- Email: [email protected]
Investigations of parity deformation in spherical nuclei
1.C
]
Nuclear Physics A133 (1969) 581--589; (~) North-Holland Publishing Co., Amsterdam
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
INVESTIGATIONS OF PARITY DEFORMATION IN SPHERICAL NUCLEI W. BURR, D. SCHOTTE and K. BLEULER
Institut Jiir Theoretische Kernphysik der Universitiit Bonn, W.-Germany Received 9 May 1969 Abstract: The possibility of parity deformation for spherical structures is investigated from various viewpoints. We use several criteria of stability for the spherical Hartree-Fock solution (Thouless condition, critical 0 - states in different approximations) as well as different types of two-body forces (a realistic effective interaction and the OPEP) and single-particle spectra (HF, oscillator, Woods-Saxon and experimental) in order to obtain a complete survey. The result is that parity deformation can be excluded in practically all cases. In addition, the positions of the two critical 0 - states of 160 are in good numerical agreement with experiments if realistic forces and experimental single-particle energies in connection with the random phase or TammDancoff approximation are used.
1. Introduction
Within the framework of Hartree-Fock (HF) calculations for nuclear structure, there exist (under well-defined conditions) self-consistent solutions (i.e. extrema in the variational scheme) with specific symmetries assumed from the outset 1) (e.g. spherically and reflection-symmetric HF solutions for closed shell (or subshell) nuclei). The main question is, however, whether these (restricted) solutions are stable, i.e. whether they represent a true minimum of the total energy with respect to the expectation values in unrestricted HF states. As an example, a spherically symmetric HF solution within the region of deformed nuclei will not satisfy this condition. In very much the same way, a HF solution with sharp parity [which exists for any particle number 2)] may be unstable with respect to parity mixture of the singleparticle states. The last question - the problem of the so-called parity deformation (PD) - was investigated by several authors 3) under various simplifying assumptions. Calculations with the one-pion-exchange potential (OPEP) and various numerical restrictions pointed to relatively strong parity deformation, self-consistent calculations 4) within the framework of a mere Hartree procedure even strengthened the tendency. On the other hand, more complete calculations with realistic two-body forces never gave an appreciable parity deformation. It is the aim of this paper to decide about the question of PD under the most general conditions which can be reached at present. We therefore investigate for a sequence of closed-shell nuclei various criteria of PD, namely the eigenvalues of the 581
w. BURRet al.
582
matrix occurring in the Thouless theorem as well as the positions of the characteristic low-lying 0- states in the Random phase approximation (RPA) and Tamm-Dancoff approximation (TDA). In both cases, two different assumptions for the two-body force (OPEP and effective realistic force) and four different assumptions for the basic single-particle energies (HF, phenomenological, oscillator-like and experimental energies) are made. The calculations include exchange terms of the matrix elements but are restricted to states of the next major shell. Combining our numerical results from all these cases, we find that appreciable PD is practically excluded for the spherical case, whereas the situation may be different in the deformed region of lighter nuclei.
2. Theoretical background Starting from a spherical symmetric H F solution for a closed-shell nucleus [its existence is guaranteed by the theorem of Ripka 1)], we investigate the possibility of an energy gain by allowing parity mixture in the single-particle states. In a number of earlier investigations 3,4), this problem was found to be rather sensitive to the choice of nuclear forces V and the assumed single-particle spectrum era. It seems, therefore, desirable to discuss all possible variations of the relevant quantities as well as a few important stability conditions. (i) A most direct test of the stability of a H F solution is given by the theorem of Thouless 6); it consists (in the most general case) in evaluating the (real) eigenvalues of the Hermitian matrix
h = (A B*
B) A*
with
Ami, nj = ( e m - e i ) S m n S i j + ( m j l V [ i n ) A s Bm~,nj = ( m n l V l i j ) A s
(1)
(i, j enumerate the occupied states and m, n the unoccupied states, AS means antisymmetrized) and yields the following local criterium (i.e. with respect to infinitesimal variations of the orbitals) for PD: There is local PD, if and only if, at least one of the eigenvalues of h is positive. This criterium cannot, however, exclude PD for a self-consistent calculation, which is based on completely different wave functions from the outset 4) (possible coexistence of several true minima, e.g. c~-decay and spallation). (ii) Another (but closely connected) method to decide about the occurrence of PD is obtained from the RPA equations (random phase approximation). They yield critical excitation energies (0- vibrations in our case) with the help of the eigenvalues E of the new non-Hermitian matrix R =
-B*
-A*
"
(2)
[The notation is the same as in expression (1).] Here the eigenvalues are real - p a i r s of opposite sign - in the case of a stable H F solution, whereas at least two become imaginary (pair of complex conjugates) in case of PD. (In this connection, it should
PARITY DEFORMATION IN SPHERICAL NUCLEI
583
be mentioned that these vibrations can be approximated in the frame work of the Tamm-Dancoff approximation (TDA) by the eigenvalues of the hermitian matrix A alone, if the corresponding RPA result is real and not too small.) In the case of PD, the RPA equations applied on the spherically symmetric ground states lose their physical meaning. The lowest excited state of the spectrum is now given in a reasonable approximation by one of the parity partners, which in this case results from the (projected) parity-deformed HF ground state solution 3,5) (in general, negative parity for the excited state). The excitation energy of this level is relatively low s) (cross matrix elements); this shows that PD always implies lowlying 0- states of the physical system. Returning to the case of stable RPA solutions, the excitation energy of the 0states tends to zero (before becoming imaginary), if the deforming potential is gradually increased. One may therefore say that the excitation energy of the lowest 0state in RPA for a given realistic potential yields additional information about possible PD. Combining this fact with the properties of the deformed structure, one might conjecture that a rigorous solution will just show a continuous decrease (and a characteristic change of the state) of the excitation energy of the 0- level to very small values with increasing deforming potential. This behaviour has in fact been proved in the frame work of an exactly solvable model s). Besides, a variational method with so-called projected wave functions a) points in the same direction in more general cases too. 3. The calculation
3.1. FORCES AND SINGLE-PARTICLE STATES In order to evaluate expression (1) or (2), we have first to define the two-body potential which occurs in the matrix elements. Since there is always a certain freedom in the choice of nuclear forces, we treat two different examples. (i) A "realistic" effective force was constructed by a Scott-Moszkowski procedure and a best fit of the low-lying energy nucleon-nucleon scattering data up to about 150 MeV, cf. ref. 7). This expression was shown to be in good agreement with a variety of characteristic nuclear properties 8). (ii) A pure OPEP was determined from the properties of the n-meson and a cutoff t tending to 0. It is used here as a test for the dependence from the force parameters and for comparison with previous model investigations of parity mixing. The single-particle states and the corresponding energies [occurring in expressions (1) and (2)] must in principle be determined from the spherical H F solution for closedshell nuclei. For the wave functions, however, we choose just oscillator functions since practically all HF calculations with all kinds of two-body interactions a) show that they represent reasonable approximations for the self-consistent orbitals. On the other t When the limiting process was not taken into account, too large (deforming) matrix elements have previously been found (even for zero cut-off).
