**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})2_{5}1*, (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 40*1_{) 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 j*1*shell and m particles to j*2 shell, outside the

*core A*0*with a total spin J*04 01*and an isospin T*0*4 (N 2 Z) O2 , the excitation energy in*

*the final nuclear state (J , T) is given by [8]:*

*E * (A*0*1 n 1 m , j*1*nj*2*mJT) 4E*B*(A*0, g.s. ) 2
(1)
*2 E*B*(A*0*1 n 1 m , g.s. ) 1 ne( j*1*) 1me( j*2*) 1E*C*1 a j*1*nj*2*m*N

### !

*l 4kEl*

*n 1m*

*V(k , l) Njn*1

*j*2

*m*b

*JT*, where

**T 4T**0

**1 T**

*p*.

**T***p* *is the extra nucleons isospin, E*B*the negative experimental binding energy and e( j)*

the single-particle energy:

*e( j) 4E*B*(A*0*1 1 , j) 2 E*B*(A*0, g.s. ) ,

(2) where

*E*B*(A*0*1 1 , j) 4 E*B*(A*0*1 1 , g.s. ) 1 E * (A*0*1 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 A*0*1 1 nucleus. The Coulomb interaction energy E*C *for Z 8 protons in j-orbit*

outside the closed core is given by [9]

*E*C*(Z 8) 4cZ 81*
1
2*Z 8(Z 821) a1*

## k

1 2*Z*

## 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 E*C

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 j*1*nj*2*m*N

### !

*1 4kEl*

*n 1m*

*V(k , l) N j*1

*nj*2

*m*b

*JT*4

### !

*J 8 T 8*

*C*1

_{J 8 T 8}a j*j*2

*NV( 1 , 2 ) Nj*1

*j*2b

*J 8 T 8*, (5)

*where a j*1*j*2*NV( 1 , 2 ) Nj*1*j*2b*J 8 T 8are the two-body matrix elements (TBME) and CJ 8 T 8*are
*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 8

*n(n 22)1T(T11) ,*

### !

*l*

*C*( 0 )

*l*4 1 8

*n(n 12)2*1 2

*T(T 11) ,*

### !

*l*

*C*( 0 )

*l*1

### !

*k*

*C*( 1 )

*k*4 1 2

*n(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 j*1*and j*2*outside the core, the total isospin T is:*

**T 4T**0**1 T***p*.

*1) Identical nucleons*

*The z-component of the isospin of the two extra nucleons is given by Tpz*4 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 * (A*0*1 2 N , j*1*j*2*JT) 4E*B*(A*0*, g.s. ) 2E*B*(A*0*1 2 N , g.s. ) 1 e( j*1*) 1e( j*2) 1

(7)

*1E*C*1 a j*1*j*2*NVNNNj*1*j*2b*JT*,
*where N is equal to p for proton or n for neutron.*

*2) Non-identical nucleons*

*The p-n pair process requires Tpz*4 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 T*0 *to give T*

according to the relation

**T 4T**0**1 T***p*.
Then instead of eq. (7) we obtain

*E * (A*0*1 p 1 n , j*1*j*2*JT) 4E*B*(A*0*, g.s. ) 2E*B*(A*0*1 p 1 n , g.s. ) 1 e( j*1*) 1e( j*2) 1

(8)

*1E*C1

1

2*]a j*1*j*2*NVpnNj*1*j*2b*JT 401 a j*1*j*2*NVppNj*1*j*2b*JT 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 4E*B*(A*0*1 p 1 n) 2 E*B*(A*0*1 p) 2 E*B*(A*0*1 n) 1 E*B*(A*0) .

(9)

**2**.*2. Three-particle states. – In the case of three particles moving in two orbits j*1and
*j*2outside the core, the last term in eq. (1) can be written as [8]:

*a j*2
1*j*2N

### !

*1 4kEl*3

*V(k , l) Nj*2 1

*j*2b

*JT4 aVj*1

*b 4 aVj*2

*b 1 aVj*1

*j*2b (10) with

### .

### `

### `

### /

### `

### `

### ´

*aVj*1

*b 4 a j*2 1

*NV( 1 , 2 ) Nj*12b ,

*aVj*2b 40 ,

*aVj*1

*j*2

*b 42(2ja1 1 )( 2 ta*1 1 )

