# Representation analysis of magnetic structures This chapter derives the magnetic structures that a nuclear structure and a propagation vector allow. [Parameterisation and constraints](parameterisation.html.md#ch-parameterisation) § Magnetic moment constraints takes a magnetic space group as given and builds the subspace of moments its own site symmetry allows. This chapter is the layer underneath that: given a nuclear structure, a propagation vector k, and nothing else, which magnetic arrangements are symmetry-allowed at all, and which of them a powder pattern can even tell apart. The construction is Bertaut’s representation analysis [[Bertaut, 1968](references.html.md#id126)], the same method BasIreps, SARAh and ISODISTORT implement; `rietx` builds its own irreps from the space-group operators rather than reading a published table, so the worked numbers below are computed, not transcribed. ## The little group and its small irreps A magnetic structure at propagation vector k breaks the full space group’s symmetry down to the subgroup that leaves k fixed, up to a reciprocal lattice vector. That subgroup is the little group, $$ G_{\mathbf{k}} \;=\; \{\, g = (R, \mathbf{v}) \in G \;:\; R^\top \mathbf{k} \equiv \mathbf{k} \pmod{L^*} \,\}, $$ *Source:* [`rietx.crystallography.representation.irreps.little_group`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/irreps.py#L641) with $L^*$ the true reciprocal lattice of the cell, which is a proper sublattice of the integer $hkl$ grid whenever the cell is centred. A moment arrangement varies only within the small (allowed) irreps of $G_{\mathbf{k}}$, so the whole problem reduces to finding those irreps and projecting the site’s moments onto them. $G_{\mathbf{k}}$ is finite, so its irreps could be read off an ordinary character table, except for one complication that the naive approach misses. Two coset representatives $g_i, g_j$ of the lattice translations compose to a third, $g_i g_j = g_l$, but the representative $g_l$ that gemmi lists may differ from $g_i g_j$ itself by a lattice translation $\mathbf{a}_{ij}$. A representation must satisfy $D(g_i)D(g_j) = D(g_i g_j)$, and pulling the extra translation’s phase out of $D(g_i g_j)$ leaves $$ D(g_i)\,D(g_j) \;=\; \omega(g_i, g_j)\, D(g_i g_j), \qquad \omega(g_i, g_j) \;=\; \exp(-2\pi i\, \mathbf{k}\cdot\mathbf{a}_{ij}), $$ *Source:* [`rietx.crystallography.representation.irreps.factor_system`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/irreps.py#L710) a projective representation with factor system $\omega$ rather than an ordinary one. $\omega \equiv 1$ away from the zone boundary, where every $\mathbf{a}_{ij}$ is a lattice vector the phase reduces to $1$. At a zone-boundary k of a non-symmorphic group, $\omega$ genuinely departs from $1$, the irreps of $G_{\mathbf{k}}$ have no ordinary counterpart, and their dimensions are larger than any point-group irrep would give [[Bradley and Cracknell, 1972](references.html.md#id127)] (§ 3.7 pp. 155–157). Kovalev tabulated exactly this case by hand; the package finds it by splitting the $\omega$-twisted regular representation of the little co-group, verifying $D(g_i)D(g_j) = \omega(g_i, g_j)D(g_i g_j)$ before returning anything. The $Pnma$ structure of LaMnO₃ is the worked case throughout this chapter. Mn sits on the 4b site $(0, 0, \tfrac12)$, whose site symmetry contains inversion. At k = 0 the magnetic representation of that orbit decomposes into the four inversion-even one-dimensional irreps at multiplicity 3 each and the four odd ones at multiplicity 0, so $\sum n_\nu d_\nu = 12 = 3N$ [[Bertaut, 1968](references.html.md#id126)]: a moment is an axial vector, and 4b’s inversion forces every mode onto the even irreps. The 4c site $(x, \tfrac14, z)$, which has no inversion in its own site symmetry, is the control: its orbit gives $(1, 2, 2, 1, 1, 2, 2, 1)$ over the same eight irreps, both even and odd occupied. At the zone-boundary k = $(0, 0, \tfrac12)$ the little group is the whole