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Phenomenological limit of the semimicroscopic description
Volume 28B, number
7
PHYSICS
PHENOMENOLOGICAL
LIMIT
OF
LETTERS
THE
20 January 1969
SEMIMICROSCOPIC
DESCRIPTION
L. SIPS Institute “Ruder BoSkovic.‘, Zagreb, Yugoslavia and DQartement de Physique Th&orique, CEN Saclay,
France
Received 25 October 1968 A simple phenomenological model of coupled proton and neutron vibrations is considered. Cubic terms in the potential are taken and the results compared with recent findings.
The use of the multiple Coulomb excitation process [l] has procuredtlie very interesting information that the quadrupole moment of the first excited 2+ state in vibration-like nuclei is remarkably large [1,2]. A number of investigations [3,4] have been undertaken with the aim to understand the coexistence of almost preserved vibrational selection rules for transition probabilities with the found magnitude of the quadrupole moment. The present attempt may be taken as the phenomenological limit of the semimicroscdpic description proposed by Alaga et al. [4,5]. There a few proton (proton hole) degrees of freedom are treated microscopically and, then, subsequently coupled to the phenomenologically treated neutron vibrations. Here both proton and neutron degrees of freedom are taken as phenomenological quadrupole vibrators (allowed to be anharmonic), which are coupled including cubic terms in the potential. The Hamiltonian of the system may then be written in the form: H = Ho + H,
+$
c,lanj2- $ k c (ap
* an+an
* ap)
@a)
and
The dot in eq. (2a) means the scalar product, and ( ) is the usual 3-j symbol. In order to reduce the number of parameters, 448
25L3
032LLZ
21 0362
2242
42
02
0223 22
(212
0
a
InpI+$Cplap12 +kn InnI+
0,
234
(I)
with Ho = j-j+ P
(0)3(2)3 U3
0, 25L3
b
-41
21
21
01
01 c
Fig. 1. The following spectra are shown: a) zeroth order, b) Ho diagonalized, and c) result of taking the total Hamiltonian H. Except in b), states are labelled in the order of appearance.
both oscillators are taken to be of the same characteristics. Then there are only k and kc left as parameters. The Hamiltonian HO of eq. (2a) may be diagonalized analytically for any k (adding simply the 7Tnterm) [6], and the subsequent diagonalivP * zation of Hc, eq. (2b) is performed on the basis thus obtained. In the numerically evaluated ex-
PHYSICS
Volume 28B, number 7
LETTERS
Table 1 B(E2) values for the case fig. lc are listed in units of B(E2:21 --t 01). The first column corresponds to no “effective charge” ascribed to valent nucleons producing vibrations, while the values in the second colnmn are the usually taken ones [3-51. (-) and (+) mean the sign of the quadrupole moment.
,eff p = e,
eff = o en
eff = 2e, ezff = e eP 1
Iiai
Ipf
21
0
22
01
0.04
2 3
01
0.43
0.041
21
2 1
0.217 (-)
0.217 (-)
22
22
0.217 (+)
0.217 (+)
22
2 1
1.356
1.356
23
21
-0
-0
23
22
-0
-0
41
21
1.52
1.52
42
21
0.76
0.084
02
21
0.10
0.10
02
22
0
0
1
1
0.04
20 January 1969
moment generated is always the same. this is highly undesirable, but is readily removed by considering the way the system is likely to become anharmonic. This is most easily done by inquiring about the sign of the anharmonic terms (Hc of eq. (2b)) from Nilsson’s diagram [7]. The crude estimates yield correct signs for the quadrupole moments. In addition one may say that the above simple considerations provide the clue how to estimate the importance and the effect of anharmonicities in the semimicroscopic description. quadrupole
Of course,
The author wishes to express his sincere gratitude to Professor G. Alaga and V. Lopac for many useful discussions. For the part of the work
done during the author’s stay at Dkpartement de Physique Theorique CEN Saclay deep gratidue is due to Professors C. Bloch and V. Gillet for their hospitality and kind interest. Financial means provided through CEN Saclay made the author’s stay possible and are gratefully acknowledged.
