Nuclear masses near N=Z from Nilsson-Strutinsky calculations with pairing corrections beyond BCS from an isospin-conserving pairing force

Nuclear masses near from Nilsson-Strutinsky calculations with pairing corrections beyond BCS from an isospin-conserving pairing force

I. Bentley Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA, Department of Chemistry and Physics, Saint Mary’s College, Notre Dame, Indiana 46556, USA    K. Neergård Fjordtoften 17, 4700 Næstved, Denmark    S. Frauendorf Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA
Abstract

A model with nucleons in a charge-independent potential well interacting by an isovector pairing force is considered. For a 24-dimensional valence space, the Hartree-Bogolyubov (HB) plus random phase approximation (RPA) to the lowest eigenvalue of the Hamiltonian is shown to be accurate except near values of the pairing-force coupling constant where the HB solution shifts from a zero to a nonzero pair gap. In the limit the HB + RPA is asymptotically exact. The inaccuracy of the HB + RPA in the critical regions of can be remedied by interpolation. The resulting algorithm is used to calculate pairing corrections in the framework of a Nilsson-Strutinsky calculation of nuclear masses near for  24–100, where and are the numbers of neutrons and protons, and . The dimension of the valence space is 2 in these calculations. Adjusting five liquid drop parameters and a power law expression for the constant as a function of allows us to reproduce the measured binding energies of 112 doubly even nuclei in this range with a root mean square deviation of 0.95 MeV. Several combinations of the masses for different , , and isospin are considered and the calculations found to be in good agreement with the data. It is demonstrated by examples how fluctuations as a function of of the constant in an expansion of the symmetry energy of the form can be understood from the shell structure.

pacs:
21.10.Hw , 21.10.Dr , 21.60.Jz , 13.75.Cs

I Introduction

Since the late 1990s the masses of nuclei near the line in the chart of nuclides, where and are the numbers of neutrons and protons, have attracted much interest from the nuclear physics community. In particular, the origin of the so-called Wigner energy Myers and Swiatecki (1966), a depression of the mass at relative to a trend described by a symmetry energy quadratic in , has been a matter of debate. For a review of this discussion, see Neergård Neergård (2009). In 1995, Duflo and Zuker Duflo and Zuker (1995) published a semiempirical mass formula with a symmetry energy proportional to , where is the isospin quantum number, in the ground states of doubly even nuclei equal to , where . This expression includes a Wigner energy in a natural manner. It was observed by Frauendorf and Sheikh Frauendorf and Sheikh () that a symmetry energy with the factor resembles the spectrum of a quantal, axially symmetric rotor. These authors identified the nuclear superfluidity as the deformation in isospace that could give rise to collective rotation in this space. The Bardeen-Cooper-Schrieffer (BCS) pair gaps are indeed components of an isovector perpendicular to the isospin. Their magnitude is a measure of the collectivity of the isorotation. Obviously the dependence of the energy on will become more regular as their magnitude increases as a result of the progressively more gradual change of the occupation numbers around the Fermi surface, which will wash out fluctuations of the level density. On the other hand the isorotational moment of inertia, which is determined by the average level density, will not change much. The results of calculations in Ref. Satuła and Wyss (2001) exemplify these generic features.

Neergård Neergård (2002, 2003, 2009) set up a microscopic theory of such a superfluid isorotation based on the Hartree-Bogolyubov (HB) plus random phase approximation (RPA). His Hamiltonian involves independent nucleons in a charge-independent potential well, an isovector pairing force, and an interaction of the nucleonic isospins, which he calls the symmetry force. The latter is shown to contribute merely a term in the total energy proportional to . In the idealized case of equidistant single-nucleon levels, the total symmetry energy is found to be proportional to provided the pairing force is sufficiently strong to produce an HB energy minimum with nonzero BCS gaps. If this condition fails to be satisfied, the model gives in this case a total symmetry energy proportional to with  Neergård (2003). In realistic cases with nonuniform single-nucleon spectra, major modifications of these simple expressions arise from shell effects Neergård (2009).

Recently, Bentley and Frauendorf Bentley and Frauendorf (2013) calculated exactly the lowest eigenvalue of Neergård’s Hamiltonian employing small valence spaces of dimensions 24 or 28. They demonstrated (see their Fig. 7) that for a sufficiently strong pairing interaction the limit of rigid isorotation is approached for various kinds of bunched single-particle level distributions. Unlike Neergård, who keeps the Hamiltonian constant along each isobaric chain, these authors take into account the variation of the nuclear shape with the isospin. In a survey of the range of mass numbers 24–100, they find that their model accounts generally for the fluctuations due to the shell structure observed in several combinations of the masses near taken as functions of : (i) the mass difference of doubly odd and doubly even nuclei, (ii) the difference in excitation energy of the lowest and states in the the doubly odd nuclei, and (iii) the constants and in an expansion of the symmetry energy of the form extracted from doubly even masses close to . However, the model underestimates when the symmetry force constant is fit to the difference of and excitation energies.

The exact results of Bentley and Frauendorf provide a background on which the accuracy of the HB + RPA may be tested. We show in the present study that the HB + RPA gives a very good approximation to the exact lowest eigenvalue of the Hamiltonian except near the values of the pairing force coupling constant where the HB solution shifts from a zero to a non-zero pair gap. We show, as well, that the HB + RPA reproduces the exact eigenvalue asymptotically in the limit . We then devise a recipe for interpolating the HB + RPA energy across the critical region of leading to an algorithm which accurately approximates the exact eigenvalue in the entire range of from zero to infinity.

This algorithm is simple enough to allow calculations with valence spaces of dimension . More specifically, we include in our present calculations all single-nucleon states below the Fermi level and equally many states above this level. To allow actual nuclear masses to be calculated from this schematic model, we add a Strutinsky renormalization. As a side effect, we can then dispense with the symmetry force. Its contribution to the total energy proportional to may thus be considered a part of the macroscopic liquid drop symmetry energy. Our microscopic Hamiltonian thus consists merely of a charge-independent independent-nucleon term and the isovector pairing force.

Using this scheme we calculate once more the combinations of masses near previously considered by Bentley and Frauendorf and show that the issue with the underestimation of the constant is resolved by the enlargement of the valence space. We also show that the present scheme allows to calculate the masses themselves with a small root-mean-square deviation from the measured ones.

The following Secs. IIIV and VI contain the formal presentation of our model. In Sec. V we compare the exact lowest eigenstate of our microscopic Hamiltonian with the HB + RPA and explain our scheme of interpolation. Then, in Sec. VII, we present and analyze our results of calculation. Sec. VIII provides arguments detailing why we disregard isoscalar pair correlations. Finally, in Sec. IX, the article is summarized and some perspectives drawn.

Ii Liquid drop energy

Following Strutinsky (Ref. Brack et al. (1972) and references therein) we assume that the nuclear binding energy for mass number and isospin is given by

(1)

Here is a deformed liquid drop energy, is the sum of occupied single-nucleon levels in a generally deformed potential well, is the pairing energy, and and are “smooth” counterparts of and . For we adopt an expression of the form proposed by Duflo and Zuker Duflo and Zuker (1995),

(2)

where and are functions of the shape. As explained below we need these functions only for axial and reflection symmetry. We then employ the expansions in the Hill-Wheeler deformation parameters and  Hill and Wheeler (1953) given by Swiatecki Swiatecki (1956) with and expressed in turn by the Nilsson deformation parameters and  Nilsson et al. (1969) by means of the expansions given by Seeger and Howard Seeger and Howard (1975). The determination of and is discussed in Sec. III.

