Pre-equilibrium statistical model: Analyses of excitation functions

Nuclear Physics A142 (1970) 559 --570; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

PRE-EQUILIBRIUM STATISTICAL MODEL: ANALYSES OF E X C I T A T I O N F U N C T I O N S M. BLANN and F. M. LANZAFAME * Department of Chemistry and Nuclear Structure Laboratory, University of Rochester, Rochester, New York, USA tt

Received 1 September 1969 Abstract: Griffin's pre-equilibrium decay model is used, with minor modifications in its derivation, to analyse a variety of excitation functions. The calculated particle spectra are taken to be a sum of pre-equilibrium and equilibrium components, the admixture of the two being determined from comparisons with experimental excitation functions. Additionally, the sensitivity of calculated results to the parameter of the initial number of particles plus holes in the compound state is investigated, and compared for consistencywith results from analyses of particle spectra for proton and helium ion induced reactions. The excitation functions analysed are for the reactions 5iV(or,n)5*Mn, 5W(cq 3n)52Mn, 51V(=t,p3n)SlCr, 56Fe(c~,3u)57Ni, 56Fe(ct,p2n)57Co, 59C0(p, 3n)57Ni, 59C0(p, p2n)57Co, and a97Au(~z,xn)2°l-xT1 (x = 1, 4). Excitation energies extend to 100 MeV. Pre-equilibrium fractions of 35 % to 65 % were required to reproduce the experimental results. 1. Introduction

A simple and physically appealing non-equilibrium statistical model for nuclear reactions has been suggested by Griffin 1). It is assumed in Griffin's model that the energy of the projectile is initially shared between some small number of particles and holes, and that this system progresses toward an equilibrium particle-hole distribution through a series of successive two-body interactions. The decay probability for each intermediate state in the equilibration sequence is calculated from phase space considerations. This model has been modified and amplified somewhat to accommodate charged particle emission 2). Early applications of the model to excitation functions for helium ion induced reactions yielded vast improvements over the equilibrium model with respect to reproducing the high-energy tails of excitation functions 3). This corresponds of course to reproducing the high-energy component of the emitted particle spectra. These results required different admixtures of pre-equilibrium and equilibrium decay for each excitation function analysed, which implies incorrect calculated spectral distributions. It was suggested that the source of the discrepancy might be the value assumed for the initial particle plus hole number in the decay sequence 3). In this work we analyse excitation functions for 4He and proton induced reactions at excitations up to 100 MeV. The sensitivity of calculated excitation funct Present address: Department of Chemistry, SUNY, Stony Brook, Long Island, USA. tt This work was supported by the U.S. Atomic Energy Commission and the National Science Foundation. 559

560

M. BLANN AND F. M. LANZAFAME

tions to the initial particle plus hole number is investigated, and results are compared with the experimental excitation functions. We check the consistency of the initial particle-hole number found from particle spectral analyses at lower excitation energies with the values required to give consistent results with excitation functions, and extract a rough estimate of the fraction of pre-equilibrium emission needed to reproduce the experimental results for different regions of target mass.

2. Pre-equilibrium decay model The rate expression for particle emission from a state characterized by n particles plus holes is given by eq. (6) of ref. 2) as o . ( e ) d e - mea p._x(U) de, g2h3 p,(E)

(1)

where e is the channel energy, m is the reduced mass of the emitted particle, a is the inverse cross section, p,_ I(U) is the level density of the residual nucleus, and p.(E) is the level density of the compound state (not an equilibrium ensemble) of n particles and holes. Eq. (1) must be multiplied by the mean lifetime of an n particle-hole state z, to give the decay probability for such a state. The statistical spin degeneracy of the emitted particle (2s+ 1) should also be included, giving the decay probability P,(~)d8 = (2s+ 1)mea p,_ I(U)z,d~. z~h 3 p.(E)

(2)

Because the level density p,(E) is a very rapidly increasing function of n, the simplifying assumption is made that transitions are predominately to higher n for n <
.=.,

An=+2

1rzha mea

.=.,

(3)

An=+2

A comment may be made concerning the termination of the series of eq. (3) at ~, since obviously the assumption that An = + 2 is not good as n approaches ~. The important point is that the series ordinarily converges for n << ~, for as the excitation becomes distributed over larger numbers of particles and holes, the chance of a particle having enough energy for emission into the continuum decreases exponentially,

STATISTICAL MODEL FOR NUCLEAR DECAY

561

Thus, the system either undergoes pre-equilibrium decay early in the reaction, or it passes toward the equilibrium distribution where multiple transitions take place prior to particle emission, i.e., the exponentially small pre-transition emission probability is multiplied by an exponentially large number of transitions. The question of the mean lifetime of an n particle hole state remains to be answered. Time dependent perturbation theory was used in ref. 1) to give the transition rate from a state n' to any one of the final states n, 27z

~o,.., = ~ Iml 2p,(E).

