Diffuse scattering

Example SAXS of a disordered composite, which generates diffuse scattering at low q.

Diffuse scattering is the scattering that arises from any departure of the material structure from that of a perfectly regular lattice. One can think of it as the signal that arises from disordered structures, and it appears in experimental data as scattering spread over a wide q-range (diffuse). Diffuse scattering is generally difficult to quantify, because of the wide variety of effects that contribute to it.

Bragg diffraction occurs when scattering amplitudes add constructively. If there is a defect in a crystal lattice (e.g. atom missing or in a slightly 'wrong' position), then the amplitude of the Bragg peak decreases. This 'lost' scattering intensity is redistributed into diffuse scattering. The diffuse scattering thus arises from the local (short range) configuration of the material (not the long-range structural order).

In the limit of disorder, one entirely lacks a realspace lattice and thus scattering does not generate any Bragg peaks. However, a disordered structure will still give rise to diffuse scattering. The Fourier transform of a disordered structure will not give any well-defined peaks, but will give a distribution of scattering intensity over a wide range of q-values. Thus samples with an inherently disordered structure (polymer blends, randomly packed nanoparticles, etc.) will only generate diffuse scattering.

Causes

This is only a partial list of sources of diffuse scattering:

• Thermal motion causes atoms to jitter about their ideal unit cell positions, which decorelates them. This suppresses the intensity of the Bragg peaks, especially the higher-order peaks (see Debye-Waller factor), and instead generates high-q diffuse scattering. (One can also think of this in terms of phonons: in ordered systems the diffuse scattering is probing phonon modes.)
• Static disorder in crystals (vacancy defects, substitutional defects, stacking faults, etc.) similarly creates diffuse scattering.
• Grain structure in otherwise ordered materials will also contribute. The grains themselves can count as 'scattering objects', but since their size is ill-defined, the grain boundaries give rise to diffuse scattering.
• Nanoscale disorder gives rise to low-q diffuse scattering. For instance, a disordered polymer blend (or a bulk heterojunction) or a random packing of nanoparticles, will generate substantial low-q diffuse scattering.
• Surface roughness in thin films measured by GISAXS gives rise to low-q diffuse scattering in GISAXS. Roughness will tend to broaden (and increase the intensity of) the specular rod, and will also generate intense low-q scattering.
• Particle size/shape polydispersity introduces a diffuse background.
• Polymer chains in solution generate scattering without a well-defined size-scale. This is normally interpreted in terms of the form factor of the polymer chain. However one can also think of it as the polymer chains having disordered arrangements and thus giving rise to diffuse scattering (c.f. definitional boundaries). Polymer clustering or gelation can also give rise to diffuse-like scattering.

Analysis: low-q

Diffuse scattering can be difficult to quantify, since so many different effects contribute to it. Nevertheless, if one has a good understanding of the expected kind of disorder, one can fit the diffuse scattering with a model.

Ornstein-Zernike model

Yields correlation length (ξ):

$I(q) \propto \frac{1}{ 1 + q^2 \xi^2 }$

Debye-Bueche random two-phase model

Yields correlation length (a):

$\frac{\mathrm{d}S}{\mathrm{d}W} = \frac{A}{ (1 + (qa)^2 )^2 }$

Guinier model

Yields average radius of gyration (Rg):

$I(q) \propto e^{ - R_g^2 q^2 / 3 }$

Analysis: high-q

Porod law

For high-q, gives specific surface area (S):

$I(q) \propto S q^{-4}$

Porod fractal law

For high-q, gives specific surface area:

$\lim_{q \rightarrow \infty} I(q) \propto S' q^{-(6-d)}$
• References:

Structured Diffuse Scattering

Even the highest-quality single-crystal materials exhibit some level of disorder, giving rise to diffuse scattering. Single-Crystal Diffuse Scattering (SCDS) can be analyzed to understand local (short-range) ordering. Experimentally, it appears as weak and diffuse 'lines' that interconnect between the bright diffraction spots. While the diffraction spots describe the idealized unit-cell, this diffuse scattering encodes the deviation from the average structure. The diffuse scattering can be quantitatively modeled.

More broadly, highly anisotropic and structured diffuse scattering indicates that the material exhibits some non-trivial combination of order and disorder in realspace. For instance, the disorder/defects may be ordered in some way (correlated disorder). Whereas diffuse scattering is traditionally seen as encoding structuring on a very local scale, structured diffuse scattering relates to order on intermediate size-scales.

Liquid Surface Diffuse Scattering

Grazing-incidence scattering can be performed on liquid interfaces. GISAXS scattering peaks may be seen if there is some kind of ordering at the interface (e.g. nanoparticles); but a pure liquid interface will, generically, not exhibit structural scattering. However, the disorder of the interface will generate diffuse scattering. In particular, the interface will fluctuate due to thermally-induced waves: i.e. capillary waves.