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In our article we wrote the three-point mass relation based on deutron separation energies in the form
$ \begin{split} \Delta_{np}^{(3)}(N,Z) =& \frac{(-1)^{N+1}}{2}\left(S_d(N+1, Z+1) - S_d(N,Z)\right) \\ =& \frac{(-1)^{N+1}}{2}\left(B(N+1,Z+1)-\right.\\ & \left.- 2B(N,Z)+B(N-1,Z-1)\right), \end{split}$
(1) which holds true for the case of even-A nuclei. Factor
$ (-1)^{N+1} $ is taken into account to reproduce the even-odd staggering (EOS) effect for even-even and odd-odd nuclei. For the case of odd-A nuclei, on the other hand, the value of$ \Delta_{np}^{(3)}(N,Z) $ was shown to oscillate near the zero value (which corresponds to EOS for odd-even and even-odd nuclei), and taking the corresponding factor into account makes no sense. The corresponding formula (39) for the case when the factor is ommited from odd-A nuclei, should properly read$ \begin{split} \Delta _{np}^{(3)}(N,Z) =& \\ = &\frac{1}{2}\left\{ {\begin{array}{*{20}{l}} {({\pi _n} - {d_n}) + ({\pi _p} - {d_p}) - 2(I' + {I^0}),}&{ee}\\ {({\pi _n} + {d_n}) + ( - {\pi _p} + {d_p}) + 2{I^0},}&{oe}\\ {( - {\pi _n} + {d_n}) + ({\pi _p} + {d_p}) + 2{I^0},}&{eo}\\ {({\pi _n} + {d_n}) + ({\pi _p} + {d_p}) - 2(I' - {I^0}),}&{oo} \end{array}} \right. \end{split} $
(2) As a result, for the case of odd-A nuclei factor
$ (-1)^{N+1} $ leads to the change of general sign for even-N nuclei:$ \begin{split} \Delta_{np}^{(3)}(N,Z) =& \frac{(-1)^{N+1}}{2}\left(S_d(N+1, Z+1) - S_d(N,Z)\right) \\=&-\frac12\left((-\pi_n + d_n)+(\pi_p + d_p)+2I^0\right), \end{split} $
(3) The inclusion of factor
$ (-1)^{N+1} $ for odd-A nuclei significantly affects the$ \Delta_{np}^{(4)} $ , resulting from the averaging of$ \Delta_{np}^{(3)}(N,Z) $ . Instead of expression (40) we get:$ \Delta _{np}^{(4)}(N,Z) = \frac{1}{2}\left\{ {\begin{array}{*{20}{l}} {\left( {{\pi _n} + {\pi _p}} \right) - 2I',}&{ee,oo}\\ {{\pi _n} - {\pi _p},}&{oe,eo} \end{array}} \right. $
(4) Since, as noted above, we are talking about values close to zero, the noted changes do not affect the main conclusions of the article. However, the ratio in Eq. (42)
$ \pi_p \approx \pi_n $
is approximate and, as can be seen from Table 2, the values of
$ \pi_p $ consistently exceed the values of$ \pi_n $ . From this point of view, the choice of the factor$ (-1)^{Z+1} $ used in the expression coinciding with$ \Delta_{np}^{(4)} $ in [1] is more reasonable.One more remark concerns the formulas for
$ \Delta_{np}^{MN} $ first introduced in [2]. In formula (19), the proper factor should be$ (-1)^{A+1} $ . In (20), the two cases correspond to even and odd values of Z rather than N, while in (21), vice versa, these are the cases of even and odd N rather than Z.
Erratum and Addendum: Empirical pairing gaps and neutron-proton correlations (Chin. Phys. C, 43(1): 014104 (2019))
- Received Date: 2018-07-18
- Accepted Date: 2018-10-08
- Available Online: 2020-06-01
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