ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 3, с. 306-313
ЯДРА
GENERALIZED TENSOR INTERACTION AND RELATIVISTIC SPIN AND PSEUDOSPIN SYMMETRIES WITH THE MANNING-ROSEN POTENTIAL
© 2014 A. N. Ikot1)*, H. Hassanabadi2), B. H. Yazarloo2), A. D. Antia1), S. Zarrinkamar3)
Received May 13,2013
We have investigated the Manning—Rosen potential under the framework of spin and pseudospin symmetries with generalized tensor interaction (GTI) using the parametric generalization of Nikiforov— Uvarov method. We have obtained the energy eigenvalues and the corresponding eigenfunction. We have also reported some numerical results and figures to show the effect of tensor interaction. The role of GTI is investigated in details. It is seen that the degenerate doublets in spin and pseudospin are changed due to the presence of GTI.
DOI: 10.7868/S0044002714020111
1. INTRODUCTION
The Dirac equation is one of the relativistic wave equations that describes the motion of spin-1/2 particles and has been widely used to solve many quantum-mechanical problems in nuclear and high-energy physics [1]. Different techniques have been employed for dealing with the Dirac equation with physically motivated potentials [2—4]. Such methods include Nikiforov—Uvarov (NU) method [5], asymptotic iteration (AI) [6], shape invariance [7], supersymmetric quantum mechanics (SUSYQM) [8], factorization [9], and others [10, 11]. The solution of the Dirac equation has successfully accounted for the experimental observation of the quasi-degeneracy in single-nucleon doublets between normal parity orbitals (n,l,j = l + 1/2) and (n — 1,l + 2,j = l + 3/2), where n, l, and j represent the radial, orbital, and total angular momentum quantum numbers, respectively. These symmetries have been used to form an effective nuclear-shell model [12] and to explain other different physical phenomena in the nuclear structure such as deformation, super-deformation, magnetic moment, and identical bands [13]. Ginocchio [14] established the
'-'Theoretical Physics Group, Department of Physics, University of Uyo, Nigeria.
2)Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran.
3)Civil Engineering Group, Alaodoleh Semnani Institute of Higher Education, Garmsar, Iran.
E-mail: ndemikotphysics@gmail.com
relationship between the pseudospin and the Dirac equation by recognizing that pseudo-orbital angular momentum l = l + 1 is nothing but the usual orbital angular momentum l of the lower component of the Dirac spinor. He showed that within the framework of the Dirac theory, the spin symmetry occurs when the difference of the potential between the repulsive Lorentz vector potential V(r) and attractive Lorentz scalar potential S(r) is a constant, i.e. A(r) = V(r) — — S(r) = const. On the other hand, the pseudopsin symmetry arises when £(r) = V(r) + S(r) = const. Pseudospin and spin symmetry concepts on a number of potentials have been previously investigated [15, 16]. On the other hand, we are now almost sure that the spin and pseudospin symmetries of Dirac equation have significant role in nuclear and hadronic spectroscopy [17, 18].The tensor interaction has attracted a great attention as it removes the degeneracy between the doublets [19]. In most of studies, due to the mathematical structure of the problem, the tensor interaction is considered as Coulomb-like [20, 21] or Cornell interaction. Hassanabadi et al. were the first who introduced the Yukawa tensor interaction [22]. Here, we consider Manning—Rosen potential [23] including a generalized tensor interaction (GTI) term and proceed on an approximate analytical manner. The Manning—Rosen potential has been used to describe diatomic molecular energy spectra [23].
The paper is organized as follows. In Section 2, we review Dirac theory for spin and pseudospin symmetries with tensor interaction. The corresponding differential equation with GTI is given in Section 3.
The solutions of the Dirac equation for pseudospin and spin symmetries are presented in Section 4. The effects of GTI are given in Section 5. Finally, we give a brief conclusion in Section 6.
