Study of Two-Proton Radioactivity Within the Relativistic Mean-Field Plus Bcs Approach
Inspired by recent experimental studies of two-proton radioactivity in the light-medium mass region, we have employed relativistic mean-field plus state dependent BCS approach (RMF+BCS) to study the ground state properties of selected even-Z nuclei in the region 20 Z 40. It is found that the effective potential barrier provided by the Coulomb interaction and that due to centrifugal force may cause a long delay in the decay of some of the nuclei even with small negative proton separation energy. This may cause the existence of proton rich nuclei beyond the proton drip-line. Nuclei Ti, Cr, Fe, Ni, Zn, Ge, Se, Kr, Sr and Zr are found to be the potential candidates for exhibiting two-proton radioactivity in the region 20 Z 40. The reliability of these predictions is further strengthened by the agreement of the calculated results for the ground state properties such as binding energy, one- and two-proton separation energy, proton and neutron radii, and deformation with the available experimental data for the entire chain of the isotopes of the nuclei in the region 20 Z 40.
PACS Nos.: 23.50.+z, 21.10.-k, 21.10.Dr
The structure and decay modes of nuclei at and beyond the proton drip-line represent one of the most active areas in both experimental and theoretical studies of exotic nuclei with extreme isospin values. At the proton drip-line, further addition of protons is not possible as the nucleus becomes unbound. Beyond the drip-line, one or more valence protons may still remain confined due to Coulomb and centrifugal barriers enabling the nucleus to acquire rather long mean life time. Subsequently, it may decay by the process wherein one or more protons tunnel through the barrier leading to observation of one or more protons radioactivity. This situation is quite different from the neutron rich side of the valley of -stability where Coulomb barrier is absent and consequently the drip-line gets extended to highly neutron rich nuclei.
Decay modes of nuclei through one- or two- proton radioactivity were theoretically proposed in the early 1960’s for the first time by Goldansky . The one-proton radioactivity predicted for odd-proton nuclei was indeed observed already in the early 1980’s in experiments carried out at GSI, Darmstadt , and presently many nuclei (more than 30) which decay in their ground state by one-proton emission are well known . However, the two-proton emission mode was experimentally verified only more than four decades after its theoretical prediction in the decay of Fe [4, 5], and subsequently in other experiments in the decay of Zn [6, 7] and Ni [8, 9]. Many more experiments are being carried out to discover the new candidates.
Two-proton radioactivity occurs when the sequential emission of two independent protons from the nuclear ground state is energetically forbidden, but the emission of a pair of protons is allowed. In this situation, due to the gain of stability from the pairing energy, the mass of the even-Z two-proton emitter is smaller than the mass of the odd-Z one-proton daughter giving rise to the negative Q value for one-proton emission. Such a situation prohibits one-proton emission and favors two-proton radioactivity. Therefore, the simultaneous two-proton emission is energetically possible only beyond the two-proton drip-line and competes with two other decay modes, namely one-proton emission and -decay. During the emission process, the two protons must tunnel through a wide Coulomb barrier. Hence, the emission probability strongly depends on the available Q value. For small Q values, -decay is more probable, whereas for the large ones the life time of the given nucleus is very short leading to a fairly small window of opportunity to observe the two-proton radioactivity. Therefore, one has to look for nuclei in which one-proton decay is energetically impossible. Consequently we have chosen to study the even-Z nuclei only. Even-Z nuclei satisfying the condition S 0 and S 0 may be the possible candidates for simultaneous two-proton emission.
Theoretical studies of one and two-proton radioactivity have been carried out within the framework of different models [3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Also, the relativistic Hartree-bogoliubov and relativistic mean field (RMF) approaches [22, 23, 24, 25, 26] have been employed with reasonable success for the description and prediction of one-proton radioactivity in the proton rich exotic nuclei in the vicinity of the proton drip-line. However, such a relativistic approach has not been used so far to describe and predict the two-proton radioactivity in the proton rich exotic nuclei as observed recently [4, 5, 6, 7, 8, 9]. This has motivated us to study the two-proton emitter nuclei within the framework of RMF+BCS approach. The main advantage of the RMF+BCS approach is that it provides the spin-orbit interaction in the entire mass region in a natural way [27, 28, 29, 30]. This indeed has proved to be very crucial for the study of unstable nuclei near the drip-line, since the single particle properties near the threshold are prone to large changes as compared to the case of deeply bound levels in the nuclear potential. In addition to this, the pairing properties are equally important for nuclei near the drip-line. As nuclei move away from stability and approach the drip-lines, the corresponding Fermi surface gets closer to zero energy at the continuum threshold. A significant number of the available single-particle states then form part of the continuum. Indeed the RMF+BCS scheme [31, 32, 33, 34] yields results which are in close agreement with the experimental data and with those of continuum relativistic Hartree-Bogoliubov (RCHB) and other similar mean-field calculations [35, 36].
