Applications / Experiments read through FDT
Giant domain walls in a cuprate
Scanning three-dimensional x-ray diffraction maps a slice through the bulk of LESCO, a one-eighth-doped cuprate, at four temperatures on cooling. At 300 K its orthorhombic strain falls to near zero in bands about 150 nm thick and 0.5 to 2 μm apart; at 100 K fine orthorhombic stripes run through a tetragonal matrix. The analysis reads the crystal and its twins as one pattern of the omnium, whose nodal architecture settles as a whole. A strain map alone does not say what a band is: a tilt that turns, a tilt that shrinks and a sharp boundary seen through the probe all leave the same near-zero band, and one convergent measurement tells them apart.
The record
What was measured
LESCO, La1.675Eu0.2Sr0.125CuO4, is a prototypical one-eighth-doped cuprate. At room temperature it takes the low-temperature orthorhombic phase, LTO (Bmab), in which the CuO6 octahedra rotate about [100]: X₃⁺(𝔱;0) in symmetry-mode notation, with 𝔱 the tilt amplitude. Upon cooling to about 160 K it passes by a first-order transition to the low-temperature tetragonal phase, LTT (P42/ncm), in which the octahedra rotate about [110] and [11̄0] in alternate layers, X₃⁺(𝔱;𝔱). The two phases have no group–subgroup relation, so a region where they coexist is necessary.
Ladbrook, Wright and Senn map a slice of the crystal by scanning three-dimensional x-ray diffraction (3DXRD) at beamline ID11 of the ESRF, from 300 to 100 K. Every reflection is indexed on a 5.35 × 5.35 × 13.2 Å cell and traced back to its place in the slice. The two orthorhombic twins, related by a fourfold rotation about [001], are A- or B-centred, with the reflection conditions 𝔨 + 𝔩 even or 𝔥 + 𝔩 even, and filtering by those absences resolves interlocking micron-scale twin domains oriented parallel to [001] and perpendicular to [110] and [11̄0], likely continuous sheets. LTT has no general absence that sets it apart, so contrast comes from the ferroelastic strain Γ₄⁺, a secondary order parameter of the tilt. Iterative reconstructions return the local average unit-cell parameters, and among the symmetry-adapted strains only Γ₄⁺ shows a pronounced stripe-like pattern. At 300 K its histogram is bimodal, negative in the A twins and positive in the B twins.
At 300 K the map of |Γ₄⁺| shows unusually wide wall-like regions of near-zero strain, about 150 nm thick and 0.5 to 2 μm apart, which the source calls effectively extended LTT-like regions coexisting within an LTO matrix. From 300 to 140 K the A-centred domain gradually dominates. By 120 and 100 K the source finds the overall microscale variation of Γ₄⁺ lost to view, consistent with a more homogeneous LTT structure, while |Γ₄⁺| keeps submicron variation at every temperature. At 100 K, within the charge-density-wave phase, fine stripe-like orthorhombic features appear on the 100–200 nm scale, repeating with a similar periodicity, and only negative Γ₄⁺ is present. The spatial resolution of the reconstructions is at least 70 nm, based on the experiment’s determined strain resolution of 2 × 10⁻⁴, both from the supplement.
| Quantity | Value | Class |
|---|---|---|
| Indexing cell | 5.35 × 5.35 × 13.2 Å | reported |
| Near-zero bands at 300 K | ≈ 150 nm thick | supplement |
| Band separation | 0.5–2 μm | supplement |
| Stripes at 100 K | 100–200 nm | reconstructed |
| Spatial resolution | ≥ 70 nm | supplement |
| Strain resolution | 2 × 10⁻⁴ | supplement |
| LTO to LTT transition | ≈ 160 K, first order | nominal |
| Change in the maps | between 140 and 120 K | reconstructed |
| Map settings | 300, 140, 120, 100 K on cooling | reconstructed |
The FDT reading
0.1b omnium patterns, I-3 profile-weighted means, P-2 nodal architecture
There are no separate particles and no boxes: there is only omnium, and omnium patterns (0.1b). The analysis reads the whole crystal as one such pattern, its twin boundaries internal support data, and the ions as closed confinement patterns of I-5, with no particle given an address. A crystal’s twin domain, an I-5 confinement domain and a universal domain are different objects; so are a structural phase, a helical phase, and the matter and spacetime phases of 0.8.
In the analysis’s symmetry reading of the source’s secondary order parameter, the orthorhombic strain couples evenly to the tilt, with no coefficient fixed,
so a near-zero band admits three structures: the tilt turning toward equal components at fixed amplitude, the LTT-like wall the source proposes; the tilt amplitude falling; and a sharp twin boundary averaged by the probe. Each voxel is a profile-weighted mean under I-3’s observation-region rule, the beam footprint composed with the reconstruction kernel for its weight, so a boundary sharper than the point spread still returns a near-zero band, with a width the kernel sets. The strain map alone leaves the three unresolved.
