Shell model calculations for two- and three-particle states
A. BOUCENNA(*)International Centre for Theoretical Physics - Trieste, Italy
(ricevuto il 31 Settembre 1996; revisionato il 25 Giugno 1997; approvato il 5 Agosto 1997)
Summary. — In this paper, we present a simplified shell model to calculate the excited state energies of two and three nucleons moving on orbits outside an inert core. The two-body matrix elements (TBME) describing the interaction between active particles outside the core are those determined experimentally. High spin states fed by the two- and three-nucleon transfer on12C ,16O ,28Si ,40Ca ,54Fe ,58Ni , 60Ni ,62Ni ,64Ni ,64Zn ,66Zn ,68Zn target nuclei are calculated and compared to the
experimental results.
PACS 21.60 – Nuclear-structure models and methods. PACS 27.20 – 6 GAG19.
PACS 27.30 – 20 GAG38. PACS 27.40 – 39 GAG58. PACS 27.50 – 59 GAG89.
1. – Introduction
Two- and three-nucleon stripping reactions (12C ,10Be ), (12C ,10B ), (12C ,10C ),
(12C ,9
Be ) performed at an incident energy in the range between 10 and 40 MeV OA on the sd, fp and g orbit nuclei feed selectively only few states in the final nucleus [1-4]. These are the experimentally observed high-spin states in which the individual spins of the transferred nucleons are maximally coupled. At the above-mentioned energies the transfer reaction cross-sections are weak, confined to small scattering angles and almost insensitive to spin and parity [2]. Coincidence measurements (particle-g) reduce drastically the yield of these reactions. For these reasons, we cannot assign unequivocally spin and parity to these observed states using classical methods. Consequently, we need to devise another approach in order to assign the correct spin and parity to these states.
An alternative method might consist in considering a theoretical model to determine the energy of the states having a given configuration. Particularly those configurations where the transferred nucleon spins are maximally coupled, such as
(*) Permanent address : Institut de Physique, Université Ferhat Abbas, 19000 Sétif, Algeria.
(d5 O2)2
41, (d5 O2)251, (d5 O2)313 O21, etc. The calculated energies are compared to those
experimentally observed and having potentially a given configuration. The achievement of a rigorous calculation in the shell model might be long, tedious and most probably impossible to realise. This is due to the fact that the active orbits, to which the nucleons are transferred, are very high and consequently the parameters necessary for the calculation are not always available. To get around this difficulty one can use a simplified theoretical model with fewer parameters, as Jahn [5] and Tsan [6,7] did in their calculation of the excitation energies of the two nucleons states. In this work we propose a simple method to compute the excitation energies of the two-and three-particle high-spin states. We assumed that the 41, 51, 61, 71, 13 O21 and
19 O22states can be described in a model where A
0is the core and transferred nucleons
are moving on the d5 O2, f7 O2or g9 O2active orbits outside the core.
In sophisticated nuclear shell models, any nucleus is assumed to be composed of an inert closed core and extra nucleons moving on the active orbits outside the core. Thus the66Ni is formed by the56Ni core (28 protons and 28 neutrons) plus 10 extra neutrons
moving on the active orbits outside the56Ni core. Consequently we have to solve the
ten-body problem. It is well known that the experimental signature of two- and three-nucleon high-spin states is given by transfer reactions. In this work, we consider that all targets (J 401) may be an inert target core and the transferred nucleons are
considered as extra nucleons, moving on the active orbits outside this target core. In this scheme the 66Ni is supposed to be formed by the 64Ni-core plus only two
transferred extra neutrons moving outside the core, consequently we have to solve the two-body problem. Our simple shell model uses the same two-body matrix elements (TBME) employed in the sophisticated nuclear shell models. However, we should note that the experimental values of the two-body matrix elements are preferable when they are available.