2 Theoretical Formulation and Model
The relativistic mean-field (RMF) approach [27, 30, 37] provides a description of the nuclear many-body system in terms of an effective Lagrangian containing mesonic and nucleonic degrees of freedom . The relativistic mean-field model of the nucleus is formulated on the basis of two approximations [29, 38]: (i) the mean-field assumption and (ii) the no-sea approximation.
2.1 Model Lagrangian Density
The Lagrangian density is written as the sum of the Lagrangian density for free nucleons , the Lagrangian density for the free mesons , and , and that of photon, and the Lagrangian density for the interaction between mesons and nucleons ,
The interaction term includes both the linear and nonlinear couplings. For the description of nonlinear couplings we follow the treatment of Boguta and Bodmer  in the case of -mesons, and that of Sugahara and Toki [39, 40] for the -mesons. These nonlinear interaction terms have been shown to play important role in various applications of the RMF theory to provide a quantitative description of nuclear matter and the ground state properties of nuclei [28, 37, 38, 39, 40]. As mentioned earlier this is an effective Lagrangian density to be used together with the mean-field and no-sea approximations. Thus when written in detail the model Lagrangian used in our present study has the following form,
where we have used throughout . Here the field tensors, H, G and F for the vector fields due to , and photon are defined through
Furthermore, the symbols , , , and , are the masses of nucleon, and that of the , , and mesons, respectively. The superscript ‘a’ labels the isospin degree of freedom and runs from 1 to 3. Similarly, , , and = are the coupling constants for the mesons, and the photon, respectively, whereas are the Pauli isospin matrices.
The set of parameters appearing in the effective Lagrangian include (i) the masses of the nucleons and the mesons , , and , (ii) the coupling constants of the meson fields to the nucleons , and , and (iii) the parameters and which describe the nonlinear coupling of the mesons among themselves, and the parameter which describes the nonlinear self coupling of the vector meson . These have been obtained in an extensive study [39, 40] which provides a reasonably good description for the ground state of nuclei and that of nuclear matter properties. This set, termed as TMA parameters, has an A-dependence and covers the light as well as medium heavy nuclei from to . The TMA force parameter set determined in Ref. [39, 40] (displayed in Table 2) has been used in the present RMF+BCS calculations.
In the literature [29, 37, 38] many sets of parameterizations are available for the Lagrangian density similar to that of Eqn. (1) containing the nonlinear terms for the mesons but without the inclusion of nonlinear potential for the mesons. The earliest sets like NL1, NL2 and NL-Z etc. have been employed extensively and further improved subsequently.
2.2 Relativistic Mean-Field Equations
The equations of motion for the RMF theory are obtained from the variational principle by varying the action integral with respect to the wave functions () for the nucleons, and fields , and for the mesons and the electromagnetic field . In general, for a given Lagrangian density the variational principle
leads for the variables , the well known E.L. equations of motion given by
Using this we can derive the equations of motion for various fields ( = , and ) appearing in our effective Lagrangian density given by Eqn. (2.1). Thus for the nucleons fields one obtains the Dirac equation given by
which may be also written in the form
with the relativistic fields and defined, respectively by
Similarly one derives the equations of motions for the meson fields. For the scalar -meson one obtains the simple Klein-Gordon equation, whereas in the case of vector mesons and , one obtains the Proca equations. However, using the Lorentz guage for the vector mesons ( etc.) the Proca equations also can be transformed as Klein-Gordon equations. Thus for the mesons and photons the field equations are given by
As mentioned earlier, the various densities appearing as source terms in the equations for the meson-fields are obtained in the mean-field and no-sea approximations whereby the nucleon field operator is expanded in terms of single particle wave functions . Thus the scalar density , the nucleon current density , the isovector current density and the electromagnetic current density are simply given by
Here the subscript ‘Pr’ has been used to denote protons. Also we have introduced the occupation weights () which would facilitate the treatment of open shell nuclei. In the context of pairing correlations, these weights are identical to the BCS factors (occupation probabilities). The RMF equations given by (4), and (7) - (10) are a set of coupled equations for the nucleon fields, meson fields and the Coulomb field which are solved by iteration after applying suitable approximations.
