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Nucleon exchange during a heavy-ion collision
Nuclear Physics QNorth-Holland
A445 (1985) 534-556 Publishing Company
NUCLEON
EXCHANGE
DURING A HEAVY-ION COLLISION R. SCHMIDT
Technische Universitiii Dresden, Sektion Ph.vsik, Dresden, German Democratic Republic Received 30 April 1985 (Revised 9 July 1985) We investigate nucleon-exchange processes during the initial, equilibration and transport stages in order to explain the anomalous mass-drift behaviour in asymmetric heavy-ion collisions. Special attention is drawn to systems where experimentally a mass drift against the static driving force of standard transport theories is found. The proposed picture from the mass transport during a heavy-ion collision allows a simultaneous interpretation of this effect and the lack of mass drift observed in other systems. In addition, we found a possible very fast mass-equilibration mode (“fast fusion”) which explains the anomalously large fusion-fission cross section for very asymmetric systems.
Abstract:
1. Introduction
For the description of the net nucleon exchange in low-energy heavy-ion collisions (HIC) transport theories? have been widely applied. In these theories the nucleon exchange between projectile and target is treated statistically by means of FokkerPlanck or master equations. The direction of the drift velocity is determined by the gradient of the potential energy, whereas its absolute value is given by the diffusion coefficient via the Einstein relation. On the basis of this fruitful concept various diffusion models have been developed and the calculations have been compared with experimental data within a broad range of target-projectile combinations (A, + AT) and incident energies2-5). F rom these extensive studies one can state the general conclusion that the experimental second moments of the mass distributions are well reproduced by nearly all models, whereas systematic deviations are found between experimental and transport-theoretical mean values. Thereby one observes increasing discrepancies with increasing initial mass asymmetry AJAr. For systems with A,/A, - 4 (a typical example is Kr + Er [ref. 2)]) the experimental data exhibit a lack of mass drift within a certain range of total kinetic energy losses (TKEL) and only the events with the largest TKEL exhibit a drift towards symmetry as expected from the driving force generated by the potential energy. For more asymmetric systems of A,/A, = & (e.g. Ar + MO [ref. 3)]) the lack of mass t For reviews,
see ref. ‘) 534
R. Schmidt /
Nucleon exchange
535
drift is experimentally realized within the total range of TKEL whereas the diffusion models predict a continuous increase of the mean values with increasing TKEL. At asymmetries of AP/AT = 4 (Ne + Cu [ref. 4)]) an excessive mass drift towards larger asymmetry, against the static driving force of standard diffusion models, is observed. For very asymmetric heavy systems A,/A, = & (e.g. Ne + Au [refs. 4*5)])the production cross section for elements with Z, > Z, is negligible as compared to that of Z, < Z,, leading to a dramatic asymmetry in the da/dZ, distributions and inducing some doubts on the general applicability of transport-theoretical methods to the interpretation of these data. For this system an anomalously large fusion-fission cross section5) has been observed, too. The lack of mass drift has been discussed by many authors [for a reference list, see ref. 6)]. Recently, Moretto ‘) has pointed out that the thermal feedback between two equilibrated Fermi gases of different temperatures could be a possible explanation. Swiatecki and Randrup 8, and independently Feldmeier 9, proposed that the lack of mass drift was a consequence of the coupling between the rate of dissipation of energy and the mass asymmetry degree of freedom which induces a freezing-out of the evolution of the mass asymmetry for small necks. Similarly to Moretto ‘) these authors treat the intrinsic structure of the nuclei as that of two Fermi gases and do not consider the initial and equilibration stage of the collision dynamics. An alternative explanation of the effect has been discussed in terms of a modified diffusion mode16) based on standard transport theories but taking into account the net nucleon exchange during earlier stages of the collision dynamics. Very few4) attempts have been made to explain the anomalous drift behaviour against the static driving force. It has been shown by Mathews et al. 4, that for Ne + Cu and Ne + Au collisions the effect of N/Z equilibration enhances the drift towards greater asymmetries but the resulting drift is too small by far to give at least a qualitative understanding of the experimental distributions. Within this work we shall extend the treatment of nucleon-exchange processes proposed for the explanation of the lack of mass drift 6, to systems where a mass drift against the static driving force has been found. We propose a simultaneous explanation of both effects, as well as a consistent interpretation of the abovementioned evolution of the mass drift with decreasing AP/AT. At the same time we present a plausible explanation of the observed anomalously large fusion-fission cross section in terms of a very fast mass equilibration mode (“fast fusion”). As an extension to ref. ‘j) we include explicitly the transfer of nucleons during the initial stage of the collision dynamics and treat nucleon exchange and excitation on a common microscopic basis (sect. 2). The change of mass asymmetry during the equilibration of intrinsic energy as well as some new arguments for its justification are discussed in sect. 3. The joining evolution of the mass asymmetry along the ridge line potential is examined in sect. 4. Model calculations are compared with experimental data and the predictions of standard diffusion models. Conclusions are drawn in sect. 5.
R. Schmidt / Nucleon exchange
536
2. Initial excitation and nucleon exchange One relative
of the interesting contributions
questions
originating
in the initial
from nucleon
dynamics
excitation
of HIC
within
concerns
the
one fragment
and
from nucleon transfer between both fragments to the total excitation energy of the ions. In this connection the change of the mass asymmetry is of special interest in the present investigations. Therefore we shall analyze the initial stage taking into account explicitly nucleon excitation and transfer as an extension to previous work 6, where we omitted an explicit treatment of exchange processes. For this purposes we shall apply an extended version of the semiclassical theory (SCT)“) based on TDHF. Details of the extended SCT will be published elsewhere ‘l). Here we briefly summarize the main ideas necessary for an understanding of the results used for the further investigations. Within
SCT the central
problem
consists
in the solution
of the time-dependent
Schriidinger equation for moving potential wells. If the particle transfer is included the non-orthogonality of the s.p. wave functions belonging to different potentials can be favourably handled by introducing the so-called dual basis 12,13) constructed from the asymptotic s.p. wave functions. Within the dual space the s.p. operators fulfil the usual anticommutation rules for fermion operators13). Therefore the generalized Thouless’ theorem”) can be applied in order to solve the many-body (m.b.) problem. Due to the non-hermiticity of the interaction fiint # hi:, within the dual space, the adjoint of the m.b. wave function is obtained by the solution of the Schrodinger equation with l?Lt implying the conservation of probability in the dual space. The expectation values of s.p. operators can be expressed in terms of infinite convergent series depending on the dual s.p. amplitudes in all orders. To achieve a more transparent insight one can neglect in these expressions the overlap matrix of the s.p. wave functions
because
its absolute
values does not exceed 10% within
the
region of interest. This leads to closed and transparent final expressions which are formally similar to that obtained in ref. lo) but now contain explicitly the different contributions excitation fragment
originating energy
from excitation
(fro)(‘)
and transfer,
and the excited particle
respectively.
number
( fil)
For the intrinsic
of the projectile-like
(l), one obtains @o>“‘=
c tep, + %,)lcp,h,12~*,h, plh, + c Ep,lCp,h,12~p,h, + c eh,lCp2h,12~p2h, 9 pzh, plh,
(‘1)
= c
lCp,h,12~plh, + c
PI h,
whereas
the change
7
(2)
plh,
of the mean mass number, (“1)
lCp,h212~p,h,
(1)
(Ail),
is given by
= c lCp,h,12~p,h, - c lCp,h,12~p2h, p-zh, plhz
(3)
R. Schmidt / Nucleon exchnnge
531
with Ed (Ed) the particle (hole) energies, c,,h the ph amplitudes, and the universal function dph which takes into account the Pauli principle. The first terms within eqs. (1) and (2) represent the contributions to the excitation energy and excited particle number which result from the excitation of nucleons within the same nucleus. The second (third) term in eq. (1) gives the net excitation energy induced by transfer of nucleons from nucleus (2) into nucleus (1) (and vice versa), whereas the second term in (2) represents the net particle flux from nucleus (2) into nucleus (1). The difference between the two fluxes induces the change of mass asymmetry in eq. (3). The time-dependent equations (l)-(3) have to be solved simultaneously with the equations of motion or the classical trajectory R(t) of the relative motion. To be consistent with earlier applications6) we generalized the derivation of a mean coupling matrix element VP,,[ref. lo)] to apply to excitation (i = k) as well as transfer (i # k) of nucleons leading to Vp,h, =
Ge- (e,,+~,,,)/A,+4-
40)
Jrn
(i,k=1,2),
where A, limits the range of s.p. energies included in single transitions, U, is the ion-ion ground-state interaction potential, and V, the common strength of the NN interaction. The explicit inclusion of the nucleon exchange via (l)-(4) does not require any additional model assumption and thus the ingredients of the actual semiclassical trajectory calculations are exactly the same as used previously 6). As a typical example we present the evolution of the intrinsic system for a representative partial wave in the *‘Ne + 64Cu collision in fig. 1. It can be seen from this figure that an appreciable amount of nucleons is exchanged between the fragments. However the fluxes into both directions are equal and thus the mass asymmetry (3) does not change, (AL,) = 0. Simultaneously the intrinsic excitation energy produced by transfer in (1) is equal in both nuclei. In fact, one can immediately prove that an effective change of the mass number (3) is given only by the different shell structure of the s.p. energies within both nuclei. Thus there does not exist a systematic trend of the initial mass drift towards symmetry or larger asymmetry as is characteristic for the drift velocity in transport theories. Moreover, in HIC the number of excited nucleons is large and therefore the individual shell structures do not show up in a substantial mass drift in one direction. This result suggests a plausible interpretation of the non-regularities and of the general smallness of the net mass change in TDHF. In the actual case any shell structure is neglected (this means that the summations within eqs. (l)-(3) are carried out over an equidistant s.p. spectrum with s.p. densities gpk= g&_= g(Ak) = &Ak MeV-’ (k = 1,2)) and thus the initial mass drift vanishes, (AA,) = 0, and the intrinsic excitation energies produced by transfer are equal in both nuclei. As can be further seen from fig. 1, excitation and transfer channels contribute nearly equally to the total excitation energy ( ko) and excited particle number (A)
R. Schmidr
538
Ne - like
30
60
/
Nucleon exchange
2oNe(170Me’/) t %u (l=60h) fragment Cu- like fragment
90
t (fm/c)
30
60
90
120
t (fm/c)
Fig. 1. Calculated excitation energy ( Golo)and excited particle number (A) witkin both fragments in the 20Ne(170 MeV) +WCu collision for the partial wave l= 60A as a function of time f (solid lines). The dotted (dashed) lines represent the corresponding contributions which result from the transfer (excitation) of nucleons.
in both nuclei although the relative tendencies are opposite within both fragments. The difference between (&,)(*) and (&a) t2) is about 5 MeV and thus small as compared to the total excitation energy of about 65 MeV. The same holds for the number of excited particles. This result which is qualitatively the same for all partial waves and target-projectile combinations considered leads to consequences on the mechanism of the relaxation of intrinsic energy towards the interfragment equilibrium discussed within the next section.
