材料科学基础作业

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Fundamentals of Materials Science 1. Determine the Miller indices for the planes shown in the following unit cell:

A:(2 1 -1) B:(0 2 -1)

2. Show that the atomic packing factor for HCP is 0.74.

Solution:

This problem calls for a demonstration that the APF for HCP is 0.74. Again, the APF is just the total sphere-unit cell volume ratio. For HCP, there are the equivalent of six spheres per unit cell, and thus

Now, the unit cell volume is just the product of the base area times the cell height, c. This base area is just three times the area of the parallelepiped ACDE shown below.

The area of ACDE is just the length of CD times the height BC. But CD is just a or 2R, and

3. For both FCC and BCC crystal structures, the Burgers vector b may be expressed as

where a is the unit cell edge length and [hkl] is the crystallographic direction having the greatest linear atomic density.

(a) Compute and compare the linear densities of the [100], [110], and [111] directions for FCC and BCC.

(b) What are the Burgers vector representations for FCC, BCC?

(c) If the magnitude of the Burgers vector b is

determine the values of b for aluminum, For Al which has an FCC crystal structure, R=0.1431nm.

solutions:

(a) In this problem we are to compute the linear densities of several crystallographic planes for the face-centered cubic crystal structure. For FCC the linear density of the [100] direction is computed as follows:

The linear density, LD, is defined by the ratio

where L l is the line length within the unit cell along the [100] direction, and Lc is line length passing through intersection circles. Now, L l is just the unit cell edge length, a which, for FCC is related to the atomic radius R according to a = 2R√2 [Equation (3.1)]. Also for this situation, Lc = 2R and therefore

4. Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is 0.414.

solutions:

In this problem we are asked to show that the minimum cation-to-anion radius ratio for a coordination number of six is 0.414. Below is shown one of the faces of the rock salt crystal structure in which anions and cations just touch along the edges, and also the face diagonals.

5. The number-average molecular weight of a polypropylene is 1,000,000 g/mol. Compute the number-average degree of polymerization. Solution:

We are asked to compute the number-average degree of polymerization for polypropylene, given that the number-average molecular weight is 1,000,000 g/mol.The mer molecular weight of polypropylene is just

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