Angular Momentum Quantum Number

A shell, is composed of subshells. One might be inclined to think of subshells as simple subdivisions of shells, as lanes dividing a road. The subshells are much stranger. Subshells are regions of space where electron “clouds” are allowed to exist, and different subshells actually have different shapes. The first subshell is shaped like a sphere, (Figure below(s) ) which makes sense when visualized as a cloud of electrons surrounding the atomic nucleus in three dimensions. The second subshell, however, resembles a dumbbell, comprised of two “lobes” joined together at a single point near the atom's center. (Figure below(p) ) The third subshell typically resembles a set of four “lobes” clustered around the atom's nucleus. These subshell shapes are reminiscent of graphical depictions of radio antenna signal strength, with bulbous lobe-shaped regions extending from the antenna in various directions. (Figure below(d) )

Orbitals: (s) Three fold symmetry. (p) Shown: s_x, one of three possible orientations (s_x, s_y, s_z ), about their respective axes. (d) Shown: d_x²-y _x² similar to d_xz, d_yz, d_xz. Shown: d_z². Possible d-orbital orientations: five.

Valid angular momentum quantum numbers are positive integers like principal quantum numbers, but also include zero. These quantum numbers for electrons are symbolized by the letter l. The number of subshells in a shell is equal to the shell's principal quantum number. Thus, the first shell (n=1) has one subshell, numbered 0; the second shell (n=2) has two subshells, numbered 0 and 1; the third shell (n=3) has three subshells, numbered 0, 1, and 2.

An older convention for subshell description used letters rather than numbers. In this notation, the first subshell (l=0) was designated s, the second subshell (l=1) designated p, the third subshell (l=2) designated d, and the fourth subshell (l=3) designated f. The letters come from the words sharp, principal (not to be confused with the principal quantum number, n), diffuse, and fundamental. You will still see this notational convention in many periodic tables, used to designate the electron configuration of the atoms' outermost, or valence, shells. (Figure below)

(a) Bohr representation of Silver atom, (b) Subshell representation of Ag with division of shells into subshells (angular quantum number l). This diagram implies nothing about the actual position of electrons, but represents energy levels.