point group, $\omega$ is projective, and the 4b orbit decomposes into two two-dimensional irreps at multiplicity 3 each, again $\sum n_\nu d_\nu = 12$. ## The projection operator and its basis vectors The object that actually decomposes is not the little group’s abstract irreps but the magnetic representation of one site orbit: the permutation of the N atoms of a primitive cell under $G_{\mathbf{k}}$, tensored with the action on the moment each atom carries, $$ \Gamma_{\mathrm{mag}} \;=\; \Gamma_{\mathrm{perm}} \otimes \Gamma_{\mathrm{axial}}, $$ *Source:* [`rietx.crystallography.representation.modes.magnetic_representation`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/modes.py#L708) a $3N$-dimensional representation of $G_{\mathbf{k}}$. Replacing $\Gamma_{\mathrm{axial}}$ (moment, $\det(R)R$) with $\Gamma_{\mathrm{polar}}$ (displacement, $R$) gives the displacive twin of the same construction [[Campbell *et al.*, 2006](references.html.md#id131)], which is why `rietx` builds the two from one function. $\Gamma_{\mathrm{mag}}$ decomposes into a sum of small irreps by the usual character inner product, and each irrep that occurs is projected out with $$ W^{\nu}_{lm} \;=\; \frac{d_\nu}{|P_{\mathbf{k}}|} \sum_{g} \overline{D_\nu(g)_{lm}}\;\Gamma_{\mathrm{mag}}(g) $$ *Source:* [`rietx.crystallography.representation.modes.basis_vectors`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/modes.py#L972) The construction is [[Izyumov *et al.*, 1991](references.html.md#id128)]’s (his equation (2.12), p. 19, and its $G_{\mathbf{k}}$ form (9.1), p. 67). The $d_\nu/|P_{\mathbf{k}}|$ prefactor is the standard Wigner normalisation, written this way by Davies & Wills (2016, *arXiv*:1610.00472, equation 6); Izyumov’s own two forms carry $1/n(G)$ and $1/N$ with no $d_\nu$. It is the ordinary group-theoretic projection operator carried over unchanged, because the same factor system $\omega$ multiplies both $D_\nu$ and $\Gamma_{\mathrm{mag}}$ and cancels between them. $W^{\nu}_{11}$ projects onto the first row of the irrep’s subspace, whose rank is the multiplicity $n_\nu$; taking exactly $n_\nu$ independent columns of it, no more, is what keeps three trial vectors at one atom from over-generating a basis larger than the multiplicity actually allows. Those columns are made orthonormal by Gram-Schmidt, but not in the ordinary Euclidean product on fractional components: a fractional rotation is an integer matrix of finite order and is generally not orthogonal, so the natural inner product is the one the cell metric $G$ defines, $\langle u, v\rangle = u^\top G v$, under which every rotation in $G_{\mathbf{k}}$ is orthogonal by construction. Only which orthonormal set comes out of a degenerate multiplicity depends on this choice; the subspace itself does not. Where $2\mathbf{k}$ is a reciprocal lattice vector and the irrep’s Frobenius-Schur indicator is $+1$, a unitary gauge exists that makes the irrep matrices real, and the projected basis vectors come out real in that gauge. Two other cases stay complex and need a real combination of a conjugate pair before they describe an actual moment: when $2\mathbf{k} \notin L^*$ the conjugate irrep belongs to $-\mathbf{k}$ rather than to $\mathbf{k}$, and the physically irreducible representation pairs the two arms of the star, $D_{\mathbf{k}} \oplus D^*_{-\mathbf{k}}$; when $2\mathbf{k} \in L^*$ but the Frobenius-Schur indicator is $0$ or $-1$ the irrep is complex or pseudoreal at the same k, and the real combination is $D \oplus D^*$ or $D \oplus D$ respectively. The free real-amplitude count follows from these cases alone, never from whether the vectors happen to come out real (Bradley & Cracknell, 1972, Definition 1.3.7 p. 20 and Theorem 4.6.2 p. 204). Where $2\mathbf{k} \notin L^*$ every irrep carries $2 n_\nu d_\nu$, even when its vectors are real, because $C$ and $iC$ give different fields (cosine against sine modulation), and no irrep at $\mathbf{k}$ is the partner of another, since $D^*$ lives