A general numerical solution of collective quadrupole surface motion

Prog. Part Nuc£ Phys., Vol. 28, pp. 405-406, 1992. Printed in Great Britain. All fishts reserved.

0146-6410/92 $15.00 • 1992 Pet'pmoa Preu Ltd

A General Numerical Solution of Collective Quadrupole Surface Motion D. TROLTENIER,J. A. MARUHNand W. GREINER lnstitutfllr Tl~oretischePhysik JohannWolfgangGoetl~-UnivcrsitlKD-6000Fra~ttOdain, C~rratmy

Abstract: We present a numerical method of solving the general Hamiltonian for quadrupole surface motion including deformation-dependent masses and moments of inertia. As an example we apply this method to the microscopically calculated potential energy surface of 2~U published in [1]. Keywords: Quadrupole surface motion, finite elements, pseudo-symplectic model.

The geometric collective model The geometric collective model [2] describes the collective qusdrupole surface motion of even-even nuclei in terms of deformation variables a2,, which determine the nuclear radius

R(0, ~,t) = Ro

1 + ~oo(~2.,0Yoo +

~ .~2.()5.(0,~) t ),

(1)

as expansion coefficients of spherical harmonics of angular momentum two in the centerof-mass system. Formulating the whole problem in the principal axis system the most general intrinsic Hamiltonian can be shown to be H

=

1 "2 ~ a Ik V(fl,'7) + ~ B ~ f l + B~.y~fl~ + B . f l 2 ~ 2 + E 4Bk/~sin2(../+ 2~_A)' k--.1

405

(2)

D. Troltenier et al.

406

238 u

E/MeV 60

Experw~nent

~c. PES

i(I÷1)

~32

5O

~0

3.0

--

--30

--21

--26

--

26

--24

--

24

--

20 - - 12

--

11 ~

--22

--22 --20

--11 --16

--16 --6

10

--26

--25

--20 2.0

--20

30

--10 --$

--|

__,,-) --t2 :2 --~

--I

10 --12

=.% 5_ ~ _17 --=|,

--I --¼

--I --6 ~42

with (~, 3') being the intrinsic quadrupole deformation variables. The seven (up to certain symmetry requirements) arbitrary deformation dependent functions - the potential V(~, 3') and the six mass functions B~ - can be determined either in a phenomenological way (i.e. by adjusting them to experimental data) or by using a microscopical model. Quantizing the above Hamitlonian we are finally left with a system of coupled partial differential equations which is solved numerically by using the finite element method. As an example we applied this model to the potential energy surface of 23sU calculated within the frame of the pseudo-symplectic model [1]. The kinetic energy has been taken to be as simple as possible (,,~ [~r × ~r]) and the corresponding mass parameter has been adjusted to the 61-state. Comparing experimental and calculated data (see figure) we find that the ~-band head nearly matches the experimental value, although its calculated moment of inertia seems to be too small. The 3'-band head is to low but the agreement for the gs-band is quite satisfactory up to L ~ 30h. On the right hand side of the figur we plotted the I(I + 1)/20-law adjusting the moment of inertia 0 to the experimental 16+-energy. Obviously this simple law does not meet the experimental data. Summarizing this report we conclude that the presented method is a powerful tool in investigating collective states and their connection to microscopical theories. Future work will concentrate on high-spin and superdeformed states. An extensive discussion of this method can be found in [3].

References [1] Castafios, 0., Hess, P.O., Proceedings of the 14.th Symposium on Nuclear Physics, World Scientific, Maria-Ester Brandan (Ed.) (1991) 46. [2] Troltenier, D., Maruhn, J.A., Hess, P.O., in Computational Nuclear Physics I, K.Langanke, J.A.Maruhn, and S.E.Koonin, Springer Verlag, Berlin, Heidelberg, New York (1991). [3] Troltenier, D., Maruhn, J.A., Greiner, W., submitted to Nucl. Phys.