The study of prompt and delayed muon induced fission - Springer Link

Zeitschrift fiJr Physik A

Z. Phys. A - Atomic Nuclei 330, 397-405 (1988)

Atomic Nuclei 9 Springer-Verlag 1988

The Study of Prompt and Delayed Muon Induced Fission II. M e a n L i f e T i m e s o f N e g a t i v e M u o n s Bound to aaTNp, 24Zpu a n d z 4 4 p u P. David, H. H~inscheid, J. Hartfiel 1, H. Janszen 2 T. Mayer-Kuckuk, R. von Mutius 3

F. Risse, Ch.F.G. R6sel, and W. Schrieder Institut ffir Strahlen- und Kernphysik, Universit/it Bonn, Federal Republic of Germany C. Petitjean and H.W. Reist Paul Scherrer Institute (PSI), formerly SIN, Villigen, Switzerland

S.M. Polikanov Gesellschaft ffir Schwerionenforschung (GSI), Darmstadt, Federal Republic of Germany J. Konijn, C.T.A.M. de Laat, and A. Taal N I K H E F - K , Amsterdam, The Netherlands T. Krogulski s Department of Mathematics and Nature, University of Warsaw, Bialystok, Poland T. Johansson 4 and G. Tibell s Gustaf Werner Institute, Uppsala University, Sweden J.F.M. d'Achard van Enschut 6 Laboratorium voor Technische Natuurkunde, Technische Universiteit Delft, The Netherlands and Philips Laboratorium Eindhoven, The Netherlands

J.P. Theobald Institut fiir Kernphysik, Technische Hochschule Darmstadt, Federal Republic of Germany N. Trautmann Institut fiir Kernehemie, Universitfit Mainz, Federal Republic of Germany C. Gugler, L.A. Schaller, and L. Schellenherg Institut de Physique, Universit6 de Fribourg, Switzerland Received March 25, 1988 Present addresses: 1 Bopp and Reuther, D-6800 Mannheim, Federal Republicof Germany 2 Physikalisch-Technische Bundesanstalt, D-3300 Braunschweig, Federal Republic of Germany

3 Fa. Rheinmetall, D-4000 D/isseldorf, Federal Republic of Germany 4 The SvedbergLaboratory, S~75121Uppsala, Sweden 5 Department of Radiation Sciences,S-75121Uppsala, Sweden 6 Philips I&E Laboratories,NL-5600 MD Eindhoven,The Netherlands

398

P. David et al.: Study of Prompt and Delayed Muon Induced Fission. II

The mean life times of negative muons bound to actinide nuclei have been measured by detecting the time difference between a stopped muon and the arrival of fragments from delayed fission after muon capture. The deduced capture rates Ar are 1.392(4). 107/s for 237Np, 1.290(7). 107/s for 242Pu and 1.240(7). 107/S for 244pu. The results are compared with published data for the fission and the neutron decay channels and for the electron decay of the bound muon. Including a former measurement of Ac for 239pu, a n isotopic dependence of the muon capture rates in the Pu isotopes is clearly observed. PACS: 25.30.-c; 25.85.-w; 25.85.Ge; 36.10.Dr

1. Introduction

tion by complex nuclei is generated by the elementary absorption process on a bound proton

The capture of negative muons by heavy nuclei has been studied before with various aims in mind, and the subject has been reviewed [1, 2]. A first step in understanding the capture rates is based on a measurement of the life time of the muon bound in the ls state. It is the purpose of this study to present and discuss measured values of the muon nuclear capture time determined from the fission decay of the actinide nuclei 23VNp, 239pu, 242pu and z44pu, excited in the nuclear muon capture process. The g--capture probability from the K-shell into the nucleus is determined by comparing mean life times of negative and positive muons. There exist two main channels for a bound negative muon to disappear 1. the free decay # - + (A, Z) --* e - + ~e + vu + (A, Z) and 2. muon absorption by the nucleus (A, Z) #-+(A,Z)--*(A,Z-1)+v

u.

