The pseudospin symmetry in Zr and Sn isotopes from the proton drip line to the neutron drip line
Abstract
Based on the Relativistic continuum HartreeBogoliubov (RCHB) theory, the pseudospin approximation in exotic nuclei is investigated in Zr and Sn isotopes from the proton drip line to the neutron drip line. The quality of the pseudospin approximation is shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential ( PSOP ). The PSOP depends on the derivative of the difference between the scalar and vector potentials . If , the pseudospin symmetry is exact. The pseudospin symmetry is found to be a good approximation for normal nuclei and to become much better for exotic nuclei with highly diffuse potential, which have . The energy splitting of the pseudospin partners is smaller for orbitals near the Fermi surface ( even in the continuum ) than the deeply bound orbitals. The lower components of the Dirac wave functions for the pseudospin partners are very similar and almost equal in magnitude.
pacs:
PACS numbers : 21.10.Hw, 21.60.n, 21.10.PC, 21.60.Jz, 27.60.jI Introduction
In a recent letter [1], by relating the pseudospin symmetry back to the Dirac equation through the framework of Relativistic Continuum HartreeBogoliubov (RCHB) theory[2], the pseudospin approximation in real nuclei was discussed. From the Dirac equation, the mechanism behind the pseudospin symmetry was studied and the pseudospin symmetry was shown to be related with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential ( PSOP ), which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a selfconsistent RCHB calculation, the pseudospin symmetry and its energy dependence have been discussed [1]. Here we will extend our previous investigation [1] to exotic nuclei. The pseudospin symmetry approximation for exotic nuclei is investigated for Zr and Sn isotopes ranging from the proton drip line to the neutron drip line. The isospin and energy dependence of the pseudospin approximation are investigated in detail.
The concept of pseudospin is based on the experimental observation that the single particle orbitals with and lie very close in energy and can therefore be labeled as pseudospin doublets with quantum number , , and . This concept was originally found in spherical nuclei 30 years ago [3, 4], but later proved to be a good approximation in deformed nuclei as well [5]. It is shown that the pseudospin symmetry remains an important physical concept even in the case of triaxiality [6] .
Since the suggestion of the pseudospin symmetry, much efforts has been made to understand its origin. Apart from the rather formal relabeling of quantum numbers, various proposals for an explicit transformation from the normal scheme to the pseudospin scheme have been made in the last twenty years and several nuclear properties have been investigated in this scheme [7, 8, 9, 10, 11]. Based on the single particle Hamiltonian of the oscillator shell model the origin of pseudospin was proved to be connected with the special ratio in the strength of the spinorbit and orbitorbit interactions [12, 10] and the unitary operator performing a transformation from normal spin to pseudospin space was discussed [10, 11, 12, 13, 14]. However, it was not explained why this special ratio is allowed in nuclei. The relation between the pseudospin symmetry and the relativistic mean field ( RMF ) theory [15] was first noted in Ref. [9], in which Bahri et al found that the RMF explains approximately the strengths of spinorbit and orbitorbit interactions in the nonrelativistic calculations. In a recent paper Ginocchio took a step further and revealed that pseudoorbital angular momentum is nothing but the “orbital angular momentum” of the lower component of the Dirac wave function [16]. He also built the connection between the pseudospin symmetry and the equality in the scalar and vector potentials [16, 17].
To understand to what extent it is broken in real nuclei, some investigation along this line has been done for square well potentials [16] and for spherical solutions of the RMF equations [18]. By relating the pseudospin symmetry back to the Dirac equation through the framework of RCHB theory, the pseudospin approximation in real nuclei was shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential ( PSOP ), which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a selfconsistent Relativistic HartreeBogoliubov calculation, the pseudospin symmetry and its energy dependence have been discussed in Ref. [1].
The highly unstable nuclei with extreme proton and neutron ratio are now accessible with the help of the radioactive nuclear beam facilities. The physics connected with the extreme neutron richness in these nuclei and the low density in the tails of their distributions have attracted more and more attention not only in nuclear physics but also in other fields such as astrophysics [19, 20]. New exciting discoveries have been made by exploring hitherto inaccessible regions in the nuclear chart. It is very interesting to investigate the pseudospin symmetry approximation both in normal and exotic nuclei. For this purpose, we will use the RCHB theory, which is the extension of the RMF and the Bogoliubov transformation in the coordinate representation, and provides not only a unified description of the mean field and pairing correlation but also the proper description for the continuum and its coupling with the bound state [2, 21]. As this theory takes into account the proper isospin dependence of the spinorbit term, it is able to provide a good description of global experimental data not only for stable nuclei but also for exotic nuclei throughout the nuclear chart [2]. It is very interesting to examine the pseudospin symmetry approximation in exotic nuclei, in which the mean field potentials are expected to be highly diffuse.