2.3 Total Energy of the System
The relation for the energy of the system within the present meson field theory is derived  from the Hamiltonian density using the relation
where the Hamiltonian density is defined in terms of as
Here the canonical momentum for the various fields ( = , and ) is given by
The mean-field energy obtained above is the major portion of the energy of a nuclear system. The total energy of the system is given by
wherein and are the pairing energy and the center-of-mass energy respectively.
In order to calculate the pairing energy [41, 42] we employ the state dependent gap equation for the single particle states wherein for the pairing interaction we have used a delta-function force. The evaluation of the pairing energy
contribution [41, 42] involves finally the calculation of the pairing interaction matrix elements and the single particle pairing gaps along with the occupation probabilities etc. For the correction to the center of mass motion we use the nonrelativistic expression 
where is the classical center of mass and is the total number of nucleons in the nucleus. A simple harmonic oscillator shell model estimate given by = MeV provides a reasonably good approximate description  and has been used in our present study.
2.4 Specialization to Spherically Symmetric Nuclei
In the case of spherical nuclei, i.e. the systems which have
rotational symmetry, the potential of the nucleon and the sources of
the meson fields depend only on the radial coordinate r. The field
equations obtained above are further simplified due to spherical
symmetry. In this case, the spinors describing nucleons are
characterized as usual in terms of the single particle angular
momentum quantum numbers and and expressed in terms of
radial functions and for the upper and lower
components, respectively, and the spinor
spherical harmonics ,
where the spinor spherical harmonic for a given nucleon with quantum numbers and is defined by
The spinors satisfy the normalization condition
where are the single particle energies, and is the Fermi energy. The particle number condition is expressed in terms of the occupation probabilities for the single particle states through
where N is the number of particles in the system, and are the occupation probabilities given by
In our calculations we use for the pairing interaction the -force, that is where denotes the strength. The final result for the paring matrix element is given by
where is the radial integral having the form
For the purpose of our RMF+BCS calculations the value of the pairing interaction strength MeV was determined by obtaining a best fit to the binding energy of isotopes. We use the same value of MeV throughout for our present studies of all the other nuclei as well. Moreover, the same strength has been used for both the protons and neutrons.
2.5 Specialization to Axially Deformed Nuclei
The relativistic mean field description has been extended for the deformed nuclei of axially symmetric shapes by Gambhir, Ring and their collaborators  using an expansion method. The treatment of pairing has been carried out in Ref.  using state dependent BCS method  as has been given by Yadav et al. [31, 32] for the spherical case. For axially deformed nuclei the rotational symmetry is no more valid and the total angular momentum is no longer a good quantum number. Nevertheless, the various densities still are invariant with respect to a rotation around the symmetry axis. Here we have taken the symmetry axis to be the z-axis. Following Gambhir et al. , it is then convenient to employ the cylindrical coordinates
The spinor with the index is now labeled by the quantum numbers and , where is the eigenvalue of the symmetry operator (the projection of on the z-axis), indicates the parity and has been used for the isospin. In terms of these quantum numbers, the spinor can now be expressed in the form:
Here the four components and obey the Dirac equations. For the axially symmetric case the spinors and are expanded in terms of the eigenfunctions of a deformed axially symmetric oscillator potential as has been described in the Refs.  and . The pairing gap appearing in the Eqn. (19) satisfies the gap equation
Here the symbols and denote the single particle and Fermi energy, whereas the pairing matrix element for the symmetrically deformed case using the zero-range -force is given by
3 Results and Discussion
In sec. 3.1 we have described the results of the calculated binding energies, one- and two-proton separation energies obtained by employing RMF+BCS approach including deformation degree of freedom assuming axially symmetric shapes of nuclei (referred to throughout as deformed RMF approach) to identify the nuclei which satisfy the criteria (S 0 and S 0) in the region 20 Z 40.