In older measurements of the electron decay of bound muons [1-5] and in a recent study [6] values of the capture rates A~ have been determined as a function of mass number A from the total disappearance rate Ar AT=Ac+Q'AD

with A T = I / z u_ and A o = l / r , + ; ~ is the mean life time of negative muons bound in the K-shell and zu + is the mean life time of positive muons. The quantity Q is the Huff-factor [7]. To measure the total capture probability A~ from the K-shell means measuring of A T and A D. The total disappearance rate Ar is determined in an experiment by measuring one of the possible decay channels e, 7, n, f or charged particles. The overall behaviour of the nuclear muon capture rate has been described in several theoretical papers. They have in common that the muon absorp-

p + # - ( l s ) ~ n + v u.

This implies that the excitation energy distribution in the nucleus after muon absorption should be described by theory, at least in principle. In reality it will be necessary to make approximations due to the high number of available states entering into the calculation of the transition amplitudes. Primakoff [8], treating muon absorption in the closure approximation, arrives at a capture rate

with 7=correction for phase Space effects in the absorption of muons by heavy nuclei compared to that by a nucleon, Ac(lH)=muon capture rate on hydrogen and 6 = factor to include the Pauli principle. From the previous formula, one obtains for the isotopic dependence of the muon capture rate A c(A, Z)

1 - 6 (A - Z)/2 A

A c(A', Z)

1 - ~ (A', Z)/2 A "

There exist only few data of muon capture rates on several enriched isotopes of the same elements, such that at present a systematic study in this respect is difficult. Due to the approximation used in its derivation, the Primakoff formula gives the dependence of muon capture rates on the bulk properties of nuclei and neglects more subtle effects. In such a way it describes a trend and allows to look for systematic deviations of muon capture values from its prediction. Improvements of this approach did not cure the basic shortcomings of its derivation, but were developed for special case. In particular the factor [ 1 - 6 ( A - Z ) / 2 A] decreases for heavy nuclei (244pu: 0.03), such that higher order terms, neglected so far, had to be included. This was done by Goulard and Primakoff [9], who derived the relation


AGp(A,Z)=K.Z4ff.(1

E,(ls)~ z m# C2 ]

9 [1 --2

m, c2--Eu(ls)]

mN7

]

{1--0"03~Z+0"25A--2Z2z

[A-Z

A-2Z\)

with K = 2 7 2 s -1, Eu(1 s)= binding energy of the muon in its 1 s orbit, 1T//~C2 = muon rest mass, mN c 2 = mass of the nucleon. In the original paper [9] the corresponding formula (22) contains a misprint. The nucleon mass mu is erroneously multiplied by a factor of two. In a number of earlier publications [31, 32] this misprint was not corrected for. Generally, the interaction between the magnetic moments of the nucleus and the muon in its orbit leads to a hyperfine structure. However, since the relative muon capture rate difference for a hyperfine doublet varies as Z -1, it should be negligible for heavy nuclei [1, 10, 11]. A different approach to the calculation of the muon capture rates was given by Foldy and Walecka [12]. Considering the strong dipole transitions induced in muon capture they calculated the transition matrix elements for muon capture from the known cross section of the isovector giant dipole resonance. By including higher order angular momenta and isospin splitting, Kozlowski and Zglinski [13] extended this model to heavy nuclei. Within this collective model they also performed a calculation of the distribution of excitation energies after muon capture by the nucleus. Such a distribution is also derived in the statistical model of Singer [14]. This ansatz was further developed by Hadermann and Junker [15] by taking more realistic momentum distributions from a Saxon-Woods potential for the nucleons involved in the capture process. A more recent study of muon absorption by Auerbach and Klein [16] is based on microscopic random phase calculations of the excitation of collective multipole strengths. The knowledge of the distribution of excitation energies plays a role in understanding the ratio of the yields of prompt and delayed fission and of the delayed fission probability per muon stop. This investigation is part of our third study [17]. Here predictions of some of the quoted models for the capture rates are discussed.

399

In Sect. 2 the experimental set up is outlined. In Sect. 3 the details of the data reduction and the results are presented. Finally, in Sect. 4 the result are discussed.

2. Experimental Set Up The measurements were performed at the Paul Scherrer Institute (PSI), formerly Schweizerisches Institut ffir Nuklearforschung (SIN). The muon beam momentum was about 50 MeV/c with a Ap/p of 3%, corresponding to a range width of AR ~ 100 mg/cm 2. The pion contamination was less than 10 -4. The size of the beam spot was 4 x 6 cmz. These conditions favoured the use of thin actinide targets (Table 1). Two kinds of set up were used, shown in Figs. I a and b, allowing to measure a triple coincidence (#-, flf2) between the muon and both fission fragments from one target (Fig. 1 a) and a double coincidence between the muon and one fission fragment from either of two targets (#-, f ) (Fig. 1 b), respectively. The triple coincidence set up was identically the same as described in [18] for the fission fragment spectroscopy.

