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Antipairing effect in the rotational states of light nuclei
I.C: I.D.I[
Nuclear Physics A201 (1973) 49--65; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ANTIPAIRING EFFECT IN THE ROTATIONAL STATES OF LIGHT NUCLEI K. G O E K E , H. MOTHER and A M A N D FAESSLER
Institut fiir Kernphysik, KFA Jiilich, 517 Jiilich, W. Germany Received 14 June 1972 (Revised 10 August 1972) Abstract: Properties of rotational levels o f s d shell nuclei are studied in the self-consistent deformed
Hartree-Bogoliubov model with a variation performed approximately after angular momentum projection. This formalism, which consists of a constrained Hartree-Bogoliubov theory with an additional pairing potential, is explained in detail. Excitation energies, moments of inertia, pairing correlations and quadrupole moments of the ground state rotational bands of 2ZNe, 24Ne and 3°Si are investigated and compared with experimental data. In the calculation all particles are treated alike without the assumption of an inert core, using the effective G-matrix elements of Barrett et al., the Yale-Shakin interaction and the B1 force of Brink and Boeker. It turns out to be important to perform the projection before the variation and not afterwards, since the pairing correlations change drastically in going to higher J-states. In the nuclei 24Ne and s°Si the calculations indicate a phase transition into the unpaired states at J : 4 + and 8 + respectively. The agreement with experiment is satisfactory and the experimental deviation from the J(J-k 1) law is quite well reproduced. In each case the Barrett G-matrix gives the best results, which are, in the case of 22Ne, in good agreement with experimental values.
1. Introduction
In the last ten years many Hartree-Fock (HF) calculations have been performed to explain the properties of light nuclei using only the assumption of reasonable twobody forces 1-3). Since the HF wave function is, for deformed nuclei, not an eigenstate of the angular momentum operator squared, Peierls and Yoccoz 4) proposed a technique to project the states of sharp angular momentum from a HF solution (HFP). However, the spectra calculated with HFP were mostly compressed 5-6) compared to the experimental values. In order to improve upon this method by the inclusion of pairing correlations, the Hartree-Bogoliubov (HB) theory 7) was formulated, first without angular momentum projection, for nuclei s) and has been frequently used 9-12). Onishi and Yoshida 13-14) first applied the Peierls-Yoccoz formalism to HB states (HBP); explicit numerical calculations for sd and pf shell nuclei have been performed only recently 1s-21). All these calculations show the importance of pairing correlations for the description of the rotational properties of light nuclei: The excitation energies are enlarged by about 100% and the agreement with experimental data is much improved; furthermore, the experimental deviations from the J(J+ 1) law are approximately reproduced. For sd shell nuclei the decrease in pairing correlation with increasing angular momentum can be shown explicitly 18, 20). 49
50
K . G O E K E e t aL
In these HBP calculations one assumes that there exists a single intrinsic wave function from which the rotational states can be projected out. For the calculation of the intrinsic wave function a variational procedure is performed in which the expectation value of the total Hamiltonian between unprojected states is minimized. It would be better to perform this variation with projected wave functions (PHB) from the outset. For H F wave functions both types of calculations have been performed to test how well the H F P method approximates the more complicated P H F method 22,23). These calculations 24) show that H F P seems to be a good approximation to PHF, at least for the low-lying rotational states of nuclei with a moderate equilibrium deformation. Only for nearly spherical nuclei is H F P different from PHF. Although all the nuclei considered here have an intrinsic deformation of more than 0.25, this consideration is not necessarily valid for HB states for the following reasons: The moment of inertia of an unpaired state is essentially determined by the equilibrium deformation of the intrinsic wave function, which is relatively stable against moderately small changes in the wave function. For a HB state the moment of inertia is additionally very sensitive to the amount of pairing correlation, and small changes in the pairing have a remarkable effect on the spectrum. In this sense the nucleus might be "soft" against pairing correlations, and their strengths can be more dependent on the rotational level than indicated by the HBP method. In this paper we perform an approximate projection before variation, using a parameter for the minimization which has a strong effect on the amount of pairing correlation. We do this by a constrained Hartree-Bogoliubov calculation with the Hamiltonian
o'= where l;'p is the pairing part of the two-body interaction. We study the decrease in pairing with increasing angular momentum and the effect of this behaviour on the spectrum. We also study by this the change in the intrinsic structure of the nucleus due to rotation. In sect. 2 we present the theoretical formalism; the forces we use and the numerical details are described in sect. 3, and our results are presented in sect. 4. 2. The theoretical formalism
The purpose of this section is to explore the constrained Hartree-Bogoliubov theory (CHB) and to collect the essential equations for the theoretical background. Details of the HB model can be found in refs. a, 12); the angular momentum projection from paired states is described in refs. 13,20.2s); we follow closely the notation of ref. 20). In the usual HB theory the HB state is defined as a product of quasi-particle annihilation operators, a~, which are defined by the HB transformation from a basis given by c~, Ck: I~) = ( l-[ a~)lO), (1)
ANTIPAIRING EFFECT
51
a+ = Z (Ak~c; + Bk~ek).
(2)
and k
This trial wave function I~b) is determined by minimizing the variation of the expectation value of the total Hamiltonian,/~, between kb):
(q~l~lq~> _ 0.
(3)
Since for finite deformations I~b) does not have a good angular momentum quantum number, one has to project out such states by the method of Peierls and Yoccoz. These projected states correspond to the true physical states and their properties can be compared with experimental data: ]q~j> _
PJI~> (q~lPJI4,) *
(4)
In this procedure one performs the variation with symmetry-violating wave functions and restores the symmetry afterwards by projection assuming one intrinsic state for all rotational levels. In a correct procedure one has to vary with wave functions which already conserve symmetry: One has to minimize the expectation value o f / ~ between projected states instead of unprojected ones: 6 (4~sl~lq~J) - 0.
(5)
(¢JI4J) For H F siates this was first formulated by Rouhaninejad and Yoccoz 22) and Zeh 23). Onishi 13) applied their theory to HB states and was able to derive theoretical expressions for the Euler-Lagrange equation of the variational principle (5) and an appropriate expansion in terms of a small parameter which is good for strongly deformed nuclei and contains only properties of the unprojected HB state. Another way of solving (5) for paired states has been given by Beck, Mang and Ring 25). By restricting themselves to cranking-model wave functions they were able to split the variational problem (5) approximately into two parts, namely the usual HB equations and a complicated system of linear equations. Because of numerical difficulties, no realistic calculation, to our knowledge, has been performed with these PHB theories. In this paper we perform an approximation to PHB, which is definitely better than HBP since we allow the variation to be performed with respect to one parameter, 2, after the angular momentum projection. According to the considerations in sect. 1 we do this by choosing a set of ).-dependent intrinsic trial wave functions, 1¢(2)), in such a way that the pairing correlations vary continuously from zero to finite values equal to their amount in the HB state, or even more. To be sure that the set of 1¢(2)) contains the H F solution as well as the HB solution and in order to
K. GOEKE et al.
52
guarantee a reasonable and physical choice, the wave functions [~b(2)) are themselves variational solutions of HB type (1) of the modified Hamiltonian / ~ ' = ~ - 21)p,
(6)
where i?p is the pairing part of the effective two-body interaction P contained i n / : / :
~p = ¼ E
--
+
+
~, ( k 1 rnl, k2~llvlkam3, k4m3>Ck,mlCk2~,Ck4~3Ck3ms.
(7)
klk2mt kak4m3
The minima with respect to 2 of the projected energies
EA;O = <~a(2)l/~l~,j(,~)>, (~J(~)l~,(~))
(8)
for the different J-values now determine the different intrinsic wave functions for the rotational levels. In the next few paragraphs we derive the CHB equations, which are the EulerLagrange equations for the variational principle (6) with trial states (1); the name is chosen in analogy to the constrained H F method 26). The H F potential F' and the pairing potential A' o f / ~ ' split into two parts corresponding respectively to V and Vp:
F' = F--2Fp, A' = A-hAp.
(9)
They are defined with help of the density matrix p and the pairing tensor ~c:
Fk~m,,k3,~ =
~
(klrnl,k2m2lv[kam3,k,m,)Pk4m4,k,m 2,
(10a)
(Fp)k,m,,k3m3 = ~ (kl ml, k2 ~,lvlk3 m3, k4m3)Pk4~3,k2~2,
(10b)
k2ra2kam4 k2k4
Ak,,.,,k2m2 = ½ ~,
( k l m l , k z m 2 [ v [ k a m 3 , k , m4)xk4m4,k3m3,
(lla)
k3m3/g4/r/4
(ap)k,mt,k2m2=~m2~½ ~, (klml,k2~llv[k3rnz,k,~3)lgk,~3,k3ms.
(lib)
k3m3k4
If we restrict the nuclear wave functions to axial symmetry with the quantum number K = 0 of the total angular momentum upon the body-fixed symmetry axis, the pairing potentials d and Ap become identical. According to eqs. (9) and (10), Fp is negligible compared to F, since it is of the order Fp ~ ~2-tlV, if f2 is the number of different m-values. This is the common approximation, that the H F part of a pure pairing potential can be neglected against the H F part of the full potential s). The energy matrix of the CHB problem is now
W(2) =
t+F (1-2)d +
(1-2)A ] --(t+F)X] '
(12)
ANTIPAIRING
EFFECT
53
where t is the kinetic energy and the superscript T indicates the transpose of a matrix. Diagonalizing IV().) in a well-known way 8) one obtains the CHB equations
Z ((tik + r,k)A~, + (1 --)-)Aik Bi, } = E~ A,,, k
Z {(tik + F'k)*Bk, + (1 --)-)A ~ Ak,} = -- E, Bi,,
(13)
k
which are, according to our considerations the Euler-Lagrange equations of the variational principle, 6 (~bl/4-)-Vpl~b) = 0.
