- Email: [email protected]
Level spectra of Ni-isotopes with realistic interaction
Volume 26B. number 12
LEVEL
SPECTRA
PHYSICS
OF
Ni-ISOTOPES
LETTERS
WITH
13 May 1968
REALISTIC
INTERACTION
Y. K. GAMBHIR International
Atomic
Energ~~ Agency
) International Trieste,
Centve
,foy Theoretical
Physics
j
Italy
Received 10 April 1968
Quasiparticle configuration-mixing calculations are made for the Ni-isotopes aith an approximate reaction matrix of Hamada-Johnston potential reported by Kuo. The results are very close to those obtained by the exact shell-model calculations. But the agreement with the experimental spectra is poor in both these cases. A remarkable improvement is achieved in the quasiparticle results when the renormalized force strength and the quasiparticle quantities extracted by the inverse gap equation procedure are used. A possible further
improvement
in the inverse
gap equation procedure
The quasiparticle description in terms of modified Tamm-Dancoff approximation (MTDA) or QSTD [l] for the low-lying excited states of spherical nuclei is quite successful. Since this description is only an approximate method for solving the configuration-mixing problem when a large number of configurations are available to a fairly large number of nucleons, the results given by such calculations can at best be a ‘caricature’ of the exact configuration-mixing results. Such approximate methods are indeed essential because the exact shell-model calculations are not practicable for most of the nuclei. However, the comparison of the exact shell-model results with the corresponding quasiparticle results measures the goodness of the approximations involved in the quasiparticle MTDA theory. Recently, Lawson et al. [2] reported the results of the exact shell-model configuration-mixing calculation for all the Ni-isotopes using the phenomenological effective two-body matrix elements of Cohen et al. (EIC) [3] and the reaction matrix of the Hamada-Johnston [4] potential corrected for core polarization calculated by Kuo [5]. The first set of results are very close to the corresponding quasiparticle configuration-mixing (MTDA) results and also reproduce equally well the observed spectra. This clearly establishes the correctness of the quasiparticle MTDA theory. The results quoted in ref. 2 for the reaction matrix of the Hamada-Johnston potential have very poor agreement with the corresponding experimental numbers. This fact did not encourage us to do the MTDA calculations earlier using the Hamada-Johnston reaction matrix elements. In all the quasiparticle calculations the occu-
is discussed.
pation probabilities (V2) and the quasiparticle energies (E) of the various single-particle shellmodel states go as input data. In the conventional (BCS) procedure these quantities are determined by feeding in the single-particle or the HartreeFock energies (E) and the pairing matrix elements, i.e., the matrix elements (G) in J = 0 state, through the pairing model or the BCS equations. Usually a fixed potential is assumed with a variable strength (V,) for calculating G. The Hartree-Fock energies (E) and the force strength (V,) are frequently treated as parameters and are adjusted so that the odd-even mass difference is reproduced and the quasiparticle energies closely resemble the first few observed states of the corresponding odd-mass nuclei. Unfortunately, such fits, which of course require a long chi-square (x2) search, are never very successful and unique. Therefore, the use of these quasiparticle quantities obtained by the conventional BCS procedure in the interacting quasiparticle (MTDA) calculations induces the ambiguities connected with these parameters. Recently, Gillet and Rho [6] proposed the inverse gap equation method for determining these quasiparticle quantities. The method assumes the first few observed levels with assigned spin and parity of single-closed-shell odd-nuclei as pure one-quasiparticle excitations. This experimental information together with the odd-even mass difference determined from the observed binding energies (BE) yield single-quasiparticle spectrum. Once the quasiparticle energies (E) are known, the BCS gap equation reduces to a real non-symmetric eigenvalue problem. Its maximum positive eigenvalue gives the inverse 695
Volume 26B. num>er 12
PHYSICS
of the force strength (V,) of the assumed interaction and the positive components (ha) of the corresponding eigenvector are related to the respective physical gaps (ha) through an overall positive real constant 5, which must satisfy 5 s Ea/Aa
,
(1)
