Nuclear Physics A252 (1975) 187--207; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
FISSION OF MEDIUM AND HEAVY NUCLEI I N D U C E D BY 4OAr F R O M 160 T O 300 M e V : Cross sections B. TAMAIN, C. N G 0 and J. PI~TER Chimie Nucldaire, IPN, BP no. 1, 91406 Orsay, France and F. HANAPPE Physique Nucldaire Expdrimentale, UniversiMLibre de Bruxelles, Bruxelles, Belgique Received 3 February 1975 (Revised 27 June 1975) Abstract: Cross sections have been measured for argon-induced fission following full momentum transfer. Two incident energies (between 200 and 300 MeV) have been used for 23sU, 2°PBi, x65HO and *atMo targets. For a natSb target, the fission excitation function has been measured from 160 to 300 MeV. The experimental method was an improved version of the angular correlation method, which takes into account kinetic energies and time-of-flight correlations. This allowed us to eliminate fission events following transfer reactions and also random events. Assuming that the angular distribution is that characteristic of fission fragments issued from a compound nucleus (close to 1/sin0) the fission cross sections range from 104-1 nab for A r + M o at 200 MeV to 10304-120 mb for A r + U at 300 MeV. When necessary, these cross sections have been added to the cross sections for the formation of evaporation-residue nuclei measured by other authors in order to obtain complete-fusion cross sections. The deduced critical angular momentum for formation of a complete-fusion nucleus are found to be an increasing function of the mass and energy of the incoming particle. These results are used to test the concept of a critical distance of approach which would govern fusion between two complex nuclei. The threshold of the excitation function measured for Ar+Sb is compared with calculations based on the statistical model for the competition between fission and evaporation. These calculations include the effects of multiple-chance fission and of angular momentum on fission widths.
E
NUCLEAR REACTIONS Mo(4°Ar, F), E -----200, 300 MeV; XeSHo('t°Ar,F), E = 226, 300 MeV; 2°9Bi, 23eU(*°Ar, F), E = 250, 300 MeV; Sb(*°Ar, F), E = 162, 179, 199, 226, 300 MeV; measured a(E(fragment mass), 0); deduced variation in critical angular momentum, reaction mechanism.
1. Introduction F o r several years, complete fusion has been k n o w n to be the most probabIe reaction i n d u c e d by heavy ions w h e n their b o m b a r d i n g energy is greater t h a n the i n t e r a c t i o n barrier. This process has been extensively studied primarily because o f the effects o f a n g u l a r m o m e n t u m b r o u g h t i n by the projectile o n the p r o b a b i l i t y o f fusion a n d o n the deexcitation properties o f the created nuclei. But other reactions which imply a n u c l e a r i n t e r a c t i o n between o n l y a few n u c l e o n s o f the projectile a n d target nuclei are also observed. These transfer reactions, or 187
188
B. TAMAIN et al.
more generally the reactions not leading to complete fusion (NCF) are considered to occur for high impact parameters, whereas the complete fusion (CF) should occur for the lowest impact parameters. This can be expressed conveniently in terms of the orbital angular momentum in the entrance channel. If erR, aCF and O'NcF are respectively the cross sections for the total reaction, for complete fusion and for non-complete fusion, then aR = ~cF +aNcF.
(1)
By using the sharp-cut-off approximation both for the angular momentum distribution l that corresponds to the total reaction (from 0 to/max), and for the values of I that lead to complete fusion (0 to/crit), one obtains: lmetx
=
E (2t + 1) =
+ x) z ,
(2)
/=0
lcrlt
acv --- lr;t2 Z ( 2 l + l ) -- zc;~z(/crit+1)z ,
(3)
1=0
where ;t is the reduced wave length. The ratio aCF/*R can be expressed by the ratio [(/cat + 1)[(I=~, + 1)] 2, and several studies have been done in the last few years in order to determine the variation of Icrtt/l=a,, when the mass of the projectile or of the target is changed, or when the bombarding energy is changed x). In the first study, projectiles lighter than 2°Ne were used. These results indicated a rather constant value of legit in the vicinity of 50h for the systems studied 2), and this was the reason why the appropriate angular momentum was named "critical". The Ima.values, however, did not exceed 90h and, moreover, the fraction of nuclei which deexcite by fission was not measured. For fusion nuclei in the rare earth region, the fission contribution was considered to be small, and for heavier nuclei (for which such an assumption could not be made) no measurements have been made except those by Sikk~land a). In this last case the projectile was 4 ° A r and the value of lma~ reached 1607,. The complete fusion nucleus was very heavy (27s110) and was presumed to deexcite only by fission. Semiconductor detectors were used to detect fission fragments, and the lo,it values deduced from these results exceeded 100h. But there were some questions regarding the interpretation of the data: the angular correlation between the two fragments indicated full momentum transfer from the projectile to the fissioning system, but it was not obvious that this implied that a compound nucleus had been formed. The fusion nucleus 27s110 is expected to have no barrier against fission, at least for values of I greater than a few h, and questions arose regarding the possibility of producing a nucleus which has no fission barrier. The aim of this work is to provide new experimental information on the following questions: how does the ratio o'cv/a R vary when projectiles heavier than 2°Ne are used and when high/-values are involved? What about the variation of l~,it? What is the fraction of fusion nuclei that deexcite by fission when the nucleus has a high
FISSION
189
