MATSCI 214 Autumn 2024 Homework #2Java

Java Python MATSCI 214, Autumn 2024

Homework #2

Due Before Midnight on 10/14/2024

Note: You may find a web-based crystal visualization application helpful for this homework. Several examples include:

https://www.chemtube3d.com/ccp-cubic-close-packing/

http://www.chemtube3d.com/solidstate/_hcp(final).htm

https://www.chemtube3d.com/_blendefinal/

https://www.chemtube3d.com/_wurtzitefinal/

1.  Body-centered cubic interstitials [14 pts]

(a)  Draw the BCC unit cell. Identify and draw the positions of the BCC octahedral interstices. Indicate the lattice vectors which comprise the BCC unit cell and basis atoms using either crystallographic or solid state conventions. Write down the octahedral interstitial coordinates within the conventional unit cell using the position point coordinate notation introduced in class. (6 pts)

(b)  Determine how many octahedral interstices are found in this unit cell. Draw the bounding coordination polyhedron for the octahedral interstitial site. Explain how these BCC octahedral interstices are different from those found in the FCC and HCP structures. (e.g. are all the sides of equal length?) (4 pts)

(c)  Using the hard sphere approximation, calculate the sizes of the interstices with respect to the host atoms, and determine how many nearest neighbor atoms would be in contact with the largest hard sphere that fits in each interstice. (4 pts)

2. α-Iron, Carbon, and Steel [6 pts]

(a)  The alpha phase of iron exists in a BCC structure, and steel is created when carbon atoms are present in low concentration. The carbon atoms are observed to reside in octahedral interstices. Take the hard sphere radius ratio of carbon to iron to be rc&n MATSCI 214, Autumn 2024 Homework #2Java bsp;/rFe = 0.6. Will carbon fit into an octahedral interstice? (2 pts)

(b)  Suppose the conventional BCC unit cell is elongated uniaxially such thÍat the C axis length is allowed to vary while the other axes remain fixed with length a. (Atom hard sphere sizes remain constant.) Note that uniaxial elongation eliminates the equivalency of the octahedral interstices – they are not all the same geometry. What is the value of c/a (> 1) for which a carbon atom in at least one of the octahedral interstices is equidistant from 6 nearest neighbors? This is important  because sparse carbon interstitial atoms contribute towards the hardness of steel by creating microscopic uniaxial strain. (4 pts)

3. Semiconductor structures [22 pts]

The diamond cubic, zincblende (sphalerite) and wurtzite structures are all closely related, each being made up of corner sharing tetrahedral units. Several important semiconductor elements (Si and Ge) and compounds (GaAs, InP, GaN, CdTe, etc.) adopt one of these structures. Each is derived from a close packed crystal structure (FCC or HCP). The purpose of this problem is to examine their similarities and differences.

(a)  Draw the unit cells of these three structures (diamond, zincblende, and wurtzite). For each, decide how many atoms there are per unit cell, and how many formula units there are per unit cell for the compound structures. (9 pts).

(b)  Re-draw these unit cells with the representative coordination tetrahedral units highlighted. (3 pts)

(c)  Calculate the atomic packing fraction of the diamond cubic structure. (4 pts)

(d)  Draw to scale the (110) projections of the two types of ZnS structures (wurtzite and sphalerite). Find and label a zone axis that is in the (110) plane and also perpendicular to the close-packed planes for each structure. Label the close-packed stacking sequence for both of these structures