Analysis of evaporation residues following light heavy-ion reactions

Nuclear Physics A341 (1980) 284- 300; @ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

ANALYSIS OF EVAPORATION RESIDUES FOLLOWING LIGHT HEAVY-ION REACTIONS

A. J. COLE, N. LON~EQUEUE, J. MENET, J. J. LUCAS, R. OST and J. B. VIANO

Institut des Sciences NuclPaires (IN2P3, USMG), 53, Avenue des Martyrs, 38026 Grenoble Cedex, France

Received 23 November 1979

(Revised 7 February 1980)

Abstract: Measurements of yields of evaporation residues identified by mass and charge following the reactions 2oNe+ “C (66.5 MeV) and ‘60+ i60 (55 MeV) have been carried out using a time-of- flight dE+E technique. These data have been analysed together with other data for compound nuclei with masses <: 32 and E, < 50 MeV in the framework of the statistical model using new computer codes which predict isotope cross sections and angular and energy distributions. A good description of all data is obtained and the determination of the critical I-value in the compound nucleus is shown to be possible using only the relative yields of the evaporation residues.

E NUCLEAR REACTIONS 160(‘*0, X), E = 55 MeV, !2C(ZoNe, X), E = 66.5 MeV;

measured relative yields evaporation residue ~(0, I?). Statistical-model calculations.

1. Introduction

A considerable amount of effort has been expended in the last five years in mea- suring and predicting cross sections for isotopic yields following compound nucleus formation in light heavy-ion reactions 1 -9). Most publications have focussed attention on one aspect of the data, namely mass yields, charge yields or fully identified isotopic yields. [In this respect the most courageous attempt appears to be that of Piihlhofer 4, who has quite successfully described mass or isotopic yields for evaporation residues from several sources using a unique model.] Thus, in the literature there exists to date little discussion of the sensitivity to model parameters and in addition energy spectra and angular distributions are usually ignored.

The main reason for this situation appears to be the difficulty of performing multistep statistical-model calculations economically. Thus it is extremely difficult to systematically explore the parameter space which may be considered physically reasonable even when marked discrepancies between predictions and data exist. With regard to energy spectra and angular distributions a further di~culty arises from the n-body nature of the kinematics and the necessity to approximate the angular dependance implied by the Hauser-Feshbach formula.

284

A. J. Cole et al. / Evaporation residues 285

We present in this work an attempt to analyse residue cross sections, angular distributions and energy spectra in a consistent way. In addition a limited investiga- tion of sensitivity to model parameters has been made.

The long computation times usually associated with such calculations were avoided since it was found to be possible to restructure the usual Hauser-Feshbach formula gaining a factor of N 10 in the calculation time. The problem of constructing angular and energy distributions was solved by use of the Ericson and Strutinsky semi- classical approach lo). The basic structure of the corresponding computer codes is described in sect. 3 and a comparison of results with those obtained using the GROG1 2 code “) is presented in sect. 4.

The above techniques have been applied to measurements carried out on the “Ne+ “C system at 66.5 MeV (“Ne) and on the 160+ 160 system at 55 MeV. The experiments are described in sect. 2. The two reactions lead, after allowing for energy losses in the target, to the 32S compound nucleus at the same excitation energy (43.75 MeV). In addition, residue cross sections from other sources have been com- bined with our measurements in an attempt to explore the sensitivity of the relative yields to the critical angular momentum in the compound nucleus. The data analysis is described in sect. 5. In sect. 6 we present a summary of the principal results of this work and suggestions for future studies.

2. Experimental procedures

The measurements were carried out at Grenoble using 160 (55 MeV) and 20Ne (66.5 MeV) beams from the ISN isochronous cyclotron. Evaporation residues were identified in mass and charge between 4O and 28” (lab) using the time-of-flight dE(gas)-E(solid state) detection system previously described in ref. 2, with the only difference being that a channel plate detector was used for the start signal.

For the “Ne+ 12C measurements the target was a self-supporting “C foil of 100 pg/cm2 thickness. Only relative cross-section measurements were made, the absolute normalization having been determined previously in an experiment using only residue charge identification 6). For the 160+ 160 experiment a 160 pg/cm2 SiO, target was used and again only relative cross sections were established, the system having already been studied by mass identification at 52 and 60 MeV where absolute cross sections accurate to within 5 % were extracted ‘). It should be stated that our final relative cross sections integrated over charge agree very well with interpolated values using the data of ref. 5).