(4)

Since not all p,,(E) levels are available in a single transition from the state n' (where n = n' _ 2), the average matrix element in eq. (4) for a two-body transition must decrease with increasing n. In eq. (4) of ref. z) it was assumed that (M)2ctl/p,,(E), which implies a constant lifetime for all n. We use this assumption throughout the remainder of this work. We point out, however, that Williams 4) has considered this problem in terms of a constant two-body matrix element, with transition rates based on a calculation of the average number of states actually available in a given transition, and finds z,,~(n + 1). The rapid variation with n of the level density ratios found in eq. (3) suggests that the differences in predicted particle spectra between the assumption ~, = constant and z,,~(n + 1) should not be great, particularly at lower excitation energies. 2.1. PARTICLE-HOLE LEVEL DENSITIES

Several expressions have been given for level densities as a function of the number of particles, p, number of holes, h, or number of particles plus holes, n = p+h. Ericson 5) gives the following expression:

pp, h(E) =

g(g~)p÷h-i p!h!(p+h-1)! '

(5)

where g is the average single-particle level density, Griffin 1) gives the result:

p.(E) - 9(gE)"-I n t(n- 1)!"

(6)

Griffin assumed that particles and holes were indistinguishable in deriving eq. (6); this is not valid since particles are defined as excitation carriers above the Fermi level, and holes as excitation carriers below the Fermi level. Thus the correction for the number of indistinguishable permutations of particles, and of holes, must be done separately as in eq. (5), and not by s u m m i n g p + h as in eq. (6). The most probable n for eq. (6) is given by n =

x/gE; for eq. (5) by ~ = x/29E. The total level density

562

M. BLANN AND F. M. LANZAFAME

obtained by applying the approximation of Laplace for (5) is p , ( E ) ~ E -1 exp(Z~/2gE),

(7)

and for (6) it is

p.(E)

exp (2,/gE).

(S)

Thus it may be seen that neither expression integrates to the proper one-fermion level density, E - 1 exp(~/~nZgE). This is due in large part to a failure to conserve energy between single fermion levels in deriving (5), or alternatively expressed, to omission of terms of lower order in n, but which become appreciable as n increases at fixed gE. Thus (5) and (6) become progressively poorer approximations as n increases. The difficulties in eq. (6) were first pointed out to us by Williams 4); the failure of (5) at high excitations was brought to our attention by Bohning 6). Substitution of (6) into (3), with the assumption % = constant, gives the preequilibrium prescription used in this work, / UV'- 2 gE

,=,,

(9)

An=+2

where all constants have been combined into the constant ~. Use of eq. (5) rather than (6) in obtaining (9) results only in the replacement of t h e n ( n - 1 ) f a c t o r in the summation b y p ( n - 1). This is a very small difference since eq. (9) is used in a relative rather than an absolute fashion in the remainder of this work.

3. Comparisons with experimental results Unlike the equilibrium statistical model 7), the pre-equilibrium model has not withstood 30 years of confrontation with a pot-pourri of experimental results. Thus, we are at a point of starting such a process on a scale of relatively coarse comparisons to test confidence in the basic assumption of the model itself. Should these comparisons prove successful, some aspects of the theoretical derivation of the model should be more closely scrutinized, followed by additional confrontations with experimental results. At this later stage, it may be possible to gain insight into the reaction dynamics, with respect to such questions as relative time of emission of the pre-equilibrium components, number of intermediate resonances contributing, degree of equilibration present in what is generally treated as the equilibrium component of particle spectra, and other related questions. In this section we consider comparisons of experimental excitation functions with those calculated using the pre-equilibrium model in conjunction with the earlier equilibrium model, looking for consistencies or inconsistencies when a broad range of comparisons are made. 3.1. C A L C U L A T I O N

OF

EXCITATION

FUNCTIONS

The first question which arises in the application of eq. (9) is the initial particle-hole number, n~, for as will be shown the calculated particle spectra are quite sensitive to