orbital angular momentum? Is it possible to produce a fusion nucleus or a compound nucleus which has no fission barrier? To answer these questions, we have measured the cross section for fission following complete fusion between projectile and target nuclei with 4°Ar projectiles at several energies available at the Orsay ALICE accelerator. Several medium and heavy targets have been studied, and some of these results have already been presented in refs. 6-s). This paper will present all of the data and also their possible interpretation. Other recent works have already given some information concerning the above questions. The value of/crit has been shown to increase with increasing projectile mass, by Zebelman and Miller 4). The fission cross section has been measured for the system A r + A g [ref. s)]. A new concept has been introduced, which relates the limits on the fusion cross section to a critical distance between the projectile and target nuclei during their initial collision ~s). In general the experimental results obtained up to now indicate that the reason for the limitation to complete fusion is related to the dynamical processes in the entrance channel rather than to the equilibrium properties of the fusion nucleus (such as the angular momentum or the fission barrier). Our results can be used to check these ideas. It is necessary to make clear what has been measured in this work and to define the meaning of "'complete fusion", since different authors have attached different meanings to this term. Let us consider a heavy projectile At approaching a target nucleus A 2 (fig. 1). A grazing trajectory leads to an interaction of only some of the nucleons of A~ and A2. This interaction generally results in the transfer of few nucleons from one nucleus to the other. Both transfer products can be left in an excited state which subsequently deexcites by particle emission. If one product is heavier than thorium, it has a high probability of undergoing fission. These fission events are of no interest to us and they will be considered as spurious events and discussed in sect. 4. At smaller impact parameters, the inter-penetration of the two nuclei is greater and all the nucleons can be involved in the interaction. This interaction can lead to the formation of a compound nucleus (~cN) whose characteristics (mass A = A t + A 2 , total energy E*) can be easily calculated. This nucleus can deexcite by particle emission yielding evaporation residues of mass somewhat lower than A. It can also undergo fission at several steps of the deexcitation chain. The average mass of the fission fragments is then somewhat lower than ½(A1 +A2), and their mass and kinetic energy distributions should be similar to distributions already observed for fission of compound nuclei at high excitation energy induced by light particles or by heavy ions such as 12C or 14N. The fragment angular distribution should be symmetric about 0 = 90° in the c.m. system, and it should have a shape close to l/sin0. Before reaching thermal equilibrium which characterizes the compound nucleus, the system can undergo fission (direct fission) especially if it has no fission barrier. Pre-equilibrium emission can also take place 9). In this latter case, it can again
surface interaction
between
nucleons
A
-
fueion
compound nucleus
complrte
corn posiie system
lquilibroted heovy tmnsf. prod. A.At* x
,
Fig. 1. Mechanisms involved in heavy-ion induced reactions.
oil
intemctions
deep interactions
quasi elastic tmnsf.
fission
evoporofion
direct
pre equil.fission
very inel. tronsfer
lvoporotion
GCF
FISSION
191
deexcite by fission or by particle emission. Few nucleons are emitted in pre-equilibrium decay, and the evaporation residues or fission fragments will be similar to those resulting from the compound nucleus. In these cases of compound nucleus formation and of pre-equilibrium deexcitation, a nucleus exists for a time that is long enough so that no nucleon can any longer be specifically attributed to either one of the two initial nuclei. To these cases applies the name "complete fusion". One can, however also imagine that the initial kinetic energy in the fusion degree of freedom is dissipated (damping effect) before complete fusion is reached. In such a case the system can be called a composite system. In this case most individual nucleons may still be assigned to either the target or the projectile nucleus. The system can then separate by Coulomb repulsion and a type of fission [called "quasi-fission" in ref. s) and fig. 1] can be observed. The characteristics (mass, energy, and angular distributions) of these events are difficult to predict. They will be either closer to those of the transfer products or closer to those of the usual compound-nucleus fission fragments, depending on the depth and duration of the overlap of the two initial nuclei. Note that if the heavy product is heavier than Th, it has a high probability of undergoing fission itself. In the experiment described here, we are interested in the fission following complete fusion. Throughout this paper, the corresponding cross section ~rr refers to fission of the compound nucleus or of another equilibrated nucleus of mass close to A, or to direct fission. Other two-body mechanisms have been eliminated as is explained in sect. 3. In sect. 4, it is shown how it has been possible to eliminate all three-body events which could perturb the results and especially fission following transfer. It must be noted that in some works, ar also includes these last type of events [see for example ref. 3,)]. The comparison between various results here must then be made carefully. The systems studied here are (energies in the lab system) Ar + °'tMo at 200 and 300 MeV, A r + S b at 162, 179, 199, 226 and 300 MeV, A r + H o at 226 and 300 MeV, and A r + U at 250 and 300 MeV. 2. Experimental method
The experimental method used has been described elsewhere ~o). We will only recall briefly that the kinetic energies of the two fission fragments are measured in coincidence with two surface barrier detectors x and y located at angles 0, and 0y (relative to the beam direction). It is then possible to calculate the characteristics of each event in the e.m. system (mass, kinetic energy, emission angle) assuming only that the sum of the masses of the two products is equal to the mass of the fusion nucleus, and provided that full momentum transfer from the projectile to the composite system has occurred 23). The knowledge of the mass and kinetic energy of the products is necessary to distinguish between fission issued from complete fusion
