Reference no: EM132718201

Problem 1 - Structure Factor
Si and Ge both have diamond cubic crystal structure, and are completely miscible in the solid state forming an alloy with the composition Si1-xGex where 0<x<1.
a. For an alloy Si1-xGex, determine the structure factor and give an expression for the structure factor for the {200}, {513} and {422} peaks.
b. Please give a general expression for the alloy form factor as a function of composition.

Problem 2- Reciprocal Lattice and Diffraction
a. Work out the structure factor of KCl, which has the rocksalt crystal structure. It is advised to use the approach where Shkl = Slattice x Sbasis.
b. Use your structure factor calculation to draw the corresponding reciprocal lattice for the (h k 0) and (h k 1) plane including lattice points up to (h2+k2+l2)=16 which is the (4 0 0) plane. Note: 7 and 15 are missing in this count, and 9 has two planes.
c. Create a schematic diffraction pattern using only the structure factor: a plot of intensity vs. scattering angle representing each peak as a line. The x-ray wavelength is 1.54 Å (Cukα radiation) over the range 0° < Θ < 45°. Normalize each peak by the intensity of the largest peak (I'hkl = Ihkl/Imax). The lattice constant a=6.2931 Å.
d. How does the diffraction pattern change if you take into account the form factor, its angle dependence, and the multiplicity? Redraw your schematic from (c) - please give a detailed account of your calculations (use of excel or other spreadsheets is permitted although not really necessary).

Problem 3 - Structure
The crystal structure of sphalerite is shown below - from Rohrer page 171 figure 4.22, table 4.25.
a. Identify the lattice and mark the basis, and specify the basis coordinates. Then draw the crystal plan in the [ 0 0 1] projection.
b. What type of rotation axis does the structure have parallel to the [001] direction?

Problem 4 - Reciprocal Lattice
Using the definitions for the reciprocal lattice vectors a*,b*, and c* obtain expressions for the three basis vectors of the reciprocal lattice of an orthorhombic C real-space Bravais lattice. Hint: use the primitive unit cell.

Problem 5 - Space Groups, eutactic system
The perovskite structure with the CaTiO3 prototype is given in table 4.28 - Rohrer
a. Identify the space group (full symbol), point group, Bravais lattice, and give the axis of the symmetry elements in the space group symbols.
b. Specify the atoms in the basis and their coordinates.

c. Draw a crystal plan for the view in the [001] direction.
d. Which type of rotation axis is parallel to the [001]? Show this axis in the crystal plan. Also add the mirror plan which is in the third position in the space group symbol.
e. Does this structure include a glide plane? If yes, please indicate the glide plane with its proper notation (dashed, dotted, combined line).
f. Describe the structure in the eutactic framework - which atoms are the framework and which dense packing does this correspond to? Which atoms sit on interstitial sites, and which sites are those?

Problem 6 - Bonding
a. What is the physical origin of the van der Waals bond?
b. The melting point of halide molecules X2 where X=F,Cl,Br,I increases with atomic weight. F2 therefore has the lowest and I2 has the highest melting point. How can you explain this observation? And which type of bonding is expressed in between and within the molecules?
c. Open table 7.1 in Rohrer and explain the following trends: (i) why does the lattice energy decrease going from fluorides to iodides? (ii) why is the lattice binding energy always higher for oxides? Use the underlying physical origin of the ionic bonding for your explanation.

Bonus Questions
a. Which symmetry elements define an enantiomorphic point group?
b. Why do a cube and an octahedron represent the same Bravais lattice?

Attachment:- Structure Factor.rar