Two-nucleon Hulthen-type interactions for few higher partial waves

P

— journal of April 2015

physics pp. 555–567

Two-nucleon Hulthen-type interactions for few higher partial waves

U LAHA^∗and J BHOI

Department of Physics, National Institute of Technology, Jamshedpur 831 014, India

∗Corresponding author. E-mail: ujjwal.laha@gmail.com

MS received 11 February 2014; revised 2 April 2014; accepted 22 April 2014 DOI: 10.1007/s12043-014-0845-z; ePublication: 29 November 2014

Abstract. By exploiting supersymmetry and factorization method, higher partial wave nucleon–

nucleon potentials(=1,2,3)for a few selected triplet and singlet states are generated from the ground-state interaction and wave function. The nuclear Hulthen potential and the corresponding wave function with the parameters of Arnold and Mackellar which fit the deuteron binding energy are used as the starting point of our calculation. The scattering phase shifts are computed for the con- structed potentials using phase function method to examine the merit of our approach to the problem.

Keywords. Nuclear Hulthen potential; supersymmetry and factorization; higher partial wave potentials; phase function method; scattering phase shifts.

PACS Nos 13.75.Cs; 30.65.Nk; 11.30.Pb; 24.10.−i

1. Introduction

The conservation laws of energy, linear momentum and angular momentum arise from the fact that space-time is homogeneous and isotropic, leading to invariance of the Hamil- tonian under translation and rotation. The invariance of the supersymmetric Hamiltonian under translation in superspace corresponds to the existence of the supercharges that com- mute with the supersymmetric Hamiltonian leading to definite relations between bosonic and fermionic spectra. Like field theory, in supersymmetric quantum mechanics (SQM), the existence of a generating operator that commutes with the Hamiltonian, leads to cer- tain specific relation between the spectra and the eigenfunctions of the component parts of the supersymmetric Hamiltonian. Witten [1] first developed the methodology to study quantum mechanical system governed by an algebra, identical to that of supersymmetry in field theory. For any Hamiltonian with one degree of freedom, a comparison Hamil- tonian can be constructed such that the resulting system as a whole is supersymmetric [1–3]. The Hamiltonian hierarchy problems in SQM lead to the addition of appropriate centrifugal barriers and consequently, higher partial wave potentials are generated fairly accurately in atomic physics.

In the case of a Coulomb potential, the so-called accidental degeneracy is recovered as a natural consequence [4]. At small values of the radial coordinater, the Hulthen potential behaves like a Coulomb potential, whereas for large values ofrit decreases exponentially so that its capacity for bound state is smaller than the Coulomb potential. Also the Hulthen potential serves as a model for the interaction between nucleons in deuteron. Because of the above similarity and points of contrasts between the Coulomb and the Hulthen potentials, it may be of considerable interest to generate supersymmetric partners of the latter and study their partner potentials, related physical observables etc. which have important applications in quantum scattering theory. In the recent past, energy eigenvalues and eigenfunctions, related to atomic cases, for Hulthen potential with non-zero angular momentum have been reported in a number of publications [5–8]. As opposed to atomic cases, nuclear potentials are basically state-dependent.

About 40 years ago, Arnold and Mackellar [9] parametrized Hulthen potential to fit the deuteron binding energy andS-wave scattering lengths. The nucleon–nucleon systems have been studied extensively and they provide a large number of reliable experimen- tal data [10–18]. These data are rather accurate for proton–proton (p-p) system while minor uncertainties are involved for the neutron–proton (n–p) system [19]. The phase shift analysis predicted by the above groups [10–18] involves many parameters. The phase shift data presented by these groups do not differ much except that the methods employed were refined in one way or the other. Thus, one can safely rely on these nucleon–nucleon phase shift values.

In the recent past [20,21] we have studied higher partial wave nucleon–nucleon scattering phase shifts by energy-dependent and independent two-nucleon interactions developed via the supersymmetry-inspired factorization method which fit well at low and intermediate energy scattering data but discern at high energy range. These higher par- tial wave phase shifts have been computed using two S-wave parameters only by the application of the phase function method. For refinement in the phase shift values for the partial waves > 0 one needs to improve the parameters involved in calculations.

