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Projected Hartree-Fock calculations for light nuclei with phenomenological and realistic effective interactions
Volume 35B, number 6
PHYSICS
PROJECTED HARTREE-FOCK PHENOMENOLOGICAL AND
LETTERS
5 July 1971
CALCULATIONS FOR REALISTIC EFFECTIVE
LIGHT NUCLEI INTERACTIONS
WITH
M. BOUTEN * International
Centre
for
Theoretical
Physics,
Trieste,
Italy
M. C. BOUTEN Theoretical
Physics
H. DEPUYDT Applied
Mathematics
Group,
SCK-CEN,
Mel,
Belgium
and L. SCHOTSMANS Group,
SCK-CEN,
Mel,
Belgium
Received 21 May 19’71
Results of PHF calculations. carried out with the phenomenological force Bl of Brink and Boeker and with the realistic matrix elements of Faessler et al., are compared, and the differences are analysed in terms of differences in the oscillator matrix elements of the two potentials. In recent years, several phenomenological effective interactions have been constructed to be used in nuclear Hartree-Fock calculations. These interactions are often chosen as a sum of Gaussians with certain exchange mixtures, the parameters being determined from conditions on energies and radii of some specially chosen nuclei. Among the more successful of these potentials, we mention the Volkov force [l] and the Brink-Boeker [Z] force Bl. Using these interactions in projected Hartree-Fock (PHF) calculations [3] yields a large equilibrium deformation for the intrinsic state in many light nuclei. This deformation enhances the E2 moments considerably, as is required by the experimental data. Another success of these calculations [4] as well as the related o-model calculations [5] is the lowering of several non-normal parity and interloper states, again in fair agreement with experiment. The Bl force, in particular, brings the 4p-4h states in I% and 160 down to their experimentally assumed position [ 51. The success of these calculations, though interesting, is somewhat darkened by the rather arbitrary choices made in constructing the effective interaction. It is not clear how much of the success is real and how much is due to a happy choice of conditions in constructing the effective force. Another line of development has been the construction of extensive tables of matrix elements of realistic effective interactions. The method
* On leave from
464
SCK-CEN,
Mel,
Belgium.
developed by Elliot and co-workers [6] for obtaining these matrix elements directly from the phase shifts is especially appealing because of its simplicity. A slightly different version of this method was used by Faessler et al. [7], who tabulated all matrix elements needed in the lowest five oscillator shells. The main difference between the Faessler and Sussex matrix elements occurs for the non-diagonal matrix elements which contain essentially off-shell information. These off-diagonal matrix elements play a rather important role in determining the equilibrium radii, since they influence strongly the dependence of the potential energy on the radius. In this letter, we first want to show that the Faessler interaction, just like the Sussex matrix elements [8], does not lead to saturation. We have carried out PHF calculations for all eveneven 2 = N nuclei up to 26Ne, using the singleparticle space of five oscillator shells. In table 1 we compare the energy and radius of the ground states, obtained with the Faessler force with those obtained with the Bl force. The Coulomb energy was taken into account. We have used for the oscillator parameter b the value b = 1.6 fm, except for 26Ne where b = 1.8 fm. The Faessler force gives a considerably smaller binding energy and a smaller radius for the lighter nuclei. In 160, however, the Faessler force gives the larger binding, but the radius becomes very small. Using a smaller b = 1.4 fm would further increase the binding energy and decrease the radius, demonstrating the lack of saturation of the Faessler force.
