Structure in the excitation functions of (3He, α) reactions on a number of alpha-type nuclei

2.C

I

Nuclear Physics A132 (1969) 9--27; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprintor microfilmwithout writtenpermissionfrom the publisher

S T R U C T U R E I N T H E E X C I T A T I O N F U N C T I O N S O F t H e , ~) R E A C T I O N S ON A NUMBER OF ALPHA-TYPE NUCLEI C. M. da SILVA t, j. O. NEWTON, J. C. LISLE and M. F. da SILVA t Schuster Laboratory, University of Manchester, Manchester 13, England Received 19 March 1969 Abstract: Excitation functions have been measured for the reactions ~60(aHe, ~)tsO and 2aSi (3He, cQ278i over a bombarding energy range 4.0-9.0 MeV and for the reaction 24Mg(aHe, e) 2aMg from 4.0-6.2 MeV. An analysis of the data using the statistical model of Ericson has been attempted, and coherence widths of 130 i 2 0 keV for 19Ne' 125 ~ 30 keV for 27Si and 102 ± l 0 keV for a ~S are found. The reactions appear to proceed predominantly by a direct mechanism. However the assumptions of the Ericson theory are not fully satisfied, since a wider structure is tbund in the experimental data. Possible interpretations of this structure are discussed.

E

NUCLEAR REACTIONS 160, 28Si(aHe, ~), E -- 4-9 MeV; 24Mg(aHe, 00, .E =4-6 MeV; 25 keV steps, measured a(E; E~, 0 ). 19Ne, 278i, 31S resonances deduced/', autocorrelations C(e), cross-correlations C(e, ct'). Natural targets.

1. Introduction O n the basis of a simplified statistical model, Ericson 1, 2) a n d Brink a n d Stephen 3) predicted that the c o m p o u n d nucleus cross section in the region of overlapping levels should fluctuate as a consequence of interference between these levels. Since that time, m a n y experiments have shown that such fluctuations do indeed occur a n d some of them show fair accordance with the results of this model. However, M o l d a u e r 4) has p o i n t e d out that the a s s u m p t i o n s o n which this model is based are n o t necessarily in accordance with those which would be required by a m o r e correct model based on the R - m a t r i x method. W h e n strongly absorbed channels are present, the two models are m a r k e d l y different, that based o n the R-matrix giving a m u c h wider d i s t r i b u t i o n of resonance pole widths a n d amplitudes t h a n is assumed ad hoc in the other model. This has the consequence that, even when the average width F of the resonance poles divided by the average spacing D is m u c h greater than unity, some of the poles will have both m u c h greater widths a n d amplitudes t h a n the average. They will therefore tend to d o m i n a t e the cross section in their vicinity a n d give rise to a " m o d u l a t i o n " of the average cross section which, a p a r t from the fine structure fluctuations, would be expected to change only rather slowly a n d smoothly with excitation energy in the simple model. Such strong poles would affect the cross section a n d have similar properties to the intermediate resonances predicted by the doorway-state models s. 6). t Present address: Laboratorio de Fisica & Engenharia Nucleares, Sacavem, Portugal. 9

10

c.M. DA SILVAet

al.

It should however be noted that the latter are based on detailed models of nuclear structure, whereas the R-matrix model has a purely statistical basis. In fact, the Rmatrix model predicts that in the presence of strongly absorbed channels, there may be a broad spectrum of "intermediate resonance widths and peak heights" which merges into the fine structure peaks without any clear demarcation. In these circumstances, any analysis on the basis of the simple model will be of no value. Unfortunately, the R-matrix model has not yet been sufficiently developed for any analysis of data to be made in terms of it, and one has still to use the Ericson model. At present, the main interest in any "fluctuation analysis" is probably to find deviations from the predictions of the simple model and to try to interpret these. The experiments reported here are concerned with (aHe, cQ reactions on 160, 24Mg and 2Ssi nuclei. Because both 3He and 4He are composite particles, one might expect that strong absorption would play an important part in these reactions and indeed one finds that one cannot interpret the results on the basis of a single fluctuation width as would be expected on the Ericson model. The 160(3He, ~)15 O reaction has been previously studied in the 2-3.2 MeV range of bombarding energy and the results interpreted on the basis of the single-level BreitWigner formalism 7). Measurements at higher energies up to 10 MeV have indicated that there is a strong and probably dominant direct component to the cross section [refs. s - 10)]. The 2Ssi(aHe, cQ27Si reactions have been studied 11-13) by a number of authors at bombarding energies above 8 MeV. The results suggest that the dominant reaction mode is direct, but that compound nuclear effects are not negligible.

