Z. Phys. D 39, 165—172 (1997)

Muonic hydrogen and deuterium in H–D mixture and muon transferin excited states

E.C. Aschenauer1, V.E. Markushin2,3

1Department of Physics, University of Gent, Proeftuinstraat 86, B-9000 Gent, Belgium2Paul Scherrer Institut, CH-5232 Villigen, Switzerland3Russian Research Center, Kurchatov Institute, Moscow 123182, Russia

Received: 6 May 1996 / Final version: 10 October 1996

Abstract. The deexcitation of kp and kd atoms in hydro-gen—deuterium mixtures has been studied with a new kinet-ics model that takes the energy dependence of the cascadeprocesses into account. The X-ray yields, the populations ofatomic states, and the muon transfer from hydrogen todeuterium during the cascade have been calculated as func-tions of density and isotope fractions. The evolution of thekinetic energy distribution during the cascade is shown toplay an important role in the transfer kinetics. The atomicenergy distribution in the ground state is significantlychanged by the transfer. The calculated X-ray yields andthe muon transfer probabilities are in fair agreement withexperimental data provided the current theoretical transferrates are reduced by a factor of about 2.

PACS: 36.10.Dr

1 Introduction

The cascade in muonic atoms of hydrogen isotopes, kp,kd, and kt, has been studied for may years both experi-mentally [1, 2, 3, 4, 5, 6, 7, 8] and theoretically [9, 10, 11, 12,13, 14, 15, 16, 17]. Recent progress in muon catalyzedfusion (kCF) in mixtures of hydrogen isotopes (see [18]and references therein) brought new interest to the prob-lems in this field which was considered for a long timemainly to be a ‘‘pure’’ probe of the deexcitation processes(contrary to hadronic atoms where the atomic cascade isstrongly influenced by nuclear absorption).

The atomic cascade in the mixtures of hydrogen iso-topes is more complicated than that in pure hydrogenbecause the muon transfer from a proton to a deuteroncan occur before the muon reaches the ground state. Themuon is initially captured by hydrogen or deuterium toform a muonic atom with probability proportional, in firstapproximation, to the corresponding isotope fractions, C

pand Cd.1 The population of muonic hydrogen in the

1There are small deviations from the Cp-to-C

dratio, see [19]

ground state constitutes only a fraction of the initialcapture probability, Q

1sC

p, and correspondingly the

probability of the muon transfer from hydrogen todeuterium during the cascade is (1!Q

1s)C

p. The factor

Q1s

is a function of density and isotope fractions.One example of the importance of the muon transfer in

excited states is muon catalyzed fusion in D/T mixtures:the number of dt fusions in dtk molecules generated byone muon depends on Q

1s[18]. Until recently there has

been no direct measurement of Q1s

in D/T mixtures, whilethe data on fusion yields in the dtk branch of kCF cycle[20] show that the earlier calculations [14, 21, 22] signifi-cantly underestimated the value of Q

1s. These controversy

is called the Q1s

problem in kCF.Since recently a novel experiment using high-resolu-

tion X-ray spectroscopy has been in progress at PSI withthe goal to measure the transfer in excited states in H/Dmixtures by observing the K-X-ray transitions in kp andkd [7, 8]. The results for H/D mixtures, which are experi-mentally much easier to study than D/T mixtures, can beused to test current models of the atomic cascade inmixtures of hydrogen isotopes. The muon transfer in ex-cited states in D/T mixtures is similar to the one in H/D,and a clear understanding of the atomic cascade in H/Dmixtures is thus important for resolving the Q

1sproblem

in D/T mixtures.Because the transfer rates are energy dependent, a cas-

cade model that takes into account the energy evolutionmust be used in order to perform reliable calculations ofthe kinetics. The assumption of constant kinetic energyduring the cascade can be one of the reasons for thediscrepancy between old theoretical calculations and ex-perimental values of Q

1sin D/T mixtures [14]. Recently

an extensive study of the muon transfer in the excitedstates was done by Czaplinski et al. [15, 16]. By treatingthe kinetic energy as a fitting parameter and comparingthe results for Q