3. Change of the mass ~~rnrne~
during the ~uilibration
One of the most outstanding problems in the understanding of HI dynamics concerns the equilibration process of intrinsic excitation energy. Up to now there
R. Schmidt / Nucleon exchange
539
have been no realistic dynamical calculations and also the experiments are somewhat contrary. In any case an interfra~ent thermal ~ui~b~urn ~st~bution is characterized by a common temperature T of the system and thus by a mass-proportional division of the excitation energies and excited particle numbers to both nuclei. In contrast, we obtained a nearly equal split of both quantities after the initial excitation (cf. last section). Therefore the equilibration process must be accompanied by energy transfer and simultan~usly by a redis~bution of excited particles to both fragments most likely realized by transfer of always excited nucleons from the light to the heavy fragment until equilibrium is established. Under this assumption the number of transferred particles Nr, can be estimated from the interfragment equilibrium conditions 6),
avoiding any dynamical calculation. In eq. (5), (fii), (fiz) and A,, A, denote the initial excited and total nucleon numbers of the light (1) and heavy (2) fragment, respectively. In Kr-induced reactions the number of “prior particles” (5) has been directly compared to experimental data6) and is found to be in good agreement. Here we shall give a complementary justification of the assumed prior-particle mechanism (5). Within a equilibrated estimated according to
Fermi gas the common temperature
T=
j6~-~(&)/g(A, +A,)
of the system can be
(6)
with (&> the total excitation energy of the system. With the temperature (6) the equilibrium number of excited particles Nk* within the Fermi gas of fragment A, (k = 1,2) is fixed: N,*( Ak) = ln2g( A,)T.
(7)
Thus the ratio between the actual excited particles and the equilibrium number (7) can be used as a measure for the deviation from the equilibrium distribution within both fragments. In table 1 we present these ratios within the fragments before and after the exchange of prior particles (5). As can be seen from this table the initial stage of the collision dynamics is characterized by a Iarge overshoot of excited particles within the light fragment whereas the heavy partner nucleus “feels” a lack of excited particles. After the exchange of excited particles via (5) the ratios approach the equilibrium value of 1 within both fragments. The smail deviations of about 15% from the equi~b~um value 1 refer to additional re~r~gement mechanisms of the s.p. occupation probabilities which of course also contribute to the estabiishment of the thermal equilibrium. So, in the Ne + Cu case the total number
540
R. Schmidt /
Nucleon exchange
TABLE1 Ratio between the actual excited particles and the equilibrium values (7) before (( kk))/Np) and after ((( fik) T Nr)/NF) the exchange of prior particles NP within the lighter (k = 1) and heavier (k = 2) fragment for representative partial waves I in 20Ne- and 40Ar-induced collisions
System
@Ar(270 MeV) + ‘O”Mo
*‘Ne(l70 MeV) +64Cu
“Ne(220 MeV) + 19’Au
A,
and
A,
120 110 100
1.81 1.88 1.96
0.67 0.69 0.72
1.00 1.04 1.07
1.00 1.03 1.07
70 66 60
2.37 2.56 2.62
0.67 0.71 0.73
1.09 1.13 1.16
1.08 1.15 1.19
111 99 90
5.60 6.28 6.58
0.38 0.42 0.43
0.94 1.05 1.09
0.78 0.86 0.89
denote the mass numbers of the projectile and target, respectively.
of excited particles should be slightly decreased within both fragments, whereas in the Ne + Au system additional particles should be excited within the heavy fragment by conserving the total excitation energy. Nevertheless, the simple estimates presented favour the exchange of initial excited particles from the light to the heavy fragment as the dominating mechanism in order to reach the interfragment thermal equilibrium in asymmetric HIC. As a typical example we present in fig. 2 the change of mass asymmetry (5) as a function of incident angular momentum and energy as well as the absolute value of the common temperature (6) for the Ne + Au system. The change of the mass asymmetry remains small for near-grazing collisions and reaches maximum values for complete damped partial waves. The temperature and number of prior particles increase with increasing bombarding energy. However, at a fixed temperature (e.g. 1.5 MeV) one obtains a decreasing number of prior particles Ivr = 8,6,4 with increasing bombarding energy E = 220,290,400 MeV, respectively. This interesting feature strongly refers to the dynamical effects imposed in (1) and (2) which effectively results in a decreasing number of total excited nucleons at fixed TKEL but increasing bombarding energy, and thus in an effective increase of the mean s.p. energies excited in single transitions. In table 2 we discuss the dependence of Nr, on the initial mass asymmetry A, - A,/(A, + AZ). Thereby we fixed the total excitation energy and select collisions with nearly the same bombarding energy per nucleon. One obtains a nearly linear increase of the numbers of prior particles with increasing mass asymmetry. It will be shown in the following that the regularities of NP as a function of incident energy and mass ~y~et~ allow for a consistent interpreta-
541
R. Schmidt / Nucleon exchange
.
dN
dN
542
R. Schmidt
/
Nucleon exchange
TABLE 2 Number
of prior particles NP at a fixed heat energy (J&i,) = 50 MeV and nearly the same incident energy per nucleon, E/A,, for different initial mass asymmetries A, - A,/( A, + AZ)
Collision
0.33 0.43 0.52 0.82
8.4 7.3 8.5 8.8
R4Kr(704 MeV) + 166Er 4oAr(290 MeV) + ‘OOMo 20Ne(170 MeV) +64Cu 20Ne(175 MeV) + 19’Au
4.2 4.9 5.5 8.5
tion of rather different phenomena of the total mass drift observed within a broad range of target-projectile combinations.
4. Evolution of the mass asymmetry along the ridge-line potential During the transport stage the change of the mass asymmetry is induced by the evolution of the system along the ridge line of the potential energy U,(A,) as a function of the mass asymmetry A,. The dynamics of the mean value (A,)(t) is determined by the drift velocity V,(A,) of the corresponding Fokker-Planck equation. In standard diffusion models (SDM) the drift equation is solved with initial conditions of projectile-target asymmetry, (A,)(? = 0) = A,. The change of the mass asymmetry during earlier stages of the collision leads to modified initial conditions and the solution of the drift equation reads (A,)(t)=&+,+
U&.-&J&
(8)
which represents the basic equation of the modified diffusion model (IL%DM)~).As in previous applications we use the drift velocity proportional to the gradient of the driving potential as well as to the diffusion coefficient DA (Einstein relation) which is proportional to the temperature (6) DA = DjT, with Dj the diffusion constant. In the following we shall discuss the multifarious consequences of the modified drift equation (8) on the mass transport in Ne- and Ar-induced collisions.
4.1. DRIFT
AGAINST
THE STATIC
DRIVING
FORCE
In fig. 3 we present for the 20Ne (170 MeV)+64C~ collision the calculated potential energy U,( A,) as a function of A, and partial waves 1. From the position of the initial configuration at A, = 20 one would expect a strong drift towards symmetry for the range of partial waves which contribute to scattering (60 Q 1 G 80). This is demonstrated in the left part of the figure where we present the predictions of the standard diffusion model (eq. (8) with ZVr= 0). As expected from the potential
R. Schmidt
/ Nucleon exchange
-t
$!
E!
--4)
‘.
I
I
8 -
(WJ)
‘.
o,
LzP/3P
‘.
-tD
‘\
‘.
_
b-i-
544
R. Schmidt / Nucleon exchange
2!Ne(170MeV)
t %u
Va calculated at the projectile-target Fig. 4. Mean diffusion time interval 7dif and drift velocity asymmetry A, = A, (solid line) and the modified initial asymmetry A, = A, - Np (dashed line) as a function of partial wave 1.
energy U,, elements with 2, > 10 are preferentially produced theoretically, whereas experimentally the opposite trend is observed. A variation of the diffusion constant Do = 5 x 1021 amu2. s-l does not change any qualitative prediction. Within the A modified diffusion model we obtain a satisfactory agreement. A variation of 0,” can now be used to improve the quantitative agreement which is beyond the scope of this work due to the simplicity of the diffusion model used. To get an impression from the timescales and absolute values of the drift velocity, we plotted in fig. 4 the diffusion time interval +rdifand VAcalculated at the initial (A, = Ar) and at the modified initial (A, = A, - NP) configuration. As can be seen, the prior particles NP increase the drift towards symmetry but do not change the qualitative behaviour of V, (in contrast to the 20Ne + ‘97Au,discussed below). The evolution of the angular distributions of elements exhibits useful information on the mean lifetime of the double nuclear system in order to reach a given asymmetry Z,. Fast processes are concerned with forward-peaked d2cr/dZ,dS2, whereas a l/sine angular distribution indicates that these products are associated with substantially longer times (2 one rotational period). Thus the evolution of the form of the angular distributions as a function of Z, provides a sensitive test of a
R. Schmidt
“Ne
-
MDF
----
SDM
/
Nucleon exchange
10 MeV) + %
__I! -!-
sine
i’
1 30
I
I 90
I
I
30
,
I
90 0 c.m
I
30
90
..
