ANGULAR DISTRIBUTIONS IN QUASI-FISSION ... - Science Direct

Nuclear Physics ONorth-Holland

A452 (1986) 351-380 Publishing Company

ANGULAR

DISTRIBUTIONS

IN QUASI-FISSION

K. LUTZENKIRCHENa Institut

REACTIONS

and J.V. KRATZ

ftirKernchemie, Universitiir Maim, D-6500 Maim, West Germany G. WIRTH,

W. BRUCHLE

Gesellschajt ftir Schwerionenjorschung

R. LUCAS,

(GSI), D-61 00 Darmstadt,

J. POITOU

CEN Saclay, DPhN/

and K. SUMMERER West Germany

and C. GREGOIREb

MF, 91191 Gij-sur- Yvette Cedex, France

Received

1 October

1985

Abstract: Angular distributions for fission-like fragments were measured in the systems “Ti, 56Fe + “*Pb by applying an off-line K X-ray activation technique. The distributions d*o/d0dZ exhibit forward-backward asymmetries that are strongly Z-dependent. They result from a process (quasifission) which yields nearly symmetric masses in times comparable to the rotational period of the composite system. A method for obtaining the variance of the tilting angular momentum, Kj, from these skewed, differential angular distributions is described. The results indicate that the tilting mode is not fully excited in quasi-fission reactions. The results are compared to the sum of the variances of all statistical spin components, measured via y-multiplicities, Integration of the angular distributions d20/d6’dZ over all values of Z yields the integral angular distributions do/d0 and da/dO symmetric around 90”. The associated unusually large anisotropies do not at all provide an adequate basis for tests or modifications of the transition-state theory. A deconvolution of d”a/d0 d Z is performed with gaussian distributions depending on rotational angles Ati extending over a range of up to 540”. From the mean values (A@) a time scale for the evolution of K, is calculated.

E

NUCLEAR REACTIONS *‘*Pb(‘“Ti, X), E = 5.0,5.5 MeV/nucleon, 20XPb(56Fe,X), E = 5.7, 6.1, 6.8, 8.3 MeV/nucleon; measured fragment K X-rays, relative o(fragment 8, Z); deduced tilting variance, tilting mode degree of relaxation, y-multiplicity, comparison, timescale.

1. Introduction Binary reaction products from the interaction of ‘!$j Pb ions with targets of f6,Mg through :zFe and $Ni have previously been studied ‘) with a large position-sensitive ring counter. When analyzed on the basis of fusion models, the cross sections for the a Also at GSI, Darmstadt. This work submitted). b Presently at GANIL, Caen, France.

forms

part

351

of a doctoral

thesis,

Universitat

Maim

(to be

352

K. Liitzenkirchen

et al. /Angular

distributions

fission-like fragments demonstrated that deformations induced at contact influence fusion of the heaviest systems in a significant and characteristic way. In particular, for the heaviest systems at energies well above the barrier, it was observed that there is a way from the entrance channel to a near-symmetric exit channel that bypasses the slow process of compound-nucleus formation’). These reactions, resulting in nearly symmetric mass yields not being evenly distributed in angle, were shown to take place in a time of about one-half revolution of the dinuclear system’). In a schematic model of nuclear coalescence and reseparation2-4), such reactions (termed “quasi-fission”) are prediced for trajectories that encompass the “conditional saddle” for capture in the entrance channel but do not encompass the “true saddle” point for fusion. In the above experiments y-ray multiplicities (M,,) were measured in coincidence with capture fission events. Capture in the sense used here refers to both fusionfission and quasi-fission. The mean fragment spin (j) was calculated from (M,) by assuming that the detected y-rays mainly came from stretched E2 transitions, and by subtracting a constant fraction of statistical y-rays:

The fragment spin (j) was taken as the vector sum of two angular momenta, the aligned spin (I,), and the tilting angular momentum (K), thus neglecting other statistical modes:

(j2)

=(l$+ (K2).

(2)

(K) corresponds to rotations of the nucleus around its symmet~ axis. ( K2) was approximated by (Kl), the variance of a gaussian K-distribution. For fissioning compound nuclei, Ki is related to the effective moment of inertia at the saddle point via K,2 = TJcrt, T being the nuclear temperature. It was reported’) that, surprisingly, the resulting values of JO/.Jcff were rather independent of 1 for a given system and, in a compound-nucleus picture, were more in agreement with the shape of the non-rotating saddle than with the actually expected rotating saddle shapes. The crucial point with this result is whether it is permissible to interpret the y-ray multiplicites this way (eq. (2)). If yes, there should be a unique connection between the y-ray multiplicities (MY) and the angular distributions W(O), and both measurements should lead to the same values of Ki and J,/J,rr. The statistical accuracy of the angular dist~butions and li~tations in angular acceptancel) made it unrealistic in the past to perform such a consistency test. However, angular distributions for a number of reactions5-‘*) leading to similar compound nuclei in more asymmetric entrance channels seemed to give some support to the validity of eqs. (1) and (2), and to the notion that larger saddle defo~ations than expected were involved. This was reflected in the experiments observation of unusually large anisotropies 5-12) as compared to predictions based on the saddle-point transition-state theory [TST13)] and on rotating liquid-drop-

K. Liitzenkirchen

model heavy

353

et ul. / Angulur distributions

[RLDM14)] saddle-point shapes. systems is being questioned15)

The validity of these expectations for very and modifications of the transition-state

theory16-18) are being vividly discussed i9- 21). Alternatively, the apparent deviations from the RLDM

expectations

have been

associated’~6*8,‘0,“,22*23) with the inability of heavy systems to form genuine compound systems inside the fission saddle point. Although fission fragments from these quasi-fission reactions tend to have kinetic energies and masses typical for fission of compound nuclei, it was suggested that the shape of the intermediate reaction complex, at which the K-distribution is determined, depends on the entrance channel. Indeed, there is growing evidence that quasi-fission reactions make their appearance whenever complete fusion is suppressed by either dynamical entrancechannel limitations’-4) or by fission barriers being as low as the nuclear temperature 1o.23) or even vanishing 24). The purpose of the present communication is to present experimental data that make it possible to study the influence of non-equilibrium fission (quasi-fission) on the anisotropies of fission-fragment angular distributions in a unique way. This influence has been made transparent by measuring differential angular distributions d2u/dBdZ (rather than integral distributions da/d8 or da/do) in reactions of ‘OTi + 208Pb and 56Fe + 208Pb at several bombarding energies. Even though compound-nucleus formation has been observed 25) or can be assumed to occur in these reactions, it is associated with a significant hindrance and vanishing cross sections, i.e. factors of = 30 smaller than the fission cross section measured in the case of “Ti + 208Pb [refs. 1,25,26)].For 56Fe + 208Pb, these hindrance factors near the barrier should be even larger. The present results fusion component: All angular distributions

are indeed lacking clear signatures of a in the centre-of-mass system except the

ones for Zsym = i(Z, + Z,) are asymmetric around 0 = 90” with forward-peaking for charges Z -C Zsym, and backward-peaking for their heavy complements. A requirement for the saddle-point TST to be applicable to fission-fragment angular distributions, i.e. that the fused system rotates many times as it slowly decays over the barrier 13), is clearly not met. It is also shown that integration over all values of Z of the skewed differential distributions d2a/dD dZ yields integral angular distributions du/dS2 symmetric around 6’= 90” with extremely large anisotropies. These anisotropies reflect the non-equilibrium features of the fast massequilibration process in quasi-fission reactions. They do not at all provide an adequate basis for tests and modifications of the transition-state theory. On the contrary, we believe that the features displayed by the angular distributions of this work are entirely characteristic of quasi-fission reactions. It is then evident that new ways for interpreting skewed angular distributions of the quasi-fission type have to be found. The present study focusses on an analysis of the differential cross sections at angles close to the beam direction in terms of a dynamical theory 27,28) for the accumulation of tilting angular momentum in direct nuclear reactions. It is important to note that, in this analysis, the extracted quantity