Volume 35A, number 6
PHYSICS
Table 1 Ground state energies and radii obtained from PHF calculations with the Bl force and with Faessler’s matrix elements. Energies Bl
4He
8Be
-27.5
-50.9
-160
% -68.2
“Ne
- 97.2
-119.0
Faessler
-13.0
-21.9
-43.9
- 97.4
-110.1
EXP
-28.3
-56.5
-92.2
-127.6
-160.6
4He
‘Be
12C
Radii
160 ~-
“Ne
Bl
1.75
2.65
2.64
2.64
3.00
Faessler
1.64
2.26
2.13
2.07
2.40
EXP
1.63
-
2.42
2.75
-
In the nuclei considered above, which have [4...4] symmetry, only the SU4-scalar part of the Faessler force contributes. In table 2 we compare the two matrix elements of this SU4scalar part which play the most important role in light nuclei, with the corresponding ones of the Bl-force, for b = 1.6 fm. The smaller value of ( OS 1V /OS) explains the smaller binding energy obtained for the lighter nuclei, where this matrix element plays the most important role. In 160 where the ( Opj V /Op) matrix element has become the most important one, the strong repulsion in the Bl force provides saturation, whereas the Faessler (Op/ VI Op) matrix element is weakly attractive and gives no saturation. In view of the non-saturating character of the Faessler force, the calculations must be constrained in some way so as to obtain reasonable radii for the nuclei considered [8]. One way to do this would be by means of a Lagrange multiplier. In this letter, we follow a simpler method by plotting the energy as a function of the deformation for a fixed radius. More quantitatively, e.g. for the ground state of 12C, we consider the function *=
9~~0
detlx$OO
x:00
LETTERS
radius, we keep (b? b,) = b3 fixed and plot the energy E = (@IfI I*) /‘(*IQ_) as a function of bZ/b, for prolate or as a function of b,/b, for oblate deformations. For b, = b,, the function Q reduces to the SUQ-function. From previous calculations with the Bl force, we know that such a procedure for minimizing the energy gives qualitatively similar results, as a complete PHF calculation, if b is chosen conveniently. Fgi. 1 shows the energy as a function of deformation for the ground state (oblate) and the 4p - 4h state (prolate) of 12C, calculated with the Bl force (curves B) and with the Faessler force (curves F). For the ground state, b = 1.6 fm and for the 4p -4h states, b = 1.8 fm. These values are convenient for using the tabulated Faessler matrix elements and are close to the values suggested by previous PHF calculations with the Bl
12
Table 2 Important matrix elements of the two potentials used, - 7.59 -10.43
(OPlVjOP) -0.31 1.65
-
12C
6 Fm)
4p
4h
(b=l.8
fm)
Drolate
0
F
-10
-20
-30
B
c
-50
(I)
x&O]
(0s p 10s)
Cg.s.(b=1 ablate
where Xnxnynz is the eigenfunction of a deformed oscillator potential with (nxnynz) oscillator quanta. The potential has an axially symmetric deformation characterized by the oscillator length parameters bx and /lZ. In order to constrain the
Faessler Bl
5 July 1971
c
!; -70 +w *
-----+ 2
x
2
Fig.1. Energy of the ground state and of the 4p-4h state in 12C as a function of deformation. The curves F are calculated with the Faessler matrix elements, the curves B with the Bl force. 465
Volume 35B, number 6
PHYSICS
LETTERS
force. The Coulomb force was not included in this calculation. Several differences between the results of the two forces can be seen which can be traced back to the difference in the (Op 1VI Op) matrix element. From the figure one sees that the equilibrium deformations are larger and that the energy gains due to deformation are much larger for the B curves than for the F curves. In fact, apart from the lowering of the kinetic energy, the potential energy also gets lowered for the Bl force, but rises for the Faessler force. This can be understood qualitatively as follows. Deforming the intrinsic states of the SU3 function replaces some s-state interactions by the weaker d-state interactions and some p-state interactions by the weaker f-state interactions. For the Faessler force, for which both s- and p-state interactions are attractive, the potential energy becomes less negative. The same is true for the even-state interacti,,,, of the Bl force, but the odd-state repulsioli becomes less repulsive. It turns out that the odd-state gain is larger than the even-state loss. Another point which one observes in fig. 1 is the difference in excitation energy of the 4p - 4h state. This state comes at about 10MeV excitation for the Bl force and at 26 MeV for the Faessler force. A small part (about 4MeV) of this difference is due to the larger gains on deformation obtained with the Bl force. In fact, both the ground state and the 4p - 4h state show larger gains, so that their difference in energy is not strongly affected. The larger part of the difference between the Bl and the Faessler result is already visible in the SU3 calculation where the excitation energy of the 4p - 4h state is respectively 34 MeV for Bl and 46 MeV for Faessler’s force. It is quite easy to analyse the importance of the different ( nil V Id) matrix elements in creating this difference in the SU3 case. We find that a strong s-state attraction shifts the 4p - 4h state up (or rather pulls the ground state down) and that a p-state repulsion shifts the 4p - 4h state down (pushes the ground state up). Cutting away the strong p-state repulsion in the Bl force would put the 4p - 4h state above 30 McV excitation. The results shown in fig. 1 for 12C appear to be quite general and are also found for other pshell nuclei (e.g., non-normal parity state in ‘Be comes much higher with Faessler’s force
