Role of pair-vibrational correlations in forming the odd-even mass difference
K. Neerg{\aa}rd, I. Bentley
TL;DR
This paper extends the RPA-amended Nilsson-Strutinskij method to odd-A nuclei, showing that pair-vibrational correlations significantly influence odd-even mass differences, especially in light nuclei, providing a more accurate nuclear binding energy model.
Contribution
It introduces the application of RPA corrections to odd-A nuclei within the nuclear binding energy calculation, highlighting their significant impact on odd-even mass differences.
Findings
RPA correction accounts for most of the odd-even mass difference in light nuclei.
The size and sign of RPA contributions vary across nuclei.
RPA corrections improve the accuracy of nuclear mass models.
Abstract
In the random-phase-approximation-amended (RPA-amended) Nilsson-Strutinskij method of calculating nuclear binding energies, the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers and of neutrons and protons and even mass number . Here it is applied to odd- nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with and . (ii) The odd- isotopes of In, Sn, and Sb with . (iii) The odd- isotopes of…
| rms | ||||||
|---|---|---|---|---|---|---|
| 15.23 | 112.5 | 16.52 | 148.9 | 0.6601 | 1.018 | |
| Around Sn | 15.37 | 115.2 | 16.97 | 157.5 | 0.6737 | 0.515 |
| Around 102Zr | 14.78 | 151.2 | 16.07 | 355.5 | 0.5774 | 0.043 |
| Nucleus | (∘) | Nucleus | (∘) | Nucleus | (∘) | Nucleus | (∘) | Nucleus | (∘) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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Role of pair-vibrational correlations in forming the odd-even
mass difference
K. Neergård
Fjordtoften 17, 4700 Næstved, Denmark
I. Bentley
Department of Chemistry and Physics, Saint Mary’s College, Notre Dame, Indiana 46556, USA
Abstract
In the random-phase-approximation-amended (RPA-amended) Nilsson-Strutinskij method of calculating nuclear binding energies, the conventional shell correction terms derived from the independent-nucleon model and the Bardeen-Cooper-Schrieffer pairing theory are supplemented by a term which accounts for the pair-vibrational correlation energy. This term is derived by means of the RPA from a pairing Hamiltonian which includes a neutron-proton pairing interaction. The method was used previously in studies of the pattern of binding energies of nuclei with approximately equal numbers and of neutrons and protons and even mass number . Here it is applied to odd- nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with and . (ii) The odd- isotopes of In, Sn, and Sb with . (iii) The odd- isotopes of Sr, Y, Zr, Nb, and Mo with . The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper sd shell this correction accounts for almost the entire odd-even mass difference for odd and about half of it for odd . The size and sign of the RPA contribution varies, which is explained qualitatively in terms of a closed expression for a smooth RPA counter term.
I Introduction
Nuclear binding energies are often calculated in mean-field approximations. The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity Bardeen et al. (1957a); *ref:Bar57b, which was applied extensively to the description of pairing in nuclei since its adaption to the nuclear system by Bohr, Mottelson, and Pines Bohr et al. (1958), Bogolyubov Bogolyubov (1958), and Solov’yov Solov’yov (1958a); *ref:Sol58b, is such an approximation. Residual interactions, which are neglected in a mean-field approximation, induce correlations, which increase the binding energy. We call this extra binding energy correlation energy (in Ref. Bang and Krumlinde (1970) this term is used differently.) The BCS theory, in particular, may be derived, for a given type of fermion (electron, neutron, proton), from the Hamiltonian
[TABLE]
Here annihilates a fermion in a member of an orthonormal set of single-fermion states which is preserved up to phases under time reversal, denoted by the bar. The single-fermion energies and the coupling constant are parameters. The second term in the expression (1) is known as the pairing interaction. The exact minimum of the Hamiltonian (1) can be calculated with any wanted accuracy for fairly large single-fermion spaces Richardson (1963).
Figure 1 shows the result of such a calculation in comparison with that obtained when the correlation energy is calculated in the random phase approximation (RPA) Bohm and Pines (1953). This approximation is seen to give a good agreement with the exact value. Appreciable deviations only occur in a narrow interval of about the threshold of BCS pairing. Because the RPA equations derived from the Hamiltonian (1) describe oscillations of the pair field about the mean field equilibrium, the correlations may thus be seen as mainly pair vibrational.
