Crystallinity Index (C-index) Overview

• Crystallinity Index is a scalar descriptor that quantifies crystalline order, varying by context and measurement model.
• In polymer crystallization, a data-driven C-index is derived from logistic regression using descriptors like q6, p2, and local entropy, yielding high validation metrics.
• The index also applies to diffraction phase-content and bond-orientational order, necessitating careful domain-specific interpretation to avoid misconceptions.

Searching arXiv for the cited C-index and related crystallinity/order-parameter papers. Crystallinity Index (C-index) denotes a scalar descriptor intended to quantify crystalline order, but its exact definition is domain dependent. In recent polymer-crystallization work, the term is defined explicitly as the logistic regression model’s predicted probability that a local atomic environment belongs to the crystalline class, using the descriptors $q_6$, $p_2$, and $\bar S_i$ (Tourani et al., 23 Jul 2025). Closely related scalar descriptors also appear as normalized X-ray diffraction phase-content parameters in multiphase samples (Gabielkov et al., 2023) and as global bond-orientational order parameters such as $Q_6$ that function as crystallinity index-like measures in polymer freezing transitions (Leitold et al., 2015). At the same time, work on semiconducting polymers stresses that crystallinity is not a single binary property and should be distinguished from degree of crystallinity, paracrystallinity, texture, and ordering quality (Peng et al., 2020).

1. Scope and nomenclature

The phrase “Crystallinity Index” is not universal across materials research. In condensed-matter and polymer contexts, it usually refers to a scalar summary of crystalline order or of the probability that a local environment is crystalline. However, different studies operationalize that summary in different ways: as a probabilistic classifier output, as a diffraction-pattern correlation statistic, or as a bond-orientational order parameter.

Context	Scalar quantity	Role
Polymer MD	$C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$	Threshold-free local crystallinity measure
Multiphase XRD	$S$, $S_{\max}$, $S_{\text{expr}}/S_{\max}$	Relative content of identified crystalline phases
Polymer folding/freezing	$Q_6$	Global crystallinity/order parameter
Semiconducting polymers	$g$, $p_2$0, $p_2$1, DoC	Multidimensional ordering framework

A recurrent source of confusion is abbreviation. “C-index” is also used outside crystallization research for mathematically unrelated quantities, including the concordance index in survival prediction (Simon et al., 5 Jun 2025) and a phase-space “crystallization” metric for Galactic globular clusters (Huang et al., 7 May 2026). In materials science, therefore, the term should be interpreted only with its local definition and measurement model.

2. Data-driven C-index in polymer crystallization

A direct, explicit definition of the Crystallinity Index is given in the machine-learning workflow for polymer crystallization from atomistic molecular dynamics. The workflow begins with a high-dimensional feature representation of each atom, combining geometric, thermodynamic-like, and symmetry-based descriptors. The initial descriptor set includes Voronoi-based local density and face count, $p_2$2, local entropy $p_2$3, local enthalpy $p_2$4, and bond-orientational order parameters $p_2$5, including $p_2$6, with locally averaged versions as well. These descriptors are standardized to zero mean and unit variance, and chain-end missing values in $p_2$7 are imputed to the lower quartile because chain ends are unlikely to crystallize and cannot define the bond vector used in $p_2$8.

The labels used to define crystallinity are not assigned by a single preset threshold. Instead, low-dimensional embeddings are used to expose latent structural fingerprints within atomic environments, with UMAP preferred because it gives the clearest separation and strong silhouette scores relative to PCA and VAE. HDBSCAN clustering on the 2D UMAP space produces the crystalline/amorphous labels, while GMM and K-means are checked as alternatives. The resulting binary $p_2$9 is then used as the target for supervised learning.

Feature selection is central to the final definition. Logistic regression, random forest, and gradient boosting all perform very well, but forward feature selection saturates after the descriptors $\bar S_i$0, $\bar S_i$1, and $\bar S_i$2, reaching an AUC of 0.9993 and then plateauing near 0.9994–0.9995 with further additions. A Fisher’s $\bar S_i$3 partial correlation screen after FDR correction likewise identifies only these three descriptors as conditionally dependent on the cluster label across all temporal regimes. The final C-index is therefore deliberately parsimonious.

