ROMANIAN JOURNAL OF PHYSICS

NUMBER-PROJECTED SPECTROSCOPIC FACTOR FOR ONE PAIR OF LIKE-NUCLEONS TRANSFER REACTION WITHIN THE FRAMEWORK OF THE PICKET-FENCE MODEL Y. BENBOUZID, N.H. ALLAL, M. FELLAH 1

Université des Sciences et de la Technologie Houari Boumediene (USTHB), Faculté de Physique, Laboratoire de Physique Théorique, BP32 El-Alia, 16111 Bab-Ezzouar – Alger – Algeria, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] Received March 8, 2015

An expression of the spectroscopic factor for one pair of like-nucleons transfer reaction is established by means of the sharp-BCS (SBCS) number-projection method. It includes pairing correlations between like-particles and the particle-number fluctuations which are inherent to the BCS theory are eliminated. This expression appears as a limit of a sequence which converges as a function of the extraction degrees of the false components. A numerical study is then performed within the framework of the picket-fence schematic model. It is shown that the convergence as a function of the extraction degrees of the false components is rapid. Moreover, it is shown that the projection effect on the spectroscopic factors is non-negligible. Indeed, the relative discrepancy between the BCS and SBCS values may reach up to 6%. The particle-number fluctuations which are inherent to the BCS approach must then be taken into account in the calculation of the spectroscopic factor of such reactions. Key words: spectroscopic factor, pairing, projection. PACS: 21.60.-n, 21.10.Jx, 21.30.Fe

1. INTRODUCTION

Spectroscopic factors have been introduced fifty years ago in the theory of nuclear transfer reactions, in order to link nuclear reactions and structure [1]. This quantity has been the subject of many works and the subject is still relevant [1–11]. The spectroscopic factor is a tool which allows the comparison either between theory and experiments or between theoretical models. This factor may be calculated when a nucleus is modified from a state to another either in knockout or stripping reactions. During this transformation, the interactions with and between the transferred nucleons play an important role in the determination of nuclear forces [9]. Rom. Journ. Phys., Vol. 61, Nos. 3–4, P. 424–434, Bucharest, 2016

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Number-projected spectroscopic factor for one pair of like-nucleons transfer reaction

425

On the other hand, pairing correlations are of major importance in nuclear structure. The spectroscopic factor may be evaluated either by neglecting these correlations [1, 9–11], or by taking them into account [2–8]. In the latter case, the inclusion of pairing effects is generally made using the BCS approach [12]. However, the main defect of the latter is the non-conservation of the particlenumber [13]. Many methods have been proposed to restore the broken symmetry, such as the Lipkin-Nogami [14–15], the generator coordinate [13] or the Jia [16] methods. The aim of the present work is to perform a theoretical and numerical study of the spectroscopic factor for one pair of paired like-particles transfer reactions, by including the pairing correlations within an approach that conserves the particlenumber. We will then use the Sharp-BCS (SBCS) method [17]. It is an exact projection method which has the advantage to be discrete and simple to use. It has been used several times during the last forty years in order to study many nuclear observables, in the pairing between like-particles case [17–21], as well as in the neutron-proton pairing case [22–26]. In what follows, we will restrict ourselves to like-particles systems with an even particle-number. The present work is organized as follows: in order to define the notations, section 2 deals with a brief recall of the BCS and SBCS formalisms. In section 3, the expression of the spectroscopic factor in the framework of the BCS theory is recalled and the one corresponding to the SBCS approach is established. The formalism has been tested using the schematic picket-fence model. Numerical results are presented and discussed in section 4. Main conclusions are summarized in last section. 2. HAMILTONIAN-DIAGONALIZATION

Let us consider a system of 2P paired like-particles (neutrons or protons). It is described by the Hamiltonian [13]:





H    a a  a~ a~  G  a a~ a~ a ,  0

(1)

 0

where a and a~ respectively represent the creation operator of a particle in the state  and of its time-reverse state ~ ,   is the energy of these single-particle states and G is the pairing-strength which is assumed to be constant. The BCS ground-state is given by [12]:





   u j  v j a j a ~j 0 j 0

(2)

426

Y. Benbouzid, N.H. Allal, M. Fellah

3

Where |0> is the actual vacuum and v j and u j respectively represent the occupation and inoccupation probability amplitudes of the state |j>. Let us note that the state (2) can only describe a system with an even particlenumber. On the other hand, it is well known that the state |ψ> is not an eigen-state of the particle-number operator [13]. Indeed, the latter is a superposition of states which correspond to different numbers of pairs of particles. This defect may be corrected by means of a particle-number projection method. In the present work, we use the Sharp-BCS (SBCS) method [17]. In the framework of the latter, the projected wave-function is given by:

m  1







 m  C m   k z k P  u j  z k v j a j a ~j  cc  0 ,  k 0

j 0



(3)

where m is a non-negative integer which corresponds to the extraction degree of the false components, Cm is the normalization constant and

