 Solution

The exponential in the Peierls stress makes a large core-width-to-Burgers vector ratio $w/b$ favourable for easy dislocation glide. Densely packed slip planes often have wider cores because the shear distortion can spread over more atomic rows, reducing the lattice-periodic barrier. Close-packed directions provide short lattice translations and hence small allowed Burgers vectors; their line energies, approximately proportional to $G{b}^2$, are also relatively low. These effects favour dense planes and directions, but the relation between packing and core width is a physical argument rather than a consequence of substituting arbitrary independent $w,b$ into the formula. Body-centred cubic metals have densely packed directions without a truly close-packed plane, and their screw-core behaviour can require additional qualifications.