Quasi-linear response theory of statistical heavy-ion collisions

Nuclear Physics A397 (1983) 141-160 ~) North-Holland Publishing Company

QUASI-LINEAR R E S P O N S E T H E O R Y OF STATISTICAL HEAVY-ION C O L L I S I O N S (II). Analysis of the friction tensor and the energy transport K. NIITA and N. TAKIGAWA

Tohoku University, Department of Physics, 980 Sendai, Japan Received 12 February 1982 (Revised 30 September 1982) Abstract: We apply the recently proposed quasi-linear response theory to the study of energy transport in deep inelastic heavy-ion collisions. By solving a master equation, we show how quickly the canonical distribution function becomes a good representation of the intrinsic state in the case of the random intrinsic excitations proposed by Weidenmiiller and co-workers. We numerically analyze the properties of the corresponding friction tensor. In addition, we demonstrate that the known fluctuation dissipation theorem in the linear response theory is considerably violated for large part of deep inelastic collisions. We then calculate the double differential cross section for three typical examples. The results agree well with the experimental data if we phenomenologically introduce a time-dependent potential. We remark on the difference of the present calculation from that of the linear response theory. We comment also on the validity of a time-dependent theory, which derives the basic equations from timeindependent equations by assuming a one-to-one correspondence between the time and the relative distance.

1. Introduction In a previous paper 1) (referred to as I) we have developed a time-dependent transport theory, the quasi-linear response theory (QLRT) of heavy-ion collisions, which consistently treats the relative and intrinsic motions based on the many-body von Neumann equation. Especially for deep inelastic collisions (DIC) we have developed a non-perturbative quantum-statistical approach by assuming Weidenmiiller's random coupling hamiltonian between the relative and intrinsic motions. We have thus discussed the effects of the Green functions that take into account the higher order effects of the coupling hamiltonian on the time evolution of the intrinsic state, the friction tensor and the fluctuation dissipation theorem. It is the purpose of the present paper to apply the Q L R T to DIC and discuss the numerical results concerning the time evolution of the intrinsic state and the double differential cross section d2tr/dEdO (Wilczynski plot). As emphasized in I, the Q L R T has the following features compared with the linear response theory (LRT) 2-s) and the random matrix theories (RMT) of Weidenmiiller 9 - 17) and of N6renberg ~s - 2 ~), which are the representative many-body theories in this field and are reviewed in refs. 22, 23 ). Similarly to R MT, it does not resort to the lowest order perturbation treatment of the coupling hamiltonian, nor presumes the 141

K. Niita and N. Taki,qawa / Quasi-linear response (H)

142

canonical distribution function for the intrinsic states of the scattering nuclei. Differently from RMT, however, the extension of the LRTin these respects is achieved by keeping the familiar structure of the basic equations of the LRT. The framework of the QLRT is thus fairly transparent and is expected to be powerful in describing the classical averages as well as the fluctuations not only in DIC, but also in other open system problems including nuclear fission. Our aim is thus to demonstrate the utility of the Q L R T in DIC as the first application. In this paper we consider only the energy transport in DIC, and leave out the mass transfer, charge equilibration and nuclear deformations. We consider only random intrinsic excitations. Furthermore, we adopt the parametrization by Weidenmiiiler and co-workers 1o). In sect. 2 we show the numerical solution of the master equation for the intrinsic state. It shows that the distribution of the intrinsic excitation energy becomes to be well represented by the canonical distribution function within a small time scale of the order of 10 22 sec. In sect. 3 the properties of the corresponding friction tensor are numerically analyzed, and are compared with those of the phenomenological friction tensor used by Gross and Kalinowski24) and those of the proximity friction 25). In addition, the validity of the known fluctuation dissipation theorem in the LRT is numerically examined. In sect. 4 we present the results of the calculation of the double differential cross section for three systems : 4°Ar + 232Th scattering at E l a b = 388 MeV, 136Xe+ 209Bi scattering at E,ab = 1130 MeV and 2°9pb + 2°9pb scatteringat 7.57 MeV/amu. Especially, we remark the difference between the present calculation and that of LRT 4). We discuss also the validity of the time-dependent approach of Weidenmiiller and co-workers ~1- ~5). We comment on the off-energy shell effect and on the phenomenological introduction of a time-dependent effective potential. We summarize the paper in sect. 5. 2. Time evolution of the intrinsic state

The time evolution of the intrinsic state is described by the coarse-grained master equation (I.4.28) with the memory kernel K~,(t, z) given by eq. (I.4.29). In this section we consider the strong-coupling regime 1), which is expected to correspond for the most part to DIC, and numerically solve the master equation in the Markov approximation. It is obtained from eq. (1.4.28) as PE(t) =

dE'Wew(t ) {a(E)Pe,(t ) - a(E')P~(t)},

(2.1)

where the transition probability WEE,(t ) is given by

f,

WEw(t ) = , dzK~,(t, z) ,,[to

2 =

^

^

,,/7

exp[-

( E - E-')2

],

(2.2)

K. Niita and N. Takioawa / Quasi-linear response (11)

143

with r(s) gl = h ( x / ~ W o f o ( t ) A ) -

½,

(2.3) (2.4)

r~ = xf2a/dl(t).