584
w. BURRet al.
hand, the H F energies emdiffer in most cases appreciably from the oscillator spectrum; in addition, they depend strongly on the choice of the forces. In view of the importance of the corresponding numerical values, we consider four different cases. (1) We take self-consistent single-particle energies computed with the same effective realistic forces s, 14). (2) The H F energies are determined from a semi-phenomenological (i.e. suitably adapted) Woods-Saxon potential well 7). (3) In order to check the e-dependence, the single-particle energies are taken oscillator-like but with variable h~o values (no spin-orbit splitting and, therefore, constant energy differences between parity partners). (4) In the special case of x60 (here the 0- states are known experimentally), we insert experimental single-particle energies as determined from neighbouring nuclei 9). We then calculate the eigenvalues of h, R and A for the following combinations of our various assumptions on the forces [(i) and (ii)] and on the e-spectrum [(1)-(4)]: (i) combinated with (1)-(3) and (ii) with (3) for the sequence of nuclei 160, 4°Ca, 56Ni, 72Ge, 9°Zr, ~2°Sn, 14°Ce, 154Gd, 19SHg and 2°spb. For 160, we consider in addition (i) with (4). In all calculations, only the states of the next major shell (socalled lh~o excitations) were taken into account.. In the case of the Thouless criterium and under assumption (3), it is also possible to determine a minimum ho9 parameter Ae which leads to PD. Table 1 contains all important data used in these calculations. In particular it contains the single-particle energy differences of the states to be used in the calculations according to cases (1) (i.e. HF) and (2) (i.e. phenomenological) as well as the ho9 of the oscillator functions which enter the corresponding matrix elements of the force. 3.2. THE MATRIX ELEMENTS As a first step, we specify the Thouless criterium [cf. expression (l)] for the case of stability with respect to PD. The relevant matrix elements to be evaluated are then of the following form: M = ~ < n l J l m l , n2J2m2lV[n3Jlml, n4J2m2> ,
(3)
ml, m2
with [njm> = f,o(r), q~(O, (p) (j and J are states with the same total angular momentum but with opposite parity and m the corresponding magnetic quantum number). Exactly the same elements occur in RPA and TDA. In order to reduce the six-fold integrations, we use instead of a generalized Moshinsky scheme a special method which has been successfully introduced in the H F problem s, 14). It consists in a major reduction of the integrations by fully using the characteristic invariance properties of the matrix element (after summation). One thus obtains an appreciable reduction of the final computer program with respect to
TABLE 1 Input data of single-particle energy differences (phenomenological and HF) and oscillator parameter used in the calculation of matrix elements Nucleus Particle
Hole
Phenomenological
HF
160
ld~ 2s~r ldk 2s~
22.0 8.0 22.0 8.0
11.7 7.2 11,0 6.7
10.9 4°Ca 9.7
56Ni 8.1
72Ge 9.9
9°Zr 9.2
12°Sn 9.0
n n p p
lp÷ lP~r lp~: lp~r
n
ld~
lf~
17.9
15,3
n n p p p
2s~: ld~ ld~r 2s~ ld~.
2p½ 2p~ lf,2 2p~r 2pg.
11.7 11.1 17.1 10.6 7.2
10,3 6,4 13,9 9,0 5.4
n n n n p p p p n n n n p p p p n n n n n p p p p
ld~. 2s,~ ld~r lf~ ld~r 2s.~ ld~. lf~ lf~ 2p~r lf~ 2p~ ld~r 2s~r lf~r 2p~ lf~. 2p~ lf~ 2p~r lg~ lfk 2p{. lf~. 2pk
lf~_ 2p~ 2P~r lg~k lf~r 2p~ 2p~ lg~ lg~_ 2d I. 2d~ 3s~ lf~ 2p~r lg k 2d~r lgk 2d~r 2d~_ 3s~: lh~_ lsff 2d~ 2d~. 3s~r
16.5 12.2 9.5 18.7 15.5 10.4 8.0 16.0 17.4 11.3 6.8 7.5 14.2 10.7 16.1 9.6 15.6 11.4 7.9 8.3 17.6 14.8 10.0 7.7 7.1
17.1 13.1 9.5 12.6 16.3 11.9 8.0 11.6 15.3 10.6 5.8 8.3 19.3 14.8 14.6 9.5 16.5 12.7 8.5 9.2 12.4 16.3 10.1 5.5 5.9
n n n n n p p p p p
lg~ lgff 2d~. 2d~_ 3s~ lfg_ 2p~ lf{. 2p~ lg~
lh?r 2f~ 2f~ 3pk 3p.~ lg k 2dg. 2d~r 3s~ lh~
15.8 6.6 11.1 6.3 7.9 12.7 9.7 8.4 7.8 14.7
14.9 5.7 9.2 4.9 6.4 17.2 12.3 9.5 9.5 13.6
Nucleus Particle
Hole
Phenomenological
HF
l'~°Ce 8.3
lS~Gd 8.2
19aHg 6.7
2°apb 7.4
n n n n
lg~ lg~ 2d~ 2d~
lh~ 2f~ 2f~ 3p~
14.6 7.1 11.0 6.9
15.4 8.0 8.9 6.9
n n p p p p p
3s~r lh¥ 2p~r lfg: 2p~ lge2_ lg k
3P~r li~, 2d~r 2d~r 3s~. lh~ 2f~
8.5 16.8 9.4 8.7 8.0 13.8 7.1
7.9 16.6 11.9 11.2 6.9 13.9 6.2
n n n n n n n p p p p p n n n n n n n p p p p p p
lg~ lg~r 2d~, 2dk 3s÷ lh~ 2f_,l_ 2p~r 2p~ lg~. lgk 2d.t. 2d~_ 3s~. lh~ lh.~ 2fk 3p~. li~ 2p~ lg~ lg~ 2d1_ 2d,1. lh¥
lh~_ 2f~ 2f~_ 3p,~ 3p~ li¥ 2g~_ 2d~_ 3s½ lh~r 2f~ 2f~. 2f~_ 3p½ li~ 2g~ 2gk 3d~ lh~ 3st lh~ 2f~ 2f~ 3p{r li~
13.9 7.4 10.8 7.2 8.6 16.2 10.1 9.0 7.9 13.0 7.4 9.3 9.9 8.5 13.9 6.6 10.7 7.0 13.0 7.6 11.1 8.1 8.8 3.7 13.6
15.6 9.1 10.2 8.0 7.6 16.0 6.4 11.6 11.6 14.0 7.8 8.5 12.1 9.8 14.5 7.5 9.2 6.6 13.0 12.1 14.2 11.3 10.5 8.2 14.0
n n n n n n n p p p p p p
lh~ lh?r 2fk 2f~r 3p~ 3p~ li~ lg{r lg k 2d t. 2d~r 3s,1. lh~.