(14)
The most interesting cases of the H F equations for unpaired states (2 = 1) and of the HB equations for paired states (2 = 0) are included in these CHB equations. The intrinsic energy E(2) = @()-)l~ql@(2)> (~(01~()-)>
(15) '
is always in a minimum at the HB solution (2 = 0), but obviously this is not necessarily true for the projected energies (8). We would like to point out that 2 is not a free parameter used for some fitting procedure but a variational parameter used in accordance with a variational principle which is a better approximation to a complete PHB calculation than the normally used HBP method. This choice of a parameter by which the rotational energies are minimized, allows the BCS degree of freedom to go continuously from an unpaired state to a state which is even more paired than the HBP state. The pairing correlations in the rotational levels should be better described in this PHB method than in the HBP method previously used, and we expect to obtain a stronger decrease of the pairing energy with increasing angular momentum quantum number than presented in earlier calculations. Independently of the agreement with experiment, the approximate PHB procedure we are using is within the model a "better" method than HBP.
3. The nuclear interactions and the numerical procedure In our calculations the nuclear wave functions are restricted to time-reversal symmetry and axial symmetry with projection K = 0 of the total angular momentum onto the symmetry axis. We use the spherical harmonic oscillator wave functions and take into account the is, lp and 2sld shells without separating off any inert core. The Coulomb force and c.m. corrections are neglected. The proton and neutron numbers are fixed in an average way by introducing two Lagrange parameters into the variational procedure and requiring the number operators to have fixed values. Nevertheless, the spectra are weighted mixtures of the spectra of neighbouring nuclei since we do not use number projected HB states. However, all of our HB solutions
54
K. GOEKE et al.
show a particle number fluctuation of about 10~o, which is not enough to noticeably perturb the spectrum of the nucleus considered2°). Proton-proton and neutron-neutron pairing are considered since only in the case N = Z is protonneutron pairing important 12). As interactions for our calculations we use the effective G-matrix elements of the Hamada-Johnston potential calculated by Barrett et aL, the Yale-Shakin elements and the effective force B1 of Brink and Boeker. Barrett, Hewitt and McCarthy 27, 28) calculate the reaction matrix G in a twoparticle harmonic oscillator basis by solving the Bethe-Goldstone equation
G(CO) = V + V
Q
--/~0
G(co).
(16)
CO
The Pauli operator Q is diagonal in this basis: Q[ab) = ( 0 for a or b in Is or lp shell for a or b in 2sld shell otherwise,
(17)
and is, therefore, especially appropriate for microscopic calculations in the sd shell. From these matrix elements, which are given for five different values of the starting energy, co, we calculated by interpolation an effective interaction V~ff. To avoid a self-consistent treatment of the starting energy, we do not fix co but take reasonable values of the single-particle energies ea, which are taken as - 4 0 . 0 , - 3 2 . 0 , - 2 9 . 0 , -23.0, - 2 0 . 0 and - 1 7 . 0 MeV for the harmonic oscillator states and treat the starting energy in the following way: (abll/~fflcd) = ½{(ablG(ea+eb+2C)lcd)+(ablG(ec+ed+2C)lcd)}. (18) Here C is a parameter which shifts the e a so that the energy difference between e, and the energies of the intermediate states used in the Bethe-Goldstone equation takes reasonable values. In our calculations C is chosen to reproduce the binding energy of the nucleus considered within 20 MeV. It is not reasonable to fit C in a more accurate way due to the intrinsic limitations of the method, and since the rotational energies in which we are interested are very insensitive compared to the binding energy (see table 1). For 22Ne and 24Ne we take C = 50 MeV and for 3°Si C = 47 MeV. The lower value of C for 3°Si indicates, that the HB single-particle states are more bound relative to the intermediate states than those of 22Ne and 24Ne. TABLE 1 The ground state energies and the spectra for HBP calculations of 22Ne with Barrett force for different values of C (eq. (18)) C (MeV)
40
B E2 £'4 E~ Ea
96.46 0.96 3.05 6.10 10.21
(MeV) (MeV) (MeV) (MeV) (MeV)
50 163.57 1.10 3.44 6.76 11.03
60 243.01 1.16 3.60 6.99 11.05
Exp 177.77 1.27 3.34 6.35 11.15
ANTIPAIRING EFFECT
55
We also use the effective G-matrix elements of the realistic Yale potential given by Shakin, Waghmare and Hull 29,3o), and an oscillator length b = 1.76 fm. The H F P and HBP results show a little improvement compared to previous ones 2o), since the starting energy parameter, A, is chosen as A = 34.2 MeV instead of A = 20 MeV as described in ref. 31). We use the effective interaction B1 of Brink and Boeker 32) with the oscillator length b = 2.0 fm of ref. as).
4. Results We performed HFP, HBP and PHB calculations for the ground states of 22Ne, 24Ne and 3°Si and compared these results with experimental data 34-36). The ground TABLE 2 Comparison o f HFP, HBP and PI-IB calculations with experimental data (a) Results for 22Ne Exp
22Ne (Barrett) HFP
E2 /?4 E6 Es
02 04 06 0s
(MeV) (MeV) (MeV) (MeV) (MeV -1) (MeV -1) (MeV -1) (MeV -1)
E4/E2 EJE2 Es/E2 B (MeV)
HBP
Z2Ne(B1) PHB
22Ne (Yale)
HFP
HBP
PHB
HFP
1.27 0.69 1.10 1.27 0.45 3.34 2.42 3.44 3.23 1.43 6.35 5.58 6.76 6.25 2.89 11.15 1 0 . 8 1 1 1 . 0 3 10.74 4.68 2.36 4.29 2.73 2.37 6.63 3.38 4.04 2.99 3.56 7.14 3.65 3.49 3.31 3.63 7.53 3.13 2.86 3.51 3.34 8.38 2.63 3.47 3.13 2.55 3.16 5.01 7.97 6.15 4.94 6.39 8.78 15.44 10.04 8.48 10.36 177.77 162.95 163.57 163.75 114.06
0.87 2.47 5.33 8.59 3.43 4.38 3.85 4.60 2.82 6.10 9.83 117.19
0.87 2.44 4.24 5.81 3.43 4.46 6.11 9.55 2.79 4.86 6.65 117.19
0.41 1.50 3.61 7.26 7.29 6.42 5.21 4.11 3.64 8.78 17.64 103.67
BHP
PHB
0.81 0.88 2.66 2.39 5.35 4.50 8.23 7.92 3.68 3.41 3.78 4.64 4.09 5.21 5.21 4.39 3.26 2.71 6.55 5.11 10.57 9.01 104.49 104.56
(b) Results for Z4Ne and a°Si ~4Ne (Barrett)
E2 E4 E6 Ea 02 04
(MeV) (MeV) (MeV) (MeV) (MeV -1) (MeV -1)
E4/Ez B (MeV)
a°Si (Barrett)
Exp
HFP
HBP
PHB
Exp
HFP
HBP
PHB
1.99 3.98
0.85 2.72 5.46 9.26 3.52 3.76 3.19 179.83
1.61 4.90 9.26 14.48 1.86 2.12 3.05 181.12
1.61 4.00 6.75 10.53 1.86 2.92 2.49 181.12
2.23 5.27
0.85 2.85 6.04 10.23 3.53 3.50 3.36 249.08
1.51 5.02 10.49 17.92 1.99 1.99 3.33 253.78
1.62 5.07 10.00 15.08 1.85 2.03 3.13 253.92
1.51 3.52 2.00 191.83
1.34 2.30 2.36 255.63
The excitation energies, Ej, of the rotational states of the ground band, the ratios Ej/Ez of the rotational states and the moments of inertia 0j according to eq. (19) are listed; B is the binding energy of the 0 + ground state.
56
K. GOEKE et al.
states of 22Ne and 24Ne are prolate deformed, that of 3°Si, oblate. For 22Ne we present the results for all three forces, for 2 4 N e and 3°Si we give only the results with the G-matrix elements of the Hamada-Johnston potential as calculated by Barrett et al., since they give in all cases by far the best energy spectrum and binding energy. All HB solutions show non-negligible pairing correlations. In table 2 the excitation energies, the ratios of the excitation energies and the moments of inertia of the different levels of the ground state rotational band are listed; see also figs. 1-3. The moments of inertia are calculated from the projected or measured excitation energies Es: 2J- 1 Oj . (19) -
Ej
-
E s_ 2
-
In all cases the HFP spectrum is strongly compressed compared to the measured data. This is a very general result which is also found in all other HFP calculations with density- and momentum-independent forces 5, 6). The 2 + states in H F P has in general only one-third or less of the experimental energy. This is drastically changed by the inclusion of pairing correlations in the HBP method as the comparison of the moments of inertia of the H F solution and of the HB solution calculated with help of the Inglis model already indicates 11, 12). The spectrum is stretched by a factor of
1211-
8"
8+
10,81~
1103 "~,8"
8+
1115
625. S*
6.35
1.27
12'I
107t. / "
10-
t .¢
9876-
/
s°
/
"\',,6"
s.~//
54IJJ
I
/
3-
/ 4+
2.42/
21-
2" 2÷
1.10
2*
2~
0.70/./-
0 HFP
HBP
PHB
EXP
Fig. 1. Comparison o f the HFP, ~ B P and PHB spectra of 22Ne with experimental data. The calculations are performed with the Barrett force.
2'
4-
5"
6"
7-
8-
HFP
2"
6"
/ / s~4
/
HBP
\~g
161 _ 2*
\\
PHB
161~/"
~o
675
g
o f t h e I-[EP, I - [ B P a n d P H B
085/
/
/ /
272/
/
/
/
/*90
~*
\
spectra of 24N¢ with
EXP
199
3.98
experimental data. The calculations are performed with the Barrett force.
F i g . 2. C o m p a r i s o n
f
U
LU
2£ .c
>~
t
/ t I I I I I
5'
HFP
;t*
¢
~5//
/
i
/ / / / /
2*
I
I I I I I t I I I I I
HBP
151 ..:.
2*
502 _ t.*
\ \ e~
PHB
'L62J
507~
/
2"
/.
EXP
223
5.27
Fig. 3. Comparison o f the H F P , HBP and PHB spectra o f a°Si with experimental data. The calculations are performed with the Barrett force.
