First Sharp Diffraction Peak (FSDP)

• FSDP is a pronounced low-Q peak in the structure factor of amorphous materials that signifies medium-range order, as seen in systems like amorphous silicon and silica.
• It arises from a combination of shell-resolved radial correlations, chemical ordering, ring topology, and void distribution, offering diverse insights across different material classes.
• FSDP serves as both an experimental signature and a computational benchmark for validating atomistic models and understanding elastic and vibrational properties in disordered systems.

Searching arXiv for recent and relevant papers on the First Sharp Diffraction Peak (FSDP). The first sharp diffraction peak (FSDP) is a low-$Q$, relatively narrow feature in the static structure factor $S(Q)$ of many amorphous materials, liquids, and glasses. It is widely treated as an experimental signature of medium-range order (MRO) or, in some systems, intermediate-range order (IRO), distinguishing structural correlations beyond nearest-neighbor coordination from crystalline long-range periodicity. Across contemporary studies, the FSDP is not a single-mechanism phenomenon: in some systems it is primarily governed by shell-resolved radial correlations, in others by chemical ordering, ring topology, void distributions, or mesoscale clustering. The modern literature therefore treats the FSDP less as a universal structural motif than as a reciprocal-space manifestation of system-dependent real-space correlations, typically on length scales larger than the first coordination shell but shorter than crystalline order (Dahal et al., 2020).

1. Definition and reciprocal-space formulation

In diffraction-based analyses of disordered matter, the FSDP is identified as the first pronounced maximum in $S(Q)$ at comparatively small wavevector. Its exact location is strongly material-dependent. In amorphous silicon it appears near $Q_0 \approx 2\ \textrm{\AA}^{-1}$ (Dahal et al., 2020); in amorphous silica the FSDP is discussed around $\sim 1.5\ \textrm{\AA}^{-1}$ (Biswas et al., 2024); in vitreous silica experimental data are described as showing an FSDP at $Q \sim 1.55\ \textrm{\AA}^{-1}$ (Balantrapu et al., 23 Apr 2026); in iron phosphate glass the reproduced FSDP is at $\sim 1.75\ \textrm{\AA}^{-1}$ (Singh et al., 2024); in isotactic poly(4-methyl-pentene-1) it occurs at $\sim 0.7\ \textrm{\AA}^{-1}$ or $\sim 0.67\ \textrm{\AA}^{-1}$ depending on the reported representation (Ogihara et al., 17 Apr 2026); and in the n-pentanol/pentanal liquid mixtures it is observed around $0.6\ \textrm{\AA}^{-1}$ (Pethes et al., 2018).

A recurring formal starting point is the Fourier relation between $S(Q)$0 and pair-correlation data. For isotropic systems one commonly encounters

$S(Q)$1

with the reduced pair-correlation function

$S(Q)$2

in amorphous silicon (Dahal et al., 2020), and closely related expressions in amorphous silica, iron phosphate glass, liquid mixtures, and polymers (Biswas et al., 2024, Singh et al., 2024, Pethes et al., 2018, Ogihara et al., 17 Apr 2026). This formalism makes the FSDP a reciprocal-space measure of constructive interference arising from selected real-space correlations.

A more general interpretation appears in work on diffraction formulas based on the true distribution of interatomic vectors (DIV). In that framework, diffracted intensity is written explicitly as a function of the distribution of interatomic vectors, and angular averaging yields a form involving $S(Q)$3. This formulation is presented as enabling a direct mapping between diffraction features such as the FSDP and specific vector populations in real space, without additional approximation beyond kinematical theory (Hadji, 2021). This suggests an exact structural interpretation of FSDP-like features is, in principle, possible whenever sufficiently complete real-space vector information is available.

2. Real-space origin: radial correlations and shell-resolved contributions

A major line of work explains the FSDP through radial atomic correlations resolved shell by shell. In amorphous silicon, the position and intensity of the FSDP are reported to be primarily determined by radial correlations on the length scale of about $S(Q)$4, with the key contributions originating from the second and fourth radial shells, a background from the first shell, and small residual corrections from distant shells (Dahal et al., 2020). The same study gives a shell decomposition

$S(Q)$5

where each $S(Q)$6 integrates $S(Q)$7 over a radial interval assigned to a shell. The second shell, at $S(Q)$8, is identified as dominant for both the FSDP intensity and its position; the fourth shell at $S(Q)$9 contributes strongly; and distant shells beyond $S(Q)$0 contribute weak residual corrections (Dahal et al., 2020).

