crystallograpy-Answers

crystallograpy-Answers - The Simple Cubic Lattice For...

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The Simple Cubic Lattice For simple cubic lattices, there is one lattice point per unit cell. The lattice may be described as a three dimensional array of lattice points by either beginning with a lattice point at the position (0,0,0), and adding additional lattice points by translating indefinitely in all three principal directions by the lattice parameter, a , or alternatively, by allotting 1/8 of a lattice point at each of the eight corners of the unit cell and stacking up unit cells in all three directions to build up the lattice. (In reality, there is one lattice point at each cube corner, with each contributing 1/8 of itself to each unit cell.) The physical surroundings of each lattice point in the unit cell of any crystal lattice are identical. The number of atoms contained within the unit cell is the number of lattice points per unit cell times the number of atoms per lattice point. (For example, table salt is a simple cubic structure with two atoms per lattice site.) The number of nearest neighbors an atom has is called the coordination number . Exercise I: A. Draw a cube. i. How many faces does the cube have? 6 ii. How many edges? 12 iii. How many corners? 8 B. Draw a simple cubic crystal lattice with atoms represented by spheres. i. What is the number of atoms per unit cell ? 1 ; the lattice point for SC is (0,0,0) ii. What is the number of nearest neighbor atoms ? 6 iii. What is the close-packed direction (direction of touching) for a simple cubic crystal structure? ( The touching directions for SC are along X, Y and Z axis ) iv. If spheres with unit volume were centered on each of the corners, what would be the net volume of sphere actually contained inside the cube? 1/8 of the sphere is inside the cube if its center is located at the corner and we have a total of 8 corners in a cube so 1 unit sphere volume is inside the cube v. If spheres with unit volume were centered on each of the edges, what would be the net volume of sphere actually contained inside the cube? 1/4 of the sphere is inside the cube if its center is located on the edge and we have a total of 12 edges per cube; so 3 units sphere volume is inside the cube vi. Draw the unit cell, indicate the position (x, y, z coordinates) of the center for the largest hole (called an interstitial position) within the unit cell. (1/2,1/2,1/2) vii. How many lattice atoms surround the interstitial position? 8 C. The edge length of the cubic unit cell is called the lattice parameter and is denoted as a , which would, for a given material, have some magnitude expressed in meters (or nanometers). Cubic unit cells have only one lattice parameter . If the unit cell were not cubic, we would need additional lattice parameters, some of which might be angles. If a simple cubic unit cell were drawn as a hard-sphere model (the hard sphere model considers the atoms to be spherical and touching their nearest neighbors), what would be i. the total edge length (in terms of the lattice parameter , a ); 12 a
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ii. the total surface area of the cube;
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This document was uploaded on 10/28/2011 for the course MSE 250 at Michigan State University.

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crystallograpy-Answers - The Simple Cubic Lattice For...

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