30
90
Fig. 5. Evolution of the form of the theoretical angular distributions as a function of Z-value of the products obtained within the standard diffusion model (dashed lines) and the modified diffusion model (solid lines). Both angular distributions have been normalized to 1 at e,,, = 30°. MDM and SDM predict the opposite trends.
diffusion model concerning the timescales involved. In the frame of any SDM one would expect more forward-peaked angular distributions for Z, > Z, as compared to that of Z, < Z, due to the strong drift towards symmetry. The MDM which takes into account the nucleon exchange within earlier (faster) stages of the collision should predict the opposite tendency. In fig. 5 we compare the predicted evolution of the angular distributions [calculated as described in ref. 14)] within both models. For AZ, = + 4 the MDM approaches the l/sin@ form, whereas for AZ, = - 4 a strong forward-peaked angular distribution is predicted. The opposite tendency is obtained within the SDM. The experiment clearly confirms the predictions of the MDM which is demonstrated in fig. 6. The form as well as the absolute values of the unnormalized theoretical distributions agree well with the experimental findings, whereas in any SDM4) both quantities behave opposite to experiment. In the spirit of the proposed MDM the general tendency that elements with Z, < Z, are substantially produced faster than elements heavier than Z, should be realized in any asymmetric HIC and thus the evolution of the angular distributions can be used as a further test on the validity of the model. We shall now discuss the 20Ne + ‘97Au collisions. The calculated potential energy for the heavy and strong asymmetric 20Ne + ‘97Aureaction is presented in fig. 7. In this case the initial configuration A, = 20 lies in the vicinity of a Bussinaro-Gallone
R. Schmidt / Nucleon exchange
546
“Ne(170 MeV) + %u Z=B
Z=6
z=12
Z=lO
z-14
Q
\ \ \ \ \ \ \ \ \ \ \
, \ \ t I \ \ \ \ \ \
~
1 S_
L
L
\
\ \ 5 \ \ \ \
x_
~
-
experiment
__--
calculated @KIM)
\
\
‘_*
i
c
I I I I 30
\ \ t
90
1
30
\ \
I
\
/
‘.
/’
L
I_-
90
Fig. 6. Comparison between experimenta14) and theoretical angular distributions of representative pro&J&s Z. In contrast to fig. 5 the theoretical distributions (dashed lines) obtained within the modified diffusion model are not normalized.
maximum of U,(A,). In such situations a nearly symmetric mass distribution around 2, = Z, is theoretically expected due to the smallness of the mean drift velocity and independent of the value of the diffusion constant Dj = 15 x lO*l amuZ +s-l. In contrast, the experiment exbibits a dramatic asymmetry in the element distribution which is qua~tatively well reproduced within the MDM. To demonstrate the origin of the enormous difference between the predictions of the SDM and MDM we present in fig. 8 the influence of the prior particles on the drift velocity for the actual case. The vanishing drift of the target-projectile combination A, = A, is changed dramatically in its absolute value as well as in sign, except in the vicinity of grazing collisions. Thus the diffusion process starts with a strong drift towards larger asymmetries for a broad range of partial waves leading to qualitative agreement with the experimental distribution. To obtain a more quantitative agreement (especially for elements Z, > Z,) the A, dependence of the transport coefficients should be taken into account, and thus numerical calculations of the probability distribution seem to be necessary althou~ the general trends will not be changed. In order to
R. Schmidt / Nucleon exchange
I
I
I t
8
pvJJy ‘ZP/9P
548
R. Schmidt
/ Nucleon exchange
20Ne(175MeV)
+ lg7Au
-z-6 i; e-8 3 -10 -12
I
-1L
I
I
I
I 80
70
I 100
90 I
Fig. 8. Calculated drift velocity VA as a function of partial wave and different asymmetries A, =A, (dotted line), A, = A, - NP (solid line) and a mean asymmetry (dashed line) centered between A, = A, - NP and A, = A, (see text).
demonstrate element between values,
v, = vJt(A,
remarkable,
of the A, dependence
we defined
A, = A, - Nr
distribution theoretical
4.2. LACK
the influence
distribution
and
a mean
drift
the a-particle
- N, + A,))
of the drift
v, calculated
configuration
(fig. 8) and present
velocity
at a mean A, = A,
V, on the asymmetry
for negative
the corresponding
VA
element
in fig. 7. Although there is no qualitative change in the behaviour of the cross section the quantitative difference between both approximations is especially
for elements
Z, > Z,.
OF MASS DRIFT
The effect of lack of mass drift in Kr + Er reactions has been discussed within a separate work6). Here we shall present a complementary analysis of the 40Ar(270 MeV) + ‘O”Mo collision. The reason is twofold: first there are qualitative differences in the lack of drift behaviour between Kr + Er and Ar + MO collisions; and second, we found tendentious deviations between transport-theoretical and experimental variances which are worth discussing. In fig. 9 we present the mean value (Z,) and variance ui as function of TKEL obtained within the SDM and MDM. In this collision the diffusion constant DoA = 7 x 1021 amu2. se1 has been determined to reproduce the absolute values of
549
R Schmidt / Nucleon exchange
“Ar (2’70 MeV) + looMo 21 / z_____---V ,8
-
--------
_________------.
.
1’
_--H -
.
l
.
l
++++++
g)M +++tMDM
+
lo’ -
+JN w
-
exp maximal mean value of TKEL
I 50
TKEL(MeV)
100
I
I
150
Fig. 9. Lack of mass drift in the ““Ar + loo M o collision. Experimenta13) and theoretical mean values (2,) and variance ai as a function of TKEL are compared. The experimental maximal mean value of TKEL is indicated by an arrow.
the experimental ui values within the deep-inelastic region. Therefore the drift velocity (8) is completely determined. The SDM predicts too large a drift towards symmetry. The MDM leads to theoretical mean values (2,) which stay at nearly constant values for all TKEL in accord with experiment?. This is the main difference from Kr + Er collisions2) where the lack of mass drift is realized only within a certain range of TKEL and a strong drift towards symmetry is observed for large energy losses, theoretically induced by the larger drift velocity and interaction times as compared to the lighter system Ar + MO. In the actual case the mean values (Z,) and variances us have been measured 3, up to TKEL larger than the maximal mean value (= 105 MeV) of the experimental spectrum. Thereby, the experimentalists found an interesting new feature. The mean + The insignificant increase of the theoretical mean value (Z,) relevance because of the large fluctuations within the TKEL.
for the largest energy losses has no real
R. Schmidt / Nucleon exchange
550
“Ne (175 MeV) t lg7Au
J
0 70
80
90
100
I Fig. 10. Mean diffusion time interval rdir and time needed in order to reach the a-particle configuration 7FFu as a function of I. Between the trapped partial wave t,, and I,,, the system reaches the a-particle asymmetry A, = A, (or equivalently the partial fused configuration A, = A, + A, -A,) during the binary collision process.
values stay at constant values whereas the variances increase up to a factor of about 2.5. This behaviour of the mass transport cannot be understood in terms of standard or modified transport theories. A possible explanation has been proposed by Grossmann and Brosa”‘) in terms of a random neck-rupture concept. The precise measurement of the variances in the Ar + MO collisions 3, reveals also a further defect of transport theories. As can be seen from the logarithmic plot of c&TKEL) in fig. 9 the form of the variance for small energy losses cannot be reproduced by the diffusion model and the absolute values are underestimated. Recently, Feldmeier 9, has stressed the importance of non-equilibrium contributions to the diffusion tensor for small TKEL. These contributions which result from the velocity mismatch of nucleons belonging to different nuclei would increase the variances, whereas at the same time the drift velocity remains unaffected and thus the conclusions about the drift behaviour in asymmetric HIC drawn within this work.
4.3. MASS EQUILIBRATION
MODE FOR VERY ASYMMETRIC SYSTEMS - FAST FUSION
Besides the very asy~et~c element ~stribution an anomalously large fission cross section as well as an unexplained large cross section of fast a-particles have been experimentally observed in the 20Ne + ‘97Au reactions 5). While these three
R. Schmidt / Nucleon exchange
551
TABLE3 Bombarding energy dependence (E = 150,220,290,400 MeV) of partial waves trapped into a potential pocket (I Q I, ) and the range of partial waves ( lTR < I < I,,“) contributing to the fast-fusion channel TFFU(IFFU)
[ML] 150 220 290 400
[lo-= 57 85 104 126
66 94 114 143
0.90 0.88 0.77 0.50
s] 5.3 1.9 10.0 13.4
are the fast-fusion times (obtained as discussed in fig. 10) and NP the number of prior ne TFFU particles of the partial wave IFFU.
features seem to be unrelated at first glance, they may well be connected as we intend to show in the following. Let us first briefly summarize the experimental statements [for a detailed discussion see ref. 5)]. The experimental critical angular momenta of fission events exceeds the critical l-value of instability of the compound nucleus against fission up to a factor of about 2. The experiments clearly rule out the “fast fission” 16) as the responsible mechanism because the mass width of fission fragments stay at low values. Most of the fast a-particles exhibit a mean energy centered at intermediate velocities between Coulomb repulsion and projectile velocity. Only about one-half of the integrated cross section of this a-particle component can be attributed to “sequential decay of the excited projectile-like fragment” which leads the authors to the conclusion that there exists an additional mechanism for producing fast a-particles, possibly allowing the remaining system to fuse. Within our MDM these experimental observations get a natural explanation. Let us first discuss qualitatively the peculiarities of the nucleon diffusion along the ridge line potential U,(A,) for negative drift velocities V, < 0. Obviously, the nucleon diffusion process stopped at the compact a-particle configuration A, = A, because for very large asymmetries A, =A, shell effects cannot be neglected. The binding energy of the o-particle produces a deep minimum within the potential U,(A,) at A, = A,. The remaining (partial fused) system A, = A, + A, - A, exhibits the total intrinsic excitation energy (Z&) because an a-particle cannot be excited. Assuming a binary process one expects the emission of a-particles whereas the remainder will undergo sequential fission due to the large excitation energy. In the following we shall check these expectations quantitatively. A mean diffusion time 7FFU during which the double nuclear system reaches the configuration A, = A, (A2 = A, + A, - A,) can be estimated according to
R. Schmidt / Nucleon exchange
552
“Ne + ‘g7Au
OL
100
/
I
4
GFFU
-
I
200
t
300
I
i
i
too
ELab(MeV)
Fig. 11. Calculated cross section of trapped partial waves on and including the fast-fusion mechanism oFfU as a function of incident energy. The calculated points are connected by straight lines. Experimental points of the fusion-fission cross section are from ref. s).