A related but more extended analysis of amorphous silicon distinguishes medium-range from extended-range radial order. Ultra-large models exhibit oscillations in $S(Q)$1 out to $S(Q)$2, yet the FSDP converges once correlations up to $S(Q)$3–$S(Q)$4 are included. The authors report that correlations beyond $S(Q)$5 do not alter the FSDP position or intensity appreciably, and that the practical error associated with truncating the Gaussian decomposition of $S(Q)$6 is $S(Q)$7 (Dahal et al., 2022). In this view, the FSDP is a marker of medium-range order rather than extended-range order.

For amorphous silica, the radial decomposition becomes chemically resolved. The FSDP is stated to be largely determined by the first two/three radial shells on a length scale of about $S(Q)$8–$S(Q)$9, with shell-by-shell analysis showing that Si–O and O–O pair correlations dominate, while Si–Si correlations contribute only small corrections (Biswas et al., 2024). The paper summarizes shell intervals such as $Q_0 \approx 2\ \textrm{\AA}^{-1}$0–$Q_0 \approx 2\ \textrm{\AA}^{-1}$1 and $Q_0 \approx 2\ \textrm{\AA}^{-1}$2–$Q_0 \approx 2\ \textrm{\AA}^{-1}$3 for Si–O, and $Q_0 \approx 2\ \textrm{\AA}^{-1}$4–$Q_0 \approx 2\ \textrm{\AA}^{-1}$5 and $Q_0 \approx 2\ \textrm{\AA}^{-1}$6–$Q_0 \approx 2\ \textrm{\AA}^{-1}$7 for O–O, as the principal FSDP-relevant ranges (Biswas et al., 2024).

A common phenomenological relation used in such analyses is that strong contributions arise when maxima in radial correlations align with constructive extrema of the Fourier kernel. In amorphous silicon, the condition

$Q_0 \approx 2\ \textrm{\AA}^{-1}$8

is used for the second shell, giving

$Q_0 \approx 2\ \textrm{\AA}^{-1}$9

for $\sim 1.5\ \textrm{\AA}^{-1}$0, and with $\sim 1.5\ \textrm{\AA}^{-1}$1 yielding $\sim 1.5\ \textrm{\AA}^{-1}$2 (Dahal et al., 2020). In amorphous silica, the more general mapping

$\sim 1.5\ \textrm{\AA}^{-1}$3

is proposed between radial peaks $\sim 1.5\ \textrm{\AA}^{-1}$4 and reciprocal-space peaks $\sim 1.5\ \textrm{\AA}^{-1}$5, with the sign distinguishing maxima and minima (Biswas et al., 2024). These relations do not imply a universal law for all amorphous materials, but they formalize how radial shells can select FSDP positions in specific network glasses.

3. Chemical order, topology, rings, and voids

Beyond purely radial descriptions, contemporary studies emphasize that the FSDP can encode chemical ordering, network topology, and free-volume organization.

In amorphous silica, the stated dominant mechanism is chemical order involving Si–O and O–O pairs, supplemented by small contributions from weak Si–Si correlations in Fourier space (Biswas et al., 2024). The implication is that the FSDP is not merely a density-fluctuation peak but a chemically weighted consequence of the binary network.

In iron phosphate glass, the FSDP is tied explicitly to ring size distribution. Hybrid MC/MD atomistic models show that configurations with a broad ring distribution peaking at 10-node rings reproduce the FSDP at $\sim 1.5\ \textrm{\AA}^{-1}$6, whereas a model described as lacking MRO by ring analysis fails to show the FSDP (Singh et al., 2024). The study constructs ring-size-specific partial structure factors $\sim 1.5\ \textrm{\AA}^{-1}$7 and fits the experimental structure factor as

$\sim 1.5\ \textrm{\AA}^{-1}$8

with fitted weights $\sim 1.5\ \textrm{\AA}^{-1}$9 that qualitatively track the ring-size percentages in the model (Singh et al., 2024). Smaller rings shift the FSDP to higher $Q \sim 1.55\ \textrm{\AA}^{-1}$0, larger rings to lower $Q \sim 1.55\ \textrm{\AA}^{-1}$1, and the total FSDP coincides with the 10-node-ring contribution (Singh et al., 2024). This provides a topological interpretation of the FSDP as a signature of the dominant repeat distance associated with the most probable ring size.

