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Angular momentum projection from non-axial intrinsic states in 28Si
Volume
35B.
number
6
PHYSICS
LETTERS
ANGULAR MOMENTUM NON-AXIAL INTRINSIC
5 July
PROJECTION STATES
IN
1971
FROM 28Si
A. WATT Department
of Natural
Philosophy,
University
Received
States of definite angular momentum projected a basis in which to diagonalize the Hamiltonian. appreciably to the low energy spectrum.
Glasgow,
UK
7 May 19’il
from seven Hartree-Foclr solution of 28Si are used It is found that each of these HF solutions contributes
The nucleus 28Si is one of the most interesting in the 2sld shell. It is the most difficult nucleus for which to perform calculations of the usual shellmodel type since twelve valence particles must be considered. Also the 2sld shell is halffilled at 28Si and this has important consequences in the structure of the Hartree-Fock (HF) solutions. It is well-known [l] that the HF solution of lowest energy, the oblate solution (Ob), can be used to generate the prolate solution (Pr) by exchanging empty and filled orbitals. The prolate and oblate solutions are nearly degenerate in energy, and the spherical HF solution (S), obtained by completely filling the ld5/2 shell, is also of low energy. Several attempts [2 - 41 have therefore been made to describe the low energy spectrum of 28Si by angular momentum projection from one or more of these three HF solutions. In this note we show that other HF solutions must also be considered if consistent projected Hartree-Fock calculations are to be performed. In the work reported here, the residual twobody interaction is taken to have a Rosenfeld exchange mixture with a Yukawa radial dependence of range parameter 1.37 fm and strength 50 MeV, and the single particle energies are e(s1i2) = -4.2, l (d3j2) = 0 and l (dg/2) = -7.OMeV. The single particle orbitals of the HF solutions are taken to be linear combinations of 2sld shell harmonic oscillator states of length parameter 1.65 fm. This force gives rise to several HF solutions in addition to the three mentioned above. Two solutions El and E2 with ellipsoidal symmetry [l, 51 are known, and we also obtained two new HF solutions Cl and C2 with the cubic symmetry first introduced in ref. [6]; cubic solutions are invariant under rotations of 3~ about the principle axes. The ener490
of Glasgow,
as
gies of the Ob, Pr, El, Cl, E2, C2 and S solutions are -154.7, -153.2, -151.7, -151.3, -150.9, -149.7 and -148.6 MeV respectively. In table 1, the single particle orbitals of the cubic solutions are given in terms of the 2sld shell basis states. The Pr, E2 and C2 solutions can be generated from the Ob, El and Cl solutions respectively by exchanging empty and filled orbitals. Also, the axial and ellipsoidal solutions are closely related to intrinsic states of the SU3 model (l), but no such interpretation seems to exist for the cubic solutions. States of definite angular momentum were projected from the seven HF solutions mentioned above. The projection method is exact and is an extension of the method of Unna [7,8]. It works for non-axial as well as for axial intrinsic states. In fig. 1, we show the spectra up to J = 6, obtained by projecting from each HF solution in turn, the mixed spectrum (labelled MIX) found by using these states as a non-orthogonal basis in which to diagonalize the Hamiltonian, and the experimental spectrum [9]. The unmixed spectra show several interesting features. The number n of states with angular momentum J which can be projected from an intrinsic wavefunction depends on the symmetry and can be deduced from group theory [lo]. For an axially symmetric state (Ob, Pr), Jn = 01, 21, state (El,E2), J” = 01, 41... ; for an ellipsoidal 22, 31, 43...; for a cubic state (Cl,C2), Jn = 01, 41, 61, 81, 91, 101 and 12l ; and for a spherical state J n = 01 only. From fig. 1 it looks as if there are in all six J = 0 states with energies lower than the spherical state. However, these six J = 0 states are not orthogonal, and it turns out that they are very nearly linearly dependent. Indeed, the J = 0 states obtained from
PHYSICS
Volume 35B, number 6
5 July 1971
LETTERS
Table 1 Expansion of single particle orbitals for the cubic HF solutions Cl and C2.
