Based on the Relativistic Mean Field (RMF) approach the existence of the
broken pseudo-spin symmetry is investigated. Both spherical RMF and constrained
deformed RMF calculations are carried out employing realistic Lagrangian
parameters for spherical and for deformed sample nuclei. The quasi - degenerate
pseudo-spin doublets are confirmed to exist near the fermi surface for both
spherical and deformed nuclei.Comment: 9 pages RevTex, 4 p.s figures, to appear in Phys. Rev. C as R.

A. Arima

A. Bohr

A. L. Blokhin

B. D. Serot

B. Mottelson

C. Bahri

C. S. Warke

G. A. Lalazissis

J. N. Ginocchio

J. P. Maharana

K. T. Hecht

O. Castanos

P. G. Reinhard

P. Ring

R. D. Ratna Raju

Y. K. Gambhir

English

arXiv

Crossref

Physical Review C

Relativistic mean field approach and the pseudospin symmetry

Based on the relativistic mean field (RMF) approach the existence of the broken pseudospin symmetry is investigated. Both spherical RMF and constrained deformed RMF calculations are carried out employing realistic Lagrangian parameters for spherical and for deformed sample nuclei. The quasidegenerate pseudospin doublets are confirmed to exist near the Fermi surface for both spherical and deformed nuclei

Lalazissis, G. A.

Gambhir, Y. K.

Maharana, J. P.

Warke, C. S.

Ring, P.