In the above equations, zg~ ts~is the decay time of the Green function qSt~(t ; t - r) [ref. ~)], and re is the decay time of the mean correlation function ( f/,,,(t)f/,,,(t - r)),,~c .... c~. We have assumed Weidenmiiller's form factor ~o) for the second moment of-the matrix elements of the coupling hamiltonian ~(t), i.e. Vm,(t)Vkl(t )) = (6mkf,t + (~ml(~nk)

1

x e - (e, - E.)a/2Az e - ~"~ - q(t'))2/2a2,

(2.5)

where a(E) is the level density. The form factorfo(q(t)) represents the degree of density overlap, and is normalized such that it equals unity at the distance where the half-density points of the colliding nuclei coincide. The values of the parameters in eq. (2.5) have been estimated 26.27) as Wo = 10 MeV, a = 3.5 fm and A = 12 MeV. As for the value of the energy correlation length A we discuss this again in the following section. In solving eq. (2.1) we have used the method ofeigenvalue problem 2s - 3o). Namely, we expanded PE(t) in terms of the eigenfunctions with respect to the energy and the time, P d t ) = ~ Czqz(t)f~(E).

(2.6)

2

We notice that 2 = 0 is always an eigenvalue, whose corresponding eigenfunction is the equilibrium distribution. The expansion coefficients C Aare determined from the initial distribution. The detailed procedure is given in ref. 29). In fig. 1 we show the calculated time evolution of PE(t) for 40Ar + 232Th scattering at E~ab = 388 MeV. For simplicity, we have disregarded the time dependence of the transition probability W~E,(t). This approximation will not cause so serious an error in the strong coupling regime. Actually, we have used Wee,(t ), which are obtained by fixing the values of the form factor and the velocity to fo = 0.15 and (1/c = 0.098. These values correspond to the relative distance r = 11.7 fm, which is 1 fm far from the half-density point. The value of the level density parameter a has been taken as 11.33 MeV-1. This corresponds to reducing the conventional value a =-~A, A being the total mass number of the scattering system, by a factor 3. This reduction is required to reproduce the realistic speed of the energy transport 29). The initial distribution has been assumed to be the gaussian distribution whose mean energy Eo and the variance a o are 20 MeV and 5 MeV, respectively. We observe that P~(t) almost reserves the gaussian shape as the time proceeds. This property holds irrespectively of the initial distribution and of the values of the parameters Wo, a, A, fo and ~.

144

K. Niita and N. Takioawa / Quasi-linear response (H) 0.08-

0.06 -

0.04 -

t=O

f

TIME UNIT 10-21SEC

-

t=0.6

0.02 -

ol o

5b

1~

1~

2oo E(MeV)

Fig. 1.TimeevolutionofPE(t).Thevaluesofparametersare Wo = 10MeV,a = 3.5fm,A = 12MeV, fo = 0.15,

(1/c = 0.098. The initial distribution is the gaussian distribution, whose mean energy Eo and variance a o are Eo = 2 0 MeVanda o = 5 MeV.

In order to investigate whether the canonical distribution function well represents the intrinsic state or not, we c o m p a r e our numerical results with the canonical distribution function with respect to the mean excitation energy E(t) and the variance aE(t ). These quantities are related by aE = V / ~ a -

1/4£3/'1-,

(2.7)

in the case of the canonical distribution function. In fig. 2 we present this relation (the dashed line) together with three our numerical results (the full lines), which were obtained for different initial distributions. We observe that the effect of the initial distribution vanishes with increasing excitation energy. Figs. 1 and 2 imply that the use of the canonical distribution becomes a g o o d a p p r o x i m a t i o n after a small time scale of the order of 10- 22 sec in the case when the intrinsic excitation proceeds t h r o u g h the r a n d o m coupling hamiltonian, t h o u g h it overestimates aE(t) at high excitation energies. The above-mentioned connection between aE(t ) and E(t) holds irrespectively of the detailed values of the parameters W0 and A, and offo and q. The energy distribution of the intrinsic state PE(t) can thus be determined to a g o o d a p p r o x i m a t i o n by specifying only the mean excitation energy. This property is utilized in the following sections when we consistently solve the coupled equations between the relative and the intrinsic motions.

K. Niita and N. Takigawa / Quasi-linear response (I1)

30'

/

145

/

/

> =E

~,=30M~/~~

~20 hl < ¢Y

lO

~',:20/MeV INITIAL DISTRIBUTION /

/

~ EXP(-E212(~2)

/

0

0

5b lOO 1so MEANEXCITATIONENERGY E (MeV)

Fig. 2. The relation between the mean excitation energy E a n d the variance a~(t). The dashed line represents the relation for the canonical distribution, eq. (2.7). The full lines are our numerical results. They differ from each other in the initial spreading, whose value is given in the figure. The other parameters are the same as in fig. 1.

