# Pseudospin Symmetry In Deformed Nuclei With

## Transcript Of Pseudospin Symmetry In Deformed Nuclei With

Vol. 41 (2010)

ACTA PHYSICA POLONICA B

No 11

PSEUDOSPIN SYMMETRY IN DEFORMED NUCLEI WITH TRIAXIAL-SYMMETRIC HARMONIC OSCILLATOR POTENTIAL

M.R. Setare†, Z. Nazari‡

Department of Science, Payame Noor University, Bijar, Iran

(Received August 2, 2010; revised version received September 29, 2010)

Pseudospin symmetry is a perfectly valid concept which can reliably be used in calculations of any deformed heavy nuclei. Therefore, we examined this symmetry in deformed nuclei with triaxial-symmetry and eigenfunctions. Energy equation were obtained for the case of triaxial-symmetric harmonic oscillator potential in the Dirac equation.

PACS numbers: 21.60.Fw, 21.60.Cs, 21.60.Ev

1. Introduction

Pseudospin symmetry is based on experimental observation of the quasidegeneracy between two single-particle orbitals with quantum numbers (nr, l, j = l+1/2) and (nr −1, l+2, j = j +1 = l+3/2), where nr, l and j are single-nucleon radials, orbital and total angular momentum quantum numbers, respectively [1,2]. The structure of the doublet is expressed in terms of a “pseudo” orbital angular momentum ˜l = l + 1 and “pseudo” spin s˜ = 1/2. For example, (nrs1/2, (nr − 1)d3/2) we have ˜l = 1, and (nrp3/2, (nr − 1)f3/2) we have ˜l = 2, etc. These doublets are almost degenerate in proportion to pseudospin s˜ = 1/2, since j = ˜l + s˜, for the two states in the doublet. This is the concept of pseudospin symmetry in the spherical nuclei.

Recently, it has been shown that the pseudospin symmetry is a good approximation in deformed nuclei, including axially and triaxially deformed nuclei, and the implementation based on non-relativistic Nilsson model have been reported [3–6]. In Ref. [4] the validity of the pseudospin concept for heavy triaxially deformed nuclei was explored using correlation coeﬃcient measure between a generalized Nilsson Model Hamiltonian and pseudospin– orbit interaction. Analysis of the correlation coeﬃcient measures for the generalized Nilsson Hamiltonian and pseudospin–orbit operator showed that

† [email protected] ‡ [email protected]

(2459)

2460

M.R. Setare, Z. Nazari

the goodness of pseudospin symmetry remains virtually unchanged from axial to triaxial deformations. This points out the importance of using pseudospin symmetry based shell-model schemes for heavy nuclei at any reasonable deformation including, in particular, triaxial shapes [4].

A study of the goodness of pseudospin dynamical symmetry in triaxial nuclei has been done in Ref. [5]. In this reference, an explicit form for the extended pseudospin transformation for arbitrary deformations has been suggested and applied to some modiﬁcations of triaxial Nilsson Hamiltonian. Nilsson-type model could be viewed as a further clariﬁcation on triaxial model which was intended to correctly reproduce the structure of the basic states in both spherical and cylindrical limits and extend this to arbitrary triaxial shapes [6].

In presence of deformation, doublets with quantum numbers

[N,

nz ,

Λ]

Ω

=

Λ

+

1 2

and

[N, nz, Λ = Λ + 2] Ω = Λ + 32

can be expressed in terms of pseudo-orbital and total angular momentum projections Λ˜ = Λ + 1, Ω = Λ˜ ± 21 [7]. The pseudospin symmetry has been used to explain features of deformed nuclei [8–10].

In the past decades, Relativistic Mean-Field Theory (RMF) achieved great success in description of nuclear properties, especially in pseudospin symmetry [11] and spin symmetry in anti-nucleon spectra [20]. In the Dirac equation of nucleon, when scalar potential S(r) and vector potential V (r) are equal in amplitudes but opposite in sign, i.e., S(r) + V (r) = 0, or more generally, d[S(r) + V (r)]/dr = 0, there is an exact pseudospin symmetry in single-particle spectra [11–14]. These conditions imply some special relations between four components of Dirac wavefunctions which have been used to test the pseudospin symmetry in spherical and axially deformed nuclei [15–19].

In the nucleus, the charge-conjugation transformation relates spin symmetry of antinucleons to pseudospin symmetry of the nucleons [20]. This has also been discussed in Ref. [21], analyzing harmonic oscillator for antinucleons with spin symmetry (S(r) = V (r)). Castro et al. [22] have solved generalized relativistic harmonic oscillator in 1 + 1 dimensions, i.e., including a linear pseudoscalar potential and quadratic scalar and vector potentials which have equal or opposite signs.

They considered positive and negative quadratic potentials and discussed in detail their bound state solutions for fermions and antifermions. Some authors studied relativistic harmonic oscillator for spin 21 particles, and obtained bound state solutions for Dirac equation whit spin and pseudospin symmetry conditions [23, 24]. In this paper, some part of Ref. [19] will be reviewed. Then, Dirac equation for triaxial-symmetric harmonic oscillator will be solved.

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2461

At ﬁrst, eigenfunctions and energy equation for Dirac equation with triaxial-symmetric harmonic oscillator under pseudospin symmetry condition were obtained. Then, the results were compared with previous results for this problem in spherical nuclei, i.e. the nuclei with spherical harmonic oscillator which was done in [23, 24]. In the previous work, this procedure for axial-symmetric harmonic oscillator was done [25]. Finally, our results were compared with those of Ginocchio’s work through chiral and chargeconjugation transformations.

