The peaks that never arrive
An earlier rung handed you the structure factor, the single sum that sets how bright each reflection is: F(hkl) = sum over atoms of f_j times exp(2 pi i (h x_j + k y_j + l z_j)), and the measured brightness is |F|^2. Most of the time that sum lands on some middling value. But every so often it lands on exactly zero, and the reflection you expected simply is not there. There are two very different reasons a peak can vanish, and telling them apart is the whole game of this guide. The good kind, the kind we can read, is a systematic absence: a whole regular family of reflections forced to zero, not by the particular atoms but by a symmetry operation that carries a translation.
Where do translation-carrying operations come from? This whole rung has been building them. The Bravais lattice can be centered, which is a translation baked into the lattice itself. Guide 1 introduced the two operations that only exist because they add a translation: the screw axis (rotate, then slide along the axis) and the glide plane (reflect, then slide along the plane). These are exactly the ingredients that turn the 32 point groups plus 14 Bravais lattices into the 230 space groups. And it turns out each of them stamps its own distinctive pattern of missing reflections. Read those absences backwards and you have identified the crystal's translational symmetry without ever solving the structure.
Why a half-step erases half the peaks
The mechanism is a tidy piece of algebra. Suppose the crystal has a translation t that maps the whole motif onto an identical copy of itself. Then the structure factor factorizes cleanly: you can write F(hkl) = F_motif times [1 + exp(2 pi i (h dot t)) + ...], one term in the bracket for each copy the translation makes. The F_motif part carries whatever the atoms are doing; the bracket is a pure geometry factor that knows nothing about chemistry. And a bracket can be zero.
Do it for body-centering, where the extra copy sits at (1/2, 1/2, 1/2). The bracket is 1 + exp(pi i (h + k + l)) = 1 + (-1)^(h+k+l). That equals 2 when h + k + l is even, and exactly 0 when h + k + l is odd. So half of all reflections vanish, whatever atoms you pour into the cell — that is body-centering's fingerprint, and it is why an I-centered crystal only shows (hkl) with h + k + l even. Body-centered-cubic iron makes it concrete: (100) has sum 1, gone; (110) has sum 2, present; (111) has sum 3, gone; (200) has sum 2, present. Face-centering plays the same trick with three half-copies, and its bracket survives only when h, k, l are all even or all odd.
Here a classic misconception is worth killing on the spot, because it is exactly the lattice-is-not-the-crystal point in disguise. Caesium chloride looks body-centered — a Cs at the corner, a Cl in the middle — but it is NOT an I lattice. The body-center translation only forces absences if it lands identical scatterers on top of one another, and here the corner is Cs while the center is Cl. So the bracket does not factor out cleanly: F(100) = f_Cs + f_Cl times exp(pi i) = f_Cs - f_Cl, which is not zero. CsCl is a primitive cubic lattice with a two-atom motif, and its (100) shows up (weakly) precisely where BCC iron's is silent. The presence or absence of that one peak is a direct readout of whether the centering is real lattice symmetry or just a motif that happens to sit in the middle.
Three sources, three geometries
The beautiful part is that the three translation-carriers prune three different-sized slices of reciprocal space. Centering is a full three-dimensional translation, so its condition applies to every reflection in the whole pattern. A glide plane is a translation living inside a plane, so it only touches the flat sheet of reflections lying in that plane — a two-dimensional zone, such as all the (h0l). A screw axis is a translation living along a single line, so it only touches reflections strung out along that line — a one-dimensional row, such as all the (00l). Smaller reach, thinner slice erased.
Watch the same 1 + (-1) cancellation do it for a screw and a glide. A 2_1 screw along c maps (x, y, z) to (-x, -y, z + 1/2). Along the (00l) row only z matters, and the screw-partner's phase picks up exp(pi i l), so the bracket is 1 + (-1)^l — zero for l odd. Hence (00l) survives only for even l: (001) gone, (002) present. A c-glide perpendicular to b maps (x, y, z) to (x, -y, z + 1/2); in the (h0l) zone this again forces l even. Same half-step, same cancellation, just aimed at a row or a sheet instead of the entire pattern. And note the corollary: a symmorphic space group has no screws or glides at all, so its only absences come from centering, whereas screw-and-glide absences are the unmistakable signature of a nonsymmorphic group.
