Cascade of muonic helium atoms

Volume 191, number 1,2

PHYSICSLETTERSB

4 June 1987

CASCADE OF M U O N I C H E L I U M ATOMS G. REIFENROTHER, E. KLEMPT and R. LANDUA Institut~r Physik, Johannes Gutenberg-Universit?it, Staudinger Weg 7, Postfach 3980,.D-6500 Mainz, Fed. Rep. Germany Received 12 December 1986;revised manuscript received 13 March 1987

The cascadeof muonichelium and its pressure dependencehas been calculatedover the wholepressure range from 1X 10-z to 1X 103arm. The calculation does not use any free parameter. The results showgoodagreementwith experimental data.

I. Introduction The atomic cascade of muonic He ions has recently attracted renewed interest because of conflicting results on the metastability of its 2s state in a gaseous He environment. A long lifetime of the 2s state is decisive for measurements of the 2s-2p level splittings [ 1 ] and for the possibility to search for parity violation in 2s-~ ls one-photon radiative transitions [2]. The natural lifetime of the 2s level of muonic He ions is given by the free-muon decay rate, 2~= 4.55 X 105 s - ' , and by the two-photon and one-photon de-excitation rates to the l s ground state, 22r = 1.18X 105 s -1 and 2MI =0.5 S-l, respectively. In an experiment at the SC at CERN negative muons were stopped in He at pressures varying from 7 atm to 50 atm. It was found that (3.4 i 0.7)% of stopped ~t- populate the 2s level, and a lifetime of (1.8 + 0.4) txs at 7 atm and of (1.43_+0.15) ~ts at 50 atm was reported [ 3,4]. These results would indicate that the 2s level is not noticeably quenched in collisions with neighbouring He atoms. More recent experiments at SIN confirmed the formation of the 2s state; but it was found that the 2s state is quenched already at pressures ranging from 100 Torr to 600 Torr; at 6 arm no two-photon deexcitation was observed [ 5 ]. In another experiment at 40 arm an upper limit of the 2s level lifetime of 46 ns was derived [ 6 ]. A discussion of collisional processes between muonic helium ions in the 2s state and helium atoms shows that the long lifetime of this level as observed in refs. [ 3,4] is not very likely [ 7]. In this letter we confirm

this conclusion, however special emphasis is devoted to the complete atomic cascade rather than to the 2s metastability. The search for 2s metastability of the muonic He ion produced a large data set on intensities of X-ray transitions over a wide range of pressures. A model calculating X-ray intensities of exotic He ions has been developed previously by two of the present authors [ 8 ]. In this letter we report on refinements of this model and present a comparison with experimental data taken at densities from 50 Torr He gas up to liquid He. 2. Cascade calculation 2.1. Cascade. Muonic He ions are formed by atomic capture of free negative muons via Auger emission of one of the He electrons. It is assumed that the muons are captured into orbits with principal quantum numbers nc = (m~,/m;)1/2

~

14

m~, me: reduced masses of the muonic and electronic He systems) for which there is optimal overlap of the wave functions of electron and muon. The second electron is ejected via internal Auger effect and a (~t-a) ÷ ion is formed. Its neutralization in pure He is not possible because of the high ionization energy of He atoms. The subsequent cascade of the (~t-tt) + ion is determined by the subtle balance of pressure dependent and independent processes: radiative transition, muon decay and nuclear cap-

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V o l u m e 191, n u m b e r 1,2

PHYSICS LETTERS B

ture rates are known and pressure independent. External Auger transitions lead to ionization of a He atom and de-excitation of the (ix-u) ÷ ion. The two ions recoil giving the (ix-u) + ion a kinetic energy of 10 eV. After a few further collisions the muonic He ion is again in thermal equilibrium. Stark mixing transitions between states with the same principal quantum number but different angular momenta occur due to the intense electric fields experienced in collisions between the ( IX-u) ÷ ion and a He atom [9]. Auger and Stark mixing rates depend, of course, on the helium density. In three-body collisions formation of molecular ions is possible because of a minimum in the (ix-u) + - H e potential. The cascade model presented here consists in a Monte Carlo simulation of (Ix-m) ÷ collisions in a He target. The simulation yields the coUisional frequency, the collisional electric fields and electron densities which are felt by the ion on collisions with He atoms. From these quantities Stark mixing and Auger transition rates per collision from all atomic levels are calculated.

