Department of Physics and pstronomy, University of Maryland, College Park, MLI 20742, U.S.A. *physics Department, George Mason University, Fairfax,

Size: px

Start display at page:

Download "Department of Physics and pstronomy, University of Maryland, College Park, MLI 20742, U.S.A. *physics Department, George Mason University, Fairfax,"

Bruno Arnold
6 years ago
Views:

1 JOURNAL DE PHYSIQUE Colloque C2, supplbment au no 6, Tome 48, juin 1987 THE QUANTAL PERMEATION CURRENT AND THE DISCREPANT INITIAL STAGE N-AND Z-DRIFTS IN NUCLEAR HEAVY ION COLLISIONS J.J. GRIFFIN, M. DWORZECKA* and A. LUKASIAK*' Department of Physics and pstronomy, University of Maryland, College Park, MLI 20742, U.S.A. *physics Department, George Mason University, Fairfax, VA 22030, U.S.A. '"BRC Associates Inc., Bethesda, MD 20814, U.S.A. Abstract A new quanta1 permeation current flowing between interacting heavy ions is identified and validated by exact numerical solutions of a model SchrBdinger system. This current is driven by the difference between the depths of the potential wells on the two sides of the dinuclear window. It is not included in the conventional descriptions of heavy ion processes, which assume a current dependent only upon the difference between the nucleonic chemical potential energies. Accumulating data on the N- and Z-drifts observed in deep inelastic collisions also contradict the conventional description. By careful exact calculation with the one-dimensional Schrodinger model, the permeation current is here quantified and then extended to three dimensions. The resulting corrections are qualitatively such as to ameliorate the discrepancies between the observations and the conventional description. 1. Observed Drifts Violate the Potential Energy Surface's Prediction In the conventional Fokker-Planck descripticn of _nucleon transfer in deep inelastic collisions [I], the time evolution of N and Z for the projectile-like fragment -is assumed to be driven by the gradient of the dinuclear energy on the N-Z plane [2]. Such an energy surface is exhibited in Fig. 1 for the reaction [3] 56Fe+'65~o, together with the drift predicted by its gradient, and the experimental drift inferred from the data after correction for neutron emission. Although the predicted proton drift is qualitatively correct, the neutron drift is nearly zero, despite the energy surface's clear preference for an increase in N,. Figure 2 exhibits another reaction [4], Fe*, in which the direction of the drift is even more dramatically opposed to the gradient of the potential energy surface (PES), here actually running up hill. Here also the correction for neutron emission (circles vs. crosses) is so small as to be irrelevant. Similar situations also have been observed in the reactions [3,4,5] Caw, Cl+Bi and Fe+Bi. On the other hand observations of drifts in experiments involving the doubly closed shell nucleus, 132~e, have been interpreted as conforming better [6], although not completely 171, to the potential energy surface. Article published online by EDP Sciences and available at

C2-260 JOURNAL DE PHYSIQUE 30 LIQUID DROP '"u + ''~e Potential Energy Surface Including The Coulomb Energy LOCUS OF KlNET ZP 20 10 + ST6.TISTICAL MODEL PREDICTION 20 30 40 NP Fig.

2 C2-260 JOURNAL DE PHYSIQUE 30 LIQUID DROP '"u + ''~e Potential Energy Surface Including The Coulomb Energy LOCUS OF KlNET ZP ST6.TISTICAL MODEL PREDICTION NP Fig MASS NUMBER Fig Exact One-Dimensional Model Validates PES Current, But Also Exhibits New Permeation Current We have studied the nucleonic currents by exact calculation of the simple one-dimensional Schrodinger "Double-Well" model [8] shown in Fig. 3. This model calculates the flow of nucleons between two one-dimensional "nuclei", each of which is assumed initially to comprise N nucleons bound in an infinite square well potential of length L. One imagines that the two "nuclei" approach one another very slowly (a non-zero velocity of approach can also be included 191, but will not be discussed here), with their edges just touching at ts0, when the potential wall between them is removed and the subsequent time-dependent flow of nucleons in both directions is calculated by the exact Schradinger equation. The problem is thus characterized by the initial numbers of nucleons (NR and NL), by the volumes (LR and LL), and by the difference, Vo, between depths of the potentials in the right and left nuclei, respectively, as illustrated in Fig. 3. As the time increases, the probability amplitudes flow in both directions, altering the average number of particles on the right, say, at a rate In the conventional model [I], these currents are ystimated classically using the Fermi gas distribution. Then the net current A j = N, depends only upon the difference between the chemical potentials, A, in the left and right wells :.L+R -.R+L L R N~(t) = Ajclass = 'class 'class = (A-A )/h = (E~V~-E:)/~, (2) L L where, e.g., EF is the Fermi kinetic energy on the left, EF = (&n)*(~~+1/2)*/2~~:. Earlier, exact quantal calculations with Double Well [8] had produced a contradiction to Eq. (2), as illustrated in Fig. 4. There a series of calculations is displayeq in which the difference, AL-AR, is kept precisely constant. Obviously, N, given by the slope of each curve in Fig. 4, depends also upon Vo, and not solely upon A=-AR. We are now able to explain this discrepancy [10,11] in terms of the "permeation current" associated with states on the right whose energy Ei is less than Vo. Although classically such states contribute nothing to the current across the window, there is at early times a quantal current associated with the

