When forbidden reflections appear

Every extra spot is telling you something.

Every diffraction pattern you record at the TEM has spots that the structure-factor selection rules say should not be there. These "forbidden" reflections are not errors: each one is a clue. It might be dynamical scattering in a thick crystal, shape-transform leakage in a thin one, planar defects that locally change the stacking, or a genuine superlattice: chemical ordering, a displacive collapse, a long-period stacking sequence. This guide covers the selection rules, every mechanism that violates them, a field guide to superlattices in metals and alloys, and how to tell one from another at the microscope.

1 · Which reflections should you see?

The structure factor, F(hkl), sums the scattering contributions of every atom in the unit cell. When F equals zero, atoms scatter in perfect antiphase and the reflection is "systematically absent." Each crystal structure has its own set of these rules, and they follow directly from the positions of atoms in the unit cell. Williams and Carter (2009, Ch. 16) derive them from first principles; what matters at the microscope is the result: a lookup table that tells you which Miller indices are allowed and which are not.

The interactive below shows diffraction patterns for 15 standard cubic zone axes, from low-index poles like [001] to higher-index orientations like [123] and [233] (Weirich 2024). Select a zone axis and crystal structure to see how the selection rules govern which reflections appear, and how the pattern geometry changes with beam direction.

Source: Williams & Carter (2009), Table 16.2. HCP indices use three-index Miller notation; the implicit fourth index i = −(h+k).

2 · Double diffraction: the sum-vector ghost

In kinematic theory every electron scatters at most once. In a real crystal of finite thickness, a beam diffracted into spot g₁ can strike another set of planes and scatter again. The result: a beam appears at position g₁ + g₂ in the pattern, even when F(g₁ + g₂) = 0. Both component reflections must individually be allowed; the forbidden spot inherits intensity from their product (Williams & Carter, 2009, §16.9, §18.9).

The classic example is silicon viewed along [011]. The (111) and (111) reflections are both allowed (all-odd indices in diamond cubic) and, just as importantly, both lie in that zone. Their vector sum is (200), which is diamond-forbidden (all-even indices, but h+k+l = 2, not a multiple of 4). Yet the 200 spot appears clearly in every Si <110> pattern taken from a specimen of normal thickness. The same pathway on the [110] pole is usually written (111) + (111) = (002); it is the same physics on a symmetry-equivalent pole.

The interactive below runs that arithmetic for five structures and their common zone axes. Nothing in it is tabulated: for whichever material and zone axis you choose it sums the structure factor over the atom basis to decide what is absent, then asks whether each absent position is the sum of two beams that are both allowed and both in the same zone, since those are the only beams a zone-axis pattern strongly excites. Switch double diffraction on to light up the positions it can reach and trace the shortest pathway.

Two results in that interactive are worth working through. Silicon lights up 200 and 222 on [011], and 222 again on [112] and [123], but on [001], [111], [012] or [114] nothing lights up at all. The absent positions on those poles are not sums of two allowed in-zone beams, so a 200 seen in a Si [001] pattern needs a different explanation: a HOLZ contribution, a thickness or bonding-charge effect, or the specimen not being where you think it is.

The second result is that aluminium, copper, nickel, α-iron, tungsten and rock salt light up nothing, in any zone. There is a clean reason, and it comes with an important limit. Absences that come from lattice centring cannot be filled by double diffraction inside a single crystal: a centred lattice's allowed reflections are themselves a lattice, closed under addition, so no pair of them can ever sum onto a gap between them. Check it by parity: in fcc, two all-odd reflections add to an all-even one, and all-odd plus all-even gives all-odd; you can never leave the set. Absences that come from the motif are not closed in that way, which is why the diamond glide of Si and Ge, and the 6₃ screw axis of hcp and wurtzite, do leave positions two allowed beams can land on.

The limit is the phrase inside a single crystal, and it matters, because a real fcc or bcc specimen very rarely is one. Put a second crystal in the beam path (an annealing or deformation twin, a stacking fault, a precipitate, a surface oxide, an overlapping grain) and a beam diffracted by the first can be diffracted again by the second. The doubly diffracted beam now lands at g_matrix + g_second, which is generally a lattice vector of neither crystal. That is a long-established reading of the odd patterns from evaporated gold films (Dickson and Pashley 1962; Pashley and Stowell 1963), and Cayron and co-workers (2009) traced the 1/3<111> extra spots in silicon <110> patterns to double diffraction between the crystal and one of its microtwins. So the honest advice is the opposite of a dismissal: in a plain fcc or bcc metal, an extra spot is unlikely to be double diffraction within the grain you are indexing, and correspondingly likely to be telling you a second crystal is overlapping it. The interactive above models the single-crystal case only.