192
B. T A M A I N e t al.
and that issued from other processes. We will see that this knowledge is also necessary to make the correct transformation of cross sections to the c.m. system. All necessary precautions have been taken so that every appropriate event is correctly detected. One of the detectors (x) was fixed at 0, with an angular aperture of 1°, and the other one (y) had an angular aperture of 10°, which allowed us in four measurements, to cover an angular correlation of 40°. (In fact, we used two y-detectors, which enabled us to perform two measurements at the same time.) In the direction perpendicular to the reaction plane, the x-detector had an angular aperture of 15° so that all correlated fragments were detected even if their direction had been changed by neutron evaporation. This angular aperture value has been determined to be sufficient by a preliminary measurement of the out-of-plane angular correlation (see fig. 2). !
i
Au + Ar ¢JD
50 MeV
/"i
300 FWHM 4°3
"5 Z 200
i
Bi + Ar
i
250MeV
FWHM 4°
I00
eB = •
e * em
•. " . ' ' T -5
I
I
-5 ~e
-
i
r
-2
=
i
0
i
t
2
=
i
4
=
-.4
,
_2
J
I
5 i
|
0
=
15 ram. i
2
=
!
4
=
deg.
Position relative to detector center Fig. 2. Angular correlation in the direction perpendicular to the plane o f the fission reaction. This result was obtained by using a position sensitive detector.
Mass and kinetic energy distributions and problems related to them are given and discussed elsewhere 11, tg). Since this paper deals with cross-section measurements, we will discuss here only those sources of error and only those corrections which are related to the counting of fission events. (i) In the transformation from the lab to the c.m. system, the recoil velocity VR of the fissioning nucleus increases the number of fission fragments detected at low lab angles and decreases this number at large angles. For fragments of velocity e.
FISSION
193
emitted at 0x in the c.m. system, let ~ be the ratio of the numbers of events in the lab and c.m. systems. For each event detected at 0x, we have calculated this ratio ~ and in the calculation of the c.m. differential cross section, d~/d~, the contribution of each event has been taken to be 1/~. (ii) Since cedepends on ~ and 0~, the errors made in the measurement of energies and angles have some effect on the differential cross section, but this effect appears to be negligeable 19). (iii) Another problem arises because the value of 0Z is also a function of vz and 0x. Thus, d0/d~ does not correspond to a well defined O~. This has no importance if the angular distribution do not change with Oz. Thus we have chosen a value of (0ffi) such that the angular distribution for fission is fiat, i.e. 0~ ~ ½~, as we will see under (v). (iv) The preceding remarks apply to any event issued from a "full momentum transfer" reaction. To distinguish fission following fusion from elastic and inelastic scattering and from transfer reactions, considerations other than kinematics have to be used, and this point will be discussed in sect. 3. More complicated cases occur when more than two main reaction products are formed such as when fission follows a transfer reaction. These cases will be discussed in sect. 4. (v) At this stage, we know d~(0 ~ ½rt)/d~. In order to obtain the total fission cross section ~f, we need to know the shape of the angular distribution. These have been measured in the case of lighter heavy-ion projectiles t2). A shape close to 1/sin0 has been observed, which is due to the effect of high orbital momentum on the deexcitation of an equilibrated (compound) nucleus. More precisely this shape is in agreement with the theory of Halpern and Strufinsky 13). Measurements have been made for the system A r + A g at 288 MeV [ref. s)] and the authors also observed a shape close to 1/sin0. For very heavy fusion nuclei such as A r + A u and Ar + Bi, similar measurements have recently been undertaken ~4). Preliminary results indicate a shape close 1]sin0 for mass ratios between 1 and 1.5. Thus we have assumed that the theory of Halpem and Strutinsky applies in all the cases studied here. The anisotropy defined by the ratio (d~/d~)oo/(d~]d{~)9oo increases with the anisotropy parameter p, as defined by Halpern and Strutinsky. This parameter is related to the rotational energy at the saddle point and to the temperature x, at the saddle point, and is given by: 2
2
2
where I: is the moment of inertia at the saddle point about an axis perpendicular to the elongation direction. Their values have been taken from refs. 2o,21). The value of p for each nucleus is indicated in table 1. The last column gives the extent to which the fission cross section would be overestimated if the angular distributions were assumed to be exactly 1/sin0. It is generally less than 7 ~o. (vi) Some uncertainties are due to the thicknesses of the targets and the intensities
194
B. TAMAIN et
aL
of the beam. These uncertainties have been determined by detection o f elastic scattering events. It was possible in most cases to compare the observed scattering cross sections with Rutherford cross sections, which were reproduced to -t-20 ~ . 3. Interpretation of experimental distributions
Fig. 3 shows some typical distributions obtained in our experiments. Fig. 3a shows a diagram o f lab kinetic energies, fig. 3b an angular correlation function (values of 0y for a given 0z value) and fig. 3c the mass-total kinetic energy diagram usually drawn in fission. Three peaks appear in all cases. Two are due to elastic and inelastic events, and one (the broadest one) is due to fission events. TABLE1 Anisotropy parameter p used in the theory of Halpern and Strutinsky to calculate the fragment angular distribution (sect. 2) and error on the fission cross section if the angular distribution was assumed to be exactly 1/sin0 System
(MeV)
p
Error (%)
Mo q-At
200 300 162 179 199 226 300 226 300 250 300
37 23 13 27 38 35 31 14 18 5 8
1 2 5 2 1 1 2 4 3 11 8
Sbq-Ar
Ho+Ar Uq-Ar
Three remarks have to be made: first, it must be pointed out that the results obtained cannot be used for an analysis of transfer events, because o f the lack o f precision in the mass calculation (_.+4 amu). Second, it can be seen in fig. 3b, that the two elastic peaks are to the left of the fission peak. This is a normal kinematic effect. The symmetric fission events are those for which 0y is maximum. F o r asymetrie events, 0r is smaller whatever the mass of the fragment (light or heavy) detected by the x-detector. Third, it may happen that only one (or even none) of the elastic + transfer peaks is present. This depends on the value of 0x as compared to the grazing angles for the projectile and target nuclei. 4. Fission following transfer reactions
When the target nucleus is very heavy, the heavy transfer products can be heavier than thorium and have a great probability to deexcite by fission. Three main flag-
FISSION [
i . . . .