1swith the data of [7], they concluded that

the average kinetic energy of the kp in H/D mixtures isabout 6 eV. The experiments on the kp and kd diffusion ingas [23, 24, 25], however, demonstrate that the kineticenergy distribution depends on the density and hasa multicomponent structure. The mean kinetic energy of

muonic hydrogen in the ground state in gas increases fromabout 2 eV at 47 mbar to more than 9 eV at 750 mbar[25]. The kinetics calculations [17] show that this in-formation can be used to determine the kinetic energydistribution in the excited states because the energy distri-bution remains practically unchanged during the lowerstage of the cascade in gas, which is dominated by theradiative deexcitation. A strong high energy component ofthe kinetic energy distribution of the lightest exotic atomsin the excited states was originally discovered for pionichydrogen [26], and because of the similarity of the pionicand muonic cascades the same effect must occur in the kpand kd atoms.

The goal of this paper is to study the kinetics of thetransfer using a new cascade model, which includes bothacceleration and deceleration of the exotic atom duringthe cascade and was recently used for the muonic andpionic atoms in pure hydrogen and deuterium [17, 27,28]. The present calculations focus on the cascade in H/Dmixtures and the muon transfer during the cascade atlarge and moderate densities.

The paper is arranged as follows. The cascade model isdescribed in Sect. 2. The results of the cascade calculationsare given in Sect. 3. Sections 4 and 5 contain the discussionand concluding remarks, respectively.

The following notations are used below: /"N/N0

isthe atomic density of the isotopic mixtures N, normalizedto liquid hydrogen density N

0"4.25 ·1022 cm~3; C

pand

Cdare, respectively, the hydrogen and deuterium fractions

(Cp#C

d"1). Until otherwise stated, all density depen-

dent rates are given for /"1. The muonic Bohr radius isak"h2/mke2"256 fm, and the Coulomb binding ener-gies of the kp and kd ground states are correspondingly,ekp"2.529 keV and ekp"2.663 keV.

2 Cascade Model

2.1 Cascade processes and kinetics

The cascade calculations were performed using a MonteCarlo computer code described in ref. [17]. The cascadestarts from a highly excited state with principal quantumnumber n+(a

e/ak)"(mk/me

)1@2+14 (see [41] and refer-ences therein). In the present calculation the initial n-distribution was taken peaked near n"11 according to[19]. The relative probability for the muon to be capturedby hydrogen or deuterium in the H/D mixtures was as-sumed to be determined by the corresponding isotopefractions C

pand C

d. A small correction to the kinetics

results due to deviations from this Cp-to-C

drule can

be applied using the relative capture probabilities from[19].

After the muon is captured the cascade develops viathe de-excitation mechanisms accepted in the standardcascade model (SCM, see [10, 11, 13, 29, 30] and refer-ences therein), with the energy dependence of the kineticsrates being treated in a straightforward way [17]. Themuonic atom energy is considered as a time dependentdistribution, thus removing one of the main limitations ofthe standard cascade model where the kinetic energy isconsidered as a tuning parameter. The basic cascade pro-

Table 1. Basic cascade processes and their energy and density de-pendence

Process Energy Density Refs.dependence dependence

Radiative no no(kp)

iP (kp)

f#c

External Auger effect no linear [12, 29](kp)

i#H

2!(kp)

f#e~#H`

2(weak)

Stark mixing moderate linear [29, 35](kp)

nl#HP (kp)

nl{#H

Coulomb collisions &1/JE linear [36](kp)

i#pP (kp)

f#p, n

f(n

iElastic scattering &1/JE linear [42](kp)

n#HP(kp)

n#H

Transfer strong linear [31](kp)

n#dP (kd)

n#p

Fig. 1. The rates of various cascade processes vs. initial state n for kpatom in liquid hydrogen. Statistical population of the nl sublevels isassumed. Also depicted are the rates of the radiative transitionsbetween the circular states (n, n!1)P (n!1, n!2) and for theK-lines (nPP1S)

cesses and their main features are listed in Table 1. Then-dependence of various cascade processes for kp is shownin Fig. 1 (all energy dependent rates were calculated fora fixed value of kinetic energy ¹kp"1 eV). The details ofthe calculations of the deexcitation rates can be found in[17, 27, 28] and the references listed in Table 1.

