Questions: X-ray Diffraction and Structure Determination

Question 1

The Bragg condition 2d sin(θ) = nλ and the von Laue condition Δk = G are two different ways of expressing the same physics. What connects them?

• A. They are independent conditions that must both be satisfied simultaneously
• B. The Bragg condition applies to X-rays while the von Laue condition applies to electrons
• C. The Bragg condition describes reflection from lattice planes with spacing d, and the perpendicular distance between planes is d = 2π/|G|, so both conditions reduce to the requirement that the path difference equals an integer number of wavelengths
• D. The von Laue condition is an approximation to the Bragg condition valid only at small angles

Answer: (C)

The Bragg picture treats the crystal as a stack of reflecting planes separated by distance d. The von Laue picture describes scattering from individual atoms with constructive interference when Δk = G. They are exactly equivalent: the reciprocal lattice vector G is perpendicular to the (hkl) planes with |G| = 2π/d, and substituting into either condition gives the same constraint on wavelength and angle. The von Laue formulation is more general and directly connects to the reciprocal lattice framework.

Question 2

In a BCC crystal, X-ray diffraction peaks with h + k + l = odd are systematically absent. What causes these extinctions?

• A. The X-ray wavelength is too long to resolve those planes
• B. The structure factor F(G) vanishes for those reflections because scattering from the body-center atom destructively interferes with scattering from the corner atoms
• C. Those Miller indices correspond to imaginary lattice planes
• D. Thermal vibrations suppress those reflections at room temperature

Answer: (B)

In BCC, the basis consists of atoms at (0,0,0) and (1/2,1/2,1/2). The structure factor is F = f[1 + e^{iπ(h+k+l)}], where f is the atomic form factor. When h+k+l is odd, the exponential equals -1 and F = 0. Physically, the body-center atom scatters exactly out of phase with the corner atoms for these reflections, producing perfect destructive interference. These systematic absences are the experimental fingerprint of a BCC lattice.

Question 3

The structure factor contains all the information about the arrangement of atoms within the unit cell, while the lattice determines which directions produce diffraction peaks.

• T. True
• F. False

Answer: True

This is the clean separation between lattice and basis in diffraction. The reciprocal lattice (determined by the Bravais lattice) specifies which scattering vectors G can produce peaks — these are the allowed diffraction directions. The structure factor F(G), which depends on the positions and types of atoms in the basis, determines the intensity at each allowed G. Some allowed G may have F = 0 (systematic absences), effectively hiding certain peaks. Determining a crystal structure means measuring peak positions (lattice) and intensities (basis) and inverting the structure factor.

Question 4

Why can't we simply inverse-Fourier-transform the measured diffraction pattern to obtain the crystal structure directly?

Model answer: Detectors measure diffraction intensities |F(G)|^2, not the complex structure factor F(G) itself. The phase information — the argument of F(G) — is lost. Since a Fourier transform requires both amplitude and phase to reconstruct the electron density, the measured intensities alone are insufficient. This is the famous 'phase problem' of crystallography. Solving it requires indirect methods: Patterson functions (which use |F|^2 directly), direct methods (statistical relationships between phases), molecular replacement (using known similar structures), or anomalous scattering techniques.

The phase problem is why crystallography remained labor-intensive even after diffraction was well understood. The 1985 Nobel Prize in Chemistry was awarded for 'direct methods' that use probability relationships among structure factor phases to solve the problem computationally.