Calculations of binding energies by the Strutinskij method Strutinskij (1966) conventionally include a pairing term based on the BCS theory. Figure 1 indicates a significance of the correlation energy which suggests that it be taken into account. For , in particular, the pairing interaction induces only correlation energy. Moreover, isobaric invariance requires that the sum of neutron and proton pairing interactions be generalized to
[TABLE]
with a pair field isovector
[TABLE]
Here is the single-nucleon isospin, and time reversal is assumed to commute with and and anticommute with . In Eq. (3) the set or of quantum numbers includes an eigenvalue of , and the span of the orthonormal set of states is isobarically invariant. The interaction (2) contains a neutron-proton term . In a doubly even nucleus the Hartree-Bogolyubov quasinucleon vacuum derived from the resulting Hamiltonian has Neergård (2009), so the neutron-proton interaction also induces only correlation energy.
In a collaboration with Frauendorf we developed an extension of the conventional Nilsson-Strutinskij scheme which takes the pair-vibrational correlations into account in the RPA Bentley et al. (2014). Minor modifications of the scheme of calculations proposed in Ref. Bentley et al. (2014) were discussed by Neergård Neergård (2016, 2017). These articles deal with nuclei with and even , where and are the numbers of neutrons and protons and . The extended Nilsson-Strutinskij scheme was found to account, with suitably chosen parameters, quite well for the pattern of even- binding energies and certain excitation energies in doubly odd nuclei in this region. We here apply it to odd- nuclei. We examine in particular the influence of the inclusion of the RPA term on the calculated odd-even mass differences. Three regions of the chart of nuclei are considered: (i) The region, previously studied with respect to the even- nuclei. (ii) A neighborhood of the Sn isotopic chain. (iii) A region of well-deformed, neutron rich nuclei around 102Zr.
The organization of the article is as follows. In Sec. II we describe the scheme of calculations. This section serves to present in one place all ingredients of the RPA-amended Nilsson-Strutinskij method in the form it has taken after several modifications since the publication of Ref. Bentley et al. (2014). Then, in each of Secs. III–V, we discuss the results for one of the regions (i)–(iii). Finally, after exploring in Sec. VI a technical matter of interpolation of the RPA energy across the threshold of BCS pairing, we summarize our results in Sec. VII.
II RPA-amended Nilsson-Strutinskij model
The binding energy is calculated by
[TABLE]
where ‘i.n.’ stands for ‘independent nucleons’. Here is a liquid drop energy, and each term has the form
[TABLE]
with a ‘smooth’ counter term . The ‘microscopic’ energy
[TABLE]
approximates the minimum of the Hamiltonian
[TABLE]
where
[TABLE]
Here, unlike in Eq. (3), the index numbers, for each for neutrons and for protons, an orthonormal set of eigenstates of a time-reversal invariant single-nucleon Hamiltonian in an order of nondecreasing eigenvalue . The numbering should be such that in the limit . In this limit then , , and in terms of components of the isovector (3) provided also all are equal. Again the set of states is supposed to be preserved under time reversal up to phases. We also assume that each pair of an odd and the following even refer to a pair of states connected by time reversal up to phases. Both of these assumptions are satisfied automatically if the eigenvalues are doubly degenerate, that is, except in spherical nuclei. In the spherical case it is satisfied if degenerate orbits are distinguished by a magnetic quantum number and pairs of an odd and the following even refer to pairs of states with opposite .
Unlike Ref. Bentley et al. (2014) strict isobaric invariance is not imposed on the microscopic model. The single-nucleon Hamiltonians and may be different, and different valence space dimension may be employed for different . We use throughout , , and so that the neutron and proton valence spaces are always half filled and . These modifications, which where introduced partly in Refs. Neergård (2016, 2017), renders the model better suited for nuclei with a large neutron or proton excess.
We also allow different coupling constants for different , writing
[TABLE]
where is the isomagnetic quantum number of the nucleus and that of the interacting pair, that is, , and 0 for , and , respectively. The parameters , , and are set separately for each region (i)–(iii). The limit where , all are equal, and all are equal will be referred to as the limit of isobaric invariance.
For each nucleus we assume a deformation, which we take from a conventional Nilsson-Strutinskij calculation Bentley (2010). It is expressed by the Nilsson parameters , , and Nilsson et al. (1969); Larsson (1973). The deformations are listed in the appendix.