Recently, by relating the pseudospin symmetry back to the Dirac equation through the framework of RCHB theory, the pseudospin approximation in real nuclei was discussed. The mechanism behind the pseudospin symmetry was studied and the pseudospin symmetry was shown to be connected with the competition between the CB and the PSOP, which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a selfconsistent RHB calculation, the pseudospin symmetry and its energy dependence have been discussed [1]. Here we will extend the previous investigation to the case of exotic nuclei. The pseudospin splitting in Zr and Sn isotopes has been studied from the proton drip line to the neutron drip line. The energy splitting of the pseudospin partners, their energy and isospin dependence will be addressed. An outline of the RCHB formalism is briefly reviewed in Sec. II. In Sec. III, the Dirac equation and the formalism leading to the pseudospin symmetry is presented. The energy splitting of the pseudospin partners and its energy dependence are given in Sec. IV. The pseudospin orbital potential, which breaks the pseudospin symmetry will be studied in Sec. V. In Sec. VI, the wavefunction of pseudospin partners will be studied. A brief summary is given in the last section.
Ii An outline of RCHB Theory
The RCHB theory is obtained by combining the RMF and the Bogoliubov transformation in the coordinate representation [21], and its detailed formalism and numerical solution can be found in Ref. [2] and the references therein. The RCHB theory can give a fully selfconsistent description of the chain of Lithium isotopes [21] ranging from Li to Li. The halo in Li has been successfully reproduced in this selfconsistent picture and excellent agreement with recent experimental data is obtained. The contribution from the continuum has been taken into account and proved to be crucial to understand the halo in exotic nuclei. Based on the RCHB, a new phenomenon ”Giant Halo” has been predicted. The ”Giant Halo” is composed not only of one or two neutrons, as is the case in the halos in light shell nuclei, but also up to 6 neutrons [22]. The development of skins and halos and their relation with the shell structure are systematically studied with RCHB in Ref. [23], where both the pairing and blocking effect have been treated selfconsistently. Therefore the RCHB theory is very suitable for the examination of the pseudospin approximation in exotic nuclei.
The basic ansatz of the RMF theory starts from a Lagrangian density by which nucleons are described as Dirac particles interacting via the exchange of various mesons and the photons. The mesons considered are the scalar sigma (), vector omega () and isovector vector rho (). The isovector vector rho () meson provides the necessary isospin asymmetry. The scalar sigma meson moves in the selfinteracting field of cubic and quadratic terms with strengths and , respectively. The Lagrangian then consists of the free baryon and meson parts and the interaction part with minimal coupling, together with the nucleon mass , and , , , , , the masses and coupling constants of the respective mesons:
(1) 
The field tensors for the vector mesons are given as:
(2) 
For a realistic description of nuclear properties, a nonlinear selfcoupling of the scalar mesons turns out to be crucial [24]:
(3) 
The classical variation principle gives the following equations of motion :
(4) 
for the nucleon spinors and
(5) 
with and for the mesons, where
(6) 
are the vector and scalar potentials respectively and the source terms for the mesons are
(7) 
where the summations are over the valence nucleons only. It should be noted that as usual, the present approach neglects the contribution of negative energy states, i.e., nosea approximation, which means that the vacuum is not polarized. The coupled equations Eq.(4) and Eq.(5) are nonlinear quantum field equations, and their exact solutions are very complicated. Thus the mean field approximation is generally used: i.e., the meson field operators in Eq.(4) are replaced by their expectation values, so that the nucleons move independently in the classical meson fields. The coupled equations are selfconsistently solved by iteration.
For spherical nuclei, i.e., the systems with rotational symmetry, the potential of the nucleon and the sources of meson fields depend only on the radial coordinate . The spinor is characterized by the quantum numbers , ,, and the isospin for neutron and proton, respectively. The other quantum number is denoted by . The Dirac spinor has the form:
(8) 
where are the spinor spherical harmonics and and are the radial wave function for upper and lower components. They are normalized according to the relation:
(9) 
The radial equation of spinor Eq. (4) can be reduced as :
(10) 
where
The meson field equations become simply radial Laplace equations of the form:
(11) 
are the meson masses for and for photon ( ). The source terms are:
(12) 
(13) 
The Laplace equation can be solved by using the Green function:
(14) 
where for massive fields
(15) 
and for Coulomb field
(16) 
The Eqs.(10) and (11) could be solved selfconsistently in the usual RMF approximation. However, Eq.(10) does not contain the pairing interaction, as the classical meson fields are used in RMF. In order to have the pairing interaction, one has to quantize the meson fields which leads to a Hamiltonian with twobody interaction. Following the standard procedure of Bogoliubov transformation, a Dirac HartreeBogoliubov equation could be derived and then a unified description of the mean field and pairing correlation in nuclei could be achieved. For the details, see Ref. [2] and the references therein. The RHB equations are as following:
(17) 
where
(18) 
is the Dirac Hamiltonian and the Fock term has been neglected as is usually done in RMF. The pairing potential is :
(19) 
It is obtained from the onemeson exchange interaction in the channel and the pairing tensor :
(20) 
The nuclear density is as following:
(21) 
As in Ref. [2], used for the pairing potential in Eq.(19) is either the densitydependent twobody force of zero range with the interaction strength and the nuclear matter density :
(22) 
or Gognytype finite range force with the parameter , , , and (): [25]
(23) 
A Lagrange multiplier is introduced to fix the particle number for the neutron and proton as and .
In order to describe both continuum and bound states selfconsistently, we use the RHB theory in coordinate representation, i.e., the Relativistic Continuum HartreeBogolyubov ( RCHB ) theory [2]. It is then applicable to both exotic nuclei and normal nuclei. In Eq. (17), the eigenstates occur in pairs of opposite energies. When spherical symmetry is imposed on the solution of the RCHB equations, the wave function can be written as:
(24) 
Using the above equation, Eq.(17) depends only on the radial coordinates and can be expressed as the following integrodifferential equation:
(25) 
where the nucleon mass is included in the scalar potential . For the force of Eq.(22), Eq.(25) is reduced to normal coupled differential equations and can be solved with shooting method by RungeKutta algorithms. For the case of Gogny force, the coupled integrodifferential equations are discretized in the space and solved by the finite element methods. The numerical details can be found in Ref. [2]. Now we have to solve Eqs.(25) and (11) selfconsistently for the RCHB case. As the calculation with Gogny force is very timeconsuming, we solve them only for one case in order to fix the interaction strength for force in Eq.(22).
Iii The pseudospin symmetry
The Dirac equation in RMF or in the canonical basis of RCHB describes a Dirac spinor with mass moving in a scalar potential and a vector potential . With , the potential , which is around MeV, and the effective mass , the relation between the upper and lower components of the wave function can be written as:
(26) 
Then the coupled equations are reduced to uncoupled ones for the upper and lower components, respectively. Effectively we get the corresponding Schrödinger equation for both components:
(27) 
In the spherical case, depends only on the radius. We choose the phase convention of the vector spherical harmonics as:
(28) 
where
(29) 
Here is nothing but the pseudoorbital angular momentum . After some tedious procedures, one gets the radial equation for the lower and upper components respectively:
(30) 
(31) 
where
(32) 
It is clear that one can use either Eq.(30) or equivalently Eq.(31) to get the eigenvalues and the corresponding eigenfunctions. Normally Eq.(31) is used in the literature and the spinorbital splitting is discussed in connection with the corresponding spinorbital potential
In Eq. (30), the term which splits the pseudospin partners is simply the PSOP. The hidden symmetry for the pseudospin approximation is revealed as , which is more general and includes discussed in [16] as a special case. For exotic nuclei with highly diffuse potentials, may be a good approximation and then the pseudospin symmetry will be good. But generally, is not always satisfied in the nuclei and the pseudospin symmetry is an approximation. However, if
In a recent letter [1], the mechanism behind the pseudospin symmetry was studied and the pseudospin symmetry was shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential ( PSOP ), which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a selfconsistent RCHB calculation, the pseudospin symmetry and its energy dependence have been discussed. Here in this paper we will extend the previous investigation to the case of exotic nuclei. The pseudospin symmetry for exotic nuclei is investigated for Zr and Sn isotopes from the proton drip line to the neutron drip line. The isospin and energy dependence of the pseudospin approximation will be investigated in detail in the following section.
Iv The energy splitting of the pseudospin partners
We use here the nonlinear Lagrangian parameter set NLSH [26] which could provide a good description of all nuclei from oxygen to lead. As we study not only the closed shell nuclei, but also the open shell nuclei, the inclusion of the pairing is necessary. The pairing interaction strength is the same as in Ref. [22]. The interaction strength in the pairing force of zero range Eq.(22) is properly renormalized by the calculation of RCHB with Gogny force. Since we use a pairing force of zero range, we have to limit the number of continuum levels by a cutoff energy. For each spinparity channel, 20 radial wave functions are taken into account, which corresponds roughly to a cutoff energy of 120 MeV for a fixed box radius fm. For the fixed cutoff energy and for the box radius , the strength of the pairing force in Eq.(22) is determined by adjusting the corresponding pairing energy to that of a RCHBcalculation using the finite range part of the Gogny force D1S [25]. We use the nuclear matter density 0.152 fm for .
