Generator coordinate method with proton–neutron pairing fluctuations and magnetic properties of N = Z odd–odd nuclei

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing correlations play an important role in a variety of nuclear phenomena. However, a quantitative understanding of proton–neutron ( pn ) pairing, especially isoscalar pn pairing ( S = 1 , T = 0) remains elusive. To clarify the property of pn pairing, we investigate the roles of pn pairing in the M 1 transition of N = Z odd–odd nuclei. We develop a theoretical model based on the generator coordinate method (GCM) in which the isoscalar and isovector pn -pair amplitudes are used as the generator coordinates. Using the particle and the angular-momentum projections, the pn -pair GCM well reproduces the M 1 transition of odd–odd nuclei for the exactly solvable SO(8) model. We apply the method to N = Z odd–odd nuclei and find that the experimental values of B ( M 1) are well reproduced. We also study the sensitivity of B ( M 1) to the strength of the isoscalar pairing interaction.

Pairing correlations play an important role in a variety of nuclear phenomena.However, a quantitative understanding of proton-neutron (pn) pairing, especially isoscalar pn pairing (S = 1, T = 0) remains elusive.To clarify the property of pn pairing, we investigate the roles of pn pairing in the M 1 transition of N = Z odd-odd nuclei.We develop a theoretical model based on the generator coordinate method (GCM) in which the isoscalar and isovector pn-pair amplitudes are used as the generator coordinates.Using the particle and the angular-momentum projections, the pn-pair GCM well reproduces the M 1 transition of odd-odd nuclei for the exactly solvable SO(8) model.We apply the method to N = Z odd-odd nuclei and find that the experimental values of B(M 1) are well reproduced.We also study the sensitivity of B(M 1) to the strength of the isoscalar pairing interaction. 1 typeset using P T P T E X.cls