at $-\mathbf{k}$; the counts close on $2 \cdot 3N$. Where $2\mathbf{k} \in L^*$ the counts close on $3N$: a real irrep carries $n_\nu d_\nu$; a complex one $2 n_\nu d_\nu$, with the pair $D, D^*$ counted once; and a pseudoreal one $n_\nu d_\nu$, because its real irreducible form has dimension $2 d_\nu$ and complexifies to $D \oplus D$, so the $n_\nu$ complex copies pair up and only $n_\nu/2$ of them are independent over the reals. P2₁2₁2₁ at $(\tfrac12,\tfrac12,\tfrac12)$ on a general site is the smallest example: one pseudoreal irrep with $n_\nu = 6$, $d_\nu = 2$, and $12 = 3N$ real amplitudes. For the LaMnO₃ 4b orbit the four one-dimensional even irreps each give one real basis vector per axis, and the twelve vectors, three irreps times three axes plus the identically-vanishing fourth, reproduce Bertaut’s own F, G, C and A modes exactly. ## The lattice return-vector phase Building $\Gamma_{\mathrm{perm}}$ needs one further piece of bookkeeping. An operation $g = (R, \mathbf{v})$ of the little group sends orbit atom $\mathbf{r}_j$ to $R\mathbf{r}_j + \mathbf{v}$, which is another orbit atom $\mathbf{r}_{\pi(j)}$ plus a lattice vector $\mathbf{a}_g = R\mathbf{r}_j + \mathbf{v} - \mathbf{r}_{\pi(j)}$ the image had to be brought back by. The permutation matrix entry carries that vector as a phase, $$ \Gamma_{\mathrm{perm}}(g)_{\pi(j)\,j} \;=\; \exp(-2\pi i\, \mathbf{k}\cdot\mathbf{a}_g), $$ *Source:* [`rietx.crystallography.representation.modes.permutation_representation`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/modes.py#L497) with every other entry of column $j$ zero. This is why an atom’s own phase enters rather than one global phase for the operation: two atoms of the same orbit are generally brought back by different lattice vectors under the same $g$, so $\Gamma_{\mathrm{perm}}(g)$ carries one phase per atom, not one phase per operation. Setting $g = \{E \mid \mathbf{a}\}$ recovers $D(\{E \mid \mathbf{a}\}) = \exp(-2\pi i\, \mathbf{k}\cdot\mathbf{a})$, so $\Gamma_{\mathrm{perm}} \otimes \Gamma_{\mathrm{axial}}$ carries the same factor system [(10.2)](#equation-rep-projective) the small irreps were built from, and the decomposition needs no sign correction between the two. The opposite sign convention is a representation too, of the conjugate factor system, and decomposes into the same multiplicities at every k tried; what tells the two apart is that [(10.2)](#equation-rep-projective) itself must hold for the composed representation, which the wrong sign breaks. ## Isotropy subgroups and the candidate models A magnetic structure is a point in one irrep’s order-parameter space, real or a realification of a complex one, and the symmetry of that point rather than of the whole space is what a candidate model actually has. For an element $h$ of the order-parameter space’s image, its fixed subspace is $\ker(D(h) - I)$; the pointwise stabiliser of a direction $V$ is $$ S(V) \;=\; \{\, h \;:\; D(h)\,\mathbf{v} = \mathbf{v} \ \ \forall\, \mathbf{v} \in V \,\}, $$ *Source:* [`rietx.crystallography.representation.order_parameter.isotropy_directions`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/representation/order_parameter.py#L337) [[Stokes and Hatch, 1988](references.html.md#id129)], and the isotropy subgroups of one irrep are the distinct subgroups this construction returns, one per direction with a different stabiliser. The whole order-parameter space is itself among them, whose stabiliser is the kernel of the representation, the subgroup that fixes every direction rather than only one; every other stabiliser sits between the kernel and the full little group. Each stabiliser, together with time reversal, is a magnetic space group in the sense [Parameterisation and constraints](parameterisation.html.md#ch-parameterisation) already parameterises: `rietx` returns it as an operator list carrying the irrep label that produced it and the parent-cell transform (magCIF’s $(P, \mathbf{p})$ pair, the magnetic cell’s basis vectors as columns in the parent’s fractional coordinates) [[Campbell *et al.