Strain, helical density and the magnetic response of h-FeS

An altermagnet, in the source’s account, is a collinear magnet with alternating local crystal environments, whose joint spin and crystal symmetries give it ferromagnet-like transport with vanishing net magnetisation. Hexagonal FeS, h-FeS, is a candidate. The source’s crystals, grown by chemical vapour transport, are close to Fe₀.₉₃S and have the NiAs-type structure of space group P6₃/mmc: the Fe ions form triangular layers, each Fe sits in an octahedron of six S, and the octahedra around the two Fe sites of the cell alternate, so that each FeS₆ octahedron has only a local three-fold symmetry while the structure keeps a six-fold one. Below a Néel temperature of about 600 K, cited from earlier work, the Fe moments order in the A-type pattern, parallel within a layer and antiparallel between neighbouring layers. Above the Morin transition they lie in the plane in three equivalent domains, I, II and III, and cant slightly towards the c axis, leaving a net moment of order 10⁻³ μ_B per Fe; below it they turn to the c axis, where a spontaneous anomalous Hall effect is forbidden by symmetry. The Morin temperature depends on the stoichiometry.

Single-crystal X-ray diffraction of a fragment refines the cell at 275 K as 𝔞 = 3.4519(1) Å and 𝔠 = 5.7619(3) Å with an Fe occupancy of 0.924(7), and at 100 K as 3.4380(2) Å and 5.7624(5) Å; no reflection of the lower-symmetry group P6̄2c is resolved at 100 K. Energy-dispersive X-ray spectroscopy of a cleaved surface gives Fe to S close to 0.93 : 1. The susceptibility at 1 T along c drops below the Morin temperature, and the in-plane resistivity kinks there.

For transport and magnetisation a small brass screw device compresses the crystal in the plane, and the strain is estimated from the screw’s advance and a Young’s modulus of about 20 GPa taken from the acoustic phonon dispersion. The Hall resistivity is antisymmetrised in field, ϱ_yx(𝖡) = [ϱ(+𝖡) − ϱ(−𝖡)]/2. At 140 K, below the main Hall crystal’s Morin temperature of about 156 K, it is linear in field; above it a spontaneous anomalous Hall effect appears and weakens as the temperature rises. The source fits the region above 2 T as ϱ_yx = ℛ_H𝖡 + Δϱ_yx, the slope ℛ_H holding the ordinary and the conventional anomalous terms and the intercept Δϱ_yx the spontaneous part. Compression along [11̄0] suppresses Δϱ_yx progressively while ℛ_H is essentially unchanged, which the source reads as a band structure the strain does not change dramatically. Compression along [110] suppresses it too, and so does about 0.08% compression along [11̄0] on the crystal used for the magnetisation.

The magnetisation along c, measured in the same device on an 8.7 mg sample with the device’s own signal subtracted, is linear in field below the Morin temperature. Above it a small hysteresis loop opens at low field, the weak ferromagnetic part of a response that stays predominantly linear. Compression along [11̄0], about 100 MPa by the source’s estimate, shrinks that ferromagnetic part, while the differential antiferromagnetic susceptibility is unchanged above and below the transition, and the susceptibility at 1 T is reduced by strain above it and nearly unchanged below; a second crystal, its Morin temperature near 178 K, repeats the result. The source takes the piezomagnetic coefficient as about 10⁻³ G MPa⁻¹ at 160 K. Since the ferromagnetic part comes from the canting, it reads the result as compression suppressing the c-axis canting, and it writes that the concurrent suppression of the anomalous Hall effect and the ferromagnetic moment suggests a close correlation between the two.

Neutron diffraction on a 60 mg crystal at 300 K, with and without about 100 MPa along [11̄0] from an aluminium screw device, the stress estimated with a modulus the source cites from earlier work, follows the (1, 1, 1) magnetic Bragg peak and its five equivalents, where the nuclear structure factor vanishes. Unstrained, the six are nearly equal; strained, (−1, 2, 1) and (1, −2, 1) drop while the other four rise slightly, and at 120 K, below the Morin temperature, the six are comparable again. Fitting the intensities with equal g-, form and Debye–Waller factors and the tiny canting set aside, the source estimates the domain shares I : II : III as 0.31(4) : 0.33(4) : 0.36(4) without strain and 0.02(4) : 0.49(3) : 0.49(3) with it: the domain whose moments lie along the strain is strongly suppressed. A change of canting would alter the intensities by order 10⁻⁷ of the antiferromagnetic scattering, against integrated intensities of 3128 ± 293 at (1, 0, 0) and 1096 ± 33 at (1, 1, 1). Under fields along c at 220 K, on a crystal with its Morin temperature near 188 K, the half of the (H, K, 1) plane the magnet leaves open shows no significant change at zero field after −3.5 T, at 1.25 T or at 3.5 T. The source considers a rotation of the moments about c unlikely, rules out one about a, whose axial state would make two peaks about 1.7 times stronger at 1.25 T, and calls a first-order transition between the planar and the flipped planar state the more likely scenario, one whose intermediate state no static probe detects.