In fig. 10 we compare this time with the diffusion, 7diI, in a binary collision. Actually, I FFU where we found r rru Q rdir and thus A, + AT - A, is realized during the binary time interval, rFFU < 10e2i s (fast fusion). section
total time interval available for there exists a certain l-range I,, the partial fused configuration collision process within a rather Therefore the theoretical fusion
uFFU = mX2(iFF” + Q2 is increased as compared to the cross section of partial waves I< I, trapped into a potential pocket uTR = nX2(Ir, + l)*.
mass -c 1 G
A, = short cross
(10) which are
(11)
We found the fast-fusion mechanism to be realized within the whole range of incident energies studied expe~ment~ly (cf. table 3). In any case we have Irru > I, and thus the theoretical fusion cross section is increased considerably by the fast-fusion mechanism. In fig. 11 we compare the cross sections (10) and (11) with the experimental fission cross section. As can be seen, the fast-fusion mechanism explains the enhanced fission cross section or the lowest and highest incident energies, whereas within the intermediate range of E there should be an additional
R. Schmidt / Nucleon exchange
553
TABLE4 Bombarding energy dependence of different contributions to the spectrum of o-particles in the *‘Ne + 19’Aucollision E
Ii
7
[lo-*1 s]
‘eq
If
VCd
E,
E,‘“P(2)
150
61
1.5
4
9
20.7
23.6
_
220 290 400
90 109 135
1.4 1.3 1.0
6 8 10
14 19 33
20.1 20.7 20.7
28.0 34.1 60.5
30 35 55
E,‘“P(l)
40 50 12
E denotes the bombarding energy, V,,,, the Coulomb energy of evaporated a-particles, EzP(1) the experimentally observed o-particle energy corresponding to projectile velocity (direct break-up), and E,‘“P(2) the experimental mean value of the intermediate component of fast particles [see fig. 11 in ref. ‘)I, r is the mean contact time for the initial partial waves li centered in the I-range leading to fast-fusion events (cf. table 3). les and I, are the calculated equilibrium (sticking) and final relative angular momenta of the relative motion, respectively (eq. (13)). E,, denotes the theoretical mean value of o-particles resulting from the fast-fusion mechanism, calculated via (12). Note the good agreement between & and E,‘“P(2). Energies are in MeV, angular momenta in h.
mechanism for enlarging the fission cross section, most probably by direct a-particle break up (cf. discussion below). Table 3 gives also an impression of the energy dependence of the mass equilibration time rrr,,. With increasing bombarding energy the fast-fusion time decreases reaching rather small values of about 7rFU = 5 x 1O-22 s for the largest energy considered. The increasing smallness of rrFU (essentially caused by the increasing number of prior particles Nr) has consequences on the energy spectrum of emitted a-particles. The final mean kinetic energy of the a-particles E, produced by the fast-fusion mechanism is given by the sum of Coulomb repulsion and centrifugal energy, E, =
A,+A,-A A,+A,
a( Goul+&)
(12)
the moment of inertia of the relative motion and I, the final relative with & angular momentum which can be estimated according to “) 03)
where Jr, J2 denote the moments of inertia of the two fragments A, = A, and respectively, and r, is the relaxation constant of angular momentum dissipation. For sufficient long contact times t > r, the final angular momentum approaches the equilibrium value I,, of the sticking configuration. Taking a constant decay time of r, = 0.6 X 10e21 s [ref. “)I we summarize in table 4 the calculated leq, I, and A?, values for representative partial waves Ii (localized in the midst of the I-range contributing to fast fusion, cf. table 3) and compare the theoretical mean values I!?, with characteristic experimental energies in the spectrum
A, = A, + A, - A,,
554
R. Schmidt
/
Nucleon exchange
of a-particles. We obtain a surprisingly good agreement with the intermediate component E,‘“P(2) for all energies considered. For the lowest bombarding energy of E = 150 MeV the theoretical mean value E, = 23.6 MeV does not remarkably exceed the energy-independent Coulomb repulsion of V,, = 20.7 MeV. In this case the a-particles originating from the fast-fusion mechanism cannot be decided from the evaporated a-particles within the experimental spectrum. This is in accord with the measured threshold for production of fast a-particles of about 200 MeV [ref. 5)] and at the same time explains the good agreement between aFFU and the experimental fission cross section at this energy (cf. fig. 11) in the absence of fast a-particles. With increasing bombarding energy the contact times 7(Zi) decreases resulting in rather large values of I, well above I,,. The emitted a-particles exhibit a mean energy centered at intermediate values between vcO”, and that of direct a-particle break up E$P(l) in agreement with experiment. Closing this section we briefly comment on the influence of direct break-up processes on the fission and deep-inelastic cross sections. This mechanism can explain the remaining difference between aFFU and the experimental fission cross section at intermediate energies (cf. fig. 11). However, with increasing bombarding energy the evolution of the projectile-like fragment (resulting from an initial (Yparticle break-up) results increasingly in an incomplete deep-inelastic reaction [INDIC18)] instead of incomplete fusion due to the increasing values of partial waves connected with that mechanism. Therefore we expect a minor influence of this mechanism on the experimental fission cross section at sufficient large incident energies, but at the same time an increasing importance of this reaction channel on the mass distribution of deep-inelastic events. This argument is in accord with the good agreement between aFFU and the experimental fission cross section at 400 MeV (cf. fig. 11).
5. Summary and outlook We have analyzed nucleon-exchange processes during three stages of a HIC. The vanishing mass drift during the initial stage obtained within a semiclassical theory in this work confirms the general observations made in TDHF calculations. We have shown that the equilibration stage must be accompanied by a redistribution of initial excited particles to both nuclei and have assumed transfer of excited nucleons to be the dominating mechanism. This leads finally to modified initial conditions for the nucleon-diffusion process along the ridge line potential during the transport phase. The gross feature of the mass transport in asymmetric HIC can be understood in terms of this concept. Partia12) and tota13) lack of mass drift, drift against the static driving force4) as well as vanishing production cross section for elements heavier than the projectile5) are explained on a common basis in terms of a continuous evolution mainly generated by the A, -A/A, + A, dependence of the number of prior particles. In addition, the anomalously large fission cross section in 20Ne + ‘97Au
R. Schmidt / Nucleon exchange
555
collisions can be attributed to a very fast mass-equilibration mode (fast fusion) connected with the emission of fast a-particles. The predictions and the validity of the MDM can be checked by further experimental studies: (i) The exchange of particles prior to the transport phase leads automatically to an asymmetry in the mean diffusion times for the production of fragments heavier and lighter than the projectile. Lighter (heavier) fragments are associated with shorter (longer) times resulting in different evolutions of the angular distributions d*u/dZ, dO for elements with the same relative mass change, Z, - AZ (Z, + AZ) (cf. subsect. 4.1). This effect should be realized in any asymmetric HIC independent of the gross behaviour of the mass drift. Experimental evidence for an asymmetry in interaction times has been found recently in Kr + Bi collisions [cf. the discussion on fig. 6 in ref. 19)] where the observed different interaction times for Z, > Z, and Z, < Z, are quite unexpected because the angle-integrated charge distribution da/dZ1 exhibits drift towards symmetry (partial lack of mass drift). (ii) For systems with partial lack of mass drift *,19)the number of predicted prior particles can be directly compared with experimental data 6). (iii) The proposed interpretation of the anomalously large fission cross section in *‘Ne + ‘97Au experiments in terms of fast fusion suggests the search of this effect in other systems which exhibit an initial asymmetry in the vicinity of the BussinaroGallone maximum of U,(A,) [and thus for target-projectile combinations which lie beyond the range of the fast-fission systematics 16)]. We expect the large fission cross section to be correlated with fast a-particles independent of a possible a-particle structure of the projectile. From the theoretical point of view an extension of the proposed treatment to the explicit inclusion of shell effects as well as N/Z degrees of freedom is desirable to achieve a detailed understanding of the puzzling mass-drift behaviour in asymmetric HIC. I am grateful to Prof. R. Reif for useful discussions and T. Gleitsmann for help in the numerical work.
References 1) A. Gobbi and Ww. Nijrenberg, Heavy ion collisions, vol. II, ed. R. Bock (North-Holland, Amsterdam, 1980) ch. 3; L.G. Moretto and R.P. Schmitt, Rep. Prog. Phys. 44 (1981) 533 2) G. Rudolf et al., Nucl. Phys. A330 (1979) 243 3) W. Bohne et al., Z. Phys. A313 (1983) 19 4) G.J. Mathews et al., Phys. Rev. C25 (1982) 300 5) Ch. Egelhaaf et al., Nucl. Phys. A405(1983) 397 6) R. Schmidt, Z. Phys. A320 (1985) 413 7) L.G. Moretto, Z. Phys. A310(1983) 61
556
R. Schmidt / Nucleon exchange
8) W.J. Swiatecki, Nucl. Phys. A428 (1984) 1990;
9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
J. Randrup and W.J. Swiatecki, Nucl. Phys. A429 (1984) 105; J. Randrup, Nucl. Phys. A307 (1979) 319; A327 (1979) 490 H. Feldmeier and H. Spangenberger, Nucl. Phys. A428 (1984) 2230 R.V. Jolos, R. Schmidt and J. Teichert, Nucl. Phys. A429 (1984) 139 R. Schmidt, to be published R.A. Broglia and A. Winther, Nucl. Phys. A182 (1972) 112; Phys. Reports 4C (1972) 154 K. Dietrich and K. Hara, Nucl. Phys. A211 (1973) 349 R. Schmidt, G. Wolschin and V.D. Toneev, Nucl. Phys. A311 (1978) 247 S. Grossmann and U. Brosa, Z. Phys. A319 (1984) 327 C. GrCgoire, C. NgB and B. Remand, Nucl. Phys. A383 (1982) 392; H. Lefort, Z. Phys. A299 (1981) 47 G. Wolschin, Nucl. Phys. A316 (1979) 146 F. Guzman and R. Reif, Nucl. Phys. A436 (1985) 294 J.R. Birkelund er al., Phys. Rev. C26 (1982) 1984
+ For reviews, see ref. ‘).