The energy spectra of ’ 6O + ’ 6O residues show a cut-off at low energies due to particles stopped in the gas AE detector. In order to estimate the missing cross section, additional spectra were taken without AE detection (gas pressure = 0, mass identification only). Low energy cut-offs were negligible in the 20Ne+ “C data. The experimental data are summarized in table 1. Error bars include statistical errors, uncer-

286 A. J. Cole et al. / Evaporation residues

TABLE 1

Experimental data

Residues % ~fwian

“Ne+ l*C 66.5 MeV 160+ I60 55 MeV

*‘Ne 23Na =Mg 26Mg 26A1 27Al “Si **Si ‘%i “Si 3oP

7.5 * 4.0 2.4+ 0.4

26.2k3.0 4.4kO.4 7.9+ 1.2

27.9+ 1.5 3.1 +0.3 0

15.6+0.7 1.9kO.2 3.2kO.3

5.2kO.4 4.6kO.3

32.6k3.3 4.OkO.6 9.5k1.2

24.0+1.8 2.8kO.8 2.5k1.3

12.3k2.6 2.5+ 1.3 0

ufusion = 1087+ 130 mb”) ,Jf”Si0” = l120&50mbb)

“) Deduced from ref. 6, for residues 11 5 Z s 15 and our measurement for *‘Ne b, Ref. ‘) interpolated between 52 and 60 MeV.

tainties in mass and charge separation and for, 160 + 160, uncertainties induced by

correcting for the missing low energy events.

3. Computer programmes

Lack of space obviously precludes detailed descriptions of the computer codes. They are described in detail in ref. i2). The main principles and ingredients can however be given.

3.1. MULTI-STEP HAUSER-FESHBACH CODE LANCELOT

The principle of the programme is identical to that of the GROG12 code ll). Absolute probabilities for competing decay modes from a given parent nucleus are obtained by assuming that the decay modes, included in any calculations, represent the total decay probability. The unnormalized probability for emission of particle b with energy E and angular momentum 1 from a parent nucleus at excitation energy E, and spin Ji is [ref. ‘“)I

J, + I S+j

PJiYEX(b, E, r) = IT;(E) S=(J,-l, Jf=;-j,p(Ef’Jf)’

c (1)

Herej is the emitted particle intrinsic spin, S the channel spin, E, the excitation energy in the daughter nucleus and J, the final spin. The normalization of this probability


is obtained by summing over discrete and integrating over the continuous variables in brackets on the left hand side of (1).

In the course of obtaining the normalization constant the two-dimensional unnormalized probability density P(b, I) obtained by integration of eq. (1) over E is obtained. After normalization Monte Carlo techniques are used to select the particle type (b) and orbital angular momentum (I). Knowledge of b and I make eq. (1) a function only of the energy of the emitted particle. It, in turn, is selected. Finally with b, 1 and E known the probability of decay to a given value of Jr is reevaluated for each J, enabling Jr to be selected.

LANCELOT is thus technically different from GROG12 since it employs Monte-Carlo techniques to simulate the evaporation process. More importantly LANCELOT is based on a parametrization of the density of states with a given projection M, W (E,, M,) related to p(E,, J,) via

p(E,, Jr) = IV,, M, = Jr) - W(E,, M, = Jr + I), (2)

instead of being based directly on a parametrization of p(E,, Jr). The advantage of this technique can be seen immediately by substitution of (2) in (1). We obtain

Vi-ll+j Ji+l+l+j

PJ1vEX(b, E, r) = IT;(E)[ 1 W(E,, K) - c W(E,, N)]. K=)IJi-II-j1 N=IJi+I+l-j(

(3)

Use of (3) rather than (1) speeds up the computation by a factor of N 1 for given 1. For y-ray emission an expression similar to (3) is used in which the quantity

e,E;’ + 1 plays the role of transmission coefficient. The parameters al have been adequately discussed by Ptihlhofer “) and by Grover and Gilat 1 ‘). It turns out that in light systems y-ray competition above particle emission thresholds is not important 14).