3. Properties of the friction tensor and the fluctuation dissipation theorem 3.1. F O R C E D U E T O T H E C O U P L I N G H A M I L T O N I A N

The force due to the coupling hamiltonian ~ ( t ) is given by eq. (1.8.2). It can be expressed as ~ ( t ) = : ~°)(t)- ~#(t)4~(t),

0.1)

if we ignore the terms involving higher powers of4#(t) and/or higher order derivatives of The momentum independent part ~ ~c)(t) reads, for WeidenmiJller's parametrization of the random matrix, as follows,

qa(t).

:~)(t)

' t~)(t)/{af:) 1 (~) (t)+fo(t)/a 2 }J/#,~ (o) (t), =~f~

(3.2)

where J/(°)is the zeroth moment of the response function X(-). In eqs. (3.1) and (3.2) and in what follows, a greek lower suffix denotes a component of a vector and f(~1~2..... ) means nth order derivative off(q(t)) with respect to q~(t), q~2(t) ..... and q~.(t). Numerical estimates show that ~ ) ( t ) becomes fairly large. Nevertheless, we do not :;(c) explicitlytake into account the effectof~" ~ (t)inthe calculationsto be reported in sect.4. Instead, as is seen in eq. (4.I I),we introduce a time-dependent effectivepotential.This makes the agreement between the data and the theoreticalcalculation of the double differential cross section much better than using : ~) as it is and using a time-

146

K. Niita and N. Takigawa / Quasi-linear response (H)

radial "Y

(I(~2,3MeV.BI f~) 200"

1~=56.9MeV

150" A=12MeV 100-

50-

0

A=8MeV '

1'1

'

'

l's

r(fm)

Fig. 3. The radial frictioncoefficient7r,(r,/~)for three differentvaluesofA.The systemis 4°Ar + 232Th. The value of A and the mean excitation energy/~are givenin the figure.The valuesof parameters W0and a are I4/0= 10 MeV and a = 3.5 fm. independent potential. We consider that the effect of.~', ~(c) is phenomenologically taken into account by the use of a time-dependent effective potential. The quantity 7~a(t) in eq. (3.1) is the friction tensor, and is given by eq. (1.8.4) as a functional of PE(t). In the following we show the numerical results ofeq. (I.8.4) by using the solution ofeq. (2.1) for PE(t). In this connection, the important result of the previous section is that Pe(t) is almost determined by specifying the mean excitation energy E. Fig. 3 shows how the radial friction coefficient ~,,,(r, E) depends on the energy correlation length A. It is independent of A in the weak coupling region, i.e. for large values of r, while in the strong-coupling region it is very sensitive to A. In order to estimate A, Weidenmiiiler et al. assumed that two states which are located at the energy E _ A differ from each other by one unit in the most likely number of particles (or holes). This idea leads to 9, lo), A = 5.6a- 1 +4.75(E/a)½,

(3.3)

where a is the level density parameter. In the following calculations we take the value of A given by eq. (3.3) for each value of/~. The non-monotonic behaviour ofv,, as a function o f r is partly due to the effect of the form factor f0 and partly due to the effect of the higher order perturbation terms, which are incorporated in terms of the Green function ~b(1). This can be understood from the

K. Niita and N. Takigawa / Quasi-linear response (H)

147

following expression of the friction tensor,

7~(t)-

"~x/-x

2h2 jo

dE f xdE'(~'(~)(t)l/~(~)(t))m~cE(E-E')(z~s))3e-{~sl(E-e')/2h!2a(E')PE(t), jo

,~cE,

(3.4)

which is obtained from eq. (I.8.4) by assuming the Green function for the strong-coupling regime and by performing the integration over z. We remark that z~s) gets smaller for smaller values of r, i.e. for larger overlap of the scattering nuclei. This causes a peaking in y,r as a function of r in addition to the peaking involved in the form factor term (r) ^ (r) (Vm.(t) V,,.(t)). This effect of the higher order perturbation terms, which appears through r gis) l ' is entirely responsible for the appearance of a peak in the tangential friction as a function of r (see fig. 5). Corresponding to eq. (3.4) the friction tensor in the weak-coupling regime can be expressed as 7~a(t)= - 4 h

for dE

0

dE'(f/~),(t)f/,t~'(t)),,~c~(E-E'),,,, n~Ce,

× a(E')Pe(t),

tw),~/z.~'