3 penetration of such states under the shallower potential floor on the left, as is illustrated in Figs. 5 and 6 by Double Well calculations for a variety of specific single particle states. DOUBLE WELL PROBLEM and INITIAL CONDITIONS ElGEN VALUES NR VS. tlme for Simulated '"Protons" Vo=O.O MeV V0=2.0 MeV Vo=4.0 MeV Vo = 8.0 MeV I I I I I I I I I I f LL (x) +LR V,*16.0 MeV TRANSLATIONAL ENERGY is Time (IO-~~SCC) characterized by E/A = C. of Moss Fig. 4 Translational Fig LR (XI SCHRODINGER CURRENT CLASSICAL CURRENT n=6; n=5; n=4; n=3; '20] SCHR~DINGER CURRENT, 0.15 VS. PERMEATION CURRENT - [ time (t)] - Fig ,,,,,,,,, a,, [time (t)] - Fig. 6

4 C2-262 JOURNAL DE PHYSIQUE Figure 5 exhibits NL(t) as calculated for four states on the right for which 6, = E,,/Vo exceeds the value 1, so that passage across the window is classically allowed. In each case a straight line with the slope implied by the classical assumption is in good agreement with the exact Schrodinger result for times less than the time required for the wave to reflect from the opposite wall. Indeed, the ratio P(n) between the best fitting value of Nn over the range where Nn(t) is approximately linear and the classical current, jclass(n) = 2En/hn, is plotted in Fig- 7 vs. 6, = En/Vo for many cases including those of Figs. 5 and 6. One sees that for 6, > 1, Pn 1, in accord with the classical assumption. On the other hand, states with 6,<1, from which the classical assumption allows no contribution to the current, do in fact contribute in the exact Schrodinger solution. This is illustrated in Fig. 6 where the Schradinger result shows a dependence linear in time also for these classically excluded states. These slopes define an approximately quantal "permeation current'' for each such state, whose value is equal to the product of the classical current and a numerical factor, P(n), which depends only upon 6, = E,/v,, as illustrated in Fig. 7. Roughly, Fig. 7 can be summarized by the relationship, P = , for 0 < 6 < 1.3, or, more simply and more approximately (f30%), by P=Ei/Vo=6, for 6 c 1. PERMEATION and CLASSICAL CURRENTS Fig. 7 Thus, one recognizes that quantal particles deeply bound in the right-hand well respond to the reduction in their binding potential (from to Vo) effected by the approach of the dinuclear partner by spreading out their penetrating tails through the window into the volume of the left-hand nucleus. The result is a net current flow at early times which moves particles from the spatial region of the deeper well into that of the shallower well. Most noteworthy is the fact that this current does not diminish exponentially with increasing Vo/Ei, but exhibits instead a linear dependence upon Ei/Vo, such that at Ei/Vo - 1, it has a value 1 appropriate for the classical picture. The results imply that the correct net Schrodinger current flowing across the window to the right is in fact approximately equal to the following function of ll, lr and Vo; where the permeation current, jperm, describes the purely quantal penetration flow of nucleons, of energy less than Vo, into the left-hand side, and AjClass is given in Eq. (2). Expressions for the three-dimensional case are given below.