2. – The model
When n particles are transferred to j1shell and m particles to j2 shell, outside the
core A0with a total spin J04 01and an isospin T04 (N 2 Z) O2 , the excitation energy in
the final nuclear state (J , T) is given by [8]:
E * (A01 n 1 m , j1nj2mJT) 4EB(A0, g.s. ) 2 (1) 2 EB(A01 n 1 m , g.s. ) 1 ne( j1) 1me( j2) 1EC1 a j1nj2mN
!
l 4kEl n 1m V(k , l) Njn 1j2mbJT, where T 4T01 Tp.Tp is the extra nucleons isospin, EBthe negative experimental binding energy and e( j)
the single-particle energy:
e( j) 4EB(A01 1 , j) 2 EB(A0, g.s. ) ,
(2) where
EB(A01 1 , j) 4 EB(A01 1 , g.s. ) 1 E * (A01 1 , j)
the occurrence of a single-particle state corresponding to a given e( j) is indicated experimentally by a large one-nucleon stripping spectroscopic factor for bound state. For unbound states, it is signalled by a large single-particle resonance amplitude in the final A01 1 nucleus. The Coulomb interaction energy EC for Z 8 protons in j-orbit
outside the closed core is given by [9]
EC(Z 8) 4cZ 81 1 2Z 8(Z 821) a1
k
1 2Zl
b , (4)where a , b , c are positive real numbers and they depend on the orbit in which the protons are moving. The factor [( 1 O2) Z] is equal to the greatest integer that does not exceed Z 8. The first term in eq. (4) represents the electrostatic interaction of the Z 8 protons in the j-orbit with the core. The second term describes the mutual interaction of the Z 8 protons and the third one is the pairing term. If we consider a harmonic-oscillator well, in the 1 d5 O2shell, then the best fit to the experimental data [9] is obtained for c 43.55 MeV, a40.375 MeV and b40.15 MeV. A modified functional form of the Coulomb energy was suggested by Cole [10]. In our calculation, we used EC
values obtained experimentally by considering the two-proton and two-neutron transfer reactions [3]. The last term in eq. (1), which describes the interaction between active particles outside the core, can be expressed in terms of two-body matrix elements (TBME) as follows:
a j1nj2mN
!
1 4kEl n 1m V(k , l) N j1nj2mbJT4!
J 8 T 8 CJ 8 T 8a j1j2NV( 1 , 2 ) Nj1j2bJ 8 T 8, (5)where a j1j2NV( 1 , 2 ) Nj1j2bJ 8 T 8are the two-body matrix elements (TBME) and CJ 8 T 8are coefficients of pure geometrical nature. If C( 0 )
l and Ck( 1 )represent the coefficients of the two-body matrix elements for two-particle states coupled to T 840 and T 841, respectively, one can get the following sum rules:
.
`
/
`
´
!
k C( 1 ) k 4 3 8n(n 22)1T(T11) ,!
l C( 0 ) l 4 1 8n(n 12)2 1 2T(T 11) ,!
l C( 0 ) l 1!
k C( 1 ) k 4 1 2n(n 21) , (6)where n is the number of particles outside the core. For two identical particles in the same orbit, the wave function should be antisymmetric under the interchange of all coordinates of the two particles, hence only values of J and T such that J 1T4odd are allowed.
2.1. Two-particle states. – In the case of two nucleons (proton and neutron) moving in two active orbits j1and j2outside the core, the total isospin T is:
T 4T01 Tp.
1) Identical nucleons
The z-component of the isospin of the two extra nucleons is given by Tpz4 21 for two protons and 11 for two neutrons, the possible isospin of the two-nucleon system is then Tp4 1 . The excitation energy of the final state (J , T), obtained from eq. (1), reads
E * (A01 2 N , j1j2JT) 4EB(A0, g.s. ) 2EB(A01 2 N , g.s. ) 1 e( j1) 1e( j2) 1
(7)
1EC1 a j1j2NVNNNj1j2bJT, where N is equal to p for proton or n for neutron.
2) Non-identical nucleons
The p-n pair process requires Tpz4 0 , and 1 O2 nucleon isospins may couple into a state having Tpz4 0 or Tpz4 1 . Both of these values of Tp may contribute to the description of a particular state, since only the total isospin T of the nucleons is a good quantum number and both values Tp4 0 and Tp4 1 can couple with T0 to give T
according to the relation
T 4T01 Tp. Then instead of eq. (7) we obtain
E * (A01 p 1 n , j1j2JT) 4EB(A0, g.s. ) 2EB(A01 p 1 n , g.s. ) 1 e( j1) 1e( j2) 1
(8)
1EC1
1
2]a j1j2NVpnNj1j2bJT 401 a j1j2NVppNj1j2bJT 41( . Relations (7) and (8) are analogous to Jahn’s [5] and Chan’s [6, 7] formulae. However, Chan assumes that for the two-identical-particle state the two-body matrix element is small, while for the p-n state he assumed it to be of the order of Dm given by
Dm 4EB(A01 p 1 n) 2 EB(A01 p) 2 EB(A01 n) 1 EB(A0) .