*, 2006](references.html.md#id131); [Perez-Mato *et al.*, 2015](references.html.md#id47)], so a candidate a refinement varies the amplitudes of is never stated except as that pair. At k = 0 in $Pnma$, LaMnO₃’s 4b site gives exactly four maximal candidates, one per inversion-even irrep. The one matching the published A-type arrangement, moments antiparallel between nearest pseudo-cubic neighbours along $c$ and parallel along $a$ and $b$, identifies as magnetic space group 62.448 in the numbering of [[Perez-Mato *et al.*, 2015](references.html.md#id47)]; the G-type arrangement, every pseudo-cubic neighbour antiparallel, identifies as the unprimed 62.441. Both come out of the same irrep decomposition that produced the twelve basis vectors above; the isotropy subgroup, not a separate calculation, is what picks the moment direction each BNS number belongs to. For a displacive order parameter the stabiliser is not the candidate’s group in the parent’s lattice: time reversal does nothing to a displacement, so the group is grey, but the coset character $\varepsilon(\Delta)$ of the anti-translations enters exactly as for a moment, so the child lattice is the sublattice on which it is $+1$ and an element $\{R \mid \mathbf{v}+\Delta\}$ belongs to the group when $\varepsilon(\Delta)\,D(R)$ fixes the displacement [[Stokes and Hatch, 1988](references.html.md#id129)]. The perovskite R-point tilt irrep then gives the six subgroups $I4/mcm$, $R\bar{3}c$, $Imma$, $C2/m$, $C2/c$, $P\bar{1}$ [[Howard and Stokes, 1998](references.html.md#id130)]. A candidate’s own group also predicts which reflections it can never show intensity at, independent of the free amplitudes. For an operation $(R, \mathbf{t}, \varepsilon)$ the magnetic structure factor obeys $$ M(\mathbf{h}) \;=\; \varepsilon\,\det(R)\,R\,\exp(2\pi i\,\mathbf{h}\cdot\mathbf{t})\; M(R^\top\mathbf{h}), $$ *Source:* [`rietx.crystallography.magnetic.isotropy.group_absences`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/magnetic/isotropy.py#L2734) [[Gallego *et al.*, 2012](references.html.md#id132)], so an operation fixing $\mathbf{h}$ ($R^\top\mathbf{h} = \mathbf{h}$) constrains $M(\mathbf{h})$ to an eigenvalue-1 subspace of the right-hand side’s matrix; where that leaves only zero, the reflection is absent whatever the moment. The MAGNEXT rule for a type IV group, one whose translations include an anti-translation $\mathbf{t}$ (Q22’s construction gives it $\varepsilon = -1$ with $R = I$), is the special case $\mathbf{h}\cdot\mathbf{t}$ an integer, since then the two terms of [(10.7)](#equation-rep-magnext) cancel identically. This absence rule follows from the group alone, before a single moment amplitude is chosen, and it is at least as restrictive as the site’s own family of allowed patterns, since the family already obeys its own group’s symmetry. ## The powder-equivalence criterion A powder pattern sums the magnetic intensity of a reflection over every domain related by the parent operations a candidate’s own group does not contain, and over the whole shell of reflections at one $d$. The intensity itself is $|M_\perp(\mathbf{h})|^2$, with $$ M(\mathbf{h}) = \sum_j \mathbf{m}_j\, \exp(2\pi i\, \mathbf{h}\cdot\mathbf{r}_j), \qquad M_\perp = M - (M\cdot\hat{\mathbf{h}})\,\hat{\mathbf{h}}, $$ *Source:* [`rietx.crystallography.magnetic.isotropy.structure_factors`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/magnetic/isotropy.py#L2575) [[Halpern and Johnson, 1939](references.html.md#id46)] over the atoms of the magnetic cell, with the moment and $\hat{\mathbf{h}}$ in one Cartesian frame, the magnetic lattice’s own: $\mathbf{m} = \sum_i m_i \mathbf{a}_i$ for the contravariant components on the cell vectors, and $\mathbf{h}$ taken through the inverse of the same row-vector lattice. A $\mathbf{k} \neq 0$ child cell is usually rotated or left-handed against the parent axes, so rebuilding either vector from the