Quantum Physics

27.4 Diffraction of X-Rays by Crystals 883Exercise 27.4What potential difference would be necessary to produce gamma rays with wavelength 1.00 10 15 m? This wavelengthis about the same size as the diameter of a proton.Solution1.24 10 9 V27.4 DIFFRACTION OF X-RAYS BY CRYSTALSIn Chapter 24 we described how a diffraction grating could be used to measurethe wavelength of light. In principle, the wavelength of any electromagnetic wavecan be measured if a grating having a suitable line spacing can be found. Thespacing between lines must be approximately equal to the wavelength of the radiationto be measured. X-rays are electromagnetic waves with wavelengths on theorder of 0.1 nm. It would be impossible to construct a grating with such a smallspacing. However, as noted in the previous section, Max von Laue suggested thatthe regular array of atoms in a crystal could act as a three-dimensional grating forobserving the diffraction of x-rays.One experimental arrangement for observing x-ray diffraction is shown inFigure 27.11. A narrow beam of x-rays with a continuous wavelength range is incidenton a crystal such as sodium chloride. The diffracted radiation is very intensein certain directions, corresponding to constructive interference from wavesreflected from layers of atoms in the crystal. The diffracted radiation is detected bya photographic film and forms an array of spots known as a Laue pattern. The crystalstructure is determined by analyzing the positions and intensities of the variousspots in the pattern.The arrangement of atoms in a crystal of NaCl is shown in Figure 27.12. Thesmaller red spheres represent Na ions, and the larger blue spheres represent Cl ions. The spacing between successive Na (or Cl ) ions in this cubic structure,denoted by the symbol a in Figure 27.12, is approximately 0.563 nm.A careful examination of the NaCl structure shows that the ions lie in variousplanes. The shaded areas in Figure 27.12 represent one example, in which theatoms lie in equally spaced planes. Now suppose an x-ray beam is incident atgrazing angle on one of the planes, as in Figure 27.13. The beam can bereflected from both the upper and lower plane of atoms. However, thegeometric construction in Figure 27.13 shows that the beam reflected fromthe lower surface travels farther than the beam reflected from the upper surfaceby a distance of 2d sin . The two portions of the reflected beam will combineto produce constructive interference when this path difference equals some integralmultiple of the wavelength . The condition for constructive interference isgiven byX-raytubeX-raysCollimatorCrystalPhotographicfilmFigure 27.11 Schematic diagramof the technique used to observe thediffraction of x-rays by a single crystal.The array of spots formed on the filmby the diffracted beams is called aLaue pattern. (See Fig. 27.7.)Figure 27.12 A model of the cubiccrystalline structure of sodium chloride.The blue spheres represent theCl ions, and the red spheres representthe Na ions. The length of thecube edge is a 0.563 nm.a2d sin m (m 1, 2, 3, . . .)[27.10] Bragg’s lawIncidentbeamReflectedbeamUpper planeLower planeud sin uuudFigure 27.13 A two-dimensionaldepiction of the reflection of an x-raybeam from two parallel crystallineplanes separated by a distance d. Thebeam reflected from the lower planetravels farther than the one reflectedfrom the upper plane by an amountequal to 2d sin .