Approximate relativistic bound states of a particle in Yukawa field

16 Pages·2013·0.17 MB·English

Approximate relativistic bound states of a particle in
Yukawa field with Coulomb tensor interaction
Sameer M. Ikhdair1
Physics Department, Near East University, 922022 Nicosia, Northern Cyprus, Turkey.
and
Physics Department, Faculty of Science, An-Najah National University,
Nablus, West Bank, Palestine.
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1
0
Babatunde J. Falaye2
2

l
u
Theoretical Physics Section, Department of Physics
J
University of Ilorin, P. M. B. 1515, Ilorin, Nigeria.
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3

]
h
t
-
l Abstract
c
u We obtain the approximate relativistic bound state of a spin-1/2 particle in the field of the
n Yukawa potential and a Coulomb-like tensor interaction with arbitrary spin-orbit coupling
[
number κ under the spin and pseudospin (p-spin) symmetries. The asymptotic iteration

1 method isusedto obtainenergyeigenvaluesandcorrespondingwavefunctions intheirclosed
v forms. Our numerical results show that the tensor interaction removes degeneracies between
6
the spin and p-spin doublets and creates new degenerate doublets for various strengths of
2
tensor coupling.
3
8
.
7 Keywords: Dirac equation; Yukawa potential; Coulomb-like tensor interaction; spin and
0
3
p-spin symmetries, asymptotic iteration method
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:
v PACS: 03.65.Ge, 03.65.Fd, 03.65.Pm, 02.30.Gp
i
X
r 1 Introduction
a
Within theframework of Dirac equation, thep-spin symmetry is used to feature deformed nuclei,
superdeformationandtoestablishaneffectiveshell-model[1,2,3,4]. However,thespinsymmetry
is relevant for mesons [5]. The spin symmetry occurs when ∆(r)= V(r)−S(r)=constant or the
scalar potential S(r) is nearly equal to the vector potential V(r), i.e., S(r) ≈ V(r). The p-spin
symmetry occurs when Σ(r) = V(r)+S(r) =constant or S(r) ≈ −V(r) [6, 7, 8, 9].
1
E-mail: [email protected]; [email protected].
2
E-mail: [email protected]
1
Furthermore, the p-spin symmetry refers to a quasi-degeneracy of single nucleon doublets
with non-relativistic quantum number (n, ℓ, j = ℓ+1/2) and (n−1, ℓ+2, j = ℓ+3/2), where
n, ℓ and j are single nucleon radial, orbital and total angular quantum numbers, respectively
[10, 11]. The total angular momentum is j = ℓ˜+s˜, whereℓ˜= ℓ+1 is pseudo-angular momentum
and s˜ is p-spin angular momentum [1]. Recently, the tensor potential was introduced into the
Dirac equation with the substitution: P~ → P~ −imωβ.rˆU(r) and a spin-orbit coupling is added
to the Dirac Hamiltonian [12, 13, 14, 15, 16, 17].
The Yukawa potential or static screened Coulomb potential (SSCP) [18, 19, 20] can be de-
fined as
e−ar
V(r)= −V , (1)
0
r
where V = αZ, α = (137.037)−1 is the fine-structure constant, Z is the atomic number and a is
0
thescreeningparameter. Thispotential is often usedto computebound-statenormalizations and
energy levels of neutral atoms [21, 22, 23]. Over the past years, several methods have been used
in solving relativistic and nonrelativistic equations with the Yukawa potential such as the shifted
large-method [24], perturbativesolution of theRiccati equation [25, 26], alternative perturbative
scheme [27, 28], the quasi-linearization method (QLM) [29] and Nikiforov-Uvarov method [30].
Thetensor coupling, ahigher ordertermin arelativistic expansion, increases significantly the
spin-orbit coupling. This suggests that the tensor coupling could have a significant contribution
to p-spin splittings in nuclei as well. This contribution is expected to be particularly relevant
for the levels near the Fermi surface, because the tensor coupling depends on the derivative of a
vector potential, which has a peak near the Fermi surface for typical nuclear mean-field vector
potentials. It has also been used as a natural way to introduce the harmonic oscillator in a
relativistic (Dirac) formalism. In a recent article, it was shown that the harmonic oscillator with
scalar and vector potentials can exhibit an exact p-spin symmetry [14, 31]. When this symmetry
is broken ( 6= 0), the breaking term is quite large, manifesting its nonperturbative behavior.