six cell parameters alone, in a standard orientation, measures the angle between them in two different frames. That average destroys some of the information the model carries: two candidate families are powder-equivalent when each can reproduce the other’s powder intensity at every shell to within a stated tolerance, over the whole range of its free amplitudes and normalised to the same total moment so a scale can never be the discriminator [[Shirane, 1959](references.html.md#id133)]. `rietx` reports the classes of that relation (`classes`, and `merged_classes` beside it where a proof splits one, below) rather than a ranked list of individually distinguishable candidates, so a refinement need only try one representative per class. “The whole range” is proved where one of two certificates applies, and sampled everywhere else. A family that lights a shell the other can never light is distinguishable whatever its amplitudes, and so is one whose shell intensities span a direction the other’s cannot reach: each shell intensity is linear in the matrix $\mathbf{b}\mathbf{b}^{\mathsf T}$ (the quadratic form below), so a family’s patterns all lie in one linear subspace, and when one family’s subspace is not inside the other’s, all but a measure-zero set of its patterns fall outside the other’s reach. “Proved” here means proved on the intensity matrices as floating point builds them, to the module’s tolerances, for almost every model of the first family. Two families are proved *equal* in two cases: when neither has any pattern at all to this $d$ limit, and when their shell matrices are related by one orthogonal change of amplitude basis, $\mathbf{Q}^{\mathsf T}\mathbf{G}^B_s\mathbf{Q} = \mathbf{G}^A_s$ on every shell, so that every pattern of one is a pattern of the other with the amplitudes rotated (an *isometry*; the two rank-1 directions of one irrep are the measured case). Because $\mathbf{Q}$ is orthogonal the condition is linear in it and is found by one singular value decomposition, but the proof is the residual of the congruence and of $\mathbf{Q}^{\mathsf T}\mathbf{Q} = \mathbf{I}$, both checked and both printed with the verdict, because an equality certificate is the direction in which a false answer silently drops a distinguishable model. Two families proved isometric then share every proved verdict: a separation proved against one of them, or from one of them, is carried to the other exactly (`propagated`, naming its source), and is not drawn again. Only proofs are carried; a sampled verdict stays where it was drawn. One family is proved *inside* another when a matrix $\mathbf{E}$ with $\mathbf{G}^A_s = c\,\mathbf{E}^{\mathsf T}\mathbf{G}^B_s\mathbf{E}$ on every shell is found, so that every pattern of the first is a pattern of the second with the amplitudes mapped and one scale absorbed (`span`; a rank-1 direction inside its irrep’s (a,b) plane is the measured case); it is found from the moment patterns and proved by the congruence residual, printed with the verdict. A proved separation then travels along the proved containments, exactly: a draw of $A$ that $B$ cannot reach is a draw of every family containing $A$, and is out of reach of every family inside $B$ (`containment`, naming its source). It also travels to a third family $C$ that reproduces the draw, by a bound that reads no sign: the certificate’s direction $\hat{\mathbf{y}}$ puts the whole of $B$’s image (where $B$’s stack has a near-kernel, its amplitudes with a kernel component no larger than the live one) inside the cone $\cos(\hat{\mathbf{y}}, \mathbf{I}) \ge \mu$, and $C$’s exact image point at the draw, formed from its fitted amplitudes with their component along $C$’s own kernel projected out, lies below $\mu - \rho$, $\rho$ a stated rounding allowance (`transfer`, stored with every number, the cosine and $\mu - \rho$ printed; a margin $\mu$ below $10^{-8}$ is labelled “proved on the float Gram stack”). A transfer that does not fire is undecided, never “equal”, and nothing sampled is ever read as a containment. Every other “equivalent”, and every “distinguishable” no certificate gives, is tested on random amplitude draws (by default 12 per direction between two irreps and 3 inside one), each fitted from a few random starts, and is statistical. “Distinguishable” can then be a fit that stopped in a local minimum on every start, and “equivalent” can be a set of draws that all missed the part of one family the other cannot reach. The amplitude basis is made canonical before any draw, so the classes do not depend on which basis of the same family the linear algebra returned, but the sampled verdicts remain a function of the seed and the number of draws and starts. Each verdict says which kind it is: `rietx.crystallography.magnetic.isotropy.powder_relations` returns one record per ordered pair of candidates, proved with its certificate named, or sampled with the number of draws actually made. The printed table marks a class P when every relation bounding it is proved and S when one rests on draws, and lists each sampled pair with its draw counts. The docstring of `rietx.crystallography.magnetic.isotropy.powder_equivalent` states what is measured about each failure and what is still open. A third certificate turns a failed fit into a proof. When the first fit of one family does not reproduce a draw of the other, `rietx` looks for a weighting $\mathbf{y}$ of the shells with $\sum_s y_s \mathbf{G}_s$ positive semi-definite and $\mathbf{y}\cdot\mathbf{t} < 0$ on the drawn pattern $\mathbf{t}$: every pattern the family can produce then has $\mathbf{y}\cdot\mathbf{I} \geq 0$, so none of them is $\mathbf{t}$ (the weak duality of semidefinite programming), and the draw is proved out of reach. Directions every shell matrix of the fitting family annihilates are projected out first, so that a family with an amplitude the data cannot determine can still be certified, and the certificate is checked in exact rational arithmetic before it counts. The check is a proof on the projected stack, with that kernel taken as structural; the record’s `kernel_residual` says how far the floating-point stack departs from it. What that departure leaves proved on the full stack is stated too. The kernel block of $\sum_s y_s \mathbf{G}_s$ is not sign-definite, so no bound on the residual makes the whole matrix positive semi-definite; what does follow is a bound on the kernel’s share of $\mathbf{y}\cdot\mathbf{I}$ under a bound on the amplitude along the kernel, and the record’s `kernel_amplitude_ratio` is that bound: no amplitude vector whose component along the kernel is at most that many times its component along the live directions reproduces the draw, on the full floating-point stack. A certificate is issued only when the ratio is at least one, which ties the kernel residual the certificate tolerates to the margin it is quoted at; on the exact stack the residual is zero and the claim covers every amplitude vector. It also measures the separation: $1/\|\mathbf{y}\|$ is a lower bound on the relative L2 distance from the draw to the other family’s patterns, and the best fit residual an upper one, and the pair record carries that bracket as `d` with the certificate itself (`witness`) so it can be re-checked without the random stream. A draw certified this way is called distinguishable only when the lower bound is large enough that no fit could have reached the tolerance `rtol` ($d \geq \mathrm{rtol}\,\sqrt{S}\,\max_s t_s/\|\mathbf{t}\|$ over the $S$ shells it uses); a smaller one is kept and printed, and the fits decide. “Proved” for this certificate is of the stored draw, not of every model of the family. An optional per-shell `weights` vector (an inverse standard deviation, say) changes the norm $d$ is measured in, not what is proved. The printed table lists every such separation with its bracket. A class that union-find joined can hold a proved separation: the join rests on a false sampled edge, or on a partial