li~t 2g?r 2g~ 3d~ 3d~r 4s~k lk~ lh~ 2fz~ 2f~ 3pk 3p~ li~
13.5 6.8 10.6 6.1 8.1 6.2 13.0 10.7 8.2 8.6 6.8 7.1 13.3
14.4 8.7 10.0 6.2 7.3 6.7 13.0 14.1 11.5 11.4 9.4 9.5 14.0
586
w . BURR e t al.
other schemes 13). From this argument, it first follows that M must be of the form
M =
fF(I,,I,
It21, (rl r2))d3rl d3r2,
(4)
where the general function F takes the following form (for the contribution of the individual terms V~ of the potential):
M = ffl(rl)f2(r2)V=(r)g=(rl, r2 ,cos O)f3(rl)f,(r2)dar, d3r2 .
(5)
The main point *a) is that the g= are determined from the structure of the corresponding potential terms. They are given explicitly by the following expressions which, however, have to be given separately for the direct and for the exchange terms: V"
Direct J1
1
0
( ~ " a2)
d~ J2 cos ,9
T
Exchange Ii 12 DllD12 J2PoPo + 8L1 1" "-'2--1 --1
-- ±2 J 1 J 2 P ohP o12- 8L 1L2P~pt12
J1J2 x (r 1 - r 2 cos `9) x (r 1 cos O-- rE)
- sin `9((r 2 + r22) cos `9-2rx r2) x (J2 L1 plo2+ L2 J1 P~2PZ01) + cos 2 `9(r2 + r 2 - ( i / c o s O)r I rE)
x (½j~j2o"ol2 ,t o ~t 0
~T L 2 - - ~.,.t.q
p~,p~2)
+terms(r I *-~ r2)
(L. S)
0
0
The following abbreviations have been used: T = (tr,(rl - r2))(az(r 1- r2)),
L, = 1 ( j ' - l'), j, = 1 (2j, + 1). Here, P~(cos 0) stands for the Legendre polynomial ~o). We dropped the Llz term of the effective potential which is not important for the matrix elements in question. Thus far these formulae are valid for general radial dependenciesf For our numerical calculation, we assume oscillator functions which allow explicit reduction to a one-dimensional integration (containing the radial parts of the various V~). For this purpose, we introduce the usual Moshinsky substitution on relative and c.m. coor-
PARITY DEFORMATION IN SPHERICAL NUCLEI
587
dinates (R and r); the matrix elements which are all of the general form (4) then contain just a characteristic polynomial (calculated explicitly) in R, r and R.r, exponential in R 2 q - r 2 and V~(r), allowing to carry out all integrations apart over r, explicitly. The possibility of an explicit computation of our transformed polynomials replaces the Moshinsky method and leads to an appreciable simplification. 4. Results and conclusions
Tables 2 and 3 contain the results for a specially selected set of spherical nuclei covering the whole periodic system. Summing up these results, we arrive at the following conclusions concerning the general behaviour: (i) For all choices of forces and single-particle energies considered here, the eigenvalues of the matrix occurring in the Thouless theorem are never negative. Therefore, RPA leads always to real excitation energies of the 0- states. (ii) Within the framework of a variable oscillator spectrum [assumption (3)], we calculated the critical hco ( = Ae), which would lead to PD (cf. the last columns in table 2). This value was found to be unphysically low for both potentials and over the entire periodic table. (iii) In all cases, however, the excitation energies of the characteristic 0- states as calculated in RPA are lower than the original single-particle values. (In order to check this fact, one must compare the energy values of table 2 to the corresponding smallest single-particle energy difference of table 1 for each nucleus.) This shows that the sign of our matrix elements is such that PD would in fact occur if the forces were increased. Comparing, in addition, these results for the two different types of forces we see that the OPEP has as expected a stronger tendency to induce PD [refs. 3,4)] than the realistic expression. (iv) From the magnitudes of the lowest eigenvalues of the matrix h [cf. expression (1)] and a comparison with previous model calculations 5), we may judge that in the case of realistic single-particle energies a supposed increase of the matrix elements (i.e. increase of all coupling constants of the forces) by a factor of about 3 for realistic forces and of about 2.5 for the OPEP would be able to cause PD (by changing the sign of e). If we consider the special nucleus 160 for which more detailed empirical data are available, it becomes possible to make a direct numerical comparison between experimental and theoretical values. Using the experimental single-particle energies of the neighbouring odd-mass nuclei (150, 15N, 17O and 17F) and our realistic effective force, we obtain extremely good theoretical values for the two lowest 0- states of this nucleus; cf. table 2 where the results for various assumptions are shown. This fact represents again a verification of the good quality of our effective force (in contrast to the OPEP; see the last columns in table 3). Our result also shows that TDA and RPA are practically equally good in this case which is far from the "critical point" of deformation 5). In this connection, it is also interesting to observe that our calculation yields the correct energy difference (and the correct assignment)