3. Extension to Three Dimensions These one-dimensional results can be extended to three dimensions by considering two adjacent volumes in which the potential depths differ by Vo, and which contain

5 3. Extension to Three Dimensions These one-dimensional results can be extended to three dimensions by considering two adjacent volumes in which the potential depths differ by Vo, and which contain Fermi gases which fill the momentum sphere out to &kl, andmkr, on the left and right respectively. This generalization then includes the appropriate weighting for the transverse states associated with a wave number k, in the direction perpendicular to the window, and allows a calculation of the permeation current per unit area as a function of Vo. Then the net current of particles from right to left through a window of finite area, ow, is given by Eq. (4) with the following expressions and jrm, based upon approximation (3a), and accurate to leading for 'jclass L R -. order in (vo/i) and (A -A )/A. and Here j-= P<v>/2 is the average particle flux impinging on the window from each side, A is the average of the two chemical potentials, 73 is the average of the two densities, and ow is the arga of the window. Also, for a Fermi distribution populated up to kinetic energy EF, one calculates <v> = (3/8)(2~~/~)~/~. Note that equations (5) show that when the chemical potentials k and lr differ by an amount comparable to Vo, then the permeation current is nearly one-half as large as the classical current. Also recall that the conventional PES description omits entirely this Vo-dependent permeation current. 4. Correlation with the Experimental Data To make an initial assessement of the differences, Vo, which might arise in actual heavy ion reactions, we have utilized the droplet model [12,13], which predicts a clear dependence of the nuclear well depths on the neutron excess, (N-Z). (a) Neutron Drifts: Since the neutron shell model potential depth diminishes with increasing neutron excess [12,13], one expects Vo to increase as (N-Z) increases in the nucleus on the left in Fig. 3, resulting in an increased permeation current of neutrons from the nucleus with the smaller neutron excess - into the nucleus with the larger. This prediction is in qualitative agreement with the neutron drifts observed in the reactions portrayed in Figs. 1 and 2, and as well with the other classically anomalous drifts cited above [3-51. In each case the light (NnZ) nucleus delivers neutrons to the heavy (N)>Z) nucleus despite the tendency of the potential energy surface to the contrary. (b) Proton Drifts: For protons the droplet model predicts an (N-Z) dependence of the non-coulombic part of the shell model potential of the same magnitude as but of opposite sign from the neutron potential. However, the Coulombic potentials for the protons here become quite important, especially that due to the Coulomb interaction between the target and projectile-like parts of the dinucleus. Upon including them, one finds that the combined result of these effects for protons is a potential difference between the two sides which is much smaller for protons than for neutrons. Thus a good prediction of the magnitude, or even of the sign of the proton permeation current, requires a more careful analysis than we have done so far. 5. Conclusions We have described calculations for a simple one-dimensional Schradinger model which identify a nonclassical permeation current which flows in a dinucleus from the deeper potential into the shallower. This current is qualitatively such as to ameliorate the discrepancies between observed drifts and the predictions of the classical potential energy surface. Further research promises to tell whether the improvement can also be made quantitative, and ultimately, whether such effects can become a tool for direct study of the detailed nature of the nuclear potentials in their deepest parts.

JOURNAL DE PHYSIQUE References RANDRUP, J., Nucl. Phys. A383 (1982) 468; (1979) 490; (1978) 319. This is also true of the alternative description via random walks, as discussed e.g. in GOKMEN, A.

6 JOURNAL DE PHYSIQUE References RANDRUP, J., Nucl. Phys. A383 (1982) 468; (1979) 490; (1978) 319. This is also true of the alternative description via random walks, as discussed e.g. in GOKMEN, A., et al., Nucl. Phys. A440 (1985) 586. BREUER, H., et al., Phys. Rev. 9 (1983) MEROUANE, C., U. of Maryland Ph.D. dissertation, report #0RC MIGPREY, A., et al., U. of Maryland report #ORO , p. 3. SCHULL, D., et al., Phys. Lett. (1981) 116. MERCHANT, A. C. and DHAR, A. K., J. Phys. G: Nucl. Phys. 9 (1983) L21. GRIFFIN, J. J. and BRONIOWSKI, W., Nucl. Phys. A428 (1984)-145c. GRIFFIN, J. J. and BRONIOWSKI, W., Proc. Winter Workshop on Nuclear Dynamics 111, Copper Mountain, Colorado, March 1984 (Indiana Univ. Report #INC ). GRIFFIN, J. J., DWORZECKA, M. and LUKASIAK, A., Proc. of the Beijing Int. Symp. on Physics at Tandem (May, 1986). GRIFFIN, J. J., DWORZECKA, M. and LUKASIAK, A., 2. f. Phys. A326 (1987) 51. MYERS, W. D., Nucl. Phys. A145 (1970) 387. MYERS, W. D. and SWIATECKI, W. J., Ann. Phys. (N.Y.) 55 (1969) 395.

Subbarrier cold fusion reactions leading to superheavy elements( )

IL NUOVO CIMENTO VOL. 110 A, N. 9-10 Settembre-Ottobre 1997 Subbarrier cold fusion reactions leading to superheavy elements( ) A. G. POPEKO Flerov Laboratory of Nuclear Reactions, JINR - 141980 Dubna,