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Direct radiative and Auger muon transfer from hydrogen to helium
PhysicsLettersA 169 (1992) 181—185 North-Holland
PHYSICS LETTERS A
Direct radiative and Auger muon transfer from hydrogen to helium W. Czapliñski and A.I. Mikhailov’ Institute ofPhysics and Nuclear Techniques, Academy ofMining and Metallurgy, Al. Mickiewicza 30, 30-059 Krakow, Poland Received 9 April 1992; revised manuscript received 29 June 1992; accepted for publication 16 July 1992 Communicated by B. Fricke
Direct muon transfer from ground state muonic hydrogen to helium accompanied by y-ray or Auger emission is considered. The transfer rates are calculated in the semi-classical approximation for different isotope combinations. At room temperature the 6s~’for Auger emission. rates range between (9.0—l0.9)x 106 s_I for the radiative process and between (2.2—3.4)x l0
1. Introduction
~
(3a) (3b)
Muon transfer from hydrogen to helium is an important process in muon catalyzed fusion (j.tCF) as it poisons the recycling of the muon in ~tCFtargets. The transfer may proceed in two distinct ways: direct and with molecular exchange [1,2]. Direct exchange was considered by Matveenko and Ponomarev [1] who calculated the rates of the muon transfer between the ground states of the initial and the final muonic atoms, (api) + He2 + —6 (He~t)+ + a,
(1)
where a = p, d and He = 3He, 4He (below “a” denotes any hydrogen isotope). In the temperature range of interest for j.tCF (below 0.1 eV) the calculated rates are ‘~‘dir~106 s_I and do not depend on the temperature. The molecular muon transfer [2] is a two-stage process. First, a molecular ion (Hei.ta ) 2 + is formed: a~.t+He—’[(He~ia)2~e]~+e,
Permanent address: St. Petersburg Nuclear Physics Institute, Gatchina 188350, Russian Federation.
(4) (a~.t),,+He—~[(He~i)j~e]+a+e,
(5)
(a~),,+He~÷(HeLt)~+a+e.
(6)
(2)
with a rate ofthe order of 108 s_I [2]. Subsequently, the [(He~ta ) 2+~]+ system decays: I
leaving the muon bound to the helium nucleus. The radiative channel (3a) dominates while the Auger contribution (3b) is estimated to be less than 15% [2]. Since decay (3) is a prompt process ~dec 10” s_I [3]), the molecular muon transfer (2), (3) is expected to be two orders of magnitude faster than the direct exchange (1). However, the existing theoretical estimates of process (1) do not include the possibility of associated y-ray or Auger electron emission. If such processes are significant they should be taken into account in the experimental estimates ofthe molecular muon transfer rates, A~[4]. In this paper we present the first calculations for the processes
2. Direct radiative muon transfer Since the probability of y-emission is suppressed by the fine structure constant a = e2/hc= i-b, one
0375-9601 1921$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
181
Volume 169, number 3
PHYSICS LET’FERS A
may be led to the conclusion that the radiative process (4) is slower than the direct transfer (1). However, in process (1) the energy difference of about 7 keY, between the initial (aLt+He) and final (Hej.t+a) states of the (Hej.ta) system (see fig. 1), is used to accelerate the reaction products, whereas in (4) it is absorbed by the y-quantum, and the kinetic energies of the heavy particles remain practically unchanged in the “instant” of y-emission. We determine the cross section of reaction (4) using the semi-classical treatment [51.The cross section can then be written as
