Description of γ-multiplicities after fusion reactions in deformed nuclei

Nuclear Physics A307 (1978) 349-364; ~ ) North-HoUandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

DESCRIPTION OF y-MULTIPLICITIES AFTER FUSION REACTIONS IN DEFORMED NUCLEI M. WAKAI *

Institut ffir Kernphysik der KernJbrschungsanlage Jfilich, D-5170 Jfilich, West German)' and AMAND FAESSLER

lnstitut .[dr Kernphysik der Kernforschungsnalge Jfilich, D-5170 Jfilich, West German)' and Physics Department, University of Bonn, D-5300 Bonn, West Germany Received 22 May 1978 Abstract: The Monte Carlo method is used for the description of particle (n, p, ~) evaporation and 7-ray

emission after fusion reactions. The theoretical description focusses on the competition between collective and statistical y-rays. It is shown that the correct side feeding pattern in final rare earth nuclei can be quantitatively understood if strong collective E2 transitions are taken into account also in bands high above the yrast line. The 7-cascades run through an irregular mixture of statistical and collective transitions before hitting the yrast line. Side feeding patterns, ~/-multiplicities and their fluctuations are described in good agreement with the data of the ~5°Nd(~60, 4n)~62Er and l'*SNd(lSO, 4n)162Er reactions at different bombarding energies using the computer program "Icarus" established according to these ideas.

1. Introduction

In (heavy ion, xn) fusion reactions, a projectile and a target are fused into a compound system and the energy and angular momentum brought into the system is spread over a large number of nucleons. Particle- and v-emissions from the compound system carry away the excess energy and angular momentum. The formation of the compound system and the emissions of particles are believed to be well described by the statistical model of nuclear reactions 1,2). In this model, a particle is emitted independently of the preceding particle emission and the formation of the compound nucleus, aside from the conservation of the energy, angular momentum and parity. The microscopic distribution of levels in a nucleus is represented by a rough distribution expressed by a level density formula as a function of the excitation energy, angular momentum and parity. The particle evaporation spectra and the emission of 7-rays depend sensitively on the level density formula used. This problem is especially severe for low excitation energies. * Permanent address: University of Osaka, Physics Department, Osaka, Japan. 349

350

M. WAKAI AND A. FAESSLER

The greater part of the excess angular momentum is carried by 7-emissions. The multiplicity of the ~/-cascade is strongly related to the excitation energy, the distribution of the angular momentum in the compound system and the relative strengths of electromagnetic probabilities with different multipolarities. The values of the strengths are averaged ones and we use experimentally fitted values, which yield good agreement with the experimental results, because it is difficult to determine the values theoretically. In addition to statistical E1 and at higher energies also statistical E2 transitions the stretched collective E2 transitions in rotational bands different from the ground-state band are very important in the case of a deformed nucleus. The role of these transitions is not so important in the higher energy region in comparison with other transitions, especially the E1 transitions. They are, however, in the low energy region and special attention has to be paid to describe them correctly. The general belief seems to be 3, 4) that a few statistical (mainly E1 ) v-rays are followed by several collective E2 transitions in a system of rotational bands close to the yrast line, which all feed into the ground-state band. But it may also be that a statistical transition is followed by one or a few collective E2 transitions and then by a statistical transition and so on. One purpose of this work is to study this question. In addition the analysis of experimental data on the v-multiplicities and related quantities also provide important information about the validity of the compound nucleus picture. The idea of this picture is simple in principle but it requires very tedious numerical calculations because of multiple integrations. The Monte Carlo method, which is very useful in overcoming these difficulties is applied here to calculations of the statistical theory of nucleon reactions and very good results are obtained. As we follow particle emissions step by step in this method, we can get a very intuitive picture of the series of particle and 7 emissions from a compound nucleus. In this way, one can also find in a straightforward manner the 7-multiplicities following a fusion reaction. In this work we analyse the v-multiplicities and related quantities following heavyion fusion reactions by applying the Monte Carlo method to the statistical theory of nuclear reactions. In sect. 2 the theoretical aspects of the statistical theory of nuclear reactions and the Monte Carlo method are summarized. In this description, we include the fact that each level in a deformed nucleus (apart from band heads) is a member of a rotational band. The corresponding collective E2 transitions turn out to be essential for reproducing the side feeding into the ground-state rotational band. We performed numerical calculations for the reactions 15°Nd(160, 4n)162Er and 14SNd(laO, 4n)162Er and obtained theoretical values of the v-multiplicity and other related quantities. They are shown and compared with experimental data in sect. 3. The results are discussed in sect. 4 and summarized in sect. 5.

