ECE 6163-1 Fall 2009

Homework Assignment #1 [70 pts] Due 09/02/09

1. (a) How many atoms are there in the unit cell characterizing the Silicon lattice? [5]

(b) Verify that there are 4.994 x1022 Atoms/cm3 in the Si lattice at room temperature. [5]

(c) Determine the center-to-center distance between nearest neighbors in the Si lattice. [5]

2. Record all intermediate steps in answering the following questions.

(a)  As shown in Fig. P1.4(a), a crystalline plane has intercepts of 6a, 3a, and 2a on the x,y, and z axes respectively, a is the cubic cell side length.

  1. What is the Miller index notation for the plane? [5]
  2. What is the Miller index notation for the direction normal to the plane? [5]

(b)  Determine the Miller indices for the cubic crystal plane pictured in Fig. P1.4(b).[5]

3. Referring to the unit cell of the Si lattice in Fig. 1.5P, and noting that the origin of coordinates is located at the lower back corner of the unit cell:

a)  What are the Miller indices of the plane passing through the points ABC? [5]

b)  What are the Miller indices of the plane passing through the points BCD? [5]

c)  What are the Miller indices of the direction vector running from the origin of the coordinates to the point D? [5]

d)  What are the Miller indices of the direction vector running from the origin of the coordinates to the point E? [5]

4. The packing fraction for a solid is defined as the fraction of the volume of a unit cell that is occupied by the atomic cores. The close packing of the atomic spheres implies that the two nearest atomic spheres (of radius R, say) touch, and this dictates the size a of the cell edges relative to the size of the atom.

a)  For a simple cubic lattice with two atoms touching along a cube edge (unit cell size equals twice the atomic radius), find the ratio R/a. Given that there is one spherical atom (or 8 1/8th atoms) inside the cell, find the packing fraction as a per-cent. [5]

b)  Repeat for a BCC structure, where the atoms touch along the body diagonal. [5]

c)  Repeat for an FCC structure, where the atoms touch along the face diagonals. [5]

d)  Repeat for silicon, where the atoms touch along the dimers shown above. [5]

Hint: You will find that the closest packing is for an FCC structure. There is another structure called hexagonal closed packing (HCP), differing just in the stacking of the atomic planes, which shares this same highest packing density as the FCC.