i
195
a) --
|
40O Gt,
t!
;oo
~6
IO 2o
J~
z
2OO
4O
50
60 Oy (degrees)
I00 0
I
0
I
I00
I
200
3o0 Ex
'"
I
_
4oo 50o (channels)
I
i
J6:SHo + 4OAr 59 °
200
1
c)
226MeV /f~,~\\\
182 ._o
1:5o o
I-
I00
t
40
,
50
. . . . . . .
,t
,
I00
150
.
t
.
"
16:5
Mass (am u.)
Fig. 3. Results for the system Ar+Ho at 226 MeV. (a) Lab kinetic energy F~ versus F~ contour diagram. (b) Angular correlation function. (c) Frasment mass - - t o t a l c . m . k i n e t i c e n e r g y contour diagram. merits are then produced in these two=step reactions. The detectors x and y can detect only two of the three products, and two possibilities exist: In the first one, these two products are the two fission fragments issued from the fission o f the heavy nucleus ("two-fragments" case). In the other case ("light transfer product + o n e fragment"), one detector receives the light transfer product and the other detector receives one of the two fission fragments. The detection of such events is possible only if the two steps of the reaction have occurred in neighbouring planes. More precisely, we have calculated that the corresponding efficiency of our experimental set-up was 20 ~o of the efficiency for binary events. If the corresponding cross section is large, these events can substantially
196
B. T A M A I N et al. !
r. O q}
/
!
!
Ima
z
I
!
90
120
150 180 e x y (degrees)
Fig. 4. Angular correlations in the plane defined by the beam axis and the detectors for three cases: fission following complete fusion, fission following transfer reaction at bombarding energies just above the interaction barrier, and fission following transfer reaction at a bombarding energy far above the interaction barrier. Here 0ffi~is the angle between the two detectors. The system is Arq-U. This drawing is schematic only. On the right part of the figure the corresponding diagrams of velocity are drawn.
increase the measured complete-fusion fission cross section. How is it possible to distinguish fission following transfer from fission following fusion? For the first ease, the angular correlations between the fragments issued from a transfer product or from the complete fusion nucleus differ very much, when the bombarding energy is just above or very high above the interaction barrier. But at the intermediate energies, the two angular correlations are not separated (see fig. 4). We have calculated the correlations for fission into two equal fragments after complete fusion and after several transfer reactions, for the system A r + T h at 250 MeV, and found no separation (table 2). Similar calculations indicate a d e a r separation at 400 MeV [ref. 3b)] and, to a lesser extent, at 300 MeV. It appears in table 2, however, that the c.m. total kinetic energies are lower for fission following transfer reactions. Moreover, in every case, the direction of the fissioning nucleus, OR, is very different from 0, and its velocity VR is small. If an event with such characteristics is treated as a two-body event (as it was the case for all events in our work) it is found that the c.m. total kinetic energy at the end of calculation is very low. Moreover, a symmetric event appears to be very asymmetric, thus allowing us to separate easily spurious events from good events in mass-energy diagrams (see fig. 6).
197
FISSION TABLE2 Fission following transfer reaction for the system Thq-Ar at 250 MeV Particle tranfferred to the target
Fissioning nucleus
TKE (MeV)
OR (deg)
Vt (MeV/amu)
0yo (
0yb (deg)
2p 2n (u)
2~eU
150
4p 4n (8Be)
24°Pu
156
lzC
z4~Cm
162
srO
2+sCf
166
full momentum transfer
27~108
220
44.6 37,6 42.8 23.6 39.5 21 35.8 22.9 0
0.424 0.504 0.364 0.604 0.320 0.540 0.276 0.440 0.272
74 62 76 46 77 50 77 57 69
69 62 71 52 71 53 72 60 69
TKE is the c.m. total kinetic energy for fission of the heavy transfer product. Two calculations are indicated for each transfer reaction depending on the Q-value assumed for the reaction (10 MeV or 50 MeV are the chosen values). Parameter 0a is the angle between the beam direction and the recoil velocity; 0. is chosen to be 65*; and 0y, and 0yb are the values of 0y obtained, depending on whether the recoil velocity Va is on the same side of the beam as the x-detector, or on the other side.
