Small-Angle Scattering. Ian W. Hamley

range 1–100 nm, which is the structural size scale corresponding to many types of soft and hard nanomaterial as well as biomolecules such as proteins in solution. Considering Bragg's law, the scattering from such large structures will be observed at small angles (less than a few degrees of scattering angle 2θ). Wide‐angle scattering covers the 0.1–1 nm range. Ultra‐small‐angle scattering (USAXS and USANS), also discussed in this book (see e.g. Chapters 3 and 5), can extend up to 1000 nm or more, which overlaps with the size scale probed by light scattering.

      SANS and SAXS have complementary characteristics (Section 5.14), which are discussed in the respective chapters (Chapters 3 and 4) dedicated to these methods. These arise from the distinct natures of neutrons and x‐rays. Neutrons are nuclear particles, with a mass 1.675 × 10⁻²⁷ kg. They have spin half (i.e. they are fermions) and a finite magnetic moment μ_n = −9.662 × 10⁻²⁷ J T⁻¹, and zero charge. In contrast, x‐rays are photons with spin 1 (they are bosons), no mass, and no magnetic moment. X‐rays are a type of electromagnetic radiation with wavelengths in the approximate range 0.01–1 nm with overlap with gamma rays at short wavelengths and the extreme ultraviolet at long wavelengths. Despite the different nature of neutrons and x‐rays, both exhibit wave‐like diffraction by matter. Using de Broglie's relationship, the associated wavelength of neutrons (this is discussed quantitatively in Section 3.6, in terms of the velocity distribution of neutrons produced by reactor and spallation sources) can be calculated.

      This chapter provides a summary of the theory that underpins SAS, starting from the basic equations for the wavenumber and scattering amplitude (Section 1.2). Section 1.3 introduces the essential theory concerning the scattered intensity and its relationship to real space correlation functions, for both isotropic and anisotropic systems. Section 1.4 discusses the Guinier approximation, often used as a first analytical technique to obtain the radius of gyration from SAS data. The separation of a SAS intensity profile into intra‐molecular and inter‐molecular scattering components, respectively termed form and structure factor is discussed in Section 1.5. These terms are discussed in more detail, Section 1.6 first considering different commonly used structure factors, then Section 1.7 focusses on examples of form factors and the effects of polydispersity on form factors. Form and structure factors for polymers are the subject of Section 1.8.

1.2 WAVENUMBER AND SCATTERING AMPLITUDE

In a SAS experiment, the intensity of scattered radiation (x‐rays or neutrons) is measured as a function of angle and is presented in terms of wavenumber q. This removes the dependence on wavelength λ which would change the scale in a plot against angle, i.e. SAS data taken at different wavelengths will superpose when plotted against q, this is useful for example on beamlines where data is measured at different wavelengths (this is more common with neutron beamlines). The wavenumber quantity is sometimes denoted Q although in this book q is used consistently. The difference between incident and diffracted wavevectors q = k_s − k_i and since

, and the scattering angle is defined as 2θ (Figure 1.1), the magnitude of the wavevector is given by

      (1.1)

Figure 1.1 Definition of wavevector q and scattering angle 2θ, related to the wavevectors of incident and scattered waves, k_i and k_f.

      In some older texts, related quantities denoted s or S are used (these can correspond to q/2 or q/2π; the definition should be checked). The wavenumber q has SI units of nm⁻¹, although Å⁻¹ is commonly employed.

      The amplitude of a plane wave scattered by an ensemble of N particles is given by

(1.2)

      Here, the scattering factors aj are either the (q‐dependent) atomic scattering factors f_j(q) (Section 4.4) for SAXS or the q‐independent neutron scattering lengths bj for SANS (Section 5.4).

For a continuous distribution of scattering density, Eq. (1.2) becomes

      (1.3)

      Here Δρ(r) is the excess scattering density above that of the background (usually solvent) scattering, which is a relative electron density in the case of SAXS or a neutron scattering length density (Eq. (5.11)) in the case of SANS.

1.3 INTENSITY FOR ANISOTROPIC AND ISOTROPIC SYSTEMS AND RELATIONSHIPS TO PAIR DISTANCE DISTRIBUTION AND AUTOCORRELATION FUNCTIONS

      1.3.1 General (Anisotropic) Scattering

In the following, notation to indicate that the intensity is ensemble or time‐averaged is not included for convenience (if the system is ergodic, which is often the case apart from certain gels and glasses etc., these two averages are equivalent).

      The intensity is defined as

      (1.4)

      Thus, using Eq. (1.2), for an ensemble of discrete scattering centres

(1.5)

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