where Ea is the energy of the lowest quasiparticle state a. This normalization constant 5 is determined through the BCS number equation usin B Sgn, the sign function of the argument (+ - Va) which is inferred from one-nucleon transfer reactions. For the values of 5 satisfying (1) the BCS number equation is usually not satisfied, yielding the finite value of the fluctuation (6%) in particle number (n). In practice, t; is determined so that 6n becomes minimum. A large value of 6n either indicates the weakness of the assumption of pure one-quasiparticle excitations or reflects the wrong feeding of the input information. The knowledge of 5 yields the occupation probabilities (V2) and consequently the singleparticle shell model energies of the Hartree-Fock energies can be determined relative to a reference state 1. Since the starting hypothesis of pure singlequasiparticle excitations is never strictly true, the extracted quantities never show their expected theoretical behaviour. The departure from this behaviour may be looked upon as a test of the independent or BCS model, if the fed input data are correct. Our MTDA results [7] of odd Ni-isotopes obtained by using EIC interaction, show that the three-quasiparticle admixtures in the first new excited states are quite small, the maximum being 90/oexcept for $- state of 5gNi and 61Ni where it amounts to Y 19%. Thus the first few excited states of odd Ni-isotopes are fairly well described in terms of pure onequasiparticle excitations, which fairly justifies the basic assumption of the inverse gap equation method in Ni-region. The slight departures of the force strength from unity and tolerably smooth variation of the Hartree-Fock energies with mass number obtained in the framework of inverse gap equation formalism, using EIC interaction, fairly justifies [8] the validity of the independent quasiparticle picture in the Ni-region. The use of the so-called experimental quasiparticle energies (exact), the renormalized force strength and the occupation probabilities obtained by the inverse gap equation method with the approximate quasiparticle (BCS) wave functions in the interacting quasiparticle calculations 0 The procedure 696
is described
in detail in [S].
LETTERS
13 Mav 1968
for single closed-shell even nuclei, directly links the theoretically obtained spectra of even nucleus with the observed data of the corresponding odd-mass nucleus. Moreover, the results of such calculations are free from any adjustable parameters. This procedure is analogous to that adopted for doubly closed shell nuclei. There one uses the observed (exact) particle-hole energies of the corresponding odd-mass nucleus with the approximate shell-model wave functions of the particle-hole type. The success already achieved there tempted us to carry out quasiparticle calculations for single closed shell even nuclei on similar lines. The quasiparticle quantities and the renormalized force strength are obtained for odd Ni-isotopes by the inverse gap equation method using the following input information: i) The shell-model interaction matrix elements are those of the reaction matrix deduced from the Hamada-Johnston potential with different core polarization corrections. Two sets of these matrix elements designated by the symbols B and C are used; they are tabulated and are also marked B and C in [2]. ii) The odd-even mass difference obtained by using the observed binding energies tabulated in [9]. iii) The experimental energies for 2p+, 2~4 and If + quoted in [2]. iv) The corresponding values of the function Sgn determined from the values of V2 given in
[lOI.
The extracted V, ranges from 0.9435 to 1.0199 for B and from 0.7759 to 0.8126 for C, thus showing right over-all pairing nature for the interaction B, but stronger nature for C. This observation agrees with the conclusions of ref. 2. The values of 6n are appreciable in some cases (63Ni) for B, but are very small (~0.3) for the interaction C. This point is discussed at length in [8]. The extracted V, and the HartreeFock energies of odd Ni-nuclei are used to obtain the relevant quasiparticle quantities for even Ni-isotopes. Quasiparticle configuration mixing calculations are made for both odd and even Ni-isotopes. A set of non-redundant orthonormal quasiparticle basis states is constructed following the procedure of [4] for the diagonalization work. The spurious states arising from the non-conservation of the particle number by the quasiparticle transformation are also removed following the prescription of [?,ll]. Two sets of quasiparticle quantities are used in the calculation. The first set corresponds to the quasiparticle quantities
Volume
26B,
Level spectra title state
number
12
PHYSICS
LETTERS
1968
Table 1 of Ni-isotopes obtained by various methods described in the text. Here + indicates the four quasiparwhile the symbols * and ** show one quasiparticle and three quasiparticle states. respectively. Excitation
Mass No.