II.1 Liquid drop energy
The liquid drop energy is written
[TABLE]
where the coefficients are parameters. The deformation dependent factors and are calculated from the Nilsson parameters in two steps. First, following Seeger and Howard Seeger and Howard (1975), we determine the coefficients in the equations in spherical coordinates of the surfaces of constant second term in the expression (16) below,
[TABLE]
where is the Legendre function of the first kind as defined by Edmonds Edmonds (1957). With and , the nonzero coefficients with are given to second order in and by
[TABLE]
This approximation is adopted. (For the expansion (12) (including results for which we do not show) should give Eqs. (10)–(13) of Ref. Seeger and Howard (1975). Some coefficients there differ from ours, which were derived by computer algebra.)
The coefficients with are not required in the second step, where and are expanded in the ’s. This expansion can be derived from Swiatecki’s results in Ref. Swiatecki (1956). Swiatecki’s expansion is restricted to , but when only terms of total rank 8 or less are retained, each term has a unique continuation into given by the requirement that it be a scalar polynomial in the spherical tensor components . The resulting expansion, which we adopt, is
[TABLE]
with
[TABLE]
For given pairing parameters , , and an RPA interpolation width defined in Sec. VI we fix the coefficients in Eq. (10) by a least-square fit of the calculated total energies (4) to the measured ones. Included in this fit are all doubly even nuclei in the considered region of the chart of nuclei whose binding energies have been measured. The limits of each region for this purpose are specified in Secs. III–V. The fit of the liquid drop parameters is done before the pairing parameters are fit to other data. Table 1 shows the results for the optimal pairing parameters. For the 102Zr region the sample of doubly even nuclei consists of only 9 nuclei.
II.2 Independent nucleons
The terms E_{\text{i.n.,\tau}} in Eq. (6) are given by
[TABLE]
with for and for . The single-nucleon energies are the eigenvalues of the Nilsson Hamiltonian Nilsson (1955); Nilsson et al. (1969); Larsson (1973),
[TABLE]
where and are the spatial coordinates and momentum, is the spin, and is the nucleon mass. The function is the Legendre polynomial. The oscillator frequencies are given by
[TABLE]
where satisfies the condition of volume conservation
[TABLE]
The ’stretched’ spherical coordinates and orbital angular momentum Nilsson (1955) correspond to Cartesian coordinates
[TABLE]
and is the number of oscillator quanta. For the parameters and we adopt the values recommended in Ref. Bengtsson and Ragnarsson (1985).
The independent-nucleon counter terms are
[TABLE]
where the smooth chemical potential is defined by
[TABLE]
and the smooth level density is given by Strutinskij (1966); Ring and Schuck (1980)
[TABLE]
in terms of the generalized Laguerre polynomial . We use smoothing width and smoothing order and include in the sum in Eq. (22) all such that and .
II.3 BCS theory
The terms E_{\text{BCS,\tau}} are given by the standard BCS theory. A derivation of the following equations is found, for example in Ref. Neergård (2009). For even one has
[TABLE]
with
[TABLE]
Here and obey
[TABLE]
For later reference we define the quasinucleon annihilators
[TABLE]
The equations (24) and (25) always have a solution with and there is a threshold G_{\text{cr,\tau}} such that no other is possible for G\leq G_{\text{cr,\tau}}. For G>G_{\text{cr,\tau}} there is a solution with and a lower E_{\text{BCS,\tau}}, which is chosen. If then G_{\text{cr,\tau}}>0 and G_{\text{cr,\tau}} is given by
[TABLE]
If , as happens in spherical nuclei when a shell is partly occupied in the absence of pairing, then G_{\text{cr,\tau}}=0.
If is odd, a Bogolyubov quasinucleon annihilated by is assumed to be present in the BCS ground state. The orbit is then fully occupied and its time reverse fully empty. The BCS energy E_{\text{BCS,\tau}} is calculated as if nucleons of type inhabited the remaining orbits. The odd nucleon is said to block the Fermi level.
