Intuition The ONE core ideaAn ionic crystal is nothing more than two grids of charged balls — a grid of + − ρ Z a r + r − which balls touch which , and how many balls live in one repeating box .
Before you can read a word of the parent note, you need the vocabulary below. We build each term from a picture, and each one leans on the one before it. Nothing is assumed.
An ion is an atom that has gained or lost electrons, so it carries a net charge. A cation (+ anion (− hard sphere — a solid ball that cannot be squished into another ball.
r radius of that ball. We write r + r − + − which ball — plus for the positive cation, minus for the negative anion.
Two balls that "touch" means their surfaces meet: the distance between their centres equals the sum of their radii. If we call one ball's radius r + r − touching = centre-to-centre distance is r + + r − . That single fact is the engine behind every formula in this chapter.
Intuition Read figure s01In the picture the blue cation (radius r + r − × . Notice the two radius arrows lie tip-to-tip along one straight line — that is why the centre-to-centre dotted line has length exactly r + + r −
Intuition Why hard spheres?Real electron clouds are fuzzy, but treating ions as hard balls is good enough to predict which structure forms. It turns chemistry into geometry: once the balls are hard, "do they fit?" becomes a ruler-and-compass question.
Cations are almost always smaller than anions (losing electrons shrinks the cloud; gaining them puffs it out). Keep that mental image: small + −
ρ ρ = r − r + measured in units of the anion . The Greek letter ρ
Intuition What the ratio picturesρ ρ = 0.4 40% tiny ρ ρ 1
Because both radii are lengths in the same units (picometres, pm), ρ no units — it is a pure number. That is why it can be compared against fixed cut-offs like 0.414
Recall Quick self-testIf r + = 100 r − = 250 ρ ρ = 100/250 = 0.40
Definition Coordination numberThe coordination number (CN) of an ion is how many oppositely-charged ions are directly touching it — its nearest neighbours.
We write CN as a pair like "6:6" or "8:4": the first number is how many anions touch each cation, the second is how many cations touch each anion.
Intuition Read figure s02Left panel: a small red cation lets only a few yellow anions crowd around before they would bump each other → low CN. Right panel: a bigger green cation shoves the anions apart, so more of them fit around it → high CN. Comparing the two panels is the reason for the golden rule bigger ρ ⇒ — the single most important trend in the chapter.
Common mistake "More neighbours is always lower energy, so CN should be as big as possible."Why it feels right: each extra + / − The fix: past a limit the anions themselves start touching and repelling. Nature stops at the largest CN that still avoids anion–anion crushing — exactly what the radius-ratio cut-offs (next section) encode. See Coordination number .
Section 3 said "bigger ρ ⇒ where exactly does CN jump from 4 to 6 to 8? At special threshold values of ρ
Intuition Why a threshold existsThe critical ρ just touch each other AND the cation just touches every anion. Below it the cation "rattles" loosely in a hole too big — unstable, so nature drops to the next-lower CN. Above it the anions are pushed apart and there is room to spare, so a higher CN becomes possible.
Intuition How to use the tableCompute ρ
Close packing means stacking spheres as tightly as possible, leaving the smallest possible gaps. The big anions usually do the packing; see Close packing FCC HCP and voids .
Even in the tightest packing, gaps remain between the balls. These gaps are called voids or holes . Two shapes of hole matter here:
Definition Tetrahedral vs octahedral hole
A tetrahedral hole is the gap where 4 spheres meet — a small pocket. A cation here has CN 4.
An octahedral hole is the gap where 6 spheres meet — a bigger pocket. A cation here has CN 6.
Intuition Read figure s03Left: the small red cation nestles where 4 blue anions meet — a tetrahedral hole, so it needs only a small ρ 0.225 0.414 6 blue anions surround it — an octahedral hole, needing the larger 0.414 0.732 ρ big anions pack, small cations fill the holes, ratio picks the hole.
A crystal is one tiny arrangement of ions copied endlessly in all directions, like tiles on a floor. The unit cell is the smallest tile — the smallest box that, stacked, rebuilds the whole crystal.
Definition FCC (face-centred cubic)A packing where identical spheres sit at each of the 8 corners of the cube and at the centre of each of the 6 faces . "NaCl: Cl⁻ forms FCC" means the chloride ions occupy exactly those positions.