Table 1. Specification of the targets used in this experiment. The

237Np and 242pu targets have been prepared at CBNM Euratom at Geel Target material

Target thickness [gg/cm z]

237NpF 4 200 2'*2PnF3 150 244p/102 45

Preparation

Backing Thickness [~tg/cm2]

Diameter [mm]

280 320 11200

30 30 32

evaporation Ni evaporation Ni drying in Ti

Sz [/.-3 l i

a

2

V

Fig. 1 a. Experimental set-up for triple coincidences between the incoming muon detected in scintillators S 1, $2 and both fission fragments coming from the target T and detected in D 1 and D 2. V= vacuum chamber, b Same as for Fig. 1a but only one fission fragment coming from target T1 and from target 72 is detected in D1 and D 2, respectively

400


The muons were identified in the scintillator telescope S 1 A S 2 and entered the vacuum chamber through a thin Al-window. Electrons were easily discriminated from muons in the telescope by their pulse height. The muons, after traversing the fragment detector D 1 struck the target. The fission fragments were identified in two Sisurface barrier detectors, D 1 and D 2. A time to amplitude converter measured the time difference t(#-, f ) between the signals of the incoming muon and the fission fragment in detector D 1. The pulse heights of both fission fragments measured in coincidence and the TAC signal t(#-, f ) were written on tape in list mode. A different set up was chosen to minimize systematic errors by simultaneously irradiating two targets (Fig. 1 b). In this set up the muon beam was again I

I

I

I

l

identified in the scintillator telescope but muons could be captured in either of two targets each facing one Si-detector and being separated by a 25 # thick A1 foil. In this way each detector identified only one fragment from one target and only double coincidences consisting of either signal from a fission detector and the corresponding time signal t(#-, f ) were written on tape. The time spectra were calibrated with a time calibrator, giving a relative accuracy of 1 x 10 -4 for measured time intervals, which is better than the internal linearity of the entire system. 3. Data Reduction and Results

In the off-line analysis, the triple and double coincidence data were evaluated in different ways. For the I

I

I

I

,,

I

,,,

I

I

a

100000

237Np

10000

244pu

10000 1000 1000

10ijI

100

I00

10

i o

0

t(p I

I

1

,f)

I

I

_

tip.-,f)Ins]

(ns) I

f

I

I

I

b 1000(

10000

242pu

242pu I00~

1000

100

100-

1o

10-

1

i

too

i t~

i 600

~ B6

i

1000

t(~',f )(ns]

f(~", f I [ns]

Fig. 2a-d. Time spectra T(/~, f ) observed in 237Np and 242'244Pu for the capture of negative muons bound in the atomic l s orbit with subsequent fission (spectra were summed up by four channels), d Illustration of the deconvolution of measured time spectra to prompt and delayed parts

P. David et al.: Study of Prompt and Delayed Muon Induced Fission. I1

triple coincidence measurement, true binary fission events were selected to form time spectra t(12-,f) by using appropriate windows in the pulse height spectra of both fragments. The time spectra, obtained in measuring only one fission fragment, were analysed by setting a window on the pulse height spectrum of the light fission fragments. In this way the lower pulse heights, originating from nuclear reactions in the detector, were excluded. This procedure gave the same results as those obtained from the double energy method (triple coincidence measurement). The resulting time spectra for 23VNp, 242Pu and ar with a resolution of 1.2 ns (FWHM) are displayed in Fig. 2a-c. This time resolution allows to determine the ratio of prompt to delayed fission yields with high accuracy.

