Nuclear shape-isomeric vibrations

Volume 171, n u m b e r 4

PHYSICS L E T T E R S B

1 May 1986

NUCLEAR SHAPE-ISOMERIC VIBRATIONS A.S. U M A R University of Pennsylvania, Department of Physics, Philadelphia, PA 19104, USA

and M.R. S T R A Y E R Oak Ridge National Laboratory, Physics Division, Oak Ridge, TN 37831, USA Received 24 February 1986

It is argued that the correlated resonances observed in low-energy collisions of light heavy-ions may be vibrational-rotational states built on the shape-isomeric configurations of the composite nuclear system. These states are calculated for the shape-isomer of the 24Mg nucleus by studying the response of the system to an external perturbation.

In recent time-dependent Hartree-Fock (TDHF) calculations [1,2] of heavy-ion collisions at low energies the long-time evolution of the collision was observed to lead to the formation of attractor regions about the shape-isomeric configurations of the composite nuclear system. These results suggest that for appropriate initial conditions the nuclear system could form a long-lived complex due to the development of a secondary minimum in the heavy-ion potential energy surface. The presence of such configurations in heavy systems strongly influences the fission decay of actinide nuclei [3], and states built on these isomers have been experimentally observed [4]. It has been suggested, both experimentally [5] and theoretically [6], that the existence of shape-isomers could be the underlying physics behind the observed correlated resonances in the collisions of light heavyions. In calculating the states built on a shape-isomer we face the difficulty of performing nuclear structure calculations for triaxially deformed nuclei. A possible solution is through the use of collective models [7,8]. However, an important simplification arises due to the fact that the configuration of a shape-isomer is considerably different than the ground-state configura0370-2693•86•$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

tions of deformed nuclei. Since a shape-isomeric configuration arises due to the redefinition of shell closures we again develop large gaps in the shell structure, iJ a way similar to closed-shell nuclei. In this sense it is physically reasonable to consider vibrational states having a particular symmetry. In what follows we will use the linear-response formalism to study the states built on an isomer of the 24Mg nucleus. This work represents the first microscopic, self-consistent calculation of resonances built on a deformed shapeisomer, using realistic effective interactions, and the formalism is general enough to be applied to other systems. The TDHF calculations mentioned earlier were restricted to axially-symmetric geometry. To establish the correct symmetries of the 24Mg shape-isomer we have performed three-dimensional constrained Hartree-Fock [9] calculations (a constraint on the quadrupole moment), using an oscillator basis expansion up to nine major shells, in cartesian coordinates. With the same force used in the TDHF calculations of ref. [ 1] the ground state of 24Mg is found to be a triaxial nucleus with a quadrupole deformation of 43 fm 2 whereas the shape-isomer corresponds to an axially symmetric configuration with a qua353

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drupole moment of 110 fm 2. These results are also in agreement with Nilsson-Strutinsky calculations [ 10] using general phenomenologieal interactions. Such calculations confirm the fact that the isomer obtained in the TDHF calculations of ref. [ I ] is a true isomer of the 24Mg nucleus and possesses axial symmetry. Below, the axially symmetric shape-isomer of 24Mg is calculated as a solution of the unconstrained Hartree-Fock equations in cylindrical polar coordinates. We have used the effective interactions Skyrme I, Skyrme II, and Skyrme M* [11] without the spin-orbit part. All of these forces yield a quadrupole deformation for the isomer of about 105 fm 2, and the resulting moment of inertia is approximately 6.2 MeV-1. The density vibrations of the isomer are calculated with the linear-response formalism [12]. Here we study the response of the nucleus to a time-dependent, one-body, external perturbation coupled to the nuclear density Hex(t) = f d3r p(r, t)F(r, t),

(1)

where p is the nuclear density which can be obtained from the one-body density matrix ~, and F represents the external field. Under the assumption that the many-body wavefunction is a Slater determinant (t~2 = ~), the time-dependent density matrix obeys the TDHF equations inb = [h(,o) +f(t), Pl.

(2)

Here h(p) is the single-particle Hartree-Fock hamiltonian and/'is taken to be the multipole operator

f(r,t)= XrLYLo(?~S(t)

L #=O,

= Xr28(t),

L = 0.

(3)

Because of the delta function in time all of the frequencies are excited by the same external field. In coordinate space, the response function is defined to be the change in the density with respect to the equilibrium density

6p(r, t) = p(r, t) -- PO(r) ,

(4)

where P0 satisfies the static Hartree-Fock equations Ih(P0), P0] = 0 .

(5)

The associated strength function which measures the transition strength from excited states to the ground 354

1 May 1986

state, for a given multipole operator F, is -1-oo

SL(E ) = f

dt

exp(-i£t'/Ti)f d3rf(r)~p(r,t).

