10. Representation analysis of magnetic structures¶
Parameterisation and constraints § 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], 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.
10.1. 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,
Source: rietx.crystallography.magnetic.irreps.little_group
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
Source: rietx.crystallography.magnetic.irreps.factor_system
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] (§ 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]: 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 \(\sum n_\nu d_\nu = 12\).
10.2. 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,
Source: rietx.crystallography.magnetic.modes.magnetic_representation
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], 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
Source: rietx.crystallography.magnetic.modes.basis_vectors
The construction is [Izyumov et al., 1991]’s (his eqn (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, eq. 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.
10.3. 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,
Source: rietx.crystallography.magnetic.modes.permutation_representation
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) 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) itself must hold for the composed representation, which the wrong sign breaks.
10.4. 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
Source: rietx.crystallography.magnetic.isotropy.isotropy_directions
[Stokes and Hatch, 1988], 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 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; Perez-Mato et al., 2015], 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]; 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.
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
Source: rietx.crystallography.magnetic.isotropy.group_absences
[Gallego et al., 2012], 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) 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.
10.5. 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
Source: rietx.crystallography.magnetic.isotropy.structure_factors
[Halpern and Johnson, 1939] 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]. rietx reports the connected
classes of that relation 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. The certificates in
this release prove no two families equal, except two that have no pattern at
all to this \(d\) limit. 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.
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].
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,
Source: rietx.crystallography.magnetic.isotropy.gram
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.