ЯДЕРНАЯ ФИЗИКА, 2015, том 78, № 10, с. 885-888

ЯДРА

SUPERSYMMETRY-GENERATED JOST FUNCTIONS AND NUCLEON-NUCLEON SCATTERING PHASE SHIFTS

©2015 J. Bhoi*, U. Laha**

Department of Physics, National Institute of Technology, Jamshedpur, India Received February 6, 2015; in final form, May 8,2015

By exploiting the supersymmetry-inspired factorization method higher partial wave Jost solutions and functions for nuclear Hulthen potential are constructed from the knowledge of the ground state wave function. As a case study the nucleon—nucleon scattering phase shifts are computed for partial waves i = 0, 1, and 2.

DOI: 10.7868/S0044002715100049

1. INTRODUCTION

Over the last few decades supersymmetric quantum mechanics [1—4] illustrates a deeper insight into the various aspects of nonrelativistic quantum mechanics. It clarifies the fact why only few considerable number of quantum mechanical potentials are exactly solvable with analytical tools and others are not. Supersymmetry in quantum mechanics provides the relationship between the energy spectra and wave functions of two potentials connected through hierarchy problems [4]. The Hamiltonian hierarchy problems in supersymmetric quantum mechanics lead to the addition of appropriate centrifugal barriers and, consequently, the higher partial wave potentials are generated fairly accurately in atomic physics [5—8]. But what will be its implication in the field of nuclear physics. In the recent past, we have applied supersymmetry to nuclear physics and studied the scattering phase shifts for both np and pp scattering in a number of publications [9—13].

Because of the similarities and points of contrasts [4, 9—13] between the Coulomb potential and Hulthen potential, it is of immense importance to construct the supersymmetric partner of the latter and study their partner potentials, related physical observables, etc., which have significant application in quantum mechanics. As an implication of Hulthen potential to the nuclear physics, Arnold and MacKellar (AM) [14] first parameterized the Hulthen potential to fit the deuteron binding energy and s-wave scattering length. The bound-state wave function for Yamaguchi potential [15] is identical to the wave function for the first bound state of Hulthen

potential with range (0 — a) 1 and depth —(ft2 — — a2). A Hulthen potential with this parameter can be considered as the nuclear Hulthen potential.

The behavior of the irregular solution f¿(k, r) of the radial Schrodinger equation near the origin determines the Jost function f¿(k) [16—23] which plays an important role in examining the analytical properties of the partial-wave scattering. The present paper is an effort to derive the expressions for the Jost solution and Jost function for higher partial waves nuclear Hulthen potential and to compute the relevant observables associated with it. In Section 2 we derive the expression for Jost solution and Jost function for nuclear Hulthen potential for few higher partial waves via supersymmetry formalism. In Section 3 we compute the scattering phase shifts for nuclear Hulthen potential for the partial waves Í = 0, 1, and 2 and put some concluding remarks on them.

2. NUCLEAR HULTHEN POTENTIAL, JOST SOLUTION AND JOST FUNCTION FOR HIGHER PARTIAL WAVE

In the supersymmetric quantum mechanics any Hamiltonian of the form

Ho = --^ + Vo(x),

(1)

which has a ground state and E000 can be factorized as

with

E-mail: jskbhoi@gmail.com E-mail: ujjwal.laha@gmail.com

Ho = A- +

^ дх дх

(2) (3)

3 ЯДЕРНАЯ ФИЗИКА том 78 № 10 2015

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The supersymmetric partner H1 with potential V1 (p-wave potential) of the Hamiltonian Ho is given by

with

d2

f)2

V1(x) = V0(x)-^ln^).