STATISTICAL MODEL FOR N U C L E A R DECAY

563

this parameter. It was shown in ref. 2) that if the ratio U/E is small, the leading term of eq. (9) is dominant. If the log of the intensity of the hard component of particle spectra is plotted against log U (assuming or(e) constant in the range of interest, and a small range of (e), the slope will be n j - 2. Griffin treated data for (e, p) reactions on seven odd Z targets in this way 8), finding nl - 2 = 3.1+0.6, or ni = 5. Thus, our first calculations were performed wtth ni = 5 for helium ion deduced reactions. Subsequently, calculations were performed with ni -- 3 and 7 to illustrate the sensitivity of results to the choice of n. All other parameters were evaluated as in the preceding article 9), with a = A/8 MeV-1 Analysis of (p, n) spectra from a variety of targets were performed by Burris and Verbinski 1o). They reproduced the high-energy portions of the spectra well with the assumption that n~ = 3. This value was used for the two proton induced reactions analysed in this work. After the question of evaluating the parameter "n~", the next question is the variation of the fraction of pre-equilibrium decay with bombarding energy for a given projectile. Intuitively, one would expect a slow increase in pre-equilibrium emission with increasing excitation. For simplicity, in making these first approximation comparisons, we have assumed the fraction to be a constant, independent of energy over the ranges considered. All nucleides in the pre-equilibrium fraction f were assumed to begin their decay sequence with the same ni, the remaining (1 - f ) decaying with an equilibrium distribution. The programs and input parameters listed in the preceding paper were used, with appropriate modification, for both sets of calculations. A more rigorous calculation where multiple particle emission is involved would require bookkeeping on the population as a function of n as well as U; thus, some of the pre-equilibrium nuclei would at each stage pass to an equilibrium distribution before further particle emission, whereas some part of the initial population would undergo pre-equilibrium emission, leaving a residual nucleus with a lower n than the parent nucleide. These residual nuclei would in turn have a higher probability for pre-equilibrium emission, and would give a "harder" particle spectrum. The two effects are in opposite directions, tending to minimize the errors in the approximations which have been made. We emphasize again that our purpose here is not to present the most detailed calculation possible of a generally untested model, but rather to present a first order test on a system at relatively high excitations to determine the advisability of more rigorous confrontations between this new model and experimental results. The excitation functions which have been analysed are for the reactions 197Au(0c, xn)Z°l-~T1, x = 1-4, S'V(e, n)S4MnSW(e, 3n)S2Mn, Slv(~, p3n)SXCr, S6Fe(~, 3n)SVNi, 59Co(p, 3n)57Ni, S6Fe(c~, p2n)57Co, and S9Co(p, p2n)STCo. Ot these reactions, the reactions in t97Au are the easiest to compare with calculated results. This is because most of the cross section goes into neutron emission, a smaller amount into proton emission, and even less into e emission. Where proton emission

564

M . B L A N N A N D F. M. L A N Z A F A M E

can compete with n e u t r o n emission (as in the lighter targets in the list above) the lack of knowledge of absolute level densities for the nucleides involved confuses the distrib u t i o n o f cross section between pre-equilibrium a n d equilibrium c o n t r i b u t i o n s on an absolute basis, a n d adds somewhat to the uncertainties on a relative basis. Where a complex particle o f mass A can be emitted (e.g. a n a-particle), the question of the a d d i t i o n a l competition must be answered. Derivation of (9) for emission of a cluster

I00

\ ,gT~u(a,n}'Z°°TI

,\ t~

i

0.1

•

e,~(MeV)

'9~u(a,2n) 199Ti

\ 100

10

Au(e,4n)1971"1

'9~u(a,3n)feBTI

1000

20

x,,,

/o

c~(MeV)

"

6o

'40

6b

c~(MeV)

8'o

40

60

80

c~(MeV)

Fig. 1. Experimental excitation functions for Z97Au(0~,xn) reactions (x ~ 1-4) compared with equilibrium and pre-equilibriumstatistical calculations. The heavy solid curves represent the experimental results; thin solid curves represents the results of the equilibrium evaporation model. The dashed curves represent calculations in which it is assumed that 35 ~ of the compound state cross section decays with a pre-equilibrium spectral distribution, with ni = 5, as discussed in the text.