(9)
2.2. Three-particle states. – In the case of three particles moving in two orbits j1and j2outside the core, the last term in eq. (1) can be written as [8]:
a j2 1j2N
!
1 4kEl 3 V(k , l) Nj2 1j2bJT4 aVj1b 4 aVj2b 1 aVj1j2b (10) with.
`
`
/
`
`
´
aVj1b 4 a j 2 1NV( 1 , 2 ) Nj12b , aVj2b 40 , aVj1j2b 42(2ja1 1 )( 2 ta1 1 )!
jutu ( 2 ju1 1 )( 2 tu1 1 ) . / ´ j1 j2 j2 J ja ju ˆ ¨ ˜ 2 Q Q./ ´ 1 O2 1 O2 1 O2 T ta tu ˆ ¨ ˜ 2 a j1j2NVNj1j2bjutu, (11)where ja4 j11 j1, ju4 j11 j2, J 4j11 ju, ta4 t11 t1, tu4 t11 t2, T 4t11 tu,
the symbol ] ( represents the 6j coefficients. Here, also two cases are to be considered, the case where the non-identical particles are on two orbits and the case where the three particles are on the same orbit.
1) Non-identical particles on two orbits
In several cases, the combination of eqs. (5), (6), (10) and (11) determines unequivocally the coefficient CJ 8 T 8. If we consider the
(
( 1 d5 O2)21 p1 O2
)
11 O221 O2state of the15
O nucleus, eq. (10) takes the following form: a(d5 O2)2p1 O2N
!
1 4kEl 3
V(k , l) N(d5 O2)2p1 O2b11 O221 O24
4 C510a(d5 O2)2NVN(d5 O2)2b5101C320ap1 O2d5 O2NVNp1 O2d5 O2b3201C321ap1 O2d5 O2NVNp1 O2f5 O2b321,
while from the sum rules (6), it is found that
C5104 1 , C3204 1 O2 , C3214 3 O2 .
For the
(
( 1 g9 O2)21 f5 O2)23 O221 O2state of the61Zn , one can writea(g9 O2)2f 5 O2N
!
1 4kEl 3 V(k , l) N(g9 O2)2f5 O2b23 O221 O24 4 C910a(g9 O2)2NVN(g9 O2)2b9101C720a f5 O2g9 O2NVN f5 O2g9 O2b7201C721a f5 O2g9 O2NVNf5 O2g9 O2b721, with C9104 1 , C7204 1 O2 , C7214 3 O2and finally for the
(
( 1 g9 O2)21 f5 O2)21 O221 O2)
state of the61Zn , we havea(g9 O2)2f 5 O2N
!
1 4kEl 3 V(k , l) N(g9 O2)2f5 O2b21 O221 O24 4 C810a(g9 O2)2NVN(g9 O2)2b8101C720a f5 O2g9 O2NVN f5 O2g9 O2b7201C721a f5 O2g9 O2NVNf5 O2g9 O2b721, with C8104 1 , C7204 3 O2 , C7214 1 O2 .2) Three particles on the same orbit
In the case of three particles moving on the same orbit j outside the core, the last term in eq. (1) can be written as
a j3 N
!
1 4kElV(k , l) Nj 3b JT4 3 3!
J 8 T 8a j 3 JTntN( j2 J 8 T 8 n8 t 8b2a j2 NVNj2bJ 8 T 8, (12)where a j3
JTntN( j2
J 8 T 8 n8 t 8b is the parentage fractional coefficient, J is the total
spin, T is the total isospin, n is the seniority and t is the reduced isospin. If two protons and one neutron in the orbit d5 O2are coupled to J 413O21and T 41O2, eq. (12) can be
rewritten as follows: a(d5 O2)3N
!