Inelastic neutron scattering on a coaligned array of about 19.5 g, its Morin transition spread from 150 to 200 K, shows dispersive spin waves rising from 𝖫 = 1 and 3 along [0, 0, 𝖫]. The gap at (0, 0, 1), fitted as an error function times the Bose factor, is 8.3(3) meV at 8 K and fills in at 250 K; with finer resolution it lies below 0.5 meV above the Morin temperature, the resolution at zero energy transfer. The spin waves reach about 80 meV, and along [H, 0, 3 + H] at 5 meV resolution the source finds no clear sign of the chiral magnon splitting reported in other altermagnet candidates. Its symmetry analysis forbids a spontaneous anomalous Hall effect in the paramagnet, allows the axial Hall response for a single easy-plane domain of group 2′m′m and keeps it allowed under in-plane strain; below the Morin temperature, in 6̄′2m′, strain leaves 2m′m′ or m′2m′ and no in-plane spontaneous linear effect. With ϱ_A = ℛ_Sμ₀𝔐 it estimates |ℛ_S| of order 100 μΩ cm T⁻¹, against less than 1 μΩ cm T⁻¹ in Fe and Ni. For the canting it adopts the magnetocrystalline anisotropy theory: strain breaks the octahedron’s local three-fold symmetry, the partly quenched orbital angular momentum of the high-spin Fe²⁺ is quenched further, the spin–orbit coupling loses effect, and the canting and the spontaneous Hall effect fall, a behaviour the source likens to the Jahn–Teller effect.

There is one substance, the omnium, and the crystal is a loaded volume of it. Its Fe and S nuclei are the saturated density cores of that volume, its anchors (P-⁠3). Loading the crystal does not compress the anchors: the load is taken by the interstitial spacetime between them, whose packing density changes by −ΔV/V, the load’s relative volume change, upward under compression. For a diagonal small strain with axial part 𝔲 and transverse parts −𝔭_in𝔲 and −𝔭_out𝔲, under a positive compressive stress 𝗉_ax,

with 𝒴 the Young’s modulus, 𝖾_el the stored elastic energy per volume and P the hydrostatic part of the stress. At the magnetisation run’s estimated 100 MPa and the source’s 20 GPa, with both Poisson ratios set to 0.3 since the source prints none, the axial strain is −0.5% and the packing rises by −ΔV/V ≈ 0.2%; the separate Hall run’s 0.08% gives 0.03% at the same ratios. These are interstitial packing changes, never a point density, and no stress is inferred for the Hall run.

Every energy of the loaded crystal enters its density ledger once (I-⁠7):

Elastic work and exchange or interaction compression enter E/c² once; the hydrostatic load enters the small PV/c² once, and lattice compression changes the geometry. The held moments, and the induced moments with the work of inducing them, enter E/c² once, and the uniform applied field is a uniform twist, a uniform level of the clock factor, which turns a held moment and pushes nothing (0.1c). With the crystal’s rest-mass density reconstructed as 𝔯_mass = 4.67 g cm⁻³ from the cell and the occupancy, the elastic energy at 100 MPa is 3 × 10⁻¹³ of the packing change’s rest-energy equivalent 𝔯_massc²(−ΔV/V). No numerical exchange or induced-energy function is supplied.

A magnetic field is a macroscopic helical density gradient of the omnium, the bivector grade of the density field coherently aligned along an axis, and imposing a physical twist on a material is establishing a magnetic field: mechanical vorticity and magnetic moments are different grade projections of one density-gradient topography, with no carrier exchange (P-⁠5, I-⁠2). The canted moment is one of these moments, and the spontaneous Hall signal remains a transport observable. In the analysis’s reading, the record’s concurrent suppression of the two under in-plane strain is experimental evidence that magnetism is geometric, as P-⁠5 states. The two Fe sites, each in its own octahedron and the octahedra alternating, are neighbours not turned alike, and their alternation is where the twist is read: copies of one pattern turned alike share E/E₀, its gradient and its torque, so that at one point the twist is invisible among them, and it is read between neighbours (0.1c, I-⁠3a). That alternation is the analysis’s geometry, not an additional observation of the source.