The five parameters in Eq. (2) are determined by fitting Eq. (1) to the 112 measured binding energies of doubly even nuclei considered in the present study according to the 2012 Atomic Mass Evaluation Audi et al. (2012) with , , , and calculated as described in Secs. IIIVI. The result is  MeV,  MeV,  MeV,  MeV, and  MeV. These parameters are similar to those obtained by Mendoza-Temis et al. in a global fit of nuclear masses with minus the binding energy given by Eq. (2) plus a phenomenological, negative definite, pairing energy and  Mendoza-Temis et al. (2010). The parameters of Mendoza-Temis et al. cannot be used in the present context because unlike their negative definite pairing energy the sum of the liquid drop deformation energy and the shell and pairing correction terms in Eq. (1) average approximately to zero. Moreover, the semiempirical formula of Mendoza-Temis et al. deviates quite a lot from the empirical binding energies locally in the present region of nuclei. This deviation increases with increasing along the line and amounts to about 20 MeV for . Since our liquid drop parameters are optimized for the present region of and they may not reproduce accurately binding energies in other regions.

Iii Single-particle plus pairing Hamiltonian

The sum in Eq. (1) is calculated as the lowest eigenvalue of the Hamiltonian

(3)

where labels orthogonal quartets of a single-proton and a single-neutron state and their time reversed. The annihilator of a nucleon in one of these states is denoted by , , , or , and

(4)

The single-nucleon energies are derived from a calculation with the Nilsson potential employing the parameters of Bengtsson and Ragnarsson Bengtsson and Ragnarsson (1985). To conserve isospin we take the average of the neutron and proton energies with given ordinal number counted from the bottom of the spectrum. The resulting quartets are labeled by in the order of increasing , and the first of them in this order included in the calculation.

The deformation parameters and are taken from a recent survey of deformations based on the Nilsson-Strutinsky plus BCS theory. The equilibrium deformation of a given nucleus was calculated in this survey by minimizing with respect to Larsson’s triaxial deformation parameters and  Larsson (1973) as well as the Nilsson-Strutinsky plus BCS energy calculated with the tac code Frauendorf (1993). The latter employs an expression for the liquid drop energy similar to Eq. (2) but with symmetry energy terms quadratic in , the Nilsson potential with the parameters of Ref. Bengtsson and Ragnarsson (1985), and a pairing correction without a smooth counterterm calculated from BCS gaps  MeV. In calculations for odd or the Fermi level is blocked. All the 136 nuclei included in our study turn out to have either or , that is, axial symmetry. The deformations are shown in Tables 2 and 3. Also see Note sup ().

The only free parameter remaining is the pairing force coupling constant . A power law for as a function of will be fit to doubly even-doubly odd binding energy differences.

Iv Hartree-Bogolyubov plus Random Phase Approximation

For an introduction to the BCS, HB, and RPA theories we refer to textbooks such as the one by Ring and Schuck Ring and Schuck (1980). The calculation of the lowest eigenvalue of the Hamiltonian (3) in the HB + RPA for even and is discussed by Neergård Neergård (2009). Since this formalism is invariant under isorotation, a nucleus with represents the entire multiplet with . To calculate the energy of the lowest state for odd we reduce this case to the even one by omitting quartet number from a HB + RPA calculation for , and adding .

The HB part of the calculation amounts to the usual BCS theory with quasinucleons annihilators

(5)

For a gap the chemical potential is taken as the limit for of the determined by or for .

The RPA part splits into separate equations for an “ space” spanned by and their Hermitian conjugates, a “ space” spanned by and their Hermitian conjugates, and an “ space” spanned by and their Hermitian conjugates. The resulting groundstate energy can be written as with and

(6)

Here and are the neutron and proton BCS gaps and

(7)

where denotes an RPA frequency and are the single-quasinucleon energies. (A term given by Eq. (35) of Ref. Neergård (2009) vanishes in the present case of .)

For nuclei with , the BCS solutions are equal for neutrons and protons and the , , and spaces have equal RPA spectra so that . If the lowest RPA frequency in the space vanishes because the Hamiltonian (3) commutes with , and the analogon of this statement holds for protons. For the lowest frequency in the space is equal to the difference of the neutron and proton chemical potentials because the Hamiltonian commutes with the components of the isospin perpendicular to the direction.

V Comparison with the exact lowest eigenvalue of the Hamiltonian

Bentley and Frauendorf calculated exactly the lowest eigenvalue of the Hamiltonian (3) in spaces with six or seven quartets Bentley and Frauendorf (2013). This allows a comparison of the HB + RPA to an exact calculation. We made this comparison in all the cases displayed in Fig. 7 of Ref. Bentley and Frauendorf (2013). The case shown in Fig. 1

Figure 1: (Color online) The energy calculated in the HB + RPA and exactly for 12-fold degenerate single-nucleon levels 0 and occupied for from the bottom by six neutrons and six protons.

is the one with the largest deviation of the HB + RPA result from the exact energy. The HB + RPA curve is seen to follow closely the exact one except in a small interval about , which is in this case the critical value of where the BCS solution (equal in this case for neutrons and protons) changes from to . While the exact groundstate energy has a smooth variation across , the HB + RPA curve shows there a prominent cusp.

The origin of this cusp can be traced to the expression (7) for the RPA contributions. Thus notice the plot in Fig. 2

Figure 2: (Color online) The two lowest RPA frequencies in any of the , , and spaces in the case of Fig. 1. These two frequencies coincide in the present case for because the single-nucleon spectrum is symmetric about the common Fermi level of neutrons and protons.

of the two lowest RPA frequencies in the case just considered in any of the , , and spaces, which have in this case identical RPA spectra because . Both frequencies are seen to go to zero for going to from below, but only one of them stays at zero for while the other one rises rapidly in this interval. The reason why two frequencies and not only one go to zero for going to from below is that the quasinucleon vacuum becomes in this limit instable against a transition to a vacuum described by a non-zero with an arbitrary complex phase. This has two real parameters. In more physical terms, Fig. 1 may be interpreted to display a shortcoming of the RPA, which is a small amplitude approximation, in a region of the parameter where the equilibrium represented by the quasinucleon vacuum changes rapidly with this parameter.

An example with less symmetry of the single-nucleon spectrum is given in Fig. 3.

Figure 3: (Color online) The two lowest RPA frequencies in each of the , , and spaces in the case of quartets at 0, , , , , and occupied for from the bottom by eight neutrons and four protons.

It is seen that while the lowest RPA frequencies in the and spaces behave as in the preceding case, the lowest RPA frequencies in the space pass almost smoothly through the critical , which are different in this case for neutrons and protons. This is general for .

These observations suggest that the HB + RPA may be improved by interpolation across the critical regions of . More specifically we have found that one gets a good approximation to the exact groundstate energies by interpolation in the interval from to . This is applied to the terms , , and for , when neutrons and protons have the same , and to and for , when may be different for neutrons and protons. No interpolation is applied if , which occurs when the Fermi level lies within a degenerate shell. The interpolating function is the polynomial of third degree in which joins smoothly the calculated values at the endpoints of the interpolation interval.

Figure 4

Figure 4: (Color online) The deviation of the HB + RPA result from the exact in the case of Fig. 1 without and with interpolation of the RPA part.

shows the result of using this recipe in the “worst case” of Fig. 1, and Fig. 5

Figure 5: (Color online) The energy relative to for the nucleus N calculated in the HB + RPA without and with interpolation of the RPA part. To enhance the details in the figure a term quadratic in is added to the calculated energies.

shows its effect for the doubly magic nucleus Ni, which resembles the case of Fig. 1 by having its Fermi level (common for neutrons and protons) within a gap in its single-nucleon spectrum. In the case of Ni we have no exact calculation for comparison, but the interpolation is seen to remove a certainly unphysical cusp from the curve of the groundstate energy as a function of .