Publications of the IAS Fellows

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98
Relativistic Mean Field Approach and the Pseudo-Spin Symmetry
G. A. Lalazissis1, Y. K. Gambhir1,2, J. P. Maharana2, C. S. Warke2, and P. Ring1
1Physik-Department der Technischen Universita¨t Mu¨nchen, D-85747 Garching, Germany
2 Department of Physics, Indian Institute of Technology Bombay
Powai, Mumbai 400026, India
February 9, 2008
Based on the Relativistic Mean Field (RMF) approach the existence of
the broken pseudo-spin symmetry is investigated. Both spherical RMF and
constrained deformed RMF calculations are carried out employing realistic
Lagrangian parameters for spherical and for deformed sample nuclei. The
quasi - degenerate pseudo-spin doublets are confirmed to exist near the fermi
surface for both spherical and deformed nuclei.
PACS numbers : 21..60Cs, 21.10.-k, 24.10.Jv
Pseudospin symmetry has been discovered in nuclear physics nearly 30 years ago [1–3].
The recent claim [4] that pseudo-spin symmetry may arise due to near equality in magni-
tude of attractive scalar and repulsive vector fields in relativistic mean theory, has revived
the activity related to the understanding of the origin of this symmetry in real nuclei. The
concept of pseudo-spin symmetry [1,2] is based on the experimental observation of the ex-
istence of quasi-degenerate doublets of normal parity orbitals (n, ℓ, j = ℓ + 1
2
) and (n − 1,
ℓ+ 2, j = ℓ+ 3
2
) such as (4s1/2, 3d3/2), (3d5/2, 2g7/2) etc., in the same major shell. Since for
spherical systems the quantum numbers jπ are conserved, the pseudo-spin angular momenta
(ℓ˜, s˜ = 1/2) satisfy j˜ = j = l˜ ± 1
2
.
In order to interpret this near degenerate pair of j = ℓ + 1/2 and j = ℓ + 3/2 states
as pseudo-spin doublets corresponding to m˜s = ±1/2, ℓ˜ has to be ℓ + 1. It then follows
for the major oscillator quantum number: N˜ = N − 1, for the radial quantum number
n˜ = (N˜ − ℓ˜)−1 and for the parity π˜ = −π. For zero pseudo-spin orbit splitting, the pseudo-
1
spin multiplet will be degenerate. Thus the pair of orbitals (4s1/2, 3d3/2) and (3d5/2, 2g7/2)
can be viewed as the (2p˜1/2, 2p˜3/2) and (1f˜5/2, 1f˜7/2) pseudo-spin doublets. The symmetry
can also be investigated in deformed nuclei. In the asymptotic Nilsson scheme one finds the
pseudo-spin quantum numbers (N˜ = N − 1 , n˜3 = n3 , Λ˜ = Λ + 1 and Ω˜ = Ω) . Therefore,
the Nilsson orbitals [N, n3,Λ,Ω = Λ+1/2] and [N, n3,Λ+2,Ω = Λ+3/2] can be viewed as
the pseudo spin-orbit doublets [N˜ , n˜3, Λ˜, Ω˜ = Λ˜± 1/2] [5].
Apart from the rather formal relabeling of quantum numbers various proposals for an
explicit transformation from the normal scheme to the pseudo-spin scheme have been made
in the last twenty years and several nuclear properties have been investigated in this scheme
[6–9]. However, the origin of pseudo-spin symmetry remained unknown until the recent
observation of Ginocchio [4,10] where for the first time the origin of this symmetry is claimed
to be revealed as due to the near equality in magnitude of the attractive scalar and repulsive
vector fields in relativistic theories. Here in this letter we follow this idea and investigate
to what extent the pseudo-spin symmetry is broken for realistic cases. For this purpose
we concentrate as well on spherical as on deformed nuclei and we use the framework of
relativistic mean field (RMF) theory [11]. It has been shown that this phenomenological
approach is very successful in describing the ground state nuclear properties of spherical,
deformed and also for nuclei far away from the beta stability line (see for example [12–14].
The RMF starts with a Lagrangian density describing the nucleons as Dirac spinors ψ,
of mass m, interacting via the meson (σ-, ω-, and ρ-) and the electromagnetic fields. The
standard Lagrangian density used in the RMF theory is written as [13]:
L = ψ¯ (γ(i∂ − gωω − gρ~ρ~τ − eA)−m− gσσ)ψ
+
1
2
(∂σ)2 − U(σ)− 1
4
ΩµνΩ
µν +
1
2
m2ωω
2
−1
4
~Rµν~R
µν +
1
2
m2ρ~ρ
2 − 1
4
FµνF
µν (1)
It includes a nonlinear self-interaction U(σ) of the σ-field:
2
U(σ) =
1
2
m2σσ
2 +
1
3
g2σ
3 +
1
4
g3σ
4. (2)
which takes into account in a phenomenological way the density dependence of the param-
eters of the model. mσ(gσ), mω(gω), mρ(gρ) are the respective meson masses (coupling
constants) and g2 and g3 are the coupling strengths of the nonlinear sigma field U(σ).
It is straightforward to write the coupled baryon spinor and the mesons mean field
equations. Starting from the Dirac equation for the single nucleon radial wave function with
the spherical attractive scalar (S = −gσσ) and the repulsive vector (V = gωω) potentials
and following the standard procedure, by eliminating the small components (gi), the large
components (fi) obey the following second order differential equation:
{
−∇2 − S
′ + V ′
2m− E − (S + V )
(
∂
∂r
+
κi + 1
r
)}
fi
= − (2m−E − (S + V )) (E − (S − V )) fi. (3)
Here the eigenvalues denoted by κi, of the operator −β(Σ · L+ 1) are given by
κi = ∓
(
ji +
1
2
)
for ji = ℓi ±
1
2
, (4)
and S ′ (V ′) are the derivatives of the potentials S (V ) with respect to r. The binding energy
E ≥ 0 is measured with respect to the nucleon mass M in natural units h¯ = c = 1.
On the other hand eliminating the large component fi we have for the small component
gi the following second order differential equation:
{
−∇2 − S
′ − V ′
E − (S − V )
(
∂
∂r
− κi − 1
r
)}
gi
= (2m− E − (S + V )) (E − (S − V )) gi. (5)
For the case of equal strengths, S = V , the Eq. (5) reduces to:
−∇2gi + E(S + V ) gi = E(2m−E) gi. (6)