However, ifPa tensor coupling is introduced, the form of harmonic-oscillator potential can still be
maintained with ( =0), but the p-spin symmetry is broken perturbatively [32].
P
Thetensorinteractionhasalsobeenconsideredtoexplainhowthespin-orbittermcanbesmall
for Λ−nucleus and large in the nucleon-nucleus case [33]. It is assumed that in the strange sector
(case of Λ ) the tensor coupling is large and the spin-orbit term obtained from this interaction
2
can cancel in part the contribution coming from the scalar and vector interactions. This result
shows that the tensor interaction can change strongly the spin-orbit term.
It is therefore the aim of the present work is to apply the asymptotic iteration method
[34, 35, 36, 37] in solving the Dirac equation with the Yukawa potential including a Coulomb-like
tensor interaction to obtain the energy eigenvalues and corresponding wave functions in view of
spin and p-spin symmetry.
The paper is organized as follows. In Section 2, the AIM is briefly introduced. In Section 3,
we present the Dirac equation with scalar and vector potentials for arbitrary spin-orbit coupling
numberincludingtensor interaction in view of spinandp-spinsymmetry. InSection 4, weobtain
the energy eigenvalue equations and corresponding wave functions. We discuss our numerical
results in Section 5. Our conclusion is given in Section 6.
2 Method of Analysis
One of the calculational tools utilized in solving the Schro¨dinger dinger-like equation including
the centrifugal barrier and/or the spin-orbit coupling term is called as the asymptotic iteration
method (AIM). For a given potential the idea is to convert the Schro¨dinger-like equation to the
homogenous linear second-order differential equation of the form:
y′′(x)= λ (x)y′(x)+s (x)y(x), (2)
o o
where λ (x) and s (x) have sufficiently many continous derivatives and defined in some interval
o o
which are not necessarily bounded. The differential Eq. (2) has a general solution [34, 10]
x x x′
y(x)= exp − α(x′)dx′ C +C exp λ (x′′)+2α(x′′) dx′′ dx′ . (3)
2 1 o
(cid:18) Z (cid:19)" Z Z ! #
(cid:2) (cid:3)
If k > 0, for sufficiently large k, we obtain the α(x)
s (x) s (x)
k k−1
= = α(x) , k = 1,2,3..... (4)
λ (x) λ (x)
k k−1
where
λ (x) = λ′ (x)+s (x)+λ (x)λ (x),
k k−1 k−1 o k−1
s (x) = s′ (x)+s (x)λ (x) , k = 1,2,3..... (5)
k k−1 o k−1
with quantization condition
λ (x) s (x)
δ (x) = k k = 0 , k = 1,2,3.... (6)
k λ (x) s (x)
(cid:12) k−1 k−1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 3 (cid:12)
The energy eigenvalues are then obtained from (6), if the problem is exactly solvable.
3 Dirac Equation with a Tensor Interaction
In spherical coordinates, the Dirac equation for fermonic massive spin−1/2 particles interacting
with arbitrary scalar potential S(r), the time-component V(r) of a four-vector potential and the
tensor potential U(r) can be expressed as [1, 7]
[α~.p~+β(M +S(r))−iβα~.rˆU(r)]ψ(~r)= [E −V(r)]ψ(~r), (7)
where E, ~p and M denote the relativistic energy of the system, the momentum operator and
mass of the particle respectively. α and β are 4×4 Dirac matrices given by
0 ~σ I 0 0 1 0 −1 1 0
α¯ = , β = , σ = , σ = , σ = ,
~σ 0 0 −I 1 1 0 2 i 0 3 0 −1
! ! ! ! !