A445 (1985) 534-556 Publishing Company
NUCLEON
EXCHANGE
DURING A HEAVY-ION COLLISION R. SCHMIDT
Technische Universitiii Dresden, Sektion Ph.vsik, Dresden, German Democratic Republic Received 30 April 1985 (Revised 9 July 1985) We investigate nucleon-exchange processes during the initial, equilibration and transport stages in order to explain the anomalous mass-drift behaviour in asymmetric heavy-ion collisions. Special attention is drawn to systems where experimentally a mass drift against the static driving force of standard transport theories is found. The proposed picture from the mass transport during a heavy-ion collision allows a simultaneous interpretation of this effect and the lack of mass drift observed in other systems. In addition, we found a possible very fast mass-equilibration mode (“fast fusion”) which explains the anomalously large fusion-fission cross section for very asymmetric systems.
Abstract:
1. Introduction
For the description of the net nucleon exchange in low-energy heavy-ion collisions (HIC) transport theories? have been widely applied. In these theories the nucleon exchange between projectile and target is treated statistically by means of FokkerPlanck or master equations. The direction of the drift velocity is determined by the gradient of the potential energy, whereas its absolute value is given by the diffusion coefficient via the Einstein relation. On the basis of this fruitful concept various diffusion models have been developed and the calculations have been compared with experimental data within a broad range of target-projectile combinations (A, + AT) and incident energies2-5). F rom these extensive studies one can state the general conclusion that the experimental second moments of the mass distributions are well reproduced by nearly all models, whereas systematic deviations are found between experimental and transport-theoretical mean values. Thereby one observes increasing discrepancies with increasing initial mass asymmetry AJAr. For systems with A,/A, - 4 (a typical example is Kr + Er [ref. 2)]) the experimental data exhibit a lack of mass drift within a certain range of total kinetic energy losses (TKEL) and only the events with the largest TKEL exhibit a drift towards symmetry as expected from the driving force generated by the potential energy. For more asymmetric systems of A,/A, = & (e.g. Ar + MO [ref. 3)]) the lack of mass t For reviews,
see ref. ‘) 534
R. Schmidt /
Nucleon exchange
535
drift is experimentally realized within the total range of TKEL whereas the diffusion models predict a continuous increase of the mean values with increasing TKEL. At asymmetries of AP/AT = 4 (Ne + Cu [ref. 4)]) an excessive mass drift towards larger asymmetry, against the static driving force of standard diffusion models, is observed. For very asymmetric heavy systems A,/A, = & (e.g. Ne + Au [refs. 4*5)])the production cross section for elements with Z, > Z, is negligible as compared to that of Z, < Z,, leading to a dramatic asymmetry in the da/dZ, distributions and inducing some doubts on the general applicability of transport-theoretical methods to the interpretation of these data. For this system an anomalously large fusion-fission cross section5) has been observed, too. The lack of mass drift has been discussed by many authors [for a reference list, see ref. 6)]. Recently, Moretto ‘) has pointed out that the thermal feedback between two equilibrated Fermi gases of different temperatures could be a possible explanation. Swiatecki and Randrup 8, and independently Feldmeier 9, proposed that the lack of mass drift was a consequence of the coupling between the rate of dissipation of energy and the mass asymmetry degree of freedom which induces a freezing-out of the evolution of the mass asymmetry for small necks. Similarly to Moretto ‘) these authors treat the intrinsic structure of the nuclei as that of two Fermi gases and do not consider the initial and equilibration stage of the collision dynamics. An alternative explanation of the effect has been discussed in terms of a modified diffusion mode16) based on standard transport theories but taking into account the net nucleon exchange during earlier stages of the collision dynamics. Very few4) attempts have been made to explain the anomalous drift behaviour against the static driving force. It has been shown by Mathews et al. 4, that for Ne + Cu and Ne + Au collisions the effect of N/Z equilibration enhances the drift towards greater asymmetries but the resulting drift is too small by far to give at least a qualitative understanding of the experimental distributions. Within this work we shall extend the treatment of nucleon-exchange processes proposed for the explanation of the lack of mass drift 6, to systems where a mass drift against the static driving force has been found. We propose a simultaneous explanation of both effects, as well as a consistent interpretation of the abovementioned evolution of the mass drift with decreasing AP/AT. At the same time we present a plausible explanation of the observed anomalously large fusion-fission cross section in terms of a very fast mass equilibration mode (“fast fusion”). As an extension to ref. ‘j) we include explicitly the transfer of nucleons during the initial stage of the collision dynamics and treat nucleon exchange and excitation on a common microscopic basis (sect. 2). The change of mass asymmetry during the equilibration of intrinsic energy as well as some new arguments for its justification are discussed in sect. 3. The joining evolution of the mass asymmetry along the ridge line potential is examined in sect. 4. Model calculations are compared with experimental data and the predictions of standard diffusion models. Conclusions are drawn in sect. 5.
R. Schmidt / Nucleon exchange
536
2. Initial excitation and nucleon exchange One relative
of the interesting contributions
questions
originating
in the initial
from nucleon
dynamics
excitation
of HIC
within
concerns
the
one fragment
and
from nucleon transfer between both fragments to the total excitation energy of the ions. In this connection the change of the mass asymmetry is of special interest in the present investigations. Therefore we shall analyze the initial stage taking into account explicitly nucleon excitation and transfer as an extension to previous work 6, where we omitted an explicit treatment of exchange processes. For this purposes we shall apply an extended version of the semiclassical theory (SCT)“) based on TDHF. Details of the extended SCT will be published elsewhere ‘l). Here we briefly summarize the main ideas necessary for an understanding of the results used for the further investigations. Within
SCT the central
problem
consists
in the solution
of the time-dependent
Schriidinger equation for moving potential wells. If the particle transfer is included the non-orthogonality of the s.p. wave functions belonging to different potentials can be favourably handled by introducing the so-called dual basis 12,13) constructed from the asymptotic s.p. wave functions. Within the dual space the s.p. operators fulfil the usual anticommutation rules for fermion operators13). Therefore the generalized Thouless’ theorem”) can be applied in order to solve the many-body (m.b.) problem. Due to the non-hermiticity of the interaction fiint # hi:, within the dual space, the adjoint of the m.b. wave function is obtained by the solution of the Schrodinger equation with l?Lt implying the conservation of probability in the dual space. The expectation values of s.p. operators can be expressed in terms of infinite convergent series depending on the dual s.p. amplitudes in all orders. To achieve a more transparent insight one can neglect in these expressions the overlap matrix of the s.p. wave functions
because
its absolute
values does not exceed 10% within
the
region of interest. This leads to closed and transparent final expressions which are formally similar to that obtained in ref. lo) but now contain explicitly the different contributions excitation fragment
originating energy
from excitation
(fro)(‘)
and transfer,
and the excited particle
respectively.
number
( fil)
For the intrinsic
of the projectile-like
(l), one obtains @o>“‘=
c tep, + %,)lcp,h,12~*,h, plh, + c Ep,lCp,h,12~p,h, + c eh,lCp2h,12~p2h, 9 pzh, plh,
(‘1)
= c
lCp,h,12~plh, + c
PI h,
whereas
the change
7
(2)
plh,
of the mean mass number, (“1)
lCp,h212~p,h,
(1)
(Ail),
is given by
= c lCp,h,12~p,h, - c lCp,h,12~p2h, p-zh, plhz
(3)
R. Schmidt / Nucleon exchnnge
531
with Ed (Ed) the particle (hole) energies, c,,h the ph amplitudes, and the universal function dph which takes into account the Pauli principle. The first terms within eqs. (1) and (2) represent the contributions to the excitation energy and excited particle number which result from the excitation of nucleons within the same nucleus. The second (third) term in eq. (1) gives the net excitation energy induced by transfer of nucleons from nucleus (2) into nucleus (1) (and vice versa), whereas the second term in (2) represents the net particle flux from nucleus (2) into nucleus (1). The difference between the two fluxes induces the change of mass asymmetry in eq. (3). The time-dependent equations (l)-(3) have to be solved simultaneously with the equations of motion or the classical trajectory R(t) of the relative motion. To be consistent with earlier applications6) we generalized the derivation of a mean coupling matrix element VP,,[ref. lo)] to apply to excitation (i = k) as well as transfer (i # k) of nucleons leading to Vp,h, =
Ge- (e,,+~,,,)/A,+4-
40)
Jrn
(i,k=1,2),
where A, limits the range of s.p. energies included in single transitions, U, is the ion-ion ground-state interaction potential, and V, the common strength of the NN interaction. The explicit inclusion of the nucleon exchange via (l)-(4) does not require any additional model assumption and thus the ingredients of the actual semiclassical trajectory calculations are exactly the same as used previously 6). As a typical example we present the evolution of the intrinsic system for a representative partial wave in the *‘Ne + 64Cu collision in fig. 1. It can be seen from this figure that an appreciable amount of nucleons is exchanged between the fragments. However the fluxes into both directions are equal and thus the mass asymmetry (3) does not change, (AL,) = 0. Simultaneously the intrinsic excitation energy produced by transfer in (1) is equal in both nuclei. In fact, one can immediately prove that an effective change of the mass number (3) is given only by the different shell structure of the s.p. energies within both nuclei. Thus there does not exist a systematic trend of the initial mass drift towards symmetry or larger asymmetry as is characteristic for the drift velocity in transport theories. Moreover, in HIC the number of excited nucleons is large and therefore the individual shell structures do not show up in a substantial mass drift in one direction. This result suggests a plausible interpretation of the non-regularities and of the general smallness of the net mass change in TDHF. In the actual case any shell structure is neglected (this means that the summations within eqs. (l)-(3) are carried out over an equidistant s.p. spectrum with s.p. densities gpk= g&_= g(Ak) = &Ak MeV-’ (k = 1,2)) and thus the initial mass drift vanishes, (AA,) = 0, and the intrinsic excitation energies produced by transfer are equal in both nuclei. As can be further seen from fig. 1, excitation and transfer channels contribute nearly equally to the total excitation energy ( ko) and excited particle number (A)
R. Schmidr
538
Ne - like
30
60
/
Nucleon exchange
2oNe(170Me’/) t %u (l=60h) fragment Cu- like fragment
90
t (fm/c)
30
60
90
120
t (fm/c)
Fig. 1. Calculated excitation energy ( Golo)and excited particle number (A) witkin both fragments in the 20Ne(170 MeV) +WCu collision for the partial wave l= 60A as a function of time f (solid lines). The dotted (dashed) lines represent the corresponding contributions which result from the transfer (excitation) of nucleons.
in both nuclei although the relative tendencies are opposite within both fragments. The difference between (&,)(*) and (&a) t2) is about 5 MeV and thus small as compared to the total excitation energy of about 65 MeV. The same holds for the number of excited particles. This result which is qualitatively the same for all partial waves and target-projectile combinations considered leads to consequences on the mechanism of the relaxation of intrinsic energy towards the interfragment equilibrium discussed within the next section.