2. Dirac Hamiltonian and pseudospin symmetry

In this section, Ref. [19] is reviewed brieﬂy. Hamiltonian of a Dirac particle of mass M in an external scalar, S(r ) , and vector, V (r ), potentials is given by

H = αˆ · p + βˆ(M + S(r )) + V (r ) ,

(1)

where α and β are Dirac matrices. Dirac equation can be written as ( = c = 1)

αˆ · p + βˆ[M + S(r )] + V (r ) Φ(r ) = EΦ(r ) .

(2)

Dirac Hamiltonian with spherically symmetric scalar and vector potentials are invariant under a SU(2) algebra for two limits: S(r ) = V (r ) + Cs and S(r ) = −V (r ) + Cps, where Cs, Cps are constants [26]. When the former limit occurs, we have spin symmetry [27]. The latter limit leads to pseudospin symmetry [9]. Generators of pseudospin are given by

S˜i = s˜0i s0i = Ups0iUp s0i , (3)

where

si

=

σi 2

,

σi

are

Pauli

matrices

and

Up

=

σ·p p

is

unitary

matrix

oper-

ator [28]. This generators commute with Dirac Hamiltonian for the limit

of S(r ) = −V (r ) + Cps, [Hps, Si] = 0. Thus, Dirac Hamiltonian and pseu-

dospin symmetry have simultaneous eigenfunction

HpsΦpk,sµ˜(r ) = EkΦpk,sµ˜(r ) ,

(4)

where k is just a label for the remaining quantum numbers besides µ˜ [19], and µ˜ = ± 21 is eigenvalue of S˜z

S˜zΦpk,sµ˜(r ) = µ˜Φpk,sµ˜(r ) .

(5)

2462

M.R. Setare, Z. Nazari

Eigenstate in pseudospin doublet will be connected by S˜± generators

S˜±Φpk,sµ˜(r ) = 12 ∓ µ˜ 32 ± µ˜ Φpk,sµ˜(r ) .

(6)

Dirac four-component wavefunction, Φpk,sµ˜(r ), is given by

gk+,µ˜(r )

ps

gk−,µ˜(r )

Φk,µ˜ (r

)

=

if + (r)

,

(7)

k,µ˜

ifk−,µ˜(r )

where gk±,µ˜(r ) are upper Dirac components, here + indicates spin up and − spin down and, fk±,µ˜(r ) are lower Dirac components, where + indicates spin up and − spin down. Pseudospin symmetry create some relations between

these components which are derived from Eqs. (5) and (6) [18]

f+

1 (r )

=

f

−

1

(r

)

=

0

,

k,− 2

k, 2

f + 1 (r ) = f − 1 (r ) ≡ fk(r ) ,

k,+ 2

k,− 2

g+ 1 (r ) = −g− 1 (r ) ≡ gk(r ) ,

k, 2

k,− 2

∂∂ −i

g− (r ) =

∂x ∂y k, 21

∂∂ +i

g+

(r ) ,

∂x ∂y k,− 21

∂ g± (r ) = ± ∂z k,∓ 12

∂∂ ∓i

∂x ∂y

g± 1 (r ) .

k,± 2

(8) (9) (10) (11)

(12)

Therefore, Dirac eigenfunction in pseudospin doublet are given by

gk(r )

Φps (r ) = gk−, 1 (r ) ,

(13)

k, 12

2

ifk(r )

0

g+ 1 (r )

k,− 2

Φps

(r ) =

−gk(r )

.

(14)

k,− 12

0

ifk(r )

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2463

3. Pseudospin symmetry for triaxially deformed nuclei

In Ref. [4] it has been shown that although near level degeneracy of pseudospin–orbit partners in axial systems is lost for triaxial geometries, pseudospin symmetry breaking induced by triaxiality is small and comparable to that found in axial cases, and therefore pseudospin symmetry remains an important physical concept.

We replace Eqs. (13) and (14) in Eq. (2) and obtain the following relations for upper and lower components of Dirac wavefunctions, respectively

g± (r ) =

−1

k,∓ 12

M −E+Σ

∂∂ ∓i

∂x ∂y

fk(r ) ,

−1 ∂ gk(r ) = M − E + Σ ∂z fk(r ) ,

−1 fk(r ) = M + E − ∆

∂∂ +i

g+

∂ (r ) +

∂x ∂y k,− 12

∂z

(15) (16) gk(r ) , (17)

where Σ = S(r ) + V (r ) and ∆ = V (r ) − S(r ).

By substituting g+ 1 (r) and gk(r ) from Eqs. (15) and (16) in Eq. (17)

k,− 2

a second order deferential equation for lower component, fk(r ), is obtained

∂2 ∂2 ∂2

1

(M + E − ∆)(M − E + Σ)fk(r ) = ∂x2 + ∂y2 + ∂z2 − M − E + Σ

∂∂

∂ ∂ ∂Σ ∂

∂x + i ∂y Σ ∂x + i ∂y + ∂z ∂z fk(r ) . (18)

By applying pseudospin symmetry condition, i.e. Σ = 0, or ∂∂Σx = 0, ∂∂Σy = 0, ∂∂Σz = 0 and replacing ∆ = 2V , this equation is reduced to

∂2 ∂2 ∂2

2

2

∂x2 + ∂y2 + ∂z2 + E − M + 2V (M − E) fk(r ) = 0 . (19)

For triaxial deformed nuclei, Eq. (19) with the following vector and scalar potentials

V (r ) = −S(r ) = 21 M ωx2x2 + ωy2y2 + ωz2z2

(20)

can be written as

∂2 ∂2 ∂2

2 2 22 22

2

2

∂x2 + ∂y2 + ∂z2 −(E −M )M ωxx +ωyy +ωz z +E −M fk(r ) . (21)

2464

M.R. Setare, Z. Nazari

This equation can be written as three separate equations for x1, y1 and z1

∂2

2

∂x2 − x1 + 2n˜x + 1 fn˜x(r ) = 0 ,

(22)

1

∂2

2

∂y2 − y1 + 2n˜y + 1 fn˜y (r ) = 0 ,

(23)

1

∂2

2

∂z2 − z1 + 2n˜z + 1 fn˜z (r ) = 0 ,

(24)

1

where

1

x1 = M ωx2(E − M ) 4 x = αxx ,

(25)

1

y1 = M ωy2(E − M ) 4 y = αyy ,

(26)

and

1

z1 = M ωz2(E − M ) 4 z = αzz .

