Microscopic investigation of one- and two-proton decay from the excited states of 10C

We present microscopic cluster model calculations for the 1p and 2p decays of the 0+2 and 2+2 states of 10C. With the R-matrix method, we have estimated the decay widths. The obtained 1p and 2p decay widths are in good agreement with the recent experimental data and support the validity of the di-proton approximation for the 2p decay of 10C(0+2). We also show the suppression of the 1p decay of the 10C(0+2) state due to the structure mismatch with the decay channel.

states of 10 C. With the R-matrix method, we have estimated the decay widths.The obtained 1p and 2p decay widths are in good agreement with the recent experimental data and support the validity of the di-proton approximation for the 2p decay of 10 C(0 + 2 ).We also show the suppression of the 1p decay of the 10 C(0 + 2 ) state due to the structure mismatch with the decay channel.

I. INTRODUCTION
The study of proton decay is particularly important because it offers a unique window into the structure of exotic nuclei near the proton dripline [1].The two-proton decay is one of the major interest.It was expected to occur when one-proton emission is energetically prohibited as firstly predicted by Goldansky in 1960 [2] and observed by experiment 20 years ago [3].Many discussions have been made about whether it is a true three-body decay (core+p+p) or a two-body decay (core+di-proton) and how it is related to the proton-proton correlation [4,5].
The 2p radioactivity has been observed not only in the ground states [6][7][8][9][10], but also in various excited states [11][12][13].In recent years, systematic measurements have been made for the excited states of 10 C [14,15].The 1p and 2p decays have been observed for the 0 + 2 and 2 + states, and the partial decay widths have been measured.There have been significant efforts to calculate the 2p decay.Some of them employ the R-matrix theory with the assumption of simultaneous emission of di-proton [16][17][18].Combining with the shell model wave functions, the 2p decay of 12 O, 18 Ne and 45 Fe have been studied [19][20][21][22].The extensions to the threebody model were also made by many others [5,23,24].However, the microscopic cluster models have not been applied to the study of 2p decay despite the importance of the cluster structure in light nuclei [25][26][27].Hence, developing the path of studying the proton decay with the microscopic cluster model is necessary.
In our previous works, a microscopic method to calculate the reduced width amplitude (RWA) has been developed [28][29][30][31].Using this method, in the present work, we investigate proton decays from the excited states of 10 C. We propose the prescriptions to determine the channel radius for the R-matrix calculation.The obtained results will be discussed in comparison with the recent experimental data [15].We found that, the molecule-like cluster of 10 C and 9 B has a strong impact on the decay pattern of the 0 + 2 and 2 + 2 states of 10 C.

A. The Hamiltonian and the wave function
We combine the real-time evolution method (REM) with the generator coordinate method (GCM) to obtain the wave function of nuclei, in which the single-particle wave function φ(r, Z) is expressed in a Gaussian form multiplied by the spin-isospin part χτ as Here the coordinates Z includes the spacial coordinates z and the spinor a and b, χ = a |↑ +b |↓ .The isospin part is τ = {proton or neutron}.The harmonic oscillator parameter is set to ν = 1/2b 2 where b = 1.46 fm, which reproduces the observed radius of 4 He and is generally used as in Refs.[32,33] and our previous works [29,30].
The wave function of the α cluster is constructed by the antisymmetrized wave function with (0s) 4 configuration as by simply set the same spatial coordinates z α for four particles.
The wave functions of nuclei composed of α-clusters plus valence nucleons are represented by With the help of the REM procedure introduced in Refs.[30,34], many wave functions with different coordinates Z are generated.The total wave function is given as the superposition of the basis wave functions after the angular momentum projection, where P J π M K is the parity and the angular momentum projector.The coefficients of the superposition f i,K and the corresponding eigen-energy E are obtained by solving the Hill-Wheeler equation.
The Hamiltonian adopted in this work is given as where ti and Tc.m. denote the kinetic energy operators of each nucleon and the center of mass, respectively.vN , vC , vLS denote the effective central nucleon-nucleon interaction, the Coulomb interaction, and the spin-orbit interaction, respectively.
For the central nucleon-nucleon interaction, we use the Volkov No.2 interaction [35], which is expressed as where W , M, B, and H denote the Wigner, Majorana, Bartlett, and Heisenberg exchange parameters.The other parameters are, V 1 = −60.65MeV, V 2 = 61.14MeV, c 1 = 1.80 fm and c 2 = 1.01 fm.We use the G3RS potential [36,37] as the spin-orbit interaction, Here P31 projects the two-body system into the triplet-odd state, which can be expressed The Gaussian range parameters d 1 and d 2 are set to be 5.0 fm −2 and 2.778 fm −2 , respectively.In this work, the exchange parameters of central interaction and the strength of the spin-orbit interaction are slightly modified to reproduce the decay Q-values.The determination of these parameters will be explained in the next section in more detail.

