Time-dependent
Hartree-Fock
andthe one-body dissipation
for
head-on collisions
D.
M. BrinkTheoretica/ Physics Department, Uniuersity ofOxford, 1,EebleR oad,,Oxford, England
F1.Stancu
Institut dePhysique, Uniuersity ofLiege, Sart Tilman, 4000Liege j,Belgium (Received 18 August 1980)
Trajectory calculations for head-on collisions of
"Kr+
'"La and '"Kr+'"Bi
are performed based on several nucleus-nucleus potentials and the window formula for friction. Results are compared to the time-dependentHartree-Pock calculations from literature inthe same energy range.
NUCLEAR REACTIONS Head-on classical trajectory calculations 86Kr+~3~La
atE&b=360-1000MeV and 8Kr+ oBiatE&ab=500-1000MeV.
I. INTRODUCTION II. DESCRIPTION OFTHE TDHF RESULTS
The aim of the present study
is
to get morein-sight into the one-body aspect ofthe dissipative mechanism of heavy-ion collisions.
For
thispur-pose we make trajectory calculations based on the
window formula''2 and compare the results with
time-dependent-Hartree-Pock (TDHF}
caicuia-tions3'4 for head-on collisions. The windowfor-mula and the time-dependent-Hartree-Fock meth-od'
are
based on the neaxly independent motion ofnucleons in
a
deformed container and ina
time-dependent mean field, respectively. In other
re-spects,
however, theyare
different. The windowformula
is
based ona classical
perturbative treat-ment; the TDHFis
based on a self-consistentquantal description. Although one hopes to get a better understanding ofthe dissipation mechanism
ofcolliding nuclei by comparing the two methods, it
is
difficult to xelate them theoxetically. 6 Vjetherefore compare the resul.
ts
of the two ap-proaches. A similar ideais
contained inRef. 7,
which includes in addition an extra excitation
me-chanism of the collective surfaces modes.The
trajectory
calculations we present in this paperare
for head-on collisions of the system"Kr+
'"I,
a
and 84Kr+'08Bi,as
in the TDHF cal-culations ofRefs.
3and 4and the resultsare
com-pared in the same energy range. The friction
is
given by the window formula and we make several choices for the conservative
force.
Some pre-liminary results have been presented elsewhere.In
Sec. II,
we summarize the TDHFresults.
InSec.
III,
we present our trajectory calculationsand discuss them in relation to the TDHF
results.
In the last section we draw some conclusionsabout the role of the conservative and nonconsex-vative
forces.
An important question in the TDHF calculations was to
see
ifheavy ion systems with composite massgreater
then 200undergo fusion.It
turnedout that the threshold behavior was rather
com-plicated and
is
different from the presentclassi-cal trajectory classi-calculations, as we shall see below.
Ez+'
I
g.
TDHF results for head-on colli-sions show two regimes of fusionlike behavior:a
narrow region just above t.he Coul. ombbarrier
atE„b-410
MeV and another broader region be-tweenE
„b=
650and 850MeV.For
energies below650MeV, except around Ey&b
—
410 Me+) thepro-jectile
is
reflected from the target, and beyond850MeV it
"passes
through" thetarget.
In a1.lthese
cases
the final kinetic energy has an almost constant value of 195 +5MeV.8Ky+
Bi.
For
this system TDHF calculationsshow only one fusion region between
E„b=
850Me& and
E „b
=
1100MeV.For E
„„&
850MeV theprojectile
"bounces"
off the target while forE„„&
1100MeV it passes "through."
In the regionjust above the
barrier
some very long-livedcon-figurations have been seen. Accoxding to Ref. 3
such dynamical resonances
recall
the fusionlike behavior which was found just above the Coulombbarrier
for the 86Kr+'"I,
a
system.Another quantity ofinterest in head-on collisions
is
the stopping time I;,.
%e
define it as the inter-val from the instant when the nuclear surfacestouch (s
=0}
to the instant ofclosest
approach. Such stopping timesare
extracted from the TDHFcalculations and
are
indicated in TableI as a
function of the bombarding energy.As
Fig.
1ofRef.
3 suggests, the nuclei approacheach other very
fast.
They sl.ow down and thenremain for
a
period at an almost constantTINK-DEPENDENT
HARTRKK-FOCK
AND THKONK-BODY. . .
TABLE
I.
Stopping times I;,in units 10 2 s asa function ofthe bombarding energy E~,bfor"Kr+
Bi. The second column indicates values of t,extracted from Ref. 4and column 3gives the stopping time obtained%1th the standard proximity potent181.
TDHF
et
al.
"
2.
Coulomb (Bondorf)+ nuclear potential of Siwek-Wilczynska and Wilczynski' (SW-W).3.