13 May
J”
energy
in MeV
Kuo C
Kuo B
Expt.
+
01’ 21’ 4;
Exact
BCS
IGE
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2.18
2.09
1.83
1.65
1.66
1.67
1.45
3.28
3.12
2.39
2.73
2.68
2.27
2.46
3.42
3.32
2.53
3.17
3.13
2.56
3.08
2.92
2.50
2.54
2.44
2.42
3.47
2.83
3.00
2.78
Exact
i
BCS
IGE
58 62’ 22’ 28
2.78 2.90
--t
3/2i
0.00
0.00
0.00
0.00
0.00
0.00
5/2;
1.27
1 .oo
0.34
1.21
1.01
0.34
0.34
1/q
0.92
0.74
0.47
0.85
1.53
0.47
0.47
3/2;
2.02
1.93
1.31
1.86
1.72
1.23
0.89
l/22
2.56
2.15
1.24
2.36
2.07
1.16
1.32
2.10
1.50
1.72
1.38
0.00
59
5/2.j
T
60
0+ 1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2;
2.45
2.28
1.84
2.02
1.97
1.73
1.33
3.21
2.94
2.49
2.91
2.46
2.53
2.16
0‘;
2.89
2.67+
2.37+
2.72
2.18
2.33
2.29
4;
3.42
3.11
2.32
2.98
2.75
2.21
2.50
3/2;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
5/2;
0.51
0.19
0.07
0.13
- 0.48
0.07
0.07
1/z;
- 0.52
- 0.25
0.28
- 0.96
- 1.10
0.28
0.28
2.02
1.80
i.24
1.46
1.15
1.25
0.91
3/Z;
1.61
0.91
0.74
1.02
l/22
1.53
0.98
1.23
0.97
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2 .a4
2.01
1.78
1.95
1.30
1.37
1.17
2.81
2.66
2.39
2.54
1.89
2.19
2.05
3.93
3.34
2.70
3.58
2.47
2.40’
4.08
3.41
2.51
3.70
2.72
2.52
2;
61 5/2.j
01 2; 62
0+ 2 2; 4;
I-
2.30 2.34
697
PHYSICS
Volume 26B, number 12
LETTERS
13 May 1968
Table 1 continued Excitation energy in MeV
Mass No.
Kuo C Exact
BCS
Kuo B IGE
BCS
Exact
Expt. IGE
-
63
64
65
l/2:
0.00
0.00
0.00
0.00
0.00
0.00
0.00
5/2i
- 0.05
0.18
0.09
-0.59
- 0.15
0.09
0.09
3/2i
1.07
0.84**
0.16
0.62
0.11**
0.16
0.16
3/22
1.41
1.20*
0.71
1.29
1.67**
0.82
0.53
14
2.87
2.46
1.02
2.37
2.42
0.87
1.01
5/2,
1.96
1.66
1.17
1.34
0.99
1.16
0;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2;
2.26
1.78
1.52
1.39
1.08
1.34
1.34
0;
3.79
3.08
2.29
3.49
2.75
2.47
2.28
41
3.31
2.93
2.47
2.31
1.98
2.46
2.62
2+ 2
3.32
2.85”
2.31+
2.59
1.91'
2.20+
2.89
5/Z;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
l/ZI
0.56
0.28
0.06
0.96
0.71
0.06
0.06
3/2i
0.84
0.57**
0.32
0.33
0.07**
0.32
0.32
3/22
2.16
1.95*
0.74
2.41
1.85**
0.45
0.70
5/22
2.68
1.96
1.41
2.36
1.72
1.08
3.34
1.35
3.39
0.90
l/22
obtained by the conventional BCS procedure, while the second represents the inverse gap equation set. The corresponding results are marked by the symbols BCS and IGE, respectively, in the table. The results presented here are without the mixing between different quasiparticle subspaces. Therefore, these results correspond to the diagonalization in three-quasiparticle space for odd isotopes and in two- and fourquasiparticle spaces separately for even isotopes. The mixing between different quasiparticle subspaces does improve the results; this is discussed in [8]. The results of exact shell-model calculations marked “Exact” are also presented in the same table for comparison. It is clear from the table that the numbers in the columns “Exact” and “BCS” are quite close, but differ appreciably from the corresponding experimental values. This is expected for the reason given earlier. The BCS results do not even reproduce the right 698
t
quasiparticle character for f- states of 63Ni and 65Ni nuclei. These levels are predicted as threequasiparticle states marked by ** in the table. The numbers in the column IGE have much better agreement with the experimental values and also re reduce the correct nature of the i- states of 6s.Nl and 65Ni nuclei. The results of both B and C interactions show the same qualitative features. However, the results of the interaction C are slightly better than those of B. Thus a remarkable improvement is achieved in the quasiparticle results when the quasiparticle quantities and the force strength extracted by the inverse gap equation method are used. The usefulness of the inverse gap equation method as such entirely depends, on one hand, on the degree of applicability of the independent quasiparticle or BCS model and, on the other hand, on the correctness of fed input information; in particular, on the realistic two-body shell-
Volume 26B.