Sources: Williams & Carter (2009), §16.9 and §18.9, for the mechanism. The two single-crystal worked cases (Si <110> 200 via {111}, and hcp <1120> 0001 via (0111) + (0110)) are given in DoITPoMS, Indexing electron diffraction patterns, and in the Practical Electron Microscopy database (after Moeck and Rouvimov, 2009) respectively. For the two-crystal case: E. W. Dickson and D. W. Pashley, Philos. Mag. 7 (1962) 1315; D. W. Pashley and M. J. Stowell, Philos. Mag. 8 (1963) 1605; C. Cayron et al., J. Appl. Cryst. 42 (2009) 242. Every other material and zone axis shown here is generated by the same computation and was cross-checked against the structure factor summed directly from atom positions.

3 · Thin specimens: the shape-transform leak

A real TEM specimen is not an infinite crystal. Its finite thickness stretches each reciprocal-lattice point into a rod (a "relrod") along the beam direction, roughly 2/t long, and the intensity along that rod goes as sin²(πts)/(πs)², where t is the thickness and s the excitation error (Williams & Carter, 2009, Eq. 17.2). This is what lets a zone-axis pattern show many spots at once: the Ewald sphere is curved and misses almost every reciprocal-lattice point exactly, but it still clips the rods growing out of them.

It is worth being precise about what this does not do. In kinematic theory a finite crystal's diffracted amplitude is the unit cell's structure factor multiplied by the shape transform of the crystal's outer shape. A rod growing from a systematically absent position is therefore multiplied by |F|² = 0 along its entire length: no thickness, voltage or tilt puts intensity there. Thin-foil relrods are often cited loosely as an explanation for forbidden spots, and for a single perfect crystal they are not one. Where they genuinely produce unexpected spots is when the thin object is a different structure from the matrix (a fault platelet, a twin lamella, a surface layer, a thin precipitate), which is section 4. The interactive shows both halves of that: drag the thickness down and watch the outer allowed reflections come up, and watch the odd-index positions stay dark whatever you do.

This is one proposed origin of the 1/3{422} spots that appear in face-centered cubic metals viewed along [111]: the thin foil's surface creates relrods from positions belonging to the two-dimensional surface lattice rather than the bulk FCC structure, and as the specimen thickens those relrods shrink and the extra spots fade. Be careful with that attribution, though: the 1/3{422} and 1/2{311} intensities in fcc metals are actively contested and almost certainly have more than one cause. Walsh and co-workers (2024) separate contributions from correlated thermal displacements, static displacements, dynamical amplification through higher-order Laue zones, and incommensurate surface terminations in Ni, Cu and CrCoNi, while Cayron (2021) argues that much of what is read as a new phase or a superstructure in fcc and hcp metals is a diffraction artefact of twins and stacking faults. Treat a 1/3{422} spot as a question, not an answer. To explore how thickness, voltage, and tilt control relrod length, see the interactive Ewald sphere guide.

4 · Stacking faults and the 1/3{422} puzzle

A stacking fault in an FCC crystal (ABCABC → ABCACABC) is, locally, a thin platelet of HCP-like stacking embedded in the FCC matrix. This platelet has its own periodicity and its own set of reciprocal-lattice points, different from the bulk FCC set. Because the platelet is extremely thin (one or two atomic layers), its reciprocal-lattice points are stretched into relrods perpendicular to the fault plane (Williams & Carter, 2009, §17.4).

These relrods appear in the diffraction pattern as streaks connecting extra spots to the fundamental reflections. The streak direction reveals the fault-plane normal: a streak running from lower left to upper right means the fault lies on a plane whose normal projects in that direction. Twin boundaries and antiphase boundaries produce analogous effects with their own characteristic spot positions and streak geometries.

The interactive below draws a fault and a twin from a single model, because they are the same construction at two thicknesses. A Σ3 twin lamella carries the matrix lattice rotated 180° about [111]; written in matrix indices its reflections land on thirds, and that is where the 1/3-type positions in an fcc pattern come from. Because the lamella is thin, each of its reflections also carries a shape-transform rod about 2/t long along [111]*. Set the lamella to one or two layers and the rods are long enough to run into one another: the continuous streaking of a stacking fault. Take it to fifteen and they contract into discrete twin spots. Nothing about the picture changes except the thickness.

5 · Chemical ordering: when the lattice splits

When atoms of different species occupy specific sublattice sites, the crystal is "chemically ordered" and its effective unit cell changes. Reflections that were forbidden by the disordered parent lattice become allowed, because atoms on different sublattice sites no longer scatter identically. Their intensity scales as |f_A − f_B|² × S², where S is the long-range order parameter (Williams & Carter, 2009, §16.7). At full order (S = 1), these "superlattice" spots are always weaker than the fundamental reflections.

Seven ordered intermetallics are built into the interactive below, three on an fcc parent and three on bcc, with one on hcp. Nothing about them is tabulated: each is an atom basis with a species on every site, and the order parameter S interpolates every site's occupancy between fully ordered and the bulk composition. At S = 0 every site scatters with the same average factor, which is the disordered parent, and the superlattice reflections vanish by construction rather than by a separate rule. Pick a structure and a zone axis, then drag S and watch the superlattice spots come up.