3.1. The Bound I2. Capture Life 71me in the Fission Channel

401 Table 2a. Life times zr of negative muons, bound in the l s orbit of the listed nuctei, as measured in this work. The values given in brackets are the differences between the values obtained for the fits with lower limits 30 ns and 6 ns Nuclide

23VNp 242pu 244pua

p/d-ratio

p/T-ratio

Number )~2 of events

[ns]

[%]

[%]

U0s]

69.8• 75.3-+0.4(-1.5) 78.2+_0.4(--1.6)

29.5+_0.2 22.8+_0.1 7.7 20.8+_0.5 17.3_+0.1 1.2 26.3• 20.8• 1.2

Table 2b. A five parameter-fit was applied to the time spectra, using the assumption of the two delayed time components. For the determination of the prompt to delayed (p/d) and prompt to total (p/T) fission yields are considered: a) The second time component is added to the delayed part. b) The second time component is added to the prompt part. c) The second time component is kept out of the ratio. The fits are obtained for the lower limit of 10 ns in the time spectra ~1 [ns]

% [ns]

23VNp

69.8_+0.2

1.55_+0.09

242pu

75.3_+0.4 14.2 +_1.4

244Ptl

78.2_+0.4

g(t) = g(0)-exp(-- t / j + B. The parameters N(0), ~ and B were varied independently to optimize )~2. A dependence of z on the lower time limit was observed, i.e. fits obtained in the intervals (6 ns, 1.6 gs) and (30 ns, 1.6 gs) gave different results. A similar dependence has been observed for 238U [-19, 20] and has been explained by the possibility of a second time component originating from an isomeric nuclear state. For this reason the time spectra have also been fit with two exponentials, using five free parameters: N (t) = NI (0) exp(-- t/z i) + N2 (0) exp(-- t/'c2) + B. When adding a second time component and when changing the lower time limit the determined life times varied within the statistical errors only. Furthermore, the life times obtained with two components agree with the results from the one-component fit for a lower limit larger than 30 ns. Table 2a gives the 12- mean life times resulting from the three-parameters fits to the experimental time spectra t(#-, f). The five-parameters fits (see Table 2 b) did not yield an unambiguous identification of an additional time component in the time spectra.

0.95 1.5 1.0

a Corrected for a 9% admixture of 242pu

Nuclide

The time spectra were first analysed by assuming an exponential decay described by

zl

3.6 +_0.2

p/d-ratio

p/T-Ratio

[%3

[%3

a b c a b c a b c

19.5--+0.3 0.95 23.1+_0.3 20.5_+0.3 16.7+_0.2 1.5 20.5-+0.4 t7.4_+_0.2 t8.3+_0.2 0.99 21.9+_0.2 19.0+_0.2

24.2+_0.4 29.9• 25.4• 20.0_+0.2 25.7• 21.1 +__0.2 22.4__+0.2 28.1• 23.5•

Za

decomposed into prompt and delayed parts. The prompt time peak is fit by an asymmetric Gauss• fp(t, to) around t=to [-21, 223. The delayed fission part is fit by an exponential, folded with the time resolution of the detector system, i.e. the width of the prompt time peak: oo

fe (t, to) = b- ~ exp ( - 2 (t' - to)).fp (t, t') d t'. to

The constant background is chosen differently for positive and negative times. The p/d-ratio of prompt to delayed fission yields is then given by: ~o

fp(t, to) dt p

d

-oo

~ fa(t, to) dt oo

3.2. Evaluation of Prompt and Delayed Fission Yields Having determined the mean life times and the constant background B, the time spectra t(#-, f ) were

Tables 2a, 2b and 3 and Fig. 2a-c contain the results from this experiment. Figure 2d displays the time spectrum t(#-, f ) for the nuclide 242pu together

402


Table 3. Mean life times of bound negative muons in Z37Np, z39pu, 242pu and z44pu as determined from different channels: (/l-, f), (#-, n) and (#-, e-). For systematical errors see Table 2a

Mean values

Decay mode

z Ins] 237Np

(#-,f) (/1-, f) (#-,f) (#-, f) (#-, n) (#-, n) (# , e) (/~-, e)

69.8__+0.2(-0.7) 72.0 + 2.0 71.34440.9

(#-,f) (/z-, f) (#-, n) (#-, e)

71.44440.8 69.9 4440.2 71.9+0.8

73.5_+1.4 71.3•

z Ins] 239pu

70.0 + 3.0 70.14440.7 76.0 + 9.0 72.64440.6 70.1_+0.7 77.5-t-2.0 73.4•

70.1_+0.7 71.5-t-0.9 76.1 • 1.7

with fits to the prompt peak, the delayed part, the background and the sum of all three components. The data analysed with a second time component, tentatively attributed to an isomeric nuclear state, give no significant change in )~2. On the basis of the present results the two-component assumption is, therefore, neither confirmed nor rejected. (Consequently, a comparison between Table 2a and b shows that there is a systematic relative uncertainty of about 20% in the p/d ratios, quoted in Table 2a. The systematical errors for the mean life times are about three to four times larger than the statistical uncertainties and are given in the tables.

