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HW05Solution - PROBLEM 3-5 B.D Cullity and S.R Stock...

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3-5 Sketch the Ewald sphere construction for a 200 diffraction with Mo K α radiation and a polycrystalline specimen of a simple cubic substance with a = 3.30 Å. Graphically determine the angular rotation required to orient the sample for 300 diffraction if a θ —2 θ diffractometer is being used. Solution There is a strong probability that this problem statement is in error, and the author wanted the sample to be a single crystal, with the x-ray beam directed along a [001] zone axis orientation. This would have made for a more meaningful calculation of a re-orientation angle, with a sketch very similar to Figure 3-6 of the text. The reason for this suspicion is because a polycrystalline specimen of a material with a simple cubic structure (all reflections are "allowed" by structure factor) will always satisfy 200 and 300 diffraction conditions simultaneously, provided the grain size is small enough. The reciprocal lattice Ewald sphere construction shows this. To establish the scale of the drawing, the wavelength of Mo K α radiation is 0.710739 Å (Appendix 7, page 629). Ewald’s sphere has a radius given by the reciprocal of the wavelength or r = 1.407 Å –1 . A simple cubic crystal with a lattice constant of 3.30 Å will exhibit diffraction from all families of planes. Reciprocal lattice vectors will therefore follow a sequence according to the interplanar spacing relation These values are 0.303 Å -1 for 100, 0.428 Å -1 for 110, 0.525 Å -1 for 111, 0.606 Å -1 , for 200, etc . including 0.909 Å -1 for 300 (as called for in the problem statement). For polycrystalline samples of small grain size in a perfectly random orientation, all of these reciprocal lattice vectors will sweep out a sphere of radius r hkl *, centered about the origin of the reciprocal lattice. Ewald’s sphere also terminates on the origin of the reciprocal lattice, generating a construction as shown in the following. r * hkl = 1 d hkl = k 2 + k 2 + l 2 a 0 = k 2 + k 2 + l 2 3 . 30 ˙ A PROBLEM 3-5 B.D. Cullity and S.R. Stock, Elements of X-Ray Diffraction, 3 rd Ed., Prentice Hall, (2001) MSE 104 Materials Characterization Professor R. Gronsky page 1 of 2
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Note that there is no significance to the “incident direction” of the Mo K α beam since the polycrystalline sample has no specific orientation, but rather all orientations. This is indicated by the reciprocal "lattice" in all orientations, sweeping out circles (spheres in 3D) with radii given by the r * values above. If the sample were single crystalline, the construction would appear as shown on the right for the 200 diffraction condition. Note that both the origin and the 200 reciprocal lattice vectors lie on the Ewald sphere. The origin always does, and in this case the 200 reciprocal lattice point does because it represents an exact Bragg diffraction condition. Drawing the construction for both the origin and a 300 diffraction condition obviously requires a reorientation of the reciprocal lattice relative to the Ewald sphere, and this is the rotation angle requested in the problem statement.
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