Polymorph: Crystal Structure Variants

• Polymorph is a distinct crystalline phase with identical chemical composition but divergent atomic arrangements and physical properties.
• It encompasses both static and dynamic forms, where free-energy landscapes and nucleation pathways govern the emergence of competing structures.
• Control over polymorphism enables tailored material properties, impacting applications in superconductivity, semiconductors, and other advanced technologies.

A polymorph is, in the crystallographic and materials-science sense, a distinct structure adopted by the same chemical composition. Across the works considered here, polymorphism denotes structural multiplicity without compositional change: $\beta$- and $\gamma$-$\mathrm{Ga_2O_3}$, $\alpha$- and $\beta$-$\mathrm{AgSO_4}$, pyrite and marcasite $\mathrm{FeS_2}$, and competing structural forms of $\mathrm{BaPd_2As_2}$, GdAlSi, and $\mathrm{La_3Ni_2O_7}$ are all treated as polymorph systems (Abdullaev et al., 2024, Domanski et al., 2022, Ma et al., 2021, Kudo et al., 2017, Averyanov et al., 2024, Sharma et al., 22 Jan 2026). The same corpus also shows that polymorphism is not only a classification of static structures: it is a problem of free-energy landscapes, nucleation pathways, synthesis history, and structure–property coupling, and in some later usages the term is extended by analogy to non-crystalline systems and software frameworks (Zarkevich, 2018, Müller et al., 12 Mar 2025, McCann, 23 May 2026, Casagrande et al., 2015).

1. Crystallographic meaning and structural scope

In the most direct usage, a polymorph is a distinct crystalline phase of one compound. The $\mathrm{Ga_2O_3}$ study defines polymorphs explicitly as different crystal structures of the same material, contrasting stable monoclinic $\gamma$0-$\gamma$1 with metastable cubic spinel $\gamma$2-$\gamma$3 (Abdullaev et al., 2024). The $\gamma$4 work likewise establishes two genuine phases, previously known $\gamma$5-$\gamma$6 in $\gamma$7 and newly identified $\gamma$8-$\gamma$9 in monoclinic $\mathrm{Ga_2O_3}$0 or equivalently $\mathrm{Ga_2O_3}$1 in the supporting setting (Domanski et al., 2022). In $\mathrm{Ga_2O_3}$2, pyrite and marcasite share trigonally distorted $\mathrm{Ga_2O_3}$3 octahedra and disulfide anions $\mathrm{Ga_2O_3}$4, but differ in connectivity: pyrite is cubic $\mathrm{Ga_2O_3}$5, whereas marcasite is orthorhombic $\mathrm{Ga_2O_3}$6 (Ma et al., 2021).

The same principle appears in intermetallic and oxide systems. $\mathrm{Ga_2O_3}$7 occurs in a superconducting ThCr$\mathrm{Ga_2O_3}$8Si$\mathrm{Ga_2O_3}$9-type polymorph with space group $\alpha$0 and PdAs$\alpha$1 tetrahedra, and a non-superconducting CeMg$\alpha$2Si$\alpha$3-type polymorph with space group $\alpha$4 and PdAs$\alpha$5 planar squares (Kudo et al., 2017). GdAlSi is presented as a thickness-controlled polymorph pair: a layered trigonal honeycomb polymorph in ultrathin films and a non-layered tetragonal bulk-stable polymorph (Averyanov et al., 2024). In nickelates, $\alpha$6-1313 is treated as a genuine polymorph of the usual bilayer compound, distinguished by alternating single-layer and trilayer stacking rather than uniform $\alpha$7 blocks (Sharma et al., 22 Jan 2026).

Polymorphism also extends to polymer crystals. A minimal united-atom zigzag model for polyethylene-like materials generates five candidate lattices—$\alpha$8, $\alpha$9, $\beta$0, $\beta$1, and $\beta$2—with triclinic, monoclinic, and orthorhombic variants arising from different combinations of the same zigzag–zigzag interaction extrema (Zubova et al., 2011). This suggests that polymorphism can emerge from local geometry and short-range interactions even in stripped-down models.

