fulltext

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 on12_{C ,}16_{O ,}28_{Si ,}40_{Ca ,}54_{Fe ,}58_{Ni ,} 60_{Ni ,}62_{Ni ,}64_{Ni ,}64_{Zn ,}66_{Zn ,}68_{Zn 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 (12_{C ,}10_{Be ), (}12_{C ,}10_{B ), (}12_{C ,}10_{C ),}

(12_{C ,}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.

(d_{5 O2})2

41, (d_{5 O2})2₅1, (d_{5 O2})3_{13 O2}1, 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_{, 5}1_{, 6}1_{, 7}1_{, 13 O2}1 _and

19 O22_{states can be described in a model where A}

0is the core and transferred nucleons

are moving on the d_{5 O2}, f_{7 O2}or g_{9 O2}active 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 the66_{Ni is formed by the}56_{Ni core (28 protons and 28 neutrons) plus 10 extra neutrons}

moving on the active orbits outside the56_{Ni 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 66_{Ni is supposed to be formed by the} 64_{Ni-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 2Z

l

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 d_{5 O2}shell, 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 C_{J 8 T 8}a 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 C_{J 8 T 8}. If we consider the

(

( 1 d_{5 O2})2_{1 p}

1 O2

)

11 O22_{1 O2}state of the

15

O nucleus, eq. (10) takes the following form: a(d5 O2)2p1 O2N

!

1 4kEl 3

V(k , l) N(d5 O2)2p1 O2b11 O22_{1 O2}4

4 C51₀a(d_{5 O2})2NVN(d_{5 O2})2b₅1₀1C₃2₀ap_{1 O2}d_{5 O2}NVNp_{1 O2}d_{5 O2}b₃2₀1C₃2₁ap_{1 O2}d_{5 O2}NVNp_{1 O2}f_{5 O2}b₃2₁,

while from the sum rules (6), it is found that

C51₀4 1 , C₃2₀4 1 O2 , C₃2₁4 3 O2 .

For the

(

( 1 g_{9 O2})21 f_{5 O2})_{23 O2}2_{1 O2}state of the61Zn , one can write

a(g_{9 O2})2_f 5 O2N

!

1 4kEl 3 V(k , l) N(g9 O2)2f5 O2b23 O22_{1 O2}4 4 C91₀a(g_{9 O2})2NVN(g_{9 O2})2b₉1₀1C₇2₀a f_{5 O2}g_{9 O2}NVN f_{5 O2}g_{9 O2}b₇2₀1C₇2₁a f_{5 O2}g_{9 O2}NVNf_{5 O2}g_{9 O2}b₇2₁, with C91₀4 1 , C₇2₀4 1 O2 , C₇2₁4 3 O2

and finally for the

(

( 1 g_{9 O2})21 f_{5 O2})_{21 O2}2_{1 O2}

)

state of the61Zn , we have

a(g_{9 O2})2_f 5 O2N

!

1 4kEl 3 V(k , l) N(g9 O2)2f5 O2b21 O22_{1 O2}4 4 C81₀a(g_{9 O2})2NVN(g_{9 O2})2b₈1₀1C₇2₀a f_{5 O2}g_{9 O2}NVN f_{5 O2}g_{9 O2}b₇2₀1C₇2₁a f_{5 O2}g_{9 O2}NVNf_{5 O2}g_{9 O2}b₇2₁, 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 3_b JT4 3 3

!

J 8 T 8a j 3 JTntN( j2 J 8 T 8 n8 t 8b2_{a j}2 NVNj2b_{J 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 d_{5 O2are coupled to J 413O2}1_{and T 41O2, eq. (12) can be}

rewritten as follows: a(d5 O2)3N

!

1 4kEl 3 V(k , l) N(d5 O2)3b13 O21_{1 O2}4 4 3 2[a(d5 O2) 2 NVN(d5 O2)2b41₁1 a(d_{5 O2})2NVN(d_{5 O2})2b₅1₀] .

For the orbit f_{7 O2, with J 419O2}2_{and T 41O2 we obtain}

a( f7 O2)3N

!

1 4kEl 3 V(k , l) N( f7 O2)3b19 O21_{1 O2}4 3 2[a( f7 O2) 2 NVN( f7 O2)2b61₁1 a( f_{7 O2})2NVN( f_{7 O2})2b₇1₀] . 3. – Results

We 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 ag_{9 O2}f_{5 O2NVNg9 O2}f_{5 O2}b72₀has 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 on12_{C ,}16_{O ,}28_{Si ,} 40_{Ca ,} 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 ,2_{He ) and}

(12_{C ,}10_{C ). 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 (12_{C ,}10_{C )}

TABLEI. – Two-body matrix elements (TBME) and Coulomb interaction energy EC.