1 4kEl 3 V(k , l) N(d5 O2)3b13 O211 O24 4 3 2[a(d5 O2) 2 NVN(d5 O2)2b4111 a(d5 O2)2NVN(d5 O2)2b510] .For the orbit f7 O2, with J 419O22and T 41O2 we obtain
a( f7 O2)3N
!
1 4kEl 3 V(k , l) N( f7 O2)3b19 O211 O24 3 2[a( f7 O2) 2 NVN( f7 O2)2b6111 a( f7 O2)2NVN( f7 O2)2b710] . 3. – ResultsWe used a simplified shell model developed in sect. 2 to calculate the excitation energies of high spin states of the sd, fp and g orbit nuclei fed by two- and three-nucleon transfer reactions. The used two-body matrix elements (TBME), describing the interaction between active particles outside the core, are those experimentally estimated. When the experimental value is not available the TBME is replaced by corresponding calculated values. Only the two-body matrix element ag9 O2f5 O2NVNg9 O2f5 O2b720has been estimated from eq. (9). In table I we show the TBME
and Coulomb energies EC used in this work. The excited-state positions have been
calculated using the mass values given in ref. [17]. Single-particle states are the states of the odd nuclei A01 1 , formed by the target A0plus one nucleon (proton or neutron).
The occurrence of a single-particle state is indicated for bound states, by a large experimental one nucleon stripping spectroscopic factor, and for unbound states, by a large single-particle resonance amplitude in the final A01 1 nucleus. The
single-particle state energies used in the present work to calculate the energies of the final states are given in table II.
1) Two-neutron states
These states are obtained by two-neutron transfer reactions on12C ,16O ,28Si , 40Ca , 54
Fe , 58Ni , 60Ni , 62Ni , 64Ni , 64Zn targets. To compute the corresponding excitation energies we consider that the nucleus is formed by the target-core and the two transferred neutrons moving on the active orbits outside the core. The model is used to predict the two neutron states fed by two-neutron transfer reactions (a ,2He ) and
(12C ,10C ). The results of the high-spin two-neutron state calculations are reported in
table III. In the same table we also report the experimental values of the energy. It should be pointed that the theoretical values obtained agree with those determined experimentally. Experimental data reported in table III [2] concerning the (12C ,10C )
TABLEI. – Two-body matrix elements (TBME) and Coulomb interaction energy EC.
Mass region TBME (MeV) EC( MeV )
12C ap 1 O2d5 O2NVNp12d5 O2b320 ap1 O2d5 O2NVNp12d5 O2b321 ad2 5 O2NVNd5 O22 b411 ad2 5 O2NVNd5 O22 b510 ad3 O2d5 O2NVNd3 O2d5 O2b411 23 .34 [ 2 , 3 , 11 ] 20 .35 [ 2 , 3 , 12 ] 20 .08 [ 2 , 3 , 12 ] 23 .86 [ 2 , 3 , 13 ] 20 .05 [ 2 , 3 , 14 ] 0.4 [2, 3] 16O ad2 5 O2NVNd5 O22 b411 ad2 5 O2NVNd5 O22 b510 ad3 O2d5 O2NVNd3 O2d5 O2b411 20 .08 [ 2 , 3 , 12 ] 23 .86 [ 2 , 3 , 13 ] 20 .05 [ 2 , 3 , 14 ] 0.4 [2, 3] 28Si ad 3 O2f7 O2NVNd3 O2f7 O2b521 a f2 7 O2NVN f7 O22 b611 a f2 7 O2NVN f7 O22 b710 ad3 O2f7 O2NVNd3 O2f7 O2b520 0 .00 [ 2 , 3 , 15 ] 20 .12 [ 2 , 3 , 16 ] 22 .40 [ 2 , 3 , 11 ] 22 .2 [ 2 , 3 , 16 ] 0.2 [2, 3] 40Ca a f2 7 O2NVN f7 O22 b611 a f2 7 O2NVN f7 O22 b710 a f7 O2f5 O2NVN f7 O2f5 O2b611 20 .12 [ 2 , 3 , 16 ] 22 .40 [ 2 , 3 , 11 ] 21 .147 [ 2 , 3 , 16 ] 0.4 [2, 3] Fe, Ni, Zn ag2 9 O2NVNg9 O22 b811 ag2 9 O2NVNg9 O22 b910 ag9 O2f5 O2NVNg9 O2f5 O2b720 ag9 O2f5 O2NVNg9 O2f5 O2b721 20 .05 [ 2 , 3 , 16 ] 21 .49 [ 2 , 3 , 15 ] 21 .062 (a) 20 .36 [ 2 , 3 , 12 ] 0.4 [2]
(a) See this text.