2. DIRAC EQUATION WITH A TENSOR COUPLING
Dirac equation with a tensor potential U(r) in the relativistic unit (h = c = 1) is written as [19—21]
[a • p + /3(M + 5(r)) — i/a • rU(r)] ^(r) = (1)
= [E — V(r)] ^(r),
where E is the relativistic energy of the system, p = — —iV is the three-dimensional momentum operator, and M is the mass of the fermionic particle. a, 3 are the 4 x 4 Dirac matrices given as
0
a
0
в =
I 0 0 —I,
(2)
where I is a 2 x 2 unit matrix and ai are the Pauli three-vector matrices defined as
(i
(2 =
V2
1 0 0 -1,
0 -i i 0
(3)
r \iGuK(r) Yj m{0, ц>)
(4)
where FnK(r) and GnK(r) represent the upper and lower components of the Dirac spinors. Yjm(d,p),
Yjm(d, p) are the spin and pseudospin spherical harmonics and m is the projection on the z-axis. With other known identities [ 19—22]
(a • A) (a • B)
a • p = a • f Г • p + i
A • B + ia • (A x B), (5) a L
as well as
(a • L) YUe,<p) = (K - 1)YUe,<p), (6)
jm
(a • L) Yjm(d,<p) = -(к - 1)Yjm(d,<p),
jm
rl
jm
(a • r) Yjm(0,<p) = -Y;m(9,V), (a • r) Yjm{e,v) = -YjmMv), we obtain the coupled equations [19—24], d
+ --U(r) dr r
FnK(r) =
The eigenvalues of the spin—orbit coupling operator are k = (j + 1/2) > 0, k = —(j + 1/2) < 0 for unaligned j = l — 1/2 and aligned spin j = l + 1/2 cases, respectively. The set (H,K,J2,Jz) forms a complete set of conserved quantities. Thus, we can write the spinors as [ 19—21]
where
= (M + EnK - A(r)) G 'ПК (r),
i - -r + m j cur) =
= (M - Епк + 4r)) FnK (r),
A(r) = V(r) - S(r),
Z(r) = V (r)+S (r).
(7)
(8)
(9)
(10)
Eliminating FnK(r) and GnK(r) in favor of each other in Eqs. (7) and (8), we obtain the second-order Schro dinger-like equations
dr2
k(k + 1) 2 KU(r) dU (r)
Q H
- U2(r) - (M + Епк - A(r)) x
r dr
dA(r) (A I « _ TT(r\)
x (M — Enli + S(r)) + i^V-AM)
é _ + МИ + ям _ _ + _ x
dr2 r2 r dr
d^irl + TT(r\)
x (M - EnK + E(r)) + Kdr-r ,
{ (M — EnK + ВД)
with к(к - 1) = 1(1 + 1), к(к + 1) = l(l + 1).
FnK (r) = 0,
GnK(r) = 0,
(11)
(12)
3. PSEUDOSPIN AND SPIN SYMMETRY LIMITS UNDER YUKAWA-LIKE TENSOR INTERACTION
In this section, we intend to investigate the Dirac equation with Manning—Rosen potential in the presence of GTI.
3.1. Pseudospin Symmetry in the Dirac Equation with Yukawa Tensor Interaction
The pseudospin symmetry occurs in the Dirac theory as dT,(r)/dr = 0 or equivalently ~E(r) = Cps = = const [11 — 14]. We take the difference of the scalar and vector potentials as the Manning—Rosen [15, 23],
A(r)= (13)
1
Jj32
-4&r
Ae
-2&r
a(a — 1)e (1 _ e~2Sr)2 (1 - e~2Sr)
where k = 2M/h2.
In addition, we proposed a novel generalized tensor interaction of the form,
U(r) = — (Uc(r) + UY(r)),
Udr) = --,
r
UY(r) = —Vo
e-
with
2
H
zazbe 4neo
IJ(r) = -i (H + Voe-^ .