than with Bl). They illustrate the very important role played by the p-state repulsion in the Bl force. It seems reasonable to surmise that the tendency for (Y-clustering found [9] with the Bl force is also a result of this p-state repulsion, since the o-model enhances the s-state interactions while keeping the particles interacting in p-states far apart. The strong p-state repulsion was introduced in the Bl force in order to obtain saturation. It is not present in any of the realistic effective two-body forces so far, which all seem to suffer from some lack of saturation. It is interesting therefore to see that different procedures for obtaining saturation (density dependence) also lead to large deformations [lo], very similar to those obtained with the BI force. The position of the interloper states appears, however, to be too high. Two of us (M.B. and M.C.B.) thank Professor C. Monsonego for his kind hospitality at the C.R.N. at Strasbourg where most of the calculations were made. We also acknowledge some useful discussions with E. Caurier, S. Khadkikar, S. C. K. Nair, G. Ripka and M. K. Pal. One author (M.B.) is indebted to Professors Abdus Salam and P. Budini, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste,
References [l) A. B. Volkov, Nucl. Phys. 74 (1965) 33. [2) D. I\iI.Brink and E. Boeker, Nucl. Phys. A 91 (1967) [3] &. Bouten, in; Theory of nuclear structure (IAEA, Vienna 1970) p. 361. [4] M. Bouten, M. C. Bouten, M. Depuydt and L. Schotsmans, Nucl. Phys. A 158 (1970) 561: Nucl. Phys. A 158 (1970) 217. In the latter paper the energy of the 4p - 4h state in I60 is in error (see erratum in Nucl. Phys.). [5] D. M. Brink, H. Friedrich, A. Weiguny and C. W. Wong. Phys. Letters 33B (1970) 143. [6] J. P. Elliot, A. D. Jackson, M. A. Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A 121 (1968) 241. [7] J. E. Galonska, A. Faessler and K. Appel, Nucl. Phys. A 155 (1970) 465. [8] J. Dey, J. P. Elliot, A. D. Jackson, H. A.Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A 134 (1969) 385. [9] H. Friedrich and A. Weiguny, Alpha clustering and angular momentum projection before and after variation, Univ. of Mffnster preprint, 1971. [lo] J. iofka and G. Ripka, Nucl. Phys. A 168 (1971) 65.
*****
466
5 July 1971
PHYSICS
PROJECTED HARTREE-FOCK PHENOMENOLOGICAL AND
LETTERS
5 July 1971
CALCULATIONS FOR REALISTIC EFFECTIVE
LIGHT NUCLEI INTERACTIONS
WITH
M. BOUTEN * International
Centre
for
Theoretical
Physics,
Trieste,
Italy
M. C. BOUTEN Theoretical
Physics
H. DEPUYDT Applied
Mathematics
Group,
SCK-CEN,
Mel,
Belgium
and L. SCHOTSMANS Group,
SCK-CEN,
Mel,
Belgium
Received 21 May 19’71
Results of PHF calculations. carried out with the phenomenological force Bl of Brink and Boeker and with the realistic matrix elements of Faessler et al., are compared, and the differences are analysed in terms of differences in the oscillator matrix elements of the two potentials. In recent years, several phenomenological effective interactions have been constructed to be used in nuclear Hartree-Fock calculations. These interactions are often chosen as a sum of Gaussians with certain exchange mixtures, the parameters being determined from conditions on energies and radii of some specially chosen nuclei. Among the more successful of these potentials, we mention the Volkov force [l] and the Brink-Boeker [Z] force Bl. Using these interactions in projected Hartree-Fock (PHF) calculations [3] yields a large equilibrium deformation for the intrinsic state in many light nuclei. This deformation enhances the E2 moments considerably, as is required by the experimental data. Another success of these calculations [4] as well as the related o-model calculations [5] is the lowering of several non-normal parity and interloper states, again in fair agreement with experiment. The Bl force, in particular, brings the 4p-4h states in I% and 160 down to their experimentally assumed position [ 51. The success of these calculations, though interesting, is somewhat darkened by the rather arbitrary choices made in constructing the effective interaction. It is not clear how much of the success is real and how much is due to a happy choice of conditions in constructing the effective force. Another line of development has been the construction of extensive tables of matrix elements of realistic effective interactions. The method
* On leave from
464
SCK-CEN,
Mel,
Belgium.
