On the models of Coulomb capture of negative particles

Nuclear Physics A407 (1983) 297-308 @ North-Holland ~blishing Company

ON THE MODELS OF COULOMB CAPTURE OF NEGATIVE PARTICLES DEZSij

HORV6ITH*

T?U!.MF, 4004 Wesbrook Mall, Vancouver, BC, Canada V6T 2A3 and FARROKH

ENTEZAMI

Department of Physics, University of British Columbia, Vancouver, BC, Canada V6T2A6 Received 14 April 1983 (Revised 30 May 1983) Abstract: Various models of atomic capture of negative mesonic particles are tested against 321 experimental Coulomb-capture ratios, measured on binary systems: gas mixtures, alloys and simple compounds. The comparison has shown that the general agreement between theory and experiment is not satisfactory. We tried to improve on the models by introducing adjustable parameters to be estimated by fitting the experimental data. The predictions of the model proposed by Schneuwly, Pokrovsky and Ponomarev (SPP) are closest to the atomic capture ratios for alloys and compounds while the data measured in gas mixtures are better approximated by the empirical formula of Vasilyev et al. The theoretical formula of Daniel fails in describing the capture in the light elements.

1. Introduction Exotic (muonic and hadronic) atoms can be considered as new nuclear probes used in practically all fields of physical sciences, from particle physics to materials sciences and biophysics. However, the basic problem of exotic atoms, their formation via Coulomb capture of heavy negative particles, has not been solved yet. The experimentalist, working in this field, needs a simple calculation method for estimating the probability WM(&) of formation of an exotic atom Z&- when the Mparticle is stopped in a sample consisting of elements Z1, Z2 , etc. (Zk denotes both the element and its atomic number.) For that purpose, several models have been suggested. A direct test of the models is the comparison of their predictions with the experimental values of the so-called Coulomb capture or atomic capture ratio: A(Zl,&)

=

wM(.&)/

w&2).

(1)

Usually, that comparison is made for a more-or-less arbitrarily chosen group of binary compounds, in most cases a subset of the data on oxides. The aim of the * On leave of absence from Central Research Institute for Physics, H-1525 Budapest, PO Box 49, Hungary. 297

298

D. Horva’th, F. Entezami / Coulomb capture

present work is to compare the predictions of the various models with the 321 available Coulomb capture ratios+ measured in simple binary gas mixtures, alloys, and compounds of composition (Z,), (Z&. First, we summarize the theoretical basis of the different models 2-22), then we make attempts to modify them by introducing adjustable parameters and fitting their predictions to the experimental data. Finally, we compare the predictions of the different models. We do not give an exhaustive bibliography on exotic atoms here, the reader is advised to consult ref. 23) for that. 2. Atomic capture - general considerations Let us summarize the common basis of the various descriptions of the atomic capture: the theoretical statements which seem to be supported by unambiguous experimental evidence or which most of the recently published theoretical works agree upon. (a) The atomic capture process is similar for all negative particles, muons or hadrons 2-577-21).In the following we shall use the general term “meson”. (b) The slowing down and atomic capture of mesons are competing processes in the kinetic energy region EM < 10 keV, where their velocities approach those of the atomic electrons 3-5S9-10).Following the pioneer work of Fermi and Teller 2, the atomic capture probability is often assumed to be proportional to the energy loss of the mesons near the corresponding atomic species. This leads to a correlation between the atomic stopping powers and the corresponding capture rates. The correlation, however, is not a proportionality; this has been predicted theoretically 9*19)and observed experimentally 24). (c) The Auger process (meson capture through electron ejection) is responsible for all the details of the meson capture because for almost every meson transition from the continuum to an atomic bound state the Auger cross sections dominate the radiative ones 3-7S9710*12-21). (d) In dense systems the mesons do not slow down to thermal velocities as they are captured at much higher energies, EM > 10 eV [refs. 376*9*10716719*21)]. (e) In the energy region 10 eV< EM < 10 keV, where most of the mesons are captured, the meson wave function has many oscillations over the dimensions of This justifies a quasi-classical treatment of the capture the atom. 2-5,7.8,10,12-18 process ). (f) In a mixture of elements of clZi + c2Z2 type (where Zi, 22 are the atomic numbers, and cl, c2 the atomic concentrations) the atomic capture rates are proportional to the concentrations. This virtually evident statement has been thoroughly an d ex p erimentally ‘). A concentration dependence tested both theoretically 1071sS19) is predicted for the atomic capture in the lighter elements 19) but was not found in the mixtures of 3He with other gases 24). ’ In ref. ‘) we have listed all the experimental corresponding references.

