Medium-Range Order in Disordered Materials
- Medium-range order (MRO) is the correlation among local structural motifs beyond immediate neighbors, characterizing intermediate-range organization in disordered materials.
- Experimental techniques like diffraction (FSDP) and variance analysis, alongside computational methods such as persistent homology, effectively uncover MRO patterns.
- MRO influences key material properties including plasticity, stability, and magnetism, guiding the design and performance tuning of metallic glasses and network formers.
Medium-range order (MRO) denotes structural correlations that extend beyond nearest-neighbor coordination yet stop short of crystalline long-range periodicity. In amorphous solids and liquids, it occupies the regime between short-range order (local bonding, coordination, and first-shell geometry) and long-range order, and has been reported on scales from roughly $1$–$6$ nm in metallic glasses to about $5$–$20$ Å in network glasses and related liquids. Depending on the material class, MRO is expressed as correlated motif networks, rings and cages, void organization, orientational coherence, or medium-range density oscillations that generate signatures such as the first sharp diffraction peak (FSDP), diffuse nanodiffraction variance, or long-ranged oscillations in and (Hilke et al., 2020, Nakamura et al., 2015, Zhang et al., 2024).
1. Definition, scale, and conceptual scope
MRO is distinguished from short-range order by the fact that it concerns correlations among local motifs rather than the motifs themselves. In metallic glasses, it has been described as correlations among connected clusters or motifs extending over nanometers, with experimentally extracted correlation lengths ranging from roughly $1$ nm up to about $6$ nm in Pd–Ni–P-based glasses (Hilke et al., 2020). In silica, silicates, and related covalent networks, MRO has been associated with rings, cages, and void arrangements on scales of about $5$–$20$ Å, with the FSDP serving as a principal reciprocal-space signature (Balantrapu et al., 23 Apr 2026, Zhang et al., 2024).
Two complementary definitions recur in the literature. One treats MRO as medium-range density coherence visible directly in the damped oscillatory tails of $6$0 and the Van Hove function $6$1, and in low-$6$2 features of $6$3 beyond the principal peak (Lieou et al., 2022). The other emphasizes that two-body functions alone are often insufficient, because many-body geometry such as ring topology, voids, or same-species connectivity can generate medium-range correlations not recoverable from pair distances alone; this is the standpoint adopted in persistent-homology and ring-resolved analyses of silica, metallic glasses, and phosphate glasses (Nakamura et al., 2015, Singh et al., 2024).
The relevant length scale is therefore not unique. In some works it is the real-space period associated with an FSDP, as in iron phosphate glass where $6$4 implies $6$5 (Singh et al., 2024). In others it is the decay length of oscillations in $6$6 or $6$7, the cutoff or correlation length of a connected motif network, or the probe-size-dependent plateau extracted by variable-resolution fluctuation electron microscopy (VR-FEM) (Hilke et al., 2020, Wu et al., 2015).
2. Experimental and computational observables
The most common two-body description begins from the pair distribution function and the structure factor,
$6$8
with MRO appearing as oscillations of $6$9 beyond the first peak, as low-$5$0 features such as the FSDP, or as shoulders and split peaks in $5$1. The Van Hove function extends this to dynamics, with $5$2; its medium-range oscillations were argued to carry essentially the same MRO information as local four-point dynamical correlations when the overlap threshold is chosen at the cage scale (Lieou et al., 2022).
FEM and VR-FEM probe higher-order correlations through the normalized variance of nanobeam diffraction intensities,
$5$3
Here $5$4 is the coherent probe size, and peaks or plateaus in $5$5 as a function of $5$6 identify characteristic medium-range correlation lengths. In Pd$5$7Ni$5$8P$5$9-based glasses, this procedure resolved discrete plateaus corresponding to multiple correlation lengths, whereas X-ray diffraction and SAED showed no differences capable of explaining the mechanical contrast (Hilke et al., 2020).
