The s, p and d orbitals

Shapes dictated by the number l

The shape of an orbital is set by the secondary quantum number l. Each type has a characteristic geometry, which can be visualized as the "silhouette" of the probability cloud.

The s orbital (l = 0): the sphere

The s orbital is spherical: the probability of presence depends only on the distance from the nucleus, not on the direction. The cloud surrounds the nucleus in a perfectly symmetric way.

   Orbitale s (vue en coupe) :

           . : ' ' : .
         :  . ' ' ' .  :
        : . '  (+)  ' . :      <- symetrie spherique
        : . '       ' . :         (aussi dense dans
         :  . ' ' ' .  :             toutes les directions)
           ' : . . : '

The p orbital (l = 1): the dumbbell

The p orbital has the shape of two opposite lobes, like a dumbbell, with a nodal plane at the level of the nucleus (probability exactly zero at the center). Three orientations exist, along the three axes:

   Orbitale p (une des trois, ex. "pz") :

              ___
             /   \\
            | lobe |      <- lobe "+"
             \\___/
              (+)         <- noeud au noyau (proba nulle ici)
              ___
             /   \\
            | lobe |      <- lobe "-"
             \\___/

   Les trois orientations :   px (axe x), py (axe y), pz (axe z)

The d orbital (l = 2): four lobes

The d orbitals are even more complex, typically with four lobes (cloverleaf shapes). Five orientations exist. We do not detail them here, but we note that they appear from n = 3 onward and play a key role for transition metals.

   Orbitale d (type "trefle") :

        lobe     lobe
           \\   /
            (+)          <- quatre lobes autour du noyau
           /   \\
        lobe     lobe

Summary of shapes

   Type  l   Forme          Nb d'orientations   Nb d'electrons max
   ----  --  -------------  -----------------   ------------------
   s     0   sphere              1                    2
   p     1   haltere             3                    6
   d     2   trefle (4 lobes)    5                   10
   f     3   complexe            7                   14

The number of electrons per type follows directly: (number of orientations) × 2 (the two spins). It is these capacities — 2, 6, 10, 14 — that draw the blocks of the periodic table (s block 2 wide, p block 6, d block 10, f block 14).

In summary

The shape of the orbital depends on l: s (l=0) is spherical, p (l=1) has the shape of a two-lobed dumbbell (3 orientations), d (l=2) presents four cloverleaf lobes (5 orientations). The number of electrons per type — 2, 6, 10, 14 — comes from the number of orientations multiplied by the two spins, and directly determines the width of the blocks of the periodic table.