S T A T I S T I C A L MODEL FOR N U C L E A R DECAY

565

o f mass A gives pA(e)de _

c~(2s+ 1)mea "f (gEa) de .=zA An =

p!(p+h-1)! [U'~"-a-' (p-a)!(p+h-A-1)! \El ,

(10)

+2

if the particle-hole level density (5) is used. The question o f preformation is not answered in (10), so that an additional normalization question arises if cluster emission is considered. In this work we did not consider e emission channels to be open for the lower Z targets; this will make the f p r e c o m p o u n d values f o u n d lower limits. ..~,k~.

1000

~ " ,

o~

L',\

L \"

/\\ i

,,, \\

/ix

, °'°° \

I00

. ~197TI

197A

19"tAu {el,3n) l~ii

l

".°0

"

7 1

"%°"

Ii 10

t

i

2o

6'o ca(MeV)

.........

4o

60

ca(MeV)

"% %'.. ~..°,~.°

4'0

6'o

8'0

ca(MeV)

Fig. 2. Experimental excitation functions as in fig. 1 compared with pre-equilibrium model predictions for several values of n~. The solid curves represent the experimental excitation functions; dotted and dashed curves represent pre-equilibrium admixtures with the equilibrium component for n~ = 3 and 7, respectively. For n~ = 3, there is a 14 70 pre-equilibrium component; for n~ = 7, 100 ~ .

In the case of the 197Au_{ - ~x reactions, the fraction o f pre-equilibrium emission ( f ) was obtained by arbitrary normalization o f the calculated pre-equilibrium excitation function for the (c~, 2n) reaction to the experimental cross section at 50 MeV incident helium ion energy. The value required was 0.35, for nl = 5. All excitation functions in fig. 1 were then constructed by taking 0.35 o f the excitation functions calculated with a pre-equilibrium energy distribution, and 0.65 o f the excitation functions calculated with equilibrium energy distributions. Total reaction cross sections were taken f r o m optical-model non-elastic cross sections, as described in the preceding article. It m a y be seen in fig. 1 that reasonable agreement results for all excitation functions w i t h f = 0.35, with an unquestionable improvement in overall fit relative to the results o f the equilibrium model, which is shown for reference. The sensitivity o f the calculated energy distributions to ni is illustrated in fig. 2, where values o f 3 and 7 were used rather than 5. Again, an arbitrary normalization

566

M. B L A N N A N D F. M. L A N Z A I ~ A M E

U (MeV) .... 2o

,40

o

,

U (MeV) ~o

4o

~

o

U (MeV) 2o

4o . ..,,, ,

P

~2'-.. "... .,,.

n x 10"~

10'

o

"....'-..,,. •.. ,,.

'

'..

"' ,.,

\

.....

\

%

/i ' 0

\

0

"

!19

7~5

20

40

i", '

II

'

\3

',:

\ 19

60

!

'

7 "S i3

20

e (MeV)

'-'

40

i 3

' '

60

'

20

~- (MeV)

"S

"

40

60

e (Me',,/)

Fig. 3. Calculated non-equilibrium particle spectra for neutrons, protons, and alpha particles as a function of ni. The system considered in this example is 197Au-I-65 MeV 4He. The abscissas are labelled both as to channel energy, e, of the emitted particles, and residual nucleus excitation U. Calculated values are for the first particle emitted, and the a is treated as changing the particle-hole number by only a single unit. The thin solid curves are the predicted equilibrium particle spectra.

~,, •

~,

IO00

,vl°,°l~,Mo

.,~:....

\~, ~'v¢o,ol"M.

,oo

, \

~oo

',,,, ~.

:\

~

1,1=3 ........

\

-

\

~

~ ~

E

t

-........... ...........

"~ io

",,. "~x rli=5 ",...

1.0

°'~o

'...

, '~. \

~

E

..............

.~

1.0

ni= 7

i

i

i

I

I

i

i

~o

~o

4o

so

~o

7o

~o

E (MeV)

910

O.I

i

~o

i

3o

i

I

so

I

t

7o

I

i

so

E(MeV}

Fig. 4. Experimental excitation function for the slV(~, n)~4Mn reaction, compared with equilibrium and pre-equilibrium calculations. The heavy curves represent the experimental results. The dotted curve labelled h represents an equilibrium model calculation, the dashed curve (hi = 5) represents the pre-equilibrium result, as do the dotted (n~ = 3) and dot dash (n~ = 7) curves. The calculated cross sections are all un-normalized, i.e. based solely on calculated yields based on total non-elastic cross sections.