4. Discussion

The determination of the mean life time of the /~in heavy elements is somewhat complicated by the possible nuclear excitation due to radiationless transitions in heavy muonic atoms. This excitation, being a prompt process, precedes nuclear #-capture and leads predominantly to prompt neutron and/or 7emission or to prompt fission [-19-27]. Thus a muon may be absorbed not only by the nucleus (A, Z) but also by the the neighbouring isotope ( A - 1 , Z) or by fission fragments from the prompt fission process. Since the mean life time depends on the mass number A as well as on the charge Z, some average value for the life time is observed. However, the change in the measured fission life times due to the absorption on the nucleus (A - 1, Z) is estimated to be within the experimental errors. In heavy nuclei, the intensities of the radiationless 3 d ~ ls ( E ~ 9 MeV) and 2p--+ ls (E,--6.5 MeV) tran-

9 [ns] 242pu

z Ins] 24~pu

Reference

75.3___0.4(-1.5) 79.0 + 5.0 75.4_+0.9

78.24440.4(--1.6)

this work [30] [31] [34] [32] [33] [37] [38]

77.2• 75.44440.9

previous work including this work

75.5_+0.9 75.3 4440.4 76.14440.7

Table 4. Mean negative muon life times for nuclear capture in actin-

ide nuclei with subsequent fission and capture rates. The Huff factors are taken from Ford and Wills [7] Nuclide

23VNp 239pl1 2*2pu 2~*pu

rf

Huff Factor

A~

%

Ins]

Q

[105 s-']

Ins]

69,9• 70.1 4440.7 75.3 -t-0.4 78.24440.4

0.818 0.816 0.816 0.816

139.2• 138.8 4- 1.4 129.0 4440.7 124.0_+0.7

71.8-t-0.2 72.0___0.7 77.5 • 0.4 80.6_+0.4

sitions are in the range of 10 to 20% p e r / t - atom formation [26-28]. Radiationless transitions from higher levels are also possible, but their total intensity is below 3%/#-stop [29]. Prompt fission occurs in heavy muonic atoms with yields of a few 10- 3/#-stop to a few 10-1/#-stop [30]. All these different processes with quoted relative weights combine with the dominant nuclear muon capture on the nucleus (A, 2,) to form the measured value of the muon mean life time. Absorption of a muon in a heavy nucleus yields an average excitation energy of about 20 MeV [17, 183 such that all the decay channels for neutrons, 7-quanta, fission and charged particles are open. Any one of these decay channels allows the determination of the mean life time of the muon in its ls orbit. To be able to compare the experimental results from our measurement with published data we have listed the relevant informations for 237Np, 239pu, 242pu and 244pu in Table 3. The determination of the # - mean life z from the fission channel is advantageous due to the unique signals as compared to measuring neutrons or elee-

403

P. David et al.: Study of Prompt and Delayed Muon Induced Fission. II Table 5a. Muonic capture rates A [10S/s] as measured in different channels and as calculated for neptunium and plutonium isotopes Isotope

237Np 239pu 2~2Pu 244Pu

Decay mode

Decay mode

Decay mode

Calculated Acp

(p-, f )

(g-, n)

(#-, e)

[105]

139.2_+0.4 138.8 _+1.4 129,0__+0.7 124.0 +_0.7

135.1 __.1.6 136.0 _+5.0 127.6 4-1.3

127.6 _ 3.5

Table 5b. Parameters used to calculate Aap Isotope

c [fm] ,i

a [fm] ")

fl"