References am le the values taken for k and kc were i and 5, respectively. The result is shown in fig. 1. As expected, the present degeneracies are removed, as the remaining ones would have been had the basis been larger. The reduced transition probabilities have been evaluated and the ones of interest are listed in table 1. The concept of “effective charge” is used to probe the coherence of the contributions to the transition and the static moments. It is interesting to note that the distribution of coherence among the levels is just the required one for the preservation of vibrational-like pattern, yet yielding large static quadrupole moments. Comparison of the B(E2:21 - 21)/B(E2:21 + 01) ratio with the one of ref. 4 for l14Cd gives the value for the quadrupole moment of the first excited state Q(21) = - 0.33 eb. The value taken for B(E2:21+ 01) is the experimental one, while Qe (21) = B(E2:21 --t 01) is the experimental one, whl?pe Qexp(21) = - 0.49 f 0.25 eb. However, it would seem that the sign of the
SP
1. A. Winther, Proc. Eleventh Summer Meeting, Herceg Novi 1966, (Zagreb 1967); H. E.Gove, ibid. 2. J.de Boer, R.G. Stokstad, G. D. Symons and A. Winther, Phys.Rev. Letters 14 (1965) 564; P.H.Stelson, W.T.Milner, J.L.C.FordJr., F.K. McGowan and R. L. Robinson. Bull. Am. Phvs. Sot. 10 (1965) 427; R.G. Stokstad. I. Hall, G.D. Symons and J. de Boer, Nucl. Phys. A94 (1967) 177. 3. T. Tamura and T. Udagawa, Phys. Rev. Letters 15 (1965) 765; M.Baranger and K.Kumar, Nucl.Phys.A92 (1967) 608; G. Do Dang, R. Dreizler, A. Klein and C. S. Wu, Phys. Rev. Letters 17 (1966) 709; B. Mrensen, Phys. Letters 23 (1966) 274; 24B (1967) 328. 4. M. Sakai, preprint INSJ-105, 1967; G. Alaga, F. KrmpotiE and V. Lopac, Phys. Letters 24B (1967) 537 and preprint IRB-TP-2-67. 5. G.Alaga and G.Ialongi, Phys. Letters 22 (1966) 619; G.Alaga and G.Ialongo, Nucl.Phys.A97 (1967) 600. 6. E. Coffou, Nuovo Cimento, to be published. 7. S.G.Nilsson, Mat.Fys.Medd.Dan.Vid.Selsk.29 no.16 (1955).
*****
449
7
PHYSICS
PHENOMENOLOGICAL
LIMIT
OF
LETTERS
THE
20 January 1969
SEMIMICROSCOPIC
DESCRIPTION
L. SIPS Institute “Ruder BoSkovic.‘, Zagreb, Yugoslavia and DQartement de Physique Th&orique, CEN Saclay,
France
Received 25 October 1968 A simple phenomenological model of coupled proton and neutron vibrations is considered. Cubic terms in the potential are taken and the results compared with recent findings.
The use of the multiple Coulomb excitation process [l] has procuredtlie very interesting information that the quadrupole moment of the first excited 2+ state in vibration-like nuclei is remarkably large [1,2]. A number of investigations [3,4] have been undertaken with the aim to understand the coexistence of almost preserved vibrational selection rules for transition probabilities with the found magnitude of the quadrupole moment. The present attempt may be taken as the phenomenological limit of the semimicroscdpic description proposed by Alaga et al. [4,5]. There a few proton (proton hole) degrees of freedom are treated microscopically and, then, subsequently coupled to the phenomenologically treated neutron vibrations. Here both proton and neutron degrees of freedom are taken as phenomenological quadrupole vibrators (allowed to be anharmonic), which are coupled including cubic terms in the potential. The Hamiltonian of the system may then be written in the form: H = Ho + H,
+$
c,lanj2- $ k c (ap
* an+an
* ap)
@a)
and
The dot in eq. (2a) means the scalar product, and ( ) is the usual 3-j symbol. In order to reduce the number of parameters, 448
25L3
032LLZ
21 0362
2242
42
02
0223 22
(212
0
a
InpI+$Cplap12 +kn InnI+
0,
234
(I)
with Ho = j-j+ P
(0)3(2)3 U3
0, 25L3
b
-41
21
21
01
01 c
Fig. 1. The following spectra are shown: a) zeroth order, b) Ho diagonalized, and c) result of taking the total Hamiltonian H. Except in b), states are labelled in the order of appearance.
both oscillators are taken to be of the same characteristics. Then there are only k and kc left as parameters. The Hamiltonian HO of eq. (2a) may be diagonalized analytically for any k (adding simply the 7Tnterm) [6], and the subsequent diagonalivP * zation of Hc, eq. (2b) is performed on the basis thus obtained. In the numerically evaluated ex-
PHYSICS
Volume 28B, number 7
LETTERS
Table 1 B(E2) values for the case fig. lc are listed in units of B(E2:21 --t 01). The first column corresponds to no “effective charge” ascribed to valent nucleons producing vibrations, while the values in the second colnmn are the usually taken ones [3-51. (-) and (+) mean the sign of the quadrupole moment.
