For full credit, show all equations, derivations, sketches plotted data in a neat and well-organized format. This cannot be over-emphasized. Place a square around your final answer. 

1.    (35) Two unlabeled boxes contain KCl and CsCl for analysis using XRD. The diffraction maxima are observed from the (111) planes, using X-rays of wavelength 0.0862 nm.

a.    What is the diffracted beam angle, q (with respect to the plane) for both crystals?

b.    Is this sufficient to identify the two materials? Please give the explanation.

c.    If not, please design a meaningful experiment to further identify the two samples. (Hint: You can assume the atomic form factors for both materials are already known, i.e. fKCl and fCsCl. Find one or two angles where the diffraction peaks are different for two materials.)

2.  (30) Ionic bonds form the alkali halide crystal. It is significant to know the exact value of energy and the approximate atomic distance at the lowest energy.

a.    What is the formation of ionic energy? Please write it down.

b.    Please derive the ionic energy of potassium chloride (KCl), i.e. energy vs. atomic distance R. (Hint: you may refer to Eq. 3.19 and some tables in the book.)

c.    Plot energy vs. R based on your answer in the previous question. Please be careful with the units. (Hint: You can refer to Figure 10 in Kittel, PP. 65. Use eV as the unit of energy.)

3.  (35) Silicon is the most common material in semiconductor industry.

a.    Please calculate the velocities of longitudinal waves along [100], [110], and [111] in silicon (in unit m/s). You can use the equations on PP. 83.

b.    Why the calculated velocities are constant? Does it mean there is only one type  of longitudinal wave along each direction? Please physically explain your idea. (Hint: The continuum approximation is made for the analysis of elastic strains on PP.73 in Kittel.)

c.    Based on the values from 4(a), what type of waves do you think they would be? (Hint: What we discuss here is the mechanical wave. You may want to search some information online to support your thoughts.)

Please keep in mind:

• You need to show the plots and label them with exact numbers.
• Some equations and parameters are given in CGS units in the book. Your plot should be in SI units.
• You can use any software to make the plot. The access of MATLAB in SEASNET has been opened for every enrolled student in this class.