The main ingredients of the calculation are therefore the spin distribution of the compound nucleus, the state densities W(E,, M,) in each nucleus encountered in the decay chains, and the transmission coefficients which here, as in other works, are supposed, for a given projectile to be obtained from the elastic scattering of the evaporated particle from the ground state of the daughter nucleus in a time reversed frame of reference.

(a) The initial spin distribution of the compound nucleus. As for the GROG12 code this distribution is specified by:

2J,+l

P(Exy Ji) = 1 + exp [(Ji - J,,)/d J] ’

The initial value of the critical angular momentum is taken from the experimentai fusion cross section

oF = aA2(JcR+ l)‘,


where 2 is the reduced wave length. Little is known about AJ. It is usually assumed to have a value similar to that obtained from optical-model calculations of the elastic scattering (Z lh).

It must be stated that eq. (4) is not firmly founded. It relies on the strong absorption assumption in the entrance channel. In light heavy-ion collisions at incident energies of the order of 100 MeV, time-dependent Hartree-Fock calculations have suggested that fusion does not take place at low impact parameters is) (low values of Ji)_ To our knowledge there exists no experimental evidence to support this conclusion. We thus use eq. (4). In LANCELOT, events for successive values of Ji are processed sequentially, the total number of events and thus the accuracy of the calculations being specified by the user.

(b) Level densities. As in other calculations we make a distinction between state densities in the discrete and continuum energy regions. At excitation energies above l&f5 MeV the Lang 16) formula is used with the parameters given by Gilbert and Cameron I’). These parameters should be adequate up to - 30 MeV excitation energy 4). We do not exclude the possibility that it may become necessary to modify the basic formula or the parameters in the light of new experience. However no convincing evidence would seem to exist at the present time precluding the use of this approach. At low energies the Taylor expansion of the Lang formula is used in conjunction with the constant temperature assumption (T = 1 MeV) and a moment of inertia “dexp determined from systematics of low-lying levels, to fit the level density parameter “a” to the experimentally known number of levels of any spin up to energy Ek (determined according to the state of knowledge of the level scheme for each nucleus considered).

The parameters a, ‘I’ = 1 MeV and <fexp are then used to generate W(E,, M,) up to E,. A smooth interpolation between the low energy and continuum regions is made between E, and E, f 5 MeV. The level density parameters used in our calculations are summarized in table 2.

(c) ~~u~s~~s~~o~ c~e~~c~e~~s~ Comparisons with the GROGI code results (see sect. 4) have led us to the conclusion that description of evaporation residue cross sections is not impaired by replacing optical-model transmission coefficients with those generated from optical-model parameters using the parabolic barrier approximation ‘*). This remark may not be true at low excitation energies where one-step evaporation processes may be influenced by diffractional effects in optical-model transmission coefficients especially for nucleon evaporation.

(d) Rotational stretching. This section would not be complete without some discussion of the influence of deformation of rotating nuclei on the evaporation process. A priori both the evaporated particle transmission coefficients and the level densities would be influenced by rotational stretching 20). For the range of nuclei, excitation energies and angular moment considered in the present work, this effect is not thought to be important; for example the liquid-drop calculations 21) of Cohen et al. for 26A1 would indicate essentially zero deformation up to J = 15 and it is rare

A. J. Cole ef al. / Evaporation residues 289

TABLE 2

Level density formula and parameters

1 exp (2JL(&3@7Z3 WE,, M,l = yj

(f/h2)1’2a3’2t3 Valid at E, > E,+5 MeV (see text)

a (MeV- ‘) u/A = O.~l~S+O.l20 best description of nuclei from Na to K [ref. “11

I/h2 (MeV-‘) 0.0096R’2A5;3 I rigid body moment of inertia [ref. 13)]

R;’ = Rf(l +jJ;)’ z R:(I +2bJ;)

rotational stretching correction to radius [see ref. ‘)]

B 0.00025 compatible with ref. *) (see text)

E yrns, J( Jf 1)

29/h2+A A pairing correction ref. “)

in the cases studied herein that evaporation from the compound nucleus leads to final angular momenta superior to this value (high angular momentum states of the compound nucleus tend to evacuate large amounts of angular momenta by Q- emission). Nevertheless this effect has been included in LANCELOT. The radius R, used to specify the moment of inertia in the continuum energy region is rnodi~~ by the angular momentum J, via

K = R,(l +&I;). (10)

A similar relation may be used (optionally) to modify transmission coefficients for evaporated particles (by m~i~cation of the radius at which the barrier is calculated). The constant p must be specified for each nucleus encountered in the deexcita- tion chains.

