Neutron multiplicities in heavy-ion-induced fission: Timescale of fusion-fission

Nuclear Physics ONorth-Holland

A452 (1986) 550-572 Publishing Company

NEUTRON

D.J. HINDE=, Department

MULTIPLICITIES IN HEAVY-ION-INDUCED Timescale of fusion-fission R.J. CHARITYb,

G.S. FOOTE, J.R. LEIGH, and A. CHATTEJEEd

J.O. NEWTON,

FISSION:

S. OGAZAC

ofNuclear

Physics, Research School of Physical Sciences, Australian National University, GPO Box 4, Canberra, 2601, Australia Received 1 August 1985 (Revised 11 October 1985)

Abstract:

Neutron

correlations have been measured for the compound nuclei Es formed at excitation energies between 45 and 90 MeV by fusion reactions induced by beams of 16,180 , 19F and 28,30Si. Analysis allows the determination of the mean multiplicities of neutrons originating from the fused system (Y,,) and the fission fragments (Zv,,,,). The total multiplicities (us, + 2v,,,,) are in good agreement with calculated values. However, calculations of Y,,, using the statistical model code ALERT1 underestimate the multiplicities, the discrepancy increasing with excitation energy and fissility. This is interpreted in terms of the dynamics of the fission process, and can be resolved by including in the calculations the effects of a delayed onset of fission (delay time 70 X lO-*l s) or a slow saddle-to-scission transition (transit time 30 X 10e21 s). Simple theoretical estimates suggest that both effects are significant, and their characteristic times may be similar. With these constraints, the data are fitted when both effects are applied, each for 20 X lO-*l s. Such long times are not inconsistent with the results of pure one-body dissipation calculations, and are supported by recent results on the timescale of quasi-fission. The quoted results are for a level-density parameter a, = &A. Use of &A (+A) results in shorter (longer) times; however, it is concluded that for all reasonable values, motion in the fission direction is overdamped. 168yb

E

fission-fragment

192.1%2~pb,210po,213~r

angular and

251

NUCLEAR REACTIONS 150Sm(‘80;F), E= 108-122 MeV; ‘64Er(2RSi,F), E= 170 MeV; ‘70Er(2XSi,F), E= 135-165 MeV; ‘70Er(30Si,F), E= 160 MeV; ts1Ta(i9F,F), E= 95-135 E = 105-138 MeV; MeV: 192Os(‘” 0, F), 19’Au(160, F), E = 95-124 MeV; 232Th(19F,F), pre-, post-fission neutron measured (fragment) n-coin, o(fragment 8, E,); deduced 168y,,, 192,198,2oOp,,,210po, 213l+., 215Es deduced fission dynamics. Statistical multiplicities. model.

1. Introduction The last decade has seen rapid development in the field of heavy-ion-induced fission. Recently, much theoretical attention has been directed towards understanding the dynamical behaviour of the fissioning system. Various aspects have been a Currently supported by a Queen Elizabeth II Fellowship. b Present address: Nucl. Sci. Division, 70A, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. ’ Present address: Inst. Nuclear Physics, ul. Radzikowskiego 152, PL-31342 Krakow, Poland. d Present address: Van de Graaff Lab., Bhabha Atomic Res. Centre, Bombay 400 085, India.

D.J. Hinde et al. / Neutron multiplicities

551

investigated, including the treatment by Weidenmiiller and coworkers of the transition from the equilibrium deformation to the saddle-point deformation lw4), and the work by Davies et al. 5, and Nix et al. 6, on the-saddle-to-session transition. The latter paper concluded that measurements of the number of neutrons emitted during fission are most likely to give information on the timescale of fission (and thus on nuclear dynamics), since more neutrons will be emitted the longer the fission process takes. Convention~y, two sources of neutrons observed in prompt coincidence with fission fragments are assumed, namely the fused system, travelling with the compound-nucleus velocity, and the fully accelerated fission fragments. A neutron angular correlation can be decomposed7) into components originating from these two sources, whose multiplicities are usually labelled pre-fission (vi,,) and post-fission (Y,,,). The tr~sition-state model of fission predicts the widths (and thus lifetimes) for fission and neutron emission on the basis of appropriate level densities. From these, using computer codes such as ALERT1 ‘) or JULIAN 9), a pre-fission multiplicity can be calculated. This model of fission is only appropriate when the calculated lifetimes are long compared to the dynamically constrained fission lifetime. If measurements of vpre give significantly greater multiplicities than those calculated, this may reflect the importance of fission dynamics. Experimental data have until now been rather limited. Measurements for the compound nuclei 170Yb [ref.l’)] and IssIr [ref. rl)], at high excitation energies (100 < E, < 300 MeV), have shown considerably larger multiplicities than expected on the basis of statistical model calculations. However, for 2ooPb [ref. 12)], at lower E, (50-=c E, < 80 MeV), reasonable agreement was obtained. We have measured preand post-fission multiplicities for seven nuclei from 16sYb to 251Es, at compoundnucleus excitation energies between 45 and 90 MeV. A comparison of results for “‘Pb with 251E~ has already been published13). Fission and evaporation-residue measurements were made for all systems except “‘Es , thus constraining the statistical model parameters, and allowing a more reliable prediction of vpre to be made. Our approach in this paper will be to ask what the experimental neutron multip~~ity data alone can tell us about fission dynamics. Then a simple application of models of pre-saddle and saddle-to-scission dynamics will be made.

2. Ex~~rn~n~l method The compound nuclei r6*Yb, 192,198,200Pb, “‘PO, 213Fr and “rEs were formed by bombardment of - 1 mg . cm-2 targets by beams of 16,180, 19F and 28.30Sifrom the ANU 14UD accelerator. The specific reactions, together with the bombarding energy ranges, are shown in table 1. The experimental apparatus used was that described by Ward et al. l*). Fission detectors were located at 0” and 90’ to the neutron detector (NE213), all detectors being in a plane perpendicular to the beam. The efficiency of the neutron detector

552

D.J. Hinde et al. / Neutron multiplicities TABLET

Compound

nucleus, target, projectile

and bombarding

Compound

target projectile energy.

energy range

nucleus

“‘Yb

19’Pb

19’Pb

‘O”Pb

‘O”Pb

*lOPo

‘13 Fr

215E~

lsoSm ls0 108-122

164Er 28Si 170

“‘Er ‘sSi 135-165

17’Er 3oSi 160

“ITa 19F 95-135

192OS lx0 95-124

‘97Au 160 95-124

232m

‘9F 105-138

WV1

was measured for a neutron velocity range of 0.6 cm. ns-’ to 4.0 cm. ns-l, both before and after the series of experiments. Further details of the experimental method will be published14).

