann equation. The equation issolved accurately by Monte Carlo simulation. The trajec-tories of a sufficiently large number NA of tp atoms arepropagated, one at a time from time zero to the largesttime of interest, and the energies and states at specifiedtimes are placed in a histogram. A.fter all trajectories arecompleted, the results are normalized, the statistical un-certainties are evaluated, and the numerical velocity distri-bution is fitted to a simple analytic function. The processfor each sample atom will now be described in detail.

The initial atomic energy is selected (at random) fromeither of two different distributions, a 5 function

Fs(E,Ep) =5(E —Ep)

or a Maxwellian

(E T )2'ttE a/kTc—1/2

(~kT, )'"

then the trial collision is rejected and the process returnsto step (i). Otherwise, the process continues and thekinematic results of the collision are determined in thenext step.

(viii) The azimuthal scattering angle P in the c.m. sys-tem is chosen from a uniform distribution in (0,2m. ) (thecross section is independent of this angle). For definite-ness the z axis is taken in the direction of the initial rela-tive velocity U„~——vz —vq and the collision is taken tooccur in the x-z plane. In this coordinate system the c.m.velocity is

+ .m. = ~c,m. ,x++ Vc.m. ,z~

where

Vc.m. , z= [UA —Ua —{ya—yA ) Vr'ei]/(2V, .i),

2Vc.m. ,x = [UA ( Vc.m. ,z + 'ya Vrei ) ]

yA mA/(mA+ma ——),eff

(12)

(13)

ya ma /(mA+ma ——) .eff eff

The collision rotates the relative velocity vector,

(i) The collision partner B is selected according to theatomic abundances in the target. To reduce the numberof collisions that will be rejected in step (vii) below, thechoice of B and final state j is also weighted by Xz, whichis at least as large as the maximum attained value ofV„idol~~(k, 8)ldll for given B, i, and j.

(ii) The translational energy of molecule BC is chosenfrom a Maxwellian distribution Fat(E, Ta) at the target("reservoir") temperature Ta.

(iii) The angle a between the incident velocities v„of A

and va of B is chosen from a uniform distribution ofcosa in ( —1, 1).

(iv) The "spectator" atom C is chosen according to theabundance of the molecule BC. The results presented inSec. IV assume high-temperature target equilibrium[Dq]:[DT]:[T2]=c~.2c~cr:cr so the probability of a givenC is the same as its atomic abundance in the target.

{v) The angle P (see Sec. II B) is chosen from a uniformdistribution of cosP in ( —1, 1). The effective target massma is then calculated as described in Sec. II B.

(vi) The polar scattering angle 8 in the c.m. system ischosen from a uniform distribution of cos8 in ( —1,1), andthe differential cross section der;~ (k, 8)/dQ is evaluated.

(vii) A random number rt E(0, 1) is generated. If

do,j (k, 8}V„) g gX,J,

where Eo———,' kTo. The hyperfIne-spin state is selected

such that initially &,',~——V„i{xsing sin8+y cosP sin8+z cos8) . (17)

[ tp( t t )]/[tp( t i)]=3 exp( &&/kTp)—(which is 3:1 triplet-to-singlet for Ep &&bE). The trajec-tory of the atom A(tp} undergoing collisions with mole-cules BC(Di,T2, or DT) is then propagated for a timet „by repeating the following Monte Carlo steps asmany times as required.

(If the collision changes the hyperfine state, the magni-tude V„] is also changed so that the relative energychanges by +b,E.) The final particle velocities are givenby

v~ =V~.m. +yaV~C

JAMES S. COHEN 34

In the case of the exchange reaction, v'„and v'tt are inter-changed.

(ix) The time elapsed since the preceding collision is as-

signed according to the probability

P(ht) =ve

where v is the total collision rate;

v= g cttcc f V„)cr,J FM(E, Ttt )dE

for atom A in state i having kinetic energy E„=—,' m„u„.

If the new time t+b, t exceeds the next tabulation time,the prior energy and state are placed in histograms. Then,unless t p t, , the process returns to step (i) with the new

energy Eg = Ttttg (Ug ) and possibly ln a new state J.After the Monte Carlo procedure has been completed

for all X„ test atoms, we have a histogram of particle en-

ergies at various times. The statistical uncertainty (onestandard deviation) of the number in a given bin, designat-ed l, 1s g1ven by

IV. RESULTS AND DISCUSSION

Results are tabulated for three different target tempera-tures, TR ——30, 100, and 300 K, and three different mix-tures, c, =0.1, 0.5, and 0.9, in Tables I—IX. For all ofthese tables the initial distribution was taken to beMaxwellian with an average energy of 1 eV. The resultsare tabulated at later times ranging from 0.05 to 15 times