I 3-
U/
w
9-
10-
\ \
10.
9-
11-
11-
[6" - - 926
12-
12-
"11 ~n rn
Z
>. =.
> Z
58
K. GOEKE
et al.
about two and is in all HBP solutions in better agreement with experiment than in the HFP solutions. In 22Ne the HBP solution with the Barrett force is in very good agreement with experiment, which is reproduced within the limits of 0.4 MeV as far as the excitation energies up to the 8+ state are concerned (see fig. 1). In this solution the 2+ and 4+ states have an excitation energy which is a little bit larger than the experimental one whereas in all other HBP solutions the spectra are still compressed even if they are improved. A similar improvement by inclusion of the pairing correlations can be seen in the moments of inertia. The experimental data show increasing moment of inertia with increasing J, which is reproduced always in the HBP solutions but in general not in the HFP solutions I*, 20). In all cases the experimental deviations from the simple J(J+ 1) law are better reproduced in HBP than in HFP, as one can see by comparing the ratios of the experimental and theoretical excitation energies. From the above results and from previous calculations “3 20) it seems to be evident, that these nuclei cannot be described without including pairing correlations in the wave function. Nevertheless the HBP theory is not the final answer and the comparison with experimental data still shows disagreement in many cases. One step in improving this can be done by performing PHB calculations whose energy spectra in the CHB approximation we are doing are given in table 2. In all cases there is a clear difference between the HBP and PHB results. This is in contradiction to the general belief that projection before variation has negligible effect for sufficiently deformed nuclei, which one concludes by comparing HFP and PHF calculations 24). The difference between PHB and HBP is very strong and has values of up to 25% of the excitation energy. In 24Ne for examp le the excitation energy of the 4+ state changes from 4.90 to 4.00 MeV by performing the projection before the variation and shows afterwards very nice agreement with the experimental value. The result of the PHB calculation in “Ne with the Barrett force is very impressive: The 2+ state is exactly reproduced, the 4+ and the 6+ states show deviations of only 0.1 MeV from experiment, and the 8+ state deviates only 0.4 MeV at an excitation energy of 11.15 MeV. The moments of inertia are also reproduced quite well and show an improvement compared to HBP; even such details as the small decrease in the moment of inertia from 6+ to 8+ is given by the PHB solution of 22Ne with the Barrett force. To our knowledge this is the best reproduction of the experimental spectrum of 22Ne with a parameter-free and self-consistent model. The energy ratios, which are sensitive not to the absolute agreement of the theoretical levels with the experimental ones but to the character of the whole spectrum and the deviations form the J(J+ 1) law, are reproduced quite well and are remarkably better than the HFP and HBP values. Here lies a general improvement of all PHB solutions compared to HBP. In all cases considered here, the energy ratios are clearly better reproduced in PHB than in HBP; this is true for all forces used in our calculations. We feel that this is more important than the agreement of the absolute values of the calculated excitation energies with experiment, since it
ANTIPAIRING EFFECT
59
reproduces the character of the total spectrum and not only single levels. Let us consider for example the H F P solution of 2aNe with the Barrett force. The 2 + and 4 + states are in total disagreement with experiment, the 6 + the reproduces roughly the experimental value and the 8 + state shows only a deviation of 0.4 MeV. F r o m the agreement in the 6 + and 8 + states one nevertheless cannot conclude that the spectrum is roughly reproduced, since this agreement is accidental. The moment of inertia of this H F P solution is too large and the spectrum shows a decreasing moment of inertia; therefore there must be levels which are in rough agreement with experiment if we go high enough in excitation energy. F r o m this point of view, the trend in the moments of inertia and the agreement in energy ratios seem to us more important than the agreement of single levels with experiment, which is sometimes worse in PHB than in HBP. In a typical example the improvement of the energy ratios can clearly be seen in fig. 4, where the HFP, HBP and PHB spectra of ZZNe calculated with the Yale force are given. These spectra are artificially normalized in such a way that the 6 + state always assumes the experimental value of the excitation energy. Additionally one should point out that the excitation energies are strongly dependent on the effective two-body interaction used in the calculations, as the comparison of the results in table 2a shows. So one should not overestimate the agreement or disagreement of the theoretical excitation energies with experiment. One should be more interested in trends which are independent of the forces. The drastic changes from HBP to PHB, the improvement in the energy ratios and the strong antipairing effect discussed below are such reliable and force-independent results. In table 3 the projected pairing energies are listed. These are a measure of the TABLE 3
The projected pairing energies Aj in the rotational levels calculated with HBP and PHB using the Barrett force 24Ne
aos i
Barrett
Barrett
22Ne Barrett
Ao d2 A4 A6 As
(MeV) (MeV) (MeV) (MeV) (MeV)
Yale
B1
HBP
PHB
HBP
PHB
HBP
PHB
HBP
PHB
HBP
PHB
2.04 1.89 1.59 1.21 0.76
2.90 2.27 0.12 0.39 0.47
1.84 1.73 1.48 1.12 0.69
2.24 2.11 0.05 0.03 0.16
4.18 3.96 3.65 2.99 1.34
4.18 3.96 2.81 1.59 1.31
2.91 2.68 2.22 1.60 1.00
2.91 2.68 0.00 0.00 0.00
2.03 1.98 1.86 1.69 1.46
2.51 2.25 1.55 1.30 0.00
amount of pairing correlation and are defined as the contribution of the pairing correlation to the total projected binding energy. In all cases the low-lying states in P H B are at least as paired as in HBP, whereas the higher states are always less paired and mostly pure H F P solutions without pairing. The antipairing effect which is already indicated in previous H B P calculations 18,20) and for heavy nuclei in theoretical
13-
8+
12.76
12118"
~25
~
635
/.+
3,16
2+
Qg7
8*
1119
8+
~15
6+
635
6*
6.35
4+
337
L,~
33/*
2+
12,'.
2+
127
109-
8>= ~E
7-
._c
LU
-8_
6,35
5Z.-
LU
I 3-
d
26~
2:" 072
0
HFP
HBP
PHB
EXP
Fig. 4. Comparison of the scaled spectra of HFP, HBP and PHB solutions of 22Ne calculated with the Yale force with experiment. All excitation energies are multiplied by a factor to normalize the 6 + energy to the experimental value to demonstrate the improvement of the energy ratios.
- 9&O-
-99.0. I -IOQO-~ -10tO.G
T -IOGK)-
°4-.....
- t~.O-
-try0
-QI
a'05
a;
K,
o.;s
o'~
Q25
Fig. 5. The binding energies of the PHB solutions with the Yale force for 22Ne v e r s u s the minimization parameter 2. The arrows indicate the minima. The HBP solutions are given by xX= 0.
24Ne
2zNe
30Si
- - - HFP - - HBP - - PHB
t
15-
J
10-
/
/
.
Ej [MeVl
5-
//
/
/
oi
2b
h
h
6
~ --J
,3 (J÷ll
2b
h
22
¢2
=
Fig. 6. The binding energies o f the rotational levels v e r s u s J(J-t-1) for the H F P , HBP and PHB solutions o f 2ZNe, 24Ne and a°Si calculated with the Barrett force. The energies are normalized so that the 0 + value o f PHB is zero. 22Ne 4
~'"~'~
- - - HFP - - - - HBP
~
. . . . . EXP 3
~
2 l
,5
t,
ej
-E
.)
t.,v )
2 I5 "'Si 3
,
1
! 1
0
g - -
~2W j2 [MeV]
lb
Fig: 7. The moments o f inertia o f 22Ne, =4He and a°Si in dependencc on (oj , see eqs. 0 9 ) and (20). The results are given f o r H F P , H B P and P H B , calculated w i t h the Barrett force, and f o r experimental values.