__-
Solution
~~~
%3
Cl
E= -151.3MeV
0
0
0.482
0
0
0.876
0
0
0
0.482
0.800
0
0.358
-0.394
0
0
1.0
0
0 -0.394 0
the Ob, El and Cl Hartree-Fock solutions are so nearly linearly dependent that the one projected from Cl had to be left out when diagonalizing the Hamiltonian. This linear dependence seems to be purely accidental and has not previously been reported in calculations of this type. In particular, it did not occur for any other value of Jused in the calculations.
Hm F
1 -P \
‘-6 \
-6
-5
-
’
-150
-L
\
-6 \ -2 t--L -. -5
-2
-2
-\
‘--\
-4
-2
-5
-3 -6
-6
-6
‘._
=;
-3,
-‘
i
E
-0
L
-6 \
\ \ -\ -L’,
Y
-2
=f
\
-6
-3 -‘
-6 -HF
--I -0
---I.
-0
-2
-&
=$ -0
-2
-0
-0 -2
-0
-2
-160 CZ
E2
Cl
El
Pr
Ob
iK”
--O EXP.
Fig.1. Projected Hartree-Fock spectra in 38si. The spectra are labelled as described in the text and the experimental
spectrum
0 0.375
0
0
0
The states of definite J obtained from El are especially interesting. TheJ = 0 component has a lower energy than any other unmixed J = 0 state, including the one projected from the oblate HF solution. Also the energies of the K = 2 band do not increase with increasing J. The order is J= 4, 3, 6, 5, 2 and its sequence persists in the E2 spectrum. Anomolous effects of this type have previously been reported in 32S and their existence is one of the manifestations of the unreliability of approximate angular momentum projection techniques [ll]. Turning now to the problem of mixing the states of definite J, we can derive some interesting results concerning the overlaps of states projected from the different HF solutions. The Peierls-YoCCoz angular momentum projector [12]
has been arbitrarily in energy.
(1) may be used [13] to find the overlaps of states projected from two intrinsic wavefunctions 1A) and /B):
(A/(pj&~~)~ p& /B) = (Aj pJK’K IW w+ 1) _/-&K(~) (AIR (a) =_871~
-0
-0
6
0.919 0
=; =j
----I
\
-2 --I
--L
-155
0 0.839
-2
\ :
Id 5/2 - 3/z 0.913
0
0
c2
E= -149.7MeV
Id 5/2 l/2 0
ld5/25/2 -0.408
l/3 0
displaced
/E) dS2
It is well-known [2-d] that ( Pr 1P$, /Ob) is very small for all J, and it has been found [4] that (Pr/ PE, /S) is also small. However, this last result can be made exact, for on expanding the single particle orbitals in the oscillator basis writing as a determinant, it and (PrlR(Sl)lS) becomes obvious that any matrix element (I%/ QR@) IS) is identically zero unless Q is an operator containing at least 4p - 4h excitations. Hence ( Prl Go IS) = 0 and for the two-body Hamiltonian H used here, (Pr/ HPgo IS) = 0. The same methods shows that (Cl] PiK jC2) = (CllHP$KIC2) =OforallJ, K, K’and (C2 /P$jo Is)=( C2lHPg, IS) = 0. These results 491
Volume
35B, number ti
PHYSICS
LETTERS
are of interest in connection with observed electromagnetic transitions as discussed in ref. [14]. It was also found that the HF solutions could be divided into two sets Sl = {Ob, El, Cl, S} and S2 = { Pr, E2, C2). Two members of the same set have large overlaps and mix strongly, but members of SI mix weakly if at all with members of S2. The levels of the mixed spectrum shown in fig. 1 therefore fall roughly into two classes according to whether they arise mainly from members of Sl or of S2. For each J, the state of lowest energy comes from Sl. However, this classification is only qualitative; for example the ground state has the following composition in terms of the J = 0 states projected from the HF solutions: Ob, 2O’h; El, 52%; S: 4%; Pr: 2%; E2: 20%; c2: 1%. It is also interesting to investigate how the predicted energy of the ground state varies when subsets of the seven HF solutions are used. We have already seen that the energy obtained from El (-158.1 MeV) is less than that got from Ob (-157.6 MeV). If the three solutions Ob, Pr, S are mixed, the result is -157.7MeV, while if all seven are used, we obtain -160.2 MeV. Thus the effect of including the non-axial solutions is very large when compared with the effect of mixing only the axial ones. Earlier calculations of this type using only the axial solutions are therefore incomplete. On the other hand, there is no guarantee from the present work that other HF solutions are unimportant. The mixed spectrum shows some encouraging
*****
492
5 July 1971
features when compared with experiment. The first J = 2 state is too high in energy, but generally speaking the spectrum is too compressed. Two J = 3 levels occur at about the correct energies, and these can only arise by projection from ellipsoidal intrinsic states. The author wishes to thank Dr. N. MacDonald for many helpful discussions.