588
W. BURR et al.
between the two states of different isospins (T = 0, T = 1), which also agrees with the experiment. Finally, the question arises whether generalized self-consistent H F calculations are able to yield PD by a finite (in contrast to infinitesimal) variation of the set of underlying single-particle functions. In fact, earlier investigations have shown that such a method m a y lead to PD despite the existence of a true local minimum without PD. This effect was found to be equivalent to a formal increase of the coupling constants (effective value) which, however, never exceeded a factor 2 in the most extreme cases. TABLE 2 Excitation energy of the lowest 0 - state in the T a m m - D a n c o f f approximation (TDA) and randomphase approximation (RPA) (in MeV) for various assumptions of the single-particle spectrum and the two-body potential S ingle-particle energy
osc.
osc.
osc.
osc.
t-IF
I-IF
phen.
phen.
osc.
osc.
Intelaction
eft.
eft.
OPEP
OPEP
eft.
eft.
eft.
eft.
eft.
OPEP
RPA
TDA
RPA
TDA
RPA
TDA
RPA
6.80 7.32 6.06 4.91 6.72 6.51 7.32 6.80 7.21 6.90
5.06 4.93 3.10 1.75 2.89 2.36 3.19 2.30 2.78 0.05
6.58 6.67 3.80 6.10 6.03 5.78 3.25 6.30 6.77 4.80
5.53 4.74 7.35 5.70 4.99 4.77 5.84 6.14 6,49 6.23
4.58 4.22 6.39 5.47 4.57 4.51 5.57 5.13 6.32 7.12
6.70 6.58 7.51 6.66 6.59 6.08 6.36 6.85 6.17 5.95
6.10 6.23 7.23 6.15 6.13 5.50 5.52 6.22 5.78 5.33
Approximation T D A 160 4°Ca 56Ni 72Ge 9°Zr 12°Sn l~°Ce 15aGd 198Hg 2°Spb
8.56 8.48 8.76 8.83 8.82 8.85 8.64 9.00 8.94 8.93
3.67 3.50 4.07 4.50 3.89 4.09 3.58 3.93 3.71 3.86
4.43 3.90 4.26 5.30 4.49 4.93 4.57 4.40 4.13 5.90
The last two columns refer to the limit of he) in the framework of the Thouless theorem. TABLE 3 0 - states of 1~O Theoretical calculation Experimental value
effective forces TDA
RPA
OPEP TDA
RPA
T = 0
10.95
10.97
10.62
8.93
7.80
T = 1
12.79
13.07
12.99
12.00
11.70
Experimental (first row) and theoretical (calculated with experimental single-particle energies and two types of forces using T a m m - D a n c o f f (TDA) as well as Random-phase Approximation (RPA).
Therefore, we can exclude appreciable PD even in this case, whereas a small amount of PD is always obtained from an extended variational principal starting with socalled parity-projected total wave functions. This effect, however, is practically
PARITY DEFORMATION IN SPHERICAL NUCLEI
589
equivalent to the c o r r e s p o n d i n g s e c o n d - o r d e r correction. I n a d d i t i o n , one should b e a r in m i n d t h a t the p a i r i n g effects also diminish the tendency o f P D (by a characteristic change o f the single-particle energies). W e m a y c o n c l u d e f r o m o u r calculation that a p p r e c i a b l e P D is n o t p r o b a b l e for realistic cases o f spherical structures t. This result is in a g r e e m e n t with the fact t h a t m o s t o f the f u n d a m e n t a l p r o p e r t i e s o f spherical nuclei (including s p i n - o r b i t splitting) were o b t a i n e d in a c c o r d a n c e with e x p e r i m e n t a l d a t a w i t h o u t assuming a n y PD. I n p a r t i c u l a r the h y p o t h e t i c a l low-lying 0 - states have never been f o u n d in the spherical regions, a n d our result for 160 shows explicitly t h a t these levels are in a m u c h higher energy region. t For deformed nuclei, the situation may be different mainly because of the drastic change of the e-values (cf. corresponding calculations ~2) on 19F).
References 1) G. Ripka, in Equilibiium shapes of light nuclei, Int. Course in nuclear physics, Trieste (1966) 2) F. Villars, in Ploc. Int. School of Physics "Enrico Fermi" Course 23 (1963) 3) K. Bleuler, Proc. Int. School of Physics "Enrico Fermi" Course 36 (1965) p. 464; J. P. Amiet and P. Huguenin, Nucl. Phys. 46 (1963) 171; J. Miiller, lJiplomarbeit Bonn (1963) unpublished; W. Ebenh6h, Z. Phys. 195 (1966) 171; J. Blomquist and A. Molinari, Nucl. Phys. A106 (1968) 545; W. H. Bassichis and J. P. Svenne, Phys. Rev. Lett. 18 (1967) 4) W. H. R~hl, Z. Phys. 195 (1966) 389 5) D. Schiitte, Proc. Liperi Summer School in theoretical physics (1966) B1, p. 25 6) D. J. Thouless, The quantum mechanics of many-body systems (New York, 1961) 7) K. Bleuler, M. Beiner and R. de Tourreil, Nuovo Cim. 52 (1967) 45 8) K. Bleuler, H. R. Perry and D. Schiitte, Nuovo Cim. 55b (1968) 296; M. Baranger, K. T. R. Davies, Nucl. Phys. 79 (1966) 403; A. K. Kerman, J. P. Svenne and F. M. H. Villars, Phys. Rev. 147 (1966) 710 9) G. E. Brown, L. Castellejo and J. A. Evans, Nucl. Phys. 22 (1961) 1 10) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, 1957) 11) T. A. Brody and M. Moshinsky, Tables of transformation brackets (Mexico, 1960) 12) B. Giraud, private communication 13) D. Schiitte and H. R. Petry, to be published 14) H. R. Petry, D. Schtitte and K. Bleuler, Energia Nucl. to be published
]
Nuclear Physics A133 (1969) 581--589; (~) North-Holland Publishing Co., Amsterdam
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
INVESTIGATIONS OF PARITY DEFORMATION IN SPHERICAL NUCLEI W. BURR, D. SCHOTTE and K. BLEULER
Institut Jiir Theoretische Kernphysik der Universitiit Bonn, W.-Germany Received 9 May 1969 Abstract: The possibility of parity deformation for spherical structures is investigated from various viewpoints. We use several criteria of stability for the spherical Hartree-Fock solution (Thouless condition, critical 0 - states in different approximations) as well as different types of two-body forces (a realistic effective interaction and the OPEP) and single-particle spectra (HF, oscillator, Woods-Saxon and experimental) in order to obtain a complete survey. The result is that parity deformation can be excluded in practically all cases. In addition, the positions of the two critical 0 - states of 160 are in good numerical agreement with experiments if realistic forces and experimental single-particle energies in connection with the random phase or TammDancoff approximation are used.