;'-MULTIPLICITIES

351

2. Theory According to the compound statistical model, the cross section for a reaction A(a, x)X is given by the following equation : o(a, x) = ~

J'~

~r~Eo,J~ Fx(E~, J'~) ¢omp --~ , ~ Fi(E¢, Jc)

(1)

where Ec, Jc and n are the energy, the angular momentum and the parity of the compound nucleus, respectively, and F~ is the partial width for the emission of a particle i. The quantity a¢omp(E¢,J~) is the formation cross section of the compound nucleus and is calculated by using transmission coefficients derived from the optical model. If the spin-orbit interaction between particles A and a in the optical potential is neglected, the formation cross section is expressed as follows: 2J~ + 1

6a°mp(Ec'S~c) = rt)~2(2Sa+ 1)(2SA+ l) [

JA+s~ E

J~+l ~

~(I)Y/(~a)],

I=IJA-Sa I l=lJ¢-I I

(2)

where 2 is the de-Broglie wavelength for the projectile in the c.m. system and z(l) (= 0 or 1) the factor which guarantees parity conservation. Tt(~) is the transmission coefficient for the lth incident particle wave of the projectile with the kinetic energy e~; S~ and Ja are the spins of the projectile a and the target A; and I is the channel spin. The expression is somewhat complicated if the spin-orbit interaction is included. It is given in ref. 5). The partial width F~(E, J) is given as follows:

Fx(E, J'~) = ~., fRx(E, J'~; E," J'[')dE v C.f J

(3)

Here Rx(E, J~; Er, Jr9 is the probability per unit time of emission of a particle x from a compound state with energy E, and angular momentum and parity J~ to give a product nucleus at excitation energy E r and angular momentum and parity J~. The principle of detailed balance gives the expression for R x by using the cross section of the inverse reaction,

Rx(E,~j,~; Ef, j.~f)dEr _ 2S~ +1 mxex~i,,~(Ef, j,~f; E,j,~)o~E~,,J'~f) dEf. ~:h 3 J'~)

(4)

Here S~ is the spin of the particle i, mx the reduced mass, ~,~the kinetic energy of the particle i, and oJ(Ev J~r) and ~(E, J~) are the densities of states of the initial and final nuclei. The probability for 7-emission of multipolarity 1is evaluated with the following equation: . . (5) Ra(Ei, Ji. ., Ef, Jr f )dEr _~_ Ct(E i _ Ef)2z+ 1 Q(Ef, J~) dEr

O(E~, JT')

where C t is a quantity related to the matrix element of the transition operator. The quantity f2(E, J") is the level density at excitation E with angular momentum and

352

M. WAKAI AND A. FAESSLER

parity J~, and is related to the density of states by co(E, J") = (2J + 1)O(E, J~).

(6)

Of course, parity should be conserved in applying eq. (5). The cross section for reactions with multiple-particle evaporation (for example, two outgoing particles x, y) is also given by eq. (1) with the numerator of the form Fy{Ef, J~,) Fxy(Ei, J~) = E f Rx(Ei, J~'; El, J~r) v, dEf. /', Fi(Er, J~)

(7)

It can be easily seen from eq. (7) that the direct numerical evaluation is very tedious and impractical. Thus, we adopted the Monte Carlo method % This method makes the numerical calculations much easier, and gives a very intuitive representation of the multiple-particle evaporation. In this way one can get also easily theoretical values of the y-multiplicities, which are the numbers of emitted y-rays after the last particle emission. A detailed explanation of the Monte Carlo method is given in the following section. For simplicity of calculations we divided the energy scale into two regions: Region I (high excitation energy): This is the region where the particle emission probabilities are equal to and larger than the decay probabilities into 7-rays. Both particle emissions and (El) v-emissions are taken into account simultaneously in this region. The actual calculation shows that the range of excitation energy of this region is E > 10 MeV. Region II (low excitation energy): The excitation energy is so low that the possibility of particle emission is much smaller than that of y-emission in this region. In this region we, therefore, need to take into account the 7-emission only, but we include different multipolarities. It is necessary to know the precise expressions and values of quantities appearing in eqs. (2)-(6) in order to evaluate the reaction cross sections according to eq. (1). We will give a somewhat detailed explanation of these points. (i) Formation cross section a¢o.,p: as is well known, the transmission coefficients Tt show a simple dependence on the angular momentum / in the case of strongly absorbing heavy-ion collisions. One can express them by a Fermi distribution to very good approximation: 1 T~ = 1 + exp [ ( / -

lo)/do]"

(8)

with two parameters l0 and d 0. If we directly use the values of d o and 1o derived from optical model calculations, the theoretical reaction cross section turns out to be larger than the measured one. This comes from the fact that the former contains also the contributions of direct reactions. We, therefore, choose a corrected value of lo which is fixed to reproduce the measured fusion cross section. As to the diffuseness d o, we use the values derived from optical model calculations.

7-MULTIPLICITIES

353

(ii) Level density: a number of different formulae have been proposed for the level densities in nuclei z). In this work, we use a modified Fermi gas level density in which the additional freedom of the rotational motion of nuclei is considered,

l~(h2"~am'~(U+t)-2(2J+ljexp{2(al°'U)~],

g2(E, a") = 48~f2 \ 0 /

(9)

Here U is the excitation energy of a nucleus from which the rotational energy and the pairing energy 6 {°) is subtracted, U = E-

{'

~OJ(J+l)-t-/i

mt

t

.