The present paper addresses itself to compute higher partial wave scattering phase shifts for supersymmetrically (SUSY) generated potentials and their modified counterparts (MOD-SUSY) with theS-wave parameters [9]. We also compute the values of the same by varying one of the two parameters involved in the potential for both n–p and p–p systems by the phase-function method (PFM) [22] and compare them with the standard works [10,11]. In §2, we briefly discuss our two-nucleon interactions with their param- eters and the PFM [22] for dealing with scattering phase shifts. We shall demonstrate the usefulness of our constructed potentials in §3 by computing the values of p–p and the associated n–p scattering phase shifts for the partial waves=0,1,2 and 3 and present concluding remarks.

2. Two-nucleon interactions and PFM

Following the approach discussed in our earlier papers [20,21], the supersymmetric part- ner potentialVi(higher partial wave potential) generated from the ground-state interaction V₀is given by

V_i(x) = V_i−1(x)− ∂²

∂x² ln_i⁽⁰⁾₋₁, i= 1,2,3. . . , (1)

where_i⁽⁰⁾₋₁ is the eigenfunction of the appropriate angular momentum state. The appli- cations of the above relations to the nuclear Hulthen potential are in order. The nuclear Hulthen potential [9,20,21] is expressed as

V_0N(r) = −(β²− α²) e^−βr

(e^−αr − e^−βr), (2)

whereβ stands for the inverse range parameter and the wave numberαis related to the strength of the interaction.

There exists an experimental situation which involves scattering by additive interac- tions like proton–proton (p–p) scattering. In charged particle scattering, the long range of the electromagnetic interaction is of great disadvantage. It has been argued that pure Coulomb potential never really occurs in nature and becomes somewhat screened at a certain distance, for example, the famous Rutherford scattering. This effect of screening should invariably affect the theory and the interpretation of data. To that end, a partic- ular screened Coulomb potential, the Hulthen potential, is considered. Thus, for p–p scattering, theS-wave effective potentialV_0P(r)is written as

V_0P(r) = V_0A(r)+V_0N(r), (3)

where

V_0A(r) = V₀ e^−r/a

(1 − e^−r/a) (4)

is theS-wave Hulthen or screened Coulomb potential with atomic parametersV₀ anda.

The corresponding higher partial wave potentials ( = 1,2,3) for the p–p system are defined as

V_1P(r) = V_0A(r) + V_1N(r), (5)

V_2P(r) = V_0A(r) + V_2N(r) (6)

and

V_3P(r) = V_0A(r) + V_3N(r) (7)

with

V_N= (+1) (β−α)²e^{−(α+β) r}

2(e^−αr − e^−βr)² , =1,2, . . . . (8) In the above formalism it is observed that the centrifugal barriers are lagging behind a factor of 2. To simulate the proper effect of the centrifugal term, a constant factor of 2 is incorporated in eq. (8) and in view of this, the modifiedP,DandF-wave potentials for p–p scattering are redefined as

V_1PM(r) = V_0A(r) +V_1NM(r), (9)

V_2PM(r) = V_0A(r) +V_2NM(r) (10)

and

V_3PM(r) = V_0A(r) +V_3NM(r), (11)

where

V_NM=V_0N+(+1) (β−α)²e^{−(α+β) r}

(e^−αr − e^−βr)² , =1,2, . . . . (12) Here the subscript M stands for the modified potentials. The above potentials are defined by twoS-wave parametersαandβrespectively. The parameterαis associated with the strength of the potential and has a strong effect on the physical observables at low and intermediate energies rather than the range parameter. In view of this, we propose another set of interactions by varying only one parameterα, keepingβ = 1.4054 fm⁻¹ fixed, with the values prescribed in the following table to achieve best fit to phase shift values within the framework of this model and examine the effectiveness of our modification in the generated potential. Associated phase shifts for supersymmetrically developed, mod- ified and one-parameter parameterized interactions will be computed by the application of PFM [22].

For a local potential, the phase functionδ(k, r)that satisfies a first-order non-linear differential equation [22] reads as

δ(k, r) = −k⁻¹V (r)[ ˆj(kr)cos δ(k, r) − ˆη(kr) sin δ(k, r)]², (13) where jˆ(kr)andηˆ(kr)are the Riccati–Bessel functions withhˆ⁽¹⁾ (x) = − ˆη(x) + ijˆ(x). The scattering phase shift δ(k) is obtained by solving the equation from the origin to the asymptotic region with the initial conditionδ(k,0) = 0.

