Structural Factor Calculation for SC, BCC & FCC

For diamond cubic (Si, Ge), F = f × [1 + (-1)^(h+k+l) + (-1)^(h+k) + (-1)^(h+l) + (-1)^(k+l) + (-1)^(h+k+l) + (-1)^(h+k) + (-1)^(h+l) + (-1)^(k+l)]. Diffraction occurs only when h,k,l are all even or all odd, and h+k+l is even. For rock salt (NaCl), F = (f_Na + f_Cl) × [1 + (-1)^(h+k+l)] when h+k+l is even, and F = (f_Na - f_Cl) × [1 + (-1)^(h+k+l)] when h+k+l is odd. The structure factor depends on atomic scattering factors.

For NaCl (FCC with two-atom basis: Na+ at (0,0,0) and Cl- at (1/2,1/2,1/2)), the structure factor is: F(hkl) = f_Na + f_Cl e^(iπ(h+k+l)). This gives: F = f_Na + f_Cl when h+k+l is even, F = f_Na - f_Cl when h+k+l is odd. Since f_Na ≈ f_Cl (similar electron counts), the odd-sum peaks have very low intensity. The first peak is (200), not (111). For KCl (same structure but with K+ and Cl- having identical electron counts), the structure factor is: F(hkl) = f_K + f_Cl e^(iπ(h+k+l)). Since f_K ≈ f_Cl, the odd-sum peaks have F ≈ 0 (systematic absences). Only even-sum peaks appear.

The structure factor for crystals with multiatomic bases can be calculated by decomposing it into a lattice part and a motive (basis) part, where the lattice part determines which HK planes produce reflections (extinction occurs for mixed indices in FCC) and the motive part introduces additional extinction conditions based on the atomic positions within the unit cell. For example, in diamond (FCC with two carbon atoms at 0,0,0 and 1/4,1/4,1/4), extra extinction occurs when H+K+L is an odd multiple of 2, while in NaCl (FCC with Cl at 0,0,0 and Na at 1/2,1/2,1/2), the structure factor depends on whether H+K+L is even or odd.

This comprehensive section covers structure factor calculations for increasingly complex crystal systems. It begins with HCP crystals defined by two atoms at (0,0,0) and (1/3, 2/3, 1/2), explaining how to handle fractional coordinates in mathematical expressions. The section then advances to crystals with basis atoms, using NaCl as the primary example where sodium and chlorine occupy different positions with different scattering amplitudes. The instructor demonstrates how to derive structure factors by recognizing that individual atom types form FCC sublattices. Key rules are established: for FCC crystals, mixed Miller indices produce zero structure factor while all-even or all-odd indices produce non-zero values. The section concludes with GaAs, showing how diamond cubic structures follow similar principles with basis vectors at (1/4, 1/4, 1/4), emphasizing that mixed indices always produce forbidden reflections regardless of the basis atoms.

For multi-element crystals like NaCl, the structure factor accounts for different atomic form factors at each position. In NaCl, sodium forms an FCC lattice while chloride ions occupy the octahedral sites at (½,0,0), (0,½,0), (0,0,½), and (½,½,½). The structure factor combines contributions from both ion types: F_HKL = f_Na[1 + (-1)^(H+K) + (-1)^(K+L) + (-1)^(H+L)] + f_Cl[(-1)^(H+K+L) + ...]. When H,K,L are all even or all odd, both ion contributions add constructively; when mixed, they may cancel partially or completely.

The atomic form factor accounts for atoms being finite electron density distributions rather than point scatterers. X-rays interact with electron clouds, not nuclei. The form factor is the Fourier transform of electron density distribution and depends on scattering angle. Different atoms have different form factors based on electron configurations. This factor must be included in structure factor calculations to accurately predict diffraction intensities.

The Atomic Scattering Factor (F) is defined as the ratio of radiation amplitude scattered from an atom to that from a single electron. For an atom with Z electrons, F = Z when each electron scatters with equal amplitude. This factor depends on: (1) number of electrons, (2) electron distribution in K, L, M shells, (3) wavelength of incident radiation, and (4) scattering angle. F is also called the form factor. At zero scattering angle, F equals Z because all scattered waves interfere constructively. As the angle increases, F decreases below Z due to destructive interference. F indicates the efficiency with which an atom scatters incident radiation.

