X-ray diffraction (XRD) is a powerful analytical technique used to study the atomic and molecular structure of crystalline materials. The fundamental principle behind XRD is the interaction of X-rays with the periodic arrangement of atoms in a crystal lattice, leading to constructive interference under specific conditions. This phenomenon is governed by Bragg's Law, crystallographic planes, and the relationship between lattice spacing and diffraction angles. Understanding these core principles requires an exploration of X-ray generation, diffraction geometry, and pattern interpretation.

X-rays are electromagnetic waves with wavelengths typically ranging from 0.01 to 10 nanometers, making them suitable for probing interatomic distances in crystals. X-rays are generated when high-energy electrons strike a metal target, such as copper or molybdenum, resulting in the emission of characteristic X-ray lines, notably Kα and Kβ. The Kα line, often used in XRD, arises from electron transitions between the L and K shells of the target material. Monochromatic X-rays are preferred for diffraction experiments to ensure a well-defined wavelength, which is critical for accurate measurements.

The interaction of X-rays with a crystal is governed by Bragg's Law, formulated by William Lawrence Bragg in 1912. The law states that constructive interference occurs when the path difference between X-rays scattered by successive planes of atoms is an integer multiple of the wavelength. Mathematically, Bragg's Law is expressed as:
nλ = 2d sinθ
where n is an integer representing the order of diffraction, λ is the wavelength of the incident X-ray, d is the interplanar spacing between crystallographic planes, and θ is the angle of incidence. This equation highlights the direct relationship between the lattice spacing and the diffraction angle, enabling the determination of crystal structures.

Crystallographic planes are defined by Miller indices (hkl), which describe the orientation of planes within a crystal lattice. These indices are inversely proportional to the intercepts of the plane with the crystallographic axes. The spacing between adjacent planes, d, depends on the crystal system and the Miller indices. For a cubic crystal system, the interplanar spacing is given by:
d = a / √(h² + k² + l²)
where a is the lattice parameter. In more complex crystal systems, such as hexagonal or tetragonal, the relationship involves additional parameters to account for anisotropic lattice dimensions.

The geometry of diffraction involves the arrangement of the X-ray source, sample, and detector. In a typical XRD setup, the X-ray beam is directed at the sample, and the detector scans through a range of angles to record the intensity of diffracted X-rays. The angle between the incident beam and the detector is 2θ, while the angle between the incident beam and the crystallographic plane is θ. When Bragg's condition is met, peaks appear in the diffraction pattern at specific 2θ values, corresponding to the constructive interference from different sets of crystallographic planes.

The diffraction pattern is a plot of intensity versus 2θ, revealing the unique fingerprint of a crystal structure. Each peak corresponds to a specific set of planes (hkl), and its position is determined by the lattice spacing. The intensity of the peaks depends on the atomic arrangement within the unit cell, including the types of atoms and their positions. Factors such as atomic scattering factors, thermal vibrations, and crystal symmetry influence peak intensities. The full width at half maximum (FWHM) of peaks provides information about crystallite size and microstrain, following the Scherrer equation:
τ = Kλ / (β cosθ)
where τ is the crystallite size, K is the Scherrer constant (approximately 0.9), λ is the X-ray wavelength, β is the FWHM in radians, and θ is the Bragg angle.

The interpretation of XRD patterns involves indexing the peaks to determine the crystal structure. For simple structures, this can be done by comparing observed d-spacings with reference data from databases such as the International Centre for Diffraction Data (ICDD). For more complex structures, Rietveld refinement is employed, which adjusts structural parameters to achieve the best fit between calculated and observed patterns. This method accounts for peak positions, intensities, and shapes, providing detailed information about lattice parameters, atomic positions, and phase purity.

The generation of X-rays for diffraction experiments requires careful consideration of the source and optics. Laboratory X-ray tubes produce polychromatic radiation, but monochromators or filters are used to isolate the Kα line. Synchrotron sources offer high-intensity, tunable X-rays, enabling advanced studies with superior resolution. The choice of wavelength affects the accessible range of d-spacings, as shorter wavelengths allow measurement of smaller lattice spacings.

The geometry of diffraction experiments can vary depending on the configuration. The most common setup is the Bragg-Brentano geometry, where the sample is flat and the detector moves in a circle centered on the sample. This configuration ensures that the focus condition is maintained, optimizing peak resolution. Other geometries, such as Debye-Scherrer or Laue methods, are used for specific applications like powder or single-crystal analysis.

The mathematical foundation of XRD extends beyond Bragg's Law to include the structure factor, which describes how X-rays scatter from a unit cell. The structure factor F(hkl) is given by:
F(hkl) = Σ f_j exp[2πi(hx_j + ky_j + lz_j)]
where f_j is the atomic scattering factor for atom j, and (x_j, y_j, z_j) are its fractional coordinates within the unit cell. The intensity of a diffraction peak is proportional to the square of the structure factor magnitude:
I(hkl) ∝ |F(hkl)|²
This relationship connects the observed diffraction pattern to the atomic-scale structure of the material.

In summary, X-ray diffraction relies on the principles of wave interference, crystallographic planes, and lattice geometry to reveal the atomic structure of materials. Bragg's Law provides the foundational relationship between diffraction angles and lattice spacings, while the structure factor links diffraction intensities to atomic positions. The generation and detection of X-rays, combined with precise geometric configurations, enable the acquisition and interpretation of diffraction patterns. These core principles form the basis for understanding crystalline materials at the atomic level, without delving into applications or advanced techniques.