3.2. SIMULATION OF ANGULAR DISTRIBUTIONS AND ENERGY SPECTRA

The isotope cross sections generated by LANCELOT are obtained using the angle-integrated Hauser-Feshbach formula. As well as relative yields of evaporation residues the programme produces for each event, a decay sequence. A sequence specifies for each particle or y-ray in the evaporation chain the mass, charge, energy, initial and final total angular momentum and the orbital angular momentum removed by the evaporation. Ericson and Strutinsky lo) have pointed out that in one-step evaporation knowledge of J, I, and J, for the evaporation of a spinless particle is


Ji A

Fig. 1. Classical angular momentum coupling. Definition of terms used in text. p, is the emitted particle momentum.

sufficient to construct an angular distribution using the “classical” concepts (see fig. 1)

1. pt = 0, e(I, p,) = 900. (8)

The validity of these approximations has been discussed in refs. “9 12). Strictly they should not be used for low-I neutrons or protons although the case 1 = 0 is correctly treated by imposing isotropic emission. However emission of such particles does not significantly affect the residue angle and energy distributions since the associated recoil momentum is small. The final angular distribution of evaporated particles is of course azimutally uniform. (Cylindrical symmetry about the beam “z” axis.) These concepts have been applied to the multistep evaporation process and incor- porated into a Monte-Carlo computer code DESTIN, which, for each event uses as input the decay sequence generated by LANCELOT. The direction and energy of the recoiling heavy nucleus and emitted light particle in the laboratory frame is followed at each stage of the evaporation. This procedure necessitates knowledge of 0 and 4 coordinates for all emitted particle momenta. These directions are obtained in DESTIN by using a more correct form 16) of eq. (7)

cos B(Ji, I) = Ji(Ji+ l)+I(I+ l)-J,(J,+ l)

2JJi(Ji + 1)1(1+ 1) ’

together with eq. (8) and the physically obvious assumptions that +(Ji, I) and &1, pJ are uniformly distributed as used in ref. lo) (see fig. 1).

The initial orientation of the compound nucleus angular momentum Ji is obtained using the Monte-Carlo method by assuming Ji to be uniformly distributed in a plane perpendicular to the beam direction. The triangle J&J, is also used to determine the orientation of Jr which in turn becomes Ji for the succeeding evaporation step. Intrinsic spin is neglected in the entrance channel in the sense that the initial Ji


is perpendicular to the beam. It is also neglected in the construction of the triangle JilJ, except that Ji and Jr are of course influenced by the presence of spin.

The influence of y-ray emission on residue and evaporated particle angular distributions and energy spectra is neglected.

4. Comparison with a GROG12 code calculation

Since the basic structure of LANCELOT is similar to that of GROG12 it is

possible to “simulate” a GROG1 2 code calculation. For this purpose we have chosen a rather detailed presentation of GROG12 results for the system 160+ “C at 60 MeV [ref. ‘)I. This choice was also influenced by the fact that the corresponding experimental data fall within the scope of the present study.

Of course we should not expect LANCELOT results to be identical with those of GROG1 2. The transmission coefficients are not specified in exactly the same way and the low-energy level densities are approximately described in LANCELOT whereas they are specifically inputted in GROG12. Finally since LANCELOT is a Monte-Carlo code its predictions contain “statistical” errors.