The atomic scattering factor (or atomic form factor), denoted by symbol 'f', is a parameter that quantifies the efficiency of an atom in scattering X-rays. It is defined as the ratio of the scattering amplitude of X-rays scattered from a single atom (or its electronic cloud) to the scattering amplitude of X-rays scattered from a single electron. Mathematically, it is expressed as F = ∫ n(r) × e^(-iG·r) dV, where n(r) represents the electronic density at position r, G is the scattering vector, and the integration is performed over the entire volume of the atom. The atomic scattering factor decreases exponentially with increasing distance from the atomic nucleus and depends on the scattering angle θ and wavelength λ (through sin(θ)/λ). Atoms with higher atomic numbers have greater atomic scattering factors because their larger electronic clouds contain more electrons capable of scattering X-rays. This concept is fundamental to understanding X-ray diffraction patterns and crystal structure determination.

The atomic scattering factor (or form factor) is a mathematical expression that describes the scattering power of a single atom when incident radiation interacts with it; it is derived by integrating the scattering amplitude of individual electrons over the entire atomic volume, considering the spherical symmetry of electron distribution, and is given by the formula f = (4π/α) ∫ ρ(r) e^(-iqr) dV, where ρ(r) is the electron charge density and q is the scattering vector, with the approximation f ≈ Z for small angles (θ < 10°) where sin(θ) is small.

X-ray scattering from atoms with multiple electrons shows reduced intensity due to interference effects. The atomic form factor describes this reduction, with values equal to electron count at zero scattering angle (e.g., O=8, Cl=17, Cl⁻=18, K⁺=18). As scattering angle increases, form factors decrease due to destructive interference. Anions show faster drop-off than isoelectronic cations because larger ions experience more significant interference effects.

Diffraction intensity is calculated as the product of structure factor modulus squared, Lorentz-polarization factor, multiplicity factor, and proportionality constants. Scale factors (proportionality constants) are individual for each phase in multiphase samples and proportional to volume fractions. Absorption effects can be included in the proportionality constant for simple geometries. The mathematical framework allows direct variation of parameters of interest, providing a more efficient path to structural and phase content information.

Peak intensity in powder diffraction depends on multiple factors: multiplicity (number of equivalent planes), polarization (electron scattering angle dependence), Lorentz (fraction of properly oriented grains), and temperature (thermal vibrations). Multiplicity: (100) family has 6 planes, (111) has 8. Polarization: I ∝ (1 + cos²(2θ))/2. Lorentz: I ∝ cosθ/sin²(2θ)—peaks weaken at higher angles. Combined as Lorentz-polarization factor. Temperature factor e^(-2M) reduces intensity while increasing background. These factors explain why identical crystals produce different peak intensities and why peak intensities decrease at higher angles.

The intensity I of diffraction peaks is calculated as the product of the geometrical structure factor with its complex conjugate: I = |F_HKL|² = F_HKL × F_HKL*. For simple cubic structure, since F_HKL = f, the intensity becomes I = f², meaning all hkl combinations are possible regardless of whether they are odd or even.

The structure factor F(hkl) determines diffraction peak intensity via F = Σ f_j exp[2πi(hx_j + ky_j + lz_j)], where f_j is the scattering factor for atom j at position (x_j, y_j, z_j). Intensity is proportional to |F|². When F = 0, no peak is observed. Scattering factor depends on atomic number, wavelength, and diffraction angle. Lattice point coordinates define atomic positions: simple cubic (0,0,0); BCC (0,0,0) and (½,½,½); FCC (0,0,0), (½,½,0), (½,0,½), (0,½,½).

Diffraction intensity depends on three factors: (1) Multiplicity factor (M) - counts equivalent planes sharing Miller indices: (100) has M=6, (110) has M=12, (111) has M=8; higher multiplicity means higher intensity; (2) Structure factor squared (|F|²) - incorporates atomic form factors with angular dependence (larger angles give smaller intensities); (3) Lorentz-polarization factor - geometric correction depending only on experimental setup, causing intensity to rise near 0° and 180°, decrease toward 90°. These factors enable intensity-based crystal structure verification beyond peak position analysis.