Synthesis of new elements within the fragmentation theory: Application to Z = 104 and 106 elements

Volume 60B, number 3

PHYSICS LETTERS

SYNTHESIS OF NEW ELEMENTS WITHIN THE FRAGMENTATION A P P L I C A T I O N T O Z = 104 a n d 106 E L E M E N T S ~

19 January 1976

THEORY:

A. SANDULESCU*, R.K. GUPTA**, W. SCHEID and W. GREINER Institut fi~r Theoretische Physik der Johann Wolfgang Goethe Universitilt, Frankfurt am Main, Germany Received 12 November 1975 From a new point of view, the fragmentation theory is used to select out the projectile and target nuclei for the production of super-heavy elements. It is suggested to choose the partners of a reaction corresponding to the minima in the potential energy, calculated as a function of the mass- and charge asymmetry coordinates. The theory is supported by the experiments already carried out, and also new experiments are suggested. An important question in the production of superheavy elements through heavy ion collisions has been the choice of the proper target and projectile combination [ 1 - 6 ] . The production cross sections are shown [5] to be very sensitive to the choice of the reaction partners. In this paper, we propose an answer to this question of the optimum choice of the reaction partners on the basis of the theory of fragmentation [ 7 - 1 2 ] . The fragmentation theory is already successfully applied to the prediction of fission mass- and charge yields [8, 11 ] and to the fragmentation process in heavy ion collisions [9, 10, 12]. In the following we discuss its application from a new point of view, giving a theoretical method to select out the projectile and target nuclei for the possible production of a superheavy element. The idea of the method is to choose the reaction partners such that the compound system is formed with a minimum of excitation energy. For a cooler compound nucleus, the number of neutrons emitted would be smaller and consequently the cross section for the formation of the nucleus in the ground state would be large [13]. Since the compound system can be reached by various combinations of the projectile and target, the excitation energy of the compound system has to be ¢~ This work is supported by the Bundesministerium ffir Forschung und Technologie, and by the GeseUschaft fiir Schwerionenforschung (GSI). * Guest Professor from Joint Institute for Nuclear Research, Dubna, USSR. ** Senior Fellow of Alexander yon Humboldt-Stfftung, and on leave of absence from Kurukshetra University, Regional Centre for Post-Grad. Studies, Rohtak, India.

calculated for all possible combinations. In this paper, we have made calculations for the compound systems A = 258, Z = 104 a n d A = 260, Z = 106 which decay, say, to the isotopesA = 2 5 6 , Z = 104, a n d A = 258, Z = 106 after the evaporation of two neutrons. Various isotopes of element Z = 104 are synthesized at Berkeley [1,2] by bombarding 249Cf with 12,13C and 248Cm with 180 and at Dubna [4, 5] by bombarding 242pu with 22Ne and 206,207,208pb with 5°Ti. Some isotopes of element Z = 106 are synthesized at Berkely [3] by bombarding 249Cf with 180 and at Dubna [6], by bombarding 206,2°7,208pb with 54Cr. Our calculations are made for the compound nucleus formed with ions of Z~> 20 and hence do not include the experiments with light projectiles. The fragmentation theory describes all two-body channels from which the compound system can be formed or into which it can decay. The coordinates of the nuclear system are: the relative distance R, the collective surface coordinates a (1) and a (2) of the individual nuclei, the neck parameter e and the massand charge fragmentation [7, 11 ] 7? --- (A 1 - A 2 ) / ( A I + A 2 ) and r/Z = (Z 1 - Z 2 ) / ( Z 1 +Z2). Under the assumption of homogenous nuclear density, the coordinates r/ and r/Z apparently reduce to the volume asymmetry coordinates ~ and ~z, respectively. The collective Hamiltonian, depending on the coordinates R , r~, r~z, a(1), and a (2) and their canonically conjugate momenta, is given by: H = Tkib(PR ' P~' P ~O )' Pa(2 ) R ' rl, r~Z, ot(1) , ot(2)) + V ( R , 7?, r/z , aft), a (2)) .

(1)

225

Volume 60B, number 3

~020[ b)

PHYSICS LETTERS

z581041R=Rc/~

0 201- b )

°0l o+ /1 + ,.~

GFU,U p,u,u ,u~

....

°1°I

~6°106R=R

I/

2GO

o)

-

106: Adiabatic Potentials

-1/51 ~ -1765 I~,13.3fm

,,~ i -1870

/'~'~j

-187G

-1875

-1880

-1875

,,+I

-lgSO '

-2O2O ~ -2120

~

R-~o / / /

201 - 212

-

- 215~

-2~5o ~

_2roof "2180-0.G-0.4-0.2 0 0.2 0.4 0.6 -o6-o4-oz o oz o4 o6 Mass~ymmelry rI 51,G 7"/.4 10&2129 1%.8180.6206.4 Woss~mberA~