$
pm~x
a~=2x
W1(p)pdp,
(7)
0
wherep is the classical impact parameter and
W~(p) 2
(8)
wy—. VR
=
Ro
w~is the probability of the y-radiation as a function of a—He separation, R, and dialthevelocity
VR
VRV~./l—UI/—p/R,
is the relative ra(9)
where and v are, respectively, the kinetic energy and
21 September 1992
U, (R) = E2,,~(R)+ 2/R + 0.5 +
—~-—
2K [K, (R) + ICK2 (R) + ic 3 (R) _fl2]
2M
(10)
where M’ =Mj-’ +M~, M1 and M2 are the hydrogen and the helium mass, respectively; ic= (M2 M1 )I (M~+M2), 6=M2/ (M1 + M2). The above notation is in units m0 = = e= 1, where m~’= m ~‘ + (M, + M2)’ and m,, is the muon rest mass. For the El-transition, the radiation probability w7 as a function of R has the following form [7], 3w3(R)D2(R), (11) w1(R)=4a where a(R)=E 2p~(R)—Eisa(R) (12) —
and R~d D(R)=
$
R~d 3r. (13) ~200(r,R) d The dipole moment, d, of the (Hej.ta) system is defined relative to its center of mass [2]: d= ar+ bR, WTsq(r, R)
where a =
—
—~—
(1 + 2m,JM~
0~), b= (mHe 3m,, m,~) /2M,01, M,0, = mHe + m,, + rn,1 and r is the muon coordinate relative to the middle of the internuclear axis. The eigenfunctions W2pa( r, R) and ~ r, R) and the eigenvalues E2pa(R) and Eisg(R) describe the muon states in the two-center Coulomb problem for a fixed R, and correspond at R—~co to the (ai.t) ~ + He and (Heij) j’ + p system, respectively [8]. Using the above equations and substituting x= e/kBT (kB is the Boltzmann constant) one obtains —
the relative velocity at R ~ U, is the interaction potential in the initial channel. The lower limit of the integration in (8), R0, is determined by the condition: v,~ = 0. According to ref. [6] one has -~
_____________
E
Zpcy
-1.0
_
-15 —2.0 00
(He~) 2.0
4.0
~
+ 60
8.0
10.0
R Fig. 1. Molecular terms for the He~spsystem as a function of the internuclear distance R.
182
0
—
—
_
0
w~(R)dR 2/R2’ (14) R ~/i U1/xk~T—p where the rate is normalized to the liquid hydrogen density N 0 and the average is taken over the Maxwell distribution. Upon performing the integral over x and —
Volume 169, number 3
PHYSICS LETTERS A
21 September 1992
Table 1 Ratesof process (4) at T= 300 K normalized to the liquid hydrogen density and expressed in units of 106 s~’.Asterisks denote the rates 8K”2 s~. obtained for process (1) in ref. [1J. The numbers in brackets are the coefficients A defined in eq. (15) expressed in 10 3He2+ p~s dit
10.9 9.6 9.0
t~t
4He2+ 6.3 l.3 —
(1.89) (1.66) (1.56)
p in (14) and neglecting the terms proportional to exp( I U 1 I IkBT) (which is justified for T~1000 K where U, I >. kBT) one obtains
10.7 9.6 9.1
5.5 1.0
(1.86) (1.66) (1.58)
—
—
A7=8~/~NoJ
w1(R)
12.0
_.
1
I
2
~ 7
~:: 0
‘-4
A (15) Table 1 presents the calculated rates A,,( T= 300 K) together with the corresponding coefficients A. Let us note that the isotope dependence of the rates in table 1 reflects the same features as the results for process (1) obtained in ref. [1]. However, in our results the differences the less ratespronounced. for the various hydrogen isotopesbetween are much The energy spectrum of the emitted 7-rays is given by =
dw
~
3
4.0 2.0 _____________________
0.0 r,,, 6.0
8.5
7.0
7.5
8.0
w (keV) 3He from the muonic atoms pit, djs (4) andfor t~t(curves 1, 2 and 3, Fig. 2. y-energy spectrum in process the muon transferto respectively) obtained for T= 300 K.