1. Introduction
Within the framework of Hartree-Fock (HF) calculations for nuclear structure, there exist (under well-defined conditions) self-consistent solutions (i.e. extrema in the variational scheme) with specific symmetries assumed from the outset 1) (e.g. spherically and reflection-symmetric HF solutions for closed shell (or subshell) nuclei). The main question is, however, whether these (restricted) solutions are stable, i.e. whether they represent a true minimum of the total energy with respect to the expectation values in unrestricted HF states. As an example, a spherically symmetric HF solution within the region of deformed nuclei will not satisfy this condition. In very much the same way, a HF solution with sharp parity [which exists for any particle number 2)] may be unstable with respect to parity mixture of the singleparticle states. The last question - the problem of the so-called parity deformation (PD) - was investigated by several authors 3) under various simplifying assumptions. Calculations with the one-pion-exchange potential (OPEP) and various numerical restrictions pointed to relatively strong parity deformation, self-consistent calculations 4) within the framework of a mere Hartree procedure even strengthened the tendency. On the other hand, more complete calculations with realistic two-body forces never gave an appreciable parity deformation. It is the aim of this paper to decide about the question of PD under the most general conditions which can be reached at present. We therefore investigate for a sequence of closed-shell nuclei various criteria of PD, namely the eigenvalues of the 581
w. BURRet al.
582
matrix occurring in the Thouless theorem as well as the positions of the characteristic low-lying 0- states in the Random phase approximation (RPA) and Tamm-Dancoff approximation (TDA). In both cases, two different assumptions for the two-body force (OPEP and effective realistic force) and four different assumptions for the basic single-particle energies (HF, phenomenological, oscillator-like and experimental energies) are made. The calculations include exchange terms of the matrix elements but are restricted to states of the next major shell. Combining our numerical results from all these cases, we find that appreciable PD is practically excluded for the spherical case, whereas the situation may be different in the deformed region of lighter nuclei.
2. Theoretical background Starting from a spherical symmetric H F solution for a closed-shell nucleus [its existence is guaranteed by the theorem of Ripka 1)], we investigate the possibility of an energy gain by allowing parity mixture in the single-particle states. In a number of earlier investigations 3,4), this problem was found to be rather sensitive to the choice of nuclear forces V and the assumed single-particle spectrum era. It seems, therefore, desirable to discuss all possible variations of the relevant quantities as well as a few important stability conditions. (i) A most direct test of the stability of a H F solution is given by the theorem of Thouless 6); it consists (in the most general case) in evaluating the (real) eigenvalues of the Hermitian matrix
h = (A B*
B) A*
with
Ami, nj = ( e m - e i ) S m n S i j + ( m j l V [ i n ) A s Bm~,nj = ( m n l V l i j ) A s
(1)
(i, j enumerate the occupied states and m, n the unoccupied states, AS means antisymmetrized) and yields the following local criterium (i.e. with respect to infinitesimal variations of the orbitals) for PD: There is local PD, if and only if, at least one of the eigenvalues of h is positive. This criterium cannot, however, exclude PD for a self-consistent calculation, which is based on completely different wave functions from the outset 4) (possible coexistence of several true minima, e.g. c~-decay and spallation). (ii) Another (but closely connected) method to decide about the occurrence of PD is obtained from the RPA equations (random phase approximation). They yield critical excitation energies (0- vibrations in our case) with the help of the eigenvalues E of the new non-Hermitian matrix R =
-B*
-A*
"
(2)
[The notation is the same as in expression (1).] Here the eigenvalues are real - p a i r s of opposite sign - in the case of a stable H F solution, whereas at least two become imaginary (pair of complex conjugates) in case of PD. (In this connection, it should
PARITY DEFORMATION IN SPHERICAL NUCLEI
583
be mentioned that these vibrations can be approximated in the frame work of the Tamm-Dancoff approximation (TDA) by the eigenvalues of the hermitian matrix A alone, if the corresponding RPA result is real and not too small.) In the case of PD, the RPA equations applied on the spherically symmetric ground states lose their physical meaning. The lowest excited state of the spectrum is now given in a reasonable approximation by one of the parity partners, which in this case results from the (projected) parity-deformed HF ground state solution 3,5) (in general, negative parity for the excited state). The excitation energy of this level is relatively low s) (cross matrix elements); this shows that PD always implies lowlying 0- states of the physical system. Returning to the case of stable RPA solutions, the excitation energy of the 0states tends to zero (before becoming imaginary), if the deforming potential is gradually increased. One may therefore say that the excitation energy of the lowest 0state in RPA for a given realistic potential yields additional information about possible PD. Combining this fact with the properties of the deformed structure, one might conjecture that a rigorous solution will just show a continuous decrease (and a characteristic change of the state) of the excitation energy of the 0- level to very small values with increasing deforming potential. This behaviour has in fact been proved in the frame work of an exactly solvable model s). Besides, a variational method with so-called projected wave functions a) points in the same direction in more general cases too. 3. The calculation
3.1. FORCES AND SINGLE-PARTICLE STATES In order to evaluate expression (1) or (2), we have first to define the two-body potential which occurs in the matrix elements. Since there is always a certain freedom in the choice of nuclear forces, we treat two different examples. (i) A "realistic" effective force was constructed by a Scott-Moszkowski procedure and a best fit of the low-lying energy nucleon-nucleon scattering data up to about 150 MeV, cf. ref. 7). This expression was shown to be in good agreement with a variety of characteristic nuclear properties 8). (ii) A pure OPEP was determined from the properties of the n-meson and a cutoff t tending to 0. It is used here as a test for the dependence from the force parameters and for comparison with previous model investigations of parity mixing. The single-particle states and the corresponding energies [occurring in expressions (1) and (2)] must in principle be determined from the spherical H F solution for closedshell nuclei. For the wave functions, however, we choose just oscillator functions since practically all HF calculations with all kinds of two-body interactions a) show that they represent reasonable approximations for the self-consistent orbitals. On the other t When the limiting process was not taken into account, too large (deforming) matrix elements have previously been found (even for zero cut-off).