The changes of the Hall intercept and of the ferromagnetic residual vanish at zero applied load. The residual’s change per load, the piezomagnetic coefficient of about 10⁻³ G MPa⁻¹, and the Hall intercept’s change per load are response coefficients, never a force or a spin-wave energy: a driven observable is a causal convolution of the static force input through its susceptibility, A(t) = ∫χ_A(t − t′)F₁₂(t′)dt′, and no response amplitude follows from the bare force alone (I-⁠1). Returning the fitted offsets and the linear residual from their own inputs is the identity map and corroborates no invariant; the P-⁠5 reading of the co-suppression is qualitative, and its prospective test is the twist comparison below, with a stated scale for detection and for exclusion.

Strain changes the boundary conditions between the Fe and S anchors, and standing-wave structure is nodal architecture the clamps set (P-⁠2). The domains are orientations of the ordered moments, never confinement domains, and at P-⁠3’s level their redistribution is boundary-driven. In the analysis’s reading the redistribution of the shares and the distortion with its quenching are each one system-wide, many-to-many settlement (0.1a), never one-to-one. The source’s unchanged diffraction cannot weigh the canted component: the squared ratio of the c-axis moment to the local moment is about 1 × 10⁻⁷, orders below the intensities’ relative errors of 0.094 and 0.030, and at the odd last index of the (H, K, 1) reflections the canting factor of the source’s model cancels exactly.

Strain changes the Fe pattern’s support and its boundary data, and the orbital levels are eigenmodes of that support, each mode costing its mass (I-⁠5),

with Ω the confinement domain, B its boundary data, χ the orientation and 𝗃_chart the position on the inside–outside chart; the operator D is to be derived (O-⁠9), so no splitting of the orbital levels is computed here. The source’s anisotropy model, rederived, returns its own statements: on the non-axial stationary branch the small canting falls as the residual orbital momentum falls and as the temperature rises, an identity map. No free energy of the source is promoted to an objective of the omnium, and the core names none for its settlement.

The acoustic phonons from which the source takes its modulus and the dispersive spin waves in the same spectra are one carrier in two regimes. Photon, graviton, gluon, phonon and magnon are the one spin-one stroke read in different density and geometry regimes of the one medium, never different substances, and the magnon is the stroke through a helical density gradient (I-⁠2, P-⁠5); the mode names stay the source’s. A wavefunction ψ = 𝔄_modee^iφ_h is the projection of a helix onto the plane normal to the cone axis (I-⁠4). In the analysis’s lift i is the unit bivector of that plane, a single-angle phase rotor; the spin waves’ phases move on that phase axis, and no strand count is assigned to a mode.

Each probe reads a weighted mean over its own region, never a point density (I-⁠3): the Hall voltage weights the current path through the contacted cuboid, the magnetometer the c-axis moment net of the device, the six Bragg intensities the in-plane domain orientations through their neutron projection factors, the inelastic spectra the array, X-ray diffraction a fragment and spectroscopy a cleaved surface. Charge-carrier density counts mobile electron patterns and mass density counts rest mass per volume, and neither is the omnium’s local density.

X-⁠1, exploratory. The concentration of the estimated domain shares, 𝖢_dom = Σ𝔳_p², is read against its value for three equal shares, 1/3. The source’s shares give 𝖢_free = 0.33 ± 0.01 and 𝖢_strained = 0.48 ± 0.04, with a covariance reconstructed under exact normalisation from the printed errors. The larger concentration reads redistribution towards two dominant shares, not a change in the strand count of matter or a local density maximum. An in-sample diagnostic of the source’s shares, never counted with the tests.

Four tests stand where a core clause supplies what the consensus calculation lacks: the field that a physical twist establishes (P-⁠5), or the density input that an isotope substitution adds at fixed geometry (I-⁠7). Each compares neighbouring preparations, so that common response cancels or the conventional odd response vanishes, and each fixes its referents, an observable solved by a stated condition, its comparator, its decision rules and its executed rates before any data. The core assigns none of these observables a magnitude: P-⁠5’s twist-to-moment forms, which run through I-⁠1 and I-⁠2 with a declared kernel, are to be derived (O-⁠6), and no isotope sensitivity is computed here (I-⁠7), so every test predicts an existence or a sign. The apparatus, the settings and the error models are the analysis’s design, not measurements in the source.

