Submitted t0 <<Nuclcar Phvsics A»OCR Output

FOR DINUCLEAR SYSTEM

MASS PARAMETERS

G.G.Adamizm, N.V.Am0nenk0, R.V.J010s

PQGEIEEBEE

imtimxt» xtwm \

(i <·~-D Y? L@©3tt/7M) " "" V°` S`\ ’¤ Z3·b*>&`+

The paper is organized as follows. In sect. 2 we obtain the general form for OCR Output

the components of the mass tensor for a dinuclear system.

simpler way, we develop in this paper a simple and mainly analytical method to find

variables makes the computations very cumbersome. To obtain the results in a

evolution of the dinuclear system . However, its dependence on various dynamical

The nondiagonal component of inertia tensor plays an important role in the

production of light isotopes increases.

the stability of the dinuclear system for large mass asymmetry decreases and the

approaches to the radial potential barrier with the increase of ry. As a consequence,

of n—-mode transforms into the kinetic energy of the radial motion and system

Due to the coupling of R- and q— modes of motion the part of the kinetic energy

motion in the R — 1; plane increases significantly with increase of mass asymmetry.

it was shown that the nondiagonal component of the inertia tensor describing the

considerably influence dynamics of the dinuclear system. For instance, in refs. IL")

The values of mass parameters and their dependence on dynamical variables

given in refs. 13*5)

two-center basis. Inertia tensor obtained in terms of the adiabatic representation is

has been found in the framework of the linear response theory using quasi-adiabatic

·8) by exploiting the diabatic two-center shell model. In refs. 9*12) the inertia tensor

basis. The approach based on the dissipative diabatic dynamics has been realized in

refs. Z`4) the calculations have been done using adiabatic two-center shell model

cranking expression and perform calculations in different single particle basis. In

are different approaches to calculate its value. These approaches mainly use the

variables. The important ingredient of the Hamiltonian is an inertia tensor. There

Therefore, the Hamiltonian of dinuclear system should depend on these dynamical

of nuclei, A = A1 + A2) and a neck radius or other characteristic of a neck *6).

mass asymmetry degree of freedom ry = (A1 - A2)/A (A1 and A2 are mass numbers

interaction of two nuclei are the distance between the centers of colliding nuclei R,

OCR OutputThe most important degrees of freedom which are necessary to describe the

T = — 7) ( OCR Outputn’ ( 6 ) 1 ( 6 ) —— d V ———-—V —— . 2m/ r p(r)6p(r) n(r) p(r)6¤(r)nian in terms of the functional derivatives of p

Using eqs. (4)—(6) we can represent the kinetic energy part T of the total Hamilto

6 ( )ih 6 · = —V ——- . J(¤‘) m (Mr) Mw)and the current operator has the following functional representation

(5)_ , , lp(r),J(r )] = —:V-(6(r — r )p(r))ih

erations. Operators p andj satisfy the commutation relation 16)

The omitted term depends only on p and is not necessary for the following consid

(4). _ . H = d¤·1(r)p ‘(r)J(r) +m 5 /

operators as given below:

(3)p(r) = ¢»"`(r)·/»(r)

and density

(2), J(¤·) = —¢“'(r)V=1»(r) — V1/»"(r)¤/¤(r)zh % (

where 1/:+, xl) are nucleon field operators, can be expressed in terms of current

I I I I H = dr V1/)+(r)V1/¤(r) + I drdr ¢+(r)·¢~(r)u(r — r)1,b+(r )1/»(r ), (1)he E/

two-body forces

It can be shown 16*17) that the nuclear Hamiltonian of the general type with the

2. Inertia tensor

are presented in sect. 4.

parameters both macroscopically and microscopically. The results of calculations

is introduced. The simple analytical expressions are obtained to calculate mass

inertia tensor and discuss some useful relations. ln sect. 3 the definition of neck

n 0

(12d) OCR Output= é§(E0 — En) < n|qj|0 >< 0]qj»|n >,

= —- 2 12 ( °)n§, @,1 — Eq2 <¤|lH,<1il10 >< 0llH7<1i·lI¤>

(12b)(B`1)u· = < 0|l<1w» [H»<1illl0 >

If the nucle0n—nucleon forces are velocity independent then

(12**)(B`1))1’ Z ij < 0lZVk91(rk)Vk9j·(Fk)l0 >

we can rewrite the formula (11) in the following way:

kzi

p(r) =< Ol Z 6(r— r;,)|0 >,/$1+/$2

Using the density of the dinuclear system

(11)_ (B ‘)n = drx»(r)Vy)(r)Vyj·(r)i E/

It is evident from (10) that the components of inverse inertia tensor are:

. 2 8q],10 ( )E -——» 12-. )J]

nz 8 0 1

@(11*Z —·; ·— l` l` ‘ l' ‘» l` T 2d<>v<>v<> p g] gi/2m 1,r 3%

(*6

we obtain the kinetic energy term as:

g ( ’.L. : . UL 6»<r> ;”’( Ge

functional derivative as given below

where gj(r) are functions required to derive qj. Then, with the expression for the

(8)qi = / <1¤¤(r)y)(r)~

some number of collective variables q_, which are donned by the relations

collective motion of two—center system. We assume that the density p depends on

ln what follows, we shall use the expression (7) to derive the inertia tensor for the

Z R" (B"l·m(B")¤R · l(B`1lRnl2(17c) OCR Output

(B_1lRn

, im (B-l)vn(B—1lRR " l(B_1lRnl2(17b)(B")nR B = --———;—

: “" <B-l>m,<B-wm — 113-%,12(Bl-1)*77I

then, all the components of inertia tensor are:

centers R and mass asymmetry parameter 7; are taken as the collective variables,

(B "l)j_,I and vice versa. For example, if the relative distance between the fragment

From eq. (16) we can express matrix elements Bj,] in terms of matrix elements

which is satisfied identically because of (14).

J1J2

BM = Z BmBm·(B`1)mz

relation

ing expression (15), we insert (13) into (15) and, using (12c), obtain the following

To demonstrate the equivalence of the inertia tensor in (11,12) and the crank

using the two-center shell model basis.

MBa = 2v 2 en 0 ' ··

<n6 6`q·0><08 8q··n> 4€’ J'

Usually, the inertia tensor Bjy is calculated with the help of cranking expression

" jj'

T Z Bjwijsiy

we can obtain from (10) the well known expression for the collective kinetic energy

(14)Bm(B")j.i· = 6m

and

jr8 F Fqj ji (13)I _ 'ZBjJ’lH=qi'l = ·{2Bn·<1r8 1

corresponding energies Eg and En. Since

where |0 > and ln > are the ground and excited states of the dinuclear-system with

A= dr p(r)0(:l;z). (19a) OCR OutputU) f2

Fragment mass numbers are defined by the relation as given below:

18 ( ¢)€(z) 9(—z)= ———- — -—~—— . yn Z ( A1 A2

follows that:

is the point where the densities of nuclei are equal to each other. From (18a) it

Here z is the axis connecting fragment centers and 6 is the step function. The z = 0

(18b)gn = ·}(6'(z} ·- 9(-z))

= —- ——-—· 18dgg 6(z) G(—z)

for gg and gn defined by the equations:

ments we get the usual definitions of R and r; if we substitute in (8) the expressions

fragment centers and the mass asymmetry parameter 17. For well separated frag

Let us firstly consider as collective variables the relative distance R between the

3. Macroscopic and microscopic considerations

(B`l)iz,, < (B`l)»¤(B`1lnR

BIQU < B,,,,BRR,

lB·mBm¤¤— B}anll(B")m(B")Rn · (3**)%,,] = L

BRn(B_1)Rn 'i` BnnlB_1)vm : BR·r1(B-1)}%: + BRR(B_1)RR : lv

1} B2.: BHRBW — = BRRBM —B

n M -1 = —B -;-1 = ——-—-——— B , "" "“<B·1>Tm \/(B`llRR(B`1lnn( )“"BRRB(B-1)R

Bnn(B—l)nn = BRH(B—1)RR¤

The solution of eq. (16) as given by eqs. (17), leads to the following useful relations:

r r ,, B w ———-—-—, B x ~———, B z ——-——+———. “*‘<B-wm "" <B-¤>.. "" <B-1>..<B-wm(B`*)R

For (B“l)R,, to be small, above components (17) can be expressed as:

`1 = -———————— d —— 22 <B hm 1 rp<»>¢xp<¤>1 A A zz mmz (MM2] ( OCR OutputTaking into account (20), the approximate value of eq. (21) is

nuclei.

separated fragments we obtain a reduced mass lt : mAlA2[A for two interacting

that BE}; decreases as the overlapping of interacting nuclei increases. For the well