62
K. GOEKE et al.
considerations using Migdal theory 37) and experimental measurements as, ag) is, according to our P H B calculations, clearly stronger than that expected from HBP. The reason for this can be explained in the following way. The unprojected HB solution is in general about 1 MeV more bound than the H F solution, but its m o m e n t of inertia is about one-half of that obtained from the H F wave function 12. z o). Therefore the H B P spectrum is expanded compared to HFP, and sometimes in higher rotational states the H F P level is more bound due to its moment of inertia which is larger than that of the H B P level. F o r this reason the nucleus "prefers" to change for this level to an unpaired intrinsic state. This behaviour can clearly be seen in fig. 6, where the binding energies of the rotational levels are plotted as a function of J ( J + 1). (The binding energy of the 0 ÷ state of P H B is normalized tO zero.) In ZaNe the gain of binding energy due to pairing correlations is very small, so only the 0 + and 2 + states are more deeply bound with the inclusion of pairing correlations. Going from the 2 + to the 4 + state is a transition to an unpaired state, and all higher-J states are also unpaired. In a°Si this transition occurs at 6 +, since the gain in binding energy due to pairing is larger. The same effect is to be seen in 22Ne, even if it is not so clear, as discussed below. In fig. 7 the moments of inertia, eq. (32), are plotted against the angular frequency, ogj, which is defined by 1
ogs = - - ~ / J ( J + 1). Os
(20)
In heavy nuclei such curves sometimes show a " b a c k w a r d " or forward bending; here a similar effect can be seen, which cannot be as clearly defined due to the small number of rotational levels in light nuclei. In 22Ne and 24Ne the PHB curve shows the forward bending, since the s-curve starts already at the 2 + state; in 3°Si one sees the backward bending. Due to numerical inaccuracies it was not worthwhile to calculate the 10 + state in these nuclei. Our calculations indicate that phenomena such as phase transitions, and backward and forward bending, which are established in the highspin states of heavy nuclei can also occur with similar properties in the low-spin states o f light nuclei. This is in agreement with the estimates of Sheline 41). This change from a paired state to an unpaired one cannot be reproduced in a HBP calculation in which we project out of one correlated state and can only approximately reproduce the decrease of pairing energy with increasing moment of inertia. F r o m this we see that the variational parameter 2, used to simulate a correct procedure of projection before variation, is the correct one to reproduce this effect. The values of 2 for the different minima of the projected solutions of one example can be obtained from fig. 5. They are a measure of how far the intrinsic wave functions differ from the different rotational levels. For the values see table 4. In contradiction to the above considerations for few solutions the projected pairing energies show very little increase from the 6 + to the 8 + state and parallel to this the cranking moments of inertia of the intrinsic state show this behaviour. We have no explanation for this
ANTIPAIRING EFFECT
63
TABLE4 The intrinsic cranking moments of inertia 0l"tr- and the values of the parameter 2 for the PHB solutions calculated with the Barrett force J 0ol"tr- (MeV -1) 0z turf" (MeV -1) 04t*t'- (MeV -1) 06 Int'' (MeV -1) 0s IntT" (MeV -1) 0HFlntr' (MeV - t ) 0HBlair" (MeV -x)
22Ne (Yale)
Z2Ne (Barrett)
1.308 (--0.025) 1.308 (--0.025) 2.818 (+0.150) 2.818 (+0.150) 2.319 (+0.100) 2.899 1.458
0.920 (--0.05) 1.015 (--0.025) 2.211 (+0.150) 1.736 (+0.100) 1.386 (+0.05) 2.392 1.124
Z4Ne (Barrett) 1.133 1.133 1.771 1.771 1.771 1.771 1.133
(--0.00) (--0.00) (+0.25) (+0.25) (+0.25)
a°Si (Barrett) 1.004 (--0.0075) 1.030 (--0.0050) 1.156 (+0.1100) 1.219 (+0.2200) 2.174 (+0.3000) 2.174 1.077
but it might be explained in the following way. For these states the energy surface Es (;t) is a very flat valley near the minimum especially in the direction towards to the
H F solution, while the pairing correlations however are still remarkably dependent on 2. For such a system one should not look for the minimum of the energy surface E s (2) but should superimpose all solutions with the help of the generator-coordinate method. It is possible that for such a solution the pairing correlations will show the expected behaviour. But, anyway, even in our states the pairing energies of these states are clearly smaller than in states with low spin. The increase of the moments of inertia due to the decrease of the pairing correlations going to higher-J values can also be seen in table 4, where the moments of inertia of the intrinsic states calculated with the cranking formula are listed. Except for the solutions we discussed in the preceding paragraph the results show a clear increase of the intrinsic moment of inertia with increasing angular momentum quantum number. This shows also, to what extent the intrinsic states differ from each other. In all cases the phase transitions can be recognized by a sudden change of the intrinsic moments of inertia by a factor of about two. Table 5 lists the projected intrinsic mass quadrupole moments Q~, which are obtained from the projected spectroscopic one Q (J) by the prescription of the rotational model: 2J+3 Q, = - - Q(J). (21) J The HBP values indicate antistretching, that means a decrease o f the intrinsic Qs with increasing J, whereas the PHB values show an increase of Qs in the phase transition due to the fact that the H F solution is more strongly deformed 42). Comparison of the results of 22Ne and of the other nuclei, which are not presented here, show that the effective G-matrix elements of Barrett et aL are very appropriate for calculations in the sd shell. For 22Ne the overall agreement is quite impressive and there is obviously no need for K-mixing to explain the ground state rotational band. For 24Ne we obtain results which are as good as the ones given by Khadkikar et al. 39). These authors describe the spectrum of Z4Ne by mixing oblate and prolate
64
K. GOEKE et aL
TAat~ 5 The projected intrinsic mass quadrupole moments calculated from the spectroscopic ones with the help of the rotational model for HBP and PHB using the Barrett force 22Ne
Q2 (e2" fm2) Q4 (e2 " fm2) Q6 (e~" fm2)
Z4Ne
aosi
HBP
PHB
HBP
PHB
HBP
PHB
62.81 61.10 57.76
61.55 60.24 61.40
60.25 58.12 54.87
60.35 65.75 64.59
-74.21 -74.52 -73.95
-73.80 -74.57 -73.47
H F P solutions with p-h excitations neglecting partially the strong effect of the pairing correlations, which are shown to be important. According to our calculations, shape mixing is not needed to obtain such agreement. It could be, however, that shapemixed, correlated H B P or P H B solutions would yield even better results. In 3°Si, P H B yields an essential improvement compared to HBP; the spectrum is a little bit better b u t still clearly compressed compared to experimental values. 5. Conclusion
In conclusion we would like to point out that the pairing correlations of sd shell nuclei are strongly dependent upon the rotational levels and, in some cases, do not already exist in the 4 + state. This requires, in the H B model, an angular m o m e n t u m projection procedure before variation (PHB) which is necessary even for strongly deformed nuclei, in contradiction to the uncorrelated projected H F model. By approximate HB variation after projection, in a constrained H B model (CHB) the excitation energies change by up to 3 0 ~ compared to projection after variation (HBP) and yield in 22Ne quite impressive agreement with experiment if one uses the effective G-matrix elements of Barrett et aL, which give in all cases the best results. In 24Ne and 3°Si the calculations indicate a phase transition into the unpaired state at d = 4 + and 8 + respectively. The CHB theory seems to be a very appropriate tool for the study of ground state rotational bands and of the effect of pairing correlations on them, especially for phase transitions into the unpaired state. We thank Dr. L. Satpathy for valuable discussions about the formulation of the constrained H B theory. We thank Dr. H. H. Wolter for the use of the HB p r o g r a m for the intrinsic wave functions. We are greatly indebted to Dr. B. R. Barrett for giving us his effective G-matrix elements. References
I) I. Kelson, Phys. Rev. 132 (1963) 2189 2) I. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269 3) G. Ripka, Advances of Nuclear Physics, vol. 1 ed. M. Baranger and E. Vogt (Plenum Press, New York, 1968) 4) R. E. Peierls and J. Yoccoz, Proe. Phys. Soc. A70 (1957) 381 5) H. A. Lamme and E. Boeker, Nucl. Phys. Alll (1968)492
A N T I P A I R I N G EFFECT
65
6) S. K. M. Wong, J. LeTourneux, N. Quang Hoc and G. Saurnier, Nucl. Phys. A137 (1969) 318 7) N. N. Bogoliubov, Usp. Fiz. Nauk 67 (1959) 236 [transl.: Sov. Phys. Usp. 2 (1959) 236] 8) M. Baranger, in 1962 Cargese lectures in theoretical physics, ed. M. Levy (Benjamin, New York~ 1963) 9) K. Dietrich, H. J. Mang and J. Pradal, Z. Phys. 190 (1966) 357 I0) A. Faessler, P. U. Sauer and M. M. Stingl, Z. Phys. 212 (1968) 1 11) L. Satpathy, D. Ooss and M. K. Banerjee, Phys. Rev. 183 (1969) 887 12) H. H. Wolter, A. Faessler, P. U. Sauer, Nucl. Phys. A167 (1971) 108 13) N. Onishi, Prog. Theor. Phys. Jap. 40 (1968) 84 14) N. Onishi and S. Yoshida, Nucl. Phys. 80 (1966) 367 15) M. R. Gunye and S. B. Khadkikar, Phys. Rev. Lett. 24 (1970) 910 16) K. R. Sandhya Devi, S. B. Khadkikar, J. K. Parikh and B. Banerjee, Phys. Lett. 32B (1970) 179 17) A. Faessler, K. Goeke and L. Satpathy, Phys. Lett. 35B (1971) 193 18) K. Goeke and A. Faessler, Phys. Lett. 35B (1971) 289 19) L. Satpathy, K. Goeke and A. Faessler, Nucl. Phys. A179 (1972) 177 20) K. Goeke, A. Faessler and 14. H. Wolter, Nucl. Phys. A183 (1972) 352 21) J. K. Parikh, Phys. Rev. C5 (1972) 153 22) H. Rouhaninejad and J. Yoccoz, Nucl. Phys. 78 (1966) 353 23) H. D. Zeh, Z. Phys. 188 (1965) 361 24) L. Satpathy and S. C. K. Nair, Phys. Lett. 26B (1968) 257 25) R. Beck, H. J. Mang and P. Ring, Z. Phys. 231 (1970) 26 26) B. Castel and J. C. Parikh, Phys. Rev. C1 (1970) 990; J. C. Parikh and B. Castel, Phys. Lett. 3713 (1971) 7 27) B. R. Barrett, R. G. L. 14ewitt and R. J. McCarthy, Phys. Rev. {23 (1971) 1137 28) B. R. Barrett, R. G. L. 14ewitt and R. J. McCarthy, Phys. Rev. C2 (1970) 1199 29) C. M. Shakin, Y. R. Waghmare and M. H. Hull, Phys. Rev. 161 (1967) 1006 30) C. M. Shakin, Y. R. Waghmare and M. H. Hull, Phys. Rev. 161 (1967) 1015 31) 14. H. Wolter, thesis 1970, Univ. of Muenster, unpublished 32) D. M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1 33) P. U. Sauer, A. Faessler, H. H. Wolter and M. M. Stingl, Nucl. Phys. A125 (1969) 257 34) P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) l 35) A. J. Howard, R. G. Hirko, D. A. Bromley, K. Bethge and J W. Olness, Phys. Rev. C1 (1970) 1446 36) W. G. Davies, C. Broude and J. S. Forster, Bull. Am. Phys. Soc. 15 (1970) 543 37) J. Meyer and J. Speth, preprint, 1972 38) R. Baader, H. Backe, R. Engler, K. Hesse, E. Kankeleit, V. Schr6der, H. K. Walther and K. Wien, Phys. Lett. 2713 (1968) 425 39) H. K. Walther, H. Backe, R. Engfer, E. Kankeleit, C. Petitjeen, H. Schneuwly and W. V. SchrOder, Phys. Lett., to be published 40) S. B. Khadkikar, S. C. K. Nair and S. P. Pandya, Phys. Lett. 3613 (1971) 290 41) R. K. Sheline, private communication 42) A. Faessler, K. Goeke and. H. MOther, Contribution to the Conf. on High spin states, Stockholm, June 1972, to be published in Physica Scripta
Nuclear Physics A201 (1973) 49--65; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ANTIPAIRING EFFECT IN THE ROTATIONAL STATES OF LIGHT NUCLEI K. G O E K E , H. MOTHER and A M A N D FAESSLER
Institut fiir Kernphysik, KFA Jiilich, 517 Jiilich, W. Germany Received 14 June 1972 (Revised 10 August 1972) Abstract: Properties of rotational levels o f s d shell nuclei are studied in the self-consistent deformed
Hartree-Bogoliubov model with a variation performed approximately after angular momentum projection. This formalism, which consists of a constrained Hartree-Bogoliubov theory with an additional pairing potential, is explained in detail. Excitation energies, moments of inertia, pairing correlations and quadrupole moments of the ground state rotational bands of 2ZNe, 24Ne and 3°Si are investigated and compared with experimental data. In the calculation all particles are treated alike without the assumption of an inert core, using the effective G-matrix elements of Barrett et al., the Yale-Shakin interaction and the B1 force of Brink and Boeker. It turns out to be important to perform the projection before the variation and not afterwards, since the pairing correlations change drastically in going to higher J-states. In the nuclei 24Ne and s°Si the calculations indicate a phase transition into the unpaired states at J : 4 + and 8 + respectively. The agreement with experiment is satisfactory and the experimental deviation from the J(J-k 1) law is quite well reproduced. In each case the Barrett G-matrix gives the best results, which are, in the case of 22Ne, in good agreement with experimental values.