584
w. BURRet al.
hand, the H F energies emdiffer in most cases appreciably from the oscillator spectrum; in addition, they depend strongly on the choice of the forces. In view of the importance of the corresponding numerical values, we consider four different cases. (1) We take self-consistent single-particle energies computed with the same effective realistic forces s, 14). (2) The H F energies are determined from a semi-phenomenological (i.e. suitably adapted) Woods-Saxon potential well 7). (3) In order to check the e-dependence, the single-particle energies are taken oscillator-like but with variable h~o values (no spin-orbit splitting and, therefore, constant energy differences between parity partners). (4) In the special case of x60 (here the 0- states are known experimentally), we insert experimental single-particle energies as determined from neighbouring nuclei 9). We then calculate the eigenvalues of h, R and A for the following combinations of our various assumptions on the forces [(i) and (ii)] and on the e-spectrum [(1)-(4)]: (i) combinated with (1)-(3) and (ii) with (3) for the sequence of nuclei 160, 4°Ca, 56Ni, 72Ge, 9°Zr, ~2°Sn, 14°Ce, 154Gd, 19SHg and 2°spb. For 160, we consider in addition (i) with (4). In all calculations, only the states of the next major shell (socalled lh~o excitations) were taken into account.. In the case of the Thouless criterium and under assumption (3), it is also possible to determine a minimum ho9 parameter Ae which leads to PD. Table 1 contains all important data used in these calculations. In particular it contains the single-particle energy differences of the states to be used in the calculations according to cases (1) (i.e. HF) and (2) (i.e. phenomenological) as well as the ho9 of the oscillator functions which enter the corresponding matrix elements of the force. 3.2. THE MATRIX ELEMENTS As a first step, we specify the Thouless criterium [cf. expression (l)] for the case of stability with respect to PD. The relevant matrix elements to be evaluated are then of the following form: M = ~ < n l J l m l , n2J2m2lV[n3Jlml, n4J2m2> ,
(3)
ml, m2
with [njm> = f,o(r), q~(O, (p) (j and J are states with the same total angular momentum but with opposite parity and m the corresponding magnetic quantum number). Exactly the same elements occur in RPA and TDA. In order to reduce the six-fold integrations, we use instead of a generalized Moshinsky scheme a special method which has been successfully introduced in the H F problem s, 14). It consists in a major reduction of the integrations by fully using the characteristic invariance properties of the matrix element (after summation). One thus obtains an appreciable reduction of the final computer program with respect to
TABLE 1 Input data of single-particle energy differences (phenomenological and HF) and oscillator parameter used in the calculation of matrix elements Nucleus Particle
Hole
Phenomenological
HF
160
ld~ 2s~r ldk 2s~
22.0 8.0 22.0 8.0
11.7 7.2 11,0 6.7
10.9 4°Ca 9.7
56Ni 8.1
72Ge 9.9
9°Zr 9.2
12°Sn 9.0
n n p p
lp÷ lP~r lp~: lp~r
n
ld~
lf~
17.9
15,3
n n p p p
2s~: ld~ ld~r 2s~ ld~.
2p½ 2p~ lf,2 2p~r 2pg.
11.7 11.1 17.1 10.6 7.2
10,3 6,4 13,9 9,0 5.4
n n n n p p p p n n n n p p p p n n n n n p p p p
ld~. 2s,~ ld~r lf~ ld~r 2s.~ ld~. lf~ lf~ 2p~r lf~ 2p~ ld~r 2s~r lf~r 2p~ lf~. 2p~ lf~ 2p~r lg~ lfk 2p{. lf~. 2pk
lf~_ 2p~ 2P~r lg~k lf~r 2p~ 2p~ lg~ lg~_ 2d I. 2d~ 3s~ lf~ 2p~r lg k 2d~r lgk 2d~r 2d~_ 3s~: lh~_ lsff 2d~ 2d~. 3s~r
16.5 12.2 9.5 18.7 15.5 10.4 8.0 16.0 17.4 11.3 6.8 7.5 14.2 10.7 16.1 9.6 15.6 11.4 7.9 8.3 17.6 14.8 10.0 7.7 7.1
17.1 13.1 9.5 12.6 16.3 11.9 8.0 11.6 15.3 10.6 5.8 8.3 19.3 14.8 14.6 9.5 16.5 12.7 8.5 9.2 12.4 16.3 10.1 5.5 5.9
n n n n n p p p p p
lg~ lgff 2d~. 2d~_ 3s~ lfg_ 2p~ lf{. 2p~ lg~
lh?r 2f~ 2f~ 3pk 3p.~ lg k 2dg. 2d~r 3s~ lh~
15.8 6.6 11.1 6.3 7.9 12.7 9.7 8.4 7.8 14.7
14.9 5.7 9.2 4.9 6.4 17.2 12.3 9.5 9.5 13.6
Nucleus Particle
Hole
Phenomenological
HF
l'~°Ce 8.3
lS~Gd 8.2
19aHg 6.7
2°apb 7.4
n n n n
lg~ lg~ 2d~ 2d~
lh~ 2f~ 2f~ 3p~
14.6 7.1 11.0 6.9
15.4 8.0 8.9 6.9
n n p p p p p
3s~r lh¥ 2p~r lfg: 2p~ lge2_ lg k
3P~r li~, 2d~r 2d~r 3s~. lh~ 2f~
8.5 16.8 9.4 8.7 8.0 13.8 7.1
7.9 16.6 11.9 11.2 6.9 13.9 6.2
n n n n n n n p p p p p n n n n n n n p p p p p p
lg~ lg~r 2d~, 2dk 3s÷ lh~ 2f_,l_ 2p~r 2p~ lg~. lgk 2d.t. 2d~_ 3s~. lh~ lh.~ 2fk 3p~. li~ 2p~ lg~ lg~ 2d1_ 2d,1. lh¥
lh~_ 2f~ 2f~_ 3p,~ 3p~ li¥ 2g~_ 2d~_ 3s½ lh~r 2f~ 2f~. 2f~_ 3p½ li~ 2g~ 2gk 3d~ lh~ 3st lh~ 2f~ 2f~ 3p{r li~
13.9 7.4 10.8 7.2 8.6 16.2 10.1 9.0 7.9 13.0 7.4 9.3 9.9 8.5 13.9 6.6 10.7 7.0 13.0 7.6 11.1 8.1 8.8 3.7 13.6
15.6 9.1 10.2 8.0 7.6 16.0 6.4 11.6 11.6 14.0 7.8 8.5 12.1 9.8 14.5 7.5 9.2 6.6 13.0 12.1 14.2 11.3 10.5 8.2 14.0
n n n n n n n p p p p p p
lh~ lh?r 2fk 2f~r 3p~ 3p~ li~ lg{r lg k 2d t. 2d~r 3s,1. lh~.
