SCATTER Formalism in Scattering Analysis

• SCATTER formalism is a mathematical and computational framework that decouples internal sub-unit structure from connectivity in composite scattering systems.
• It employs three key functions—form factor, amplitude, and phase—to provide explicit recursive models for acyclic, hierarchically assembled structures.
• The approach enhances the interpretation of small-angle scattering data by incorporating flexible connectivity and distributed reference points.

The SCATTER formalism refers to a family of related mathematical and computational frameworks for modeling scattering phenomena, typically in the context of soft condensed matter, hierarchical composite structures, and small-angle scattering experiments. The central innovation of these approaches is the analytical decoupling of internal sub-unit structure from inter-sub-unit connectivity, producing closed-form expressions for the total scattering intensity of arbitrary (but loopless) branched or linear assemblies, based on properties of their constituent "sub-units" and the topology of their connections. The formalism supports recursive construction, distributed link positions, and accommodates arbitrary, hierarchically organized, acyclic architectures. The development, generalization, and applications of the SCATTER formalism are detailed in works by Svaneborg and Pedersen (Svaneborg et al., 2011, Svaneborg et al., 2011).

1. Theoretical Foundations and Assumptions

The SCATTER formalism is predicated on four principal assumptions regarding the composite object being modeled:

• Sub-unit Decoupling: All sub-units are mutually non-interacting; their internal conformations and positional distributions are statistically independent, with no inter-sub-unit correlations.
• Flexible Connectivity: All links (vertices) connecting sub-units are completely flexible, neglecting orientational constraints at the joint level.
• Acyclic Connectivity Graph: The overall connectivity graph of sub-units is a tree or loopless branched structure; cycles are not permitted at the coarse-grained (sub-unit) scale.
• Reference Points (Vertices): Each sub-unit possesses one or more reference points (vertices) through which it may be joined to other sub-units; these can be either fixed ("regular") or distributed according to a prescribed spatial probability distribution ("distributed reference points").

Given these stipulations, the formalism prescribes that the total scattering intensity, $I(q)$, for a structure comprised of sub-units can be calculated exactly (in the statistical sense), with the full spatial complexity of the assembly encoded in analytic, recursive algebraic expressions.

2. Threefold Decomposition: Form Factor, Form Factor Amplitude, Phase Factor

Each sub-unit $I$ is characterized by three $q$-dependent functions, which serve as "propagators" for assembling the total response:

• Form Factor $F_I(q)$:

$$F_I(q) = \frac{1}{\beta_I^2} \left\langle \sum_{j,k} b_{Ij} b_{Ik} e^{i\mathbf{q}\cdot(\mathbf{r}_{Ij}-\mathbf{r}_{Ik})} \right\rangle$$

This is the normalized Fourier transform of the site-site distribution inside sub-unit $I$, with excess scattering length $\beta_I$.

• Form Factor Amplitude $A_{I\alpha}(q)$:

$$A_{I\alpha}(q) = \frac{1}{\beta_I} \left\langle \sum_{j} b_{Ij} e^{i\mathbf{q}\cdot(\mathbf{r}_{Ij}-\mathbf{R}_{I\alpha})} \right\rangle$$

This function describes the amplitude from all scatterers relative to reference point (vertex) $\alpha$ in sub-unit $I$.

• Phase Factor $\Psi_{I\alpha\omega}(q)$:

$$\Psi_{I\alpha\omega}(q) = \left\langle e^{i\mathbf{q}\cdot(\mathbf{R}_{I\alpha}-\mathbf{R}_{I\omega})} \right\rangle$$

This captures the orientationally averaged propagation between two reference points within the sub-unit.

All higher-level constructions build upon these fundamental quantities.

3. Structural Scattering Equations for Composite Hierarchies

For a composite structure $S$ composed of sub-units $I = 1,\dots,N$, the total form factor (normalized scattering intensity per unit squared scattering length), form factor amplitude, and phase factor are given by (Svaneborg et al., 2011, Svaneborg et al., 2011):

$$F_{S}(q) = \beta_S^{-2} \left[ \sum_{I} \beta_I^2 F_I(q) + \sum_{I \neq J} \beta_I\beta_J A_{I\alpha}(q)A_{J\omega}(q) \prod_{(K,\tau,\eta)\in P(\alpha,\omega)} \Psi_{K\tau\eta}(q) \right]$$

$$A_{S\alpha}(q) = \frac{1}{\beta_S} \sum_{I,\omega} \beta_I A_{I\omega}(q) \prod_{(K,\tau,\eta)\in P(\alpha,\omega)} \Psi_{K\tau\eta}(q)$$

$$\Psi_{S\alpha\omega}(q) = \prod_{(K,\tau,\eta)\in P(\alpha,\omega)} \Psi_{K\tau\eta}(q)$$

where $\beta_S = \sum_I \beta_I$, and $P(\alpha,\omega)$ denotes the unique ordered path of sub-units and their entry/exit vertices connecting reference points $\alpha$ and $\omega$ in the structure (defined by its acyclic graph topology).