2. Stability, metastability, and dynamic polymorphism

The cited works treat polymorphs not merely as alternative labels, but as minima or competing regions on an energy landscape. One formulation distinguishes conventional polymorphism from a regime of dynamic polymorphism. When barriers between local potential energy minima are high compared with $\beta$3, each polymorph is stable or metastable and effectively stationary; when the barriers are low, atoms may visit multiple local potential energy minima within a basin by ergodic motion, producing a dynamically polymorphic solid (Zarkevich, 2018). In that framework, the hierarchy

$\beta$4

organizes intra-basin motion, symmetry breaking, and the predicted first-order transition on cooling near $\beta$5 (Zarkevich, 2018).

Several materials examples make this energetic language concrete. The superconducting ThCr$\beta$6Si$\beta$7-type polymorph of $\beta$8 is described as metastable, while the CeMg$\beta$9Si$\mathrm{AgSO_4}$0-type is stable; the authors connect this metastability to soft phonons and low-energy lattice fluctuations (Kudo et al., 2017). For $\mathrm{AgSO_4}$1-$\mathrm{AgSO_4}$2, static total-energy ordering depends somewhat on the functional, but lower zero-point energy and Gibbs free-energy trends make the $\mathrm{AgSO_4}$3 form increasingly favored with temperature, so it is treated as the high-temperature polymorph (Domanski et al., 2022). In GdAlSi, the layered trigonal polymorph is stabilized only in the ultrathin limit, with a critical thickness around $\mathrm{AgSO_4}$4–$\mathrm{AgSO_4}$5 monolayers; above that, the system reverts toward the tetragonal bulk form (Averyanov et al., 2024). In $\mathrm{AgSO_4}$6-1313, orthorhombic $\mathrm{AgSO_4}$7 is lower in enthalpy than $\mathrm{AgSO_4}$8 at ambient pressure, whereas pressure suppresses octahedral tilts and stabilizes tetragonal $\mathrm{AgSO_4}$9 (Sharma et al., 22 Jan 2026).

A plausible implication is that polymorphism is best understood as a competition among structurally distinct but often closely spaced minima whose accessibility depends on temperature, strain, pressure, surface energy, and barrier topology rather than on composition alone.

3. Polymorph selection in nucleation and growth

A central research theme is polymorph selection: which competing form actually appears during a finite-time process. In $\mathrm{FeS_2}$0 nanoparticle freezing, slow cooling yields the equilibrium FCC $\mathrm{FeS_2}$1 phase, faster cooling increases the probability of metastable BCC $\mathrm{FeS_2}$2, and very fast cooling leads to amorphous outcomes (Amodeo et al., 2020). The key mechanistic result is that every crystallizing nanoparticle first forms a BCC $\mathrm{FeS_2}$3-like nucleus, even when the final crystal becomes FCC-rich through later reverse martensitic transformation (Amodeo et al., 2020).

Soft-colloid simulations place this problem in a broader nucleation framework. In Gaussian Core Model and Hard-Core Yukawa systems, tuning thermodynamic state points from FCC-stable to BCC-stable causes a transition from FCC-dominated to BCC-dominated nucleation, with an intermediate regime near the triple point where both phases nucleate selectively or competitively and the growing cluster exhibits a critical-like composition fluctuation measured by

$\mathrm{FeS_2}$4

Near that crossover, FCC- and BCC-like particles form interpenetrating arrangements rather than a core-shell morphology (Kumari et al., 17 Jun 2025). A related Yukawa-colloid study finds a two-stage route in which a hexagonally ordered, BOOP-HCP-like precursor appears first, and the stable polymorph—BCC in the long-ranged case, FCC in the shorter-ranged cases—emerges only in a second step (Royall, 6 Jan 2026).