/.~n/Zg, ) ± ( E -

E,)2,2 (3.5)

where r ~ )is the decay time of the Green function ~b~1~in the weak-coupling regime, and is given by eq. (I.4.24). Eq. (3.5) reduces to the expression of the friction tensor in the LRT 8) if we replace PE(t) and h/r~ ) with the canonical distribution function and the smearing width, respectively. Figs. 4 and 5 show the radial and the tangential friction coefficients, ~'rrand 7o0,for three different excitation energies. We remark that the values of both friction coefficients become smaller in the outer region and larger in the inner region as the excitation energy increases. These behaviours can be understood from eqs. (I.8.5), 0.8.6) and (3.3). Two additional friction tensors are given on these figures. One of them (the dashed line) is the proximity friction of Randrup et al. 2s). This model assumes that the energy transfer is purely due to the particle exchange. This mechanism would be important especially at the stage when the scattering nuclei are in close vicinity to each other and the excitation energy is moderate or high. Correspondingly, a characteristic of the proximity friction is that it is confined within the region of small distances. We consider only intrinsic excitations of each nucleus as the source of the friction tensor. Nevertheless, the present energy-dependent friction tensor carries similar properties as those of the proximity friction at high excitation energies. The other friction tensor in figs. 4 and 5 (the dotted line) has been used in the friction model of Gross et al. 24). Though it resembles the proximity friction and ours with respect to the tangential component, its radial component is much stronger. In order to obtain such a strong radial friction in our model, we have to assume a much larger value for A than that given by eq. (3.3) at low excitation energies (see fig. 3).

K. Niita and N. Takioawa / Quasi-linear response (H)

148

\

i

"~ (162.3MeV.S/f rn2)

tangential Y

radial

i

200'

2ooJ, \

150-

150

\\

\\

i i

\\

100"_5=

100-

50" E:=31.8Me¥~ , , ,

50-

0

_2,3MeV.S/frn2)

'

~

x

'.,,

E=31.SM~~ ~- ~'~,.~.._ 0 ~=3.2MeV~ b ' li ' 1~ ' 1~ r(fm)

¢=3;2MeV \-_'~-.._~..L~.. ' 1'~ ' 1~ ' 15

'

r(fm) Fig. 4. The radial friction coefficient 7,,(r, E) for three different excitation energies. The system is 4°Ar + 232Th. The values of parameters Wo and tr are the same as in fig. 3. A has been taken as given in eq. (3.3}. The dashed line is the radial part of the proximity friction of Randrup et al. 2s), while the dotted line is the radial friction coefficient used by Gross et al. 24).

Fig. 5. The tangential friction coefficient ),~(r, E). The system, notations and parameters are the same as in fig. 4.

3.2.. ,~(o) AND THE F L U C T U A T I O N DISSIPATION T H E O R E M

The zeroth m o m e n t of the correlation function .,V (°) is the most i m p o r t a n t to determine the fluctuations of the relative m o t i o n a r o u n d the classical m e a n trajectory. As shown in eq. (I.9.1), it is given by Jr~r(oq , a , t ) =_ W o { ¼ f ( o ' P ) ( t ) + 6 , a f o ( t ) / a 2 }

;'

o

dz~b(1)(t, t - r )

fo;o dE

/

h

dE' , c r ( E ' ) , ½ \,~(e)/

x e~- ( e - E)V2a2: cos {(E - E ' ) ( t - z ) / h } P E ( t ) .

(3.6)

Similarly to the friction tensor, this formula allows us to determine X ~ ) as a function o f r for each value of the m e a n excitation energy E. In figs. 6a, b, c we present the results ofeq. (3.6) for JV'~°) by choosing three different values of the m e a n excitation energy (solid lines), and c o m p a r e them with the predictions (the dashed lines) of the conventional

K. Niita and N. Takigawa / Quasi-linear response (II)

a)

b)

~(°)(16~.M@s.~)

200"

149

/ N<°b°'~M@S/fd)

200-

E=15.6MeV

=56,9MeV

150"

150"

100"

100

50-

50' /"

' 6 ' 1'1 ' 1'3

r(tm)

\ ~ , _ i

1~

9

|

i

11

i

i

i

13

|

15

r(frn)

200" //~l 150"

/

/

E=IO6.5MeV

/\

100"

l\

50" 0

9

I1

13

r(fm)

15

Fig. 6. The zeroth moment of the correlation function ./if{o)calculated from eq. (3.6) (the full lines) and the prediction of the conventional fluctuation dissipation theorem given by eq. (3.7) (the dashed lines). The values of/~ are given in the figures.

K. Niita and N. Takiyawa / Quasi-linear response (H)

150

fluctuation dissipation theorem in the linear response theory, in which ~o~ is given by

f.~o, = T?',,,

(3.7)

using the values ofT,, calculated in the previous subsection. In applying eq. (3.7) we have estimated the temperature parameter T for each E by assuming the reduced level density parameter mentioned before. Fig. 6 shows that, for low excitation energies, the conventional fluctuation dissipation theorem almost holds in the weak-coupling region, but it underestimates ,I/'~°~in the strong-coupling region. This could be a reason of the failure of LRT in reproducing the large spreading of the double differential cross section in the quasi-elastic region (see subsect. 4.4a), though a quantal effect has to be properly taken into account as well in discussing the fluctuations in this region 31). For the high excitation energy, our numerical result of. U~o) and the prediction of the conventional fluctuation dissipation theorem do not agree over a very broad region of r.