To simplify notation we let without an argument mean and write
[TABLE]
The BCS counter terms are then given by Strutinsky (1967); Neergård (2016)
[TABLE]
II.4 Random-phase approximation
The calculation of E_{\text{RPA,\tau}} is based on the theory in Ref. Neergård (2009). It involves linear relations in the space spanned by the terms in the sums in Eq. (8). A linearly independent set of terms in the expression for may be labeled by the odd single-nucleon indices from 1 to . When both and are even, we denote this set of by . Modifications of this definition when one or both of and are odd are discussed below. It is convenient to introduce at this point labels alternative to and synonymous with and vectors and matrices with components or element indexed by the set . A diagonal matrix is defined by its elements
[TABLE]
and column vectors and by their components
[TABLE]
Let
[TABLE]
Then
[TABLE]
where are the eigenvalues of
[TABLE]
The terms are the RPA frequencies.
For and, in the limit of isobaric invariance, for and , one RPA mode is, for G_{\tau\tau^{\prime}}>G_{\text{cr,\tau\tau^{\prime}}} (with G_{\text{cr,np}}=G_{\text{cr,n}}=G_{\text{cr,p}} in the isobarically invariant limit), a Nambu-Goldstone mode with zero frequency Neergård (2009); Hinohara and Nazarewicz (2016). That is, in this degree of freedom vibration turns into rotation. This is what gives rise to the singularity at in Fig. 1 Bentley et al. (2014). To circumvent this singularity we interpolate the calculated E_{\text{RPA,\tau\tau^{\prime}}} across the region of G_{\tau\tau^{\prime}}=G_{\text{cr,\tau\tau^{\prime}}} for or and with G_{\text{cr,np}}\approx G_{\text{cr,n}}\approx G_{\text{cr,p}} in the latter case. Details are given in Sec. VI.
The expression (33) results from the expansion of the ground state energy in Feynman diagrams formed as closed bubble chains; see Eq. (36) in Ref. Neergård (2009). Each bubble represents a virtual creation and subsequent annihilation of a pair of Bogulyubov quasinucleons. When, say, is odd, the presence of the unpaired nucleon in the BCS ground state blocks the creation of quasinucleon pairs by the terms in and proportional to . Therefore should be and is omitted from for odd . The remainder exhausts the set of excitations of the BCS ground state mediated by the fields and .
The case of is more involved for odd . The fields and have terms proportional to and , which, respectively, adds a pair of quasinucleons and scatters the quasineutron in the Fermi level orbit into a quasiproton. The latter excitation, in particular, may have negative energy, which inhibits the use of the RPA. Even when the energy is positive, it is small in comparison to that of the genuine two-quasinucleon excitations, which may render the RPA calculation unstable anyway. For in the limit of isobaric invariance, both these excitations have zero matrix elements when one assumes, as we do, cf. Sec. II.5, that the unpaired neutron and the unpaired proton combine to isospin . This allows using Eq. (33), omitting from like it is omitted from . To avoid the troubles just described, we have chosen to do so also when is even. That is, we generally omit from and when is odd, and analogously for odd . In physical terms this amounts to extending to the RPA the assumption in the BCS theory with the Fermi level blocked that the unpaired nucleon acts as a spectator to interactions among the paired nucleons in a valence space that excludes the half occupied single-nucleon level. A more satisfactory treatment of the neutron-proton pair vibrational correlations for odd might be based on the theory of (quasi-) particle-vibration coupling.
For even and the RPA energy as given by Eq. (33) gets contributions from fluctuations of the quasinucleon vacuum in every direction generated by an operator with . Vaquero, Egido, and Rodríguez take an different path to study pairing fluctuations Vaquero et al. (2013). A combination of the variances of and is used (for a given deformation) as a generator coordinate to obtain a wave function that describes the distribution of quasinucleon vacua in the single degree of freedom associated with this coordinate. The quasinucleon vacua are generated by the constrained Hartree-Fock-Bogolyubov method with a Gogny two-nucleon interaction.
For the calculation of the RPA counter terms \tilde{E}_{\text{RPA,\tau\tau^{\prime}}} we define by replacing by in the expression (22). This definition coincides with Eq. (22) for . A function is defined by
[TABLE]
In particular by Eq. (21). We let without an argument mean and generalize Eq. (28) to
[TABLE]
and the definition of in Eq. (29) to
[TABLE]
Then \tilde{E}_{\text{RPA,\tau\tau^{\prime}}} is given by Neergård (2016)
[TABLE]
with
[TABLE]
II.5 Isobaric analogs
The scheme presented so far describes states with isospin . This relation is satisfied empirically by nearly all ground states. The exception is that for odd most ground states have while the lowest states with are excited. For odd the lowest states with are mostly excited. We denote the energies of these states by to distinguish them from the energies of the states. For odd the states are the isobaric analogs of the ground states of the doubly even nuclei with neutron and proton numbers . Accordingly we set
[TABLE]
where is calculated from the deformation of the doubly even nucleus.
