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Time dependent Hartree-Fock and the one-body dissipation for head-on collisions

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Time-dependent

Hartree-Fock

and

the one-body dissipation

for

head-on collisions

D.

M. Brink

Theoretica/ 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-dependent

Hartree-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 more

in-sight into the one-body aspect ofthe dissipative mechanism of heavy-ion collisions.

For

this

pur-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 window

for-mula and the time-dependent-Hartree-Fock meth-od'

are

based on the neaxly independent motion of

nucleons in

a

deformed container and in

a

time-dependent mean field, respectively. In other

re-spects,

however, they

are

different. The window

formula

is

based on

a classical

perturbative

treat-ment; the TDHF

is

based on a self-consistent

quantal 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 Vje

therefore compare the resul.

ts

of the two ap-proaches. A similar idea

is

contained in

Ref. 7,

which includes in addition an extra excitation

me-chanism of the collective surfaces modes.

The

trajectory

calculations we present in this paper

are

for head-on collisions of the system

"Kr+

'"I,

a

and 84Kr+'08Bi,

as

in the TDHF

cal-culations of

Refs.

3and 4and the results

are

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 TDHF

results.

In

Sec.

III,

we present our trajectory calculations

and discuss them in relation to the TDHF

results.

In the last section we draw some conclusions

about the role of the conservative and nonconsex-vative

forces.

An important question in the TDHF calculations was to

see

ifheavy ion systems with composite mass

greater

then 200undergo fusion.

It

turned

out that the threshold behavior was rather

com-plicated and

is

different from the present

classi-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. omb

barrier

at

E„b-410

MeV and another broader region

be-tween

E

„b=

650and 850MeV.

For

energies below

650MeV, except around Ey&b

410 Me+) the

pro-jectile

is

reflected from the target, and beyond

850MeV it

"passes

through" the

target.

In a1.l

these

cases

the final kinetic energy has an almost constant value of 195 +5MeV.

8Ky+

Bi.

For

this system TDHF calculations

show only one fusion region between

E„b=

850

Me& and

E „b

=

1100MeV.

For E

„„&

850MeV the

projectile

"bounces"

off the target while for

E„„&

1100MeV it passes "through.

"

In the region

just above the

barrier

some very long-lived

con-figurations have been seen. Accoxding to Ref. 3

such dynamical resonances

recall

the fusionlike behavior which was found just above the Coulomb

barrier

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 surfaces

touch (s

=0}

to the instant of

closest

approach. Such stopping times

are

extracted from the TDHF

calculations and

are

indicated in Table

I as a

function of the bombarding energy.

As

Fig.

1of

Ref.

3 suggests, the nuclei approach

each other very

fast.

They sl.ow down and then

remain for

a

period at an almost constant

(2)

TINK-DEPENDENT

HARTRKK-FOCK

AND THK

ONK-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 proximity

po-tential" (SP),

4.

Coulomb (Bondorf)+ modified proximity

poten-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

and

Kr+

Bi.

'Zhey

are

given

as a

function of the separation distance R between the nuclear centres related to8 by

mum separation distance. In estimating the

stop-ping time we consider the

first

instant when the xelative motion appears to have gtopped, and

therefore the values of

t,

given in Table

I

might

be underestimated. One can

see

that the stopping

time

is

ofthe order of

a

few units of 10

s

and that, it

is

shorter for higher

energies.

In the next

section we shall make

a

comparison with the stopping times resulting from c]lassical

trajectory

calculations.

For

each potential the nuclear radii

R,.

axe taken

according to the corxesponding definition given

either in Ref. 11

or

12.

But in fact the numerical values of

C, are

almost the same for the three po-tentials. These give C&+C2

-

—10.

7 fm for 86Kr

+'39I.a

and

C,

+C,

=11.

8fm for 84Kr+20'Bi. The

standard pxoximity potential hag

a

shallow pocket

of

-5.

5MeV deep with xespect to the baxriex for

Kr

+

Qa.