As the the kinetic energy distribution is very critical forthe muon transfer from hydrogen to deuterium, its evolu-tion was taken into consideration below n"6. Aboven"6 only the collisional cascade processes are importantunder the experimental conditions concerned, thereforethe energy distribution developed at this stage does notdepend on the density. A Maxwellian kinetic energy distri-bution was assumed at n"6, with the energy ¹M

*/*5being

treated as parameter. This assumption is in fair agreementwith results from the muonic hydrogen diffusion experi-ment [23, 24], as it was discussed in [17]. Contrary toradiative and Auger processes, the transition energy in

166

Coulomb collisions transforms to the recoil of heavy par-ticles, and a kp can gain about 28 eV as a result oftransition (n"5)P (n"4) and 61 eV for (n"4)P(n"3). Muonic atoms with kinetic energy ¹<1 eV formthe so-called high energy component of the kinetic energydistribution. Evidence that a large fraction of atoms havehigh kinetic energies (&50%) was found for the first timefor n-p atoms at the instance of absorption (n"3, 4) inliquid hydrogen [26]. Recently the high energy compon-ent was also observed for pionic hydrogen in gaseouspressures of 17 and 40 bar [27]. Due to the similarity ofthe deexcitation of pionic and muonic atoms one canexpect a significant high energy component in muonicatoms as well.

The rates of Coulomb deexcitation calculated accord-ing to [36] were used in the present paper. The validity ofthe model used in [36] was doubted in the later calcu-lations [37, 38, 39]. However, the rates obtained in[37, 38, 39] are too small to explain the observed highenergy component in pionic hydrogen, as it was demon-strated in [27]. The latest calculations of the Coulombdeexcitation [40] (available at the moment in the limitedrange n"3—5) give the rates which are significantly largerthan the ones from [37, 38] and for n"5 only a factorof about 3 smaller than the rates from [36]. It is notexcluded that Coulomb deexcitation proceeds via somenew mechanism, e.g. formation of resonant states, as itwas suggested recently for n"2 [43] (if this mechanismwould be important, the rate would be maximal at lowkinetic energies).

Taking into account the current uncertainty in theCoulomb deexcitation rates, one can investigate its influ-ence on the results of the cascade calculations by usinga Coulomb-deexcitation coefficient k

C(a factor multiply-

ing all Coulomb deexcitation rates calculated accordingto [36]). The comparison of the cascade calculations withthe data on pionic hydrogen suggests the value k

C+

0.5$0.2 [27]. The rates shown in Fig. 1 were calculatedat ¹"1 eV and k

C"1. Their energy dependence is

shown in Fig. 2 for n"3 and 4.During the cascade the muonic atom can also lose

its kinetic energy in elastic collisions. The elastic scatter-ing differential cross sections were calculated in theclassical motion approximation, using the exact terms forthe Coulomb three body problem, with the code from[42]. The deceleration cross sections, as well as theCoulomb ones, calculated in the semiclassical approxima-tion are inversely proportional to the collision energy atlow ¹. In order to regularize the low energy behaviour ofthese rates a constant cutoff below the c.m. collisionalenergy, ¹

#65"0.1 eV, was imposed. The cutoff for the

deceleration rates does not directly influence the transferkinetics because the transfer rates are smooth at lowenergy.

The deceleration rates shown in Fig. 1 were calculatedusing the transport cross sections at ¹"1 eV. The energydependence of the deceleration rates is shown in Fig. 3 forn"3 and 4. The energy dependent differential cross sec-tion were used in the kinetics calculations. The calculatedtransport cross sections are similar to the results of therecent calculations [44, 45] in their n-dependence andenergy dependence at ¹'0.1 eV, with our values being

Fig. 2. The Coulomb deexcitation rates calculated according to[36] for n"3, 4 vs. the laboratory energy of the muonic atom atLHD

Fig. 3. The deceleration rates calculated with the transport crosssections from [42] for various states n vs. the laboratory energy ofthe muonic atom at LHD

typically larger by a factor of (1.5—2) than the ones from[44].

2.2 Muon transfer

In mixtures of hydrogen isotopes the muon can be trans-ferred from the light nucleus to the heavier one during theatomic cascade

(kp)n#dP(kd)

n{#p (1)

with the rate /Cdjnpd

. The normalized muon transfer ratesjnpd

were calculated in [15, 16, 21, 22, 31]. These rates de-pend on the collision energy and are comparable for thestates n"2—5 to the Auger deexcitation rates (see Fig. 4).