Nuclei satisfying the above criteria have been predicted as potential candidates for exhibiting two-proton radioactivity. It is found that barring a few cases most of the potential candidates for the two proton radioactivity are well deformed proton rich nuclei as has been discussed in sec. 3.2. A detailed examination of the results for the calculated value of the deformation obtained for the potential candidates in sec. 3.2 shows that, for example, the candidate Ni has spherical shape, whereas the potential nuclei Ge and Cr are slightly deformed. These spherical or near spherical potential candidates are especially useful in our theoretical studies since these can be described in detail within the spherical RMF framework in terms of spherical single particle wave functions and energies. This in turn enables us to discuss the nuclear structure aspects of these nuclei in greater details in terms of relativistic mean field (RMF) potential, wave function, proton single particle spectrum and density distributions etc. as has been elucidated in sec. 3.2.
In order to obtain the one- and two-proton separation energies using Eqns. (33) and (34) given below, systematic calculations are carried out within the framework of deformed RMF approach for the even-Z nuclei in the region 20 Z 40. Thus for a nucleus with proton and neutron number (Z, N), the calculations for S and S require the deformed RMF results for the chain of isotopes with proton number Z, Z-1 and Z-2, as is evident from the form of expressions (33) and (34). In order to proceed towards the proton drip-line and obtain the S and S values for the proton rich nucleus, similar calculations are repeated in a systematic way for decreasing value of N to obtain the results for the nuclei of the isotopic chain (Z, N-1), (Z, N-2), (Z, N-3)… etc. The calculated deformed RMF results for the binding energies then yield the desired S and S values up to the proton drip-line and beyond. These results are then plotted as a function of decreasing mass number A for different values of Z and analysed to predict the potential candidates for two-proton radioactivity as has been described in the subsection below. Calculation for neutron rich nuclei (N Z) have not been shown and discussed as they are not relevant to our study of the phenomenon of two-proton radioactivity.
3.1 Analysis of separation energy for the identification of two-proton emitters
The results of our extensive calculations obtained for the nuclei constituting the Z = 20 to Z = 40 isotopic chains by employing the deformed RMF approach have been shown in Fig. 1 as a function of increasing mass number A for all the isotopic chains of nuclei with even Z values in the region 20 Z 40 along with the available experimental data .
From the figure it is observed that the calculated binding energy values are very close to the available experimental data . However, due to large scales involved in this figure, small differences between the calculated and experimental values of the binding energy (0.1 MeV to 3 MeV) are not distinctly visible in Fig. 1. The maximum difference of 3 MeV occurs in the case of few proton rich nuclei located in the vicinity of Z = 26 (Fe) isotopic chain. This difference is about 1 percent of the total binding energy of the respective nucleus in that region.
Due to close agreement between the theoretical and experimental results for the binding energy, we can accurately determine one- and two-proton separation energy values which are very crucial for making reliable prediction of the two-proton emitters. Thus, the deformed RMF approach is expected to describe the phenomena of two-proton radioactivity in the region 20 Z 40 in a more realistic manner as compared to other theoretical approaches wherein the accuracies are rather not really large.
As mentioned earlier, since the nuclei satisfying the condition S 0 and S 0 might be the possible candidates for simultaneous two-proton emission, we obtain reliable and accurate calculations for the S and S values. In fact these are obtained from the calculated binding energies using the expressions,
where B(Z,N) is the binding energy of the nucleus with Z protons and N neutrons. The calculated results of the one- and two-proton separation energy for the nuclei belonging to the isotopic chains of nuclei with Z = 20 to 40 have been displayed in Figs. 2 and 3, respectively, as a function of decreasing mass number A. In these figures, the filled circles denote the experimental value of one- and two-proton separation energies, whereas the open circles represent the calculated results.
A comparison of the results obtained in the deformed RMF approach with the available experimental data for the one- and two-proton separation energies shows that in general theoretical results are in fairly good agreement with the measurements, though for some cases the calculated values are rather appreciably different from the experimental data. In a few cases this marked difference between the calculated results and the data can be attributed to the large uncertainty in the measurements shown by the error bars in Figs. 2 and 3.