The GROG1 2 calculation of ref. ‘) used Gilbert and Cameron level density parameters above particle emission threshold. The same densities were used in LANCELOT. The authors of ref. ‘) used energy dependant optical-model parameters to generate nucleon transmission coefticients. Since LANCELOT uses energy independent parameters they were obtained from the energy-dependent formulae

160+lZc

60 McV

w

n GROG/ 2 (ril: 11 x LAMELOT

Fig. 2. Comparison of GROG12 [ref. ‘)I and LANCELOT calculations. (a) Emission probability in the first evaporation step as a function of compound nucleus angular momentum. (b) Relative residue cross

sections.


by supposing that E,,,, = 3 MeV and Eproton = 6 MeV. Finally as in ref. ‘) we have included only neutron, proton, alpha and y-emission. The comparison of the two calculations is shown in fig. 2. As expected the probability of particle emission in the first evaporation step is similar but not identical in the two calculations. The main difference (an increase in the a/p ratio in the LANCELOT calculation) was traced to diffractional effects in the 1 = 3 and 4 proton transmission coefficients which increases the relative proton emission probability in the GROG1 2 calculation. This difference does not have important effect in the relative yields as can be seen in fig. 2b where the calculations of the isotope cross section are seen to agree rather well with those of ref. ‘). It should be remarked that the calculation in fig. 2 consumes ‘V 200 set CP CDC 6600 and is thus considerably faster than the GROG1 2 calculation.

5. Analysis of data

5.1. ISOTOPIC YIELDS

The data which has been compared with the statistical model calculations is summarized in table 3. As well as the restriction on excitation energy we have rather arbitrarily limited the analysis to compound nuclei with mass =< 32 for which residues were either identified in mass or in mass and charge. Furthermore measurements involving residue identilication by y-ray detection were excluded since as shown in ref. ‘) residues may be produced with sizeable cross sections in their ground states. Even with these restrictions a large body of experimental data falls within the bounds of the analysis.

Preliminary tests showed that the description of the data was improved by slight modification on the nucleon optical-model parameters obtained from the literature. This is not surprising since the usual “global” optical-model parameters are for masses > 40 and also because the transmission barriers for nucleons obtained from optical-model calculations are somewhat lower than those calculated in the parabolic barrier approximation due to the shape resonance effects discussed in sect. 4. The parameters used for the various channels are displayed in table 4.

Since it was impossible to systematically explore all the parameter space of the model for all systems it was decided to focus attention on the sensitivity of the data to the angular momentum cut-off, JcR, in the initial compound nucleus in order to see if it was well determined by relative yields of evaporation residues. It was thus necessary to specify a quality of fit criterion which was taken to be

This criterion is identical to the usual chi-square criterion except for the absence in the denominator of a factor depending on the experimental error which was

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TABLE 4

Optical model parameters and gamma multipole strengths

Particle

n

V,, (MeV)

55

& (fm)

1.4

a (fm)

0.66

Ref.

22 )

Remarks

obtained from typical parameters (see comment in text)

P 50 1.25 0.65 22 ) typical parameters for 5-10 MeV protons from light nuclei

d

a

109.4 1.12 0.76 23 )

109.2 1.8 0.54 24 )

tit to 24Mg data at 10 MeV

fit to averaged data over energy range 12.3 to 14.0 MeV for 24Mg

y(l= 1)

Y (1 = 2)

El = 5.0x 10-s

a* = 3.0x lo-‘0

V, - real potential depth

obtained from lifetimes of low lying states in s-d shell nuclei

R, - radius parameter r. = R,A ‘P V(r) = - VO I+ exp C(r - r&l

a - diffuseness parameter

omitted because of the absence of experimental error bars in some of the published data.

In order to carry out these grid searches it was of course necessary to fix the values of the other parameters since a complete search over all parameters was considered to be too time consuming.

We choose /I = 0.00025 consistent with the liquid-drop model and with ref. ‘). At low values of JCR it would not be expected to have a marked effect on the calculations. This can easily be seen by noting that for JCR 5 20 the highest spin encountered in the first daughter nucleus is x 12/z which with the above value of /I produces a change of only N 7 % in the moment of inertia. A verification of this lack of sensitivity was made in the case of 160 + ’ 2C at 60 MeV.

The diffuseness parameter was chosen as AJ = lh. In the systems studied the reaction cross section differs little from the fusion cross section (- 10%). Thus values of AJ must be consistent with the fall off of the transmission coefficient produced by optical-model calculations 25). Otherwise one obtains fusion cross sections at the same J-value greater than the corresponding reaction cross section.