1  if k  0 or k  m  1  k  . k   2 , zk  exp i m  1   1 if 1  k  m

(4)

The notation cc means the complex conjugate with respect to zk. As soon as the condition

2m  1  max P,   P 

(5)

is satisfied (Ω being the total degeneracy of states), all the false components included in the state |ψ> are eliminated. The constant Cm is evaluated using the relation:

m1

1  2m  1

 

 k z k P  u 2j  z k v 2j   cc  0   

 C m2 

 k 0

j 0



.

(6)

3. SPECTROSCOPIC FACTOR

In what follows, we will establish the expression of the spectroscopic factor for one pair of paired like-particles transfer reactions: neutron-neutron (nn) or proton-proton (pp). Since the calculation of the spectroscopic factor in both cases

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427

(nn and pp transfer) is analogous, the calculation will be performed only in one case, the other one may be easily deduced. The spectroscopic factor is defined as the square of the matrix element which represents the overlap between the wavefunctions of the final state f (nucleus with (A+2) or (A2) nucleons) and the initial state i (target nucleus with A nucleons). Several definitions have been proposed in the literature [2], [5–8]. In the present work, we have chosen the definition used by Chasman [6]. The spectroscopic factor is then respectively given for Stripping (A→A+2) and Pick-up (A→A2) reactions by:

S strip   f  A  2  

ak ak~

2

i  A

(7)

k 2

S pick   f  A  2   ak~ ak i  A

(8)

k

|f(A+2)> and |f(A–2)> are the wave-functions of the final state, and |i(A)> is that of the initial state of the target nucleus. 3.1. BEFORE PROJECTION

Within the framework of the BCS theory, if one considers an even-even nucleus, the total wave-function which describes the nucleus (either in the initial or final state) is given by:

i  f    i  f  Ni  f   . i  f  Zi  f  

(9)

,

where the notations N and Z respectively refer to the neutron and proton systems and where  i  f  t i  f  (t = n, p) is given by Eq. (2), that is:





 i  f  t i  f    u ijt f   vijt f a jt a~j t 0

.

(10)

j 0

If one assumes that the neutron (respectively proton) system is not affected by the transfer of one pair of protons (respectively neutrons), one has, respectively:

 f N  i N   1 , where N f  N i  N ,

(11)

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5

 f Z   i Z   1, where Z f  Z i  Z .

(12)

For stripping and pickup reactions, respectively, Eqs. (7) and (8) may then be written:





(13)





(14)

BCS i f i f i f S strip tt    ut vt  u jt u jt  v jt v jt , t  n , p



j

BCS f i i f i f S pick tt    ut vt  u jt u jt  v jt v jt , t  n , p .



j

Some authors [2, 5, 7–8] consider that the reaction is very fast and the difference between the single-particle states of the initial and final systems is negligible, in particular in the term u ijt u jtf  vijt v jtf in Eqs. (13) and (14), which then





j

becomes equal to one. In this case, Eqs. (13) and (14) respectively become: BCS i f S strip tt    ut v t ,

BCS f i S pick tt    ut v t , t  n, p.



(15)



For our part, we have chosen to not use this approximation. Its validity will be studied in section 4 using numerical results. 3.2. AFTER PROJECTION

In this section, new expressions of the spectroscopic factor will be established, still by means of the definitions (7) and (8), but this time by using the projected wave-functions of the form (3); that is (for t = n, p):

 i  f  t 

proj

m1    Pi  f  i f   i f  i f     C mt    k z k t  u jt  z k v jt a jt a ~j t  cc  0 (16)   j 0  k 0 





Eqs. (9) and (11–12) then respectively become:

i  f    i  f  Ni  f   proj

proj



. i  f  Zi  f 

 proj ,

(17)

 f N   i N 

proj

 1 , where N f  N i  N ,

(18)

 f Z   i Z 

proj

 1, where Z f  Z i  Z .