B. Reduced width amplitude and decay width
The reduced width amplitude (RWA) can be regarded as the wave function of a cluster in a nucleus, which can be used for the calculation of many other quantities through the Rmatrix theory [38].It is defined as the overlap amplitude between the A-body wave function of the mother nucleus Ψ and the decay channel composed of the residue nuclei with mass numbers A 1 and A 2 , Here Ψ A 1 and Ψ A 2 are the wave functions of the two residues and l represents the relative angular momentum between them.Eq. 8 is calculated by using the Laplace expansion method [28].In this work, the wave functions of the mother and daughter nuclei are calculated by the GCM explained above.The two-proton wave function is approximated by a single Slater determinant projected to where d denotes the distance between two protons which is set to be d = 0.5 fm describing a compact diproton state.Hence, we estimate the 2p decay width by assuming the two-body decay of 10 C(0 + 2 )→ 8 Be(0 + )+2p.According to the R-matrix theory, the partial width for the two-body decay A → A 1 + A 2 is given by the square of the RWA and the reduced mass µ as In this equation, P l,η (a) is the penetrability factor given as where F l,η and G l,η are the regular and irregular Coulomb functions.The wave number k and the dimensionless Sommerfeld parameter η are defined by the decay Q-value and the reduced mass as follows.
From Eq. 10, the partial width is obtained as a function of the channel radius a between two nuclei.Theoretically, the channel radius is the position where the nuclear force between the decay residues and the decay particle is negligible.Therefore, it should be chosen as the point where the RWA is smoothly connected to the Coulomb function.

III. RESULTS
A. Decay of the 0 + 2 and 2 + 2 states of 10 C The experiments [15] showed that the 10 C(0 + 2 ) predominantly decays to the ground state of 8 Be by the 2p emission and the 1p decay is suppressed, although both decay channels are open.This surprising result was explained by the mismatch of the valence proton configurations in 10 C(0 + 2 ) and 9 B(3/2 − ) within the context of the shell model [40].Alternatively, the molecular orbit model might be able to give a more reasonable explanation for this structure mismatch as the 10 C(0 + 2 ) has a pronounced cluster structure.In the molecular orbit model, 10 C, which is the mirror nucleus of 10 Be, is modeled as two α clusters coupled with two valence protons occupying so-called molecular orbits in analogy with the atomic molecules.It has been discussed that the valence protons occupy the π-orbit (negative-parity and mainly composed of p-shell) in the ground, 2 + 1 and 2 + 2 states, whereas they occupy the σ-orbit (positive-parity, mainly composed of sd-shell) in the 0 + 2 state.This structural difference of the 0 + 2 and 2 + 2 states impacts their decay pathways.First, let us consider the decay of the 10 C(0 + 2 ) state.After the emission of 1p occupying the σ-orbit, the residual nucleus 9 B also has a proton in the σ-orbit, whose wave function largely overlaps with the first excited state (1/2 + 1 ) of 9 B, but almost orthogonal to the ground state.Therefore, the 1p decay to the ground state of 9 B should be suppressed due to the structural mismatch.Note that the energy of 9 B(1/2 + ), which is a broad resonance, is higher than that of 10 C(0 + 2 ), and hence, the decay to the 9 B(1/2 + ) may also be suppressed.Consequently, the decay to the 8 Be by the simultaneous 2p emission might be the dominant decay pathway.
The situation is quite different for the 10 C( 2 + 2 ) state.After the 1p emission, the wave function of the residual nucleus largely overlaps with the 9 B ground state.Therefore the 1p decay should be dominant.This argument is based on qualitative expectations and has not been quantitatively confirmed by the nuclear structure model calculations.Therefore, in this study, we examine it numerically by using a microscopic cluster model that can properly describe the molecular structure of 10 C and 9 B.