354 K,

K. Liitzenkirchenet al. / Angulurdistributions represents

the dynamically

accumulated

excitation

of the tilting mode which has

not necessarily achieved some sort of steady state in relation to an effective decision-point configuration. As reported in a preliminary communicationz9), we find

that

the relaxation

of tilting

angular

momentum

in quasi-fission

reactions

is

incomplete. This paper also considers the correlation of the tilting angular momentum with the previously measured’) y-ray multiplicities. A systematic analysis of the angular distributions in terms of Z-dependent reaction times as obtained from fits to the angular distributions involving an unfolding procedure for rotations of the intermediate complex through 540” provides a timescale for the evolution of the tilting mode towards full relaxation.

2. Experimental

procedure and data analysis

The experimental method is based on the measurement of K X-rays which are emitted after a reaction product has decayed via electron capture or internal conversion. This allows for a very high product Z-resolution, and offers the unique possibility to measure angular distributions I%‘(e) very close to the beam direction, i.e. 0” and 180”. The detection of fission-like reaction products close to the beam direction is of crucial importance, if one wants to study the influence of the tilting fluctuations on W( f3). The detection range of scattering angles is only limited by the diameter of the beam, and by the requirement that the beam must not hit the catcher foils mounted on the inside of the detection chamber. The experimentally accessible range of scattering angles in the laboratory system was 5” < ela,, I 178.2” for the “Ti-induced reactions, and 3.5” I 8,, I 177.6” for the 56Fe-induced reactions. A high Z-resolution is needed in order to unravel Z-dependent changes in the shape of W(0).

Such changes

may be anticipated

if fission-like

products

are formed

via a

non-compound reaction mechanism. The measurements were performed at the UNILAC accelerator at GSI. Bombardments were carried out with ::Ti beams at Elab = 250 and 275 MeV, and :iFe beams at E,, = 321.4, 343.3, 379.1 and 464.9 MeV. The energies were controlled with high accuracy by a time-of-flight method monitoring the microbunches of the UNILAC beam, and were corrected to represent the energies in the middle of the target. [The energies represent as close as possible those of ref. ‘), so that we can use the same capture cross sections and associated l-distributions.] The *tiPb metal targets had a thickness of 525 pg/cm* with 30 pg/cm* carbon coating on both sides to improve radiative cooling. They were mounted on a wheel which was rotated synchronous to the UNILAC pulsing. Beam intensities of up to 3 x 1011 particles per second were used for typically 8 h of experiment. The target wheel was rotated through the centre of a cylindrical chamber of 12 cm length and diameter. The chamber was covered inside by mylar and aluminum foils of sufficient thickness to stop the reaction products emerging from the target. The beam charge was collected in a Faraday cup

K. Liitzenkirchen

et al. / Angular distributions

355

placed behind the cylindrical chamber. At a few MeV/u beam energies, targets may be burnt by the beam. In order to ensure that this does not happen, the elastically scattered beam ions were measured by a silicon surface-barrier detector placed at 25”. After the irradiations the catcher foils were removed, and cut into pieces representing 5” slices of laboratory angles. For the simultaneous off-line measurements of K X-ray intensities, 14/15 catcher foil segments representing 14/15 laboratory angles were selected, and folded in a reproducible way to allow for subsequent corrections of transmission losses (typically 3%) during passage of the X-rays through a given number of catcher-foil layers. The folded foils were mounted in front of 14/15 absolutely calibrated Ge low-energy photon spectrometers. The simultaneous measurements lasted typically for 8 h. The K X-ray spectra were analyzed with an interactive version of the code TAXI 30). There are at least four, and for heavier elements up to eight, natural K X-ray lines. Lines of y-rays with energies below 200 keV can be included in the fit. For each element a response is generated which consists of the energies and relative intensities of the natural K X-rays, calibrated line shapes, and, possibly, the y-rays. The numerical results, namely the intensities of each element and the corresponding uncertainties, are generated by a least-squares analysis of the data. The decay curves were not decomposed into contributions from individual isotopes, but integrated over the full counting time. The results are therefore differential only with respect to the charge 2. The integration involves the assumption that the isotopic composition of the Z-yields is independent of the scattering angle. This assumption is well justified for compound-nucleus fission, and was also shown to be valid for quasi-fission reactions3*). The isotopic composition of a fragment Z determines the time evolution of its K X-ray intensity. Thus, an additional proof of the assumption consists in showing that the time evolution of the K X-ray intensity is independent of the scattering angle. Two fragments, the charge-symmetric Z = 52 and the more target-like Z = 72 in the reaction “Ti + *“Pb at E/A = 5.5 MeV, were chosen to perform such an analysis. Fig. 1 depicts the relative K X-ray intensity of Z = 52, 72 as a function of time for the angular intervals 10-15’ and 160-165” in the laboratory

system. The decrease of intensity

as a function

of time is the same for

the two intervals with both Z-values. The result clearly shows that the isotopic composition of fragments Z formed in a quasi-fission reaction is independent of the scattering angle. As a result we obtained K X-ray intensities of individual elements integrated over all contributing masses; they correspond to relative cross sections d*a/df3,,dZ in the laboratory system. The measured elements are not necessarily the primary ones produced in the reactions. They may also originate from neighbouring elements which decay into the measured ones via P-decay and electron capture (EC). Thus one should keep in mind that the angular distributions shown later correspond to a finite Z-range (AZ = l-2 [ref. 32)]) usually above the measured Z. If the products

K. Liitzenkirchen

356

I,

I

0

2

*

8

et al. /Angular

x

4

Time after

a

I

6

1

8

distributions

*!

8

10

8

12

I

I1 16

Irradiation / hours

intervals 10” 5 Blab 2 15’ and Fig. 1. Relative K X-ray intensities of 2 = 52, 72 in two laboratory 160’ i eiah I 165’ versus the time after the irradiation. The intensities are normalized to values measured 1 h after the irradiation.