(4)

(5)

V0(x) = -{i32-a2) —

o-Pr

(e-ar - e-Pr)

and

^i.0)(r) = e-ar - e

-Pr

Vi(x) = Vo(x) -

(/3 - a)2e~(a+^r (e~ar - e_/3r)2

0o(k,r) =

„ikr

-(1 - e-

(P - a)

x 2F1 (l + A, 1 + B;2; 1 - e-(P-a)r

i(P - a) (P - a)

and

rp ( o ^ r(7)r(7 - a - P) 2F1(a,/3;7;^) = r(7_a)r(7_/3) x

x 2F1 (a, P; a + p - 7 + 1; 1 - z) +

x 2F1 (y - a,Y - P; Y - a - P + 1; 1 - z),

The nuclear Hulthen potential and its corresponding bound-state wave function [9—13] with AM [14] parameters can be written as

we get

Mk,r) = ±[2F1(A*,B*-,C*-,l)eikr x (IT

x 2Fi(A,B; C; e-(P-a)r) -- 2Fi(A, B; C; 1)e-ikr2F1A ,B*; C*; e-(P-a)r)]

with

(6)

(6a)

C = 1 + -

2k

i(P - a)'

(12)

The two quantities in Eq. (11) are complex conjugate of one another. Therefore the Jost solution f0(k,r) and Jost functionf0(k) are given by [17]

In view of Eqs. (5) and (6), the wave function in Eq. (6a) leads to the supersymmetric partner potential

fo(k,r) = eikr 2F1 (A,B; C; e-(P-a)r) (13)

(6b)

and

Equation (6b) is regarded as the approximate p-wave nuclear Hulthen potential. Similarly, following the above formalism the d-wave nuclear Hulthen potential [9—13] reads as

(P _ a)2e-(a+p)r

V2(x) = V0(x)+3 [IJ{e_J_e_,r)2 ■ (6c)

The regular solution 0o(k, r), for nuclear Hulthen potential to the continuous eigenvalue spectrum E = = k2 > 0 is given by [24]

fo(k)= 2 F1 (A,B; C; 1) =

r(c)

T(1 + A)T(1 + B)'

(14)

Using the methodology of supersymmetric quantum mechanics we have found that the regular solution of the partner potential in Eq. (6b) is in the form

ikr

01(k,r) =

(1 - e

-(P-a)r )2

)2 x (15)

) x (7)

where

A = -J^— + [k2 - (I32 - a2)]1/2 (8)

(P - a)2

x 2F1 (A + 2, B + 2; 4; 1 - e-(P-a)r) .

In order to arrive at the result in Eq. (15) we have made use of the method of [26] to generate the ladder operator relations for Gaussian hypergeometric function. Using the recurrence relation (10) and

2F1 (a,P; y; z) = (1 - z)Y-a-P x (16) x 2F1 (Y - a,Y - P; Y; z)

Eq. (15) can be rewritten as

3

(9)

Transforming the 2F1 ( ) function in Eq. (7) by the recurrence relation [25]

Mk,r) =

2ikr(C *)

(P - a)r(2 + A*)r(2 + B*)

(17)

ikr

(1 - e-(p-a)r)

2F1A - 1, B - 1; C; e-(p-a)r) -

2ikr(C)

ikr

(10)

(P - a)r(2 + A)r(2 + B) (1 - e-(—)r) x 2F1 (A* - 1,B* - 1; C*; e-(P-a)r) .

ftOEPHAfl OH3HKA tom 78 № 10 2015

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SUPERSYMMETRY-GENERATED JOST FUNCTIONS

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From Eq. (17) the p-wave Jost solution and Jost function for the Hulthen potential are identified as

Ahr

fi(k,r) =

(1 _ e-{ß-a)r )

(18)

x 2Fi (A _ 1,B _ 1; C; e-(ß-a)r )

and

fi(k) =

2ikr(C )

(ß _ a)r(2 + A)r(2 + B)'

(19)

For d-wave potential the regular solution, Jost solution, and Jost function for Hulthen potential are as below

h (k,r) =

2ik3

8k2 r(C *)