3. Change of the mass ~~rnrne~
during the ~uilibration
One of the most outstanding problems in the understanding of HI dynamics concerns the equilibration process of intrinsic excitation energy. Up to now there
R. Schmidt / Nucleon exchange
539
have been no realistic dynamical calculations and also the experiments are somewhat contrary. In any case an interfra~ent thermal ~ui~b~urn ~st~bution is characterized by a common temperature T of the system and thus by a mass-proportional division of the excitation energies and excited particle numbers to both nuclei. In contrast, we obtained a nearly equal split of both quantities after the initial excitation (cf. last section). Therefore the equilibration process must be accompanied by energy transfer and simultan~usly by a redis~bution of excited particles to both fragments most likely realized by transfer of always excited nucleons from the light to the heavy fragment until equilibrium is established. Under this assumption the number of transferred particles Nr, can be estimated from the interfragment equilibrium conditions 6),
avoiding any dynamical calculation. In eq. (5), (fii), (fiz) and A,, A, denote the initial excited and total nucleon numbers of the light (1) and heavy (2) fragment, respectively. In Kr-induced reactions the number of “prior particles” (5) has been directly compared to experimental data6) and is found to be in good agreement. Here we shall give a complementary justification of the assumed prior-particle mechanism (5). Within a equilibrated estimated according to
Fermi gas the common temperature
T=
j6~-~(&)/g(A, +A,)
of the system can be
(6)
with (&> the total excitation energy of the system. With the temperature (6) the equilibrium number of excited particles Nk* within the Fermi gas of fragment A, (k = 1,2) is fixed: N,*( Ak) = ln2g( A,)T.
(7)
Thus the ratio between the actual excited particles and the equilibrium number (7) can be used as a measure for the deviation from the equilibrium distribution within both fragments. In table 1 we present these ratios within the fragments before and after the exchange of prior particles (5). As can be seen from this table the initial stage of the collision dynamics is characterized by a Iarge overshoot of excited particles within the light fragment whereas the heavy partner nucleus “feels” a lack of excited particles. After the exchange of excited particles via (5) the ratios approach the equilibrium value of 1 within both fragments. The smail deviations of about 15% from the equi~b~um value 1 refer to additional re~r~gement mechanisms of the s.p. occupation probabilities which of course also contribute to the estabiishment of the thermal equilibrium. So, in the Ne + Cu case the total number
540
R. Schmidt /
Nucleon exchange
TABLE1 Ratio between the actual excited particles and the equilibrium values (7) before (( kk))/Np) and after ((( fik) T Nr)/NF) the exchange of prior particles NP within the lighter (k = 1) and heavier (k = 2) fragment for representative partial waves I in 20Ne- and 40Ar-induced collisions
System
@Ar(270 MeV) + ‘O”Mo
*‘Ne(l70 MeV) +64Cu
“Ne(220 MeV) + 19’Au
A,
and
A,
120 110 100
1.81 1.88 1.96
0.67 0.69 0.72
1.00 1.04 1.07
1.00 1.03 1.07
70 66 60
2.37 2.56 2.62
0.67 0.71 0.73
1.09 1.13 1.16
1.08 1.15 1.19
111 99 90
5.60 6.28 6.58
0.38 0.42 0.43
0.94 1.05 1.09
0.78 0.86 0.89
denote the mass numbers of the projectile and target, respectively.
of excited particles should be slightly decreased within both fragments, whereas in the Ne + Au system additional particles should be excited within the heavy fragment by conserving the total excitation energy. Nevertheless, the simple estimates presented favour the exchange of initial excited particles from the light to the heavy fragment as the dominating mechanism in order to reach the interfragment thermal equilibrium in asymmetric HIC. As a typical example we present in fig. 2 the change of mass asymmetry (5) as a function of incident angular momentum and energy as well as the absolute value of the common temperature (6) for the Ne + Au system. The change of the mass asymmetry remains small for near-grazing collisions and reaches maximum values for complete damped partial waves. The temperature and number of prior particles increase with increasing bombarding energy. However, at a fixed temperature (e.g. 1.5 MeV) one obtains a decreasing number of prior particles Ivr = 8,6,4 with increasing bombarding energy E = 220,290,400 MeV, respectively. This interesting feature strongly refers to the dynamical effects imposed in (1) and (2) which effectively results in a decreasing number of total excited nucleons at fixed TKEL but increasing bombarding energy, and thus in an effective increase of the mean s.p. energies excited in single transitions. In table 2 we discuss the dependence of Nr, on the initial mass asymmetry A, - A,/(A, + AZ). Thereby we fixed the total excitation energy and select collisions with nearly the same bombarding energy per nucleon. One obtains a nearly linear increase of the numbers of prior particles with increasing mass asymmetry. It will be shown in the following that the regularities of NP as a function of incident energy and mass ~y~et~ allow for a consistent interpreta-
541
R. Schmidt / Nucleon exchange
.
dN
dN
542
R. Schmidt
/
Nucleon exchange
TABLE 2 Number
of prior particles NP at a fixed heat energy (J&i,) = 50 MeV and nearly the same incident energy per nucleon, E/A,, for different initial mass asymmetries A, - A,/( A, + AZ)
Collision
0.33 0.43 0.52 0.82
8.4 7.3 8.5 8.8
R4Kr(704 MeV) + 166Er 4oAr(290 MeV) + ‘OOMo 20Ne(170 MeV) +64Cu 20Ne(175 MeV) + 19’Au
4.2 4.9 5.5 8.5
tion of rather different phenomena of the total mass drift observed within a broad range of target-projectile combinations.
4. Evolution of the mass asymmetry along the ridge-line potential During the transport stage the change of the mass asymmetry is induced by the evolution of the system along the ridge line of the potential energy U,(A,) as a function of the mass asymmetry A,. The dynamics of the mean value (A,)(t) is determined by the drift velocity V,(A,) of the corresponding Fokker-Planck equation. In standard diffusion models (SDM) the drift equation is solved with initial conditions of projectile-target asymmetry, (A,)(? = 0) = A,. The change of the mass asymmetry during earlier stages of the collision leads to modified initial conditions and the solution of the drift equation reads (A,)(t)=&+,+
U&.-&J&
(8)
which represents the basic equation of the modified diffusion model (IL%DM)~).As in previous applications we use the drift velocity proportional to the gradient of the driving potential as well as to the diffusion coefficient DA (Einstein relation) which is proportional to the temperature (6) DA = DjT, with Dj the diffusion constant. In the following we shall discuss the multifarious consequences of the modified drift equation (8) on the mass transport in Ne- and Ar-induced collisions.
4.1. DRIFT
AGAINST
THE STATIC
DRIVING
FORCE
In fig. 3 we present for the 20Ne (170 MeV)+64C~ collision the calculated potential energy U,( A,) as a function of A, and partial waves 1. From the position of the initial configuration at A, = 20 one would expect a strong drift towards symmetry for the range of partial waves which contribute to scattering (60 Q 1 G 80). This is demonstrated in the left part of the figure where we present the predictions of the standard diffusion model (eq. (8) with ZVr= 0). As expected from the potential
R. Schmidt
/ Nucleon exchange
-t
$!
E!
--4)
‘.
I
I
8 -
(WJ)
‘.
o,
LzP/3P
‘.
-tD
‘\
‘.
_
b-i-
544
R. Schmidt / Nucleon exchange
2!Ne(170MeV)
t %u
Va calculated at the projectile-target Fig. 4. Mean diffusion time interval 7dif and drift velocity asymmetry A, = A, (solid line) and the modified initial asymmetry A, = A, - Np (dashed line) as a function of partial wave 1.
energy U,, elements with 2, > 10 are preferentially produced theoretically, whereas experimentally the opposite trend is observed. A variation of the diffusion constant Do = 5 x 1021 amu2. s-l does not change any qualitative prediction. Within the A modified diffusion model we obtain a satisfactory agreement. A variation of 0,” can now be used to improve the quantitative agreement which is beyond the scope of this work due to the simplicity of the diffusion model used. To get an impression from the timescales and absolute values of the drift velocity, we plotted in fig. 4 the diffusion time interval +rdifand VAcalculated at the initial (A, = Ar) and at the modified initial (A, = A, - NP) configuration. As can be seen, the prior particles NP increase the drift towards symmetry but do not change the qualitative behaviour of V, (in contrast to the 20Ne + ‘97Au,discussed below). The evolution of the angular distributions of elements exhibits useful information on the mean lifetime of the double nuclear system in order to reach a given asymmetry Z,. Fast processes are concerned with forward-peaked d2cr/dZ,dS2, whereas a l/sine angular distribution indicates that these products are associated with substantially longer times (2 one rotational period). Thus the evolution of the form of the angular distributions as a function of Z, provides a sensitive test of a
R. Schmidt
“Ne
-
MDF
----
SDM
/
Nucleon exchange
10 MeV) + %
__I! -!-
sine
i’
1 30
I
I 90
I
I
30
,
I
90 0 c.m
I
30
90
..