4. Mean trajectory and the double differential cross section 4.1. T H E O F F - E N E R G Y S H E L L E F F E C T

Figs. 4 through 6 show that the magnitudes of the friction tensor and the zeroth moment of the correlation function strongly depend on the mean intrinsic excitation energy E. It is therefore very important to properly estimate E(t)in calculating the double differential cross section d2a/dEdO. In this connection, we remark that the total energy conservation law can be expressed as 1), E(t) = E ....

(p(t)y 2~

U(q(t))- TrB(f~(t)lSB(t)) = EB(t)-- TrB(V(t)~B(t)), (4.1)

where

EB(t) = E~.m.

(p(t)) 2 2#

U(q(t)).

(4.2)

The interesting term concerning the present problem is the term due to the coupling hamiltonian Try{ F'(t)lSa(t)}. In the same approximation as that for eq. (3.1), it can be expressed as, TrB(V(t)~B(t)) = --fo(t )/ {¼f~o~)(t ) + fo(t )/az } J/~°)(t )

+½fCo~)(t)/{¼fo¢~)(t) +fo(t)/az}Jff~l~¥1~(t ),

(4.3)

where J # ~ is the first moment of the response function Z~), and corresponds to the friction tensor ?,p [ref. 1)].

K. Niita and N. Takigawa / Quasi-linear response (11)

151

In the weak-coupling regime, the value ofTr B{ 9 (t)t5B(t)} must be small, so that one can estimate/~(t) by the knowledge ofp(t) and U(t) alone. In the strong-coupling regime, on the other hand, the value ofTr B{ ~'(t)t)B(t)} becomes fairly large. For example, it becomes as large as - 100 MeV or even larger in the 4°Ar + 232Th scattering to be discussed later. This is known as the off-energy shell effect ~6). In the following calculations, however, we do not explicitly take into account the contribution from Tra{ V(t)t~8(t)} in estimating E(t), and identify the friction tensor at the time t with the one calculated according to eq. (I.8.4), where P e ( t ) is assigned such that its mean excitation energy is not E(t), but Ea(t). The idea underlying this prescription is the same as that for ff~¢~.Namely, we consider that the first term of eq. (4.3) is phenomenologically taken into account by using a time-dependent effective potential for U(q(t)) (see eq. (4.11)). On the other hand, the second term of eq. (4.3) is much smaller than the first term, at least, in the examples to be discussed later in this section. 4.2. EXPRESSION O F T H E D O U B L E D I F F E R E N T I A L CROSS S E C T I O N

We deal with only two degrees of freedom r and 0 concerning the relative motion. The equations determining the classical mean trajectory then explicitly read,

~p,

=

- ~ u(r)+

d

+~,(t),

dr _ p, dt la'

Po = ~ o ( t ) ,

(4.4)

dO _ Po dt lar 2"

The solution of the Fokker-Planck equation (1.2.26), which determines the fluctuations around the classical mean trajectory, is a gaussian distribution function. We therefore need to determine only the mean values and the mean-square deviations. Instead of the mean-square deviations, we define the half-variances by

O'~#(t)

--

1TrA(~#DA(t)),

(4.5)

where/)A(t) is the density operator for the subspace of the relative motion in the Galilei transformed coordinate system, which is specified by the mean trajectory 1). The coupled equations to determine them are obtained from eq. (1.2.26) as d

~o" 2 =

4 vrPr

rr

+ ~PotTp,po

6 12r4

z

152

K. Niita and N. Takioawa / Quasi-linear response ( H )

d

1

d t a PrP° -

~¢,/(1)____1

re

~ r 2 O p,po"~" O0

2

~_ _ _

,, ~

~1)

2

3

]2 if P°P'J[rr q- t t r 3 POif popo

2

i~r4 Poif {po,I,

/H(1}

igr 3 POV {p,rl ~¢" 00,

d dt

i f PoPs

= ~ , o ~ _ 2 if J [ ~a}-a-. 4 ~ - tr //¢'~} "~' oo /ar ~ popo oo - #r 3 ~'o {pot}' 00,

d 1 1 ~l~a}j_ 2 dttr,p,,} = ~ a p , p - ~,p,rl.~,,~, . ~3potr,po, }

d

if{prO } :

d

1

~

O'pop,. -

1

~ - if{por} =

1

~ O'prpo

d

1

-- fr

2

//1(1)

/l(l)

3 itr4P2if,r,

2

__

2

n ,-r

//~(1)