2. Experimental method The experiments were carried out using the Manchester University 6 MeV Van de Graaff accelerator in the bombarding energy range 4-6 MeV and the Liverpool Universit3 Tandem accelerator in the range 6-9 MeV. The 3He beam was analysed in both cases with a 90 ° deflecting magnet and collimated to produce a spot approximately 2 m m in diam. The magnetic field was measured using a proton resonance magnetometer, the energy calibration having been established previously. Self-supporting targets of silicon monoxide of approximately 40 tlg • c m - 2 thickness were used for the 160 and 2Ssi experiments. They were prepared by vacuum evaporation onto glass slides covered with a thin layer of "teepol"; this acted as a release agent when the thin films were floated off in water. The magnesium targets were approximately 28/~g • c m - 2 in thickness and were obtained by vacuum evaporation of known quantities of pure metal onto very thin 5-10 #g • c m - 2 carbon foils previously outgassed by gentle heating 14). For both target% thickness measurements were carried out using the ~-gauge technique 15). The targets used were placed in the centre of a scattering chamber at 45 ° to the beam producing a beam energy spread of less than 30 keV in the SiO target and less than 20 keV in the Mg target.

(3He, 30 REACTIONS

I1

The reaction products were detected with silicon surface-barrier counters placed at angles of 55 °, 125 ° and 165 ° with respect to the beam direction. Each counter was provided with a set of defining and field limiting collimators, and it was estimated that the solid angle subtended by the defining collimators was 6.1 x 10 . 4 st. The pulses developed in the detectors were fed through charge-sensitive preamplifiers to the main amplification stage and then recorded in a 1024-channel pulse-height analyser. The detector bias and the integration and differentiation time constants were kept low to minimize the pulse height of the protons associated with competing reactions, thus reducing the background in the region of the alpha-particle spectra. Typical particle spectra are shown in fig. 1. 24Mg (3He, O. )23Mg

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(3He,~x) REACTIONS

15

Runs were normalized to the integrated beam current; the consistency of different runs was ensured by making check runs at a standard energy at regular intervals throughout the experiment and making use of the intense elastic peaks for comparison. Each excitation function was first measured in 50 keV steps. The same range of energy was then again covered in 50 keV steps but shifted by 25 keV. This procedure, therefore, gave both a step interval of 25 keV and also acted as a further reproducibility check on the excitation functions. In the case of the SiO target, about 200 keV of overlap between the separate data from the 6 MeV Van de Graaff and that fi'om the tandem was allowed for normalization purposes. The counting dead time was kept below 5 ~o, and corrections were made whenever it exceeded 2 %. 3. Results and analysis of data The cross sections for the (SHe, ~) reactions on 160, Z4Mg and 28Si shown in figs. 2 and 3 fluctuate considerably as functions of energy. In all cases one sees a rapidly fluctuating component with peaks having widths of about 100 keV. This component appears to be superposed on an "average cross section" which also varies with energy. In some cases this average cross section also fl,~ctuates with energy, though with a much larger characteristic width. In others, as for example the Si c~0 group at 5 5 , it rises fairly smoothly with bombarding energy and can be well accounted for by a Hauser-Feshbach calculation 16). The level spacings in the nuclei ~gNe, 27Si and 31S at the excitation energies with which they are formed in these reactions are such that one would not expect to see individual resonances of the compound systems. The ranges of excitation energies spanned in these experiments are 11.8-16.0, 17.1-18.8 and 16.0-20.5 MeV for ~gNe, 27Si and 3~S, respectively. One can make rough estimates for the level spacings D in these regions of energy from Newton's formula ~7) suitably modified to take into account the variation of level spacing with angular momentum. One finds, for example, maximum level density in a ~S for J = ~; witia a corresponding spacing of about 10 keV and in 19Ne for J = ~ o r ~ with spacing of about 27 keV. Thus if the narrower fluctuation width is a good indication of the average level with F, then it seems likely that in all of our three cases the quantity F/D is greater than unity but not by more than an order of magnitude. In view of this, one might expect that the narrow structure arises mainly from interference between the overlapping states of the compound system in a similar way to that proposed by Ericson t). The small values of F/D make the strict application of Ericson's theory somewhat dubious, though Dallimore and Hall 18) have shown that such an analysis is useful even down to values of 2 for F/D. The presence of the wider structure indicates moreover that Ericson's assumption of random phases for the matrix elements is incorrect. In spite of this we have, as have others in other cases, analysed our data in terms of the simple fluctuation model of

c.M. DA SILVAet al.