(10)

Also t is the thermodynamic temperature and the parameter a ~°1 is a constant called the level density parameter. The moment of inertia 0 is assumed to have the following energy dependence 6): 0 = 0rig{1

-.l'exp(-qE)~,

(111

where 0rig is the rigid-body value of the moment of inertia, and f and ,q are constants choosen so as to reproduce the experimental dependence of the moment of inertia on excitation energy. In the lower energy region (region II), it is difficult to reproduce the experimental level distributions with eq. (9). Thus, we employed another formula,

f2(g,J ~) = ~

a~(g'+t)-2(2j+

l)exp

{2(aU')~l,

(12)

where U' is the excitation energy from which the energy of the yrast level Eyr~t{J") has been subtracted 7),

U'= E-Eyr,~,t(J").

(13)

In this case the level density parameter a has an energy dependence, a = a~°~{1 - :~ exp ( - flU')}.

(14)

We adjusted the values of the constants ~ and fl so that the expression (12) reproduces the experimental level distribution near the yrast levels. The value of (E, d~) for U(U') < 0 is zero in both the case of eq. (9) and (12). Recently, some authors have used the level density formula derived with level counting 8) or exact level distributions deduced from experiments 9). This is a more realistic method for discussing phenomena near the ground state, but complicates the application to different nuclei. We, therefore, use in our calculations the formula (12) with the energy dependent level density parameter (14). (iii) The 7-e~ission probability: In region II we take into account El, MI and E2 transitions. The reduced matrix elements in eq. (5) are assumed to be constant. In the region I, only El transitions are considered, which seems to be sufficient in this

354

M. WAKA| AND A. FAESSLER

high energy region. We adopted Sperber's 10) approximation to the E1 reduced matrix element for region I, CEI = K(2Ji+ 1)(2Jr+ 1),

(15)

which is, with a constant K, a good approximation for high angular momentum states. But in region I, the emission of 7-rays is not important. Only in about 2 of all cascades is a ~;-ray emitted in this region for the fusion reactions considered in this work. It is well known that E2 transitions between states in the same rotational band with an angular momentum difference 61 = 2 are strongly collective. Thus we need to consider the effect of these transitions separately and added the partial width F~°~l given by Fcon c¢ontE ~, ~5, E2 E2 ~ i--~f~ (16) =

where C~°)~ is a constant and E i - Er is the energy difference between the two states. The effect of this modification reduces rapidly with the increase of the initial excitation energy E i and becomes negligible for the high energy in region L (We included also the special M1 transitions in K -¢ 0 rotational bands. We found no appreciable modifications of the results and therefore shall not discuss this effect further.) The fact that an electromagnetic transition operator is a one-body operator and cannot reach all final states at energy Ef is included in an approximate way in eq. (5) according to Blatt and Weisskopf 1~). Liotta and Sorensen 12) gave recently a more accurate expression to take this effect into account, Ra(E ~, JT'; El, JTf)dEf

=

Q(E~ - El)Z2 +2 (2(E~ J'~) aE f2(E~, j~i) ~ f'

(17)

This effect has the largest influence on the results if included for the E1 transitions. It is clear from eq. (17) that these transitions may be reduced in the high energy region, as shown in ref. 12). We performed calculations based on this formula, too. The results are presented later on. (iv) Cross sections of the inverse reaction: The o~,Vare evaluated by using the optical model. The optical parameters are those which give good fits to elastic scattering data. The dependence of the parameters on the excitation energy of the target is neglected. 3. Calculations and results

As was mentioned above, we employed the Monte Carlo method 5) instead of direct numerical evaluations of the formulae. Here we will give a short outline of the method. In the region where mainly particles (n, p. a) are evaporated from the compound nucleus (region I), our computer program Icarus uses an extended version of the program Roulette. The treatment of the 7-emission in region II in Icarus is described below * • Icarus: The nucleus arrives finally in the ground state like Icarus falls into the sea from a "'highly excited state".

7-MULTIPLICITIES

355

At first, we choose a random number Z~ (values of random numbers are always non-negative and not larger than 1 in the subsequent discussions), which determines the value of the angular momentum Ji of the compound nucleus formed in the fusion reaction by Ji

~-

Jcm a x ZI.