In this formulation, the C-index is the probability from a logistic regression model: $\bar S_i$4

$\bar S_i$5

Values near 0 indicate amorphous-like environments and values near 1 indicate crystalline-like environments. The authors emphasize that this avoids discrete thresholds and provides a continuous local measure of order. Validation is extensive: logistic regression gives an AUC of about 0.9718, random forest about 0.9791, and gradient boosting about 0.9846, while a gradient boosting classifier using only $\bar S_i$6 reaches an AUC of 0.979 and an adjusted Rand index of 0.93 relative to the full model (Tourani et al., 23 Jul 2025).

3. Relation to established crystallinity descriptors

The data-driven C-index is best understood as part of a longer tradition of scalar order parameters for crystallization. In the polymer freezing study of a single flexible homopolymer chain, the global bond-orientational order parameter $\bar S_i$7 is described as a global order parameter, sensitive to close-packed structures, and therefore informative about the crystallinity of a configuration. It is the best-performing second coordinate when combined with the energy $\bar S_i$8, outperforming $\bar S_i$9, $Q_6$0, $Q_6$1, anisotropy, and moments of inertia in the string reaction-coordinate analysis (Leitold et al., 2015).

In that framework, $Q_6$2 is not a direct geometric fraction of crystalline monomers. It is a bond-orientational order parameter, so it quantifies symmetry and long-range orientational order rather than simply counting crystalline particles. This distinction is important because it clarifies what a crystallinity index can and cannot mean: some indices measure the degree to which a structure resembles a crystal, while others measure the amount of crystalline material.

The semiconducting-polymer literature makes this distinction explicit. There, crystallinity is not treated as a single scalar “index” but as a multidimensional concept involving the degree of crystallinity (DoC), the quality of ordering within ordered domains, paracrystalline disorder, coherence length, and stacking extent. The paracrystalline disorder parameter $Q_6$3 is used as the principal ordering metric, with the classification $Q_6$4 for a perfect crystal, $Q_6$5 for crystalline ordering, $Q_6$6 for paracrystalline ordering, and $Q_6$7 for amorphous ordering. The same work recommends reserving “crystallinity” for the volume fraction of crystalline domains and using terms such as “molecular ordering,” “aggregate,” and “crystallite” for packing quality and range (Peng et al., 2020).

A plausible implication is that a C-index is most informative when its target is specified explicitly: local crystalline class membership, orientational order, diffraction-phase content, or volume-fraction crystallinity are not interchangeable quantities.

4. Diffraction-based phase-content indexing

A conceptually different but closely related index is developed for X-ray diffraction analysis of multiphase samples. Here the objective is not local classification in molecular dynamics, but estimation of the relative content of identified crystalline phases from diffraction intensities. The method introduces a numerical correlation parameter that uniquely characterizes a diffraction pattern and compares patterns using computational statistics and Monte Carlo or permutation-test style resampling rather than a direct deterministic formula alone.

The experimental diffraction intensities are denoted by $Q_6$8 and the tabulated or reference intensities by $Q_6$9, with

$C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$0

Correlation is treated probabilistically, and the resulting $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$1 is interpreted in relative units. The method relies on the fact that $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$2 reaches its maximum $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$3 when the compared patterns are identical. Because tabulated diffraction patterns are normalized so that the strongest line has intensity 1000, $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$4 differs from phase to phase and provides a characteristic numerical descriptor of each reference diffraction pattern. The paper also proposes that the sum of all line intensities of the normalized reference pattern,

$C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$5

can be used as a phase-characteristic parameter.

In practical use, the observed experimental value $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$6 is compared with the self-correlation maximum $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$7 for the same phase. If the experimental intensities are reduced by a factor, then $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$8 decreases proportionally, and the ratio $C=P(\text{crystalline}\mid p_2,\bar S_i,q_6)$9 is used to estimate the relative amount of that phase. The method is formulated for the relative content of identified crystalline phases, not for absolute crystallinity or amorphous fraction. It assumes phases with similar absorption coefficients and neglects self-absorption, texture, and other correction factors in the basic derivation.

