A fluctuation analysis of the reaction 31P(p, α)28Si

Nuclear Physics AI08 (1968) 150--176; (~) North-Holland Pubhshing Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permtsslon from the publssher

A FLUCTUATION ANALYSIS OF THE REACTION 31p(p, ~)2SSi P. J. D A L L I M O R E t

and B. W. A L L A R D Y C E

Nuclear Physics Laboratory, Oxford Received 27 July 1967 Abstract: The reaction 3up(p, c~)..sSt has been mvestsgated for alpha particles detected at 13 angles in the proton energy range 8.50-12.30 MeV, corresponding to excitations in the compound nucleus ~2S of between 17.36 and 21.16 MeV. The excitation functions were measured in ten keV steps. The two ~-particle groups ~0 and ~1 were recorded. All of the data, i.e. 26 excitation functions each o f 380 points, are quantitatively in agreement w~th the Hauser-Feshbach stat~stlcal assumptions including statistical fluctuations. N o evidence ~s found for direct interaction or for doorway states. The mean level widths of the compound nucleus for decay to the ground and first excited states o f zsS1 have been found to be 47-.:7 keV and 4 2 - 5 keV, respectively. The statistical dependences of these values on the range of data n and on the experimental values of the autocorrelation are discussed and found to be important considerations.

E I

I

N U C L E A R R E A C T I O N S 31p(p, ~), Ep = 8.50-12.30 MeV; measured a(Ep; E,, 0). ] Fluctuation analysis. '~aS levels deduced 1, autocorrelations, cross correlations. Natural targets. I

1. Introduction At low excitation energies, nuclear reaction cross sections exhibit resonances as a function of energy. In general, these resonances are well separated and the mean level width F is less than the mean level spacing D. As the excitation energy of the compound nucleus increases, the level widths increase and the level spacings decrease so that eventually a region of excitation is reached where the levels are overlapping. With further increase in excitation energy, the overlapping becomes extreme so that F >> D; this occurs at about 20 MeV excitation energy in nuclei of around mass 30. At these energies sevcral authors ~t have observed the differential cross section to fluctuate rapidly with energy with a width characteristic of the mean level width in the compound nucleus. The existence of these fluctuations was predicted by Ericson (refs. 2.3)) and by Brink and Stephen 4) using the assumption of random phases between the scattering amplitudes. Excitation functions have been measured for the reaction 31p(p, ~)28Si using the proton beam from the tandem Van de Graaff at the Oxford University Nuclear Physics Laboratory to test the predictions of the theory of fluctuations and to investigate the effects of a finite range of data (FRD) on the statistical analysis. The N o w at Austrahan National Laboratory, Canberra. *'r See ref 1) for earher references. 150

alp(p, d)2sSl REACTION

151

excitation functions were measured in 10 keV steps with protons in the energy range 8.5-12.3 MeV, corresponding to excitation energies in the compound nucleus a2S of between 17.36 and 21.16 MeV. Since one of the basic requirements for the fluctuation theory to be valid is that F > D, it is necessary to know whether this condition applies at these excitation energies. Hooton 5) has investigated this reaction for proton energies up to 9.5 MeV and has carried out a fluctuation analysis for the range 7.5-8.5 MeV. The results obtained for the autocorrelation and fol the channel and angular cross-correlations all indicate that the fluctuation analysis is applicable in this region; i.e., the requirement that F > D is satisfied at these energies. Katsanos and Huizenga 6) have also investigated the reaction in the three energy ranges 8.37-9.00, 10.00-10.90 and 10.90-11.87 MeV using the Ericson theory of fluctuations. Another requirement of the theory is that the mean cross section remains constant with energy. However, for the results of the present experiment this is not so, and corrections have had to be made to the basic experimental excitation functions. Energy variations of the mean cross section may be due to either a varying direct component, intermediate structure or the Hauser-Feshbach (HF) decay of the compound nucleus. The analysis of the results of this experiment indicate that the energy dependence is due only to the H F decay, and theoretical estimates of its effect have used the Hauser-Feshbach theory T); this enabled the basic experimental excitation functions to be corrected for their energy dependence by being divided by the calculated Hauser-Feshbach cross sections. In the following sections the basic experimental differential cross sections are defined as acxp(E)and the cross sections corrected for the Hauser-Feshbach decay are defined as a¢(E). The ranges of data are expressed in terms of either the number of data points p or the number of level widths n.

2. Experimental procedure The experimental technique is similar to that described elsewhere s). The proton beam entered the scattering chamber through a series of tantalum collimators and an annular counter. After passing through the target it was collected in a Faraday cup and the integrated current measured. Beam currents of from 0.1 to 1.0/~A were used and the total charge accumulated for each data point was usually 150/~C. Alpha particles up to an energy of 13.5 MeV were detected by Au/Si surface barrier detectors. The data were recorded in two runs. The first covered the proton energy range 8.5-11.6 MeV with the alpha particles detected at angles of 90 °, 120 °, 135 °, 150 ° and 177 ° in the lab system. The second run cove~ed the range 8.5-12.3 MeV with the alpha particles detected at 44 °, 59 '~, 74 '~, 90", 105 °, 143 °, 157 ° and 169 °. The repetition of the 90 ° (lab) results enabled the reproducibility to be checked and as different targets were used in each run it allowed the target thicknesses to be normalised. The angles could be set to +0.5 °, and the angular resolution was less than 2 °. At the forward angles only the ground and first excited state alpha-particle

152

P . J . DALLIMORE AND B. W. ALLARDYCE

groups could be resolved from the background. However, for the backward angles the ground and first three excited states could be analysed, although here only the results for the ground and first excited states are discussed. A typical spectrum obtained at 143 ° (lab) with 10.50 MeV protons is shown in fig. 1. The targets were made by vacuum evaporation 9) of natural phosphorus onto carbon backings ( ~ 10/~g). They were 1.1 cm in diam., and the thicknesses were measured by weighing a glass slide before and after evaporation. Two sets of targets were made, the first having thicknesses of about 90 pg/cm 2 and the second 150 pg/cm 2. However a comparison of the two sets of data taken at 90 ° (lab) from the

=lp(p,o)=lS=

1200

Ep = 10 5 0 H E Y etab = 143 (/)

z

800

0 t.)