The quality of pseudospin symmetry can be understood more clearly by considering the microscopic structure of the wave functions and the single particle energies in the canonical basis. As shown in Ref. [2], the particle levels for the bound states in canonical basis are the same as those by solving the Dirac equation with the scalar and vector potentials from RCHB. Therefore Eqs. (30) and (31) are valid in canonical basis after the pairing interaction has been taken into account and are very suitable for the discussion of the pseudospin symmetry.
The neutron single particle levels in Sn and Zr are given in Fig. 1a and 1b, respectively. The four sets of pseudospin partners, i.e., and , and , and , and , are marked by boxes. As seen in the figure, the energy splitting between pseudospin partners decreases with the decreasing binding energy. The single particle energy of in is MeV, and its partner is MeV, the splitting is MeV. While is MeV, Mev, the splitting is MeV, which is bigger than the former one by a factor of . The same situation is found for the energy splitting between pseudospin partners in Sn: The single particle energies of and are and MeV, respectively. The single particle energies for other pseudospin partners in Sn are, and MeV for and partners, and MeV for and partners, and for and partners, respectively. Although we show only the neutron single particle levels in Sn and Zr as examples here, the same are found in other Sn and Zr isotopes. It is usually seen that the pseudospin symmetry approximation becomes better near the Fermi surface, which is in agreement with the experimental observation.
In Fig.1 there are also two pairs of pseudospin partners ( and partners and and partners ) near the threshold, apart from the fours pairs of pseudospin partners below the Fermi level. The energies for these two pairs of pseudospin partners are and Mev for the and partners, and and MeV for the and partners, respectively. Considering their pseudospin orbital angular momentum and , their splittings are only MeV and MeV, respectively. This is due to the energy dependence and the diffuseness of the potential in exotic nuclei, which we will discuss in the following. As it is seen in Fig.1, the normal splitting is such that the orbital is below the orbital , except for and partners. The same also happens for and partners in Zr isotopes. The pseudospin splitting depends on the derivative of the difference between the scalar and vector potentials , which is small for the exotic nuclei with highly diffuse potential. The integration of over gives the splitting of the pseudospin partners, whose sign will decide the normal splitting or the reverse. The subtle details of the potential are crucial for the pseudospin splitting.
To see the behavior of the pseudospin partners around the Fermi level and the isospin dependence of the pseudospin splitting, we show the single particle levels near the Fermi surface in the canonical basis for the Sn and Zr isotopes with an even neutron number as a function of the mass number in Fig. 2. The Fermi level is shown by the dashed line. The pseudospin splitting for the pseudospin partners, and , remains small in Zr and Sn isotopes from the proton drip line to the neutron drip line. The pseudospin symmetry remains even valid for exotic nuclei. The pseudospin symmetry near the neutron drip line becomes better than that near the stability line. In Fig. 2a, there is a kink for the single particle levels in the continuum, as the contribution from the continuum becomes important and the potential becomes diffuse around Sn. But the splitting for and partners, and and partners in Sn isotopes is small and the pseudospin symmetry approximation is very good, independent of whether they are in the continuum or near the threshold. Therefore we can see that the pseudospin symmetry is very well reserved for the orbital near the threshod energy and in the continuum region.
In order to see the energy dependence and the isospin dependence of the pseudospin orbital splitting more clearly, we plot versus for the bound pseudospin partners in Sn and Zr isotopes in Fig. 3. In both isotopes, a monotonous decreasing behavior with the decreasing binding energy is clearly seen. The pseudospin splitting for and is more than times smaller than that of the and . As far as the isospin dependence of the pseudospin orbital splitting is concerned, the splitting in Sn isotopes gives a monotonous decreasing behavior with the increasing isospin. Particularly for and partners, the pseudospin splitting in Sn is only half of that in Sn. Just as we expected, the pseudospin symmetry in neutronrich nuclei is better. In Zr isotopes, although the situation is more complicated ( e.g., the effect of the deformation which is neglected here ) , the pattern is more or less the same, i.e., a monotonous decreasing behavior with the decreasing binding energy and a monotonous decreasing behavior with the isospin. From these studies, we see that the pseudospin symmetry remains a good approximation for both stable and exotic nuclei. A better pseudospin symmetry can be expected for the orbital near the threshold, particularly for nuclei near the particle dripline.
V The pseudospin orbital potential
To understand why the energy splitting of the pseudospin partner changes with different binding energies and why the pseudospin approximation is good in RMF, the PSOP and CB should be examined carefully. Unfortunately, it is very hard to compare them clearly, as the PSOP has a singularity at . As we are only interested in the relative magnitude of the CB and the PSOP, we introduce the effective CB, , and the effective PSOP, respectively. , for comparison. They correspond to the CB and the PSOP multiplied by a common factor
