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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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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