INTRODUCTION
The pair correlation has significant impacts on a variety of nuclear properties, such as odd-even mass difference, rotational moments of inertia, fission dynamics, and so on [1].
The pairing in particles of the same kind (like-particle pairing), namely, correlations between two protons or between two neutrons that couple to spin singlet (S = 0, T = 1), has been extensively studied.However, protons and neutrons can be regarded as different isospin components of the same kind of particles "nucleons."It is natural to expect a Cooper pair composed of a proton and a neutron, leading to the pn-pair condensation [2].
The like-particle pairing is a part of the isovector (IV) pairing with (T = 1, T z = ±1).In the isospin-triplet pairing, there exists a (T = 1, T z = 0) pn channel.The isospin symmetry of the nuclear force implies that the IV pn pairing may play an important role in N = Z nuclei.In fact, experimental data of the binding energy of N = Z odd-odd nuclei in the 0 + 1 states suggest the occurrence of the T = 1 pair condensation [3].
On the other hand, our knowledge of the isoscalar (IS) pairing remains limited.Despite the strong attractive interaction in the (S = 1, T = 0) channel, there is no clear experimental evidence for the IS pair condensation.Nevertheless, some proton-rich nuclei are suggested to be close to the critical point [4], in which the quantum fluctuation associated with the IS-pair vibrations [1] plays an important role.Low-energy collective modes have been observed in the Gamow-Teller (GT) energy spectrum in N = Z odd-odd nuclei [5,6].Theoretical calculations suggest that the low-energy GT peaks develop as the IS pairing strength increases [7].
The deuteron-knockout reaction is also known to provide useful insights into the IS pair correlation.In Ref. [8] the triple differential cross section of the proton-induced deuteron-knockout reaction 16 O(p, pd) 14 N * and its sensitivity to the IS pairing strength are studied based on the nuclear density functional theory and the distorted-wave impulse approximation.
Since the IS pn Cooper pair has a nonzero spin (S = 1), we expect that the IS pairing correlation influences nuclear magnetic properties, such as the nuclear magnetic moment and the magnetic transition, which has been studied theoretically [9][10][11][12].In this paper, we analyze the role of the IS pairing on the nuclear magnetic properties based on a generator coordinate method (GCM) [13] with both the IV and the IS pn-pair amplitudes as the generator coordinates.We call this method as "pn-pair GCM."The collective wave functions obtained in the pn-pair GCM help us to visualize the quantum pn-pairing fluctuations.In order to remove undesirable mixing in the mean-field states, we combine the GCM with the projection technique on good total angular momentum (J) and good particle numbers of protons (Z) and neutrons (N ).In this paper, we apply the method to N = Z odd-odd nuclei.
If the ground states of proton-rich N = Z nuclei are close to the critical point of the IS pair condensation, large-amplitude fluctuation beyond the mean field is important.In addition, the small-amplitude approximation fails, as the quasiparticle random-phase approximation (QRPA) collapses at the critical point.The GCM adopted in the present study is suitable for the treatment of such large-amplitude collective motion [14,15].
Among the magnetic properties, we especially focus our study on the M 1 transition.Although the importance of (S = 1, T = 0) interaction in the M 1 transition of N = Z oddodd nuclei has been discussed based on the three-body model [9], the relation between the M 1 transition and IS-pair condensation has not been perfectly clarified, due to the limitation of the three-body model.The pn-pair GCM with the projection is able to provide an intuitive and quantitative answer to the question.Moreover, the M 1 transition is dominated by the GT operator of the IV-spin type (στ z ), and the present analysis on the spin-flip excitation may lead to the origin of the collective behavior observed in the GT transition [5,6].This paper is organized as follows.In Sec. 2, we introduce a pairing model Hamiltonian and describe the pn-pair GCM based on the projected pn-mixed mean-field states.In Sec. 3, we give a brief description of the calculation of B(M 1).Section 4 provides numerical results.
First, we use the SO(8) (single-shell) Hamiltonian and compare the result of the pn-pair GCM with that of the exact solutions.Then, we apply the method to the sd-shell nuclei and analyze how the pn pairing affects its nuclear structure and magnetic property.In Sec. 5, the summary and the conclusion are given.
2 pn-pair GCM where ĥspe represents the spherical single-particle energies, G IV is the IV pairing strength, G IS is the IS pairing strength, and G ph is the GT interaction strength.The IV pair operators R † ν , the IS ones P † µ , and GT operators F µ ν are defined as follows.
Here, the single-particle states c † k are labeled by the LS coupling indices with α denoting the radial quantum number, (l, m) denoting the orbital angular momentum, and (s, t) indexing the spin s = ±1/2 and isospin t = ±1/2.The index k stands for (α, l, m, s, t).The L = 0 coupling products are defined as where cαlmst = (−1) l+1+m+s+t c αl−m−s−t .R † 1 and R † −1 are commonly used as the pair creation operators of neutrons and protons, respectively.R † 0 corresponds to the IV pn-pair creation operator.P † µ creates IS pn pair where two nucleons couple to (S = 1, T = 0), and µ is the label of a spin component S z .F µ ν are the GT operators.In addition to these one-body operators, the spin operators Ŝµ , the isospin operators Tν , and the number operator N constitute so(8) algebra.When the single-particle-energy term ĥspe is omitted, all the matrix elements of the Hamiltonian are analytically evaluated with the basis belonging to the irreducible representation of SO (8) group.This Hamiltonian is called the SO(8) Hamiltonian [14,16,17].

