Muonic conversion as a multipole meter of γrays and muon distribution of fragments from

prompt nuclear fission of U and Pu

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1990 J. Phys. G: Nucl. Part. Phys. 16 1195

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J. Phys. G: Nucl. Part. Phys. 16 (1990) 1195-1202. Printed in the UK

Muonic conversion as a multipole meter of y rays and muon distribution of fragments from prompt nuclear fission of U and Pu

F F Karpeshin Institute of Physics, Leningrad State University, Ulyanovskaya 1, Petrodvorets, Leningrad 198904, USSR

Received 29 January 1990

Abstract. The multipolarity of the muonic conversion observed in prompt fission induced by radiationless transitions in p atoms is investigated theoretically. It is shown by comparison with experiment that the E l and E2 transitions make approximately equal contributions to the observed conversion probabilities. This means that in the energy range from 3 to 6 MeV about 10% of the radiative transitions in the fragments are of E2 type. Possible manifestations of the fission dynamics in muonic conversion are discussed.

1. Introduction

Prompt nuclear fission induced by radiationless transitions to the 1s state in ,U atoms (Wheeler 1948, Zaretsky 1958) presents unique possibilities for studying the fission process. As distinct to delayed fission occurring as a result of p capture by the nucleus, prompt fission occurs from beginning to end in the presence of a muon. The muon is bound in the K orbit of a fissile nucleus, from where it transfers to one of the fragments, usually the heavy one, forming a ,U atom. Hence it is attractive to use this process for probing both fission dynamics and the properties of the fission fragments as neutron-rich nuclei.

Internal conversion of y rays from prompt-fission fragments is a process of great interest. It has been studied experimentally by Belovitsky et af (1978, 1982), Belovitsky and Petitjean (1987) and Ganzorig et af (1978, 1980). It is noted here for a clearer understanding of the problem under consideration that the experiments do not determine from which fragment the muonic conversion has taken place. If electronic methods are used (Ganzorig et af 1978, 1980), only the probability of muonic conversion per prompt fission is extracted based on the measured lifetime of muons with respect to the decay. If the conversion is observed by means of nuclear emulsions (Belovitsky et a1 1978, 1982), this is because the characteristic time of muonic conversion determined from the electromagnetic interaction is about s. Although fully accelerated for this time, the fragments have a range of around lO-’cm, which is too short a distance to be resolved. Only indirect information concerning the role of each fragment in the conversion can be obtained from analysis of the angular distribution and conversion probabilities per prompt fission.

0954-3889/90/081195 + 08 $03.50 0 1990 IOP Publishing Ltd 1195

1196 F F Karpeshin

Many theoretical investigations of muonic conversion have been undertaken (Karpeshin et a1 1976, Karpeshin 1980, 1984, 1986, Barit et a1 1977), and this is also the main subject of the present paper. The binding energy of a muon in the K orbit of a fragment is about 3-6MeV. Muonic conversion therefore offers a way to explore high-energy y rays from fission fragments. The role of muonic conversion in this method is quite analogous to the part usually played by internal conversion in the electronic shell in investigations of low-lying nuclear states. Studies of the angular distribution of the conversion muons are promising, as comparison between preliminary experimental data (Belovitsky et a1 1988) and theoretical predictions (Karpeshin 1984, 1986) shows. Furthermore, the probability of muonic conversion seems also to depend on fission dynamics. This dependence is twofold (Karpeshin et a1 1987). Firstly, direct muon emission can take place at the fission stage due to a shake-off process. Secondly, the shake-up process can contribute to the probability of muon entrainment by a light fragment. This would also reveal itself immediately in both the probability and angular distribution of muonic conversion.

Here we search for some characteristic features of muonic conversion which are inherent in El and E2 transitions. These peculiarities mean that the multipolarity of the muonic conversion as well as a possible fission dynamics contribution can be determined in the experiment. While prompt fission of U is the best subject for study, we also consider the case of muon fission in Pu. Experimenting with Pu is desirable since its probability of prompt fission per muonic atom is 20 times as large as that of U. Furthermore, comparison of the experimental probabilities and angular distribution of the muonic conversion for these two elements, as well as comparison of their muon attachment probabilities, is of great interest for investigating their mechanisms and extracting any possible fission dynamics contributions to their rates.