Both twist tests use twelve truncated-cone bars cut from crystals grown by the source’s method, the c axis along the bar within 1°, end diameters 0.60 and 0.85 mm and 3 mm long, so that the cross-section halves from one end to the other. An external torsion–load insert in the source’s magnetometer, a high-temperature one for T-⁠2, turns identical end clamps by +Θ_tw/2 and −Θ_tw/2, with Θ_tw = ±1° and no added axial displacement, and an axial force gives a mean compression of 30 MPa, graded along c by the taper. The analysis reads the tapered load as grading the interstitial packing and the twist as winding that gradient. Each bar is mounted wide end up and wide end down, its volume centroid at the clamps’ mid-plane and the magnet centre in both, the narrow end gripped 0.34 mm deeper; a coin fixes the mounting order, and at each mounting another coin fixes the order of four loops, + − − + or − + + −. Each hand’s two loops are averaged, and the hand-by-taper contrast is

The proposed helical field is odd in the hand and odd in the taper. Static offsets are even in the hand and cancel in the hand difference; the shared actuator field is odd in the hand and even in the taper, and cancels in the mounting difference; linear drift cancels in the loop order, and the two coins randomise a drift curvature and a between-mounting change. Detection is p_flip ≤ 0.0027 in the exact sign-flip test over all 2¹² = 4096 sign vectors of the twelve contrasts. An interaction of 0.5 mT or more is excluded when the mean lies between −0.5 and +0.5 mT and the tests on the contrasts shifted by +0.5 and by −0.5 mT both reject; otherwise the result is inconclusive. A detection is attributed to the load-graded field only when the twelve same-bar load-on minus load-off contrasts also reject zero by the same sign-flip test, the load-off contrasts reported beside them, since P-⁠5 assigns a field to twist alone.

1. T-⁠1
   In the ordered phase, the hand and the taper together shift the loop centre. At 250 K, above every Morin temperature the source reports, the observable is the centre 𝖡_c of the ferromagnetic residual’s loop after the source’s linear antiferromagnetic subtraction, half the sum of its two zero crossings. FDT: 𝗊 ≠ 0. Comparator: zero. Hand, taper and load are even under time reversal, which reverses moment and field and centres each static consensus loop at zero field; the primed vertical mirror of the source’s group 2′m′m reverses the hand at fixed taper and keeps the c-axis moment and field on the conical bar, forbidding a hand-odd moment. In a uniform cylinder the unprimed horizontal mirror already forbids a hand-odd axial field, so the taper is essential to the comparison.
2. T-⁠2
   In the paramagnet, the hand and the taper together shift the zero crossing. At 650 K, above the source’s cited Néel temperature of about 600 K, sweeps over ±20 mT and a local linear fit give the applied field 𝖡₀ at which the c-axis magnetisation vanishes, and the contrast is formed from 𝖡₀. FDT: 𝗊 ≠ 0. Comparator: zero. The paramagnet preserves time reversal, so its conventional crossing is zero in each condition, and a transverse twofold of the uniform bar reverses an axial field at fixed hand.

I-⁠7: for purely density-coupled equilibrium observables a heavier isotope gives a larger R_s, a higher α and a larger response, opposite in sign to conventional reduced-mass scaling. The isotope tests grow crystals by the source’s method from ⁵⁴Fe and ⁵⁷Fe powders of at least 95% enrichment, one powder grade and common sulfur and iodine batches: eight pairs, each two back-to-back growths in one furnace with one crystal per growth, a coin for each pair’s order, and each crystal’s occupancy measured on that crystal by single-crystal X-ray diffraction. The density input is the Fe-source exterior A-⁠1 coordinate at fixed pressure, radius and packing, and its contribution is the integral along the light-to-heavy path,

positive under the declared positive sensitivity, which is a scope and not a measured derivative. The comparator ℭ joins the three consensus channels, each integrated along the same path: a volume term of either sign, a reduced-mass term −∫η d ln M_Fe of nonpositive sign, with Δln M_Fe = 0.0541 at full enrichment, and a composition term of either sign. Each test reads the residual 𝔶_j, the heavy-minus-light log difference less the comparator’s upper end, over the eight pairs, and detects when the mean residual exceeds ℌ = 4.530 𝔢/√8, with 𝔢 the residuals’ sample standard deviation: the one-sided Student criterion at seven degrees of freedom and a rate of 0.00135.