Validation illustrates both the promise and the limitations. For a model 50:50 mixture of nantokite and eriochalcite, the method yields 50.68% nantokite and 49.32% eriochalcite, with an error no worse than about $S$0 in the idealized model case. For an experimental calibration mixture claimed to be $S$1, it yields approximately 61.7% and 38.9%. The paper states that control measurements give uncertainty not exceeding $S$2, while a Monte Carlo propagation of a $S$3 uncertainty in reflection intensities gives a spread in $S$4 with width at half-height also around $S$5 (Gabielkov et al., 2023).

5. Mechanistic interpretation during crystallization

A notable feature of several crystallinity indices is that their physical meaning changes across stages of the transformation. In the polymer-chain freezing study, the optimal reaction coordinate is a curved string in $S$6 space rather than a linear interpolation. Initially, the system changes mainly by increasing its overall crystallinity as quantified by the $S$7 parameter; later on in the crystallization, the strongest change is seen in the potential energy. This leads to a two-stage interpretation: an ordering or nucleation stage followed by a growth or compaction stage (Leitold et al., 2015).

The machine-learning C-index for polyethylene crystallization yields an analogous staged picture, but with a different set of variables. At nucleation onset, especially for longer, entangled chains, entropy dominates: low-entropy fluctuations are the most informative signal of early crystallinity. As crystallization proceeds into growth, the entropy weight declines while $S$8 and $S$9 become more important, and by mid-growth $S_{\max}$0 generally becomes the primary discriminator across all chain lengths. The authors interpret this as symmetry taking over from entropy once nuclei are formed (Tourani et al., 23 Jul 2025).

This suggests that a single scalar C-index may remain useful across an entire trajectory only if it is derived from descriptors whose relative weights can accommodate stage dependence. In both cases, the scalar does not merely label end states; it tracks the changing structural logic of nucleation and growth.

6. Limitations, misconceptions, and acronym overload

A common misconception is to treat any C-index as a direct percentage of crystallinity. That interpretation is not generally valid. In the polymer $S_{\max}$1 study, the relevant scalar is a bond-orientational order parameter rather than a direct geometric fraction of crystalline monomers (Leitold et al., 2015). In the XRD phase-content method, the normalized correlation statistic is explicitly aimed at the relative content of identified crystalline phases and includes an additional term $S_{\max}$2 for unidentified reflections, halos, amorphous background, and related contributions (Gabielkov et al., 2023). In semiconducting polymers, the distinction between degree of crystallinity and the quality of order is central, and the recommended framework combines GIWAXS and DSC with $S_{\max}$3, $S_{\max}$4, and $S_{\max}$5 rather than collapsing the problem into a single number (Peng et al., 2020).

There are also methodological limits. The machine-learning C-index inherits the quality and biases of the UMAP plus HDBSCAN labels, is calibrated for the specific polyethylene system and force field studied, requires imputation for chain-end atoms in $S_{\max}$6, and depends on choices such as cutoff distance, smoothing, and neighborhood definition in the entropy descriptor $S_{\max}$7. Although the model is transferable across time in the reported study, the coefficients evolve and should be validated before universal use is assumed (Tourani et al., 23 Jul 2025).

Finally, the acronym itself is overloaded. In survival analysis, the C-index is a discrimination metric for pairwise ranking of survival times, with extensions such as the expected C-index, SUBCI, and SUBECI (Simon et al., 5 Jun 2025). In a Gaia-based study of Galactic globular clusters, $S_{\max}$8 is a quadrature combination of positive standardized radial and kinematic anomaly scores and measures “phase-space crystallization” rather than condensed-matter crystallinity (Huang et al., 7 May 2026). For technical writing, therefore, the term “C-index” is only meaningful when accompanied by its full operational definition, target variable, and domain of validity.