Global phenomenological descriptions of nuclear odd-even mass staggering

We examine the general nature of nuclear odd-even mass differences by employing neutron and proton mass relations that emphasize these effects. The most recent mass tables are used. The possibility of a neutron excess dependence of the staggering is examined in detail in separate regions defined by the main nuclear shells, and a clear change in this dependency is found at $Z=50$ for both neutrons and protons. A further separation into odd and even neutron (proton) number produces very accurate local descriptions of the mass differences for each type of nucleons. These odd-even effects are combined into a global phenomenological expression, ready to use in a binding energy formula. The results deviate from previous parametrizations, and in particular found to be significantly superior to a recent two term, $A^{-1}$ dependence.

Figure 1

The structure of ${\Delta }_n$, ${\Delta }_p$, ${\Delta }_n^{\left(4\right)}$, ${\Delta }_p^{\left(4\right)}$, ${\Delta }_n^{\left(5\right)}$, and ${\Delta }_p^{\left(5\right)}$ given in the $NZ$ plane. The relative weight of each individual isotope, not including normalizing factors, is also included.

Figure 2

The odd-even staggering ${\Delta }_n$ and ${\Delta }_p$ as a function of $A$ evaluated using Eqs. (5) and (6).

Figure 3

The $(N,Z)$ dependence of ${\Delta }_n$. Figure (a) shows $49<N<84$ and $30<Z<69$, and Figure (b) shows $81<N<127$ and $49<Z<91$.

Figure 4

The extent of shell effects. Values calculated using Dieperink and Van Isacker's expression Eq. (9) are combined as in Eq. (2) using $Q(1,0)$ to estimate the remaining shell contribution for ${\Delta }_n$.

Figure 5

The ${\left(\frac{N-Z}{A}\right)}^2$ dependency for all even-even nuclei with $A>50$. The line is the best possible fit based on Eq. (11). The nuclei indicated in red (fat and full) are influenced either by the Wigner effect or by shell effects and are not included in the calculations. The dashed, blue line is the best result when using a model with a linear dependence on the absolute value of neutron excess, $|\frac{N-Z}{A}|$.

Figure 6

The ${\left(\frac{N-Z}{A}\right)}^2$ dependency of even-even nuclei divided into regions defined by major nuclear shells as given in the figures.

Figure 7

The ${\left(\frac{N-Z}{A}\right)}^2$ dependency for nuclei with $50<N<82$ and $50<Z$ divided according to whether $(N,Z)$ is even-odd, odd-even or odd-odd. The even-even nuclei are included in Fig. 6.

Figure 8

The ${\left(\frac{N-Z}{A}\right)}^2$ dependency for all even-even nuclei with $A>50$ with a shell effect correction given by Eq. (9).

Figure 9

Best global fit with Eq. (16) for even $Z$ nuclei and $A>50$. The nuclei marked in blue are restricted by $82<N$ and $50<Z$. Nuclei influenced by shells or the Wigner effect are not shown.

Figure 10

The equivalent to Fig. 5 only with respect to $|\frac{N-Z}{A}|$. Here the dashed, blue line is the best fit quadratic in neutron excess.