E 45 - Fall 2010 - Gronsky - Midterm 1 (solution)

# Referring to the plane to the right of the origin in

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Referring to the plane to the right of the origin in this figure, its intercepts with the coordinate axes are , ½ , – ½ , . Taking the reciprocals, clearing fractions (unnecessary here) and enclosing in parentheses, the answer is Check : the parallel plane to the left of the origin has intercepts , – ½ , ½ , , confirming that it belongs to the same family. d . The figure below shows one member of the (0001) family of planes in an hexagonal close-packed structure. Specify in Miller-Bravais index notation the lattice directions connecting the following pairs of atoms. Check : h + k = –i ! " # " \$ " % ! " # " \$ " % ! " # \$ −→ OA = [ ¯ 2110] −−→ OB = [ ¯ 1100] −−→ OC = [ ¯ 12 ¯ 10] −−→ AB = [ ¯ 12 ¯ 10] −→ AC = [11 ¯ 20] E45 Fall 10 Midterm 01 Solutions Professor R. Gronsky page7 of 11 (02 ¯ 20)

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4. Crystal Structure a . Refer to the following drawing 1 by M.C. Escher entitled "Study of Regular Division of the Plane with Reptiles" produced with pencil, India ink and watercolor in 1939. Show directly on this drawing the points of a lattice , and a primitive unit cell that fully describes the "structure" of the periodic image. b . Recall that a periodic (for example, "crystal") structure can be described by the combination of a lattice and a "motif" that must be assigned to each and every lattice point in the same orientation. Now specify the " motif " associated with the periodic structure of this Escher image. Both the motif and the unit cell must fully describe the "crystal." The motif can be therefore discerned from the contents of the unit cell: the unit cell contains 3 reptiles, so must the motif. The motif above is: 1 yellow reptile, 1 red reptile and 1 green reptile arrayed in clockwise fashion in the same fixed orientation shown at each lattice point. E45 Fall 10 Midterm 01 Solutions Professor R. Gronsky page8 of 11 1 The World of M.C. Escher , J.L. Locher, (Ed.), H.N. Abrams, Inc. Publisher, New York (1971).
4. Crystal Structure c . The cubic unit cell describing one of the oxides of uranium used as a fuel in nuclear reactors has uranium cations at locations 0,0,0; 0, ½ , ½ ; ½ ,0, ½ ; and ½ , ½ ,0, with oxygen anions at ¼ , ¼ , ¼ ; ¼ , ¾ , ¼ ; ¾ , ¼ , ¼ ; ¾ , ¾ , ¼ ; ¼ , ¼ , ¾ ; ¼ , ¾ , ¾ ; ¾ , ¼ , ¾ ; and ¾ , ¾ , ¾ . Specify the lattice and motif that defines this structure. What is its chemical formula ? The uranium cations occupy FCC sites and the oxygen anions occupy all of the tetrahedral interstices (the corners of the interior cube indented by ¼ of the lattice constant from all faces). There are 4 uranium ions per unit cell and 8 oxygen ions per unit cell (given); this establishes the stoichiometry of the crystal, 2 oxygens for each uranium, which must also be true of the motif. Consequently, the lattice is face-centered cubic (FCC) and the motif must contain at least three ions, 1 uranium and 2 oxygens. There are several options, but the most common one is to site them symmetrically about each lattice point, so an acceptable motif is 1 uranium cation at 0,0,0 and 2 oxygen anions, one at + ¼ ,+ ¼ ,+ ¼ , and the other at – ¼ ,– ¼ ,– ¼ . Finally, the same ratio must be preserved in the chemical formula: UO 2 . Note the charge on the ions and the need to preserve charge neutrality.

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• Spring '10
• Various
• Materials Science, ........., Tensile strength, Chemical bond

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