(27)

Here we consider

fk(r ) = N fn˜x (x)fn˜y (y)fn˜z (z) ,

(28)

where N is normalization constant which is determined by

2

dx dy dz Φps 1 (r ) = 1 .

(29)

k,± 2

Energy equation becomes

En˜2x,n˜y,n˜z − M 2 = αx2 (2n˜x + 1) + αy2(2n˜y + 1) + αz2(2n˜z + 1) , (30)

where n˜x, n˜y and n˜z are quantum numbers of oscillator of lower component in x, y and z directions, respectively.

In the spherical harmonic oscillator each level has a (N + 2)(N + 1) degeneracy because of pseudospin symmetry and because allowed pseudoorbital angular momenta are ˜l = N, N − 2, . . . 0 or 1 and allowed pseudoorbital angular momentum projections are m = ˜l, ˜l − 1, . . . , ˜l.

Each group of N in deformed harmonic oscillator contains the levels for n˜z = 0, 1, . . . , N with excitation energy increasing with decreasing n˜z. Each level has a 2(N − n˜z) + 1 degeneracy for (N − n˜z) even and a 2(N − n˜z + 1) degeneracy for (N − n˜z) odd due to spin symmetry and because the allowed orbital angular momentum projections are Λ˜= (N −n˜z), (N −n˜z −2), 1 or 0. The splitting of levels within each N appears to be approximately linear with n˜z (see Fig. 2 from Ref. [21]).

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2465

Eqs. (22)–(24) are Hermite equations and solution of these equations can be written in terms of Hermite functions, thus we can obtain lower component wavefunction, fk(r ), from Eq. (28)

α2xx2+α2y y2+α2z z2

fk(r ) = e−

2

Hn˜x (αxx)Hn˜y (αyy)Hn˜z (αzz) .

(31)

Now, we are going to obtain energy spectrum in the spherical symmetry case.

For spherical harmonic oscillator potential we have ωx = ωy = ωz = ω1 then, by using Eqs. (25)–(27), one can obtain αx = αy = αz = α1. Therefore, the energy equation of Eq. (30) can be rewritten as

En˜2x,n˜y,n˜z − M 2 = 2α12

n˜x

+

n˜y

+

n˜z

+

3 2

.

(32)

On the other hand, energy spectrum of spherical harmonic oscillator corresponding to Eq. (62) of Ref. [23], becomes as following

E2 − M 2 = 2 M ω12(E − M ) 12 2n˜ + ˜l + 32 = 2α12 2n˜ + ˜l + 32 , (33)

where the potential, used in our paper, is twice the potential introduced in Ref. [23]. The ˜l and n˜ are the orbital angular momentum and the number

of nodes of lower component of radial wavefunction.

By using the Eqs. (32) and (33), we have

n˜x + n˜y + n˜z = 2n˜ + ˜l .

(34)

Then the energy equation of triaxial-symmetric harmonic oscillator for ωx = ωy = ωz = ω1 is equal to that of spherical harmonic oscillator.

Now, we derive upper components from Eqs. (15) and (16)

gk(r ) = αz 1 M −E Hn˜z (αzz)

Hn˜z+21(αzz) −n˜zHn˜z−1(αzz) fk(r ) , (35)

g± (r ) = 1

k,∓ 12

M −E

αx Hn˜x (αxx)

Hn˜x+12(αxx) − n˜xHn˜x−1(αxx)

∓ iαy Hn˜y (αyy)

Hn˜ y +1 (αy y)

2

− n˜yHn˜y−1(αyy)

fk(r ) .

(36)

In an interesting paper Castro et al. [22] found that solutions for zero pseudoscalar potential are related to spin and pseudospin symmetry of Dirac equation in 3 + 1 dimensions. They showed how charge conjugation and

2466

M.R. Setare, Z. Nazari

chiral transformations are related to spectra of spin and pseudospin symmetries. They also found that there is the same spectrum, but diﬀerent spinor solutions for massless particles of the spin and pseudospin symmetries.

And now, our results are compared with those of Ginocchio’s work [21] through chiral and charge-conjugation transformations. At ﬁrst, we introduce charge-conjugation operation:

The charge-conjugation operation changes the sign of the vector potential in (1). This is performed by the transformation [29]

Φ → Φc = α Φ∗ .

(37)

After applying this charge-conjugation operation to Dirac equation (2), time independent Dirac equation becomes

HcΦ˜c = −E Φ˜c ,

(38)

where Φ˜c = α Φ˜∗, Φ˜(x) = e i Etφ (x, t) and Hc is given by

Hc = αˆ · p + βˆ(M + S(r )) − V (r ) .