B. Decay Q-values and interaction parameters
We first determine the parameters of the central and spin-orbit interactions to reproduce the experimental decay Q-values.The 0 + 2 state of 10 C has two decay pathways: one is to 9 B(3/2 − ) via emission of a single proton with the decay Q-value of 1.21 MeV, and the other is to 8 Be(0 + ) via emission of two protons with the Q-value of 1.40 MeV.Despite the negative Q-value, the decay to the 9 B(1/2 + ) might be also allowed, because it is a broad resonance.However, we will not investigate this pathway in the present paper.We also calculate the decay pathways of 10 C( 2 + 2 )→ 9 B(3/2 − )+p with the Q-value of 1.37 MeV.An ordinary set of the parameters, which we call set 1 in the following, is W = 0.4, M = 0.6, B = H = 0.125, and V ls = 2000 MeV.The set 1 has been widely used in previous calculations [30,33,39].However, it cannot reproduce the Q-values as shown in Fig. 1.
Hence, we introduce a slightly modified parameter set denoted by set 2 that is W = 0.44, 2 )→ 8 Be(0 + )+2p.The Q-value for 1p decay of 10 C(0 + 2 ) still cannot be reproduced, but it will not strongly affect the calculating result as we will see later.We also calculate the 10 C( 2 + 2 )→ 9 B(3/2 − )+p channel.For this case, set 1 is more appropriate.In short, we use set 2 for all calculations, except for the 1p decay of the 2 + 2 state.

C. RWA and decay width
In the decay channel of 10 C(2 + 2 )→ 9 B(3/2 − )+p, several relative angular momenta between 9 B and the proton are allowed, i.e.J π = 1/2 − , 3/2 − , 5/2 − , and 7/2 − .In Fig. 2, we compare the RWAs for the J π = 1/2 − and 3/2 − channels.It shows that the contribution of the J π = 1/2 − channel is small, and we found that its partial width is only a few keV.We also confirmed that the other angular momenta, 5/2 − , and 7/2 − are negligible.Therefore, in the present discussions, we only consider the J π = 3/2 − case for the 10 C( 2 + 2 )→ 9 B(3/2 − )+p channel.Fig. 3 shows the RWAs with respect to the decays of the 10 C(0 + 2 ) and 10 C(2 + 2 ) to the ground (3/2 − ) and first excited (1/2 + ) states of 9 B as well as the decay of the 9 B(1/2 + ) to 8 Be(0 + ).Since the RWA is the overlap of the wave functions between the mother nucleus and the decay residues, the amplitude of the RWA reflects the structural similarity between them.As already explained, 10 C(0 + 2 ) has similar structure with the 9 B(1/2 + ) but not 9 B(3/2 − ).Consequently, the amplitude of the RWA for the 10 C( 2 + 2 )→ 9 B(3/2 − )+p and 10 C(0 + 2 )→ 8 Be(0 + )+2p channels are large, whereas that of the 10 C(0 + 2 )→ 9 B(3/2 − )+p channel is negligible.These indicate that the 10 C(0 + 2 ) decays to 8 Be(0 + ) via 2p emission, or 1p emission to 9 B(1/2 + ).Furthermore, because of the negative Q-value of the decay to 9 B(1/2 + ), the 2p emission becomes the main decay pathway of 10 C(0 + 2 ).Contrary, 10 C(2 + 2 ) has the similar structure to the 9 B(3/2 − ), which makes the amplitude of the RWA for the 10 C( 2 + 2 )→ 9 B(3/2 − )+p channel large.Similarly to the decay of 10 C(0 + 2 )→ 8 Be(0 + )+2p, the valence proton in 9 B(1/2 + ) occupies the same orbit as in 10 C(0 + 2 ), so that the amplitude of 9 B(1/2 + )→ 8 Be(0 + )+p is large too.The longer tail of the RWA for 9 B(1/2 + )→ 8 Be(0 + )+p 2 )→ 8 Be(0 + )+2p channels.For these cases, we cannot determine the channel radius, and hence, we estimate an average of the expected decay widths.Comparing the RWA 9 B(1/2 + ) have been calculated.Within the diproton assumption, the 2p decay in 10 C(0 + 2 ) is also calculated.The width results are in good agreement with the recent experimental data in order of magnitude.These results demonstrate that the suppression of 1p decay in 10 C(0 + 2 ) is due to the mismatch structure of the valence protons, and the 2p decay in this state can be well explained by the two-body decay model.This work is the first time to apply the microscopic cluster model calculation to obtain the partial decay width.The calculation procedure still needs to be further developed, for example by applying it to the three-body decay model.

FIG. 1 .
FIG.1.Energy spectra measured from the energy of 8 Be(0 + )."Expt."denotes the experimental data [44].The 5.22 MeV excited state in the experiment is denoted as the 0 + 2 state of 10 C. "Set 1" and "Set 2" denote the results calculated using two sets of interaction parameters (see text).The arrow lines are the results adopted for the following discussions.

6 FIG. 5 .
FIG. 5. Same as previous figure but for the 10 C(2 + 2 )→ 9 B(3/2 − )+p channel (upper figure) and 10 C(0 + 2 )→ 8 Be(0 + )+2p (lower figure), respectively.The channel radii indicate the upper and lower limit for the width result.The average value in this region is derived as the width result shown in the gray box.