(6)

--oo

The numerical solution of eq. (2) is described in ref. [13]. As a check of the numerical procedure we have calculated the isosealar quadrupole, octupole, and hexadecupole giant resonances for the 160 nucleus. The Hartree-Fock density for the spherical 160 was obtained by using the Skyrme II force. The parameter X was chosen to assure an undamped motion. During the time evolution the mass number was conserved to better than one part in 107 and we have used h = 0.005. Decreasing X does not change any of the results presented here whereas larger values lead to the dampening of the motion due to the emission of particles which reflect from the walls of the numerical box [14]. The box size used was 25 fm in z and 10 fm in r l direction with a uniform mesh spacing of 0.5 fm. The time discretization employed a 0.5 fm/c time-step and the density oscillations were followed up to 214 fm/c. This time evolution allows a Fourier transform with an accuracy of 75 KeV. The comparison to full continuum RPA calculations of ref. [15] was excellent, the largest difference between the centroid energies being 0.5 MeV for the isoscalar quadrupole resonance (the error was better than 2% for all three resonances.) The calculation of the resonances built on the isomer of 24Mg proceeds along the same lines. Since the TDHF equations are solved in axial geometry this symmetry is always preserved. Because of this restriction the direction of the symmetry axis does not change and the projection of the total angular momentum onto the symmetry axis, K, remains a constant of the motion [7] (K ~r= 0+). The possible vibrational modes are vibrations that are parallel and orthogonal to the symmetry axis of the nucleus. Because of the restriction to axial symmetry we only study the L = 0 states. In fig. la we show the time development of the nuclear radius for the Skyrme I force. The evolution is nearly periodic with a slight dampening at very large times and it corresponds to a very slow motion. This time behavior has a characteristic frequency of about 1 MeV as can be seen from the strength function shown in fig. 1b. We also note that the spurious zero-frequency peak appears

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Table 1 Comparison of the experimental (L = 0) energies with the resuits of the present calculation and the vibrational energies of the U(4) model. The data and U(4) energies are from ref. [16]

3 4'9 548

i

i 9 s4s a: 3 4 3 3.41

3 39

0

35

t0 °

'

'140

70 tC5 TIME x 'qO5 (fro/C)

i

I

i

'

'

i

'

:u_.,S

1 May 1986

[

V

'

:!.

0

,I

2

3

4

'V

~1,,

5 6 7 ENERGY (MeV)

i 8

9

Skyrme M* (MeV)

Experimental (MeV)

U(4) (MeV)

0.98 1.97 3.01 4.00 4.80 5.75

1.01 2.12 3.08 4.13 5.06 5.90

3.17-3.35 4.25 5.80

1.04 2.58 3.44 4.44 5.20 5.84

(D)

I0 -310-z :~A il. l-.," :. ,o-,

Skyrme I (MeV)

:,

t0

Fig. 1. (a) The time evolution of the nuclear radius for the oscillating ~Mg isomer with the Skyrme I force. (b) The strength function (fro4) of eq. (6) for the ~'Mg isomer (Skyrme I). The peaks are normalized to unity. exactly at zero energy which is an indication of the approximate removal of the zero-point energy and the overall convergence of the mesh spacing and the box size used in calculations. This is a consequence of having continuum states with high enough energy to achieve the convergence of the particle-hole algebra in calculating the states built on the isomer. The centroid energies of the peaks are tabulated in table 1. Due to the large static deformation of the isomer the spectrum is considerably compressed as compared to the vibrations based on the ground states of the spherical nuclei. In table 1 we also show the same quantity for the Skyrme M* force together with the experimental levels (L = 0) and the vibrational energies of the U(4) mode [16]. At this point we note that the calculated spectrum can be shifted by a constant energy due to the uncertainty in the excitation energy of the isomer with respect to the

ground state. This uncertainty arises because of the center-of-mass motion in Hartree-Fock theories. This spurious energy, in principle, will depend on the density distribution of the nuclear configuration, and for strongly deformed nuclei it produces a significant rotational correlation energy (in the case of the isomer of I2C the relative spurious energy is about 7 MeV.) [17] In table 1 the lowest energy was assumed to coincide approximately with that of the U(4) model, and the other five resonances have been shifted by the same constant. Overall, the energy spacings are in good agreement with the vibrational spacings of the U(4) model and the data. In the case of Skyrme II force the isomer was found to be unstable against large amplitude axially symmetric vibrations. In principle we could not decouple the vibrations that are parallel (longitudinal) and perpendicular (transverse) to the symmetry axis. In order to gain further insight about the nature of the vibrations we have computed the response functions for longitudinal and transverse modes by choosingf as r± and (z - Zcm)2 respectively. The subsequent motion in the longitudinal direction oscillates with large amplitude (Az 4.2 fm) whereas the amplitude in the transverse direction is small (~r± ~ 0.02 fm). This result is in agreement with the fact that for strongly deformed nuclei the vibrations in the direction of smallest incompressibility (for prolate nuclei the longitudinal direction) will produce the low-energy components of the collective excitation. The resonance energies shown in table 1 are approximately equally spaced. Also, the crossover transition strengths from excited states to the ground state of the isomer (fig. lb) fall rapidly with increasing excitation. These characteris355

