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A shape isomeric state in 32S from an unconstrained variation
Volume 94B, number 3
PHYSICS LETTERS
11 August 1980
A SHAPE ISOMERIC STATE IN 32S FROM AN UNCONSTRAINED VARIATION cr W. BAUHOFF!, H. SCHULTHEIS and R. SCHULTHEIS lnstitut ffir Theoretische Physik, Universitiit Tiibingen, D-7400 Tiibingen, West Germany Received 20 Junary 1980
An unconstrained variational search has been performed for a shape isomeric state in 32S. The effect of parity and angular momentum projection has been studied. The resulting many-body state has a double-peaked density distribution with an 160-160 structure.
Isomeric fission associated with the existence of shape isomeric states has been observed in many actinide nuclei. In 32S a similar shape isomerism has been predicted, but not observed. The predictions are based on the occurrence of a second minimum in Strutinsky-type, constrained H a r t r e e - F o c k and other variational calculations [ 1 - 3 ] . There is, however, some controversy about the prediction because constrained Hartree-Fock results with and without a second minimum exist [ 2 - 4 ] . Moreover, these calculations are for the intrinsic state only and the effect of parity and angular momentum projection has not been considered. In the present note we give the results of a numerical search for a shape isomeric state in 32S using standard finite range nucleon-nucleon forces. In contrast to the usual procedure we have performed an unconstrained variation o f all parameters of the many-body wave function. This avoids the arbitrariness of the customary constrained variation the outcome of which depends to some extent on the choice of the constraint and may even be at variance with it. In th e present calculation the variation has been performed for the Brink-Boeker force [5] B 1 and Volkov's force [6] no. 1. using states of the alphacluster model as introduced by Brink [7]. For the 328 system there are 19 variational parameters: the Supported by the Bundesministerium fiir Forschung und TechnologJe and the Deutsche Forschungsgemeinschaft. 1 Present adress: I. Institut ftir Experimentalphysik, Universit~/t Hamburg, D-2000 Hamburg, Fed. Rep. Germany.
oscillator width b = ( h / m ~ ) 1/2, the ¼A = 8 a-cluster positions R1, ..., R 8, with three coordinates each, minus the three center-of-mass coordinates and the three Euler angles for the orientation of the system. The variational 32-particle state is a Slater determinant of ls harmonic oscillator orbitals around R 1 , ..., R 8 , each occupied by four nucleons with anti-aligned spins and isospins, i.e. q~= Nq~-~ ~1 (x 1) ...~OA(XA),
where ~i(xi) = (blrl/2)-3/2 exp [ - ( x i - R/)Z/Zb 2 ] , i = 1. . . . . A ,
1 j = 1. . . . . g A ,
with the normalization factor NO and the antisymmetrizer M . The method of the present note is similar to that described in refs. [8,9] except that the Coulomb energy has been included and a full variation of all 19 parameters has been performed. The resulting parameter values, excitation energy, rms radius and quadrupole moment are given in table 1 for the B 1 force. It is remarkable that the resulting a-cluster centers R i are located in the corners of two elongated tetrahedrons at a center-to-center distance of d = 3.9 fm. In the a-particle model the groundstate wave function of 16 O is associated with a regular tetrahedron of cluster centers (cf. e.g. ref. [7]). Therefore, the calculated isomeric state can be interpreted as consisting of two distorted 160 states at a short separation, that are anti-symmetrized with respect to all 32 particles. 285
Volume 94B, number 3
PHYSICS LETTERS
11 August 1980
Table 1 Parameter values R i and b, excitation energy E, rms radius Rrm s and quadrupole moment Qo of the shape isomeric state for the B1 before, as resulting from an unconstrained variation of the intrinsic state. The quantities labelled 0÷ result from the restricted variation after parity and angular-momentum projection described in the text. The excitation energies Z~Eare relative to the Ba ground state of ref. [10].