1

~1}

2

2 ~'o~or ~

d

z

~ r 2 °{por}J~t 00 -~- ~ r 3 FO~rr ~'~ 00,

1

d

3

oo - - ~

Poif {por!,

2

1 1 a,o = f'7'if {p,O~+ i~r---~if {po,}

d ~ [ ifoo = ~

2

2 t~r3 Poar,,

4 tr{po!o}

# r 3 P°a'°"

(4.6)

In obtaining eq. (4.6), we have neglected the terms involvingJ¢ {°)and the first moment of the correlation function yfft x).The effect of the Jff~l) terms is in fact negligibly small, while we ignore the d'¢ '°} term in correspondence to the neglect of ~,c) in determining the mean trajectory (see subsect. 3.1). We remark here the difference between the Present formalism represented by eqs. (4.4) and (4.6) and the time-dependent approach of Weidenmfiller and co-workers 14). Their equations are very similar to ours. The trip,,}, tr{~,~, tr,,, and tr,o terms are, however, absent in their equations. This is because they start from a time-independent formalism. The time t is later introduced in place of r. Consequently, the fluctuation of r and the other associated fluctuations do not exist. As will be pointed out in subsect. 4.4a, these fluctuation terms cause significant effects on the double differential cross section. Let us denote the Wigner transform of the density operator for the relative motion

K. Niita and N. Takigawa / Quasi-linear response (H)

153

PA(t) by PAW'It is given by the gaussian distribution function with respect to 0, Po, r and p,. After integrating over Po and r, we obtain the following expression for the part which is needed to calculate the double differential cross section, PAW(0, P,, t) -- 4 ~ , ~ exp --

[(0--O(t))Eap,,,(t)

-- 2(0 -- O(t))(p, -- p,(t))a{p,o!(t ) + (p, - pr(t))2%o(t)]l

)

(4.7)

with

D = trpw,(t)troo(t ) - (tr{p,o}(t))2,

(4.8)

where 0(t), p,(t), trp,p,(t), a~p,o~(t) and troo(t) are obtained by solving eqs. (4.4) and (4.6). Using eq. (4.7), we obtain the following expression for the double differential cross section,

dEfd0f- ~n

~xdEfdOf}n+÷[xd~E--fdOf)n_ ~'

(4.9)

where d2a ~ T = 2nvf 1 f o b d b 47zx/D l ~ _ e x p ~[,- ~1 [(Of-T-O(b)+ 2nrt)2trl,,p, (b) dEfd0fj,

T-2(OfTO(b)+ 2nz)(pf-p(b))a~p,oi(b)+(pf -p(b))2troo(b)]},

(4.10)

with p2/21~ = Ef, and vf = Pf/l~. 0(b), p(b) and {tr(b)} are the values of0(t), p(t) and {tr(t)} at t = + ~ for a given impact parameter b. The sum over n corresponds to the Poisson sum 32), while the lower suffices - and + the near side and the far side contributions respectively known, for example, in the semi-classical theory of elastic scattering of heavy ions. (d2tr/dEfdOf)o - and (dZtr/dEfdOf)o+ correspond to the positive angle and the negative angle scattering discussed by Wilczynski 33), respectively. 4.3. CHOICE OF THE POTENTIAL The potential U(r) consists of the sum of a Coulomb term and a nuclear term. We assume the Coulomb potential for the uniform charge distribution of radius R0, whose value is to be given later in eq. (4.17). The choice of the nuclear potential is crucial in obtaining a good agreement between the experimental data and the theoretical calculations. In this paper, we follow Nfrenberg and co-workers 34-36), and assume a time-dependent nucleus-nucleus potential, UN(r, t) = Udi~b(r)z(t)+ U,d(r)[ 1 -- z(t)],

(4.i i)

154

K. Niita and N. Taki#awa / Quasi-linear response (H)

where Udiaband Uad are the diabatic and the adiabatic potentials, respectively. As stated before, we suppose that the effects of the off-energy shell and of.y" ~tc) r are phenomenologically taken into account by the use of the time-dependent effective potential. In the original idea of N6renberg and co-workers 34), X(t) is the form factor that measures the degree of maintenance of the coherency of the entrance channel. We assume the following simple form for Z(t). (4.12)

Z(t) = e -'/~,

where the transition time z is a parameter of the order of the equilibration time. In the practical applications to be reported in the next subsection, we fix the value of r to 5 × 10 -22 sec for all systems. We choose the potential o f N g 6 et al. 37) as the diabatic potential. It has been calculated by the energy density formalism and is based on the sudden approximation. The analytic form reads

Udiab(/') ---- UED(r)

A ~1A 2~ - - A ~1 + A~2 VN(S),

(4.13)

with Voe-°'27s2

(s > 0)

(4.14a)

- Vo+6.3 s 2

(s < 0),

(4.14b)

-

-

VN(s ) =

where s = r--ro(A~a+A~),

(4.14c)

and Vo = 30 MeV,

r o = 1 fm.

