We examine in the light of recent experimental data smooth mass surfaces which may be placed in the form $M-A={\Delta }_m\mathrm{}\left(A\right)+J\left(A\right)\mathrm{}{[D-D_m\mathrm{}(A\left)\right]}^2,$ where ${\Delta }_m\mathrm{}\left(A\right)\mathrm{}$, $D_m\mathrm{}\left(A\right)\mathrm{}$, and $J\left(A\right)\mathrm{}$ are key functions which characterize the mass surface and $D=N-Z$ is the neutron excess. We here attempt to find an optimum set of key functions and to evaluate various semiempirical key functions now in use.

In this study we introduce the reference key functions ${\Delta }_m\mathrm{}\left(A\right)=\frac{{\mathrm{}(A-100\mathrm{})\mathrm{}}^2}{100}-64, \left(\mathrm{mMU}\right)$ $J\left(A\right)=\frac{25}{A}, \left(\mathrm{mMU}\right)$ and $D_m\mathrm{}\left(A\right)=\frac{0.4A^2}{\mathrm{}(A+200\mathrm{})\mathrm{}}.$ We use these reference functions in such a way as to effectively subject the data and various semi-empirical functions to microscopic examination, so that "fit" becomes immediately apparent. We note that most of the semi-empirical mass surfaces in current use give rise to large systematic errors in nuclear masses. The large errors are not inherent properties of the semi-empirical equation since a set of constants can be found which reduce these errors to within the range of uncertainty caused by shell effects.

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