li~t 2g?r 2g~ 3d~ 3d~r 4s~k lk~ lh~ 2fz~ 2f~ 3pk 3p~ li~
13.5 6.8 10.6 6.1 8.1 6.2 13.0 10.7 8.2 8.6 6.8 7.1 13.3
14.4 8.7 10.0 6.2 7.3 6.7 13.0 14.1 11.5 11.4 9.4 9.5 14.0
586
w . BURR e t al.
other schemes 13). From this argument, it first follows that M must be of the form
M =
fF(I,,I,
It21, (rl r2))d3rl d3r2,
(4)
where the general function F takes the following form (for the contribution of the individual terms V~ of the potential):
M = ffl(rl)f2(r2)V=(r)g=(rl, r2 ,cos O)f3(rl)f,(r2)dar, d3r2 .
(5)
The main point *a) is that the g= are determined from the structure of the corresponding potential terms. They are given explicitly by the following expressions which, however, have to be given separately for the direct and for the exchange terms: V"
Direct J1
1
0
( ~ " a2)
d~ J2 cos ,9
T
Exchange Ii 12 DllD12 J2PoPo + 8L1 1" "-'2--1 --1
-- ±2 J 1 J 2 P ohP o12- 8L 1L2P~pt12
J1J2 x (r 1 - r 2 cos `9) x (r 1 cos O-- rE)
- sin `9((r 2 + r22) cos `9-2rx r2) x (J2 L1 plo2+ L2 J1 P~2PZ01) + cos 2 `9(r2 + r 2 - ( i / c o s O)r I rE)
x (½j~j2o"ol2 ,t o ~t 0
~T L 2 - - ~.,.t.q
p~,p~2)
+terms(r I *-~ r2)
(L. S)
0
0
The following abbreviations have been used: T = (tr,(rl - r2))(az(r 1- r2)),
L, = 1 ( j ' - l'), j, = 1 (2j, + 1). Here, P~(cos 0) stands for the Legendre polynomial ~o). We dropped the Llz term of the effective potential which is not important for the matrix elements in question. Thus far these formulae are valid for general radial dependenciesf For our numerical calculation, we assume oscillator functions which allow explicit reduction to a one-dimensional integration (containing the radial parts of the various V~). For this purpose, we introduce the usual Moshinsky substitution on relative and c.m. coor-
PARITY DEFORMATION IN SPHERICAL NUCLEI
587
dinates (R and r); the matrix elements which are all of the general form (4) then contain just a characteristic polynomial (calculated explicitly) in R, r and R.r, exponential in R 2 q - r 2 and V~(r), allowing to carry out all integrations apart over r, explicitly. The possibility of an explicit computation of our transformed polynomials replaces the Moshinsky method and leads to an appreciable simplification. 4. Results and conclusions
Tables 2 and 3 contain the results for a specially selected set of spherical nuclei covering the whole periodic system. Summing up these results, we arrive at the following conclusions concerning the general behaviour: (i) For all choices of forces and single-particle energies considered here, the eigenvalues of the matrix occurring in the Thouless theorem are never negative. Therefore, RPA leads always to real excitation energies of the 0- states. (ii) Within the framework of a variable oscillator spectrum [assumption (3)], we calculated the critical hco ( = Ae), which would lead to PD (cf. the last columns in table 2). This value was found to be unphysically low for both potentials and over the entire periodic table. (iii) In all cases, however, the excitation energies of the characteristic 0- states as calculated in RPA are lower than the original single-particle values. (In order to check this fact, one must compare the energy values of table 2 to the corresponding smallest single-particle energy difference of table 1 for each nucleus.) This shows that the sign of our matrix elements is such that PD would in fact occur if the forces were increased. Comparing, in addition, these results for the two different types of forces we see that the OPEP has as expected a stronger tendency to induce PD [refs. 3,4)] than the realistic expression. (iv) From the magnitudes of the lowest eigenvalues of the matrix h [cf. expression (1)] and a comparison with previous model calculations 5), we may judge that in the case of realistic single-particle energies a supposed increase of the matrix elements (i.e. increase of all coupling constants of the forces) by a factor of about 3 for realistic forces and of about 2.5 for the OPEP would be able to cause PD (by changing the sign of e). If we consider the special nucleus 160 for which more detailed empirical data are available, it becomes possible to make a direct numerical comparison between experimental and theoretical values. Using the experimental single-particle energies of the neighbouring odd-mass nuclei (150, 15N, 17O and 17F) and our realistic effective force, we obtain extremely good theoretical values for the two lowest 0- states of this nucleus; cf. table 2 where the results for various assumptions are shown. This fact represents again a verification of the good quality of our effective force (in contrast to the OPEP; see the last columns in table 3). Our result also shows that TDA and RPA are practically equally good in this case which is far from the "critical point" of deformation 5). In this connection, it is also interesting to observe that our calculation yields the correct energy difference (and the correct assignment)