t>( I o<

400

h

0

25

,

50

~J

k.,

75

CHANNEL

Fig. 1. Typical or-particle spectrum at 10.50 MeV. two runs indicates that the actual target thicknesses were in the ratio 1.0 to 1.9. It has therefore been assumed that the thicker target was of the order of 180 pg/cm2; with the target at 60 ° to the beam this corresponds to an energy loss of about 7 keV for 8.5 MeV protons. The energy resolution of the tandem beam is expected to be less than 5 keV at these proton energies, which gives a combined overall energy resolution of less than 10 keV; this is considerably less than the mean level width of approximately 45 keV (see sect. 7) and therefore should enable all the fine structure to be observed. Also since the excitation function measured at 90 ° (lab) with the thin target does not show any more structure than that obtained with the thicker target, it is reasonable to assume that the overall resolution was sufficient to enable a fluctuation analysis to be carried out.

alp(p, d)2aSl REACTION

153

3. Mean cross-section analysis

To estimate the effect on the mean cross section of the increasing number of exit channels opening up with increasing proton energy, a Hausel-Feshbach (HF) calculation was performed for the incident proton energy range 8.5-12.5 MeV. The H F expression for the differential compound cross section averaged over energy is given by 10)

~d-~

~/ (0)

~

Z Ace(O)

4(2I,+ 1)(2i~+ 1) #c'

"CT< ~Tc,' ~c.

where 2~ is the wavelength of the incoming particle, I~ and i~ are spins of the incoming particle and target nucleus, respectively, c and c' are all the quantum numbers of the entrance channel and exit channel and A isa geometrical factor; ~..~,,is overall possible exit channels.

>-

20 n," LLI Z LLI

16

b--

8

X tad 0")

4

~

0 a25

Fig 2. Decay scheme for the compound nucleus asS. To evaluate ~c,,T¢, exactly, it is necessary to know the spins and parities of all the final states. Fig. 2 shows the decay scheme for the compound nucleus 32S, and it is seen that an exact calculation of ~c.,Tc,, is impossible for incident proton energies in the range of the experiment. As may be seen from the excitation functions (figs. 3 and 4) the differential cross sections show a marked decrease with increase in excitation energy. It is partly due to the decrease in 2a with increase in proton energy but is mainly due to the increasing number of exit channels opening up. An estimate for the number of open channels was obtained by using the statistical level density formula derived from the Fermi gas model 1,). The level density p(E, J) is given by

p(E, J )

= (2J+ l ) p ( E ) exp

(-J(J+ I)/2CT),

| 54

P , .!° D A L L I M O R E

AND B. W. ALLARDYCE

where C is in essence the moment of inertia and p(E) the density of states with angular momentum zero (or if it is an odd nucleus then p ( E ) is one half the density of states with angular momentum J = ~). The expression for p ( E ) is

(1)

p(E) = 18.5 at a~rA - ~ ( U + T) -2 exp (2(aU)~).

~t is a constant which allows for the leduction of the actual moment of inertia below that of a rigid sphere, and Preson ]~) has given 6.0 as a reasonable value. The level

10000

3 1 p ( p , 01 )2~Sb

,~ i'

eern =91.*

5000

~-' , ~ " '

! ~ ~',, ~v,

',~ ',, --~'~-~-~-~---..~_l!'~/" :

Z 3.

,

•,

~

-

~.

I.¢,.,~

hl

OW I

-

3000

I

I

,

I

,.i

=lp(p, ao)~,S, Oc m = 93"

lt!l

2oo0 'il

i

C;i

90

'

10.0

11 0 Ep

12.0

(HEY)

Fig. 3. Excitation functnons for the ~ and % transztions to the ground and first excited states m 28St at 0x~b = 90 °. The solid lines are the energy dependent Hauser-Feshbach calculations of the mean cross section normahsed to the total experzmental mean cross section.

density parameter a is fixed empirically. The values used here for the nuclei 31p and 28Si were calculated from the empirical rule a =

,//7.4.

The parameter U is the excitation energy less the pairing energy 6, and the values for 6 have been taken from Cameron 12), Finally, the nuclear temperature T is obtained from the expression 13) U = aT 2 - T .

alp(p, d)2sSl REACTION

155

It should be noted that for the calculation of the relative energy dependence of the mean cross section in the present experiment, it is only necessary to know the relative variations of the level densities with energy; i.e. it is unnecessary to have an accurate estimate of the constant of proportionality used in eq. (l). Turkiewicz et al. 14) have shown that although the nucleus is not a Fermi gas, and that shell structure does have some effect on the level density, the formulae used here for the estimation of the parameters a and 6 are good approximations for nuclei around these mass numbers.

rl

~

:31p(PoO1)11SI

10000111

eo,,, -- 177"

~5oooP'~L~; 'ml

'

II

I

~

~

~ ~.,.

~'

'/I F

;

~'..~

,

'~',~

za

',

~,

~

i

~,1 'I

/,/

'

|

, ool I

' e c r u =177"

I~l

'

'

J ,

,~',;

',!

9

,

'~ I

I

,

1

,.

,~,

I

,rl!,, ,~'

4

I

,

10.0 Ep (HeV)

11 0

Fig. 4 The same results as for fig. 3 but w~th 0~ab = 177 °. The dashed curve in the ~0 excitation function is a least-squares fit of a s m o o t h function to the experimental data.

Thus eq. (1) should give a reasonable estimate for the relative dependence of the level density on the excitation energy even though the absolute values may be in error. With the above parameters, the level densities in the final nuclei 31p and 28Si were found up to excitation energies of 12.5 MeV. Using these approximations it was possible to calculate the dependence of the mean cross section on the incident proton energy as well as obtaining an approximate result for the absolute value of the mean differential compound nucleus cross section. A programme written by Wilmore 27) was used for this latter calculation. The optical potentials for calculating the transmission coefficients were from Percy for protons 15), from Weiss for alpha particles 16) and from Bjorklund for neutrons 17). For the proton exit channels, the known levels up to an excitation energy of 2.5 MeV in a l p w e r e used, together with

156

P . J . DALLIMORE AND B . W . ALLARDYCE

the level density results for higher energies. For the alpha-particle exit channels, the known levels up to an excitation energy of 7.5 MeV in 2sSi were used together with the level density results for the higher energies. For neutrons all the open channels are known and therefore may be used directly. The resulting energy dependence of the average differential cross section is shown on the excitation functions in figs. 3 and 4. To obtain these results, the H F calculations were performed at proton energies of 8.5, 9.5, 10.5, 11.5 and 12.5 MeV. A least-

c~ o OE

Exper,rnental

Points

Hauser-Feshbach Calcul.atlon ~0

E ~0~

o

Z 0 I-c) O:' LU if) m 0 r~" 0

Q

01

0o

,

0

30 '