number
12
____
PHYSICS
model interaction. It is well established that the latter model is very approximate in many nuclear regions where the contributions from the residual quasiparticle interaction are significant. In view of this fact, the following improved or secondstep procedure is suggested [8]. One now starts repeating the whole calculation by feeding in the modified single-quasiparticle energies Ed given by
E; = Ea + “Ea ,
(2)
where 6Ea is the contribution from the residual quasiparticle interaction calculated by the secondorder perturbation theory in which the obtained values of V2, Vo, etc., are used. The Sgn function should also be determined through these V2. It is expected that new 6n would be less than that obtained previously. The calculations are repeated again with the modified El, obtained by using new values of V2, Vo, etc., by the perturbation method. This procedure is continued until the correction 6E, reduces (- 0) considerably. In practice, it is enough to stop even after two iterations. This follows from the smallness of 6E, which can be checked by doing the MTDA calculations using the final set of extracted quantities. The correction 6E is very small (-‘4%) for both the interactions except for the f- states of 5gNi and 61Ni nuclei in the case of B interaction. The corrections to V, are also very small (-” 2%) for both the interactions. Therefore, the results reported here will not change appreciably with the use of quasiparticle quantities obtained through the improved inverse gap equation procedure. However, considerable improvement is obtained for the other interactions and is fully discussed in ref. 8.
LETTERS
13 May 1968
The author is grateful to Professors M. K. Pal, J.Sawicki and V. Gillet and Drs. S. C. K. Nair, M. Rho and Ram Raj for valuable comments and continuous interest in the work. The author is very grateful to the Saha Institute of Nuclear Physics, Calcutta, India, where most of the work reported here was completed, for providing the facilities and financial assistance. He also wishes to thank Professors Abdus Salam and P. Budini as well as the IAEA for kind hospitality at the International Centre for Theoretical Physics, Trieste.
References 1. M. K. Pal. Y. K.Gambhir and Ram Raj. Phys. Rev. 155 (1967) 1144; I’. L. Ottaviani, M. Savoia. J. Sav,icki and A. Tomasini, Phys. Rev, 153 (1967) 1138. 2. R. D. Lawson, M. H. MacFarlane and T. T. S. Kuo. Phys. Letters 22 (1966) 168. S. P. 3. S. Cohen, R.D. Lawson, M. H. MacFarlane. Pandya and M.Soga, Phys. Rev. 160 (1967) 903. 4. T. Hamada and J. D. Johnston, Nucl. Phys. 34 (1962) 382. Nucl. Phys. A90 (1967) 199. 5. T.T.S.Kuo, 6. V.Gillet and M.Rho. Phys. Letters 21 (1966) 82. Ram Raj and M. K. Pal. Phys. Rev. 7. Y.K.Gambhir, 162 (1967) 1139. to appear as a preprint. 8. Y. K.Gambhir, 9. J. H. E. Mattauch. W. Thiele and A. H. R’apstra, Nucl. Phys. 67 (1965) 1. Phys. Rev. 10. R. H. Fulmer and W.W.Dackmick, B139 (1965) 579. and M.K. Pal. Phys. Rev. 11. Ram Raj, Y.K.Gambhir 163 (1967) 1004.
