N=Z nuclei: A laboratory for neutron-proton collective mode

# N=z nuclei: A laboratory for neutron-proton collective mode

Chong Qi and Ramon Wyss
Department of Physics
Royal Institute of Technology (KTH)    SE-10691 Stockholm    Sweden
###### Abstract

The neutron-neutron and proton-proton pairing correlations have long been recognized to be the dominant many-body correlation beyond the nuclear mean field since the introduction of pairing mechanism by Bohr, Mottelson and Pines nearly 60 year ago. Nevertheless, few conclusion has been reached concerning the existence of analogous neutron-proton (np) pair correlated state. One can see a renaissance in np correlation studies in relation to the significant progress in radioactive ion beam facilities and detection techniques. The np pairs can couple isospin T = 1 (isovector) or 0 (isoscalar). In the isovector channel, the angular momentum zero component is expected to be the most importance one. On the other hand, as one may infer from the general properties of the np two-body interaction, in the isoscalar channel, both the np pairs with minimum (J=1) and maximum (J=2j) spin values can be important. In this contribution, we will discuss the possible evidence for np pair coupling from different perspective and analyze its influence on interesting phenomena including the Wigner effect and mass correlations in odd-odd nuclei. In particular, we will explain the spin-aligned pair coupling scheme and quartet coupling involving pairs with maximum (J=2j) spin values.

###### pacs:
21.10.-k, 21.10.Pc, 21.30.-x, 21.60.-n, 21.60.Ka

## 1 Introduction

It is well known that, along the discovery of the uncharged particle neutron by Chadwick in 1932 [1], Heisenberg immediately introduced the idea that the nucleus is composed of protons and neutron as well as the concept of isospin [2]. These mark the beginning of nuclear structure physics. The isospin has been extensively applied in explaining many aspects of the nuclei [3], even though it is not a fundamental symmetry since the masses of the proton and neutron and the interactions involving the two are not exactly the same. Another glorious event is the suggestion of nuclear pairing mechanism by Bohr, Mottelson and Pines [4] one year after the introduction of the Bardeen-Cooper-Schrieffer (BCS) ansatz in superconductors [5]. The neutron-neutron (nn) and proton-proton (pp) pairing correlations between identical particles have been shown to be crucial in explaining a wealth of experimental data has including odd-even staggering in binding energies and charge radii, nuclear deformation as well as moments of inertia [6, 7, 8]. It is natural to expect that the pairing correlation between the neutron and the proton can be equally important. The neutron and proton can be coupled to both isoscalar pair, with a symmetric wave function for the radial spin part, and isovector pair in analogous to the nn and pp pair coupling. However, there is no conclusive evidence for either np pair.

There has been a long history and extensive efforts studying the importance of np correlations from many different perspectives. Recently, the np has attracted renewed interest in relation to the advances in experimental techniques and availability of radioactive beam facilities. Overviews on np pairing correlations are published recently in Refs. [9, 10, 11]. Extensive discussions can also be found in workshops organized recently (see, e.g., Ref. [12]). The importance of np correlation in the development of collective correlation and nuclear deformation as well as in the evolution of the shell structure has been generally accepted. This is related to the fact that the np two-body interaction contribute significantly to the nuclear mean field. The remaining controversy is whether it is necessary to include the residual np pairing coupling on top of the nuclear mean field and, if so, how to separate it from the mean field channel of the two-body interaction. Besides the dynamic effects of the residual correlation, another challenging task to understand the different predictions of the approximate methods and exact solutions within the shell-model context in treating the np correlation or the np coupling scheme. In the present contribution we will review briefly a few aspects along that direction that are not fully covered in Ref. [9]. Then we will give a more detailed explanation on the works done at Stockholm during the past two decades.

## 2 Systematics of nuclear binding energy and the residual correlations

In a broader context, one may state that nuclear physics is an emergent phenomenon which is created by the complicated interplay among its constitutes: protons and neutrons. The understanding of its emergent behavior progresses by systematic experimental observations and the construction of models to interpret them [13], in particular their local fluctuations. Studies on the nuclear mass and other ground state properties reveal strikingly systematic behaviors including the nuclear liquid, shell structure well as the nuclear deformation. It is thus natural to expect that the differences of binding energies can be used to isolate specific correlations. As mentioned above, the zigzag behavior of one-body separation energies and the nuclear binding energy has long been well known. It provides clues to the pairing correlation between like nucleons. It may also be possible to extract the residual interactions between protons and neutrons from the binding energy differences.