A different class of observables treats MRO as a many-body topological problem. Persistent homology maps rings and cavities into persistence diagrams whose birth and death coordinates encode the scale and robustness of medium-range features; in amorphous silica this resolved distinct ring families such as $20$0, $20$1, $20$2, and $20$3, each tied to different geometric constraints (Nakamura et al., 2015). Ring statistics and void analysis play a similar role in vitreous silica, amorphous arsenic, and iron phosphate glass, where ring-size distributions, void radii, and void anisotropy were explicitly linked to the FSDP and to MRO evolution under pressure or processing (Jena et al., 2023, Liu et al., 2 Sep 2025, Singh et al., 2024).
Several studies also define MRO through connectivity graphs. In Cu$20$4Zr$20$5, interpenetrating icosahedra are taken as connected when their central atoms are nearest neighbors, and clusters of these nodes define an icosahedral MRO network whose size distribution obeys a percolation-like scaling law (Wu et al., 2015). In chemically resolved persistent-homology analyses of Cu$20$6Zr$20$7, same-species connectivity produces distinct short-, intermediate-, and medium-range topological regions in persistence diagrams, with a chemical correlation length $20$8 Å at $20$9 K, 0 Å at 1 K, and 2 Å at 3 K (Liu et al., 2023).
3. Structural realizations across material classes
In metallic glasses, one important realization of MRO is a network of connected locally favored structures. For Cu4Zr5, icosahedral short-range order identified by Voronoi index 6 forms interpenetrating clusters whose size distribution follows
7
with 8, 9, and 0 K. The inferred fractal dimension is 1, and the correlation length exponent is 2 (Wu et al., 2015). In Al3Tb4, by contrast, MRO is Bergman-type packing built from Tb-centered ‘3661’ and ‘15551’ short-range motifs that extend coherently through second and third coordination shells; about 5 of neighboring SRO pairs share faces or vertices, whereas only about 6 are interpenetrating (Tang et al., 2020).
Covalent and network glasses exhibit a different structural vocabulary. In amorphous silica, MRO has been resolved as a hierarchy of rings and cavities in the Si–O network, with persistence diagrams distinguishing primary Si–O rings, tetrahedral distortions, and oxygen-only medium-range ring families (Nakamura et al., 2015). In vitreous SiO7, FSDP-based metrics yield 8, 9, $1$0, and $1$1 for an AIMD melt-quench model, while ring statistics peak at 12-membered rings (Jena et al., 2023). In amorphous arsenic, the FSDP was linked closely to the size and spatial distribution of voids, with void morphology becoming an explicit microscopic origin of MRO (Liu et al., 2 Sep 2025).
Other systems emphasize chemically specific topologies. In amorphous GeTe, severe chemical disorder generates homopolar bond chain-like polyhedral clusters of Ge-centered tetrahedra interpenetrating through Ge–Ge bonds; these clusters, rather than a chemically ordered network, carry the prepeak/FSDP signature of MRO (Liu et al., 2017). In iron phosphate glass, the total FSDP coincides with the ring-resolved $1$2, and glass models that reproduce the experimental $1$3 all exhibit a bell-shaped ring-size distribution peaking at 10-member rings (Singh et al., 2024). In chemically resolved metallic-glass analyses, same-species connectivity was interpreted as a Turing-pattern-like activator–inhibitor topology whose MRO depends on the relative depths of interatomic potential wells rather than on $1$4 alone (Liu et al., 2023).
4. MRO, dynamical heterogeneity, and glass transition
A recurrent theme is that MRO is tied to slow relaxation and heterogeneous dynamics. In Cu$1$5Zr$1$6, increasing connectivity of icosahedral motifs both lowers the local formation energy and makes the bond-orientational invariant $1$7 more negative, indicating more regular icosahedra. At the same time, the relaxation time of ISRO central atoms grows approximately exponentially with node degree,
$1$8
and the approach to $1$9 is associated with a growing characteristic cluster size and correlation length (Wu et al., 2015).