points (and some theoretical predictions) with the

D. Horva’th, F. Entezami / Coulomb capture

299

(g) A chemical compound is usually treated, in first approximation, as a mixture of elements. However, molecular effects have been observed in mesic X-rays and atomic capture rates [see e.g. reviews 4S5*16)]. Therefore, at least for compounds of light elements, the influence of the chemical bond has to be considered. (h) In the case of exotic hydrogen atoms a transfer of the meson from the hydrogen nucleus (proton, deuteron, or triton) to the heavy atom was observed 1see reviews 4*5*‘6)].The transfer is assumed to proceed via collisions of the small, neutral, and fast exotic hydrogen atom with the heavy nucleus 25). According to the experimental observations 24) and the present theoretical picture of the transfer process, there is no particle transfer from atom ZM- if 2 > 1.

3. Experimental

information

The Coulomb-capture ratio seems to be the best observable quantity for testing the models of the atomic capture process. In ref. ‘) we made an attempt to collect all available atomic capture ratios measured in binary mixtures of elements and simple binary compounds. We rejected, however: (i) the hydrogen-containing compounds and mixtures where the transfer effect obscures the atomic capture information; (ii) the compounds with complicated chemical bond structure as e.g. C6C16 or the peroxides; (iii) the data which are related but not atomic capture ratios in the strict sense, e.g. line intensity ratios. Unfortunately, there are systematic differences among the data measured at different places and times, using different methods, for the same compounds ‘l). Due to these and other inherent systematic errors, it is very difficult to perform a proper statistical test of the various theories. Following the generally accepted method, we shall compare the goodness of agreement between theory and experiment using the quantities

xr’=x?;h -PI,

(2b)

where Aexp* gexpis the measured value, At,,_ is a theoretical prediction and p is the number of fitted parameters. Quantities (2a) and (2b) are called-somewhat inaccurately - total chi-square and reduced chi-square 8,11S17). The “best” parameters of the various models were estimated by minimizing (2). The computations have been performed using the MINUIT program 26) on the VAX-l l/780 computer at TRIUMF. In table 1 we list our efforts to improve the models in order to obtain a better goodness of fit (x2). In ref. ‘) the experimental data are presented in comparison

TABLE 1 Comparison of atomic capture models

Function

Degrees of freedom

model: original Z-law model: Z-law A(Zt, Z2) = 4.Wz2)b

321 319

model: Vasilyev et al. W(z)=z”3-1 model: Vasilyev et al. modified version (eq. (25))

321

model: Daniel W(Z)=Z”31n(aZ+b) model: Daniel W(Z)=Z1’31n(aZ+6) model: Daniel W(Z)=Z”31n(aZ+b)

321

model: SPP rigid boundary

320

320

319

Parameters

-

2

X”

2

Xr

42 152.6 12 877.8

131.32 40.37

21905.3

68.24

18 145.8

56.88

192 280.7

599.01

20 332.9

63.54

20 202.1

63.33

E,=92.13*0.27 x=2

8342.8

26.07

E,=92.60*0.44

8773.1

27.42

6267.8

19.65

6201.4

19.50

5386.3

16.88

4972.5

15.64

4578.6

14.44

4540.3

14.37

a = 0.69kO.0004 6 = 0.86k 0.0008

a = 1.00*0.000 b =0.28*0.004

320 319

a = 0.57 b = 0.00

original values a = 1.28*0.0015 b = 0.00 a =0.83*0.009 b =0.69*0.002

01

model: SPP rigid boundary

x=2

02

model: SPP smooth boundary gaussian distribution

319

E,=13.11~0.02 E,=75.68&0.44

x=2

01

model: SPP smooth boundary gaussian distribution

318

Z,,= 18 x=2

01

model: SPP smooth boundary gaussian distribution

E. = 10.59 f 0.44 El = 79.98 f 0.51 E2 = 73.00 f 0.74

319

E. = OO.OOk 0.01 E, = 88.76 i 0.15

x=2

02

model: SPP smooth boundary gaussian distribution

318

02

model: SPP smooth boundary gaussian distribution

317

02

model: SPP smooth boundary gaussian distribution 02

316

E,, = 15 original El = 70 parameters E2 = 100

Z,=18 x=2 E,=00.00*0.19 El = 86.12*0.31 E2= 119.06*0.60 Z0 = 15 (fitted) x=2 E. = OO.OOkO.02 El = 85.74kO.25 E2= 121.09+0.56 Z0 = 15 (fitted) x = 1.69kO.01

301

D. Horva’th, F. Enterami / Coulomb capture TABLE

1 (cont.)