30
90
Fig. 5. Evolution of the form of the theoretical angular distributions as a function of Z-value of the products obtained within the standard diffusion model (dashed lines) and the modified diffusion model (solid lines). Both angular distributions have been normalized to 1 at e,,, = 30°. MDM and SDM predict the opposite trends.
diffusion model concerning the timescales involved. In the frame of any SDM one would expect more forward-peaked angular distributions for Z, > Z, as compared to that of Z, < Z, due to the strong drift towards symmetry. The MDM which takes into account the nucleon exchange within earlier (faster) stages of the collision should predict the opposite tendency. In fig. 5 we compare the predicted evolution of the angular distributions [calculated as described in ref. 14)] within both models. For AZ, = + 4 the MDM approaches the l/sin@ form, whereas for AZ, = - 4 a strong forward-peaked angular distribution is predicted. The opposite tendency is obtained within the SDM. The experiment clearly confirms the predictions of the MDM which is demonstrated in fig. 6. The form as well as the absolute values of the unnormalized theoretical distributions agree well with the experimental findings, whereas in any SDM4) both quantities behave opposite to experiment. In the spirit of the proposed MDM the general tendency that elements with Z, < Z, are substantially produced faster than elements heavier than Z, should be realized in any asymmetric HIC and thus the evolution of the angular distributions can be used as a further test on the validity of the model. We shall now discuss the 20Ne + ‘97Au collisions. The calculated potential energy for the heavy and strong asymmetric 20Ne + ‘97Aureaction is presented in fig. 7. In this case the initial configuration A, = 20 lies in the vicinity of a Bussinaro-Gallone
R. Schmidt / Nucleon exchange
546
“Ne(170 MeV) + %u Z=B
Z=6
z=12
Z=lO
z-14
Q
\ \ \ \ \ \ \ \ \ \ \
, \ \ t I \ \ \ \ \ \
~
1 S_
L
L
\
\ \ 5 \ \ \ \
x_
~
-
experiment
__--
calculated @KIM)
\
\
‘_*
i
c
I I I I 30
\ \ t
90
1
30
\ \
I
\
/
‘.
/’
L
I_-
90
Fig. 6. Comparison between experimenta14) and theoretical angular distributions of representative pro&J&s Z. In contrast to fig. 5 the theoretical distributions (dashed lines) obtained within the modified diffusion model are not normalized.
maximum of U,(A,). In such situations a nearly symmetric mass distribution around 2, = Z, is theoretically expected due to the smallness of the mean drift velocity and independent of the value of the diffusion constant Dj = 15 x lO*l amuZ +s-l. In contrast, the experiment exbibits a dramatic asymmetry in the element distribution which is qua~tatively well reproduced within the MDM. To demonstrate the origin of the enormous difference between the predictions of the SDM and MDM we present in fig. 8 the influence of the prior particles on the drift velocity for the actual case. The vanishing drift of the target-projectile combination A, = A, is changed dramatically in its absolute value as well as in sign, except in the vicinity of grazing collisions. Thus the diffusion process starts with a strong drift towards larger asymmetries for a broad range of partial waves leading to qualitative agreement with the experimental distribution. To obtain a more quantitative agreement (especially for elements Z, > Z,) the A, dependence of the transport coefficients should be taken into account, and thus numerical calculations of the probability distribution seem to be necessary althou~ the general trends will not be changed. In order to
R. Schmidt / Nucleon exchange
I
I
I t
8
pvJJy ‘ZP/9P
548
R. Schmidt
/ Nucleon exchange
20Ne(175MeV)
+ lg7Au
-z-6 i; e-8 3 -10 -12
I
-1L
I
I
I
I 80
70
I 100
90 I
Fig. 8. Calculated drift velocity VA as a function of partial wave and different asymmetries A, =A, (dotted line), A, = A, - NP (solid line) and a mean asymmetry (dashed line) centered between A, = A, - NP and A, = A, (see text).
demonstrate element between values,
v, = vJt(A,
remarkable,
of the A, dependence
we defined
A, = A, - Nr
distribution theoretical
4.2. LACK
the influence
distribution
and
a mean
drift
the a-particle
- N, + A,))
of the drift
v, calculated
configuration
(fig. 8) and present
velocity
at a mean A, = A,
V, on the asymmetry
for negative
the corresponding
VA
element
in fig. 7. Although there is no qualitative change in the behaviour of the cross section the quantitative difference between both approximations is especially
for elements
Z, > Z,.
OF MASS DRIFT
The effect of lack of mass drift in Kr + Er reactions has been discussed within a separate work6). Here we shall present a complementary analysis of the 40Ar(270 MeV) + ‘O”Mo collision. The reason is twofold: first there are qualitative differences in the lack of drift behaviour between Kr + Er and Ar + MO collisions; and second, we found tendentious deviations between transport-theoretical and experimental variances which are worth discussing. In fig. 9 we present the mean value (Z,) and variance ui as function of TKEL obtained within the SDM and MDM. In this collision the diffusion constant DoA = 7 x 1021 amu2. se1 has been determined to reproduce the absolute values of
549
R Schmidt / Nucleon exchange
“Ar (2’70 MeV) + looMo 21 / z_____---V ,8
-
--------
_________------.
.
1’
_--H -
.
l
.
l
++++++
g)M +++tMDM
+
lo’ -
+JN w
-
exp maximal mean value of TKEL
I 50
TKEL(MeV)
100
I
I
150
Fig. 9. Lack of mass drift in the ““Ar + loo M o collision. Experimenta13) and theoretical mean values (2,) and variance ai as a function of TKEL are compared. The experimental maximal mean value of TKEL is indicated by an arrow.
the experimental ui values within the deep-inelastic region. Therefore the drift velocity (8) is completely determined. The SDM predicts too large a drift towards symmetry. The MDM leads to theoretical mean values (2,) which stay at nearly constant values for all TKEL in accord with experiment?. This is the main difference from Kr + Er collisions2) where the lack of mass drift is realized only within a certain range of TKEL and a strong drift towards symmetry is observed for large energy losses, theoretically induced by the larger drift velocity and interaction times as compared to the lighter system Ar + MO. In the actual case the mean values (Z,) and variances us have been measured 3, up to TKEL larger than the maximal mean value (= 105 MeV) of the experimental spectrum. Thereby, the experimentalists found an interesting new feature. The mean + The insignificant increase of the theoretical mean value (Z,) relevance because of the large fluctuations within the TKEL.
for the largest energy losses has no real
R. Schmidt / Nucleon exchange
550
“Ne (175 MeV) t lg7Au
J
0 70
80
90
100
I Fig. 10. Mean diffusion time interval rdir and time needed in order to reach the a-particle configuration 7FFu as a function of I. Between the trapped partial wave t,, and I,,, the system reaches the a-particle asymmetry A, = A, (or equivalently the partial fused configuration A, = A, + A, -A,) during the binary collision process.
values stay at constant values whereas the variances increase up to a factor of about 2.5. This behaviour of the mass transport cannot be understood in terms of standard or modified transport theories. A possible explanation has been proposed by Grossmann and Brosa”‘) in terms of a random neck-rupture concept. The precise measurement of the variances in the Ar + MO collisions 3, reveals also a further defect of transport theories. As can be seen from the logarithmic plot of c&TKEL) in fig. 9 the form of the variance for small energy losses cannot be reproduced by the diffusion model and the absolute values are underestimated. Recently, Feldmeier 9, has stressed the importance of non-equilibrium contributions to the diffusion tensor for small TKEL. These contributions which result from the velocity mismatch of nucleons belonging to different nuclei would increase the variances, whereas at the same time the drift velocity remains unaffected and thus the conclusions about the drift behaviour in asymmetric HIC drawn within this work.
4.3. MASS EQUILIBRATION
MODE FOR VERY ASYMMETRIC SYSTEMS - FAST FUSION
Besides the very asy~et~c element ~stribution an anomalously large fission cross section as well as an unexplained large cross section of fast a-particles have been experimentally observed in the 20Ne + ‘97Au reactions 5). While these three
R. Schmidt / Nucleon exchange
551
TABLE3 Bombarding energy dependence (E = 150,220,290,400 MeV) of partial waves trapped into a potential pocket (I Q I, ) and the range of partial waves ( lTR < I < I,,“) contributing to the fast-fusion channel TFFU(IFFU)
[ML] 150 220 290 400
[lo-= 57 85 104 126
66 94 114 143
0.90 0.88 0.77 0.50
s] 5.3 1.9 10.0 13.4
are the fast-fusion times (obtained as discussed in fig. 10) and NP the number of prior ne TFFU particles of the partial wave IFFU.
features seem to be unrelated at first glance, they may well be connected as we intend to show in the following. Let us first briefly summarize the experimental statements [for a detailed discussion see ref. 5)]. The experimental critical angular momenta of fission events exceeds the critical l-value of instability of the compound nucleus against fission up to a factor of about 2. The experiments clearly rule out the “fast fission” 16) as the responsible mechanism because the mass width of fission fragments stay at low values. Most of the fast a-particles exhibit a mean energy centered at intermediate velocities between Coulomb repulsion and projectile velocity. Only about one-half of the integrated cross section of this a-particle component can be attributed to “sequential decay of the excited projectile-like fragment” which leads the authors to the conclusion that there exists an additional mechanism for producing fast a-particles, possibly allowing the remaining system to fuse. Within our MDM these experimental observations get a natural explanation. Let us first discuss qualitatively the peculiarities of the nucleon diffusion along the ridge line potential U,(A,) for negative drift velocities V, < 0. Obviously, the nucleon diffusion process stopped at the compact a-particle configuration A, = A, because for very large asymmetries A, =A, shell effects cannot be neglected. The binding energy of the o-particle produces a deep minimum within the potential U,(A,) at A, = A,. The remaining (partial fused) system A, = A, + A, - A, exhibits the total intrinsic excitation energy (Z&) because an a-particle cannot be excited. Assuming a binary process one expects the emission of a-particles whereas the remainder will undergo sequential fission due to the large excitation energy. In the following we shall check these expectations quantitatively. A mean diffusion time 7FFU during which the double nuclear system reaches the configuration A, = A, (A2 = A, + A, - A,) can be estimated according to
R. Schmidt / Nucleon exchange
552
“Ne + ‘g7Au
OL
100
/
I
4
GFFU
-
I
200
t
300
I
i
i
too
ELab(MeV)
Fig. 11. Calculated cross section of trapped partial waves on and including the fast-fusion mechanism oFfU as a function of incident energy. The calculated points are connected by straight lines. Experimental points of the fusion-fission cross section are from ref. s).