A more direct structural-dynamical equivalence was proposed for liquids and metallic glasses through the Van Hove function. In that view, cooperative rearrangements and dynamical heterogeneities originate from the same medium-range density correlations that appear as oscillations in $6$0 and $6$1. The short-time identity $6$2, together with the empirical coincidence of $6$3 and $6$4 at cage-scale overlap thresholds, implies that the four-point susceptibility does not introduce a fundamentally new medium-range structural field beyond that already present in two-point correlations (Lieou et al., 2022).
Silicate glass-formers show that MRO may evolve non-monotonically with thermodynamic control parameters. In Na$6$5O–$6$6SiO$6$7, many-body orientational correlators reveal a hidden transition in tetrahedral relative orientation around $6$8 Å that is invisible in two-point correlators, while the MRO correlation length extracted from many-body and pair functions varies non-monotonically with temperature due to competition between energetic and entropic terms. Increasing Na broadens the ring-size distribution, drives the Na arrangement from blob-like to channel-like, and correlates reduced MRO lengths with higher fragility and higher Poisson’s ratio (Zhang et al., 2024).
The Kob–Andersen binary Lennard-Jones model provides a complementary case because it has often been treated as lacking medium-range crystalline order. There, a local structural order parameter derived from the mean-field caging potential shows little bare spatial growth, but after spatial coarse-graining over an optimal scale $6$9 that maximizes structure–dynamics correlation, the extracted structural correlation length grows with cooling and tracks the dynamical lengthscale. This growth is strongest for A particles in the 80:20 mixture and for both species in the 60:40 mixture, implying that intermediate-range structural descriptors can recover a growing static length even in models previously labeled MRCO-free (Kumawat et al., 10 Jul 2025).
5. Mechanical, magnetic, and functional consequences
In bulk metallic glasses, MRO has been directly linked to plasticity. For $5$0, minor alloying with Co or Fe leaves the Poisson ratio unchanged at $5$1 but changes both MRO and deformation behavior drastically. The Co-added alloy $5$2–$5$3 exhibits three distinct MRO correlation lengths, $5$4 nm, $5$5 nm, and $5$6 nm, together with plastic strain to failure $5$7. The Fe-added alloy $5$8–$5$9 shows only $20$0 nm and fails at $20$1. The operative conclusion is not that larger MRO or larger MRO volume fraction alone improves ductility, but that multiple discrete correlation lengths and “rich structural diversity” facilitate easier shear banding (Hilke et al., 2020). A related VR-FEM comparison between ductile Pd$20$2Ni$20$3P$20$4 and brittle Vit105 likewise associated a second MRO correlation length near $20$5 nm with larger shear transformation zones and improved deformability (Davani et al., 2020).
MRO has also been implicated in glass stability itself. In ultra-stable Cu$20$6Zr$20$7 prepared by high-temperature physical vapor deposition, crystal-like MRO is built from Cu-centered $20$8 and Zr-centered $20$9 Voronoi polyhedra that align with local translational symmetry over about $6$00–$6$01 nm. This state shows a distinct sub-peak in $6$02 at $6$03, a higher glass transition temperature by $6$04 K relative to the ordinary glass, $6$05 K, hardness $6$06 GPa versus $6$07 GPa, and an HAADF-STEM autocorrelation length $6$08 nm. The crystal-like MRO remains subcritical for crystallization because the domains are smaller than the estimated critical nucleus size of $6$09 nm at $6$10 K (Lu et al., 2021).
Magnetic and magnetoelastic functionalities likewise correlate with MRO. In amorphous Tb$6$11Co$6$12, the FEM variance peak near $6$13–$6$14 increases from $6$15 at $6$16C growth to about $6$17 at $6$18C, while the uniaxial anisotropy constant rises from $6$19 to $6$20; annealing reverses both trends, and tilt-dependent nanodiffraction indicates stronger orientation-dependent bond-length anisotropy in the higher-temperature film (Kennedy et al., 2023). In $6$21, a liquid–liquid phase transition near $6$22 K transiently enhances edge-sharing MRO, produces field-induced hexagonal SANS patterns with $6$23, raises the spin-correlation length to about $6$24 Å, and is accompanied by increases in saturation magnetization and stress-impedance response (Ge et al., 2023).
