# Because each of the atoms may scatter wave the total

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is the position of the detector. Because each of the atoms may scatter wave, the total amplitude at the detector is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k r k r r k r k r k r r G k k r k r k r G G r r i f f d f d i s i s i f f d i s i i i i whole whole crystal crystal i i whole crystal Ae n e dV Ae n e dV A e n e dV + = = Because the exponential function ( ) G k k r i f i e + is a rapidly varying function in the crystal with both negatives and positives, the above integral will be close to zero except when the exponent vanishes, i.e., when G k k 0 i f + = . This is called the Bragg condition for diffraction. Because the wavevectors k f and k i are determined by the relative positions of the source, the sample, and the detector. The reciprocal lattice vectors G, and thus the crystal structure, can be determined from diffraction experiments. x Crystal Specimen dV Source Detector z r r s r d ( ) k r r i s i e ( ) (r) k r r f d i n e y θ a

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2.57 Fall 2004 – Lecture 8 6 Consider the special set of crystal planes separated by a distance a as shown in the following figure and an incident wave (photon or electron) of wavelength λ at an angle θ . Constructive interference between waves reflected from crystal planes occurs when the phase difference of the waves scattered between two consecutive planes is 2 π n. From the figure, we see that the path difference is 2asin θ. Thus diffraction peaks will be observed when the path difference is multiples of the wavelength, i.e., λ = θ n sin a 2 . In the following lecture, we will continue the discussion of energy dispersion for electrons and phonons. For phonons, in the acoustic waves a linear energy dispersion function is used as the approximation. For electrons, two parabolic curves are connected to approximate the energy dispersion. Debye ω E k ω Einstein π /a dE/dk=0 k E
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