A plane is never alone — it comes as a stack
Guide 2 taught you to pin a label (hkl) on a single plane by reading its intercepts, taking reciprocals, and clearing fractions. But a lone plane cutting through a crystal is a bit of a fiction. Because the lattice repeats forever, that plane has infinitely many identical parallel copies marching through the crystal, all with the same tilt and all evenly spaced. The perpendicular distance between two neighbouring members of that stack is the interplanar spacing, written d_hkl — and it is the quantity this guide is about.
Picture a deck of cards, the floors of a tall building, or the slats of a venetian blind. The Miller index (hkl) names the orientation of the deck — which way it tilts — while d_hkl names how tightly the cards are stacked. Everything in a whole family {hkl} of symmetry-equivalent planes shares one value of d in a cubic crystal, because symmetry makes them geometrically interchangeable. So d is not a property of one plane; it is a property of the family, and of the lattice that generates it.
The cubic case — the one formula worth memorising
For a cubic crystal the spacing formula is as clean as it gets: d = a / sqrt(h^2 + k^2 + l^2), where a is the single cube edge. It is easier to remember in its squared, reciprocal form, 1/d^2 = (h^2 + k^2 + l^2) / a^2. The geometry behind it is quick to see: the (hkl) plane nearest the origin cuts the axes at a/h, a/k and a/l, and the perpendicular distance from the origin to that plane works out to exactly this d — which is also the gap to the next plane in the stack.
Put a real number to it. Take a cubic crystal with a = 4.00 angstrom. Then d_100 = 4.00 / sqrt(1) = 4.00 angstrom, d_110 = 4.00 / sqrt(2) = 2.83 angstrom, d_111 = 4.00 / sqrt(3) = 2.31 angstrom, and d_200 = 4.00 / sqrt(4) = 2.00 angstrom. Notice the trend: as the indices grow, the sum h^2 + k^2 + l^2 grows, so d shrinks. Higher-index planes are packed more closely together — a pattern the plane-spacing equation makes exact, and one we will cash in twice more below.
- Square each Miller index and add them up: for (111) that is 1 + 1 + 1 = 3; for (210) it is 4 + 1 + 0 = 5.
- Take the square root of that sum: sqrt(3) = 1.73, sqrt(5) = 2.24.
- Divide the edge length by it: with a = 4.00 angstrom, d_111 = 4.00 / 1.73 = 2.31 angstrom and d_210 = 4.00 / 2.24 = 1.79 angstrom.
- Sanity-check the trend: a larger index sum always gives a smaller d, so higher-index planes are stacked more tightly — never the other way around.
Beyond cubic — the general plane-spacing equation
Cubic is a gift because one length, a, sets everything. In lower-symmetry crystals more lattice parameters enter, and the formula grows accordingly — one term per independent axis. The shape of each expression follows straight from the crystal system: the fewer the equalities among the axes, the more separate pieces the equation carries, until the fully general triclinic case drags in all three edge lengths and all three angles.
SYSTEM 1 / d^2 =
------------ -------------------------------------------
Cubic (h^2 + k^2 + l^2) / a^2
Tetragonal (h^2 + k^2) / a^2 + l^2 / c^2
Orthorhombic h^2 / a^2 + k^2 / b^2 + l^2 / c^2
Hexagonal (4/3)(h^2 + h k + k^2) / a^2 + l^2 / c^2
Triclinic full expression in a, b, c and all
three angles alpha, beta, gamma (messy)Two honesties here. First: the cubic formula d = a / sqrt(h^2 + k^2 + l^2) is ONLY for cubic — reusing it for a tetragonal or hexagonal crystal is a classic and costly slip, because those need their own extra c-terms. Second, for hexagonal crystals you met the four-index Miller-Bravais scheme (hkil) in guide 3; note the d-formula uses only h, k and l. The redundant third index i = -(h+k) carries no new information, so it simply never appears in the spacing equation.
The bridge: from d to Bragg's law
Why obsess over d? Because it is the hinge that connects an invisible lattice to a number you can read off a bench instrument. Bragg's law says a stack of planes reflects an X-ray strongly only when lambda = 2 d sin(theta), where lambda is the wavelength and theta is the glancing angle. Think of echoes bouncing off a set of evenly spaced cliff walls: the reflected waves only add up in step at special angles, and those angles are set entirely by the spacing d between the walls.
Run the numbers on our d_200 = 2.00 angstrom plane using copper K-alpha radiation, lambda = 1.54 angstrom. Rearranging Bragg's law, sin(theta) = lambda / (2 d) = 1.54 / (2 times 2.00) = 0.385, so the Bragg angle is theta = arcsin(0.385) = 22.6 degrees. A diffractometer actually reports the total bend of the beam, the angle 2-theta = 45.3 degrees. Large-d planes light up at small angles, small-d planes at large angles — so a diffraction pattern is really a list of d-spacings in disguise, and from those d's you work backwards to the edge length a and the crystal system.
One tidy point about the order of reflection. The fuller statement is n times lambda = 2 d sin(theta), with n a whole number. A second-order (n = 2) reflection from the (100) planes lands at the very same angle as a first-order reflection from (200), because d_100 is exactly twice d_200. Rather than juggle a separate n, the modern convention folds it straight into the indices — every reflection is written as first order off the planes (nh nk nl). So (200) already means 'second order off (100)', and you never carry n around loose.
The same fact in reciprocal language
There is a second way to carry d around, and it is the one the next rung makes central. Instead of picturing the whole stack of parallel (hkl) planes in real space, represent that entire family by a single point. Place the point along the direction of the plane normal, at a distance from the origin of exactly 1/d_hkl. That point is a reciprocal-lattice point, and the vector reaching it is g_hkl, with length |g_hkl| = 1/d_hkl. A fat stack of widely spaced planes becomes a point close to the origin; a thin stack of closely spaced planes becomes a point far out.
This turns the awkward reciprocal in the spacing formula into something linear and easy to plot. It is the inverse-size relationship: small real distances map to large reciprocal distances, and a big real unit cell makes a small reciprocal one. And it is not a mere bookkeeping trick — a diffraction camera literally photographs the reciprocal lattice, capturing a crystal's shadow directly. The next rung builds this shadow world and the Ewald sphere that shows which reciprocal points a given wavelength can actually reach.
That is the whole job of d-spacing, and it is a workhorse: index a powder pattern, measure strain (stretch the crystal and every d shifts, moving the peaks), or fingerprint an unknown by its set of d's. Guide 5 closes this rung by turning from spacing to the other everyday geometry — the zone axis [uvw] shared by a set of planes, the atomic density packed onto a plane (which picks slip and cleavage), and the angle between two planes or directions. Carry forward the honest boundaries: the cubic formula is cubic-only, d depends on the lattice not the motif, positions give the cell while intensities give the motif, and the hexagonal i-index never enters the d-formula.