1. Introduction
In the last ten years many Hartree-Fock (HF) calculations have been performed to explain the properties of light nuclei using only the assumption of reasonable twobody forces 1-3). Since the HF wave function is, for deformed nuclei, not an eigenstate of the angular momentum operator squared, Peierls and Yoccoz 4) proposed a technique to project the states of sharp angular momentum from a HF solution (HFP). However, the spectra calculated with HFP were mostly compressed 5-6) compared to the experimental values. In order to improve upon this method by the inclusion of pairing correlations, the Hartree-Bogoliubov (HB) theory 7) was formulated, first without angular momentum projection, for nuclei s) and has been frequently used 9-12). Onishi and Yoshida 13-14) first applied the Peierls-Yoccoz formalism to HB states (HBP); explicit numerical calculations for sd and pf shell nuclei have been performed only recently 1s-21). All these calculations show the importance of pairing correlations for the description of the rotational properties of light nuclei: The excitation energies are enlarged by about 100% and the agreement with experimental data is much improved; furthermore, the experimental deviations from the J(J+ 1) law are approximately reproduced. For sd shell nuclei the decrease in pairing correlation with increasing angular momentum can be shown explicitly 18, 20). 49
50
K . G O E K E e t aL
In these HBP calculations one assumes that there exists a single intrinsic wave function from which the rotational states can be projected out. For the calculation of the intrinsic wave function a variational procedure is performed in which the expectation value of the total Hamiltonian between unprojected states is minimized. It would be better to perform this variation with projected wave functions (PHB) from the outset. For H F wave functions both types of calculations have been performed to test how well the H F P method approximates the more complicated P H F method 22,23). These calculations 24) show that H F P seems to be a good approximation to PHF, at least for the low-lying rotational states of nuclei with a moderate equilibrium deformation. Only for nearly spherical nuclei is H F P different from PHF. Although all the nuclei considered here have an intrinsic deformation of more than 0.25, this consideration is not necessarily valid for HB states for the following reasons: The moment of inertia of an unpaired state is essentially determined by the equilibrium deformation of the intrinsic wave function, which is relatively stable against moderately small changes in the wave function. For a HB state the moment of inertia is additionally very sensitive to the amount of pairing correlation, and small changes in the pairing have a remarkable effect on the spectrum. In this sense the nucleus might be "soft" against pairing correlations, and their strengths can be more dependent on the rotational level than indicated by the HBP method. In this paper we perform an approximate projection before variation, using a parameter for the minimization which has a strong effect on the amount of pairing correlation. We do this by a constrained Hartree-Bogoliubov calculation with the Hamiltonian
o'= where l;'p is the pairing part of the two-body interaction. We study the decrease in pairing with increasing angular momentum and the effect of this behaviour on the spectrum. We also study by this the change in the intrinsic structure of the nucleus due to rotation. In sect. 2 we present the theoretical formalism; the forces we use and the numerical details are described in sect. 3, and our results are presented in sect. 4. 2. The theoretical formalism
The purpose of this section is to explore the constrained Hartree-Bogoliubov theory (CHB) and to collect the essential equations for the theoretical background. Details of the HB model can be found in refs. a, 12); the angular momentum projection from paired states is described in refs. 13,20.2s); we follow closely the notation of ref. 20). In the usual HB theory the HB state is defined as a product of quasi-particle annihilation operators, a~, which are defined by the HB transformation from a basis given by c~, Ck: I~) = ( l-[ a~)lO), (1)
ANTIPAIRING EFFECT
51
a+ = Z (Ak~c; + Bk~ek).
(2)
and k
This trial wave function I~b) is determined by minimizing the variation of the expectation value of the total Hamiltonian,/~, between kb):
(q~l~lq~> _ 0.
(3)
Since for finite deformations I~b) does not have a good angular momentum quantum number, one has to project out such states by the method of Peierls and Yoccoz. These projected states correspond to the true physical states and their properties can be compared with experimental data: ]q~j> _
PJI~> (q~lPJI4,) *
(4)
In this procedure one performs the variation with symmetry-violating wave functions and restores the symmetry afterwards by projection assuming one intrinsic state for all rotational levels. In a correct procedure one has to vary with wave functions which already conserve symmetry: One has to minimize the expectation value o f / ~ between projected states instead of unprojected ones: 6 (4~sl~lq~J) - 0.
(5)
(¢JI4J) For H F siates this was first formulated by Rouhaninejad and Yoccoz 22) and Zeh 23). Onishi 13) applied their theory to HB states and was able to derive theoretical expressions for the Euler-Lagrange equation of the variational principle (5) and an appropriate expansion in terms of a small parameter which is good for strongly deformed nuclei and contains only properties of the unprojected HB state. Another way of solving (5) for paired states has been given by Beck, Mang and Ring 25). By restricting themselves to cranking-model wave functions they were able to split the variational problem (5) approximately into two parts, namely the usual HB equations and a complicated system of linear equations. Because of numerical difficulties, no realistic calculation, to our knowledge, has been performed with these PHB theories. In this paper we perform an approximation to PHB, which is definitely better than HBP since we allow the variation to be performed with respect to one parameter, 2, after the angular momentum projection. According to the considerations in sect. 1 we do this by choosing a set of ).-dependent intrinsic trial wave functions, 1¢(2)), in such a way that the pairing correlations vary continuously from zero to finite values equal to their amount in the HB state, or even more. To be sure that the set of 1¢(2)) contains the H F solution as well as the HB solution and in order to
K. GOEKE et al.
52
guarantee a reasonable and physical choice, the wave functions [~b(2)) are themselves variational solutions of HB type (1) of the modified Hamiltonian / ~ ' = ~ - 21)p,
(6)
where i?p is the pairing part of the effective two-body interaction P contained i n / : / :
~p = ¼ E
--
+
+
~, ( k 1 rnl, k2~llvlkam3, k4m3>Ck,mlCk2~,Ck4~3Ck3ms.
(7)
klk2mt kak4m3
The minima with respect to 2 of the projected energies
EA;O = <~a(2)l/~l~,j(,~)>, (~J(~)l~,(~))
(8)
for the different J-values now determine the different intrinsic wave functions for the rotational levels. In the next few paragraphs we derive the CHB equations, which are the EulerLagrange equations for the variational principle (6) with trial states (1); the name is chosen in analogy to the constrained H F method 26). The H F potential F' and the pairing potential A' o f / ~ ' split into two parts corresponding respectively to V and Vp:
F' = F--2Fp, A' = A-hAp.
(9)
They are defined with help of the density matrix p and the pairing tensor ~c:
Fk~m,,k3,~ =
~
(klrnl,k2m2lv[kam3,k,m,)Pk4m4,k,m 2,
(10a)
(Fp)k,m,,k3m3 = ~ (kl ml, k2 ~,lvlk3 m3, k4m3)Pk4~3,k2~2,
(10b)
k2ra2kam4 k2k4
Ak,,.,,k2m2 = ½ ~,
( k l m l , k z m 2 [ v [ k a m 3 , k , m4)xk4m4,k3m3,
(lla)
k3m3/g4/r/4
(ap)k,mt,k2m2=~m2~½ ~, (klml,k2~llv[k3rnz,k,~3)lgk,~3,k3ms.
(lib)
k3m3k4
If we restrict the nuclear wave functions to axial symmetry with the quantum number K = 0 of the total angular momentum upon the body-fixed symmetry axis, the pairing potentials d and Ap become identical. According to eqs. (9) and (10), Fp is negligible compared to F, since it is of the order Fp ~ ~2-tlV, if f2 is the number of different m-values. This is the common approximation, that the H F part of a pure pairing potential can be neglected against the H F part of the full potential s). The energy matrix of the CHB problem is now
W(2) =
t+F (1-2)d +
(1-2)A ] --(t+F)X] '
(12)
ANTIPAIRING
EFFECT
53
where t is the kinetic energy and the superscript T indicates the transpose of a matrix. Diagonalizing IV().) in a well-known way 8) one obtains the CHB equations
Z ((tik + r,k)A~, + (1 --)-)Aik Bi, } = E~ A,,, k
Z {(tik + F'k)*Bk, + (1 --)-)A ~ Ak,} = -- E, Bi,,
(13)
k
which are, according to our considerations the Euler-Lagrange equations of the variational principle, 6 (~bl/4-)-Vpl~b) = 0.