TABLE I. Results of Monte Carlo simulation of hot tpatoms in a target mixture with 10% tritium (c,=0.1) and 90%deuterium (c~ ——1 —c,) at temperature Tq ——30 K. The molecu-lar composition corresponds to high-temperature chemicalequilibrium, i.e. , [D,]:[DT]:[Tq]=cq.2cqc, :c,'. The tp atoms ini

tially have a Maxwellian distribution of velocities with averagekinetic energy Ep ——1.0 eV, and a Boltzmann distribution oftriplet and singlet states at energy Ep. Results are tabulated atlater times in units of a reduced time at liquid-hydrogen density

(tp ——2.34 ns). The fraction of tp atoms remaining in the tripletstate is designated f3 (at t =0, f3 ——0.6764). The fit of thetime-dependent velocity distribution is given by the sum of twoMaxwellian functions at temperatures T, and Tb with coeffi-cients C, and Cb, respectively.

hX;=X;1/2

Xq —X;

Clearly, the finer the energy resolution (i.e., the morebins), the larger Nz must be to achieve the same relativeaccuracy. In the present work N& ——20000 and 50 energybins (49 nonlinearly spaced for 0 ~E„&1 eV and one forEq & 1 eV) were used. The energy histograms can be uti-lized directly; however, for many purposes the results aremore conveniently approximated by an analytic function.A good representation was usually obtained by fitting thenormalized distribution function at each time to the sumof two Maxwellian functions,

F(E)=C,FM(E, T, ) +CbF~(E, Tb ) (23)

with C, „T„Cb,and T~ determined by a nonlinear least-squares procedure. The sum of the squares of the residu-als,

—G(E; ),E;) (24)

G(E; i,E;)=J F(E)dE,i —l

(2&)

There is no sense in running xnore trajectories once 7 /nbecomes significantly greater than unity unless an im-proved fit (or the histogram itself) is utilized.

is minimized. The adequacy of the fit and of the numberof test particles Nz was judged by a X-squared test,

N~G(E; i,E;)—n n;

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.67050.66390.65660.64870.64150.63240.62520.61780.61040.60350.58910.57530.56130.54690.53400.50890.48570.46260.44110.42320.40440.38540.36810.35170.33550.26280.20700.16500.12800.10050.07810.06120.04730.03860.03030.02460.0195

C,

1.0000.6500.6990.6850.6720.6440.6350.5890.5720.5420.5070.5050.4470.4250.4090.3800.3190.3450.2850.2290.2120.1860.1590.1490.1370.0810.0740.0640.0440.0370.0500.0250.0170.0100.0080.0060.004

T, (K)

229017041095863707613539502453424364303288259236200191157158169162169181177181290226171219211124169243289258

321

0.3SO

0.3010.3150.3280.3560.3650.4110.4280.4580.4930.4950.5530.5750.5910.6200.6810.6550.7150.7710.7880.8140.8410.8510.8630.9190.9260.9360.9560.9630.9500.9750.9830.9900.9920.9940.996

Tb (K)

459275214174157138130118112978278726760574948474543424140383533333231313131313131

34 THERMALIZATIGN OF THE MUONIC TRITIUM ATOM IN. . . 2723

a reduced time unit to .This reduced time unit is definedas the thermal (temperature Tz) collision time of ground-state tp at liquid-hydrogen density multiplied by the aver-

age number of collisions required for a fast atom to losehalf of its kinetic energy (the latter number is only slightlygreater than unity since the projectile and target havesimilar masses). The second column gives f3, the fractionof tp atoms remaining in the excited triplet state. Thenext four columns give the coefficients and temperaturesof the two Maxwellian functions that fit the Monte Carlohistogram. A typical fit is shown in Fig. 2. Plots of theauerage rp energy as a function of time are made in Fig.3.

8

0.00 0.05I

0,15

E (ev)0.20

, = 0.5T = 300 K

0,30

The most outstanding feature is the dependence on c, .As the tritium fraction increases, the thermalization rateslows down, but the hyperfine relaxation rate speeds up.Both of these effo:ts are quite pronounced so the non-thermal properties of targets are a strong function of thetritium fraction. At low-to-moderate c, (&0.5) thermali-

FIG. 2. A typical Monte Carlo histogram of the tp energydistribution and its fit by the sum of two Maxwellian functions.This plot is for a target with c, =0.5 and T=300 K at a time of2 ns. The g /n of the fit is 1.6.

TABLE II. c,=0.1, Ep ——1.0 eV, T~ ——100 K, tp ——1.19 ns.The notation is the same as in Table I.

TABLE III. c, =0.1, F.p ——1.0 eV, T& ——300 K, tp=0. 60 ns.The notation is the same as in Table I.

