Masses and Pairing Energies of Deformed Nuclei

Abstract

A description is considered of the mass surfaces and pairing energies of a number of even–even and odd deformed nuclei with mass numbers in the range of 150 to 190 using polynomials no higher than the second order. An approach in which pairing energy depends on the mass of one odd nucleus is applied. It is shown that the main features of the behavior of pairing energies are preserved in this approach.

Appendices

Parameters of Second Order Surfaces for Describing Energies of Even–Even Nuclei (Even N and Z)

Let us determine functions $e\left( {s,t} \right)$ and $e\left( {2,0} \right)$:

$$\begin{gathered} e\left( {s,t} \right) \equiv E\left( {N + s,Z + t} \right) + E\left( {N - s,Z - t} \right) \\ = 2\mathcal{E}\left( {N,Z} \right) + 2\sum\limits_{\begin{subarray}{l} i,k \geqslant 0 \\ i + k = 2\mu \\ \mu = 1;2;3; \ldots \end{subarray}} {{{d}_{{in,kp}}}\frac{{{{s}^{i}}{{t}^{k}}}}{{i!k!}}} {\kern 1pt} {\kern 1pt} ; \\ \end{gathered} $$

$$\begin{gathered} o\left( {s,t} \right) \equiv E\left( {N + s,Z + t} \right) - E\left( {N - s,Z - t} \right) \\ = 2\sum\limits_{\begin{subarray}{l} i,k \geqslant 0 \\ i + k = 2\nu + 1 \\ \nu = 1;2;3; \ldots \end{subarray}} {{{d}_{{in,kp}}}\frac{{{{s}^{i}}{{t}^{k}}}}{{i!k!}}{\kern 1pt} {\kern 1pt} } . \\ \end{gathered} $$

s-approximation:

$$\begin{gathered} E\left( {N,Z} \right) - \mathcal{E}\left( {N,Z} \right) \\ = E\left( {N,Z} \right) - \frac{1}{6}\left[ {4e\left( {2,0} \right) - e\left( {4,0} \right)} \right]; \\ \end{gathered} $$

$${{d}_{{1n}}} = {{o\left( {2,0} \right)} \mathord{\left/ {\vphantom {{o\left( {2,0} \right)} 4}} \right. \kern-0em} 4};\,\,\,\,{{d}_{{2n}}} = \frac{1}{{12}}\left[ { - e\left( {2,0} \right) + e\left( {4,0} \right)} \right].$$

t-approximation:

$$\begin{gathered} E\left( {N,Z} \right) - \mathcal{E}\left( {N,Z} \right) \\ = E\left( {N,Z} \right) - \frac{1}{6}\left[ {4e\left( {0,2} \right) - e\left( {0,4} \right)} \right]. \\ \end{gathered} $$

$${{d}_{{1p}}} = {{o\left( {0,2} \right)} \mathord{\left/ {\vphantom {{o\left( {0,2} \right)} 4}} \right. \kern-0em} 4};\,\,\,\,{{d}_{{2p}}} = \frac{1}{{12}}\left[ { - e\left( {0,2} \right) + e\left( {0,4} \right)} \right].$$

st-approximation:

$$\begin{gathered} E\left( {N,Z} \right) - \mathcal{E}\left( {N,Z} \right) \\ = E\left( {N,Z} \right) - \frac{1}{6}\left[ {4e\left( {2, - 2} \right) - e\left( {4, - 4} \right)} \right]. \\ \end{gathered} $$

$$\begin{gathered} {{d}_{{1n}}} = \frac{1}{8}\left[ {o\left( {2, - 2} \right) + o\left( {4, - 2} \right) - o\left( {2, - 4} \right)} \right]; \\ {{d}_{{1p}}} = \frac{1}{8}\left[ { - o\left( {2, - 2} \right) + o\left( {4, - 2} \right) - o\left( {2, - 4} \right)} \right]. \\ \end{gathered} $$

$$\begin{gathered} {{d}_{{2n}}} = \frac{1}{{24}}\left[ {e\left( {4, - 4} \right) - \frac{5}{2}e\left( {2, - 2} \right)} \right. \\ \left. { + \,\,\frac{3}{2}e\left( {2,2} \right) + e\left( {4, - 2} \right) - e\left( {2, - 4} \right)} \right]. \\ \end{gathered} $$

$$\begin{gathered} {{d}_{{2p}}} = {{d}_{{2n}}} - \frac{1}{{12}}\left[ {e\left( {4, - 2} \right) - e\left( {2,4} \right)} \right]; \\ {{d}_{{2n}}} + {{d}_{{2p}}} - 2{{d}_{{1n1p}}} = \frac{1}{2}\left[ {e\left( {4, - 4} \right) - e\left( {2, - 2} \right)} \right]. \\ \end{gathered} $$

Parameters of Second Order SUrfaces for Describing Energies of Odd Nuclei

Neutron Odd Nuclei

s-approximation:

$${{d}_{{1n}}} = {{o\left( {1,0} \right)} \mathord{\left/ {\vphantom {{o\left( {1,0} \right)} 2}} \right. \kern-0em} 2};\,\,\,\,{{d}_{{2n}}} = \frac{1}{8}\left[ {e\left( {3,0} \right) - e\left( {1,0} \right)} \right].$$

(st)_n-approximation:

$$\begin{gathered} {{d}_{{1n}}} = \frac{1}{4}\left[ {o\left( {3, - 2} \right) - o\left( {1, - 4} \right)} \right]; \\ {{d}_{{2n}}} = \frac{1}{8}\left[ {e\left( {1,2} \right) + e\left( {3, - 2} \right) - 2e\left( {1, - 2} \right)} \right]; \\ \end{gathered} $$

$${{d}_{{1n}}} - {{d}_{{1p}}} = \frac{1}{8}\left[ {o\left( {3, - 2} \right) + o\left( {1, - 2} \right)} \right].$$

Here, $e\left( {s,t} \right)$ and $o\left( {s,t} \right)$ are functions$E\left( {{{N}_{{{\text{odd}}}}} + s,Z + t} \right)$, where ${{N}_{{{\text{odd}}}}}$ is an odd number and Z is an even number.

Proton Odd Nuclei

t-approximation:

$${{d}_{{1p}}} = {{o\left( {0,1} \right)} \mathord{\left/ {\vphantom {{o\left( {0,1} \right)} 2}} \right. \kern-0em} 2};\,\,\,\,{{d}_{{2p}}} = \frac{1}{8}\left[ {e\left( {0,3} \right) - e\left( {0,1} \right)} \right].$$

(st)_p-approximation:

$$\begin{gathered} {{d}_{{1p}}} = \frac{1}{4}\left[ {o\left( { - 2,3} \right) - o\left( { - 2,1} \right)} \right]; \\ {{d}_{{2p}}} = \frac{1}{8}\left[ {e\left( {2,1} \right) + e\left( { - 2,3} \right) - 2e\left( { - 2,1} \right)} \right]; \\ {{d}_{{1n}}} - {{d}_{{1p}}} = - \frac{1}{8}\left[ {o\left( { - 2,3} \right) + o\left( { - 2,1} \right)} \right]. \\ \end{gathered} $$

Here, $e\left( {s,t} \right)$ and $o\left( {s,t} \right)$ are functions $E\left( {N + s,{{Z}_{{{\text{odd}}}}} + t} \right)$, where $N$ is an even number and Z_odd is an odd number.