2) Two-proton states
These states are obtained by two-proton transfer reactions on 12C , 16O , 28Si , 40Ca , 54
Fe , 58Ni , 60Ni , 62Ni , 64Ni , 64Zn , 66Zn , 68Zn targets. To compute the corresponding excitation energies we consider that the nucleus is formed by the target-core plus two transferred protons moving on the active orbits outside the core. The model is used to predict the two-proton states fed by two-proton transfer reaction (12C ,10Be ). The
results of the high-spin two-proton state calculations are reported in table IV. In the same table we also report the experimental values of the energy. In general, it should be pointed that the theoretical values obtained agree with those determined experimentally.
TABLEII. – The single-particle state energies (MeV). 1 p1 O2 1 d5 O2 2 s1 O2 1 d3 O2 1 f7 O2 2 p3 O2 1 f5 O2 2 p1 O2 1 g9 O2 2 d5 O2 13C [ 18 ] 13N [ 19 ] 17O [ 20 ] 17F [ 20 ] 29Si [ 21 ] 29P [ 21 , 22 ] 41Ca [ 21 ] 41Sc [ 21 ] 55Co [ 23 ] 55Fe [ 24 ] 59Ni [ 25 ] 61Ni [ 26 ] 63Ni [ 27 ] 65Ni [ 28 ] 59Cu [ 25 ] 61Cu [ 26 ] 63Cu [ 27 ] 65Cu [ 28 ] 59Zn [ 25 , 29 ] 61Zn [ 26 ] 63Zn [ 27 ] 65Zn [ 28 ] 65Ga [ 28 ] 67Ga [ 30 ] 69Ga [ 31 ] 65Ge [ 32 ] 67Ge [ 30 ] 69Ge [ 31 ] 0.00 0.00 3.85 3.55 0.00 0.00 3.88 2.36 0.87 0.49 0.00 0.00 8.20 8.00 5.08 5.00 1.27 1.38 10.17 10.50 3.62 3.45 0.00 0.00 0.00 1.94 2.17 1.72 0.00 0.00 0.00 0.155 0.692 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.155 0.00 0.00 0.00 0.00 0.123 0.233 8.27 7.46 5.65 5.71 3.30 0.93 0.339 0.067 0.087 0.00 0.914 0.970 0.962 1.115 0.90 0.124 0.193 0.00 0.191 0.359 0.574 0.111 0.018 0.00 4.17 0.41 0.465 0.283 0.00 0.063 0.491 0.475 0.670 0.771 0.54 0.088 0.248 0.054 0.062 0.167 0.319 0.00 0.087 6.07 3.081 3.81 2.122 1.292 1.013 3.043 2.721 2.506 2.534 2.68 2.002 1.704 1.065 2.040 2.074 1.970 1.216 0.752 0.398 4.462 2.697 2.297 1.920 3.580 3.406 3.476 3.391 1.370 2.822 2.746 2.148 3) Proton-neutron states
These states are obtained by proton-neutron transfer reactions on 12C , 16O , 28Si , 40Ca targets. To compute the corresponding excitation energies we consider that the
nucleus is formed by the target-core and transferred proton and neutron moving on the active orbits outside the core. The model is used to predict the deuteron states fed by proton-neutron transfer reaction (12C ,10B ). The results of the high-spin proton-neutron state calculations are reported in table V. In the same table we also report the experimental values of the energy. It should be pointed that the theoretical values obtained agree with those determined experimentally. Experimental data reported in table III [2] concerning the (12C ,10B ) reaction are obtained using the same
experimental arrangement as in ref. [4]. 4) Three-particle states
These states are obtained by three nucleons (two neutrons plus one proton or two protons plus one neutron) transfer reactions on12C ,16O ,28Si ,40Ca ,54Fe ,58Ni ,64Ni and
TABLEIII. – Excitation energies for two-neutron states.