It is seen from Fig. 2 for the two-proton separation energy that nuclei Fe, Ni and Zn which have been shown to be two-proton emitters in recent experiments [4, 5, 6, 7, 8] are located beyond the two-proton drip-line with negative two-proton separation energy values -0.90 MeV, -1.49 MeV and -1.97 MeV, respectively. Out of these, nuclei Fe and Ni have positive one-proton separation energy (S 0) and therefore fulfill the criteria of being a two-proton emitter (S 0 and S 0). In contrast to the above mentioned two nuclei, Fe and Ni, for the nucleus Zn both one- and two-proton separation energies are found to be negative. Interestingly, the nucleus Zn which belongs to the Z = 30 isotopic chain is found to be a two-proton emitter according to the results of our deformed RMF calculations.
Moreover, the result of our extensive calculations further show that the nuclei Ti, Cr, Ge, Se, Kr, Sr and Zr satisfy the criteria S 0 and S 0. These nuclei are, therefore, expected to be the potential candidates for exhibiting the two-proton radioactivity. It may be emphasized that the experimental data  for the separation energies for the nuclei identified above (with the exception of Zn and Ge) are consistent with the criteria S 0 and S 0. The difference between the calculated and measured results for nuclei Zn and Ge can be attributed to the appreciable errors in the experimental binding energies . Results of the one- and two-proton separation energies for the nuclei identified above along with the available experimental data have been listed in Table 3 to facilitate an easy comparison.
It may be mentioned that the separation energy values near the drip-line are close to zero and therefore the sign of the separation energy value is very sensitive and at times model dependent.
In order to check the force parameter dependence, we have carried out the RMF+BCS calculations for some nuclei using the NL-SH force parameters which are equally popular for the relativistic mean-field calculations. It is found that generally our RMF+BCS results for the two force parameters are similar though there are always some differences in finer details. However, it is worthwhile to note that our predictions of potential two-proton emitters are not affected if we use NL-SH force parameters instead of TMA force parameters[39, 40] as evident from Table 4.
3.2 Representative Examples of Two-Proton Emitters Located Beyond the Two-Proton Drip-Line
The results of the calculated quadrupole deformation values of the predicted two-proton emitters and their respective daughter nuclei are summarized below Table 5, especially to highlight and identify the nuclei which are spherical in shape as these may be studied in greater detail within the framework of spherical RMF.
From the table it is seen that nuclei Fe, Ni and Ge as well as their corresponding daughter nuclei have spherical or near spherical shapes. In contrast, the nuclei Zn and Se, and also their respective daughter nuclei are seen to have prolate deformation, whereas the nuclei Cr, Kr, Sr and Zr, and their respective daughter nuclei have oblate shapes. From these results, it is evidently seen that besides the nuclei Ti and Se, all other identified two-proton emitters and their corresponding daughter nuclei have similar shapes. Thus, it appears that the shape of the two-proton emitter nucleus is almost preserved in the process of two-proton emissions.
From amongst the predicted two-proton radioactive nuclei mentioned above, we have chosen the nuclei Cr, Ni and Ge which have spherical or near spherical shapes as can be seen from Table 4, to describe their detailed properties within the framework of the spherical RMF approach in order to obtain greater insight into the structural properties of these two-proton emitters.
It should be emphasized that though the nuclei Cr and Ge are slightly deformed, these have been treated here within the spherical RMF approach only as an approximation. On the other hand not only Ni is spherical but most of the Ni isotopes are found to be spherical in shape indicating Z = 28 remains a good magic number for all the isotopes with neutron number ranging from N = 20 to N = 70. It should also be stated that the nuclei Cr, Ni and Ge which have been identified as two-proton emitters are extremely proton rich nuclei located beyond the two-proton drip-line. Since the Ni isotopes are found to be spherical, these are ideally suited to be studied within the spherical RMF approach without any approximation. Thus these isotopes have been studied in rather great detail in order to show that the physical characteristics exhibited by bound proton rich isotopes remain intact even beyond the two-proton drip-line. It is found that the nucleus Ni which lies beyond the two-proton drip-line and thus is unbound, due to the centrifugal and Coulomb barrier develops finite life time and eventually decays via two-proton radioactivity. In the following we have also presented some detailed results for the proton rich Ni isotopes to demonstrate a systematic variation in the mean-field potential, wave function, energy of the single particle states and the proton density with increasing neutron number as we move towards the line of stability. The most interesting result of these study of Ni isotopes is that the highest proton single particle state 1f, for example, which remains completely filled and bound for all the Ni isotopes up to the proton drip-line, preserves its characteristics even beyond the drip-line when the nucleus Ni becomes unbound as it exhibits negative two-proton separation energy. This explains the long life time of this isotope even when it lies beyond the two-proton drip-line and eventually decays. This is manifested in the form of two proton radioactivity. This conclusion is reinforced by observing a similar characteristics of the radial density distribution for the Ni isotope as compared to those of the bound isotopes Ni. The shell closure or magicity of the Ni isotope also remains preserved as for the bound isotopes Ni. These are significant characteristics explaining why the nucleus Ni even while lying beyond the two-proton drip-line has positive one-proton separation energy and decays via two-proton radioactive mode.