emission processes (5), (6). Because muon transfer N 0 w~(R
.~/i1i~i ~o) R2(a) kBT dw I dl~’~
(16) where R=R(co) is determined by eq. (12). 3He2~, Figure 2 shows the obtained 7-energy spectraspectra for aj.t +for 4He as an example. The corresponding are nearly identical and are not shown in the figure. Finally, let us note that the rates and spectra for helium nuclei, atoms and He~ions are practically identical because the dominant contribution to integral (15) comes from 3 ~ R ~ 8 ~< a~(a~is the Bohr radius), where the influence of the electron shell is negligible,
3. Direct muon transfer accompanied by Auger emission A similar approach can be applied to the Auger
takes place at R ~ ae, the interaction between the hehum electrons and the (He~ia)system can be described in the dipole approximation. The probability of the Auger process is then d r~ c~~’ dPe W~, 0(r,R)Ø~(r~) —i--dwA(R)=2 C
$
I
2
XØi,(re)~2pa(r,R) ó(Eg+Ie_~(R)) (2)~’ I (17) where r~is the electron coordinate relative to the 8q are center ofmomentum mass of the and (Hep.a) system, q and electron kinetic energy, 4 is the ionization energy for the ls helium electron, Ø 15(r~)and Ø~(r6) are the wave functions of the bound and the Auger electron, respectively. Using Coulomb wave functions one obtains wA(R)=t~rn~Z~ffQ(R)D2(R),
(18) 183
Volume 169, number 3
PHYSICS LETTERSA
21 September 1992
Table 2 Rates ofthe processes (5) and (6) defined as in table 1. The numbers in brackets are now the coefficients Bof eq. (19). 3He
p~ d~t
3He’~
3.4 3.0 2.7
tLt
(5.94) (5.12) (4.76)
4He
2.7 2.3 2.2
(4.67) (4.03) (3.75)
where
2.6 2.3 2.2
(5.75)
(5.06) (4.75)
(4.53) (3.98) (3.74)
together with the corresponding coefficients B. The electron energy spectrum of the Auger elec—
2
3.3 2.9 2.7
V =
exp[—4varcctg(v)] (l+v2)(l—e~2’~”)eq(R)
trons has the form
‘
~
I R2(Cq),
~
=8~NowA(R)I~~~
dtq
Z~ffrnC 2Cq(R)
dCq (20)
rn~is the electron mass and D (R) is defined by eq. (13). For He~ions one has ~= 1, Zeff=2, Ic= 54.42 eY, and for He atoms ~= 2, Zeff= 1.69 and = 24.58 eV. Calculations analogous to those described above lead to the following expression for the rate of muon transfer accompanied by Auger electron emission,
where R(q) is the internuclear distance at which electron emission takes place. Figures 3a, 3b show the obtained electron energy spectra for processes (5) and (6), respectively, calculated for 3He. The corresponding spectra for 4He practically coincide due to a very small helium-isotope effect.
AA=8~/~NOJ WA(R)~/~iTR2dR
=
B 7=.
4. Conclusions As is seen from tables 1 and 2 the rates of the radiative process (4) and Auger-emission processes (5), (6) are rather surprisingly high. They exceed the rates of the direct muon transfer (1) calculated
(19)
Table 2 presents the calculated rates AA( T= 300 K)
5.0
5.0
a) 0)
4.0
b) 4.0
2
3.0
3.0
2
10
C ‘-4
-d -d
2.0 00 1.0 6.0
6.5
7.b
~ 7.5
e’ectron
00 2oN 1.0 8.0
‘
6.0
energy
6.5
Eq
7.0
7.5
8.0
(keV)
Fig. 3. Energy spectra of the Auger electrons in the processes (5) and (6) ((a) and (b), respectively) for T= 300 K. The curves are labelled as in fig. 2.
184
Volume 169, number 3
PHYSICS LETTERS A
by Matveenko and Ponomarev [1]by a factor of about two for p~i+ He and ofabout nine for dj.t + He. We also present the results for t~.t+ He which were missing in ref. [1]. It should be noted that our results for process (4) show little variation with the hydrogen isotope, whereas in ref. [1] the rates for d~.t and p~.tdiffer by a factor of about six. Let us note here the importance of the muontransfer rates for the description of ~.tCFkinetics [9] and the consideration of possible practical applications of muon-catalyzed fusion [10]. The obtained rates of the processes (4)— (6) coincide with the present accuracy of the experimental determination of the molecular muon exchange rate 2~ [4,11]. With somewhat higher statistics they will become essential in the analysis of j.tCF experiments with hydrogen— helium mixtures. It should be pointed out that our 7-ray energy spectra peak at energies similar to those for decay (3a) [3,12]. Therefore, the muon transfer rates obtained in the 7-ray experiment [12] reflect the sum of the resonant transitions (3) and non-resonant transitions. Let us remark here that although our result for p~.t+4He(10.7 x 1065_I) is relatively close to the experimental muon transfer rate of ref. [12] ((3.2 ±1.3) x 106s’) it does not resolve the discrepancy between this result and the theoretical resonant transfer rate which is an order of magnitude higher.