### !

*jutu*

*( 2 ju1 1 )( 2 tu*1 1 ) . / ´

*j*1

*j*2

*j*2

*J*

*ja*

*ju*ˆ ¨ ˜ 2 Q Q./ ´ 1 O2 1 O2 1 O2

*T*

*ta*

*tu*ˆ ¨ ˜ 2

*a j*1

*j*2

*NVNj*1

*j*2b

*jutu*, (11)

where
**j***a***4 j**1**1 j**1,
**j***u***4 j**1**1 j**2,
**J 4j**1**1 j***u*,
**t***a***4 t**1**1 t**1,
**t***u***4 t**1**1 t**2,
**T 4t**1**1 t***u*,

*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(d*5 O2)2*p*1 O2N

### !

*1 4kEl*
3

*V(k , l) N(d*5 O2)2*p*1 O2b11 O22_{1 O2}4

*4 C*51_{0}*a(d*_{5 O2})2*NVN(d*_{5 O2})2b_{5}1_{0}*1C*_{3}2_{0}*ap*_{1 O2}*d*_{5 O2}*NVNp*_{1 O2}*d*_{5 O2}b_{3}2_{0}*1C*_{3}2_{1}*ap*_{1 O2}*d*_{5 O2}*NVNp*_{1 O2}*f*_{5 O2}b_{3}2_{1},

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

*C*51_{0}4 1 , *C*_{3}2_{0}4 1 O2 , *C*_{3}2_{1}4 3 O2 .

For the

### (

*( 1 g*

_{9 O2})2

*1 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(g*9 O2)2

*f*5 O2b23 O22

_{1 O2}4

*4 C*91

_{0}

*a(g*

_{9 O2})2

*NVN(g*

_{9 O2})2b

_{9}1

_{0}

*1C*

_{7}2

_{0}

*a f*

_{5 O2}

*g*

_{9 O2}

*NVN f*

_{5 O2}

*g*

_{9 O2}b

_{7}2

_{0}

*1C*

_{7}2

_{1}

*a f*

_{5 O2}

*g*

_{9 O2}

*NVNf*

_{5 O2}

*g*

_{9 O2}b

_{7}2

_{1}, with

*C*91

_{0}4 1 ,

*C*

_{7}2

_{0}4 1 O2 ,

*C*

_{7}2

_{1}4 3 O2

and finally for the

### (

*( 1 g*

_{9 O2})2

*1 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(g*9 O2)2

*f*5 O2b21 O22

_{1 O2}4

*4 C*81

_{0}

*a(g*

_{9 O2})2

*NVN(g*

_{9 O2})2b

_{8}1

_{0}

*1C*

_{7}2

_{0}

*a f*

_{5 O2}

*g*

_{9 O2}

*NVN f*

_{5 O2}

*g*

_{9 O2}b

_{7}2

_{0}

*1C*

_{7}2

_{1}

*a f*

_{5 O2}

*g*

_{9 O2}

*NVNf*

_{5 O2}

*g*

_{9 O2}b

_{7}2

_{1}, with

*C*81

_{0}4 1 ,

*C*

_{7}2

_{0}4 3 O2 ,

*C*

_{7}2

_{1}4 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 j*3
N

### !

*1 4kElV(k , l) Nj*3

_{b}

*JT*4 3 3

### !

*J 8 T 8a j*3

*JTntN( j*2

*J 8 T 8 n8 t 8b*2

*2*

_{a j}*NVNj*2b

*, (12)*

_{J 8 T 8}*where a j*3

*JTntN( j*2

*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 O2}* _{are coupled to J 413O2}*1

_{and T 41O2, eq. (12) can be}rewritten as follows:
*a(d*5 O2)3N

### !

*1 4kEl*3

*V(k , l) N(d*5 O2)3b13 O21

_{1 O2}4 4 3 2

*[a(d*5 O2) 2

*NVN(d*5 O2)2b41

_{1}

*1 a(d*

_{5 O2})2

*NVN(d*

_{5 O2})2b

_{5}1

_{0}] .

*For the orbit f*_{7 O2}* _{, with J 419O2}*2

_{and T 41O2 we obtain}*a( f*7 O2)3N

### !

*1 4kEl*3

*V(k , l) N( f*7 O2)3b19 O21

_{1 O2}4 3 2

*[a( f*7 O2) 2

*NVN( f*7 O2)2b61

_{1}

*1 a( f*

_{7 O2})2

*NVN( f*

_{7 O2})2b

_{7}1

_{0}] .