(14)
The most interesting cases of the H F equations for unpaired states (2 = 1) and of the HB equations for paired states (2 = 0) are included in these CHB equations. The intrinsic energy E(2) = @()-)l~ql@(2)> (~(01~()-)>
(15) '
is always in a minimum at the HB solution (2 = 0), but obviously this is not necessarily true for the projected energies (8). We would like to point out that 2 is not a free parameter used for some fitting procedure but a variational parameter used in accordance with a variational principle which is a better approximation to a complete PHB calculation than the normally used HBP method. This choice of a parameter by which the rotational energies are minimized, allows the BCS degree of freedom to go continuously from an unpaired state to a state which is even more paired than the HBP state. The pairing correlations in the rotational levels should be better described in this PHB method than in the HBP method previously used, and we expect to obtain a stronger decrease of the pairing energy with increasing angular momentum quantum number than presented in earlier calculations. Independently of the agreement with experiment, the approximate PHB procedure we are using is within the model a "better" method than HBP.
3. The nuclear interactions and the numerical procedure In our calculations the nuclear wave functions are restricted to time-reversal symmetry and axial symmetry with projection K = 0 of the total angular momentum onto the symmetry axis. We use the spherical harmonic oscillator wave functions and take into account the is, lp and 2sld shells without separating off any inert core. The Coulomb force and c.m. corrections are neglected. The proton and neutron numbers are fixed in an average way by introducing two Lagrange parameters into the variational procedure and requiring the number operators to have fixed values. Nevertheless, the spectra are weighted mixtures of the spectra of neighbouring nuclei since we do not use number projected HB states. However, all of our HB solutions
54
K. GOEKE et al.
show a particle number fluctuation of about 10~o, which is not enough to noticeably perturb the spectrum of the nucleus considered2°). Proton-proton and neutron-neutron pairing are considered since only in the case N = Z is protonneutron pairing important 12). As interactions for our calculations we use the effective G-matrix elements of the Hamada-Johnston potential calculated by Barrett et aL, the Yale-Shakin elements and the effective force B1 of Brink and Boeker. Barrett, Hewitt and McCarthy 27, 28) calculate the reaction matrix G in a twoparticle harmonic oscillator basis by solving the Bethe-Goldstone equation
G(CO) = V + V
Q
--/~0
G(co).
(16)
CO
The Pauli operator Q is diagonal in this basis: Q[ab) = ( 0 for a or b in Is or lp shell for a or b in 2sld shell otherwise,
(17)
and is, therefore, especially appropriate for microscopic calculations in the sd shell. From these matrix elements, which are given for five different values of the starting energy, co, we calculated by interpolation an effective interaction V~ff. To avoid a self-consistent treatment of the starting energy, we do not fix co but take reasonable values of the single-particle energies ea, which are taken as - 4 0 . 0 , - 3 2 . 0 , - 2 9 . 0 , -23.0, - 2 0 . 0 and - 1 7 . 0 MeV for the harmonic oscillator states and treat the starting energy in the following way: (abll/~fflcd) = ½{(ablG(ea+eb+2C)lcd)+(ablG(ec+ed+2C)lcd)}. (18) Here C is a parameter which shifts the e a so that the energy difference between e, and the energies of the intermediate states used in the Bethe-Goldstone equation takes reasonable values. In our calculations C is chosen to reproduce the binding energy of the nucleus considered within 20 MeV. It is not reasonable to fit C in a more accurate way due to the intrinsic limitations of the method, and since the rotational energies in which we are interested are very insensitive compared to the binding energy (see table 1). For 22Ne and 24Ne we take C = 50 MeV and for 3°Si C = 47 MeV. The lower value of C for 3°Si indicates, that the HB single-particle states are more bound relative to the intermediate states than those of 22Ne and 24Ne. TABLE 1 The ground state energies and the spectra for HBP calculations of 22Ne with Barrett force for different values of C (eq. (18)) C (MeV)
40
B E2 £'4 E~ Ea
96.46 0.96 3.05 6.10 10.21
(MeV) (MeV) (MeV) (MeV) (MeV)
50 163.57 1.10 3.44 6.76 11.03
60 243.01 1.16 3.60 6.99 11.05
Exp 177.77 1.27 3.34 6.35 11.15
ANTIPAIRING EFFECT
55
We also use the effective G-matrix elements of the realistic Yale potential given by Shakin, Waghmare and Hull 29,3o), and an oscillator length b = 1.76 fm. The H F P and HBP results show a little improvement compared to previous ones 2o), since the starting energy parameter, A, is chosen as A = 34.2 MeV instead of A = 20 MeV as described in ref. 31). We use the effective interaction B1 of Brink and Boeker 32) with the oscillator length b = 2.0 fm of ref. as).
4. Results We performed HFP, HBP and PHB calculations for the ground states of 22Ne, 24Ne and 3°Si and compared these results with experimental data 34-36). The ground TABLE 2 Comparison o f HFP, HBP and PI-IB calculations with experimental data (a) Results for 22Ne Exp
22Ne (Barrett) HFP
E2 /?4 E6 Es
02 04 06 0s
(MeV) (MeV) (MeV) (MeV) (MeV -1) (MeV -1) (MeV -1) (MeV -1)
E4/E2 EJE2 Es/E2 B (MeV)
HBP
Z2Ne(B1) PHB
22Ne (Yale)
HFP
HBP
PHB
HFP
1.27 0.69 1.10 1.27 0.45 3.34 2.42 3.44 3.23 1.43 6.35 5.58 6.76 6.25 2.89 11.15 1 0 . 8 1 1 1 . 0 3 10.74 4.68 2.36 4.29 2.73 2.37 6.63 3.38 4.04 2.99 3.56 7.14 3.65 3.49 3.31 3.63 7.53 3.13 2.86 3.51 3.34 8.38 2.63 3.47 3.13 2.55 3.16 5.01 7.97 6.15 4.94 6.39 8.78 15.44 10.04 8.48 10.36 177.77 162.95 163.57 163.75 114.06
0.87 2.47 5.33 8.59 3.43 4.38 3.85 4.60 2.82 6.10 9.83 117.19
0.87 2.44 4.24 5.81 3.43 4.46 6.11 9.55 2.79 4.86 6.65 117.19
0.41 1.50 3.61 7.26 7.29 6.42 5.21 4.11 3.64 8.78 17.64 103.67
BHP
PHB
0.81 0.88 2.66 2.39 5.35 4.50 8.23 7.92 3.68 3.41 3.78 4.64 4.09 5.21 5.21 4.39 3.26 2.71 6.55 5.11 10.57 9.01 104.49 104.56
(b) Results for Z4Ne and a°Si ~4Ne (Barrett)
E2 E4 E6 Ea 02 04
(MeV) (MeV) (MeV) (MeV) (MeV -1) (MeV -1)
E4/Ez B (MeV)
a°Si (Barrett)
Exp
HFP
HBP
PHB
Exp
HFP
HBP
PHB
1.99 3.98
0.85 2.72 5.46 9.26 3.52 3.76 3.19 179.83
1.61 4.90 9.26 14.48 1.86 2.12 3.05 181.12
1.61 4.00 6.75 10.53 1.86 2.92 2.49 181.12
2.23 5.27
0.85 2.85 6.04 10.23 3.53 3.50 3.36 249.08
1.51 5.02 10.49 17.92 1.99 1.99 3.33 253.78
1.62 5.07 10.00 15.08 1.85 2.03 3.13 253.92
1.51 3.52 2.00 191.83
1.34 2.30 2.36 255.63
The excitation energies, Ej, of the rotational states of the ground band, the ratios Ej/Ez of the rotational states and the moments of inertia 0j according to eq. (19) are listed; B is the binding energy of the 0 + ground state.
56
K. GOEKE et al.
states of 22Ne and 24Ne are prolate deformed, that of 3°Si, oblate. For 22Ne we present the results for all three forces, for 2 4 N e and 3°Si we give only the results with the G-matrix elements of the Hamada-Johnston potential as calculated by Barrett et al., since they give in all cases by far the best energy spectrum and binding energy. All HB solutions show non-negligible pairing correlations. In table 2 the excitation energies, the ratios of the excitation energies and the moments of inertia of the different levels of the ground state rotational band are listed; see also figs. 1-3. The moments of inertia are calculated from the projected or measured excitation energies Es: 2J- 1 Oj . (19) -
Ej
-
E s_ 2
-
In all cases the HFP spectrum is strongly compressed compared to the measured data. This is a very general result which is also found in all other HFP calculations with density- and momentum-independent forces 5, 6). The 2 + states in H F P has in general only one-third or less of the experimental energy. This is drastically changed by the inclusion of pairing correlations in the HBP method as the comparison of the moments of inertia of the H F solution and of the HB solution calculated with help of the Inglis model already indicates 11, 12). The spectrum is stretched by a factor of
1211-
8"
8+
10,81~
1103 "~,8"
8+
1115
625. S*
6.35
1.27
12'I
107t. / "
10-
t .¢
9876-
/
s°
/
"\',,6"
s.~//
54IJJ
I
/
3-
/ 4+
2.42/
21-
2" 2÷
1.10
2*
2~
0.70/./-
0 HFP
HBP
PHB
EXP
Fig. 1. Comparison o f the HFP, ~ B P and PHB spectra of 22Ne with experimental data. The calculations are performed with the Barrett force.
2'
4-
5"
6"
7-
8-
HFP
2"
6"
/ / s~4
/
HBP
\~g
161 _ 2*
\\
PHB
161~/"
~o
675
g
o f t h e I-[EP, I - [ B P a n d P H B
085/
/
/ /
272/
/
/
/
/*90
~*
\
spectra of 24N¢ with
EXP
199
3.98
experimental data. The calculations are performed with the Barrett force.
F i g . 2. C o m p a r i s o n
f
U
LU
2£ .c
>~
t
/ t I I I I I
5'
HFP
;t*
¢
~5//
/
i
/ / / / /
2*
I
I I I I I t I I I I I
HBP
151 ..:.
2*
502 _ t.*
\ \ e~
PHB
'L62J
507~
/
2"
/.
EXP
223
5.27
Fig. 3. Comparison o f the H F P , HBP and PHB spectra o f a°Si with experimental data. The calculations are performed with the Barrett force.
