Moments of inertia of fissioning isomers

Moments of inertia of fissioning isomers

Volume 77B, number 1 PHYSICS LETTERS 17 July 1978 MOMENTS OF INERTIA OF FISSIONING ISOMERS Manfried F A B E R Institut ffir Kernphysik der TU Wien,...

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Volume 77B, number 1


17 July 1978

MOMENTS OF INERTIA OF FISSIONING ISOMERS Manfried F A B E R Institut ffir Kernphysik der TU Wien, Austria Received 29 December 1977 Revised manuscript received 25 May 1978

The moments of inertia of ground states and fissioning isomers of 236U and 24°pu are calculated within the cranking model by modifying the pairing interaction. In both cases the measured and calculated values agree if the pairing strength is independent of the surface area.

The ground state moments of inertia taking account o f superfluidity have been discussed by many authors within the cranking model [ 1 - 6 ] . The results are systematically smaller then the experimental values 9 exp. The pairing strength is usually obtained by fitting the gap parameter A to the empirical o d d - e v e n mass differences pexp. Since A is about 10% larger than pcalc this inaccuracy estimates the experimental moments o f inertia better than the correct choice of G. Sobiczewski et al. [5] and B r a c k e t al. [6] reproduce the moments o f inertia of fissioning isomers better than the ground state moments whereas Hamamoto [7] - including the Migdal term - could explain the ground states but neither for G = const, nor for G c~ S obtained the moments o f isomer bands. In the present letter we calculate the moments o f inertia for the ground state and shape isomers of 24°pu [8] and the recently measured nucleus 236U [9]. We show that the ground state moments are reproducible by a simple choice of the pairing matrix elements. Bey o n d it we obtain accurate moments o f fissioning isomers for a surface independent pairing interaction. We start from the idea that for identical pairing matrix elements G ( k l , k2) = const, the occupation probabilities _


with rlk = e k - ~ - G(k, k ) v 2 and A k = E k l G ( k l , k2) X Ukl Vkl, converge too slowly to the trivial values. Therefore it follows that the interactiOn zone must be 18

limited. This limitation is avoidable if the o2 converge faster. In this case the m o m e n t o f inertia increases also if we keep the e v e n - o d d effect constant. By the simple choice of a gaussian dependence of the pairing matrix elements on energy, G ( k l ' k2 ) = G o e x p ( _ [ ( e k I _ ek2)/Gside ] 2),


the residual interaction of distant levels is small. Now the parameters A k are no more constant. Their largest values are near the Fermi energy. With increasing distance from the Fermi energy e k goes to zero and o2 converges fast if Gside is small enough. The special dependence of the pairing matrix elements on energy is unimportant but the described effect on the occupation probabilities is essential. Similar functional dependences like a triangular dependence: G ( k 1, k 2) = G0(1 - [ekl - ek2 [/(;side)


X u(Gside - [ekl - ek2[ ), where u is the unit step function, affect the moments of inertia in the same way. We determine the pairing strength G O by comparison o f the calculated pairing energies pcalc - n , p for neutrons or protons. The Pne~p~ value we obtain by quadrat. ic interpolation between the binding energies [10] o f three neighbouring nuclei with odd neutron (proton) numbers. If we know the binding energies of four neighbouring nuclei with odd neutron numbers: N - 3, N - 1, N + 1, N + 3 we can fit two parabolas and esti-

Volume 77B, number 1


17 July 1978

Table 1 Moments of inertia for the ground state deformations (I) and the shape isomers (II) calculated with the modified pairing interaction (1). All energies are in MeV. Nuclei


-nPeXp-- -nPCalc pexp p -- pcalc _p





236u I

0.635 _+0.015 0.906

0.325 -+0.005 0.419

90.1 _+1.8l J 135.0 44.9


0.592 _+0.011 1.172

0.325 ± 0.005 0.419

183.5 ± 1.0-t J 291.7 108.2


0.564 ± 0.018 0.740 ± 0.008

0.281 _+0.005 0.425 _+0.003

16 tj 93.0 ± 0.4143.3 50.3 + .