167

Fig. 4. The normalized rates of the muon transfer from hydrogen todeuterium [31] for various states n vs. laboratory kinetic energy

In the present calculations the transfer rates from [31]were used. Transitions with n@(n were neglected becausethey are much less probable than transitions with n@"n.The energy difference between the states (kp)

nand (kd)

n,

DEn"(ekd!ekp)/n2, is transformed to the recoil of the

particles in the final state and thus provides an additionalacceleration mechanism (DE

2"33.7 eV, DE

3"15.0 eV,

DE4"8.4 eV, DE

5"5.4 eV).

At low collision energies the reaction (1) is irreversible.But for large kinetic energies of the kd the reverse transfer,i.e. from the heavier nucleus to the lighter one, can occur

(kd)n#pP (kp)

n{#d (2)

with the rate /Cpjndp

, where the reduced rate jndp

is connec-ted with jn

pdvia detailed balance.

In the present calculations we did not include thetransfer in the states with n56. The transfer at high n wasconsidered in [15, 16] where it was found, in particular,that the transfer kpPkd at n"8 brings a significantcontribution because the corresponding rate of Augerdeexcitation is suppressed for the Dn"1 transition, asthey are forbidden by the energy conservation. This con-clusion, however, strongly depends on the importanceof other deexcitation mechanisms. It is known that theAuger deexcitation alone is insufficient to provide theright cascade time [29]. Since in our model the Coulombcascade rates are much larger than in [15, 16], the transferat n56 is comparatively small. The recent results on thekp drift in gas at very low density [46] evidence for largecontribution of the Coulomb transitions to the deexcita-tion at high n. It will be reasonable to include the transferat n56 in our calculations when more detail informationabout the corresponding energy distribution becomesavailable. It is worthwhile to notice that at high n theinverse transfer is expected to be energetically allowed fora significant fraction of the atoms and, therefore, it bal-ances the direct process [16].

2.3 Results

2.3.1 X-ray yields in pure hydrogen and deuterium. Thecalculations of the X-ray yields for pure hydrogen anddeuterium as functions of density were presented in [17].In order to investigate the influence of the Coulombdeexcitation on the X-ray yields in more detail, the presentcalculations were performed for scaling factor k

C" 0, 0.5

and 1. Results are shown in Fig. 5. The ratio of the yieldsof the Ka and Kb lines is rather sensitive to the rates ofcollisional deexcitation. At density /'0.04 this ratio ismainly determined by the competition between the radi-ative transition 3PP1S and the collisional deexcitationn"3Pn"2 followed by the 2PP1S radiativetransition. This leads to an approximately linear depend-ence of the Ka/Kb ratio on pressure, as it is shown inFig. 6.

The best agreement between calculations and experi-mental data [8] for muonic deuterium was found forkC+0.5$0.2 while the data for muonic hydrogen sug-

gest somewhat smaller value. This estimate of kC+0.5

was obtained from the analysis [27] of the high energycomponent measured for pionic hydrogen [26], which isthe most direct way to observe the Coulomb acceleration.The value k

C"0.5 is thus used in the calculations of the

muon transfer discussed below. When our calculationswere finished, the latest calculations of the Coulomb deex-citation rates in the advanced adiabatic approach [40]became available for n45, the results confirming theimportant role of the Coulomb deexcitation (for example,the calculated rate for n"5 and E'1 eV corresponds tokC+0.3).Good agreement with the data could also be achieved

by increasing the Auger rates n"3Pn"2 by a factor ofabout 2 without Coulomb deexcitation. However, theAuger rates at small n calculated in Born approximation[29] and eikonal approximation [12] are in very goodagreement with each other, thus we assume that the uncer-tainties on the Auger rates are much smaller than those onthe Coulomb rates.

2.3.2 Muon transfer in excited states. The results for thekp fraction arriving at the ground state, Q

1s, in H—D

mixtures are shown in Figs. 7, 8 and 9. Excited state muontransfer is more sensitive to the kinetic energy of the kpatom than X-ray yields, because of the strong energydependence of the transfer rates (Fig. 4). Figure 7 demon-strates the influence of the Coulomb deexcitation onmuon transfer: the acceleration of kp atoms due to theCoulomb transition depopulates the low energy regionE&1 eV, where the transfer rate is maximal, thus result-ing in a significant increase of Q

1sin comparison to the

case which neglects Coulomb deexcitation. This resultconfirms an earlier estimate [13] of the influence of theacceleration due to Coulomb deexcitation on the transferprobabilities.