I 3-
U/
w
9-
10-
\ \
10.
9-
11-
11-
[6" - - 926
12-
12-
"11 ~n rn
Z
>. =.
> Z
58
K. GOEKE
et al.
about two and is in all HBP solutions in better agreement with experiment than in the HFP solutions. In 22Ne the HBP solution with the Barrett force is in very good agreement with experiment, which is reproduced within the limits of 0.4 MeV as far as the excitation energies up to the 8+ state are concerned (see fig. 1). In this solution the 2+ and 4+ states have an excitation energy which is a little bit larger than the experimental one whereas in all other HBP solutions the spectra are still compressed even if they are improved. A similar improvement by inclusion of the pairing correlations can be seen in the moments of inertia. The experimental data show increasing moment of inertia with increasing J, which is reproduced always in the HBP solutions but in general not in the HFP solutions I*, 20). In all cases the experimental deviations from the simple J(J+ 1) law are better reproduced in HBP than in HFP, as one can see by comparing the ratios of the experimental and theoretical excitation energies. From the above results and from previous calculations “3 20) it seems to be evident, that these nuclei cannot be described without including pairing correlations in the wave function. Nevertheless the HBP theory is not the final answer and the comparison with experimental data still shows disagreement in many cases. One step in improving this can be done by performing PHB calculations whose energy spectra in the CHB approximation we are doing are given in table 2. In all cases there is a clear difference between the HBP and PHB results. This is in contradiction to the general belief that projection before variation has negligible effect for sufficiently deformed nuclei, which one concludes by comparing HFP and PHF calculations 24). The difference between PHB and HBP is very strong and has values of up to 25% of the excitation energy. In 24Ne for examp le the excitation energy of the 4+ state changes from 4.90 to 4.00 MeV by performing the projection before the variation and shows afterwards very nice agreement with the experimental value. The result of the PHB calculation in “Ne with the Barrett force is very impressive: The 2+ state is exactly reproduced, the 4+ and the 6+ states show deviations of only 0.1 MeV from experiment, and the 8+ state deviates only 0.4 MeV at an excitation energy of 11.15 MeV. The moments of inertia are also reproduced quite well and show an improvement compared to HBP; even such details as the small decrease in the moment of inertia from 6+ to 8+ is given by the PHB solution of 22Ne with the Barrett force. To our knowledge this is the best reproduction of the experimental spectrum of 22Ne with a parameter-free and self-consistent model. The energy ratios, which are sensitive not to the absolute agreement of the theoretical levels with the experimental ones but to the character of the whole spectrum and the deviations form the J(J+ 1) law, are reproduced quite well and are remarkably better than the HFP and HBP values. Here lies a general improvement of all PHB solutions compared to HBP. In all cases considered here, the energy ratios are clearly better reproduced in PHB than in HBP; this is true for all forces used in our calculations. We feel that this is more important than the agreement of the absolute values of the calculated excitation energies with experiment, since it
ANTIPAIRING EFFECT
59
reproduces the character of the total spectrum and not only single levels. Let us consider for example the H F P solution of 2aNe with the Barrett force. The 2 + and 4 + states are in total disagreement with experiment, the 6 + the reproduces roughly the experimental value and the 8 + state shows only a deviation of 0.4 MeV. F r o m the agreement in the 6 + and 8 + states one nevertheless cannot conclude that the spectrum is roughly reproduced, since this agreement is accidental. The moment of inertia of this H F P solution is too large and the spectrum shows a decreasing moment of inertia; therefore there must be levels which are in rough agreement with experiment if we go high enough in excitation energy. F r o m this point of view, the trend in the moments of inertia and the agreement in energy ratios seem to us more important than the agreement of single levels with experiment, which is sometimes worse in PHB than in HBP. In a typical example the improvement of the energy ratios can clearly be seen in fig. 4, where the HFP, HBP and PHB spectra of ZZNe calculated with the Yale force are given. These spectra are artificially normalized in such a way that the 6 + state always assumes the experimental value of the excitation energy. Additionally one should point out that the excitation energies are strongly dependent on the effective two-body interaction used in the calculations, as the comparison of the results in table 2a shows. So one should not overestimate the agreement or disagreement of the theoretical excitation energies with experiment. One should be more interested in trends which are independent of the forces. The drastic changes from HBP to PHB, the improvement in the energy ratios and the strong antipairing effect discussed below are such reliable and force-independent results. In table 3 the projected pairing energies are listed. These are a measure of the TABLE 3
The projected pairing energies Aj in the rotational levels calculated with HBP and PHB using the Barrett force 24Ne
aos i
Barrett
Barrett
22Ne Barrett
Ao d2 A4 A6 As
(MeV) (MeV) (MeV) (MeV) (MeV)
Yale
B1
HBP
PHB
HBP
PHB
HBP
PHB
HBP
PHB
HBP
PHB
2.04 1.89 1.59 1.21 0.76
2.90 2.27 0.12 0.39 0.47
1.84 1.73 1.48 1.12 0.69
2.24 2.11 0.05 0.03 0.16
4.18 3.96 3.65 2.99 1.34
4.18 3.96 2.81 1.59 1.31
2.91 2.68 2.22 1.60 1.00
2.91 2.68 0.00 0.00 0.00
2.03 1.98 1.86 1.69 1.46
2.51 2.25 1.55 1.30 0.00
amount of pairing correlation and are defined as the contribution of the pairing correlation to the total projected binding energy. In all cases the low-lying states in P H B are at least as paired as in HBP, whereas the higher states are always less paired and mostly pure H F P solutions without pairing. The antipairing effect which is already indicated in previous H B P calculations 18,20) and for heavy nuclei in theoretical
13-
8+
12.76
12118"
~25
~
635
/.+
3,16
2+
Qg7
8*
1119
8+
~15
6+
635
6*
6.35
4+
337
L,~
33/*
2+
12,'.
2+
127
109-
8>= ~E
7-
._c
LU
-8_
6,35
5Z.-
LU
I 3-
d
26~
2:" 072
0
HFP
HBP
PHB
EXP
Fig. 4. Comparison of the scaled spectra of HFP, HBP and PHB solutions of 22Ne calculated with the Yale force with experiment. All excitation energies are multiplied by a factor to normalize the 6 + energy to the experimental value to demonstrate the improvement of the energy ratios.
- 9&O-
-99.0. I -IOQO-~ -10tO.G
T -IOGK)-
°4-.....
- t~.O-
-try0
-QI
a'05
a;
K,
o.;s
o'~
Q25
Fig. 5. The binding energies of the PHB solutions with the Yale force for 22Ne v e r s u s the minimization parameter 2. The arrows indicate the minima. The HBP solutions are given by xX= 0.
24Ne
2zNe
30Si
- - - HFP - - HBP - - PHB
t
15-
J
10-
/
/
.
Ej [MeVl
5-
//
/
/
oi
2b
h
h
6
~ --J
,3 (J÷ll
2b
h
22
¢2
=
Fig. 6. The binding energies o f the rotational levels v e r s u s J(J-t-1) for the H F P , HBP and PHB solutions o f 2ZNe, 24Ne and a°Si calculated with the Barrett force. The energies are normalized so that the 0 + value o f PHB is zero. 22Ne 4
~'"~'~
- - - HFP - - - - HBP
~
. . . . . EXP 3
~
2 l
,5
t,
ej
-E
.)
t.,v )
2 I5 "'Si 3
,
1
! 1
0
g - -
~2W j2 [MeV]
lb
Fig: 7. The moments o f inertia o f 22Ne, =4He and a°Si in dependencc on (oj , see eqs. 0 9 ) and (20). The results are given f o r H F P , H B P and P H B , calculated w i t h the Barrett force, and f o r experimental values.