(8)
whereI isthe2×2unitarymatrixand~σ arethethree-vector paulispinmatrices. Theeigenvalues
of the spin-orbit coupling operator are κ = j+ 1 > 0 and κ = − j + 1 < 0 for unaligned
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
spin j = ℓ− 1 and the aligned spin j = ℓ+ 1 respectively. The set (H2,K,J2,J ) can be taken
2 2 Z
as the complete setof conservative quantities with J~beingthetotal angular momentum operator
and K = (~σ.L~ +1) is the spin-orbit where L~ is the orbital angular momentum of the spherical
nucleons that commutes with the Dirac Hamiltonian. Thus, the spinor wave functions can be
classified according to their angular momentum j, the spin-orbit quantum number κ and the
radial quantum number n. Hence, they can be written as follows:
ψ (~r) = fnκ(~r) = 1 Fnκ(r)Yjℓm(θ,φ) , (9)
nκ gnκ(~r) ! r iGnκ(r)Yjℓ˜m(θ,φ), !
whereF (r)andG (r)aretheradialwavefunctionsoftheupper-andlower-spinorcomponents
nκ nκ
respectively and Yℓ (θ,φ) and Yℓ˜ (θ,φ) are the spherical harmonic functions coupled to the
jm jm
total angular momentum j and it’s projection m on the z axis. Substitution of equation (7) into
equation (2) yields the following coupled differential equations:
d κ
+ −U(r) F (r) = (M −E −∆(r))G (r)
nκ nκ nκ
dr r
(cid:18) (cid:19)
d κ
− +U(r) G (r) = (M −E −Σ(r))F (r) (10)
nκ nκ nκ
dr r
(cid:18) (cid:19)
where ∆(r) = V(r)− S(r) and Σ(r) = V(r) +S(r) are the difference and sum potentials re-
spectively. After eliminating F (r) and G (r) in equations (10), we obtain the following two
nκ nκ
4
Schro¨diger-like differential equations for the upper and lower spinor components:
d2 κ(κ+1) 2κ dU(r) d∆(r) d κ
− + U(r)−U2(r)− + dr + −U(r) F (r)
"dr2 r2 r dr M +Enκ−∆(r)(cid:18)dr r (cid:19)# nκ
= [(M +E −∆(r))(M −E +Σ(r))]F (r), (11)
nκ nκ nκ
d2 κ(κ−1) 2κ dU(r) dΣ(r) d κ
− + U(r)−U2(r)+ + dr − +U(r) G (r)
"dr2 r2 r dr M −Enκ+Σ(r)(cid:18)dr r (cid:19)# nκ
= [(M +E −∆(r))(M −E +Σ(r))]G (r), (12)
nκ nκ nκ
where κ(κ−1) = ℓ¯(ℓ¯+1) and κ(κ+1) =ℓ(ℓ+1)
3.1 P-spin Symmetry Limit
d[V(r)+S(r)] dΣ(r)
The pseudospin symmetry occurs when = = 0 or Σ(r) = C =constant. Here
dr dr ps
we are taking ∆(r) as the Yukawa potential and the tensor as the Coulomb-like potential, i.e.
e−ar A
∆(r)= −V and U(r) = − , r ≥ R , (13)
o c
r r
with
Z Z e2
a b
A = , (14)
4πǫ
o
where R is the Coulomb radius, Z and Z respectively, denote the charges of the projectile a
c a b
and the target nuclei b [1, 7]. Under this symmetry, equation (12) can easily be transformed to
d2 δ˜ γ˜e−ar
− + −β˜2 G (r)= 0, (15)
"dr2 r2 r # nκ
where κ =−ℓ˜and κ = ℓ˜+1 for κ< 0 and κ> 0 respectively and
γ˜ = (E −M −C )V ,
nκ ps 0
β˜ = (M +E )(M −E +C ), (16)
nκ nκ ps
q
δ˜ = (κ+A)(κ+A−1),
have been introduced for simplicity.
3.2 Spin Symmetry Limit
d∆(r)
In the spin symmetry limit, = 0 or ∆(r) = C = constant [1, 7]. Similarly to section 3.1,
dr s
we consider
e−ar A
Σ(r)= −V and U(r)= − , r ≥ R . (17)
o c
r r
5
With equations (17), equation (11) can be transformed to
d2 δ γe−ar
− + −β2 F (r) = 0, (18)
"dr2 r2 r # nκ
where κ = ℓ and κ = −ℓ−1 for κ < 0 and κ > 0, respectively. We have also introduced the
following parameters
γ = (M +E −C )V ,
nκ s 0
β = (M −E )(M +E −C ), (19)
nκ nκ s
q
δ = (κ+A)(κ+A+1),
for simplicity.