3. Analysis Neutron time spectra were deconvoluted with the neutron-detector time resolution, and converted into velocity spectra in a manner similar to that described by Ward et al. 12) [see ref. 14) for further details]. Typical deconvoluted neutron velocity spectra at 0” and 90” to the fission-fragment direction are shown in fig. 1. From such spectra, components assigned to “pre-fission” emission and “post-fission” emission are extracted. It is conventionally assumed that pre-fission neutrons are emitted isotropically from a source moving with the compound-nucleus velocity, whilst post-fission neutrons are emitted from the fully accelerated fission fragments. With this assumption, the iterative method described by Bishop et al. 15) and Ward et al. 12) (the “free-fit” method) was used to deduce the pre- and post-fission neutron velocity spectra, and thus multiplicities. The fits to the data in fig. 1 show the pre-fission

component

(labelled

Y,,,), and the summed

components

from both the

detected and complementary fission fragments (labelled Y,,,,). Included are the effects of the angular acceptance of the neutron (k 6”) and fission-fragment (_t 4”) detectors. Although a range of mass-splits and total kinetic energies for fission were also included in the calculated curve, there is still a slight discrepancy in the 0” curve at the mean fission-fragment velocity, corresponding to the small probability of neutrons of low velocity being emitted from the observed fragment. This is due to the inability of the deconvolution to regenerate a feature which is smaller than the experimental error bars, and is quite sharp compared to the time resolution of - 0.5 MeV neutrons. From the deduced velocity spectra in the centre-of-mass frame of each neutron source, the pre-fission multiplicity v,,, and the post-fission multiplicity vpost were determined. The latter is defined as the multiplicity per fragment; hence the total multiplicity ( vt,,) is given by

D.J. Hinde et al. / Neutron multiplicities

I

I

I

I50 MeV ‘*Si

1.0

2.0

3.0

I

AU

I

I

I

DATA \

+ 17’Er

1.0

4.0 NEUTRON

553

VELOCITY

2.0

3.0

4.0

(cm ns-‘1

Fig. 1. Neutron velocity spectra at 0” and 90” to the fission-fragment direction, for the reaction of 150 fit to the 0” and 90” data, MeV 2XSi with i7’Er, giving the compound system 19’Pb From a simultaneous components associated with the pre-fission (Y,,,,) and post-fission ( v,,,~) multiplicity are indicated, together with the total (v,,,). Inset are shown the x*/n envelopes obtained when fitting all data (upper curve) and when data near vr are omitted (lower curve). See text for further details.

4. Experimental results and their interpretation 4.1. TOTAL

NEUTRON

MULTIPLICITY

The total multiplicity vtot for a particular reaction can be compared with the number expected from the energy balance equation. This can be useful in checking the consistency of experimental data, and in determining whether large contributions from reactions other than complete fusion are present. To compare many sets of data, with widely differing neutron binding energies, it proves useful to define the total available decay energy E,(f). This quantity is defined in the appendix. E,(f) can be equated with the neutron kinetic energies E,i, and multiplicity x, and the total y-ray energy E,(f) by the equation E,(f)

= E,(f)

+ f

(8.07 +

E;),

(2)

i=l

assuming that all the available decay energy is removed by neutron and y-ray emission. Note that the variation in neutron binding energies has been taken out of the right-hand side, and put into E,(f) (see appendix). Experimental total neutron multiplicities are shown in fig. 2, plotted against E,(f). The ANU data are shown by crosses, whilst those of Gavron et al. are shown by

554

D.J. Hinde et al. / Neutron multiplicities

,,-14

/

13

t 12 II IO 9

utot

;

+ GAVRON etol.

17’Yb

4 HOLUB et.ol.

18?r

6 5 4

f 3

A.N.U. *“Es ,2’3Fr,2’0Po, 200,198,192pb,m3Yb,

2

CALCULATION

-

I

0

100

200

300

400

E,(f) Fig. 2. Total neutron m~tip~citi~s calculated curve is derived assuming

for fission, plotted against the total decay energy (see text).‘The fission following complete fusion, with subsequent decay by neutron and y-ray emission only.

circles, and those of Holub et al. by triangles. The multiplicities follow a consistent trend, increasing rapidly until .&(f) = 150 MeV, then showing a much slower rise. The data can be compared with eq. (2) if the dependence of E,(f) and Ej, can be estimated. Neutron evaporation will on average populate the fission products at an excitation energy of - half their neutron binding energies. However, angular momentum in the compound system gives angular momentum to the fragments (in the sticking limit, each fragment carries one-seventh of the compound-nucleus angular momentum). Excitation energy also causes the fragments to rotate, through the thermal excitation of angular momentum bearing collective modes16-1s). Thus E;(f) will rise with increasing bombarding energy, which is related to E,(f). Taking a mean y-ray energy of 1 MeV [refs. 19,20)],for this purpose we can fit the experimental

555

D.J. Hinde et al. / Neutron multiplicities

y-ray

multiplicity

empirical

data18)

for 19F + 17’Er, i9F + lslTa

and

19F +232Th

with

the

formula E,,(f) = 8 + 0.07&.(f)

MeV.

(3)

The second term corresponds to rotation of the fragments, so is subtracted from the E,(f) to give the thermal energy, which is assumed to be divided equally between the fragments, each with U,(f). From this, the nuclear temperature T can be determined a n a, = &A and A = 105). For an evaporation-type (givenby T=IIU,o/a,, t ki g neutron energy spectrum, the mean kinetic energy (equated with EA) is equal to 2T. for the second Subtracting (8.07 + 2T) from U.(f) allows E, to be calculated neutron. The number of steps required to reduce the thermal energy to 4 MeV in each fragment (eq. (3)) is equated with the total neutron multiplicity. The curve in fig. 2 was calculated by this method, and the experimental data below E,(f) = 150 MeV agree with it very well. Thus for these data, the assumption of fission following complete fusion, with subsequent decay by neutron and y-ray emission, appears to be essentially correct. However, the multiplicities of Holub et al. and Gavron et al. taken at E,(f) > 150 MeV are considerably lower than the calculated curve. Although the pre-equilibrium neutrons observed at these energies would reduce the expected multiplicity slightly, because of their higher kinetic energy, it would seem that substantial charged-particle emission, during or after fusion, must be the explanation for the discrepancy. This might be expected at the high bombarding energies in question’l). Thus these reactions probably do not proceed by the simple complete fusion/neutron evaporation mechanism, and so should not be analyzed as if they did. Consequently the data above E,(f) = 150 MeV will not be further discussed in this work. 4.2. PRE-FISSION

The pre-fission

NEUTRON

neutron

MULTIPLICITY

multiplicities

DATA

(v,,)

for all the reactions

shown in fig. 3, plotted against the excitation E,(CN); this is defined in the appendix, using the M CN. The data of Gavron et al. for 176 MeV “Ne in the same figure as the uj8Yb data. The results

considered

are

energy in the compound system liquid-drop model mass defect for + lsoNd (giving 17’Yb) are shown

of Ward et al. are shown with the *“Pb data (see also fig. 9). In all cases the pre-fission multiplicity rises monotonically with E,(CN). Such a monotonic rise is in contrast with the predictions of the statistical model, as will be seen in the next section. The possibility that this disagreement is due to experimental errors must be considered. An exhaustive x2 analysis 14) of several randomly selected data sets was made, to gain information on both systematic and statistical uncertainties. Using the deduced spectra of vpre and vpost, and including the solid angles of the fission and neutron detectors and the range of fission-fragment velocities, envelopes of x2 per degree of freedom (x2/n) were determined as the multiplicities vpre and