(18)

Here jm~x is the maximum value of angular momentum of the compound nucleus. It has been estimated by using the excitation energy of the compound nucleus E c, 1

A second random number Z 2 is chosen to decide if this selection of the angular momentum should be accepted (if Z 2 =( O'ji/(~j . . . . . ) or rejected (if Z 2 > aj,/(~j~). If rejected, a new number Z~ is chosen. If it is accepted, the reduced widths Fi(E i, Ji) are calculated for all particles and 7-rays. The type of particle (or 7-ray) emitted is determined by a third random number Z 3 according to the magnitudes of the F r The fourth and fifth numbers Z 4 and Zs give the energy Et and angular momentum Jr of the residual nucleus after particle emission, by using relations similar to eq. (18). This choice of energy and angular momentum is accepted if the sixth number Z 6 is smaller than Rx(E i, Ji; E t, Jr)~ Rx(E i, J~; E, j)m~x, and is rejected otherwise. If the choice is rejected, new numbers Z,~ and Z~ are chosen. When the choice is accepted, the widths F i at the final energy are calculated again and it is checked if the probability of particle emission is much smaller (10 -4) than that of 7-emission. If this is the case, the residual nucleus is in region II where only ~,-ray emission is considered. Otherwise, a new third number Z~ is chosen. This procedure is repeated until the residual nucleus reaches region II. In region II, a slightly modified procedure is used. First we check if the residual nucleus reached is the one we want to calculate. If yes, we repeat the steps described above but for 7-emission only. The parity selection rules are included in these steps. The process is repeated until the energy Ef of the residual nucleus with angular momentum J'" (~0: the parity of the ground band) satisfies the following condition: Eg.b.(J) <_--Ef < Eg.b.(J)',

(20)

where Eg.b .(J) is the excitation energy of a ground-state band with angular momentum J and Eg.b.(J)' is a constant deduced from Ug.b!S)' 1= ] Q(E,J'°)dE. (21) d Eg.b .(J)

Eq. (20) formulates the test that the residual nucleus reached the ground rotational band. It means exactly that the final level lies within an energy interval from the yrast energy up in which only one level (the yrast level) is found. The side feeding population to a state of the ground band is evaluated as the

356

M. WAKA1 A N D A. FAESSLER

product of the total fusion cross section and the fraction of the cascades leading to this state. The ),-multiplicity My,; is given by averaging over the number of 7-rays emitted in region II via the grohnd band state ]I). The higher moments ~t,; are defined as follows: /t,1 = (M I --_b31)".

(22)

The standard deviation a and skewness s are given as a = ~,~,

/23)

S = ll3/ff 3.

(24)

The numerical calculations were performed for the reactions 14SNd (180, 4n) 162Er and 15°Nd(160, 4n)162Er [refs. 13, 14)]. The emissions of neutrons, protons, a-particles and 7-rays are taken into account. The cascade calculations show that in our specific examples only neutrons and 7-rays are emitted. The formation cross section acomp in eq. (1) was evaluated with eq. (8) using the optical parameters given in ref. 13) and l0 adjusted to the total experimental fusion cross section. The level density parameters #o) and pairing energies 3 t°) are those deduced experimentally by Holmes et al. 15). The rigid-body value of moment of inertia is calculated with 15) r0 = 1.25 fin. The value of K (in region I) is fitted to give an overall good agreement for the bombarding energy dependence of the cross sections of the three-, four- and five-neutron emissions ~3.1,). The most suitable value is K = 0.1 x 10 -8. To get the values of C t in region lI, we evaluated the side feeding populations of ~6ZEr with several sets of the C v The following choice gives the best agreement with the experimental data: CEI " CMI " CE2 " "~E2/'~¢°11=

50 " (25) " (1) ' 200.

Here C~°~~is the strength of the E2 transitions between members of the same rotational band. It is strongly increased due to collective effects. The explicit treatment of these rotational transitions turns out to be essential. If one translates the above numbers into single-particle units one finds: S(E1): S(M1) : S(E2) : Sc°H(E2) = 0.0023 : (0.0012): (1): 308. The small El strength reflects the depletion of the strength in other states by the giant dipole resonance. The results for the M 1 and the single-particle E2 strengths (in brackets) are unreliable since the final results do not depend sensitively on these numbers. The results would not be appreciably modified including only the statistical E1 and the collective E2 transitions. The ratio of the coefficients C is the primary ingredient for the program. The ratio of the strengths can only be calculated approximately from these numbers. One needs to know among others the numbers of levesls N c which can be reached by a one-body operator from the final state at Ef at the

y-MULTIPLICITIES I

i

~°Nd+ ~0 Etab: 77HeV

1.0

~

c

357

o

I

I

I

lS°Nd(160,4n 1162Er m

p

°cog,('~)/ornox

Elob=77 MeV

10

....

Ex p. Theory

r-

-4" 0.5

~s's C)

0

10

20

30

0

40

Angular Momentum ~ [h ]

Fig. 2. Experimental and theoretical angular momentum distribution of the entry state in the reaction 15°Nd(t60, 4n)162Er with Elab = 77 MeV. The experimental values are deduced from the ;,-multiplicities L3,~4). The entry state is defined as the nuclear state after the last particle is evaporated and a cascade of only 7-rays starts.