### 2.1 Like-particle pairing and odd-even staggering

The pairing energy or pairing gap implies that the energies of even-even nuclei are systematically lower than those of odd-odd and odd- nuclei. As commented in P. 169 of Ref. [7], one can extract the empirical pairing gap by comparing the local average of the masses of odd- nuclei with the masses of the corresponding even-even nuclei, where one assumes that the masses are a smooth function of and except for the pairing effect.

The four-point formula used to extract the neutron empirical pairing gap from the odd-even staggering (OES) of the binding energies is defined as [7]

 Δ(4)(N)=14[−B(N+1,Z)+3B(N,Z)−3B(N−1,Z)+B(N−2,Z)], (1)

where is the (positive) binding energy and and are assumed to be even numbers. The proton pairing gap can be defined in a similar way. The trend of the extracted pairing gaps can be well approximated as MeV. However, as already noted in Ref. [7], above filter shows large local fluctuations and appears to be correlated with the shell structure.

The three-point formula is a simpler expression one can use to extract the empirical pairing gap from the binding energy [7, 14], which, for systems with even neutrons, has the form [14]

 (2)

where is the one-neutron separation energy. Above formula indicates that measures the additional binding gain by the last neutron in the even- system relative to the odd system with one more neutron. It should be mentioned that, besides pairing, a number of other mechanisms may contribute to the OES [14, 15, 16, 17]. This includes effects induced by the mean field in deformed nuclei (or the Kramers degeneracy) and the contribution from the diagonal interaction matrix elements of the two-body force. As discussed in detail in Refs. [14, 18], in even systems where the last neutrons occupy different orbitals the single-particle energy contributes substantially to .

The contribution from the quickly varying single-particle structure of the mean field to the empirical pairing gap is minimized in odd systems. One can propose another version of the three-point formula as

 Δ(3)C(N)=12[Sn(N,Z)−Sn(N−1,Z)]=12[B(N,Z)+B(N−2,Z)−2B(N−1,Z)]=12[S2n(N,Z)−2Sn(N−1,Z)], (3)

which actually corresponds to for the case of odd nuclei [14]. The values of extracted from experimental binding energies [19, 20] are plotted in Fig. 1. One may state that measures the pairing effect in the odd nuclei, whereas is impacted by single particle states. The value of has often been compared to the theoretical pairing gap calculated for the even systems [21, 22] and to the OES derived from theoretical binding energies [23, 24]. The direct comparison between the theoretical pairing gap and empirical OES is convenient from a computational point of view since only one single calculation is required, which avoids the complicated handling of the blocking effect in the odd nuclei.

The two three-point formulas are often written together as

 Δ(3)(N)=(−1)N+12×[B(N−1,Z)+B(N+1,Z)−2B(N,Z)] (4)

where takes both even and odd numbers and defines the number parity. However, it should be emphasized that, as we understand now, the physics behind the two quantities are very different.

The four-point formula can be rewritten as

 Δ(4)(N)=12[Δ(3)(N)+Δ(3)C(N)]. (5)

That is, it measures the average value of in adjacent even and odd systems.

There are other formulas available for the pairing gap. The five-point formula is given by [25, 26, 27]

 Δ(5)(N)=18[B(N+2,Z)−4B(N+1,Z)+6B(N,Z)−4B(N−1,Z)+B(N−2,Z)]=14[Δ(3)C(N+2)+2Δ(3)(N)+Δ(3)C(N)]. (6)

It indicates that the shell effect is still present in . The five-point formula is also used in Refs. [17, 28, 29]. and show quite similar results for most nuclei since varies smoothly [30, 31]. In Refs. [32, 33], the experimental pairing gap is taken as the average of adjacent ones deduced through the three-point formula as

 Δ(3)ave(N)=12[Δ(3)C(N)+Δ(3)C(N+2)], (7)

which is actually also a five-point formula involving the same group of nuclei as but with different weights for each nucleus. Again there is no significant difference between the results derived from and for open-shell nuclei where the pairing gap is a smooth function of . Noticeable differences between and may be seen where abrupt changes in pairing correlations are expected to happen, e.g., around shell closures, which is smoothed out in the former case. A quite sophisticated version of is used in Ref. [15] by subtracting the liquid-drop and shell effect contributions to the binding energy. The results are similar to those of . In particular, they show a quite similar isospin dependence.

show a much weaker dependence than other formulae. Actually it can be reasonably fitted as a constant value. This agrees with the suggestion in Ref. [16] that the pairing gap may not show any dependence. In [7] it is commented that the pairing energy derived from Eq. (2) is systematically too small. However, this is definitely not the case for which are systematically larger than all the other three cases. show the smallest values due to the reason that they largely remove the contribution from the mean field. In fact, the differences between and largely reflects the gap between the corresponding neighboring orbitals [14]. Another essential difference between and all other mentioned OES formulas is that diminish for closed-shell nuclei, which is in agreement with our common expectation that the pairing effect diminish at shell closure due to the reduced level density.

is smaller than , as well as in most cases. The differences between the various gap formulae and the 3-point formula are plotted in Fig. 2. The dispersal of the data below is apparent. This can be an indication of the significant mean-field contribution to the three formulae in this region, which is expected to show a dependence [16].

