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EMS162-2010-Homework5_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 162, page 1/4 Prof. Y. Takamura, 2010 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 162: Structure and Characterization of Engineering Materials Homework 5 - SOLUTIONS Due Date: Tuesday, Feb. 9th, 2010, in class – NO LATE HOMEWORK Note: You must show all the steps in order to receive full credit. Be sure to include units! 1. Interference Function [23 pts] Consider the scattering of x-rays from a one dimensional periodic arrangement of electrons as shown in Figure 1: In this case, the intensity of the diffracted x-rays at a scattering angle, 2 θ , is given by the interference function: ( ) ϕ ϕ ϕ 2 2 sin sin N I where N is the number of electrons in the line, and ϕ is the phase angle given by: ( ) λ θ π λ θ π ϕ 2 sin 4 2 cos 1 2 a a = = where a is the spacing between electrons and λ is the wavelength of the x-rays. (a) Using your favorite plotting software (Excel, Origin, Matlab, Mathematica), plot the interference function as a function of phase angle (from zero to 4 π ) for the case of six, 20, and 1000 electrons in a 1D line. ( Hint : Be sure to use a small enough step size between data points as N gets large). Hand drawings will not be given ful l credit. (b) Comment on what happens to the shape of the curves as N gets large. The interference function tells us that we expect strong diffracted intensity for certain phase angles and therefore certain scattering angles, and low/no diffracted intensity for all other scattering angles.
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