16

Ericson and of Brink and Stephen, after attempting to allow for the effect of the wider structure on the average cross section. This has been done because the more accurate R-matrix model is very difficult to apply. Some limited justification for this procedure is given by the fact that the values for F obtained by this analysis are in reasonable agreement with those expected from the systematics of the variation of F with A and excitation energy 19). Nevertheless, any conclusions drawn from this and similar analyses must be taken with some reserve. The excitation functions on oxygen and silicon were taken between bombarding energies of 4 and 9 MeV, whereas that on magnesium was only taken between 4.2 and 6.1 MeV. Because of the limited range of the latter, only the first two have been analysed in full detail. It is convenient to analyse separately the higher and lower energy data, partly because of overlap of oxygen and silicon alpha groups and partly because there are indications that the relative amplitude of the finer structure may be reduced in the higher energy part of the range. The data were, therefore, analysed in two sets, e.g. one ranging from 4 to 7 MeV in bombarding energy and the other from 7 to 9 MeV. In cases where there were data available from 4 to 9 MeV, an analysis was made of the whole range as well, and the results compared with the partial-range analyses. 3.1. METHOD OF ANALYSIS

3.1.1. Autocorrelation functions The values of F were derived from the autocorrelation widths and from the method of peak counting. The autocorrelation function C(e) is frequently estimated from the expression C(e)= - 1 . (1) This estimate is only useful if the cross section averaged over a number of the fine structure peaks is approximately independent of bombarding energy over the energy range of interest. I f this energy range were infinite, then the autocorrelation function would be expected to have the Lorentzian form F2 C(e) = C ( O ) - - . F 2+ e 2

(2)

This tends to zero as e becomes much larger than F. When the range is finite, as in all practical cases, C(e) is expected to follow the Lorentzian form for e < F, whilst for e >> F, it is expected to oscillate about zero. I f however the averaged cross section is not constant but modulated, then C(e) will oscillate around some non-zero value which will itself be a function of e. In this case, there m a y be difficulty both in determining the values of F and of C(O) from the plot of C(e). It is then sometimes convenient to try to separate out the effects of the modulation from the statistical fluctuations. If we can sensibly describe the observed cross section in the form

cr'(E) = a(E)~r(E),

(3)

(3He,~t) REACTIONS

17

where a(E) represents the non-statistical modulating function and a(E) the statistically fluctuating cross section, it is easy to show 20,21) that the experimental autocorrelation function C(e) is related to the "true" autocorrelation function by

1 + C' (~) = A(E)[1 + C(e)].

(4)

The quantity A(E) is expected to be fairly constant over a range provided that the period of the modulating function is much greater than that of the statistical fluctuations. If this is so, then it can be shown 2o) that

C'(O)- C'(~) 2

1 C(e) + 1 - A(~) F2

F2

(5)

Hence, to the extent that A(e) can be considered to be constant, we obtain a linear relation if we plot the left-hand side of expression (5) against C(e). Since we are now dealing with differences in C(e), it is most important that errors introduced by the counting statistics are properly taken into account 20). If a linear plot is obtained we can derive values for F and A. A value for C(0) can then be calculated with the aid of eq. (4). If the plot is not linear, the "period" of the modulating function is comparable to that of the statistical fluctuations in which case an analysis in terms of the simple Ericson model has no significance. In the simple Ericson model C(0) is approximately related to the fluctuation damping coefficient N and to the proportion of direct reaction 7 by c(o) =

1 (1-72).

(6)

The quantity N is related to the number of independent contributing modes in the compound nucleus cross section. The maximum number of such modes which can be contributed to a given reaction is given by 22) Nma x =

O.5(2J+1)(2S+1)(2j+1)(2s+1),

or

(Nma,+k),

if Nmax is half integral,

(7)

where J, S and j, s are the spins of the nucleus and particle in the ingoing and outgoing channels, respectively. At certain angles of observation, the number of magnetic substates which can be populated in the reaction may be severely limited in which case N will be much less than Nma x [ref. 23)]. A n example of this is given by the 28Si(3He, ~) 27Si reaction when the alpha particles are observed at 180 ° to the beam direction. Here only the m = + ½ substates of the final nucleus can be populated, because both the incoming 3He and outgoing alpha particles have zero component of angular momentum along the beam axis. In this case the value of Nis unity, whereas Nmax= 6. One of the interesting quantities which can, in principle, be derived from a fluctuation analysis is 7, e.g. the proportion of the reaction which goes by direct interaction.