Coulomb (Bondorf)+ standard proximitypo-tential" (SP),
4.
Coulomb (Bondorf)+ modified proximitypoten-tial ofRandrup'3 (MP). 4.9 4.0 3.0 2.5 2.1 5.2 4.0 3.4 3.0 2.8
Figures 1 and 2 show the various potentials for
Kr+
I a
andKr+
Bi.
'Zheyare
givenas a
function of the separation distance R between the nuclear centres related to8 bymum separation distance. In estimating the
stop-ping time we consider the
first
instant when the xelative motion appears to have gtopped, andtherefore the values of
t,
given in TableI
mightbe underestimated. One can
see
that the stoppingtime
is
ofthe order ofa
few units of 10s
and that, itis
shorter for higherenergies.
In the nextsection we shall make
a
comparison with the stopping times resulting from c]lassicaltrajectory
calculations.For
each potential the nuclear radiiR,.
axe takenaccording to the corxesponding definition given
either in Ref. 11
or
12.
But in fact the numerical values ofC, are
almost the same for the three po-tentials. These give C&+C2-
—10.
7 fm for 86Kr+'39I.a
andC,
+C,
=11.
8fm for 84Kr+20'Bi. Thestandard pxoximity potential hag
a
shallow pocketof
-5.
5MeV deep with xespect to the baxriex forKr
+
Qa.For
Kr +
Bl
the potential lg evenshallower, the pocket being only
-2
MeV deep. InIn this section we write an equation for the
rela-tive motion ofthe two heavy iong ina
head-on col-lision.%e
assume that the nuclei retain their spherical shape throughout the collision, that themass associated with the relative motion is just
the reduced mass p,, and the force acting between
the two nuclei has
a
conservative part anda
fxic-tionalpart.
If
we calls
the separation between the surfaces ofthe interacting nuclei, the equation to be solvedin head-on collisions
is
400-
300-X 200-86K&+139L&On the right-hand side
(rhs)
thefirst
termis
theconservative
force
derived from the potentialV(s) discussed below. The second term
is
theradial component ofthe window friction formula defined as the product oftwice the radial velocity
ds/dt and the total flux N(s) passing through the window, for which we use the analytic expression ofHandrup.
Four different choices have been made for
V(s):
1.
Pure Coulomb potential for overlappingcharge distributions using the formula ofBondorf
/
0R (fm)
FIG.1. Interaction potential of Kr+ La as a function ofthe separation distance between nuclear
centers
(-")
—
Coulomb, (—
)—
8W-W+Coulomb, (x—
x—
x)—
SP+Coulomb, (——
-)—
MP+Coulomb.0.
M.BQINK
ANDFL.
STANCU I 5QQ-"""""
4QQ-I )I I X ' *. . & ~ X X~ X 84 209 Kf'+ Bt ..500 ~$200 .1200 X~X~X~
~X~X ~X .X~X~X 500 X X ~X~X~X X «X~ 500 1200 -3QQ-(U t 8IK+209Bi 500 2QQ-~ ~ 100y'
4 8 t2 't6 20 24 t (in 10 s)FIG.4. Trajectories for Kr+ BiatE&b=500 MeV and El~=1200MeVfor the potentials ofFig. 1(same Iegend). The separation distance g between surfaces is 8
=g
—11.
63for SW-Vf and 8=P
—11.
66for8Por MP potentials.8 R {fIn)
FIG.2. Sam.easFig. 1,but for Kr+
Bi.
both
cases
the 3Vjf-Vf potential hasa
deepex' pocketthan the proximity potential. Typical
for
the mod-ified pxoximity potentialis a
broad bax'riex beyond which the potential decreases linearly towards thecenter.
In
Figs.
3and 4 we show results of EII.(&)solved wllll tile illltiRI collditlons
s(0)
= 3.
2 fnl Rlld=
[~(&.
—I'(3-3))/'V]'"
&0
fol Kl
+
Qa ancl Kx'+81,
x'espectlvely. The initial pointis
chosen because the frictionI I I I I I I I I I 360 ~X
~)c~
e~~ 910 ~~~—
x-~ 960 950 -8-86K t39L~
910zero separation betvreen rentres
t 1 a ~~~ a
8 t2 't6 20 24
t (in 10 s)
FIG.
3.
Trajectories for S~Kr+ La atEl~b=360MeVRnd El~b 910MeV forthe potentlRls ofFig.'1
(SRIne
legend). The separRtlon dlstRnce 8 between surfaces is 8=& —10.70for
8%-
Wand s=8
—10.73for SPorMP potentials.
stax'ts to act at
s
= 3.
2 fm.9 Calculations mere done in the same energy range asfor
the TDHF study. 3'4For
each potential we show results forthe t.omest and highest val,ues of the energy
con-si
de 1"ed.86'
Ey+
f88I
a.