3. Results and discussion

In figures 1–5, we portray the p–p and n–p potentials as functions of distance for the par- tial wave states¹S₀,³P₀,³P₁,¹D₂and³F₃withλ = −5.237 fm⁻³andβ =1.4054 fm⁻¹ for singlet state scattering,λ = −7.533 fm⁻³ andβ = 1.4054 fm⁻¹ for triplet state scattering [9]. It is seen that each pair of n–p and p–p potential curves in figures 1–5 are not resolved distinctly in the scale of the figures. However, to have a clear view a smaller part of it with the scales of higher resolution is projected on the top right-hand corner of the respective figures. These sets of potentials are termed as SUSY and MOD-SUSY according as their centrifugal terms are associated to eqs (8) and (12) respectively. We also plot the p–p and n–p potentials in figures 2–5 with the parameters given in table 1 for respective states as well. These potentials are designated as one-parameter variation potentials (SUSY-PRM or MOD-SUSY-PRM). It is observed that in figures 2–5 repulsive cores develop in the generated potentials. These potentials, generated from¹S₀parameters correspond to¹D₂state, while those from³S₁state correspond to the³P₀,³P₁and³F₃states respectively. It is noticed that the potentials with the parameters of table 1 produce better agreement with the nature and values of phase shifts of the respective states. The corre- sponding singlet and triplet state phase shifts have been computed using the PFM and we have presented them in figures 6–10 as a function of laboratory energy up to 300 MeV along with the values of Arndt et al [10] and Gross and Stadler [11]. Note that the results for the pure nuclear phase shifts have been obtained by turning off the atomic Hulthen interactionV_0A(r)in the associated numerical routine for generating p–p phase shifts.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 -500

-400 -300 -200 -100 0

0.2 0.4

-300 -200 -100

V (r) (M eV )

r (fm)

Arnold-Mackellar-np Arnold-Mackellar-pp

Figure 1. ¹S₀potentials as a function ofr.

In figure 6, the¹S₀phase shiftsδ_ppandδ_npagree well with that of Arndt et al [10], Gross and Stadler [11] and Lacombe et al [16] forE_Lab ≤25 MeV. Beyond 25 MeV the phase shifts differ significantly with energy. In our earlier publications [20,21], we observed that our phase shift values for ³S₁ state with the parameters of Arnold and Mackellar [9] also agree well with that of the standard values [10,16] up toE_Lab = 25 MeV. As a consequence, one may expect reasonable fit to p–p and n–p scattering phases at low energies for higher partial waves too.

The sets of potentials, namely SUSY, SUSY-PRM and MOD-SUSY-PRM for both n–p and p–p systems are quite capable of producing correct nature of phase shifts for the³P₀ state. Looking at figure 2 closely, we notice that the SUSY-PRM potential has a strong attractive part compared to other interactions. In figure 7, the phase shiftsδ_ppandδ_npfor pure SUSY potential contribute positive values to phase shifts up to 175 MeV and 275 MeV respectively. Beyond that they change sign. These phase shifts correspond to the

3P₀state but with lower numerical values. The other phase shiftsδ_pp andδ_np for SUSY- PRM produce correct values for³P₀state up toE_Lab =75 MeV, but do not change sign within 300 MeV due to their strong attraction compared to SUSY and MOD-SUSY-PRM interactions. Our phase-shift values for both p–p scattering and n-p scattering produce better agreement with those of Arndt et al [10] up to 175 MeV and up to 60 MeV with Gross and Stadler [11]. They change sign at 300 and 275 MeV respectively. In this state, it is noticed that MOD-SUSY-PRM potential is superior to pure SUSY or SUSY-PRM potentials.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0

100 200 300 400 500

0.5 1.0 1.5

0 25

V (r) (M eV )

r (fm)

SUSY-np SUSY-pp SUSY-PRM-np SUSY-PRM-pp MOD-SUSY-PRM-np MOD-SUSY-PRM-pp

Figure 2. Supersymmetry generated along with its one-parameter variation ³P₀ potentials as a function of distancer.