In amorphous arsenic, the origin of the FSDP is linked to the size and spatial distribution of voids in the amorphous network (Liu et al., 2 Sep 2025). Machine-learned-potential simulations that reproduce the experimental structure factor show that the FSDP height is strongly and linearly correlated with the average equivalent void radius across pressures, with

$Q \sim 1.55\ \textrm{\AA}^{-1}$2

The study further relates this void organization to ring topology and dihedral-angle distribution: amorphous arsenic exhibits a broad and nearly uniform dihedral-angle distribution, a significant population of large rings $Q \sim 1.55\ \textrm{\AA}^{-1}$3, and an anisotropic, partly layered void arrangement, whereas red amorphous phosphorus exhibits more compact 5-membered-ring motifs and nearly isotropic voids (Liu et al., 2 Sep 2025). The FSDP is therefore interpreted as a fingerprint of MRO generated by topologically enabled density inhomogeneity.

In isotactic poly(4-methyl-pentene-1), the FSDP is also attributed to internal voids rather than to crystal-like order (Ogihara et al., 17 Apr 2026). Here the polymer’s bulky side chains generate nanovoids and low density, and the FSDP behaves as a void-sensitive scattering signature. This parallel with silica and amorphous arsenic indicates that void-related FSDP mechanisms are not restricted to inorganic covalent glasses.

4. Medium-range order versus long-range periodicity and common misconceptions

A central misconception in the older literature is that the FSDP should be interpreted as a diffuse Bragg peak or as direct evidence of incipient crystallinity. The papers consistently reject that interpretation.

For amorphous silicon, the FSDP is stated to arise not from long-range periodicity, but from medium-range order in a continuous random network (Dahal et al., 2020). The 2022 extended-range-order study strengthens this point by showing that even when oscillatory order persists to $Q \sim 1.55\ \textrm{\AA}^{-1}$4, the FSDP is still governed by correlations within $Q \sim 1.55\ \textrm{\AA}^{-1}$5, not by any genuine long-range periodic structure (Dahal et al., 2022).

In silica glass, the 2026 study on machine-learning interatomic potentials defines the FSDP as the principal experimental signature of MRO and shows that even models reproducing the local $Q \sim 1.55\ \textrm{\AA}^{-1}$6 tetrahedral geometry can fail at the level of the FSDP if medium-range network flexibility is incorrect (Balantrapu et al., 23 Apr 2026). This directly separates short-range tetrahedral order from the MRO encoded by the FSDP.

The liquid-mixture study of n-pentanol and pentanal highlights a different misconception: that low-$Q \sim 1.55\ \textrm{\AA}^{-1}$7 pre-peaks must be reproducible from short-range pair statistics or classical force fields. Experimentally, the FSDP intensity around $Q \sim 1.55\ \textrm{\AA}^{-1}$8 changes non-linearly and non-monotonically with composition, whereas classical MD with OPLS-AA force fields predicts a monotonic decrease and thus fails in the low-$Q \sim 1.55\ \textrm{\AA}^{-1}$9 region (Pethes et al., 2018). Because the relevant O–O and O–H partial radial distribution functions do not change with composition in a way that tracks the FSDP, the authors argue that short-range hydrogen-bond distributions alone cannot explain the effect (Pethes et al., 2018). A plausible implication is that, in some liquids, the FSDP reports on clustering or supramolecular organization inaccessible to conventional local descriptors.

5. Materials-specific manifestations

The FSDP is a cross-material phenomenon, but its microscopic origin varies substantially.