Degrees of freedom

Function

model: SPP smooth boundary Fermi distribution

Parameters

z X”

2

XC

319

E,=71.37*0.23 E, = 18.72ztO.13 x=2

6238.1

19.56

319

E. = 48.21 f 0.04 El = 38.79 f 0.02 x=2

5262.2

16.50

317

E,=63.65*0.29 E,=33.98*0.16 EZ= 19.15ztO.22 2, = 17 (fitted) x=2 E. = 64.61*0.39 El = 33.29k0.12 E2 = 18.58 f 0.42 Z,, = 17 (fitted) x = 1.84ztO.04

4738.0

14.95

4128.0

14.96

Wl

model: SPP smooth boundary Fermi distribution 02

model: SPP smooth boundary Fermi distribution w2

316

model: SPP smooth boundary Fermi distribution w2

with the predictions of five different models for various groups of chemical systems: mixtures, oxides, sulphides, halides, alloys and nitrides (BN). Table 2 summarizes the x2 values for the various groups and models.

4. Discussion

of the atomic capture models

As mentioned earlier, our aim is to aid the experimentalists to choose among calculation methods when estimating atomic capture probabilities. To start with, we try to reproduce the gross atomic number dependence (Z-dependence) of the atomic capture probabilities, possibly including the quasi-periodic oscillations first observed by Zinov et al. 27) on oxides (see fig. 1); the consideration of molecular and solid-state effects comes next. In their classic paper 2, Fermi and Teller estimated the capture probabilities of mesons in atoms to be roughly proportional to their atomic numbers (Z-law): A(ZI,

22)

=

Wz2.

Relation (3) has been deduced using the assumption that the capture rates are proportional to the stopping powers of the atoms. This is in contradiction with consideration (b) when studied in detail, but has been commonly used for estimating the capture rates.

302

D. Horvdth, F. Entezami / Coulomb capture TABLE

2

Total chi-squares for the groups Group mixtures oxides halides sulphides alloys nitrides all

Data no.

Model 1

Model 2

Model 3

16 127 131 15 27 5

910.3 3005.6 3897.6 747.8 2340.5 1982.7

247.3 5194.8 6222.5 497.6 2087.2 7655.9

321

12 884.4

21905.3

Model 4

Model 5

610.6 6265.4 4129.7 342.5 1891.1 6963.0

588.8 1708.5 1661.6 205.0 369.0 45.7

583.3 1642.8 1653.0 206.7 338.9 116.0

20 202.4

4578.6

4540.7

Models: 1: Modified Z-law: 0.69(Z1/Z2)os6. 2: z”3 - 1 by Vasilyev et al. 3: Daniel’s model modified: Z1’3 In (0.832 + 0.69). 4: SPP model: w2.with gaussian cut-off (E. = 0, Er = 86 eV, E2 = 119 eV, ZO = 15, q(Z) = 2’. 5: SPP model: w2 with gaussian cut-off (E, = 0, El = 86 eV, E2 = 121 eV, Zc = 15, q(Z) = Z’ 6g.

The Z-law underwent

numerous experimental tests i) and modifications. Baijal capture ratios could be described by

et al. **) observed that the experimental

A(Zi,

(4)

z2)=GdZ2)",

and most of the results happened to be in the region 0.5 s n s 1.5 (n = 1 corresponds to the (3) Z-law). Zinov et al. proposed a modified, empirical Z-law for metallic halides and alloys *‘): A(Zi, 22) = O.66Zi/Z2

.

(5)

Vogel et al. lo), using a semi-classical approximation with the Thomas-Fermi model of the atom, deduced a Z-dependence similar to eqs. (3) and (4): A(Zr,

Z2)=

GWZ2)"".

(6)

It has been shown in several works 11*27-29) that the Z-law cannot give a satisfactory prediction of the atomic capture ratios in any of its forms (3)-(6). We tried to find an “optimal” Z-law by fitting parameters a and b of the general expression A(Zi,

Z2)=4WZ21b.