Proton-neutron-mixing HFB + GCM
To incorporate the pn-pair condensation, we solve the generalized Hartree-Fock-Bogoliubov (HFB) equation [18], In the generalized HFB equation, the quasiparticle operators are mixtures of proton and neutron operators, that is, h and ∆ become 2 × 2 matrices in the isospin space.We solve the generalized HFB equation neglecting Fock terms.The solutions of the generalized Hartree-Bogoliubov (HB) equation are used as the basis state of the GCM, and the Fock terms are re-introduced when computing the Hamiltonian kernel in the GCM calculation.Adding the linear constraint terms with Lagrange multipliers, we obtain the constrained HB states |P 0 , R 0 ⟩ with a positive parity having the expectation values ⟨ P0 ⟩ = P 0 and ⟨ R0 ⟩/i = R 0 . 1 In general, the generalized HB solutions break the time-reversal and the axial symmetries.In the following calculation, we impose the axial symmetry by the conditions ⟨ P±1 ⟩ = 0. Here, ⟨ R±1 ⟩ are variationally determined under the proton and neutron number conditions.Even if the minimum-energy HB solution has ⟨ P0 ⟩ = 0 and ⟨ R0 ⟩ = 0, the pn-pair correlation may play an important role in the beyond-mean-field level, especially when the potential energy surface (PES) is relatively flat in the plane of (P 0 , R 0 ).In order to take into account the beyond-mean-field correlations, we use a trial wave function in the GCM ansatz [13] with the weight functions f J (P 0 , R 0 ), P N , P Z , and P J M K are neutron-number, proton-number, and angular-momentum projection operators, respectively.The values P max and R max /i are the maximum values in the adopted model space.To the best of our knowledge, for describing odd-odd systems, no attempts have been made before to perform the two-dimensional GCM calculation with both P 0 and R 0 as generator coordinates.It should be noted that odd-particle-number states are constructed as the one-quasiparticle-excited states in the HFB theory with the conventional like-particle pairing only.In contrast, the present pn-mixed states contain components with both even and odd numbers of particles.We are able to extract the odd-particle-number states by performing the number projections.
The variation of the energy ⟨Ψ JM | H |Ψ JM ⟩ with respect to the weight functions f J leads to the Hill-Wheeler equation, ) Solving the generalized eigenvalue equation, we obtain the eigenenergy and the weight function f (k) J corresponding to the k-th eigenstate with the good quantum numbers (N, Z) and (J, M ).In the numerical calculation, the integration is discretized with the mesh sizes of ∆P 0 = ∆R 0 = 0.5.For calculations of the Hamiltonian and the norm kernels, we use 11-points and 16-points Gauss-Legendre quadrature for the particle-number and the angularmomentum projections, respectively.In order to avoid the numerical instability associated with the overcompleteness of the GCM basis, we exclude the components corresponding to the small eigenvalues of the norm kernel [13].