21 September 1992
Acknowledgement The authors are grateful to Professor Adam Gula for helpful discussions and his interest in this work.
References [1] A.V. Matveenko and L.I. Ponomarev, Zh. Eksp. Teor. Fiz. 63 (1972) 48 [Soy. Phys. JETP 36 (1973) 24]. [21Yu.A. Aristov et al., Yad. Fiz. 33 (1981)1066 [Soy. Nucl. Phys. 33 (1981) 564]. [3] A.V. Kravtsov et al., Phys. Lett. A 83 (1981) 379. [4] A.A. Vorobyov, Muon Catal. Fusion 2 (1988)17; D.V. Balm et al., Report INT 250/PS, Krakow (1991). [5) L.D. Landau and E.M. Lifshitz, Quantum mechanics (Moscow, 1963) [in Russian]. [6] L.I. Ponomarev and T.P. Puzynina, Preprint JINR P4-3405, Dubna (1967). [7] H.A. Bethe and E.E. Salpeter, Quantum mechanics of oneand two-electron atoms (Springer, Berlin, 1957). (8] I.V. Komarov, L.I. Ponomarev and S.Yu. Slavyanov, Spheriodal and Coulomb spheroidal functions (Nauka, Moscow, 1973). [9] A. Gula, Acta Phys. Pol. B 16 (1985) 589. [10] Yu.V. Petrov, Nature 285 (1980) 466; L.A. Alimova et al., Muon Catal. Fusion 3 (1988) 607. [11] D.V. Balm et al., Report INT KrakOw (1992), to be published. [12] H.P. von Arb et al., Muon Catal. Fusion 4 (1989) 61.
185
PHYSICS LETTERS A
Direct radiative and Auger muon transfer from hydrogen to helium W. Czapliñski and A.I. Mikhailov’ Institute ofPhysics and Nuclear Techniques, Academy ofMining and Metallurgy, Al. Mickiewicza 30, 30-059 Krakow, Poland Received 9 April 1992; revised manuscript received 29 June 1992; accepted for publication 16 July 1992 Communicated by B. Fricke
Direct muon transfer from ground state muonic hydrogen to helium accompanied by y-ray or Auger emission is considered. The transfer rates are calculated in the semi-classical approximation for different isotope combinations. At room temperature the 6s~’for Auger emission. rates range between (9.0—l0.9)x 106 s_I for the radiative process and between (2.2—3.4)x l0
1. Introduction
~
(3a) (3b)
Muon transfer from hydrogen to helium is an important process in muon catalyzed fusion (j.tCF) as it poisons the recycling of the muon in ~tCFtargets. The transfer may proceed in two distinct ways: direct and with molecular exchange [1,2]. Direct exchange was considered by Matveenko and Ponomarev [1] who calculated the rates of the muon transfer between the ground states of the initial and the final muonic atoms, (api) + He2 + —6 (He~t)+ + a,
(1)
where a = p, d and He = 3He, 4He (below “a” denotes any hydrogen isotope). In the temperature range of interest for j.tCF (below 0.1 eV) the calculated rates are ‘~‘dir~106 s_I and do not depend on the temperature. The molecular muon transfer [2] is a two-stage process. First, a molecular ion (Hei.ta ) 2 + is formed: a~.t+He—’[(He~ia)2~e]~+e,
Permanent address: St. Petersburg Nuclear Physics Institute, Gatchina 188350, Russian Federation.
(4) (a~.t),,+He—~[(He~i)j~e]+a+e,
(5)
(a~),,+He~÷(HeLt)~+a+e.
