A powder diffraction pattern shows peaks at 2-theta values of 38.2, 44.4, 64.5, and 77.5 degrees using Cu K-alpha radiation. These peak positions are consistent with which crystal structure?
ASimple cubic — all hkl reflections are allowed
BBody-centered cubic — only reflections where h+k+l is even are allowed
CFace-centered cubic — only reflections where h,k,l are all odd or all even are allowed
DHexagonal close-packed — peak positions follow no simple selection rule
The ratios of sin^2(theta) for these peaks follow the sequence 3:4:8:11, which corresponds to the FCC allowed reflections (111), (200), (220), (311). In FCC, only planes where h,k,l are all odd or all even produce diffraction — the face-centering causes systematic absences for mixed indices. This pattern matches aluminum (a = 4.05 Angstroms). The ability to distinguish BCC from FCC from the pattern of present and absent reflections is one of the most basic applications of XRPD.
Question 2 True / False
The width of a diffraction peak increases as crystallite size decreases, according to the Scherrer equation.
TTrue
FFalse
Answer: True
The Scherrer equation relates peak broadening to crystallite size: t = K-lambda / (B cos(theta)), where t is crystallite size, B is the peak width at half maximum (in radians), and K is a shape factor near 0.9. Smaller crystallites have fewer lattice planes contributing to diffraction, which reduces the destructive interference that would otherwise sharpen the peak. Below about 100 nm, broadening becomes measurable; below 5 nm, peaks may be so broad they merge into the background. This makes peak width analysis a routine method for estimating nanoparticle size.
Question 3 Short Answer
Why does a powder sample produce a complete diffraction pattern while a single crystal at a fixed orientation typically shows only a few reflections?
Think about your answer, then reveal below.
Model answer: A powder contains millions of tiny crystallites oriented randomly in all directions. For every set of lattice planes (hkl), some crystallites will be oriented at exactly the Bragg angle relative to the incident beam. This means all allowed reflections are observed simultaneously. A single crystal has only one orientation, so only the few planes that happen to satisfy Bragg's law at that orientation will diffract. To get a complete pattern from a single crystal, you must rotate it through many orientations.
This is the fundamental advantage of powder diffraction for routine phase identification — no special sample preparation or orientation is needed. The disadvantage is that 3D structural information is compressed into a 1D pattern (intensity vs. 2-theta), which causes peak overlap and makes structure solution harder than with single-crystal data. Rietveld refinement addresses this by fitting the entire pattern simultaneously using a structural model, extracting maximum information from the overlapping peaks.