4 Approximate Relativistic Bound States
In this section, within the framework of the AIM, we shall solve the Dirac equation with the
Yukawa potential in the presence of the tensor potential.
4.1 P-Spin Symmetric Solution
To obtain analytical approximate solution for the Yukawa potential, an approximation has to be
made for the centrifugal term 1/r2, which is similar to the one taken by other authors [37−43]
1 4a2e−2ar
≈ , (20)
r2 (1−e−2ar)2
which is valid for ar << 1. With this approximation, equation (15) can be written as
d2G (r) 2aγ˜e−2ar 4a2δ˜e−2ar
nκ + − −β˜2 G (r)= 0. (21)
dr2 "(1−e−2ar) (1−e−2ar)2 # nκ
In order to obtain the solution of equation (21), we introduce a transformation of the form
z = e−2ar −1 −1, as a result, equation (21) can be rewritten as
(cid:0) d(cid:1)2Gnκ(z) dGnκ(z) δ˜z2+ δ˜+ 2γ˜a z+ 2γ˜a + 4β˜a22
z(z+1) +(1+2z) − G (r). (22)
dz2 dz (cid:16) (cid:16) z(1(cid:17)+z)(cid:16) (cid:17)(cid:17) nκ
According to theFrebenius theorem, the singularity points of theabove differential equation play
an essential role in the form of the wave functions. The singular points of the above equation
(22) are at z = 0 and z = −1. As a result, we take the wave functions of the form
Gnκ(z) = zρ˜(1+z)−2β˜agnκ(z), (23)
6
where
β˜2 γ˜
ρ˜= + . (24)
s4a2 2a
Substituting equations (23) and (24) into equation (22) allows us to find the following second-
order equation
d2gnκ(z) 2z 2β˜a −ρ˜−1 −(2ρ˜+1) dgnκ 2β˜a −ρ˜ − 2β˜a −ρ˜ 2+δ˜
− − g (z), (25)
dz2  (cid:16) z(1+(cid:17)z)  dz (cid:16) (cid:17)z(z(cid:16)+1) (cid:17)  nκ
   
   
which is suitable to an AIM solutions. In order to use the AIM procedure, we compare equation
(25) with equation (2) and obtain λ (z) and s (z) equations as
0 0
2z β˜ −ρ˜−1 −(2ρ˜+1)
2a
λ (z) = ,
0 (cid:16) z(1+(cid:17)z)
β˜ −ρ˜ − β˜ −ρ˜ 2+δ˜
2a 2a
s (z) = . (26)
0 (cid:16) (cid:17)z(z(cid:16)+1) (cid:17)
By using the termination condition of the AIM given in equation (6), we obtain
λ (z) s (z) β˜ 1 1
δ (z) = 1 1 = 0 ⇒ ρ˜ − 0 = − − 1+4δ˜
0 (cid:12)(cid:12) λ0(z) so(z) (cid:12)(cid:12) 0 2a 2 2q
δ (z) = (cid:12)(cid:12)(cid:12) λ2(z) s2(z) (cid:12)(cid:12)(cid:12) = 0 ⇒ ρ˜ − β˜1 = −3 − 1 1+4δ˜ (27)
1 (cid:12)(cid:12) λ1(z) s1(z) (cid:12)(cid:12) 1 2a 2 2q
δ (z) = (cid:12)(cid:12)(cid:12) λ3(z) s3(z) (cid:12)(cid:12)(cid:12) = 0 ⇒ ρ˜ − β˜2 = −5 − 1 1+4δ˜
2 (cid:12)(cid:12) λ2(z) s2(z) (cid:12)(cid:12) 2 2a 2 2q
.(cid:12)..etc. (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The above expressions can be generalized as
β˜ 2n+1 1
ρ˜ − n = − − 1+4δ˜. (28)
n
2a 2 2
q
If one insert the values of ρ˜, β˜ and δ˜ into equation (28), the energy spectrum equation can be
obtained as
(M +E )(M −E +C )−2a(κ+A+n) = (M −E +C )(M +E −2aV ). (29)
nκ nκ ps nκ ps nκ o
q q
On squaring both sides of equation (29), we obtain a more explicit expression for the energy
spectrum as
2a(n+κ+A)2+V (M −E +C ) 2
o nκ ps = (n+κ+A)2. (30)
4(M +E )(M −E +C )
(cid:2) nκ nκ ps (cid:3)
7
For special case when A= 0, our result is exacly identical with the one obtained by Ikhdair [41]