Since (0(;l:z) — 02(;hz)) and 0(z)0(—z) are always positive, it is obvious from (21)

_ _ — _ 0(z) — 02(z) 0(—z) — 02(—z) 0(z)0(—z)

B-] = ———- -—-—- ( )RR mA] + THA;l l

Substituting (18a,b) with 0 instead of 0 into (11) and using (19b) we obtain

A= dr p(r)0(;l;z).v) f2

norm of 0 is defined as in eq. (19a)

interval a amounts to the diffusion of nucleon density distribution in nucleus. The

(Function 0 is schematically depicted in Hg.1.). ln the limit a —> O, 0 —> 0. Physically

where a is the small interval over which the value of 0 jumps from zero to one.

(20)— 1 z 0(Z) Z - (1 + err 2 a

nuclei. This function 0 is defined in the following way

into account the mutual interpenetration of nucleons belonging to the overlapping

We generalize the definitions (18a,b) by introducing a function 0 instead of 0 to take

0.5 tion 0(z).

Fig.1. Schematic form of the func

I.O

1!(z)

we are interested in the description of the dinuclear system in the noncquilibrium OCR Output

can find A,,,ck and, therefore, the components of mass tensor microscopically. Since

the results of microscopic calculations for the nucleon occupation numbers 18) we

Substituting in (23) an expression for p in the second quantized form and using

V BRRBUU W 7V AIAZ " AAncck/4.BRUI I \/5 AAneck

· ccBA U = -——-—, ( )R TTL \/7FbA]Ag1 Ank

( A)B-] uu : —-`_`°—_» ( l m \/§b2A21 Aneck

In the same way as above we obtain

(24)v : A,,,ck/A.

in the system A:

the ratio of the number of particles in the neck Auger to the total number of particles

lt is convenient to introduce a degree of freedom ·u connected with the neck as

the mass asymmetry parameter 1; increases.

BR,, is very small for almost symmetric configurations but increases significantly if

Here A; 2 A2. As it is seen from (22c,d) the nondiagonal component of inertia tensor Y

BRRBW ,/(B—¤)RR(3—¤)W V N/27r A(AiA2 ~ A/har/4)L Ancck (A1 _ A2) (.).7d)lBRnl Z l(B_l)Rnl Z

and

(B·)Hn I -...m 2\/Fb AAIA2 (22c)1 A A — A 1i5’iLJ

22b ( )*

B·: -;; ( )YV’7 /27rb2A2

VVe obtain the other components of inverse inertia tensor as

Amr =<1rr>(r<¢x1>— (23):2 /)(§>in the neck between two fragments Amckt

The integral which appears in (22a) can be interpreted as the number of particles

where bz = vraz /4 which demonstrates that values of b and a are close.

matrix elements in (27) can be obtained analytically. For the frequency parameter OCR Output

particle wave functions of harmonic oscillator in the cylindrical representation, the

are calculated exploiting the expressions (17), (22), (25) and (27). Using the single

”Ni+5sNi—>“°Ba and “8Pd+“8Pd—>°3°U. The components of the inertia tensor

To illustrate the results of the previous sections we have considered the systems

4. Results of calculations

variables.

to elucidate the values of mass parameters and its dependences on the collective

of the elements of inertia tensor. However, the obtained results give possibility

(22), (25) and (27) in numerical calculations allows us to obtain the smoothed parts

rectly the cranking expression to determine the inertia tensor. Thus, use of (17),

definitions of Amd,. In the consistent microscopical calculation we should use di

only partly taken into account. This influence manifests through the microscopical

tensor the influence of the interacting nuclei shell structure on the value of Bjy is

Because of the approximations used to derive the components of inverse inertia

(26)Ama = E mcdr exp —|<.¤k(r)l2Z2 /(§) ke1,2where ni, are the Fermi occupation numbers. Ultimately, eq. (23) becomes:

km,2

< 0|p(r)l0 >= Z ¤»l<x¤k(r)|’»

neglecting their contribution we obtain from (26)

Assuming the chaoticity of the phases of the nondiagonal matrix elements of p and

§(<pI(r)s¤¤(r)¤¥¤¤ + M-)2,2’1,\’

n(r) = X =»¤I(r)s¤¤·(r)¤T¤¤· + Z s¤;(r)w(r)¤£`¤¤·

overlapping of the colliding nuclei, p can be written as

nuclei i.e. projectile (gal) and target (cp;). Thus, taking into account the small

stage we shall use a.s a, basis the single particle wave functions of the noninteracting

