Alpha factor

The α-factor is a dimensionless quantity used to predict the solid–liquid interface type of a material during solidification. It was introduced by physicist Kenneth A. Jackson in 1958. In his model, crystal growth with larger values of α is smooth, whereas crystals growing at smaller α (below the threshold value of 2) have rough surfaces.

Method
According to John E. Gruzleski in his book Microstructure Development During Metalcasting (1996):
 * $$\alpha = \frac{L}{kT_\mathrm{E}}\cdot\frac{\eta}{v} $$

where $$L$$ is the latent heat of fusion; $$k$$ is the Boltzmann constant; $$T_\mathrm{E}$$ is the freezing temperature at equilibrium; $$\eta$$ is the number of nearest neighbours an atom has in the interface plane; and $$v$$ is the number of nearest neighbours in the bulk solid.

As $$\frac{L}{T_\mathrm{E}} = \Delta S_f$$, where $$\Delta S_f$$ is the molar entropy of fusion of the material,
 * $$\alpha = \frac{\Delta S_f}{k} \cdot \frac{\eta}{v}$$

According to Martin Glicksman in his book Principles of Solidification: An Introduction to Modern Casting and Crystal Growth Concepts (2011):
 * $$\alpha = \frac{\Delta S_f}{R_\mathrm{g}} \cdot\frac{\eta_1}{Z}$$

where $$R_\mathrm{g}$$ is the universal gas constant. $$\frac{\eta_1}{Z}$$ is similar to previous, always $$\frac{1}{4} < \frac{\eta_1}{Z}$$ < 1.