Periodic Trends in Atomic Radius, Ionic Radius and van der Waals' Radius

"Atomic size" is not a single fixed number — an atom has no sharp outer boundary, since the probability of finding an electron never quite reaches zero even far from the nucleus. Chemists instead use three distinct, operationally defined radii, each suited to a different situation.

Covalent (atomic) radius is half the distance between the nuclei of two identical atoms joined by a single covalent bond — e.g. half the $Cl-Cl$ bond length in $Cl_2$ gives the covalent radius of chlorine. This is the radius normally meant by "atomic radius" for a non-metal.

Ionic radius is the effective distance from the nucleus of an ion up to which it exerts its influence on the electron cloud of a neighbouring, oppositely charged ion in an ionic crystal, obtained by dividing the measured inter-nuclear distance between the two ions in a crystal lattice appropriately between them.

Van der Waals' radius is half the distance between the nuclei of two identical, non-bonded atoms of neighbouring molecules that are simply touching — as found, for example, between two adjacent molecules of a solid noble gas, or between non-bonded atoms of neighbouring molecules in a molecular crystal. Because it measures atoms that are only weakly touching rather than sharing a bonding electron pair, van der Waals' radius is always larger than the covalent radius of the same atom — there is far less overlap of electron clouds in a non-bonded contact than in a shared covalent bond.

Trend across a period. Atomic radius decreases on moving left to right across a period. Each successive element adds one more proton to the nucleus and one more electron to the same outermost shell (the principal quantum number $n$ does not change within a period). The added proton increases the effective nuclear charge felt by the outer electrons, pulling the same shell inward, so the atom shrinks — e.g. across Period 3, $Na > Mg > Al > Si$ in atomic radius.

Trend down a group. Atomic radius increases on descending a group. Each period adds a completely new, higher principal shell ($n$ increases by 1), and the increase in distance from a new outer shell far outweighs the accompanying increase in nuclear charge — e.g. $Li < Na < K$ in atomic radius.

Ionic radius rules. A cation is always smaller than its parent atom — removing one or more electrons reduces electron-electron repulsion (and, when an entire outer shell is emptied, removes a shell entirely), so the remaining electrons are pulled in closer by the unchanged nuclear charge; e.g. $Na^{+}$ is smaller than $Na$. An anion is always larger than its parent atom — adding electrons increases electron-electron repulsion in the same shell without adding any extra nuclear charge to compensate, so the electron cloud expands; e.g. $Cl^{-}$ is larger than $Cl$. …