To study the dependence of the results for Q1s

on thetransfer rates we used a constant scaling factor k

Tfor all

the transfer rates [31]. Figure 8 shows the dependence ofQ

1son the deuterium fraction at LHD calculated for

kT"0.5 and 1 in comparison with the experimental data

[8]. While the theoretical values of Q1s

for kT"1 are

168

Fig. 5. The calculated absolute K-X-ray yields from muonic hydro-gen (left) and muonic deuterium (right) vs. density. The Coulombdeexcitation scaling factor k

C"0 (solid), 0.5 (dashed), 1 (dotted)

Fig. 6. The ratio ½Ka

/½Kb

for muonic hydrogen (left) and muonicdeuterium (right) vs. density, calculated with k

C"0 (solid), 0.5

(dashed), 1 (dotted)

significantly smaller than the experimental ones, the re-sults obtained for k

T"0.5 are in fair agreement with the

data. The dashed-dotted curve depicts the dependence ofQ

1son the deuterium fraction at 780 bar (k

T"0.5) with-

out taking the reverse transfer (kd)n#pP (kp)

n{#d into

account. The comparison of this curve with the solidcurves in Fig. 8 shows that the strong change in slope atC

d'0.9 of the solid curves is due to the reverse transfer.

The density dependence of Q1s

changes from weak atC

d(0.1 to strong at C

d'0.5 as shown in Fig. 9.

The interplay between kinetic energy distribution andmuon transfer in excited states can be illuminated bycomparing the kinetic energy distributions of kp atoms inpure hydrogen and in H/D mixtures, as it is shown in Fig.10. The fraction of low energetic kp atoms in the ground

Fig. 7. The conditional probability of the kp arrival in the groundstate, Q

1s, in H/D mixtures vs. pressure at deuterium fraction

Cd"0.5 calculated with k

C"0 (solid), 0.5 (dashed), 1 (dotted)

Fig. 8. The dependence of Q1s

on the deuterium fraction at differentdensities. The theoretical results are for the transfer rates from [31]and the scaling factor k

T"0.5 and 1 (curves) and the experimental

data are from [8] (points). The dashed-dotted curve shows thedependence of Q

1son the deuterium fraction at 780 bar (k

T"0.5)

without taking the reverse transfer ((kd)n#pP(kp)

n{#d) into ac-

count

state (¹-08

48 eV) decreases with increasing deuteriumfraction because transfer takes predominantly place in theenergy region near the maximum of the transfer rates.

Muon transfer in excited states also influences theenergy distribution of kd atoms. In particular, a significantfraction of the transfer takes place at n"3, and since thetypical initial energy is small (about 1 eV) and the energyrelease is only 15 eV, the final kinetic energy of the kdatoms is of the order of a few electron-volt. Moreover, thesubsequent deexcitation can produce only a limited frac-tion of high energetic muonic atoms, therefore the energydistribution of thus produced kd atoms in the groundstate is less energetic than that following the direct cascadeinvolving the Coulomb acceleration.

169

Table 2. The relative contribution (perinitial kp) of the transfer kpPkd fromdifferent states n at different pressures anddeuterium concentration (k

C"0.5)

kT"1.0 k

T"0.5

Cd"0.2 C

d"0.5 C

d"0.2 C

d"0.5

n 32 bar 780 bar 32 bar 780 bar 32 bar 780 bar 32 bar 780 bar

2 0.012 0.215 0.018 0.216 0.007 0.164 0.014 0.2153 0.125 0.265 0.214 0.378 0.075 0.166 0.145 0.2914 0.086 0.091 0.176 0.184 0.056 0.061 0.125 0.1345 0.035 0.036 0.080 0.084 0.018 0.018 0.044 0.045

Table 3. The relative contribution (perinitial kd) of the transfer kdPkp fromdifferent n states at different pressures anddeuterium concentration (k