Magnetic moment and M 1 transition
The magnetic dipole operator μ is defined as where µ N is the nuclear magneton, Lq and Ŝq (q = n, p) are the orbital angular momentum and the spin operators, respectively.g q (q = n, p) are the spin g-factor with g p = 5.586 and g n = −3.826.The nuclear magnetic moment is given by The M 1 reduced transition rate between the initial |J i ⟩ and the final state For convenience, we decompose μ into the IS and the IV parts, Utilizing the GCM wave functions based on the axially symmetric basis states, the reduced M 1 transition probability is evaluated as [19] |⟨J where Here, N and Ẑ are the neutron and the proton number operators, respectively, and Ĵy = Ly + Ŝy .by N and Z.The isospin symmetry is exact but is broken in the basis wave function of the pn-pair GCM.We do not perform the isospin projection to restore it.The IS and IV pairing strengths in Eq. ( 1) are parameterized as In the SO(8) Hamiltonian, g is the only parameter with the dimension of energy and determines the energy scale.The parameter x is varied and the GT interaction G ph is set to zero for simplicity.
In Fig. 1, the eigenvalues of the norm kernel for the S = 0 state and the S = 1 state are plotted in descending order.The clear gaps are observed around 10 −4 ; therefore we set the lower limit of the norm eigenvalues to 10 −4 .In the following sections for studies with multilevel models, since we encounter a severe overcompleteness problem, we use a larger value for the norm cutoff.See section 4.2.The energy of the lowest states of S = 0 and 1 are shown in Fig. 2. We compare the results of the pn-pair GCM with the exact ones.The minimum energy of the projected states P N P Z P J=S M 0 |P 0 , R 0 ⟩ in the (P 0 , R 0 ) plane is also shown by the dashed line, which in the followings, we call "minima in PAV-PES."The pn-pair GCM almost perfectly reproduces the exact energy, while there are some deviations in that of the minima in PAV-PES, especially in x > 0. Reference [20] shows that the HFB state obtained with the variation-after-projection (VAP) calculation well reproduces the exact solutions.
Although we perform neither the isospin projection nor the VAP for the particle numbers and angular momentum, the pn-pair GCM effectively incorporates full correlations for the entire range of the parameter x, from x = −1 (pure IS pairing) to x = 1 (pure IV pairing).We have confirmed that the isospin symmetry is practically restored in the pn-pair GCM states, e.g., the S = 0 (1) states have T = 1 (0) at high accuracy; see Fig. 3.The present pnpair GCM has an advantage in its applicability to realistic models.In addition, the collective wave functions provide an intuitive picture of the quantum fluctuation in the pairing degrees of freedom.
Next, we analyze the M 1 transition strength.In the SO(8) model, the orbital angular momentum does not contribute to the M 1 transition and it is sufficient to take the spin part where the IV pairing is dominant.Together with the observation in Fig. 3, this suggests that the isospin symmetry breaking may be a main source of the deviation.
The accuracy of the pn-pair GCM results suggest that the GCM with the two-dimensional generator coordinates (P 0 , R 0 ) is suitable for describing the J = 0 and J = 1 lowest-energy states of the N = Z odd-odd nuclei.The pn-pair GCM also reproduces properties of lowlying excited states, which may be useful for the analysis of the collective excitation modes such as pn-pair vibrations.In N = Z nuclei, the protons and neutrons occupy similar orbits.Particularly, the pnpair correlation is expected to play an important role in odd-odd nuclei where unpaired neutrons and protons exist in the like-particle pairing.We apply the pn-pair GCM to the N = Z odd-odd nuclei.For this purpose, the single-particle-energy term, ĥspe = i ε i c † i c i , that breaks the SO(8) symmetry, is activated in the Hamiltonian (1).The jj-coupling singleparticle energies with different values for protons and neutrons also break the spin and the isospin symmetry.
Among those N = Z nuclei, 18 F shows a large M 1 transition strength between the lowest 0 + and the 1 + states.The large B(M 1) was reproduced in the three-body model calculation, assuming an 16 O core + p + n structure with the residual pn interaction [9].The antisymmetrized molecular dynamics (AMD) calculation also reproduced the M 1 transition strength, and the spatial behavior of the pn correlation was analyzed in detail in Ref. [21].
In the following calculation, in order to describe the N = Z sd-shell nuclei including 18 F, the s + p + sd shells are used for the single-particle model space.We use the canonical single-particle energies of 16 O for the single-particle energies ε i of 18 F and those of 24 Mg for ε i of 22 Ne and 26 Al.The canonical single-particle energies are obtained with the Skyrme SkP energy density functional [22].We solve the HFB equation using the hfbtho (v1.66p) code [23] within the spherical symmetry.The obtained single-particle energies are summarized in Table 1.Although the d 3/2 orbitals are unbound, for a proper description of the IS pairing correlations, it is important to include those resonance states in the model space as the spinorbit partner of d 5/2 [9].We fix the IV pairing strength G IV = 1 MeV that is determined to reproduce the empirical pairing gap of 26 Ne.The GT interaction strength is set to G ph = 20.8/A 0.7 MeV to reproduce the position of the energy peak of the GT giant resonance of 48 Ca, 90 Zr, and 208 Pb in the pn-QRPA calculation [24].We examine the role of the IS pairing, by changing the IS pairing strength G IS .In this subsection, instead of the x parameter, we introduce a parameter g pp which is the ratio between the IS pairing and the IV pairing strengths, g pp = G IS /G IV .
In the following calculation, we set the value of the norm cutoff, 0.65, which results in the collective dimension (the number of the norm eigenvalues larger than the cutoff) equal to 8 for states projected on J = 0 and 1 in 18 F, 22 Ne, and 26 Al.The reason for this choice is as follows.We have found that, when we increase the collective dimension by reducing the value of the norm cutoff, the calculated collective wavefunction g J (P 0 , R 0 ) [13] exhibits a double peak structure even for the ground state.Since the diagonal elements of the Hamiltonian kernel do not show multiple local minima, this is most probably due to the overcompleteness of the adopted collective space.
The calculated energy difference between 1 + 1 and 0 + 1 states is plotted in Fig. 5.The energy difference E(1 + 1 ) − E(0 + 1 ) decreases as a function of g pp .This is because, even though the spin symmetry is partially broken, the IS (S = 1, T = 0) pair correlation is stronger in the J π = 1 + state than in the 0 + state.The clear correlation in Fig. 5 suggests that the energy (a) GCM, E(1 Energy difference between the 1 + 1 and the 0 + 1 states in 18 F obtained with the pn-pair GCM as a function of g pp (the left panel) or G ph (the right panel).G ph and g pp are fixed to be 2.72 MeV and 1.51 in the left and right panels, respectively.The dashed line shows the experimental value.difference E(1 + 1 ) − E(0 + 1 ) may serve as a possible observable to determine the values of g pp in the Hamiltonian.The experimental value in 18 F, E(1 + 1 ) − E(0 + 1 ) = −1.04MeV, leads to g pp = 1.51.This ratio is consistent with the estimated value in the previous studies [25].On the other hand, the energy difference is less sensitive to the GT interaction strength G ph .The same behaviors of the E(1 + 1 ) − E(0 + 1 ) are obtained for the 22 Na and 26 Al.The calculated energy differences are given in Table 2.Although the agreement with experiments is not quantitatively perfect, they are qualitatively well reproduced with the common interaction parameters: G IV = 1 MeV, g pp = 1.51, and G ph = 20.8/A 0.7 MeV.
Exp. (MeV) pn-pair GCM (MeV) 18  1 states in 18 F as a function of g pp .The blue solid line shows the total magnetic moment.The orange dashed and the green dash-dotted lines show the orbital and the spin parts, respectively.The black dotted line indicates µ = (g p + g n )/2 = 0.88µ N .