In the desired case, if the even nucleus of concern and the intermediate odd nucleus show the same mean field property, is not expected to contain any contribution from the mean field [14]. However, as pointed out by Bohr and Mottelson in P. 171 of Ref. [7], the average binding energy contains significant terms that are not linear in . As a result, from a macroscopic point of view, there may be a residual contribution to from the symmetry energy (expected to be negative) and other non-linear terms. To illustrate this point, in Fig. 3 we evaluated the values from the liquid drop model (LQM) with no shell or pairing energy corrections and the Duflo-Zuker (DZ) shell model mass formula [34] by removing the pairing term. The parameters of the LQM and DZ models are taken from Ref. [35] and [36], respectively. Calculations with the DZ model show more local fluctuations than those of the LQM, which has not been understood yet. But both cases show a vanishing behavior as increases. For heavy nuclei, this residual contribution is around -60 keV. This amount is acceptable by taking into account the fact the in practice the OES is a result of the delicate interplay between the mean field and the pairing correlation. It is hoped that, by using , the smooth non-linear terms in the binding energy can be canceled up to the fourth order. However, as mentioned in Ref. [15], the odd-even effects may be diminished as a result of the averaging over nuclei further apart.

contains fruitful information on the pairing effects. It is the best to remove the contribution from the varying part of the nuclear mean field as well as contributions from other shell structure details and can serve as a good measure of the pairing effect in in even- systems. Moreover, by using one can make it more convenient to extract the neutron-proton interaction from binding energy differences [37].

The possible isospin dependence of the empirical odd-even staggering was discussed in Ref. [38].

Within the shell model context, one has to separate the contribution from the monopole channel of the two-body interaction when studying the residual two-body correlations. For simple systems within a single- shell with a monopole pairing coupling , the total energy of a system with particles can be written as [3, 37]

 E = εn+2a−G4n(n−1) +b−2G2[T(T+1)−3n4] +(j+1)G(n−v)+G[v24−v+s(s+1)],

where is the single-particle energy, and defines the monopole interactions for the orbital , is the seniority quantum number, is the total spin, the reduced isospin. The monopole interaction is defined as the angular momentum weighted average of all two-body interaction matrix elements. They are independent of angular momentum and their contribution to the energy (or diagonal matrix element of the Hamiltonian matrix) are only related simply to the particle number and total isospin. The reduced isospin is related to the isospin of the states with seniority in the configuration, from which one can realize that we always have . For systems with the maximum isospin ( as in above case), the reduced isospin of any seniority state is . The (ground) state with is uniquely defined with reduced isospin for any configuration. In addition, the state is a state with and . From above equation it can be seen that it is the term that may result in an odd-even staggering in nuclear binding energies. This suggested that the residual pairing term in macroscopic mass formulas may be written as

 Ep∝2−v, (9)

where for odd- nuclei and for the /2 ground state of odd-odd nuclei. There should be no additional gain in pairing energy when crossing the line.

### 2.2 The empirical neutron-proton interaction

The (phenomenological) average interaction between the last protons and the last neutrons in even-even nuclei can be extracted from the double difference of binding energies as [39]

 Vee(Z,N) = 14[B(Z,N)+B(Z−2,N−2) (10) −B(Z−2,N)−B(Z,N−2)],

where is the (positive) binding energy of a nucleus with protons and neutrons. The factor takes into account the fact that four additional pairs are formed by the last two protons and neutrons. extracted from experimental nuclear binding energies [20] evolve rather smoothly as a function of mass number . In fact, this average behavior of also probes the symmetry energy term (i.e., the isospin-dependence of the binding energy) in the macroscopic mass formula. The overall trend of can be well approximated by a smooth relation of [40, 37]. Above formula has been extensively analyzed in Refs. [41, 42, 43, 44, 45, 46, 47].

The average np interaction in odd- and odd-odd nuclei can be extracted in a similar way as

 Veo(Z,N−1)=12[B(Z,N−1)+B(Z−2,N−2) −B(Z−2,N−1)−B(Z,N−2)], (11)
 Voe(Z−1,N)=12[B(Z−1,N)+B(Z−2,N−2) −B(Z−2,N)−B(Z−1,N−2)], (12)

which involve two np pairs and

 Voo(Z−1,N−1)= B(Z−1,N−1)+B(Z−2,N−2) −B(Z−1,N−2)−B(Z−2,N−1), (13)

involving one np pair. We have assumed that and only take even numbers in above equations.

