An absence is a structure-factor zero
Guide 4 built the structure factor F_hkl = sum over the motif of f_j times exp(2 pi i (h x_j + k y_j + l z_j)), and showed that a reflection's measured strength is proportional to |F_hkl|^2. Each atom in the unit cell sends out a scattered wavelet; the structure factor is the running total of those wavelets, added with their phases, for one particular reflection (hkl). Usually the sum lands on some nonzero number and the reflection shows up. But sometimes the wavelets are arranged so that they cancel each other perfectly — the phases spread evenly around the clock and the sum collapses to zero. When that happens F_hkl = 0, the intensity |F|^2 = 0, and the reflection simply is not there.
Here is the crucial twist. If a reflection disappeared only for one random (hkl), that would be a curiosity. What makes this powerful is that the cancellation is often forced by symmetry to happen for a whole infinite family of reflections at once — every (hkl) with h + k + l odd, say, or every (0 k 0) with k odd. That systematic pattern of missing reflections is called a systematic absence (or an extinction), and it is one of the most useful things a diffraction pattern gives you. The peaks that are present tell you the shape and size of the cell; the peaks that are pointedly absent tell you about the hidden translational symmetry — the centering and the screw axes and glide planes — that no single strong peak could reveal.
Centering absences: the BCC and FCC fingerprints
Start with the cleanest case, lattice centering. A body-centered cell carries an extra lattice point smack in the middle, so an identical atom sits at (0,0,0) and at (1/2,1/2,1/2). Feed just those two into the structure factor: F_hkl = f times [1 + exp(pi i (h + k + l))] = f times [1 + (-1)^(h+k+l)]. Read off the two outcomes. When h + k + l is even, (-1)^even = +1 and F = 2f — full-strength peak. When h + k + l is odd, (-1)^odd = -1 and the bracket is 1 minus 1 = 0 — dead silence. The centre atom's wave is exactly half a wavelength out of step with the corner atom's for every odd-sum reflection, and they annihilate.
So the rule for a body-centered lattice is blunt: (hkl) is present only if h + k + l is even. Face-centering plays the same trick with four atoms — corner plus three face centres — and the arithmetic gives a different survivor rule: for a face-centered lattice, (hkl) survives only when h, k and l are all even or all odd (all the same parity, 'unmixed'), and any mixed set like (1 0 0) or (2 1 0) vanishes. These conditions apply to every reflection in the pattern, which is why they are called general or integral conditions — they are the signature of the centering itself.
LATTICE-CENTERING ABSENCES (identical atoms on the lattice points)
lattice present only when ... first powder peaks (h k l)
--------- ------------------------- ---------------------------
P simple (no condition) 100 110 111 200 210 211 ...
I body h + k + l = even 110 200 211 220 310 222 ...
F face h,k,l all even or all odd 111 200 220 311 222 400 ...
C base h + k = even (kills the mixed-hk rows)
s = h^2 + k^2 + l^2
BCC metal -> s = 2 4 6 8 10 12 14 16 (all even)
FCC metal -> s = 3 4 8 11 12 16 19 20
The MISSING peaks are the fingerprint of the Bravais lattice:
read the sequence of present peaks and you name P vs I vs F at a glance.This is why a powder pattern lets you tell BCC iron from FCC copper in a heartbeat, long before you know a single atomic position. The observed peaks march through s = h^2 + k^2 + l^2 = 2, 4, 6, 8 for the body-centered metal, but 3, 4, 8, 11, 12 for the face-centered one. The absences did the sorting. A lovely further example: diamond is face-centered but carries a two-atom motif, and that motif adds its own extra cancellation — the (2 0 0) and (2 2 2) reflections, allowed by FCC, go dark in diamond and silicon. So the motif can pile absences on top of the centering's, and each layer of missing peaks peels back another layer of the structure.
Screw axes and glide planes: absences in rows and zones
Centering was the easy case because the extra lattice point applies everywhere. The subtler and more thrilling absences come from the two symmetry operations that carry a built-in fractional translation: the screw axis (rotate, then slide part-way along the axis — the spiral car-park ramp of the last rung) and the glide plane (reflect, then slide part-way along the plane — a footprint trail where every left print is a right print shifted forward). Because these operations bury a fraction of a lattice translation inside them, they force certain motif atoms into positions whose waves cancel — but only for a restricted slice of reciprocal space, not the whole thing.