2. Muon conversion probabilities

A method for evaluating muonic conversion probabilities using an experimental fission y-ray spectrum and the muonic conversion coefficients (MCC) was described by Karpeshin et a1 (1976) and Karpeshin (1980). Barit et a1 (1977) attempted to apply this method to their calculation but they failed to calculate the E2 conversion probabilities for light fragments. We use the simple expression obtained by Karpeshin (1980) for the conversion probability p of a transitiun of type and multipolarity zL:

p ( z L ) = X S / ( X + S) (1)

where x = (a , ( zL ) )Nc , (a,(zL)) is the mean MCC value averaged over an experimental y-ray spectrum of the fragment and N ; is the number of y quanta in the spectrum with energy above the conversion threshold. S is the probability that the excitation energy of the nucleus remains above the conversion threshold after the neutron evaporation has finished. In practice the assumption is usually adopted that the experimental fission y-ray spectrum belongs entirely to the fragments. Furthermore, the spectrum of each fragment is supposed to be the same to a first approximation and hence can be taken to be half of the experimental spectrum. Then, assuming that all the quanta are defined (either El or E2), one evaluates the conversion probabilities from equation (1). Comparing the conversion probabilities

Muonic conversion in y atoms 1197

so obtained with the experimental values, one can determine the percentage of El and E2 transitions in the original y-ray spectrum.

From equation (1) we note that the conversion probability may not be greater than S, and it obviously tends to S in the limit of large (a ; ( zL ) ) such that x* S . This is reflected in equation (1). In the opposite limiting case of small (a,(zL)) with x e s , we also arrive by means of equation (1) at the physically clear result p ( t L ) = ( a P ( t L ) ) N ~ , in accordance with the definition of the conversion coefficients. These details were discussed specifically by Karpeshin (1980).

The calculated p ( 2 ) values for the E l and E2 transitions are plotted in figure 1 as a function of the atomic number of the fragment. The experimental fission y-ray spectrum given by Peelle and Meienschein (1971) was used. The values of S were taken from Terre1 (1958), allowing for the possibility of neutron evaporation from fragments up to four inclusive. Two characteristic features of the curves in figure 1 stand out.

(i) The conversion probabilities for the E2 transitions are generally an order of magnitude larger than those for the El transitions. This makes their contribution to the muonic conversion comparable to that of the E l transitions, even though their part in the radiative transitions of the same energy does not exceed 10% of that of the El transitions.

(ii) The difference between the E l and E2 probabilities is significantly smaller for the light fragments than for the heavy ones. This is a consequence of the fact that the limiting case x 9 S occurs for El transitions both in heavy and light fragments, as well as for E2 transitions in heavy fragments. In contrast, for E2 transitions in light fragments the opposite limiting case x + S is relevant, restricting the conversion probability. This means that the contribution of the light fragments to the conversion depends on the multipolarity, which is larger for the E l transitions. This fact is of most significance to the study of the angular distribution of converted muons. If used it makes the determination of the multipolarity of fission y rays considerably easier.

10 1

P

1 0 - 3 - I I I I I

Figure 1. Muon conversion probabilities p various fragments for the El and transitions.

for E2

40 50 60

Z

1198 F F Kurpeshin

3. Muon distribution of fragments

Figure 1 shows that the muonic conversion probability in the light fragments is significantly larger than that in their heavy partners. It is concluded from this fact that the conversion probability per prompt fission is sensitive to the muon fraction entrained by the light fragments. It is therefore reasonable to discuss the muon distribution of the fragments, W(Z), before comparing with the experiment.