In fig. 10 we compare this time with the diffusion, 7diI, in a binary collision. Actually, I FFU where we found r rru Q rdir and thus A, + AT - A, is realized during the binary time interval, rFFU < 10e2i s (fast fusion). section
total time interval available for there exists a certain l-range I,, the partial fused configuration collision process within a rather Therefore the theoretical fusion
uFFU = mX2(iFF” + Q2 is increased as compared to the cross section of partial waves I< I, trapped into a potential pocket uTR = nX2(Ir, + l)*.
mass -c 1 G
A, = short cross
(10) which are
(11)
We found the fast-fusion mechanism to be realized within the whole range of incident energies studied expe~ment~ly (cf. table 3). In any case we have Irru > I, and thus the theoretical fusion cross section is increased considerably by the fast-fusion mechanism. In fig. 11 we compare the cross sections (10) and (11) with the experimental fission cross section. As can be seen, the fast-fusion mechanism explains the enhanced fission cross section or the lowest and highest incident energies, whereas within the intermediate range of E there should be an additional
R. Schmidt / Nucleon exchange
553
TABLE4 Bombarding energy dependence of different contributions to the spectrum of o-particles in the *‘Ne + 19’Aucollision E
Ii
7
[lo-*1 s]
‘eq
If
VCd
E,
E,‘“P(2)
150
61
1.5
4
9
20.7
23.6
_
220 290 400
90 109 135
1.4 1.3 1.0
6 8 10
14 19 33
20.1 20.7 20.7
28.0 34.1 60.5
30 35 55
E,‘“P(l)
40 50 12
E denotes the bombarding energy, V,,,, the Coulomb energy of evaporated a-particles, EzP(1) the experimentally observed o-particle energy corresponding to projectile velocity (direct break-up), and E,‘“P(2) the experimental mean value of the intermediate component of fast particles [see fig. 11 in ref. ‘)I, r is the mean contact time for the initial partial waves li centered in the I-range leading to fast-fusion events (cf. table 3). les and I, are the calculated equilibrium (sticking) and final relative angular momenta of the relative motion, respectively (eq. (13)). E,, denotes the theoretical mean value of o-particles resulting from the fast-fusion mechanism, calculated via (12). Note the good agreement between & and E,‘“P(2). Energies are in MeV, angular momenta in h.
mechanism for enlarging the fission cross section, most probably by direct a-particle break up (cf. discussion below). Table 3 gives also an impression of the energy dependence of the mass equilibration time rrr,,. With increasing bombarding energy the fast-fusion time decreases reaching rather small values of about 7rFU = 5 x 1O-22 s for the largest energy considered. The increasing smallness of rrFU (essentially caused by the increasing number of prior particles Nr) has consequences on the energy spectrum of emitted a-particles. The final mean kinetic energy of the a-particles E, produced by the fast-fusion mechanism is given by the sum of Coulomb repulsion and centrifugal energy, E, =
A,+A,-A A,+A,
a( Goul+&)
(12)
the moment of inertia of the relative motion and I, the final relative with & angular momentum which can be estimated according to “) 03)
where Jr, J2 denote the moments of inertia of the two fragments A, = A, and respectively, and r, is the relaxation constant of angular momentum dissipation. For sufficient long contact times t > r, the final angular momentum approaches the equilibrium value I,, of the sticking configuration. Taking a constant decay time of r, = 0.6 X 10e21 s [ref. “)I we summarize in table 4 the calculated leq, I, and A?, values for representative partial waves Ii (localized in the midst of the I-range contributing to fast fusion, cf. table 3) and compare the theoretical mean values I!?, with characteristic experimental energies in the spectrum
A, = A, + A, - A,,
554
R. Schmidt
/
Nucleon exchange
of a-particles. We obtain a surprisingly good agreement with the intermediate component E,‘“P(2) for all energies considered. For the lowest bombarding energy of E = 150 MeV the theoretical mean value E, = 23.6 MeV does not remarkably exceed the energy-independent Coulomb repulsion of V,, = 20.7 MeV. In this case the a-particles originating from the fast-fusion mechanism cannot be decided from the evaporated a-particles within the experimental spectrum. This is in accord with the measured threshold for production of fast a-particles of about 200 MeV [ref. 5)] and at the same time explains the good agreement between aFFU and the experimental fission cross section at this energy (cf. fig. 11) in the absence of fast a-particles. With increasing bombarding energy the contact times 7(Zi) decreases resulting in rather large values of I, well above I,,. The emitted a-particles exhibit a mean energy centered at intermediate values between vcO”, and that of direct a-particle break up E$P(l) in agreement with experiment. Closing this section we briefly comment on the influence of direct break-up processes on the fission and deep-inelastic cross sections. This mechanism can explain the remaining difference between aFFU and the experimental fission cross section at intermediate energies (cf. fig. 11). However, with increasing bombarding energy the evolution of the projectile-like fragment (resulting from an initial (Yparticle break-up) results increasingly in an incomplete deep-inelastic reaction [INDIC18)] instead of incomplete fusion due to the increasing values of partial waves connected with that mechanism. Therefore we expect a minor influence of this mechanism on the experimental fission cross section at sufficient large incident energies, but at the same time an increasing importance of this reaction channel on the mass distribution of deep-inelastic events. This argument is in accord with the good agreement between aFFU and the experimental fission cross section at 400 MeV (cf. fig. 11).
5. Summary and outlook We have analyzed nucleon-exchange processes during three stages of a HIC. The vanishing mass drift during the initial stage obtained within a semiclassical theory in this work confirms the general observations made in TDHF calculations. We have shown that the equilibration stage must be accompanied by a redistribution of initial excited particles to both nuclei and have assumed transfer of excited nucleons to be the dominating mechanism. This leads finally to modified initial conditions for the nucleon-diffusion process along the ridge line potential during the transport phase. The gross feature of the mass transport in asymmetric HIC can be understood in terms of this concept. Partia12) and tota13) lack of mass drift, drift against the static driving force4) as well as vanishing production cross section for elements heavier than the projectile5) are explained on a common basis in terms of a continuous evolution mainly generated by the A, -A/A, + A, dependence of the number of prior particles. In addition, the anomalously large fission cross section in 20Ne + ‘97Au
R. Schmidt / Nucleon exchange
555
collisions can be attributed to a very fast mass-equilibration mode (fast fusion) connected with the emission of fast a-particles. The predictions and the validity of the MDM can be checked by further experimental studies: (i) The exchange of particles prior to the transport phase leads automatically to an asymmetry in the mean diffusion times for the production of fragments heavier and lighter than the projectile. Lighter (heavier) fragments are associated with shorter (longer) times resulting in different evolutions of the angular distributions d*u/dZ, dO for elements with the same relative mass change, Z, - AZ (Z, + AZ) (cf. subsect. 4.1). This effect should be realized in any asymmetric HIC independent of the gross behaviour of the mass drift. Experimental evidence for an asymmetry in interaction times has been found recently in Kr + Bi collisions [cf. the discussion on fig. 6 in ref. 19)] where the observed different interaction times for Z, > Z, and Z, < Z, are quite unexpected because the angle-integrated charge distribution da/dZ1 exhibits drift towards symmetry (partial lack of mass drift). (ii) For systems with partial lack of mass drift *,19)the number of predicted prior particles can be directly compared with experimental data 6). (iii) The proposed interpretation of the anomalously large fission cross section in *‘Ne + ‘97Au experiments in terms of fast fusion suggests the search of this effect in other systems which exhibit an initial asymmetry in the vicinity of the BussinaroGallone maximum of U,(A,) [and thus for target-projectile combinations which lie beyond the range of the fast-fission systematics 16)]. We expect the large fission cross section to be correlated with fast a-particles independent of a possible a-particle structure of the projectile. From the theoretical point of view an extension of the proposed treatment to the explicit inclusion of shell effects as well as N/Z degrees of freedom is desirable to achieve a detailed understanding of the puzzling mass-drift behaviour in asymmetric HIC. I am grateful to Prof. R. Reif for useful discussions and T. Gleitsmann for help in the numerical work.
References 1) A. Gobbi and Ww. Nijrenberg, Heavy ion collisions, vol. II, ed. R. Bock (North-Holland, Amsterdam, 1980) ch. 3; L.G. Moretto and R.P. Schmitt, Rep. Prog. Phys. 44 (1981) 533 2) G. Rudolf et al., Nucl. Phys. A330 (1979) 243 3) W. Bohne et al., Z. Phys. A313 (1983) 19 4) G.J. Mathews et al., Phys. Rev. C25 (1982) 300 5) Ch. Egelhaaf et al., Nucl. Phys. A405(1983) 397 6) R. Schmidt, Z. Phys. A320 (1985) 413 7) L.G. Moretto, Z. Phys. A310(1983) 61
556
R. Schmidt / Nucleon exchange
8) W.J. Swiatecki, Nucl. Phys. A428 (1984) 1990;
9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
J. Randrup and W.J. Swiatecki, Nucl. Phys. A429 (1984) 105; J. Randrup, Nucl. Phys. A307 (1979) 319; A327 (1979) 490 H. Feldmeier and H. Spangenberger, Nucl. Phys. A428 (1984) 2230 R.V. Jolos, R. Schmidt and J. Teichert, Nucl. Phys. A429 (1984) 139 R. Schmidt, to be published R.A. Broglia and A. Winther, Nucl. Phys. A182 (1972) 112; Phys. Reports 4C (1972) 154 K. Dietrich and K. Hara, Nucl. Phys. A211 (1973) 349 R. Schmidt, G. Wolschin and V.D. Toneev, Nucl. Phys. A311 (1978) 247 S. Grossmann and U. Brosa, Z. Phys. A319 (1984) 327 C. GrCgoire, C. NgB and B. Remand, Nucl. Phys. A383 (1982) 392; H. Lefort, Z. Phys. A299 (1981) 47 G. Wolschin, Nucl. Phys. A316 (1979) 146 F. Guzman and R. Reif, Nucl. Phys. A436 (1985) 294 J.R. Birkelund er al., Phys. Rev. C26 (1982) 1984
+ For reviews, see ref. ‘).