**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 O2}_{NVNg}_{9 O2}*f*_{5 O2}b72_{0}has been estimated from eq. (9). In table I we show the TBME

*and Coulomb energies E*C 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 A*0*1 1 , formed by the target A*0plus 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 *A*01 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 )}

TABLE*I. – Two-body matrix elements (TBME) and Coulomb interaction energy E*C.

Mass region TBME (MeV) *E*C( MeV )

12_{C} * _{ap}*
1 O2

*d*5 O2

*NVNp*12

*d*5 O2b32

_{0}

*ap*1 O2

*d*5 O2

*NVNp*12

*d*5 O2b32

_{1}

*ad*2 5 O2

*NVNd*5 O22 b41

_{1}

*ad*2 5 O2

*NVNd*5 O22 b51

_{0}

*ad*3 O2

*d*5 O2

*NVNd*3 O2

*d*5 O2b41

_{1}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}

*2 5 O2*

_{ad}*NVNd*5 O22 b41

_{1}

*ad*2 5 O2

*NVNd*5 O22 b51

_{0}

*ad*3 O2

*d*5 O2

*NVNd*3 O2

*d*5 O2b41

_{1}20 .08 [ 2 , 3 , 12 ] 23 .86 [ 2 , 3 , 13 ] 20 .05 [ 2 , 3 , 14 ] 0.4 [2, 3] 28

_{Si}

*3 O2*

_{ad}*f*7 O2

*NVNd*3 O2

*f*7 O2b52

_{1}

*a f*2 7 O2

*NVN f*7 O22 b61

_{1}

*a f*2 7 O2

*NVN f*7 O22 b71

_{0}

*ad*3 O2

*f*7 O2

*NVNd*3 O2

*f*7 O2b52

_{0}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}

*2 7 O2*

_{a f}*NVN f*7 O22 b61

_{1}

*a f*2 7 O2

*NVN f*7 O22 b71

_{0}

*a f*7 O2

*f*5 O2

*NVN f*7 O2

*f*5 O2b61

_{1}20 .12 [ 2 , 3 , 16 ] 22 .40 [ 2 , 3 , 11 ] 21 .147 [ 2 , 3 , 16 ] 0.4 [2, 3] Fe, Ni, Zn

*ag*2 9 O2

*NVNg*9 O22 b81

_{1}

*ag*2 9 O2

*NVNg*9 O22 b91

_{0}

*ag*9 O2

*f*5 O2

*NVNg*9 O2

*f*5 O2b72

_{0}

*ag*9 O2

*f*5 O2

*NVNg*9 O2

*f*5 O2b72

_{1}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.

TABLE*II. – 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}

TABLE*III. – Excitation energies for two-neutron states.*

Core Nucleus Configuration *Jp _{T}*

_{E *}The(MeV) (*a*) *E **Exp(MeV)
12_{C}
14_{C} * _{(np}*
1 O2

*nd*5 O2)32

_{1}

*(nd*

_{5 O2})2 41

_{1}

*(nd*

_{5 O2}

*nd*

_{3 O2})41

_{1}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}

*5 O2)241*

_{(nd}_{1}

*(nd*

_{5 O2}

*nd*

_{3 O2})41

_{1}41

_{1}41

_{1}3.82 8.93 3.56 [5] 9.0 [5] 28

_{Si}30

_{Si}

*3 O2*

_{(nd}*nf*7 O2)52

_{1}

*(nf*

_{7 O2})2 61

_{1}52

_{1}61

_{1}7.03 9.25 7.04 [5] 8.95 [5] 40

_{Ca}42

_{Ca}

*7 O2)261*

_{(nf}_{1}

*(nf*

_{7 O2}

*nf*

_{5 O2})61

_{1}61

_{1}61

_{1}3.00 7.63 3.19 [5] 7.40 [5] 54

_{Fe}56

_{Fe}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}72

_{1}81

_{1}6.62 9.51 6.03 [33] 9.28 [33] 58

_{Ni}60

_{Ni}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}72

_{1}81

_{1}5.78 8.50 5.30 [2, 33] 8.53 [2, 33] 60

_{Ni}62

_{Ni}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}72

_{1}81

_{1}4.97 7.02 4.66 [2, 33] 7.19 [2, 33] 62

_{Ni}64

_{Ni}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}72

_{1}81

_{1}4.11 5.40 4.52 [2, 33] 5.81 [2, 33] 64

_{Ni}66

_{Ni}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}72

_{1}81

_{1}3.89 4.91 4.08 [2, 33] 5.14 [2, 33] 64

_{Zn}66

_{Zn}

*5 O2*

_{(nf}*ng*9 O2)72

_{1}

*(ng*

_{9 O2})2 81

_{1}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.