0.612 ± 0.019 1.276 ± 0.010

0.281 ± 0.005 0.425 +_0.003

189.2± 10 i}297. 8 108.6 + 0 3

mate also the uncertainties of the experimental pairing energies, see table 1. The single-particle energies and wave functions used in our calculations are evaluated in a S a x o n Woods potential with a constant skin thickness [11] and necked and diamond like shapes as described by Bracket al. [12]. For rotational and reflection symmetric shapes the minima of the Strutinsky renormalised energy surface [13] of 236U and 240pu have the deformation/3 = 0.34, r = 1.10 and/3 = 0.58, r = 0.90. The parameter r means the ratio: neck cross section to cross section of the ellipsoid parametrised by 13and ~/ = 0 [14] with the same volume and same length of symmetry axis. In table 1 we show the numerical results of the moments of inertia of 236U and 24°pu for our modified pairing interaction (1) (Gside = 1.6 MeV) both for the ground state band and the shape isomeric band by using the cranking model. The Migdal term vanishes since the residual interaction acts only between pairs. The uncertainties in the calculated moments of inertia Qcalc and in the strength of the pairing interaction G O are due to the just mentioned uncertainties in the experimental o d d - e v e n mass differences pexp. Since we do not know the binding energy of 233Ac - a neighbouring nucleus of 2 3 6 U - the uncertainty in the proton pairing energy of 236U cannot be given. On the magnitude of the pairing energy in the shape-isomeric state we have no direct experimental information. We show the results for G O = const. It has been argued that G O should be proportional to the surface area [15]. For G O cx S the moments of inertia


132 297.6 ± 0.9 140 300.3±0.9

of the shape isomers however are about 10% smaller. Thus G O seems to be surface independent. In table 2 we show for comparison the moments of inertia for constant pairing matrix elements G(kl, k2) = G O which corresponds to Gside = oo. The calculated moments are systematically too low by 1 0 - 2 0 % if the pairing strength, as in table 1, is fitted to the o d d - e v e n mass differences. At the end we would like to mention that a variation of the deformation/3 by ~0.01 changes 2 9/h 2 by around 4 MeV. An increment of Gside by 0.5 MeV for the gaussian form of the pairing matrix elements (1) decreases 2 ~/h 2 in the ground state by 7 - 8 MeV and in the fission isomeric state by 2 - 3 MeV. Generally, moments of inertia vary at most by 3% under variations b f the functional form of the pairing matrix ele-

Table 2 Moments of inertia for constant pairing matrix elements G(kl, k2) = Go. Nuclei


pexp = PnCalcG0n


p~xp _- p~alc G0p


0.635 0.906

0.067 0.084

0.636 0.966

0.067 170.0]. 0.084 94.7 J 264.7


~°Pu I

0.564 0.740

0.065 77.7-b 0.083 41.0 J 118.7



0.596 1.008

0.065 175.1) 0.083 97.2 ~ 272.3



74.8?. 37.3 ~ 112.1




Volume 77B, number 1


ments. That is true only if Gside is chosen correctly, i.e. the occupation probabilities u2 are nearly unchanged. For the triangular dependence of the pairing matrix elements (2) with Gside = 3.0 MeV the results differ not more than ~ 3 % from the experimental values, similar to the results for the gaussian dependence (1).

References [1] S.T. Beliaev, Nucl. Phys. 24 (1961) 322. [2] S.G. Nilsson and O. Prior, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 32 (1961) No. 16. [3] J.J. Griffin and M. Rich, Phys. Rev. 118 (1960) 850. [4] J. Krumlinde, Nucl. Phys. A160 (1971) 471. [5] A. Sobiczewski, S. Bjornholm and K. Pomorski, Nucl. Phys. A202 (1973) 274.


17 July 1978

[6] M. Brack, T. Ledergerber, H.C. Pauli and A.S. Jensen, Nucl. Phys. A234 (1974) 185. [7] I. Hamamoto, Phys. Lett. 56B (1975) 431. [8] H.J. Specht, J. Weber, E. Konecny and D. Neunemann, Phys. Lett. 41B (1972) 43. [9] J. Borggreen, J. Pedersen, S. Sletten, R. Heffner and E. Swanson, Nucl. Phys. A279 (1977) 189. [10] At. Data Nucl. Data Tables 19 (1977) 431. [I 1] J. Damgaard, H.C. Pauli, V.V. Paskhehvich and V.M. Strutinsky, Nucl. Phys. A135 (1969)432. [12] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 (1972) No. 2. [13] M. Faber, A. Faessler, M. Ploszajczak and H. Toki, Phys. Lett. 70B (1977) 399. [14] K. Neergard, H. Toki, M. Ploszajczak and A. Faessler, Nucl. Phys. A287 (1977) 48. [15] S.G. Nilsson et al., Nucl. Phys. A131 (1969) 1.