(39)

In terms of the potentials ∆ and Σ, this Hamiltonian becomes

Hc = αˆ · p + βˆM c2 − I +2 βˆ∆ − I −2 βˆΣ . (40)

We can see that charge-conjugation operation changes the sign of the energy and of the potential V (r). This means that Σ turns into −∆ and ∆ into −Σ. Therefore, to be invariant under charge conjugation, the Hamiltonian must contain only a scalar potential [22]. Now, we introduce chiral transformation: Chiral operator for a Dirac spinor is the matrix γ5. Transformed Dirac spinor under chiral transformation is given by Φχ = γ5Φ and transformed Dirac Hamiltonian Hχ = γ5Hγ5. Chiral transformed Dirac equation is

HχΦ˜χ = E Φ˜χ ,

(41)

where Hχ is given by

Hχ = αˆ · p − βˆ M c2 + S(r ) + V (r ) ,

(42)

in terms of Σ and ∆, this Hamiltonian becomes

Hχ = αˆ · p − βˆM c2 + I +2 βˆ∆ + I −2 βˆΣ . (43)

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2467

This means that the chiral transformation changes the sign of the mass and that of scalar potential, thus turning Σ into ∆ and vice versa. A chiralinvariant Hamiltonian needs to have zero mass and S(r) zero everywhere [22].

With respect to this fact that charge-conjugation transformation performs the changes of ∆ → −Σ, Σ → −∆ and E → −E, we can conclude that charge-conjugation transformation relates the spin symmetry of the negative bound-state solutions (antinucleons) to pseudospin symmetry of positive bound-state solutions (nucleons) [20]. Therefore, under chargeconjugation transformation, the nucleons energy spectrum, Eq. (30), becomes

En2x,ny,nz − M 2 = αx2 (2nx + 1) + αy2(2ny + 1) + αz2(2nz + 1) , (44)

where

1

αx = M ωx2(E + M ) 4 ,

(45)

1

αy = M ωy2(E + M ) 4 ,

(46)

and

1

αz = M ωz2(E + M ) 4 .

(47)

Here, nx, ny and nz are the number of nodes for upper component in x, y and z directions, respectively.

Eq. (44) is the same energy equation (Eq. (19), Ref. [21]) which is ap-

plied for the study of antinucleons. Ginocchio got this equation for triaxial harmonic oscillator for the case ∆ = 0, Σ = M (ωx2x2 + ωy2y2 + ωz2z2), i.e., spin symmetry.

Moreover, since under chiral transformation we have Σ → ∆, ∆ → Σ

and M → −M changes, so we can obtain the conclusions related to ∆ = 0, Σ = M (ωx2x2 + ωy2y2 + ωz2z2), by changing M sign in relevant parameters. Therefore, under chiral transformation, nucleons energy spectrum, Eq. (30),

becomes

En2x,ny,nz − M 2 = αx2 (2nx + 1) + αy2(2ny + 1) + αz2(2nz + 1) . (48)

Eq. (48) is the same as Eq. (44) but for nucleons with spin symmetry. By comparing Eqs. (44) and (48), one may note that the spectra of nucleons and antinucleons are the same for spin symmetry. It means that, for Dirac harmonic oscillator which we considered, spectra of nucleon ∆ = 0 states are degenerate with antinucleon ∆ = 0 states.

2468

M.R. Setare, Z. Nazari

The number of nodes of the function gk(r ), in the z direction is diﬀerent from that of lower component, fk(r ), by one unit, but they are the same in x and y directions. Although, the number of nodes of g± 1 (r ) functions

k,∓ 2

is the same as that of lower component in z direction, they are diﬀerent in x and y direction (by one unit).

In fact, this shows the structure of nodes in the pseudospin doublet of deformed nuclei with triaxial-symmetric harmonic oscillator potential.

The structure of nuclear rotational bands has been investigated in Ref. [30]. Szymanski considered single-particle motion of a nucleon in a highly deformed and fast rotating mean ﬁeld by the rather well known procedure of the cranking model. Pseudospin picture which will be used extensively throughout this work is mainly connected to cranking treatment of the rotation. The cranking Hamiltonian [30]

Hω = H − ωj1

(49)

will, therefore, be employed here together with the pseudospin picture. Here, the angular momentum j1 is the sum of the pseudo-orbital angular momentum ˜l1 and pseudospin s˜1. In Ref. [30], the whole dynamics has been described in terms of a simple picture of a Rotating Harmonic Oscillator (RHO) in the coordinate space. An exact solution to the cranking Hamiltonian Hω exists [31, 32] and may be employed to investigate explicitly the single-particle Routhians. The solution has the form of three independent normal modes of the Harmonic Oscillator (HO) type and obtained the onenucleon Routhians as [30]

eων =

n1

+

1 2

ω1 +

n2

+

1 2

Ω2 +

n3

+

1 2

Ω3 .

(50)

Here, ω1, ω2, ω1 are the three original harmonic oscillator frequencies. The two modiﬁed (normal) frequencies Ω2 and Ω3 are simple functions of ω2 and ω3, and rotational frequency ω [31, 32]. Integers n1, n2 and n3 are the three quantum numbers of the RHO. It seems to be a remarkable result of such a

model that whenever the condition

n2 = n3

(51)

is fulﬁlled, the orbit (n1, n2, n3) becomes almost a ﬂat line in the eωn1,n2,n3 = f (ω) representation in a rather large interval of ω (see Fig. 1 of Ref. [30]).