( I , , .8.0 0.2 0.4 0.6 0.8 1.0` {

$ \ "—_ i `\\ \

aol- ‘ \

~»,1g_g for the system “8Pd+u8Pd. OCR Output

Fig.3. The same as in fig. 2, but5 F Ai"

15.0 F XH

20.0 RA H8Pd+’18Pd system

0.0 0.2 · 0.4 0.6 0.8 1.00.0

N ‘· `\ .` \ " 3 dashed lines, respectively.\ __ _ \`\.1 by solid, short dashed and long

ELOF \ \ at d = -1,0,2 fm are presented

8Ni+58Ni. The calculated resultsR

R1 + R2 + d for the system10.0 t `~_ `

ues of fragment separations R =

mass asymmetry ry at various val

. l5·O E R NH, N1 System Q Fig.2. Dependence of Ama;. on55 . 5.5

of nuclei in contact.

of AMC;. are larger at smaller R i.e. Amd. is proportional to the overlapping volume

It is seen that the nucleon number in neck decreases with increasing ry. The values

1.15/I}? fm are the nuclear radii and d = -1, 0, 2 fm) are presented in figs. 2 and 3.

fm in our calculations. The dependences of AMC;. on n at R = R1 + R; + d (HL2 =

O.7)`1 fm'2 which reproduces the systematics of nuclear radii. We have used b = 0.8

of the harmonic oscillator wave functions we have taken the value *9) v = (0.9/11/ 3 +

I I OCR Output

nuclear shell structure. The values of Bm, and BW increase with increasing 1]. In the

of mass parameters which are seen in figs. 4 and 5 are the consequences of the

fragment separation R are presented in figs. 4 and 5. The oscillations of the values

The mass parameters Bjj. calculated as functions of 1] for various values of the

Fig.5. The same as in fig. 4, but for the system u8Pci+“8Pd.

°‘g.o 0.2 0.4 0.6 0.8 1.0

¤¤ 10.0

E 20.0ifA’ r

I / ,·: :0.0 3* 1

5_ 8.0 0.2 0.4 0.6 0.8 1.00.00

[D -2100.20

E »E -21.0\%E0.40 xx ___·* T. O E _/' `¤,\

"--"\*\C —I9.0

/\ YX0.60 \ é

_] * ` _' —»»` v~

0.00.0

.... .---..--,..--·"{/fi - 4 ‘’

s.¤E /·” x} / [ '2.0/_ 1 4, / ’ [

/ I, ' / I$10.0

I :4.0E 15.0

{*20.0

6.0

E 25.0

8.0OCR Output30.0

I3 OCR Output

yield of light particles in fusion-type reactions observed in the experiments 2°·21·“).

The coupling between R- and 7;- modes of motion can be the reason of the enhanced

and the nondiagonal components of the inertia tensor should be taken into account.

condition BR, < \/BRRB,,,, is not correct for strongly asymmetric dinuclear systems

7] ——» 1. This behavior of BR,, is in agreement with the results of refs. ’3·“‘). The

significantly with increasing mass asymmetry 1] and can approach to 0.4 in the limit

motion vanishes. However, the value of the ratio Bm,/\/BRRBM (figs.6,7) increases

For the symmetric configurations the coupling between R- and ry- modes of

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2E "———/"“

0.:E "`··"""'N

"all

0.4

for the system “°Pd+“8Pd.

up Fig.7. The same as in fig. 6, but

xt //

{IIE [*,/4* / 4l,»· f_A- ’»’ /

0.2" ’

» Im0.3

0.4

"°Pd+`1°Pd system0.5

formation is necessary to investigate the dinuclear system transition to mononucleus.

OCR Outputfor the description of the nuclear fusion process because the consideration of the neck

14 OCR Output

2 (North—Holland, Amsterdam, 1980)

J. Maruhn, W. Scheid and W. Greiner, in Heavy ion collisions, ed. R. Bock, vol.

O. Zohni, J. Maruhn, W. Scheid and W. Greiner, Z. Phys. A275 (1975) 235

H.J. Fink, J. Maruhn, W. Scheid and W. Greiner, Z. Phys. A268 (1974) 321

Wong, Rev. Mod. Phys. 44 (1972) 320

M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y.