In addition to the family of mass relations shown above, there is another way to extract the average np interaction as

 V1n−1p(Z,N)=B(Z,N)+B(Z−1,N−1) −B(Z,N−1)−B(Z−1,N), (14)

which is irrespective to the oddness of the proton and neutron numbers. This was proposed in Ref. [48] and applied recently in Refs. [49, 50, 51, 52]. This equation is identical to Eq. (2.2) studied above for the cases of odd-odd nuclei, but involves, in the other cases, the breaking of the proton and/or neutron pairs for which one intentionally avoid in the construction of the first family of average np interaction. By comparing Eqs. (2.2) and (2.2) one interesting thing we notice is that one can re-interpret for even-even nuclei as a measure of the np interaction between the two neutron and proton holes relative to the even-even ’core’ nucleus. This is illustrated in the lower-right panel of Fig. 4. Indeed, the values for for even-even nuclei follow roughly the same trend as those for the odd-odd nuclei with one less np pair. The latter case coincide with Eq. (2.2). This is also related to the fact that, as mentioned in Ref. [49], the average np interactions for even-even and odd-odd nuclei as extracted from the binding energies predicted by the Garvey-Kelson mass relations [53] satisfy

 V1n−1p(Z,N)−V1n−1p(Z+1,N+1) ≅ 0 (15)

One more thing that one should notice is that both above values are larger than for the neighboring even-even nuclei. The reason will be analyzed in the next section.

Stoitsov et al. showed that the global properties of can be reproduced by Hartree-Fock-Bogoliubov (HFB) calculations with the Skyrme functional plus a density-dependent pairing interaction [40]. A detailed calculation was also done in Ref. [54] where the effects of the deformation and collective fluctuation on were analyzed. It is still quite interesting to explore the local fluctuations of around the average values which large-scale HFB calculations fail to explain [40], which may carry further nuclear structure information and serve as a constraint in future developments of nuclear structure models.

It is understood that also measures the extra binding gained by the neutron (proton) pair when two additional protons (neutrons) are added. The two-nucleon separation energies in even-even nuclei can be written as

 S2n(Z,N)=2Sn(Z,N−1)+2Δ(3)C(Z,N). (16)

Eq. (10) can be rewritten as

 Vee(Z,N) = 12[Sn(Z,N−1)−Sn(Z−2,N−1)] (17) +12[Δ(3)C(Z,N)−Δ(3)C(Z−2,N)].

The quantities and measure the isospin dependences of the one-body separation energy (the mean-field) and pairing interaction, respectively. One can easily see that (and ) also reveals the average proton-neutron interaction between the last proton pair and odd neutron as

 Veo(Z,N−1) = 12δSn(Z,N−1). (18)

Contributions from the two basic ingredients and on the empirical proton-neutron interaction can be extracted from experimental nuclear masses. It is seen that is dominated by the contribution from . The and values are comparatively small, mostly within  keV. This indicates that the empirical proton-neutron interaction can to a large extent be understood as a mean-field or symmetry energy effect. It is also consistent with the observation of Ref. [40] that HFB calculations on are insensitive to the different choices of pairing forces.

Empirical studies of the nuclear masses suggest that the average np interactions for even-even and neighboring odd- nuclei thus extracted from experimental data are roughly the same and show a rather smooth behavior as a function of in most cases [37]. If the local fluctuations in the pairing interactions are negligible, it should be

 Vpn(Z,N)≈Vpn(Z,N−1)≈Vpn(Z−1,N), (19)

whereas those for the odd-odd nuclei are systematically larger than the former ones.

For a , system with three particles in a single- shell, we have and . The for such a nucleus can be expressed in the same form as above. The empirical relation of Eq. (19) still holds for these self-conjugate nuclei. In reality we have

 Vpn(Z,Z)≈Vpn(Z,Z−1)≈Vpn(Z−1,Z), (20)

where takes even values.

The empirical interactions between the odd proton and odd neutron in odd-odd nuclei can be extracted from binding energies in a way similar to those of even-even and odd- systems. The ground state of odd-odd nuclei may carry isospin quantum numbers or 1. For the lowest state one may extract the proton-neutron interaction as

 Vpn(Z−1,Z−1) =B(Z−1,Z−1)+B(Z−2,Z−2) −B(Z−1,Z−2)−B(Z−2,Z−1) =3b4−a. (21)

It indicates that the in odd-odd nuclei are three times as large as the average values in nuclei while those in even-even nuclei and the adjacent odd- nuclei with one less nucleon are roughly twice as large as those in neighboring nuclei. In reality should be positive. In medium mass and heavy nuclei, it should also be much larger than the pairing strength . In the spin-isospin SU(4) symmetry limit, the of nuclei are four times larger than those for [55]. There was also no difference between in even-even and odd-odd nuclei.