N o w , let us consider h o w to identify events of the type "light transfer p r o d u c t + one fragment". This is very easy because such events have a high lab kinetic energy: the light transfer p r o d u c t has a kinetic energy nearly equal to that o f the scattered projectile, and the fission fragment has a velocity which results f r o m the addition o f its c.m. velocity and the recoil velocity o f the fissioning nucleus (fig. 4). Such spurious events were eliminated in the E,-E I distributions (see fig. 5). Thus a careful study o f all the experimental informations allows us to discriminate between fission following transfer a n d that following fusion reactions. The results given below are thus not perturbed.
5. Experimental results and discussion 5,1. CROSS SECTIONS I n table 3 the fission cross section following fusion is denoted by , f . First o f all, it must be pointed out that ~f has a high value even for light targets. This is due to the angular m o m e n t u m effect which is equivalent to a lowering o f the fission barrier [ref. 21)], This will be discussed in m o r e detail in sect. 6. Let us n o w c o m p a r e ~rf with the total reaction cross section *x. We have calculated *x by using the classical f o r m u l a ax = ~ ( R t + R 2 ) 2 ( I - Vtnt/E) where R t and R 2 are the radii o f the interacting nuclei, Vint is the interaction barrier, and E
Nd
<16a>Tm
2osAt
=*110
-‘MO
“‘Sb
I-Ho
Z3SU
WO
13
28.5
39
200 300 162 179 199 226 300 226 300 250 300
140 209.5 122 135 150 170 226 182 242 214 257 172
136
110
94
98 169 58 72 86 106 164 97 160 82 125 141& 30 160* 15 lo* 1 184h 20 505% 80 530* 50 610& 60 800% 90 1350*140 516*150 1030&120 195 630 1045 1040 1130 800 1350 516 1030
1380 2320 420 830 1230 1600 2340 1330 2270 1210 2015
91 145 48 71 91 111 155 108 163 116 164
33&t 62& 5 84&6 89&t: 107&10 841t5 126& 7 76&11 117* 7
The interaction barrier Riot and the reaction cross section uR have been calculated with the formulas and parameters given in subsect. 5.1. For the system Ar+Sb, the complete fusion cross section used to calculate Iollt include the evaporation residue cross sections given in ref. *‘lb).
Compound nucleus
Target
TABLE 3 Cross sections for fission of complete fusion nuclei
Pj 5 g z g
2
FISSION
199
|
70O
U ÷ Ar 2 5 0 M e V
~600 I
C e-
(D 400 300 2O0 I00 I
I00
200
300
400 500 Cl~onnels
Fig. 5. Lab kinetic energy F~ versus Ex contour diagram for the system U-[-Ar 250 MeV. The three hills are due to elastic (and quasielastic) events and fission following complete fusion. The wide bubble (high values for both F.ffi and F~) is due to "light transfer p r o d u c t ~ o n e fragment" events (see text).
.o_
|
300
- ey-
I
8 4 o. 7 4 •
o
!
300
!
ey- 74 e. 104"
(2),
200 I
I00
I
200 (o.m.u.)
Fig. 6. Mass versus total c.m. kinetic energy contour diagrams for the system U-{-Ar 300 MoV. The three diagrams correspond to different positions of the y-detector. At low 0~ values, elastic scattering, quasi elastic events and asymmetric fission events are observed; at medium and high angles, symmetric fission events. A t high 0y value asymmetric events of very low kinetic energy are also present: they are due to fission following transfer reactions.
200
B. T A M A I N etal.
is the incident energy in the c.m. system. The parameters V~nt, R1 and given by:
R 2 are
Vi~, = Z1 Z2 e2/r,(A~, + A~), Ri = ro A~.