STATISTICAL

MODEL

FOR

NUCLEAR

567

DECAY

was made based on the (~, 2n) excitation function at 50 MeV; pre-equilibrium fractions of 0.14 and 1.00 were required for ni = 3 and 7, respectively. It may be seen that a consistent result is no longer possible, for any f, ifn~ ~ 5. Particle spectra calculated for different n~ for ~97Au + 65 MeV 4He are shown in fig. 3 to emphasize the sensitivity 400

I000

5~V( a, 5n)SZMn

5TV(a,pSn)5=Cr

IOO

/ /'"

'. ~

//

-

"~ E

IOO

/ i

i

r.! i..'

I

I

4OO .#

"-.\

,'/I I

~

I

I

I

I

v

i

T

I000

°J" ,% ~,%

t/

,. t

10

•\

.i

.,,.ni-7

,.-"

~=V{a,3n) r~Mn i:7 [%'x

I00

i i

x.

.13

~4o b

"~"

"~

•

!

..:

i

-

i

.:"

E "X

IO0

........ .

x.

" - .....

~,%

""

/// i

, [, 4

L 40

,

I 60

E(MeV)

i

i 80

t

I 100

I0 40

;

i I 60

I

| 80

•

_

t

IO0

E(MeV)

Fig. 5. E x p e r i m e n t a l a n d calculated excitation f u n c t i o n s for (~, 3n) a n d (~, p3n) reactions in s l v . T h e thick solid curves represent t h e experimental excitation functions; t h e curves labelled ~ represent u n - n o r m a l i z e d e q u i l i b r i u m m o d e l calculations. Curves labelled with nt values o f 3 a n d 7 represent u n - n o r m a l i z e d calculations for t h o s e initial particle-hole n u m b e r s . T h e dot d a s h curves for nj = 5 are n o r m a l i z e d calculations, a n d t h e d a s h e d curves represent normalized s u m s o f ni = 5 a n d equil i b r i u m calculations, as discussed in the text.

of spectral distributions to this parameter (the e-spectra shown in fig. 3 are not strictly valid since the residual nucleus level densities were calculated using p._ 1 rather than

P.-s).

568

M . B L A N N A N D F. M . L A N Z A F A M E

A n a l y s e s o f excitation functions for helium ion i n d u c e d reactions in v a n a d i u m are shown in figs. 4 a n d 5. A g a i n it m a y be seen that ni = 5 seems to be better than either 3 or 7 in consistently r e p r o d u c i n g all results. A s in the case o f 19VAu(~ ' n), the 51V(c~, n) results give the p o o r e s t agreement, b u t a f a c t o r o f 3 to 4 e r r o r at 90 MeV, for an excitation function p e a k i n g at 20 M e V is well within the a p p r o x i m a t i o n s involved in

10

~/ ~3~Nylb~O~_ ~' / 1,0

Z b 1000

,oof

e56(a,p2n)Co 57

/ /

Io

4~

5'o

6b

7'o

E(MeV) Fig. 6. Experimental and calculated excitation functions for decay of 6°Ni compound states. The thick solid curves with experimental points represent the S6Fe(cq 3n)S7Ni and 56Fe(g, p2n)STCo excitation functions [ref. 11)]. The heavy solid curves represent the ~9C0 (p, 3n)s 7Ni and SgCo (p, p2n) STCo excitation functions of ref. 12). The calculated (dotted) curves represent a mixture of 50 equilibrium emission with 50 % pre-equilibrium emission, n~ = 3. The dashed curves are for the same mixture of equilibrium and pre-equilibrium components, but for n~ = 5.

the calculation. D e t e r m i n a t i o n o f f for these results is c o m p l i c a t e d by the factors previously described; results in ref. 3) illustrate the difficulties in trying to calculate absolute cross sections for these reactions. We therefore f o u n d the best relative a d m i x ture o f e q u i l i b r i u m a n d p r e - e q u i l i b r i u m excitation functions for each o f the reactions (~, 3n) a n d (c~, p3n), to g e t f . The s u m m e d excitation functions were then n o r m a l i z e d