Fig. I. Transmission coefficients Tt and fusion cross sections a~omp(l) for different incoming orbital angular momenta / of the fusion reaction 15°Nd+ 160 ~ 166Er*.The transmission coefficients T~ are calculated according to a fit (8) to the optical model data with /o adjusted afterwards to reproduce the total experimental fusion cross section.

i

"

10 20 30 Angular Momentum [ [ h ]

-q-

i

1

30

40

--

2O

LtJ 10

~s 10

Angular

20

Momentum t [h]

Fig. 3. Entry distribution of the reaction ~5°Nd(~60, 4n)~62Er with /:'hh = 77 MeV in the excitation energy and angular momentum plane. The solid line indicates the yrast energy. The dashed line shows the position of the yrast states with odd spins and of those with even spins and negative parity. The number on a contour line shows the permillage value (°/0o) of the entry distribution in the energy range E and E + 1 (MeV) with fixed angular momentum 1.

358

M. WAKAI A N D A. FAESSLER i

30

i

Elob = 65MeV Exp . . . . Theory

i

Etob=71 MeV -Exp. . . . . Theory

Elob=77MeV - ....

Exp Theory

"6

-20 .c_ 'o I1 t/3 . . . . . . . . .

10

20

lo

i

,I Ill

,

20

Angular Momentum

,

i

. . . .

1o

i

,

20

I (h)

Fig. 4. Side feeding population of the ground-state band in percent as a function of the angular momentum into which the ;'-rays feed for the reaction 15°Nd(160, 4n)162Er for different bombarding energies in the lab system. The statistical theoretical error is indicated.

initial energy E i [refs. 11,12)], We used the expression of ref.

12)

for N c,

N c = ½(2]+ 1)- lff2(E i - Ef). Here ] is the average single-particle angular momentum and 9 the single-particle level density. To estimate the above ratios of the single-particle strengths values for the N = 5 oscillator shells and a transition energy orE i - Er = 0.5 MeV are employed. The excitation energies of yrast levels in eq. (13) are chosen to be the experimental ones. If no experimental data are available, the ground band energies are extrapolated: 1 Eyrast(J r~) ~ ~ O J ( J +

1 ) + 6 I°),

for J --, 30.

(25)

Optical parameters used for the evaluation of ai, ~ in eq. (4) are from refs. 16-~8) for neutrons, protons and c~-particles, respectively. The results of the calculations are plotted in the figures and compared with the experimental data 13,14). In fig. 1 we present the transmission coefficient T~ and the formation cross section acor,p(/) as functions of the angular momentum of the incoming particle for the case l S°Nd + ~60 (El, b = 77 MeV). The formation cross section shows a maximum value at / ~ 22, This value is very close to the l,,,x(~ 22) which is given by a semi-classical sharp cut-off model 19). The angular momentum distribution of the entry states is plotted in fig. 2 for the same reaction as in fig. 1. The experimental values are deduced from the 7-multiplicities 13.14) under assumptions stated in refs. 13.14). The agreement is quite good. Fig. 3 shows the entry distribution for the reaction 150Nd(~ 60, 4n)163Er in the residual nucleus at the beginning of v-emission. In fig. 4, we plot the relative side feeding population for the reaction ~5°Nd(160,4n)t62Er for three bombarding energies. They show satisfactory agreement with the experimental data. A projectile

,

i

I

i

5o

>, 15

.4.

(.3

-[

15

359

f°" L,,-"

.• /

?,eV+S'I>'

_ca.

6 ' 62 ' Ndt 0,Z,n) Er~ ,,-, Eqab=Bg.7MeV,,~/

i

15°Nd( 180,4n)162Er EtQb=77MeVo Eiab =77MeV/ _ [ .... .,".,~:

lO °. E×p.

10

.,i-{" 65 MeV

-

>, O

"--" 65MeV

cr

--o-- Theory

" ~ "

Exp.

Theory o

...... /

10

20

10

20

+ 15

l

---

~'°'°'°"

'_-{"

.o.°"

L

L

I

=

20

Fig. 6. G a m m a multiplicities < M I > ,

standard

deviations o] and skewness .Yl, as defined in eqs. (23) and (24) for *,'-cascades through angular momentum I for the reaction ~5"Nd('O+ 4n) "z Er wit h El:,,,( ~60) = 69.7 MeV. The theoretical results are calculated using eq. (5) for the ;.-transition probability in which the l p l h nature of the transition operator is only approximately taken into account ~'). The improved expression tz) (17) reduces only the ;,-multiplicity by about one half to one 7-ray. It therefore improves the agreement. But the modification lies still within the statistical error of the theoretical results.

15 / Eexc = 56,2 MeV

10

-

~8

~50Nd + ;60

~Z'SNd+ 180

(,.

,

Angdar NomentumI[h]

I

10 //~l~Nd + 160

,

' • Exp~ ..... T h e o r y

le" ,o E exc=49.5Me~ o.I< M>


i

lO

Angular Momentum I [ h ] Fig. 5. Gamma-ray multiplicities for the reaction IS°Nd(160, 4n)16ZEr for three bombarding energies 65, 71 and 77 MeV in the lab system. The left-hand side gives the experimental data [refs. ls. 14)], while the right-hand side shows the theoretical results including the statistical error. The abscissa disp;ays the angular momentum through which the ;,-cascade has to go.