We finish this section with a discussion of the limit . This discussion is restricted to the case of even and . It is not restricted to . Without loss of generality the centroid of the single-nucleon spectrum is supposed to vanish. First consider the case of degenerate single-nucleon levels, that is, for all so that the Hamiltonian (3) has only the second term, the pairing force. Our -dimensional valence space is then equivalent to a -shell with . From the formulas in Ref. Neergård (2009) one gets in this case

(8)

Exactly this expression for the lowest eigenvalue of the pairing force results from the formulas derived by Edmonds and Flowers Edmonds and Flowers (1952) by means of group theory. In other words, for degenerate single-nucleon levels the HB + RPA gives the exact result. Now assume a spreading of the levels . Due to their vanishing centroid the first term in the expression (3) is then, in the -shell analogy, a sum of spherical tensor components of rank higher than zero. Since its expectation value in the angular momentum zero ground state of the degenerate case then vanishes, its contribution to the energy vanishes in the Born approximation, and the leading term in this contribution in an expansion in powers of is the linear term. In other words, for a general single-nucleon spectrum the HB + RPA result converges asymptotically to the exact one in the limit .

Vi Smooth terms

The smooth sum of single-nucleon energies in Eq. (1) is calculated separately for neutrons and protons by a standard third order Strutinsky smoothing Brack et al. (1972) with smoothing parameter  MeV including energies until approximately , where is the smooth Fermi level. Corresponding to the spitting of the pairing energy into a BCS and RPA part discussed in Sec. IV we write

(9)

Here the part is calculated separately for neutrons and protons essentially as suggested by Brack et al. Brack et al. (1972): We relate the paring strength to a smooth pair gap by considering a half-filled single-nucleon spectrum of Kramers doublets with equal distance , where is the smooth level density at the smooth Fermi level, and by replacing the sum in the gap equation with an integral. When this integral is evaluated more accurately than in Ref. Brack et al. (1972) one arrives at

(10)

The BCS correlation energy can be expressed by a sum over the Kramers doublets, which can be approximated in a similar way by an integral. By evaluating also this integral more accurately than in Ref. Brack et al. (1972), one gets

(11)

A derivation of an expression for a smooth RPA correlation energy can be based on Eq. (38) of Ref. Neergård (2009). We treat first the case of an or space and consider again a half-filled single-nucleon spectrum of Kramers doublets with equal distance . In Eq. (38) of Ref. Neergård (2009) the sum over Kramers doublets can be replaced with an integral in the propagators etc. with . Some mathematics then leads to

(12)

In the derivation of this expression the integral in Eq. (38) of Ref. Neergård (2009) is displaced to the imaginary axis. This is allowed because etc. are asymptotically proportional to .

For the space one can take into account the discussion in Sec. IV A of Ref. Neergård (2009) of the case of a half-filled infinite spectrum of equidistant quartets. It is shown there that in a very good approximation the RPA correlation energy in the space deviates from that of an or space only by a term , where is the chemical potential. Totally we then have

(13)

Here is taken equal to for or equal to .

Vii Comparison with experimental data

As in Ref. Bentley and Frauendorf (2013), we fit an expression for the pairing force coupling constant proportional to a power of to the empirical binding energy differences of the even-even and odd-odd nuclides defined by

(14)

for odd , cf. the definition of in the beginning of Sec. II. (Note that this is different from the BCS pair gap considered in Secs. IV and V.) We take the groundstate binding energies from the 2012 Atomic Mass Evaluation and the excitation energies for odd from the NNDC Evaluated Nuclear Structure Data Files Tu1 (). The best fit, shown in Fig. 6,

Figure 6: (Color online) The even-even-odd-odd binding energy difference defined by Eq. (14). The empirical groundstate binding energies are taken from Ref. Audi et al. (2012) and the excitation energies in the doubly odd nuclei from Ref. Tu1 (). The solid line shows the results of our calculations and the purple (gray) dashed line is MeV.

is obtained for

(15)

The calculation is seen to reproduce the observed pattern of fluctuations due to shell structure of as a function of about a trend line MeV. For a given , the calculated and observed values thus lie consistently both above or both below this line.

Our further analysis involves Coulomb reduced binding energies. We assume that the electromagnetic contribution to the total energy is given by the last term in Eq. (2) and thus write for the remainder

(16)

This reduction is applied to both the calculated and the measured binding energies. The error bars shown in Figs. 69 include the uncertainties of the empirical mass differences involved and the uncertainty of in the fit of Eq. (2) to the observed masses. When no error bar is shown, the uncertainty is less than the size of the symbol.

To the extend that the last term in Eq. (16) may be assumed to account for all contributions to the total energy from non-isospin-conserving interactions, is independent of . On the right hand side of Eq. (16), we mostly choose . The only exception is that we compare the difference for odd measured for with the one calculated for .

Figure 7

Figure 7: (Color online) Energy difference of the first and states in odd-odd nuclei. The solid line shows the results of our calculations and the experimental data are from from Ref. Tu1 (). The green (gray) dashed line is the zero line, so when points are below it there is an isospin inversion.

shows the measured and calculated values of this difference. The measured is the difference in excitation energy of the lowest and states of the nucleus and is taken from its Evaluated Nuclear Structure Data File. The observed trend of a shift from ground states of the lighter doubly odd nuclei in the range  26–98 to ground states of the heavier ones as well as the average slope of the excitation energy difference as a function of are well reproduced. As discussed by Vogel Vogel (2000) and Macchiavelli et al. Macchiavelli et al. (2000), can be interpreted as a difference between the symmetry energy and the cost in energy of breaking a Cooper pair. Its downslope as a function of then results from an increase of the latter relative to the former. It may be noticed that is the only one of the four combinations of energies displayed in Figs. 69 where measured and calculated energies with different are compared and the difference of the energies compared is thus influenced by the Coulomb reduction (16).

We finally consider the quantities and defined by

(17)

where the constant is a function of , and is even. The constant can be interpreted as an isorotational moment of inertia Frauendorf and Sheikh (), and is related to the Wigner energy and may be called the Wigner . In this analysis, is chosen throughout on the right hand side of Eq. (16), so only doubly even nuclei, and therefore only groundstate energies, are involved. The empirical groundstate binding energies are taken from the 2012 Atomic Mass Evaluation. The constants and are extracted from the calculated and measured for in the case of even and in the case of odd .

Figures 8

Figure 8: (Color online) The reciprocal isorotational moment of inertia defined by Eq. (17). The empirical values are extracted from the binding energies in Ref. Audi et al. (2012). The solid line shows the result of our calculations.

and 9

Figure 9: (Color online) The Wigner defined by Eq. (17). The empirical values are extracted from the binding energies in Ref. Audi et al. (2012). The solid line shows the result of our calculations. The orange (gray) dashed line is .

show the result of this analysis. The measured is very well reproduced including, generally speaking, the features seen in the experimental values. This is a vast improvement from the calculations in Ref. Bentley and Frauendorf (2013), which considerably underestimated . This confirms what was suggested in Ref. Bentley and Frauendorf (2013), namely that the underestimate there originated in the small size of the valence space. For the calculation is very accurate, indicating that the deformations used in the calculations are accurate.

As already mentioned in the introduction, several effects contribute according to Neergård Neergård (2009) to the linear term in the expansion (17). One such contribution, corresponding to , comes from the expression for an isorotational energy. This is an average contribution, represented in our present model by the linear term in the liquid drop symmetry energy in Eq. (2). Microscopically, a symmetry energy proportional to is shown in Ref. Neergård (2009) to emerge from the HB + RPA in the idealized case of equidistant single-nucleon levels. It is dependent in this case on the spontaneous breaking of the isobaric invariance by the isovector pairing force. In fact, in the case of equidistant single-nucleon levels the symmetry energy is proportional to with in the absence of static pair fields Neergård (2003). The numerical solutions for this case in Fig. 7 of Ref.Bentley and Frauendorf (2013) show that approaches one from below with increasing pair coupling strength . Equidistant levels are a particular favorable case. The same figure demonstrates that is approached for uneven level distributions as well, because a large represents the limit of strong deformation in isospace, which results in rigid isorotation.