E,(I s) [MeV] b

Ze*ff [106] b

237Np z39pu 242pu z44Pu

7.001 7.037 7.063 7.089

0.523 0.498 0.499 0.432

0.283 0.261 0.277 0.281

12.319 12.526 12.488 12.468

1.454 1.485 1.468 1.458

a Charge distribution parameters taken from Ref. [42, 43]; for 2*4Pu the values were extrapolated u Values calculated with the computer code MUON 2 (obtained from Los Alamos)

trons. In particular, this method is free of background originating from m u o n capture in the materials surrounding the target. It is also not influenced by the decay or absorption of those muons attached to fission fragments after prompt fission. A precise correction of the mean life times as determined from electron decay and neutron measurements for those muons attached to one of the fission fragments is possible only if the number of prompt fissions per muon stop is known. A lower limit of 90% for the probability of /~--attachment to the heavy fragment has been determined [39, 40]. With the known fission probability per muon stop it is then possible to correct the measured life time "c. Since the mean life time of a muon attached to a heavy fragment is about 130 ns [40], the measured life times -c for the isotopes 237Np, 239pu, 2 4 2 p u and 244pu should be too high by 0.6 to 1.2 ns when detecting neutrons and by about 2 to 4 ns when detecting the decay electrons, respectively. Values obtained with these corrections to the mean lives z are in agreement with the experimental results of Wilcke et al. [32], who corrected the measured values ~, for 2 3 9 p u and 242pu, respectively. The value for 237Np in Table 3 from [32] could not be corrected in this way, and is therefore too high by about 1 ns. The experimental results of Schr6der et al. [33] are not influenced by neutrons originating from the absorption of muons by the heavy fragments, because these neutrons could be discriminated against the others.

111.5 115.5 103.2 95.5

Calculated Refs. 13, 31 [105]

Calculated Ref. 41 [105]

138 139 128

134.9 139.1 120.0 123.1

Regarding the measured mean lives r from the e--decay of bound muons, experimental results are, for the isotopes under discussion, only published for z39pu [37, 38] (Table 3). The value from Hashimoto et al. [37] is higher by about 7 ns than the results from fission experiments. It is also higher in comparison with the result of Johnson et al. [38] by about 4 ns, although these results agree within statistical errors. Johnson et al. [38] have corrected their life time -ce for copper contaminations in the target; by this correction the influence of decay electrons from muons attached to fission fragments was also included. A reduction of the value of z(239pu) obtained by Hashimoto et al. [37] by about 4 ns, as discussed above, gives good agreement with the result from Johnson et al. [38] as well as with the data from the fission fragment measurements. This correction has to be considered when comparing the mean values of the results from the electron and from the fission decay measurements. F r o m the averaged values of the mean life times, as determined from the fission channel zz, listed in Table 3, and using the corresponding total disappearance rates it is now possible to determine the mean life times for nuclear capture, taking into account the mean life of positively charged muons (r,+ = 2.19703 gs). The results are listed in Table 4. Table 5 a shows the averaged experimental results of muon capture life times zc for neptunium and plutonium isotopes together with the calculated values of AGp from the formula of Goulard and Primakoff given above (see also Table 5 b for parameters used to calculate Amp). The numbers obtained with this formula are too low for all the measured actinide isotopes by about 20% (see also [41]). Also included are calculated results obtained by Wilcke et al. [31] from the giant resonance (GR) model [13], although the special assumptions on which these calculations are based, like position and strength of the giant resonances, have not yet been published. In [41], using the formula of Goulard and Primakoff [-9] a threeparameter fit is performed to the data compiled by Suzuki et al. [6]. The values given in the last column of Table 5 a are obtained with this set of parameters. As seen in Table 5a there is evidence for an

404


Table 6. Ratios of muonic capture rates determined from the fission channel for the plutonium isotopes (see Table 5a) Experiment