For amorphous tantala, annealing increases MRO without crystallization. FEM variance peaks are significantly larger after $6$25C annealing than after $6$26C or $6$27C, and virtual dark-field imaging reveals ordered patches up to about $6$28 nm in the highest-temperature sample. These structural changes track changes in mechanical loss: little difference between $6$29C and $6$30C, but reduced room-temperature mechanical loss and a pronounced $6$31 K loss peak after $6$32C annealing (Hart et al., 2015).
6. Conceptual tensions, unresolved issues, and modeling limits
A central unresolved issue is whether MRO is fundamentally a two-body density-correlation phenomenon or whether it requires many-body and topological descriptors. The density-correlation viewpoint argues that $6$33, $6$34, and $6$35 already contain the relevant medium-range information and make four-point measures largely redundant (Lieou et al., 2022). By contrast, persistent-homology analyses of silica, same-species topological analyses of metallic glasses, and ring-resolved decompositions of phosphate-glass diffraction all show that many-body topology can separate structural classes and reveal medium-range organization not transparent in pair functions alone (Nakamura et al., 2015, Liu et al., 2023, Singh et al., 2024). This divergence does not imply contradiction so much as a difference in the chosen coarse-graining of the same disordered structure.
Another recurring lesson is that no universal motif defines MRO across all amorphous systems. In Cu–Zr it may be an icosahedral network; in Al–Tb, Bergman-type superclusters; in GeTe, homopolar-bond chain-like tetrahedral clusters; in silicates and silica, rings, cages, and voids; in Tb–Co, orientation-dependent bond ordering; and in liquid gallium, two overlapping medium-range oscillatory components with $6$36, $6$37, common coherence length $6$38, and fluctuation time $6$39 ps. In that gallium case, the two-component MRO picture was proposed specifically to challenge a fluctuating metallic/insulating domain interpretation (Hua et al., 18 Dec 2025).
A further misconception is that larger MRO or higher MRO fraction necessarily improves a target property. In Pd–Ni–P-based glasses, both Fe and Co additions reduce the relative MRO volume fraction, yet only the Co alloy becomes exceptionally ductile; the decisive variable is the heterogeneity and discreteness of the correlation-length spectrum, not a monotonic increase in size or volume fraction (Hilke et al., 2020). Likewise, ring populations, void sizes, and FSDP intensity can correlate strongly in some systems, but those relations remain system-specific rather than universal laws (Singh et al., 2024, Liu et al., 2 Sep 2025).
Current atomistic modeling has not eliminated these ambiguities. In silica glass, a short-range MACE potential over-structures the network and yields an overly intense FSDP, while a long-range extension with reciprocal-space gated attention improves the liquid structure but still fails to recover the experimental glassy MRO after quenching. Both models preserve tetrahedral local geometry yet retain excessive memory of the parent liquid network, implying that explicit long-range interactions are necessary but not sufficient; the liquid-to-glass transition must also be represented adequately in the training data and sampling protocol (Balantrapu et al., 23 Apr 2026). By contrast, machine-learning-driven modeling of amorphous arsenic reproduced the experimental FSDP height with relative error below $6$40, illustrating that quantitative MRO prediction is achievable when medium-range environments are included effectively in the workflow (Liu et al., 2 Sep 2025).
Taken together, the literature defines MRO not as a single motif or metric but as a family of intermediate-length-scale correlations whose concrete expression depends on chemistry, topology, and the observable used to interrogate them. The strongest consensus is not over one universal structural picture, but over one empirical fact: properties as different as plasticity, fragility, ultra-stability, magnetic anisotropy, nanodomain formation, and mechanical loss all vary with structural organization beyond the first coordination shell, and that organization is experimentally and computationally resolvable once the appropriate medium-range descriptor is chosen.