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.00S.OO

6.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.67710.67390.67040.66690.66280.65860.65440.65170.64800.64450.63710.629S0.62310.61520.60800.59390.58150.56880.55530.54180.53000.51750.50500.49240.48160.42810.37790.33250.29470.26170.23210.20860.18570.16390.14330.12790.1143

0.7230.6170.5930.6610.6670.6690.6460.6350.6010.5950.5390.4890.4610.4100.3520.2600.2350.1810.1310.1040.0850.0800.0730.0610.0550.0510.0470.0430.0400.0340.0290.032O.S220.0200.0180.0200.012

T, {K)

46123247234616451330112010008978317S6670618560538525S194724935365696346396567127236867097SO

688711794557114

689591731

0.2770.3830.4070.3390.3330.3310.3540.3650.3990.4050.4610.5110.5390.5900.6480.7400.7650.8190.8690.8960.9150.9200.9270.9390.9450.9490.9530.9570.9600.9660.9710.9680.1780.9800.9820.9800.988

Tb (K)

117086365845837532430227326124122721219618918617616015314914413913312912612311310810710610410410369

103103103103

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.67650.67510.67410.67270.67130.66930.66680.66500.66310.66100.65770.65330.64960.64570.64240.63290.62510.61760.60950.60310.59630.58920.58260.57530.56880.5353O.S0330.47280.44770.42160.39630.37480.35440.33450.31630.29850.2806

0.7920.6910.6360.5830.5480.5420.5210.5980.6000.5780.5260.4750.4380.3850.3460.2820.2470.1500.0370.0100.0110.0101.0001.0001.0001.0000.0250.0500.1040.8730.1050.1080.1600.0521.0001.0001.000

T, (K)

597547943916333S2876244621391729154814461288116710701000946858764747

1010141312671213366360355331886715571337547538476614314311310

0.2080.3090.3640.4170.4520.4580.4790.4020.4000.4220.4740.5250.5620.6150.6540.7180.7530.8500.9630.9900.9890.990

0.9750.9500.8960.1270.8950.8920.8400.948

Tb (K)

164613451095984879777727605568547525510486477463439414410409399385376

318310298190295293290302

JAMES S. COHEN 34

0.12

O.Q9

O.Q6

~ ~

303030

300300300

C)0. 1

0.50.90.1

0.50.9

0.03

&0

FIG. 3. Average tp energy as a function of time. Thechained curves are denoted as follows: dot for 30 K and dashfor 300 K; single dot (or dash) for c,=0.1, double dot (or dash)for c, =0.5, and triple dot (or dash) for c, =0.9.

zation is almost entirely due to collisions with d. At lowc, (&0.1) a large fraction of triplet tlJ, atoms persist afterthe velocity distribution becomes near thermal. On theother hand, at high c, the triplet atoms are virtually allquenched while the atoms are still hot. Comparison ofthe results for low and high c, at 30 K shows that the as-sumed definition of to provides only a rough measure ofthe thermalization time. This inexactitude is due largelyto the different energy dependences of the crosssections —the tp+t elastic cross section is extremelysmall at very low energies Be.cause the rate (true) of elas-tic scattering decreases greatly as the effective tempera-ture decreases, the thermalization process becomes corre-spondingly less efficient, especially for large tritium frac-tions.

In order to determine the sensitivity of these results tovarious aspects of the treatment, some test calculationswere made for comparison. The modifications (madeseparately in independent runs) include the following:

TABLE IV. c, =0.5, Ep ——1.0 eV, Tz ——30 K, tp ——3.95 ns.The notation is the same as in Table I.

TABLE V. cg =0.5, Ep= 1.0 eV, Tq ——100 K, tp ——1.98 ns.The notation is the same as in Table I.

t/to

0,050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.QQ

8.009.00

10.0011.0012.0013.0014.0015.00

0.63270.57870.52230.47190.42930.38710.34760.31530.28320.2S520.20960.16950.14050.1}450.09240.06290.04150.02810.01920.01270.00810.00540.00390.00280.00160.00030.00000.00000.00000.00000.00000.00000.00000.0000Q.00000.00000.0000

0.557D.7490.7650,7S20.6970.6520.6170.5810.5500.5230.4680.4320.4080.3870.3800.37S0.3720.3170.2730.2240.2040.1810.1340.0920.0760.0081.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000

3166135597981274070567264662359856252746942237327321}19517516614413113113713119834323231

30

30

0.4430.2510.2350.2480,3030,3480.3830.4190.4500.4770.5320.5680.5920.6130.6200.625D.6280.6830.7270.7760.7960.8190.8660.9080.9240.992

Tb (K)

9143202121661561491421361301231151069688816859565250474544434237

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.65510.63030.60290.57410.54780.52020.49450.46980.44590,42510.38480.34650.31370.28070.25460.20880.17190.14}50.11550.09510.07660.06250.05120.04180.03410.01310.00480.00160.00050.00020.00000.00000.0000O.DDOO