Most ofour discussion mill concernthe standaxd px'oximity potential. Vfith the other
potentials the xesults
are
qualitatively independentof energy as one can
see
fromFig.
3.
Inparticu-lar
for the pure Coulomb the fragments alwayssepaxate, for the
3'-%
potential the projectileis
always captuxed in the pocket and in the Mppo-tential the projectile always goes to the
center.
Vjfith the proximity potential me obtaina
lax'gefusion window from just above the
barrier
(2„,
=
370Me&) until about1000 MeV, much wider that the lower fusion region in the TDHF calcula-tions.'Below and above this region the projectile
bounces back. In the fusion x'egion fouod in our
calculations me observe oscil.lations ib
s as a
function of time, the amplitude ofwhich increasesmith the bombarding enexgy.
8
Az+
Bi.
Inthe energy range
500-1200
MeV,none ofthe considered potentials gives the
be-havior observed in TDHF calculations. AsFig.
4 shoms, the projectileis
captured in the pocket of the3%-%
potentialor
near the center ofthe Mp potential over the whole enexgy range. The3P
potential givesa
fusion windom from relativeen-ergies
just above thebarrier
or
B„b-440
MeVun-tit.about
E„~-700
MeV. This type of fusionis
more reminiscent ofthe low energy fusionlikebe-havlox' seen 1n TDH,
F
ealculat1on of Kx'+ Qa, and itmould correspond to the long-livedconfig-urations found at
E„=
5j.oand 525 MeV in the TDHF studv of84Kr+20~Bi. In our calculationsTINK-DEPENDENT
HARTREE-POCK
AND THKONE-BODY.
..
the fusion appears as
a
combined effect of theexistence of
a
shallow pocket in the SP potentialand the strong
character
ofthe windowfriction.
For
both systems"Kr+
'39I.a
and"Kr+"'B»
wefind some similarity between
trajectory
calcula-tions and the TDHF
results.
All the potentials weuse, except the MP potential. , stop the
projectile.
The stoppingtimes,
as defined in the previoussection,
are
sensitive to the conservativeforce.
For
'4Kr+'"Bi
the values we foundare
compar-able but somewhat larger than the TDHF stopping
times.
Except for the Coulomb potential,it
alsodecreases
with the energy like in TDHF calcula-tions. The nearest to TDHF resultsare
those of theSP
potential and theyare
also indicated inTable
I.
The difference between TDHF andtra-jectory calculations essentially appears after the instant of
closest
approach.IV. CONCLUSIONS
The trajectory calculations include the effects ofboth the conservative and the nonconservative
forces.
Therefore we can at most draw somequalitative conclusions about the dissipation.
For
high energies the slowing down of the rela-tive motion in the approach phaseis
initial.ly dueto'the frictional
force,
but in the last stages of the slowing downprocess
the conservativeforce
makesa
difference. At lower energies the fric-tional.force
is
notso
dominant. The conservativeforce
is
more important at all stages and makesa
difference to the stopping times and stoppingdis-tances.
The stopping times calculated using the standard proximity potentialare
consistent withthe results of the TDHF calculations.
In the late stages of the collision our
trajectory
calculations give quite different results fromTDHF. Calculations with the standard proximity
potential lead to fusion except at very high incident
energies. The modified proximity and the
8%-%
potential always give fusion when the incidenten-ergy
is
about the Coulombbarrier.
An importantreason for this difference
is
that the nucleiare
constrained to be spherical in ourtrajectory
cal-culations. Vfe do not allow other degrees offree-dom such as a neck formation to develop in the
separation phase of the reaction. Swiateeki' has
shown that neck formation gives xise to important changes in the potential energy landscape, and
is
crucial in deciding whether the nuclei separateagain after the collision. According tothe model.
ofRef. 14the value of the
"effective fissility"
pa-rameterMw&(R,
+a
)a,
Ris
important for determining whether the nucleireseparate.
IfX
&0.
57 the nuclei should separate unl.ess
the incident energyis
sufficiently high, while ifX
&0.
57,
the composite system shoul.d normally fuse. Inour examplesX= 0.
81for 8Kr+
'"I.
a
andX=0.
97for'4Kr+'"B».
The resultsof the TDHF calculations
are
consistent with this picture in bothcases.
From this discussion it seems reasonable to conclude that
a
more complex description ofclas-sical trajectory
cal.culations would be necessary inorder
to cover certain aspects ofTDHFresults.
A possibility
is
to incl. ude more degrees of free-dom, in particulara
neck radius. Anotherpos-sibility
is
to introducea
timeor
energy dependentpotential. An interesting problem would be to
study the relation between these procedures. This work has been. supported by NATO
Re-search
Grant No.1782.
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J.
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