The angular momentum operator j1 from Eq. (48) couples all the states |N˜ n3Λ˜Ω in the pseudospin picture so that expansion of any RHO state (n1, n2, n3) into the states |N˜ n3Λ˜Ω is inﬁnite. Nevertheless, for slow rota-

tion a certain correspondence between the two representations can be estab-

lished approximately (see Table II from Fig. [30]). The states (n1, n2, n3) are

ACTA PHYSICA POLONICA B

No 11

PSEUDOSPIN SYMMETRY IN DEFORMED NUCLEI WITH TRIAXIAL-SYMMETRIC HARMONIC OSCILLATOR POTENTIAL

M.R. Setare†, Z. Nazari‡

Department of Science, Payame Noor University, Bijar, Iran

(Received August 2, 2010; revised version received September 29, 2010)

Pseudospin symmetry is a perfectly valid concept which can reliably be used in calculations of any deformed heavy nuclei. Therefore, we examined this symmetry in deformed nuclei with triaxial-symmetry and eigenfunctions. Energy equation were obtained for the case of triaxial-symmetric harmonic oscillator potential in the Dirac equation.

PACS numbers: 21.60.Fw, 21.60.Cs, 21.60.Ev

1. Introduction

Pseudospin symmetry is based on experimental observation of the quasidegeneracy between two single-particle orbitals with quantum numbers (nr, l, j = l+1/2) and (nr −1, l+2, j = j +1 = l+3/2), where nr, l and j are single-nucleon radials, orbital and total angular momentum quantum numbers, respectively [1,2]. The structure of the doublet is expressed in terms of a “pseudo” orbital angular momentum ˜l = l + 1 and “pseudo” spin s˜ = 1/2. For example, (nrs1/2, (nr − 1)d3/2) we have ˜l = 1, and (nrp3/2, (nr − 1)f3/2) we have ˜l = 2, etc. These doublets are almost degenerate in proportion to pseudospin s˜ = 1/2, since j = ˜l + s˜, for the two states in the doublet. This is the concept of pseudospin symmetry in the spherical nuclei.

Recently, it has been shown that the pseudospin symmetry is a good approximation in deformed nuclei, including axially and triaxially deformed nuclei, and the implementation based on non-relativistic Nilsson model have been reported [3–6]. In Ref. [4] the validity of the pseudospin concept for heavy triaxially deformed nuclei was explored using correlation coeﬃcient measure between a generalized Nilsson Model Hamiltonian and pseudospin– orbit interaction. Analysis of the correlation coeﬃcient measures for the generalized Nilsson Hamiltonian and pseudospin–orbit operator showed that

† [email protected] ‡ [email protected]

(2459)

2460

M.R. Setare, Z. Nazari

the goodness of pseudospin symmetry remains virtually unchanged from axial to triaxial deformations. This points out the importance of using pseudospin symmetry based shell-model schemes for heavy nuclei at any reasonable deformation including, in particular, triaxial shapes [4].

A study of the goodness of pseudospin dynamical symmetry in triaxial nuclei has been done in Ref. [5]. In this reference, an explicit form for the extended pseudospin transformation for arbitrary deformations has been suggested and applied to some modiﬁcations of triaxial Nilsson Hamiltonian. Nilsson-type model could be viewed as a further clariﬁcation on triaxial model which was intended to correctly reproduce the structure of the basic states in both spherical and cylindrical limits and extend this to arbitrary triaxial shapes [6].

In presence of deformation, doublets with quantum numbers

[N,

nz ,

Λ]

Ω

=

Λ

+

1 2

and

[N, nz, Λ = Λ + 2] Ω = Λ + 32

can be expressed in terms of pseudo-orbital and total angular momentum projections Λ˜ = Λ + 1, Ω = Λ˜ ± 21 [7]. The pseudospin symmetry has been used to explain features of deformed nuclei [8–10].

In the past decades, Relativistic Mean-Field Theory (RMF) achieved great success in description of nuclear properties, especially in pseudospin symmetry [11] and spin symmetry in anti-nucleon spectra [20]. In the Dirac equation of nucleon, when scalar potential S(r) and vector potential V (r) are equal in amplitudes but opposite in sign, i.e., S(r) + V (r) = 0, or more generally, d[S(r) + V (r)]/dr = 0, there is an exact pseudospin symmetry in single-particle spectra [11–14]. These conditions imply some special relations between four components of Dirac wavefunctions which have been used to test the pseudospin symmetry in spherical and axially deformed nuclei [15–19].

In the nucleus, the charge-conjugation transformation relates spin symmetry of antinucleons to pseudospin symmetry of the nucleons [20]. This has also been discussed in Ref. [21], analyzing harmonic oscillator for antinucleons with spin symmetry (S(r) = V (r)). Castro et al. [22] have solved generalized relativistic harmonic oscillator in 1 + 1 dimensions, i.e., including a linear pseudoscalar potential and quadratic scalar and vector potentials which have equal or opposite signs.

They considered positive and negative quadratic potentials and discussed in detail their bound state solutions for fermions and antifermions. Some authors studied relativistic harmonic oscillator for spin 21 particles, and obtained bound state solutions for Dirac equation whit spin and pseudospin symmetry conditions [23, 24]. In this paper, some part of Ref. [19] will be reviewed. Then, Dirac equation for triaxial-symmetric harmonic oscillator will be solved.

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2461

At ﬁrst, eigenfunctions and energy equation for Dirac equation with triaxial-symmetric harmonic oscillator under pseudospin symmetry condition were obtained. Then, the results were compared with previous results for this problem in spherical nuclei, i.e. the nuclei with spherical harmonic oscillator which was done in [23, 24]. In the previous work, this procedure for axial-symmetric harmonic oscillator was done [25]. Finally, our results were compared with those of Ginocchio’s work through chiral and chargeconjugation transformations.