4 J u n e 1987

E ~3

-

! fit

•

-

2.2. Collision processes. The collisional processes between muonic He ions and He atoms were investigatednumerically. Thirty He atoms were confined in a confinement sphere of radius rile defining the average He density. A second sphere of radius rion=0.6 rue confined the ion in order to ensure a constant He density arounfthe ion. The ion starts its motion at the center, the He atoms are randomly distributed. Their initial energies follows a Boltzmann distribution at room temperature. The classical trajectories of all 31 particles defined by 93 second order differential equations .were calculated using the known He-He-potential [ 10], and the H+-He-potential [ 11 ] to approximate the ( g - a ) +-He-potential. The CERN library program DRKSTP was used for the integration; step sizes between 0.25 fs and 1 fs were chosen, depending on the particles distances. The sum of kinetic and potential energies of all particles was constant to better than 1 0 - 3 , indicating the high precision of the Monte Carlo simulations. The calculations were done at He densities corresponding to 175 arm, 350 atm, 700 atm and 1000 arm of He pressure. Fig. 1. shows the electric field strengths seen by the ion for a typical simulation of the collision processes 16

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)

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i'~~

~

Fig. 1. Electric field strength seen b y a (#-c~) + ion i n a typical collision.

in liquid helium. A three-body collision occurs after two two-body collisions, leading to molecular ion formation. A second three-body collision breaks up the molecular ion, etc. From the number of collisions per time interval the collisional frequency was determined to be Scoll = (3.54_+0.26) . p × 1 0 1 ° s - l a t m -1 . Molecular ion formation occurred with a frequency MMIF-----(1.01 + 0.09) .p2 X 1 0 7 S

- 1a t m - 2

.

Experimental data [ 5 ] indicate a molecular ion formation frequency of MEXP MIF = (4.26 + -- 0.58)-p2 ×

107S-

Iatm-2

This high molecular ion formation rate would lead within the present model to an extreme short lifetime of the 2S metastable state. In a molecular ion the classical collision frequency is

SMIv = (3.35 + 0.9) X 1013s - 1

•

PHYSICSLETTERSB

Volume 191, number 1,2

The errors are of statistical nature. Individual collisions differ by the m a x i m u m electric field strengths while the collision time is constant. The distribution of maximum electric fields depends on the He density. At low pressures the deep minimum of the ( W a ) + - H e potential completely dominates the collision and the collisions are mostly central. At higher pressures long-range three-body collisions lead to a considerable fraction of peripheral collisions. Therefore the distribution of the maximum electric field seen in collisions depend on the pressure, and Stark mixing rates have to be calculated with their density dependence. The electron density experienced by the (~t-a) + ion as a function of ion-He distance was calculated by Zurbov and Bolotin [ 12 ]. For central collisions the integrated electron density per collision is Rcou = (5.29 + 0.28)pofs, where po = (n a 3) - t is the electron density at the position of a hydrogen nucleus. For peripheral collisions the integrated electron density falls off rapidly. In a molecular ion the integrated electron density was found to be

RMIV = ( 5.61 + 0.3)pofS, consistent with the density in free collisions.

2.3. Stark mixing rates. The interaction of a external static electric field E with an induced electric dipole m o m e n t / t leads to a transition matrix element for Stark mixing of Ml.l+ 1 ~ - ~ / ( n 2 _/z)(12 _ m 2)

=~aenlEI

4~i--f_~

,

(1)

where a is the Bohr orbit of a muon. During a collision the electric fields change adiabatically with respect to the precession of the electric dipole moment, hence the magnetic quantum number is conserved and only AI= _+ 1 transitions are allowed. The sign changes of the electric field at the beginning and the end of a collision are very fast, leading to m - - * - m transitions and no change of the Stark mixing rates. In between two collisions, the electric dipole moment precesses only slowly and

4 June 1987

does not follow the directions of the residual electric fields. Hence the magnetic quantum numbers have to be redefined for each collision. Because of the variation of the electric field during a collision it is necessary to solve the time-dependent SchriSdinger equation ih 8 T ( t ) / S t = H ( t ) T ( t ) ,

(2)

with the hamiltonian H(t)=Ho+Hs(t). The operator Ho describes the undisturbed system and the operator Hs the disturbance

H~(t) = / t . E ( t ) .