1. T-⁠3
   Under a positive gap sensitivity, the ⁵⁷Fe crystal has the larger low-temperature gap. Far-infrared antiferromagnetic resonance at 8 K gives each crystal’s zero-wave-vector gap, E_gap = ħω_res, independently of the source’s array fit, and 𝗒 = ln(E_gap,57/E_gap,54). Comparator: the volume, reduced-mass and composition bounds join to ℭ_g ∈ [−0.04, +0.03], rounded outward. At the error model’s pair scatter of 0.0301 the design threshold is 0.048, and a density contribution of 0.17 is excluded when 𝔶̄ + ℌ < 0.10, the interval’s full width of 0.07 protecting the exclusion at its lower end.
2. T-⁠4
   Under a positive ordering-temperature sensitivity, the ⁵⁷Fe crystal has the higher Néel temperature. Neutron diffraction of the (0, 0, 1) magnetic Bragg intensity against temperature, the source’s Morin method applied to the Néel transition, gives each crystal’s T_N, where the fitted ordered contribution vanishes on heating, and 𝗒 = ln(T_N,57/T_N,54). Comparator: ℭ_N ∈ [−0.01, +0.01]. At a pair scatter of 0.00374 the design threshold is 0.0060, and a contribution of 0.04 is excluded when 𝔶̄ + ℌ < 0.02.

A detection is attributed to the density channel only when it survives recomputing the composition term from the measured mean pair occupancy difference plus three standard errors, an allowance of 0.01 from the per-crystal 0.007 of the source’s refinement. Executed under the declared error models, the twist rules reject a symmetric null at 0.0025 ± 0.0003, against an exact size of 10/4096 = 0.0024, and hold at 0.0025 and 0.0028 under a between-mounting change and a drift curvature with their coins in place; without the mounting coin the change rejects at 0.139, and without the loop-order coin the curvature at 0.008. With no effect present the exclusion succeeds at 0.9974, and at the boundary effect it errs at 0.0012. At an instantiated contrast of 0.5 mT on every bar, p_flip = 2/4096, and the mean exceeds the attainable threshold of 5(0.5)/6 = 0.417 mT. The isotope rules reject at 0.00135 and 0.00130 at the comparators’ upper ends, and at 0.00197 and 0.00186 with an enlarged drift, rows that require an updated criterion before a detection is claimed; with no density contribution they exclude at 0.99896 and above 0.99999. Every block’s ablation is live: removing the added field returns a zero contrast, and removing the density contribution returns the joined consensus intervals.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. For the same source and data it returns the record’s numbers unchanged: the Hall antisymmetrisation and its high-field fit, the linear magnetic subtraction, the domain-intensity fit, the error-function gap fit, the estimate of canting against intensity, the Hall proportionality ϱ_A = ℛ_Sμ₀𝔐, the symmetry constraints, the anisotropy model and the spectroscopic ratio return exactly their source meanings, identity maps on the source’s inputs, each corroborating no invariant. The printed lattice lengths give 𝔠/𝔞 = 1.6692(1) at 275 K and 1.6761(2) at 100 K, the atomic percentages 48.26 and 51.74 give the ratio 0.93, and the two approximate moments give the canting ratio 1 × 10⁻⁷; values the source prints without their ordinates stay its fitted or estimated values at its precision, the resolution bounds stay bounds, and the close correlation stays the source’s statement. In the gravitational window the crystal and its nuclei read laboratory-small compactness on A-⁠1, and for the same stress tensor and data FDT and general relativity solve the same field equation, G_μν = κT_μν (I-⁠3), with I-⁠4’s weak-field, slow-motion reduction at leading Newtonian order. In the consensus limit the tests set the added twist field to zero, which returns a zero hand-by-taper contrast, and the added density contribution to zero, which returns the joined consensus intervals.

A nuclear clock completes the comparison. An external ⁵⁷Fe Mössbauer absorber at 250 K is read against a single-line source outside any field, on the 14.4 keV transition. The six consensus line positions, with their isomer, hyperfine and quadrupole shifts, absorb a factor common to all lines, and only the differential clock factor of the two ground sublevels survives the fit: with the induced term set to zero, the held term of 0.1c gives 𝔇_clock = 2𝐦_nuc|𝖡|/E_0,nuc. At a ground-state moment of 0.0906 nuclear magnetons, a nuclear rest energy of 5.30 × 10¹⁰ eV and a site-field bound of 100 T it is about 1 × 10⁻¹⁷, below the three-standard-error threshold 3𝗏_line/c ≈ 1 × 10⁻¹⁴ at a line-position precision of 0.001 mm s⁻¹. The nucleus’s transition runs on its own rest energy in the field where it sits, and the common fit with the differential comparison recovers the conventional line positions at the stated precision.