556

D.J. Hinde et al. / Neutron mubiplicities

_ *“Si + lwEr _“*pb

5 _ ‘80+‘MSm-‘Yb

*‘Ne +“‘Nd -.‘70Yb

43_

x =

I

_ x = 0.72

0.60 +/--/

---

-

z-

I 5_

I

100 I,,,,,,,,,,,

I

*%i +‘“Er -‘%Pb

pre3

140

_ ?$i +170Er _200pb _ *F+‘“T~

4 _ x =O.?O u

I20

- x = 0.70

_.. 200pb

_ x = 0.70

I+

-

+

+

+ -_-_-

-//e-r--;=

:_+-==

5 _ Ia0

+‘Q’os _.2’oPo

4 _ x =@?I

: _/_./-zz

0

‘60 + ‘97’” __2’3~~ - x = 0.74

+ ++

_ x = 0.03

+ + a _!____-_

I,,

60

(

(

,

-25’~~

++

+ _~~.--_--79 - **_.(/.-

II,

‘9,Z +232T,,

(

80

t

i

_/-_ _._._.--Q __--. I , I , 60

80

E, cc.;;@Ae~j Fig. 3. Pre-fission nuclei. The fissility with af/cln = 1.00, curves indicate the

neutron multiplicities (Q_) plotted against the excitation energy of the compound parameter x is given for each case. The full curves show statistical model calculations and k, adjusted to fit the experimental fission probability data (see fig. 5). The dashed effect of neutron emission during the fission-fragment acceleration time, for u, = 5.4. The effect of varying

a, to iA and &A is shown in the “*Es panel.

or,* were independently varied. This was done for the full data sets, and also when the 0” data between O.SSv, and l.lSv, were omitted (where vf is the mean fragment velocity). The reason for such omission is discussed in sect. 3. The x2/n envelopes for 150 MeV 28Si +170Er are shown inset in fig. 1. A sharp minimum is observed in both cases, giving vPre= 2.73$~~ and 2.63itj,,, respectively, for a 70% confidence level. The minima in x2/n were 1.02 and 0.92, respectively. Similar results were obtained for the other cases studied, with a lower minimum x2,/n when the 0” data around v, were omitted (average 0.99) and percentage errors being smaller where more data was available. Analysis of the many m~tiplicity measurements repeated during the experimental program gave a random error of - 50.2, which is in agreement with the errors for

D.J. Hinde et al. / Neutron multiplicities

these points

determined

the data points

from the x2 analysis.

are an accurate

raw data and systematic

representation

errors in the spectral

557

It is concluded

that the error bars on

of uncertainties

due to statistics

decomposition.

Together

data (fig. 2) the good x2/n values suggest that other systematic and are estimated to be < _t 15%.

in the

with the vtot

errors are not large,

4.3. STATISTICAL MODEL CALCULATION OF vpre

Detailed statistical model calculations are necessary for quantitative comparisons with experimental data. However, a simple calculation of the ratio of the fission-toneutron emission width (r,/l?,) will first be made, in order to make clear the origins of the results shown later from the Hauser-Feshbach statistical model computer code ALERT1 8)_ Assuming that the neutron inverse cross section takes its geometrical value, and using a level density of the form p(U) CCU-*exp(2m) [following Morettoz2)],

The level-density parameters at the saddle-point and equilibrium deformations have been taken to be equal (ar = a, = a). The thermal excitation energies above the saddle point (US), and in the daughter nucleus following neutron emission (U,), are given by u,=E,(CN)-E,,-%(J), U, = E, (CN) - E,, - B,, with E,, being and E,(J) the Calculations MeV and the

(5)

the rotational energy of the nucleus at its equilibrium deformation, angular-momentum-dependent fission barrier. using eq. (4) have been made for 16*Yb [using a = $A, B, = 7.8 finite-range liquid-drop-model fission barrierz3)]. The results are

shown in fig. 4 as contours

of constant

T,/r,,

as a function

of E,(CN)

and angular

momentum (Jlz). Nuclei populated at E,(CN) = 60 MeV and J = 35 (see fig. 4) will generally neutron emission the survive first-chance fission since r/r, - 10P4. Following excitation energy is reduced by - 10 MeV and r,/r, is substantially lower; after the emission of two neutrons r,/r, < 10e5. Thus the fission which does occur is predominantly “first chance”, so the pre-fission neutron multiplicity is low. For nuclei with E,(CN) = 90 MeV and J = 60 (close to the angular momentum at which E,(J) = B,), it is evident that r,/r, is nearly independent of E,(CN). Approximately 95% of nuclei survive first-chance fission (r/r, - 0.05) and the same is true for second- and third-chance fission, etc. Thus vr_ is large. Finally, at E,(CN) = 110 MeV and J = 7,5, the fission probability increases with each neutron emitted. However, since only 50% of nuclei survive first chance fission (r,/r, - 1.0) vpre is again small. These three points illustrate the variation of v,,, expected in an

558

D.J. Hinde et al. / Neutron multiplicities

16*Yb

rf

160 -

I

0

fin

I

10-a

I

1

I

IO

20

30

I

10-a

I

40

50

10-l

I

I

I

I

60

70

80

-

J Fig. 4. Contours of &‘r, as a function of excitation energy E,(CN) and angular momentum (Jh) for 16’Yb. The thermal excitation energies US and U, are indicated (see text). Nuclei populated at J = 35, 60 and 75 experience a considerably different variation of r,/I’,, with Ex(CN), and thus different pre-fission neutron multiplicities will be expected (see text).

excitation function. This extremely simple picture does not take into account angular momentum carried by the neutrons, or the contribution to the total fission yield from many J-values in the experimental situation. Nevertheless, similar characteristics are seen in the excitation function of ~r~~calculated using ALERTl. In the computer simulation of the decay of a compound nucleus (as in ALERTl), many parameters need to be defined. Values based on theoretical expectations may be used, but the use of values constrained by experimental results should give the most reliable calculation. The most important variables defining fission decay are the fission barrier height, the ratio of level-density parameters ar/a,, and the angular momentum distribution of the nuclei. These, and other parameters, are discussed for example in refs. 24-26), and in references therein. A detailed description of the effect of the angular momentum distribution on fission probabilities will be published shortly27). These three important variables will now be briefly discussed. Fission and evaporation-residue excitation functions have been measured for all the reactions presented in this paper, except 19F + 232Th --f 251Es. However, in this case, fission is expected to occur with almost unit probability. The measured fission and evaporation-residue excitation functions are shown in fig. 5. Those for 192.198JooPb and 17’Yb have previously been published24,28-30), whilst for 168Yb and 210Po, a detailed discussion is in preparation27). The experimental data were used to define the fusion excitation functions for all cases but 17’Yb and 251Es, where Bass

559

D.J. Hinde et al. / Neutron multiplicities

2oNe?50Nd_“‘Yb

IO -

kf = 1.00, I

70

90

_--. _____ ____-.