0.00000.00000.0000

C,

0.6810.5780.6000.7460.7560.7390.7380.7130.6950.6720.6100.5560.5040.4610.4210.3610.3140.2780.2420.2060.1900.17D

0.1480.1120.1020.0291.0001.0001.0000.9430.9851.0001.0001.0001.0001.0001.000

T, (K)

455031062095138211341005900840790754718693677673668643611594578569525491478507467514111105102104103100101101101101101

0.3190.4220.4000.2540.2440.2610.2620.2870.3050.3280.3900.4~0.4960.5390.5790.6390.6860.7220.7580.7940.8100.8300.8520.8880.8980.971

0.0570.015

Tb (K)

1244914644355286264231218206196192189184182178172164157151149141136132131127117

5937

34 THERMALIZATION OF THE MUONIC TRITIUM ATOM IN. . . 2725

(i) Use of an initial 5 function tp, energy distribution at18.8 eV instead of a Maxwellian at l eV;

(ii) Use of the atomic rather than the approximatemolecular cross sections (i.e., neglect of binding effects);

(iii) Neglect of the triplet state of tp (i.e., all rp atomsare initially put in the ground state and the hyperfine-transition cross section is set to zero);

(iv) Neglect of hyperfine transitions (so that the tp,

atoms that are initially triplet do not relax).

20

1 eV18.8 eV

OMIC TARGETSOUND-STATE ONLYSTIC ONLY

UILISRIUM

The resulting energy distributions at a time of 2 ns areshown in Fig. 4. The equilibrium distribution is alsoshown for reference.

The distribution at 2 ns can be seen to be only slightlyhotter if the tp atoms start with 18.8 eV rather than anaverage of only 1 eV; in fact, the early slowing from 18.8to —l eV is very rapid. Because this case is of some ex-perimental interest for low c, targets [the initial distribu-tion of tp energies will be close to 5(E 18.8 eV—) if tp is

'"-.- ~ ~------ I

0.20 0.250

0 0.05 O. I 0 O.I 5 O.ME {ev)

FIG. 4. Comparison of alternative treatments of a targetwith c,=0.5 and T=300 K at a time 2 ns with the results ofthe method described in Sec. III (heavy solid curve) and with theequilibrium distribution (light solid curve). The results of modi-fIcations (i)—(iv} described in the text are shown by the {i)short-dashed curve. (ii) dotted curve, (iii) medium-dashed curve, and(iv} long-dashed curve, respectively.

TABLE VI. c,=0.5, Ep ——1.0 eV, Tg ——300 K, tp ——0.98 ns.The notation is the same as in Table I.

TABLE VII. e, =0.9, E =1.0eV,The notation is the same as in Table I.

Tg ——30 K, tp ——17.49 ns.

t/tp

0.050.100.150.200.250.300.3S0.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.66560.65590.64370.63150.61770.60310.58970.57550.5624O.S4920.52290.49680.47330.44970.42690.38450.34850.31660.28550.25770.23350.21180.19170.17460.15760.09740.05890.03620.02190.01360.00790.00450.00300.00200.00160.00100.0005

0.7900.6580.5780.5200.494OA550.5540.5790.6910.7040.6850.6840.7270.7050.6370.5880.53S0.4620.3960.5310.579O.S20OA47

0.4040.4980.0830.0240.0060.0051.0001.0001.0001.0001.0001.0001.0001.000

T, {K)

58714732383632242757242418661597132112071071961862815798756723708711619587582582579530773

104917241069313308307304304301303303

0.2100.3420.4220.4800.5060.5450.4460.4210.3090.2960.3150.3160.2730.2950.3630.4120.4650.5380.6040.4690.421OA800.5530.5960.5020.9170.9760.9940.995

Tb (K)

1560134311761043940878719656533493457427381374387374369371367324295302317318299337330324314

tltp

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.36200.16430.07220.03100.01490.00600.00260.00120.00020.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000

C,

0.9160.8800.8500.7930.7170.6650.6080.5590.5460.5550.5460,5330.5280.4970.4790.4660.3920.2990.1990.1400.0850.0520.0280.0071.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000

1245971823709621547486432376326262219186166149121109103104101104102110151393432

30303030

3030

0.0840.1200.1500.2070.2830.3350,3920.4410.4540.4450.4540.4670.472D.S030,5210.5340,6080.7010.8010.8600.9150.9480.9720.993

Tb (K)

13311811812713413212812311210087766965615351494846

4341

JAMES S. COHEN 34

formed by transfer from ground-state dp], the fits of thecorresponding time-dependent distributions for c, =0.1'

and Tz ——30, 100, and 300 K are given in Tables X—XII.The remaining comparisons are of more academic in-

terest. Thermalization calculated with the atomic crosssections is slower than that calculated with the molecularcross sections. The longer thermalization time is due tothe fact that at low energies the molecular cross sectionsare larger than the corresponding atomic cross sections.On the opposite side, thermalization calculated neglectinghyperfine transitions is faster. In the case that onlyground-state tp was treated this quickness is just due tothe avoidance of the kinetic boost resulting from the in-elastic transition. In the case that included nonrelaxingexcited states, thermalization is also enhanced by the trip-let elastic cross section, which is much larger than thesinglet cross section at low energies.