. . . .

Nd+

~8

0

• " Exp - - - - - Theory

o

' - { ' - ~

0 ~__l.l.l~,.o.o. .... •

~,..... e........ "T,/'L.i

20

S

=.e-e-o-e~e-e

....

=.o-e

S

r.......... ,,I,, 10

:

I

10

20

Angutar Momentum I [h] Fig. 7. Gamma multiplicities , standard deviations o, and skewness is), defined in eqs. (23) and (24), for the reactions 15°Nd(160, 4n) 162Er (left-hand side) and 148Nd(lSO, 4n)162Er (righthand side) at 49.5 MeV excitation energy of the compound system. For the 7-transitions eq. (5) has been used (see caption of fig. 6).

L

,

*

I

i

t

10 Angular

I

,

I

,

20 10 Momentum I

20

[hl

Fig. 8. The corresponding experimental ~4) and theoretical results as in fig. 6 at 56.2 MeV excitation energy of the compound system 166Er*.

360

M. WAKAI AND A. FAESSLER

with higher bombarding energy excites the target to a compound nucleus with higher angular momentum. Thus, we can expect that the angular momentum receiving the maximum population shifts to higher values with increasing bombarding energy. This is supported by the experimental and theoretical results. The entry distribution in the final nucleus shown theoretically in fig. 3 can until now not be checked experimentally in a direct way. (A direct experimental test may be possible in the near future, when subdivided NaI 4zt counters with an inserted Ge(Li) will be able to measure the total energy release in a 7-cascade, the ;,-multiplicity and an identification of the final nucleus by a coincidence with the Ge(Li), all at the same time.) Naturally, all the quantities measured in the residual nucleus, like the side feeding pattern (fig. 4), depend sensitively on the entry distribution. On the other hand, this quantity is essentially determined by the model for the description of the compound nucleus and the evaporation of particles. The experimental quantity depending most directly on the energy and angular momentum in the entry state of the final nucleus is the distribution of the 7-multiplicities. We calculated the multiplicity of the emitted 7-rays in the residual nucleus 162Er for both reactions 14SNd(180,4n)16ZEr and 15°Nd(160,4n)162Er. In fig. 5 the calculated results of (160, 4n) are plotted for three bombarding energies on the right and compared with the data on the left hand side. The average 7-multiplicity (M r) increases with the energy of the projectile and the angular momentum I of the groundstate band through which the 7-cascade is going. The rate of increase with I is reduced for larger bombarding energies. This is connected with the fact that the entry states shift to higher angular momenta with increasing projectile energy. The experimental tendencies mentioned above are well reproduced by the theoretical calculations. The width (standard deviation) a and skewness s of the 7-multiplicity distribution defined in eqs. (23) and (24) are shown in fig. 6 for the case of the excitation with 160 ( E l a b = 69.7 MeV). The comparison with the experimental data shows a quantitatively good agreement. This suggests that our treatment is sufficiently reliable in spite of the fact that we used level density formulae and not the experimental level densities. Anderson et al. 14) studied the y-multiplicity distribution from forming the compound nucleus at excitation energies 49.5 and 56.2 MeV by two different projectiles 160 and 180. In figs. 7 and 8 the experimental data are plotted with our theoretical results for the excitation energies 49.5 and 56.2 MeV, respectively. One can see that at low spins the experimental values of the 7-multiplicities for the projectile ~sO are one or two units higher than those with the projectile ~60. This is due to the fact that the ~80 projectile brings a higher angular momentum into the compound nucleus 13). The theoretical calculations can reproduce the experimental tendency. The half-width ~r for the projectile 180 is almost constant, even for the smaller excitation energy. The theoretical calculations cannot explain this trend too well, but there is no large discrepancy between theory and experiment. The agreement of the skewness is not satisfactory for the projectile 180 and the excitation energy of

7-MULTIPLICITIES

361

of 49.5 Me.V. The experimental and also the theoretical value (calculated with the Monte Carlo method) of the skewness has large error bars. As was mentioned above, the different projectiles 160 and t s o give different experimental trends for the y-multiplicities and other quantities, in spite of giving the same compound nucleus at the same excitation energy. This comes from differences in the mechanism for the fusion processes 148Nd+ 180 ~ 166Er and lS°Nd+ 160 166Er. The projectile 180 for example brings in larger angular momenta than 160. From the viewpoint of the compound statistical model, this difference can be reflected only in the angular momentum distribution of the formation cross section atomy To investigate this effect, we calculated using two other models for the O'compin the reaction 150Nd + x6O with Eex~ = 49.5 MeV. In one, we calculate a~ompusing eqs. (2) and (8) with the diffuseness d o of the transmission coefficients twice as large as suggested by the optical model. In addition, we calculate Tt in the sharp cut-off model (d o = 0). In both cases the values of 1o are fixed to reproduce the experimental data of the fusion reaction. The results are presented in fig. 9.