### 2.3 Residual neutron-proton interaction in odd-odd nuclei

On P. 171 of Ref. [7], Bohr and Mottelson pointed out that there is a systematic tendency for an extra binding of the odd-odd nuclei. It may be a result from the residual interaction between the last unpaired neutron and the unpaired proton in those nuclei. The average np interaction in Eq. (2.2) were often compared with the proton-proton and neutron-neutron pairing interactions. In reality, is a mixture of the mean field effect and the re-coupling effect due to the residual np interaction between the two unpaired particles. The mean field effect has to be properly filtered out if one aims at studying the residual np coupling. This is important for our eventual clarification of the role played by np pairing correlation in nuclei.

From a phenomenological point of view, it is understood that , and are dominated by contributions from the nuclear symmetry energy which is induced by the monopole np interaction. The non-vanishing values for as extracted from experimental data can be a measure of the residual/additional np re-coupling energy, which can be rewritten as

 δnp = Δ(3)n(Z,N)−Δ(3)n(Z−1,N) (22) = 12[B(Z,N)+B(Z,N−2) −B(Z−1,N)−B(Z−1,N−2) −2B(Z,N−1)+2B(Z−1,N−1)] = Voo−[2Veo+Voe−2Vee] = 12[V1n−1p(Z,N)−V1n−1p(Z,N−1)].

That is, it corresponds to the difference between and an weighted average of those for odd- and even-even nuclei or half the difference between for even-even nuclei and odd- nuclei. A very similar result can be obtained by taking the difference between the pairing gaps of the even-even and corresponding even--odd- nucleus. There are also different ways to extract the residual np interaction [56, 57, 58, 25, 26].

The residual np interaction as extracted from experimental binding energies [19] by using above three formulas are plotted in Fig. 5. It is seen that, as expected, the extracted values are positive for almost all known nuclei. The values of all above three formulas show a weak dependence on the mass number with large fluctuations. The mean values are around 300 keV in all cases. The values for available superheavy nuclei reduces to below 200 keV.

The positive contribution from the residual np interaction to the total binding energy is the origin of the odd-even staggering in that was studied in Ref. [49].

The additional binding in odd-odd nuclei is due to the residual interaction between the odd proton and neutron. That enhancement can be extracted from binding energy differences and is beyond the mean-field treatment. The challenge, however, is how to differentiate between a correlated np pair (in the BCS sense) and those uncorrelated ones.

### 2.4 The Wigner energy

Wigner noticed that there are large changes in binding energy in nuclei with approximately the same numbers of neutrons and protons [59]. This has often been referred to as the Wigner effect.

The residual np interaction discussed above have been applied in the study of the Wigner energy. In Ref. [55], van Isacker, Warner, and Brenner showed that the large double binding energy differences for even-even nuclei can be a consequence of Wigner’s SU(4) symmetry. Ref. [60] studied the separated contributions of neutron-proton pairs of a given angular momentum and isospin to the Wigner energy. It is also suggested that the Wigner term can be traced back to the isospin part of nuclear interaction. It cannot be solely explained in terms of correlations between the neutron-proton , pairs.

In Ref. [61] it is argued that the discrepancies between empirical shell gaps determined from binding energy differences and the gaps calculated with mean-field models can be resolved by taking into account the Wigner energy in the even-even nuclei and the corresponding nuclei with one less nucleon as induced by the diagonal correlation energy due to nn, pp, and np pairing interactions. The large difference between observation and mean-field calculation was already noticed by Bohr and Mottelson (see, Fig. 3-5 and P. 328 of Ref. [7]). It was pointed out that the calculated spectra reproduce approximately the observed positions of the single-particle levels above the Fermi surface, but underestimate the binding of deep-lying hole states. The increased binding of these states may be interpreted in terms of a velocity dependence of the mean field [7].

There are still extensive efforts trying to determine the Wigner energy in a precise way. One thing has to be considered is the effect of the symmetry energy on the extraction of the Wigner effect. The extracted Wigner energy can be very different if one takes the symmetry energy of the from instead of the normal .