Clearly Eq. (6) has an energy dependent potential (E(V + S)) and has the eigenvalue
E(2m− E). After scaling the radial variable r = x/(
√
E), the potential has a complicated
3
(
√
E) dependence i.e., S
(
x/
√
E
)
+ V
(
x/
√
E
)
. In such a situation this equation (6) is
no longer a normal Schro¨dinger eigenvalue equation. Further, it is obvious that in this
equation all solutions with “bound” states in the Fermi sea with E ≥ 0 are shifted to one
degenerate eigenvalue with E = 0, which, in fact, is not bound. The corresponding wave
functions are not normalizable. This indeed is an unphysical situation. This equation is the
same as the equation (3) of Ref. [4] in the scaled variable x when written in terms of the
partial waves and using the relation ℓ(ℓ + 1) = κ(κ − 1). Here ℓ, the angular momentum
of the lower component gi is identified with the pseudo-spin angular momentum (ℓ˜). This
is the pseudo-spin symmetry limit of Ref. [4], where the doublets j = ℓ˜ ± 1/2 with the
same ℓ˜ are degenerate. However, in this limit only the Dirac sea states exist and no Dirac
valence bound states and therefore contradicts reality. According to these considerations in
all realistic situations the pseudo-spin symmetry must be broken. Therefore the question
arises, to which extent it is broken in real nuclei. So far only the spherical case has been
investigated for square well potentials [4] and for spherical solutions of the RMF equations
[15,16].
In the present letter we investigate the broken pseudo-spin symmetry both for the spheri-
cal and deformed nuclei within the relativistic mean field approach. For our study, we choose
208Pb as a representative of spherical nuclei and 154Dy as a representative of deformed nu-
clei. We use in our calculations the Lagrangian parameter set NL3 [17] which successfully
reproduces the ground state properties of nuclei, spread over the entire periodic table. The
other parameter sets like NL1 and NLSH (see [14]) are expected to give almost identical
results for these nuclei.
First, spherical RMF calculations in the coordinate space are carried out for 208Pb. The
calculated binding energy and the charge radius agree remarkably well with the experiment.
The calculated single particle energies for the bound orbitals near the fermi surface are shown
in Fig. (1a) for neutrons and protons. It is clear from the figure that the pairs of bound
neutron valence orbitals (2g7/2,3d5/2) and (1i11/2,2g9/2) which correspond to pseudo-spin
4
doublets (2f˜7/2,2f˜5/2) and (1h˜11/2,1h˜9/2) respectively, are quasi-degenerate indicating only
a small breaking of pseudo-spin symmetry. The same is more or less true for the pairs of
neutron hole ((2f5/2,3p3/2),(1h9/2,2f7/2)), proton valence (particle) (1h9/2,2f7/2)), and proton
hole ((2d3/2,3s1/2), (1g7/2,2d5/2)), orbitals forming the pseudo-spin doublets. But here the
energy separation between the partners of the respective doublets is relatively larger. The
larger is the binding energy the larger is the separation. This indicates that the concept
of the pseudo-spin symmetry becomes better and better for the orbitals as their energies
approach closer and closer to the continuum. This is consistent with the results found in
Ref. [4] for the square well potentials. In addition, the energy separation becomes larger, if
the pseudo-orbital angular momentum (ℓ˜) increases. The dependence of the energy splitting
of the pseudo-spin partners on the energy E and on the pseudo-orbital angular momentum
ℓ˜ can easily be understood from Eq.(5). For a given pseudo-orbital angular momentum ℓ˜
the term in Eq.(5) which splits the pseudo-spin partners is:
S ′ − V ′
(S − V )− E
κi
r
(7)
It has the energy dependence (E − (S − V )) in the denominator. Now (S − V ) is about
50 MeV. Bound states in the Fermi sea have a binding energy E < 50 MeV. For increasing
binding energy E , i.e. going to more deeply bound states, the denominator decreases. This
then results in a larger energy splitting between the pseudo-spin partners. For example for
the orbit ℓ˜ = 3 the energy splitting between the pseudo-spin partners (1g7/2 and 2d5/2))
will be relatively larger as compared to that between (2g7/2 and 3d5/2)). In addition, the
bigger is the value of ℓ˜ the larger is the splitting. For instance,the energy splitting between
the pseudo-spin partners (1i11/2 and 2g9/2) corresponding to ℓ˜ = 5 is relatively larger as
compared to that between the partners (2g7/2 and 3d5/2) which corresponds to ℓ˜ = 3, in the
same major shell. Interestingly, the sign of the energy splittings between the partners of
the neutron valence doublets is opposite to those of the neutron hole, proton particle and
proton hole doublets.
The normalized single nucleon wave functions (both large (f) and small (g) components)
5
are plotted for the pseudo-spin partners corresponding to the valence neutron pairs, the
neutron hole pairs and valence proton pairs in Figs. 1 (b), (c) and (d) respectively. The
phase of the lower components (g) of one of the partners is reversed while plotting, in order
to exemplify the differences in the lower components of the pseudo-spin partners. Clearly,
the lower components are much smaller in magnitude as expected and are almost equal in
magnitude. In the case of exact pseudo-spin symmetry, the lower component of the pseudo-
spin partners should be identical (except for the phase). The very small differences between
these g’s which mainly appear around the surface are negligible for the pseudo-spin partners