6'0

i

i

i

go

120

150

180

cm

Fig. 5. Angular distribution of the mean cross section in the range Ep = 8.5-11.5 MeV. The circles are the experimental points and the sohd line the Hauser-Feshbach calculation. See the text for an explanation of the errors revolved. squares fitting programme was then used to fit the arbitrary but convenient function to the results. Also plotted (dashed curve) on the excitation function for ato at 177 ° is the average energy dependence obtained by fitting the same function to the experimental points. It is in good agreement with the theoretical calculations (solid curve) and indicates that all the energy dependence of the mean cross section has been satisfactorily explained by the H F calculation. As has been noted previously, an exact determination of the absolute magnitude of the differential cross section is difficult. Experimentally this is due to the large errors in the target thickness together with smaller errors in the solid angle deter-

a+b/(x+c)

alp(p, d)zSSt REACTION

157

mination and in the counting statistics, and theoretically it is due to the error in the absolute value of the constant used in eq. (l). However assuming the thicker target to have been 180/~g/cm 2 and using eq. (l), absolute values have been calculated both experimentally and theoretically for the average differential cross sections. The results are shown in fig. 5. It must be noted that the errors on these results are of the order of a factor of 2. Fig. 5 gives the average results for the cto group in the range 8.5-11.6 MeV. The experimental results are shown as circles and the H F calculation is shown as a solid curve. The angular distributions obtained from the experiment are again compared with the computed results in figs. 6 and 7. As the error on the absolute value of the mean cross section is large the results have been normalised at 177 °, where the reaction is assumed to go completely through the compound nucleus process. The error bars I'

L

z

c~0

o

l

I

,

o m

o

~

o w

.4 L_ m

t~ 1

Z

8 5-11 6 HeV

O

3

1

8 5 - 1 1 6 HeV

O

2 J~

ne

i,i I

0 0

30

I 60

| 90

~, 120

I 150

(~cm.

Fig. 6. A v e r a g e d a n g u l a r d i s t r i b u t i o n for the r e a c t i o n atp(p, ~ ) 2 a S i m t h e e n e r g y i n t e r v a l 8.5-11.6 MeV. T h e e r r o r s o n the e x p e r i m e n t a l p o i n t s are d u e to the F R D effects. T h e s o h d c u r v e is the H a u s e r - F e s b b a c h c a l c u l a t i o n w i t h t h e e x p e r i m e n t a l a n d t h e o r e t i c a l results norm a l i s c d a t 177 °.

0 180

0

a

I

I

I

30

60

90

120

1 0

180

0era. Frog. 7. A s i m i l a r a n a l y s i s to t h a t in fig. 6 b u t for the r e a c t i o n 3~p(p, cq)2aSl"

are those expected from the finite range of data 18,19). From the good agreement between theory and experiment for all angles and the relative symmetry between backward and forward angles, it is concluded that no direct interaction is present. Furthermore the evidence of the autocorrelation analysis of sect. 5 suggests that the direct reaction component is negligible. Richter et al. 1o) have shown that if there is no direct interaction present then esti2 where O'r¢ 2 s is the spin cut-off parameter of the residual numates of F o / D o and a .... cleus, may be obtained from the angular distributions. The present data have been analysed by Richter and give a value for O'r¢ 2 s between 4 and 7. Since the compound

158

p. ,1. D A L L I M O R E A N D B. W . A L L A R D Y C E

nucleus 32S decays mainly through proton emission, are s2 is approximately the value for 3~p. 4. Correction of excitation functions

As has been shown in the previous section, the excitation functions exhibit a marked energy dependence of the mean cross section due to the HF decay of the compound nucleus. In order that correlation functions and level widths may be calculated, some correction for this energy dependence must be made. There are two possible methods; the first is the by "modulation" technique which assumes no correlation between the local mean cross section and the actual cross section. It gives l+Ce~p(~)+p(e) = ( l + K ( e ) )

1+ N F ~ + ~ ~ '

(2)

where Cexp(~) is the autocorrelation function for the experimental excitation function, ~(e) the bias associated with the finite range of data (FRD) involved and K(~) the autocorrelation function for the curve representing the mean cross section. As will be shown in sects. 5 and 6, this method has serious disadvantages because of the approximations employed. The second method is by the correction of the experimental excitation functions by dividing by the calculated local mean cross sections 2t). It gives

(1+

~o'n~ (E) O'nF( E + e ) / 1 F2F2e2 ) - / ' a ' x p ( E ) ' ~ ___

facto(E+e)

+'~(e),

(2a)

where oe,v(E) is the experimental cross section and aaF(E) the theoretically calculated mean value. 5. Autocorrelation analysis

As the FRD errors on the autocorrelation are large for small ranges of data, it is important that a sufficiently large energy range be covered to give meaningful results. However, if any non-statistical processes are present they will become more predominant the larger the range of data. For the reaction investigated here the non-statistical effect due to the increasing number of exit channels opening with increasing proton energy gives results for the correlation functions which cannot be explained by the simple fluctuation theory. The long-range effect of this non-statistical process is seen in figs. 8 and 9. In fig. 8, both the uncorrected and HF corrected progressive autocorrelations ts) for the ~to and 0q groups at 123 ° c.m. are compared with the theoretical predictions. For the

alp(p, d)=8Sl REACTION

159

present, only the uncorrected results are considered. They are seen to increase very rapidly for the first few data points and then to show a gradual increase with increase in the range of data. From a consideration of the excitation functions (figs. 10 and 11 ), it is seen that the initial rise is due to one or two very large peaks at the low-energy end and the gradual increase over the whole range is due to the slow variation in the mean cross section.

/ 12 t Z

t ....

o

I-<

123"

I

.'"''"'----"'-'''""~---.

0/~o

Experimental

0 8/

. . . . . ..:

/" "'~"

HF C o r r e c t e d . , '

_

w n," 0 0 0 I--

20

LLJ > 03 W n," r,.D 0 n," n

4O

6'0 j"

123" 0.3 . . . .

~1

s

Experlrnental HF Corrected

L

0

•

,

.-

--

210

r

RANGE (n) Fig. 8. P r o g r e s s i v e a u t o c o r r e l a t i o n c u r v e s for the ~t0 a n d cq e x c i t a t i o n f u n c t i o n s at 123 ° cm. T h e r a n g e n is in u n i t s o f t h e m e a n level w i d t h F.