decay by EC or pi, the physical Z-distribution is necessarily at larger Z-values than the measured one. The laboratory distributions were transformed into the centre-of-mass system by assu~ng fully relaxed average kinetic energies in the exit channe133) and equilibrated N/Z ratios. In most cases the isotopes produced after the deexcitation are on the neutron-deficient side of the line of P-stability 31). Decay via EC (and p’) therefore plays an important role. Since most EC decays are followed by X-rays, the X-ray technique is well suited to detect the product nuclei, and to take a representative sample thereof. The widths of the kinetic energy and isotope distributions were taken to be I’rKE = 50 MeV [ref. ‘)I and I” = 7 mass units 31). These widths are incorporated into the error bars of the tr~sformed ~st~butions. So are the statistical uncertainties from the measured K X-ray spectra and the finite size of the beam spot on the target (diameter = 3 mm) which influences the angular resolution. For the 5oTi + 208Pb reaction as well as for the 56Fe + 208Pb reaction all angular dist~butions are clearly asymmetric around 90”, except for 2 2: Zsym. Fig. 2 gives some examples of distributions d2u/df?dZ in the reaction 50Ti + “‘Pb at E/A = 5.5 MeV. The solid lines are fits with eq. (4) which will be explained in detail below. The results for the lower bombarding energy and for 56Fe -t “*Pb are very similar to the ones shown in fig. 2 [ref. 32)]. The conspicuous asymmetry demonstrates that the scission time is comparable to the rotation time, and suggests that theories which assume the existence of an equilibrated compound nucleus may not be applicable. In order to demonstrate further that the TST is inappropriate to account for the results in fig. 2, we performed fits with the TST, treating the forward and backward hemispheres

K. Liitzenkirchen

et al. / Angular

357

distributions

50Ti + *o*pb 5.5 MeV/u

a p

5

b

1.3

5 3

L g

.A d

10

\ ; F

a.3

D "rr 0.0 0

0

20

an

SO

20

10

60

60

80 %m

/

100

120

140

160

180

1QO

120

140

160

180

w

1.2 1.0 OB Q,6

E: $

Q.6

g

0.2 DO

0

20

AR

30

0

20

60

60

80

80 @,

I

1QQ

126

I(0

13D

l8Q

100

120

110

160

180

Oeg

Fig. 2. Angt~lar distributions d%/d@dZ in the cm. system The solid lines are fits with eq. (41~The distributions are asymmetric around 90” except for Z = Zsym= 52. The curves are normalized to the data point closest to 90°. The dashed line in (c) indicates the behaviour of the cross-section function f(8). The arrows in (c) represent typical cross sections near O0 and 180’.

separately. (One common fit for the whole angular range from 0’ to 180” would not be sensible, due to the obvious asymmetry around 900.) The idea behind this approach is that products from different reaction mechanisms might be concentrated in different angular regions: A target-like fragment resulting from a fast mass equilibration (quash-fission) would be concentrated in the backward hemisphere, whereas products from eompo~d-nucl~s fission would be spread more isotropitally, thus being well separated from quasi-fission products only in the forward hemisphere. As an example, the dist~bution for 2 = 64 in the reaction 5oTi + 20RPb at E/A = 5.5 MeV is shown in fig. 3. The solid line represents a separate fit for both .he~spheres with the TST, Some support for the idea of products from compoundnucleus fission being dominant in the forward hemisphere comes from the resulting values of K,. For 8 2 90°, one finds K, = 27A. This comes close to K, = 33A which

K. Liitrenkirchen et al. / Angular distributions

358

1 ’

1.5

I

I ’

II

II

r * t r I r I

Z=64

+!I

_:

J:

1.0

/ TST

0.5

50Ti + zoepb 5.5 MeV/u

0.0

Fig.

0

20

40

60

80

100

120

140

160

180

3. d20/d0 dZ for Z = 64. The solid line is a separate fit for B < 90” and 19> 90” using transition-state theory’3). The curve is normalized to the data point closest to 90”.

the

is the value computed for a mean angular momentum of (I) = 46t2 from the RLDM saddle shapei4). The cross section for all fragments with Z > Zsym (“target-like”), however, is systematically overestimated around 60” and underestimated around 150”. For all projectile-like fragments these systematic deviations are also present but go in the opposite direction. We conclude that a separate fit for both hemispheres with the TST is not a suitable way to reflect the trend of the data. Further evidence for the non-equilibrium nature of the process producing fission-like fragments in the present reactions is presented in the discussion. A theoretical framework for the description of the skewed angular distributions W(0) was presented in ref. *‘). It is based on an analysis of IV(e) “near the beam direction” **), i.e. near 0” and 180”. It yields exclusive information about the tilting mode, i.e. the rotation of the dinucleus around its symmetry axis. One should note that the tilting angular momentum K in a direct reaction is not characterized by a Instead, it is a dynamically accumulated statistical equilibrium distribution. quantity

28) which is time dependent.

The measurement

of y-ray multiplicities,

on

the other hand, may also reflect a time dependence, but it probes all spin components of intrinsic rotational excitation that are possibly excited in a nuclear reaction. These are the forementioned tilting mode, which corresponds to a common rotation of both spheres of a dinucleus around the symmetry axis, the twisting mode, where the two spheres rotate oppositely around the dinuclear axis, the two degenerate wriggling modes, where the two spheres rotate in the same sense around an axis perpendicular to the dinuclear one, and the two bending modes, where the spheres turn oppositely around an axis perpendicular to the dinuclear axis. The ansatz in ref. 28) can be regarded as a generalization of the theory for fission angular distributions to direct reactions, such as quasi-fission, (eq. (3)). An analysis

K. Liitzenkirchen

et al. / Angular distributions

359

of I+‘(0) near the beam direction for a mean angular momentum I, is achieved by a factorization of the cross section d2a/dlI dZ:

& =M@ +f~-~)lCo(f~o/2~o)sin~) -+j+“(0)B2[CO((Z0/2K,)sin8)

- C,((E,/2K,)sinB)].

(3)

The cross section close to the beam is mainly determined by the degree of excitation of the tilting mode which is described by C,(x) = (277)r/*x eeX21n(x2), with 1, being the modified Bessel functions of the order n = 0,1, and x = I,sin 0/(2K,). The cross section in the central range (30’ I 0 I 150’), on the other hand, is do~nated by a function f(6) which describes the cross section with tilting ignored; f( - 8) is the corresponding cross section for rotations through 0”. The distribution of K-values is taken to be gaussian. The analysis of the data has actually been performed with an expression that differs slightly from eq. (3): The mean angular momentum I, is replaced by a summation over all partial waves, and a normalization to the total number of K-values is introduced. This was to ensure that, for f(S) = const, which corresponds to many rotations of a fissioning system, eq. (4) is equivalent to the result of the TST’3134):

X { LW

+f(-6)lCo((~/2~o)sin~)

-~f”(0)f32[C,((I/2K,)sin8)

- C,((1/2K,)sinB)]}.

(4)

The number of partial waves contributing to the region of fission-like products is taken from the measured cross section22). Tt is the trans~ssion coefficient for partial wave 1. The fits to the data are performed in the sharp cut-off approximation except for the lowest energies (E/A = 5.0 MeV for S0Ti + *08Pb and E/A = 5.74 MeV for 56Fe + 208Pb), where barrier fluctuations are expected to be significant. This phenomenon is known to be important in the fusion process at energies near the barrier 35). It may be suspected that the quasi-fission cross section near the barrier is influenced by the same barrier fluctuation phenomena as is the fusion cross section. The barrier fluctuations are simulated in an approximate way by the expression T,=l-(l+exp((l’-/)/A))-* from ref. 36). The parameters used are 1’ = 9, A = 11.5

for “Ti + 208Pb,

1’ = 17, A = 13

for 5”Fe + 208Pb.