On top of this isorotational contribution major contributions to the linear term in the symmetry energy arise from shell effects. This is discussed by Neergård Neergård (2009) and further elaborated by Bentley and Frauendorf Bentley and Frauendorf (2013). They relate the deviations of from one to the deviation of the distance between the last occupied and first free level in the absence of pair correlation (cf. Fig. 8 of Ref. Bentley and Frauendorf (2013)). The realistic pair correlation is too weak to wash out this consequence of the level bunching. Neergård shows, in particular, that is large when the Fermi level lies within a gap in the single-nucleon spectrum. The reason is that the isospin is then produced by promotion of pairs from proton levels below the gap to neutron levels above the gap, and each such promotion costs approximately the same amount of energy equal to twice the gap energy. That the empirical is mostly larger than one may thus be seen as the result of the level density at the Fermi level being generally at equilibrium shape lower than corresponding to a uniform spectrum. By analogy with spatial rotation of nuclei it may be seen as evidence for a “softness” of isorotation. The ratio , which is 10/3 for a rigid rotor and 2 for a harmonic vibrator, is commonly used as a measure of how “rotational” a nucleus is. The energy of the first few yrast levels of even-even transitional nuclei can be very well parametrized by the expression . This gives , so is equivalent to .

As seen from Fig. 9, our model reproduces very well these fluctuations of due to shell effects. For  90, 92, 98, and 100, the calculated valued are markedly below the empirical ones. It should be noted that the corresponding values of in Fig. 8 are markedly above the empirical ones. If one would take the product of both numbers to obtain , which is twice the coefficient of the term linear in in the expansion (17), a much better agreement would result.

As anticipated from the discussion above, large occur in the isobaric chains containing the doubly magic nuclei Ca, Ni, and Sn. Both chains with odd neighboring each of these chains with even also have large . This is because in the odd- chains one pair of neutrons is passive in producing isospin. This pair just sits for all in the last quartet below the gap or the first quartet above the gap while other nucleons are promoted across both the gap and this quartet repeating the mechanism described above. A somewhat similar mechanism gives rise to a large for . There one neutron pair occupies for all the shell while other nucleons are promoted from the to the shell.

For and , deformation is involved in producing a large . In these chains, the nuclei thus have large deformations while the and nuclei are essentially spherical. Since isospin is produced by promotion of nucleons within the or shell (except that in O one neutron pair occupies the shell), the two spherical nuclei have roughly equal . We therefore have a large increase of from to due to the departure from the deformed shape and essentially no such increase from to . This gives rise to a large . It also gives a small as seen in Fig. 8.

Tables 2 and 3 show the calculated and measured binding energies of the individual nuclei and components of the calculated ones. Also see Note sup (). The root mean square deviation is 0.95 MeV for the 112 doubly even nuclei with a measured binding energy.

Viii Omission of isoscalar pairing and Coulomb interaction

It has been proposed Satuła and Wyss (); Goodman (); B Cederwall et. al. (2011); Qi et al. (2011) that the strong attraction of isoscalar nucleon pairs exhibited by effective shell model interactions, especially in channels with maximally aligned nucleonic angular momenta, could give rise, in nuclei, to a condensation of such pairs coexisting with or replacing the BCS type of condensation of isovector pairs. We feel that there are points of contention with this proposal: (i) The energy of a condensate depends smoothly on its number of constituents. In the presence of a condensate of isoscalar pairs the mass of the lowest state of an nucleus should therefore depend smoothly on . In reality, these masses show a staggering with the parity of , the doubly odd masses being elevated above the doubly even ones by an amount approximately twice the typical BCS pair gap; see Fig. 6. (ii) Bentley and Frauendorf, in their aforesaid study, examine the effect of adding to the Hamiltonian a schematic interaction of isoscalar pairs of a neutron and a proton in time-reversed orbits with an separable structure similar to that of the isovector pairing force. They find that a weak interaction of this form does not significantly alter their results, while a stronger one would not allow the model to reproduce the data. (iii) In the single--shell model, seniority zero represents condensation of isovector pairs in the BCS sense. Neergård analyzed the ground states of nuclei with two neutrons and two protons or two neutron holes and two proton holes in the or shell calculated in the single--shell approximation with effective interaction from the literature Neergård (2013). He found these states to have by about 80% seniority zero. As pointed out in Neergård’s study, since the seniority zero state has a considerable contingent of isoscalar pairs, the attractive interaction of such pairs stabilizes the seniority zero component of the state vector rather than competing with it.

For these reasons, we do not consider the possibility of condensation of isoscalar pairs in our present work. Some studies, for example Refs. Martínez-Pinedo et al. (1999); Qi et al. (2011), infer a pairing structure of a shell model state from counts of nucleon pairs with given angular momentum. Since Neergård demonstrates in Ref. Neergård (2013) that such a count is not a reliable tool for this purpose, we have not taken such work into account in the discussion in the preceding paragraph.

In Strutinsky-type calculations including schematic isoscalar and isovector pair correlations, Głowacz et al. Głowacz et al. (2004) achieved results for in doubly odd nuclei in agreement with the data similar to that of our results displayed in Fig. 7. This is consistent with the findings of Bentley and Frauendorf mentioned in the first paragraph of this section. Although such a mixed scenario cannot be excluded we follow the principle of Occam’s razor, assuming pure isovector pair correlations.

For we made a supplemental shell model calculation using the interaction “model I” of Zamick and Robinson Zamick and Robinson () with a normalization to zero of the largest matrix element (two-nucleon angular momentum ) so as to make the total interaction attractive. Like Satuła et al. Satuła et al. (1997), we switched off successively the interactions in individual channels. The results for and are shown in Table 1.

of included interactions

(MeV)

(MeV)

0 1 2 3 4 5 6 7 2.41 3.17 1.31
0 2 3 4 5 6 7 2.23 2.37 1.07
0 2 4 5 6 7 1.86 2.29 1.23
0 2 4 6 7 1.38 2.42 1.75
0 2 4 6 0.25 0.42 1.71
0 2 4 0.25 0.42 1.71
0 2 0.43 0.61 1.41
0 0.81 0.81 1.00
Table 1: shell model calculation for .

Note that because the matrix element is normalized to zero, it makes no difference whether it is included or not. Like and the quantity displayed in Fig. 2 of Ref. Satuła et al. (1997) is a function of for . The relation is . Like Satuła et al., who include in their calculations the shells , , and and employ an interaction appropriate for this larger valence space, we find that and therefore decreases when the isoscalar interactions are switched off successively and very much so when the interaction is switched off finally. It is seen, however, that this is due not to a decrease of , which actually increases, but to a decrease of the symmetry energy coefficient . The isoscalar shell model interactions thus contribute significantly to the entire symmetry energy and not just its Wigner term. The reduction of the Wigner energy when the isoscalar interactions are switched off is only a side effect of this general reduction of the symmetry energy. The symmetry energy coefficient reaches its minimum when all the isovector interactions and none of the isoscalar interactions are present, and the calculations confirm the well known result derived analytically by Edmonds and Flowers Edmonds and Flowers (1952) that is exactly one for the pure pairing force, . In this case as seen from Eq. (8).