Work the screw first. A 2_1 screw axis along b means: rotate 180 degrees about b, then translate by b/2. It maps an atom at (x, y, z) to (-x, y + 1/2, -z). Ask when the pair cancels for reflections that run straight along the b* axis — the (0 k 0) row. Along that row only the y-coordinate matters, and the two atoms sit at y and y + 1/2; their path difference is exactly half a wavelength whenever k is odd, so they annihilate. The result is a serial condition: (0 k 0) is present only when k is even. Odd (0 k 0) reflections are systematically wiped out — but the screw touches nothing else, so the rest of the pattern is untouched.
A glide plane casts a wider shadow. Take an 'a' glide perpendicular to b — reflect across the plane and slide by a/2. It relates atoms across a whole zone of reflections, the (h 0 l) plane in reciprocal space, and its half-cell slide of a/2 kills every (h 0 l) with h odd. So the reflection condition is (h 0 l) present only when h is even. Notice the tidy hierarchy the two elements produce: centering strikes out reflections everywhere (an integral condition on all hkl), a glide plane strikes out a two-dimensional zone (like h 0 l), and a screw axis strikes out a one-dimensional row (like 0 k 0). Read backwards, the geographic reach of an absence — everywhere, a plane, or a line — tells you which kind of symmetry element wrote it.
Reading the absences: from missing peaks to the space group
Put it together and the absences become a decoding procedure. Every letter and number in a space group symbol that involves a translation — the centering letter P/I/F/C, each screw axis like 2_1 or 4_3, each glide plane a/b/c/n/d — leaves its own reflection condition. The International Tables for Crystallography print these conditions as a lookup dictionary, running the other way: you observe which classes of reflections are missing, match the pattern against the table, and it hands you back the translational symmetry elements. In effect the crystal writes its space-group symbol in the language of absent reflections, and you read it off.
- Collect the reflections and note which classes are systematically zero — not just weak, but zero for the whole class.
- Check the general condition on all hkl first: h+k+l even points to I-centering, all-same-parity to F, no condition to primitive P.
- Look at the zonal conditions (planes like h0l): a missing class there flags a glide plane and tells you its glide direction.
- Look at the serial conditions (rows like 0k0): a missing class there flags a screw axis and its pitch (2_1, 3_1, ...).
- Assemble the fragments into the space-group symbol and match against the tables — often landing on a unique group, sometimes a short list.
The fine print: multiplicity, thermal blur, and what absences cannot say
For the reflections that do survive, two more factors shape how tall the peak actually stands in a powder pattern. The first is the multiplicity. In a powder the crystallites point every which way, so all the symmetry-equivalent planes of a family diffract to the very same angle and pile their intensities into one peak. The {1 0 0} family of a cube has 6 members, {1 1 1} has 8, {1 1 0} has 12 — so a peak inherits a multiplicity factor p equal to how many equivalent planes feed it. That is why an intrinsically modest reflection can still stand tall: many equivalent planes are shouting in unison.
The second is thermal motion. Atoms are not frozen dots; they jitter around their sites, and the hotter the crystal the wider they blur. That smearing weakens the high-angle reflections most, because short-wavelength, high-angle scattering probes the finest detail and is the first thing a fuzzy atom loses. The Debye-Waller factor exp(-2M), with M proportional to (sin theta / lambda)^2, is the damping term that dims those peaks; heat a crystal and its high-angle reflections fade. Be clear on what this does and does not do: thermal motion only lowers intensities, gently and smoothly. It never removes a whole class of reflections and never creates a new one, so it can never mimic or erase a systematic absence — the absences stand on symmetry, which vibration cannot touch.
One last piece of honesty ties the whole rung together. Even after the absences hand you the lattice and the translational symmetry, and the present intensities are scaled by multiplicity and the Debye-Waller factor, the measurement has still only ever recorded |F|^2 — the amplitude squared. The phase of each structure factor, the timing information that says where in the cell the atoms actually sit, was thrown away the instant the detector counted photons. That is the phase problem, the deep obstacle every structure solution must climb around, and no catalogue of absences can rescue it. Absences are gloriously informative about the crystal's symmetry and lattice, yet completely silent about phase. Guide 4 gave the amplitudes; this guide read the symmetry from the zeros; the phase — the last missing ingredient — is the story the next rung takes up.