TABLE*IV. – 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 O2

*pd*5 O2)32

_{1}

*(pd*

_{5 O2})2 41

_{1}

*(pd*

_{5 O2}

*pd*

_{3 O2})41

_{1}32

_{1}41

_{1}41

_{1}6.28 10.10 13.36 6.27 9.90 14.1 16

_{O}18

_{Ne}

*5 O2)241*

_{(pd}_{1}

*(pd*

_{5 O2}

*pd*

_{3 O2})41

_{1}41

_{1}41

_{1}3.64 8.67 3.4 7.9 28

_{Si}30

_{S}

*3 O2*

_{(pd}*pf*7 O2)52

_{1}

*(pf*

_{7 O2})2 61

_{1}52

_{1}61

_{1}6.76 8.63 6.7 8.3 40

_{Ca}42

_{Ti}

*7 O2)261*

_{(pf}_{1}

*(pf*

_{7 O2}

*pf*

_{5 O2})61

_{1}61

_{1}61

_{1}2.96 7.64 3.04 7.5 54

_{Fe}56

_{Ni}

*7 O2*

_{(pf}*pf*5 O2)61

_{1}

*(pf*

_{7 O2}

*pg*

_{9 O2})72

_{1}

*(pf*

_{5 O2}

*pg*

_{9 O2})72

_{1}

*(pg*

_{9 O2})2 81

_{1}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}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}5.70 8.13 5.30 7.98 60

_{Ni}62

_{Zn}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}5.40 7.46 5.19 7.54 62

_{Ni}64

_{Zn}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}5.10 6.94 5.30 6.70 64

_{Ni}66

_{Zn}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}5.16 6.88 5.20 6.85 64

_{Zn}66

_{Ge}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}4.59 6.73 4.59 6.63 66

_{Zn}68

_{Ge}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}4.59 6.61 4.61 6.30 68

_{Zn}70

_{Ge}

*5 O2*

_{(pf}*pg*9 O2)72

_{1}

*(pg*

_{9 O2})2 81

_{1}72

_{1}81

_{1}4.51 6.2 4.33 5.73

*(a) This work.*

TABLE*V. – Excitation energies for neutron-proton states.*

Core Nucleus Configuration *Jp _{T}*

_{E *}The(MeV) (*a*) *E **Exp(MeV)

12_{C} 14_{N} * _{(np}*
1 O2

*pd*5 O2)32

_{0}

*(pd*

_{5 O2}

*nd*

_{5 O2})51

_{0}32

_{0}51

_{0}5.92 8.75 5.83 [34] 8.96 [34] 16

_{O}18

_{F}

*5 O2*

_{(pd}*nd*5 O2)51

_{0}51 0 1.17 1.12 [1, 2] 28

_{Si}30

_{P}

*7 O2*

_{(pf}*nf*7 O2)71

_{0}71 0 5.75 6.40 [1, 2] 40

_{Ca}42

_{Sc}

*7 O2*

_{(pf}*nf*7 O2)71

_{0}71 0 0.79 0.617 [24]

*(a) This work.*

TABLE*VI. – 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
41*pd*_{5 O2})_{13 O2}1_{1 O2}
(*(nd*_{5 O2})2
41*nd*_{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
41*pd*_{5 O2})_{13 O2}1_{1 O2}
(*(nd*_{5 O2})2
41*nd*_{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
61*pf*_{7 O2})_{19 O2}2_{1 O2}
(*(pf*_{7 O2})2
61*nf*_{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
61*pf*_{7 O2})_{19 O2}2_{1 O2}
(*(pf*_{7 O2})2
61*nf*_{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 O2

*pg*9 O2)72

*ng*

_{9 O2})

_{23 O2}2

_{1 O2}23 O22 1 O2 7.38 (6.92) [36] 64

_{Ni}67

_{Zn}

_{(}

*5 O2*

_{(pf}*pg*9 O2)72

*ng*

_{9 O2})

_{23 O2}2

_{1 O2}23 O22 1 O2 4.53 (4.60) [36] 64

_{Zn}67

_{Ge}

_{(}

*5 O2*

_{(pf}*pg*9 O2)72

*ng*

_{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})261

*pf*

_{7 O2})

_{19 O2}2

_{1 O2}

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

_{(}

_{(pf}5 O2*pg*9 O2)72*ng*_{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 E*R 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 E*R 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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