15

.

Eexc :Z'9"5 MeV ~'-~ / ~8 la x,,x;-~" ,, Nd + 0 x x'~..'° J 7 ~- -o,,'" < M > I _°.o;..

10

---x-----o--.......

5

'

0

CASEA) " CASEB l T h e o r y CASEC ! Exp.

10

20

Angular Momentum][h] Fig. 9. Gamma multiplicites , standard deviations rr~ and skewness st calculated for different sets of transmission coefficients (8) for the reaction ~*SNd(~SO, 4n) ~62Er with an excitation energy 49.5 MeV in the compound system ~66Er*. The diffuseness d o is calculated from the optical model potential (case B: circles). Doubling the diffuseness (case A; crosses) increases the ;,-multiplicities since more angular momentum is brouhgt into the nucleus. The sharp cut-off model (case C: triangles) is obtained in the limit d o = 0. The experimental data ~4) are indicated by a solid line.

The sharp cut-off model gives smaller values for the y-multiplicities, because part of the higher angular momenta are suppressed in a¢omp.An increase in the diffuseness (do --* 2do) enlarges the y-multiplicities. The theoretical trends for the width ~ and the skewness s are not so different from the original results. We also made calculations by using the expression (17) of Liotta and Sorensen 12) for all electromagnetic transitions in region II. We obtained values of y-multiplicities

362

M. W A K A I A N D A. FAESSLER

smaller by 0.5-1.0 units, which give better agreement with the experimental data. This decrease comes from the fact that a higher power of the transition energy (E i - E r ) favours 7-transitions of higher energies and therefore smaller ?-multiplicities. The influence on the standard deviation and the skewness is not remarkable. It can be said that the effect of the correction to the electromagnetic transition proposed by eq. (17) is not large, at least in the case of the ?-multiplicities. Finally, we want to make a remark on the computational error of the calculated values. We estimated roughly the relative error to be (N = number of cascades calculated): 1/N ½. In our evaluations, 500 cascades are calculated for each reaction. Such an evaluation gives errors of 20 30% to calculated values of ?-multiplicities and the width a. Errors of the skewness s are larger than 100 %. We must, however, take notice of the following point: The above estimates give too large errors especially in the case of the skewness. These values are very small due to cancellations. We did computations of 2000 cascades to compare with the results obtained with 500 cascades. Some of the results are presented in fig. 10 as an example. It is notable that the results of the computation with 2000 cascades do not show any remarkable differences from those of 500 cascades. Thus we conclude that theoretical values obtained with 500 cascades are sufficiently reliable. i

i

LI.. x

<.> 15 - 1. . . . . .......

.,..~:,.~.;~'JT 1 N=5CO I -

..... x . . , " N = 2 0 0 0 i T n e o r y

10

,

Exp.

15oNd * 160 Elob=77MeV 5

0

x . x . x _ ,.., Q...:R .-.R .-. R -'I£-' ~ .

.

.

.

I

.

.

.

.

10

I

S ,

20

Angular MomentumI[hl Fig. 10. G a m m a multiplicities ,standard deviations a I of the numbers of ?-rays in a cascade and skewness as defined in eqs. (23) and (24) for 500 (usual number used above) and 2000 cascades calculated for the reaction lS°Nd(160, 4n)t6ZEr with Ela ~ = 77 MeV. The two theoretical r e s u l t s s h o w no appreciable difference. Therefore, 500 cascades seem to be sufficient.

5. Conclusion The evaporation of particles and the emission of 7-rays from a compound nucleus after fusion reactions has been described successfully 5,9, zo, 13) in the past for the high excitation region (region I). Here we extended the Monte Carlo treatment 5)