Our approach makes further simplifying assumptions: (i) The Coulomb interaction can be treated as a first-order perturbation, that is, its contribution to the total energy may be approximated by its expectation value with the wave function determined by the strong interaction only. This contribution can then be incorporated in the form of the Coulomb term in the smooth liquid drop binding energy formula (or subtracted from the experimental binding energies, as done in this paper). (ii) Isospin-breaking terms of the strong interaction Hamiltonian (as the difference between the proton and neutron masses) are neglected. (iii) The difference between proton and neutron mean fields generates only a constant shift of the proton single-particle levels relative to the neutron ones. This constant shift drops out in the shell correction procedure so that one can assume the same single proton and neutron energies from the outset. These assumptions lead to our isospin-invariant Hamiltonian (3) used to calculate shell and pairing corrections.

Sato et al. Sato et al. (2013) studied nuclei in the framework of the density functional mean field theory. They describe the states by isocranking about an axis in isospace that is tilted with respect to the axis. The resulting quasiparticles are mixtures of proton and neutron particles and holes. As discussed by Frauendorf and Sheikh Frauendorf and Sheikh (), for an isospin invariant Hamiltonian all mean field solutions that correspond to the same cranking frequency but a different orientation of the cranking axis (the “semicircle” of Ref. Sato et al. (2013)) have the same energy. They can be generated by rotation in isospace from the solution obtained by cranking about the axis, which has pure proton and pure neutron quasiparticles. The rotation generates a mixing of the proton and neutron quasiparticles (cf. the example of isocranking about the axis discussed in Ref. Frauendorf and Sheikh ()). Hence if our assumptions hold, it is sufficient to study isosrotation about the axis, which generates solutions and avoids proton-neutron mixing. The solutions are given by rotation in isospace. If the mean field theory includes the Coulomb interaction, as the study by Sato et al. Sato et al. (2013) does, the different orientations of the cranking axis are no longer equivalent, and the orientation of minimal energy has to be calculated. However the finding of Sato et al. that such solutions lie with a good accuracy on a shifted semicircle indicates that our assumptions are good approximations.

Ix Summary and outlook

A model with nucleons in a charge independent potential well interacting by an isovector pairing force has been discussed. For a 24-dimensional valence space, the Hartree-Bogolyubov (HB) plus random phase approximation (RPA) to the lowest eigenvalue of the Hamiltonian was shown to be accurate except near the values of the pairing force coupling constant where the HB solution shifts from a zero to a non-zero pair gap. The HB + RPA was shown to be asymptotically exact in the limit . To remedy the inaccuracy of the HB + RPA in the critical regions of we devised a scheme of interpolation across the these regions. It is described in Sec. V.

The resulting algorithm was used to calculate with a valence space of dimension twice the mass number pairing corrections in the framework of a Nilsson-Strutinsky calculation. For this purpose we derived in Sec. VI expressions for smooth counterterms to the Bardeen-Cooper-Schrieffer (BCS) and RPA parts of the pair correlation energy. The deformations and corresponding single-nucleon energies for the Nilsson-Strutinsky calculation were taken from a previous Nilsson-Strutinsky plus BCS calculation with the code tac Frauendorf (1993). To enforce charge independence the average of the calculated single-neutron and single-proton energies was employed. Our expression (2) for the macroscopic liquid drop energy was taken from the work of Duflo and Zuker Duflo and Zuker (1995) with the omission of a phenomenological pairing energy and has symmetry energy terms proportional to , where is the isospin. Its five parameters were fit to the empirical masses according to the 2012 Atomic Mass Evaluation Audi et al. (2012) of the 112 doubly even nuclei with a measured binding energy considered in the present study.

In this model we calculated the binding energies of the ground states of the doubly even nuclei with and and the lowest isospin states of the doubly odd nuclei with and , where and are the numbers of neutrons and protons. These calculated binding energies were compared to the empirical ones from the 2012 Atomic Mass Evaluation with excitation energies from the NNDC Evaluated Nuclear Structure Data Files Tu1 (). In terms of both the calculated and the empirical binding energies , where , a Coulomb reduced energy was defined by Eq. (16) with on the right-hand side with one exception to be told later. The following combinations were then extracted and compared: (i) 2 as defined by Eq. (14). (ii) for odd . In this case the measured was defined with on the right hand side of Eq. (16). (iii) The constants and in the expansion (17) for all .

Comparisons of the calculated and measured values of these combinations are shown in Figs. 69. The expression (15) adopted for the pairing force coupling constant was fit to the empirical 2. The present enlargement of the valence space resolved an issue with the constant , which was underestimated in the previous exact calculation by Bentley and Frauendorf with a 28-dimensional valence space. The fluctuations of with were discussed in Sec. VII. They are well understood from the shell structure. The root mean square deviation of the calculated and measured binding energies of the 112 doubly even nuclei with a measures binding energy is 0.95 MeV.

We anticipate a generalization of the present method to the more realistic case when neutrons and protons move in different potential wells due to the Coulomb force. The chief obstacle to this generalization is the RPA calculation in the space, which will be more complex because neutron and proton stationary states no longer form time-reversed pairs. Simplifying approximations might be warranted in this step of the procedure, however. The result would be a method for including pairing correlations beyond a mean field approximation (BCS, Hartree-Fock-Bogolyubov, relativistic mean field) that would be simple enough to go on top of any state-of-the-art mean field approach. This would eliminate the need for the phenomenological Wigner term often employed in present mean field calculations such as, for example, those of Refs. Möller et al. (1995); Goriely et al. (2010); Wang and Liu (2013).

This work was supported by the DoE Grant DE-FG02-95ER4093.