*****
699
LEVEL
SPECTRA
PHYSICS
OF
Ni-ISOTOPES
LETTERS
WITH
13 May 1968
REALISTIC
INTERACTION
Y. K. GAMBHIR International
Atomic
Energ~~ Agency
) International Trieste,
Centve
,foy Theoretical
Physics
j
Italy
Received 10 April 1968
Quasiparticle configuration-mixing calculations are made for the Ni-isotopes aith an approximate reaction matrix of Hamada-Johnston potential reported by Kuo. The results are very close to those obtained by the exact shell-model calculations. But the agreement with the experimental spectra is poor in both these cases. A remarkable improvement is achieved in the quasiparticle results when the renormalized force strength and the quasiparticle quantities extracted by the inverse gap equation procedure are used. A possible further
improvement
in the inverse
gap equation procedure
The quasiparticle description in terms of modified Tamm-Dancoff approximation (MTDA) or QSTD [l] for the low-lying excited states of spherical nuclei is quite successful. Since this description is only an approximate method for solving the configuration-mixing problem when a large number of configurations are available to a fairly large number of nucleons, the results given by such calculations can at best be a ‘caricature’ of the exact configuration-mixing results. Such approximate methods are indeed essential because the exact shell-model calculations are not practicable for most of the nuclei. However, the comparison of the exact shell-model results with the corresponding quasiparticle results measures the goodness of the approximations involved in the quasiparticle MTDA theory. Recently, Lawson et al. [2] reported the results of the exact shell-model configuration-mixing calculation for all the Ni-isotopes using the phenomenological effective two-body matrix elements of Cohen et al. (EIC) [3] and the reaction matrix of the Hamada-Johnston [4] potential corrected for core polarization calculated by Kuo [5]. The first set of results are very close to the corresponding quasiparticle configuration-mixing (MTDA) results and also reproduce equally well the observed spectra. This clearly establishes the correctness of the quasiparticle MTDA theory. The results quoted in ref. 2 for the reaction matrix of the Hamada-Johnston potential have very poor agreement with the corresponding experimental numbers. This fact did not encourage us to do the MTDA calculations earlier using the Hamada-Johnston reaction matrix elements. In all the quasiparticle calculations the occu-
is discussed.
pation probabilities (V2) and the quasiparticle energies (E) of the various single-particle shellmodel states go as input data. In the conventional (BCS) procedure these quantities are determined by feeding in the single-particle or the HartreeFock energies (E) and the pairing matrix elements, i.e., the matrix elements (G) in J = 0 state, through the pairing model or the BCS equations. Usually a fixed potential is assumed with a variable strength (V,) for calculating G. The Hartree-Fock energies (E) and the force strength (V,) are frequently treated as parameters and are adjusted so that the odd-even mass difference is reproduced and the quasiparticle energies closely resemble the first few observed states of the corresponding odd-mass nuclei. Unfortunately, such fits, which of course require a long chi-square (x2) search, are never very successful and unique. Therefore, the use of these quasiparticle quantities obtained by the conventional BCS procedure in the interacting quasiparticle (MTDA) calculations induces the ambiguities connected with these parameters. Recently, Gillet and Rho [6] proposed the inverse gap equation method for determining these quasiparticle quantities. The method assumes the first few observed levels with assigned spin and parity of single-closed-shell odd-nuclei as pure one-quasiparticle excitations. This experimental information together with the odd-even mass difference determined from the observed binding energies (BE) yield single-quasiparticle spectrum. Once the quasiparticle energies (E) are known, the BCS gap equation reduces to a real non-symmetric eigenvalue problem. Its maximum positive eigenvalue gives the inverse 695
Volume 26B. num>er 12
PHYSICS
of the force strength (V,) of the assumed interaction and the positive components (ha) of the corresponding eigenvector are related to the respective physical gaps (ha) through an overall positive real constant 5, which must satisfy 5 s Ea/Aa
,
(1)