(5)

360

K. Liitzenkirchen

et al. / Angular distributions TABLE1

Maximum angular momenta of sharp cut-off distributions SoTi t2’s Pb E/A [MeV] 4 ihl

5.5 65

56Fe +2o8 Pb 6.1 65

6.8 91

8.3 132

The sharp cut-off distributions used for the higher energies 22) are characterized by maximum Z-values, I,. They are listed in table 1. In the first part of our analysis, the explicit form of the cross section function f(0) is taken to be exponential, Be. This particular choice has virtually no influence on the values of the f(e)=Ae tilting variance Ki calculated below. f( 0) determines the central part of d2a/dtr dZ (30’ I 0 I 150”), whereas the angular range near the beam, where the,influence of the tilting is strongest, is mainly dependent on C,(x). The data points are normalized to the one closest to 0 = 90”. The fits with eq. (4) (see fig. 2) are performed by a minimization of x2 in a four parameter space, the coefficients A and B for the exponential function, and two values of K, for the forward and backward hemisphere, respectively. The reason for allowing two possibly different values of K, for 8 I 90” and 8 2 90” has already been discussed. In general, angles near 0” probe reaction times different from those near 180”. For a target-like fragment emission near 180” would correspond to a shorter reaction time than emission near 0”. If the relaxation time of the tilting degree of freedom is of the order of the rotational period, there is reason to suspect that K, will be different near 0” and 180”. 3. Results and discussion 3.1. THREE DIFFERENT

REACTION

CHANNELS

The quantity measured in the experiments reported here is the fragment charge Z. Other experiments which are based on a time-of-flight technique, are sensitive to the fragment mass A. For the sake of a common terminology we loosely use the words “mass” and “charge” in many expressions interchangeably. This seems well justified, because in highly damped collisions there is a strong correlation between mass and charge flow. One of the aims of the present study has been to substantiate the evidence’) for the coexistence of three distinct reaction channels (complete fusion-fission, quasifission, and deep-inelastic scattering). Here, the angular distributions of near symmetric fragment-charges already reveal the presence of a process, where the mass asymmetry equilibrates in a time comparable to the rotational period of the nuclear system (see fig. 2). Nevertheless, one may wonder whether this process is any

361 5GFe + 2oepb

E/A

= 6.1 MeV

/

-0

20

10

60

100

80 e,,

1

120

110

160

Relative

Units

~0

Deg

c

30

25 20 -

20

16 -

Z=67

I

10 -

I

10 50

,//

o --**-_. I 0

20

40

60

60

8,

100

/ Deg

120

140

160

180

a

20

10

60

80 %,

100 /

I

r_,_,lr_u,my~-m-;": 120

140

160

160

Deg

Fig. 4. Distributions d*o/dQdZ for fragments with Z> Zsym in the reaction 56Fe + *ORPb at 6.1 MeV/u. The dashed lines are to guide the eye. The peak for the target charge Z = 82 gradually decreases with increasing charge flow. In contrast to this, the cross section near 18Oo dncreases with increasing charge flow.

different from a usual deep-inelastic reaction. To this question we show four angular distributions in fig. 4 for the elements Z = 67, 72, 78 and 82 from the reaction 56Fe + *08Pb at E/A = 6.1 MeV. The transformation into the centre-of-mass system was performed by assuming total kinetic energies close to the fully relaxed, i.e. fission-fragment-like, energies 33). The gross features of the distributions are little affected even by changes in the values of TKE as large as 70 MeV. For Z = 82 one notices a peak around 90-110” which gradually decreases for lighter elements. The grazing angle for target-like products is 91”. It seem quite natural to conclude that the pronounced peak in the angular dist~bution originates

362

K. Liitzenkirchen

et al. / Angular distributions

from the deep-inelastic process, and that this process reaches down to elements around Z = 70. As one goes towards lower Z, one notices a pronounced increase in da/d0 close to 180”. This different behaviour is seen as a signature of an additional process, quasi-fission. It contributes predominantly to the cross section in the near-symmetric mass region. The two processes overlap considerably, resulting in a common element range around Z = 70 f 5. The angular distributions therefore indicate the coexistence of at least two distinct reaction channels for the same Z-range: sideward-peaked deep-inelastic processes, and quasi-fission reactions populating essentially all angles but with a large probability (in do/dQ) for backward angle scattering. We have searched for the third reaction channel, i.e. compound-nucleus fission, in the following way: As mentioned above we have performed fits with eq. (4) (cf. fig. 2) to the distributions d2a/d8dZ. These distributions were then compared with angular distributions expected for fissioning compound nuclei of charges Z = 104,108. The distributions were calculated on the basis of the TST and the RLDM, and were normalized to the data points at 90”. The maximum possible contributions by compound-nucleus fission to the total cross section extracted this way were of the order of 60% for both 5oTi, 56Fe + 208Pb, being rather independent of the incident energy, and, in particular, practically the same for all values of Z. Detailed measurements of mass and charge distributions in these reactions 37) show that the elemental, angle-integrated yields are constant for a large central range of Z-values (30 I Z I 74). A constant percentage of a constant cross section, however, is incompatible with the expected peaking at Zsym for compound fission. In addition there is the poor quality of the TST fits (fig. 3), and the much better reproduction of the experimental trends in the frame of a non-compound ansatz, fig. 2. Our conclusion is therefore that the data rule out the existence of a compound-nucleus fission component on the 50-608 level. On the other hand, it is impossible from our data to exclude a minor contribution below the lo-20% level. This is better decided on the basis of measurements of evaporation-residue cross sections. In the “Ti + 208Pb reaction near the barrier 25), the cross sections are reported to be dramatically suppressed by dynamical entrance channel limitations, e.g. by a factor of 30 for the 3n channel 25). Thus, only 3% of the observed fission-like cross section at this energy is due to compound-nucleus fission. (The maximum of the 3n channel occurs at 5.0 MeV/u which is the lower energy of this work. Using the extra-push formalism2-4) with slope parameters a = 18 [ref. 38)] for true fusion-fission and a = 12 for quasi-fissionl), and a common threshold xeff = 0.70 for both processes, one can show that the 3% contribution of true compound-nucleus fission at 5.0 MeV/u [ref. 25)] will increase to about 5% at 5.5 MeV/u.) For 56Fe +208 Pb, the hindrance factor at the barrier should be even larger. In addition, compound-nucleus formation is suppressed in this case by exit-channel limitations; the partial waves populating the near symmetric masses are usually much larger (table 1) than the Z-value for a vanishing fission barrier, ZB,=O2: 37A.

K. Liitzenkirchen

et al. /Angular

distributions

363

TABLE 2

K,

values of nearly complementary

fragments

for 56Fe + 2oRPb at E/A

= 6.1 MeV

Z

Kcwm 0 [h] K”>wo

42

41

50

52

54

56

58

64

65

4.0 25.8

4.8 9.0

7.1 10.0

9.8 1.6

9.6 8.0

11.9 5.5

11.7 8.6

11.5 6.4

16.1 6.3

Thus the contribution by compound-nucleus negligible at all energies, and the features characteristic of quasi-fission reactions.