(19)

proj

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429

The spectroscopic factor for stripping and pick-up reactions respectively become in this case:

m1 m' 1   Pti  Pti SBCS i f  i f i f i f S strip  C C tt  m' t mt     k  k' z k z k'  u t v t  u jt u jt  z k z k' v jt v jt  CC   k 0 k' 0   j







(20)



where the fact that Pt  Pt  1 has been taken into account, and: f

i

m 1m' 1 f f  Pt  Pt SBCS i f  S pick  C C zk' tt  m' t mt     k  k' zk  k 0 k' 0



 f i i f i f u v u u  z z v v  CC   t t  jt jt k k' jt jt   j





(21)



where the fact that Pt  Pt  1 has been taken into account. f

i

The notation CC means that we added similar terms where

 zk , zk ' 

is

replaced by zk , zk'  , then by zk , zk'  and finally by zk , zk'  . If one assumes, as suggested by some authors (see previous section), that the

difference between u ijt (respectively v ijt ) and u jtf (respectively v jtf ) is negligible in the product

 u ijt u jtf  zk zk' vijt v jtf , one has (for t = n,p): j

m 1m' 1   Pti  Pti SBCS i f  i f 2 2 S strip  C C tt  m' t mt     k  k' zk zk'  ut vt  u jt  zk zk' v jt  CC  (22)  k 0 k' 0   j m 1m' 1  f f  Pt  Pt SBCS i f  f i 2 2 S pick  C C   z z u v u  z z v  CC   (23)  t t  jt k k' jt tt  m' t mt   k k' k k'  k 0 k' 0   j









where u jt (respectively v jt ) represents u ijt  u jtf (respectively vijt  v jtf ) when j. The validity of this approximation will be also studied in next section, using numerical results. 4. NUMERICAL RESULTS-DISCUSSION

Calculations have been performed within the picket-fence schematic model [28] in which each level is doubly degenerate and the single-particle energies are

430


7

given by εk = k for k = 1, ..., Ω. We have first considered the BCS approach and then the SBCS one. Table 1 Values of the spectroscopic factor for one pair of paired like-particles transfer reactions (stripping and pick-up) calculated within the BCS approach for given values of the initial nucleon number (Ni), Ω and G. Columns 4 and 5 (S) correspond to Eqs. (13) and (14), columns 6 and 7 (S') correspond to Eq. (15) Ni

Ω

G (MeV)

BCS S strip

BCS S pick

BCS S' strip

BCS S' pick

8 16 32

8 16 32

0.800 0.475 0.375

11.5978 19.7344 38.1940

11.5978 19.7344 38.1940

13.8333 23.7823 45.0431

13.8333 23.7823 45.0431

4.1. CALCULATIONS WITHIN THE FRAMEWORK OF THE BCS THEORY

Values of the spectroscopic factor for one pair of paired like-particles transfer reactions (stripping and pick-up ) calculated within the BCS approach are reported in Table 1 for given values of the initial nucleon number (Ni), G and Ω. In the table, the notation S corresponds to Eqs. (13) and (14) (values obtained without approximation) and the notation S′ corresponds to Eq. (15) (values obtained using the approximation). The approximation effect will be discussed later. One notes in each case that the spectroscopic factors of stripping and pick-up reactions are the same. This result could be foreseeable since the definitions (7) and (8) are in fact equivalent. This observation is also valid in the case of the SBCS method, this is the reason why, we will use in what follows only one value denoted S (respectively S′). 4.2. CALCULATIONS WITHIN THE SBCS APPROACH

a. Test of the convergence

We have first tested the convergence of the SBCS method. We have then calculated the spectroscopic factor values for the stripping reaction of two paired nucleons as a function of the extraction degrees of the false components m and m′ for Ni = 8, Ω = 8 and G = 0.800 MeV, chosen as an example, in the framework of the picket-fence model (Table 2). One then notes that the convergence is rapid: the stability is reached as soon as m = m′ = 2. On the other hand, m and m′ must, in principle, simultaneously satisfy the condition (5) in order that all the false components will be eliminated in the wavefunction. In what follows, the m and m′ values predicted by the condition (5) will be denoted mth and mth’, respectively. These values are compared in Table 3 to those obtained after the effective convergence of the calculated SSBCS values, for

8


431

given values of Ni, Ω and G. One then observes that the effective convergence is clearly more rapid than the one predicted by the condition (5) (except in the case of Ni = 8). In what follows, in order to insure the convergence, we will use the values m = m′ = 4. Table 2 Variations of the values of the spectroscopic factor for the stripping reaction of two paired nucleons as a function of the extraction degrees of the false components m and m' for Ni = 8, Ω = 8 and G = 0.800 MeV m\m’ 0 1 2 3

0 11.6186 8.7829 8.7874 8.7874

1 8.6085 12.2309 12.2371 12.2371

2 8.6065 12.2280 12.2605 12.2605

3 8.6065 12.2280 12.2605 12.2605

Table 3 Comparison of the m and m’ values predicted using the condition (5) (mth and mth') with those obtained after the effective convergence of the SSBCS values for given for given values of the initial nucleon number (Ni), G and Ω Ni 8 16 32