;,-MULTIPLICITIES

363

of the decay cascades to the low energy and angular momentum region (region II: up to about 10 MeV above the yrast line) where the level densities cannot be described by the Fermi gas model. By counting the experimental levels we interpolated the level density parameter a = a~°t~l - ~ exp ( - f l U ) } from the experimental values at low energies to the Fermi gas value a ~°1 at higher energies. The main purpose of this work was to study the competition between statistical El, M1 and E2 transitions and collective E2 ,,-rays in rotational bands above the yrast line. We assumed therefore that each level which decays in region II is a member of a rotational band and can also decay by a collective E2 transition. The general assumed belief about the ),-ray deexcitation process is the following 24.3,4): High energy photons, probably mostly dipole, will first be emitted down to the vicinity of the yrast levels. There it follows close-lying roughly parallel bands until it feeds into the ground-state rotational band. The result which emerges from this work is different: A statistical ),-ray may be followed by one or several collective E2 transitions still high above the yrast line and even further down collective and statistical transitions may be mixed in an irregular way until the yrast line is reached. Only in this way is it possible to explain the observed side feeding pattern with a minimal amount of assumptions. We assumed averaged constant values for the reduced statistical El, M1 and E2 transitions taking into account the fact that the ),-transition operator is a one-body operator and can therefore reach only states which differ by l pl h excitations. The approximate expression given by Blatt and Weisskopf ~1) for this effect turned out to be appropriate. The use of an improved expression given by Liotta and Sorensen ~2) reduced the "/-multiplicity by one half to one ",,-ray from a total of more than 10, a change in the right direction. The deexcitation pattern of the ),-rays is mainly affected by the R = (7F1/('~ n ratio. (i) A large value of R yields a behaviour as indicated by Newton et al. 24t: A few statistical ",,-rays deexcite the nucleus without loosing too much angular momentum close to the yrast line. The residual deexcitation goes mainly through collective E2 transitions. (iij A small ratio R favours already high above the yrast line collective E2 transitions parallel to this line and produces the large side feeding observed at low angular momenta. The variance cr2 of the ),-multiplicities and its dependence on the state II) of the ground-state rotational band through which the ",,-cascade is forced to go by a coincidence requirement is reproduced correctly. The skewness (24) is at least in one case not reproduced in a satisfactory way. One of the main reasons is that for this sensitive quantity the number of cascades taken into account in the Monte Carlo calculations are insufficient. The y-multiplicities and their fluctuation a (23) show only small changes when we increased the number of cascades from 500 to 2000, but the value of the skewness changed appreciably. One should, however, notice that even for this quantity the theoretical values are not in large disagreement with the data. The experimental trends with increasing coincidence angular momentum in the ground band and increasing bombarding energies are reproduced well. The

364

M. WAKAI AND A. FAESSLER

enlargement of the y-multiplicities going from an 160 to an 180 projectile is also reproduced by the theoretical treatment. It is connected with the fact that a larger radius for 180 allows larger angular momenta in the fused compound system. The comparison between theory and experiment for the side feeding pattern indicates that at higher bombarding energies the agreement gets worse. This may be connected with the fact that the level density formulae are not reliable enough for higher angular momenta. But it may also be caused by an incorrect extrapolation (10) and (11) of the yrast energies on which the level densities sensitively depend. Further precompound processes which are not included in this description may become important. We would like to thank Profs. Mayer-Kuckuk and Ernst from the University of Bonn for making available to us their version of the "'Roulette" program 5). It was used in our program ~'Icarus" in a slightly extended version to describe the particle evaporation and the E1 y-emission in region I.

References I) 2) 3) 4) 5) 6) 7t 8) 9) 10) II) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

22) 23) 24)

V. F. Weisskopf, Phys. Rev. 52 (1937) 294 T. Ericson, Adv. in Phys. 9 (1960) 425 A. Bohr and B. R. Mottelson, Nucl. structure, vol. 2 (Benjamin, NY, 1975) R. M. Diamond, Nukleonika 21 (1976) 29 D. G. Sarantites and B. D. Pate, Nucl. Phys. A93 (1967) 545 IV. H. Ruddy and B. D. Pate, Nucl. Phys. AI27 (1969) 323 J. R. Grover, Phys. Rev. 127 (1962) 2142 M. Hilman and J. R. Grover, Phys. Rev. 185 (1969) 1303 D. G. Sarantites el al., Nucl. Phys. AIS0 (1972) 177 D. Sperber, Nuovo Cim. 36 (1965) 1164: Phys. Rev. 142 (1966) 578 J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, NY, 1952) R. J. Liotta and R. A, Sorensen, Nucl. Phys. A297 (1978) 136 G. B. Hagemann e t a / . , Nucl. Phys. A245 (1975) 166: R. Broda e t a / . , Nucl. Phys. A248 (1975) 356 O. Anderson e t a / . , Nucl. Phys. A295 (1978) 163 J. A. Holmes et al., Atomic Data and Nucl. Data Tables 18, no. 4 (1976) D. Wihnore and P. E. Hodgson, Nucl. Phys. 55 (1964) 673 F, D. Becchetti, Jr. and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 J. H. Huizenga and G. lgo, Nucl. Phys. 29 (1962) 462 J. S. Blair. Phys. Rev. 95 (1954) 1218, 1957:108 (1954) 827 J. R. Grover and J. Gilat, Phys. Rev. 157 (1967)802 H, E. Kurz, E. W. Jasper, K. Fischer and F. Hermes, Nucl. Phys. AI68 (1971) 129: F. Hermes, E. W. Jasper, H. E. Kurz, T. Mayer-Kuckuk, P. IV. A. Goudsmit and H. Arnold, Nucl. Phys. A228 (1974) 175, 165 J. Bisplinghofl\ J. Ernst, T. Mayer-Kuckuk, P. Jahn and C. Mayer-BOricke, Nucl. Phys. A228 (1974) 180 Th. Lindblad and J. Garrett, private communication J.O. Newton, IV.S. Stephens, R. M. Diamond, W. H. Kelly and D. Ward, Nucl. Phys. AI41 (1970) 631