x y z intr 0÷
R1 (fm)
Ra(fm)
R3(fm)
R4(fm)
R5 ( f r o )
R6(fm)
RT(fm)
Rs(fm)
0.0 0.0 -3.28
0.05 0.09 -1.95
0.05 -0.09 -1.95
-0.10 0.0 -1.95
-0.05 0.09 1.95
-0.05 -0.09 1.95
0.10 0.0 1.95
0.0 0.0 3.28
b = 1.78 fm b = 1.72 fm
E = --175.40 MeV E = -183.16 MeV
AE = 0.70 MeV zXE= 3.71 MeV
In order to investigate the possibility of mixing with the ground state [10] of 32S the overlap of the two states has to be calculated. Since both states are strongly deformed the overlap will depend on their relative orientation. The largest value obtained is of the order of 10 -13 so a mixing is negligible. It is further an indication that a variation with the constraint of orthogonality to the ground state, as used in H a r t r e e Fock calculations (e.g. ref. [11]), would lead to the same result. Within the point symmetry, that results from the unconstrained calculation, an additional variation after parity and angular m o m e n t u m projection has been performed. It turned out that the variation after projection leads only to small changes in the parameters of table 1. Since a second m i n i m u m is present in the potential energy surface of both the intrinsic state and the projected 0 + state the prediction of the existence of a shape isomeric state in 328 should be reliable. The calculated excitation energy of 3.71 MeV, however, may be at variance with the choice of the n u c l e o n - n u c l e o n force (see below). The single-particle density of the isomeric state A
p(r)= <41~3 6(r-x,)14>, i=1
is plotted in fig. 1 for a cut along the "fission" axis connecting the 160 "fragments". It exhibits a doublepeaked structure corresponding to two distorted 160 states. We note that the Pauli exclusion principle is important in forming such substructures at a rather short separation. No double-peaked density distribution results if the exchange terms between both 160 "fragments" are left out in the antisymmetrization. 286
Rrm s = 3.67 fm Rrm s = 3.69 fm
Qo = 199 fm2 Qo = 208 fm2
'~f~ Fig. 1. Density distribution of the shape isomeric state in 32S. The state has distinct clustering into two tetrahedral 160 substructures (R 1..... R4 and R s ..... R8 in table 1) that give rise to the two pear-shaped density peaks in the center.
In this case the densities of the strongly overlapping fragments add up in a single peak at the center. A density distribution similar to fig. 1 has been found in a generator-coordinate 160 + 160 calculation [12] for the same force but under the sudden assumption (see fig. 4c of ref. [12]). In this case, however, the state is no equilibrium state, it occurs at a smaller separation and at an energy much above the 160 + 160 quasi-molecular state of 24.5 MeV. Our results for the V1 force are very similar to the plot of fig. 1 except that the V1 densities are more compact leading to a larger maximum density. Thus the behaviour of the system under distortion should not depend on pecularities of the n u c l e o n - n u c l e o n interaction chosen. We find, however, a large difference
Volume 94B, number 3
PHYSICS LETTERS
in the excitation energy of the shape isomeric state. It is 22.11 MeV for the intrinsic state and 24.08 MeV for the 0 + state which is much higher than for the B 1 force. This discrepancy may be caused by the different form of the potentials in the odd state: The B 1 force is strongly repulsive in the odd state, contrary to the V1 force [5]. The odd-state repulsion is known to favour energetically the formation of c~ clusters [13]. By analogy, we expect it also to favour the formation of larger (closed-shell) clusters like 160. In order to study the occurrence of substructures between first and second m i n i m u m we have also performed constrained variations for a sequence of increasing elongations of the system (as in ref. [8]). According to our results the substructure formation sets in between the first and the second m i n i m u m in the vicinity of the barrier. Beyond the second minimum there is no such rearrangement in the many-body state. The further separation is associated only with gradually increasing distortion but with no structural changes in the density distribution. As regards the old controversy of early versus late fragmentation in fission our results clearly support the former (to the extent to which a many-body system heavier than 32S can be expected to behave in a similar way under deformation).
11 August 1980
References [1] R.K. Sheline, I. Ragnarsson and S.G. Nilsson, Phys. Lett. 41B (1972) 115. [2] S.J. Krieger and C.Y. Wong, Phys. Rev. Lett. 28 (1972) 690. [3] P.G. Zintand U. Mosel, Phys. Lett. 58B (1975) 269; Phys. Rev. C14 (1976) 1488. [4] H. Flocard, Phys. Lett. 49B (1974) 129. [5] D.M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1. [6] A.B. Volkov, Nucl. Phys. 74 (1965) 33. [7] D.M. Brink, Intern. School of Physics Enrico Fermi, Course XXXVI (Academic Press, New York, 1966) p. 247. [8] H. Schultheis, R. Schultheis, K. Wildermuth, A. Faessler and F. Griimmer, Z. Phys. A286 (1978) 65. [9] W. Bauhoff, H. Schultheis, R. Schultheis and K. Wildermuth, Proc. Fourth Intern. IAEA Symp. on Physics and chemistry of fission (J/ilich, 1979), (IAEA, Vienna, 1979). [10] W. Bauhoff, H. Schultheis and R. Schultheis, Phys. Rev. C, to be published. [11] G. Do Dang, K.W. Schmid, R.M. Dreizler and H.G. Miller, Phys. Lett. 62B (1976) 1. [12] D. Baye and G. Reidemeister, Nucl. Phys. A258 (1976) 157. [13] S.B. Khadkikar, Phys. Lett. 36B (1971)451.
