IL NUOVO CIMENTO

Nucleus-Nucleus M.A. S ~ _ ~

VoL 56 A, N. 2

21 1Karzo 1980

Potentials (*).

(*')

Physics Department, Faculty o] Science, King Abdul-Aziz University POB 1540 Jeddah, Saudi Arabia (ricevuto il 29 Ottobre 1979)

Summary. - - The real part of the heavy-ion optical potential is calculated by the folding technique, by dropping the contribution of the inner nucleons instead of using density-dependent potentials. An expression is derived for the potential when one of the nuclei is considered as two fragments.

l.

-

Introduction.

Nuclear potentials have been studied b y several people, starting f r o m an effective t w o - b o d y interaction. The simplest a p p r o a c h is to fold the effective nucleon-nucleon interaction into the densities of b o t h nuclei. SA~C~[LE~ (~) gives a review of t h e simple folding, t o g e t h e r with the double folding. T h e i n t e r a c t i o n p o t e n t i a l between the nuclei is calculated b y folding the nucleardensity distribution function of a nucleus with the real p a r t of the singlenucleon optical p o t e n t i a l of t h e other nucleus. This model for the i n t e r a c t i o n p o t e n t i a l has been used in the studies of ~-nucleus scattering (2). The real p a r t of nucleus-nucleus i n t e r a c t i o n for m a g i c nuclei has been derived (a) f r o m (*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (**) Permanent address: Physics Department, Faculty of Science, Cairo University, Cairo, Egypt. (1) G. R. SATCHL]~n: Proceedings of the Inte~'national Con]erence on .Eeactions between Complex Nuclei (Nashville, Tenn., 1974). (2) D. F. JACKSON: Nucl. Phys. A, 123, 273 (1969); G. R. SATCHL]~g: Nucl. Phys., 77, 481 (1966). (3) D . M . B g I ~ : and FI. STA~CU: .Nucl. Phys. A, 243, 175 (1975); F1. STAnCe and D. M. B ~ K : •ucl. Phys. A, 270, 236 (1976). 129

~ . A. SH.AI:~A~ "

130

the Skyrme interaction density function by taking into account the exchange effects. This real part of the potential is tested against the elastic-scattering data by adopting an imaginary part with the same geometry and variable strength and good agreement is found (4). GOLa)~A~B(5) gives analytic forms for the real part of the heavy-ion optical potential through the folding technique 9 considering Yukawa terms for ~he effective potential between the nucleons. The elastic scattering is fairly sensitive to the nucleon distortion. C ~ A ~ et al. (~) obtained an excellent fit for the elastic data of 2sSi-[-x~O adopting the potential labelled E18 in ref. (6). SA~C'~LER(7) used a combination of Gaussian potentials for the effective nucleon-nucleon interaction to calculate the ~sSi+ ~eO potential by the double-folding technique and he ha4 to multiply his results by a factor of about 0.5 to get the potential of Cramer et al. (s) in the scattering region (6--10) fro. However, S I ~ (s) shows that density-dependent potentials are appropriate for the effective interaction in eases in which the densities of the colliding nuclei overlap significantly, so that the saturation property of this interaction prevents the total density from increasing beyond a certain value and he shows t h a t at the very tail of the potential the saturation term is negligible. But for heavier target-projectile systems the saturation term is not negligible, even for a separation larger t h a n the touching radius. I n general, he finds that the potential is weaker than the potential obtained by simple folding calculations, without density-dependent terms (9). Furthermore, the exchange terms were found to give a small contribution (s.9). I n the present work, the nucleus-nucleus potential is calculated by adopting the double-folding technique and accounting for the saturation property by dropping the contribution of the inner nucleons. Moreover, an expression for the nucleus-nucleus potential is derived by considering one nucleus as two fragments. This expression is useful for the calculations of heavy-ion transfer reactior~s and of recoil effects.

2. - M o d e l for n u c l e u s - n u c l e u s

potential.

The double-folding form for the real part of the optical potential between two nuclei separated by a distance R is (1,5,9) (1)

V(R) ~-(~v~(x)q~2(y) V ( I R + x -- y[) d x d y , J

(4) D. VAu~rH]~RIN and D. M. BRINK: Phys. Rev. C, 5, 626 (1972). (5) L. J. B. GOLDFARB:•UCl. Phys. A, 301, 497 (1978). (6) J.G. CRAM]~R,R. M. D]~¥mv,S, D. A. GOLDB]~]~G,M. S. ZISMANand C. F. MAGum]~: Phys. l~ev. C, 14, 2158 (1976). (7) G. R. SATCHL~: ~Vud. Phys. A, 279, 493 (1977). (s) B. C. SINI~A: Phys. Rev. Lett., 33, 600 (1974); Phys. Rev. C, 11, 1546 (1975). (~) D. M. B~INK and N. ROWL]~Y:Nucl. Phys. A, 219, 79 (1974).