III region
Our calculations for even in the region follow the scheme previously applied in Refs. Bentley et al. (2014); Neergård (2017). Again we consider the doubly even nuclei with and and the doubly odd ones with and . Unlike Ref. Neergård (2017) we use different for different and a considerably smaller interval of interpolation of the RPA energies as discussed in Sec. VI. Further, the deformations were recalculated, all oscillator shells with being included in the calculation by the scheme of Ref. Bentley (2010) instead of just four shells close to the neutron or proton Fermi level for and , respectively. For the doubly even nuclei this only changed the deformations of 84Zr and 86Mo, which went from spherical to oblate. For the states of the doubly odd nuclei, the deformations were determined in the prior work by averaging over the deformations of the adjacent doubly even nuclei. In the present work these deformations are calculated independently by blocking the Fermi levels. This resulted in significant changes of the individual deformations, while the overall pattern of variation along the chain of these states remains the same.
Again we set in Eq. (9) so that one pair coupling constant covers the cases , , and . The parameters and are fit to the following data for odd .
(1) The doubly odd–doubly even mass differences
[TABLE]
(2) The differences of the lowest energies for and , that is,
[TABLE]
The set of data is the same as in Refs. Bentley et al. (2014); Neergård (2017) and thus includes extrapolated masses of 82Nb and 86Tc, but all mass data were updated from AME12 Audi et al. (2012) to AME16 Huang et al. (2017). Again excitation energies are taken from the Evaluated Nuclear Structure Data File ref . A least-square fit gives
[TABLE]
with an rms deviation of 0.789 MeV. Plotting the doubly odd–doubly even mass differences, the to energy splittings, the symmetry energy coefficients, and the ‘Wigner ’ as functions of results in figures grossly similar to Figs. 6–9 of Refs. Bentley et al. (2014) and Fig. 1 of Ref. Neergård (2017). As for the Wigner , more detail is given in Sec. VI.
With the parameters thus set we consider the odd- nuclei with and . The odd-even mass difference is defined as the mass of the odd- nucleus relative to the average mass its two doubly even neighbors. The calculated are shown in Fig. 2 in comparison with the data. The model is seen to reproduce the typical size of the measured values. This is remarkable because and were fit, not to these data but to energies in doubly odd nuclei. This supports an interpretation of the lowest states of such nuclei as essentially two-quasinucleon states.
The figure also displays the individual contributions to the calculated from , \delta E_{\text{i.n.}}=\sum_{\tau=n,p}\delta E_{\text{i.n.,\tau}}, \delta E_{\text{BCS}}=\sum_{\tau=n,p}\delta E_{\text{BCS,\tau}}, and \delta E_{\text{RPA}}=\sum_{\tau=n,p,np}\delta E_{\text{RPA,\tau}}. The liquid drop contribution is negative except for with an average about MeV. The contribution from the independent-nucleon shell correction fluctuates wildly as a function of or . These fluctuations are reduced by the pairing, which also renders the total mostly positive in accordance with the data. Very low and, for odd , even negative values are calculated, however, for and and for , not the least induced by anomalously low contributions of . These low contributions, as well as one at , are correlated with G_{\text{cr,n}} or G_{\text{cr,p}} being close to for odd and , respectively, so that the accuracy of the RPA is uncertain, cf. Sec. VI. The measured odd-even mass difference actually decreases when or is approached from below, but this decrease is much exaggerated in the calculation.
The RPA contribution is positive for all odd except and 49 and for all odd except . In the upper sd shell it gives almost the entire for odd and about half of it for odd . For odd the RPA contribution is negative, and both for odd and for odd it is numerically smaller in the heavier than in the lighter nuclei.