,eff p = e,
eff = o en
eff = 2e, ezff = e eP 1
Iiai
Ipf
21
0
22
01
0.04
2 3
01
0.43
0.041
21
2 1
0.217 (-)
0.217 (-)
22
22
0.217 (+)
0.217 (+)
22
2 1
1.356
1.356
23
21
-0
-0
23
22
-0
-0
41
21
1.52
1.52
42
21
0.76
0.084
02
21
0.10
0.10
02
22
0
0
1
1
0.04
20 January 1969
moment generated is always the same. this is highly undesirable, but is readily removed by considering the way the system is likely to become anharmonic. This is most easily done by inquiring about the sign of the anharmonic terms (Hc of eq. (2b)) from Nilsson’s diagram [7]. The crude estimates yield correct signs for the quadrupole moments. In addition one may say that the above simple considerations provide the clue how to estimate the importance and the effect of anharmonicities in the semimicroscopic description. quadrupole
Of course,
The author wishes to express his sincere gratitude to Professor G. Alaga and V. Lopac for many useful discussions. For the part of the work
done during the author’s stay at Dkpartement de Physique Theorique CEN Saclay deep gratidue is due to Professors C. Bloch and V. Gillet for their hospitality and kind interest. Financial means provided through CEN Saclay made the author’s stay possible and are gratefully acknowledged.
References am le the values taken for k and kc were i and 5, respectively. The result is shown in fig. 1. As expected, the present degeneracies are removed, as the remaining ones would have been had the basis been larger. The reduced transition probabilities have been evaluated and the ones of interest are listed in table 1. The concept of “effective charge” is used to probe the coherence of the contributions to the transition and the static moments. It is interesting to note that the distribution of coherence among the levels is just the required one for the preservation of vibrational-like pattern, yet yielding large static quadrupole moments. Comparison of the B(E2:21 - 21)/B(E2:21 + 01) ratio with the one of ref. 4 for l14Cd gives the value for the quadrupole moment of the first excited state Q(21) = - 0.33 eb. The value taken for B(E2:21+ 01) is the experimental one, while Qe (21) = B(E2:21 --t 01) is the experimental one, whl?pe Qexp(21) = - 0.49 f 0.25 eb. However, it would seem that the sign of the
SP
1. A. Winther, Proc. Eleventh Summer Meeting, Herceg Novi 1966, (Zagreb 1967); H. E.Gove, ibid. 2. J.de Boer, R.G. Stokstad, G. D. Symons and A. Winther, Phys.Rev. Letters 14 (1965) 564; P.H.Stelson, W.T.Milner, J.L.C.FordJr., F.K. McGowan and R. L. Robinson. Bull. Am. Phvs. Sot. 10 (1965) 427; R.G. Stokstad. I. Hall, G.D. Symons and J. de Boer, Nucl. Phys. A94 (1967) 177. 3. T. Tamura and T. Udagawa, Phys. Rev. Letters 15 (1965) 765; M.Baranger and K.Kumar, Nucl.Phys.A92 (1967) 608; G. Do Dang, R. Dreizler, A. Klein and C. S. Wu, Phys. Rev. Letters 17 (1966) 709; B. Mrensen, Phys. Letters 23 (1966) 274; 24B (1967) 328. 4. M. Sakai, preprint INSJ-105, 1967; G. Alaga, F. KrmpotiE and V. Lopac, Phys. Letters 24B (1967) 537 and preprint IRB-TP-2-67. 5. G.Alaga and G.Ialongi, Phys. Letters 22 (1966) 619; G.Alaga and G.Ialongo, Nucl.Phys.A97 (1967) 600. 6. E. Coffou, Nuovo Cimento, to be published. 7. S.G.Nilsson, Mat.Fys.Medd.Dan.Vid.Selsk.29 no.16 (1955).
*****
449