The effective PSOP does not depend on the binding energy of the single particle level, but depends on the angular momentum and parity. On the other hand the effective CB depends on the energy. Comparing these two effective potentials one could see the energy dependence of the pseudospin symmetry. They are given in Fig. 4 for ( lower ) and ( upper ) of Zr in arbitrary scale.
The pseudospin approximation is much better for the less bound pseudospin partners, because the effective CB is smaller for the more deeply bound states. This is in agreement with the results shown in Fig.3. The effective PSOP and the effective CB are also given as inserts in Fig.4 in order to show their behavior near the nuclear surface.
In order to examine this carefully, we compare the effective CB ( dashed lines or dotdashed lines ) and the effective PSOP ( solid lines ) multiplied by the squares of the lower component wave function , which are given in Fig. 5, for ( upper left ), ( lower left), ( upper right ), and ( lower right ) of Zr in arbitrary scale. The pseudospin approximation is much better for the less bound pseudospin partners, because the effective CB is smaller for the more deeply bound states. This is in agreement with the results shown above. The integrated values of the potentials in Fig.2 with are proportional to their contribution to the energy after some proper renormalization. It is clear that the contribution of the effective CB ( dashed lines or dotdashed lines ) is much bigger than that of the effective PSOP ( solid lines ). Generally the effective PSOP is two orders of magnitude smaller than the effective CB.
In Fig. 1 and 2 we notice that the orbital is generally below the orbital , except for and partners in Sn isotopes. The same situation also happens for and partners in Zr isotopes. As the pseudospin splitting depends on PSOP, which depends on the subtle radial dependence of the potentials, sometimes the PSOP may have positive or negative regions as a function of which cancell each other. The integration of over gives the splitting of the pseudospin partners, whose sign will decide the normal splitting or the reverse. That is the reason why the orbital is above the orbital for and partners in Sn isotopes and for and partners in Zr isotopes.
Vi The wavefunction of pseudospin partners
In the above discussion, we have seen that the PSOP is much smaller than the CB. Therefore if we neglect the PSOP in Eq.(30), the lower component of the Dirac wave functions for the pseudospin partners will be the same, i.e., in the case of the exact pseudospin symmetry, the lower component of the pseudospin partners should be identical (except for the phase). The upper component of the Dirac wave functions can be obtained from the transformation in Eq.(26), which depends on the quantum number . Therefore the study of the Dirac wave functions for the pseudospin partners will provide a check for the pseudospin approximation in nuclei. As examples, the normalized single nucleon wave functions for the upper ( ) and lower ( ) components of the Dirac wave functions for the pseudospin partners and , and , and , and in Zr are given in Fig. 6. Of course, the lower components are much smaller in magnitude compared with the upper component in Eq.(26). The phase of the Dirac wave functions for one of the pseudospin partners has been reversed in order to have a careful comparison. It is seen that the lower components of the Dirac wave functions for the pseudospin partners are very similar and are almost equal in magnitude, as observed also for Pb in Ref. [18]. The similarity in the lower components of the wave function for the pseudospin partners near the Fermi surface is better than for the deeply bound ones. The lower components for the pseudospin partners with small pseudospin orbital angular momentum are better than for the ones with large pseudospin orbital angular momentum. As seen in Fig.6, the similarity for pseudospin partners and is better than for pseudospin partners and . The similarities for pseudospin partners, and , and , are better than for the pseudospin partners and , and .
Although the lower components for the pseudospin partners are very close to each other, the difference for the upper components is very big. The upper component of the Dirac wave functions can be obtained from the transformation in Eq.(26), which for the sperical case can be reduced to the follows:
(33) 
As seen in Fig.6, in the case of exact pseudospin symmetry, where both and are identical for the pseudospin partners, the upper conpnonents will be different due to the term . For the pseudospin partners with small , the contribution of the term becomes less important for larger and a similarity between the upper components can happen in the nuclear surface. As for examples, for fm, the upper components for the pseudospin partners and , and , and , in Fig. 6 are very similar.
Vii Summary
In conclusion, the pseudospin symmetry is examined in normal and exotic nuclei in the framework of RCHB theory. Based on RCHB theory the pseudospin approximation in exotic nuclei is investigated in Zr and Sn isotopes from the proton drip line to the neutron drip line. The quality of the pseudospin approximation is shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential ( PSOP ), which is mainly decided by the derivative of the difference between the scalar and vector potentials . If the derivative of the difference between the scalar and vector potentials vanishes, the pseudospin symmetry is exact. The condition may be a good approximation for the exotic nuclei with highly diffuse potential. Further the new condition