and by Aydogˇduand Sever [42]. Now we shall obtain theeigenfunction usingthe AIM. Generally
speaking, the differential equation we wish to solve should be transformed to the form [35]:
ΛxN+1 m+1 WxN
y′′(x) = 2 − y′(x)− , (31)
1−bxN+2 x ! 1−bxN+2
where a, b and m are constants. The general solution of equation (31) is found as [2]
y (x) = (−1)nC (N +2)n(σ) F (−n,t+n;σ;bxN+2), (32)
n 2 n2 1
where the following notations have been used
Γ(σ+n) 2m+N +3 (2m+1)b+2Λ
(σ) = , σ = and t = . (33)
n Γ(σ) N +2 (N +2)b
Now, comparing equations (32) and (22), we have Λ = β˜ − 1, b = −1, N = −1, m = ρ˜− 1,
2a 2 2
σ = 2ρ˜+1, t = 2 ρ˜− β˜ and then the solution of equation (25) can easily found as
2a
(cid:16) (cid:17)
Γ(2ρ˜+1) β˜
g (z) = (−1)n F −n,2 ρ˜− +n+1;2ρ˜+1;−z , (34)
nκ 2 1
Γ(2ρ˜) 2a! !
where Γ and F are the Gamma function and hypergeometric function, respectively. By using
2 1
equations (23) and (34) we can write the corresponding lower spinor component G (z) as
nκ
Gnκ(z) = Nnκzq4β˜a22+2γ˜a(1+z)−2β˜a 2F1−n,2s4β˜a22 + 2γ˜a − 2β˜a+n+1;2s4β˜a22 + 2γ˜a +1;−z,
   (35)
where N is the normalization constant.
nκ
4.2 Spin Symmetric Solution
Following the previous section 4.1, approximation (20) is used instead of the centrifugal term
1/r2 and we rewrite equation (18) as
d2F (r) 2aγe−2ar 4a2δe−2ar
nκ + − −β2 F (r)= 0. (36)
dr2 "(1−e−2ar) (1−e−2ar)2 # nκ
We have decided to use the same variable so as to avoid repetition of algebra. It is clear that
equation (36) is similar to equation (21); therefore substituting for γ, δ and β in equation (28),
the relativistic energy spectrum turns out as
2a(n+κ+A+1)2 −V (M +E −C ) 2
0 nκ s = (n+κ+A+1)2, (37)
4(M −E )(M +E −C )
(cid:2) nκ nκ s (cid:3)
8
and the associated upper spinor component F (z) as
nκ
Fn,κ(z) = Cn,κzq4βa22+2γa(1+z)−2βa 2F1−n,2s4βa22 + 2γa − 2βa+n+1;2s4βa22 + 2γa +1;−z,
   (38)
where C is the normalization constant.
n,κ
Unlike the non relativistic case, the normalization condition for the Dirac spinor combines
the two individual normalization constants N and C in one single integral. The radial wave
n,κ n,κ
functions are normalized according to the formula ∞ψ†ψr2dr = 1 which explicitly implies for
0
f (r) and g (r) in equation (9) that [44] R
n,κ n,κ
∞ ∞
f2 (r)+g2 (r) r2dr = F2 (r)+G2 (r) dr = 1. (39)
n,κ n,κ n,κ n,κ
Z0 (cid:16) (cid:17) Z0 (cid:16) (cid:17)
wherethe upperand lower spinor components of the total radial wave functions can beexpressed
in terms of the confluent hypergeometric functions as [41]
Γ(n+2β +1)
F (s)= N sβ(1−s)κ+A+1 F (−n,2(β +κ+A+1)+n;2β+1;s), (40)
nκ 2 1
Γ(2β +1)n!
and
Γ(n+2γ +1)
G (s)= N sγ(1−s)κ+A F (−n,2(γ +κ+A+1)+n;2γ+1;s), (41)
nκ 2 1
Γ(2γ +1)n!
with
β = (M −E )(M +E −C ), γ = (M +E )(M −E +C ), s= e−2ar. (42)
nκ nκ s nκ nκ ps
q q
Hence, equation (39) can be re-written in terms of variable s as
1 ds
F2 (s)+G2 (s) = 2a, (43)
s n,κ nκ
Z0 h i
where s → 1 when r → 0 and s → 0 when r → ∞. The method to compute N is given in Ref.