developed by Elliot and co-workers [6] for obtaining these matrix elements directly from the phase shifts is especially appealing because of its simplicity. A slightly different version of this method was used by Faessler et al. [7], who tabulated all matrix elements needed in the lowest five oscillator shells. The main difference between the Faessler and Sussex matrix elements occurs for the non-diagonal matrix elements which contain essentially off-shell information. These off-diagonal matrix elements play a rather important role in determining the equilibrium radii, since they influence strongly the dependence of the potential energy on the radius. In this letter, we first want to show that the Faessler interaction, just like the Sussex matrix elements [8], does not lead to saturation. We have carried out PHF calculations for all eveneven 2 = N nuclei up to 26Ne, using the singleparticle space of five oscillator shells. In table 1 we compare the energy and radius of the ground states, obtained with the Faessler force with those obtained with the Bl force. The Coulomb energy was taken into account. We have used for the oscillator parameter b the value b = 1.6 fm, except for 26Ne where b = 1.8 fm. The Faessler force gives a considerably smaller binding energy and a smaller radius for the lighter nuclei. In 160, however, the Faessler force gives the larger binding, but the radius becomes very small. Using a smaller b = 1.4 fm would further increase the binding energy and decrease the radius, demonstrating the lack of saturation of the Faessler force.
Volume 35A, number 6
PHYSICS
Table 1 Ground state energies and radii obtained from PHF calculations with the Bl force and with Faessler’s matrix elements. Energies Bl
4He
8Be
-27.5
-50.9
-160
% -68.2
“Ne
- 97.2
-119.0
Faessler
-13.0
-21.9
-43.9
- 97.4
-110.1
EXP
-28.3
-56.5
-92.2
-127.6
-160.6
4He
‘Be
12C
Radii
160 ~-
“Ne
Bl
1.75
2.65
2.64
2.64
3.00
Faessler
1.64
2.26
2.13
2.07
2.40
EXP
1.63
-
2.42
2.75
-
In the nuclei considered above, which have [4...4] symmetry, only the SU4-scalar part of the Faessler force contributes. In table 2 we compare the two matrix elements of this SU4scalar part which play the most important role in light nuclei, with the corresponding ones of the Bl-force, for b = 1.6 fm. The smaller value of ( OS 1V /OS) explains the smaller binding energy obtained for the lighter nuclei, where this matrix element plays the most important role. In 160 where the ( Opj V /Op) matrix element has become the most important one, the strong repulsion in the Bl force provides saturation, whereas the Faessler (Op/ VI Op) matrix element is weakly attractive and gives no saturation. In view of the non-saturating character of the Faessler force, the calculations must be constrained in some way so as to obtain reasonable radii for the nuclei considered [8]. One way to do this would be by means of a Lagrange multiplier. In this letter, we follow a simpler method by plotting the energy as a function of the deformation for a fixed radius. More quantitatively, e.g. for the ground state of 12C, we consider the function *=
9~~0
detlx$OO
x:00
LETTERS
radius, we keep (b? b,) = b3 fixed and plot the energy E = (@IfI I*) /‘(*IQ_) as a function of bZ/b, for prolate or as a function of b,/b, for oblate deformations. For b, = b,, the function Q reduces to the SUQ-function. From previous calculations with the Bl force, we know that such a procedure for minimizing the energy gives qualitatively similar results, as a complete PHF calculation, if b is chosen conveniently. Fgi. 1 shows the energy as a function of deformation for the ground state (oblate) and the 4p - 4h state (prolate) of 12C, calculated with the Bl force (curves B) and with the Faessler force (curves F). For the ground state, b = 1.6 fm and for the 4p -4h states, b = 1.8 fm. These values are convenient for using the tabulated Faessler matrix elements and are close to the values suggested by previous PHF calculations with the Bl
12
Table 2 Important matrix elements of the two potentials used, - 7.59 -10.43
(OPlVjOP) -0.31 1.65
-
12C
6 Fm)
4p
4h
(b=l.8
fm)
Drolate
0
F
-10
-20
-30
B
c
-50
(I)
x&O]
(0s p 10s)
Cg.s.(b=1 ablate
where Xnxnynz is the eigenfunction of a deformed oscillator potential with (nxnynz) oscillator quanta. The potential has an axially symmetric deformation characterized by the oscillator length parameters bx and /lZ. In order to constrain the
Faessler Bl
5 July 1971
c
!; -70 +w *
-----+ 2
x
2
Fig.1. Energy of the ground state and of the 4p-4h state in 12C as a function of deformation. The curves F are calculated with the Faessler matrix elements, the curves B with the Bl force. 465