2. Dirac Hamiltonian and pseudospin symmetry

In this section, Ref. [19] is reviewed brieﬂy. Hamiltonian of a Dirac particle of mass M in an external scalar, S(r ) , and vector, V (r ), potentials is given by

H = αˆ · p + βˆ(M + S(r )) + V (r ) ,

(1)

where α and β are Dirac matrices. Dirac equation can be written as ( = c = 1)

αˆ · p + βˆ[M + S(r )] + V (r ) Φ(r ) = EΦ(r ) .

(2)

Dirac Hamiltonian with spherically symmetric scalar and vector potentials are invariant under a SU(2) algebra for two limits: S(r ) = V (r ) + Cs and S(r ) = −V (r ) + Cps, where Cs, Cps are constants [26]. When the former limit occurs, we have spin symmetry [27]. The latter limit leads to pseudospin symmetry [9]. Generators of pseudospin are given by

S˜i = s˜0i s0i = Ups0iUp s0i , (3)

where

si

=

σi 2

,

σi

are

Pauli

matrices

and

Up

=

σ·p p

is

unitary

matrix

oper-

ator [28]. This generators commute with Dirac Hamiltonian for the limit

of S(r ) = −V (r ) + Cps, [Hps, Si] = 0. Thus, Dirac Hamiltonian and pseu-

dospin symmetry have simultaneous eigenfunction

HpsΦpk,sµ˜(r ) = EkΦpk,sµ˜(r ) ,

(4)

where k is just a label for the remaining quantum numbers besides µ˜ [19], and µ˜ = ± 21 is eigenvalue of S˜z

S˜zΦpk,sµ˜(r ) = µ˜Φpk,sµ˜(r ) .

(5)

2462

M.R. Setare, Z. Nazari

Eigenstate in pseudospin doublet will be connected by S˜± generators

S˜±Φpk,sµ˜(r ) = 12 ∓ µ˜ 32 ± µ˜ Φpk,sµ˜(r ) .

(6)

Dirac four-component wavefunction, Φpk,sµ˜(r ), is given by

gk+,µ˜(r )

ps

gk−,µ˜(r )

Φk,µ˜ (r

)

=

if + (r)

,

(7)

k,µ˜

ifk−,µ˜(r )

where gk±,µ˜(r ) are upper Dirac components, here + indicates spin up and − spin down and, fk±,µ˜(r ) are lower Dirac components, where + indicates spin up and − spin down. Pseudospin symmetry create some relations between

these components which are derived from Eqs. (5) and (6) [18]

f+

1 (r )

=

f

−

1

(r

)

=

0

,

k,− 2

k, 2

f + 1 (r ) = f − 1 (r ) ≡ fk(r ) ,

k,+ 2

k,− 2

g+ 1 (r ) = −g− 1 (r ) ≡ gk(r ) ,

k, 2

k,− 2

∂∂ −i

g− (r ) =

∂x ∂y k, 21

∂∂ +i

g+

(r ) ,

∂x ∂y k,− 21

∂ g± (r ) = ± ∂z k,∓ 12

∂∂ ∓i

∂x ∂y

g± 1 (r ) .

k,± 2

(8) (9) (10) (11)

(12)

Therefore, Dirac eigenfunction in pseudospin doublet are given by

gk(r )

Φps (r ) = gk−, 1 (r ) ,

(13)

k, 12

2

ifk(r )

0

g+ 1 (r )

k,− 2

Φps

(r ) =

−gk(r )

.

(14)

k,− 12

0

ifk(r )

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2463

3. Pseudospin symmetry for triaxially deformed nuclei

In Ref. [4] it has been shown that although near level degeneracy of pseudospin–orbit partners in axial systems is lost for triaxial geometries, pseudospin symmetry breaking induced by triaxiality is small and comparable to that found in axial cases, and therefore pseudospin symmetry remains an important physical concept.

We replace Eqs. (13) and (14) in Eq. (2) and obtain the following relations for upper and lower components of Dirac wavefunctions, respectively

g± (r ) =

−1

k,∓ 12

M −E+Σ

∂∂ ∓i

∂x ∂y

fk(r ) ,

−1 ∂ gk(r ) = M − E + Σ ∂z fk(r ) ,

−1 fk(r ) = M + E − ∆

∂∂ +i

g+

∂ (r ) +

∂x ∂y k,− 12

∂z

(15) (16) gk(r ) , (17)

where Σ = S(r ) + V (r ) and ∆ = V (r ) − S(r ).

By substituting g+ 1 (r) and gk(r ) from Eqs. (15) and (16) in Eq. (17)

k,− 2

a second order deferential equation for lower component, fk(r ), is obtained

∂2 ∂2 ∂2

1

(M + E − ∆)(M − E + Σ)fk(r ) = ∂x2 + ∂y2 + ∂z2 − M − E + Σ

∂∂

∂ ∂ ∂Σ ∂

∂x + i ∂y Σ ∂x + i ∂y + ∂z ∂z fk(r ) . (18)

By applying pseudospin symmetry condition, i.e. Σ = 0, or ∂∂Σx = 0, ∂∂Σy = 0, ∂∂Σz = 0 and replacing ∆ = 2V , this equation is reduced to

∂2 ∂2 ∂2

2

2

∂x2 + ∂y2 + ∂z2 + E − M + 2V (M − E) fk(r ) = 0 . (19)

For triaxial deformed nuclei, Eq. (19) with the following vector and scalar potentials

V (r ) = −S(r ) = 21 M ωx2x2 + ωy2y2 + ωz2z2

(20)

can be written as

∂2 ∂2 ∂2

2 2 22 22

2

2

∂x2 + ∂y2 + ∂z2 −(E −M )M ωxx +ωyy +ωz z +E −M fk(r ) . (21)

2464

M.R. Setare, Z. Nazari

This equation can be written as three separate equations for x1, y1 and z1

∂2

2

∂x2 − x1 + 2n˜x + 1 fn˜x(r ) = 0 ,

(22)

1

∂2

2

∂y2 − y1 + 2n˜y + 1 fn˜y (r ) = 0 ,

(23)

1

∂2

2

∂z2 − z1 + 2n˜z + 1 fn˜z (r ) = 0 ,

(24)

1

where

1

x1 = M ωx2(E − M ) 4 x = αxx ,

(25)

1

y1 = M ωy2(E − M ) 4 y = αyy ,

(26)

and

1

z1 = M ωz2(E − M ) 4 z = αzz .

