Answer to Question #164779 in Electrical Engineering for Mital sosa

Question

(a) A thin-film layer of thickness d is composed of contacting upright cylindrical grains of radius r. If these columnar grains pack in a square array (in plane view) what is the packing factor for this film?

(b) Suppose smaller cylindrical grains exactly fill in all the interstices between the larger original grains to create a denser bimodal grain-size structure. What is the packing factor for such a film?

Answer 1

In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity. In atomic systems, by convention, the APF is determined by assuming that atoms are rigid spheres. The radius of the spheres is taken to be the maximum value such that the atoms do not overlap. For one-component crystals (those that contain only one type of particle), the packing fraction is represented mathematically by

d - thickness r - radius in a square array

$APF = \large\frac{N_{particle}V_{particle}}{V_{unit \space cell}}$ $=$ $\large\frac{1*\frac{1}{3}\pi r^2H}{(2d)^3} = \frac{\pi}{24} *(\frac{r}{d})^3$