Level-density parameters “a” were taken from Gilbert and Cameron ’ ‘). As stated previously these parameters are expected to be valid at least up to - 30 MeV which is approximately the excitation energy encountered in the first daughter nucleus in these calculations.

A problem was however posed by the choice of R,, the radius used to specify the

A. J. Cole et at. / Evaporation residues 295

t 1 I , 1 I ,

0 9 10 11 12 13 14 JCR

16 17 18 19 20 21 22J 13 14 l5 16 17 16 19 - CR JCR

* _ 19F+12C

50 MoV

14 15 17 18 19 20 21 JCR

300- 160*% 32 MoV

Fig. 3. Variation of the goodness of tit parameter x with Jca for di!Terent values of R,. m R, = 1.2 fm 6R, = 1.3 fm l R, = 1.4 fm. Arrows represent thevalues of JCR used for the ~~cuiations shown in fig. 4.

Energies shown in the figure are those used in the calculations.

rigid-body moments of inertia for all nuclei in the d~xcitation chains. A priori one would expect this parameter to be between - 1.1 and - 1.4 fm consistent with electron and nucleon scattering results. The results of test calculations showed a rather high sensitivity to this parameter. It was therefore decided to perform grid searches on JCR for three values of R,, 1.2, 1.3 and 1.4 fm.

The results of these calculations are shown in fig. 3 and summarized in table 3. Several important conclusions may be drawn:

296 A. J. Cole et al. 1 Evaporation residues

(a) For any value of R, the variation of x with JCR shows a pronounced minimum. (b) The value of J,, at the minimum increases with increasing R,. (c) The value of J,, at the minimum for R, = 1.3 is in rather good agreement with

the experimental value determined from the absolute fusion cross section measurement.

For two of the systems studied further comments are necessary. In the I60 + “C systems at 60 MeV (160) no measurement of the 160 yield was made and, for the grid searches was taken to be zero. In fact the calculation gives x 10 “/ In the 19F+ 12C system at 63.2 MeV the trends cited above are not reproduced and there is a tendancy for a double minimum structure. In our opinion this may indicate a slight problem with the data since the values of x for this case are dominated by the contribution from mass 26.

The predicted mass or isotope relative yields are compared with the experimental

Fig. 4. Histogram plots of percentage yields for residues compared with Lancelot calculations hatched histograms). Dotted lines give experimental error bars where available.


(a) For any value of R, the variation of x with Jc, shows a pronounced minimum. (b) The value of Jc, at the minimum increases with increasing R,. (c) The value of J,, at the minimum for R, = 1.3 is in rather good agreement with

the experimental value determined from the absolute fusion cross section measurement.

For two of the systems studied further comments are necessary. In the 160 + i2C systems at 60 MeV (160) no measurement of the 160 yield was made and, for the grid searches was taken to be zero. In fact the calculation gives x 10 “/ In the “F+ “C system at 63.2 MeV the trends cited above are not reproduced and there is a tendancy for a double minimum structure. In our opinion this may indicate a slight problem with the data since the values of x for this case are dominated by the contribution from mass 26.

The predicted mass or isotope relative yields are compared with the experimental

%. %

Fig. 4. Histogram plots of percentage yields for residues compared with Lancelot calculations (cross hatched histograms). Dotted lines give experimental error bars where available.

A. J. Cole et al. 1 Evaporation residues 291

values in fig. 4 at the optimum values of J,, for R, = 1.3 fm. (For 19F + “C at 63.2 MeV the experimental value was used.) In all cases a very satisfying description of the data is obtained. As a further test of the performance of the model we have calculated the 160 + ‘*C residue distributions at Elab = 35, 45 and 52 MeV. Using J,, values derived from the experimental fusion cross sections ‘). The comparison of these predictions with experiment, also shown in fig. 4, shows that once again, a good description of the data is obtained except perhaps at the lowest energy. We emphasize that no search was made for these last three cases which thus give some indication of the predictive power of the calculations using only fusion cross section values as input.