Inserting Eqs. (13) and (17) into Eq. (12) yields,
d2 k(k - 1)
dr2 r2
2kh H H
2
r
2
+ O 2
r
r
2HVo e
—&r
2nVoe-Sr 5Vo e-Sr -^-+ —-+
Voe
—&r
+
+ I —e2ps +
V2e-2&r
GZJr) +
(M - Ep% + Cp8) kf32
—4&r
Ae
-2&r
a(a — 1)e (1 _ e~2Sr)2 (1 — e~2&r)_
GpnsJr) = 0,
where e2ps = (M + EPstK) (M — EPstK + Cps) and
a _ (M-EK%+Cps)
Pps — w- '
Again, Eq. (18) cannot be exactly solved by any known method because of the centrifugal term 1/r2. In order to get the approximate solution, we use the following Pekeris approximations [26]:
1
1
462
-2&r
(1 — e-2*r)2'
462
—&r
r2 (1 — e-2Sr)2'
Substituting Eqs. (19) and (20) into Eq. (18) in view of the transformation y = e-2Sr, we get
(19)
(20)
dGp% | (1 -y) dGp% | dy2 V0--V) dy
(21)
(14)
+
y2 (1 — y)2
(—xTy2 + XPpSy — xf) Gpnl(y) = 0
where Uc (r) and UY(r), respectively, are the Coulomb-like and Yukawa-like potentials [24, 25] defined as,
where
XT = Vo I Vo +
1
/3psA
Id2'
(22)
(15)
f%sa(a — 1) eps
452
+
452'
XP2s = -r],(r]K-l)-^2K + 2H-^jVo- (23)
(16)
where Re is the Coulomb radius, za and zb, respectively, denote the charges of the projectile a and the target nuclei b and V0 is the depth of the potential. Substituting Eq. (15) into Eq. (14), we obtain the GTI as,
ftps14 £ps ~252
ps _
X3 4S2 '
(24)
(17)
3.2. Spin Symmetry in the Dirac Equation
In the spin symmetry limit, we use the scalar, vector and tensor potentials:
(18)
nr) =
kf 2
A(r) = Cs, q(q - l)e~4Sr
1
r
(25)
Ae
-2&r
(1 — e-2&r)
U{r) = — {H + VQe-Sr)
which, with z = e 2Sr, transform Eq. (11) into
d2FS
dz2
n,K (1 z) dF£K
z (1 — z) dz
(26)
x
2
r
1
r
2
1
2
r
r
r
2
2
r
r
X
+
z2 (1 - z)2
-xlz2 + x2z - Xs) FL(z) = 0,
where
+
AA el 4Ó2 2Ó2'
X3
'-s
4Ó2'
where e* = M - E^) (M + E^ - Ca), AK =
= H + n + I md/3S = {M+E¿rCs).
c4 = 0, c5 = —
1
1
ce
+xpa
C7
XPa x2,
pa
c8 = xs,
pspsps C9 = J + Xl +X3 - X2 »
Cío = 1 + 2\fx[s,
cn =2 + 2
ps ps ps ps 4+Xi +X3 -*2 +v%3
ps
C12 = \ Xp
ps ps ps ps + Xp + Xp - Xp + V xp
XP2a +
= (27)
l3sa(g - 1) e%s_ 4Ó2 4Ó2 '
XS2 = -Ak(Ak-1)- Í2K + 2H + ^)V0+ (28)
+ 2xls + 2^xls ^+xT + XPs-XP2S) =0.
From Eq. (A.2) the lower and upper component of the wave function are obtained as follows:
GpnsJr) = x
(33)
x 1 - e"
— 2&r \ 2 V 4
Wi+xf+xf-xf
X
X P.
(2^T;2^+Xr+Xr-Xr) (1 _ 2e~2Sr
(29)
FPp%(r) =
1
m - epsk + c
(34)
ps
4. PSEUDOPSIN AND SPIN SYMMETRY SOLUTIONS
In this section, we intend to investigate the solutions of Eqs. (21) and (26) for pseudospin and spin symmetries using the NU method [27] in Appendix.
4.1. Pseudospin Symmetry Solution Now comparing Eq. (21) with Eq. (A.4), we obtain C1 = 1, C2 = 1, C3 = 1, (30)
6 = XPlS, 6 = X/pS, 6 = xf-
Other parameters can be obtained as follows:
l--r+U(r))Gpn%(r),
and Ep% = M + C
ps
4.2. Spin Symmetry Solution
The energy eigenvalues for the spin symmetry case with GTI for the Manning—Rosen potential in the Dirac theory becomes
n2 + ( n + \ ) + (2n + !) x (35)
\j\+x\ + xl~xs2 + Vxi
- x2 +
(31)
+ 2x1 + 2^x1 (j+Xt+Xf"X!j =0.
Finally
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.