These differences in the size and sign of the RPA contribution may be understood qualitatively from the expression (38). Thus for , which holds by Eq. (39) for and approximately for and , Eqs. (36)–(38) give
[TABLE]
with
[TABLE]
This function is displayed in Fig. 3. The contribution of to stems mainly from the microscopic term . In fact, because the counter term is a smooth function of , , and deformation, with no distinction between even and odd , its contribution is small. Consider the case of odd . The difference between E_{\text{RPA,n\tau}} for odd and even is roughly a result of the effective dilution in the odd case of the single-neutron spectrum by the blocking of the Fermi level. The impact on E_{\text{RPA,n\tau}} of this decrease of level density near the Fermi level is similar to the impact on \tilde{E}_{\text{RPA,n\tau}} of a decrease of . By Eqs. (28) and (36) the latter increases and thus gives rise to an increase of \tilde{E}_{\text{RPA,n\tau}} proportional to with a positive coefficient. The case of odd is analogous. The calculated decrease from about 3.8 for to about 2.6 for . Thus in the lighter nuclei we have and accordingly expect a large positive RPA contribution to , while in the heavier nuclei we have and accordingly expect a small contribution, which can take either sign.
Also shown in Fig 2 are the calculated gap parameters for both the odd- nucleus and its doubly even neighbors. It is seen that often in the lighter nuclei, , most often for odd . The BCS approximation to is seen to follow roughly the fluctuating gap parameters as a function of or .
IV Neighborhood of the isotopes
In the neighborhood of the Sn isotopic chain we consider all nuclei with and even in the interval and all Sn isotopes with odd in the interval . In Eq. (9), we keep the exponent which resulted from the analysis of data for , cf. Eq. (43), but adjust and so as to reproduce the average of the measured separately for odd and odd . The result is
[TABLE]
For 100Sn, Eq. (43) gives MeV for all , while Eq. (46) gives MeV for all . We thus have two determinations of the pair coupling constant in 100Sn, the higher one 24% greater than the lower one. They result from extrapolation from different directions in the chart of nuclei, one from the line and one from the neighborhood of the Sn isotopic chain. Because the data in the fit (43) include extrapolated masses and interpretations of incomplete spectra of 82Nb and 86Tc, the lower value is likely to be most reliable.
Figure 4 illustrates the need of both the nonzero and the smaller . The quantities plotted in the upper left, upper right, and lower right panels are the total calculated shell correction and its empirical counterpart , where is the measured binding energy. They are displayed for the doubly even Sn isotopes as functions of . Different sets of liquid drop parameters give rise to a difference of between the panels. In the upper left panel, the pairing parameters are inherited from the region, cf. Eq. (43). They describe fairly well the empirical binding energies near the shell closure but not at all near the shell closure. Because the Sn isotopes have constant proton configuration, the that most significantly influences the isotopic variation is . When is positive, decreases more with increasing than by the factor . The upper right panel shows the result when MeV is kept—so that Eq. (43) would be retained for —but is set to 0.0170. Now is equally well described at both shell closures, but the empirical is seen in the lower left panel to be vastly overestimated. The top panel of of Fig. 5 shows that this discrepancy is eliminated when is reduced to MeV. As seen from the lower right panel of Fig. 4 this also improves the reproduction of the measured doubly even binding energies near both shell closures.
We notice in passing that, in particular, a discontinuity of the measured two-neutron separation energy at is reproduced. Togashi* et al.* Togashi et al. (2018) describe this discontinuity as a second order phase transition. In our calculations it is correlated with an onset of oblate deformation at the entrance at of the highly degenerate shell, cf. the appendix. This concurs with a finding of Togashi* et al.*, based on an analysis of the result of a large-scale shell model calculation, that these nuclei have oblate deformations. In the upper panels of Fig. 4, the plots of behave differently at . Pairing thus contributes to the formation of the discontinuity in our calculations.
Also shown in Fig. 4 is the neutron-proton RPA energy E_{\text{RPA,np}}(N,50). It increases with increasing neutron excess because the products in Eq. (31) decrease with increasing distance between and . It is seen, however, that in 142Sn with almost twice as many neutrons as protons, it is only reduced numerically to about two thirds of its value in the nucleus 100Sn.
Figure 5 shows the measured and calculated odd-even mass differences and the decompositions of the latter. The RPA contribution to the calculated is positive with few exceptions. On average it makes up 7, 31 and 14 per cent of the total for the odd- isotopes of Sn, In and Sb. This dominantly positive sign is qualitatively consistent with the values of . For they are approximately equal, about 3.3, and they decrease slightly to about 3.2 for . When increases further, increases to about 4.0 while and continue decreasing to about 2.7 and 3.0, respectively. That the of 100Sn are larger here than in the calculation discussed in Sec. III is due to the smaller .