3.2. COMPARING

INTEGRAL

AND

fission to the present data is likely to be displayed by the data can be viewed as

DIFFERENTIAL

ANGULAR

DISTRIBUTIONS

The information contained in an integral distribution da/d0 is clearly less than in differential distributions d2a/d8dZ. In order to demonstrate this, we have added the values of the fit functions describing d*a/dB dZ of a number of complementary fragment charges (see fig. 2 and table 2) in the reaction 56Fe + *‘*Pb at E/A = 6.1 MeV. Since the K X-ray measurements yield only relative cross sections d2a/d0dZ, the addition had to rest on the observed3’) constant cross section du/dZ in the central Z-range 30 I Z I 74; all relative cross sections were normalized to the same integral value for each Z, and summed. The resulting distribution, converted

to values

of du/dS2

(= (1/2n

sin8).

practically symmetric around 90”, a trivial binary reaction. Distributions of this type ments where no differentiation according to Returning to the differential cross sections to the above reaction using eq. (4). Values 8 = 180” are obtained under the assumption

do/de),

is shown

in fig. 5. It is

result because we are dealing with a are bound to be observed in measureZ or A is made. d%/dB dZ, table 2 shows results of fits K,< 90’ near 8 = 0” and K,’ 900 near of a sharp cut-off Z-distribution with a

maximum value I, = 65fi for all Z-values. It is seen from table 2 that K, is a function of Z, possibly indicating that the statistical generation of tilting angular momentum is a slow process which accumulates gradually, parallel to a relaxation of the mass asymmetry. Analysis of the summed angular distribution (fig. 5) in terms of the TST with the same sharp cut-off I-distribution yields the value K, = 7.Oft. It represents a mean value for the Z-dependent quantities K,< 90’ and K,’ 90” of table 2. This value does not tell anything about the saddle point shape of a compound nucleus. Instead it is just the mean value of the dynamically accumulated tilting variance. We conclude that integral angular distributions da/d9 for fission-like fragments in systems with suppressed fusion contributions are inadequate as a basis for tests of the TST, whatever transition state is assumed.

364

K. Liitzenkirchen

et al. / Angular distributions

E/A

01

0

= 6.1 MeV

I

50

I

1

100

150

O,, / Deg Fig. 5. do/dQ, obtained by summing the fit values of angular distributions (cf. fig. 2). The summation was performed for nearly complementary fragments Z= 42, 47, 50, 52, 54, 56, 58, 64, 65 in the 6.1 MeV/u 56Fe + “‘Pb reaction. Analysis with the TST yields a value of K, = 7.OA for both the forward and backward hemispheres.

In the following we shall discuss the differential distributions d20/de dZ in order to learn more about the Z-dependence, and also about system and bombarding energy dependences of the statistically accumulated tilting angular momentum in quasi-fission reactions.

3.3.

K,

VALUES

FROM

ANGULAR

DISTRIBUTIONS

W (0)

In figs. 6a-f the values of K, resulting from fits with eq. (4) are depicted as a function of Z. The quantity K, was calculated separately for 0” I 8 I 90” and 90” I 8 I 180” to take into account the possibility of different values for fragments emitted at forward and backward angles, respectively. We used sharp cut-off f-distributions common to all Z-values except for the lowest energies in the two systems (see sect. 2). The critical I-values are listed in table 1. The target-like products (Z > ZSF) at backward angles (u) have small K, values, usually K, I 10A. There is little dependence of K, on Z, except perhaps for a general trend for K, to increase slightly as one approaches mass symmetry. To further see the trend of K,, one would have liked to have the results for projectile-like fragments (Z < Z_,) at backward angles. Unfortunately, it has only been possible to analyze a limited number of the projectile-like fragments. This was due to unfavourable decay characteristics or presence in the X-ray spectra of L-lines from heavy fragments. On the other hand, in order to observe the dependence of K, on Z beyond mass symmetry, one can simply switch from looking at events with Z -CZsym emitted at backward angles to events with Z > Zsym emitted at forward angles.

K. Liirzenkirchen et al. / Angular distrihurions

50

=Ti

40

E/A

+ 266pb

0 *Ion

= 5.0 MsV

0 >30°

50

365

50Ti + 266pb E/A

40

. (900

- 5.5 MeV

•ISO0

$ \

30

Z

C 50

, “Fe E/A

40

$

,

+ 20*Pb - 5.7 MeV

l

I

a00

o >9o”

i

10 0 30

35

LO

L5

50

55

60

65

70

30

75

35

40

65

50

55

60

65

70

75

I

I

I

I

60

65

70

75

Z f 7, 56Fe 40

9

+ 2oePb l 400 - 6.8 MeV

10 _

q 'goo

1

56Fe E/A

I

+ *OsPb

I

I .

(900

- 8.3 MeV q ‘goo

$

1c \

E/A

50 -

30 t 20 10

\ y"

3020 10 -

rli 30

35

40

A5

50

55

Z Fig. 6. K, values in the forward (0) and backward (0) hemispheres for fragment charges 2. The solid lines are equilibrium values for KO calculated with eq. (6). The stars at ZSym = 52, 54 correspond to the sum of the variances of the statistical spins measured via the y-multiplicity I).

In all such cases it is then evident (from figs. 6a-f, solid circles) that K, further increases with increasing fragment charge. For one fragment charge there are thus two quite different variances K i, depending upon whether the fragment is emitted at forward or at backward angles. This difference between the variances increases as one goes away from charge symmetry towards the target, Z = 82 (see e.g. fig. 6e). As the mass asymmetry is equilibrated, K, rises very slowly from Z = 74 towards Z sym = 52. This is seen particularly clearly by following the open squares in fig. 6e.

366

K. Liitzenkirchen

et al. /Angular

distributions

These correspond to events at backward angles, making up the main part of the cross section for Z 2 ZsYm.The events at forward angles which make up the smaller portion of the cross section for 2 2 Zsym, are represented by the full circles. The variances are larger than those for events at backward angles, and may be indicative of longer interaction times. This is not surprising, because different scattering angles may be associated with different reaction times, as will be detailed in subsect. 3.5. Such different trajectories may reflect the occurrence of very large fluctuations associated with the mass drift, leading to quite broad distributions in interaction times and rotation angles for each given fragment charge. One contribution to such fluctuations may be related to random neck rupture. Such a mechanism was proposed in refs. 39,40) to account for the large mass variances u,’ in deep-inelastic reactions. In quasi-fission, a target-like fragment could be created analogously in two different ways. On the way to charge symmetry it is emitted at backward angles within some short reaction time (small K,, open squares). From an already mass-equilibrated dinucleus it is emitted due to random neck rupture after a much longer interaction time (large K,, solid circles). If, as we have seen, the saddle-point transition-state model is not relevant for 50Ti 56Fe + 208Pb, one does, of course, need some other reference for the approach of Ki ;o equilibrium. As suggested by Freifelder et al. 43) this new type of equilibrium might be associated with an effective “decision-point” which might be located somewhere between the saddle-point (or the turning point of the trajectory if a saddle point does not exist) and the scission point. A theory predicting such decision points is simply not developed at present. Next, for purposes of orientation we have decided to use a scission-point model 27) to calculate “equilibrium values” for the tilting variance. It should be noted that this is a massive step beyond the nature of the equilibration assumed for rigid-rotor effective transition states13) in that it allows the fission decay motion to carry orbital angular momentum due to other, thermally excited Z-bearing modes. That these modes do indeed play a role in the quasi-fission reactions considered here will be discussed in subsect. 3.4. in connection with the y-ray multiplicity results. Equilibrium values for the tilting variance within such a scission-point model for two spheres in contact are indicated in fig. 6 as solid lines (see also fig. 7). They were calculated according to2’)

where JA, B are the moments of inertia of spherical nuclei with mass numbers A, and A BY JA,B = $m,A,,,Ri,,