HOW the edge a The rule is always the same: find the straight line inside the cube along which a + − r + + r −
NaCl (CN 6): Na⁺ sits at each edge midpoint, Cl⁻ at the corners, so a Na⁺ and its two neighbouring Cl⁻ lie along the cube edge . Walking the edge you cross r − + 2 r + + r − a = 2 ( r + + r − ) . Why the edge and not the diagonal? Because that is where the octahedral holes (Na⁺) actually sit.CsCl (CN 8): Cs⁺ is at the body centre, Cl⁻ at the corners, so the touching line is the body diagonal of the cube. A cube of edge a 3 a r + + r − 3 a = 2 ( r + + r − ) .
Intuition Why a cube and not the whole crystal?The crystal has trillions of ions — impossible to draw. But it repeats , so if we understand one box we understand all of them. Every count and density in the parent note is done inside this one box.
Z Z
The subtlety: an ion sitting on a corner or a face is shared with neighbouring boxes, so only a fraction of it counts for this box.
Adding these fractions up gives Z Z = 4 Unit cell and Z calculation .
HOW the top line is built — step by step.
Mass of one formula unit. M N A just one formula unit is N A M N A N A Mass of everything in the box. The box holds Z Z × N A M = N A Z M Divide by the box's volume a 3 Density is always mass ÷ volume, giving ρ crystal = N A a 3 Z M
Intuition Read the formula like a sentence"Mass of everything in the box, divided by the volume of the box." Z M / N A a 3 Density of crystals formula .
Common mistake Confusing the twoρ ρ ρ crystal
Sections 2–4 covered ρ 0 1
ρ = 1 r + = r −
ρ > 1 bigger than the anion (rare)
A few salts (e.g. with tiny anions or huge cations) have ρ > 1 swap : the cation becomes the packing sphere and the smaller anion fills the holes. You just compute the ratio the other way, r − / r + < 1
ρ Why it feels right: ρ The fix: ρ > 1 bigger ion as the cation. Flip the ratio and treat the larger ion as the packer.
Ion as hard sphere, radius r
Close packing of big anions
Tetrahedral and octahedral holes
Which structure NaCl CsCl ZnS fluorite
Intuition Walk the map in words (for non-graph readers)Start at the hard-sphere ion — it splits two ways. Down one branch it gives you the radius ratio ρ ; down the other it lets the big anions close-pack , which creates tetrahedral and octahedral holes . The ratio and the holes together fix the coordination number CN , and the CN names the structure (NaCl, CsCl, ZnS, fluorite). Separately, close packing sets the unit-cell edge a , from which you count Z Z a density formula and you predict a number you can check against a lab measurement.
Test yourself — you are ready for the parent note only if every line comes instantly.
What does it mean, physically, for two ions to "touch"? Their centre-to-centre distance equals the sum of their radii, r + + r −
What is ρ The radius ratio r + / r −
Why is a cation usually smaller than its anion? Losing electrons shrinks the electron cloud; gaining electrons expands it.
Define coordination number in one sentence. The number of oppositely-charged ions directly touching a given ion.
Which way does CN go as ρ CN increases — a bigger cation pushes anions apart so more can crowd around it.
List the three key radius-ratio cut-offs and the CN they switch to. 0.225 0.414 0.732
What physically happens exactly at a critical ρ Anions just touch each other AND the cation just touches every anion simultaneously.
How many spheres surround a tetrahedral hole? An octahedral hole? 4 for tetrahedral, 6 for octahedral.
What does ρ > 1 The cation is bigger than the anion; swap roles and use r − / r +
What is a unit cell? The smallest repeating box that rebuilds the whole crystal when stacked.
In FCC, where do the spheres sit? On the 8 corners and at the centre of each of the 6 faces.
Edge–radius relation for NaCl, and why the edge? a = 2 ( r + + r − )
Contact relation for CsCl, and along what line? 3 a = 2 ( r + + r − ) , along the cube's body diagonal.
A corner ion counts as what fraction of a cell? An edge ion? A face ion? 1/8 1/4 1/2
What does Z The number of complete formula units belonging to one unit cell.
Why does the density formula divide M N A To turn mass-per-mole into mass-per-single-formula-unit.
Write the crystal density formula and name every symbol. ρ = Z M / ( N A a 3 ) Z M N A a
Which two different quantities share the symbol ρ Radius ratio (dimensionless) and crystal density (g/cm³).
Hinglish version Coordination number — the CN idea built here in depth.Close packing FCC HCP and voids — how anions pack and where the holes are.Unit cell and Z calculation — the sharing-fraction counting.Density of crystals formula — the final formula this page assembles.