Role of pair-vibrational correlations in forming the odd-even mass difference

In the random phase approximation (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 $N$ and $Z$ of neutrons and protons and even mass number $A = N + Z$. Here it is applied to odd-$A$ nuclei. Three sets of such nuclei are considered: (i) The sequence of nuclei with $Z = N - 1$ and $25 \le A \le 99$. (ii) The odd-$A$ isotopes of In, Sn, and Sb with $46 \le N \le 92$. (iii) The odd-$A$ isotopes of Sr, Y, Zr, Nb, and Mo with $60 \le N \le 64$. The RPA correction is found to contribute significantly to the calculated odd-even mass differences, particularly in the light nuclei. In the upper \textit{sd} shell this correction accounts for almost the entire odd-even mass difference for odd $Z$ and about half of it for odd $N$. 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 meanfield approximations. The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [1], which was applied extensively to the description of pairing in nuclei since its adaption to the nuclear system by Bohr, Mottelson, and Pines [2], Bogolyubov [3], and Solov'yov [4], 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. [5] 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 Here a k annihilates a fermion in a member |k 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 ǫ k = ǫ k and the coupling constant G 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 [6]. 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) [7]. This approximation is seen to give a good agreement with the exact value. Appreciable deviations only occur in a narrow interval of G about the threshold G cr of BCS pairing. Because the RPA equations derived from the Hamiltonian (1) describe oscillations of the pair field P about the mean field equilibrium, the correlations may thus be seen as mainly pair vibrational.  [5]. The exact minimum E of the Hamiltonian (1), normalized to zero for G = 0, is shown as a function of G in comparison with the approximations BCS and BCS+RPA. The single-fermion space accommodates 32 equidistant doublet levels ǫ k = ǫ k spaced by 1/g and is inhabited by n = 32 fermions. The expectation value −Gn/2 of the pairing interaction in the G = 0 ground state is subtracted from the exact and RPA energies. The threshold Gcr of BCS pairing is indicated. We turned the figure upside down to display energy rather than binding energy.
Calculations of binding energies by the Strutinskij method [8] 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 G < G cr , 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 with a pair field isovector Here t = (t x , t y , t z ) is the single-nucleon isospin, and time reversal is assumed to commute with t x and t z and anticommute with t y . In Eq. (3) the set k or l of quantum numbers includes an eigenvalue of t z , and the span of the orthonormal set of states |k is isobarically invariant. The interaction (2) contains a neutron-proton term −GP † z P z . In a doubly even nucleus the Hartree-Bogolyubov quasinucleon vacuum derived from the resulting Hamiltonian has P z = 0 [9], 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 [10]. Minor modifications of the scheme of calculations proposed in Ref. [10] were discussed by Neergård [11,12]. These articles deal with nuclei with N ≈ Z and even A, where N and Z are the numbers of neutrons and protons and A = N + Z. The extended Nilsson-Strutinskij scheme was found to account, with suitably chosen parameters, quite well for the pattern of even-A binding energies and certain excitation energies in doubly odd nuclei in this region. We here apply it to odd-A 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 N ≈ Z region, previously studied with respect to the even-A nuclei. (ii) A neighborhood of the Sn isotopic chain. (iii) A region of well deformed, neutron rich nuclei around 102 Zr.
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 RPAamended Nilsson-Strutinskij method in the form it has taken after several modifications since the publication of Ref. [10]. 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 −E(N, Z) is calculated by where 'i.n.' stands for 'independent nucleons'. Here E LD is a liquid drop energy, and each term δE x has the form with a 'smooth' counter termẼ x . The 'microscopic' energy approximates the minimum of the Hamiltonian where P n = 1 2 2Ωn k=1 a kn a kn , a kp a kp , (a kp a kn + a kn a kp ).
Here, unlike in Eq. (3), the index k numbers, for each τ = n for neutrons and τ = p for protons, an orthonormal set of eigenstates |kτ of a time-reversal invariant single-nucleon Hamiltonian h τ in an order of nondecreasing eigenvalue ǫ kτ . The numbering should be such that |kp = t − |kn in the limit h p = h n . In this limit then P n = −P − / √ 2, P p = P + / √ 2, and P np = P z in terms of components of the isovector (3) provided also all Ω τ are equal. Again the set of states |kτ is supposed to be preserved under time reversal up to phases. We also assume that each pair of an odd and the following even k 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 m and pairs of an odd and the following even k refer to pairs of states with opposite m.
Unlike Ref. [10] strict isobaric invariance is not imposed on the microscopic model. The single-nucleon Hamiltonians h n and h p may be different, and different valence space dimension 2Ω τ may be employed for different τ . We use throughout Ω n = N , Ω p = Z, and Ω np = ⌈A/2⌉ so that the neutron and proton valence spaces are always half filled and Ω np ≈ (Ω n + Ω p )/2. These modifications, which where introduced partly in Refs. [11,12], renders the model better suited nuclei with a large neutron or proton excess.
We also allow different coupling constants G τ for different τ . Writing G n = G 1 , G p = G −1 , and G np = G 0 so that the subscript M ′ T is the isomagnetic quantum number of the interacting pair, we set where M T = (N −Z)/2 is the isomagnetic quantum number of the nucleus. The parameters G, ζ, and α are set separately for each region (i)-(iii). The limit where h p = h n , all G τ are equal, and all Ω τ 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 [13]. It is expressed by the Nilsson parameters ǫ 2 , γ, and ǫ 4 [14,15]. The deformations are listed in the appendix.