Three of them are worth heading for. L1₀ (CuAu I, γ-TiAl, FePt) stacks alternating A and B layers on (002), which makes the cell tetragonal and puts superlattice spots at 001 and 110. D0₂₂ (Al₃Ti, and the γ″ of nickel superalloys) is L1₂ with a ½[110] antiphase shift on every (002) plane; the extra reflections with h, k of opposite parity and l odd are what distinguish it from L1₂ in a pattern. And L2₁, the Heusler structure, shares D0₃'s geometry but with three species, so its all-odd and all-even superlattice families carry different intensities, which is exactly how B2 and L2₁ disorder are told apart in practice.

6 · A field guide to superlattices in metals and alloys

Section 5 covers one route to a superlattice, and it is the one most people mean by the word: atoms of different species take up specific sites, the effective cell grows, and reflections the parent lattice forbade become allowed. But it is not the only route. A superlattice is simply a periodicity longer than the parent lattice's, and there are several distinct physical ways for a metal to acquire one. They give different-looking patterns, respond differently to heat treatment, and are told apart by different tests, so it is worth knowing the whole list rather than reaching for "ordering" every time.

The most important division is between superlattices that come from who sits where and superlattices that come from where things sit. Chemical ordering is the first kind. The ω phase is the second: in a β-titanium alloy nothing about the chemistry changes at all, yet {111}_β planes collapse in pairs and a new set of reflections appears. Both are in the interactive above, and the slider means something different for each: occupancy in one case, displacement in the other. That is also why ω spots are strong, often approaching a fundamental in intensity, while chemical superlattice spots are weak: one is scaled by an atomic displacement, the other by a scattering-factor difference |f_A − f_B|.

Three cautions worth carrying. First, a diffuse blob is not a superlattice. Short-range order gives diffuse intensity because there is no long-range periodicity to give a sharp spot, and several unrelated effects (correlated thermal displacements, static displacements, incommensurate surface terminations, dynamical amplification through higher-order Laue zones) put diffuse intensity in the same region of reciprocal space. Walsh and co-workers (2024) untangle exactly this in fcc metals, and it is the reason a 1/3{422} intensity should not be read as an ordering measurement without more evidence.

Second, not every extra spot is a new periodicity at all. Sections 2 to 4 cover the impostors: double diffraction between a matrix and a twin or an overlapping grain, shape-transform streaking from a thin second phase, and moiré from two superposed crystals. All three produce spots at positions that belong to no single lattice. Cayron's work on silicon and on fcc metals is largely an argument that these get mistaken for genuine new phases more often than the literature admits.

Third, the strength of a spot is evidence. A chemical superlattice reflection carries a few per cent of a fundamental, because |f_A − f_B| is small compared with the average. A displacive one such as ω can approach a fundamental outright. If an extra spot is nearly as bright as the reflections around it, chemical ordering is the less likely explanation.

Sources for this section: Williams & Carter (2009), §16.7 and §16.9; J. Ballor, T. Li, F. Prima, C. J. Boehlert and A. Devaraj, Int. Mater. Rev. 68 (2023) 26, for the ω phase, its {111}_β collapse mechanism, its ⅓⟨112⟩_β and ⅔⟨112⟩_β positions in β [110] and [113], and the orientation relationship ⟨0001⟩_ω ∥ ⟨111⟩_β; F. Walsh et al., Sci. Adv. (2024), on the several origins of diffuse and extra intensity in fcc metals; C. Cayron, Scripta Mater. (2021), on twin and stacking-fault artefacts mistaken for new phases. The ω structure in the interactive was not taken from a table: it is built as a hexagonal supercell of the bcc lattice holding the six (111)_β layers, with the collapse applied as a displacement, and its reflections were confirmed to land at ⅔⟨111⟩_β, which, reduced by a bcc reciprocal-lattice vector, is the ⅓⟨112⟩_β position the literature quotes. The two descriptions are the same set of spots.

7 · Spotting a forbidden reflection: the diagnostic checklist

When you see a spot where the selection rules say there should not be one, four questions help identify the mechanism. First: does it vanish under precession electron diffraction? If so, double diffraction is almost certainly the source. Second: does it fade when you move to a thicker area of the foil? That points to the thin-foil relrod effect. Third: are there streaks connecting the extra spot to the fundamental reflections? Look for a planar defect, whether a stacking fault, twin boundary, or antiphase boundary; the streak direction reveals the defect-plane normal. Fourth: are the extra spots at specific fractional positions of the fundamental spacing, persistent regardless of thickness or precession? Consider chemical ordering or a structural modulation, and check whether heat treatment above the transition temperature removes them.

@misc{tripathy2026forbiddenreflections,
  author = {Tripathy, Manisha},
  title  = {When Forbidden Reflections Appear},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/tem/forbidden-reflections-guide}},
  note   = {Interactive teaching resource}
}