## 3 Seniority and np coupling schemes

The low-lying yrast states in Pd were recently reported [62]. This is the heaviest nucleus with measured spectrum so far. It was suggested that in this nucleus, as well as in neighboring nuclei like Cd, the properties of the low-lying states can be classified by a spin-aligned pair coupling scheme [62, 63]. That is, the ground state wave functions do not consist mainly of pairs of neutrons () and protons () coupled to zero angular momenta, but rather of isoscalar pairs () coupled to the maximum angular momentum , which in the shell is [62, 63]. A detailed shell-model analysis of the spin-aligned pair coupling was performed in Refs. [63, 64] based on coefficients of fractional parentage and multistep shell model calculations.

In Fig. 6 we plotted the experimental spectra of three even- neutron-deficient Pd isotopes. As one would expect, in Pd with four proton holes with respect to Sn, the positions of the energy levels correspond to a isovector pairing (or seniority) spectrum. In particular, the yrast state, which has the largest spin among the states, shows a typical pattern for isomeric states as induced by the seniority coupling. When the number of neutron holes increases, approaching , the levels tend to be equally separated. The lowest excited states in Pd exhibit a particularly regular pattern. The regular pattern was first noticed by J. Blomqvist. He explained that in terms of aligned np pairs in a scheme that is similar to the stretched scheme as described in Ref. [65]. He also expected that systems with such couplings should be deformed.

In the following we will briefly describe the formalism used in above papers [63, 64] and in Refs. [66, 67, 68, 69]. We will construct the basis in the neutron-proton representation. We will show that it is a quite natural extension of the seniority coupling within the shell-model context. There are quite many recent publications following the same direction or re-interpret the results in slight different ways (see, Refs. [70, 71, 72, 73, 74, 75] and references contained therein). A detailed investigation on the applications of the nucleon-pair approximation of the shell model on this subject is presented in Ref. [76]. In any case the isospin symmetry is exactly conserved, as it is included in the interaction matrix elements that we use [63].

### 3.1 The seniority coupling

Many features in nuclear structure physics can be understood in term of the seniority coupling scheme. It showed to be extremely useful for the classification of nuclear states in the -scheme [3], particularly in semi-magic nuclei with only one type of nucleons. The driving force behind the dominance of seniority coupling is the strong pairing interaction between like particles. The study of the seniority coupling in single- systems is still an very active field [77, 78, 79, 80, 81], in particular in relation to the possible existence of partial dynamic symmetry.

The seniority coupling scheme can be generalized to systems within many shells. The diagonalization of the Hamiltonian matrix within the seniority model space provides a way to solve the pairing Hamiltonian in an exact way [82, 83, 84, 85, 30]. It is known that there is one nontrivial solution from the BCS ansatz which is interpreted as the nuclear ground state. Within the exact pairing scheme, there are as many eigenstates as the number of seniority states as defined within the model space. The state with the lowest energy is usually linked to the nuclear ground state. But it can still be interesting to understand the fundamental differences between the lowest-lying states and the excited states.

### 3.2 Number of pairs

An interesting quantity is the so-called number of pairs or number of interaction links which can give a simple idea on the relative importance of different two-body interactions and pairs on the total energy as well as on the nuclear wave function. For nucleons in a single- shell, the correlation energy can be written as [86]

 EI=CIJVJ, (23)

where is the total angular momentum, are two-body matrix elements and is the number of pairs with angular momentum . If isospin symmetry is conserved in the two-body interaction , then one has a total number of matrix elements with angular momenta to . The total number of nucleon pairs is given by [87, 86],

 ∑JCIJ=n(n−1)/2, (24)

and

 ∑J,oddCIJ=12[n2(n2+1)−T(T+1)], (25)

where is the total number of nucleons and is the total isospin quantum number.

For a fully-filled single- shell, one has . This is simply related to the contribution from the monopole interaction and there is no correlation. The monopole interaction is defined as the weighted average of the two-body interaction matrix elements as

 Vjj′ = ∑J(2J+1)VJjj′jj′∑J(2J+1)(1−δjj′(−1)J) (26) = ∑J(2J+1)VJjj′jj′(2j+1)1+δjj′2j′+1−δjj′.

Their contribution to the total energy corresponds to

 Em=∑jj′Vjj′⟨nj(nj′−δjj′)1+δjj′⟩, (27)

where denotes the number of particles instead of pairs. If only the pairing interaction is considered for the particle-particle channel, we have and .

