9780030839931, Chapter 4, Problem 6P

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Chapter Problem 6P Problem The face-centered the most dense and the simple cubic least dense of the three cubic Bravais The diamond less dense than any of these One measure of this that coordination numbers diamond Another following: Suppose identic solid spheres distributed through space in such that points of each of these four spheres on neigh boring points just (Such close-packing Assuming that the spheres have unit density show that density of a set of close packed spheres on each of the four structures (the fraction fee se 0.52 diamond: Step solution Step The relative packing density the packing defined as the ratio of the volume of the atoms occupying the unit cell the volume the unit cell relating that structure So, Volume atoms per unit cell(v) Packing factor= Volume of Step Face-centered cubic each corner's atoms are shared by surrounding cubes and each the centered atoms shared b surrounding cubes So the total number atoms cube be Total number of atoms =1+3 =4 Atomic radius Here lattice constant Volume volume one atom Volume Step Thus the packing factor Volume the atoms p unit cell(v) Packing factor Volume o 6 =0.74 From above observation conclude that the packing factor for the lattice would be 0.74 Step Body -centered cubic the bcc one atom present each corner and one atom the center the cube Each atom has only nearest neighbors number cube will be Total number of atoms =1+1 =2 Atomic Here constant Volume atoms is volume one atom Volume the packing factor Volume atoms unit cell(v) Packing factor Volume o =0.68 From above observation we conclude that the packing factor for lattice would be 0.68 Step Simple (sc) each the corners of cubes The contribution of each atom to the As there are comers each unit cell the number atoms per unit cell 8 Total number of atoms =1 Atomic radius Here is the lattice constant Step Volume one atom Volume of unit cell Step Thus, the packing factor cell(v) Packing factor Volume of 6 =0.52 From above observation we conclude that the packing factor for the 0.52 Step Diamond structure (dc) he dc there are spheres the and of each contained the there are 6 spheres and of each the cube center each tetrahedron there spheres So the number the dc Total number of atoms =1+3+4 =8 Here the lattice constant Volume volume o 16 Volume the packing factor will be Volume the atoms Packing factor Volume o 16 16 34 From above observation we conclude tha the packing factor for the would be 0.34