(3)

The wave function ~(t) can be written in terms of hydrogen eigenfunctions ~K- The solution of (1) then leads to a system of first order differential equations describing the variation in time of the level population Bj

ihi~j( t ) = Bjl ( t)mj,jexp ( - iogj,jt ) + Bj2(t)Ms2jexp ( - icoj2jt),

(4)

j~ is the level with AI= + 1,j2 the level with Al= - 1, Msd is defined in (1) and ¢ojij=(Eji-Ej)/h is the energy difference of the two states Ji and j. During the collision time several transitions may occur successively allowing AI= +_2, + 3, ... transitions. E.g., an electric field of IEh = 15V/~ needs 2 fs to mix the levels In = 14, li = 0) and In = 14, lf= 1 ) completely while the typical collision time is 40 fs. Equation system (4) was solved for all levels Inilimi)(ni=2 .... , no; /i=0 .... , n i - 1 , m i = - l i ..... +li), giving the Stark transition probabilities F st (n~limi--*nilfmi) for a transition from a given level Inilimi) to a level ] ni/rmi) (lf=0, ..., l l - 1, li+ 1, ..., ni-1). All Stark transition probabilities were calculated for five types of collisional processes, differing by their electric maximum field strength. The final Stark transition probabilities were then determined as the weighted average at a given pressure.

2.4. External Auger effect. In a collision between a (gt-~t) + ion and a He atom the electronic ls level of the (gt-a) + is partially occupied with a He electron. This electron can be ejected leading to deexcitation of the (gt-a) + ion. 17

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PHYSICSLETTERSB

(~t- a ) ~ +He-+ (~t- ct)~z, + H e + + e - . The external Auger rates were calculated by scaling the interval Auger rates [ 13 ] p~ -

3 2 Z C m e e 4 ( ~ ) ' y 2 max(li,10 3h 3 l+y2 2li+I

4 June 1987

the mean molecular ion formation time. Collision rate and electron densities saturate at the values experienced in a molecular ion. The metastable 2slevel requires consideration of two additional effects: ( a ) Two-photon transition (Ix- a ) L - + (~t- ~t) ~ +2~, (Fp = 1.18)< 10SS -~) .

)< exp (4y arctan y - n y ) ~o nr#S2 sinh n y ~'at~'nili] ' where mr is the electron mass, Z~ the effective nuclear charge seen from the Auger electron,

(,8) Nuclear capture (~t- ct)~--, 3 n + p + +v~ ( F c = 4 5 s -1) .

ZeO~ Y = [ ( T/meC2)2 + 2 T / m e c 2] l/2 ,

T the kinetic energy of the electron and ,a~.nil , r # i the radial matrix element r between the states [n J i ) and Inf/r), with the electron density p taken from the Monte Carlo simulations. Note that the Auger transition probability has a high probability for small kinetic energies of the ejected electron. 2.5. The cascade. The cascade calculations started with the principal quantum number nc = (m'~/m'e) 1/2 = 14.

The initial distribution of the/-quantum number was chosen to be a statistical one ( P o c ( 2 l + l ) e '~z, a t [0.0,0.2]) [ 14], Since the Stark effect mixes the highly excited levels very fast, the initial angular momentum distribution has no essential influence on the cascade results. The cascade is then calculated by following a sequence of collisions. During the flight through vacuum the levels are depopulated by radiative transitions and muon decay. The radiative transition probabilities are proportional to A E 3 ( A E = E i - E f : energy differences between initial and final state) and prefer therefore large An transitions (An = ni-- nf). After the mean collision time, a collision takes place, causing Auger transitions and Stark mixing. Free flight through vacuum and collisions are repeated until the residual population of all levels with the given principal quantum number is below 10-3. The formation of molecular ions is introduced by an exponential increase of the collision rate and the electron density with a time constant given by 18