(7)

As seen from table 1, the best fit was obtained at a = 0.69 and b = 0.86 with a much less xz than for (3). This “empiric” Z-law is close to (5) but with a somewhat weaker Z-dependence. Having considered the energy loss of the meson in the atom along a trajectory which is close enough to the nucleus for the meson to be subsequently captured

D. Hodth,

0

.5

1

t-5

303

F. Entezami f Coulomb capture

2

2.5

3

3-5

4

4 5

5

Zl izi

Fig. 1. Atomic capture ratios A(Z1, Z2) measured in oxides, fluorides and chlorides ‘) against the ratio of the corresponding atomic numbers Z1/Z2. Note the characteristic oscillations in each case. The upper soiid line corresponds to the Fermi-Teller Z-law and the tower one to the (7) modified Z-law. The dashed curve represents the predictions of the SPP model with the (26) parameter values (i.e. modef 4 in table 2).

304

D. Horva’th, F. Entezami / Coulomb capture

in an atomic orbit, Daniel ‘) deduced a Z-dependence

A (21, Zd =

of the form

2:” In (0.572,) 2$‘3 In (0.57&) ’

which fitted the atomic capture ratios measured in metal halides rather well. Later, in order to describe the oscillations observed in oxides 27), Daniel ‘*) revised eq. (8) by including the atomic radii R (2):

A(&, 22) =

2:‘3 In (0.57Zr)R (22) Z:‘” In (0.57Z2)R(Zr) *

(9)

This expression predicts the oscillations in the oxides fairly well 29). However, its use is hindered by the somewhat undefined nature of the R(Z) radius. In his original paper “) Daniel used the metallic radii of the metal atoms and the ionic radius R(O-‘) of oxygen. At the same time most of the oxides have a highly covalent chemical bond, and there are considerable differences between the atomic, metallic, covalent and ionic radii. Even the best-defined crystal ionic radii have high systematic uncertainties as: “Numerical values of the radii of the ions may vary depending on how they were measured. They may have been calculated from wave functions and determined from the lattice spacings or crystal structure of various salts. Different values are obtained depending on the kind of salt used or the method of calculating.” [Quotation from ref. ‘“).I Eq. (9) is highly instructive for the theory of atomic capture but - in our opinion - cannot be used for predicting atomic capture ratios. Daniel’s formula (8) gives a poor agreement with the experimental data; the surprisingly high x2 is mostly due to the systems with light elements such as He, Li, Be, B. We made attempts to improve eq. (8) by using the form A

(Z,,Z2)

=

Z:‘3 In (a21 +b) .Z:‘3 In (aZ2 + b) ’

(10)

As shown in table 1, the best fit was obtained at a = 0.83 and b = 0.69 but with a reduced x2 still considerably higher than that for the Z-law (7). A Z-dependence, somewhat similar to eq. (8), has been empirically found by Petrukhin and Suvorov 31) in an experimental study of pion capture in mixtures of hydrogen with noble gases. After the removal of the contribution of pion transfer, the observed atomic capture ratios A(& H) could be well approximated by A(Z, H)~(.2?‘~-1).

(11)

Vasilyev et al. 11) have generalized expression (11) in the form AGC,

and compared

its predictions

Z2)=

A(Zr,H)=Z:‘3 -1 A(&, H) Z:‘3 - 1’

(12)

with the Coulomb capture ratios available at that

305

D. Horvdth, F. Enterami / Coulomb capture

time. Eq. (12), as shown in table 1, approximated the experimental values much better than the Z-law (3) and somewhat better than (8). Another line in describing the atomic capture data has been initiated by Schneuwly, Pokrovsky and Ponomarev (SPP) 16*17).They formulated a model where the p (I?) efficiency of an electron in atomic capture depends on the electron binding energy E : (13) [rigid boundary 16>],or forEC& p(E) = ct,

[-((E -E0),E$]

(14)

for-E>&

[smooth boundary 17)], where EO and E, are parameters. Following the idea of Gershtein et al. 4), after the meson has knocked off an electron it is presumably trapped on an atomic or a molecular orbit. Thus the capture ratio is defined as A(Zr,Z2)=

(15)

n1+2v1w n2+2v2(1 -w) ’

where rzI and n2 are the effective electron numbers of atoms Z1 and Z2: ni

=~p(Ej)n;. i

(16)

Here Ef is the energy and nf is the population of level j in atom Zi ; v1 and v2 are the corresponding valencies. w is the transition probability of the meson from the molecular orbit to the atomic orbit of atom Z1, and is assumed to have one of the following forms: w1=

(17)

(1 +p2q2/p1d

for a long-lived mesic molecular state, or a2

=p1P1+

(1

-P1P1

-P2P2)@

(18)

for a short-lived mesic molecular state 17).The pi’s are the valence electron densities expressed in terms of u, the ionicity of the Z1-Z2 bond: p1=$(1-a)

;

p2=&+Cr).