Experimentally, glycine provides a competing-risks formulation of polymorph selection. The cumulative incidence functions $\mathrm{FeS_2}$5 and $\mathrm{FeS_2}$6, survival probability $\mathrm{FeS_2}$7, and cause-specific hazards

$\mathrm{FeS_2}$8

show that $\mathrm{FeS_2}$9 nucleation starts fast and then slows, whereas $\mathrm{BaPd_2As_2}$0 nucleation starts slowly and accelerates (Little et al., 2017). Exploiting that time dependence by interrupting nucleation after $\mathrm{BaPd_2As_2}$1 h increased the final $\mathrm{BaPd_2As_2}$2 fraction from $\mathrm{BaPd_2As_2}$3 to $\mathrm{BaPd_2As_2}$4 (Little et al., 2017).

For ZIF polymorphs, the mechanistic question is shifted even earlier in self-assembly. Path-collective-variable metadynamics and neural-network classification indicate that both pre-nucleation clusters and amorphous intermediates are already polymorph-dependent, suggesting that selection can occur as early as the pre-nucleation cluster stage rather than only during later crystallization (Méndez et al., 30 Apr 2026).

4. Property divergence across polymorphs

Because the atomic arrangement changes while composition is fixed, polymorphs can display sharply different physical properties.

System	Polymorph contrast	Reported consequence
$\mathrm{BaPd_2As_2}$5	ThCr$\mathrm{BaPd_2As_2}$6Si$\mathrm{BaPd_2As_2}$7-type vs CeMg$\mathrm{BaPd_2As_2}$8Si$\mathrm{BaPd_2As_2}$9-type	Superconducting $\mathrm{La_3Ni_2O_7}$0 K vs normal metal (Kudo et al., 2017)
$\mathrm{La_3Ni_2O_7}$1	$\mathrm{La_3Ni_2O_7}$2 vs $\mathrm{La_3Ni_2O_7}$3	$\mathrm{La_3Ni_2O_7}$4–$\mathrm{La_3Ni_2O_7}$5 vs anisotropic $\mathrm{La_3Ni_2O_7}$6 values $\mathrm{La_3Ni_2O_7}$7–$\mathrm{La_3Ni_2O_7}$8 (Abdullaev et al., 2024)
GdAlSi	layered trigonal vs tetragonal	ferromagnetic layered metal vs bulk reference described as antiferromagnetic (Averyanov et al., 2024)
$\mathrm{La_3Ni_2O_7}$9	marcasite vs pyrite	band gaps $\mathrm{Ga_2O_3}$0 eV vs $\mathrm{Ga_2O_3}$1 eV by XAS/XES (Ma et al., 2021)
SQIB	monoclinic $\mathrm{Ga_2O_3}$2 vs orthorhombic $\mathrm{Ga_2O_3}$3	H-type blue-shifted vs J-type red-shifted aggregate behavior (Balzer et al., 2022)

The $\mathrm{Ga_2O_3}$4 case is particularly direct: the ThCr$\mathrm{Ga_2O_3}$5Si$\mathrm{Ga_2O_3}$6-type polymorph has $\mathrm{Ga_2O_3}$7 K, $\mathrm{Ga_2O_3}$8, and strong-coupling superconductivity, while the CeMg$\mathrm{Ga_2O_3}$9Si$\gamma$00-type has $\gamma$01 K and no superconductivity down to $\gamma$02 K (Kudo et al., 2017). In double-polymorph $\gamma$03-$\gamma$04, the cross-plane thermal conductivity contrast can reach an order of magnitude across a chemically uniform interface, making polymorphism a route to thermal functionalization without composition change (Abdullaev et al., 2024). In layered GdAlSi, graphitization creates a trigonal polymorph with ferromagnetism, negative magnetoresistance, and anomalous Hall effect, in contrast to the non-layered tetragonal polymorph (Averyanov et al., 2024).

Organic semiconductors show the same principle in excitonic form. SQIB thin films exhibit monoclinic $\gamma$05 and orthorhombic $\gamma$06 polymorphs with different preferred planes parallel to the substrate, different aggregate types, and different spectral signatures, including orthorhombic peaks around $\gamma$07 nm and $\gamma$08 nm and monoclinic peaks at $\gamma$09 nm and $\gamma$10 nm on KCl (Balzer et al., 2022).