### 3.3 Simple systems with two np pairs

The neutron-proton () correlation breaks the seniority symmetry in a major way. Correspondingly, the wave function is a mixture of many components with different seniority quantum numbers. It is not clear yet how this kind of states can be classified in the -scheme. The stretch scheme, which corresponds to the maximally aligned intrinsic angular momentum, was proposed in the 1960s to describe the rotational-like spectra of open-shell nuclei [65]. The quasi-spin formalism was applied in Refs. [3]. For a system with two np pairs in a single- shell, it is natural and very convenient to decompose the system into proton and neutron blocks. The wave function of a given state with total angular momentum can be written as [86],

 |ΨI⟩=∑Jp,JnXI(JpJn)|j2π(Jp)j2ν(Jn);I⟩, (28)

where is the amplitude of the four-body wave function and and are even numbers denoting the angular momenta of the proton and neutron pairs, respectively. For example, in the hole-hole channel, the ground state wave function of Cd is calculated to be [63],

 |Ψ0(gs)⟩ = 0.76|[π2(0)ν2(0)]I⟩+0.57|[π2(2)ν2(2)]I⟩ (29) + 0.24|[π2(4)ν2(4)]I⟩+0.13|[π2(6)ν2(6)]I⟩ + 0.14|[π2(8)ν2(8)]I⟩.

The four nucleons can couple to spin to and isospin , 1 and . The single- Hamiltonian can be written as,

 ⟨j2π(Jp)j2ν(Jn);I|^V|j2π(J′p)j2ν(J′n);I⟩ =(VJp+VJn)δJpJ′pδJnJ′n+∑JMIJ(JpJn;J′pJ′n)VJ, (30)

where the spin can take both even and odd values ( to ). The symmetric matrix is given as

 MIJ(JpJn;J′pJ′n)=∑λ4^Jp^Jn^J′p^J′n^J2^λ2{JpJnIλjj} ×{J′pJ′nIλjj}{jjJλjJn}{jjJλjJ′n}, (31)

where and and are half integers. The number of nucleon pairs in the system can be calculated as,

 CIJ=|XI(JpJn)|2(δJpJ+δJnJ) +∑JpJn;J′pJ′nXI(JpJn)MIJ(JpJn;J′pJ′n)XI(J′pJ′n), (32)

where the first and second terms in the right-hand side give the numbers of identical nucleon pairs and proton-neutron pairs, respectively.

The state in Fe and in the four-hole system of Cd below the double magic Sn have long been expected to be a spin-trap isomers. The latter case was measured recently [88]. That means their energies are lower than those of the corresponding and states. In shell, the correlation energy of the four-hole system is,

 E12(52Fe)=613¯V5+3¯V6+3313¯V7, (33)

where denotes two-hole matrix elements. The position of the state relative to the corresponding states is sensitive to the strength of interaction . The number of nucleon pairs for the 2 np system in the shell is,

 E16(96Cd)=817¯V7+3¯V8+4317¯V9. (34)

Again, the position of the state relative to the first state is sensitive to the strength of the aligned interaction matrix element .

### 3.4 Spin-aligned np pair coupling

To illustrate the idea of the spin-aligned pair mode we will start with the simple example of a - system within a single shell. One may re-express the wave function in Eq. (28) in an equivalent representation in terms of pairs. This can be done analytically with the help of the overlap matrix as

 ⟨[νπ(J1)νπ(J2)]I|[π2(Jp)ν2(Jn)]I⟩ =−2√NJ1J2^J1^J2^Jp^Jn⎧⎪⎨⎪⎩jjJpjjJnJ1J2I⎫⎪⎬⎪⎭, (35)

where denotes the normalization factor. The overlap matrix automatically takes into account the Pauli principle. With interactions taken from experimental data and Ref. [89], we examined a few shells with a high degeneracy, i.e., , and , values in the range for the ground states of these even-even nuclei. This means that the normal isovector pairing coupling scheme accounts for only about half of the ground state wave functions. Instead, we found that for these wave functions it is , i.e., they virtually can be represented by the spin-aligned coupling scheme. An even more striking feature is that the low-lying yrast states are calculated to be approximately equally spaced and their spin-aligned structure is the same for all of them. Moreover, the quadrupole transitions between these states show a strong selectivity, since the decay to other structures beyond the pair coupling scheme is unfavored. It should be emphasized that only states with even angular momenta can be generated from the spin-aligned coupling for systems with two pairs. The maximum spin one can get is . For a given even spin , only one state can be uniquely specified from the coupling of two aligned pairs. The other states (and also states with odd spins or total isospin ) involve the breaking of the aligned pairs.

Shell-model calculations for the nuclei Cd are plotted in Fig. 7. In the former case, the yrast states are all found to be dominated by the spin-aligned np coupling except the state. In that case the normal seniority coupling is favored in relation to the low energy of the isovector pair in Cd. On the other hand, it is the second state in Cd that favors the spin-aligned np coupling. The overlap between the full wave function and the spin-aligned np pair wave funciton is given in Fig. 8.