F

(mb) IO

130

110

100 CT‘Oool

8, = 5

c

I

2eSi +‘%

-. lgBPb

kf = 0.87,

sJ = 9

C>

F

mSi +lmEr -2wPb

f I

kf= 0.89,

I

1000

a,= 9

/

I

I

70

50

19 F+‘81To

_

200Pb

f kf=O.89.

I

I

50

6,= 6

I

I

70

t 19F +=Th _=‘Es

,,

il-:::;‘,’

/’

,’

100

8’

,

/’

IO

/’

/ ;

‘90

+ ‘9Qj

a

2’0

po

I60 +lg7Au ew213Fr

~f=l;oo.,8,=~

k,= 1.05,

I

50

I

I

50

70

&=4

kf =l.OO, BJ = 6

I

I

70

50

I

I

I

70

E, (c.N.) (MeV) Fig. 5. Experimental excitation functions for fission (filled circles) and evaporation residues (hollow circles) for each system studied. The full lines show the statistical model (ALERTl) calculations of the excitation functions, for aJa, = 1.00 a, = $4 and with kr as indicated. The adopted diffuseness parameter S,, used to define the population distribution in the compound nucleus, is given for each case (see text). The dashed lines show the effect of delaying the onset of fission by 70 x IO-” s. For 251Es, the calculated a-particle evaporation excitation functions are shown.

mode131) fusion cross sections were used. The fusion cross section ufUSis given by q”,=n&2J+l)T,.

(6)

0

The transmission coefficient T, was parameterised as T,=

[l +e~p((J-J,)/6,)]-~

(7)

560

following

D.J. Hinde et ul. / Neutron multiplicities

Vigdor

et al. 32). When the diffuseness

the compound-nucleus

angular

momentum

parameter

distribution

8, is fixed, knowing is uniquely

defined.

ofUs, The

values of 8, for each case are shown in fig. 5. For the purposes of this paper, a full discussion of angular momentum distributions is not necessary, since pre-fission multiplicities are not very sensitive to the details of the distribution, unlike the fission probability 33). The parameter aJa, was set to unity, since in the previous analysis of or,, data for *O”Pb the statistical model codes gave reasonable agreement with the data for = 1.00. aJa, The final variable is the fission barrier height. For convenience, the rotating liquid-drop-mode134) barrier was used, scaled by a constant factor k,. This parameter was adjusted to give a good fit to the fission and evaporation-residue excitation functions, using the code ALERTl. The value of k, is shown in each panel, and the fit is.shown by the full lines in fig. 5. In each case the agreement between experiment and calculation is excellent. The calculated pre-fission neutron multiplicities are compared with the experimental data in fig. 3, where the ALERT1 calculations are shown by the full curves. The fissility parameter34) x is shown for each compound system. It is apparent that although there is agreement at low E,(CN) and fissility, at high values of either the disagreement is very substantial. Possible sources of the discrepency are discussed below.

4.4. POSSIBLE

ORIGINS

OF EXCESS

NEUTRONS

Previous theoretical and experimental papers have identified that may contribute to an increased “pre-fission” multiplicity. saddle-to-scission, not bound convenient

scission and fission-fragment

acceleration.

four stages of fission These are pre-saddle, Of course neutrons

by such divisions, and will be emitted continuously; however to consider each one separately, starting with the last stage.

are it is

4.4.1. Neutrons emitted during fragment acceleration. The conventional analysis of fission-neutron angular correlations relies upon the postulate of two sources of neutrons; the compound system and the fully accelerated fission fragments. Any neutrons

emitted

during

acceleration

will be identified

as originating

partly from the

compound system, and partly from the fragments. Thus the “pre-fission” multiplicity will apparently be increased. It had been suggested as early as 1965 by Eismont 35) that such emission could influence several types of experiment. Skarv%g36) has since argued that this mechanism could explain the observation of scission neutrons 37) in spontaneous and low-energy fission. Recently it was shown 13) that for 251E~, emission from the accelerating fragments could account for a large part of the discrepancy between experiment and calculation [if short saddle-to-scission times were assumed as predicted by two-body dissipation models5)]. Here, a simpler method of calculating the apparent increase in vpre has been used. It has been found that in the time taken for the fragments to reach 0.82 of their asymptotic

D.J. Hinde et al. / Neutron multiplicities

velocity,

the number

vPre resulting

of neutrons

evaporated

from the more rigorous

0.82 is dependent both are almost

on the relative constant

is the same as the apparent

but laborious

fragment

561

method

and neutron

for the systems studied,

velocities;

this approach

increase

in

of ref. 13). This factor however,

since

was considered

to be

adequate. The excitation-energy, distributions (multi-chance fission distribution) were taken from the ALERT1 calculations, and no saddle-to-scission time was allowed. Otherwise the assumptions of ref. 13) were used in calculating neutron lifetimes. Fig. 3 shows the increase in vPre, being the difference between the dashed lines and full lines. As expected from previous results, a large effect is evident for 251Es, but for l”Yb, even at the highest excitation energies, the calculated multiplicity is increased by less than one. The reason for this difference is apparent on studying how E, varies on passing from the compound system to the fission fragments (for 251Es E increases by 46 MeV, whilst for “‘Yb it is reduced by 20 MeV), and on comiarmg the average binding energies in the fragments (6.0 MeV and 7.2 MeV, respectively). These calculations were made for a level-density parameter a, = $4. The effect of using a, = $4 and &4 is shown in the “‘Es panel; the multiplicities are obviously very sensitive to this parameter, which determines the rate at which the level density increases with excitation energy. Despite uncertainty in the value of u,, from fig. 3 it is apparent that although neutron emission during acceleration can explain much of the discrepancy for “lEs, it is not sufficient to explain

the other experimental

data.

4.4.2. Scission neutrons. Analysis of spontaneous 37) shows that the yield of neutrons at 90” to the fission assuming emission only from fully accelerated fragments. as being due to neutrons emitted during the “snapping”

and low-energy fission 38) axis exceeds that expected This is generally explained of the neck connecting the

nascent fragments at the moment of scission39), whose intensity is - 10% of the total multiplicity. However, Skarsvlg has shown that neutron evaporation during fragment acceleration can explain the previous experimental data. Subsequently, results for 252Cf [ref.40)] have shown a rise in excess yield at 90” with decreasing fragment total kinetic energy (increasing excitation energy), supporting this hypothesis. However, conflicting results exist 41), showing the excess yield occurring at high total kinetic energy, and only when one of the fragments is spherical. If the latter data were correct, this could be taken to imply that scission neutrons from the “snapping” of the neck are only produced in shell-affected fragments, and would not be expected in heavy-ion induced fission. In view of the considerable uncertainties in experimental data and their interpretation, the effect of scission neutrons will be ignored here. Furthermore, the scission neutron multiplicity is very unlikely to be sufficiently large to explain the experimental data. 4.4.3. Emission during the saddle-to-se&ion transition. In the transition-state picture of fission, once the saddle-point deformation is reached, the nucleus is committed to fission. Although this is not true for real nuclei with finite viscosity6,42), nevertheless a mean time from saddle to scission ( ts,,) can be defined. To calculate