C"0.5)

kT"1.0 k

T"0.5

Cd"0.2 C

d"0.5 C

d"0.2 C

d"0.5

n 32 bar 780 bar 32 bar 780 bar 32 bar 780 bar 32 bar 780 bar

2 0.003 0.107 0.001 0.061 0.001 0.040 0.001 0.0293 0.005 0.009 0.004 0.010 0.002 0.003 0.003 0.0054 0.001 0.002 0.002 0.002 0.001 0.000 0.001 0.0015 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Fig. 9. The calculated conditional probability of the kp arrival at theground state, Q

1s, in H/D mixtures vs. pressure for deuterium

fractions Cd"0.1 (solid), 0.2 (dashed), 0.5 (dotted). All curves are for

kT"0.5, k

C"0.5

In principle a direct observation of the high energycomponent in low lying excited states is possible by meansof high resolution X-ray spectroscopy [17]. By measuringthe Doppler broadening profile of the X-ray linesnPP1S one can determine the fraction and charac-teristic energy of slow and fast components. Also theinfluence of the muon transfer on the kinetic energy distri-bution in the ground state can be investigated. In the caseof H/D mixtures the effect of Doppler broadening for kpatoms must be enhanced in comparison with pure hydro-gen because the low energy component is reduced by thetransfer to deuterium.

The relative contribution of different states n to thetransfer kpPkd is shown in Table 2. Table 4 gives a brief

Fig. 10. The fraction of the ground state population with lowkinetic energy, ¹(8 eV, for kp and kd after the cascade withouttransfer (solid line) and with transfer at C

d"0.2 (dashed line), 0.5

(dotted line)

summary of the calculated X-ray yields and the kp frac-tion arriving at the ground state, Q

1s, at various densities

and deuterium fractions. The reverse transfer kpPkdplays a minor role at n(6, and even at large densitiesits effect on Q

1sdoes not exceed a few percent, compare

Table 3.

3. Discussion

Our cascade calculations demonstrate that the Q1s

frac-tion can be reliably determined by measuring the ratio ofK X-ray yields for muonic hydrogen and deuterium in

170

Table 4. The absolute K-X-ray yields ofmuonic hydrogen and deuterium and thekp ground state arriving fraction Q

1s(per

kp formed) at various deuterium fractionsC

din liquid (780 bar) and gas at 32 atm

(kC"0.5, k

T"0.5)

p (bar) Cd

½kpKa

½kpKb

½kpK505

½kdKa

½kdKb

½kdK505

Q1s

32 0.0 0.652 0.308 0.998 0.0 0.0 0.0 1.0032 0.2 0.438 0.209 0.675 0.203 0.109 0.323 0.8532 0.5 0.216 0.108 0.339 0.409 0.223 0.659 0.6832 0.8 0.070 0.036 0.112 0.547 0.300 0.887 0.5632 1.0 0.0 0.0 0.0 0.616 0.336 0.998 0.49

780 0.0 0.920 0.031 0.952 0.0 0.0 0.0 1.00780 0.2 0.439 0.019 0.459 0.480 0.017 0.498 0.60780 0.5 0.158 0.009 0.168 0.759 0.030 0.790 0.35780 0.8 0.043 0.003 0.046 0.875 0.037 0.914 0.24780 1.0 0.0 0.0 0.0 0.918 0.041 0.961 0.17

H/D-mixtures. In liquid mixture the ground states aremainly populated by the radiative transitions 2PP1S athigh target pressures, the following equation holds truewith good accuracy:

Ka(kp)

Ka (kd)"

CpQ

1sC

d#C

p(1!Q

1s). (3)

Corrections to Eq. (3) come from the transitions to the 1Sstate with Dn'1 and the collisional deexcitationn"2P1S which are both small in comparison with theradiative transition 2PP1S and were found to be lessthan 5%.