The nucleus Cr
The predicted two-proton emitter nucleus Cr is found to be somewhat deformed with = -0.17. However, as mentioned above, in order to learn its detailed structure in terms of spherical single particle energies and wave functions, we have studied it within the framework of the spherical RMF approach. This approximate treatment, though not fully justified, is expected to shed light as regard to its finite life time even though it is located beyond the two-proton drip-line.
As mentioned earlier, this nucleus acquires long life time against two-proton decay due to the combined barrier provided by the Coulomb and centrifugal effects, even though it has negative two proton separation energy. In order to demonstrate the physical situation of proton rich Cr nucleus, we have plotted in Fig. 4 the RMF potential and the single particle energy spectrum for the bound proton single particle states. The figure also depicts a few positive energy proton states in the continuum including the resonant states 1f at energy 0.77 MeV. We have also shown in Fig. 4 the total mean-field potential for the resonant state, obtained by adding the centrifugal potential energy.
It may be emphasized that besides the resonant state 1f, other positive energy proton single particle states do not play any significant role in contributing to the total pairing energy as only this state has substantial overlap with the bound states near the Fermi level. This can be inferred from Fig. 5 wherein we have displayed the radial wave functions of some of the proton single particle states close to the Fermi surface, the proton Fermi energy being MeV. These include the bound states 2s and 1, and the continuum states 2p and 3, in addition to the resonant state 1f.
The wave function for the 1f state plotted in Fig. 5 is clearly seen to be confined within a radial range of about 7 fm and has a decaying component outside this region, characterizing a resonant state. In contrast, the main part of the wave function for the non-resonant states, e.g. 2p and 3s, is seen to be spread over outside the potential region, though a small part is also contained inside the potential range. This type of states thus has a poorer overlap with the bound states near the Fermi surface leading to small value for the pairing gap . Further, the positive energy states lying much higher from the Fermi level, for example, 1g, 1g etc. have a negligible contribution to the total pairing energy of the system.
This is also clearly evident from Fig. 6 which shows the calculated single particle pairing gap energy for some of the proton single particle states in the nucleus Cr. However, we have not shown in the figure the single particle states having negligibly small values as these do not contribute significantly to the total pairing energy. One observes indeed in Fig. 6 that the proton resonant single particle state 1f at energy 0.77 MeV has the pairing gap energy of about 1 MeV which is close to that of the bound states , and . Also, Fig. 6 shows that the pairing gap value for the non-resonant states and lying in the continuum is negligibly small.
From the characteristics of the proton resonant 1f single particle state, it is evidently clear that this state behaves similar to a bound state. This property also enables the nucleus Cr to have finite life time even with negative two-proton separation energy while lying beyond the two-proton drip-line as is elucidated below. In the nucleus Cr, the last 4 protons outside the closed s-d shells occupy the 1f single particle state. If we consider the neighboring isotones of Cr, that is Ca and Ti with increasing number of protons in the 1f proton single particle state, calculations show that while the nucleus Ca for which the 1f state is completely empty, is bound and has spherical shape, the nucleus Ti is also bound though slightly deformed. On further addition of two more protons to Ti, the next isotone Cr is formed in which now the 1f resonant state is occupied by four protons. Due to the resonant nature of the proton single particle 1f state the pairing energy is increased and the nucleus tends to remain bound, whereas the Coulomb interactions amongst the increasing number of protons acts in a disruptive way to make the nucleus unbound with negative two-proton separation energy. However, the centrifugal barrier provided by high angular momentum resonant 1f state together with the Coulomb barrier enables the nucleus Cr to gain finite life time, and eventually it decays by two proton emission. It should be emphasized that the contribution of pairing energy plays an important role for the stability of nuclei and consequently in deciding the position of the neutron and proton drip-lines. Also it is remarkable that in contrast to the proton rich nucleus such as Cr, if we consider the neutron rich nucleus like Ca, it is found that further addition of neutrons while approaching the extremely neutron rich nucleus Ca, the single particle neutron states 3s, 1g, 2d and 2d which lie near the Fermi level gradually come down close to zero energy, and subsequently the 1g and 3s neutron single particle states even become bound states. This helps in accommodating more and more neutrons which are just bound. In the case of neutrons we do not have the disruptive Coulomb force anymore. Similar explanation holds good for other two-proton emitters like Ti and Ge.