System	FSDP region	Principal origin emphasized
Amorphous silicon	near $\sim 1.75\ \textrm{\AA}^{-1}$0	second and fourth radial shells; MRO to $\sim 1.75\ \textrm{\AA}^{-1}$1–$\sim 1.75\ \textrm{\AA}^{-1}$2
Amorphous silica	around $\sim 1.75\ \textrm{\AA}^{-1}$3	Si–O and O–O chemical/radial order within $\sim 1.75\ \textrm{\AA}^{-1}$4–$\sim 1.75\ \textrm{\AA}^{-1}$5
Vitreous silica	$\sim 1.75\ \textrm{\AA}^{-1}$6	network motifs, rings, voids; sensitive to MLIP limitations
Iron phosphate glass	$\sim 1.75\ \textrm{\AA}^{-1}$7	ring-size distribution, especially 10-node rings
Amorphous arsenic	small-$\sim 1.75\ \textrm{\AA}^{-1}$8 pronounced peak	void size and spatial distribution; ring topology and dihedrals
n-pentanol/pentanal mixtures	around $\sim 1.75\ \textrm{\AA}^{-1}$9	composition-dependent clustering or transient complexes
P4MP1 polymer	$\sim 0.7\ \textrm{\AA}^{-1}$0	internal voids in the amorphous phase

In amorphous silicon, sufficiently large models with $\sim 0.7\ \textrm{\AA}^{-1}$1 atoms or $\sim 0.7\ \textrm{\AA}^{-1}$2 yield an FSDP position converged to $\sim 0.7\ \textrm{\AA}^{-1}$3–$\sim 0.7\ \textrm{\AA}^{-1}$4, with height matching experiment within uncertainty when correlations up to $\sim 0.7\ \textrm{\AA}^{-1}$5 are included (Dahal et al., 2020). The width also converges only for large models and is reported to correlate inversely with peak intensity (Dahal et al., 2020).

In amorphous arsenic, the FSDP is quantitatively matched by refined machine-learned-potential simulations to within <1% relative error in intensity, and its evolution with pressure tracks shrinkage of voids (Liu et al., 2 Sep 2025). In contrast, for silica glass, both short-range and long-range MACE-based MLIPs fail to fully recover the experimental glassy FSDP after quenching, despite improved liquid-state agreement from the long-range model (Balantrapu et al., 23 Apr 2026). This contrast underscores that reproducing the FSDP can be either a validation success or a diagnostic of model insufficiency, depending on the material and methodology.

In the n-pentanol/pentanal mixture, the FSDP is strongest not in the pure alcohol but at intermediate compositions, especially 25% and 50% pentanal, and diminishes only above 75 mol % pentanal (Pethes et al., 2018). Because this behavior is absent from the MD simulations, the study speculates about reversible hemiacetal formation or mesoscale clustering (Pethes et al., 2018). The term “pre-peak” is used interchangeably with FSDP in that liquid-state context.

In P4MP1, pressure lowers the FSDP intensity in the melt, decompression restores it, helium suppresses the pressure-induced reduction, and decane immersion decreases the FSDP by filling internal voids (Ogihara et al., 17 Apr 2026). This system provides an unusually direct experimental manipulation of the structural contrast responsible for the FSDP.

6. Experimental, computational, and analytical uses of the FSDP

The FSDP functions both as an observable and as a structural benchmark. Its utility is evident in several distinct methodological contexts.

First, it is used for model validation. Iron phosphate glass models were validated against short-range descriptors such as pair correlation functions, bond-angle distributions, and coordination numbers, and against medium-range descriptors including ring statistics, void size distributions, and especially the FSDP position and intensity in $\sim 0.7\ \textrm{\AA}^{-1}$6 (Singh et al., 2024). In amorphous arsenic, MLIP refinement was explicitly judged by the ability to reproduce the FSDP (Liu et al., 2 Sep 2025). In silica glass, failure to match the FSDP despite correct local tetrahedral geometry exposed deficiencies in learned potentials and quench protocols (Balantrapu et al., 23 Apr 2026).

Second, it is used as a real-space inversion target. Shell-by-shell and Gaussian-decomposition analyses in amorphous silicon and amorphous silica calculate the contributions of distinct radial intervals directly to the FSDP (Dahal et al., 2020, Biswas et al., 2024). In amorphous silicon, the reduced pair-correlation function is approximated by

$\sim 0.7\ \textrm{\AA}^{-1}$7

leading to a semi-analytical structure factor

$\sim 0.7\ \textrm{\AA}^{-1}$8

for the Gaussian components (Dahal et al., 2022). In amorphous silica, Gaussian shell models are likewise used to derive semi-analytical expressions for shell contributions (Biswas et al., 2024).