(6)
(2)
with a rate ofthe order of 108 s_I [2]. Subsequently, the [(He~ta ) 2+~]+ system decays: I
leaving the muon bound to the helium nucleus. The radiative channel (3a) dominates while the Auger contribution (3b) is estimated to be less than 15% [2]. Since decay (3) is a prompt process ~dec 10” s_I [3]), the molecular muon transfer (2), (3) is expected to be two orders of magnitude faster than the direct exchange (1). However, the existing theoretical estimates of process (1) do not include the possibility of associated y-ray or Auger electron emission. If such processes are significant they should be taken into account in the experimental estimates ofthe molecular muon transfer rates, A~[4]. In this paper we present the first calculations for the processes
2. Direct radiative muon transfer Since the probability of y-emission is suppressed by the fine structure constant a = e2/hc= i-b, one
0375-9601 1921$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
181
Volume 169, number 3
PHYSICS LET’FERS A
may be led to the conclusion that the radiative process (4) is slower than the direct transfer (1). However, in process (1) the energy difference of about 7 keY, between the initial (aLt+He) and final (Hej.t+a) states of the (Hej.ta) system (see fig. 1), is used to accelerate the reaction products, whereas in (4) it is absorbed by the y-quantum, and the kinetic energies of the heavy particles remain practically unchanged in the “instant” of y-emission. We determine the cross section of reaction (4) using the semi-classical treatment [51.The cross section can then be written as
$
pm~x
a~=2x
W1(p)pdp,
(7)
0
wherep is the classical impact parameter and
W~(p) 2
(8)
wy—. VR
=
Ro
w~is the probability of the y-radiation as a function of a—He separation, R, and dialthevelocity
VR
VRV~./l—UI/—p/R,
is the relative ra(9)
where and v are, respectively, the kinetic energy and
21 September 1992
U, (R) = E2,,~(R)+ 2/R + 0.5 +
—~-—
2K [K, (R) + ICK2 (R) + ic 3 (R) _fl2]
2M
(10)
where M’ =Mj-’ +M~, M1 and M2 are the hydrogen and the helium mass, respectively; ic= (M2 M1 )I (M~+M2), 6=M2/ (M1 + M2). The above notation is in units m0 = = e= 1, where m~’= m ~‘ + (M, + M2)’ and m,, is the muon rest mass. For the El-transition, the radiation probability w7 as a function of R has the following form [7], 3w3(R)D2(R), (11) w1(R)=4a where a(R)=E 2p~(R)—Eisa(R) (12) —
and R~d D(R)=
$
R~d 3r. (13) ~200(r,R) d The dipole moment, d, of the (Hej.ta) system is defined relative to its center of mass [2]: d= ar+ bR, WTsq(r, R)
where a =
—
—~—
(1 + 2m,JM~
0~), b= (mHe 3m,, m,~) /2M,01, M,0, = mHe + m,, + rn,1 and r is the muon coordinate relative to the middle of the internuclear axis. The eigenfunctions W2pa( r, R) and ~ r, R) and the eigenvalues E2pa(R) and Eisg(R) describe the muon states in the two-center Coulomb problem for a fixed R, and correspond at R—~co to the (ai.t) ~ + He and (Heij) j’ + p system, respectively [8]. Using the above equations and substituting x= e/kBT (kB is the Boltzmann constant) one obtains —
the relative velocity at R ~ U, is the interaction potential in the initial channel. The lower limit of the integration in (8), R0, is determined by the condition: v,~ = 0. According to ref. [6] one has -~
_____________
E
Zpcy
-1.0
_
-15 —2.0 00
(He~) 2.0
4.0
~
+ 60
8.0
10.0
R Fig. 1. Molecular terms for the He~spsystem as a function of the internuclear distance R.
