= NaCl conventional cell and coordination
{c}
Ordinary solid <sodium chloride> has the <rock salt structure>. Choose a conventional cube of side $a$ with <chlorine> <anions> at the corners and face centers, and <sodium> <cations> at the edge centers and cube center. Corner sites are shared by eight cubes, face sites by two and edge sites by four. The <unit cell> counts are therefore
$$
N_{\mathrm{Cl}^-}=8\left(\frac18\right)+6\left(\frac12\right)=4,\qquad
N_{\mathrm{Na}^+}=12\left(\frac14\right)+1=4.
$$
There are four <formula units> per conventional <unit cell>, rather than four separate NaCl <molecules>.<Chemistry 2e>[§10.6, “Unit Cells of Ionic Compounds”, NaCl paragraph and Figure 10.60: corner/face chloride sites, edge/center sodium sites and four ions of each kind]{ref}
\Image[/nacl-conventional-cell-and-coordination.png]
{title=NaCl conventional-cell accounting and the six nearest unlike neighbors}
{id=image-nacl-conventional-cell-and-coordination}
{description=Original ideal <rock salt structure> geometry. The left panel includes all boundary sites before sharing is accounted for; the right panel isolates one <sodium> <cation> and its six nearest <chlorine> <anions>. Grey segments indicate distances, not discrete <molecules>. Marker sizes do not encode ionic radii.}
The <Bravais lattice> is a <face-centered cubic lattice>. One choice of <primitive lattice translation vectors> and <atomic basis> is
$$
a_1=\frac a2(0,1,1),\quad a_2=\frac a2(1,0,1),\quad a_3=\frac a2(1,1,0),\qquad
r_{\mathrm{Cl}^-}=0,\quad r_{\mathrm{Na}^+}=\frac a2(1,0,0).
$$
The <determinant> gives a <primitive unit cell> <volume> of $a^3/4$; that cell represents one <cation> and one <anion>. Four such volumes make the conventional cube. The union of all <ion> positions resembles a <primitive cubic lattice> of step $a/2$ only if the chemical labels are discarded: translation by $a(1,0,0)/2$ exchanges unlike <ions> and fails to preserve the chemical structure.
Each <ion> has six nearest unlike neighbors at distance $a/2$, along the positive and negative directions of the three cube axes. Thus the <coordination number> is six. Within either same-species <face-centered cubic lattice>, the nearest same-ion neighbors instead number twelve: their displacements are permutations of $(\pm a/2,\pm a/2,0)$, all of length $a/\sqrt2$. Since $a/\sqrt2>a/2$, those twelve are a more distant shell in the complete <sodium chloride> structure. The number twelve describes the same-ion <FCC lattice> neighborhood; six describes the nearest unlike-ion neighborhood.
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