Third, the FSDP can serve as a contrast-sensitive probe of void occupancy. In P4MP1, the use of uniaxially stretched samples and 2D-XRD separates the isotropic amorphous FSDP ring from directional crystalline Bragg reflections, allowing clean observation of FSDP suppression upon decane uptake (Ogihara et al., 17 Apr 2026). SANS with deuterated decane then confirms solvent localization in amorphous regions (Ogihara et al., 17 Apr 2026).

Fourth, the FSDP may link to dynamical anomalies. A 2025 study on the boson peak reports a strong correlation between the FSDP wavenumber $\sim 0.7\ \textrm{\AA}^{-1}$9 and the maximum possible coarse-graining wavenumber $\sim 0.67\ \textrm{\AA}^{-1}$0 in heterogeneous elasticity theory, with correlation coefficient $\sim 0.67\ \textrm{\AA}^{-1}$1 across many glasses (Kyotani et al., 10 Jan 2025). The proposed two-step picture is that the FSDP sets the unit size of elastic modulus heterogeneity, while the magnitude of elastic-modulus fluctuations determines boson-peak frequency and intensity (Kyotani et al., 10 Jan 2025). This suggests the FSDP may act not only as a structural fingerprint but also as a scale-setting parameter for glassy vibrational response.

7. Open issues and current research directions

Despite its ubiquity, the FSDP remains a structurally plural phenomenon. One open issue is whether a universal explanatory framework exists at all. The reviewed studies point in different directions: radial shell interference in amorphous silicon (Dahal et al., 2020), chemical ordering in amorphous silica (Biswas et al., 2024), ring-size topology in iron phosphate glass (Singh et al., 2024), void morphology in amorphous arsenic (Liu et al., 2 Sep 2025), and mesoscale clustering or reversible chemistry in alcohol–aldehyde mixtures (Pethes et al., 2018). This suggests that “FSDP” denotes a common diffraction signature whose microscopic origin is material-specific.

A second active question concerns simulation fidelity. In amorphous arsenic, automated ML workflows and refined MLIPs reproduce the FSDP with high accuracy (Liu et al., 2 Sep 2025). In silica glass, however, even explicit long-range interactions are reported as necessary but not sufficient: the long-range model improves the liquid structure but still fails for the glass after quenching, indicating that training data and sampling strategies for the liquid-to-glass transition are also decisive (Balantrapu et al., 23 Apr 2026). The FSDP has thus become a stringent criterion for the realism of atomistic models of disorder.

A third issue concerns the precise role of voids versus network connectivity. In amorphous arsenic and P4MP1, the evidence for void-controlled FSDP behavior is direct (Liu et al., 2 Sep 2025, Ogihara et al., 17 Apr 2026). In amorphous silica, by contrast, the 2024 shell-analysis paper emphasizes Si–O and O–O chemical/radial order rather than a void-centered mechanism (Biswas et al., 2024). This does not necessarily imply contradiction: a plausible implication is that void distributions and chemically resolved pair correlations may represent complementary descriptions of the same medium-range network organization, with their relative explanatory value depending on scattering weights, topology, and material class.

A final issue is the extent to which the FSDP can be elevated from an empirical signature to a predictive parameter. The boson-peak correlation work argues that the FSDP wavenumber represents the largest structural correlation in glass and sets the scale of elastic heterogeneity (Kyotani et al., 10 Jan 2025). If that relationship proves robust beyond the surveyed systems, the FSDP would have significance extending beyond static structure into mesoscale elasticity and dynamics.

In current usage, the FSDP is therefore best understood as a reciprocal-space fingerprint of nonlocal structural organization in disordered matter. It is sharp enough to be diagnostically useful, but not universal enough to admit a single microscopic interpretation. Its position, width, and intensity encode medium-range correlations; the challenge is to determine, for each material class, whether those correlations are most faithfully described in terms of shells, chemical order, rings, voids, clusters, or coupled structural and dynamical heterogeneity.