131

l q U C L E U S - N UCL]~IYS P O T E N T I A L S

where ~v~(x), ~ ( y ) are the nucleon density distribution functions of nuclei 1 and 2 measured from their own centres and v is the nucleon-nucleon potential. I t can be seen from eq. (1) t h a t each nucleon of the projectile nucleus is t r e a t e d essentially as free and, therefore, the saturation p r o p e r t y of the two-body interaction, which prevents the nuclear density from increasing b e y o n d a certain limit, is ignored. Thus, the potential calculated in this way is found to be over-estimated (7,s). An approach was suggested b y S I ~ A (s), to get the correct order of m a g n i t u d e of the potential, b y using a density-dependent two-body interaction to account for the saturation p r o p e r t y of the two-body interaction, lqow the density function ~o2(y) of nucleus 2 can be written in terms of the single-particle wave function ~v~m~, of a nucleon in the state

(nil~j~) as i=1 m~

i=1

47~

where N2 is the n u m b e r of the nucleons in nucleus 2. But, as the wave function of the nucleus is an antisymmetric one, t h e n each nucleon contributes the same to ~o~(y), so one can drop the summation in eq. (2) and multiply the singleparticle density b y the n u m b e r of nucleons in the state (n~l~j~), thus 21-F1

(3)

~(y) = ~ ~

~v~l~.,(y)l-"

with ~ N~ ~ N2. Substituting eq. (3) in eq. (1) and for the region of light nuclei, one can use the oscillator wave function for ~n~(Y) and one gets for V(R) the following form: 2l+1 nt

•fq~(X)jo(2ifl~yr) exp [-- (fl~ H- 2~)y2]Y "~+~exp [-- D ~ r : J d r d y

i

Here ~ = A~r ~e x p [ - - ~ r 2 ] , and for v ( [ r - Yl) a combination of Gaussian potentials is used (7), i.e. v(Ir

-

Yl) = -

~ v°

exp[-/~lr

--

yp].

E q u a t i o n (4) is evaluated for lp-nuclei and adopting a Gaussian form for ~(x) (nucleus i is also in the region of light nuclei, although one can adopt a F e r m i shape for ~vl(x) and apply the m e t h o d to heavier target nuclei) and finally the following expression is obtained for the nucleus-nucleus potential:

(5)

[(

V(R) = 2Z2~o ~ exp - - a. - -

/~o. (A. + B . R ~)

132

~.

A. S H A R A ~

with

i =

[

L ot. ol

L .1.411

s[a~(2~ + #.)]~ '

~" = #"

9p N.

2~ + / ~ . '

,~. = a . + y

and @i = ~o exp [-- 7x~]. E q u a t i o n (5) is adopted to calculate the real p a r t of the 160-28Si potential, and the results are shown in fig. 1. I n the numerical calculation, the s t a n d a r 4

5

f /

/

15

/

I

7 r

9 (fro)

I

11

Fig. 1. - The 160-28Si optical potential: the solid line is Cramer et al. (6) potential, the broken line gives the present results, taking nil nucleons in consideration, while the dashed line gives those taking only the nucleons of the last state in consideration.

values of ~, fi, y and @0 are used, while the form used b y SATCm~E~ (v) and others (lo) for the nucleon-nucleon effective potential, v i z . v(r) = --

5.44 exp [-- 0.292r ~] -- 12.548 exp [-- 0.415r~],

is adopted. As said before, the value of the potential calculated b y this m e t h o d is found to be over-estimated, indeed, it is found t h a t one has to multiply the resulting values of the potential when all of the nucleons are considered (broken line in fig. 1) b y a normalization factor of the order of 0.5 to get the values of the Cramer et al. potential (solid line) in the scattering region (6 + 1 0 ) f m . This factor is nearly the same as t h a t found b y SAmC~LE~ (7). I n the work of Sinha (8) the nuclear density is t a k e n into account b y using a two-body density-dependent effective interaction. Thus, the internal nucleons i n b o t h nuclei are shielded b y the outer nucleons and, for the internal ones to participate in the interaction process, the two nuclei must get close to each

(lo) I. R]~ICHSm~IN and Y. C. TANG: .N~el. P h y s . A , 139, 144 (1969).

NUCLEUS-NUCLEUS

133

POTENTIALS

other in such a way t h a t a significant overlap occurs and the nuclear density increases up to the violation of the saturation property. So, if the contribution from these inner nucleons of the projectile nucleus is dropped and t h e y are treated as an inert core (these inner nucleons were treated similarly in the theories of direct nuclear reactions) and only the contribution from the outer nucleons is taken into account, the nuclear density for all the considered nucleons does n o t exceed the proper value and keeps the saturation property unviolated. I n fig. 1 the dashed line gives the potential when only the lp½ nucleons are considered. I~o appreciable difference between the Cramer et al. potential and the present results is observed. This result shows t h a t it is possible b y simple calculations rather t h a n other approaches (3,8) to obtain the right order of magnitude of the real part of the optical potential.

3. - N u c l e u s - n u c l e u s potential considering o n e n u c l e u s as t w o fragments.