Core Nucleus Configuration JpT E *
The(MeV) (a) E *Exp(MeV) 12C 14C (np 1 O2nd5 O2)321 (nd5 O2)2 411 (nd5 O2nd3 O2)411 32 1 41 1 41 1 6.73 10.85 15.22 6.73 [5] 10.72 [5] 14.9 [5] 16O 18O (nd 5 O2)2411 (nd5 O2nd3 O2)411 41 1 41 1 3.82 8.93 3.56 [5] 9.0 [5] 28Si 30Si (nd 3 O2nf7 O2)521 (nf7 O2)2 611 52 1 61 1 7.03 9.25 7.04 [5] 8.95 [5] 40Ca 42Ca (nf 7 O2)2611 (nf7 O2nf5 O2)611 61 1 61 1 3.00 7.63 3.19 [5] 7.40 [5] 54Fe 56Fe (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 6.62 9.51 6.03 [33] 9.28 [33] 58Ni 60Ni (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 5.78 8.50 5.30 [2, 33] 8.53 [2, 33] 60Ni 62Ni (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 4.97 7.02 4.66 [2, 33] 7.19 [2, 33] 62Ni 64Ni (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 4.11 5.40 4.52 [2, 33] 5.81 [2, 33] 64Ni 66Ni (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 3.89 4.91 4.08 [2, 33] 5.14 [2, 33] 64Zn 66Zn (nf 5 O2ng9 O2)721 (ng9 O2)2 811 72 1 81 1 4.14 5.21 4.26 [2, 33] 5.25 [2, 33] (a) This text.
64Zn targets. To compute the corresponding excitation energies, we consider that the
nucleus is formed by the target core plus three transferred nucleons, moving on the active orbits outside the core. The model is used to predict the two-neutrons one-proton states fed by two-neutrons one-proton transfer reactions (12C ,9B ) and the two-protons
one-neutron states fed by two-protons one-neutron transfer reactions (12C ,9Be ). The
results of the high-spin three-nucleon state calculations are reported in table VI. In the same table we also report the experimental values of the energy. In general, the theoretical values obtained agree with those determined experimentally.
TABLEIV. – Excitation energies for two-proton states.
Core Nucleus Configuration JpT E *
The(MeV) (a) E *Exp(MeV)
[1-4] 12C 14O (pp 1 O2pd5 O2)321 (pd5 O2)2 411 (pd5 O2pd3 O2)411 32 1 41 1 41 1 6.28 10.10 13.36 6.27 9.90 14.1 16O 18Ne (pd 5 O2)2411 (pd5 O2pd3 O2)411 41 1 41 1 3.64 8.67 3.4 7.9 28Si 30S (pd 3 O2pf7 O2)521 (pf7 O2)2 611 52 1 61 1 6.76 8.63 6.7 8.3 40Ca 42Ti (pf 7 O2)2611 (pf7 O2pf5 O2)611 61 1 61 1 2.96 7.64 3.04 7.5 54Fe 56Ni (pf 7 O2pf5 O2)611 (pf7 O2pg9 O2)721 (pf5 O2pg9 O2)721 (pg9 O2)2 811 61 1 72 1 72 1 81 1 4.65 8.57 11.51 14.6 5.31 (8.1) (8.9) — 58Ni 60Zn (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 5.70 8.13 5.30 7.98 60Ni 62Zn (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 5.40 7.46 5.19 7.54 62Ni 64Zn (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 5.10 6.94 5.30 6.70 64Ni 66Zn (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 5.16 6.88 5.20 6.85 64Zn 66Ge (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 4.59 6.73 4.59 6.63 66Zn 68Ge (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 4.59 6.61 4.61 6.30 68Zn 70Ge (pf 5 O2pg9 O2)721 (pg9 O2)2 811 72 1 81 1 4.51 6.2 4.33 5.73 (a) This work.
TABLEV. – Excitation energies for neutron-proton states.
Core Nucleus Configuration JpT E *
The(MeV) (a) E *Exp(MeV)
12C 14N (np 1 O2pd5 O2)320 (pd5 O2nd5 O2)510 32 0 51 0 5.92 8.75 5.83 [34] 8.96 [34] 16O 18F (pd 5 O2nd5 O2)510 51 0 1.17 1.12 [1, 2] 28Si 30P (pf 7 O2nf7 O2)710 71 0 5.75 6.40 [1, 2] 40Ca 42Sc (pf 7 O2nf7 O2)710 71 0 0.79 0.617 [24]
(a) This work.