### 3.5 The over-complete basis

Calculations in Ref. [63] were done within the standard shell-model framework with the help of two-body coefficients of fractional parentage. In the following we go through the formalism as used in Ref. [64] within the so-called Multistep shell model approach (MSM), where the nn, pp and np pairs are considered on the same footing. We will use the Greek letter to label the -particle states. The states will be where the creation operator is and () is the neutron (proton) single-particle creation operator. In the same fashion the two-proton (two-neutron) creation operator will be denoted as (). The four-particle state, , is

 P+(γ4)=∑α2,β2X(α2β2;γ4)P+(α2)P+(β2)+∑γ2≤γ′2X(γ2γ′2;γ4)P+(γ2)P+(γ′2), (36)

where all possible like-particle and pairs are taken into account. In the two-pair case the basis elements and may be proportional to each other. The over-counting thus occurring is a result of describing the and like-particle excitations at the same time. Since the number of MSM basis vectors is larger than the dimension of the shell model space, the wave function amplitudes are not well defined in our case and, therefore, they are not meaningful physically. The meaningful quantities are the projections of the basis vectors upon the physical vector, which we denote as [90]

 F(α2β2;γ4) = ⟨γ4|P+(α2)P+(β2)|0⟩, F(γ2γ′2;γ4) = ⟨γ4|P+(γ2)P+(γ′2)|0⟩. (37)

 δγ4γ′4=∑α2,β2X(α2β2;γ4)F(α2β2;γ′4)+∑γ2≤γ′2X(γ2γ′2;γ4)F(γ2γ′2;γ4). (38)

The norm of the MSM basis , i.e., , may not be unity. Therefore the interesting quantity is not the projection but rather the cosine of the angle between the basis vector and the physical vector, i.e., and

 x(γ2γ′2;γ4)=F(γ2γ′2;γ4)/N(γ2γ′2;γ4). (39)

If we would have taken as basis elements the complete set of orthonormal states (which is the standard shell model basis as used in Ref. [63]) then the second term in Eq. (38) would not have appeared and one would have obtained , as expected in an orthonormal basis. One thus sees that the advantage of the MSM basis is that one can extract the physical structure of the calculated states just by examining the quantity .

The dynamic matrix of the two-neutron two-proton system is given as

 (W(γ4)−W(γ2)−W(γ′2))⟨γ4|(P†(γ2)P†(γ′2))γ4|0⟩=∑γ′′2⩽γ′′′2{∑p1p2n1n2(−1)W(γ′′2)+W(γ′′′2)−εp1−εp2−εn1−εn21+δγ′′2γ′′′2×(A1+A2)}⟨γ4|(P†(γ′′2)P†(γ′′′2))γ4|0⟩+∑α2β2{∑p1p2n1n2(W(α2)+W(β2)−εp1−εp2−εn1−εn2)×B}⟨γ4|(P†(α2)P†(β2))γ4|0⟩, (40)

and

 (W(γ4)−W(α2)−W(β2))⟨γ4|(P†(α2)P†(β2))γ4|0⟩=∑γ′′2⩽γ′′′2{∑p1p2n1n2W(γ′′2)+W(γ′′′2)−εp1−εp2−εn1−εn21+δγ′′2γ′′′2C}×⟨γ4|(P†(γ′′2)P†(γ′′′2))γ4|0⟩, (41)

where denotes the corresponding -particle energy. To obtain above equation we have assumed that the four-particle system was decomposed into two different blocks in terms of and .

The , and matrix elements are defined as,

 A1=(−1)2p1+n1+n2+γ′2+γ′′′2^γ2^γ′2^γ′′2^γ′′′2×X(p1n1;γ2)X(p2n2;γ′2)X(p1n2;γ′′2)X(p2n1;γ′′′2)×⎧⎪⎨⎪⎩p1n1γ2n2p2γ′2γ′′2γ′′′2γ4⎫⎪⎬⎪⎭, (42)
 A2=(−1)2p1+n1+n2+γ′2+γ′′′2+γ4^γ2^γ′2^γ′′2^γ′′′2×X(p1n1;γ2)X(p2n2;γ′2)X(p2n1;γ′′2)X(p1n2;γ′′′2)×⎧⎪⎨⎪⎩p1n1γ2n2p2γ′2γ′′′2γ′′2γ4⎫⎪⎬⎪⎭, (43)
 B=^γ2^γ′2^α2^β2X(p1n1;γ2)X(p2n2;γ′2)Y(p1p2;α2)×Y(n1n2;β2)⎧⎪⎨⎪⎩p1n1γ2p2n2γ′2α2β2γ4⎫⎪⎬⎪⎭, (44)

and

 C=^α