(27)

Here we consider

fk(r ) = N fn˜x (x)fn˜y (y)fn˜z (z) ,

(28)

where N is normalization constant which is determined by

2

dx dy dz Φps 1 (r ) = 1 .

(29)

k,± 2

Energy equation becomes

En˜2x,n˜y,n˜z − M 2 = αx2 (2n˜x + 1) + αy2(2n˜y + 1) + αz2(2n˜z + 1) , (30)

where n˜x, n˜y and n˜z are quantum numbers of oscillator of lower component in x, y and z directions, respectively.

In the spherical harmonic oscillator each level has a (N + 2)(N + 1) degeneracy because of pseudospin symmetry and because allowed pseudoorbital angular momenta are ˜l = N, N − 2, . . . 0 or 1 and allowed pseudoorbital angular momentum projections are m = ˜l, ˜l − 1, . . . , ˜l.

Each group of N in deformed harmonic oscillator contains the levels for n˜z = 0, 1, . . . , N with excitation energy increasing with decreasing n˜z. Each level has a 2(N − n˜z) + 1 degeneracy for (N − n˜z) even and a 2(N − n˜z + 1) degeneracy for (N − n˜z) odd due to spin symmetry and because the allowed orbital angular momentum projections are Λ˜= (N −n˜z), (N −n˜z −2), 1 or 0. The splitting of levels within each N appears to be approximately linear with n˜z (see Fig. 2 from Ref. [21]).

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2465

Eqs. (22)–(24) are Hermite equations and solution of these equations can be written in terms of Hermite functions, thus we can obtain lower component wavefunction, fk(r ), from Eq. (28)

α2xx2+α2y y2+α2z z2

fk(r ) = e−

2

Hn˜x (αxx)Hn˜y (αyy)Hn˜z (αzz) .

(31)

Now, we are going to obtain energy spectrum in the spherical symmetry case.

For spherical harmonic oscillator potential we have ωx = ωy = ωz = ω1 then, by using Eqs. (25)–(27), one can obtain αx = αy = αz = α1. Therefore, the energy equation of Eq. (30) can be rewritten as

En˜2x,n˜y,n˜z − M 2 = 2α12

n˜x

+

n˜y

+

n˜z

+

3 2

.

(32)

On the other hand, energy spectrum of spherical harmonic oscillator corresponding to Eq. (62) of Ref. [23], becomes as following

E2 − M 2 = 2 M ω12(E − M ) 12 2n˜ + ˜l + 32 = 2α12 2n˜ + ˜l + 32 , (33)

where the potential, used in our paper, is twice the potential introduced in Ref. [23]. The ˜l and n˜ are the orbital angular momentum and the number

of nodes of lower component of radial wavefunction.

By using the Eqs. (32) and (33), we have

n˜x + n˜y + n˜z = 2n˜ + ˜l .

(34)

Then the energy equation of triaxial-symmetric harmonic oscillator for ωx = ωy = ωz = ω1 is equal to that of spherical harmonic oscillator.

Now, we derive upper components from Eqs. (15) and (16)

gk(r ) = αz 1 M −E Hn˜z (αzz)

Hn˜z+21(αzz) −n˜zHn˜z−1(αzz) fk(r ) , (35)

g± (r ) = 1

k,∓ 12

M −E

αx Hn˜x (αxx)

Hn˜x+12(αxx) − n˜xHn˜x−1(αxx)

∓ iαy Hn˜y (αyy)

Hn˜ y +1 (αy y)

2

− n˜yHn˜y−1(αyy)

fk(r ) .

(36)

In an interesting paper Castro et al. [22] found that solutions for zero pseudoscalar potential are related to spin and pseudospin symmetry of Dirac equation in 3 + 1 dimensions. They showed how charge conjugation and

2466

M.R. Setare, Z. Nazari

chiral transformations are related to spectra of spin and pseudospin symmetries. They also found that there is the same spectrum, but diﬀerent spinor solutions for massless particles of the spin and pseudospin symmetries.

And now, our results are compared with those of Ginocchio’s work [21] through chiral and charge-conjugation transformations. At ﬁrst, we introduce charge-conjugation operation:

The charge-conjugation operation changes the sign of the vector potential in (1). This is performed by the transformation [29]

Φ → Φc = α Φ∗ .

(37)

After applying this charge-conjugation operation to Dirac equation (2), time independent Dirac equation becomes

HcΦ˜c = −E Φ˜c ,

(38)

where Φ˜c = α Φ˜∗, Φ˜(x) = e i Etφ (x, t) and Hc is given by

Hc = αˆ · p + βˆ(M + S(r )) − V (r ) .