0ihr

(1 _ e-(ß-a)r )2

(ß _ a)2r(3 + A*)r(3 + B*)

(20)

2Fi (A _ 2, B _ 2; C; e-(ß-a)r) _

8k2r(C )

ihr

(ß _ a)2r(3 + A)r(3 + B) (1 _ e-(ß-a)r)2 x 2Fi(A* _ 2, B* _ 2; C*; e-(ß-a)r)

ihr

Í2(k,r) - -¡j_ e_(ß_a)ry2 -x 2Fi(A _ 2, B _ 2; C; e-(ß-a)r)

(21)

and

f2(k) =

8k2r(C )

(ß _ a)2r(3 + A)r(3 + B)'

(22)

3. RESULTS, DISCUSSION AND CONCLUSION

It is well known that the phase of Jost function [17] is negative of the scattering phase shift. We have computed the scattering phase shifts for triplet states for the partial waves i = 0, 1, and 2 for two different sets of parameters. For the AM [14] parameters, f3 = = 1.4054 fm"1 and a = 0.232 fm"1 and for Laha and Bhoi (LB) parameters (designated in figures as LB) /3 = 1.1 fm"1 and a = 0.2316 fm"1 we have plotted the scattering phase shifts along with the standard results [27] for the partial waves i = 0, 1, and 2 in Figs. 1, 2, and 3.

Looking closely into Fig. 1 it is quite clear that the scattering phase shifts for AM parameters [14] agree well with those of Arndt et al. [27] for low energy (up to 25 MeV) but gradually diverge for energies beyond 25 MeV. The scattering phase shifts for the s wave as depicted in Fig. 1 for LB parameters are

S, deg 150-

100

50

— AM

........LB

* Arndt

100

200 300

¿Lab, MeV

Fig. 1. 3si nuclear Hulthen phase shifts as a function of

¿Lab-

S, deg 15 -

10

-5

-10

-AM

.........LB

* Arndt

100

200

300

¿Lab, MeV

Fig. 2. Supersymmetry-generated 3p0 nuclear Hulthen phase shifts as a function of ELab.

also comparable with that of standard result [27] up to energy 75 MeV.

For 3p0 states numerical values of the scattering phase shifts with AM parameters [14] as in Fig. 2 differ significantly from those of Arndt et al. [27]. The phase shifts with these parameters attains the maximum values of 2.68° at FLab = 68 MeV and changes its sign at the laboratory energy 275 MeV, whereas the same for Arndt et al. [27] becomes maximum of 12.66° at FLab = 75 MeV and changes the sign at

x

0

0

x

x

5

0

0

an,EPHA^ OH3HKA TOM 78 № 10 2015

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BHOI, LAHA

8, deg

¿Lab, MeV

Fig. 3. Supersymmetry-generated 3d1 nuclear Hulthen

phase shifts as a function of ELab.

the energy around 225 MeV. For LB parameters, the scattering phase shift attains the maximum values of 3.902 77° at the laboratory energy 56 MeV and change its sign at 237 MeV. The result of 3di state in Fig. 3 for AM parameters [14] is systematically lower than that of Arndt et al. [27], however the same with LB parameter possess some improved result and is quite comparable with standard result [27].

The nuclear potentials are highly state dependent [28—30] and are parameterized by the strength and range parameters. The higher partial waves, as generated via supersymmetric quantum mechanics from their ground state, are obtained only by the addition of centrifugal barrier. Therefore, it is unexpected that the higher partial wave potential generated from their ground state can produce correct numerical values for phase shifts. However, they are able to produce the correct nature of the scattering phase shifts for different states. It is noticed that our parameters (LB) are superior to the AM parameters [14] as is clearly visible in the results for p- and d-wave scattering phase shifts.

Arndt et al. [27] have computed the scattering phase shifts for np and pp scattering with a number of parameters. Our method for computing the scattering phase shifts for higher partial waves via supersymmetric quantum mechanics and J

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