Magnetic model for alkali and noble metals: From diatoms to the solid state

An effective Hamiltonian of the Heisenberg type is derived from the alkali-metal diatom potential curves. Its qualitative relevance is exemplified on small clusters. Approximate solutions for the metal in various types of crystallization are obtained by consideration of one of the most-ordered spin distributions and by perturbation under its coupling with the less-ordered spin distributions. The results for Na are in good agreement with experiment for the lattice parameter (4.27 Å, experiment 4.225 Å at 5 K), the cohesive energy (1.19 eV, experiment 1.113 eV), and for the bulk modulus (0.0065 × ${10}^{12}$ ${\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^2$, experiment 0.0068 × ${10}^{12}$ ${\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^2$. The close-packed fcc and hcp structures are nearly degenerate, which explains the low-temperature martensitic transformation. The model may be extended to noble metals, provided that diabatic ${\left(d^{10}{s}^1\right)}^2$ potential curves are used for the ${}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ state of the diatoms, as illustrated for copper, giving reliable results. The connection of the present model with band models, and possible extensions to open $d$ shells, are discussed.