The same effect is also observed in fig. 9 where the data have been divided into samples of varying sizes and the resulting autocorrelations averaged. For the small samples, the experimental and theoretical results are in reasonable agreement, but for the larger samples the disagreement is well outside the estimated F R D errors Is). It should be noted that in these and subsequent figures the errors are due to the finite ranges of data involved and not to the errors on the counting statistics. The agreement for small sample sizes and the disagreement for large sample sizes is explained by the effect the energy dependence has on the autocorrelation. For small samples this energy dependence is negligible and therefore does not effect the autocorrelation, but as the sample size increases, the energy dependence becomes more

160

P.J.

DALLIMORE

AND

B. W.

ALLARDYCE

pronounced and hence has a greater effect on the autocorrelation. In fig, 12 the average autocorrelation values for sample sizes of p -- 25 data points in the proton energy range 8.5-11.5 MeV are plotted against angle for both the ~to and ~tt groups. The solid lines are the theoretical calculations of 1IN less the F R D bias corrections [?(e = 0) ref. is)]. For the calculation of~(e = 0), F was taken as 47 keV for the ~o group and 42 keV for the ctt group (see sect. 7). It must be noted that when predicting the results

~o

123" 1.2

" Expertmentat Z 0

• HF C o r r e c t e d

0e

,< ..J

L~ Q~ Q::

0.4

0 0 O0

i

123"

LU

I

1

2O

,<

I

I

|

60

40

~1

Exper,mentat

03

nLU

•

HF Corrected

02 It

0.0

I !

•

01

L

0

|

I

2o

I

I

RANGE

|

|

60

40

(n)

Fig. 9. Average a u t o c o r r e l a t i o n s for varying ranges n for the ct0 and cq excitation functions at 123°c.m.

for small sample sizes, considerable error may occur due to the inaccuracy in the value of the level width used. This is because of the rapid variation of the bias on the autocorrelation for small n. From these results, it is seen that for small ranges of data the variation of the mean cross section has negligible effect on the values of the autocorrelation. However, as fig. 9 illustrates, the Hauser-Feshbach dependence does have an effect for ranges greater than about 1 MeV. When considering the full range of data, two methods of correcting the experimental results for varying mean cross sections have been shown to exist (see sect. 4). In the present analysis, the energy-dependence of the mean cross section due to the

atp(p, d)=sSt REACTION

161

Hauser-Feshbach theory [any(E)] was first removed from the experimental excitation function [tT=xp(E)]; i.e. new excitation functions at(E) were calculated using the equation c ¢ ( E ) = ~.xp(E)/aHv(E). (3) Here ac~,,(E) is a rapidly fluctuating curve and trnF(E ) a smoothly varying curve, therefore ire(E) is still rapidly fluctuating. Figs. 10 and 11 show the HF corrected and

I,

EWI'ERtN|Nr,I,L EXCn'Al't0X

='P(p.al~ =lSi

FUNCTION

~

ecru

1200

= 123 °

30 OlD

800

m

i'I

i,

c~

" ,~!I

~.00

r

HF C O R R E C T E D

>-

FXCITATION

FUMCT;ON

.J bJ

800

' 1'

u~

:

I

!

_J

400 nO 7

,

~

,,

•~: ,'ll;

',", '

, ~. ,

,'I

,~.',

j,./',

90

~.,,

100 Ep

,'V~

.

!?~- .~',

$ ~( tV

~ ,! ',;

,~, i~ 't ".!~,

110

(MeV)

F~g. 10. T h e c x p e r ] m c n t a l and H a u s e r - F e s h b a c h

corrected e×cltation s,p(~, p0)28S1 at 123 ° c.m.

funct=ons

for the reaction

uncorrected excitation functions for the ~to and at groups at 123 °. It is evident that correction of the experimental excitation functions by eq. (3) satisfactorily explains all the energy dependence of the mean cross section. The progressive autocorrelation and varying sample size effects have also been plotted on figs. 8 and 9 for these HF corrected excitation functions. The progressive autocorrelations still exhibit a relatively sharp rise for small ranges of data because of the presence of the large peaks in the cross sections. However, as the range increases they tend towards the theoretical values and for the full range of data are within the estimated FRD errors. The HF corrected and the uncorrected average autocorrelations for varying sample sizes are nearly identical for ranges up to about l MeV. However, for larger ranges

162

P.J.

DALLIMORE AND B.W.

ALLARDYCE

the H F corrected results now agree with the theoretical predictions. If some form of intermediate structure were present, then it should still result in values for the autocorrelations being higher than predicted by the fluctuation theory for the larger ranges. In fig. 13 and tables I and 2, the results for the corrected and uncorrected autocorrelations at all angles are compared with the theoretical predictions of 1/Nobtained by the method of Brink, Stephen and Tanner 22). All the autocorrelation values have been

EXPERINENTAL

EXCITATION

IIP(p,O~)2ISl

FUNCTION

/~!

gc'm'=123"

::)o o

,o',4000

J:

0

tu 2000 >-

I

I

NF C O R R E C T E D

W

I

L

I

EXCITATIOR

FUNCTION a W

i

4000

/

/',

,% •

... 90 Ep

10.0 (MeV)

11 0

Fig. 11. T h e experimental and H a u s e r - F e s h b a c h corrected excitation functions for the reaction 3W(p, ~1)~8S1 at 123 ° c.m

corrected for F R D biases using the values of the mean level widths obtained in sect. 7. In these tables Cc~p are the autocorrelations obtained from the uncorrected excitation functions, CnF those obtained from the Hauser-Feshbach corrected excitation functions and Cb~ those obtained from the base line shifts (see below) of the autocorrelation functions. From these results, it is once again evident that the H F calculation explains all the energy dependence of the mean cross section. The large values for the uncorrected autocorrelations of the e0 group at 62 °, 108 °, 123 ~ and 138 ° are explained by the chance occurrence of very large peaks in the excitation functions. As the cross sections are correlated over an angle approximately given by (kR) -1, it must be expected that if a high value for C(e = O) occurs at 123 °

31p(p, d)28Sl REACTION

163

TABLE 1 Autocorrelation results calculated from the excitation functions for the % group Oe m

46 62 78 93 108 123 138 145 152 159 166 169 177

1

--

Cexp(t = O)

+ {~}~

--~?~ -- 0)

CHF(e = 0) -~-~,(~ = 0)

Cbl(a = 0) +~(e, = 0)

0.650 1.069 0.673 0 761 1.361 1.216 0.997 0.734 0.701 0.764 0.822 1.184 1.022

0.407 0.671 0.732 0.535 0.576 0.565 0.464 0.697 0.420 0.480 0.571 0.676 0.701

0.435 0.799 0.455 0.531 1.089 0.927 0.610 0.445 0.466 0.470 0.719 0.899 0.992

0.5104-0.106 0.501 4-0.104 0.505 ~ 0 105 0.530A-0.111 0.506_-L0.105 0.500 ~ 0.113 0.511 ~_0.115 0.511 ___0.107 0.510 zk0.115 0.545 ~0. ! 15 0.721 zk0.170 0.805 zk0.178 0.956 zk_0.234

Cexp(e = 0) Is from the experimental excitation functions, CHF(e = 0) from the Hauser-Feshbach corrected excRation functions and Cb~(e = 0) from the base-line shift of the experimental excitation functions. All results have been corrected for the finite range of data bias effects ~(e = 0). The quantity/~i is from ref. 18) and for large n becomes n ( N + I ) / n N . TABLE 2 Autocorrelation results calculated from the excitation functions for the gl group 0¢ m.