The nucleus Ni
In order to illustrate the case of proton rich Ni nucleus, we have plotted in Fig. 7, the RMF potential and the wave functions of its few representative proton single particle states in the lower and upper panels respectively. The lower panel also shows the spectrum of bound proton single particle states along with the positive energy state 2p in the continuum. We have also depicted in lower panel of Fig. 7 the total mean-field potential for the highest bound and fully occupied proton single particle state 1f, obtained by adding the centrifugal potential energy. The combined effect of Coulomb barrier and centrifugal barrier prevents the protons from quickly leaving the proton rich nucleus Ni located beyond the two-proton drip-line. The delay associated with the tunneling process allows for the observation of two-proton radioactivity. In the upper panel of Fig. 7 we have displayed the radial wave functions of some proton single particle states, the proton Fermi energy being = 2.86 MeV. These include the bound 1d, 2s, and the continuum 2p proton single particle state.
It is seen that the radial wave function of the proton single particle state 1f remains mainly confined to the region of the potential well. In contrast, the wave function for a typical continuum state 2p is spread over to large distances outside of the potential region. Therefore, the important outcome from the above discussions is that the highest proton single particle state 1f remains completely occupied and bound even for the nucleus Ni lying beyond the two-proton drip-line with negative two-proton separation energy.
Since Ni isotopes have the spherical shape, it would be interesting to employ spherical RMF approach to investigate the behaviour of single particle spectrum, potential, wave function and density distribution of Ni isotopes with increasing Z/N ratio. With this in view, first we plot in Fig. 8 the RMF potential (lower panel), and radial wave function of the proton single particle state 1f for the Ni isotopes as a function of radius in two different scales (middle and upper panel). We note from the lower panel of Fig. 8 that although the two-proton separation energy of nucleus Ni is negative, its potential behaves similar to the Ni isotopes which have positive two-proton separation energy.
Moreover, it is clearly seen that similar to the case of Ni isotopes, the proton single particle state 1f in the nucleus Ni remains confined to its potential region. In order to demonstrate the difference in the wave function for various isotopes at large distances, we have plotted in the upper panel of Fig. 8 the radial wave function of proton single particle state 1f for the Ni isotopes in logarithmic scale. It is seen that the wave function of the proton single particle state 1f starts spreading slightly outside the potential region with increasing Z/N ratio. Nevertheless, a large part of the wave functions is contained inside the potential range. From these results it is clear that the nucleus Ni lying beyond the two-proton drip-line behaves similar to the Ni isotopes for which the two-proton separation energy is positive.
In Fig. 9 we have shown the variation of the proton single particle levels and position of the Fermi level of Ni isotopes with decreasing neutron number N (or in other words with increasing Z/N ratio). It is readily seen from the figure that the existence of large energy gap between proton single particle levels 1 and 1 explains the tradition proton shell closure at Z = 28 in Ni isotopes. The proton Fermi energy which lies at = -4.91 MeV in the Ni nucleus moves to = 2.86 MeV in the nucleus Ni having maximum Z/N value.
It is evident from the Fig. 9 that despite the fact that the nucleus Ni is located beyond the two-proton drip-line, the energy gap between proton single particle levels 1 and 1 remains significant enough to maintain the Z = 28 shell closure. This is due to the fact that the highest proton single particle state 1f preserves its characteristics beyond the two-proton drip-line. A similar conclusion can be drawn from the variation of the proton density distribution as a function of radial distances. Result of such a variation have been depicted in Fig. 10 for Ni isotopes with N = 20 to 28. As the proton shell in Ni isotopes remains closed we observe sharply falling asymptotic density distribution for these isotopes. It is remarkable that the proton density distribution of the unbound Ni isotope has the radial dependence similar to the other isotopes Ni which are bound. From the various results presented for the Ni isotopes, it is observed that the unbound proton rich nucleus preserves properties very similar to the other bound states and this enables it to have finite life time and decay by emission of two protons.
