Spectroscopic factors with generator coordinate method and application to 25Mg states

Nuclei can be studied through the analysis of their shell structure, that provides important information on the wavefunctions useful for interpreting many phenomena, from excitations to transfer reactions. Spectroscopic factors provide the degree of single particle behaviour of a nuclear state and are used to represent the shell structure of a nucleus. In this manuscript, we expand our recently developed model, which uses the generator coordinate method with an effective Hamiltonian, to calculate spectroscopic factors of states of odd deformed nuclei. Here we provide the results for 25Mg.


Introduction
Spectroscopic factors shows the degree of single-particle behaviour of nuclear states, and are therefore a useful tool for investigating shell closures, which characteristically show a high degree of single-particle behaviour, and other effects in shell evolution.Spectroscopic factors can also be used to construct optical potentials for use in scattering calculations.Constructing these potentials from microscopic theories has been challenging, especially for deformed nuclei [1,2].Therefore, calculating the spectroscopic factors for deformed nuclei would open up new possibilities for calculating scattering cross-sections.
In the following, we will provide a short overview of the method that we have developed for calculating spectra using a deformed generator coordinate basis, and go into some of the details in the procedure of using it to calculate spectroscopic factors.Since the Hamiltonian is constructed from a microscopic origin, and deformation is explicitely included in the model, it can be used to calculate spectroscopic factors of a wide range of nuclei.Finally, the results of this method applied on 25 Mg are presented and discussed.

Model
We use the model described in [3] that is based on calculating the generator coordinate method solutions to an effective Hamiltonian fitted to a density functional.The effective Hamiltonian is used to create a basis of HFB states using the quadrupole moment β and γ, pairing strengths g n and g p , and cranking frequency ω as generator coordinates.This gives a basis of HFB states |φ a , but since the fundamental symmetries has been broken by the quadrupole and pairing, we project the state to good angular momentum and particle number.
for the even and odd cases respectively, where P A = P Z P N , P Z , and P N are the projection operators for protons and neutrons, P J M K = |JM JK| is the projection operator for angular momentum, and i = (K, a).Finally, the Hill-Wheeler equation, is solved, where , E is the energy, and h are the coefficients of the solution in the projected basis.More details can be found in [3].This procedure then finally gives the solutions with A either even or odd, and n the label of the state.

Spectroscopic factors
The spectroscopic factors used within the definition of the one-particle Green's function, as described in [1,4], are is the ground state of the even system, and a † b is the single particle creation operator in the basis, with b = (n b , l b , j b , m b ).
Inserting the form of the solutions from (3) we get where the β quasiparticle operators are defined with respect to |Φ a .The terms are only nonzero when J = j b and M = m b .With these restrictions, we can use the relation by Enami [5] to get P J KM a † b P 0 00 = a † b K P 0 00 , where b K means the same quantum numbers as b but with m replaced by K. We also have that P A+1 a † b P A = a † b P A .This gives us the relation Writing the creation operator in terms of the quasiparticle operators, Since

Calculating the quasiparticle overlaps
To calculate the overlap (8), we use the technique described in [6], which uses the pfaffian, pf.Here we will describe a way to speed up the calculation of (8) for many different quasiparticles, but the same two even vacua.This is a general technique to calculate pfaffians.We first define |Φ to be the HFB state described by the U and V matrices U = D Ū C and V = D * V C, while |Φ is HFB state described by U = D Ū C and V = D * V C .We also define to be arbitrary quasiparticle operators, where U k , V k , U k , and V k are completely general and do not need to be related to U , V , U , or V .Then, the overlap where where A and C are skew-symmetric, Λ is a diagonal matrix with the diagonal equal to √ v 0 , √ v 0 , √ v 1 , • • • , with v i being the elements of V , and σ is a block diagonal matrix with all blocks equal to 0 1 −1 0 .The notation [•] n×n denotes the trunction of the matrix to n × n dimensions, which can be done since the high-energy quasiparticle states have very low occupation and does therefore not contribute to the overlap.The formula (10) can be adapted to calculate Φ|β k β k |Φ simply by substituting U k , V k → V k * , U k * in the expressions for A, B and C.However, applying the relation (10) directly would require the calculation of the pfaffian of a large matrix for each pair of quasiparticle operators.Fortunately, the structure of the matrix allows us to separate the part that depends on the quasiparticles.For the pfaffian of a block matrix, we have that if A is invertible.Since the overlap of the even states Φ|Φ is given by the pfaffian of A, and pf (A) 2 = det (A), A is invertible if and only if Φ|Φ = 0. Since A only depends on the even vacua, not on which quasiparticles we excite, and C + B T A −1 B is a skew-symmetric 2 by 2 matrix (since C and A are both skew-symmetric), the pfaffian of which is easily calculated as pf 0 x −x 0 = x, we can calculate the overlaps of many different quasiparticle excitations on the same even vacua, by simply calculating pf (A) and A −1 once, and then calculate the top-right element of the matrix C + B T A −1 B for each quasiparticle pair.

Results
The shell evolution of Magnesium isotopic chain has long been studied for the possibility of the "island of inversion" [7, 8, 9] Furthermore, 24 Mg and 25 Mg isotopes are of interest for astrophysical processes [10].We have, therefore, calculated the spectroscopic factors of 25 Mg.
The neutron spectroscopic factors of the even parity states of 25 Mg were calculated using the parameters used for 24 Mg in [3].That is, using SLy4-H effective Hamiltonian calculated for 25 Mg with 138 reference HFB vacua (cf.also [11]).The results are shown in Figure 1.The largest spectroscopic factors are seen for the low energy states 1/2 + and 5/2 + and a 1/2 + at ≈ 8.3 MeV.The spectroscopic factors of higher spin are increasingly smaller, since the g-shell and above are at a higher energy, so the lower energy states are of collective nature.
This manuscript provides the first application of model [3] to the calculation of spectroscopic factors.This result will lead to the analysis of shell structure using generator coordinate method and the calculation of optical potentials in deformed nuclei [4].

Figure 1 .
Figure 1.Neutron spectroscopic factors for positive parity states at different spins.For each spin the neutron spectroscopic factor of the first radial harmonic oscillator state is shown, except 2s 1/2 .The circled points for 1/2 + and 5/2 + are larger in magnitude than the rest, with values of 0.27 and 0.20 respectively.