In figure 8, we plotted three sets of phase shifts for³P₁state with MOD-SUSY, SUSY- PRM and MOD-SUSY-PRM potentials for both n–p and p–p systems. Out of these three sets, the values for MOD-SUSY-PRM produce better agreement with refs [10,11] up to 150 MeV, while those with MOD-SUSY interactions match up to 75 MeV. The phase-shift values with SUSY-PRM, although produce the correct nature; differ from the standard values due to the strong repulsive character of the potential. This observation establishes the need of modification in the associated interaction.

Our constructed¹D₂ potentials, both SUSY and MOD-SUSY, plotted in figure 4, are completely repulsive in nature which is not consistent with the real nature of the same.

This arises due to the addition of centrifugal barriers to the¹S₀potential. Thus, we obtain negative scattering phase shifts (figure 9) which are in complete disagreement with stan- dard values [10–18]. However, our SUSY-PRM and MOD-SUSY-PRM interactions are quite capable of producing correct nature of ¹D₂ phase shifts at low and intermediate energy ranges which agree well with refs [10,11] up to 125 MeV exceptδ_pp for SUSY- PRM. In this case also, the modified parametrized interaction proves its superiority over pure supersymmetry-generated potential.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0

100 200 300 400 500

1.0 1.5

0 50

V (r) (M eV )

r (fm)

MOD-SUSY-np MOD-SUSY-pp SUSY-PRM-np SUSY-PRM-pp MOD-SUSY-PRM-np MOD-SUSY-PRM-pp

Figure 3. Supersymmetry generated along with its one-parameter variation ³P₁ potentials as a function of distancer.

In figure 10, we portray both p–p and n–p scattering phase shifts for our constructed potentials forE_Lab between 20 and 300 MeV. All these phase shifts produce correct³F₃ nature. Among these, the numerical values for SUSY phase shifts permit comparison with that of ref. [10] up to 50 MeV whereas those for SUSY-PRM compare well with refs [10,11] up to 150 MeV. For the modified parts, MOD-SUSY-PRM values match up to 75 MeV whereas MOD-SUSY phase values at very low energies. For this partial wave state we note that SUSY interactions are more acceptable compared to their modified counterpart. This is because of the fact that for higher partial wave states the strong repulsive centrifugal barriers play a crucial role in the effective interaction.

From our observation, it is reflected that our constructed potentials, namely SUSY, MOD-SUSY (except¹D₂), SUSY-PRM and MOD-SUSY-PRM are quite capable of pro- ducing the nature of phase shifts of respective higher angular momentum states. To produce the correct nature of interaction and refinement in phase-shift values, a simple one-parameter(α)variation technique has been adopted here. Here the higher partial wave supersymmetric potentials are generated by adding a centrifugal barrier-like term. Among these sets of potentials, the modified interactions (MOD-SUSY and MOD-SUSY-PRM) are the most effective and efficient ones that justify our modification for the addition of a

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0

100 200 300 400 500

2.0 2.5

0 30

V (r) (M eV )

r (fm)

SUSY-np SUSY-pp SUSY-PRM-np SUSY-PRM-pp MOD-SUSY-np MOD-SUSY-pp MOD-SUSY-PRM-np MOD-SUSY-PRM-pp

Figure 4. Supersymmetry generated along with its one-parameter variation ¹D₂ potentials as a function of distancer.

normalization constant in the interaction up to=2. However, for angular momentum states > 2, SUSY and SUSY-PRM interactions are superior to MOD ones. Conse- quently, our p–p interactions become more repulsive than n–p interactions along with the addition of repulsive electromagnetic interaction. The supersymmetric transforma- tion used in the present paper eliminates the ground state leaving rest of the spectrum unaltered. There exist other transformations which add new ground states or maintain the same spectrum with additional singularities in the potential. The singular potentials originating from such transformations have proved useful in understanding the relation between deep and shallow potentials that describe a wide variety of scattering problems in nuclear physics [23,24]. Baye [23] has explained that singular shallow (α–α) potential of Ali and Bodmer [25] is a supersymmetric partner to the deep potential of Buck et al [26]. Combining inverse scattering theory and algebra of SQM, Sparenberg and Baye [27] have also constructed phase-equivalent shallow potential with a repulsive core to the deep³S₁–³D₁n–p potential.