having very small binding energies.
Next we consider deformed systems and impose constraint on the quadrupole moment.
Constrained relativistic Hartree calculations have been carried out for the nucleus 154Dy.
The coupled differential equations for the spinors and the meson fields are given in Ref.
[13]. They have been solved by expanding the spinors and the meson fields in terms of
anisotropic oscillator wave functions. Numerical details are given in Refs. [13] and [18].
Pairing correlations are treated in the constant gap approximation and the Lagrangian
parameter set NL3 [17] is used. The calculated potential energy surface is shown in Fig. 2.
The value of the calculated ground state deformation parameter β2 is 0.202 which is to be
compared with 0.237, the corresponding experimental value. The calculated ground state
binding energy 1262.95 MeV differs from the corresponding experimental value by merely
1.2 MeV.
The energies of the bound neutron pairs of orbitals corresponding to pseudo-spin doublets
are plotted against the deformation β2 ranging from 0.0 to 0.5 in Fig. 3. The asymptotic
Nilsson quantum numbers [N, n3,Λ,Ω] are good for large values of the deformation β2. The
pseudo-spin doublets [N˜, n˜3, Λ˜, Ω˜ = Λ˜ ± 1/2] [5] are indicated by [N˜, n˜3, Λ˜] ↑ and ↓ in
the figure. For zero deformation (β2 = 0) the orbitals are indicated by the corresponding
spherical states. The figure reveals the following:
(i) The energy splitting between the pseudo-spin partners is smaller for the valence orbitals
6
and for the partners just below the Fermi surface.
(ii) This energy difference is relatively larger for the partners having larger pseudo-spin
angular momentum (ℓ˜).
(iii) In general, this separation stays almost constant and does not vary with deformation
after reasonable value of β2.
(iv) The energy difference between the ↓ and the ↑ partners always remains positive except
for [ ˜404], where there is crossing at around β = 0.3. Such a crossing is not very
unusual, it has also been observed in Ref. [5].
These systematics are consistent with those observed in the spherical case above. A
similar plot for the proton pseudo-spin doublets shown in Fig. 4 reveals identical systematics
as those observed for the neutron case (Fig. 3). It is interesting to note that in Ref. [5] the
energy difference between the valence neutron pseudo-spin partners is negative (opposite
to ours) while it has the same sign as ours for protons. This may be due to the negative
value obtained for V ℓs , the strength of the pseudo-spin orbit interaction, from the Nilsson
parameterization for 82 < N < 126.
Similar calculations have also been carried out for other spherical and deformed nuclei
and they show identical systematics. The conclusions presented here, are therefore rather
general.
In conclusion, it is shown in the relativistic mean field framework that quasi-degenerate
pseudo-spin doublets do exist near the fermi surface for both spherical and deformed nuclei.
The pseudo-spin symmetry is restored better and better as one moves closer to the continuum
limit. These conclusions confirm the findings of Ginocchio [4,10].
One of the authors (G.A.L) acknowledges support from the DAAD. The work is also
supported in part by the Bundesministerium fu¨r Bildung und Forschung under the project
06 TM 875.
7
[1] K.T. Hecht, A. Adler; Nucl. Phys. A137, 129 (1969).
[2] A. Arima, M. Harvey, K. Shimizu; Phys. Lett. B30, 517 (1969).
[3] R.D. Ratna Raju, J.P. Draayer, K.T. Hecht; Nucl. Phys. A202, 433 (1973).
[4] J. N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997).
[5] A. Bohr, I. Hamamoto and B. R. Mottelson, Phys. Scr. 26, 267 (1982).
[6] B. Mottelson, Nucl. Phys. A522, 1 (1991).
[7] C. Bahri, J. P. Draayer ans S. A. Moszkowski, Phys. Rev. Lett. 68, 2133 (1992).
[8] O. Castanos, M. Moshinsky, and C. Quesne, Phys. Letts. B277, 238 (1992).
[9] A. L. Blokhin, C. Bahri, J. P. Draayer, J. Phys. A29, 2039 (1996).
[10] J. N. Ginocchio and A. Leviatan, Los Alamos Preprint LA-UR-97-4357
[11] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986), and references cited therein.
[12] P. G. Reinhard, Rep. Prog. Phys. 52, 439 (1989).
[13] Y. K. Gambhir, P. Ring and A. Thimet, Ann. Phys. (N.Y.) 198, 132 (1990), and references
cited therein.
[14] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).
[15] J. N. Ginocchio and D. G. Madland, Los Alamos Preprint LA-UR-97-3921
[16] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, and A. Arima, preprint RIKEN 1997
[17] G. A. Lalazissis, J. Ko¨nig and P. Ring, Phys. Rev. C55, 540 (1997).
[18] P. Ring, Y.K. Gambhir, and G.A. Lalazissis Comp. Phys. Comm. 105, 77 (1997).
8
FIG. 1. Pseudo-spin splitting in the spherical nucleus 208Pb: (a) single particle spectra in the
vicinity of the Fermi surface for neutrons ν and protons (pi) and large (f) and small (g) components
of the Dirac wave functions for the pseudo-spin doublets ν2d˜ (b), ν2f˜ (c) and pi2g˜ (d)
FIG. 2. Energy surface of the deformed nucleus 154Dy as function of the quadrupole moment q
in units of barn
FIG. 3. Single particle energies of the deformed Dirac equation for the neutrons in the nucleus
154Dy as a function of the quadrupole deformation parameter β2. Asymptotic pseudo-spin quantum
numbers are given and the pseudo-spin partners are indicated by arrows ↑ and ↓
FIG. 4. Single particle energies for protons in 154Dy, for details see Fig. 3
9
0 5 10 15
r  (fm)
-1.0
-0.5
0.0
0.5
1.0
w
.
f
.
  f: 2f5/2
  f: 3p3/2
-g: 2f5/2
  g: 3p3/2
-16
-12
-8
-4
0
s
.
p
.
 