562

D.J. Hinde et al. / Neutron multiplicities

the number nuclei

of neutrons

at the saddle

emitted

point

during

t,,,, the distribution in excitation energy of For this purpose, the statistical model

must be known.

code ALERT1 was used, with the parameters as previously defined (see fig. 5). The mean excitation energy during the saddle-to-scission transition is equal to the excitation

energy

above the saddle point

at the average fissioning

angular

momen-

tum, plus the time-averaged increase during the descent to scission. This increase was taken to be half of the total gain in going from the saddle to the spherical primary fragments, which is undoubtedly an overestimate, corresponding to motion at a constant velocity. The other extreme would be to take the excitation energy at the saddle point; this approximation obviously reduces the calculated saddle-to-scission multiplicity, typically by - 30%. For each excitation energy, the lifetimes (ti> for successive neutron emission were calculated from the approximate expression of Moretto22). The probability of emitting i neutrons was then found by integrating to for the probability Pi of the nucleus having emitted i time t,,, the expressions neutrons:

Pi_1 ---dtt,_1 dPi

where i ranged multiplicity for population of transition. The

Pi ti ’

from zero to four. Thus for each populated energy bin, the neutron any saddle-to-scission time could be calculated. Weighting by the each bin gave the total multiplicity during the saddle-to-scission acceleration multiplicity was then determined in a similar manner.

Choosing a level-density parameter a, of $A, it was could be fitted reasonably well for a saddle-to-scission 6). Taking a, = hA, t,,, is reduced to 20 x 10F21 s, 60 X 10p21 s is required. This sensitivity is expected,

found that the whole data set time of 30 x 10p21 s (see fig. whilst for a, = iA, a time of since the rate of increase

of

level density with excitation energy is governed by a,. In fig. 6, the panel for 251E~ shows both the total calculated multiplicity, and the value when emission during acceleration is ignored. The latter appears to give a better fit; however, the overestimate of the excitation energy is greatest for 251E~, so this result might be expected. (Using the excitation-energy at the saddle-point reduces the saddle-to-scission multiplicity by - 35%.) Because of the energy removed during saddle-to-scission emission, the acceleration multiplicity is reduced considerably (compare fig. 3); however it still makes a substantial contribution to vpre. Although the data can be fitted with t,,, = 30 x 10p21 s, it has been recognized that experimental data include emission from the initiation of fusion onwards6); thus the actual mean saddle-to-scission time must be shorter. However, unless there is substantial emission of neutrons before reaching the saddle point, a slow saddleto-scission transition is implied. Pre-saddle emission will now be discussed. 4.4.4. Pre-saddle neutron emission. This aspect of fission is not quite as simple as the saddle-to-scission transition, since the system can “escape” from this region by passing over the saddle. Thus neutron emission for a fixed time is not appropriate,

D.J. Hinde et al. / Neutron multiplicities

5

“O+Tm

_

--‘=Y b 2ok +“‘Nd .elnYb

4x =

0.60

t

_ x = 0.72

,,+/

2-

,,y-----

-

---____-

‘/

II

I

I

I

=0.70

‘80 +‘92&

100 ,

I

28Si +lnEr -lgBPb

4 _ x

5-

_ 2*Si + ‘64Er -.‘g2Pb

/

3 _

5 _

/

,

563

_210,3

,

120 ,

,

140 ,

,

,

__+=-

,

,

,

_ ?Si +‘*Er -mPb

_ Is F+‘*‘Ta _ 200Pb

-

_ x = 0.70

x = 0.70

‘60

+ ‘97&

-2’3~~

_ lgF

_ x = 0.74

+232Th

-2J’E~

_ x i 0.,3,/q+-‘X’ -

;y,:+‘.

-

+

I I

I

60

I

,

,

,

,

,

,

80

,

I

60

E, 6hAe:)

I

1

,

80

Fig. 6. As fig. 3, however, the dashed curve in this case shows the effect of allowing neutron evaporation during a saddle-to-scission time of 30 X 10Wzl s. Emission during fragment acceleration is included. Neglecting it gives the dot-dashed curve shown in the 251Es panel. For the other cases the reduction is smaller.

and modification to the statistical model expression for I’,/r, must be made. Describing fission by the diffusion model of Kramers 43), Grange and Weidenmtillerl) found that the full fission decay rate (or width) takes a finite time (r) to become established. For simplicity, in this work, the variation of rr with time is taken to be of the form

G(f) =M4[1-

ed-Vdl.

(9)

Here r,(co) is taken to be transition-state (ALERTl) value of r,. 7d is related to the r of ref. ‘) by the approximate relation rd = $7 ( rd is shorter since it corresponds to I’,(t) = 0.63I’r(co), whilst r corresponds to I’,(r) = 0.9rr(co)). The relationship between the delay time, nuclear friction and Tr(cc) will be discussed later.

564

D.J. Hide et ai. / Neutron ~ulti~ii~ities

The application of eq. (9) was treated as realistically as possible within the statistical model code ALERTL At a given E,(CN) and J, the neutron lifetime was calculated (from the internal value of r,), and a suppression factor averaged over this lifetime was applied to the competing fission process. For every angular moments, the mean neutron lifetime for each emission was evaluated, and added to the elapsed time, so that at least on average, the time evolution of the decaying nucleus was taken into account. Thus the fission suppression factor was reduced for each emitted neutron, both due to the increasing neutron lifetimes as the excitation energy became less, and due to the increasing time after compound-nucleus formation. The total m~tiplicity was calculated for various values of rd (including emission during fragment acceleration but not during the saddle-to-scission transition). A reasonable fit to all data sets was obtained with rd = 70 X 1O--21s, shown in fig. 7.

5

_

_ 28Si+ ls4Er+‘*‘pb

‘80+‘MSm+‘?‘b *ONe+‘soNd~t70yb - i -

4-

_x

= 0.72

x=060

I 5 _

I

I

I

2*Si+lmEr-‘gBPb

4-x

5-

I

=0.70

‘80 + ‘920s _ 2t0Po

100 120 I,,,,,,,,,

140

_ 3oSi +lmEr +mPb

_ Is F+‘*ITc-J _. 200Pb

- x = 0.70

- x = 0.70

‘60 + ‘*7&

_ 213~~

'9F +232T,, _25!& -X=0.83

-x

= O.74

tt

+__L :'__-+---

II' __r. -+<* ,* r_C.C.*.*. Cr.C.

li

I,,

60

Fig.

.'