1 -7¢ = { ~ ) ~

46 62 78 94 109 123 138 146 152 159 166 169 177

0.125_--.0.024 0.116-A_0.022 0.117±0.022 0.120±0.023 0.120±0.023 0.115~0.023 0.134+0.027 0.155:k0.029 0.171 :k0.035 0.189 5_0.036 0.263 __0.056 0.329_4-0.065 0.473 -2_0.106

Cexp(e = O) + f , ( e = O)

0.338 0.373 0.412 0.499 0.578 0.399 0.364 0.467 0.327 0.441 0.345 0 548 0.508

C n v ( e = O) + f , ( e = O)

Cb~(e = O) + f , ( e = O)

0.147 0.108 0.118 0.102 0.096 0,113 0,111 0,134 0.121 0.152 0.260 0 316 0.470

0.070 0.107 0.121 0 078 0.111 0.110 0.082 0 193 0 134 0.242 0 245 0 370 0 436

See table 1 for explanation of headings. it will a l s o o c c u r a t n e a r b y angles. F o r t h e t h r e e r e s u l t s a t I 0 8 °, 123 ° a n d 138 °, t h e l a r g e p e a k s in t h e c r o s s s e c t i o n o c c u r a t a p p r o x i m a t e l y 8.60 a n d 9.07 M e V (see fig. 10), a n d it is t h e c o m b i n a t i o n o f t h e s e l a r g e p e a k s w i t h t h e r e l a t i v e l y s m a l l c r o s s s e c t i o n s in n e i g h b o u r i n g e n e r g y r e g i o n s t h a t r e s u l t s in t h e h i g h v a l u e s f o r t h e u n c o r r e c t e d a u t o c o r r e l a t i o n s . T h e s a m e e x p l a n a t i o n is a l s o g i v e n f o r t h e r e s u l t s a t 62 ° w h e r e h e r e t h e r e is a l a r g e c r o s s s e c t i o n a r o u n d 9.10 M e V .

164

P.J.

DALLIMORE

AND

B. W . A L L A R D Y C E

The second method of analysing the experimental results is to look at the autocorrelation function C(e). Assuming the local mean cross section at an energy E is not correlated to the experimental cross section at that energy then eq. (2) applies. If the

X EXPERINENTAL

p : 25

o
e

I

H F CORRECTEO

~ 0

x

o 61 x

A 0

" 04

\

¢.J

. . .. I..j

.o,\

~

,

~

6 '0

90

~ o

I

120

30

:S

0

*

I

3 '0

"

°.

I!

to 0 / ,

O;

o

I I

60

9

ecru

b

I

120

t

150

ecru.

1

p = 25

o<1

O~

×

EXPE R r V ~ N T A L

o

HF C O R R E C T E D

c>< 1

06 x

r~

x

0

. 0.2

4,

0.4

x

x

x

V

01

02

;I 1

30

I

60

9b

do

1~o

3'0

ecru

6b

9b

~;;o

ld0

ecru

Fig. 12. Exper=mental average autocorrelatlons for data samples of 25 points The solid curve is the theoretical value corrected for F R D effects. The errors are due to F R D effects.

Fig. 13. Experimental and Hauser-Feshbach corrected autocorrelatlons for the full range o f data for the % and cq excitatmn functions. The values have been corrected for F R D effects and are compared wzth the theoretical values of I/N (sohd line). The error bars are those due to the fimte range of data involved.

non-statistical process is sufficiently long ranged, then K(e) can be approximated by a smooth curve, and for ~ >> F the experimental autocorrelation function will be C(~)+~(e) ,.~ K(a). Thus K(~) can be found from the upward shift of the base line for C(~)+'7(~) at large

sip(p,

d)2gSl REACTION

165

~, a n d by e x t r a p o l a t i o n to e = 0 the value o f K(e = 0) is obtained. I f however there is s o m e s h o r t - r a n g e effect influencing the a u t o c o r r e l a t i o n function, as for e x a m p l e very large p e a k s in the differential cross section, then e x t r a p o l a t i o n o f K(e >> F ) back to e = 0 will not give the correct value for K(e = 0). In fig. 14, the a u t o c o r r e l a t i o n functions for cto at 62 ° and 152 ° are shown together with the estimates o f K(e) from the

o, 0

e = 62*era

0.2

0 = 152"cm

ccc° ' o

0.0

0

I

I

/

250

500

"750

8

I

1000

I

1250

(keV)

Fig. 14. Autocorrelation functions for the reaction ,~lp(p, %)~aSl at angles of 62° and 152~ c.m. The dashed line is the estimated base-hne shift. base-line shift. A t 152 ° the effect o f the col relation between the local mean cross section a n d the experimental cross section has little effect on the correlation function d u e to the relatively small peaks at the low-energy end. H o w e v e r at 62 ° there is the a d d i t i o n a l effect o f a very large p e a k in the excitation function at 9. I0 MeV. T h e a u t o c o r r e l a t i o n functions are similar for large e where only the long range effect is evident b u t they

166

P. J. DALLIMORE AND B. W. ALLARDYCE

show marked differences for e 2 F. Thus the value for C(e = 0)+~(~ = 0) obtained by using eq. (2) is in reasonable agreement with the theoretical value of 1IN for 152 ° but not for 62 °. The results for all angles and both groups are tabulated in tables 1 and 2, and the only ones which do not agree with the theory are where the short range effects are predominant; i.e. for the ao group at angles of 62 °, 108 °, 123 ° and 138 °.