A. Liquid drop energy
The liquid drop energy is written where the coefficients a are parameters. The deformation dependent factors B s and B c are calculated from the Nilsson parameters in two steps. First, following Seeger and Howard [16], we determine the coefficients α lm in the equations in spherical coordinates (r, θ, φ) of the surfaces of constant second term in the expression (16) below, where P m l (x) is the Legendre function of the first kind as defined by Edmonds [17]. With ǫ 20 = ǫ 2 cos γ and ǫ 22 = (−ǫ 2 sin γ)/ √ 2, the nonzero coefficients with l ≤ 4 are given to second order in ǫ 2 and ǫ 4 by This approximation is adopted. (For ǫ 22 = 0 the expansion (12) (including results for l > 4 which we do not show) should give Eqs. (10)-(13) of Ref. [16]. Some coefficients there differ from ours, which were derived by computer algebra.) The coefficients with l > 4 are not required in the second step, where B s and B c are expanded in the α's. This expansion can be derived from Swiatecki's results in Ref. [18]. Swiatecki's expansion is restricted to γ = 0, but when only terms of total rank 8 or less are retained, each term has a unique continuation into γ = 0 given by the requirement that it be a scalar polynomial in the For given pairing parameters G, ζ, α and an RPA interpolation width w defined in Sec. VI we fix the coefficients a x 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 a x is done before the pairing parameters are fit to other data. Table I shows the results for the optimal pairing parameters. For the 102 Zr region the sample of doubly even nuclei consists of only 9 nuclei.

B. Independent nucleons
The terms E i.n.,τ in Eq. (6) are given by with N τ = N for τ = n and N τ = Z for τ = p. The single-nucleon energies ǫ kτ are the eigenvalues of the Nilsson Hamiltonian [14,15,19], where r = (x 1 , x 2 , x 3 ) and p are the spatial coordinates and momentum, s is the spin, and M τ is the nucleon mass. The function P l (x) is the Legendre polynomial. The oscillator frequencies ω q are given by where ω 0 satisfies the condition of volume conservation The 'stretched' spherical coordinates (ρ, θ t , φ t ) and orbital angular momentum l t [19] correspond to Cartesian coordinates and N sh is the number of oscillator quanta. For the parameters κ N sh ,τ and µ N sh ,τ we adopt the values recommended in Ref. [20]. The independent-nucleon counter terms arẽ where the smooth chemical potentialλ τ is defined by and the smooth level densityg τ (ǫ) is given by [8,21] in terms of the generalized Laguerre polynomial L(n, a, x). We use smoothing width γ Str = • ω and smoothing order m Str = 3 and include in the sum in Eq. (22) all such k that ǫ kτ < 47.5 MeV + 5 γ Str and N sh ≤ 9.