*'

I

80

/

f

)

8

,

,

,

60

,

*

t

80

7. As fig. 6, for a fission delay time of 70 X l.O-21 s, and no saddle-to-scission including emission during fragment acceleration.

emission,

but

D.J. Hinde et al. / Neutron multiplicities

565

The change in I’,/& necessary to make this calculation means that the overall fission probability, to be compared with that measured experimentally, may be rather different. The fission and evaporation-residue cross sections are of course calculated by ALERTl. In fig. 5, the original calculations are denoted by full lines. The dashed lines show the effect of a delay time of 70 x 1O-21 s. Although in all cases the fission cross section is reduced, and the residue cross section increased, the change is rather small. Only for “‘Es and 16sYb is a substantial effect seen. In the former case, the charged-particle emission cross sections are increased by a factor 2.5, whilst in the latter case, the fission cross section is reduced by - 40%. The reason for the small change in cross sections for the intermediate-mass cases is again related to multi-chance fission. Fig. 8 shows the distribution of fission with the number of neutrons emitted, for 123 MeV 19F + ‘*‘Ta (E,(CN) = 80.7 MeV). The full points show that first-chance fission is dominant in the statistical model picture. The introduction of a delay time suppresses this fission (hollow points in fig. 8). However, since r,(cc)/& does not change rapidly with excitation energy for total (measured) fission probabilities of - 0.5 [ref. 12)], the increased population for later chances simply results in an increased fission cross section after two or three neutrons have been emitted, giving only a slight reduction in the total fission cross section. Further study of fission excitation functions for light compound nuclei may

123 MeV “F + ‘*‘Ta 200

t

0 \ 0

DELAY TIME OF 70 x 1o-2’s

Ofis = 730

NEUTRONS

EMITTED

BEFORE

mb

i

SADDLE

Fig. 8. ALERT1 calculation, for zooPb at EX(CN) = 80.7 MeV, showing the dependence of the cross section for fission on the number of neutrons emitted, up to the commitment of the system to fission (assumed to be the saddle point in the statistical model). The full points show the statistical model prediction, whilst delaying the onset of fission by 70 x 10W2’ s gives the hollow points. There is only a small reduction in the total cross section from 779 to 738 mb. The lines guide the eye.

566

D.J. Hinde et al. / Neutron multiplicities

help to define the delay time TV.There are problems, however, in calculating such fission cross sections reliably 27,33).As in the saddle-to-scission analysis, choice of a different value of a, results in a different rd. For a, = &4, rd = 140 X 1O-21 s, whilst for a, = kA, 7d= 40 X 10e21 s gives reasonable fits to the multiplicity data. It is perhaps surprising that the required delay time is so much longer than the saddle-to-scission time. Three reasons can be identified. Firstly, for a given E,, the neutron lifetime calculated using the approximate expression is typically 0.7 of that calculated within the modified ALERTl. This factor has been retained since the surface areas of the highly deformed saddle and scission shapes are greater than that of the nucleus before reaching the saddle, so a shorter neutron lifetime might be expected. Secondly, fission is not suppressed totally, but only by a factor 0.37 up to rd, thus for r/I’, = 1, the neutron multiplicity is only - 0.7 of that for a saddle-toscission transition of the same time. Finally, at high fission probability, r/I’, increases as neutrons are emitted (see fig. 4), tending to counteract the fission suppression of later fission-neutron competition. A detailed comparison of the saddle-to-scission and pre-saddle emission is given for the 19F + lslTa reaction in fig. 9. This shows that the above comments are borne out, since the pre-saddle emission gives a higher multiplicity at low fission probability (E,(CN)), but is lower than for saddle-to-scission emission at high fission probability. Overall, the agreement with the experimental data is slightly worse than for saddle-to-scission emission (compare figs. 6 and 7). This may not be significant in view of the simplicity of the model.

‘F+““Ta -mPb WARDET AL THIS

WORK

STATISTICAL ---I -

SSC

~OXIO-~‘S.

---.-

DELAY

TIME

-

BOTH TlMES

SSC AND DELAY PO x 1021 5

I

50

MODEL

TIME

70x10*’

5. -

I

60

E,

70 (c.N)

80

90

(MeV)

Fig. 9. Pre-fission neutron excitation function for r9F +lslTa +‘OOPb. The results for a fission delay time of 70 X 10m21 s, a saddle-to-scission time of 30 x 10e2’ s, and a combination of 20 X 10e2r s delay time and 20 x 1O-21 s saddle-to-scission time are shown. All fit the experimental data better than the statistical model calculation, even though the times are not adjusted to fit the 2ccPb data alone, but rather the whole data set.

D-J. Hinde et al. / Neutron multiplicities

567

5. Discussion Statistical model (tr~sition state) calculations of pre-fission neutron multiplicities do not reproduce experimental excitation functions. With increasing excitation energy and fissility the disagreement becomes larger. This difference can be removed either by introducing a delayed onset of fission, with a delay time rd of 70 X lO-*l s, or by allowing neutron emission during the saddle-to-scission transition, with a transit time of 30 X 10e2i s. The derivation of these times relies upon the pPFeexcitation functions calc~ated using the statistical model code ALERTl. It is clear that such a sophisticated model must be used for quantitative calculations, and that by fitting measured fission probabilities the calculated pre-fission multiplicities are more likely to be correct. Nevertheless, if it could be shown that vital physical effects were omitted from the statistical model description, it is conceivable that all the expe~ment~ fission probability and neutron multiplicity data could be explained without calling on dynamical effects. This does not, however, seem likely. In this work, scaled liquid-drop-model fission barriers, in conjunction with diffuse fusion angular momentum distributions (see subsect. 4.3) have been used to fit the fission and evaporation-residue excitation functions, and then predict the upre excitation functions. The use of finite-range ~quid-drop-mode123} barriers may well be more appropriate; however, it has been shown that the effect on pi,,, of varying the barrier height is negligible when E;,(J) - B,. This is the angular momentum region in which vPre measurements have been made, since for E,(J) Z+ B,, the fission yield is very small. Varying the angular momentum dist~bution, and more ~portantly a/a,, may have a significant effect on the extracted fission timescales. These questions, together with the effect of using finite-range-model fission barriers, will be discussed in a forthcoming paper. As has been shown, the fission timescale is most sensitive to the parameter a,. It may be possible to determine a, for a hot rotating nucleus by a precise measurement of the energy spectrum of evaporated neutrons, extending to higher energies than in this work. However, until this is done, the fission timescale will not be well defined by neutron multiplicity measurements. With the above uncertainties, only the bralad qualitative agreement of the present calculations with the experimental data should be considered. Thus it is not possible to assess the relative ~port~ce of pre-saddle and saddle-to-scission emission, despite the slightly better fit when considering only the latter process. It would be surprising if both were not important, and one might expect the relative importance of each to change with fissility, due to changes in barrier heights and saddle-point deformations. To gain further insight, existing models of nuclear “viscosity” will be briefly considered. An “empirical” formula for the fission delay time 7 (7 = $r,} is4) T= 1.4,F11n(10EJT)

+ 1.4(/3/T)

x (10m4’ MeV-s*)

s.