References

  • Myers and Swiatecki (1966) W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81, 1 (1966).
  • Neergård (2009) K. Neergård, Phys. Rev. C 80, 044313 (2009).
  • Duflo and Zuker (1995) J. Duflo and A. P. Zuker, Phys. Rev. C 52, R23 (1995).
  • (4) S. Frauendorf and J. A. Sheikh, Nucl. Phys. A 645, 509 (1999); Phys. Scr. T 88, 162 (2000).
  • Satuła and Wyss (2001) W. Satuła and R. Wyss, Phys. Rev. Lett. 86, 4488 (2001).
  • Neergård (2002) K. Neergård, Phys. Lett. B 537, 287 (2002).
  • Neergård (2003) K. Neergård, Phys. Lett. B 572, 159 (2003).
  • Bentley and Frauendorf (2013) I. Bentley and S. Frauendorf, Phys. Rev. C 88, 014322 (2013).
  • Brack et al. (1972) M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky,  and C. Y. Wong, Rev. Mod. Phys. 44, 320 (1972).
  • Hill and Wheeler (1953) D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953).
  • Swiatecki (1956) W. J. Swiatecki, Phys. Rev. 104, 993 (1956).
  • Nilsson et al. (1969) S. G. Nilsson, C. F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I. L. Lamm, P. Möller,  and B. Nilsson, Nucl. Phys. A 131, 1 (1969).
  • Seeger and Howard (1975) P. A. Seeger and W. M. Howard, Nucl. Phys. A 238, 491 (1975).
  • Audi et al. (2012) G. Audi, M. Wang, A. Wapstra, F. Kondev, M. MacCormick, X. Xu,  and B. Pfeiffer, Chin. Phys. C 36, 1287 (2012).
  • Mendoza-Temis et al. (2010) J. Mendoza-Temis, J. G. Hirsch,  and A. P. Zuker, Nucl. Phys. A 843, 14 (2010).
  • Bengtsson and Ragnarsson (1985) T. Bengtsson and I. Ragnarsson, Nucl. Phys. A 436, 1 (1985).
  • Larsson (1973) S. A. Larsson, Phys. Scr. 8, 17 (1973).
  • Frauendorf (1993) S. Frauendorf, Nucl. Phys. A 557, 259 (1993).
  • (19) In Tables 23, . Extrapolated empirical binding energies are indicated by an asterix.
  • Ring and Schuck (1980) P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980).
  • Edmonds and Flowers (1952) A. R. Edmonds and B. H. Flowers, Proc. R. Soc. London, Ser. A 214, 515 (1952).
  • (22) National Nuclear Data Center, Brookhaven National Laboratory, http://www.nndc.bnl.gov, data retrieved December 21, 2011.
  • Vogel (2000) P. Vogel, Nucl. Phys. A 662, 148 (2000).
  • Macchiavelli et al. (2000) A. O. Macchiavelli, P. Fallon, R. M. Clark, M. Cromaz, M. A. Deleplanque, R. M. Diamond, G. J. Lane, I. Y. Lee, F. S. Stephens, C. E. Svensson, K. Vetter,  and D. Ward, Phys. Rev. C 61, 041303 (2000).
  • (25) W. Satuła and R. A. Wyss, Phys. Lett. B 393, 1 (1997); Nucl. Phys. A 676, 120 (2000).
  • (26) A. L. Goodman, Phys. Rev. C 58, R3051 (1998); 60, 014311 (1999).
  • B Cederwall et. al. (2011) B Cederwall et. al. , Nature (London) 469, 68 (2011).
  • Qi et al. (2011) C. Qi, J. Blomqvist, T. Bäck, B. Cederwall, A. Johnson, R. J. Liotta,  and R. Wyss, Phys. Rev. C 84, 021301 (2011).
  • Neergård (2013) K. Neergård, Phys. Rev. C 88, 034329 (2013).
  • Martínez-Pinedo et al. (1999) G. Martínez-Pinedo, K. Langanke,  and P. Vogel, Nucl. Phys. A 651, 379 (1999).
  • Głowacz et al. (2004) S. Głowacz, W. Satuła,  and R. Wyss, Eur. Phys. J. A 19, 33 (2004).
  • (32) L. Zamick and J. Q. Robinson, Yad. Fiz. 65, 773 (2002) [Phys. At. Nucl. 65 740 (2002)].
  • Satuła et al. (1997) W. Satuła, D. J. Dean, J. Gary, S. Mizutori,  and W. Nazarewicz, Phys. Lett. B 407, 103 (1997).
  • Sato et al. (2013) K. Sato, J. Dobaczewski, T. Nakatsukasa,  and W. Satuła, Phys. Rev. C 88, 061301(R) (2013).
  • Möller et al. (1995) P. Möller, J. R. Nix, W. D. Myers,  and W. J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995).
  • Goriely et al. (2010) S. Goriely, N. Chamel,  and J. M. Pearson, Phys. Rev. C 82, 035804 (2010).
  • Wang and Liu (2013) N. Wang and M. Liu, J. Phys. Conf. Ser. 420, 012057 (2013).
(Error) Audi et al. (2012)
(MeV) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)
12 12 0.284 0.014 -196.494 -1.599 0.000 -0.208 -15.263 -15.966 0.911 197.182 198.257(0.000)
14 10 0.091 0.000 -191.739 2.999 -0.874 -0.254 -13.751 -13.448 -0.923 189.663 191.840(0.001)
16 8 0.000 0.000 -162.435 -4.428 0.000 -0.315 -12.576 -10.691 -1.570 168.433 168.952(0.110)
14 12 0.201 0.012 -216.049 1.347 -0.358 -0.279 -14.924 -15.121 0.118 214.584 216.681(0.000)
16 10 0.000 0.000 -201.290 2.183 -1.966 -0.313 -13.362 -12.708 -2.307 201.414 201.551(0.018)
18 8 0.000 0.000 -163.793 -2.082 -1.178 -0.382 -12.341 -10.059 -3.078 168.953 168.862(0.156)
14 14 -0.222 -0.003 -233.778 -1.686 0.000 -0.310 -15.841 -16.543 1.012 234.452 236.537(0.000)
16 12 0.000 0.000 -231.608 3.086 -2.013 -0.333 -14.332 -14.370 -1.642 230.164 231.627(0.002)
18 10 0.000 0.000 -207.636 4.304 -3.107 -0.374 -13.146 -12.042 -3.837 207.169 206.882(0.096)
16 14 0.000 0.000 -254.613 -0.669 -0.035 -0.365 -15.489 -15.782 0.623 254.659 255.620(0.000)
18 12 0.000 0.000 -242.838 5.142 -3.097 -0.386 -14.000 -13.686 -3.025 240.721 241.635(0.003)
20 10 0.000 0.000 -211.426 2.649 -1.803 -0.451 -13.216 -11.451 -3.117 211.894 211.276(0.280)
16 16 0.000 0.000 -271.276 -2.921 0.000 -0.399 -16.240 -17.023 1.182 273.015 271.780(0.000)
18 14 0.000 0.000 -270.577 1.426 -1.172 -0.411 -15.080 -15.088 -0.753 269.904 271.407(0.000)
20 12 0.000 0.000 -251.164 3.507 -1.834 -0.456 -14.041 -13.076 -2.343 250.000 249.723(0.003)
18 16 0.000 0.000 -291.799 -0.823 -1.108 -0.438 -15.741 -16.322 -0.089 292.711 291.839(0.000)
20 14 0.000 0.000 -283.337 -0.096 -0.016 -0.473 -15.038 -14.466 -0.115 283.548 283.429(0.014)
22 12 0.000 0.000 -256.998 7.279 -4.331 -0.557 -13.666 -12.533 -4.907 254.626 256.713(0.029)
18 18 0.000 0.000 -307.277 1.193 -2.144 -0.471 -16.301 -17.438 -0.536 306.620 306.717(0.000)
