Sodium Diffusion in Sodium Tungsten Bronze

Diffusivity has been measured in single crystals of the metallic sodium tungsten bronze, Na_0.78WO₃, at 664°C, 752°C, and 832°C. Concentration gradients were established by effusion of sodium from single crystals into a vacuum and were measured by determining the variation in the lattice constant. The diffusivities were calculated by a method which is based upon Fick's first law and which, to the author's knowledge, has not been used previously. The data obtained from several crystals held at constant temperature for varying periods of time gave a family of curves showing the sodium concentration as a function of depth with time as a parameter. From these curves the mass of sodium transferred through a plane parallel to the surface and at any particular depth could be obtained; this mass was plotted as a function of time. The slope of this curve divided by the concentration gradient, both evaluated at the same time and depth, was taken as the diffusivity for the corresponding concentration. The diffusivity of sodium in Na_0.78WO₃ was found to be represented by the equation D=D₀ exp (—ΔH/RT), where D₀=0.87 cm²/sec and ΔH=51.8 kcal/mole. In order to calculate the diffusivity it was necessary to determine the coefficients of linear expansion. For Na_0.8WO₃ the length L at temperature T_c was found to be $${\mathrm{L=L}}_0{\text{[1+(8.81×10}}^{\text{−6}}{\mathrm{)T}}_c{\text{−(1.92×10}}^{\text{−9}}{\mathrm{)T}}_c^2\mathrm{],}$$ where L₀ is the length at 0°C and T_c is the temperature in degrees centigrade.