182
0
—
—
_
0
w~(R)dR 2/R2’ (14) R ~/i U1/xk~T—p where the rate is normalized to the liquid hydrogen density N 0 and the average is taken over the Maxwell distribution. Upon performing the integral over x and —
Volume 169, number 3
PHYSICS LETTERS A
21 September 1992
Table 1 Ratesof process (4) at T= 300 K normalized to the liquid hydrogen density and expressed in units of 106 s~’.Asterisks denote the rates 8K”2 s~. obtained for process (1) in ref. [1J. The numbers in brackets are the coefficients A defined in eq. (15) expressed in 10 3He2+ p~s dit
10.9 9.6 9.0
t~t
4He2+ 6.3 l.3 —
(1.89) (1.66) (1.56)
p in (14) and neglecting the terms proportional to exp( I U 1 I IkBT) (which is justified for T~1000 K where U, I >. kBT) one obtains
10.7 9.6 9.1
5.5 1.0
(1.86) (1.66) (1.58)
—
—
A7=8~/~NoJ
w1(R)
12.0
_.
1
I
2
~ 7
~:: 0
‘-4
A (15) Table 1 presents the calculated rates A,,( T= 300 K) together with the corresponding coefficients A. Let us note that the isotope dependence of the rates in table 1 reflects the same features as the results for process (1) obtained in ref. [1]. However, in our results the differences the less ratespronounced. for the various hydrogen isotopesbetween are much The energy spectrum of the emitted 7-rays is given by =
dw
~
3
4.0 2.0 _____________________
0.0 r,,, 6.0
8.5
7.0
7.5
8.0
w (keV) 3He from the muonic atoms pit, djs (4) andfor t~t(curves 1, 2 and 3, Fig. 2. y-energy spectrum in process the muon transferto respectively) obtained for T= 300 K.
emission processes (5), (6). Because muon transfer N 0 w~(R
.~/i1i~i ~o) R2(a) kBT dw I dl~’~
(16) where R=R(co) is determined by eq. (12). 3He2~, Figure 2 shows the obtained 7-energy spectraspectra for aj.t +for 4He as an example. The corresponding are nearly identical and are not shown in the figure. Finally, let us note that the rates and spectra for helium nuclei, atoms and He~ions are practically identical because the dominant contribution to integral (15) comes from 3 ~ R ~ 8 ~< a~(a~is the Bohr radius), where the influence of the electron shell is negligible,
3. Direct muon transfer accompanied by Auger emission A similar approach can be applied to the Auger
takes place at R ~ ae, the interaction between the hehum electrons and the (He~ia)system can be described in the dipole approximation. The probability of the Auger process is then d r~ c~~’ dPe W~, 0(r,R)Ø~(r~) —i--dwA(R)=2 C
$
I
2
XØi,(re)~2pa(r,R) ó(Eg+Ie_~(R)) (2)~’ I (17) where r~is the electron coordinate relative to the 8q are center ofmomentum mass of the and (Hep.a) system, q and electron kinetic energy, 4 is the ionization energy for the ls helium electron, Ø 15(r~)and Ø~(r6) are the wave functions of the bound and the Auger electron, respectively. Using Coulomb wave functions one obtains wA(R)=t~rn~Z~ffQ(R)D2(R),
(18) 183
Volume 169, number 3
PHYSICS LETTERSA
21 September 1992
Table 2 Rates ofthe processes (5) and (6) defined as in table 1. The numbers in brackets are now the coefficients Bof eq. (19). 3He
p~ d~t
3He’~
3.4 3.0 2.7
tLt
(5.94) (5.12) (4.76)
4He
2.7 2.3 2.2
(4.67) (4.03) (3.75)
where
2.6 2.3 2.2
(5.75)
(5.06) (4.75)
(4.53) (3.98) (3.74)
together with the corresponding coefficients B. The electron energy spectrum of the Auger elec—
2
3.3 2.9 2.7
V =
exp[—4varcctg(v)] (l+v2)(l—e~2’~”)eq(R)
trons has the form
‘
~
I R2(Cq),
~
=8~NowA(R)I~~~
dtq
Z~ffrnC 2Cq(R)
dCq (20)
rn~is the electron mass and D (R) is defined by eq. (13). For He~ions one has ~= 1, Zeff=2, Ic= 54.42 eY, and for He atoms ~= 2, Zeff= 1.69 and = 24.58 eV. Calculations analogous to those described above lead to the following expression for the rate of muon transfer accompanied by Auger electron emission,
where R(q) is the internuclear distance at which electron emission takes place. Figures 3a, 3b show the obtained electron energy spectra for processes (5) and (6), respectively, calculated for 3He. The corresponding spectra for 4He practically coincide due to a very small helium-isotope effect.