C. BCS
The terms E BCS,τ are given by the standard BCS theory. A derivation of the following equations is found, for example in Ref. [9]. For even N τ one has with Here λ τ and ∆ τ obey These equations always have a solution with ∆ τ = 0 and there is a threshold G cr,τ such that no other ∆ τ is possible for G ≤ G cr,τ . For G > G cr,τ there is a solution with ∆ τ > 0 and a lower E BCS,τ , which is chosen. If ǫ (Nτ +2)τ > ǫ Nτ τ then G cr,τ > 0 and G cr,τ is given by If ǫ (Nτ +2)τ = ǫ Nτ τ , as happens in spherical nuclei when a j shell is partly occupied in the absence of pairing, then Nτ τ is assumed to be present in the BCS ground state. The orbit |N τ τ is then fully occupied and its time reverse |(N τ + 1)τ fully empty. The BCS energy E BCS,τ is calculated as if N τ −1 nucleons of type τ inhabited the remaining orbits. The odd nucleon is said to block the Fermi level.
To simplify notation we letg τ without an argument meang τ (λ τ ) and write The BCS counter terms are then given by [11,22]

D. RPA
The calculation of E RPA,τ is based on the theory in Ref. [9]. 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 P τ may be labeled by the odd single-nucleon indices k from 1 to 2Ω τ − 1. When, say, N is odd, the blocking of the Fermi level inhibits both a N n a (N +1)n and its Hermitian conjugate from exciting the BCS ground state. The term with k = N may then be omitted for τ = n. If N = Z this also holds for τ = np. For simplicity we omit in general k = N for τ = n or np when just N is odd. We thus assume that, like in the BCS approximation, the odd neutron stays in the orbit |N n as a spectator to interactions among the other nucleons. The procedure is analogous when Z is odd. The remaining set of k is denoted by S τ .
It is convenient to introduce at this point labels τ τ ′ = nn, pp, np alternative to and synomynous with τ = n, p, np and vectors and matrices with components or element indexed by the set S τ τ ′ . A diagonal matrix E τ τ ′ is defined by its elements and column vectors U τ τ ′ and V τ τ ′ by their components Let Then where z τ τ ′ ,k are the eigenvalues of The terms √ z τ τ ′ ,k are the RPA frequencies. For τ = τ ′ and, in the limit of isobaric invariance, for τ τ ′ = np and N = Z, one RPA mode is, for G τ τ ′ > G cr,τ τ ′ (with G cr,np = G cr,n = G cr,p in the isobarically invariant limit), a Nambu-Goldstone mode with zero frequency [9]. That is, in this degree of freedom vibration turns into rotation. This is what gives rise to the singularity at G = G cr in Fig. 1 [10]. To circumvent this singularity we interpolate the calculated E RPA,τ τ ′ across the region of G τ τ ′ = G cr,τ τ ′ for τ = τ ′ or τ τ ′ = np and N = Z with G cr,np ≈ G cr,n ≈ G cr,p in the latter case. Details are given in Sec. VI.

E. Isobaric analogs
The scheme presented so far describes states with isospin T ≈ |M T |. This relation is satisfied empirically by nearly all ground states. The exception is that for odd N = Z > 20 most ground states have T ≈ 1 while the lowest states with T ≈ 0 are excited. For odd N = Z < 20 the lowest states with T ≈ 1 are mostly excited. We denote the energies of these T ≈ 1 states by E * (N, Z) to distinguish them from the energies of the T ≈ 0 states. For odd N = Z the T ≈ 1 states are the isobaric analogs of the ground states of the doubly even nuclei with neutron and proton numbers (N ′ , Z ′ ) = (N + 1, Z − 1). Accordingly we set where B c is calculated from the deformation of the doubly even nucleus.