Except for the largest we get when is magic or magic 1. These are the cases when the Fermi level lies within the magic gap in the single-nucleon spectrum. Otherwise . The emergence of in 90Sn, 92Sn, and 92Sb reflects that G_{\text{cr,p}} is close to for the heaviest isotopes of In, Sn, and Sb. This is correlated with low RPA contributions to the calculated in the isotopes of In and Sb with and .
V region
In the region around 102Zr we consider all doubly even and odd- nuclei with and . As in the Sn region, we keep the exponent from Eq. (43) but adjust and in Eq. (9) so as to reproduce the average of the measured separately for odd and odd . The result is
[TABLE]
Thus is practically the same as in the Sn region, cf. Eq (46), but is significantly smaller.
The measured and calculated odd-even mass differences are compared and the decompositions of the latter shown in Fig. 6. The sign of the RPA contribution varies with a slight predominance of the positive sign, which occurs in 8 out of 12 cases. This is consistent with the values of , which are and . On average the RPA contribution makes up 6% of the total calculated .
The gap parameters are almost constant with averages about 1.1 MeV for even and and 0.8 MeV for odd . The latter is close to the average of the calculated .
VI Interpolation
We mentioned that the RPA energies E_{\text{RPA,\tau\tau^{\prime}}} are interpolated across intervals of about the thresholds G_{\text{cr,\tau}} of BCS pairing to avoid the singularities there. The interpolating function is the polynomial of third degree in which joins the calculated values smoothly at the interval endpoints. Interpolation is done for and for and . In terms of the interpolation width mentioned in Sec. II.1, the interval is G_{\text{min,\tau\tau^{\prime}}}<G_{\tau\tau^{\prime}}<G_{\text{max,\tau\tau^{\prime}}} with
[TABLE]
If G_{\text{max,\tau\tau^{\prime}}}=0 no interpolation is done.
For even the threshold G_{\text{cr,\tau}} increases with increasing . It is therefore particularly large when is magic. As a result both G_{\text{cr,\tau}} are close to the common value of , , and in the doubly magic nuclei 56Ni and 100Sn. For 100Sn, Fig. 7 shows the energy given by Eq. (6) as a function of upon interpolation with different . A figure for 56Ni is very similar. In this calculation we used the levels for both neutrons and protons so that G_{\text{cr,n}}=G_{\text{cr,p}}:=G_{\text{cr}}. It is seen that the choice of can make a difference of 1–2 MeV in when is close to .
In Refs. Bentley et al. (2014); Neergård (2017), was chosen. This choice was based on a comparison with a result of diagonalization of the Hamiltonian (7) in a small valence space Bentley and Frauendorf (2013). Also Fig. 1 seems to suggest a fairly large interpolation interval. In the latter calculation, however, the Hamiltonian is given by Eq. (1), not Eq. (7). Probably more importantly, the single-nucleon levels are equidistant. The behavior of the exact energy may be different when the Fermi level lies in a gap in the single nucleon spectrum. In an early study, Feldman indeed observed an approach of the exact result for the lowest excitation energy to that of the RPA with increasing degeneracies of two separate shells the lower of which is closed for Högaasen-Feldman (1961). There is no way of determining the which best approximates the exact minimum of any such Hamiltonian other than calibrating the interpolation against an exact calculation, which is beyond our capacity. Dukelsky* et al.* calculated the exact lowest energies for isospin , 1 and 2 given by the Hamiltonian (7) in the limit of isobaric invariance as functions of for the single nucleus 64Ge with a different valence space and different single-nucleon energies Dukelsky et al. (2006), and even in this elaborate calculation the dimension of the valence space (pf shell plus subshell) is little greater than half of ours for 56Ni.
With the large employed in Refs. Bentley et al. (2014); Neergård (2017), quite a few calculated binding energies depend on this parameter. This is unsatisfactory because the choice of is largely arbitrary. We prefer to trust the actual RPA energies unless there is a clear reason not to do so. Such a reason is given by the observation that the exact minimum of the Hamiltonian (7) must decrease as a function of because the interaction is negative definite. As shown in Fig. 7, for the interpolated of 100Sn to similarly decrease as a function of it is necessary that . The same approximate limit results for 56Ni. Therefore was used in the present calculations.