Values of W have been evaluated by many authors (Karnaukhov 1978, Demkov et a1 1978, Zaretsky et a1 1980, Karpeshin and Ostrovsky 1981, Karpeshin et a1 1982, 1987, Olanders et a1 1980, Maruhn et a1 1980, Ma et a1 1980, 1981, Boi et a1 1983, Bracci and Fiorentini 1984), and they agree qualitatively with experiment (Ganzorig et a1 1978, 1980, Schroder et a1 1979, Belovitsky and Petitjean 1983, d’Achard van Enschut et a1 1988) in that the entrainment probability by a light fragment is a few per cent, the influence of fission dynamics on this value being studied.

For the present purposes we rely upon the results of Karpeshin et a1 (1987), who considered the most realistic model. These authors calculated the muon distributions of a number of fragments from U prompt fission with kinetic energies in the range 140 S Ekin S 200 MeV. In the calculations any possible fission dynamics contribution was deliberately omitted with a view to extracting it by comparison with experiment.

The problem was considered in the adiabatic two-state quasimolecular approach. Initially a muon is in the 1 so molecular orbital which in the limit of small intranuclear distance R-, 0 corresponds to the 1 s state of the united atom (U or Pu). For large R this orbital correlates with the 1 s state of muons in the heavy fragment which is the final state of a muon in the limit of very slow nuclear motion. Due to the finite velocity of the fragments there is some probability of non-adiabatic transitions to the nearest 2 p o orbital which corresponds to the 2 p state of the united atom and to the 1 s state of a muon in the light fragment in the separated-atom limit. Using Schrodinger’s equation, it is readily shown in such a picture that W(Z) for the light fragments can be considered as a function of two scaled variables:

w ( z ) = w(z/zh, Ekin/Zt) (2) where Zh is the atomic number of the heavy partner. Strictly speaking, the scaling equation (2) is valid under two assumptions which are nevertheless slightly broken: (i) the equivalent electromagnetic radii of the fragments have to be proportional to their atomic numbers; (ii) equality of the reduced masses of all the pairs of fragments is also needed. However, W(Z) first of all depends on A Z = Z h - Z and it depends less strongly on the other parameters, so equation (2) holds fairly well for our purposes. This equation allows one to obtain the values of W(Z) for prompt fission of every actinoid nucleus by virtue of simple interpolation, if the results for the prompt fission of U are known for various values of Z and .!&in. The results interpolated in this way for Pu are plotted in figure 2 together with those for U given by Karpeshin et a1 (1987).

It is in fact the independent mass yield which is known for the prompt fission of 238U (David et a1 1983). We assume a constant charge-to-mass ratio in primary fragments, although this is known not to be quite so. This assumption allows us to give the relative contributions of every light fragment Y W ( Z ) to the total muon attachment probability of the light fragment. These contributions are presented in figure 3. It is seen that the muons are most often caught from the heavy partner by

Muonic conversion in FL atoms 1199

1 Figure 2. Muon attachment probabilities to the light fragments W as a function of atomic number 2 from prompt fission of U and Pu.

L

the light fragments with Z = 43 and 44 for U and Pu respectively. Summing W ( Z ) over 2, we obtain the average light-fragment muon entrainment probability w. This turns out to be about 0.05 and 0.07 for the prompt fission of U and Pu respectively. Thus for the prompt fission of Pu U: happens to be approximately 1.4 times as large as that for U. This difference is also apparent in figure 3.

We underline that the latter result is based on fairly general assumptions. Namely, that the W ( Z ) values depend sharply (exponentially) on AZ, and that the heavy-fragment yields of U and Pu fission are close to each other. The former condition means that for a given heavy fragment the probability of muon transition to the light fragment is obviously larger the higher the atomic number of the light partner. These assumptions predetermine the main features of the results.

Finally, we stress that the results obtained in this section make no allowance for the influence of fission dynamics on the muon distribution. Including the dynamics breaks the scaling relation (2) and smears out differences between the muon distributions from prompt fission of U and Pu. Hence comparison of the latter values with each other provides independent information on the fission dynamics.