(10)

568

D.J. Hinde et al. / Neutron multiplicities

For critical a barrier which

damping

of a typical nucleus4),

E, = 8 MeV and temperature is very

experimental

small

compared

the friction

coefficient

fi = 5 x 1021. For

T = 1.5 MeV, eq. (10) gives rd y 4 X 10P2r s

to the value

of 70 x 10P21 s required

data with this process alone. If, somewhat

arbitrarily,

critical

to fit the damping

is also required for the saddle-to-scission transition, the calculations of ref. 6, for 16*Yb show that t,, 2: 4 x 10P21 s. Applying both the fission delay time and the saddle-to-scission time (each = 4 x 10P21 s) to the calculation results in multiplicities much below the experimental data. However, extrapolation to five times critical damping (giving times of - 20 X 10e21 s) gives a reasonable fit to the data, as is shown in fig. 9 for 19F + ‘*lTa. (It should be noted‘ that for these times, 5 of the increase in vPre occurs during the saddle-to-scission transition). Thus according to these results the motion is heavily overdamped, in agreement with the preliminary conclusion of ref. 6). This is so even for a, = &A, which gives the shortest times. More rigorous calculations would have to take into account the potential energy surface for each nucleus as a function of angular momentum, and any variation of “viscosity” with temperature and deformation. Ideally pre- and post-saddle shape evolution would not be separated; thus the possibility of passing back over the saddle point 42) would be included. This latter process should allow the emission of more neutrons; however, it is unlikely to change the conclusion of this work, namely that at moderate excitation energies, where pairing correlations are broken, motion in the fission direction is overdamped. This is the situation predicted by the theory of one-body dissipation45), which is expected to be valid for the systems studied here at least in the saddle-to-scission regime, if not in the pre-saddle regime where the greater symmetry may reduce the one-body energy dissipation45). For pure one-body dissipation, a saddle-to-scission time of 18 X 10P21 s is calculated by Schiitte et al. 5, for 238U at zero angular momentum. However, for less fissile systems, shorter times are expected, due mainly to the smaller change in deformation between the saddle and scission configurations (the angular momentum introduced in the heavy-ion fusion reactions reduces but does not eliminate the expected change in saddle-to-scission times). The experimental data show that overall, a constant time is sufficient to explain the data. Thus a longer delay time may be required for lighter nuclei, with a dependence on barrier height

stronger

than

that

suggested

in eq. (10). These,

however,

are speculative

problems at the moment. Although at present experimental results do not favour either pre- or post-saddle emission, our simple calculation together with the expectations for one-body dissipation suggest that post-saddle emission is the major contributor to the large multiplicities observed. If it can be concluded that both pre-saddle and saddle-to-scission motion are heavily overdamped, several questions arise which must be addressed. For very high pre-saddle viscosity, Kramer’s stationary solution gives a lower fission width4) than the transition-state formula, with obvious implications for the extraction of fission

D.J. Hinde et al. / Neutron multiplicities

barriers

from

collective barrier46)

heavy-ion

enhancement and

saddle-to-scission

data [further

uncertainty

of level densities,

the diffuseness

of fusion

exists due to problems

temperature angular

times would result in increased

569

dependence

momentum

such as

of the fission

distributions].

charged-particle

emission,

Long which

may be substantially enhanced by the highly deformed nuclear shape47). Finally, it is conventionally assumed that nuclei fission after passing through the equilibrium deformation. However, for high viscosity, with the resultant large fluctuations, fission may bypass this configuration, in analogy with quasi-fission, even at relatively low angular momentum. Support for the long fission timescales required to fit the neutron multiplicity data comes from the recent work of Take et al. 48), who conclude that for quasi-fission, corresponds to a full mass equilibration takes > 10 x lo-*I s. Since fusion-fission longer path in deformation space [see fig. 2 of ref.48)], this suggests that fusion-fission should

take substantially

longer than 10 X lO-*l

s.

6. Conclusion A large body of pre- and post-fission neutron multiplicity data has been presented. A simple comparison of the total multiplicity with that expected from energy balance has been made. This shows that most of the experimental data are consistent with a simple picture of complete fusion followed by neutron evaporation. However, at high bombarding energy the comparison suggests that considerable energy is removed by charged particles. In all cases the pre-fission multiplicity data rise monotonically with compoundnucleus excitation energy (bombarding energy). In comparison, statistical model calculations (using the computer code ALERTl) based on the static transition-state picture of fission show a rise followed by a fall as the bombarding energy increases. Including in ALERT1 the effect of delaying the onset of fission and/or allowing neutron evaporation during the transition from saddle-to-scission gives reasonable fits to all the experimental

data. When both are allowed, a viscosity

corresponding

to

five times critical damping is required to fit the data, giving 7d = t,,, = 20 x 10M21 s for a, = $A. Shorter (longer) times are necessary if a, = &A (;A), but according to this simple model, motion is overdamped for all reasonable values of a,. The effect of neutron evaporation during fragment acceleration is of importance only for the most fissile nuclei (in this case, 251Es). It would be valuable if a, (or neutron lifetimes) at high excitation energy could be measured, to reduce the uncertainty in the extracted fission timescales. Further development of the theoretical model, treating pre- and post-saddle behaviour in a complete picture of the fission process is essential. However, it is stressed that the model must be applied within the framework of a sophisticated computer simulation of the fusion-fission process, and an extensive range of data should be fitted (for example, fission probabilities).

570

D.J. Hinde et al. / Neutron multiplicities

The data presented here appear to require a very significant excess neutron emission before scission for all nuclei studied, with low to very high fissility. Together

with further

may allow the dynamics

data, and in conjunction

with theoretical

developments,

they

of the fission process to be elucidated.

Appendix For complete-fusion E,(CN) is given by

reactions

the excitation

E&N)

energy

in the compound

Q&N) >

= EC.,.+

nucleus

(11)

where E,.,,

is the centre-of-mass bombarding energy, and Q(CN) is the Q-value for complete fusion, defined in terms of the mass defects of the projectile, target and compound nucleus: Q(CN)=M,+MM,-Mc,.

(12)

Assuming decay only by neutron and y-ray emission, the number of neutrons emitted (x) is related to E,(CN) by the well-known energy balance equation: E,(CN)

= E,(PN)

-I- f

(B; + E;),

(13)

i=l

where B6 and Ei are the binding and kinetic energy for neutron i, and E,(PN) is the energy removed by y-ray emission from the product nucleus after neutron emission. For different reactions, at the same E,(CN), x will not necessarily be the same, due mainly to the different values of BA. There is thus no common relationship between E,(CN) and x for different reactions. However, for a known x, the available

decay

energy

than the compound

relative

nucleus,

to the ground

E,(PN) where nucleus

Q(PN)

is found

state of the product

nucleus,

rather

is given by

by replacing

by that of the product E,(PN)

= E,.,. +

Q@‘N>7

(14

in eq. (12) the mass defect of the compound

nucleus.

Then

= E,(PN)

t 5 (8.07 + E;), i=l

(15)

where now, except for minor variations in E,(PN) and EA, x can be directly related to E,(PN), independent of the reaction. For a system that undergoes fission, the product nucleus of eqs. (14) and (15) is replaced by the fission fragments, thus the total available decay energy can be defined in a similar manner to eq. (14):

Ex(f)=E,.,.+Q(f)-E,, where

Q(f) = M, + M, - (Mrl + M,J

(+

and Mf, being the fission product

06) mass

D.J. Hinde et al. / Neutron multiplicities

E, is the total kinetic

defects)

and

analogy

with eq. (15), the energy balance

energy

(corrected

equation

E,(f)=E,(f)+

t

571

for neutron

emission).