SYMMETRY ELEMENT AFFECTS REFLECTION APPEARS ONLY IF (translation part) which reflections -------------------- ---------------- -------------------------- I body-centering ALL (hkl) [3-D] h+k+l = even F face-centering ALL (hkl) [3-D] h,k,l all even OR all odd C base-centering ALL (hkl) [3-D] h+k = even 2_1 screw along c row (00l) [1-D] l = even 3_1 screw along c row (00l) [1-D] l = 3n c-glide (perp b) zone (h0l) [2-D] l = even a-glide (perp b) zone (h0l) [2-D] h = even n-glide (perp b) zone (h0l) [2-D] h+l = even d-glide (perp b) zone (h0l) [2-D] h+l = 4n
Fingerprinting a powder: BCC or FCC?
For a cubic powder this becomes a party trick. Peak positions obey Bragg plus the cubic spacing rule, so sin^2(theta) is proportional to h^2 + k^2 + l^2, a single integer we can call N. List the allowed N for each lattice and you get three unmistakable sequences. A primitive cubic lets every N through: 1, 2, 3, 4, 5, 6, 8, 9, ... (7 is missing because no three squares sum to 7). A body-centered lattice keeps only h+k+l even: N = 2, 4, 6, 8, 10, 12, 14, 16. A face-centered lattice keeps only all-even-or-all-odd: N = 3, 4, 8, 11, 12, 16, 19, 20. The bare sequence of peaks tells you the lattice before you have refined a single atom.
The two closest-packed metals separate at a glance. Body-centered's first two peaks sit at N = 2 and 4, a clean 1 : 2 in sin^2(theta). Face-centered's first two sit at N = 3 and 4, a 3 : 4 ratio, with a tell-tale close pair (111 then 200) followed by a gap before 220. Once you have spotted the lattice, one peak recovers the size: a = lambda times sqrt(N) / (2 sin theta). This is the everyday bread and butter of powder indexing — positions give you the cell and its centering, and the absences did half the work for free.
- Measure the angle theta for every peak, and compute sin^2(theta) for each.
- Divide every sin^2(theta) by the smallest one; the ratios should come out as near-integers — that integer is N = h^2 + k^2 + l^2.
- Match the sequence of N to a lattice: all integers (bar 7,15,...) means primitive; 2,4,6,8,... means body-centered; 3,4,8,11,... means face-centered.
- Read the smallest N off as its (hkl), then get the cell edge from a = lambda times sqrt(N) / (2 sin theta) using any indexed peak.
Reading the fingerprint backwards, and its limits
Now the payoff. The International Tables for Crystallography print, for every one of the 230 groups, its full list of reflection conditions — the same rules from our reference card, worked out for each element. Solving a structure starts by running this in reverse: you note which classes of reflections show systematic absences (an all-hkl condition, plus any zone conditions like h0l, plus any row conditions like 00l), and you look that combination up. It collapses the 230 possibilities down to a short list, and sometimes to a single space group. That is the true first step of nearly every crystal-structure determination.
Be honest about what the fingerprint can and cannot see. Absences reveal ONLY the translational symmetry — centering, screws, glides. Pure rotations, mirrors and the inversion center carry no translation, leave no absence, and are therefore invisible to this test. On top of that, diffraction imposes an apparent center (Friedel's law), blurring centrosymmetric and non-centrosymmetric groups within one Laue class. The upshot: absences give you an extinction symbol that usually narrows things to a handful of candidate space groups, not one. Only about fifty of the 230 are pinned down uniquely by absences alone; the rest need the intensities of the surviving reflections, the chemistry, and sometimes anomalous scattering to break Friedel's law and settle the tie.
One honest caveat before we close the rung. In electron diffraction the samples are thin and the scattering is strong, so beams bounce more than once on the way through — dynamical scattering. Multiple scattering can quietly pump intensity into a spot that X-rays or neutrons would leave perfectly dark (double diffraction), so a kinematic absence can look filled-in in a selected-area electron pattern. The screw-and-glide absences do have a genuine defence — the Gjonnes-Moodie dark lines that survive even dynamically — and tilting the crystal separates real absences from double-diffraction artefacts. With that, this rung is complete: from the screw axes and glides of guide 1, through the 230 space groups, the symbol you can read, and the Wyckoff positions, all the way to reading a crystal's translational symmetry straight off the peaks that never arrive.