(19)

The transition from molecular state to atomic states is described by the quantities q and@: q1=

[1+

Pl=nll(nl+2Vl),

(z,/-m’1-’

,

q2=l-q1; P2

=

n2/h2

(20) +

2Y2)

.

(21)

306

D. Horva’th, F. Entezami / Coulomb capture

It is shown in ref. “) that eqs. (14)-(16) and (18)-(21), i.e. the smooth boundary approximations with u2 and parameters Eo=15eV,

EC=

E1=70eV I E2 = 100 eV

ifZCZO= 18 ifZ>Zo=18,

(22)

can describe the atomic capture ratios measured in oxides better than any of the simple relations (3)-(6), (S), (12). However, it is mentioned in ref. “) that the (22) parameter values are not optimized in the least-squares sense. Our attempts to optimize the SPP model are summarised in table 1. We have tried to find an improvement for both the rigid- and smooth-boundary approximations, using both forms, (17) and (18), of the w transition probability. The rigid-boundary model has only one parameter, the E. cut-off energy. The smooth-boundary model has been fitted with two different p(E) efficiency functions: the original gaussian (14) and a Fermi function 1 p(E) = 1 + exp {(E -Ed/E,}

*

(23)

In table 1, if ZO is not presented among the parameters, then we have used E, = El with no change in the width. We also tried to find improvement in the q1 probability by fitting the exponent x in q1=

u

+

Gxm”l-’ *

(24)

It is quite clear from table 1 that the best agreement is given by the SPP model in any form. Furthermore, the agreement between predictions and experimental values is much better for the smooth-boundary approximation. The best fits are obtained when using the original SPP model with modified parameters. The cut-off energy E. tends to be zero in every case. The increase in the E, width at Z > 18 suggested in ref. “) seems to be well established; its removal increases the xr’ value, and the fitting changes the Z0 value only slightly (from Z0 = 18 to 15-17). The fitted values of exponent x in (24), x = 1.7-1.8, are not far from the original 2, and these changes do not affect the obtained & value very much. The goodness of fit is not sensitive to the actual form of the p(E) distribution, whether it is gaussian or fermian. It is, however, sensitive to the actual form of the transition probability of the meson from the molecular to atomic orbit, preferring the shortlived state described by eq. (18). In table 2 the total x2 values are presented for the six groups of binary systems and five chosen models: the generalised Z-law (7) with a = 0.69 and b = 0.86; the modified model of Daniel (10) with a = 0.83 and b = 0.69; the model (12) of Vasilyev et al.; and two versions of the SPP model with gaussian efficiency function (14) and w2 probability, We did not feel any justification for the removal of experimental data from the comparison as removing a dozen of those with the largest deviations does not affect the final picture. For the mixtures where no

D. Horvdth,

F.

Entezami

f

Coulomb capture

307

chemical effects are involved, formula (12) gives the best predictions; for the compounds the SPP model is the best, Therefore, we made an attempt to improve on formula (12) by including the terms of the molecular orbits from eq. (15). With adjustable parameters a and 6, it assumed the following form: 2y3 4G,Z,i=Z;,3

‘fb(2rQW) -a +b(2v,(l-w))’ -iz

For the probability w of the transition from the molecular to an atomic orbit on 21 we used eq. (18), the fast transition approximation. From the results presented in table 1 it is clear that we recovered the initial form feq, (12)) for the atomic capture in the mixtures, and including molecular effects does not improve the xz considerably. 5. Conclusion

We tested the various descriptions of the atomic capture of mesons against 321 experimental atomic capture ratios. The comparison shows that there is no adequate model for the atomic capture yet. The model proposed by Schneuwly, Pokrovsky and Ponomarev gives the best agreement with the measured data, with zero boundary and slightly modified parameters:

Eo=O;

for26125 E = E1=86eV E~=119eVforZ>lS’ c

For mixtures of elements (gases) the use of the (12) empirical formula is recommended. To facilitate the application of the SPP model for the estimation of atomic capture probabilities, in ref. ‘) we provided a list of the effective electron numbers Zes, calculated by using models 4 and 5 of table 2 for the elements of the periodic system. Finally, we emphasize the need for an adequate method of estimating the probability of Coulomb capture of mesons in atoms. As shown by the reduced ,& value obtained in the best case (xf = 14 >>1), at present the agreement between theory and experiment is far from being satisfactory. The authors are indebted to Dr. V.I. Petrukhin for suggesting this work and encouraging it in its early stage, to Dr. D.F. Measday for useful discussions and to Dr. J.H. Brewer and Mr. A. Bagheri for help in the computations. D.H. expresses his gratitude to Dr. E.W. Vogt and the staff of T~I~MF for kind hospitality. References 1) D. Horvgthand F. Entezami,TRIUMFreport TRI-83-1, Vancouver, 1983 2) E. Fermi and E. Tel&, Phys Rev. 72 (1947) 399