AA=8~/~NOJ WA(R)~/~iTR2dR
=
B 7=.
4. Conclusions As is seen from tables 1 and 2 the rates of the radiative process (4) and Auger-emission processes (5), (6) are rather surprisingly high. They exceed the rates of the direct muon transfer (1) calculated
(19)
Table 2 presents the calculated rates AA( T= 300 K)
5.0
5.0
a) 0)
4.0
b) 4.0
2
3.0
3.0
2
10
C ‘-4
-d -d
2.0 00 1.0 6.0
6.5
7.b
~ 7.5
e’ectron
00 2oN 1.0 8.0
‘
6.0
energy
6.5
Eq
7.0
7.5
8.0
(keV)
Fig. 3. Energy spectra of the Auger electrons in the processes (5) and (6) ((a) and (b), respectively) for T= 300 K. The curves are labelled as in fig. 2.
184
Volume 169, number 3
PHYSICS LETTERS A
by Matveenko and Ponomarev [1]by a factor of about two for p~i+ He and ofabout nine for dj.t + He. We also present the results for t~.t+ He which were missing in ref. [1]. It should be noted that our results for process (4) show little variation with the hydrogen isotope, whereas in ref. [1] the rates for d~.t and p~.tdiffer by a factor of about six. Let us note here the importance of the muontransfer rates for the description of ~.tCFkinetics [9] and the consideration of possible practical applications of muon-catalyzed fusion [10]. The obtained rates of the processes (4)— (6) coincide with the present accuracy of the experimental determination of the molecular muon exchange rate 2~ [4,11]. With somewhat higher statistics they will become essential in the analysis of j.tCF experiments with hydrogen— helium mixtures. It should be pointed out that our 7-ray energy spectra peak at energies similar to those for decay (3a) [3,12]. Therefore, the muon transfer rates obtained in the 7-ray experiment [12] reflect the sum of the resonant transitions (3) and non-resonant transitions. Let us remark here that although our result for p~.t+4He(10.7 x 1065_I) is relatively close to the experimental muon transfer rate of ref. [12] ((3.2 ±1.3) x 106s’) it does not resolve the discrepancy between this result and the theoretical resonant transfer rate which is an order of magnitude higher.
21 September 1992
Acknowledgement The authors are grateful to Professor Adam Gula for helpful discussions and his interest in this work.
References [1] A.V. Matveenko and L.I. Ponomarev, Zh. Eksp. Teor. Fiz. 63 (1972) 48 [Soy. Phys. JETP 36 (1973) 24]. [21Yu.A. Aristov et al., Yad. Fiz. 33 (1981)1066 [Soy. Nucl. Phys. 33 (1981) 564]. [3] A.V. Kravtsov et al., Phys. Lett. A 83 (1981) 379. [4] A.A. Vorobyov, Muon Catal. Fusion 2 (1988)17; D.V. Balm et al., Report INT 250/PS, Krakow (1991). [5) L.D. Landau and E.M. Lifshitz, Quantum mechanics (Moscow, 1963) [in Russian]. [6] L.I. Ponomarev and T.P. Puzynina, Preprint JINR P4-3405, Dubna (1967). [7] H.A. Bethe and E.E. Salpeter, Quantum mechanics of oneand two-electron atoms (Springer, Berlin, 1957). (8] I.V. Komarov, L.I. Ponomarev and S.Yu. Slavyanov, Spheriodal and Coulomb spheroidal functions (Nauka, Moscow, 1973). [9] A. Gula, Acta Phys. Pol. B 16 (1985) 589. [10] Yu.V. Petrov, Nature 285 (1980) 466; L.A. Alimova et al., Muon Catal. Fusion 3 (1988) 607. [11] D.V. Balm et al., Report INT KrakOw (1992), to be published. [12] H.P. von Arb et al., Muon Catal. Fusion 4 (1989) 61.
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