MeV ’ fm2/c2,

(7)

and m, equals 931 MeV/c2. The relative moment of inertia p( R, + RB)2 is given

K. Liitzenkirchen et al. /Angular distributions

367

with radii calculated according to R = 1 .28A1/3 - 0.76 + 0.8A-1/3 fm . The temperature

W

T is the one near the scission point, calculated according to T2=+(AA+AB)E*

MeV.

f IQ>

The excitation energy E* is given by E*=Qgp+Ec,-Ekin

MeV,

01)

with Q,, representing the ground-state reaction Q-value, E,, the incident energy in tbe centre-of-mass system, and Ekin the kinetic energy of fission-like fragments 33) in the exit channel. These variances should be lower limits to the equilibrium value, since a composite system which eventually decays by quasi-fission is likely to have formed a large neck near the turning point, thus having been more compact than the ~~fig~ation of two touching spheres ‘w3). The more compact the system has become, the larger equilibrium values for the tilting variances are to be expected. Looking at the experimental results, the K, values lie below the equilibrium value for most fragment charges 2, i.e. the accumulation of tilting angular momentum is incomplete. The slight indication of an increase of K,(Z) with increasing charge flow at backward angles, and the equivalent but more evident rise of It;‘@(Z)for Z > ZSymat the forward angles suggest that the tilting mode is developing towards equilibrium on a timescale comparable to the rotation period which again is comparable to the relaxation time of mass asymmet~, 7 = (5.5 f 2) X lU-‘l s [ref. 41)]. In several cases K, goes beyond the equi~b~um value for touching spheres (e.g. fig. 6f, solid circles for Z 2 67). These events, representing only a small portion of the cross section, indicate that the configurations involved have probably been more. compact than two touching spheres. The amount of the tilting relaxation can best be judged at symmetry, where 2 = ZSYrn= 52, 54. For the lower energies, K,’ 90” and K,’ “” lie clearly below the line indicating the equilib~um value for touching spheres (figs. 6a-d). For the two highest energies in 56Fe + 208Pb, E/A = 6.8, 8.3 MeV, the K, values at Zsym are very close to the corresponding equilibrium values. This may qualitatively be understood in the frame of a model *‘) describing angular momentum dissipation in direct nuclear reactions. It involves nucleon exchange as the underlying dissipative mechanism. The tilting mode, i.e. rotation of the dinucleus around its symmetrji axis, is not directly excited by nucleon exchange. Its excitation happens indirectly, through coupling to the wriggling mode via Coriolis forces induced by the orbital

368

K. Liitzenkirchen et al. / Angular distributions

~30

35

40

45

50

55

60

65

70

75

50 $iizi+

y” ”

,’

10

-.-

i

40

E/A

\ h

30 t

+ *O6pb - 6

1 MeV

----.____

,__=’

20

5"Fe

G

&&j$-.,“:::. ; ob 30

10 35

1’1 40

45

50

0

1

’

0

55

60

65

70

01 75

Z

50

30

35

40

45

50

55

Z Fig. solid sum i of

60

65

70

75

rF

4O

\ h

30

56Fe

E/A

+ 3O6&

- 8.3 MeV _*-..

,_--

Ol” 30

---__

.'

i

35

40

I1 45

’

0

50

55

--__

11

60

8

65

70

11

7S

Z

7. K, values averaged over the forward and backward hemispheres for fragment charges Z. The lines are equilibrium values for Kc calculated with eq. (6). The dashed lines are deduced from the of the variances of the statistical spins measured via the y-multiplicity’). The dashed-dotted lines are the variances from the (M,) measurements; this involves the idea that the six statistical modes may share the total excitation about equally.

K. Liitzenkirchen

rotation.

The amount

becomes

larger,

of excitation

the more

angular

369

et al. / Angulur distrihutions

of the tilting momentum

induces an Z-dependence for the relaxation becoming shorter for higher angular momenta.

mode

through

is involved

Coriolis

coupling

in the reaction.

This

time of the tilting mode, the time Reactions with high incident energy,

i.e. a large number of partial waves, should therefore reach the equilibrium value faster than those with a lower incident energy. This tendency is clearly visible in fig. 6. The bombarding energy dependence of K,(Z) is seen even more transparently in fig. 7. Average values of K, for the whole angular range 0” I B I 180” were formally calculated. In obtaining an average value (K,(Z)), K,’ 90” and K,’ 9oo were weighed by typical magnitudes of the cross section near 0” and 180”, respectively (eq. (12)). As an example, typical cross sections are marked by arrows for Z = 64 in fig. 2c. (K,(Z))=

L(Ko&,,o~ ‘Jtotal

+ Ko%..,Iso~).

(12)

Within error there is now very little dependence of (K,) on Z. The equilibrium values K;q for two touching spheres (solid lines in fig. 7) are clearly not reached for the lower energies, whereas for the two highest energies the average values of (K,) are quite close to KEq. I-fractionation. The angular distribution is sensitive to the tilting angle $J, the angle between the entrance- and exit-channel reaction planes. It is related to K, via Thus, lY(t?) sin 4 = K,/(l), (1) re P resenting the mean orbital angular momentum. is not a direct measure of K, but a measure of the ratio K,/(f). The magnitude of K, obtained in the analysis depends crucially upon what angular momenta are assumed to contribute to the cross section. In the analyses discussed above, the K, values were determined throughout with the Z-distributions from table 1; i.e. each fragment Z was assumed to be associated with the whole range of l-values, consistent

with the idea that the evolution

of the system

in the multidimensional

1,13,A space is dominated by fluctuations. The converse possibility of a correlation between fragment charge and I-value was suggested recently 22). Assuming a rough Z-fractionation, quasi-fission mass and 1 distributions

were divided 22) into three parts of equal cross sections.

The symmetric

masses were assigned to the most central l-bin, the intermediate masses to an intermediate l-bin, and the entrance channel-like masses to the highest l-bin. In order to test the effect of Z-fractionation upon our values of K,(Z), we chose the reaction 56Fe + 208Pb at E/A = 6.1 MeV. The Z-values corresponding to three Z-intervals are shown in table 3. These Z-distributions were then used to deduce the K, values of fig. 8. Comparing fig. 6d and fig. 8 shows that the assumption of l-fractionation changes the magnitude of K, in a number of cases. The global Z-dependence of K,, however, has not changed drastically; the K, values in the forward and backward hemispheres still differ considerably.

370 TABLE 3

I- and Z-intervals for 56Fe +‘08Pb at E/A = 6.1 MeV resulting from I-fractionation following ref. 22) AZ Al

54-60 o-37

61-67 38-53

68-14 54-65

The crude association (see table 3) of a given Z-interval with a given i-bin obviously ignores any dependence of 6 on 1. On the other hand, it is clear from fig. 2 that each 2 is formed with a large variance in scattering angles. It will be shown in subsect. 3.5 that the range of rotation angles is actually two to three times larger than the O”-180” interval of scattering angles that is experimentally accessible. This suggests that there is a very broad range of l-values contributing to a fragment Z. In the case a random neck rupture is of importance in these reactions, then each Z could actually be associated with the whole Z-dist~bution as initially assumed in our analysis. Since some I-fractionation cannot be ruled out with certainty, it seems not worthwhile to discuss local trends in the dependence of K, on Z. We stress instead the following conclusions which are not affected by the different assumptions about the ~-dist~butions: (i) The equilib~um value lugq for two touching spheres is not reached at Zsym, except possibly at the highest bombarding energies. (ii) The K, values in the forward and backward hemispheres are clearly different from each other.