Ω G (MeV) 8 0.800 16 0.475 32 0.375

mth 2 4 8

mth′ 2 4 8

m 2 3 3

m′ 2 3 3

b. Projection effect

We have then calculated the spectroscopic factor values for the same reactions and with the same parameters as those considered within the BCS approach. These values are reported in Table 4. Let us note that in this table the notation S corresponds to Eqs. (20) and (21) (values obtained without approximation), the notations S′ and S′′ respectively correspond to Eqs. (22) and (23) when u jt  u ijt and u jt  u jtf (values obtained using the approximation). The approximation effect will be discussed later. In order to evaluate the projection effect on the spectroscopic factor we defined the relative discrepancy: S  S SBCS  S BCS / S BCS . The values of δS(%) are reported in Table 5. One observes that the projection effect varies from one system to another. It is not negligible since it may reach up to about 6%. One may then conclude that the effect of the particle-number fluctuations in the BCS wavefunction must be taken into account when one calculates the spectroscopic factor of the considered reactions.

432


9

Table 4 Values of the spectroscopic factor for one pair of paired like-nucleons transfer reactions calculated within the SBCS for given values of the initial nucleon number (Ni), Ω and G. Column 4 (S) corresponds to Eqs. (20) and (21), columns 5 and 6 (S’ and S’’) correspond to Eqs. (22) and (23) where u jt  u ijt and u jt  u jtf respectively Ni

Ω

G (MeV)

S SBCS

S' SBCS

S' ' SBCS

8 16 32

8 16 32

0.800 0.475 0.375

12.2605 19.2805 36.4917

12.0913 19.7637 37.6520

12.8693 20.1861 37.9214

Table 5 Values of the relative discrepancy between the values of the spectroscopic factor calculated before and after projection, for one pair of like-nucleons transfer reactions, for given values of the initial nucleon number (Ni), Ω and G Ni 8 16 32

Ω 8 16 32

G (MeV) 0.800 0.475 0.375

δS(%) 5.71 2.30 4.45

c. Approximation effect

We have then studied the effect of the approximation used in Eqs. (15), (22) and (23). We have defined the relative discrepancy between the values obtained using the approximation (S′ and S") and without approximation (S): BCS ) Sappi( f )  S  S' S' '  / S . The corresponding values obtained before ( S app

SBCS SBCS and after ( S app ) projection are reported in Table 6. We denoted S app ( i ) the

SBCS discrepancy which corresponds to S′ and S app ( f ) the one which corresponds to

S". One observes that the approximation effect varies from one system to another. However, it may be very important in the case of the BCS theory since it may reach up to about 21%. It is also non-negligible after the projection since it may reach up to around 5%. This effect thus diminishes after the projection for each system. One may then conclude that this approximation is not justified in both BCS and SBCS approaches. Let us however recall that the present work calculations have been performed within the framework of a schematic model. These conclusions have to be confirmed in the case of a realistic model which allows comparison with experiment.

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433

Table 6 Values of the relative discrepancy between the values of the spectroscopic factor calculated with and SBCS without approximation (see the text for notations) in the framework of the BCS ( S app ) and SBCS SBCS SBCS ( S app and S app( f ) ) approaches, for one pair of like-nucleons transfer reactions for given values of (i)

the initial nucleon number (Ni), Ω and G Ni

Ω

G (MeV)

8 16 32

8 16 32

0.800 0.475 0.375

SBCS SBCS SBCS (%) S app (%) S app (%) S app (i) (f )

19.27 20.51 17.93

1.38 2.50 3.17

4.96 4.69 3.91

5. CONCLUSION

An expression of the spectroscopic factor for one pair of like-nucleons transfer reaction has been established by means of the (SBCS) projection method. It includes pairing correlations between like-particles but the particle-number fluctuations which are inherent to the BCS theory have been eliminated. This new expression appears as a limit of a sequence which converges as a function of the extraction degrees of the false components. A numerical study has then been performed within the framework of the picket-fence schematic model. The convergence of the method has first been checked. It was shown that the convergence as a function of the extraction degrees of the false components is rapid. By comparing the values of the spectroscopic factor obtained using the BCS and SBCS approaches for various sets of parameters, it was shown that the projection effect is non-negligible. Indeed, the relative discrepancy between the BCS and SBCS values may reach up to 6%. A particle-number projection is thus necessary. The effect of the approximation in the spectroscopic factor expression proposed by some authors has then been studied. It was shown that the relative discrepancy between the approached and exact values may reach up to 21% in the case of the BCS theory and 5% in the case of the SBCS method. This approximation seems thus to be not justified in both approaches. Let us however recall that the present work calculations have been performed in the framework of a schematic model. These conclusions have to be confirmed using a realistic model which allows comparison with experiment. REFERENCES 1. 2.

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