I t was found t h a t the optical potential of the deuterons and of the helions can be calculated in terms of the potentials of their constituents at the appropriate energies (1~). I n this section, the single-folding form is adopted to write the reaI part of the nucleus-nucleus potential, b y considering one nucleus as two fragments, in terms of the potentials between each fragment and the other nucleus. The single-folding form for the nucleus-nucleus potential is (1)

Vl~(r) = f

(6)

V (rl -- r)

,

here V~(rl -- r) is the potential between the single nucleon of nucleus 1 and nucleus 2. If one considers nucleus 2 as two fragments A and B, t h e n the potential Vl~(r) m a y be written in the form (11) (if one neglects spin complications)

(7)

Vlz(r) =fl~(x)p(Ox(rx) + Os(rs) ) dx = VA(r ) + VB(r) ,

where r A and r~ are the separation distances of the centres of nuclei A and B, respectively~ measured from the centre of nucleus 1; and ~b(x) is the relative wave function describing the motion of nuclei A and B in nucleus 1. F r o m eq. (6) one writes for the potential VA(r )

(8)

VA(r)

=f [q~(x)l~l(rl)OA(rl- r-m~mSx) d x d r l

.

(~) P. E. HODGSON: Nuclear Reactions and Nuclear Structure (Oxford, 1971); J. R. RooK: Nuel. Phys., 61, 219 (1965); A. Y. ABUL-1VIAGDand M. EL-NADI: Prog. Theor. l~hys., 35, 798 (1966).

134

~. A. S ~ A F

B y expanding 0A as c x p [ - - - -mA X ' V ( F 1 m2

F) ] 0A(F 1 -

F),

where m A and ms are the masses of A and 2, and using the Saxon-Woods form for A

'

then

where VA°, RA° and d are the depth, the half-way radius and the diffuseness of the potential between the single nucleon of nucleus 1 and nucleus A (fragment of nucleus 2). In hears-ion scattering the important partial waves have a large angular m o m e n t u m compared to the relative angular m o m e n t u m between the fragments of nucleus 2, hence one can use a Gaussian form for qS(x), i.e. q~(x) = (N/4u) exp [-- axe], and the potential VA(rA) becomes

[;

]

with y2 = m~/8m~. The factor exp [y~n~/A 2] can be, to a good approximation, put equal to cosh (*~/2yn/gl), and substitution from eq. (9) gives

here 0- and 0+ mean the half-way radius /~A0 of the single-nucleon potential replaced, respectively, b y RA° -- y and RA. ~ y. The integrals in eq. (11) are evaluated b y B u i c k and I~ow~LY (9) who, analytically adopting reasonable approximations and using the Saxon-Woods form for both 71 and O(rl -- r) with the same diffuseness, found the following result: (12)

- ~V A. /±(r) fell(r,) O(±)(rl -- r) dr1 -- :~¢fo

with l(±)(r) = 2 A ( r -}- d)2[r q-/I - - (R(A~~ q- R A ) ] e x p lq_ r- - (I R ~- °

A

RA)]I

Equations (7), (11) and (12) show that the nucleus-nucleus potential, when

NUCL]~US-NUCL]~US

135

POT]~NTIALS

one of the nuclei is considered as two fragments, can be expressed as a s u m of four terms, t w o for each f r s g m e n t . These results are applied to the eases of 56Ni-2°sPb

as

(160+4°Ca)-~°Tb

(fig. 2)

~Ni-D°Zr

as

(160-F 4°Ca)-9°Zr

(fig. 3).

and I n the nlmlerical calculations the values given b y BnI~I~ a n d STA~CI5 (8) for Vo, Ro a n d A are adopted. As m a y be seen f r o m t h e figures, the a g r e e m e n t o

/Z

lO

lO

20 2o

:E 3O

?

I

¢o

4O

5o

5O

6o

6O

/ I

I

I

12

1~

/

IIi6

r(fm)

Fig. 2.

I ~1 8

I

10 r(fm)

I

12

Fig. 3.

Fig. 2. - The solid line represents the 56Ni-2°spb potential, the d~shed line gives the potential when the ~°Ni nucleus is considered as ~°Ca+160. Fig. 3. - The solid line represents the ~6Ni-9°Zr potential, the dashed line gives the potential when the 56Ni nucleus is considered as ~9Ca+160. is excellent for t h e scattering region of the potential. The separation of t h e p o t e n t i a l into the potentials of t h e constituents of one nucleus has a big adv a n t a g e in t h e calculations of t r a n s f e r reaction between h e a v y ions, specially in the s t u d y of recoil effects (12). (12) M. A. SI~AI~Ar: Phys. I~ett., to be published.

136

•

~. A. SHARA~

RIASSUNTO

(*)

Si ealcola la p a r t e reale del potenziale oCtico degli ioni pesanti per mezzo della tecnica a strati, t r a s c u r a n d o il c o n t r i b u t o dei nucleoni pifi in¢erni i n v e c e di usare p o t c n z i a l i d i p e n d e n t i dalla densitY. Si d e r i v a l'espressione per il potenziale q u a n d o uno dei nuclei considerato c o m e s o m m a di due f r a m m e n t i .

(*)

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