In the two last formulae, we have chosen rc = r o = 1.44 fm. These values have been obtained from the analysis of elastic scattering data and/or reaction threshold energies [see ref. x) and references therein]. 5.1.1. Fusion nuclei heavier than uranium. This refers to the system U + A r . In this case acP = af and it is very easy to calculate the critical angular momentum from relation (3). The results appear in table 3. It must be noted that l~tt is an increasing function of the incident energy, and that its value can be very high (more than 100h). 5.1.2. Fusion nuclei lighter than uranium. In these cases, the knowledge of af is not sufficient to calculate lc~,, but we have used experimental results already published ~7, xs) which give cross sections ~R for evaporation residue products. This has been possible in the Ar + H o and Ar + S b cases. The calculated l~m values are given in table 3. We can make the same remarks as in subsect. 5.1.1 : l~r~tcan be very high and is an increasing function of incident energy. 5.2. I N F L U E N C E O F T H E E N T R A N C E C H A N N E L
For a given fusion nucleus, lcrit is an increasing function of the mass of the projectile or of the value of lmaz. This appears clearly in fig. 7 which refers to fusion nuclei close to 17°Yb. The black points give the lcr, values and the white ones the lmaz values. The four 17Oyb results are from Miller and Zebelman 4) and are compared to le2Tm formed by A r + S b . All these results involve the same excitation energy and the same value of (/~- V~nt). It appears that the difference between/max and lcr~t is approximately constant, suggesting that the non-fusion processes are surface reactions involving an overlap of nuclear matter smaller than a critical overlap. This idea supports the concept of critical approach distance introduced by Galin et al. 15) which is illustrated in fig. 8. This figure refers to our results and to data quoted in table 1 of ref. 16). Let us assume that the critical approach distance concept is correct. At this distance R~ the potential energy of the system, which is equal to the incident c.m. energy, can be written as: V(Ro) = t = VN(Ro)+ Vc(Ro)+
h'lo,t,(lo,i,+ 1) ~4522
2/z(A1 +A2) refit
'
where VN(R~) and Vc(R~) are the nuclear and Coulomb potentials at the critical distance R~, A 1 and ,4 2 are the mass numbers of the two nuclei,/~ the reduced mass and rcm the critical approach distance parameter defined in ref. 15). The relation-
FISSION
I Ecru- Binf
E
E*
201
m
I
I
~ 65MeV = 107MeV
E
0
E
I00
21
c-
'~moxJ/~
<~
/ / /
o
17..Oyb:
l'o/
/
/ / 50
J
/
I
I
I0
I
20
II
30
40
Projectile moss (o.m.u.) Fig. 7. Evolution o f l=,x and l©rlt values for neighbouring fusion nuclei formed in four different ways. In four cases, the excitation energy and the difference ~ - - Vl,t are approximately the same. This figure shows clearly how lc, lt is a function o f the entrance-channel parameters.
,
i
l
I
w ~ r . U
l
t~/'"-
refit-1.01
Ar~,U 2 0
Ar+Ho2
Ar+Ag 1,08
.42 -'_ A r + o b 0 Ar+Ag O 2
( rcrit). 1,03
N "~" Ar+Se 02 2 N+Rh 0
f
C+Cu
~
±0,08
0.93 C.~Ni 1.07
C+Cu O 2
C+Ti
C+Ni
0,96
1.15
C+Ti (~ C+AI. (~ 50
100
150
I 200
I 250
I 3OO
I 350
Ecm (MeV) Fig. 8. For the quantity plotted: see text. This figure shows to which extent the concept o f critical distance agreo with experimental results.
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B. TAMAIN et al.
ship between E and h21¢rtt(l,,,t+l)/2#(A~+A~2)2 is linear, and the slope gives the value of refit. The main interest of such a figure is due to the fact that it is based on purely experimental data and is independent of calculated interaction potentials. The assumptions that are made are first that each nucleus is spherical, second that the sudden approximation is sufficiently valid to enable us to write down the centrifugal term in the way given above, and third that there is no damping of the incident kinetic energy. Since it is possible that the above assumptions are not correct, the numerical values of rom should be regarded with some reservation. A straight line can indeed be drawn through the data for each system of fig. 8. The values of r,~it obtained range from 0.90 to 1.15 (see fig. 8). No systematic variation of rcm from the light to the heavy systems can be seen. The average value is 1.03 4- 0.08 fm (standard deviation). It is shown in the figure that this average value can be used for most of the systems, taking into account the experimental uncertainties in the fusion cross sections. This figure should thus not be regarded as proof of the validity of the rorit concept, but only as an indication that most of the experimental results are not at variance with this concept. To really check if (i) a value of rcm can be defined for a given system, and if (ii) the value is unique for all the systems (or if it varies regularly from light to heavy systems), more data of good accuracy are needed. 5.3. HIGH VALUES OF Icm The high values of lcrit obtained here are surprising, and very much higher than those calculated by Cohen, Plasil and Swiatecki 21). In fact, the comparison has to be made carefully, since the meaning of critical angular momentum is different in ref. 21) from that used here. In their calculation, Cohen, Plasil and Swiatecki refer to the point at which the compound nucleus has no fission barrier. Our work, on the other hand, refers to the fusion processes. Using the Swiatecld picture 22) we can imagine that the trajectory followed by the system (see fig. 9) can go from the fusion valley to the fission valley without reaching the shape of stable equilibrium H. If this jump occurs immediately after reaching the ridge AC, dynamical calculations are necessary to predict whether or not the time and the energy dissipated are sufficient to obtain the symmetric mass distribution and the kinetic energies characteristic of fission. In other words, it is difficult to know if these paths, which never go through a compound-nucleus-like shape, contribute to the cross section for pre-equilibrium fission (fig. 1) and hence to Crc~, or if they contribute to the "quasi-fission" cross section. What happens if the dynamical path goes through a compound-nucleus-like shape? Cohen et al. postulate that a nucleus having no fission barrier cannot be a compound nucleus. It must be noted, however, that the vanishing of the fission barrier is an angular momentum effect, which holds only for one given deformation direction. Before "finding" this direction, the nucleus can "try" other directions
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and can thus survive a longer time than would be expected on the simple consideration of B f ( l ) ----0 made by Cohen et al. In such a case at least a partial equilibrium can be obtained.
Fig. 9. Potential energy surfaces for fusion and fission. Here, == and =, are the usual deformation parameters. This figure is for a fusion nucleus having no fission barrier.