containment, one family reproducing every draw of another while a draw of its own is proved out of the other’s reach. The consistency rule then does three things. Every pair the loop had skipped as already joined, whose join the proofs have since undone, is drawn as it would have been, so that no join and no separation rests on a pair nobody tested. Every sampled direction a proof contradicts, between the two irreps holding the proof, is drawn once more with the pair’s own budget; a draw certified or not reproduced removes the edge on that direction’s own evidence. What is still joined is split at the irreps holding the proof: every edge with a sampled half between them is dropped, and the verdict names the proof that refuted it. `classes` is that split partition, and `merged_classes` the union-find one beside it; they agree exactly when nothing was dropped, and where they differ the split rests on sampled “⊆” halves, which the table prints with their draw counts. Nothing sampled is upgraded by this, and a proved edge is never dropped, so a class can still hold a proved separation inside one irrep, or between two irreps joined only through a third no proof reaches; the table names it. A wrong split costs one extra trial refinement, a wrong merge silently drops a distinguishable model, which is why the split is the partition a refinement reads. Shirane’s own example is mechanical here. His criterion is on the configurational symmetry (the symmetry of the signed moment arrangement, which he distinguishes from the chemical one; his MnO case is cubic in the chemical cell and rhombohedral in the configurational symmetry, and there the direction is measurable). For a collinear structure whose configurational symmetry is cubic, every direction of the order-parameter space gives the same powder intensity, and the [100], [110] and [111] isotropy subgroups of that irrep come out one equivalence class. Lowering the parent symmetry to tetragonal splits the same three directions into at least two classes, because the angle a moment makes with the unique axis is then a measurable quantity a powder pattern does carry. Neither result is asserted; both are the measured outcome of fitting one family’s amplitudes against the other’s intensities. LaMnO₃’s four $Pnma$ candidates above are the opposite case: each is told apart from every other by the reflections it does or does not extinguish, so all four stay in separate classes, proved rather than sampled, and a ranked refinement has to try them all. Shirane’s MnO case itself, `F m -3 m` site 4a at $\mathbf{k} = (\tfrac12, \tfrac12, \tfrac12)$, comes out two classes: the candidate with its moment along [111] on its own, and the three with moments in the (111) plane together. For the [111] candidate the $(\tfrac12\,\tfrac12\,\tfrac12)$ reflection is absent, because there the moment is parallel to $\mathbf{Q}$. This is the reflection that places MnO’s moments in the (111) planes [[Roth, 1958](references.html.md#id134)]. The fit itself never goes back to the reflections. Because $M_\perp$ is linear in a family’s real amplitudes $\mathbf{b}$, the powder intensity of shell $s$ is a quadratic form, $$ I_s(\mathbf{b}) = \mathbf{b}^{\mathsf T} G_s\, \mathbf{b}, \qquad (G_s)_{qr} = \operatorname{Re} \sum_{\text{domains}} \sum_{\mathbf{h}\in s} \overline{M_{\perp,q}(\mathbf{h})}\cdot M_{\perp,r}(\mathbf{h}), $$ *Source:* [`rietx.crystallography.magnetic.isotropy.gram`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/magnetic/isotropy.py#L2630) with $M_{\perp,q}$ the perpendicular structure factor of amplitude $q$ alone, so each candidate is reduced once to one small positive semi-definite matrix per shell and the least-squares fit, and its exact Jacobian $2G_s\mathbf{b}$, work on those. A shell where $G_s$ vanishes is one the family can never light: if the other family’s intensity is zero there too the shell carries no information and is left out, and if it is not, the pair is distinguishable without a fit.