One convergent measurement on the same slice at 300 K separates them, inside the source’s own diffraction dictionary and with no FDT parameter added. Map the reflections that LTT allows and both LTO twins forbid, 𝔥 + 𝔩 and 𝔨 + 𝔩 both odd with 𝔥 and 𝔨 nonzero, such as (110), (112) and (221) in P42/ncm. Beside them, map the reflections allowed by only one twin, each normalised within its own twin. A turning tilt lights the LTT reflections and keeps the twin sum at one; a blurred sharp boundary leaves them dark and keeps the sum at one; a falling amplitude pulls the sum below one. The LTT signal separates rotation from blur, and the summed intensity separates rotation from amplitude loss. The kernel of a sharp edge is reconstructed through the same procedure before any wall is called extended.
The analysis reads the evolving twins through P-2 as a standing network conditioned by its boundaries. The clamps and the internal boundary data set the mode geometry; excitations migrate along density gradients and lock transversely at nodal channels; the allowed network changes as the whole configuration settles, and the change from LTO to LTT is a global eigenmode snap of that network. On the existing cooling maps the reading is in-sample. Every step is a system-wide settlement (0.1a), and events do not commute: a warming leg from 100 to 300 K, at the same settings, is its own settlement, measured separately, and the cooling maps fix no warming map. The core names no objective for the settlement, and none is assigned here.
P-3: a load does not compress the anchors. It is taken by the interstitial spacetime between them, whose packing density changes by −ΔV/V. The analysis reads the strain relative to the aristotype as that load, and to first order the geometric identities behind I-3a’s trace and shear split give
To first order the volume trace changes the packing, and the trace-free orthorhombicity does not. The sense in which the octahedra rotate is a structural sign: neither P-4’s circulation of a cone nor a P-5 torsion. Rotations that alternate between layers leave a cone’s intrinsic sense unchanged, and the Néel-like ferroelectric walls the source cites in systems such as BaTiO₃, where polarisation rotates, supply no physical twist either. FDT reads a Cooper pair as P-4’s tip-to-tip overlap of two opposite-helicity electron cones and pairing as P-2’s shared scattering-free nodal channels, and the analysis reads the source’s superconductivity discussion through them; the source presents no electronic measurement of these walls.
On the A-1 chart a laboratory piece of the crystal reads a small compactness, a correct reading under I-1. Its formula unit weighs 401.554 u with natural isotopes, the indexing cell holds 378 ų and the rest density is 7.06 g cm⁻³. An illustrative sphere of 1 mm³, radius 0.620 mm, reads αext = 1.7 × 10⁻²⁹ at its own surface, and two such spheres in contact attract with 2.16 × 10⁻¹⁵ N: I-1’s correct bare force, not the binding scale, which needs its own response kernel. At 300 K the radiation term adds 9.66 × 10⁻²⁷ of the rest content, one positive channel at fixed volume and never the whole response to cooling (I-7). No chart is assigned to a strain, a tilt amplitude, a histogram, a wall width or a critical tilt.
One strain, three walls. One strain, three walls: a tilt that turns through the LTT direction at fixed amplitude, a tilt whose amplitude falls to zero without turning, and a sharp twin boundary, each drawn as it is (thin) and as a Gaussian probe weight returns it (thick). With Γ₄⁺ ∝ 𝔱1² − 𝔱2² every wall reads zero at its centre, and wall and weight set the band’s width together: through a 150 nm weight a 121 nm turning wall and a 129 nm falling one both return a 150 nm band, while a sharp boundary returns 86 nm, and 150 nm only through a 262 nm weight. DM-1 tells them apart on the same slice: the LTT-allowed reflections, present only where both tilt components share one region, and the summed intensity of the reflections allowed by one twin only, each normalised within its twin, which stays at 1 across a turning tilt and a blurred boundary and falls below 1 where the tilt is reduced. Two settled networks: a 10 × 6 μm field at the source’s four settings on cooling, each map with its histogram, the analysis’s P-2 reading as an in-sample overlay (X-1), and the warming leg, DM-2, to be measured. Wall profiles, reflection presence and maps are schematic, drawn to the printed scales, and the reflection rules are the standard structural account’s.