References [II G. Ripka, in Advances in Nuclear Physics Vol. 1. (Plenum Press, New York and London, 1968) [21 S. N. Tewari and D. Grillot, Phys. Rev. 177 (1969) 1717. [31 B. Caste1 and J. C. Parikh, Phys. Rev. Cl (1970) 990. [41 G.Do Dang, Nucl. Phys. Al57 (1970) 231. [51 J. Bar-Touv and I. Kelson, Phys. Rev. 138 (1965) B 1035. [‘51 J. A. Glen and N. MacDonald, to be published; J. A. Glen, Ph.D. Thesis, University of Glasgow, 1969. 132 (1963) 2225. [71 I.Unna, Phys.Rev. ISI A. Watt, to be published. PI D. M. Endt and-C. van der Leun, Nucl. Phys. Al05 (1967) 1. Group Theory (Addison Wesley, [lOI M. Hamermesh, 1952). [Ill H. A. Lamme and E. Boeker, Nucl. Phys. 136 (1969) 609. Proc. Phys. Sot. A 70 [I21 R. E. Peierls and J. Yoccoz, (1957) 381. Adv. in Physics 19 (1970) 371. P31 N.MacDonald, [I41 J. B.ar-Touv and A. Goswami, Phys. Letts. 28 B (1969) 391.
35B.
number
6
PHYSICS
LETTERS
ANGULAR MOMENTUM NON-AXIAL INTRINSIC
5 July
PROJECTION STATES
IN
1971
FROM 28Si
A. WATT Department
of Natural
Philosophy,
University
Received
States of definite angular momentum projected a basis in which to diagonalize the Hamiltonian. appreciably to the low energy spectrum.
Glasgow,
UK
7 May 19’il
from seven Hartree-Foclr solution of 28Si are used It is found that each of these HF solutions contributes
The nucleus 28Si is one of the most interesting in the 2sld shell. It is the most difficult nucleus for which to perform calculations of the usual shellmodel type since twelve valence particles must be considered. Also the 2sld shell is halffilled at 28Si and this has important consequences in the structure of the Hartree-Fock (HF) solutions. It is well-known [l] that the HF solution of lowest energy, the oblate solution (Ob), can be used to generate the prolate solution (Pr) by exchanging empty and filled orbitals. The prolate and oblate solutions are nearly degenerate in energy, and the spherical HF solution (S), obtained by completely filling the ld5/2 shell, is also of low energy. Several attempts [2 - 41 have therefore been made to describe the low energy spectrum of 28Si by angular momentum projection from one or more of these three HF solutions. In this note we show that other HF solutions must also be considered if consistent projected Hartree-Fock calculations are to be performed. In the work reported here, the residual twobody interaction is taken to have a Rosenfeld exchange mixture with a Yukawa radial dependence of range parameter 1.37 fm and strength 50 MeV, and the single particle energies are e(s1i2) = -4.2, l (d3j2) = 0 and l (dg/2) = -7.OMeV. The single particle orbitals of the HF solutions are taken to be linear combinations of 2sld shell harmonic oscillator states of length parameter 1.65 fm. This force gives rise to several HF solutions in addition to the three mentioned above. Two solutions El and E2 with ellipsoidal symmetry [l, 51 are known, and we also obtained two new HF solutions Cl and C2 with the cubic symmetry first introduced in ref. [6]; cubic solutions are invariant under rotations of 3~ about the principle axes. The ener490
of Glasgow,
as