50 _

Fig. 8. K,

56Fe + 208Pb o 90” are averaged in order to obtain an angle-independent (R,) (see subsect. 3.3), which is to be compared to the respective values derived from the angle independent (M,). These average values of (K,) are shown as open squares in figs. 7a-f. They all lie clearly below the corresponding variances obtained from (MY) [ref. ‘)I (dashed lines). If all six modes contribute equally, then the tilting variance should be of the order of i of the total one, 6

‘i2=i C

Uj2,

(15)

j=l

with uf as the variance of the i th statistical mode. Using this relation, revised K, values follow straightforwardly from the (M,) results. They are shown as dasheddotted lines in figs. 7b-f. In all cases they match the results from W(e) almost exactly. The result seems to suggest that the six modes share the total excitation to roughly equal portions, as suggested by eq. (15). 3.5. DECONVOLUTION

OF W (B)

In the foregoing sections we have paid little attention to the cross-section function f(B). For ease of handling, and because the actual choice of f(s) has virtually no influence on the results concerned with the tilting angular momentum, we have so far used a simple exponential function for f(0). In the following we shall try to interpret the angular distributions d*u/d6dZ in terms of gaussian distributions depending on rotation angles A0 and the associated reaction times. Because the rotation during contact (plus the Coulomb deflection) Afl may well be much larger than the maximum observable scattering angle, its determination requires a deconvolution procedure. There are two obvious motivations for trying to unfold the angular distributions d%/de dZ: (i) For a target-like fragment (see figs. 2c,d) the cross section decreases as one comes closer to 0”. It vanishes near 0” due to the fluctuations induced by the tilting mode. The function f( 8), on the other hand, which describes d2u/dS dZ with tilting ignored, is still quite large (dashed line in fig. 2~). Thus, a considerable portion of the dinuclei rotates through 0’ before decaying after rotation through angles A# larger than 180’. In the following, the cross section will therefore be displayed as a function of the rotation angle which is defined for all values A6 2 0”. (ii) Fragments with charges near ZSYmdo not evolve exclusively either from the projectile or from the target. Due to fluctuations in the mass drift, the measured

K. Liitzenkirchen

cross

section

projectile The

of a nearly

et al. / Angular distributions

symmetric

fragment

results

313

from

a superposition

of

and target contributions.

unfolded

projectile

angular

and

distribution

distribution

one from

depending

f(A;,

consists

the target.

Each

on the rotation Ai,(A

of two contributions, one can be represented

one

from

the

by a gaussian

angle AO,

A&) =A;exp[

-A;(AO’-

(AO)i)z],

(16)

with i =p, t for the projectile and target contributions, respectively. The physical picture behind the gaussian ansatz is that during rotation there is a progressing, highly viscous mass and charge flow toward symmetry for which the correlation of A and 2 with a given scattering angle is weakened due to fluctuations. For a fixed value of Z this is equivalent to a statistical distribution of associated rotation angles Atl. Eq. (16) is folded with a function f(l, K,; A@‘) [ref. 29)] which effect of the tilting fluctuations in an approximate way, (~+l/2)sin(A@‘)exp( _K2/2~;)

simulates

the

dK

/ f( I, K,; de’)

= i T,(21+ I=0

1)

O (‘+“*‘exp(

-K2/2K,f)dK

The parameters Ai and A; were determined by a least-squares anlaysis of K, for the forward and backward hemispheres taken with reference

’

07)

with values to fig. 6, I,

values from table 1, and fixed mean rotation angles (de)’ as parameters. The analysis was performed several times with different angles (AtI)’ in order to obtain the best fit. The deconvolutions were performed for “Ti + *08Pb at E/A = 5.5 MeV, and for 56Fe + ‘08Pb at E/A = 6.8 MeV. Two examples are given in fig. 9. The scale of rotation angles ABP for the projectile contribution (solid line) starts on the left side, the scale AB’ for the target contribution (dashed line) on the right side. The AtI scale ranges from 0” to 540”. (In most cases an interval of 360” would have been sufficient to display the unfolded distributions; the cross section contained in the 360-540” interval was usually very small.) If one assigns a scattering angle B to each rotation angle de’, and then adds all cross sections with the same 8, one obtains the dotted curve in fig. 9. It reproduces the experimental data quite well. Remarkably, the deconvolution reveals that there is a contribution of = 20% from the projectile even for fragments as heavy as Z = 70. Thus, the fluctuations in mass asymmetry indeed seem to be extremely large (see also subsect. 3.3), as one would also expect from the constant cross section da/dZ between 2 = 30 and 2 = 74 [ref. 37)]. The gaussian distributions for projectile and target contributions are characterized by the mean rotation angles (A8)i and the variances uid’. The Z-dependence of the two quantities is depicted in fig. 10 for “Ti + 208Pb at E/A = 5.5 MeV. The error bars for (A@)’ indicate estimated uncertainties. The angles (AO)i actually include the Coulomb deflection angles of initial and final trajectories. The results for 56Fe + 20RPb are qualitatively the same. The mean angles of rotation (All) as well as

K. Liitzenkirchen et al. / Angular distributions

374

50Ti + *O*pb 5.5 MeV/u

a

b Z=6L

0

200

100

300

bO0

500

140

LO

--+AOP / Deg 540

2b0

350

b40 -

-----

. .

A0’ / Deg Projectile Target

Sum

Fig. 9. Unfolded angular distributions d*a/d(AfI)dZ as a function of the rotation angle A0 for (a) Z = 56 and (b) Z = 64 in the 5.5 MeV/u 5oTi + *OSPb reaction. The solid lines are gaussians representing the contributions evolving from the projectile; the respective scale A8P starts on the left side. The dashed lines are gaussians representing the contributions evolving from the target; the respective scale Al3’ starts on the right side. At integer multiples of 180” the cross section vanishes due to the tilting fluctuations. The dotted line results upon assigning a scattering angle t’ E [0”, 180”] to each rotation angle AtiP,‘, and subsequently adding all cross sections with the same scattering angle 19. From the figures mean rotation and variances D&( Z)P,’ can be deduced. angles (Ae(Z))p~’

K. Liitzenkirchen

et al. /Angular

375

distributions

50Ti + *o*pb a,

LO

5.5 MeV/u t

r

I

1

1

r

I

1

I

t

I

I

25

50

55

60

65

70

I

r

,

Z b

’

I

’ 40

I

I

4

I

1

I

45

50

55

60

65

70

10000 F-

O

1

I

1

Z Fig. 10. (a) Mean rotation angles (A@( Zf)P.’ and (b) gaussian variances u&( Z)P.’ of unfolded distributions for the projectile (e) and the target (0) contributions (cf. fig. 9) in the reaction “Ti at E/A = 5.5 MeV.

angular + ‘O’Pb

the variances CT:*increase roughly linear with increasing mass flow. This is in line with expectations based on statistical theories of mass and charge flow. The curves for the projectile and the target contributions intersect near Zsyrn, and have approximately the same slope, as is anticipated for a binary reaction. Of course, the deconvolution always faces the problem of distinguishing between positive and negative scattering angles. Instead of (AB)r = 230” (fig. 9a) which results in a (negative) scattering angle of 8 = (AB)P - 180” = 50”, one could have also performed the fit with a rotation angle (A6)P = 130” which