Magnetic moment and M 1 transition strengths
In Fig. 6, we plot the magnetic moment of the 1 + 1 state in 18 F as a function of g pp .With a larger value of g pp , the IS pair configuration becomes dominant.The magnetic moment reaches the IS spin g-factor value (g p + g n )µ N /2 = 0.88µ N .An IS pair of the valence proton and neutron (S = 1, T = 0) dominantly contributes to the magnetic moment.A partial restoration of the SO(5) symmetry, which becomes exact at the pure IV-pair model (g pp = ∞), takes place by the strong IS pairing strength.
The obtained values of the magnetic moment in 18 F, 22 Na, and 26 Al are summarized in Table 3.Although the magnetic moment of 1 + 1 state in 18 F has not been measured, the present value well agrees with the value 0.834µ N obtained with the three-body model [9].The pnpair GCM overestimates the magnetic moment for 22 Na, µ exp = 0.535µ N .The correlations Table 3 Magnetic moment µ, its orbital part µ L , and the spin part µ S in units of µ N in 1 + 1 states.The interaction strengths are G IV = 1 MeV, g pp = 1.51, and G ph = 20.8/A 0.7 MeV.
Next, we present the M 1 transition strengths between the 1 + 1 and 0 + 1 states.We plot the reduced M 1 transition probability B(M 1) as a function of g pp and G ph in Fig. 7(a) and (b), respectively.In addition to the total transition probability, we decompose the contributions from each term, the spin M 1 strength B(M 1) Fig. 8 PES without the angular momentum projection in the unit of MeV, in the twodimensional (P 0 , R 0 ) plane for 18 F.The number projection is performed.The g pp values are 0, 2, and 4 for the left, the center and the right panels, respectively.The star represents the energy minima in each graph.P 0 = R 0 = 0 solution does not have N = Z = 9 component; therefore the origins are excluded in these plots.function of g pp up to g pp ≲ 3, then, increases beyond that.In the range of realistic values of g pp < 2, B(M 1) may not be a good probe of the IS pairing strength.On the other hand, B(M 1) is less sensitive to the G ph , similarly to the energy difference in Fig. 5.
In the GCM framework, in addition to the experimental observables such as the excitation energies and the transitions, one can visualize the collective wave function in the collective space, the (P 0 , R 0 ) space in the present case.We analyze the interplay between the IS and IV pn-pairing correlations on the magnetic property in these odd-odd nuclei.The PESs defined as ⟨P 0 , R 0 |HP N P Z |P 0 , R 0 ⟩/⟨P 0 , R 0 |P N P Z |P 0 , R 0 ⟩ are shown in Fig. 8, and the collective wave functions g J=0 (P 0 , R 0 ) and g J=1 (P 0 , R 0 ) are shown in Fig. 9.
In the case that the IS pairing is switched off [Fig.8(a) and Fig. 9(a) or (d)], the energy minimum is located at P 0 = 0, and the collective wave functions also localize around P 0 = 0.In Fig. 8(b) for the case with g pp = 2, the PES has the shallow minimum around P 0 = 5.Corresponding to it, the collective wave functions have the peak around the minimum, though it is widely spread in the P 0 direction.In Fig. 8(c), the PES has a well-developed energy minimum around P 0 = 6 indicating a complete transition to the IS-pair condensation phase.
In Fig. 9(c) or (f), the collective wave functions are also localized in the same region.The prominent increase of P 0 ≈ 6 configurations in the g pp > 3 is reflected to the increases of B(M 1) S in Fig. 7.In the present model, g pp = 1.51 is the value that reproduces the energy difference between 0 + 1 and 1 + 1 states in 18 F. g pp = 4 is too strong to explain the experimental data.However, we should note that there is a possibility that the realistic N = Z nuclei are states, |g J=0 (P 0 , R 0 )| 2 .Lower panels: Those of the 1 + states, |g J=1 (P 0 , R 0 )| 2 .The g pp values are 0, 2, and 4 for the left, the center, and the right panels, respectively.Note that the projection on J = 0 (1) produces no state at R 0 = 0 (P 0 = 0).located close to the critical point of the IS-pair condensation [4], which may be probed by the low-energy peak in the GT transition strength in 18 O( 3 He, t) 18 F [6].
The concave behavior of B(M 1) as a function of g pp in Fig. 7 is due to the decrease of the interference term B(M 1) LS , which is associated with a monotonic reduction of B(M 1) L , from B(M 1) L = 0.42µ 2 N at g pp = 0 to 0.01µ 2 N at g pp = 4.At large values of g pp , since the strong IS S = 1 pairing configuration becomes dominant, we expect that the angular momentum of the J = 1 state is mainly carried by the spin.This explains the reduction of B(M 1) L .The spin dominance is confirmed by ⟨ Ŝ2 ⟩, shown in Fig. 10.Increasing g pp , the expectation value ⟨ Ŝ2 ⟩ becomes closer to the eigenvalues, S(S + 1) = 0 and 2 for J = 0 and 1 states, respectively.The single-particle energies with the spin-orbit splitting explicitly break the spin symmetry, however, the spin-SO(5) symmetry is approximately realized at large g pp .
We compare the M 1 transition strength in 18 F, 22 Na, and 26 Al.Their g pp dependence is shown in Fig. 11.All the lines decrease in small values of g pp , then, increase in large g pp .