For

Kr +

Bl

the potential lg even

shallower, the pocket being only

-2

MeV deep. In

In this section we write an equation for the

rela-tive motion ofthe two heavy iong in

a

head-on

col-lision.

%e

assume that the nuclei retain their spherical shape throughout the collision, that the

mass associated with the relative motion is just

the reduced mass p,, and the force acting between

the two nuclei has

a

conservative part and

a

fxic-tional

part.

If

we call

s

the separation between the surfaces ofthe interacting nuclei, the equation to be solved

in head-on collisions

is

400-

300-X 200-86K&+139L&

On the right-hand side

(rhs)

the

first

term

is

the

conservative

force

derived from the potential

V(s) discussed below. The second term

is

the

radial 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 overlapping

charge distributions using the formula ofBondorf

/

0

R (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.

(3)

0.

M.

BQINK

AND

FL.

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-~ ~ 100

y'

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 has

a

deepex' pocket

than the proximity potential. Typical

for

the mod-ified pxoximity potential

is a

broad bax'riex beyond which the potential decreases linearly towards the

center.

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 point

is

chosen because the friction

I I I I I I I I I I 360 ~X

~)c~

e~~ 910 ~~~

x-~ 960 950

-8-86K t39L

~

910

zero 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=360MeV

Rnd 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 SPor

MP potentials.

stax'ts to act at

s

= 3.

2 fm.9 Calculations mere done in the same energy range as

for

the TDHF study. 3'4

For

each potential we show results for

the t.omest and highest val,ues of the energy

con-si

de 1"ed.

86'

Ey+

f88

I

a.

Most ofour discussion mill concern

the standaxd px'oximity potential. Vfith the other

potentials the xesults

are

qualitatively independent

of energy as one can

see

from

Fig.

3.

In

particu-lar

for the pure Coulomb the fragments always

sepaxate, for the

3'-%

potential the projectile

is

always captuxed in the pocket and in the Mp

po-tential the projectile always goes to the

center.

Vjfith the proximity potential me obtain

a

lax'ge

fusion 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 increases

mith the bombarding enexgy.

8

Az+

Bi.

In

the energy range

500-1200

MeV,

none ofthe considered potentials gives the

be-havior observed in TDHF calculations. As

Fig.

4 shoms, the projectile

is

captured in the pocket of the

3%-%

potential

or

near the center ofthe Mp potential over the whole enexgy range. The

3P

potential gives

a

fusion windom from relative

en-ergies

just above the

barrier

or

B„b-440

MeV

un-tit.about

E„~-700

MeV. This type of fusion

is

more reminiscent ofthe low energy fusionlike

be-havlox' seen 1n TDH,

F

ealculat1on of Kx'+ Qa, and itmould correspond to the long-lived

config-urations found at

E„=

5j.oand 525 MeV in the TDHF studv of84Kr+20~Bi. In our calculations

(4)

TINK-DEPENDENT

HARTREE-POCK

AND THK

ONE-BODY.

..

the fusion appears as

a

combined effect of the

existence of

a

shallow pocket in the SP potential

and the strong

character

ofthe window

friction.

For

both systems

"Kr+

'39I.

a

and

"Kr+"'B»

we

find some similarity between

trajectory

calcula-tions and the TDHF

results.

All the potentials we

use, except the MP potential. , stop the

projectile.

The stopping

times,

as defined in the previous

section,

are

sensitive to the conservative

force.

For

'4Kr+'"Bi

the values we found

are

compar-able but somewhat larger than the TDHF stopping

times.

Except for the Coulomb potential,

it

also

decreases

with the energy like in TDHF calcula-tions. The nearest to TDHF results

are

those of the

SP

potential and they

are

also indicated in

Table

I.

The difference between TDHF and

tra-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 some

qualitative conclusions about the dissipation.