18

C . M . DA SILVA

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(3He,~) REACTIONS

02I

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20

C . M . D A SILVA

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(3He, ~) REACTIONS

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C(E) Fig. 5a, b. E x a m p l e s o f auto-correlation functions. O n the left-hand side C(e) is plotted against e in the n o r m a l way for a n u m b e r o f cases. T h e c o r r e s p o n d i n g plots o f (C(O)--C(e))/e z against C(e) are s h o w n on the r i g h t - h a n d side. T h e latter d i a g r a m s are expected to exhibit a linear relationship if Ericson fluctuation theory holds.

¢. M. DA SILVAet aL

22

Since C(0) depends both on N and 7 [see eq. (6)], it is not usually possible to obtain values for these two quantities from C(0) alone. However, in principle it is possible to derive values for N and 7, if one also studies the probability distribution P(x), where x = a(E)/
=

< j(E +

_ 1.

According to the simple theory, Cij should be zero both when i a n d j refer to different groups of particles and when they refer to the same group of particles taken at angles 01 and 0z with 0t - 0 a >> (kR) -~ [ref. 23)]. Here k is the wave number of the incident particle and R the radius of the target nucleus. The effect of the finite range of data will be to cause the function to oscillate about zero. For the case of no direct interaction, the standard deviation a of F R D oscillation has been given by Allardyce et al. 21) as

where the data energy range is given by nF. No expression has been given, to our knowledge, for the case where a direct interaction component is also present. How-

(3He, o:) REACTIONS

23

ever, such a c o m p o n e n t will reduce the a m p l i t u d e of the F R D oscillations. Since there is little difference in most cases between the p r o b a b i l i t y distributions P ( x ) for large N and zero 7 a n d for smaller N a n d finite y, we m i g h t expect that we could get a n approximate value for a from the above expression for cr in terms of C(0), even when a direct c o m p o n e n t is present. I n our cases, the value of n is a b o u t 30, whereas C ( 0 ) varies from a b o u t 0.1 to 0.8. Thus if we take C1(0) = C2(0) = 0.4, for example, we would expect the s t a n d a r d deviation of the fluctuations to be a b o u t 0.09 if thv simple theory applied. A few typical cross correlations, uncorrected for the effects of m o d u l a tion, are shown in fig. 4. It can be seen that the oscillations are m u c h larger than would 55 ° }

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24

c.M.

D A S I L V A et al.

be expected on this basis. They are moreover much wider than would be expected on the basis o f F ~ 100 keV. Both of these results can be explained if a component with larger F, say 0.5 to 1 MeV, is also present. The amplitude of the oscillations for this component would be larger because the sample size n is smaller. The presence of such components in some of the excitation functions can be seen by visual inspection and rather better by averaging them over an energy interval. 3.2. ANALYSIS OF THE 4-7 MeV DATA FOR THE REACTIONS 160(aHe, ct)lsO AND 2sSi(3He, ~)z7si

The data on the ~60(3He, ~0)150 and 2Ssi(3He, c~1,2)275i reactions at the three angles of observation were analysed by these methods. Of the 12 cases, ten show three or more points in a straight line on tile plots of eq. (5) before the autocorrelation breaks down. Some of the results are shown in fig 5. Values for A and F were obtained by least-squares fits to the points which appeared to lie on straight lines. It should be noted that only the points corresponding to the range of e from zero to the value when C ( e ) becomes zero are plotted in fig. 5. As a check on the above procedure, values for A(e) were estimated by an alternative method, which involved the averaging of the cross sections over intervals which were large compared with F. With averaging intervals of 500 and 1000 keV good agreement was found between the two methods with one exception. Moreover when plots were obtained with four or more points on a straight line, it was found that the A(e) thus obtained varied only by about 0.l ~o over a range of 3F. However, when non-linear plots occurred the A(e) varied by as much as 1 ~ over this range. Some of the contour plots from which the values o f N a n d ? allowed by the measurements of C(0) and P(x) can be obtained are shown in fig. 6. In most cases, there is an ambiguity which allows N = 1 and a large value of ? or else ? ~ 0 with some range of N. It is possible, however, to rule out some of the cases where ? ~ 0 because the values of N required are greater than those allowed by eq. (7). In the 160(3He ' Cto)l 50 reaction at 55 ° and 125 °, Nmax = 2 whereas, for ? ,.~ 0, values for N > 4 would be required. In the 28Si(3He, c~0)27Si reaction at 55 °, Nmax = 6 whereas N > 10 would be required for ? ~ 0. The others are the 165 ° cases where N must be approximately unity owing to the proximity to 180~; that N is indeed close to unity is well supported by the plots. The numerical results obtained from the analysis of the 4 to 7 MeV data are given in table 1. In all of the cases where the N, ~ ambiguity could be overcome, a rather high value for ? is indicated, and it seems fairly safe to assume that there is a rather high direct contribution to all of these reactions. Values for F of about 100 keV and 135 keV for the 3~S and 19Ne compound nuclei were obtained. The corrections for F R D biases [ref. 21)], finite-energy resolution and spacing of measurements 24), were found to be small compared to the F R D errors. There were estimated to be of order 20 ~ and 40 ~ for F and C (0), respectively, when obtained from the autocorrelation analysis. The agreement between the values of F obtained by the latter method, and the peak