 
e
n
e
r
g
y
 
(
M
e
V
)
0 5 10 15
r  (fm)
-1.0
-0.5
0.0
0.5
1.0
w
.
f
.
  f: 1h9/2
  f: 2f7/2
-g: 1h9/2
  g: 2f7/2
0 5 10 15
r  (fm)
-1.0
-0.5
0.0
0.5
1.0
w
.
f
.
  f: 2g7/2
  f: 3d5/2
-g: 2g7/2
  g: 3d5/2
a)
b)
c) d)
Neutron Wavefunctions 208Pb
Neutron Wavefunctions 208Pb Proton Wavefunctions 208Pb
ν pi
1g7/2
2d5/2
2d3/2
3s1/2
1h9/2
2f7/2
1h9/2
2f7/2
2f5/2
3p3/2
1i11/2
2g9/2
4s1/2
2g7/2
3d5/2
0 10 20 30
q(b)
-8.21
-8.20
-8.19
-8.18
E
/
A
 
 
(
M
e
V
)
154Dy
RMF/NL3
0.0 0.1 0.2 0.3 0.4 0.5
β2
-25
-15
-5
5
s.
p.
 e
ne
rg
y 
(M
eV
)
1g7/2
2d5/2
2d3/2
3s1/2
1h9/2
2f7/2
2f5/2
3p3/2
[321]
[312]
[303]
[301]
[431]
[422]
[413]
[411]
[404]
[402]
~
~
~
~
~
~
~
~
~
~
↑
↑
↑ ↑
↑
↑
↑
↑
↑↑
↓
↓
↓
↓
↓
↓
↓
↓↓
↓
Neutron: pseudo-spin
partners
RMF/NL3
0.0 0.1 0.2 0.3 0.4 0.5
β2
-25
-15
-5
5
s.
p.
 e
ne
rg
y 
 (M
eV
)
1f5/2
[211]
[202]
[321]
[312]
[303]
[301]
2p3/2
1g7/2
2d5/2
3s1/2
2d3/2
Proton : pseudo-spin
↓
↓
↓
↓
↓↓
~
~
~
~
~
~partners
↑
↑
↑
↑
↑↑
RMF/NL3


Relativistic Mean Field Approach and the Pseudo-Spin Symmetry

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