As mentioned in the introduction the 32S compound nucleus is formed at the same excitation energy (43.75 MeV) by the ‘ONe+ “C and 160+r60 reactions. The main difference in the isotope spectra occurs for 24Mg and is well described by a difference of J,, of lh between the two systems. This difference has already been discovered from cross section measurements 6). Another difference in the two systems is that the “S formed by 160+ 160 contains only even partial waves 6). A simple calculation shows that the data should not be sensitive to this effect at the level of accuracy presently available.

5.2. ANGULAR DISTRIBUTIONS AND ENERGY SPECTRA

Predictions of residue angular distributions and energy spectra have been made using the Monte-Carlo code DESTIN based on the Ericson and Strutinsky semi- classical method outlined in sect. 3.

A comparison is made with the experimental data for the *‘Ne + ‘*C and I60 + ’ 6O cases in fig. 5. Without exception the shape of the angular distributions are very well reproduced indicating that the semi-classical method is adequate for the problem under study. A representative selection of energy spectra is shown in fig. 6 which confirms the above conclusion. To the extent that angular distributions and energy spectra are dominated by kinematic considerations this is not surprising. However the fact that a uniformly good description of all the data is obtained does seem to constitute support for the supposition that no mechanisms other than multistep evaporation are contributing. Contributions from direct reaction mechanisms, that

should show up as additional counts at high energies at small angles are not observed for isotopes with masses greater than that of the projectile (160, *‘Ne). The yield of lighter evaporation residues is negligible (< 1%) at the energies of the present study, with exception of “Ne (from *‘Ne+ “C) where strong elastic and inelastic scattering contributions do not allow us to extract a reliable yield.

An important technical point in the calculations must be mentioned since the predictions of figs. 5 and 6 were made with > lo5 events. This is possible due to the fact that a given evaporation sequence may be reused many times in the construction of angular distributions and energy spectra due to the random selection on the


do/d8 Irehnitsl 160+16~ 1001

55tleV

5 4 s 300

Iq

0 + 2'Al

30 t I”,c-‘, *t

T\

27Si

100 29Si

50

t

du/d8+[rehnitsJ 2oNe+12C 66.5MeV

\

f

r, ml

Z'AI E”‘: 0

100

moo 'Si

500

I?

29si

100 .

5

10

f-l

I'"a Osi

50 '3OP

10" 20. 30' e 10' 20. 0

Fig. 5. Residue angular distributions for ‘60-‘60 at 55 MeV and ‘ONe+ “C at 66.5 MeV. Solid curves are statistical-model calculations.

various azimuthal angles. Thus an amplification of the number of events may be obtained at this stage of the calculation. These statements are equivalent to the statement that kinematic effects dominate. An amplification of 100 was used in the present calculations.

In spite of the dominance of kinematic effects it was considered to be of interest to determine whether or not angular distributions and energy spectra were sensitive to the moment of inertia radius R,. A priori we might expect that the moments of inertia influence residue angular distributions and energy spectra since the spin cut-off parameter determines the degree of in-plane alignment of successive evaporated particles.

Unfortunately test calculations carried out on the 20Ne+ “C system at 66.5 MeV showed only a slight sensitivity to this parameter probably indicating a predominance of the first particle emission on the recoil residue angular distributions.

A. J. Cole et al. 1 Evaporation residues 299

counts

I *ONe+'*C Elnb=66.5MeV &,,.l"

50 26Al

10 20 30 40 50 E PleVl

Fig. 6. Energy spectra for residues from the *ONe + 12C reaction at 66.5 MeV. Solid curves are statistical model calculations.

6. Summary and conclusions

The application of a fast Monte-Carlo multistep evaporation computer code in the analysis of evaporation residue data from low-mass compound nuclei with E, < 50 MeV has shown that in all cases considered a very good description of all the data can be obtained with a single set of model parameters.

The most remarkable aspect of the analysis is that the fit to the data is sensitive to the critical angular momentum Jc, which specifies the initial compound nucleus angular momentum distribution. The optimum values of Jc, are in good agreement with those determined from cross section measurements when a moment of inertia radius parameter of 1.3 fm is used in the calculations.

