Abstract
The single-particle spectrum obtained from the relativistic mean field (RMF) theory is used to extract the shell correction energy with the Strutinsky method. Considering the delicate balance between the plateau condition in the Strutinsky smoothing procedure and the convergence for the total binding energy, the proper space sizes used in solving the RMF equations are investigated in detail by taking 208 Pb as an example. With the proper space sizes, almost the same shell correction energies are obtained by solving the RMF equations either on basis space or in coordinate space.

The liquid drop model of the nucleus was historically the first model to be proposed as an explanation of the properties of the nucleus. Then the occurrence of the magic numbers has been one of the strongest evidences for the nuclear shells. The shell effect presents a fluctuation in the binding energy, the so-called shell correction energy, which can be supplemented to the liquid drop model to improve the descriptions of the nuclear masses.
One of the most practical and effective methods for extracting the shell correction energy was proposed by Strutinsky [1, 2] . The Strutinsky method has been widely applied to the calculations of the masses, shapes and fission barriers of nuclei [3, 4] .
Two key points in the Strutinsky smoothing procedure are the so-called plateau condition as well as the reliable single-particle spectrum.
On one hand, the plateau condition means that the shell correction energy should be insensitive to the smoothing range γ and the order of generalized Laguerre polynomial M, since these two variables are only parameters in the smoothing procedure which have no physical meanings [2, 5] . The plateau condition has been investigated in the past several decades. The Strutinsky method was first combined with the Nilsson model. In this case a good plateau could appear due to the infinity of the potential [1] . However, no plateau appears in the realistic finite potential because of the discontinuity of the level density at the upper edge of the potential [5] . A possible solution for this problem is to complete the spectrum outside the bound states region, using the resonances in the continuum region [6] or using the positive eigenvalues obtained by diagonalizing the Hamiltonian in a harmonicoscillator basis [4, 7] . The first method is cumbersome, especially for deformed nuclei, although the plateau could appear if resonances up to 60 MeV are included [6] . The second one is more practical, however, the shell correction energy and even the appearance of the plateau condition depend on the number of harmonic oscillator shells included in the basis [4, 7] . This requires a delicate adjustment to the shell number N 0 . In the non-relativistic framework, the plateau condition for 208 Pb has been checked with the phenomenological folded Yukawa potential [3, 4] , where the shell correction energy was found to be stable in the range of 8 ≤ N 0 ≤ 13, thus N 0 = 12 was recommended and a good plateau was found.
On the other hand, the self-consistent relativistic mean field (RMF) theory [8] , which has received wide attention due to its successful description of lots of nuclear phenomena [9, 10, 11] , could provide the single-particle spectrum microscopically. With this single-particle spectrum thus obtained, the shell correction energies in superheavy nucleus have been inves-tigated and it is found that the shell correction energy at the saddle point is too important to be neglected [12] , which will influence the synthesis of the superheavy nuclei [13, 14] .
The RMF equations can be solved by diagonalizing the Hamiltonian in a harmonicoscillator basis space [15] or by solving the coupled channel differential equations within a finite box [0, R] in coordinate space [16] , meanwhile the single-particle spectrum outside the bound states region can also be obtained. In this way, the level density in the continuum depends on the space size, i.e. the shell number for fermions N f (bosons N b ) or the box size R. Therefore, the space size used in solving the RMF equations is one of the crucial quantities for the Strutinsky smoothing procedure. Meanwhile, the choice of the space size should guarantee the convergence for the total binding energy.
In this Letter, based on the single-particle spectrum obtained from the RMF theory, the shell correction energy will be extracted by the Strutinsky method. The efforts will be focused on determining the proper space size used in solving the RMF equations via the delicate balance between the plateau condition and the convergence for the total binding energy.
The basic ansatz of the RMF theory is a Lagrangian density where nucleons are described as Dirac spinors that interact via the exchange of several mesons (the scalar σ, the vector ω, and isovector vector ρ) and the photon [8] .