Einc lob. (MeV) 200 240 280
160 I
I
I
I
I
I
I
I
!
3 2000 E
8~ ooo o u
~6 5 0 0 tO
IOO
r
50
I0
I
I00
140
,
=
)
=
180 220 Einc c.m ( MeV}
Fig. 10. System A r + S b : excitation functions for fission following fusion (~v, open points), evaporation residue formation [black points dashed curve; energies up to 150 MeV, ref. 17=); higher energies, ref. 17v)]. The calculated total reaction cross section ¢rR is calculated with the relation given in subsect. 5.1.
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|
i
J
i
i
i
i
>= 20 IO
o
o
O-cF50
Ioo
g
I =o
"
°_
Angular momentum
(~ units)
Fig. 11. Competition between fission and evaporation for the system Sbq-Ar 200 MeV. The upper part of the figure shows the variation of B~(l) with L The solid line is obtained when the shapes of the nucleus in its fundamental state and at the saddle point are assumed to be independent of L The dashed line is for the opposite hypothesis 21). The lower part of the figure gives the distribution of partial waves for the total reaction cross section and the fission cross section corresponding to the solid curve of the upper part. The results corresponding to the dashed curve are nearly the same.
6. Competition between fission and evaporation for the system Ar + Sb In fig. 10, we have plotted the excitation function for fission after fusion for At" + S b , together with the experimental excitation function for evaporation residue products 17) and the calculated total reaction cross sections. The reaction and fission thresholds are different. A reason for this could be that the fusion barrier is greater than the reaction threshold. But these two quantities have been found to be very close to each other, or identical, for argon induced reactions 19). Furthermore in fig. 10, and from ref. 17=), the threshold for the evaporation residue cross section aEx is lower than the fission threshold, and is approximately equal to the calculated interaction barrier. To reproduce by calculation this result, we use the statistical theory of deexcitation by evaporation of particle or fission. Thus it is assumed that all the fission events we have observed are compound-nucleus fission events. The lowering of the fission barrier due to angular m o m e n t u m effects is given by the difference between the rotational energies of the nucleus in its ground state ERO and in its saddle point state ERs [ref. 25)]. The value of the fission barrier Bf is taken f r o m ref. 26) the level density parameter a, has been taken equal to ~0 A, the moments of inertia are those o f rigid bodies, and we have focussed our attention on the two factors which are the
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most important: the value of the ratio allan (level density parameters for fission and evaporation, (taken to be 1 or 1.2), and the/-dependence of the ground state and of the saddle-point shapes [we have considered two cases, one in which there was no/-dependence as in ref. 2o), and one in which the/-dependence was given by ref. 21)]. The combination of the above factors gives four cases. In each case, the calculations have be done along a whole deexcitation chain of evaporated neutrons, by varying E* from the compound-nucleus value down to Em(l)+Bf below which value no fission is possible. For a given value of E* and for allan = 1, the results are shown in fig. 11. The most striking result is that the fission probability increases very rapidly in a limited range of/-values. This result was also found by Plasil 24) in making similar calculations. We have then defined two values of angular momentum, ls% and 195%: at 15%, only 5 70 of the compound nuclei undergo fission at any step of the deexcitation chain and at 19s%, 95 7o undergo fission. TABLE 4 Angular momentum effects on the competition between fission and neutron evaporation
Shape dependence on aug. momentum
at
15%--19s%
an
o(195%)--o(15%)
ot (mb)
acv
(%) no
1
no
1.2
yes yes
1 1.2
53 --62 (44--55) 22--49 (4--37) 50--63 3--40
15 (15) 25 (20) 22 23
510 (630) 800 (880) 520 880
Results of calculation done with the statistical model. For notation, see text. Four cases are considered depending on the choice made for the two factors: effects o f angular momentum on the shapes o f the nuclei; value of the ratio o f the level density parameters at and an. The calculations have been done for two lab beam energies: 200 and 300 MeV. All the steps o f the deexcitation have been taken into account. We have assumed that the fission barrier Bt and the neutron binding energy So do not vary along the deexcitation chain: Bt = 28.5 MeV [ref. 26)], S~ = '7.5 MeV. The values in parentheses are for So = 12 MeV.
In table 4 the values of 15% and 195% are given for 200 MeV beam energy and for the four cases defined above. By comparing cases 1 and 3 or cases 2 and 4, it can be seen that the shape dependence on the angular momentum has very little effect on 18% and 195%. These calculations explain why the fission threshold is higher than the interaction barrier: indeed, for beam energies corresponding to critical angular momentum values lower than/5% very few compound nuclei undergo fission: the fission threshold can be defined as the energy for which lcrit becomes equal to 15%. The experi-
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m e n t a l fission t h r e s h o l d f o u n d in this w o r k is close to 162 MeV. T h e c o r r e s p o n d i n g /cat value is 31. This m e a n s t h a t 15% is also close to this value. T a b l e 4 indicates t h a t a n alia, o f 1.2 r a t h e r t h a n 1 is necessary to o b t a i n 15% = 31. A t b o m b a r d i n g energies higher t h a n 200 MeV, the value o f the fission cross section is n o t o n l y d e t e r m i n e d b y the e v a p o r a t i o n - f i s s i o n c o m p e t i t i o n , b u t also b y the lcrit value f o r c o m p l e t e fusion. I n d e e d , this lcrit value exceeds t h e / - v a l u e a t which t h e fission b a r r i e r vanishes [Bf(I) = 0], a n d all nuclei in this r a n g e o f / - v a l u e s u n d e r g o fission. A calculation w h i c h r e p r o d u c e s the value o f the e v a p o r a t i o n cross section is given in ref. 17b). T h e a u t h o r s wish to express their a p p r e c i a t i o n to C. C a b o t a n d B. Borderie f o r e x p e r i m e n t a l assistance a n d t o J. P o u t h a s f o r c o n s t r u c t i o n o f a p a r t o f the electronic system. T h e y a r e p a r t i c u l a r l y i n d e b t e d to Prof. M . L c f o r t a n d D r . F. Plasil f o r s t i m u l a t i n g discussions a n d f o r their helpful c o m m e n t s o n the manuscript.