III. N ≈ Z REGION
Our calculations for even A in the N ≈ Z region follow the scheme previously applied in Refs. [10,12]. Again we consider the doubly even nuclei with 24 ≤ A ≤ 100 and 0 ≤ N − Z ≤ 10 and the doubly odd ones with 26 ≤ A ≤ 98 and N = Z. Unlike Ref. [12] we use different Ω τ for different τ 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 N sh ≤ 9 being included in the calculation by the scheme of Ref. [13] instead of just four shells close to the neutron or proton Fermi level for τ = n and p, respectively. For the doubly even nuclei this only changed the deformations of 84 Zr and 86 Mo, which went from spherical to oblate. For the T ≈ 0 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 α = 0 in Eq. (9) so that one pair coupling constant G covers the cases τ = n, p, and np. The parameters G and ζ are fit to the following data for odd N = Z.
(1) The T ≈ 0 doubly odd-doubly even mass differences N +1)]. (40) (2) The differences of the lowest energies for T ≈ 1 and T ≈ 0, that is, The set of data is the same as in Refs. [10,12] and thus includes extrapolated masses of 82 Nb and 86 Tc, but all mass data were updated from AME12 [23] to AME16 [24]. Again excitation energies are taken from the Evaluated Nuclear Structure Data File [25]. A leastsquare fit gives with an rms deviation of 0.789 MeV. Plotting the T ≈ 0 doubly odd-doubly even mass differences, the T ≈ 0 to T ≈ 1 energy splittings, the symmetry energy coefficients, and the 'Wigner x' as functions of A results in figures grossly similar to Figs. 6-9 of Refs. [10] and Fig.  1 of Ref. [12]. As to the Wigner x, more detail is given in Sec. VI.
With the parameters thus set we consider the odd-A nuclei with Z = N − 1 and 25 ≤ A ≤ 99. The odd-even mass difference ∆ oe (N, Z) is defined as the mass of the odd-A nucleus relative to the average mass its two doubly even neighbors. The calculated ∆ oe (N, Z) 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 G and ζ were fit, not to these data but to energies in doubly odd nuclei. This supports an interpretation of the lowest T ≈ 0 states of such nuclei as essentially two-quasinucleon states.
The figure also displays the individual contributions to the calculated ∆ oe (N, Z) from E LD , δE i.n. = τ =n,p δE i.n.,τ , δE BCS = τ =n,p δE BCS,τ , and δE RPA = τ =n,p,np δE RPA,τ . The liquid drop contribution is negative except for N = 43 with an average about −0.4. The contribution from the independentnucleon shell correction δE i.n. fluctuates wildly as a function of N or Z. These fluctuations are reduced by the pairing, which also renders the total ∆ oe (N, Z) mostly positive in accordance with the data. Very low and, for odd N , even negative values are calculated, however, for N and Z = 25 and for N = 49, not the least induced by anomalously low contributions of δE RPA . These low contributions, as well as one at Z = 49, are correlated with G cr,n or G cr,p being close to G for odd N and Z, respectively, so that the accuracy of the RPA is uncertain, cf. Sec. VI. The measured odd-even mass difference actually decreases when N or Z = 25 is approached from below, but this decrease is much exaggerated in the calculation.
The RPA contribution is positive for all odd Z except Z = 25 and 49 and for all odd N < 30 except N = 25. In the upper sd shell it gives almost the entire ∆ oe (N, Z) for odd Z and about half of it for odd N . For odd N > 30 the RPA contribution is negative, and both for odd N and for odd Z 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 (37). Thus for l τ τ ′ = 0, which holds by Eq. (38) for τ = τ ′ and approximately for τ τ ′ = np and N ≈ Z, Eqs. (35)-(37) givẽ This function is displayed in Fig. 3. The contribution of δE RPA to ∆ oe (N, Z) stems mainly from the microscopic term E RPA . In fact, because the counter termẼ RPA is a smooth function of N , Z, and deformation, with no distinction between even and odd N τ , its contribution is small. Consider the case of odd N . The difference between E RPA,nτ for odd and even N 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 RPA,nτ of this decrease of level density near the Fermi level is similar to the impact onẼ RPA,nτ of a decrease of g nτ . By Eqs. (27) and (35) the latter increases χ nτ and thus gives rise to an increase ofẼ RPA,nτ proportional to f ′ (χ nτ ) with a positive coefficient. The case of odd Z is analogous. The calculated χ τ τ ′ decrease from about 3.8 for A = 24 to about 2.6 for A = 100. Thus in the lighter nuclei we have f ′ (χ τ τ ′ ) > 0 and accordingly expect a large positive RPA contribution to ∆ oe (N, Z), while in the heavier nuclei we have f ′ (χ τ τ ′ ) ≈ 0 and accordingly expect a small contribution, which can take either sign. Also show in Fig 2 are the calculated gap parameters ∆ τ for both the odd-A nucleus and its doubly even neighbors. It is seen that often in the lighter nuclei, ∆ τ = 0, most often for odd A. The BCS approximation to ∆ oe (N, Z) is seen to follow roughly the fluctuating gap parameters as a function of N or Z.