287
PHYSICS LETTERS
11 August 1980
A SHAPE ISOMERIC STATE IN 32S FROM AN UNCONSTRAINED VARIATION cr W. BAUHOFF!, H. SCHULTHEIS and R. SCHULTHEIS lnstitut ffir Theoretische Physik, Universitiit Tiibingen, D-7400 Tiibingen, West Germany Received 20 Junary 1980
An unconstrained variational search has been performed for a shape isomeric state in 32S. The effect of parity and angular momentum projection has been studied. The resulting many-body state has a double-peaked density distribution with an 160-160 structure.
Isomeric fission associated with the existence of shape isomeric states has been observed in many actinide nuclei. In 32S a similar shape isomerism has been predicted, but not observed. The predictions are based on the occurrence of a second minimum in Strutinsky-type, constrained H a r t r e e - F o c k and other variational calculations [ 1 - 3 ] . There is, however, some controversy about the prediction because constrained Hartree-Fock results with and without a second minimum exist [ 2 - 4 ] . Moreover, these calculations are for the intrinsic state only and the effect of parity and angular momentum projection has not been considered. In the present note we give the results of a numerical search for a shape isomeric state in 32S using standard finite range nucleon-nucleon forces. In contrast to the usual procedure we have performed an unconstrained variation o f all parameters of the many-body wave function. This avoids the arbitrariness of the customary constrained variation the outcome of which depends to some extent on the choice of the constraint and may even be at variance with it. In th e present calculation the variation has been performed for the Brink-Boeker force [5] B 1 and Volkov's force [6] no. 1. using states of the alphacluster model as introduced by Brink [7]. For the 328 system there are 19 variational parameters: the Supported by the Bundesministerium fiir Forschung und TechnologJe and the Deutsche Forschungsgemeinschaft. 1 Present adress: I. Institut ftir Experimentalphysik, Universit~/t Hamburg, D-2000 Hamburg, Fed. Rep. Germany.
oscillator width b = ( h / m ~ ) 1/2, the ¼A = 8 a-cluster positions R1, ..., R 8, with three coordinates each, minus the three center-of-mass coordinates and the three Euler angles for the orientation of the system. The variational 32-particle state is a Slater determinant of ls harmonic oscillator orbitals around R 1 , ..., R 8 , each occupied by four nucleons with anti-aligned spins and isospins, i.e. q~= Nq~-~ ~1 (x 1) ...~OA(XA),
where ~i(xi) = (blrl/2)-3/2 exp [ - ( x i - R/)Z/Zb 2 ] , i = 1. . . . . A ,
1 j = 1. . . . . g A ,
with the normalization factor NO and the antisymmetrizer M . The method of the present note is similar to that described in refs. [8,9] except that the Coulomb energy has been included and a full variation of all 19 parameters has been performed. The resulting parameter values, excitation energy, rms radius and quadrupole moment are given in table 1 for the B 1 force. It is remarkable that the resulting a-cluster centers R i are located in the corners of two elongated tetrahedrons at a center-to-center distance of d = 3.9 fm. In the a-particle model the groundstate wave function of 16 O is associated with a regular tetrahedron of cluster centers (cf. e.g. ref. [7]). Therefore, the calculated isomeric state can be interpreted as consisting of two distorted 160 states at a short separation, that are anti-symmetrized with respect to all 32 particles. 285
Volume 94B, number 3
PHYSICS LETTERS
11 August 1980
Table 1 Parameter values R i and b, excitation energy E, rms radius Rrm s and quadrupole moment Qo of the shape isomeric state for the B1 before, as resulting from an unconstrained variation of the intrinsic state. The quantities labelled 0÷ result from the restricted variation after parity and angular-momentum projection described in the text. The excitation energies Z~Eare relative to the Ba ground state of ref. [10].