I-3, P-2, 0.1a
What FDT predicts
Four tests, each against its comparator’s zero
The maps measure no isotope response and no field from a twist, so each test adds one FDT input (Use 2). The torsion test compares opposite hands, arranged so that the conventional response cancels and the comparator reads zero (Use 7); the isotope tests compare matched partners and compute and subtract the conventional isotope response. The isotope tests read I-7: a heavier isotope enters 𝓜 with more rest content and, at a common support radius, raises the crystal’s exterior compactness; the analysis reads the domain strain and the wall width as density-coupled equilibrium observables of the structure, which respond to that input. The torsion test reads P-5: imposing a physical twist on a material is establishing a magnetic field, and the analysis reads that field along the twist axis, reversing with the hand. A candidate built on the local unit-cell volume is retired, because the core supplies no map from the local omnium profile to the measured volume contrast; that profile is to be derived (O-4).
For a positive structural observable 𝔔 and its full conventional logarithmic isotope correction 𝒩, the corrected shift, integrated from the light crystal’s compactness to the heavy one’s, is
with S = ∂(ln 𝔔 − 𝒩)/∂ ln αext at fixed temperature, pressure, shape, packing, composition apart from the isotope, and kernel. The conventional comparator is zero once the correction is subtracted: classically the configurations are mass-independent, and nuclear quantum motion, terms beyond the Born–Oppenheimer surface, and nuclear field-shift and hyperfine terms all sit in 𝒩. The input’s sign is fixed and the observable’s sign is the integral’s. At fixed packing, full copper substitution, ⁶³Cu to ⁶⁵Cu, raises ln αext by 0.00497, and full oxygen substitution, ¹⁶O to ¹⁸O, by 0.01977, or 0.01781 at an assumed exchange fraction of 0.90. The actual inputs come from weighing each crystal and measuring its support radius before the response is measured.
- T-1Copper-isotope domain strain. Grow matched LESCO crystals from one feed and one schedule, one enriched in ⁶³Cu and its partner in ⁶⁵Cu. Anneal them alike, assay the enrichment, weigh each crystal and measure its support radius, cut each pair to a common shape, and map several slices at 300 K by the source’s scanning 3DXRD. The observable is the mean magnitude of the A- and B-twin peaks of the strain histogram, and the crystal pair is the replication unit. FDT predicts a nonzero corrected shift; the conventional comparator is zero, and copper sits on an inversion centre of the parent structure, so the even X-point tilt displaces no copper. With seven controls and eight pairs the median threshold is 1.69 design units and the test point 3.38; the simulated comparator rejects 0.0103 ± 0.0007 of the time.
- T-2Oxygen substitution and a species-blind contrast. Anneal pairs of natural-copper crystals from the same growth in ¹⁸O and in ¹⁶O on one schedule; measure the exchange fraction, its depth profile and the oxygen stoichiometry after exchange, weigh each crystal, measure its radius, and map the pair in the copper sessions. FDT predicts a nonzero corrected oxygen shift. A response that is one species-blind function of log compactness makes the contrast ℭ = (ΔO ln αext)𝔶Cu − (ΔCu ln αext)𝔶O vanish when it is linear, so a contrast beyond its threshold rejects linearity and species-blindness together. The contrast is also zero under the conventional comparator, where both shifts vanish, so it tests the joint response model, and the oxygen shift carries the test against the comparator. Oxygen moves in the tilt mode, so its quantum correction is computed in full. The shift’s threshold is 2.09 and its test point 4.18; the contrast’s are 0.0350 and 0.0700; the comparator rejects 0.0084 ± 0.0006 and 0.0075 ± 0.0006 of the time.
- T-3Physical torsion and a hand-odd field. Cut a LESCO rod along [001] from the source’s growth, clamp one end at 300 K, and impose equal static elastic torsions of opposite hand at the other, inside a magnetic shield, with a coaxial axial SQUID gradiometer beyond the free end; map the twist along the rod by scanning 3DXRD, with the beam off for the magnetic reads. FDT predicts a hand-odd axial field per mapped twist, 𝔟odd/𝒴twist ≠ 0. The conventional comparator is zero: static torsion is even under time reversal, and once the controls pass, no magnetic order, applied field, or axial heat or charge current supplies a hand-odd axial field. Ten controls run from zero-field muon spin rotation for magnetic order to circumferential remanence, heat flow and drift, and the hands run +−−+ and −++− in alternating cycles. The threshold is 2.02 and the test point 4.04; the comparator rejects 0.0057 ± 0.0005 of the time.
- T-4Copper-isotope wall width. Use the copper pairs of T-1, several slices per crystal and both wall orientations, on the 300 K wall class that DM-1 identifies as extended. Deconvolve each signed profile with a sharp reference edge reconstructed alike, then take the full width at half depth of its magnitude. FDT predicts a nonzero corrected width shift; the conventional comparator is zero after the full correction for the wall, computed under each twin’s own zero-point stress, since inside the wall the tilt leaves the parent X point and no inversion argument holds copper fixed. The threshold is 1.83 and the test point 3.66; the comparator rejects 0.0097 ± 0.0007 of the time.