IV. NEIGHBORHOOD OF THE Sn ISOTOPES
In the neighborhood of the Sn isotopic chain we consider all nuclei with 48 ≤ Z ≤ 52 and even N in the    Fig. 5 shows that this discrepancy is eliminated when G is reduced to 5.818 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 N = 66 is reproduced. Togashi et al. [26] 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 N = 68 of the highly degenerate 1h 11/2 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 δE behave differently at N ≈ 66. Pairing thus contributes to the formation of the discontinuity in our calculations.
Also shown in Fig. 4 is the neutron-proton RPA energy E RPA,np (N, 50). It increases with increasing neutron excess because the products in Eq. (30) decrease with increasing distance between λ n and λ p . It is seen, however, that in 142 Sn with almost twice as many neutrons as protons, it is only reduced numerically to about two thirds of its value in the N = Z nucleus 100 Sn. Figure 5 shows the measured and calculated odd-even mass differences and the decompositions of the latter. slightly to about 3.2 for N = 54. When N increases further, χ n increases to about 4.0 while χ n and χ np continue decreasing to about 2.7 and 3.0, respectively. That the χ τ τ ′ of 100 Sn are larger here than in the calculation discussed in Sec. III is due to the smaller G. Except for the largest N we get ∆ τ = 0 when N τ is magic or magic ± 1. These are the cases when the Fermi level lies within the magic gap in the single-nucleon spectrum. Otherwise ∆ τ > 0. The emergence of ∆ p > 0 in 90 Sn, 92 Sn, and 92 Sb reflects that G cr,p is close to G p for the heaviest isotopes of In, Sn, and Sb. This is correlated with low RPA contributions to the calculated ∆ oe (N, Z) in the isotopes of In and Sb with N = 90 and 92.

V. 102 Zr REGION
In the region around 102 Zr we consider all doubly even and odd-A nuclei with 60 ≤ N ≤ 64 and 38 ≤ Z ≤ 42. As in the Sn region, we keep the A exponent ζ = −0.7461 from Eq. (42) but adjust G and α in Eq. (9) so as to reproduce the average of the measured ∆ oe (N, Z) separately for odd N and odd Z. The result is Thus G is practically the same as in the Sn region, cf. Eq (45), but α 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 χ τ τ ′ , which are χ n ≈ 3.4 and χ p ≈ χ np ≈ 3.2. On average the RPA contribution makes up 6% of the total calculated ∆ oe (N, Z).
The gap parameters ∆ τ are almost constant, equal to about 1 MeV for even N and Z and about 13% less for odd A. The odd-A gap parameter is often a good approximation to the total calculated ∆ oe (N, Z).