where Ea is the energy of the lowest quasiparticle state a. This normalization constant 5 is determined through the BCS number equation usin B Sgn, the sign function of the argument (+ - Va) which is inferred from one-nucleon transfer reactions. For the values of 5 satisfying (1) the BCS number equation is usually not satisfied, yielding the finite value of the fluctuation (6%) in particle number (n). In practice, t; is determined so that 6n becomes minimum. A large value of 6n either indicates the weakness of the assumption of pure one-quasiparticle excitations or reflects the wrong feeding of the input information. The knowledge of 5 yields the occupation probabilities (V2) and consequently the singleparticle shell model energies of the Hartree-Fock energies can be determined relative to a reference state 1. Since the starting hypothesis of pure singlequasiparticle excitations is never strictly true, the extracted quantities never show their expected theoretical behaviour. The departure from this behaviour may be looked upon as a test of the independent or BCS model, if the fed input data are correct. Our MTDA results [7] of odd Ni-isotopes obtained by using EIC interaction, show that the three-quasiparticle admixtures in the first new excited states are quite small, the maximum being 90/oexcept for $- state of 5gNi and 61Ni where it amounts to Y 19%. Thus the first few excited states of odd Ni-isotopes are fairly well described in terms of pure onequasiparticle excitations, which fairly justifies the basic assumption of the inverse gap equation method in Ni-region. The slight departures of the force strength from unity and tolerably smooth variation of the Hartree-Fock energies with mass number obtained in the framework of inverse gap equation formalism, using EIC interaction, fairly justifies [8] the validity of the independent quasiparticle picture in the Ni-region. The use of the so-called experimental quasiparticle energies (exact), the renormalized force strength and the occupation probabilities obtained by the inverse gap equation method with the approximate quasiparticle (BCS) wave functions in the interacting quasiparticle calculations 0 The procedure 696
is described
in detail in [S].
LETTERS
13 Mav 1968
for single closed-shell even nuclei, directly links the theoretically obtained spectra of even nucleus with the observed data of the corresponding odd-mass nucleus. Moreover, the results of such calculations are free from any adjustable parameters. This procedure is analogous to that adopted for doubly closed shell nuclei. There one uses the observed (exact) particle-hole energies of the corresponding odd-mass nucleus with the approximate shell-model wave functions of the particle-hole type. The success already achieved there tempted us to carry out quasiparticle calculations for single closed shell even nuclei on similar lines. The quasiparticle quantities and the renormalized force strength are obtained for odd Ni-isotopes by the inverse gap equation method using the following input information: i) The shell-model interaction matrix elements are those of the reaction matrix deduced from the Hamada-Johnston potential with different core polarization corrections. Two sets of these matrix elements designated by the symbols B and C are used; they are tabulated and are also marked B and C in [2]. ii) The odd-even mass difference obtained by using the observed binding energies tabulated in [9]. iii) The experimental energies for 2p+, 2~4 and If + quoted in [2]. iv) The corresponding values of the function Sgn determined from the values of V2 given in
[lOI.
The extracted V, ranges from 0.9435 to 1.0199 for B and from 0.7759 to 0.8126 for C, thus showing right over-all pairing nature for the interaction B, but stronger nature for C. This observation agrees with the conclusions of ref. 2. The values of 6n are appreciable in some cases (63Ni) for B, but are very small (~0.3) for the interaction C. This point is discussed at length in [8]. The extracted V, and the HartreeFock energies of odd Ni-nuclei are used to obtain the relevant quasiparticle quantities for even Ni-isotopes. Quasiparticle configuration mixing calculations are made for both odd and even Ni-isotopes. A set of non-redundant orthonormal quasiparticle basis states is constructed following the procedure of [4] for the diagonalization work. The spurious states arising from the non-conservation of the particle number by the quasiparticle transformation are also removed following the prescription of [?,ll]. Two sets of quasiparticle quantities are used in the calculation. The first set corresponds to the quasiparticle quantities
Volume
26B,
Level spectra title state
number
12
PHYSICS
LETTERS
1968
Table 1 of Ni-isotopes obtained by various methods described in the text. Here + indicates the four quasiparwhile the symbols * and ** show one quasiparticle and three quasiparticle states. respectively. Excitation
Mass No.
13 May
J”
energy
in MeV
Kuo C
Kuo B
Expt.