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0.10

,-~

= 1.0 fn

0.08 0.06

×

t2

52.0

x

39.0

I

1

' 26.0

~q~, 13.0

0.02 0 46

~o

'~

0.04

ro

ZI m

65.0

032

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o ~o o

,fm

I

I

I

8

'~-~' 2~

4

ct.j

2

) A \ \

o 0

2

4 Z (frn)

0

6

2

4 q. (fro)

Fig. 2. Slices of the transition densities of eq. (7) plotted for different resonance energies, in cylindrical polar coordinates the radial distance is given by r = (r~ + z2) 112. The force used is Skyrme M*.

tics suggest that these states could be harmonics built on the ground state of the isomer. This can be quantified by considering the transition densities. The structure o f S t ( E ) seen in fig. lb implies

(at least for L < 10) is physically reasonable since we are dealing with a strongly deformed rotational type nucleus. In the case of axial symmetry the wavefunction representing the rotational motion of the nucleus is

6p(r, t) = ~ p(~)(r) e x p ( - i E o t ~ ) ,

*LMK=0(0, ~b, ")') = (1/V~)YLM(O, q~),

(7)

(8)

o

where the summation runs over the vibrational energies o f table 1 and/9(°) are the transition densities [18]. Consequently the transition densities can be obtained by taking the Fourier transform of eq. (7). In fig. 2 we plot slices of the transition densities for the first two resonances of table 1 using the Skyrme M* force. The most important feature of these densities is their similarity for different resonances (the small differences arise due to the fact that in doing the finite Fourier transform we get energies which are slightly off the peak energy). This indicates that the resonances are multiple excitations of an harmonic based on the ground state of the isomer. The vibrations mainly occur on the nuclear surface. Although there are only 3 experimental L = 0 states to compare with our findings, we can use all six of our resonances to produce a fit for the experimental states having L > 0. The experimental states (28 resonances) given in ref. [16] can be fitted by building a rotational band onto each of the sixL = 0 states. The approximate decoupling of the intrinsic and rotational motions 356

where (0, ¢, 3,) denote the three Euler angles, and M is the component of the total angular momentum L on the space fixed z-axis. The energies of a rota~io-nal band built on a particular vibrational state with energy E o can be written as E(o,L) =E a + (hE/29)L(L

+ 1).

(9)

Here 9 denotes the moment o f inertia of the isomer (6.2 MeV-1), and the values o f E a run through the six L = 0 resonances in columns 1 or 2 o f table 1. The values o f L are restricted to even integers because o f the invariance of the nucleus under 180 ° rotations about an axis perpendicular to the symmetry axis. The resulting fit to the 28 resonances, with either set o f the six L = 0 states, is as good as the fit given in fig. 2 of ref. [16]. In summary, we have performed calculations of resonances built on the axially-symmetric shape-isomer of the 24Mg nucleus. The L = 0 states are found to be harmonics which correspond to density virbations along the symmetry axis. A simple assumption of decoupling the intrinsic and rotational motions seems

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to work well in fitting the experimental states with L > 0. These results together with the findings of refs. [1,2] indicate that such calculations have the possibility for describing the correlated resonances observed for light systems. Furthermore, we believe that the computational methods used in the present work may provide a useful tool for the study of states built o n other isomers and deformed configurations. We wish to thank Dr. G.F. Bertsch and Dr. K.A. Erb for many fruitful discussions. The research was sponsored in part by the United States Department of Energy under contract 40132-5-20441 and under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.

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[4] H.J. Specth, J. Weber, E. Konecny and D. Heunemann, Phys. Lett. B 41 (1972) 43. [5] K.A. Erb and D.A. Bromley, in Treatise on heavy-ion science, Vol. 3, ed. D.A. Bromley (Pleura, New York, 1985). [6] H. Chandra and U. Mosel, Nucl. Phys. A298 (1978) 151. [7] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II, (Benjamin, New York, 1975) p.7. [8] F. Arickx, J. Broeckhove, E. Caurier, E. Deumens and P. Van Leuven, Nucl. Phys. A398 (1983) 467. [9] H. Flocard, P. Quentin, A.K. Kerman and D. Vautherin, Nucl. Phys. A203 (1973) 433. [10] G. Leander and S.E. Larsson, Nucl. Phys. A239 (1975) 93. [11] J. Bartel, P. Quentin, M. Brack, C. Guet and H.B. Hankansson, Nucl. Phys. A386 (1982) 79. [12] G.F. Bertsch and S.F. Tsai, Phys. Rep. 18 (1975) 125; J. Blocki and H. Flocard, Phys. Lett. B 85 (1979) 163. [13] K.T.R. Davies and S.E. Koonin, Phys. Rev. C23 (1981) 2042. [14] S. Stringari and D. Vautherin, Phys. Lett. B 88 (1979) 1. [15] S. Krewald, V. Klemt, J. Speth and A. Faessler, Nucl. Phys. A281 (1977) 166. [16] K.A. Erb and D.A. Bromley, Phys. Rev. C23 (1981) 2781. [17] R.Y. Cusson, R. Hilko and D. Kolb, Nucl. Phys. A270 (1976) 437. [18] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berfin, 1980) p. 290.

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