VI. INTERPOLATION
We mentioned that the RPA energies E RPA,τ τ ′ are interpolated across intervals of G τ τ ′ about the thresholds G cr,τ of BCS pairing so as to avoid the singularities there. The interpolating function is the polynomial of third degree in G τ τ ′ which joins the calculated values smoothly at the interval end points. Interpolation is done for τ = τ ′ and for τ τ ′ = np and N = Z. In terms of the interpolation width w mentioned in Sec. II A, the interval is G min,τ τ ′ < G τ τ ′ < G max,τ τ ′ with G min,τ τ ′ = (1 − w) min(G cr,τ , G cr,τ ′ ), If G max,τ τ ′ = 0 no interpolation is done. For even N τ the threshold G cr,τ increases with increasing ǫ (Nτ +2)τ −ǫ Nτ τ . It is therefore particularly large when N τ is magic. As a result both G cr,τ are close to the common value G of G n , G p , and G np in the doubly magic nuclei 56 Ni and 100 Sn. For 100 Sn, Fig. 7 shows the energy E mic given by Eq. (6) as a function of G upon interpolation with different w. A figure for 56 Ni is very similar. In this calculation we used the levels (ǫ kn + ǫ kp )/2 for both neutrons and protons so that G cr,n = G cr,p := G cr . It is seen that the choice of w can make a difference of 1-2 MeV in E mic then G cr is close to G.
In Refs. [10,12], w = 0.5 was chosen. This choice was based on a comparison with a result of diagonalization of the Hamiltonian (7) in a small valence space [27]. 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 G = 0 [28]. There is no way of determining the w 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 T = 0, 1 and 2 given by the Hamiltonian (7) in the limit of isobaric invariance as functions of G for the single nucleus 64 Ge with a different valence space and different single-nucleon energies [29], and even in this elaborate calculation the dimension of the valence space (pf shell plus 1g 9/2 subshell) is little greater than half of ours for 56 Ni.
With the large w employed in Refs. [10,12], quite a few calculated binding energies depend on this parameter. This is unsatisfactory because the choice of w 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 G because the interaction is negative definite. As shown in Fig. 7, for the interpolated E mic of 100 Sn to similarly decrease as a function of G it is necessary that w > ∼ 0.035. The same approximate limit results for 56 Ni. Therefore w = 0.035 was used in the present calculations.
This diminishing of w relative to the calculations in Refs. [10,12] has implications for the calculated 'Wigner x', defined by [27]  Sec. II A. As a function of A the empirical x has local maxima at the mass numbers of the doubly magic nuclei 40 Ca, 56 Ni, and 100 Sn. This is seen in Fig. 8 (and also in the plots of x in Refs. [10,12], which resemble the mid panel in Fig. 8 in this respect) to be reproduced with w = 0.035 but not with w = 0.5. It is also not reproduced for 56 Ni and 100 S, and barely for 40 Ca, when the RPA correction is omitted entirely. The latter observation may be understood from the fact that the BCS approximation gives little or no pairing energy then the gap parameters ∆ τ are small or zero, as happens in the doubly magic nuclei. It thus supports the assumption of significance of the pair-vibrational correlations in the formation of the total binding energy. The small w is similarly decisive for the sharpness of the calculated shell correction minimum at 100 Sn in the lower right panel of Fig. 4. These successes of the small w 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).
In the calculation without the RPA correction the pairing parameters were optimized separately (with G τ τ ′ := G as in Sec. III), giving G = 47.36A

VII. SUMMARY
The random phase approximation (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 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 N and Z of neutrons and protons and the mass number A = N + Z they are: (i) The sequence of nuclei with Z = N − 1 and 25 ≤ A ≤ 99. (ii) The odd-A isotopes of In, Sn, and Sb with 46 ≤ N ≤ 92. (iii) The odd-A isotopes of Sr, Y, Zr, Nb, and Mo with 60 ≤ N ≤ 64. 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 Z and about half of it for odd N . In the heavier part of the set (i) it is less significant and the contribution is negative for odd N > 30. 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 G n , G p , and G np of neutron, proton and neutron-proton pairing interactions were expressed by Eq. (9) in terms of parameters G, ζ, and α, which were set independently for regions of the chart of nuclei each containing one of the sets (i)-(iii) of odd-A nuclei. In region (i), following previous studies of even-A nuclei in this region, we took α = 0 and adjusted G and ζ to data on doubly odd nuclei with N = Z. Remarkably, the resulting parameters reproduce the typical size of the odd-even mass difference. In the regions (ii) and (iii) the parameters G and α were fit directly to the odd-even mass differences with ζ kept from region (i). Essentially the same G but different α resulted. The value of G derived from the data on doubly odd N = Z 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 100 Sn, which belongs to both regions (i) and (ii), two values of the common value of G n , G p , and G np differing by these 24%. It was suggested that this difference be due to uncertainty of a part of the data on doubly odd N = Z nuclei.
An investigation of the binding energies of the Sn isotopes with even N showed that our model reproduces a discontinuity of the two-neutron separation energy at N = 66 discussed recently by Togashi et al. [26]. 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 1h 11/2 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 142 Sn, which has almost twice as many neutrons as protons, it was it was found to be reduced anyway only to about two thirds of its value in the N = Z nucleus 100 Sn.
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 A of the measured and calculated 'Wigner x'.

ACKNOWLEDGMENTS
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: Deformations
Tables II shows the deformations used in the calculations. For odd N = Z these are the deformations assumed for the lowest states with T ≈ 0.