These equations support recursive application: any composite structure obeying them can serve as a sub-unit in larger assemblies, enabling hierarchical modeling of, e.g., dendrimers, micelles, bottle-brushes, and multi-generation networks.

4. Distributed Reference Points and Generalization

The generalization to distributed reference points accounts for situations where link positions are not fixed, but follow a prescribed spatial distribution $Q_{I\alpha}(r)$. In this case, the relevant functions are averaged over the link distribution:

• For a distributed reference point $\langle \alpha \rangle$ on sub-unit $I$, the average form factor amplitude is

$$A_{I\langle\alpha\rangle}(q) = \sum_m Q_{I\alpha, m} A_{I\alpha}(q;\mathbf{R}_{I\alpha m})$$

• The phase factor between two distributed reference points $\langle \alpha \rangle$, $\langle \omega \rangle$ is

$$\Psi_{I\langle\alpha\rangle\langle\omega\rangle}(q) = \sum_{m,n} Q_{I\alpha, m} Q_{I\omega, n} \Psi_{I\alpha\omega}(q;\mathbf{R}_{I\alpha m}, \mathbf{R}_{I\omega n})$$

This formalism preserves structural modularity: once the functions for each sub-unit (including the effect of reference-point distributions) are determined, the recursive summation and product structure of the total scattering remains unchanged. The ability to specify arbitrary $Q$ enables the modeling of tethers, patchiness, or surface distributions critical for, e.g., polymer micelles and bottle-brushes (Svaneborg et al., 2011).

5. Canonical Examples and Analytical Forms

A variety of standard sub-units possess closed-form expressions for $(F,\,A,\,\Psi)$:

• Rigid Rods: $$F_\text{rod}(q) = 2\,\mathrm{Si}(x)/x - 4\,[\sin^2(x/2)]/x^2$$, $A_\text{rod}(q) = \mathrm{Si}(x)/x$, $\Psi_\text{rod}(q) = \sin x / x$ with $x = qL$.
• Gaussian Chains: $F_\text{pol}(q) = 2[e^{-x} - 1 + x]/x^2$, $A_\text{pol}(q) = [1-e^{-x}]/x$, $\Psi_\text{pol}(q) = e^{-x}$ with $x = q^2 R_g^2$.
• Solid Spheres: $A_\text{sph,c}(q) = 3[\sin x - x\cos x]/x^3$, $F_\text{sph}(q) = A_\text{sph,c}(q)^2$, $\Psi_\text{sph,cc}(q) = 1$, $x = qR$.
• Other Units: Loops, disks, cylinders, and combinations thereof also have tabulated forms (Svaneborg et al., 2011).

For distributed reference points, e.g., uniform surface distributions on spheres or Gaussian clouds, analytic or numerical integration over the probability density can be performed, altering the resulting $A$ and $\Psi$ (and thus $F_S$) correspondingly. This representation is indispensable for interpreting "corona" or "patchy" architectures in soft-matter systems.

6. Diagrammatic Representation and Recursive Construction

A key feature of the SCATTER formalism is its correspondence with diagrammatic topology:

• Each sub-unit is represented graphically by an ellipse with marked reference points.
• Internal lines correspond to $F_I$ (site-site correlators), rim-to-interior lines to $A_{I\alpha}$ (site-vertex), and rim-to-rim traversals to $\Psi_{I\alpha\omega}$ (vertex-vertex).
• For any pair of scatterers in the full composite, the unique path traversed through the connectivity tree maps directly onto an algebraic product of propagators.

This diagrammatic tool allows for both the conceptual visualization and the systematic algebraic computation of scattering from arbitrarily branched, hierarchically constructed architectures. Substitution of complex sub-assemblies as "super-sub-units" through their own propagators enables rapid construction and updating of models as the underlying molecular architecture evolves.

7. Practical Application, Fitting, and Model Selection

The SCATTER formalism is routinely employed in small-angle neutron and x-ray scattering (SANS/SAXS) analysis of synthetic polymers, biological macromolecules, and colloidal aggregates. In application:

1. Candidates for sub-units are selected based on the known or hypothesized structure; relevant propagators $F,\,A,\,\Psi$ are tabulated or computed for each.
2. The topological connectivity graph and choice of reference point distributions are established (e.g., star "arms" attached at random points on a spherical core).
3. The composite scattering function $F_S(q)$ is constructed recursively, optionally including distributions for link positions.
4. Experimental intensity data $I(q)$ is fit to $\beta_S^2 F_S(q)$ by varying sub-unit attributes (size, flexibility, density), link distributions, and populations (arms, generations, etc.).
5. The hierarchy of limiting behaviors—Guinier regime, Porod scaling, intermediate power laws—can be validated analytically.
6. Comparative model selection between possible architectures or link distributions is facilitated by the modular and decoupled structure of the formalism.

These capabilities make SCATTER a powerful and extensible mathematical foundation for quantitative analysis and model fitting of complex soft-matter systems, particularly for systems with high degrees of compositional and topological sophistication (Svaneborg et al., 2011, Svaneborg et al., 2011).