5. Control, identification, and prediction

Polymorph control is achieved by manipulating thermodynamic and kinetic boundary conditions. Hydrothermal $\gamma$11 synthesis maps phase purity against $\gamma$12, temperature, and reaction geometry: phase-pure marcasite appears in a narrow $\gamma$13–$\gamma$14, $\gamma$15–$\gamma$16 window, while very low $\gamma$17 and high temperature favor pyrite (Ma et al., 2021). In space-separated hydrothermal growth, phase-pure marcasite single crystals form only above the solution surface under the involvement of $\gamma$18 and sulfur vapor (Ma et al., 2021). In SQIB films, temperature selects the polymorph after spin-casting, whereas substrate templating controls the polymorph during vapor deposition (Balzer et al., 2022). In GdAlSi, the decisive variables are epitaxial stabilization on $\gamma$19 and thickness below the critical ultrathin limit (Averyanov et al., 2024).

Identification relies on complementary structural and spectroscopic methods. The cited studies use powder X-ray diffraction, Raman, FTIR/FIR, X-ray absorption and emission spectroscopy, magnetization, TDTR, RBS-C, AFM, ellipsometry, polarized spectro-microscopy, and in situ annealing microscopy (Domanski et al., 2022, Little et al., 2017, Ma et al., 2021, Abdullaev et al., 2024, Balzer et al., 2022). The methodological pattern is consistent: symmetry, connectivity, and local coordination must be resolved alongside transport, magnetic, optical, or thermal measurements.

Prediction has become a data and optimization problem. "Polymorphism Crystal Structure Prediction with Adaptive Space Group Diversity Control" introduces ParetoCSP2, a multi-objective genetic algorithm with adaptive space-group diversity control, PyXtal initialization, and iterative M3GNet or CHGNet relaxations; on a benchmark of formulas with two polymorphs and the same number of unit-cell atoms, it reports $\gamma$20 space-group coverage and $\gamma$21 StructureMatcher coverage (Omee et al., 12 Jun 2025). A complementary large-scale study of 19,049 polymorphic structure entries from the Materials Project argues that topology, not symmetry alone, is central: polyhedron connectivity graphs $\gamma$22 and t-SNE embeddings cluster polymorphs across different space groups, revealing recurrent local coordination frameworks (Dey et al., 14 Aug 2025).

6. Extended uses of “Polymorph” and “PolyMorph”

Although crystallography dominates, the term is used more broadly in the cited literature. In mechanistic interpretability, “polymorphism” denotes a relation between independently trained transformers that compute the same function in residual-stream bases differing by an orthogonal rotation $\gamma$23; the paper summarizes this as “same function, mutually unintelligible interior coordinates” (McCann, 23 May 2026). Orthogonal Procrustes alignment restores sparse-autoencoder transfer, and the fitted rotations are reported as statistically consistent with Haar-random orthogonal matrices (McCann, 23 May 2026). This is an analogical extension of polymorphism from multiple crystal structures to multiple internal coordinate realizations of one computation.

“PolyMorph” also appears as a proper name in two software systems. In developmental biophysics, "PolyMorph: Extension of PolyHoop for tissue morphogenesis coupled to chemical signaling" is a lightweight standalone C++11 program for 2D tissue morphogenesis that extends PolyHoop with a finite-difference solver for multi-component reaction-advection-diffusion equations, coupling polygonal cell mechanics and chemical signaling bidirectionally (Müller et al., 12 Mar 2025). In brain–computer interfaces, "PolyMorph: Increasing P300 Spelling Efficiency by Selection Matrix Polymorphism and Sentence-Based Predictions" names a P300 speller in which “selection matrix polymorphism” means that the active symbol matrix changes dynamically by removing impossible symbols and adding prediction symbols derived from sentence context (Casagrande et al., 2015).

These extended usages do not erase the crystallographic meaning. Rather, they show that “polymorph” and “polymorphism” have become general descriptors for multiple realizations of one underlying entity—most rigorously in crystal chemistry, but increasingly also in dynamical models, computation, and adaptive interfaces.