+
01’ 21’ 4;
Exact
BCS
IGE
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2.18
2.09
1.83
1.65
1.66
1.67
1.45
3.28
3.12
2.39
2.73
2.68
2.27
2.46
3.42
3.32
2.53
3.17
3.13
2.56
3.08
2.92
2.50
2.54
2.44
2.42
3.47
2.83
3.00
2.78
Exact
i
BCS
IGE
58 62’ 22’ 28
2.78 2.90
--t
3/2i
0.00
0.00
0.00
0.00
0.00
0.00
5/2;
1.27
1 .oo
0.34
1.21
1.01
0.34
0.34
1/q
0.92
0.74
0.47
0.85
1.53
0.47
0.47
3/2;
2.02
1.93
1.31
1.86
1.72
1.23
0.89
l/22
2.56
2.15
1.24
2.36
2.07
1.16
1.32
2.10
1.50
1.72
1.38
0.00
59
5/2.j
T
60
0+ 1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2;
2.45
2.28
1.84
2.02
1.97
1.73
1.33
3.21
2.94
2.49
2.91
2.46
2.53
2.16
0‘;
2.89
2.67+
2.37+
2.72
2.18
2.33
2.29
4;
3.42
3.11
2.32
2.98
2.75
2.21
2.50
3/2;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
5/2;
0.51
0.19
0.07
0.13
- 0.48
0.07
0.07
1/z;
- 0.52
- 0.25
0.28
- 0.96
- 1.10
0.28
0.28
2.02
1.80
i.24
1.46
1.15
1.25
0.91
3/Z;
1.61
0.91
0.74
1.02
l/22
1.53
0.98
1.23
0.97
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2 .a4
2.01
1.78
1.95
1.30
1.37
1.17
2.81
2.66
2.39
2.54
1.89
2.19
2.05
3.93
3.34
2.70
3.58
2.47
2.40’
4.08
3.41
2.51
3.70
2.72
2.52
2;
61 5/2.j
01 2; 62
0+ 2 2; 4;
I-
2.30 2.34
697
PHYSICS
Volume 26B, number 12
LETTERS
13 May 1968
Table 1 continued Excitation energy in MeV
Mass No.
Kuo C Exact
BCS
Kuo B IGE
BCS
Exact
Expt. IGE
-
63
64
65
l/2:
0.00
0.00
0.00
0.00
0.00
0.00
0.00
5/2i
- 0.05
0.18
0.09
-0.59
- 0.15
0.09
0.09
3/2i
1.07
0.84**
0.16
0.62
0.11**
0.16
0.16
3/22
1.41
1.20*
0.71
1.29
1.67**
0.82
0.53
14
2.87
2.46
1.02
2.37
2.42
0.87
1.01
5/2,
1.96
1.66
1.17
1.34
0.99
1.16
0;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2;
2.26
1.78
1.52
1.39
1.08
1.34
1.34
0;
3.79
3.08
2.29
3.49
2.75
2.47
2.28
41
3.31
2.93
2.47
2.31
1.98
2.46
2.62
2+ 2
3.32
2.85”
2.31+
2.59
1.91'
2.20+
2.89
5/Z;
0.00
0.00
0.00
0.00
0.00
0.00
0.00
l/ZI
0.56
0.28
0.06
0.96
0.71
0.06
0.06
3/2i
0.84
0.57**
0.32
0.33
0.07**
0.32
0.32
3/22
2.16
1.95*
0.74
2.41
1.85**
0.45
0.70
5/22
2.68
1.96
1.41
2.36
1.72
1.08
3.34
1.35
3.39
0.90
l/22
obtained by the conventional BCS procedure, while the second represents the inverse gap equation set. The corresponding results are marked by the symbols BCS and IGE, respectively, in the table. The results presented here are without the mixing between different quasiparticle subspaces. Therefore, these results correspond to the diagonalization in three-quasiparticle space for odd isotopes and in two- and fourquasiparticle spaces separately for even isotopes. The mixing between different quasiparticle subspaces does improve the results; this is discussed in [8]. The results of exact shell-model calculations marked “Exact” are also presented in the same table for comparison. It is clear from the table that the numbers in the columns “Exact” and “BCS” are quite close, but differ appreciably from the corresponding experimental values. This is expected for the reason given earlier. The BCS results do not even reproduce the right 698
t
quasiparticle character for f- states of 63Ni and 65Ni nuclei. These levels are predicted as threequasiparticle states marked by ** in the table. The numbers in the column IGE have much better agreement with the experimental values and also re reduce the correct nature of the i- states of 6s.Nl and 65Ni nuclei. The results of both B and C interactions show the same qualitative features. However, the results of the interaction C are slightly better than those of B. Thus a remarkable improvement is achieved in the quasiparticle results when the quasiparticle quantities and the force strength extracted by the inverse gap equation method are used. The usefulness of the inverse gap equation method as such entirely depends, on one hand, on the degree of applicability of the independent quasiparticle or BCS model and, on the other hand, on the correctness of fed input information; in particular, on the realistic two-body shell-
Volume 26B.