The quality of the pseudospin approximation is connected with the competition between the CB, and the PSOP which is mainly proportional to the derivative of the difference between the scalar and vector potentials ;

The pseudospin symmetry is a good approximation for normal nuclei and become much better for exotic nuclei with highly diffuse potentials;

The pseudospin symmetry has strong energy dependence. The energy splitting between the pseudospin partners is smaller for orbitals near the Fermi surface.

The energy difference between the orbital and the orbital is always negative, except for and partners. The same situation also happens for and partners in Zr isotopes. The integration of over gives the splitting of the pseudospin partners, whose sign will decide the normal splitting or the reverse.

The lower components of the Dirac wave functions for the pseudospin partners are very similar and almost equal in magnitude. The similarity in the lower components of the wave function for the pseudospin partners near the Fermi surface is closer than for the deeply bound ones.
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Figure Captions
 Fig. 1

The single particle levels in the canonical basis for the neutron in Zr and Sn. The Fermi surface is shown by a dashed line. The bound pseudospin partners are marked by boxes
 Fig. 2

The single particle energies of the neutron in the canonical basis as a function of the mass number for Zr and Sn isotopes. The dashed line indicates the chemical potential.
 Fig. 3

The pseudospin orbit splitting versus the binding energy for Zr and Sn isotopes. From left to right, the pseudospin partners correspond to (), (), () and (), respectively.
 Fig. 4

The comparison of the effective centrifugal barrier ( CB ) ( dashed lines and dotdashed lines ) and the effective pseudospin orbital potential ( PSOP ) ( upper ) and ( lower ) in Zr. The dashed lines are for and , and the dotdashed lines are for and . The inserted boxes show the same quantities, but the ordinate is magnified and the abscissa is reduced to show the behaviors of the effective CB and the effective PSOP near the nuclear surface. ( solid line ) in arbitrary scale for
 Fig. 5

The comparison of the effective centrifugal barrier ( CB ) ( dashed lines and dotdashed lines ) and the effective pseudospin orbital potential ( PSOP ) of the lower components in arbitrary scale for ( upper ) and ( lower ) in Zr. The dashed lines are for and , and the dotdashed lines are for and . ( solid line ) multiplied by the square of the wave function
 Fig. 6

The upper component and lower component of the Dirac wave functions for the pseudospin partners in Zr. The phase of the Dirac wave functions for one of the pseudospin partners has been reversed in order to have a careful comparison.