x y z intr 0÷
R1 (fm)
Ra(fm)
R3(fm)
R4(fm)
R5 ( f r o )
R6(fm)
RT(fm)
Rs(fm)
0.0 0.0 -3.28
0.05 0.09 -1.95
0.05 -0.09 -1.95
-0.10 0.0 -1.95
-0.05 0.09 1.95
-0.05 -0.09 1.95
0.10 0.0 1.95
0.0 0.0 3.28
b = 1.78 fm b = 1.72 fm
E = --175.40 MeV E = -183.16 MeV
AE = 0.70 MeV zXE= 3.71 MeV
In order to investigate the possibility of mixing with the ground state [10] of 32S the overlap of the two states has to be calculated. Since both states are strongly deformed the overlap will depend on their relative orientation. The largest value obtained is of the order of 10 -13 so a mixing is negligible. It is further an indication that a variation with the constraint of orthogonality to the ground state, as used in H a r t r e e Fock calculations (e.g. ref. [11]), would lead to the same result. Within the point symmetry, that results from the unconstrained calculation, an additional variation after parity and angular m o m e n t u m projection has been performed. It turned out that the variation after projection leads only to small changes in the parameters of table 1. Since a second m i n i m u m is present in the potential energy surface of both the intrinsic state and the projected 0 + state the prediction of the existence of a shape isomeric state in 328 should be reliable. The calculated excitation energy of 3.71 MeV, however, may be at variance with the choice of the n u c l e o n - n u c l e o n force (see below). The single-particle density of the isomeric state A
p(r)= <41~3 6(r-x,)14>, i=1
is plotted in fig. 1 for a cut along the "fission" axis connecting the 160 "fragments". It exhibits a doublepeaked structure corresponding to two distorted 160 states. We note that the Pauli exclusion principle is important in forming such substructures at a rather short separation. No double-peaked density distribution results if the exchange terms between both 160 "fragments" are left out in the antisymmetrization. 286
Rrm s = 3.67 fm Rrm s = 3.69 fm
Qo = 199 fm2 Qo = 208 fm2
'~f~ Fig. 1. Density distribution of the shape isomeric state in 32S. The state has distinct clustering into two tetrahedral 160 substructures (R 1..... R4 and R s ..... R8 in table 1) that give rise to the two pear-shaped density peaks in the center.
In this case the densities of the strongly overlapping fragments add up in a single peak at the center. A density distribution similar to fig. 1 has been found in a generator-coordinate 160 + 160 calculation [12] for the same force but under the sudden assumption (see fig. 4c of ref. [12]). In this case, however, the state is no equilibrium state, it occurs at a smaller separation and at an energy much above the 160 + 160 quasi-molecular state of 24.5 MeV. Our results for the V1 force are very similar to the plot of fig. 1 except that the V1 densities are more compact leading to a larger maximum density. Thus the behaviour of the system under distortion should not depend on pecularities of the n u c l e o n - n u c l e o n interaction chosen. We find, however, a large difference
Volume 94B, number 3
PHYSICS LETTERS
in the excitation energy of the shape isomeric state. It is 22.11 MeV for the intrinsic state and 24.08 MeV for the 0 + state which is much higher than for the B 1 force. This discrepancy may be caused by the different form of the potentials in the odd state: The B 1 force is strongly repulsive in the odd state, contrary to the V1 force [5]. The odd-state repulsion is known to favour energetically the formation of c~ clusters [13]. By analogy, we expect it also to favour the formation of larger (closed-shell) clusters like 160. In order to study the occurrence of substructures between first and second m i n i m u m we have also performed constrained variations for a sequence of increasing elongations of the system (as in ref. [8]). According to our results the substructure formation sets in between the first and the second m i n i m u m in the vicinity of the barrier. Beyond the second minimum there is no such rearrangement in the many-body state. The further separation is associated only with gradually increasing distortion but with no structural changes in the density distribution. As regards the old controversy of early versus late fragmentation in fission our results clearly support the former (to the extent to which a many-body system heavier than 32S can be expected to behave in a similar way under deformation).
11 August 1980
References [1] R.K. Sheline, I. Ragnarsson and S.G. Nilsson, Phys. Lett. 41B (1972) 115. [2] S.J. Krieger and C.Y. Wong, Phys. Rev. Lett. 28 (1972) 690. [3] P.G. Zintand U. Mosel, Phys. Lett. 58B (1975) 269; Phys. Rev. C14 (1976) 1488. [4] H. Flocard, Phys. Lett. 49B (1974) 129. [5] D.M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1. [6] A.B. Volkov, Nucl. Phys. 74 (1965) 33. [7] D.M. Brink, Intern. School of Physics Enrico Fermi, Course XXXVI (Academic Press, New York, 1966) p. 247. [8] H. Schultheis, R. Schultheis, K. Wildermuth, A. Faessler and F. Griimmer, Z. Phys. A286 (1978) 65. [9] W. Bauhoff, H. Schultheis, R. Schultheis and K. Wildermuth, Proc. Fourth Intern. IAEA Symp. on Physics and chemistry of fission (J/ilich, 1979), (IAEA, Vienna, 1979). [10] W. Bauhoff, H. Schultheis and R. Schultheis, Phys. Rev. C, to be published. [11] G. Do Dang, K.W. Schmid, R.M. Dreizler and H.G. Miller, Phys. Lett. 62B (1976) 1. [12] D. Baye and G. Reidemeister, Nucl. Phys. A258 (1976) 157. [13] S.B. Khadkikar, Phys. Lett. 36B (1971)451.
287