Mass region TBME (MeV) EC( MeV )

12_{C 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] 16_{O ad}2 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] 28_{Si 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] 40_{Ca a f}2 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 12_{C ,} 16_{O ,} 28_{Si ,} 40_{Ca ,} 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 (12_{C ,}10_{Be ). 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 p_{1 O2} 1 d_{5 O2} 2 s_{1 O2} 1 d_{3 O2} 1 f_{7 O2} 2 p_{3 O2} 1 f_{5 O2} 2 p_{1 O2} 1 g_{9 O2} 2 d_{5 O2} 13_{C [ 18 ]} 13_{N [ 19 ]} 17_{O [ 20 ]} 17_{F [ 20 ]} 29_{Si [ 21 ]} 29_{P [ 21 , 22 ]} 41_{Ca [ 21 ]} 41_{Sc [ 21 ]} 55_{Co [ 23 ]} 55_{Fe [ 24 ]} 59_{Ni [ 25 ]} 61_{Ni [ 26 ]} 63_{Ni [ 27 ]} 65_{Ni [ 28 ]} 59_{Cu [ 25 ]} 61_{Cu [ 26 ]} 63_{Cu [ 27 ]} 65_{Cu [ 28 ]} 59_{Zn [ 25 , 29 ]} 61_{Zn [ 26 ]} 63_{Zn [ 27 ]} 65_{Zn [ 28 ]} 65_{Ga [ 28 ]} 67_{Ga [ 30 ]} 69_{Ga [ 31 ]} 65_{Ge [ 32 ]} 67_{Ge [ 30 ]} 69_{Ge [ 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 12_{C ,} 16_{O ,} 28_{Si ,} 40_{Ca 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 (12_{C ,}10_{B ) 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 on12_{C ,}16_{O ,}28_{Si ,}40_{Ca ,}54_{Fe ,}58_{Ni ,}64_{Ni and}

TABLEIII. – Excitation energies for two-neutron states.

Core Nucleus Configuration Jp_{T E *}

The(MeV) (a) E *Exp(MeV) 12_C 14_{C (np} 1 O2nd5 O2)32₁ (nd_{5 O2})2 41₁ (nd_{5 O2}nd_{3 O2})41₁ 32 ₁ 41 ₁ 41 ₁ 6.73 10.85 15.22 6.73 [5] 10.72 [5] 14.9 [5] 16_O 18_{O (nd} 5 O2)241₁ (nd_{5 O2}nd_{3 O2})41₁ 41 ₁ 41 ₁ 3.82 8.93 3.56 [5] 9.0 [5] 28_Si 30_{Si (nd} 3 O2nf7 O2)52₁ (nf_{7 O2})2 61₁ 52 ₁ 61 ₁ 7.03 9.25 7.04 [5] 8.95 [5] 40_Ca 42_{Ca (nf} 7 O2)261₁ (nf_{7 O2}nf_{5 O2})61₁ 61 ₁ 61 ₁ 3.00 7.63 3.19 [5] 7.40 [5] 54_Fe 56_{Fe (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 6.62 9.51 6.03 [33] 9.28 [33] 58_Ni 60_{Ni (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 5.78 8.50 5.30 [2, 33] 8.53 [2, 33] 60_Ni 62_{Ni (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 4.97 7.02 4.66 [2, 33] 7.19 [2, 33] 62_Ni 64_{Ni (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 4.11 5.40 4.52 [2, 33] 5.81 [2, 33] 64_Ni 66_{Ni (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 3.89 4.91 4.08 [2, 33] 5.14 [2, 33] 64_Zn 66_{Zn (nf} 5 O2ng9 O2)72₁ (ng_{9 O2})2 81₁ 72 ₁ 81 ₁ 4.14 5.21 4.26 [2, 33] 5.25 [2, 33] (a) This text.