V. CONCLUSIONS

The reduced time unit defined in Sec. IV is useful forfollowing the thermalization state from initial formationto equilibration. However, a more relevant experimentaltime constant is established by the rate of molecular (dip)formation. Depletion of tp atoms by molecular forma-tion is not expected to affect seriously the velocity-distribution function. (This process could easily be in-

cluded in the Monte Carlo simulation if the molecular-formation rate was known with sufficient certainty. )

However, the remaining population of tp atoms at lowenergies may be greatly diminished if the molecular-formation rate exceeds the therm alization rate. Forc,=0.5 the experimentally observed molecular-formationtime (reduced, as usual, to liquid-hydrogen density) is -6ns. Of course, this is the average rate and does not pre-

TABLE VIII. c, =0.9, Eo ——1.0 eV, T~ ——100 K, to ——8.24ns. The notation is the same as in Table I.

TABLE Ix. c, =0.9, Eo=1.0 eV, Tg ——300 K, to=3.64 ns.The notation is the same as in Table I.

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700,800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.53420,37950.26260.17990.12570.08690.05830.04080.02720.01860.00860.00340.00170.00090.00030.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000

0.4200.9160.9110.8980.8840.8560.8390.8040.7680.7200.6250.5320.4810.4190.3750.3020.1670.0610.0220.0021.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000

T, (K)

343112881094996912845785733692656595547492448406339329368422698140134129124121112106103101

101

101

102101

Cb

0.5800.0840.0890.1020.1160.1440.1610.1960.2320.2800.3750.4680.5190.5810.6250.6980.8330.9390.9780.998

1217185157146149158164173180190199202195189185173167162154147

t /to

0.050.100.150.200,250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.62280.55710.48700.42590.36520.31250.27060.23200.20010.16880.12300.08810.06560.04760.03460.01820.00970.00590.00350.00160.00080.00030.00030.00040.00030.00000.00040.00020.00000.00010.00010.00010.00010.00010.00010.00020.0000

C,

0.5840.3790.2610.2880.9510.9530.9570.9600.9610.9620.9550.9580.9630.9670.9540.9670.9461.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000

T, (K)

54294008315021891255116210871028983946885825764708663578522466434413394377366356348322315310306304303306307304304303

0.4160.6210.7390.7120.0490.0470.0430.0400.0390.0380.0450.0420.0370.0330.0460.0330.054

Tb (K}

1687142912771130234205198187191188183182193198217227336

34 THERMALIZATION OF THE MUONIC TRITIUM ATOM IN. . .

elude a higher epithermal rate at early times. In anyevent, dtp, formation in low-temperature (e.g., 30 K) tar-gets clearly may occur mainly before the rp atoms arethermalized (see Fig. 5). This lack of thermalizationpresumably explains the observed absence of the predictedtelnperature dependence as targets are cooled to tempera-tures below -50 K. In practice, the molecular-formationrates calculated for equilibrium distributions at such tar-

get temperatures are simply irrelevant to the actual exper-imental conditions. This imp1ies that, in order to deter-

00 O.OI

cg = 0.5T= 30K

'l~~~ ~ ol&& w% C

0.03E (ev3

0.05

TABLE X. Results of Monte Carlo simulation of hot tpatoms in a target mixture with 10% tritium (e, =0.1) and 90%deuterium (cq ——1 —c,) at temperature T~ ——30 K. In this calcu-lation the tp atoms were all started with kinetic energy

Ep ——18.8 eV. Results are tabulated at later times given in units

of a reduced time at liquid-hydrogen density (tp ——2.35 ns). Thefraction of tIJ, atoms remaining in the triplet state is designated

f3 (at t =0, f3 ——0.7464). The fit of the time-dependent velocitydistribution is given by the sum of two Maxwellian functions attemperatures T, and Tb with coefficients C, and Cq, respec-tively, and a third term (1—C, —Cb)5{E—Ep).

FIG. 5. Energy distributions of tp atoms for a target with

c,=0.5 and T=30 K at reduced times t/tp ——O. S, 1, 2, 3, 4, 5,and 00 (tp ——3.95 ns).

TABLE XI. c'g =0.1, Ep= 18.8 eV Tg = 100 K, 4= 1 19 ns

The notation is the same as in Table X.