For

high energies the slowing down of the

rela-tive motion in the approach phase

is

initial.ly due

to'the frictional

force,

but in the last stages of the slowing down

process

the conservative

force

makes

a

difference. At lower energies the

fric-tional.

force

is

not

so

dominant. The conservative

force

is

more important at all stages and makes

a

difference to the stopping times and stopping

dis-tances.

The stopping times calculated using the standard proximity potential

are

consistent with

the results of the TDHF calculations.

In the late stages of the collision our

trajectory

calculations give quite different results from

TDHF. 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 incident

en-ergy

is

about the Coulomb

barrier.

An important

reason for this difference

is

that the nuclei

are

constrained to be spherical in our

trajectory

cal-culations. Vfe do not allow other degrees of

free-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 separate

again after the collision. According tothe model.

ofRef. 14the value of the

"effective fissility"

pa-rameter

Mw&(R,

+a

)a,

R

is

important for determining whether the nuclei

reseparate.

If

X

&

0.

57 the nuclei should separate unl.

ess

the incident energy

is

sufficiently high, while if

X

&

0.

57,

the composite system shoul.d normally fuse. Inour examples

X= 0.

81for 8Kr

+

'"I.

a

and

X=0.

97for

'4Kr+'"B».

The results

of the TDHF calculations

are

consistent with this picture in both

cases.

From this discussion it seems reasonable to conclude that

a

more complex description of

clas-sical trajectory

cal.culations would be necessary in

order

to cover certain aspects ofTDHF

results.

A possibility

is

to incl. ude more degrees of

free-dom, in particular

a

neck radius. Another

pos-sibility

is

to introduce

a

time

or

energy dependent

potential. An interesting problem would be to

study the relation between these procedures. This work has been. supported by NATO

Re-search

Grant No.

1782.

~W.

J.

Swiatecki,

J.

Phys. (Paris) 33, Suppl. 8-9, C5 (1972).

J.

Blocki, Y.Boneh,

J.

R.Nix,

J.

Randrup, M. Robel, A.

J.

Sierk, and %.

J.

Swiatecki, Ann. Phys. (N.Y.) 113,'

330 (1978).

K.

T.

R.Davies, K.R.Sandhya Devi, and M.R.Strayer,

Phys. Rev. Lett. 44, 23 (1980).

K.

T.

R.Davies et

al.

, private communication. SP. Bonche, S.

E.

Koonin, and

J.

Negele, Phys. Rev.

C 13,1226 (1976).

S.

E.

Koonin, Lecutres given atthe International School of Physics "Enrico Fermi,

"

Varenna, Italy, 1979.

VR.A.Broglia, A.K.Dhar, H. Esbensen,

B.

S.Nilsson,

and C.H. Dasso, Phys. Lett. 83B, 301 (1979).

SD.M.Brink and

Fl.

Stancu, Intet'national W'orkshoP

oe Cross Properties of

¹clei

andNuclem Excitations VIII, Hixschegg, Austria, edited by H. Feldmeier (Technische Hochschule Darmstadt, Darmstadt, 1980), p. 162.

J.

Randrup, Ann. Phys. (N.Y.)112,356(1978). OJ.

P.

Bondorf, M.

I.

Sobel, and D.Sperber, Phys.

Rep. C15, 83(1974).

K.Siwek-Wilczydska and

J.

%'ilczynski, Phys. Lett. 74B, 313 (1978).

J.

Blocki,

J.

Randrup, Vf.

J.

Swiatecki, and C.

F,

Tsang, Ann. Phys. (N. Y.)105, 427 (1977).

J.

Randrup, Nucl. Phys. A307, 319 (1978). ~4.

J.

Swyatec@, phys.

Scr.

(tobe publ1.shed).

Figure

FIG. 1. Interaction potential of Kr+ La as a function of the separation distance between nuclear centers (-") — Coulomb, ( —) —8 W-W+ Coulomb, (x — x — x) — SP+ Coulomb, ( ——-) — MP+ Coulomb.
FIG. 2. Sam. e as Fig. 1, but for Kr+ Bi.

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