55 125 165

55 125 165

55 125 165

55 125 165

27Si~o

27Si~1

275i~2

(deg.)

0~ab

1~0~o

Group

0.675 0.457 0.960

0.446 0.464 0.344

0.389 0.304 1.259

0.330 0.247 0.478

C(O)

115 142 132 100 103 94 111 88 91 116 97 122

0.092 0.152 0.838

0.367 0.436 0.449

0.312 0.213 0.687

(keV)

F

0.145 0.117 0.454

C(O)

Fit

1.277 1.20l 1.162

1.058 1.019 0.927

1.272 1.132 1.229

1.161 1.117 1.016

A

A

1.276 1.279 1.280

1.181 1.163 1.005

1.257 1.131 1.480

1.160 1.121 1.015

with zl = 1 MeV

Computed

4-7 MeV

size

Sample

103 110 116

97 114 94

103 111 110

127 137 110

30 30 30

30 25 13

30 28 30

24 22 10

peak approxic o u n t i n g mately

from

l'(keV)

0.112 * 0.254

0.232 * 0.320

0.051 0.246 0.171

0.072 0.157 0.255

C(O)

F

97-131 * 114

101-125 * 90

94 60-162 67-168

111 177 <169

(keV)

7-9 MeV

Measured parameters and related i n f o r m a t i o n for the (3He, ~) reactions on 160 and 28Si

TABLE 1

74-114 94-112 106-117

97-136

0.690 0.365

0.408

123 > 137

F (keV)

0.368

0.376 0.443

C(O)

4-9 MeV

26

C. M. D A S I L V A e t

al.

counting method is good. It has been suggested that the peak counting method is more reliable and accurate than the autocorrelation method 24). According to this, the error for the peak counting method would be about 15 ~ . However, the error will almost certainly be greater than this value, because the statistical errors on the individual points of the excitation functions make it sometimes difficult to decide whether a peak is present. 3.3. ANALYSIS OF THE 7-9 MeV DATA FOR THE REACTIONS 160(aHe, ~¢)150 AND

28Si(aHe, a¢)27Si

Autocorrelations were computed for this range of data. The values of C (0) together with graphical estimates for F derived by the method of eq. (5) are compared in table 1 with those derived from the 4-7 MeV data. The values for C(0) are clearly lower than those for the lower range of energy indicating, as can be seen from the excitation functions themselves, that the fluctuations are more damped in this region. Presumably this is because there is a higher proportion of direct contribution to the cross section in this region. Such an effect might be expected since the higher the excitation energy, the greater the number of open channels over which the compound nucleus cross section must be spread, whereas the direct reaction cross section to a particular final state will be expected to vary only slowly with bombarding energy. In only two cases were reasonably straight lines obtained for the plots (see fig. 5). This may be partly due to the small values for C(0) resulting in the F R D errors being relatively more serious and possibly because the conditions required for this type of analysis to be useful are not satisfied. It is certainly clear that in neither of the two regions are the conditions required for the application of Ericson's simple theory fully satisfied since the average cross section itself is fluctuating. However, it would seem that an analysis in terms of this theory might have somewhat more meaning in the lower energy region than in the higher one. 3.4. THE 24Mg(aHe, et)2aMg REACTION The excitation functions are shown in fig. 3. Their general appearance is similar to that for the other reactions. The very prominent peak in the 165 ° excitation function is notable. Only the peak counting method was used to estimate a value for F. Allowing for the experimental energy resolution and spacing of the measured points, a value of about 125 keV was obtained. The error on this value is expected to be not less than 20 ~ . 4. Conclusions