6. Cross-correlation analysis 6.1. A N G U L A R

CROSS CORRELATIONS

The definition of the angular cross correlation is

C(O, 0') = a(O)a(O')

1 = I/M.

a(0) o(0') Including finite range of data effects, this becomes as)

C(O, 0') -

1-a

,

(4)

M+a where 2 1 a = -- tg- i n - - 2 ln(1 - n 2 ) , n n and n = A/F (d is the range covered by the excitation function). As in the previous section a correction must be made for the varying mean cross section before the results can be compared with the theoretical predictions. Again this is possible by either the direct correction of the excitation functions or by an estimate of the base-line shift. The modulation equation is a generalization of eq. (2) and is

l+C(O,O';e)+~(O,O';e)=

1+

~tr2 + e~ (l+K(O,O;e)).

For two uncorrelated excitation functions (i.e. IO- O'l >> (kR)- 1), this equation reduces to

C(O, 0'; e) ~ K(O, 0'; e), and hence K(O, 0'; e) may be found. In fig. 15, two angular cross-correlation functions for the ~o group with 0 = 123 ° c.m. and O' = 62 ° and 169 ° c.m. are shown. They exhibit two distinct patterns. For O' = 169 °, the base-line shift is caused by the longrange effect of the Hauser-Feshbach decay of the compound nucleus. For O' = 62 °, the base-line shift is caused by a combination of the long-range H F decay and the short-range effect of the chance overlap between very large peaks in the excitation functions; these occur at 8.58 MeV for 123 ° and at 8.60 MeV for 62 °.

alp(p,

d)eaSl

167

REACTION

E x t r a p o l a t i o n of the base-line fits for e >> F to e = 0 gives values for K(O, 0';~ = O) which are nearly identical in each case; i.e. the values of K(O, 0'; e = 0) obtained by this method do n o t take a c c o u n t of any short-range m o d u l a t i o n present in the excitation functions. Fig. 16 shows the a n g u l a r cross-correlation values with 0 --- 123 °

0.(

~o2

oo I

200

,`o0

6'00

8'o0

•

I

1000

I

1200

C(keV)

o~e

06

e = 123 °

e'= 169"

~04

U

02

0.0

200

400

600

i 800

1000

1200

¢ (keV)

Fig. 15. Angular cross correlation functions for the reaction a~p(p, :t0)~sSi with 0 = 123° c.m. and 0' = 62~ and 169° c m. The straight hnes are the estimated base-hne shifts obtained from the data. c.m. for the ceo group. The crosses are the uncorrected results, the circles the H F corrected results and the triangles the values corrected by the base-line shift. A comparison of the two methods for correcting the experimental data shows that the H F correction is the more satisfactory. W h e n there is no chance overlap of large peaks, the two methods give results for the a n g u l a r cross correlations which are within the esti-

168

P.J.

DALLIMORE

A N D B. W . A L L A R D Y C E

mated F R D errors of the theoretical values. However, when there is a chance overlap between large peaks the base-line shift m e t h o d is completely unsatisfactory as it is unable to correct for these short range effects. Fig. 17 shows the uncorrected and H F corrected results for both groups with 0 = 177 ° c.m. a n d the theoretical estimates obtained from eq. (4). The coherence angles calculated from the half widths at half height of the H F corrected angular cross

~

1.2

X

i

i

i

i

i

EXPERINENTAI HF

X

CORRECTED

C~(~0

BL CORRECTED 1.0 e = 123"cm 08

~-

06 •

~

X

X

X

04

0.2

O0

0

I

I

I

1

I

I

I

I

20

40

60

80

100

120

140

160

180

9'c m

Fig. 16. Angular cross correlations for the reaction SW(p,ct0)28Si with 0 = 123° c.m. The crosses are the experimental points, the circles the Hauser-Feshbach corrected points and the solid triangles the base-line shifted points. The solid curve is the theoretical calculation and the error bars are due to FRD effects.

correlations are 6 = 21 ° for ~to a n d 6 -- 17 ° for ct 1. F r o m the relationship derived by Brink, Stephen and T a n n e r 22) that

6 ,.~ ( k R ) - l , values are f o u n d for k R of 2.7 and 3.4, respectively, compared with k R = 3.0 for protons of 11 MeV and r o = 1.3 fm. The results are consistent with a m a x i m u m angular m o m e n t u m in the initial channel of approximately 3; indeed the p r o t o n transmission coefficients used in the H a u s e r - F e s h b a c h calculation are small for l > 3.

169

31p(p, d)HS] R E A C T I O N

6.2. G R O U P

CROSS CORRELATIONS

For purely statistical reactions in which the mean cross section is constant with energy, the cross correlations between particle groups are assumed to be zero. However, when the mean cross sections are not constant but have the same energy dependence for each group, then the cross correlations must be large and positive. i

O = 177"

12

o~ o

IK Experimental

~J

HF Corrected

0.8

X

X;

O0 I

30

I

60

I

|

go

120

i

150

~c.m.

06

e = 177" X

0.4

0<: 1

Experlmentll|

• HF Corrected

~/

.f

(D ~

3'o

8'o

9'o

x

xx =

1do

e'c.m F~g. 17. A n g u l a r c r o s s c o r r e l a t i o n s f o r t h e ~t0 a n d ctI e x c i t a t i o n f u n c t i o n s w i t h 0 = 177 ~ c . m . T h e crosses are the experimental points and the circles the Hauser-Feshbach corrected points. The sohd c u r v e ~s t h e t h e o r e t i c a l c a l c u l a t i o n a n d t h e e r r o r b a r s a r e d u e t o F R D effects.

In fig. 18 the uncorrected and H F corrected group cross correlations are plotted against angle for the proton energy range 8.5-11.6 MeV. The uncorrected experimental results are all positive and well outside the estimated F R D errors. However, after the energy dependence of the cross sections has been removed, the values are distributed about zero and within the estimated errors. The results again show that in the excitation region covered in this experiment the pure statistical fluctuation theory corrected for Hauser-Feshbach decay is a good description of the compound nucleus.

170

P.

J. DALLIMORE

AND

B. W.

ALLARDYCE

7. Level widths

As Ericson 3) and Brink and Stephan 4) have pointed out, the level width in the compound nucleus in the region of overlapping levels may be found from the autocorrelation function 1 F2 c(~)

-

N F2+e 2

Generally the value of e at half height is used for estimating F although in theory any value of the ratio C(e = O)/C(e) may be used. However, because of the large F R D errors on C(e) for large ~, the values are limited to e ~ F.

=tp(p,a)=lS= 0.4

C(ao,a,) i

02

x

R

~

s

i.