Another important point is that the r9F + “C prediction is similar to that made using the program CASCADE of Ptihlhofer [ref. ‘)I despite the use of somewhat different prescriptions for the level densities. This indicates a low sensitivity of these data to the relevant parameters.

Use of the Ericson and Strutinsky semi-classical approach led to good description of angular distributions and energy spectra and confirms the consistency of the data


with a multistep evaporation mechanism. Unfortunately, these data are dominated by kinematic effects and are not very sensitive to model parameters. The same remark should not be true of correlation data.

Clearly it would be desirable to extend the analysis to higher energies and angular momenta and also to include higher mass compound nuclei. However in the light of the present analysis we feel that priority should be given to studies of correlations between successive evaporated particles since in principal they should show increased sensitivity to model parameters and thus should lead to improved estimates of critical J-values 23). The consistency of JcR values extracted from cross sections and residue distributions would then become a very useful test of the data.

Another justification of the present work was emphasized in ref. 20); at high angular momentum the deformation induced in the compound nucleus should have notice- able effects on the evaporation process and thus the “standard’ calculation would no longer be expected to account for the data. In view of the “giant backbend” recently discovered by Diamond 26) a careful search for this effect in light nuclei would be highly desirable.

References

1) A. Weidinger, F. Busch, G. Gaul, W. Trautmann and W. Zipper, Nucl. Phys. A263 (1976) 51 I 2) J. Menet, A. J. Cole, N. Longequeue, J. J. Lucas, G. Mariolopoulos, J. B. Viano, J. C. Saulnier and

D. H. Koang, J. de Phys. 38’(1977) 1051 3) J. P. Coffin, P. Engelstein, A. Gallmann, B. Heusch, P. Wagner and H. E. Wegner, Phys. Rev. Cl7

(1978) 1607 4) F. Ptihlhofer, Nucl. Phys. AU#I(l977) 267 5) B. Fernandez, C. Gaarde, J. S. Larsen, S. Pontoppidan and F. Videbaek, Nucl. Phys. A306 (1978) 259 6) F. Saint Laurent, M. Conjeaud, S. Harar, J. M. Loiseaux, J. Menet and J. B. Viano, Nucl. Phys. A327

(1979) p. 517 7) J. Gomez del Campo, R. A. Dayras, G. A. Biggerstaff, D. Shapira, A. H. Snell, P. M. Stelson and

R. G. Stokstad, Phys. Rev. Lett. 43 (1979) 26 8) M. N. Namboodiri, E. T. Chulick and J. B. Natowitz, Nucl. Phys. A263 (1976) 491 9) R. G. Stokstad, J. Gomez de1 Campo, J. A. Biggerstaff, A. H. Snell and P. H. Stelson, Phys. Rev.

Lett. 36 (1976) 1529 10) T. Ericson and V. Strutinsky, Nucl. Phys. 8 (1958) 284 11) Computer Programme GROGI2; see J. R. Grover and J. Gilat, Phys. Rev. 157 (1967) 802 12) A. J. Cole, Internal Report ISN 79-14, unpublished 13) See for example: T. D. Thomas, Ann. Rev. Nucl. Sci. 18 (1968) 343 14) R. A. Dayras, private communication 15) P. Bonche, B. Grammaticos and S. Koonin, Phys. Rev. Cl7 (1978) 1700 16) D. W. Lang, Nucl. Phys. 77 (1966) 545 17) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 18) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 19) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, 1960)

appendix 2 20) M. Beckermann and M. Blann, Phys. Rev. Lett. 42 (1979) 156 21) S. Cohen, F. Plasil and W. I. Swiatecki, Ann. of Phys. 82 (1974) 557 22) C. M. Perey and F. G. Perey, Nucl. Data Tables 17, no. 1 (1976) 23) U. Scheib, A. Hofmann and F. Volger, Phys. Rev. Lett. 34 (1975) 1586 24) S. S. So, C. Mayer-Biiricke, R. H. Davis, Nucl. Phys. 84 (1966) 641 25) See for example: A. J. Cole, N. Longequeue and J. F. Cavaignac, J. de Phys. 38 (1977) 1043 26) R. M. Diamond, Summer School, Serre Chevalier, Sept. 1979

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Analysis of evaporation residues following light heavy-ion reactions