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700,800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.73750.72970.72110.71270.70450.69600.687S0.67920.67100.66270.64710.63230.61800.60430.59140.56480.53890.51380.49090.46870.44760.42660.40700.38620.36890.28850.22700.17750.13760.10960.08700.06920.05420.04290.03470.02720.0216

0.6660.5880.6600.7090.6960.6670.6290.5910.572O.S440.5170.4770.4550.4220.4060.3530.2850.2460.2410.2230.2270.1810.1670.1510.1470.0970.0920.0670.0600.0530.0230.0250.0170.0110.0060.0160.975

32572129127489S732631571523472438375334295271244220220211186182160181187197183225171213

14127120623422137310731

0.2070.3960.3370.2900.3040.3330.3710.4090.4280.4560.4830.5230.5450.5780.5940.6470,7150.7540.7590,7770.7730.8190.8330.8490.8530.9030.9080.9330.9400.9470.9770.9750.9830.9890.9940.9840.025

Tg (K)

8215953472251811591471381251169989807468625854504744434241403734343331323131313130

t/tp

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.74130.73740.73270.72840.72370.71930.71530.71160.70650.70180.69320.68470.67730.66910.66020.64570.62950.61510.60250.58760.57480.56200.55050.53830.52630.47120.41730.36920.32670.29100.25660.22900.20230.17960.15860.14040.1246

0.5700.6400.6580.6580.6860.6730.6510.6470.6300.6110.5820.5180,4970.4570.4120.3460.2860.2250.1730, 1300.1040.0930.0920.0840.0720.0580.0570.0520.0550.0380.0360.0280.0290.0240.9430.9680.925

T, (K)

4030326824131867146912541106963869799688635565528510458441

461516558578544585605662693665570673667649694753108105107

0.0420.2430.3040.3290.3080.3250.3480.3530.3700.3890.4180.4820.5030.5430.5880.6540.7140.7750.8270.8700.8960.9070.9080.9160.9280.9420.9430.9480.9450.9620.9640.9720.9710.9760.0570.0320.075

Tb (K)

955937675524396343309273251239218207190183176166157150144141137131126123121113109108106107105104103103

4056

JAMES S. COHEN 34

mine experimentally the specific energy-dependent rate,time-dependent nonthermal distribution functions that de-pend on rand c, should be used. On the other hand, ex-perirnents may be possible that more directly probe thenontherrnal tp atoms.

The most important theoretical uncertainties thatremain come from incomplete knowledge of the ground-state elastic cross sections for tp collisions with molecularhydrogen and of the excited-state d-to-t muon-transfercross sections. The present method of Monte Carlo simu-lation can easily be adapted to make full use of more com-plete and accurate cross sections as they become available.

TABLE XH- c~=O ls So=18.8 eV, Tz ——300 K to=0 60ns. The notation is the same as in Table X.

t/to

0.050.100.150.200.250.300.350.400.450.500.600.700.800.901.001.201.401.601.802.002.202.402.602.803.004.005.006.007.008.009.00

10.0011.0012.0013.0014.0015.00

0.74260.74230.74060.73930.73700.73530.73350.73140.72960.72820.72310.71890.71460.71000.70570.69670.68850.68050.67290.66500.65720.64900.64160.63390.62500.58800.5535O.S2020.49030.46370.43650.41190.38980.36590 3AAA

0.32470.3081

0.3180.5710.6530.6420.6100.5830.5760.5510.5160.5100.4770.4390.4030.3610.2790.2400.1960.1940.0850.0480.0280.0220.0250.0180.0230.0250.0170.0520.0580.0490.0740.0280.0200.0660.0400.0140.658

480540073591323129002584226820561905171814721308118311021077931861739863

102611581220112012141117992

1169722679647624905952604752

342

0.0380.1330.2390.3220.3780.4010.4360.4760.4850.5200.5600.5960.6390.7210.7600.8040.8060,9150.9520.9720.9780.9750.9820.9770.9750.9830.9480.9420.9510.9260.9720.9800.9340.9600.9860.342

Tb (K)

891921940906837763729699657604552524503498461432413410393380370358353346329322308311310299306307296305309252

ACKN0%'I EDGMENTS

The author is grateful to Dr. M. Leon for many valu-able discussions. This work was supported by the U. S.