[45]. Thus we have
Γ(n+2β +1) 2 1
N2 s2β−1(1−s)2(κ+A+1)[ F (−n,2(β +κ+A+1)+n;2β +1;s)]2ds
2 1
"(cid:18) Γ(2β +1)n! (cid:19) Z0 #
Γ(n+2γ+1) 2 1
+N2 s2γ−1(1−s)2(κ+A)[ F (−n,2(γ +κ+A)+n;2γ+1;s)]2ds =2a,(44)
2 1
"(cid:18) Γ(2γ +1)n! (cid:19) Z0 #
with the confluent hypergeometric function which is defined by
∞ (a ) ...(a ) si
1 i p i
F (a ,......a ;b ,.....b ;s) = , (45)
p q 1 p 1 q
(b ) ...(b ) i!
i=0 1 i q i
X
9
where (a ) and (b ) are Pochhammer symbols. We can obtain the normalization constant as
1 i 1 i
[46]:
Γ(n+2β +1) 2 ∞ (−n) (2β +2κ+2A+2+n) (2β)
N2 i i iB(2β,2κ+2A+3)
((cid:18) Γ(2β +1)n! (cid:19) i=0 (2β +1)i(2β +2κ+2A+3)ii!
X
× F (−n,2(β +κ+A+1)+n,2β+i;2β +1,2(β +κ+A+1)+1+i;1) (46)
3 2
Γ(n+2γ +1) 2 ∞ (−n) (2γ +2κ+2A+n) (2γ)
i i i
+ B(2γ,2κ+2A+1)
Γ(2γ +1)n! (2γ +1) (2γ +2κ+2A) i!
(cid:18) (cid:19) i=0 i i
X
× F (−n,2(γ +κ+A)+n,2γ +i;2γ +1,2(γ +κ+A)+1+i;1)} = 2a,
3 2
where
1 Γ(x)Γ(y)
B(x,y)= tx−1(1−t)y−1dt = , Re(x),Re(y)> 0,
Γ(x+y)
Z0
1 1
B , = π,B(x,y) = B(y,x). (47)
2 2
(cid:18) (cid:19)
Incomplete beta function is given as
x
B(x;a,b) = ta−1(1−t)b−1dt, (48)
Z0
and the regularized beta function as
B(x;a,b)
I (a,b) = (49)
x
B(a,b)
In the special case when A = 0, our result is identical to the one obtained by Ikhdair [41] and
also by Setare and Haidari [43] by means of the Nikiforov-Uvarov method.
5 Numerical Results
By taking V = 1 and C = −5.0fm−1, we found that the particle is strongly attracted to the
0 ps
nucleus. In the absence of tensor interaction, i.e., A = 0, we noticed that the set of p-spin sym-
metrydoublets: 1s ,2p , 1p ,2s ,2d , 1d ,2f , 0d ,1f , 0f ,1d ,
1/2 3/2 3/2 1/2 5/2 5/2 7/2 3/2 7/2 5/2 3/2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
0g ,1f and 0h ,1g havesameenergiesas−4.500000000, −4.497559159, −4.490219111,
7/2 5/2 9/2 7/2
(cid:16) (cid:17) (cid:16) (cid:17)
−4.477926642, −4.460590892, −4.438679893 and −4.410215299fm−1, respectively.
It is also noticed that the presence of tensor interaction, say A = 0.5, removes the degeneracy
between the states in the above doublets and creates a new set of p-spin symmetric doublets
1s ,1p ,2p ,2d , 1d ,2f ,2s , 1f , 0d , 0f ,1d , 0g ,1f
1/2 3/2 3/2 5/2 5/2 7/2 1/2 7/2 3/2 5/2 3/2 7/2 5/2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
and 0h ,1g having identical energies as −4.499390062, −4.494504009, −4.484696693,
9/2 7/2
(cid:16) (cid:17)
−4.469896410, −4.449992282, −4.424829942 and −4.394205177fm−1, respectively.
10