20 16 0.000 0.000 -308.862 -2.273 0.000 -0.494 -15.640 -15.694 0.548 310.587 308.714(0.000)
22 14 0.000 0.000 -293.328 3.674 -2.546 -0.566 -14.649 -13.912 -2.717 292.371 292.008(0.071)
20 18 0.000 0.000 -328.496 -0.268 -1.041 -0.521 -16.290 -16.806 -0.004 328.768 327.343(0.000)
22 16 0.000 0.000 -322.915 1.468 -2.503 -0.580 -15.225 -15.133 -2.015 323.462 321.054(0.007)
24 14 0.132 -0.005 -300.467 0.803 -0.674 -0.683 -14.796 -13.436 -1.351 301.015 299.928(0.070)
20 20 0.000 0.000 -342.868 -1.679 0.000 -0.565 -17.132 -17.828 1.261 343.286 342.052(0.000)
22 18 0.000 0.000 -346.496 3.375 -3.474 -0.600 -15.784 -16.242 -2.416 345.537 343.810(0.000)
24 16 0.000 0.000 -334.329 2.113 -3.206 -0.703 -14.997 -14.634 -2.866 335.082 333.173(0.004)
22 20 0.000 0.000 -364.689 1.929 -2.409 -0.637 -16.675 -17.264 -1.183 363.943 361.896(0.000)
24 18 0.000 0.000 -361.658 3.980 -4.121 -0.716 -15.506 -15.739 -3.172 360.850 359.336(0.006)
26 16 0.000 0.000 -343.414 -0.002 -2.383 -0.862 -15.000 -14.188 -2.333 345.749 344.116(0.003)
22 22 0.000 0.000 -378.017 5.424 -4.720 -0.701 -16.899 -18.207 -2.711 375.304 375.475(0.001)
24 20 0.000 0.000 -383.498 2.548 -3.088 -0.745 -16.292 -16.759 -1.876 382.826 380.960(0.000)
26 18 0.000 0.000 -374.304 1.869 -3.295 -0.868 -15.502 -15.288 -2.641 375.076 373.729(0.002)
24 22 0.000 0.000 -400.365 5.983 -5.354 -0.800 -16.595 -17.702 -3.447 397.829 398.196(0.000)
26 20 0.000 0.000 -399.626 0.490 -2.316 -0.889 -16.242 -16.305 -1.364 400.500 398.772(0.002)
28 18 0.000 0.000 -384.707 -2.675 -0.920 -1.047 -15.575 -14.881 -0.567 387.949 386.929(0.041)
24 24 0.150 -0.014 -411.947 -0.464 -0.768 -0.862 -18.238 -18.561 0.417 411.994 411.469(0.007)
26 22 0.000 0.000 -419.885 3.908 -4.566 -0.936 -16.599 -17.247 -2.982 418.959 418.703(0.000)
28 20 0.000 0.000 -413.352 -3.969 0.000 -1.061 -16.281 -15.893 0.673 416.648 416.001(0.000)
26 24 0.100 -0.002 -435.206 0.153 -1.545 -0.997 -17.592 -18.108 -0.032 435.085 435.051(0.001)
28 22 0.000 0.000 -436.862 -0.538 -2.241 -1.100 -16.670 -16.833 -0.978 438.378 437.785(0.000)
30 20 0.000 0.000 -424.920 -1.037 -1.451 -1.249 -16.231 -15.517 -0.916 426.873 427.508(0.002)
26 26 0.000 0.000 -446.900 2.481 -4.416 -1.134 -17.820 -18.915 -2.187 446.606 447.700(0.007)
28 24 0.000 0.000 -455.667 0.084 -2.878 -1.172 -17.139 -17.701 -1.144 456.727 456.350(0.001)
30 22 0.000 0.000 -451.546 2.310 -3.602 -1.280 -16.545 -16.453 -2.414 451.650 451.966(0.007)
28 26 0.000 0.000 -470.139 -1.832 -2.155 -1.280 -17.881 -18.500 -0.256 472.227 471.764(0.000)
30 24 0.000 0.000 -473.400 2.888 -4.205 -1.344 -16.977 -17.319 -2.519 473.031 474.008(0.001)
32 22 0.000 0.000 -464.150 2.970 -3.055 -1.466 -16.399 -16.102 -1.886 463.066 464.237(0.125)
28 28 0.000 0.000 -480.607 -6.040 0.000 -1.416 -18.436 -19.234 2.214 484.433 483.995(0.001)
30 26 0.000 0.000 -490.853 0.962 -3.480 -1.443 -17.634 -18.117 -1.554 491.445 492.259(0.000)
32 24 0.000 0.000 -488.938 3.533 -3.662 -1.523 -16.813 -16.965 -1.987 487.392 488.499(0.002)
30 28 0.000 0.000 -504.227 -3.230 -1.350 -1.571 -18.371 -18.851 0.701 506.756 506.459(0.000)
32 26 0.000 0.000 -509.265 1.623 -2.963 -1.613 -17.428 -17.761 -1.017 508.659 509.950(0.000)
34 24 0.087 0.002 -502.230 2.988 -2.722 -1.706 -16.779 -16.626 -1.169 500.411 501.195(0.203)
30 30 0.000 0.000 -513.812 -0.485 -2.653 -1.717 -18.900 -19.528 -0.308 514.605 514.982(0.001)
32 28 0.000 0.000 -525.451 -2.529 -0.889 -1.733 -18.075 -18.493 1.262 526.718 526.846(0.000)
34 26 0.000 0.000 -525.572 4.579 -4.999 -1.784 -17.159 -17.428 -2.946 523.939 525.350(0.003)
32 30 0.000 0.000 -537.780 0.177 -2.182 -1.871 -18.621 -19.170 0.238 537.365 538.119(0.001)
34 28 0.000 0.000 -544.476 0.440 -2.958 -1.896 -17.762 -18.159 -0.665 544.701 545.262(0.000)
36 26 -0.043 0.001 -539.889 4.484 -4.551 -1.961 -17.011 -17.114 -2.487 537.892 538.959(0.003)
32 32 0.000 0.000 -546.508 0.826 -1.732 -2.017 -19.131 -19.799 0.953 544.729 545.845(0.004)
34 30 0.000 0.000 -559.467 3.087 -4.221 -2.026 -18.291 -18.834 -1.652 558.032 559.098(0.001)
36 28 0.000 0.000 -561.482 1.511 -3.579 -2.060 -17.540 -17.844 -1.215 561.186 561.757(0.001)
34 32 0.091 0.004 -570.523 2.342 -2.801 -2.173 -18.838 -19.464 -0.002 568.183 569.279(0.002)
36 30 -0.037 0.001 -579.006 3.232 -3.995 -2.189 -18.078 -18.519 -1.365 577.139 578.136(0.001)
38 28 0.000 0.000 -576.625 0.810 -2.899 -2.231 -17.420 -17.548 -0.540 576.355 576.808(0.001)
34 34 -0.171 -0.002 -577.748 1.959 -2.569 -2.277 -19.364 -20.050 0.394 575.395 576.439(0.000)
36 32 -0.113 0.002 -592.486 3.372 -3.622 -2.323 -18.566 -19.145 -0.720 589.834 590.793(0.002)
38 30 0.000 0.000 -596.694 3.400 -4.112 -2.349 -17.911 -18.221 -1.453 594.747 595.386(0.001)
36 34 -0.213 -0.002 -601.773 1.828 -2.874 -2.428 -19.205 -19.731 0.080 599.865 600.322(0.002)
38 32 -0.121 0.005 -612.512 3.318 -3.691 -2.506 -18.403 -18.848 -0.740 609.934 610.519(0.001)
40 30 0.000 0.000 -612.537 2.383 -3.551 -2.532 -17.808 -17.939 -0.888 611.042 611.086(0.002)
36 36 -0.273 -0.003 -607.918 -0.804 -1.434 -2.570 -19.937 -20.289 1.488 607.234 606.911(0.008)
38 34 -0.200 0.002 -624.444 2.469 -3.375 -2.646 -18.947 -19.436 -0.240 622.215 622.403(0.002)
40 32 0.000 0.000 -631.231 2.986 -3.122 -2.653 -18.215 -18.564 -0.120 628.365 628.686(0.000)
38 36 -0.248 0.001 -633.186 1.431 -3.102 -2.802 -19.574 -19.994 0.120 631.635 631.445(0.002)
40 34 -0.190 0.008 -645.134 1.908 -3.019 -2.866 -18.820 -19.156 0.183 643.043 642.891(0.000)
42 32 0.000 0.000 -647.719 4.019 -4.995 -2.859 -18.078 -18.296 -1.918 645.618 645.665(0.000)
38 38 -0.238 0.006 -639.705 2.508 -4.048 -3.007 -19.917 -20.524 -0.434 637.631 637.939(0.034)