In general all radiative transitions must be taken intoaccount and the following relation can be used:

+iK

i(kp)

+iK

i(kd)

"

CpQ

1sC

d#C

p(1!Q

1s). (4)

One of the problems of the transfer kinetics is whether thesum of the X-ray yields (K

i(kp)#K

i(kd)) is weakly depen-

dent on the isotope composition of the H/D mixture (inother words, whether it is little influenced by the trans-fer).2 Because the bulk of the transfer occurs between thestates with the same n, the most important factor is thecompetition between the radiative and collisional deexci-tation. The radiative rates are proportional to the effectivemass of the muonic atom, M, and the collisional crosssection are proportional to M~2, therefore the ratio of theradiative to collisional rates for the kd is larger than forthe kp by a factor of (Mkd/Mkp)3"1.17. Thus one canexpect that the X-ray yields for the kd are enhanced bya similar factor in comparison with the kp for those initialstates for which collisional processes dominate the deexci-tation. The exact calculations confirm this simple estima-tion: the variation of the (Kb(kp)#Kb(kd)) for 04C

d41

is about 25% in liquid and 10% in gas at 32 bar.The relative yields of the Ka and Kb lines at high

densities depend on the deuterium fraction due to thetransfer from n"3 (see Tables 2 and 4). Experimentaldata on this ratio would be very desirable for a moredetailed test of the transfer kinetics in the atomic cascade.As the density decreases, the radiative deexcitation stageof the cascade starts earlier. This can be seen in theKa/K505!-

ratio which diminishes by about one half when

2The constraint Ki(kp)#K

i(kd)"const would be very useful for

separating the individual lines in the experiment [8]

liquid mixtures are replaced by gaseous (STP) ones. Inhigh density gaseous mixtures (/&0.1) the Kb yields areabout one third of the total while the higher lines are stillweak, and this case seems to be very attractive for detailedexperimental studies of the Q

1sproblem (along with liquid

mixtures).In comparison with the earlier calculations [14, 16,

21] the assumption about constant kinetic energy of themuon atom was not used in the present calculations. Onthe contrary, our calculations show that there are differentenergy components in the kinetic energy distribution, anddue to the energy dependence of the transfer rates it is thelow energy component of kp that mainly contributes tothe transfer to deuterium, while the transfer from the highenergy component is significantly reduced.

Our results for the population of the ground state kpcalculated with the transfer rates from [31] are noticeablysmaller than is indicated by the experimental data [8].However, a fair agreement between theory and experi-ment can be achieved provided the transfer rates arereduced by a factor of about 2.

4 Conclusion

Our cascade calculations taking the kinetic energy distri-bution into account reveal that the interplay between theevolution of the energy distribution and the energy de-pendence of the transfer rates plays an important role inthe kinetics of muon transfer in excited states. Since thetransfer rates are peaked at low energy (¹&1 eV) the lowenergy component of kp gives the main contribution tothe reaction kp#dPkd#p, while the high-energeticatoms (¹'10 eV) are influenced by the transfer withrelatively smaller probability. Thus acceleration mecha-nisms, like Coulomb deexcitation result in an increase ofthe conditional probability, Q

1s, for the initially formed

muonic hydrogen atom to reach the ground state in H/D-mixtures.

Our calculations also show that the Q1s

fraction can bereliably determined from the ratio of Ka X-ray yields frommuonic hydrogen and deuterium. The comparison ofthe cascade calculations with the data from [7, 8] showsthat the transfer rates calculated in [31] are a factor ofabout two too big. We also evaluated the effect if wewould use the transfer rates from [16]. Also these transferrates would have to be reduced by a factor of 3 to give a

171

reasonable agreement between the calculations and thedata from [7, 8] on Q

1s. Contrary to the previous treat-

ments of the Q1s

problem [14, 15, 16, 21, 22] we did notconsider the kinetic energy as a fitting parameter, but useda kinetic energy distribution which appears to be consis-tent with all other experiments using the lightest muonicand pionic atoms.

The present model of the atomic cascade kinetics inmixtures of hydrogen isotopes can be used for detailed testof the transfer rates when the final results of the X-rayyields measurements in H/D mixtures at various pressuresand isotope fractions are available. After fine tuning withthe H/D data the cascade model considered can be usedfor reliable calculations of the Q

1sfraction for D/T-mix-

tures which are much more difficult to study experi-mentally with the X-ray method.

The development of the cascade code was supported in part (V.M.)by the National Science Foundation under the NSF Grant MJR000.We acknowledge L.I. Ponomarev, V.I. Savichev and the group ofProf. W.H. Breunlich at the Austrian Academy of Science, especiallyB. Lauss and P. Ackerbauer for useful discussions.

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