0

Z

Figure 3. Effective catch probabilities YW by the light fragments.

1200 F F Karpeshin

4. Comparison with experiment and discussion

It was found by Ganzorig et a1 (1980) that the muon conversion probability per prompt fission is about 1% for muons ejected from the atom with kinetic energy Ep 3 0.5 MeV (in the laboratory system). This value was considered to be the effective threshold for converted muons to escape from the target layers contained in the fission chamber. Similar results were obtained by Belovitsky et a1 (1988). For comparison with experiment the energy spectrum of the muons is needed.

An accurate energy spectrum was given by Karpeshin (1980) in the centre-of- mass system of the fragment. It agrees well with the experimental spectrum obtained by Belovitsky et a1 (1982, 1988). The transformation to the laboratory frame also requires knowledge of the angular distribution of the conversion muons. This was

- 3 0

x - Y

- 2 5 4

1

0 35

c

4 -

- 3 - z X -

Z

L

Figure 4. (a ) Contributions of various fragments P to the muon conversion probability for El transitions for muons having kinetic energies E,, 5 0.5 MeV. ( b ) As in (a) but for E2 transitions.

Muonic conversion in ,U atoms 1201

Table 1. Calculated conversion probabilities in prompt-fission fragments for muons with kinetic energy E, 3 0.5 MeV

El E2

Light Heavy fragments fragments Total

Light Heavy fragments fragments Total

U 2.1 x 1 0 - ~ 2.2 x 4.3 x 1 0 ~ 1.1 X lo-’ 3.1 X lo-* 4.2 X lo-’ pU 2.1 x 1 0 - ~ 2.1 x 4.2 x 1.3 X lo-’ 3.1 X lo-’ 4.4 X lo-’

investigated by Karpeshin (1984) and was proved to be mainly proportional to JYLo(8, q)12 (for EL transitions in the centre-of-mass system of a fragment) relative to the fragment velocity direction. This direction is chosen due to the alignment of spins of both fragments in the plane perpendicular to their velocity direction.

Integrating the laboratory-frame spectrum over the desired energy interval, we can find the observable part s(Z) of the converted muons. For muons with kinetic energy E, 3 0.5 MeV this part happens to be approximately 1/2 (Karpeshin 1980).

This produces a total conversion probability of around 2% (Karpeshin 1980). The calculated contributions of every fragment, P ( Z ) = s ( Z ) p ( Z ) Y ( Z ) W ( Z ) , for the muons with E, a 0.5 MeV are plotted in figure 4 for the E l and E2 transitions in the prompt fission of U and Pu. It can be seen that the conversion probabilities in heavy fragments are essentially the same for U and Pu, in accordance with our assumption of their close yields. The corresponding light-fragment contributions are similar, but are shifted relative to each other quite analogously to the plots in figures 2 and 3. The most remarkable feature, however, is that the contributions of the light fragments to the conversion probability for El and E2 transitions are essentially different. In the former case the total conversion probabilities in the light and heavy fragments are close to each other. In contrast, for the E2 transitions the total contribution of the light fragments turns out to be significantly smaller than that of heavy fragments.

Summing the contributions of all the fragments presented in figure 4 provides the total conversion probabilities listed in table 1. It can be seen that the assumption of 100% E l multipolarity of the converted muons fails to explain the experimental value of the conversion probability, underestimating it by a factor of about two. An admixture of the E2 component is to be adopted at the level of 10-15% in the radiative spectrum of the fragments. This supplies the missing half needed for agreement of the calculated muonic conversion probability with the experimental value.

Analogous results have been obtained in an analysis of the angular distribution of the converted muons. This is the subject of a separate paper.

Acknowledgments

The author is grateful to Professor G E Belovitsky for stimulating discussions of the experimental situation. He is also indebted to Professor P David for sending a number of papers, especially that by d’Achard van Enschut et a1 (1988), and for detailed discussions at the Conference ‘50th Anniversary of Nuclear Fission’ (Leningrad, 1989).

1202 F F Karpeshin

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