, , i~,s,s ,Asymmetry,q. . . . 52 78 104 130 15G 182 208

HOSS NumberA~

Fig. 1. (a) Potential energy as a function of mass-asymmetry and the relative separation R for the compound system 2s8104 and 260106. Curves for R ~ R c include the minimization in charge-asymmetry coordinate nz, except for the dashed part. Energy scales used forR ~JR c, where R c is some critical distance at 226

which the two nuclei come in close contact with each other, the potential is given simply by the Coulomb interaction plus the sum of the ground state binding energies of the two fragments [9, 10]

V(R, f/, rIz, t31,132) = ZlZ2e2/R

L_.

o) ZsBlO4 : ~iobotic Potentials -1751

~-m65

19 January 1976

B(A1, Z1,131) B(A2, Z2,132)"

(2)

-

The binding energies are taken from the atomic mass tables, given by Seeger [18] for Z/> 20, where the liquid drop energy is smoothed out with the shell corrections determined with the Nilsson model. Since for separated fragments the two-centre shell model approaches the Nilsson model for each fragment, we obtain smooth continuation of V from the interaction (R < Rc) to the asymptotic region (R/> Rc). Figs. l(a)give the result of our calculation for the potential V(R, ~7)for the compound nuclei 258104 and 260106. These plots, therefore, represent the potential V(R, r~)for various combinations of the target and projectile nuclei leading to the same compound nucleus. In the overlapping region (R < Rc), a full three dimensional minimization in e, 131 and 132 is carried out, thereby assuming adiabaticity in the coordinates e, 1~1 and 132"The minimization procedure is very time consuming and hence we have made our calculations for only two R-values in the case of the compound nucleus 258104. Since for the fixed length of the nucleus the distance R changes with ~7, we have labelled these curves by an average value of R. The coordinate rIz is not included in these calculations. In the asymptotic region (R ~>Rc) we have applied our earlier method [10] of minimizing V(R, 7, rtz) for each possible fragmentation in masses and charges. The minimization in the 13-coordinates is then automatically carried out. This method thus involves the calculation of the charge dispersion potential V(r~z) for each R- and q-value. The distance R c of closest approach is calculated by using the empirical relation of Gutbrod et al. [19] and the calculations shown in figs. l(a) are carried out for an average value o f R c. We notice from figs. l(a) that for each compound nucleus deep minima in the potential energy occur at only a few ~?-values. As the two nuclei overlap (R 7 - 8 fm), the potential V(R, rl) becomes more or less flat in rl. This happens because for R ~ 8 fm the neck has already started to disappear so that the placing of a dividing plane which determine r~, is no more signifi-

Volume 60B, number 3

PHYSICS LETTERS

19 January 1976

Table 1 Projectile and target combinations for the production of elements Z = 104 and Z = 106, corresponding to the potential energy minima and its neighbourhood. Only those combinations are given where both the partners of the reaction are stable in nature. Element Z

104

Ir/I

0.039 0.054* 0.360 0.610

106

0.031 0.046 0.062* 0.338* 0.369 '0.585

Projectile and target nuclei

Quadrupole deform. [18]

A1

A2

A1

A~

lgo4Sn 12~Sn 8~Se ~OTi

134.

l~6Te soan 122~so~n 86 36Kr 82 34Se 124.,

54 24Cr

Experiment performed

s4xe

0.0

0.0

No

136.. 54A.e

0.0

0.0

No

t~6ybm) 208 82Pb

0.052 0.0

0.183 0.0

No Yes [51

134.~

0.037

0.0

No

136~ 56t~a 138~

54~xe

0.0

0.0

No

0.0

0.0

No

174 70Yb 178 m)

0.0

0.185

No

0.052 0.047

0.173 0.0

No Yes [61

56t~a

72Hf 206 ~ . 82ro

* Values that correspond to the minima in V (R, ~, nz)" m) Stable isomer. cant. The interesting point is that these deep minima in V ( R , rl) are not only stable in ~7but also no new minima appear after the two nuclei overlap to form the compound system. Therefore, the potential V(r/, r/z, R = R c ) w h i c h is easily computable through eq. (2) gives already the positions of the minima with respect to r~ and rIz. Evidently, the potential minima in figs. l(a) are related to shell effects, with at least one of the two nuclei being a spherical nucleus. This fact is demonstrated in figs. l(b) where the static deformations [18]/31 and/~ 2 are plotted f o r R = R c. Next, in the dynamical fragmentation theory applied to nucleus-nucleus collision [10], one has to solve the stationary Schr6dinger equation H ( R , rl, r?Z , a (1), a (2)) ~ = E f t ,

(3)

with the following initial state (R ~ ~o) ffi ~ exp (ikz) 61/207 - r~0) 61/207z+- rlZo)

(4)