•

O0 I

30

1

60

"t910

•

120

e 9

150

Ocm F~g. 18. G r o u p cross correlations for the cto and ctI excitation functions. The crosses are the experimental results and the circles the H a u s e r - F e s h b a c h corrected results. The error bars are those due to F R D effects.

The autocorrelation functions for both the 0to and ctt groups have been analysed at all angles, and the values of F taken from the widths at half height. The uncorrected results are shown as Fexp in tables 3 and 4. The lesults from the Hauser-Feshbach corrected excitation functions are shown as Fur and the values obtained after the base-line shift is removed are shown as Fbt (i.e. Fbt is the value of e at the half height for the function { C ( e ) - K ( e ) } ) . All the results tabulated here have been corrected for F R D bias effects. Because of the difficulty in estimating the base-line shift K(e) for small e, the values of Fb~ are not considered to be as accurate as those obtained from the H F corrected excitation functions. However, it is seen that this method does correct for the overestimate of the level width due to a slowly varying mean cross section at those angles where this is the only effect. For results where intermediate structure is present s.,4), this method is the only satisfactory way of estimating the level width from the autocorrelation function.

alp(p, d)2aSt REACTION

171

TABLI~ 3 Mean level widths (keV) calculated from the excitation functions for the ~ group

46 62 78 93 108 123 138 145 152 159 166 169 177

66 90 74 63 58 64 42 43 45 73 50 56 42

51 88 93 57 42 44 33 51 32 41 37 39 34

42 59 52 49 52 53 28 31 33 48 45 46 42

42 41 37 39 41 39 33 36 38 35 37 38 36

The subscripts exp, HF, and bl have the same meaning as in tables 1 and 2. _P=~ is the value obtained from counting the number of maxima in the excitation functions. TABLE 4 Mean level widths (keV) calculated from the excitation functions for the ~x group

0o.=.

-re.p

rnF

rp~

_rmu

46 62 78 94 109 123 138 146 152 159 166 169 177

570 300 280 410 390 295 275 235 100 92 44 51 36

55 39 41 35 35 55 33 36 30 38 40 41 41

39 42 50 44 44 35 28 48 31 52 33 41 33

30 31 32 33 31 30 35 32 31 28 32 32 31

See table 3 for explanation of headings.

I n fig. 19, t h e a u t o c o r r e l a t i o n f u n c t i o n s c o r r e s p o n d i n g t o t h e u n c o r r e c t e d a n d c o r r e c t e d e x c i t a t i o n f u n c t i o n s f o r t h e ~tt g r o u p a t 123 ° c.m, a r e s h o w n w i t h t h e e s t i m a t e s o f t h e level w i d t h s Fexv,/"llF a n d Fbl. T h e H F c o r r e c t e d a u t o c o r r e l a t i o n f u n c t i o n n o w o s c i l l a t e s a b o u t z e r o f o r l a r g e e in a g r e e m e n t w i t h t h e p r e d i c t i o n s o f t h e s t a t i s t i c a l theory. T h e v a l u e s o f -FHF f o r b o t h t h e ct o a n d ~ g r o u p s a r e n o w m o r e c o n s i s t e n t t h a n t h e v a l u e s f o r Fexp e x c e p t a t t h e t w o a n g l e s o f 62 ° a n d 78 ° c . m . f o r t h e % g r o u p . H o w e v e r , a s w a s p o i n t e d o u t in sect. 5 t h e s e t w o e x c i t a t i o n f u n c t i o n s h a v e h i g h a u t o c o r r e l a t i o n s

P. J. DALLIMOREAND B. W. ALLARDYCE

172

due to the presence of an extremely high cross section around 11.70 MeV. It has been shown la) that if a high value of the autocorrelation is obtained, a high value of the level width should be expected. In fig. 20, the value of Fur for cto and for cq are plotted against the corresponding values for N(C(e = 0 ) + ~(e = 0)), where C (e = 0) is the H F corrected autocorrelation. The solid curves are the theoretical calculations from Dallimore and Hall la). By comparing the theoretical calculations for various values of F with the experimental results, the final values are 47 +__7 keV for F(cto) and 42 + 5 keV for F(cq).

~

k

0

c(¢) 0

2~o

~o

"' 7~

lo~o

12Eo

(keV) Fig. 19. T h e experimental a n d H a u s e r - F e s h b a c h corrected autocorrelation functions for the reaction sip(p, ~t)2ssi at 123 ° c.m. See the text for an explanation o f the extraction o f the width F.

A completely independent method of calculating the mean level width is by counting the number of maxima in the excitation functions and by using the expression of Brink 23) r = 0.55/K,

where K is the number of maxima per unit energy range. This method is independent of any long range, non-statistical process and should give good results if the experimental resolution is satisfactory is). The values of the level widths obtained by this method are shown as Fma, in tables 3 and 4, and the average results give F(cto) = 41 keV and F(ctl) = 31 keV. The values are lower than those obtained from the autocorrelation functions, but Put et al. 24) have pointed out that limitations in counting statistics and uncertainties in background subtractions will give too many peaks per unit energy range; this means that the value of the level width so obtained will be too small. No estimate is available for the magnitude of this effect although in practice it should be possible to calculate it by a Monte-Carlo method.

173

a l p ( p , d)~SSl R E A C T I O N

The variation of the mean level width with alpha-particle group is due to the different spin values for the ground and first excited states in 28Si (0 and 2, respectively). The higher spin of the first excited state means that the reaction can proceed through higher spin states in the compound nucleus 32S. Vonach and Huizenga 25) have shown the dependence of the level width Fj on the spin J of the nucleus, and it is seen that

1-' = 4.7 T

ii

i~. 14

Io It

~Io Z

06

'

1.2

r' = 42

,b

'

6'o ' 8'o rexp (keY)

c~1 ~ ,

I!

i~1o Io II

o.8

06 i

t

60

rexp (keY) Fig. 20. Plots o f the dependence o f the mean level widths (obtained from the H a u s e r - F e s h b a c h corrected a u t o c o r r e l a t l o n functions) on the normalised a u t o c o r r e l a t i o n { N [ C ( e = 0)-r-(7 = 0)l}. The solid curves are the theoretical calculations with F = 47 for the ~0 group a n d / ' = 42 keV for the cq group. The error bars are those due to F R D effects.

for higher spins the level width decreases. As the values of the level widths obtained by fluctuation analyses are mean values for all the contributing levels in the compound nucleus it is expected that there should be a small dependence of the level width on the spin of the final nucleus. The values obtained for F(0(o) and F(~I) are in agreement with this qualitative result.