In

becomes

@.07+E:),

(17)

i=l

again assuming that all available decay energy is removed by neutron and y-ray emission. Equating vtot with x, eq. (17) can be compared with the experimental data to check its consistency. To evaluate E,(f) for a given bombarding energy and system, Q(f) and E, must be defined. The experimental data include all mass splits, so an effective average was obtained by using the liquid-drop model mass defects for symmetric fragmentation. E, was evaluated from the empirical formula of Viola49):

E, = 0.107122/A1/3 The errors introduced by using estimated to be - _t 5 MeV.

these

+ 22.2 MeV.

approximations

(I@ in determining

E,(f)

are

References 1) 2) 3) 4) 5)

6) 7) 8) 9) 10)

11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

P. Grange and H.A. Weidenmtiller, Phys. Lett. %B (1980) 26 P. Grange, Li Jun-Qing and H. A. Weidenmliller, Phys. Rev. C27 (1983) 2063 S. Hassani and P. Grange, Phys. Lett. 137B (1984) 281 H.A. Weidenmiiller and Zhang Jing-Shang, Phys. Rev. C29 (1984) 879 J.W. Negele, SE. Koonin, P. MBller, J.R. Nix and A.J. Sierk, Phys. Rev. Cl7 (1978) 1098; K.T.R. Davies, A.J. Sierk and J.R. Nix, Phys. Rev. Cl3 (1976) 2385; G. Schiitte, P. Mliller, J.R. Nix and A.J. Sierk, Z. Phys. A297 (1980) 289 J.R. Nix, A.J. Sierk, H. Hofmann, F. Schemer and D. Vautherin, Nucl. Phys. A424 (1984) 239 E. Cheifetz and Z. Fraenkel, Phys. Rev. Lett. 21 (1968) 36 M. Blann and T. A. Komoto, Lawrence Livermore National Laboratory report UCID 19390 A. Gavron, Phys. Rev. C21 (1980) 230 A. Gavron, J.R. Beene, B. Cheynis, R.L. Ferguson, F.E. Obenshain, F. Plasil, G.R. Young, G.A. Petitt, M. Jtiskel’tinen, D.G. Sarantites and CF. Maguire, Phys. Rev. Lett. 47 (1981) 1255 [erratum: 48 (1982) 8351 E. Holub, D. Hilscher, G. Ingold, U. Jahnke, H. Orf and H. Rossner, Phys. Rev. C28 (1983) 252 D. Ward, R.J. Charity, D.J. Hinde, J.R. Leigh and J.O. Newton, Nucl. Phys. A403 (1983) 189 D.J. Hinde, R.J. Charity, G.S. Foote, J.R. Leigh, J.O. Newton, S. Ogaza and A. Chattejee, Phys. Rev. Lett 52 (1984) 986 [erratum: 52 (1984) 22751 J.O. Newton et cd.,to be published C.J. Bishop, I. Halpem, R.W. Shaw Jr. and R. Vandenbosch, Nucl. Phys. A198 (1972) 161 L.G. Moretto and R.P. Schmitt, Phys. Rev. C21 (1980) 204 R.P. Schmitt and A.J. Pacheco, Nucl. Phys. A379 (1982) 313 J.R. Leigh, W.R. Phillips, J.O. Newton, G.S. Foote, D.J. Hinde and G.D. Dracoulis, Phys. Lett. 159B (1985) 9 F. Pleasonton, R.L. Ferguson and H.W. Schmitt, Phys. Rev. C6 (1972) 1023 J.R. Leigh, private communication See e.g. L.C. Vaz, D. Logan, E. Duek, J.M. Alexander, M.F. Rivet, M.S. Zisman, M. Kaplan and J.W. Ball, Z. Phys. A315 (1984) 169 L.G. Moretto, Nucl. Phys. A180 (1972) 337 H.J. Krappe, J.R. Nix and A.J. Sierk, Phys. Rev. C20 (1979) 992; A.J. Sierk. private communication

512

24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49)

D.J. Hinde et al. / Neutron multiplicities D.J. Hinde, J.R. Leigh, J.O. Newton, W. Galster and S.H. Sie, Nucl. Phys. A385 (1982) 109 B. Sikora, W. Scobel, M. Beckerman, J. Bisplinghoff and M. Blann, Phys. Rev. C25 (1982) 1446 M. Blann and T.T. Komoto, Phys. Rev. C26 (1982) 472 R.J. Charity et al., to be published D.J. Hinde, J.O. Newton, J.R. Leigh and R.J. Charity, Nucl. Phys. A398 (1983) 308 M.L. Halbert, R.A. Dayras, R.L. Ferguson, F. Plasil and D.G. Sarantites, Phys. Rev. Cl7 (1978) 155 A.M. Zebelman, L. Kowalski, J. Miller, K. Beg., Y. Eyal, G. Jaffe, A. Kandil and D. Logan, Phys. Rev. Cl0 (1974) 200 R. Bass, Phys. Rev. Lett. 39 (1977) 265 SE. Vigdor, H.J. Karwowski, W.W. Jacobs, S. Kailas, P.P. Singh, F. Soga and T.G. Throwe, Phys. Rev. C26 (1982) 1035 R.J. Charity, Ph.D. thesis, Australian National University (1984). unpublished S. Cohen, F. Plasil and W.J. Swiatecki, Ann. of Phys. 82 (1974) 557. V.P. Eismont, Atom. Energi 19 (1965) 113 [transl.: Sov. J. At. Energy 19 (1965) lOOO] K. Skarsv%g, Phys. Scripta 7 (1973) 160 H.R. Bowman, S.G. Thompson, J.C.D. Milton and W.J. Swiatecki, Phys. Rev. 126 (1962) 2120 K. Skarsvgg and K. Bergheim, N&l. Phys. 45 (1963) 72 R.W. Fuller, Phys. Rev. 126 (l-962) 684 V.M. Piksaikin, P.P. D’yachenko and L.S. Kutsaeva, Yad. Fiz. 25 (1977) 723 [transl.: Sov. J. Nucl. Phys. 25 (1977) 3851 Yu. S. Zamyatnin, D.K. Ryazanov, B.G. Basova, A.D. Rabinovich and V.A. Korostylev, Yad. Fiz. 29 (1979) 595 [transl.: Sov. J. Nucl. Phys. 29 (1979) 3051 L.G. Moretto and G. Guarino, Nuovo Cim. 79 (1984) 149 H.A. Kramers, Physica 7 (1940) 284 J.R. Leigh, D.J. Hinde, J.O. Newton, W. Galster and S.H. Sie, Phys. Rev. Lett. 48 (1982) 527 J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk and W.J. Swiatecki, Ann. of Phys. 113 (1978) 330 M. Pi, X. Viiias and M. Barranco, Phys. Rev. C26 (1982) 733 M. Blann and T.T. Komoto, Phys. Rev. C24 (1981) 426 J. TGke, R. Bock, G.X. Dai, A. Gobbi, S. Gralla, K.D. Hildenbrand, J. Kuzminski, W.F.J. Muller, A. Olmi, H. Stelzer, B.B. Back and S. Bjornholm, Nucl. Phys. A440 (1985) 327 V. E. Viola Jr., Nucl. Data Tables Al (1966) 391