(4.14d)

As for the adiabatic potential, on the other hand, we use the potential proposed by SiwekWilczynska and Wilczynski 3s). It well describes the data of fusion, and is given by Uaa(r ) = Uws(r ) = - Uo[1 + e x p ( ( r - C 1 - C2)/a ] - 1,

(4.15)

where U o = 17[a~/3 + A 2 / 3 - (A x +A2)2/3], a = (C 1 +C2)Uo/16r~yC1C2, C i = Ri-b2/Ri,

(i = 1, 2)

(4.16a) (4.16b) (4.16c)

K. Niita and N. Takigawa / Quasi-linear response (H)

(4.16d)

R i = (1.28 A ~ - 0 , 7 6 + 0 . 8 A~-+) fro,

7=0.9517

-1.7826

1-2AI+A~j

155

],

b = 1 fm.

(4.16e)

(4.16f)

The previously mentioned Coulomb radius is then assumed to be given by, Ro = RI +R2.

(4.17)

4.4. R E S U L T S O F T H E C A L C U L A T I O N

We have calculated the mean trajectories and the double differential cross section for the following typical examples in three different mass regions: (a) the medium heavy system 4°Ar+ 2aZTh at E~ab = 388 MeV; (b) the heavy system 136Xe+2°9Bi at E,ab = 1130 MeV; (c) the very heavy system 2°9pb + 2°9pb at 7.57 MeV/amu. For a given impact parameter b, the initial time t = 0 was assigned to the moment when the target and the projectile approach to the distance r = C1 + C2 + 5 fm. The initial values of the momentump(t) and the deflection angle O(t) were determined from the Coulomb trajectory. Eqs. (4.4) and (4.6) were then numerically solved until r attains the value 100 fm. The final deflection angle was then determined by matching to the Coulomb trajectory at that point. The asymptotic values of the half-variances ap,p,(b), troo(b) and a{p,o}(b ) almost coincide with their values at r = 100 fm. In all of the calculations, the friction tensor 7,~ and the zeroth moment of the correlation function JV t°) were taken from the numerical results in sect. 3. We used especially the results which correspond to the initial condition that the mean value and the variance are given by E o = 10 MeV and a o = 7 MeV, respectively. For the region where the excitation energy g is still lower than E o, the values of 2:and JV t°) for/~ = E o have been used. We then used the values of~ and :V t°) that correspond to the excitation energy at each time t (see subsect. 4.1). (a ) The medium heavy system 4° Ar + 23z Th at Eta b = 388 Me V. Fig. 7 shows the angleenergy correlation of the mean trajectories (the dashed line) and the double differential cross section (Wilczynski plot) d2a/dEdO(mb/MeV'rad) (the full lines). The mean trajectories reproduce quite well the ridge observed in the experimental data 33) as for the region of small energy loss, i.e. as for the region of positive deflection angles. Concerning the ridge in the negative angle region, however, the present calculation fails to reproduce the experimental value of the energy loss by about 30 MeV. The same failure has been

156

K. Niita and N. Taki#awa / Quasi-linear response (H) 30040Ar • 232Th

250-

200-

150-

z'o

3'o

4b

S'Oo(,) 6'o

Fig. 7. The angle-energycorrelation of the mean trajectory (the dashed line), and the double differentialcross section (Wilczynskiplot) d2~r/dEdO(mb/MeV • rad) (the full lines) for 4°Ar+ 232Th scattering at Elab = 388 MeV.

encountered with the classical friction model in early days, and has been later attributed to the effect of the deformations of the scattering nuclei a, 15, 39). We remark that the different choice of the value ofz in eq. (4.12) mainly affects the angle-energy correlation in the quasi elastic region. The present calculation of the double differential cross section gives fairly broad distribution in the quasi elastic region. In this respect the present calculation is more in line with the experimental data than the calculation based on the linear response theory 4). This is a consequence of the facts that the friction tensor in the present theory depends on the intrinsic excitation energy, and that we do not use the conventional fluctuation dissipation theorem to estimate ,j~,to~, but calculate it by taking account of higher order terms of the coupling hamiltonian. The present calculation fails to reproduce the experimental data of the double differential cross section in the region of negative angles. Though there exist trajectories which lead to negative deflection angles, their contributions to the cross section are too small. This is because the range of the initial angular m o m e n t a that contribute to negative deflection angles is very small. In other words, the calculated classical deflection function shows too sharp orbiting phenomenon. This situation seems to occur in general for medium heavy systems unless we use a specific friction tensor such as that of Gross and Kalinowski 24). The a6Kr -4- 166Er scattering at 8.18 MeV/amu is an another example of such cases. We finally wish to give a comment concerning the difference between the present time dependent approach and that of Weidenmiiller et al. 14). Comparing fig. 7 with the