gies of the Ob, Pr, El, Cl, E2, C2 and S solutions are -154.7, -153.2, -151.7, -151.3, -150.9, -149.7 and -148.6 MeV respectively. In table 1, the single particle orbitals of the cubic solutions are given in terms of the 2sld shell basis states. The Pr, E2 and C2 solutions can be generated from the Ob, El and Cl solutions respectively by exchanging empty and filled orbitals. Also, the axial and ellipsoidal solutions are closely related to intrinsic states of the SU3 model (l), but no such interpretation seems to exist for the cubic solutions. States of definite angular momentum were projected from the seven HF solutions mentioned above. The projection method is exact and is an extension of the method of Unna [7,8]. It works for non-axial as well as for axial intrinsic states. In fig. 1, we show the spectra up to J = 6, obtained by projecting from each HF solution in turn, the mixed spectrum (labelled MIX) found by using these states as a non-orthogonal basis in which to diagonalize the Hamiltonian, and the experimental spectrum [9]. The unmixed spectra show several interesting features. The number n of states with angular momentum J which can be projected from an intrinsic wavefunction depends on the symmetry and can be deduced from group theory [lo]. For an axially symmetric state (Ob, Pr), Jn = 01, 21, state (El,E2), J” = 01, 41... ; for an ellipsoidal 22, 31, 43...; for a cubic state (Cl,C2), Jn = 01, 41, 61, 81, 91, 101 and 12l ; and for a spherical state J n = 01 only. From fig. 1 it looks as if there are in all six J = 0 states with energies lower than the spherical state. However, these six J = 0 states are not orthogonal, and it turns out that they are very nearly linearly dependent. Indeed, the J = 0 states obtained from
PHYSICS
Volume 35B, number 6
5 July 1971
LETTERS
Table 1 Expansion of single particle orbitals for the cubic HF solutions Cl and C2.
__-
Solution
~~~
%3
Cl
E= -151.3MeV
0
0
0.482
0
0
0.876
0
0
0
0.482
0.800
0
0.358
-0.394
0
0
1.0
0
0 -0.394 0
the Ob, El and Cl Hartree-Fock solutions are so nearly linearly dependent that the one projected from Cl had to be left out when diagonalizing the Hamiltonian. This linear dependence seems to be purely accidental and has not previously been reported in calculations of this type. In particular, it did not occur for any other value of Jused in the calculations.
Hm F
1 -P \
‘-6 \
-6
-5
-
’
-150
-L
\
-6 \ -2 t--L -. -5
-2
-2
-\
‘--\
-4
-2
-5
-3 -6
-6
-6
‘._
=;
-3,
-‘
i
E
-0
L
-6 \
\ \ -\ -L’,
Y
-2
=f
\
-6
-3 -‘
-6 -HF
--I -0
---I.
-0
-2
-&
=$ -0
-2
-0
-0 -2
-0
-2
-160 CZ
E2
Cl
El
Pr
Ob
iK”
--O EXP.
Fig.1. Projected Hartree-Fock spectra in 38si. The spectra are labelled as described in the text and the experimental
spectrum
0 0.375
0
0
0
The states of definite J obtained from El are especially interesting. TheJ = 0 component has a lower energy than any other unmixed J = 0 state, including the one projected from the oblate HF solution. Also the energies of the K = 2 band do not increase with increasing J. The order is J= 4, 3, 6, 5, 2 and its sequence persists in the E2 spectrum. Anomolous effects of this type have previously been reported in 32S and their existence is one of the manifestations of the unreliability of approximate angular momentum projection techniques [ll]. Turning now to the problem of mixing the states of definite J, we can derive some interesting results concerning the overlaps of states projected from the different HF solutions. The Peierls-YoCCoz angular momentum projector [12]
has been arbitrarily in energy.