The nucleus Ge
The predicted two-proton emitter nucleus Ge is found to have slightly deformed shape with = 0.13 and has been described here in an approximate manner within the framework of spherical RMF approach as in the case of nucleus Cr. The calculated two-proton separation energy for the nucleus Ge is found to be negative, though very close to the zero energy. Thus, theoretically this nucleus appears to be unbound against the two-proton decay. On the other hand, through the measured value of the two-proton separation energy for this nucleus is also found to be close to zero, it has positive sign but with large error bars.
In the case of proton rich Ge nucleus the proton resonant states are found to be 1f, 2p and 2p as has been shown in Fig. 11. The figure displays together the RMF potential and the spectrum for the bound proton single particle states. We have also depicted in Fig. 11 the total mean-field potential for the resonant state, obtained by adding the centrifugal potential energy. It may be emphasized that amongst the many proton single particle states in the continuum only the resonant state has sizable overlap with the bound single particle states. Thus the resonant states 1f, 2p and 2p make significant contribution to pairing energy.
This can be inferred from Fig. 12 wherein we have displayed the radial wave functions of some of the proton single particle states close to the Fermi surface, the proton Fermi energy being MeV. These include the bound states and , and the continuum states and , in addition to the resonant 1f, 2p and 2p single particle states. The wave functions for the 1f, 2p and 2p states plotted in Fig. 12 are clearly seen to be confined within a radial range of about 7 fm and have a small decaying component outside this region, characterizing the resonant states. In contrast, the main part of the wave function for the non-resonant states, e.g. 3s and 3p, is seen to be spread over outside the potential region, though a small part is also contained inside the potential range. This type of states thus has a poorer overlap with the bound states near the Fermi surface leading to small value for the pairing gap .
This is clearly evident from Fig. 13 which shows the calculated single particle pairing gap energy for some of the proton states in the nucleus Ge. One observes indeed in Fig. 13 that the gap energies for the resonant , 2p and 2p states have values close to MeV which is quantitatively similar to that of bound states and . Also, Fig. 13 shows that the pairing gap value for the non-resonant states and lying in the continuum is negligibly small.
The explanation of this nucleus acquiring long life time despite lying close to the two-proton drip-line with negative two-proton separation energy can be given by considering the neighboring isotones like Ni, Zn in a manner similar to the case of nucleus Cr described earlier.
In order to show the effect of addition of protons to a fixed number of neutrons which is equivalent to highlighting the changes as one moves towards the proton drip-line and beyond, we have shown in Fig. 14, the variation in radial density distributions for the isotones corresponding to N = 20 covering the Z range from 20 to 28.
It is evidently seen that the proton density has a long extended tail indicating loosely bound protons. Further increase in number of protons make the system unbound and in some cases such as Cr, Ni and Ge discussed here, we have the phenomena of two proton radioactivity.
To, summarize, we have employed deformed relativistic mean-field plus state dependent BCS (RMF+BCS) approach to study the ground state properties of proton rich nuclei in the region 20 Z 40. We have found that the potential barrier provided by the Coulomb interaction and that due to centrifugal force may cause a long delay in the decay of some of the nuclei with small negative proton separation energy. This may cause the existence of proton rich nuclei beyond the proton drip line. Nuclei Ti, Cr, Fe, Ni, Zn, Ge, Se, Kr, Sr and Zr are expected to be the possible candidates for exhibiting two-proton radioactivity in the region 20 Z 40. The calculated two-proton separation energy and other relevant ground state properties are found to be in good agreement with the available experimental data. This demonstrates the validity and usefulness of the deformed RMF+BCS approach for the description of proton rich nuclei near the drip lines.
The above mentioned results of our investigations for the two-proton radioactivity are expected to provide additional impetus for more experimental studies to verify the potential candidates predicted above for the two-proton radioactivity in the near future.
Support through a grant (SR/S2/HEP-01/2004) by the Department of Science and Technology (DST), India, is acknowledged. The authors are indebted to Dr. L. S. Geng, RCNP, Osaka University, Osaka, Japan, for valuable correspondence.
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