Measurement of the Equilibrium Net Vacancy Concentration in Sodium

Simultaneous measurements of length and lattice thermal expansions have been made on two single-crystal rods of sodium, over a temperature range of -26°C to near the melting point of 97.8°C. The relative length changes were measured by means of two filar micrometer microscopes, and the lattice parameter changes were measured by a back-reflection, translated-sample method. The length and lattice expansions begin to diverge at approximately 10°C, with least-squares-fitted curves leading to a net atomic fraction of vacancies at the melting point of (0.94±0.11)×${10}^{-3\mathrm{}}$ and (0.66±0.15)×${10}^{-3\mathrm{}}$ for the two samples. These values are lower than some that exist in the literature but agree with one set of electrical resistivity measurements and with one set of specific-heat measurements. In the latter case, anharmonic contributions had been subtracted from the specific-heat data before arriving at a defect concentration. Proceeding on the assumption that only vacancy formation is present, energies of 0.157±0.014eV and 0.117±0.023 eV have been obtained by fitting straight lines to semilog plots of the concentration versus reciprocal temperature. These values are lower than any existing in the literature with the possible exception of that of one set of electrical resistivity measurements. Entropies of formation have also been calculated by extrapolating the plots to infinite temperature. These appear to be negative well within the errors. It is quite possible that interstitial formation is present, thus lowering the semilog plots at the high-temperature end and giving erroneously low and negative entropies. Thus these values may be less meaningful than the values for melting-point concentrations. A value is given for the lattice parameter of sodium at 25°C, and the fitted thermal expansion coefficients are presented.