As observed in the present paper, the higher partial wave triplet state potentials and consequently the associated phase shifts are more consistent than the singlet states. There- fore, one may conclude that the triplet series of two-nucleon forces possesses a special kind of symmetry compared to singlet series to facilitate supersymmetry operation. Also

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0

100 200 300 400 500

2.0 2.5

0 50

V (r) (M eV )

r (fm)

SUSY-np SUSY-pp SUSY-PRM-np SUSY-PRM-pp MOD-SUSY-np MOD-SUSY-pp MOD-SUSY-PRM-np MOD-SUSY-PRM-pp

Figure 5. Supersymmetry generated along with its one-parameter variation ³F₃ potentials as a function of distancer.

our method of computing scattering phase shifts by using variable phase method deserves serious attention. By comparing our results for phase shifts with those of refs [10,11,16]

it can be concluded that this simple-minded combined approach of SQM and PFM will be quite interesting for dealing with nucleon–nucleon scattering at low and intermediate energies. Complete parametrization (four parameters) of the nuclear Hulthen potentials

Table 1. Parameterαfor various angular momentum states.

αfor supersymmetrically αfor modified potentials

Partial waves generated potentials (fm⁻¹) (fm⁻¹)

3P₀ 0.565 0.894

3P₁ −0.863 −0.232

1D₂ 1.023 1.4053

3F₃ 1.161 1.4053

0 50 100 150 200 250 300 -10

0 10 20 30 40 50 60 70

Phase shift (Degree)

ELab (MeV)

np pp

Gross-Stadler(WJC-1-np) Arndt (Energy dependent-np) Arndt (Energy dependent-pp) Arndt (Single energy-np) Arndt (Single energy-pp)

Figure 6. ¹S₀phase shifts as a function ofE_Lab.

0 50 100 150 200 250 300

-10 -5 0 5 10 15

Phase shift (Degree)

E_Lab (MeV)

np(SUSY)

pp(SUSY)

np(SUSY-PRM)

pp(SUSY-PRM)

np(MOD-SUSY-PRM)

pp(MOD-SUSY-PRM) Arndt(Energy dependent-np) Arndt(Single energy-np) Arndt(Energy dependent-pp) Arndt(Single energy-pp) Gross-Stadler (WJC-1-np)

Figure 7. Supersymmetry generated along with its one-parameter variation³P₀phase shifts as a function of laboratory energyE_Lab.

0 50 100 150 200 250 300 -30

-20 -10 0

Phase shift (Degree)

E_Lab (MeV)

np (MOD-SUSY)

pp (MOD-SUSY)

np (SUSY-PRM)

pp (SUSY-PRM)

np (MOD-SUSY-PRM)

pp (MOD-SUSY-PRM) Arndt (Energy dependent-np) Arndt (Single energy-np) Arndt (Energy dependent-pp) Arndt (Single energy-pp) Gross-Stadler (WJC-1-np)

Figure 8. Supersymmetry generated along with its one-parameter variation³P₁phase shifts as a function of laboratory energyE_Lab.

0 50 100 150 200 250 300

-40 -30 -20 -10 0 10

Phase shift (Degree)

E_Lab (MeV)

np (SUSY)

pp (SUSY)

np (SUSY-PRM)

pp (SUSY-PRM)

np (MOD-SUSY)

pp (MOD-SUSY)

np (MOD-SUSY-PRM)

pp (MOD-SUSY-PRM) Arndt (Energy dependent-np) Arndt (Single energy-np) Arndt (Energy dependent-pp) Arndt (Single energy-pp) Gross- Stadler (WJC-1-np)

Figure 9. Supersymmetry generated along with its one-parameter variation¹D₂phase shifts as a function of laboratory energyE_Lab.

0 50 100 150 200 250 300 -40

-30 -20 -10 0

Phase shift (Degree)

E_Lab (MeV)

np (SUSY)

pp (SUSY)

np (SUSY-PRM)

pp (SUSY-PRM)

np (MOD-SUSY)

pp (MOD-SUSY)

np (MOD-SUSY-PRM)

pp (MOD-SUSY-PRM) Arndt (Energy dependent-np) Arndt (Single energy-np) Arndt (Energy dependent-pp) Arndt (Single energy-pp) Gross-Stadler (WJC-1-np)

Figure 10. Supersymmetry generated along with its one-parameter variation ³F₃ phase shifts as a function of laboratory energyE_Lab.

for various angular momentum states (up to = 3) are in our active consideration and will be communicated in a future correspondence.

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