Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

6.3. The Experimental Determination of g(r)*

In this section we provide some background on how to obtain the correlation function from scattering experiments using X-ray diffraction (XRD) and/or neutron ray diffraction (NRD). Since these experiments are loaded with many experimental pitfalls, this topic requires significant technical and professional skills, and only the bare essentials are touched on here.

For scattering experiments using monochromatic radiation, part of the incident radiation is scattered in various directions, characterized by the angle θ between the incoming and outgoing radiation. A scattering parameter s = (4π/λ) sin(θ/2) is defined, where λ is the wavelength of the radiation used and s = |s| = |s_sca − s_inc| is the length of the difference between the wave vector of the scattered and incident radiations. In the first Born approximation, representing single scattering, the amplitude of the outgoing radiation A(s) is directly related to the Fourier transform of the scattering potential Φ(r) – that is, A(s) ∼ V⁻¹ ∫Φ (r)exp(−is·r)dr, where V is the volume of the sample. This theory is general and can be applied to X-rays, neutrons and electrons, although the scattering matter for each is different. X-rays are scattered by electrons, neutrons are mainly scattered by nuclei, while for electron radiation both electrons and nuclei cause scattering. For X-rays, typically Kα radiation is used (Cu, λ = 0.1541 nm or Mo, λ = 0.07093 nm), while λ for thermal neutrons (kT ≅ 4−5 × 10⁻²¹ J) is ∼0.1 nm. For electrons, λ = h/[2m₀eV(1 + eV/2m₀c²)]^1/2, where V is the acceleration voltage, m₀ and e are the rest mass and charge of the electron, respectively, and c is the speed of light; this results in 1.97 pm using 300 kV.

In general, to describe structure of a mono-atomic sample we need atomic coordinates and it is assumed that Φ(r) = Σ_iϕ(r − r_i) with r_i the atomic coordinates. For elastic scattering |s_sca| = |s_inc|, the intensity is I ∼ |A(s)²| and this leads to, dropping geometric factors and introducing ρ = N/V,

(6.22)

The factor |ϕ(s)| is the Fourier transform of the atomic scattering potential, usually called the atomic form factor, while S(s) is the structure factor. For a perfect crystal the latter represents an array of delta functions at the reciprocal lattice points. Evaluating S(s) by taking its canonical average yields

(6.23)

where the term 1 is due to i = j contributions. Introducing the total correlation function h(r) = g(r) − 1 (see Section 7.2), this expression becomes

The last integral represents a δ(s) contribution (see Appendix B) originating from the fact that for r → ∞, g(r) → 1 and has to be subtracted. Otherwise the Fourier inversion

(6.24)

would not converge because S(s) → 1 for s → ∞. The last step can be made if the system is isotropic. The accuracy of g(r) depends heavily on accurate knowledge of S(s) for an as-wide range of s-values as possible, since the Fourier transform may introduce serious defects when performed with a too-limited data range.

For neutrons, an atom is characterized by ϕ(r) = bδ(r) with the scattering length b, typically 0.01 pm and θ-independent. As the length for isotopes is slightly different, the average value ⟨b⟩ is used. The scattered intensity I(s) is given by I(s) = α(θ)⟨b⟩²[S(s) + Δ], where α(θ) is a proportionality factor that is instrument-dependent but sample-independent, converting to absolute intensities. The structure factor S(s) is due to coherent scattering, while Δ is due to incoherent scattering. The limiting values of S(s) are S(0) = ρkTκ_T, with a value of about 0.01, and S(∞) = 1. Intensities at large angle I(∞) and small angle I(0) can be used to eliminate α(θ) and Δ, leading to

For X-rays, the quantity corresponding to ⟨b⟩ is the atomic scattering factor a(s), defined by the sine transform of the atomic electron density ρ(r),

This quantity is θ-dependent. The structure factor becomes

where N is the number of atoms with atomic number Z. The value of I(∞)/I₀ → Na²(s)/Z², so that S(s) approaches 1; this can be used to normalize the data.

For molecules, one generally assumes that Φ(r) = Σ_i,αϕ_α(r − r_i,α) with r_i,α the atomic coordinates of the atom of type α at coordinate r_i,α. The Fourier transform of the scattering potential Φ(r) becomes

(6.25)

If we separate the terms with i = j and i,α = j,β, these contribute a structure-independent term, weighted with the concentrations c_α, that is, using leads to

(6.26)

The second term refers to the pairs of distinct atoms (both often of the same type). Defining a partial structure factor S_αβ, again with removal of the singularity,

(6.27)

the scattered intensity becomes

(6.28)

Combining Eqs (6.26) and (6.28), we obtain

(6.29)

where the first term represents the coherent scattering and the second term the incoherent scattering. In the case of NRD from an isotopic mixture, the incoherent term is just proportional to the variance of b, that is, ⟨Δb⟩² = ⟨b²⟩ − ⟨b⟩². For X-rays, ϕ_i,α(r − r_i,α) represents the atomic electron density, while for crystals deviations from total electron density as measured, and the superposition of atomic densities due to the presence of chemical bonds, can be detected experimentally [4]. Finally, it will be clear from Eq. (6.29) that from a single measurement, whether using NRD nor XRD, the various S_αβ cannot be determined.