X [Xnl (a (1), 770, r/Zo) ® Xn2(a(2),r/o , r/zo) ] . The 6-type wave functions describe the mass and charge states. The wave functions Xnl and Xn2 represent the intrinsic states of the incoming nuclei, specified by r/0 and rlZo and are also assumed to be the surface vibrational states described by aft)and a (2), respectively. Since the static deformations/31 and/32 depend strongly

on the fragmentation coordinates 77 and r/z (see figs. 1(b)), the Hamiltonian couples the surface vibrations and the mass- and charge fragmentations strongly. Then, if the transfer of mass and charge occurs, the shapes of the two incoming nuclei would change which would apparently cause a strong transfer of energy into the surface degrees of freedom. Thus, two different cases of initial fragmentation can be distinguished which would lead to a different amount of excitation of the compound system: i) The initial fragments lie outside the potential energy minima in figs. l(a), which according to figs. l(b) correspond to both ¢11 and/~2¢0, and ii) the initial fragments lie on the energy minima, which refer then to either one or both nuclei to be spherical. We might remind here that for reasons of large fusion probability, we are interested in the case of minimum excitation energy of the compound system. In case i), a large mass and charge transfer would occur in the direction to the minima of potential V(r~) as the two nuclei start to overlap. The forces which drive the system are given by - 3 V / 3 r l and - ~ V/Orlz according to classical mechanics. Since the nuclei would change shapes during the running of the system in the direction to the potential minima, a large amount of energy is transferred into the excitation of the surface vibrations. On the other hand in case ii) no mass and charge transfer would occur since then the 227

Volume 60B, number 3

PHYSICS LETTERS

driving forces - 3 V/3~? and - b V/3rlz are zero. In that chase, quantum mechanically the r/-dependence o f the wave functions is approximately given by the zeropoint motion around the potential minima. Hence, we find that if the target and projectile are chosen with respect to the minima in the potential V(R, 7, rlz) and further if the bombarding energy is so chosen that the compound system is reached straight in a central collision, the excitation would be minimum. For non-central collisions, the compound system carries a large amount of angular m o m e n t u m and is then highly excited. As a first test of our suggested selection rule, we have given in table 1 the possible combinations of the projectile and target, corresponding to the minima o f Vin ~ and 77Z of figs. l(a), which we consider could synthesize to produce the elements Z = 104 and 106. Since, in general due to the charge dispersion effects a transfer o f 2 protons could produce a gain or loss of 7 - 1 0 MeV we have also listed the neighbouring combinations with -+2 protons/neutrons. It is interesting to find that the combinations 50Ti + 208 54 206 -82Pb and 24Cr + 82Pb are used in Dubna experiments [5,6] for the synthesis of Z = 104 and 106 elements, respectively. Secondly, on the basis o f the estimates for fission half-life, Bengtsson et al. [20] have suggested 112 136 the combination 50Sn + 54Xe for the production o f Z = 104 element. It is encouraging that b o t h the studies stress on Sn and Xe beams for their use as target and projectile for the production of Z = 104 in the laboratory In table 1, we have also given the quadrupole deformation parameters [18]. As already pointed out, all these combinations have at least one o f the two nuclei with a spherical shape. The choice o f at least one sphe-

228

19 January 1976

rical nucleus in scattering experiments is in agreement with the successful experiments at Berkeley and Dubna.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

A. Ghiorso et al., Phys. Rev. Lett. 22 (1969) 1317. A. Ghiorso et al., Phys. Lett. 32B (1970) 95. A. Ghiorso et al., Phys. Rev. Lett. 33 (1974) 1490. G.N. Flerov et al., Phys. Lett. 13 (1964) 73. Yu.Ts. Oganessian et al., preprint JINR, D2-8224, Dubna 1974. Yu.Ts. Oganessian et al., preprint JINR, D7-8099, Dubna 1974. H.J. Fink et al., Z. Physik 268 (1974) 321. J. Maruhn and W. Greiner, Phys. Rev. Lett. 32 (1974) 548. H.J. Fink et al., Proc. Int. Conf. on Reaction Between Complex Nuclei, Nashville 1974, Vol. II (North-Holland Pub. Co.) p. 21. H.J. Fink et al., Proc. Int. Summer School on Nucl. Phys., Predeal Rumania 1974, to be published, R.K. Gupta, W. Greiner and W. Scheid, ibid. R.K. Gupta, W. Scheid and W. Greiner, Phys. Rev. Lett. 35 (1975) 353. O. Zohni, J. Maruhn, W. Scheid and W. Greiner, preprint, Frankfurt 1975. Yu.Ts. Oganessian et al., preprint, JINR D7-8194, Dubna 1974. V.M..Strutinsky, Nucl. Phys. A95 (1967) 420; A122 (1968) 1. W.D. Myers and W.J. Swiatecki, Arkiv Fizik 36 (1967) 343. T. Johansson, S.G. Nilsson and Z. Szymanski, Ann. Phys. 5 (1970) 377. J. Maruhn and W. Greiner, Z. Physik 251 (1972) 431. P.A. Seeger, CERN report No. 70-30, Vol. 1, p. 217, Geneva 1970. H.H. Gutbrod, W.G. Winn and M. Blann, Phys. Rev. Lett. 30 (1973) 1259. R. Bengtsson et al., Phys. Lett. 55B (1975) 6.