K. Niita and N. Takioawa / Quasi-linear response (11)

157

corresponding figure in ref. 14) we observe that they differ very much from each other. This sounds curious at first, because both theories are based on the same assumption as for the coupling hamiltonian. We have, however, numerically studied that the difference originates from the fact that ref. xg) totally ignores the fluctuation of the relative distance r, and the other associated fluctuations during the collision time (see subsect. 4.2). (b ) The heavy system 136Xe + 209B/at Ela b = 1130 M e V. Fig. 8 shows the same figure as fig. 7, but for the 1 3 6 X e + 2 ° 9 B i scattering at Ela b : 1130 MeV. The angle-energy correlation of the mean trajectories very well reproduces the ridge in the experimental data, which is strongly focussing at 0 .... ~ 50 ° for a wide region of the energy loss 40). The fluctuations around the mean trajectories also over all well agree with the experimental data (see fig. 9). A problem is that the calculated cross section for very damped events (E .... < 500 MeV) is smaller than the experimental value by a factor of 2 ~ 4. (c) The very heavy system 2°9pb + 2°9pb at7.57 M e V / a m u . Since this is a symmetric system, one has to modify eq. (4.10) by adding the contribution from the deflection angle 7t- 0(b). Fig. 10 shows the resultant numerical result of the double differential cross section (the full lines) as well as the classical energy-angle correlation (the dashed line). Comparing with the experimental data in ref. 41 ), we observe that both the energy-angle correlation and the double differential cross section reproduce very well the experimental data over whole region on the energy-angle plane.

6. Summary We have applied the Q L R T to analyzing the characteristics of DIC concerning the friction tensor and the time evolution of the intrinsic state and to calculating the double 700 -

t36Xe

.,. 209 Bi

600 >= 5

E soo

400

300

3'o

s'o

60

70

80

9b e(°)

Fig. 8. The same as fig. 7, but for 136Xe-l-2°gBiscattering at El=b = 1130 MeV.

158

K. Niita and N. Takigawa / Quasi-linear response (H) 136Xe " 209Bi

1130 MeV

105

104

>~

3;

103 E LU "o. E 102 0

"10

101

,j io:,\ ,7 f

100

10-I 30

40

.:,o kk.

'9"

5()

6'0

xlO 0

7()

8()

e(°) Fig. 9. The angular distribution (d2~/dQdE) of the light fragment for different final kinetic energy given in the figure (energy in MeV ). The full lines represent the experimental data 39), while the dashed lines are the results of our calculation. The cross sections shown are multiplied by the factors indicated on the right-hand side.

208Pb + 208pb

800-

,,,.oo

7.57MeVlamu

(

500

60

7b

8b

9b

O(°)

1do

Fig. 10. The same as fig. 7, but for ;~°gpb+2°9pb scattering at 7.75 MeV/amu.

K. Niita and N. Takioawa / Quasi-linear response (H)

159

differential cross section d2a/dEdO based on the m a n y body von N e u m a n n equation. We have considered only the random intrinsic excitations as the source of the energy dissipation from the relative motion and of the associated fluctuations. Furthermore, we have adopted Weidenmiiller's parametrization of the random coupling hamiltonian. We have first solved the master equation for the intrinsic state, and found that the distribution of the intrinsic excitation energy at each time t is well represented by a gaussian distribution function. An important observation is that the average and the width of the gaussian distribution function soon become to be correlated in the same way as in the canonical distribution function. This property has enabled to calculate the friction tensor and the zeroth moment of the correlation function as functions ofr for each average excitation energy E. We found that the friction tensor corresponding to the random coupling hamiltonian of Weidenmialler and co-workers has a surface peak as a function of r, and strongly depends on the average excitation energy. Naturally, the magnitude of the friction tensor strongly depends on the energy correlation length. Concerning the fluctuation, we numerically demonstrated that the conventional fluctuation dissipation theorem, which is based on the lowest order perturbation theory, considerably underestimates the fluctuation in the strong coupling region, especially at low excitation energies. We have then calculated the double differential cross section d2a/dEdO for typical examples in three different mass regions. We obtained a good agreement with the experimental data, especially for heavy systems. This implies that the present formalism, which is based on the random intrinsic excitations and on the assumption of the short m e m o r y time, becomes more reliable as the degree of freedom of the system increases 42). A problem is that we have phenomenologically introduced a time-dependent effective potential in order to obtain the good agreement with the data. On the other hand, we have not explicitly dealt with the problem of the off-energy shell effect and the related momentum-independent force ~c~. The relation between the effective potential and the off-energy shell effect as well as ~-~c~ is left to be clarified in the future. The authors would like to thank Prof. S. Yoshida for many valuable discussions and for the useful comments in preparing the paper. We are grateful to Prof. K. H a r a for invaluable discussions. Thanks are due to Dr. K. Sato and Dr. H. O k u n o as well for useful discussions. The computer calculation for this work has been financially supported by Research Center for Nuclear Physics, Osaka University.

References 1) 2) 3) 4)

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