308

D. Harvdth, F. Entezami / Coubmb capture

3) M.Y. Au-Yang and M.L. Cohen, Phys. Rev. 174 (1968) 468 4) S.S. Gershtein, V.I. Petrukhin, L.I. Ponomarev and Yu.D. Prokoshkin, Usp. Fiz. Nauk 97 (1969) 3; Sov. Phys. Usp. 12 (1970) 1 5) L.I. Ponomarev, Ann. Rev. Nucl, Sci. 23 (1973) 395 6) PK. Haff and T.A. Tombrello, Ann. of Phys. 86 (1974) 178 7) M. Leon and R. Seki, Phys. Rev. Lett. 32 (1974) 132 8) H. Daniel, Phys. Rev. Lett. 35 (11375) I649 9) G.Ya. Korenman and S.I. Rogovaya, Yad. Fix. 22 (1975) 754; Sov. J. Nucl. Phys. 22 (1976) 389 IO) P. Vogel, P.K. Ha& V. Akylas and A. Winther? Nucl. Fhys. A254 (1975f 445 11) V,A. Vasilyev, V.I. Petrukhin, V.E. Risin, V.M. Suvorov and D. Horvath, Dubna report JINR-Rl10222,1976 12) M. Leon and R. Seki, Nucl. Phys. A282 (1977) 445 13) M. Leon and J.H. Miller, Nucl. Phys. AZ82 (X977) 461 14) P. Vogel, A. Winther and V. Akylas, Phys. Lett. 70B (1977) 39 1.5) M. Leon, Phys. Rev. Al7 (1978) 2122 16) II. Schneuwly, Exotic atoms, Proc. 1st Course of Int. School of Physics of Exotic Atoms, ed. G. Fiorentini, G. Torelli (Piss, 1977) p, 255 17) H. Schneuwly, V.N. Pokrovsky and L.I. Ponomarev, Nucl. Phys. A312 (1978) 419 18) II. Daniel, Z. Phys. A291 (1979) 29 I9) GYa. Korenman and S.I. Rogovaya, Radiat. EL 46 {198O) 189 20) J.S. Cohen, R.L. Martin and W.R. Wadt, Phys. Rev. A24 (1981) 33 21) V.S. Evseev, T.N. Mamedov, V.S. Roganov and N.I. Kholodov, Dubna report JINR-E4-$I-237, I981 22) ‘I’.von Egidy and F-J. Hartmann, Phys. Rev, A26 (1982) 2355 23) D. Horv&h and R.M. Lambrecht, Exotic atoms: A bibliography 1939-1982 (Elsevier, Amsterdam, 1983) 24) A.V. Bannikov, B. Livay, V.I. Petrukhin, V.A. Vasilyev, L.M. Kochenda, AA. Markov, V.I. Medvedev, G.L. Sokolov, II. Strakovsky and D. Horvfth, Dubna preprint JINR-RI-82-789,1982, Nucl. Phys. A, to be published 25) S.S. Gershtein, ZhETF 43 (1962) 706; YETP (Sov. Phys.) 16 (1963) 501 26) F. James and M. Roos, MINUIT, CERN Program Library 27) V.G. Zinov, A.D. Konin and A.I, Mukhin, Yad. Fix 2 (1965) 859; SOV.J. Nucl. Phys. 2 (1966) 613 28) J.S. Baijal, J.A. Diaz, S.N. Kaplan and R.V. Pyle, Nuovo Cim. 30 (1963) 711 29) ‘I’, von Egidy, W. Denk, R. Bergmann, II. Daniel, F.J. Hartmann, J.J. Reidy and W. Wilhelm, Phys, Rev. A23 (1981) 427 39) Handbook of chemistry and physics (CRC Press, Boca Raton, FL, 1980) p. F-2 14 31) V.I. Fetrukhin and V.M. Suvorov, ZbETF 70 (1976) I145; Sov. Phys+ JETP 43 (1976) 595