Fusion Dynamics at the Extremes

International Workshop on

Fusion Dynamics at the Extremes


— i — » -

N

100

Figure 1. Contour map of the shell correction to the ground-state energy, E3^. at the contour lines give the values of Ea^.

Numbers

correction to the ground-state energy, Esh, calculated for a wide region of nuclei with proton number Z=82-120 and neutron number iV=126-190.5 One can see that the correction has three minima in this region. The first one (—14.3 MeV) is obtained for the doubly magic spherical nucleus 2 0 8 Pb. The second one (—7.2 MeV) appears for the nucleus 270 Hs, which is predicted to be doubly magic deformed nucleus.3 The third one (—7.2 MeV) is obtained for the nucleus 296 114, which is close to the nucleus 298 114 predicted to be doubly magic spherical nucleus. 15 Shapes of nuclei considered in the even larger region of nuclides, Z=82-130

23

and ./V=126-190, are illustrated in Fig. 2. They are calculated by minimizing the ground-state energy of each nucleus in the 4-dimensional deformation space {/?A}> A = 2,4, 6, 8. One can see that most of the nuclei in the considered region are deformed, in particular those around 270 Hs. The values of the deformation parameters /3°, A = 2,4, 6, 8, in the equilibrium points are shown in Fig. 3 for deformed nuclei with Z=94-114 and JV=146-168.

120

130

U0

150

160

170

180

190

N Figure 2. Shapes of nuclei plotted for a wide region of Z=82-130 and iV=126-190.

Thus, the calculations illustrated in Fig. 1 reproduce the experimentally known extra stability of spherical nuclei around 2 0 8 Pb and predict two regions of increased stability of superheavy nuclei. According to Fig. 2, one is the region of deformed nuclei around 270 Hs and the other is the region of spherical nuclei around 298 114.

24

U5

150

155

160

165

170

H5

150

N

155

160

165

170

N

Figure 3. Contour maps of the equilibrium deformations /3°, A=2,4,6,8, plotted as functions of proton Z and neutron TV numbers. Numbers at the contour lines give the values of the deformations.

3 3.1

Problem of deformed superheavy nuclei Energy of the lowest 2+ state

Contour map of the energy of the first 2+ state, E2+, calculated for even-even nuclei with Z—94-114 and 7V=146-168, is given in Fig. 4. 16 According to the calculations, these nuclei are well deformed (cf. Figs. 2 and 3). Thus, their first 2+ state is of the rotational nature and its energy is low. The energy is obtained from the usual formula for a rotational band EI+ =

(h2/2J)I{I+l),

(1)

25

where J is the moment of inertia of a nucleus and / is spin of a rotational state. Moment of inertia is calculated in the cranking approximation. 17 It has been shown in a number of papers (e.g. Refs.18>19>20'21'22) that this approach leads to a good description of the ground-state moments of inertia of well deformed nuclei, especially of the heaviest ones. 22 In this paper, a multidimensional deformation space, particularly important for heaviest nuclei, is used for the first time for the calculation of moments of inertia. Also a finaldepth (Woods-Saxon) single-particle potential is used instead of an infinite (modified oscillator) one, taken in older studies (e.g. Refs. 18 ' 19 ' 20 ' 21 ' 22 ).

J

I

I

I

L

U5

150

155

160

165

N Figure 4. Contour map of calculated energy E2+ of the first rotational state 2 + .

170

26 One can see in Fig. 4 t h a t the calculated energy E2+ is low, it is in the range of 40-50 keV for most of the considered nuclei. T h e calculated values reproduce the existing experimental d a t a for actinide nuclei with a very good average accuracy of about 4 keV. In particular, the value 41.6 keV calculated for the nucleus 2 5 4 No is close to this (44 keV), deduced from recent measurements of the ground-state rotational band of this nucleus. 1 1 ' 1 2 One can also see t h a t a rather unusual systematics of the energies E2+, with two minima of it at the nuclei 2 5 4 No (41.6 keV) and 2 7 0 H s (40.2 keV), is obtained in the region of heaviest nuclei. In regions of lighter deformed nuclei, as those of light-barium and rare-earth nuclei, only one m i n i m u m of £"2+ in each region, situated around its center, is obtained. 2 0 We connect this unusual systemetics with specific structure of heaviest deformed nuclei. In particuar, with the appearance of strong closed deformed shells at 7V=152 and 162 and a weaker shell at Z=IQ8 and a subshell at Z=1Q2 (e.g. Ref. 3 ). T h e energy gap appearing at a shell closure weakens the pairing correlations and increases the m o m e n t of inertia of a nucleus, which is a sensitive function of these correlations. By this mechanism, the m i n i m a of £2+ in Fig. 4 are obtained. 1 6 3.2

Probability

of a decay to the lowest 2+

state

To estimate the chance of measuring the energy of the lowest 2 + state in a decay, it is important to know the probability of the decay to this state, P2+As a m a t t e r of fact, it is sufficient to know the branching ratio P2+/P0+, where po+ is the probability of the decay to the ground state 0-f of a nucleus, as po+ (more exactly a half-lives) has been already calculated for superheavy nuclei in a number of papers (e.g. Refs. 3 ' 5 ' 7 > 8 ) and also measured for some of these nuclei. T h e probability pj+ may be presented as pi+=wI+-PI+,

(2)

where wj+ is the reduced decay probability and Pj+ is the probability to penetrate the potential-energy barrier by a particle with angular m o m e n t u m / . Thus, the branching ratio p 2 +/po+i in which we are interested, is P2+/P0+ = {w2+/w0+)

• (P2+/P0+).

(3)

T h e penetration probabilities P2+ and Po+ are calculated in the quasiclassical W K B approximation, while the ratio of the reduced probabilities, W2+/wo+, is treated phenomenologically. We find t h a t it may be well described by the expression lQ{*A+b) w h e r e A i s t h e mass number of a nucleus.

27

Thus, the ratio P2+/P0+ finally is p2+/p0+

= 10(aA+6> • (P 2 + /Po+).

(4)

Adjustment of the parameters a and b to experimental values of P2+/P0+ 10 for 26 nuclei with Z=88-98 leads to the values a = -0.02687,

6 = 6.3608

(5)

and reproduces the measured values of P2+/P0+ with a good accuracy (rms=0.027). This is shown in Fig. 5. One can see a strong isotopic dependence of the branching ratio. T

"—

0.8 - calc

P2+/P