TABLEVI. – Excitation energies for three-particle states.
Core Nucleus Configuration JpT E *
The (MeV) (a) E *Exp(MeV) 12C 15N 15O ((nd5 O2)2 41pd5 O2)13 O211 O2 ((nd5 O2)2 41nd5 O2)13 O211 O2 13 O21 1 O2 13 O21 1 O2 16.84 16.00 15.41 [35] 15.05 [1, 2] 16O 19F 19Ne ((nd5 O2)2 41pd5 O2)13 O211 O2 ((nd5 O2)2 41nd5 O2)13 O211 O2 13 O21 1 O2 13 O21 1 O2 5.38 4.9 4.64 [35] 4.64 [1, 2] 28Si 31P 31S ((nf7 O2)2 61pf7 O2)19 O221 O2 ((pf7 O2)2 61nf7 O2)19 O221 O2 19 O22 1 O2 19 O22 1 O2 13.58 12.91 — 13.2 [1, 2] 40Sc 43Sc 43Ti ((nf7 O2)2 61pf7 O2)19 O221 O2 ((pf7 O2)2 61nf7 O2)19 O221 O2 19 O22 1 O2 19 O22 1 O2 3.18 2.81 3.12 [21] 3.2 [1, 2] 58Ni 61Zn ((pf 5 O2pg9 O2)72ng9 O2)23 O221 O2 23 O22 1 O2 7.38 (6.92) [36] 64Ni 67Zn ((pf 5 O2pg9 O2)72ng9 O2)23 O221 O2 23 O22 1 O2 4.53 (4.60) [36] 64Zn 67Ge ((pf 5 O2pg9 O2)72ng9 O2)23 O221 O2 23 O22 1 O2 4.81 (5.25) [36]
(a) This work.
The defined model also allows to predict the two- or three-particle state positions when the experimental values are unknown. In this respect the
(
(nf7 O2)261pf7 O2)19 O221 O2state of31P is predicted to be at 13.58 MeV. Moreover, the
(
(pf5 O2pg9 O2)72ng9 O2)23 O221 O2
states of 61Zn , 67Zn and 67Ge are predicted at 7.38 MeV, 4.53 MeV and 4.81 MeV,
respectively. These three states may correspond to the highly excited states observed at E * 46.92 MeV, E *44.60 MeV and E *45.25 MeV respectively in the (12C ,9Be )
reaction induced by the 112 MeV12C beam on the 58Ni ,64Ni and64Zn targets [36]. The corresponding spectra are shown in figs. 1, 2 and 3 respectively. These spectra are obtained using the same experimental arrangement as in ref. [4], where the 112 MeV
Fig. 1. – Three-nucleon transfer on58Ni .
Fig. 2. – Three-nucleon transfer on64Ni .
12C beam was provided by the Strasbourg MP accelerator. The 9Be ejectiles were
analysed with a three-dipoles and one-quadrupole (Q3D) spectrometer coupled to the incident beam in an energy-loss mode. A gaseous hybrid counter composed of three proportional counters separated by two ionisation chambers was placed along the focal plane. Two mylar foils separated the isobutane of the counter from the vacuum of the detection chamber, while allowing the particles to reach a 2 cm thick NE110 scintillator. The ejectiles were identified by their energy loss DE in the ionisation
Fig. 3. – Three-nucleon transfer on64Zn .
chambers and residual energy ER in the scintillator. Position spectra in the first and
third proportional counters, separated by 211.5 mm, were obtained by charge division. The positions, which conditioned by the DE and ER requirements, allowed
reconstruction of the trajectories of the studied ejectiles, the determination of the particle entrance angle and construction of momentum spectra on the focal plane.
4. – Conclusion
In this work we presented a simplified shell model to calculate the excitation energies of two- and three-nucleon states. This model has been used first to compute a series of known state positions in order to confirm its validity. Second, it has been used to locate unknown states populated by two- and three-particle transfer reactions. Our theoretical results are in reasonable agreement with the available experimental data. In addition, this simple method has the potential to locate easily some two- and three-nucleon high-spin states, which are reached by two- and three-particle transfer reactions at high enough energies.
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