Table 4
The reduced M 1 transition probability B(M 1; 0 + 1 → 1 + 1 ) for the N = Z oddodd nuclei in units of µ 2 N .The experimental values are taken from Refs.[26,27], among which those of 22   The magnitude of variation is the largest for 26 Al and the smallest for Fig. 11 The reduced M 1 transition probability as a function of g pp .The blue solid, orange dashed, and green dash-dotted lines correspond to B(M 1) in 18 F, 22 Na, and 26 Al, respectively.The interaction strengths are G IV = 1 MeV, g pp = 1.51, and G ph = 20.8/A 0.7 MeV.

Summary and Conclusions
We have developed the pn-pair GCM for describing quantum states of odd-odd nuclei based on the IS and IV pn-pair amplitudes as the generator coordinates (P 0 , R 0 ), combined with the particle-number and the angular-momentum projections.To assess the feasibility of the method, we have examined the GCM calculations for the spin 0 and 1 states of an odd-odd nucleus using the SO (8) Hamiltonian in which the exact solutions are available.The GCM solutions reproduce both the energy and the M 1 transition properties over the entire range of the parameters between the pure IV and the pure IS pairing limits.On the other hand, the single HB state with the projections fails to reproduce the M 1 transitions.
We have calculated N = Z odd-odd nuclei, 18 F, 22 Na, and 26 Al, introducing the singleparticle-energy term into the Hamiltonian, with six spherical orbits (s, p, and sd shells) both for protons and neutrons.A strong correlation between the energy differences E(1 + 1 ) − E(0 + 1 ) and the IS pn pairing strength is found.The IS pairing strength is determined by fitting the energy difference in 18 F.The same Hamiltonian well reproduces the energy differences in 22 Na and 26 Al systematically.
Increasing the IS pairing strength, the calculated PES in the (P 0 , R 0 ) plane becomes flatter and eventually leads to the IS pair condensation.In between the IS and the IV limits, where most probably the real N = Z nuclei are situated, the pn-pair GCM calculation predicts the coexistence of the different phases due to beyond-mean-field correlations.
The calculated M 1 transition strength in 18 F is not so sensitive to the IS pairing strength.
Increasing the number of the valence nucleons in the sd shell, the sensitivity grows.However, the calculated B(M 1) does not show a monotonic behavior as a function of the IS pairing.It shows a decreasing trend in the small IS pairing regime, then changes into an increasing behavior in the stronger IS pairing regime.In order to fix the IS pairing strength, the experimental information on the B(M 1) values for different N = Z odd-odd nuclei is valuable.
In the present work, we focus our studies on the role of the IS and IV pn pairings in oddodd N = Z nuclei.Performing the GCM calculations with other degrees of freedom, such as the quadrupole deformation, would be an important extension to understand beyondmean-field dynamics of both N = Z and N ̸ = Z nuclei.Some deviations for the pn-pair transfer reaction between experiments and calculations of the particle-particle random-phase approximation [4] were reported in Ref. [2].It may be interesting to apply the pn-pair GCM to the description of nuclei close to the critical point and to investigate properties of the pn-pair transfer.