An analysis of these (3He, ~) reactions within the framework of the Ericson model gives only moderate agreement with the expectations of the model. In nearly all cases, in addition to the narrow structure usually attributed to Ericson fluctuations, a structure with greater widths not expected on the Ericson model is also present. In some cases it appears to be possible to separate the wide and narrow structure but in other

(3He,~) REACTIONS

27

cases it does not. The presence of the wider structure m a y be explainable o n the more complete statistical theory suggested by M o l d a u e r 4) or on the basis of d o o r w a y states [refs. 5, 6)]. It is worthy of note that in the (3He, c~) reactions on the alpha-type model 160, 24Mg a n d 28Si a n d also on 12C [ref. 25)], a very p r o m i n e n t peak occurs in the 165 ° excitation functions in the vicinity of 5 MeV b o m b a r d i n g energy. In all cases the width is a b o u t 400 keV. It is not, at the m o m e n t , clear whether there is any significance in the occurrence of these peaks in all four cases and, if there is any, what is the exp l a n a t i o n of them. Insofar as the m e t h o d of analysis is valid, we find a value of 1 3 0 + 2 0 keV for the coherence width F of the 19Ne c o m p o u n d system in the range of excitation energy 12--15 MeV. For 278i in the range of excitation energy from 17-19 MeV, we find F = 1 2 5 + 3 0 keV a n d for 3IS, F = 102+ I0 keV in the range of excitation energy 16-19 MeV. The values for 27Si a n d 31S are rather higher t h a n those obtained for n e i g h b o u r i n g nuclei at similar excitation energies where values of 50-60 keV have been o b t a i n e d 19, 2t). In the cases where it was possible to come to an u n a m b i g u o u s conclusion, the direct interaction c o m p o n e n t was f o u n d to be high in agreement with other experimental data. We would like to t h a n k D. J. Jacobs for help in the early measurements on the 24Mg(3He, c~)Z3Mg reaction. Two of us (C.M. da S. a n d M. F. da S.) carried out this work while oil leave of absence from the L a b o r a t o r i o de Fisica & E n g e n h a r i a Nucleares, Sacavem, Portugal. References

11 T. Ericson, Advan. Phys. 9 (19601 425 2) T. Ericson, Ann. of Phys. 23 (19631 390 3) D. M. Brink and R. O. Stephen, Phys. Lett. 5 (19631 77 4) P. A. Moldauer, Phys. Rev. Len. 18 (19671 249 5) B. Block and H. Feshbach, Ann. of Phys. 23 (19631 47 6) A. K. Kerman, L. S. Rodberg and J. E. Young, Phys. Rev. Lett. 11 (1963) 422 7) D. A. Bromley et aL, Nucl. Phys. 13 (19591 1 8) S. Hinds and R. Middleton, Proc. Phys. Soc. 74 (19591 775 9) W. P. Alford eta[., Nucl. Phys. 61 (19651 368 10) R. L. Hahn and E. Ricci, Phys. Rev. 146 (19661 650 ll) B. H. Wildenthal and P. W. M. Glaudemans, Nucl. Phys. A92 (19671 353 121 S. Hinds and R. Middleton, Proc. Phys. Soc. 75 (19601 444 13) L. W. Swenson et al., Nucl. Phys. A90 (19671 232 141 A. H. F. Muggleton, private communication (1965) 151 D. I. Porat and K. Ramavataran, Nucl. Instr. 4 (19591 239 16) H. Feshbach, in Nuclear spectroscopy, Part B (Academic Press, New York, 19601 p. 665 171 T. D. Newton, Can. J. Phys. 34 (19561 804 181 P. J. Dallimore and I. Hall, Phys. Lett. 18 (19651 138 19) T. Ericson and T. Mayer-Kuckuk, Ann. Rev. Nucl. Sci. 16 (19661 183 20) C. M. da Silva, Ph.D. thesis, University of Manchester (19671 unpublished 211 B. W. Allardyce et aL, Nucl. Phys. 85 (19651 193 22) T. Ericson, Phys. Len. 4 (19631 258 23) D. M. Brink, R. O. Stephen and N. W. Tanner, Nucl. Phys. 59 (19641 577 24) A. van tier Woude, Nucl. Phys. 80 (19661 14 25) R. S. Blake, D. J. Jacobs, J. O. Newton and J. P. Shapira, Nucl. Phys. 77 (19661 254