z42Pu/Z39pu 244pu/Zr

244pu/e39pu

0.929 + 0.016 0.961 _+0.011 0.893 +0.016

Calculation Goulard Primakoff

Calculation

Calculation

Refs. 13, 31

Ref. 41

0.894 0.925 0.827

0.921 -

0.927 0.954 0_885

i s o t o p e effect o f the m u o n c a p t u r e rate in the a c t i n i d e region, see also [41]. This is p r e d i c t e d b y the f o r m u l a o f G o u l a r d a n d Primakoff. It is m a i n l y d u e to the b l o c k i n g of n e u t r o n states b y e x t r a n e u t r o n s for t h o s e o r i g i n a t i n g b y n e g a t i v e m u o n c a p t u r e on a p r o t o n . A n increase in the n e u t r o n n u m b e r reduces the # - - c a p t u r e p r o b a b i l i t y . F o r the P u isotopes, the ratios of the c a p t u r e rates listed in T a b l e 4 as determ i n e d f r o m the fission c h a n n e l a n d f r o m the m o d e l c a l c u l a t i o n s are c o m p a r e d in T a b l e 6. N e i t h e r a r e the a b s o l u t e values well r e p r o d u c e d b y the G o u l a r d P r i m a k o f f f o r m u l a [-9, 32] with the o r i g i n a l p a r a m e ters, n o r a r e the r a t i o s of the c a p t u r e rates well predicted. H o w e v e r , using the fit p a r a m e t e r s given in [41] values are c a l c u l a t e d w h i c h a r e in m u c h b e t t e r a g r e e m e n t with the e x p e r i m e n t a l results as is s h o w n in T a b l e s 5 a a n d 6. T h e results o b t a i n e d in the f r a m e w o r k of t h e G R m o d e l [13, 31, 32], b y c h o o s i n g p r o p e r r e s o n a n c e p a r a m e t e r s , a r e in g o o d a g r e e m e n t with the e x p e r i m e n tal values. A d e s c r i p t i o n of m u o n c a p t u r e in t e r m s of the statistical m o d e l of Singer [14, 15] a n d of the m i c r o s c o p i c m o d e l of A u e r b a c h a n d K l e i n [16] has n o t been d e v e l o p e d far e n o u g h as to p r e d i c t the isotopic dependence. T h e d a t a are, h o w e v e r , t o o scarce to d r a w a n y firm c o n c l u s i o n a n d d i s t i n g u i s h b e t w e e n the a p p l i c a bility o f these models. Therefore, a d d i t i o n a l s y s t e m a t ic m e a s u r e m e n t s o n i s o t o p i c effects in m u o n c a p t u r e are r e q u i r e d as well as m o r e refined t h e o r e t i c a l des c r i p t i o n s o f the m u o n c a p t u r e p r o c e s s are necessary.

It is a pleasure to thank Professor J.P. Blaser and his staff for their encouraging support and for the excellent working conditions at PSI. We are indebted to the following institutes or organizations for financial support: Delft University of Technology (Jd'AvE), Bundesministerium fiir Forschung und Technologic der Bundesrepublic Deutschland contract number 06 BN 271 (PD, HH, JH, HJ, TM-K, RvM, FR, CFGR, WS), the Swedish Natural Science Research Council (TJ, GT), Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization of the Advancement of Pure Research (ZWO), (JK, CTAMdL, AT), the University of Warsaw, Bialystok (TK) and the Schweizer Nationalfonds (CG, LAS, LS). We are grateful to Professor H. Schopper and Dr. A.M. Wetherell, CERN, for putting to our disposal a Hewlett Packward HP 1000 Computer during the first stage of this experiment.

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405 P. David, H. Hfinscheid, J. Hartfiel, H. Janszen, T. Mayer-Kuckuk, R. von Mutius, F. Risse, Ch.F.G. R6sel, W. Schrieder Institut ffir Strahlen- und Kernphysik Universit~it Bonn Nussallee 14-16 D-5300 Bonn 1 Federal Republic of Germany C. Petitjean, H.W. Reist Paul Scherrer Institute (PSI) formerly SIN CH-5234 Villigen Switzerland S.M. Polikanov Gesellschaft ffir Schwerionenforschung, GSI D-6100 Darmstadt Federal Republic of Germany J. Konijn, C.T.A.M. de Laat, A. TaM NIKHEF-K, Amsterdam Postbus 41882 1009 DB Amsterdam The Netherlands T. Krogulski Department of Mathematics and Nature University of Warsaw Branch in Bialystok 41 ul. Lipowa PL-15-424 Bialystok Poland T. Johansson The Swedberg Laboratory S-75121 Uppsala Sweden G. Tibell Department of Radiation Sciences S-75121 Uppsala Sweden J.F.M. d'Achard van Enschut Laboratorium voor Technische Natuurkunde Technische Universiteit Delft 2600 GA Delft The Netherlands and Philips Laboratorium Eindhoven, The Netherlands J.P. Theobald Institut ffir Kernphysik Technische Hochschule Darmstadt D-6100 Darmstadt Federal Republic of Germany N. Trautmann Institut ffir Kernchemie Universit/it Mainz D-6500 Mainz Federal Republic of Germany C. Gugler, L.A. Schaller, L Sehellenberg Institut de Physique Universit6 de Fribourg CH-1700 Fribourg Switzerland