174

P. J. DALLIMORE

AND

B . ~.V. A L L A R D Y C E

As was stated previously, all the values tabulated in tables 3 and 4 have been corrected for F R D bias effects. For the large ranges considered here these biases are only of the order of 4 ~o or approximately 2 keV. To illustrate the effect of the range of data on the determination of the level width FHF,the data for c% at 93 ° c.m. were divided into sections of varying sizes and the average level widths calculated. The results are shown in fig. 21 with the theoretical calculation obtained from ref. is) with F = 47 keV. The error bars have been calculated from the expression given by Gibbs [ref. ~9)],

50 .

.

.

.

.

"

'

'

40 ~ , ( k e y ) 30 2O O = 93°em 10

%

J" = /,7 keV

io

'

io

RAN6E

'

'

(n)

Fig. 21. Dependence of the Hauser-Feshbach corrected mean level width on the range of data for the reaction 3tp(p, %)asSi at 0 = 93° c.m.

8. Summary and conclusion The investigation of the reaction 3~p(p, ot)2sSi in the proton energy range 8.5-12.3 MeV has shown that at these energies the process is satisfactorily explained by a combination of the Hauser-Feshbach decay of the compound nucleus and the pure fluctuation theory. There is no evidence for any direct interaction or for the existence of any doorway states; the reaction is purely statistical. The direct-reaction contribution was assumed to be zero because of the symmetry of the average angular distributions and the agreement with the Hauser-Feshbach theory. The assumption was further justified by the agreement between the H F corrected correlation functions and the theoretical values. If any direct reaction component was present it would decrease the values of the autocorrelation below the theoretical predictions of (1/N)-~(e = 0); this was not so, and all the results agreed with theory within the estimated F R D errors. Further evidence for the assumption of negligible direct interaction is shown in fig. 22, which compares the probability distribution for the HF corrected excitation function of the ~o group at 123 ° c.m. with the theoretical calculation when N = 2 and YD = 0.

31p(p, d)~SS1 REACTION

175

When the only effect contributing to the energy dependence of the mean cross section is the H F decay of the compound nucleus, new excitation functions may be generated which fulfill the requirements of the statistical theory. However if intermediate structure is present, its form cannot be theoretically predicted and therefore it cannot be removed from the excitation functions. In this case, the analysis must be performed using the modulation theory. To test the applicability of this method to excitation functions with varying mean cross sections, the present uncorrected data were analysed using the base-line shift technique. It was found that when the only effect was the long-range H F decay of the compound nucleus it gave reasonable results, but when any short-range effects were present it was unsatisfactory. The short-range effects were caused by the chance oc-

'~

o~. 0 123"

II I (

1

Yo = 0 N =2

0

f

0

1

2

3

Fig. 22. T h e probability distribution for the reaction 31p(p, %)~sSi at 123 ° c.m. T h e experimentally f o u n d h i s t o g r a m is fitted with a theoretical curve a s s u m i n g no direct interaction contribution.

currence of large peaks in the cross sections at the low-energy end of the excitation functions. Although they may be only three or four times larger than the local mean cross section, they may be more than ten times the overall mean. Therefore, before the base-line shift method is used, the excitation functions must be corrected for the HF decay of the compound nucleus to remove these short-range effects. If intermediate structure is present, the correlation functions should still exhibit base-line shifts, but provided this structure is sufficiently long-ranged compared to the statistical fluctuations, it should be possible to obtain accurate estimates for the mean level width and the fluctuation damping coefficient. The average level width in the compound nucleus 32S has been found to be 47-t-7 keV for decay to the ground state of 28Si and 4 2 + 5 keV for decay to the first excited state. Although these results are within the estimated errors they qualitatively indicate the J-dependence of the level widths. The results correspond to a lifetime in the com-

176

P.J.

DALLIMORE AND B. W. ALLARDYCE

pound nucleus 32S at excitation energies between 17 and 20 MeV of approximately 1.4 x 10 -20 sec, which is in agreement with the general results for nuclei of around mass 30 as reported by Mayer-Kuckuk 26). Several methods for obtaining the mean level width have been investigated, and the most satisfactory results were obtained from the autocorrelation functions of the HF corrected excitation functions. The authors gratefully acknowledge the many discussions with Drs. N. W. Tanner and D. M. Brink. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)

W. von Wltsch et al., Nuclear Phystcs 80 (1966) 394 T. Ericson, Advan. Phys. 9 (1960) 425 T. Ericson, Ann. of Phys. 23 (1963) 390 D. M. Brink and R. O. Stephen, Phys. Lett. 5 (1963) 77 B. W. Hooton, Oxford Conference (1966) A. Katsanos and T. R. Hmzenga, Bull. Am Phys. Soc. 9 (1964) 667 H. Feshbach, m Nuclear spectroscopy, Part B, ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) B. W. Allardyce et al., Nuclear Physics 85 (1966) 197 B. W. Hooton, Nucl. Instr. 27 (1964) 338 A. Richter et al., Phys.Lett. 14 (1965) 121 M. A. Preston, Physics of the nucleus (Addtson-Wesley Publ. Co., Reading, 1962) A. G. W. Cameron, Can. J. Phys. 36 (1958) 1040 J. M. B. Lang and K. J. Le Couteur, Proc. Phys. Soc. A67 (1954) 586 !. M. Turkevtch et aL, Nuclear Phystcs 77 (1966) 276 F. G. Perey, Phys. Rev. 131 (1963) 745 P. B. Weiss and R. H. Davies, Bull. Am. Phys. Soc. 8 (1963) 47 E. Bjorklund and S Fernbach, Phys. Rev. 109 (1958) 1295 P. J. Dalhmore and 1. Hall, Nuclear Physics 88 (1966) 193 W. R. Gibbs, Los Alamos Report LA 3266 (1965) 1. Hall, Phys. Lett. 10 (1964) 199 P. J. Dalhmore, S. M. Perez and B. W. Allardyce, to be published D. M. Brink, R. O. Stephen and N.W. Tanner, Nuclear Physics 54 (1964) 577 P J. Dalllmore and 1 Hall, Phys. Lett. 18 (1965) 138 L. W. Put, J. D. A. Roeders and A. van der Woude, Phys Lett., to be published H. K. Vonach and J R. Huizenga, Phys. Rev. 138 (1965) BI372 T. Mayer-Kuckuk, Herceg Nov1 Lectures (1964) D. Wdmore, Umted Kingdom Atomic Energy Authority Research Group Report AERE-R5053