References 1) M. Lefort, Y. LeBeyec and J. P~ter, Riv. Nuovo Cim. 4 (1974) 79 2) J. B. Natowitz, Phys. Rev. CI (1970) 623 3) a. T. Sikkeland, Ark. Fys. 36 (1966) 539 b. T. Sikkeland, Phys. Lett. 2713 (1968) 277 4) A. M. Zebelman, K. Beg, Y. Eyal, G. Jaffe, D. Logan, J. Miller, A. Kandil and L. Kowalski, Prec. 3rd Syrup. on the physics and chemistry of fission, Rochester, 1973, vol. 2 (IAEA, Vienna) p. 335 5) H. H. Gutbrod, F. Plasil, H. C. Britt, B. H. Erkkila, R. H. Stokes and M. Blann, Prec. 3rd Symp. on the physics and chemistry of fission, Rochester, 1973, voL 2 (IAEA, Vienna) p. 309 6) F. Hanappe, C. Ng0, J. P6ter and B. Tamain, Prec. 3rd Syrup. of the physics and chemistry of fission, Rochester, 1973, vol. 2 (IAEA, Vienna) p. 289 7) J. P~ter, F. Hanappe, C. Ng6 and B. Tamain, Prec. Int. Conf. on nuclear physics, Munich, 1973, vol. 1, ed. J. de Boer and H. J. Mang (North-HoUand, Amsterdam, 1973) p. 611; and internal report IPNO-RC-73-07, Orsay (1973) 8) M. Lefort, C. Ng6, J. P6ter and B. Tamain, Nucl. Phys. ,4,216 (1973) 166 9) J. J. Griffin, Phys. Rev. Lett. 17 (1966) 488; M. Blann and F. M. Lanzafame, Nucl. Phys. A142 (1970) 559 10) C. Cabot, C. Ng6, J. P6ter and B. Tamain, Nucl. Instr. 114 (1974) 41 11) B. Borderie, F. Hanappe, C. Ng0, J. P~ter and B. Tamain, Nucl. Phys. A220 (1974) 93 12) H. Britt and A. Quinton, Phys. Rev. 120 (1960) 1768; A. M. Zebelman, L. Kowalski, J. Miller, K. Beg, J. Eyal, G. Jaffe, A. Kandil and D. Logan, Phys. Rev. C10 (1974) 200 13) L Halpern and V. M. Strutinsky, Prec. 8th Int. Conf. on the peaceful uses of atomic energy 15P/1513 408 (1958) 14) B. Tamain, C. Ng6, J. P~ter, R. Lucas, J. Poitou and H. Nifenecker, Institut de Physique Nucl6aire Orsay report IPNO-RC-75-04 (1975) 15) J. Galin, D. Guerreau, M. Lefort and X. Tarrago, Phys. Rev. 69 (1974) 1018 16) M. Lefort, Y. LeBeyec and J. P6ter, Prec. Int. Conf. on reactions between complex nuclei, Nashville 1974, vol. 2, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 91 17) a. H. Gauvin, Y. LeBeyec and N. T. Porile, Nucl. Phys. A223 (1974) 103 b. H. Gauvin, D. Guerreau, Y. LeBeyec, M. Lefort, F. Plasil and X. Tarrago, to be published 18) Y. LeBeyec, M. Lefort and A. Vigny, Phys. Rev. C3 (1971) 1268
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B. Tamain, Ph.D. thesis, Clermont Ferrand, 1974 S. Cohen and W. J. Swiatecki, Ann. of Phys. 22 (1963) 406 S. Cohen, F. Plasil and W. J. Swiatecki, Ann. of Phys. 82 (1974) 557 W. J. Swiatecki, Suppl. J. de Phys. 33 (1972) CS-45 T. Sikkeland, E. L. Haines and V. E. Viola, Jr., Phys. Roy. 12,5 (1962) 1350 F. Plasil, Prec. Int. Conf. on reactions between complex nuclei, Nashville 1974, vol. 2, od. R. L. Robinson et al. (North-Holland, Amsterdam, 1974) 25) D. W. Lang and K. J. Le Couteur, Nucl. Phys. 14 (1960) 21 26) W. D. Myers and W. J. Swiatecki, UCRL 11980, Berkeley 1965