40 36 -0.220 0.008 -656.480 2.144 -3.574 -3.035 -19.272 -19.715 -0.096 654.432 654.270(0.004)
42 34 0.000 0.000 -665.112 6.732 -6.951 -2.978 -18.393 -18.882 -3.484 661.864 662.072(0.000)
40 38 -0.218 0.013 -665.086 2.461 -4.005 -3.224 -19.695 -20.244 -0.232 662.857 663.007(0.007)
42 36 -0.201 0.014 -677.723 2.205 -3.861 -3.268 -19.114 -19.451 -0.256 675.774 675.578(0.001)
44 34 0.058 0.000 -682.075 3.823 -5.372 -3.246 -18.361 -18.633 -1.854 680.106 679.989(0.000)
40 40 -0.212 0.020 -670.705 1.787 -3.677 -3.427 -20.231 -20.743 0.262 668.656 669.929(1.490)
42 38 -0.205 0.018 -688.423 2.006 -3.898 -3.450 -19.556 -19.978 -0.026 686.443 686.288(0.003)
44 36 0.063 0.001 -698.213 4.544 -5.757 -3.375 -18.704 -19.186 -1.900 695.569 695.434(0.001)
42 40 -0.204 0.025 -696.104 1.021 -3.372 -3.646 -20.086 -20.478 0.666 694.417 694.458(0.164)*
44 38 -0.073 0.003 -711.058 4.148 -5.495 -3.525 -19.101 -19.713 -1.358 708.268 708.129(0.006)
46 36 0.051 0.002 -715.895 2.810 -5.457 -3.637 -18.703 -18.943 -1.580 714.665 714.274(0.001)
42 42 -0.200 0.031 -700.894 0.124 -3.045 -3.861 -20.630 -20.952 1.138 699.632 699.636(0.420)*
44 40 0.000 0.000 -721.028 5.488 -7.106 -3.607 -19.427 -20.208 -2.718 718.258 718.117(0.006)
46 38 0.000 0.000 -730.971 4.551 -7.171 -3.733 -19.009 -19.463 -2.984 729.404 728.911(0.001)
44 42 0.000 0.000 -727.856 6.454 -8.833 -3.781 -19.771 -20.688 -4.135 725.537 725.385(0.004)
46 40 0.000 0.000 -742.815 3.596 -6.638 -3.873 -19.389 -19.965 -2.189 741.408 740.808(0.004)
48 38 0.000 0.000 -749.119 1.310 -5.577 -4.047 -19.169 -19.233 -1.466 749.275 748.927(0.001)
44 44 0.000 0.000 -731.868 5.965 -9.440 -3.992 -20.144 -21.146 -4.446 730.349 730.224(0.264)*
46 42 0.000 0.000 -751.670 4.570 -8.370 -4.041 -19.729 -20.444 -3.614 750.714 750.104(0.004)
48 40 0.000 0.000 -762.974 0.388 -5.074 -4.182 -19.546 -19.734 -0.704 763.290 762.610(0.005)
46 44 0.000 0.000 -757.674 4.099 -8.991 -4.247 -20.106 -20.902 -3.948 757.523 756.879(0.004)
48 42 0.000 0.000 -773.811 1.378 -6.820 -4.345 -19.887 -20.213 -2.149 774.582 773.733(0.004)
50 40 0.000 0.000 -781.602 -4.065 -2.113 -4.529 -19.296 -19.514 2.634 783.033 783.898(0.002)
46 46 0.000 0.000 -760.954 2.262 -8.560 -4.495 -20.605 -21.339 -3.331 762.023 761.668(0.460)*
48 44 0.000 0.000 -781.766 0.934 -7.460 -4.545 -20.268 -20.670 -2.513 783.345 782.439(0.003)
50 42 0.000 0.000 -794.376 -3.049 -3.878 -4.686 -19.645 -19.993 1.156 796.269 796.510(0.001)
48 46 0.000 0.000 -786.965 -0.867 -7.052 -4.787 -20.771 -21.107 -1.929 789.761 788.817(0.004)
50 44 0.000 0.000 -804.240 -3.458 -4.545 -4.881 -20.038 -20.450 0.748 806.950 806.864(0.003)
52 42 0.000 0.000 -813.453 1.390 -7.108 -5.056 -19.673 -19.783 -1.942 814.005 814.258(0.000)
48 48 0.000 0.000 -789.526 -3.952 -5.585 -5.074 -21.534 -21.523 -0.522 794.000 792.864(0.384)*
50 46 0.000 0.000 -811.319 -5.216 -4.180 -5.118 -20.571 -20.887 1.254 815.281 815.041(0.004)
52 44 0.000 0.000 -825.186 0.953 -7.762 -5.246 -20.028 -20.240 -2.304 826.537 826.502(0.000)
50 48 0.000 0.000 -815.732 -8.249 -2.763 -5.399 -21.388 -21.303 2.551 821.430 821.073(0.052)
52 46 0.000 0.000 -834.107 -0.824 -7.395 -5.478 -20.498 -20.676 -1.739 836.670 836.322(0.005)
54 44 0.000 0.000 -844.686 4.129 -9.659 -5.625 -19.618 -20.039 -3.613 844.170 844.791(0.006)
50 50 0.000 0.000 -817.586 -12.489 0.000 -5.718 -21.073 -21.699 6.344 823.731 825.297(0.302)
52 48 0.000 0.000 -840.335 -3.866 -5.985 -5.753 -21.186 -21.092 -0.326 844.527 843.774(0.002)
54 46 0.000 0.000 -855.412 2.343 -9.305 -5.852 -20.060 -20.475 -3.038 856.107 856.371(0.018)
Table 2: Deformations and Binding Energy Contributions for Even-Even Nuclei
(Error) Audi et al. (2012)
(MeV) (MeV) (MeV) (MeV) (MeV) (MeV) (MeV)
13 13 0.031 0.006 -216.587 6.083 0.000 -0.281 -14.087 -16.296 2.490 208.014 211.894(0.000)
15 15 -0.111 -0.002 -252.841 0.034 0.000 -0.365 -14.224 -16.801 2.942 249.865 250.605(0.000)
17 17 0.000 0.000 -289.325 0.086 0.000 -0.434 -14.951 -17.235 2.718 286.521 285.419(0.000)
19 19 0.000 0.000 -325.126 0.559 0.000 -0.514 -15.694 -17.635 2.455 322.112 320.646(0.000)
21 21 0.000 0.000 -360.499 2.682 0.000 -0.626 -16.181 -18.019 2.464 355.353 354.076(0.000)
23 23 0.075 -0.007 -395.239 3.580 -0.673 -0.784 -16.664 -18.389 1.836 389.823 389.560(0.000)
25 25 0.075 -0.007 -429.677 1.752 -0.440 -0.994 -17.252 -18.742 2.044 425.881 426.409(0.001)
27 27 0.000 0.000 -463.816 -1.258 0.000 -1.272 -17.336 -19.078 3.014 462.060 462.540(0.000)
29 29 0.000 0.000 -497.273 -2.700 0.000 -1.566 -17.626 -19.384 3.324 496.649 497.116(0.001)
31 31 0.000 0.000 -530.224 0.671 -0.326 -1.868 -18.442 -19.666 2.766 526.787 527.584(0.001)
33 33 -0.086 -0.001 -562.430 3.069 -0.678 -2.159 -18.818 -19.927 2.590 556.771 558.078(0.006)
35 35 -0.222 -0.003 -592.989 1.898 -0.470 -2.414 -19.125 -20.171 2.990 588.101 587.700(0.015)
37 37 -0.256 0.002 -623.894 1.593 -0.574 -2.789 -19.534 -20.409 3.090 619.211 619.241(0.003)
39 39 -0.225 0.013 -655.291 2.738 -1.475 -3.218 -19.911 -20.635 2.467 650.086
41 41 -0.206 0.026 -685.874 1.341 -0.875 -3.642 -20.002 -20.849 3.614 680.919 680.814(0.328)*
43 43 -0.100 0.016 -716.766 3.893 -3.156 -3.896 -19.915 -21.049 1.874 710.999 710.298(0.258)*
45 45 0.000 0.000 -746.475 4.439 -6.121 -4.234 -19.961 -21.243 -0.605 742.641
47 47 0.000 0.000 -775.304 -0.543 -4.287 -4.775 -20.659 -21.432 1.261 774.586
49 49 0.000 0.000 -803.620 -7.941 0.000 -5.389 -20.418 -21.613 6.584 804.977
Table 3: Deformations and Contributions to the Binding Energy for Odd-Odd Nuclei
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
49368
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description