IL NUOVO CIMENTO

VOL. 103A, N. 3

Marzo 1990

NOTE BREVI

The Extended Elastic Model II Applied to the Reactions 2H(d, n)3He, 2H(d, p)~H. A. SCALIA and R. GIORDANO Dipartimento di Fisica dell'Universita - Catania I N F N - S e z i o n e di C a t a n i a

P. FIGUERA and S. PIRRONE INFN INFN

- L a b o r a t o r i o N a z i o n a l e del S u d - S e z i o n e di C a t a n i a

(ricevuto il 21 Novembre 1989)

Summary. - - Recent fusion cross-section data for the reaction 2H(d, n)3He, 2H(d, p)3H are compared with the predictions of the ,,extended elastic model II,. A comparison with cluster-impact fusion results is also given. PACS 25.70 - Heavy-ion-induced reactions and scattering.

In previous papers, the sub-barrier fusion for the reactions 2H(d, n)SHe, and 2H(d, p)3H was investigated by using ,,the elastic model, [1] and the ,,extended elastic model I, [2]. In these works, a comparison with experimental data reported by Arnold et. a l . [3] and Preston et. a l . [4] has been performed. In the present paper, an extension of the previous models is applied to the above reactions and a comparison with the experimental data reported by Krauss et. a l . [5] is given. In the following, this approach is referred as the ,~extended elastic model I I , (EEM II) [6]. We note that the experimental data of ref. [5] are more recent and extensive than those of ref. [3, 4] so that the EEM II is tested in a more satisfactory way. Further, we determine the value of fusion cross-section at E .... = 300 eV and we find that this value is more than 12 orders of magnitude larger than that computed by using the standard expression for the fusion cross-section [7]. We 465

466

A. SCALIA, R. GIORDANO, P. FIGUERA and s. PIRRONE

note that our result is more than 2 orders of magnitude larger than clusterimpact fusion estimate [8]. Following E E M II, the fusion cross-section can be written as [6] (1)

zf = $f[1 - g~(y)] = -~-~[1- g(y)] [1 - g~(y)] = = ~:

G(y) [1 + G(y)] [1 - g(y)] [1 - g~(y)],

where 2V 2

(2)

2V 2

r

ex,

/d_y\r

,

7

-

k is the wave number, v is the Coulomb parameter, E is the centre-of-mass energy, EB and Es are two parameters expressed in MeV, which are determined by comparing the experimental values of fusion cross-section with ~f[9], Ym is the value of y at which 5-~ attains the minimum value [2], Ym is obtained by using z-~ in the expression of cross-section factor[10], ~.1 and ~2 are determined by comparing the experimental values of fusion cross-section with zf[6]. To determine Ym we consider the solution of the equation [2]

(3)

2 - MI(y) 1 + 2G(y) - [exp [exp [y]]] (exp [y]) 1 + 2G(y) d- y 1 + G(y) 1 + G(y)

which satisfies the inequality (4)

Ym > O. 19.

The value Ym is obtained by solving the equation [10]

(5)

F

(d - y)2

M,(y) 7 2 . . . . - - - i c~Ly) = Ml(y)[1 + f ( y ) ] 1 + 2G(y) 1 + G(y)

1 + G(y)j

with (6)

f ( y ) = exp [y].

467

THE E X T E N D E D ELASTIC MODEL ]I APPLIED ETC.

TABLE

[.-

System 2H + 2H.

E m(keV) ffm Em(keV) Ym E B(keV) Es(keV) 7] ~'z

24.4487 0.7645 31.3502 0.7338 196.4839 225.0297 3 3.25

The values of EB, E s , E m = E B - f f m E s , E m = E ~ - y , , E s , ]'1, ~'z for the considered reactions are r e p o r t e d in table I. A comparison b e t w e e n the e x p e r i m e n t a l values of fusion cross-section and those obtained from eqs. (1)-(6) is shown in fig. 1.

10-1

/." 10.3

§

-~

10-~.

lO-~

10-71

I 20

i

I 60

i

I

,

100

I I/-0

i

I 180

, 22 0

Ec.m.(keV)

Fig. 1. - Comparison between the experimental values of ref. [5] and the calculated ones by using eqs. (1)-(6).

468

A. SCALIA, R. GIORDANO, P. FIGUERA and S. PIRRONE

B'y extrapolating eqs. (1)-(6) at energy E = 300 eV, we obtain for the fusion cross-section the value ~f= 5.179.10-nb. On the other hand, the standard expression [7] for ~f is

S(E) [ - 31.281 z,(E) = ----if- exp [ ~ j , where E is expressed in keV and S(E) is derived from independent data on higher-energy D-D reactions[6]. By using for S(E) the value S ( E ) = = 111.62- 10-~ cm 2keV, the resulting fusion cross-section is ~f-- 5.867" 10-~b. This value is 12 orders of magnitude smaller than the value obtained from the EEM II and 2 orders of magnitude larger than the value reported in ref. [8].

REFERENCES [1] A. SCALIA:Nuovo Cimento A, 101, 795 (1989). [2] A. SCALIA:Nuovo Cimento A, 102, 1105 (1989). [3] W. R. ARNOLD, J. A. PHILLIPS, G. A. SAWYER, E. J. STOVALjr. and J. L. TUCK: Phys. Rev., 88, 159 (1952). [4] G. PRESTON, P. F. D. SHAWand S. A. YOUNG:Proc. R. Soc. London, Set. A, 226, 206 (1954). [5] A. KRAUSS, H. W. BECKER, H. P. TRAUTVETTERand C. ROLFS:Nucl. Phys. A, 465, 150 (1987). [6] A. SCALIA:Nuovo Cimento A, 103, 85 (1990); 102, 953 (1989). [7] D. D. CLAYTON:Pri~ciples of Stellar Evolution (McGraw-Hill, New York, N.Y., 1968). [8] R. J. BEUHLER, G. FRIEDLANDERand L. FRIEDMAN:Phy8. Rev. Lett., 63, 1292 (1989). [9] A. SCALIA:Nuovo Cimento A, 98, 571 (1987). [10] A. SCALIA:Nuovo Cimento A, 100, 559 (1988); 101, 355 (1989).