Department of Energy, in large part by the Division ofAdvanced Energy Projects.

k cotgi ————+ck +3 = 1 2

a(A3)

fits the calculated phase shift very well for E & 10 eV ex-cept at E HEI, . This observation is readily understoodwhen Eq. (A3) is rewritten to emphasize the role of theweakly bound level,

I2(Eb+E) ' (A4)

where E=k /2p, I =k /(pc), and Eb ———1/(2pac).The least-squares procedures does indeed choose EI,-0.64eV, the actual energy of the bound state. 27 The fit is fur-ther improved at very low energies by incorporating thequadratic term, given by Eq. (A2), with the result

1 i' k2a 151——+ck

where P=6.97, a = —2100, and c=0.194.The atomic differential cross section is given in terms

of the above partial-wave phase shifts by

der(k, O) 1[(cosgo singo+ 3 cosg, sing

&cos6) )

k

+(sin go+3 sin g, cos8) ] .

APPENDIX A: t@+dSCATTERING

For tp+d s-wave scattering' the effective-range ex-pansion as modified by O' Malley, Spruch, and Rosen-berg' works very well; the phase shift go is expanded interms of the wave nuinber k as

1 n. 4 Pkkcotgo ————+ k+ k ln +ck +

a 3a 3a 4

(A 1)

for the reduced potential (2p/iri )V(r)- p—lr describ-ing ion-atom scattering. The parameter P is given interms of the atomic polarizability a (=4.5 in atomic unitsfor hydrogen isotopes) by I3 =pa/fi; for tp+d, P=6.97.The scattering length a and coefficient c were deducedfrom the scattering calculations, ' yielding a=3.8 andc =5.5 for the s wave.

The tp+d p-wave scattering' is anomalous in that it isdominated by the weakly bound state of dtp with one unitof angular momentum and binding energy Eb ——0.64 eV.Target polarizability plays a much less comprehensiverole, and the leading term in the modified effective-rangeexpansion'

tang i—— P'k '+

15

is not important except at very low energies (E &&Eh).On the other hand, the usual expansion (for a short-rangepotential)

34 THERMAL. IZATION OF THE MUONIC TRITIUM ATOM IN. . . 2729

The integrated cross section

o(k)= (sin i)o+3sin rii)4m

2

is shown ln Fig. 1.

APPENDIX 8: tp+ t SCATTERING

For the scattering of tp, by t, the hyperfine splittingand spin coupling must be taken into account. In the caseof the doublet (J= —,) interaction this coupling results inan effective two-state representation. For 1=0 hyperfinequenching has been explicitly treated. A 2X2 reactancematrix (designated R;)~ here, with subscript 1 for theground state and 2 for the excited state) that fits the pub-lished cross sections' ' was determined and provides areasonable extrapolation where necessary. For I = 1 thehyperfine quenching was approximated using the calculat-ed g and u phase shifts. ' In the case of the quartet(J= —,) interaction, only elastic scattering is possible andboth the i =0 and i = 1 phase shifts were obtained fromthe corresponding g, u phase shifts. '

Considerations for fitting the diagonal elements R ii'

and R22 ' are similar to those in Appendix A for fittingthe tangent (or cotangent) of an elastic phase shift. Forstate 1 the scattering length is very short, and the inverseof the usual expansion was found to be better, namely,

1/2, 0811 '4 2 2 pk) z= —ai, ——,mP k, ——,P ai)kiln +ci)ki

1

—W3

1(86)

2l 1jfe 0

for even 1

or

e2lg

2lfor odd I . (88)

This approximation is poorest for s-wave scattering nearthreshold; fortunately we have Eqs. (81)—(84) and do nothave to use it there. The J=—', state can be formed only

from triplet tp; because J is conserved, only elasticexcited-state scattering can occur in this state. Makingthe wave function antisymmetric under exchange of theidentical spin- —, nuclei requires

Q

S '=e ' forevenl (89)

well as the elastic scattering in the quartet state, using gand u phase shifts calculated neglecting the splitting. Forthe J= —,

' states this approximation yields

S'/'~=~as,where

(81) and

with aii ——0.05, P=7.81, and cii ——400. For R», analo-gous to Eq. (Al),

S'"'=.""for ~d i . (810)

k2

g 1/2, 022

'

~p' 4p', Pkt, k, + k', ln +c»k,'

3&22

To implement the g-u phase-shift approximation, weneed fits of the phase shifts. The fits extracted fromavailable calculations follow:

(82)

with a» ——3.87, p=7.81, and c» ———333. In these ex-pressions ki is the wave number relative to tp(tt)+t andk2 is the wave number relative to tp, (tt}+t The off-.diagonal element is

k2R i/20 R l/20 g(k )1/2

12 21 2 'k1

—tanriI%=6. 5 —63.8k+528k in(1.95k)+850kk

(811)

k cotri0 ———0.42+11.1k+33.8k ln(1.95k) —70k2,

(812)

with 8=0.674 and c&2 ——2. 15. The scattering matrix isgiven in terms of the R matrix by and

tani) f= 12.8k 2 70k i, — (813}

S=(I+iR }(I—iR }

For the /=1 contribution to tp+t scattering in thedoublet state, existing calculations have not explicitly tak-en the hyperfine splitting into account. However, we canstill approximate the scattering matrix for this state, as