(1) may be used [13] to find the overlaps of states projected from two intrinsic wavefunctions 1A) and /B):
(A/(pj&~~)~ p& /B) = (Aj pJK’K IW w+ 1) _/-&K(~) (AIR (a) =_871~
-0
-0
6
0.919 0
=; =j
----I
\
-2 --I
--L
-155
0 0.839
-2
\ :
Id 5/2 - 3/z 0.913
0
0
c2
E= -149.7MeV
Id 5/2 l/2 0
ld5/25/2 -0.408
l/3 0
displaced
/E) dS2
It is well-known [2-d] that ( Pr 1P$, /Ob) is very small for all J, and it has been found [4] that (Pr/ PE, /S) is also small. However, this last result can be made exact, for on expanding the single particle orbitals in the oscillator basis writing as a determinant, it and (PrlR(Sl)lS) becomes obvious that any matrix element (I%/ QR@) IS) is identically zero unless Q is an operator containing at least 4p - 4h excitations. Hence ( Prl Go IS) = 0 and for the two-body Hamiltonian H used here, (Pr/ HPgo IS) = 0. The same methods shows that (Cl] PiK jC2) = (CllHP$KIC2) =OforallJ, K, K’and (C2 /P$jo Is)=( C2lHPg, IS) = 0. These results 491
Volume
35B, number ti
PHYSICS
LETTERS
are of interest in connection with observed electromagnetic transitions as discussed in ref. [14]. It was also found that the HF solutions could be divided into two sets Sl = {Ob, El, Cl, S} and S2 = { Pr, E2, C2). Two members of the same set have large overlaps and mix strongly, but members of SI mix weakly if at all with members of S2. The levels of the mixed spectrum shown in fig. 1 therefore fall roughly into two classes according to whether they arise mainly from members of Sl or of S2. For each J, the state of lowest energy comes from Sl. However, this classification is only qualitative; for example the ground state has the following composition in terms of the J = 0 states projected from the HF solutions: Ob, 2O’h; El, 52%; S: 4%; Pr: 2%; E2: 20%; c2: 1%. It is also interesting to investigate how the predicted energy of the ground state varies when subsets of the seven HF solutions are used. We have already seen that the energy obtained from El (-158.1 MeV) is less than that got from Ob (-157.6 MeV). If the three solutions Ob, Pr, S are mixed, the result is -157.7MeV, while if all seven are used, we obtain -160.2 MeV. Thus the effect of including the non-axial solutions is very large when compared with the effect of mixing only the axial ones. Earlier calculations of this type using only the axial solutions are therefore incomplete. On the other hand, there is no guarantee from the present work that other HF solutions are unimportant. The mixed spectrum shows some encouraging
*****
492
5 July 1971
features when compared with experiment. The first J = 2 state is too high in energy, but generally speaking the spectrum is too compressed. Two J = 3 levels occur at about the correct energies, and these can only arise by projection from ellipsoidal intrinsic states. The author wishes to thank Dr. N. MacDonald for many helpful discussions.
References [II G. Ripka, in Advances in Nuclear Physics Vol. 1. (Plenum Press, New York and London, 1968) [21 S. N. Tewari and D. Grillot, Phys. Rev. 177 (1969) 1717. [31 B. Caste1 and J. C. Parikh, Phys. Rev. Cl (1970) 990. [41 G.Do Dang, Nucl. Phys. Al57 (1970) 231. [51 J. Bar-Touv and I. Kelson, Phys. Rev. 138 (1965) B 1035. [‘51 J. A. Glen and N. MacDonald, to be published; J. A. Glen, Ph.D. Thesis, University of Glasgow, 1969. 132 (1963) 2225. [71 I.Unna, Phys.Rev. ISI A. Watt, to be published. PI D. M. Endt and-C. van der Leun, Nucl. Phys. Al05 (1967) 1. Group Theory (Addison Wesley, [lOI M. Hamermesh, 1952). [Ill H. A. Lamme and E. Boeker, Nucl. Phys. 136 (1969) 609. Proc. Phys. Sot. A 70 [I21 R. E. Peierls and J. Yoccoz, (1957) 381. Adv. in Physics 19 (1970) 371. P31 N.MacDonald, [I41 J. B.ar-Touv and A. Goswami, Phys. Letts. 28 B (1969) 391.