linear atomic density
Suppose you walk in a straight line through a crystal along some direction and count how many atom centres you actually step on per unit of distance. That count is the linear atomic density. It is a way of measuring how crowded a particular direction is — some directions run straight through a dense row of touching atoms, others thread through mostly empty space.
The number is defined as the number of atoms whose centres lie on the direction vector, divided by the length of that vector: LD = atoms per length. A related and often more telling quantity is the linear packing fraction, the share of the line actually covered by atoms, which for a close-packed direction reaches 1. Take face-centred cubic along [110], the face diagonal: within one repeat of length a times sqrt(2) the line passes through two atom centres (a corner atom, a face-centre atom, and the next corner), so LD = 2 / (a times sqrt(2)) = sqrt(2) / a. Because the atoms touch along that diagonal (4r = a times sqrt(2)), the line is fully covered — a packing fraction of 1.
Why bother counting? Because the direction of highest linear density is the one along which atoms are packed shoulder to shoulder, and that is the direction a dislocation prefers to glide. Linear density is how you pick out the slip directions of a metal without any experiment — you simply find the most crowded lines.
In face-centred cubic, [110] has linear density sqrt(2)/a and a packing fraction of 1 (atoms touch), while [100] passes through only corner atoms — one per length a, with the atoms not touching. The denser [110] direction is exactly the observed slip direction in FCC metals.
Count only atom centres that lie on the line; the densest line is the slip direction.
Count only atoms whose centres actually lie on the line — an atom merely near the line does not count. Values depend on both the direction and the crystal structure.