2. 1
Multishell L = 0 pair-coupling model In order to treat the IV pairing and the IS pairing on equal footing, we use the following Hamiltonian, with L ≡ Ln + Lp , Ŝ ≡ Ŝn + Ŝp , Ľ ≡ Ln − Lp , and Š = Ŝn − Ŝp .The values g p − g n = 9.412 and g p + g n = 1.76 indicate that the IV spin part is dominant.It corresponds to the T z = 0 component of the GT operator ( Šµ = F µ 0 ), and the properties of the IV spin M 1 transition may be qualitatively applicable to the GT transition in the charge-exchange reactions and β decay.

4 Results 4 . 1
Validity of the pn-pair GCM for N = Z odd-odd systems We first show the validity and usefulness of the pn-pair GCM for a description of odd-odd nuclei, by comparing the results with those of the exact solution of the SO(8) Hamiltonian of the degenerate l shells without ĥspe .In this subsection, we treat N = Z = 5 nuclei with the degeneracy of the two l shells, Ω = (2l 1 + 1) + (2l 2 + 1) = 12 (p and g shells).The eigenstates of the SO(8) Hamiltonian have the following quantum numbers; the mass number A = N + Z, the total spin (S, S z ), and the isospin (T, T z ).A and T z = (N − Z)/2 can be replaced

Fig. 1 Fig. 2
Fig. 1 Eigenvalues of the norm kernel for the S = 0 state (the upper panel) and the S = 1 state (the lower panel).The horizontal lines indicate the value of the norm cutoff, 10 −4 .

Fig. 3 Fig. 4 B 4 . 2
Fig.3Calculated expectation values of isospin T 2 for the states obtained with the pn-pair GCM and the minima in PAV-PES states with projection on good N , Z, and J = S.

Fig. 6
Fig. 6 Magnetic moment µ of 1 + 1 states in 18 F as a function of g pp .The blue solid line shows the total magnetic moment.The orange dashed and the green dash-dotted lines show

Fig. 7
Fig. 7 The reduced M 1 transition probability, B(M 1; 0 + 1 → 1 + 1 ) in 18 F as a function of g pp (the left panel) or G ph (the right panel).The blue solid line shows the total M 1 transition probability.The orange dashed line and the green dash-dotted line show the contribution from the orbital part B(M 1) L or spin part B(M 1) S , respectively.The interference term B(M 1) LS is shown in the red dotted line.The black dashed line with the shaded region shows the experimental data and its error.

2
and the interference term B(M 1) LS =

Fig. 9
Fig. 9Square of the collective wave functions for 18 F. Upper panels: Those of the 0 +

Fig. 10
Fig. 10 Calculated ⟨ Ŝ2 ⟩ as a function of g pp .The blue solid and the orange dashed lines correspond to the J = 0 and J = 1 states, respectively.

Na and 26
Al are summed values over several transitions.The calculated total B(M 1) is decomposed into the orbital (B(M 1) L ), the spin (B(M 1) S ), and the interference (B(M 1) LS ) contributions.The interaction strengths are G IV = 1 MeV, g pp = 1.51, and G ph = 20.8/A 0.

18 F
, reflecting the number of valence particles.The variation originates from the decline of B(M 1) LS and the increases of B(M 1) S .The greater the number of the valence particles is, the smaller the g pp value at the minimum of B(M 1) becomes.Therefore, their relative values may possibly provide information about the IS pairing strength.The values of B(M 1) with g pp = 1.51 are also summarized in Table4.The B(M 1) value in18 F is consistent with the experimental values within the standard error.For 22 Na and26 Al, the decreasing trend of the B(M 1) as a function of the mass number is reproduced.

Table 1
Single-particle energies ε i for 18 F,22Ne, and26Al in the unit of MeV.