588
W. BURR et al.
between the two states of different isospins (T = 0, T = 1), which also agrees with the experiment. Finally, the question arises whether generalized self-consistent H F calculations are able to yield PD by a finite (in contrast to infinitesimal) variation of the set of underlying single-particle functions. In fact, earlier investigations have shown that such a method m a y lead to PD despite the existence of a true local minimum without PD. This effect was found to be equivalent to a formal increase of the coupling constants (effective value) which, however, never exceeded a factor 2 in the most extreme cases. TABLE 2 Excitation energy of the lowest 0 - state in the T a m m - D a n c o f f approximation (TDA) and randomphase approximation (RPA) (in MeV) for various assumptions of the single-particle spectrum and the two-body potential S ingle-particle energy
osc.
osc.
osc.
osc.
t-IF
I-IF
phen.
phen.
osc.
osc.
Intelaction
eft.
eft.
OPEP
OPEP
eft.
eft.
eft.
eft.
eft.
OPEP
RPA
TDA
RPA
TDA
RPA
TDA
RPA
6.80 7.32 6.06 4.91 6.72 6.51 7.32 6.80 7.21 6.90
5.06 4.93 3.10 1.75 2.89 2.36 3.19 2.30 2.78 0.05
6.58 6.67 3.80 6.10 6.03 5.78 3.25 6.30 6.77 4.80
5.53 4.74 7.35 5.70 4.99 4.77 5.84 6.14 6,49 6.23
4.58 4.22 6.39 5.47 4.57 4.51 5.57 5.13 6.32 7.12
6.70 6.58 7.51 6.66 6.59 6.08 6.36 6.85 6.17 5.95
6.10 6.23 7.23 6.15 6.13 5.50 5.52 6.22 5.78 5.33
Approximation T D A 160 4°Ca 56Ni 72Ge 9°Zr 12°Sn l~°Ce 15aGd 198Hg 2°Spb
8.56 8.48 8.76 8.83 8.82 8.85 8.64 9.00 8.94 8.93
3.67 3.50 4.07 4.50 3.89 4.09 3.58 3.93 3.71 3.86
4.43 3.90 4.26 5.30 4.49 4.93 4.57 4.40 4.13 5.90
The last two columns refer to the limit of he) in the framework of the Thouless theorem. TABLE 3 0 - states of 1~O Theoretical calculation Experimental value
effective forces TDA
RPA
OPEP TDA
RPA
T = 0
10.95
10.97
10.62
8.93
7.80
T = 1
12.79
13.07
12.99
12.00
11.70
Experimental (first row) and theoretical (calculated with experimental single-particle energies and two types of forces using T a m m - D a n c o f f (TDA) as well as Random-phase Approximation (RPA).
Therefore, we can exclude appreciable PD even in this case, whereas a small amount of PD is always obtained from an extended variational principal starting with socalled parity-projected total wave functions. This effect, however, is practically
PARITY DEFORMATION IN SPHERICAL NUCLEI
589
equivalent to the c o r r e s p o n d i n g s e c o n d - o r d e r correction. I n a d d i t i o n , one should b e a r in m i n d t h a t the p a i r i n g effects also diminish the tendency o f P D (by a characteristic change o f the single-particle energies). W e m a y c o n c l u d e f r o m o u r calculation that a p p r e c i a b l e P D is n o t p r o b a b l e for realistic cases o f spherical structures t. This result is in a g r e e m e n t with the fact t h a t m o s t o f the f u n d a m e n t a l p r o p e r t i e s o f spherical nuclei (including s p i n - o r b i t splitting) were o b t a i n e d in a c c o r d a n c e with e x p e r i m e n t a l d a t a w i t h o u t assuming a n y PD. I n p a r t i c u l a r the h y p o t h e t i c a l low-lying 0 - states have never been f o u n d in the spherical regions, a n d our result for 160 shows explicitly t h a t these levels are in a m u c h higher energy region. t For deformed nuclei, the situation may be different mainly because of the drastic change of the e-values (cf. corresponding calculations ~2) on 19F).
References 1) G. Ripka, in Equilibiium shapes of light nuclei, Int. Course in nuclear physics, Trieste (1966) 2) F. Villars, in Ploc. Int. School of Physics "Enrico Fermi" Course 23 (1963) 3) K. Bleuler, Proc. Int. School of Physics "Enrico Fermi" Course 36 (1965) p. 464; J. P. Amiet and P. Huguenin, Nucl. Phys. 46 (1963) 171; J. Miiller, lJiplomarbeit Bonn (1963) unpublished; W. Ebenh6h, Z. Phys. 195 (1966) 171; J. Blomquist and A. Molinari, Nucl. Phys. A106 (1968) 545; W. H. Bassichis and J. P. Svenne, Phys. Rev. Lett. 18 (1967) 4) W. H. R~hl, Z. Phys. 195 (1966) 389 5) D. Schiitte, Proc. Liperi Summer School in theoretical physics (1966) B1, p. 25 6) D. J. Thouless, The quantum mechanics of many-body systems (New York, 1961) 7) K. Bleuler, M. Beiner and R. de Tourreil, Nuovo Cim. 52 (1967) 45 8) K. Bleuler, H. R. Perry and D. Schiitte, Nuovo Cim. 55b (1968) 296; M. Baranger, K. T. R. Davies, Nucl. Phys. 79 (1966) 403; A. K. Kerman, J. P. Svenne and F. M. H. Villars, Phys. Rev. 147 (1966) 710 9) G. E. Brown, L. Castellejo and J. A. Evans, Nucl. Phys. 22 (1961) 1 10) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, 1957) 11) T. A. Brody and M. Moshinsky, Tables of transformation brackets (Mexico, 1960) 12) B. Giraud, private communication 13) D. Schiitte and H. R. Petry, to be published 14) H. R. Petry, D. Schtitte and K. Bleuler, Energia Nucl. to be published