64_{Zn 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 (12_{C ,}9_{B ) and the two-protons}

one-neutron states fed by two-protons one-neutron transfer reactions (12_{C ,}9_{Be ). 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 Jp_{T E *}

The(MeV) (a) E *Exp(MeV)

[1-4] 12_C 14_{O (pp} 1 O2pd5 O2)32₁ (pd_{5 O2})2 41₁ (pd_{5 O2}pd_{3 O2})41₁ 32 ₁ 41 ₁ 41 ₁ 6.28 10.10 13.36 6.27 9.90 14.1 16_O 18_{Ne (pd} 5 O2)241₁ (pd_{5 O2}pd_{3 O2})41₁ 41 ₁ 41 ₁ 3.64 8.67 3.4 7.9 28_Si 30_{S (pd} 3 O2pf7 O2)52₁ (pf_{7 O2})2 61₁ 52 ₁ 61 ₁ 6.76 8.63 6.7 8.3 40_Ca 42_{Ti (pf} 7 O2)261₁ (pf_{7 O2}pf_{5 O2})61₁ 61 ₁ 61 ₁ 2.96 7.64 3.04 7.5 54_Fe 56_{Ni (pf} 7 O2pf5 O2)61₁ (pf_{7 O2}pg_{9 O2})72₁ (pf_{5 O2}pg_{9 O2})72₁ (pg_{9 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) — 58_Ni 60_{Zn (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 5.70 8.13 5.30 7.98 60_Ni 62_{Zn (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 5.40 7.46 5.19 7.54 62_Ni 64_{Zn (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 5.10 6.94 5.30 6.70 64_Ni 66_{Zn (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 5.16 6.88 5.20 6.85 64_Zn 66_{Ge (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 4.59 6.73 4.59 6.63 66_Zn 68_{Ge (pf} 5 O2pg9 O2)72₁ (pg_{9 O2})2 81₁ 72 ₁ 81 ₁ 4.59 6.61 4.61 6.30 68_Zn 70_{Ge (pf} 5 O2pg9 O2)72₁ (pg_{9 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 Jp_{T E *}

The(MeV) (a) E *Exp(MeV)

12_C 14_{N (np} 1 O2pd5 O2)32₀ (pd_{5 O2}nd_{5 O2})51₀ 32 ₀ 51 ₀ 5.92 8.75 5.83 [34] 8.96 [34] 16_O 18_{F (pd} 5 O2nd5 O2)51₀ 51 0 1.17 1.12 [1, 2] 28_Si 30_{P (pf} 7 O2nf7 O2)71₀ 71 0 5.75 6.40 [1, 2] 40_Ca 42_{Sc (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 Jp_{T E *}

The (MeV) (a₎ E *Exp(MeV) 12_C 15_N 15_O ((nd_{5 O2})2 41pd_{5 O2})_{13 O2}1_{1 O2} ((nd_{5 O2})2 41nd_{5 O2})_{13 O2}1_{1 O2} 13 O21 _{1 O2} 13 O21 _{1 O2} 16.84 16.00 15.41 [35] 15.05 [1, 2] 16_O 19_F 19_Ne ((nd_{5 O2})2 41pd_{5 O2})_{13 O2}1_{1 O2} ((nd_{5 O2})2 41nd_{5 O2})_{13 O2}1_{1 O2} 13 O21 _{1 O2} 13 O21 _{1 O2} 5.38 4.9 4.64 [35] 4.64 [1, 2] 28_Si 31_P 31_S ((nf_{7 O2})2 61pf_{7 O2})_{19 O2}2_{1 O2} ((pf_{7 O2})2 61nf_{7 O2})_{19 O2}2_{1 O2} 19 O22 _{1 O2} 19 O22 _{1 O2} 13.58 12.91 — 13.2 [1, 2] 40_Sc 43_Sc 43_Ti ((nf_{7 O2})2 61pf_{7 O2})_{19 O2}2_{1 O2} ((pf_{7 O2})2 61nf_{7 O2})_{19 O2}2_{1 O2} 19 O22 _{1 O2} 19 O22 _{1 O2} 3.18 2.81 3.12 [21] 3.2 [1, 2] 58_Ni 61_{Zn ((pf} 5 O2pg9 O2)72ng_{9 O2})_{23 O2}2_{1 O2} 23 O22 1 O2 7.38 (6.92) [36] 64_Ni 67_{Zn ((pf} 5 O2pg9 O2)72ng_{9 O2})_{23 O2}2_{1 O2} 23 O22 1 O2 4.53 (4.60) [36] 64_Zn 67_{Ge ((pf} 5 O2pg9 O2)72ng_{9 O2})_{23 O2}2_{1 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

(

(nf_{7 O2})261pf_{7 O2})_{19 O2}2_{1 O2}

state of31_{P is predicted to be at 13.58 MeV. Moreover, the}

₍

_(pf

5 O2pg9 O2)72ng_{9 O2})_{23 O2}2_{1 O2}

states of 61_{Zn ,} 67_{Zn and} 67_{Ge 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 (12_{C ,}9_{Be )}

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 on58_{Ni .}

Fig. 2. – Three-nucleon transfer on64_{Ni .}

12_{C beam was provided by the Strasbourg MP accelerator. The} 9_{Be 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 on64_{Zn .}

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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