Each nonzero integral is an existence statement. Its magnitude is to be derived through the response sensitivity, and the twist field’s through a twist-to-field kernel. A test point is twice the median threshold, in design units of one replication unit’s standard deviation: experimental reach, never an FDT amplitude, and a finite null excludes the named effect while leaving smaller amplitudes untested. With 𝔳̄ the mean corrected response of the replication units and 𝓕* the test point, detection is |𝔳̄| > ℌ against the zero comparator and exclusion |𝔳̄ − 𝓕*| > ℌ, where the threshold ℌ adds every control’s weighted bias bound to the Student quantile times the standard error; attribution needs the independent positive input and the block’s sign or co-variation to survive the conventional corrections.
Two measurements test the structural reading with no FDT input added: DM-1, the reflection maps of the three walls, and DM-2, the same temperature settings on warming from 100 to 300 K, each map equilibrated on its own and measured as its own settlement. A warming map incompatible with the boundary-selected architecture falsifies that mechanism reading; hysteresis of the first-order transition belongs to the conventional comparator.
X-1, exploratory. The broad near-zero bands at 300 K and the fine stripes at 100 K read, in the analysis’s P-2 reading, as two globally settled nodal networks of one pattern: boundary-selected support at 300 K and a different settled architecture at 100 K, along whose channels excitations lock. An in-sample re-description of the source’s maps, with no chart position, thermal direction or local density inferred, never counted with the tests.
Two readers of an additional input. Isotope reader: the corrected response against logarithmic exterior compactness at fixed packing, from the ⁶³Cu base. Full copper substitution spans 0.00497, and oxygen exchange at an assumed fraction of 0.90 spans 0.01781 from a base 0.00150 higher. A linear species-blind response puts both shifts on one secant and the contrast ℭ at zero, and curvature moves it off zero. Each decision rule shows its median threshold ℌ and its test point 𝓕* = 2ℌ, in design units of one replication unit’s standard deviation, with the simulated rates. Torsion reader: a LESCO rod twisted to equal opposite hands before an axial SQUID gradiometer, eighty readings under the assumed drift, and ten unit combinations (𝓠1 − 𝓠2 − 𝓠3 + 𝓠4 − 𝓠5 + 𝓠6 + 𝓠7 − 𝓠8)/8, which cancel constant, linear and quadratic drift exactly. The response’s secant and curvature and the field’s size are to be derived, so their sliders explore; noise and drift are the analysis’s design assumptions.
I-7, P-5, Use 2, Use 7
Readings declined
Use 6 Data yes, interpretation not necessarily
Standard reading
The ions of the structure, its A-site cations and the anions that migrate, are distinct things.
FDT reading
The ions and their crystal are omnium patterns (0.1b, I-5), and the analysis reads the crystal as one support with its own boundary data; the structural data stand.
The fine stripes may be nanoscale inclusions of LTO in the LTT matrix, likely forming to minimise the elastic energy of antiphase boundaries related to the tilt orientation.
The change is a system-wide settlement with no objective named by the core (0.1a); the boundary-selected nodal reading returns the same strain maps.
The extensive heterogeneities are distinct coexisting phases.
One omnium pattern with its own support and boundary data, the analysis’s application of 0.1b to the crystal; every reconstructed heterogeneity stands.
The source anticipates that the symmetry breaking of the walls shapes the competition between superconductivity and charge order, rather than being shaped by it.
Structure and electronic order settle together at every step (0.1a); no one-way chronology follows from the cited correlation lengths of 2–15 nm.
Convergence
Use 2 Same source, same data
On the source’s own reflections, centering filters, reconstruction and symmetry-mode decomposition the limiting map is the identity: with every added constitutive input at its conventional value it returns the same maps and histograms. The reconstructed temperature sequence, the near-zero bands and the stripe widths keep their classes and values; the 150 nm and 0.5–2 μm dimensions stay supplement-reported and the resolution supplement-based, at least 70 nm. The cited 10⁻⁴ and the determined 2 × 10⁻⁴ strain resolutions stay distinct, and so do the nominal 160 K transition and the bracket between the 140 and 120 K maps. For the same stress tensor and data, I-3 returns the curvature of general relativity, and I-4’s weak-field, slow-motion expansion recovers the pair force with corrections of order the compactness and v²/c². A chart, a branch tag and the conventional diffraction dictionary produce no divergence. The added density response and the P-5 twist source do, and setting them to their conventional values restores the isotope and torsion comparators exactly: on the source’s own inputs the map is the identity, and the four tests above are where FDT’s added inputs meet new data.