B. S. Ishkhanov ^1,2,
, S. V. Sidorov ^1,
, T. Yu. Tretyakova ^2,,
, E. V. Vladimirova ^1,2,
,

Corresponding author: T. Yu. Tretyakova, tretyakova@sinp.msu.ru

1.Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia
2.Skobeltzyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Received Date:2018-07-18
Accepted Date:2018-10-08
Available Online:2020-06-01
Abstract:

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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.