References

2-61—13—28

This work was supported partly by the Russia Ministry of Education under Grant

We are grateful to Dr.R.P.Malik for discussions and reading of the manuscript.

useful in the consideration of nuclear fusion process.

the condition Bm, < \/BRRBW, is justified 4). The results of this paper can be

of the strongly asymmetric systems. However, for almost symmetric configurations

account the nondiagonal matrix elements of the inertia tensor to consider dynamics

increases strongly if mass asymmetry increases. Thus, it is important to take into

R- and 1]- modes of motion is small for almost symmetric configurations but it

clusion of refs. 21*22) that the nondiagonal component of inertia tensor connecting

on the nuclear Hamiltonian of a general form. The results obtained confirm the con

in the dissipative heavy ion collisions, have been obtained. The derivation is based

and nondiagonal components of inertia tensor, describing a dinuclear system formed

On the basis of approach suggested above the general expressions for the diagonal

5. Summary

evolution of asymmetric dinuclear system.

well (figs.6,7). S0, all components of inertia tensor are necessary to describe the

The role of the coupling between R- and v—modes of motion is considerable as

I5 OCR Output

1988) p.216

Conference on Clustering Aspects in Nuclear and Subnuclcar Systems (Kyoto,

V.L.Mikheev, J.Szmiger and V.V.Volkov, Contributed Papers of International

A.N.Mezentsev, A.G.Artul»;h, G.F.Gridnev, W.Karcz, S.Kliczewski, M.Madeja,20)

J.Blomqvist and A.M0linari, Nucl. Phys. A106 (1968) 54519)

A551 (1993) 321

G.G. Adamian, N.V. Antonenko, R.V. Jolos and A.K. Nasirov, Nucl. Phys.18)

S.P. Ivanova and R.V. Jolos, Nucl.17) Phys. A530 (1991) 232

R.F. Dashen and D.H. Sharp, Phys. Rev. 165 (1968) 185716)

15) F.A. Ivanyuk, Z. Phys. A334 (1989) 69

14) S. Ayik, Z. Phys. A292 (1979) 257

13) S. Ayik, Phys. lett. 63B (1976) 22

12) V.M. Kolomietz and I.Yu. Tsekhmistrenko, Sov. J. Nucl. Phys. 45 (1987) 1279

11) S. Yamaji, H. Hofmann and R. Samhammer, Nucl. Phys. A475 (1987) 487

(1977) 152

10) P..I. Johansen, P.J. Siemens, A.S. Jensen and H. Hofmann, Nucl. Phys. A288

H. Hofmann and P.J. Siemens, Nucl. Phys. A257 (1976) 165

A. Lukasiak and W. Niirenberg, Z. Phys. A326 (1987) 79

A. Lukasiak, W. Cassing and W. Norenberg, Nucl. Phys. A426 (1984) 181

Bromley, v01. 2 (Plenum, New York, 1984)

W.U. Schréder and J.R. Huizcnga, in Treatise on heavy-ion science, ed. D.A.

W.J. Swiatccki, Progr. Part. Nucl. Phys. 4 (1980) 383

I6 OCR Output

on August 31, 1993.Received by Publishing Department

Kaplan, M.S. Zisman and E. Duek, Phys. Rev. C22 (1980) 1080

24) D. Logan, H. Delagrange, M.F. Rivet, M. Rajagopalan, J.M. Alexander, M.

N.V. Antonenko and R.V. Jolos, Z. Phys. A341 (1992) 45923)

N.V. Antonenko and R.V. Jolos, Z. Phys. A339 (1991) 45322)

1991) p.39

1991, ed. E.Gadi0li (Ricerca Scientifica ed Educazione Permanente, Milano,

V.V.V01k0v, Proc. 6th Int. Conf. on Nuclear Reaction Mechanisms, Varenna,21)

Preprint of the Joint Institute for Nuclear Research. Dubna, 1993 OCR Output

Physics, JINR.The investigation has been performed at the Laboratory of Theoretical

asymmetry increases.

is weak for almost symmetric configuration but it enhances significantly as theproposed. It is shown that the coupling of the radial and mass asymmetry modesis analyzed. The definition of a neck parameter for the dinuclear system isdifferent modes of motion for various configurations and fragment separationscomponents of the inertia tensor for a dinuclear system. The coupling between

A microscopical method is proposed to derive the diagonal and nondiagonal

Mass Parameters for Dinuclear SystemAdamian G.G., Antonenko N.V., Jolos R.V.

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