This diminishing of relative to the calculations in Refs. Bentley et al. (2014); Neergård (2017) has implications for the calculated’Wigner ’, defined by Bentley and Frauendorf (2013)
[TABLE]
for a constant and when and when . Here, besides , also and are constants. The value of is the one that results from the fit of liquid drop parameters described in Sec. II.1. As a function of the empirical has local maxima at the mass numbers of the doubly magic nuclei 40Ca, 56Ni, and 100Sn. This is seen in Fig. 8 (and also in the plots of in Refs. Bentley et al. (2014); Neergård (2017), which resemble the bottom panel in Fig. 8 in this respect) to be reproduced with but not with . The small is similarly decisive for the sharpness of the calculated shell correction minimum at 100Sn in the lower right panel of Fig. 4. These successes of the small in reproducing qualitative features of the patterns of binding energies near closed shells should evidently not be seen as a proof that it best approximates the exact minimum of the Hamiltonian (7).
VII Summary
The random-phase-approximation-amended (RPA-amended) Nilsson-Strutinskij method of calculating nuclear binding energies was reviewed in the form it has taken after modifications in the preceding literature and in our present work. It was then applied in a study of odd-mass nuclei. Three sets of such nuclei were considered. In terms of the numbers and of neutrons and protons and the mass number they are: (i) The sequence of nuclei with and ; (ii) the odd- isotopes of In, Sn, and Sb with ; (iii) the odd- isotopes of Sr, Y, Zr, Nb, and Mo with . An RPA based part of the total shell correction which accounts for the pair-vibrational correlation energy was found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper sd shell it thus gives almost the entire odd-even mass differences for odd and about half of it for odd . In the heavier part of the set (i) it is less significant and the contribution is negative for odd . In the sets (ii) and (iii) it is dominantly positive and makes up 6–31% of the total calculated odd-even mass difference in various cases. These differences were explained qualitatively in terms of a closed expression for a smooth RPA counter term.
The coupling constants , , and of neutron, proton and neutron-proton pairing interactions were expressed by Eq. (9) in terms of parameters , , and , which were set independently for regions of the chart of nuclei each containing one of the sets (i)–(iii) of odd- nuclei. In region (i), following previous studies of even- nuclei in this region, we took and adjusted and to data on doubly odd nuclei with . Remarkably, the resulting parameters reproduce the typical size of the odd-even mass difference. In the regions (ii) and (iii) the parameters and were fit directly to the odd-even mass differences with kept from region (i). Essentially the same but different resulted. The value of derived from the data on doubly odd nuclei is 24% greater than the one derived from odd-even mass differences in the regions (ii) and (iii). As a result we got for 100Sn, which belongs to both regions (i) and (ii), two values of the common value of , , and differing by these 24%. It was suggested that this difference be due to uncertainty of a part of the data on doubly odd nuclei.
An investigation of the binding energies of the Sn isotopes with even showed that our model reproduces a discontinuity of the two-neutron separation energy at discussed recently by Togashi* et al.* Togashi et al. (2018). Like in their analysis of results of a large-scale shell-model calculation, it is associated in our calculation with an onset of oblate deformations at the entrance of the neutron shell. Pairing was found to contribute to the formation of the discontinuity.
The RPA neutron-proton pair-vibrational correlation energy is expected to decrease numerically with increasing neutron excess due to an increasing mismatch of the occupations of single-neutron and single-proton levels. In 142Sn, which has almost twice as many neutrons as protons, it was found to be reduced anyway only to about two thirds of its value in the nucleus 100Sn.
The RPA-amended Nilsson-Strutinskij method involves an interpolation of RPA energy terms across the thresholds of the pair coupling constants for Bardeen-Cooper-Schrieffer pairing in the neutron or proton system. Arguments were given for choosing the interpolation interval substantially smaller than in previous applications of the method, and such a smaller width was applied in our present calculations. As a side effect, diminishing the width of the interpolation interval resulted in an improved qualitative correspondence between the variations with of the measured and calculated ‘Wigner ’.
Acknowledgements.
We would like to thank Stefan Frauendorf for providing access to the tac code that was used to calculate the deformations shown in the appendix and the corresponding single-nucleon levels used in this work.
Appendix A Deformations
Tables A shows the deformations used in the calculations. For odd these are the deformations assumed for the lowest states with .
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