I

1.3ktan(gf —rii }=

0.0192—k

The cross sections for tp+t elastic scattering and hyper-fine transitions are calculated from the above S matrices.The differential cross section is given by

2

z x; g(2l+1)[5;J S~i' Pi(cos8)] + ——35; 25' 2 g(2!+1)[1 S / 'Pi(cos8)]-

4k; l(815)

where x1 ——1 and x2 ———,. At the scattering energies considered in the present work only s and p waves are important,

JAMES S. COHEN 34

i.e., only the terms in Po(cosO) =1 and Pt(cos8) =cos8. The integrated cross sections

2

o;J(k;)= 2 x;Q(21+1) 5; 1S—~j'

' + —', 5;25J2+(21+1) 1 —5'r ' (B16)

are shown in Fig. 1.

V. M. Bystritsky et al. , Zh. Eksp. Teor. Fiz. 80, 1700 (1981)[Sov. Phys. —JETP 53, 877 (1981)].

~(a) S. E. Jones et a/. , Phys. Rev. Lett. 51, 1757 (1983); {b) 56,588 (1986).

3%. H. Breunlich et al. , Phys. Rev. Lett. 53, 1137 (1984).4S. I. Vinitsky et al. , Zh. Eksp. Teor. Fiz. 74, 849 (1978) [Sov.

Phys. —JETP 47, 44" (1978)].5M. Leon, Phys. Rev. Lett. 52, 605 {1984);J. S. Cohen and R. L.

Martin, ibid. , 53, 738 (1984).6J. S. Cohen and M. Leon, Phys. Rev. Lett. 55, 52 (1985).7S. E. Jones, in Proceedings of the Workshop on Fundamental

Muon Physics: Atoms, Nuclei, and Particles, Los AlamosNational Laboratory, Los Alamos, New Mexico, January,1986 [Los Alatnos National Laboratory Report No. LA-10714-C (unpublished)].

8K. Koura„J. Chem. Phys. 65, 3883 (1976).See, e.g. , J. J. Janik and A. Kowalska, in Thermal neutron

Scattering, edited by P. A. Egelstaff {Academic, New York,1965), Chap. 9.

' L. I. Ponomarev, Atomkernenergie 43, 175 (1983)."J.S. Cohen, Phys. Rev. A 27, 167 (1983).'2L. I. Menshikov and L. I. Ponomarev, Z. Phys. 0 2, I (1986).

P. Kammel et al. , Phys. Rev. A 28, 2611 {1983).'4P. Kammel, Lett. Nuovo Cimento 43, 349 (1985).'~S. S. Gershtein and L. I. Ponomarev, in Muon Physics, edited

by V. W. Hughes and C. S. %'u (Academic, New York, 1975),Vol. III, p. 141.

' T. F. O' Malley, L. Spruch, and L. Rosenberg, J. Math. Phys.

2, 491 (1961).'~V. S. Melezhik, L. I. Ponomarev, and M. P. Faifman, Zh.

Eksp. Teor. Fiz. 85, 434 (1983) [Sov. Phys. —JETP 58, 254(1983)].

'8A. V. Matveenko and L. I. Ponomarev, Zh. Eksp. Teor. Fiz.59, 1593 (1970) [Sov. Phys. —JETP 32, 871 (1971)].

' A. V. Matveenko, L. I. Ponomarev, and M. P. Faifman„Zh.Eksp. Teor. Fiz. 68, 437 (1975) [Sov. Phys. —JETP 41, 212(1975)].

zoS. S. Gershtein, Zh. Eksp. Teor. Fiz. 34, 463 (1958) [Sov.Phys. —JETP 7, 318 (1958)].

2'A. V. Matveenko and L. I. Ponomarev, Zh. Eksp. Teor. Fiz.58, 1640 (1970) [Sov. Phys. —JETP 31, 880 (1970)].

2~A. Bogan, Phys. Rev. A 9, 1230 (1974).R. G. Sachs and E. Teller, Phys. Rev. 60, 18 (1941).

2~K. P. Huber and G. Herzberg, Molecular Spectra and Molecu-lar Structure IV. . Constants of Diatomic Molecules (VanNostrand, New York, 1979).

25In the terminology used here, thermalization refers to the ki-netic energy, and relaxation {or excitation) refers to the tripletstate of tp.

2 Muonic atomic units, R=e =m„=1, are used in the Appen-dixes.

27S. I. Vinitsky et al , Zh. Eks.p. Teor. Fiz. 79, 698 (1980) [Sov.Phys. —JETP 52, 353 (1980)].

2sN. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, 3rd ed. (Oxford University Press, London, 1965), pp.379—385.