The classical variation principle gives equations of motion for the nucleon, mesons and the photon. As in many applications, we study the ground-state properties of nuclei with time reversal symmetry, thus the nucleon spinors are the eigenvectors of the stationary Dirac
and equations of motion for mesons and photon are
where ω 0 and A 0 are timelike components of the vector ω and the photon fields and ρ 0 the third component of the timelike component of the isovector vector ρ meson. Eq. (1) and Eq. (2) are coupled to each other by the vector and scalar potentials
and various densities
The above RMF equations can be solved in basis space or coordinate space as mentioned above. The single-particle spectrum thus obtained will be used to calculate the shell correction energies with the Strutinsky method.
The Strutinsky method is based on the assumption that the realistic shell correction energy can be extracted uniquely from a mean field model. The shell correction energy, representing the fluctuating part of the binding energy, is defined as the difference between the total single-particle energy and its smooth part [5] ,
where N (Z) is the neutron (proton) number, ǫ i is the single-particle energy. The smoothed Fermi levelλ is determined by the conservation of the particle number,
The smoothed level densityg(ǫ) takes the form,
with the smoothing range γ. The folding function is usually taken as
where
. In practice, a cutoff is needed in the summation of the single-particle energy, and the shell correction energy is stable when the cutoff energy is large enough.
The Strutinsky method, as shown above, has two additional variables, the smoothing range γ and the order of generalized Laguerre polynomial M. Since neither γ nor M has physical meaning, the value of the shell correction should be insensitive to these quantities within a certain range of values, i.e. the so-called plateau condition,
This forms an important criterion for the feasibility of the Strutinsky method.
Taking the nucleus 208 Pb as an example, the neutron single-particle spectrum is calculated by the RMF theory with the parameter set PK1 [17] . In the Strutinsky smoothing procedure, the unit of the smoothing range γ is adopted as ω 0 = 41A −1/3 (1 + First of all, solving the RMF equations in basis space, it is found that the shell correction energies are almost independent on the shell number for bosons N b , which could be understood as follows. The shell correction energies are determined by the single-particle level density. Since the single-particle energies are the eigenvalues of the Hamiltonian matrix for fermions (nucleons), changing the shell number for fermions N f will change the singleparticle level density in the positive energy region, whereas changing N b doesn't influence it. Thus, the shell correction energies are not sensitive to N b . In the following investigation, N b = 20 is fixed as a proper number [18] , and the efforts will be focused on the influence of N f on the shell correction energies.
In the upper panel of [6] . When N f increases, the single-particle levels in the positive energy region become much denser than those in the bound region. The discontinuity in the single-particle level density makes the plateau vanish.
Another important criterion for choosing the shell numbers is that N f should be large enough to achieve the convergence for the total binding energy. According to this point, N f ≥ 12 was proposed [18] . Specifically, in the present calculation, the accuracy for the total binding energy is 0.04% when N f = 12 is chosen.
Taking the balance between the above criteria, the size of basis space used in solving the RMF equations should be taken as N f = 12 and N b = 20, which is in accordance with the space size for folded-Yukawa potential [4] . The neutron shell correction energy reads −11.92 ∼ −11.74 MeV within the range 1.2 ≤ γ ≤ 1.8.
In the lower panel of Fig. 1 , the neutron shell correction energies as a function of the smoothing range γ for the nucleus 208 Pb are shown, where the RMF equations are solved in coordinate space with different box sizes R box = 11 ∼ 19 fm. The plateau condition is well fulfilled in the cases of R box ≤ 15 fm. Especially, the shell correction energies are stable between R box = 11 fm and R box = 13 fm. It is also found that the plateau vanishes gradually as R box > 15 fm. The reason is the same as that in the basis space.
On the other hand, R box should be large enough to achieve the convergence for the total binding energy. It is found that the accuracy for the total binding energy is 0.01% when R box = 13 fm is chosen.
Therefore, based on the above two criteria, the box size R box = 13 fm is recommended for solving the RMF equations in coordinate space. The neutron shell correction energy reads −12.00 ∼ −11.80 MeV within the range 1.2 ≤ γ ≤ 1.8.
In order to verify the choice of the above space sizes, the dependence of the plateau condition on the order of generalized Laguerre polynomial M will be investigated.
In 

{'paper_id': '119226693', 'title': 'Stability of Strutinsky Shell Correction Energy in Relativistic Mean Field Theory', 's2_url': 'https://api.semanticscholar.org/CorpusID:119226693', 'source': 'S2ORC'}