Volume 35B, number 6
PHYSICS
LETTERS
force. The Coulomb force was not included in this calculation. Several differences between the results of the two forces can be seen which can be traced back to the difference in the (Op 1VI Op) matrix element. From the figure one sees that the equilibrium deformations are larger and that the energy gains due to deformation are much larger for the B curves than for the F curves. In fact, apart from the lowering of the kinetic energy, the potential energy also gets lowered for the Bl force, but rises for the Faessler force. This can be understood qualitatively as follows. Deforming the intrinsic states of the SU3 function replaces some s-state interactions by the weaker d-state interactions and some p-state interactions by the weaker f-state interactions. For the Faessler force, for which both s- and p-state interactions are attractive, the potential energy becomes less negative. The same is true for the even-state interacti,,,, of the Bl force, but the odd-state repulsioli becomes less repulsive. It turns out that the odd-state gain is larger than the even-state loss. Another point which one observes in fig. 1 is the difference in excitation energy of the 4p - 4h state. This state comes at about 10MeV excitation for the Bl force and at 26 MeV for the Faessler force. A small part (about 4MeV) of this difference is due to the larger gains on deformation obtained with the Bl force. In fact, both the ground state and the 4p - 4h state show larger gains, so that their difference in energy is not strongly affected. The larger part of the difference between the Bl and the Faessler result is already visible in the SU3 calculation where the excitation energy of the 4p - 4h state is respectively 34 MeV for Bl and 46 MeV for Faessler’s force. It is quite easy to analyse the importance of the different ( nil V Id) matrix elements in creating this difference in the SU3 case. We find that a strong s-state attraction shifts the 4p - 4h state up (or rather pulls the ground state down) and that a p-state repulsion shifts the 4p - 4h state down (pushes the ground state up). Cutting away the strong p-state repulsion in the Bl force would put the 4p - 4h state above 30 McV excitation. The results shown in fig. 1 for 12C appear to be quite general and are also found for other pshell nuclei (e.g., non-normal parity state in ‘Be comes much higher with Faessler’s force
than with Bl). They illustrate the very important role played by the p-state repulsion in the Bl force. It seems reasonable to surmise that the tendency for (Y-clustering found [9] with the Bl force is also a result of this p-state repulsion, since the o-model enhances the s-state interactions while keeping the particles interacting in p-states far apart. The strong p-state repulsion was introduced in the Bl force in order to obtain saturation. It is not present in any of the realistic effective two-body forces so far, which all seem to suffer from some lack of saturation. It is interesting therefore to see that different procedures for obtaining saturation (density dependence) also lead to large deformations [lo], very similar to those obtained with the BI force. The position of the interloper states appears, however, to be too high. Two of us (M.B. and M.C.B.) thank Professor C. Monsonego for his kind hospitality at the C.R.N. at Strasbourg where most of the calculations were made. We also acknowledge some useful discussions with E. Caurier, S. Khadkikar, S. C. K. Nair, G. Ripka and M. K. Pal. One author (M.B.) is indebted to Professors Abdus Salam and P. Budini, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste,
References [l) A. B. Volkov, Nucl. Phys. 74 (1965) 33. [2) D. I\iI.Brink and E. Boeker, Nucl. Phys. A 91 (1967) [3] &. Bouten, in; Theory of nuclear structure (IAEA, Vienna 1970) p. 361. [4] M. Bouten, M. C. Bouten, M. Depuydt and L. Schotsmans, Nucl. Phys. A 158 (1970) 561: Nucl. Phys. A 158 (1970) 217. In the latter paper the energy of the 4p - 4h state in I60 is in error (see erratum in Nucl. Phys.). [5] D. M. Brink, H. Friedrich, A. Weiguny and C. W. Wong. Phys. Letters 33B (1970) 143. [6] J. P. Elliot, A. D. Jackson, M. A. Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A 121 (1968) 241. [7] J. E. Galonska, A. Faessler and K. Appel, Nucl. Phys. A 155 (1970) 465. [8] J. Dey, J. P. Elliot, A. D. Jackson, H. A.Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A 134 (1969) 385. [9] H. Friedrich and A. Weiguny, Alpha clustering and angular momentum projection before and after variation, Univ. of Mffnster preprint, 1971. [lo] J. iofka and G. Ripka, Nucl. Phys. A 168 (1971) 65.
*****
466
5 July 1971