(39)

In terms of the potentials ∆ and Σ, this Hamiltonian becomes

Hc = αˆ · p + βˆM c2 − I +2 βˆ∆ − I −2 βˆΣ . (40)

We can see that charge-conjugation operation changes the sign of the energy and of the potential V (r). This means that Σ turns into −∆ and ∆ into −Σ. Therefore, to be invariant under charge conjugation, the Hamiltonian must contain only a scalar potential [22]. Now, we introduce chiral transformation: Chiral operator for a Dirac spinor is the matrix γ5. Transformed Dirac spinor under chiral transformation is given by Φχ = γ5Φ and transformed Dirac Hamiltonian Hχ = γ5Hγ5. Chiral transformed Dirac equation is

HχΦ˜χ = E Φ˜χ ,

(41)

where Hχ is given by

Hχ = αˆ · p − βˆ M c2 + S(r ) + V (r ) ,

(42)

in terms of Σ and ∆, this Hamiltonian becomes

Hχ = αˆ · p − βˆM c2 + I +2 βˆ∆ + I −2 βˆΣ . (43)

Pseudospin Symmetry in Deformed Nuclei with Triaxial-symmetric . . . 2467

This means that the chiral transformation changes the sign of the mass and that of scalar potential, thus turning Σ into ∆ and vice versa. A chiralinvariant Hamiltonian needs to have zero mass and S(r) zero everywhere [22].

With respect to this fact that charge-conjugation transformation performs the changes of ∆ → −Σ, Σ → −∆ and E → −E, we can conclude that charge-conjugation transformation relates the spin symmetry of the negative bound-state solutions (antinucleons) to pseudospin symmetry of positive bound-state solutions (nucleons) [20]. Therefore, under chargeconjugation transformation, the nucleons energy spectrum, Eq. (30), becomes

En2x,ny,nz − M 2 = αx2 (2nx + 1) + αy2(2ny + 1) + αz2(2nz + 1) , (44)

where

1

αx = M ωx2(E + M ) 4 ,

(45)

1

αy = M ωy2(E + M ) 4 ,

(46)

and

1

αz = M ωz2(E + M ) 4 .

(47)

Here, nx, ny and nz are the number of nodes for upper component in x, y and z directions, respectively.

Eq. (44) is the same energy equation (Eq. (19), Ref. [21]) which is ap-

plied for the study of antinucleons. Ginocchio got this equation for triaxial harmonic oscillator for the case ∆ = 0, Σ = M (ωx2x2 + ωy2y2 + ωz2z2), i.e., spin symmetry.

Moreover, since under chiral transformation we have Σ → ∆, ∆ → Σ

and M → −M changes, so we can obtain the conclusions related to ∆ = 0, Σ = M (ωx2x2 + ωy2y2 + ωz2z2), by changing M sign in relevant parameters. Therefore, under chiral transformation, nucleons energy spectrum, Eq. (30),

becomes

En2x,ny,nz − M 2 = αx2 (2nx + 1) + αy2(2ny + 1) + αz2(2nz + 1) . (48)

Eq. (48) is the same as Eq. (44) but for nucleons with spin symmetry. By comparing Eqs. (44) and (48), one may note that the spectra of nucleons and antinucleons are the same for spin symmetry. It means that, for Dirac harmonic oscillator which we considered, spectra of nucleon ∆ = 0 states are degenerate with antinucleon ∆ = 0 states.

2468

M.R. Setare, Z. Nazari

The number of nodes of the function gk(r ), in the z direction is diﬀerent from that of lower component, fk(r ), by one unit, but they are the same in x and y directions. Although, the number of nodes of g± 1 (r ) functions

k,∓ 2

is the same as that of lower component in z direction, they are diﬀerent in x and y direction (by one unit).

In fact, this shows the structure of nodes in the pseudospin doublet of deformed nuclei with triaxial-symmetric harmonic oscillator potential.

The structure of nuclear rotational bands has been investigated in Ref. [30]. Szymanski considered single-particle motion of a nucleon in a highly deformed and fast rotating mean ﬁeld by the rather well known procedure of the cranking model. Pseudospin picture which will be used extensively throughout this work is mainly connected to cranking treatment of the rotation. The cranking Hamiltonian [30]

Hω = H − ωj1

(49)

will, therefore, be employed here together with the pseudospin picture. Here, the angular momentum j1 is the sum of the pseudo-orbital angular momentum ˜l1 and pseudospin s˜1. In Ref. [30], the whole dynamics has been described in terms of a simple picture of a Rotating Harmonic Oscillator (RHO) in the coordinate space. An exact solution to the cranking Hamiltonian Hω exists [31, 32] and may be employed to investigate explicitly the single-particle Routhians. The solution has the form of three independent normal modes of the Harmonic Oscillator (HO) type and obtained the onenucleon Routhians as [30]

eων =

n1

+

1 2

ω1 +

n2

+

1 2

Ω2 +

n3

+

1 2

Ω3 .

(50)

Here, ω1, ω2, ω1 are the three original harmonic oscillator frequencies. The two modiﬁed (normal) frequencies Ω2 and Ω3 are simple functions of ω2 and ω3, and rotational frequency ω [31, 32]. Integers n1, n2 and n3 are the three quantum numbers of the RHO. It seems to be a remarkable result of such a

model that whenever the condition

n2 = n3

(51)

is fulﬁlled, the orbit (n1, n2, n3) becomes almost a ﬂat line in the eωn1,n2,n3 = f (ω) representation in a rather large interval of ω (see Fig. 1 of Ref. [30]).

The angular momentum operator j1 from Eq. (48) couples all the states |N˜ n3Λ˜Ω in the pseudospin picture so that expansion of any RHO state (n1, n2, n3) into the states |N˜ n3Λ˜Ω is inﬁnite. Nevertheless, for slow rota-

tion a certain correspondence between the two representations can be estab-

lished approximately (see Table II from Fig. [30]). The states (n1, n2, n3) are