3. Results a n d discussion

The cascade calculations have been performed at densities correspofiding to pressures from 1 )< 10 -2 atm to 1000 atm. Selected results, intensities of radiative transitions, the population of the metastable 2s level, and its lifetime are shown in figs. 2a-2d and compared to experimental data. The errors of the cascade results were estimated by introducing + 10% changes of ion He collision rate, of the rate of molecular ion formation, or of the electron density. The changes had an effect of only less than __+1% in the cascade results. In order to demonstrate the stability of the results, cascade calculations were made with two extreme (pressure independent) Stark mixing rates and with the molecular ion formation switched off. The results are shown in figs. 2a-2c as dotted verge curves. These are, of course, not the true errors but extreme values. A special case is the 2s lifetime (fig. 2d), where the agreement between cascade calculations with or without MIF and the data is only fair. The very high molecular ion formation rate suggested by the experimental results [ 5 ] would lead to a even shorter lifetime of the 2S level in clear contradiction to the measured lifetimes, This discrepancy has, however, only a negligible effect on all short-lived levels. In any case, the 2s lifetime decreases even without M I F very fast with increasing pressure. 4. Conclusion

The cascade of muonic He and its pressure dependence has been calculated without use of any free parameter. The calculated X-ray intensities agree

Volume 191, number 1,2

PHYSICS LETTERS B

0.8

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4 June 1987

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Fig. 2. X-ray intensities, population and lifetime of the 2s level. (o, ~ =ref. [ 5 ] , / x =ref. [6], [] =ref. [15], at =ref. [3], • =ref. [4]. (a) KJKtot and K#/Ktot. (b) LdLtot and L#/Ltot. (e) Population of the 2s level. (d) Lifetime of the 2s level

reasonably well with measured intensities over the whole density range from 50 Torr He gas up to liquid helium. The lifetime of thee metastable 2s level decrreases rapidly with increasing pressure. At pressures between 7 atm and 50 arm, the lifetime is less then 1/100 of the vacuum lifetime. This is striking

in contrast to the results of the CERN experiment [ 1,3,4 ] but in agreement with recent SIN data [ 5,6 ].

References [ 1 ] G. Carboni et al., Nuel. Phys. A 278 (1977) 381.

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Volume 191, number 1,2

PHYSICS LETTERS B

[2] J. Bernabeu, T.E.O. Ericson and C. Jadskog, Phys. Lett. B 50 (1974) 467; G. Feinberg and M.Y. Chert, Phys. Rev. D 10 (1974) 190. [3] A. Placci et al., Nuovo Cimento A 1 (1971) 445. [4] A. Bertin et al., Nuovo Cimento B 26 (1975) 433. [5] H.P. von Arb et al., Phys. Lett. B 136 (1984) 232; F.B. Dittus, Thesis (1985 ), unpublished. [6] M. Eckhause et al., Phys. Rev. A 33 0986) 1743. [7] J.S. Cohen, Phys. Rev. A 25 (1982) 1791; J.E. Russell, Phys. Rev. A 34 (1986) 3865. [ 8 ] R. Landua and E. Klempt, Phys. Rev. Lett. 48 (I 982 ) 1722. [9] R.O. Mueller et al., Phys. Rev. A 11 (1975) i 175; G. Carboni and O. Pitzurra, Nuovo Cimento B 25 (1975) 367; J.S. Cohen and J.N. Bardsley, Phys. Rev. A 23 (1981) 46.

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[ 10] L.W. Bruch and I.J. McGee, J. Chem. Phys. 46 (1967) 2959; 52 (1970) 5884. [ i 1 ] H.H. Michels, J. Chem. Phys. 44 (1966) 3834. [12] V.B. Zurbov and A.B. Bolotin, Lit. Fiz. Sbornik 15 (1975) 235. [13] G.R. Burbidge and A,H. de Borde, Phys. Rev. 89 (1983) 189. [ 14 ] N.A. Cherepkov and L.V. Chernysheva, Soy. J. Nucl. Phys. 32 (1980) 366; J.S. COhen et al., Phys. Rev. A 27 (1983) 1821. [ 15] G. Backenstoss et al., Nucl. Phys. A 232 (1974) 519.