STATISTICAL MODEL FOR NUCLEAR DECAY

569

for the comparisons shown in fig. 5. The values found w e r e f = 0.50 for the (~, 3n) reaction, a n d f = 0.65 for the (~, p3n) reaction. As stated above, the assumption of no a-emission in the pre-equilibrium case makes these values lower limits. No normalization was attempted for the (0~, n) reaction since the poor agreement in shape for the pre-equilibrium component would make this meaningless. The final examples for comparison were selected to show the dependence of decay mode on mode of compound state formation where pre-equilibrium emission is involved. The four excitation functions chosen result from the decay of the 6°Ni compound state, formed with helium ions in one case 11) and with protons in the other 12). The calculated excitation functions for 57Co and s 7Ni production are compared with experimental results in fig. 6. All results correspond to sums of equilibrium plus preequilibrium calculations such that f = 0.50, but with n i = 3 for proton induced reactions, and nl = 5 for helium ion induced reactions. These values o f n i, as stated earlier, are consistent with values derived either explicitly or implicitly from analyses of particle spectra. Thus there seems to be preliminary evidence for a consistency in the requirements for fixed ni for each projectile, and of a pre-equilibrium fraction of the order of tens of percent over the excitation range investigated in this work. The absolute values found for the pre-equilibrium components will change as the model is modified, either by using different assumptions for z,, or by changing level density expressions, etc. Nonetheless, it seems reasonable that values will remain in the range of tens of percent in the excitation range of the data analysed in this work. The results of these comparisons encourage further work with the prequilibrium model. (For additional comparisons, see ref. 13); see also the pre-equilibrium approach of Harp et al. a 4).) More desirable data for analysis would be spectral data for various light projectile induced reactions, integrated over angle, for increasing bombarding energies, and over a broad sampling of target mass. One could then hope to get quantitative answers to such questions as (i). Does the model predict the energy distributions well? (ii) How does n i vary with projectile and odd-even character of the target? (iii) How does the fraction of precompound emission vary with the excitation energy for a given target mass and (iv) How does the precompound emission fraction vary with target mass for a given excitation? If these questions can be answered empirically, it may then be possible to begin interpretation of data in terms of reaction dynamics. The fact that the fraction of pre-equilibrium emission needed to fit the data of this work is as large as it is, has implications for analysis of spectral data in terms of an equilibrium model. Where multiple particle emission may be involved, it has been customary to analyse the high-energy portion of the particle spectrum to extract nuclear temperatures or level spacing parameters. This is precisely the region where preequilibrium contributions will be greatest relative to equilibrium contributions. Thus, the parameter values extracted from such data analysis could very well be found to vary with U/E and U as was found by Siderov 15), Lassen and Siderov 16), and others. Additionally, since nj seems to be a function of the projectile used to form the corn-

570

M, B L A N N A N F F. M. L A N Z A F A M E

p o u n d state, the statistical model parameters extracted for a given U/E a n d U may well be a f u n c t i o n of the mode of f o r m a t i o n of the c o m p o u n d state, in contradiction to the Bohr hypothesis which is based on an assumed e q u i l i b r i u m situation. These ideas are n o t qualitatively new or original, but it is hoped that this model m a y provide a quantitative basis with which to u n d e r s t a n d the i m p o r t a n c e of such effects. One of the authors (MB) acknowledges with thanks, helpful discussions o n the p r e - e q u i l i b r i u m decay model with Professors J. M. Miller a n d J. B. French, a n d with Drs. F. C. Williams, a n d J. C. Parikh.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

J. J. Griffin, Phys. Rev. Lett. 17 (1966) 488 M. Blann, Phys. Rev. Lett. 21 (1968) 1357 W. W. Bowman and M. Blann, Nucl. Phys. A131 (1969) 513 F. C. Williams, Univ. of Rochester, private communication, and UR 3591-13 (1969) T. Ericson, Adv. in Phys. 9 (1960) 425 M. Bohning, Technische ttochschule MiJnchen, private communication V. F. Weisskopf, Phys. Rev. 52 (1937) 295 J. J. Griffin, Phys. Lett. 24B (1967) 5 F. M. Lanzafame and M. Blann, Nucl. Phys. A142 (1969) 545 V. V. Verbinski and W. R. Burrus, Phys. Rev. 177 (1969) 1671 A. Ewart and M. Blann, J. Inorg. Nucl. Chem. 27 (1965) 967 R. A. Sharp, R. M. Diamond and G. Wilkinson, Phys. Rev. I01 (1955) 1493 E. Kondaiah and K. Parthasaradhi, Nuovo Cim. 53 (1968) 486 G. D. Harp, J. M. Miller and B. J. Berne, Phys. Rev. 165 (1968) 1166 V. A. Siderov, Nucl. Phys. 35 (1962) 253 N. O. Lassen and V. A. Siderov, Nucl. Phys. 19 (1959) 579