number
12
____
PHYSICS
model interaction. It is well established that the latter model is very approximate in many nuclear regions where the contributions from the residual quasiparticle interaction are significant. In view of this fact, the following improved or secondstep procedure is suggested [8]. One now starts repeating the whole calculation by feeding in the modified single-quasiparticle energies Ed given by
E; = Ea + “Ea ,
(2)
where 6Ea is the contribution from the residual quasiparticle interaction calculated by the secondorder perturbation theory in which the obtained values of V2, Vo, etc., are used. The Sgn function should also be determined through these V2. It is expected that new 6n would be less than that obtained previously. The calculations are repeated again with the modified El, obtained by using new values of V2, Vo, etc., by the perturbation method. This procedure is continued until the correction 6E, reduces (- 0) considerably. In practice, it is enough to stop even after two iterations. This follows from the smallness of 6E, which can be checked by doing the MTDA calculations using the final set of extracted quantities. The correction 6E is very small (-‘4%) for both the interactions except for the f- states of 5gNi and 61Ni nuclei in the case of B interaction. The corrections to V, are also very small (-” 2%) for both the interactions. Therefore, the results reported here will not change appreciably with the use of quasiparticle quantities obtained through the improved inverse gap equation procedure. However, considerable improvement is obtained for the other interactions and is fully discussed in ref. 8.
LETTERS
13 May 1968
The author is grateful to Professors M. K. Pal, J.Sawicki and V. Gillet and Drs. S. C. K. Nair, M. Rho and Ram Raj for valuable comments and continuous interest in the work. The author is very grateful to the Saha Institute of Nuclear Physics, Calcutta, India, where most of the work reported here was completed, for providing the facilities and financial assistance. He also wishes to thank Professors Abdus Salam and P. Budini as well as the IAEA for kind hospitality at the International Centre for Theoretical Physics, Trieste.
References 1. M. K. Pal. Y. K.Gambhir and Ram Raj. Phys. Rev. 155 (1967) 1144; I’. L. Ottaviani, M. Savoia. J. Sav,icki and A. Tomasini, Phys. Rev, 153 (1967) 1138. 2. R. D. Lawson, M. H. MacFarlane and T. T. S. Kuo. Phys. Letters 22 (1966) 168. S. P. 3. S. Cohen, R.D. Lawson, M. H. MacFarlane. Pandya and M.Soga, Phys. Rev. 160 (1967) 903. 4. T. Hamada and J. D. Johnston, Nucl. Phys. 34 (1962) 382. Nucl. Phys. A90 (1967) 199. 5. T.T.S.Kuo, 6. V.Gillet and M.Rho. Phys. Letters 21 (1966) 82. Ram Raj and M. K. Pal. Phys. Rev. 7. Y.K.Gambhir, 162 (1967) 1139. to appear as a preprint. 8. Y. K.Gambhir, 9. J. H. E. Mattauch. W. Thiele and A. H. R’apstra, Nucl. Phys. 67 (1965) 1. Phys. Rev. 10. R. H. Fulmer and W.W.Dackmick, B139 (1965) 579. and M.K. Pal. Phys. Rev. 11. Ram Raj, Y.K.Gambhir 163 (1967) 1004.
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