62
K. GOEKE et al.
considerations using Migdal theory 37) and experimental measurements as, ag) is, according to our P H B calculations, clearly stronger than that expected from HBP. The reason for this can be explained in the following way. The unprojected HB solution is in general about 1 MeV more bound than the H F solution, but its m o m e n t of inertia is about one-half of that obtained from the H F wave function 12. z o). Therefore the H B P spectrum is expanded compared to HFP, and sometimes in higher rotational states the H F P level is more bound due to its moment of inertia which is larger than that of the H B P level. F o r this reason the nucleus "prefers" to change for this level to an unpaired intrinsic state. This behaviour can clearly be seen in fig. 6, where the binding energies of the rotational levels are plotted as a function of J ( J + 1). (The binding energy of the 0 ÷ state of P H B is normalized tO zero.) In ZaNe the gain of binding energy due to pairing correlations is very small, so only the 0 + and 2 + states are more deeply bound with the inclusion of pairing correlations. Going from the 2 + to the 4 + state is a transition to an unpaired state, and all higher-J states are also unpaired. In a°Si this transition occurs at 6 +, since the gain in binding energy due to pairing is larger. The same effect is to be seen in 22Ne, even if it is not so clear, as discussed below. In fig. 7 the moments of inertia, eq. (32), are plotted against the angular frequency, ogj, which is defined by 1
ogs = - - ~ / J ( J + 1). Os
(20)
In heavy nuclei such curves sometimes show a " b a c k w a r d " or forward bending; here a similar effect can be seen, which cannot be as clearly defined due to the small number of rotational levels in light nuclei. In 22Ne and 24Ne the PHB curve shows the forward bending, since the s-curve starts already at the 2 + state; in 3°Si one sees the backward bending. Due to numerical inaccuracies it was not worthwhile to calculate the 10 + state in these nuclei. Our calculations indicate that phenomena such as phase transitions, and backward and forward bending, which are established in the highspin states of heavy nuclei can also occur with similar properties in the low-spin states o f light nuclei. This is in agreement with the estimates of Sheline 41). This change from a paired state to an unpaired one cannot be reproduced in a HBP calculation in which we project out of one correlated state and can only approximately reproduce the decrease of pairing energy with increasing moment of inertia. F r o m this we see that the variational parameter 2, used to simulate a correct procedure of projection before variation, is the correct one to reproduce this effect. The values of 2 for the different minima of the projected solutions of one example can be obtained from fig. 5. They are a measure of how far the intrinsic wave functions differ from the different rotational levels. For the values see table 4. In contradiction to the above considerations for few solutions the projected pairing energies show very little increase from the 6 + to the 8 + state and parallel to this the cranking moments of inertia of the intrinsic state show this behaviour. We have no explanation for this
ANTIPAIRING EFFECT
63
TABLE4 The intrinsic cranking moments of inertia 0l"tr- and the values of the parameter 2 for the PHB solutions calculated with the Barrett force J 0ol"tr- (MeV -1) 0z turf" (MeV -1) 04t*t'- (MeV -1) 06 Int'' (MeV -1) 0s IntT" (MeV -1) 0HFlntr' (MeV - t ) 0HBlair" (MeV -x)
22Ne (Yale)
Z2Ne (Barrett)
1.308 (--0.025) 1.308 (--0.025) 2.818 (+0.150) 2.818 (+0.150) 2.319 (+0.100) 2.899 1.458
0.920 (--0.05) 1.015 (--0.025) 2.211 (+0.150) 1.736 (+0.100) 1.386 (+0.05) 2.392 1.124
Z4Ne (Barrett) 1.133 1.133 1.771 1.771 1.771 1.771 1.133
(--0.00) (--0.00) (+0.25) (+0.25) (+0.25)
a°Si (Barrett) 1.004 (--0.0075) 1.030 (--0.0050) 1.156 (+0.1100) 1.219 (+0.2200) 2.174 (+0.3000) 2.174 1.077
but it might be explained in the following way. For these states the energy surface Es (;t) is a very flat valley near the minimum especially in the direction towards to the
H F solution, while the pairing correlations however are still remarkably dependent on 2. For such a system one should not look for the minimum of the energy surface E s (2) but should superimpose all solutions with the help of the generator-coordinate method. It is possible that for such a solution the pairing correlations will show the expected behaviour. But, anyway, even in our states the pairing energies of these states are clearly smaller than in states with low spin. The increase of the moments of inertia due to the decrease of the pairing correlations going to higher-J values can also be seen in table 4, where the moments of inertia of the intrinsic states calculated with the cranking formula are listed. Except for the solutions we discussed in the preceding paragraph the results show a clear increase of the intrinsic moment of inertia with increasing angular momentum quantum number. This shows also, to what extent the intrinsic states differ from each other. In all cases the phase transitions can be recognized by a sudden change of the intrinsic moments of inertia by a factor of about two. Table 5 lists the projected intrinsic mass quadrupole moments Q~, which are obtained from the projected spectroscopic one Q (J) by the prescription of the rotational model: 2J+3 Q, = - - Q(J). (21) J The HBP values indicate antistretching, that means a decrease o f the intrinsic Qs with increasing J, whereas the PHB values show an increase of Qs in the phase transition due to the fact that the H F solution is more strongly deformed 42). Comparison of the results of 22Ne and of the other nuclei, which are not presented here, show that the effective G-matrix elements of Barrett et aL are very appropriate for calculations in the sd shell. For 22Ne the overall agreement is quite impressive and there is obviously no need for K-mixing to explain the ground state rotational band. For 24Ne we obtain results which are as good as the ones given by Khadkikar et al. 39). These authors describe the spectrum of Z4Ne by mixing oblate and prolate
64
K. GOEKE et aL
TAat~ 5 The projected intrinsic mass quadrupole moments calculated from the spectroscopic ones with the help of the rotational model for HBP and PHB using the Barrett force 22Ne
Q2 (e2" fm2) Q4 (e2 " fm2) Q6 (e~" fm2)
Z4Ne
aosi
HBP
PHB
HBP
PHB
HBP
PHB
62.81 61.10 57.76
61.55 60.24 61.40
60.25 58.12 54.87
60.35 65.75 64.59
-74.21 -74.52 -73.95
-73.80 -74.57 -73.47
H F P solutions with p-h excitations neglecting partially the strong effect of the pairing correlations, which are shown to be important. According to our calculations, shape mixing is not needed to obtain such agreement. It could be, however, that shapemixed, correlated H B P or P H B solutions would yield even better results. In 3°Si, P H B yields an essential improvement compared to HBP; the spectrum is a little bit better b u t still clearly compressed compared to experimental values. 5. Conclusion
In conclusion we would like to point out that the pairing correlations of sd shell nuclei are strongly dependent upon the rotational levels and, in some cases, do not already exist in the 4 + state. This requires, in the H B model, an angular m o m e n t u m projection procedure before variation (PHB) which is necessary even for strongly deformed nuclei, in contradiction to the uncorrelated projected H F model. By approximate HB variation after projection, in a constrained H B model (CHB) the excitation energies change by up to 3 0 ~ compared to projection after variation (HBP) and yield in 22Ne quite impressive agreement with experiment if one uses the effective G-matrix elements of Barrett et aL, which give in all cases the best results. In 24Ne and 3°Si the calculations indicate a phase transition into the unpaired state at d = 4 + and 8 + respectively. The CHB theory seems to be a very appropriate tool for the study of ground state rotational bands and of the effect of pairing correlations on them, especially for phase transitions into the unpaired state. We thank Dr. L. Satpathy for valuable discussions about the formulation of the constrained H B theory. We thank Dr. H. H. Wolter for the use of the HB p r o g r a m for the intrinsic wave functions. We are greatly indebted to Dr. B. R. Barrett for giving us his effective G-matrix elements. References
I) I. Kelson, Phys. Rev. 132 (1963) 2189 2) I. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269 3) G. Ripka, Advances of Nuclear Physics, vol. 1 ed. M. Baranger and E. Vogt (Plenum Press, New York, 1968) 4) R. E. Peierls and J. Yoccoz, Proe. Phys. Soc. A70 (1957) 381 5) H. A. Lamme and E. Boeker, Nucl. Phys. Alll (1968)492
A N T I P A I R I N G EFFECT
65
6) S. K. M. Wong, J. LeTourneux, N. Quang Hoc and G. Saurnier, Nucl. Phys. A137 (1969) 318 7) N. N. Bogoliubov, Usp. Fiz. Nauk 67 (1959) 236 [transl.: Sov. Phys. Usp. 2 (1959) 236] 8) M. Baranger, in 1962 Cargese lectures in theoretical physics, ed. M. Levy (Benjamin, New York~ 1963) 9) K. Dietrich, H. J. Mang and J. Pradal, Z. Phys. 190 (1966) 357 I0) A. Faessler, P. U. Sauer and M. M. Stingl, Z. Phys. 212 (1968) 1 11) L. Satpathy, D. Ooss and M. K. Banerjee, Phys. Rev. 183 (1969) 887 12) H. H. Wolter, A. Faessler, P. U. Sauer, Nucl. Phys. A167 (1971) 108 13) N. Onishi, Prog. Theor. Phys. Jap. 40 (1968) 84 14) N. Onishi and S. Yoshida, Nucl. Phys. 80 (1966) 367 15) M. R. Gunye and S. B. Khadkikar, Phys. Rev. Lett. 24 (1970) 910 16) K. R. Sandhya Devi, S. B. Khadkikar, J. K. Parikh and B. Banerjee, Phys. Lett. 32B (1970) 179 17) A. Faessler, K. Goeke and L. Satpathy, Phys. Lett. 35B (1971) 193 18) K. Goeke and A. Faessler, Phys. Lett. 35B (1971) 289 19) L. Satpathy, K. Goeke and A. Faessler, Nucl. Phys. A179 (1972) 177 20) K. Goeke, A. Faessler and 14. H. Wolter, Nucl. Phys. A183 (1972) 352 21) J. K. Parikh, Phys. Rev. C5 (1972) 153 22) H. Rouhaninejad and J. Yoccoz, Nucl. Phys. 78 (1966) 353 23) H. D. Zeh, Z. Phys. 188 (1965) 361 24) L. Satpathy and S. C. K. Nair, Phys. Lett. 26B (1968) 257 25) R. Beck, H. J. Mang and P. Ring, Z. Phys. 231 (1970) 26 26) B. Castel and J. C. Parikh, Phys. Rev. C1 (1970) 990; J. C. Parikh and B. Castel, Phys. Lett. 3713 (1971) 7 27) B. R. Barrett, R. G. L. 14ewitt and R. J. McCarthy, Phys. Rev. {23 (1971) 1137 28) B. R. Barrett, R. G. L. 14ewitt and R. J. McCarthy, Phys. Rev. C2 (1970) 1199 29) C. M. Shakin, Y. R. Waghmare and M. H. Hull, Phys. Rev. 161 (1967) 1006 30) C. M. Shakin, Y. R. Waghmare and M. H. Hull, Phys. Rev. 161 (1967) 1015 31) 14. H. Wolter, thesis 1970, Univ. of Muenster, unpublished 32) D. M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1 33) P. U. Sauer, A. Faessler, H. H. Wolter and M. M. Stingl, Nucl. Phys. A125 (1969) 257 34) P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) l 35) A. J. Howard, R. G. Hirko, D. A. Bromley, K. Bethge and J W. Olness, Phys. Rev. C1 (1970) 1446 36) W. G. Davies, C. Broude and J. S. Forster, Bull. Am. Phys. Soc. 15 (1970) 543 37) J. Meyer and J. Speth, preprint, 1972 38) R. Baader, H. Backe, R. Engler, K. Hesse, E. Kankeleit, V. Schr6der, H. K. Walther and K. Wien, Phys. Lett. 2713 (1968) 425 39) H. K. Walther, H. Backe, R. Engfer, E. Kankeleit, C. Petitjeen, H. Schneuwly and W. V. SchrOder, Phys. Lett., to be published 40) S. B. Khadkikar, S. C. K. Nair and S. P. Pandya, Phys. Lett. 3613 (1971) 290 41) R. K. Sheline, private communication 42) A. Faessler, K. Goeke and. H. MOther, Contribution to the Conf. on High spin states, Stockholm, June 1972, to be published in Physica Scripta