316

K. Liitzenkirchen

et al. / Angular distributions

leads to the same (positive) scattering angle, 0 = 180” - (de)” = 50”. In this case the cross section is already very large for 0” 5 Af?PI 100”. This is not realistic, however, because the grazing angle for 5oTi + ‘08Pb at E/A = 5.5 MeV amounts to 100” which corresponds to ABP = 80” in the frame of fig. 9. During the first 80-100” of rotation, where there is little net mass flow between the two nuclei, one would expect the cross section to be quite small. For this reason we performed the fits with rotation angles usually exceeding 180”. From the angles (Ae>i one can calculate the corresponding reaction times after having subtracted the Coulomb deflection angles of initial and final trajectories42). Average reaction times (T,!_) are calculated via

08) with (f > = &Zc taken from table 1, and a mean moment of inertia equal to’) (J) = 1.34

09) J, is the rigid moment of inertia of the spherical compound nucleus using a radius parameter of r,, = 1.16 fm. For a given Z each reaction time (T,&) was then related to a value Koc900, and each reaction time ( T:~,,) to a value K,’ 90”. The results are shown in fig. 11. One sees that K, increases with the average reaction time. The way it increases (fig. lib) is not incompatible with an exponential approach of K, towards a saturation value, perhaps with some initial time delay involved. With the precision achieved here, and without knowing the actual saturation value (shape dependent) it is not considered meaningful to try to extract a relaxation time constant. The largest K, values are reached after (r,,,,)

= 13 x 1o-21 s

for 50Ti + 208Pb at 5.5 MeV/u, and after (&)

= 9 x 1o-21 s

for 56Fe + *“Pb at 6.8 MeV/u. The excitation occurs faster in the reaction with the larger amount of angular momentum. This is consistent with theoretical considerations 27,28) which expect the relaxation time of the tilting mode to be inversely proportional to the square of the orbital frequency of the dinucleus. 4. Conclusions

We have measured angular distributions d2a/dti dZ for binary reaction products in the systems “Ti + *08Pb at E,, = 202 and 222 MeV, and 56Fe + 208Pb at E,, = 253, 270, 301 .and 366 MeV. All distributions except the ones for symmetric charge splits Zsym= :( Z, + Z,) are forward-backward asymmetric, those for the projectilelike fragments being forward-peaked, and those for the target-like fragments being backward-peaked. The Z-dependent asymmetries show that the fission fragments are

K. Liitzenkirchen

50Ti

+ *o6pb

E/A

6

0 (90”

= 5.5 MeV

7

9

6

t /

b

I

4

= 6.8 MeV/A

5

‘goo

q

10

6

t /

11

13

12

lo-*'s

I

,

E/A

311

et (11./ Angular drstrihutions

I

I

8

9

I

o ‘goo

7

lo-*'s

Fig. 11. K, values in the forward (0) and backward (0) hemispheres as a function of time in the reactions (a) 50Ti +20”Pb at E/A = 5.5 MeV and (b) 56Fe + 20RPb at E/A = 6.8 MeV. The reaction time was calculated from the mean rotation angles (AtJ( 2))P.l of gaussi an distributions for the projectile and target contributions. The Coulomb deflection angles of initial and final trajectories were subtracted beforehand. The time deduced from an angle (AO( Z))P was related to the corresponding K,< 9o0(Z), (e), and the time deduced from an angle (A8( Z))’ to the corresponding K,,’ 9o”(Z), (IX).

produced in a reaction (quasi-fission), whose lifetime is of the order of the rotational period of the combined nuclear system. The cross section for compound-nucleus fission is at most of the order of a few percent of the fission-like cross section for “Ti + “‘Pb [ref. 25)], and presumably even smaller for 56Fe +208Pb, i.e. much too small to be of significance in the angular distributions. We have analyzed the quasi-fission angular is based on the factorization of the probability

distributions with an ansatz 28) which density for a given exit direction into

378

K. Liitzenkirchen

et al. / Angular distributions

(i) an angular distribution function f(e), and (ii) a gaussian distribution of projections K of the total angular momentum vector on the symmetry axis with a variance Ki,

reflecting

the other hand, fission reactions.

the physical contains

effect of the tilting the physics

fluctuations.

The function

of the mass equilibration

f(e),

process

on

in quasi-

As to the values of K, we conclude that, irrespective of the assumed I-distribution, the relaxation of the tilting mode is incomplete, i.e. the values (K,,(Z)) lie below the equilibrium value for two touching spheres. This is more pronounced for the lower energies (50Ti + ‘08 Pb at E/A = 5.0, 5.5 MeV, 56Fe + 208Pb at E/A = 5.7, 6.1 MeV), whereas for the higher energies (E/A = 6.8,8.3 is very close to this particular equilibrium value. The reason for this MeV), (K,(Z)) might be that the relaxation time for the tilting mode is I-dependent 27), being shorter for higher angular momenta. Reactions with higher incident energy, i.e. a large number of partial waves, should therefore reach the equilibrium value earlier. Other observations such as the occurrence of different values of K,(Z) for forward or backward angles, corresponding to longer or shorter reaction times, respectively, as well as the dependence of K, on the mass drift, depend in their quantitative interpretation on the (unknown) shape of the l-distribution for each Z-value. However, these observations still underline that the relaxation of the tilting degree of freedom occurs on a timescale comparable to that for the mass equilibration. The results reported here are consistent with earlier measurements of y-multiplicities (M,) [ref. ‘)I. The quantity (M,) probes the variances of the six statistical spin modes together, whereas W(8) is only sensitive to the tilting mode. If all six modes are excited about equally, the tilting mode should have a variance of about t of the total one. The values calculated along these lines starting consistent with the variances from W(e). In order

to provide

a time-scale

for the relaxation

from

of the tilting

(M,)

are indeed

angular

momen-

tum in quasi-fission reactions we have unfolded the angular distributions into distributions of rotation angles ascribing to each Z a contribution originating from the target, and one originating from the projectile. The mean rotation angles (de) as well as the dispersions in rotation angles u& are seen to increase regularly as the mass and charge flow progresses. From the mean rotation angles and plausible assumptions about the moments of inertia and the angular momentum distribution estimates for the mean reaction times can be made. The times required to reach the largest K, values in the reactions 5.5 MeV/u “Ti +*O* Pb and 6.8 MeV/u 56Fe + 208Pb are 13 x 10e21 s and 9 x 10e21 s, respectively, the shorter time being associated with the reaction involving higher angular momenta. This is in line with theoretical predictions 27,28). We thank the staff of the UNILAC accelerator for delivering beams of high quality, as well as H. Folger and his coworkers for target preparation. We greatly

K. Liitzenkirchen

appreciate experiments,

the help

given

et al. /Angular

by L. D&r

and

379

distributions

E. Schimpf

during

and by H.J. Schott, who got the target wheel running

all stages

of the

just in time. We

wish to thank T. Dossing and J. Randrup for helping us to understand many details of their model, and S. Bjomholm, A. Gobbi, K.D. Hildenbrand and W.Q. Shen for an illuminating series of discussions. manuscript by S. Bjornholm.

We also acknowledge

the critical reading

of the

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