PHY455
Spring, 2014
Exam #1
Name
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Total
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Given a 2D centered rectangular lattice with sides 2Å and 3Å. (i.e. the sides of
the rectangle have those dimensions).
1. Sketch the lattice including a coordinate system and write down a set of
primitive lattice vectors in vector notation.
2. Sketch the Wigner-Seitz primitive cell for this lattice. (Hint: the cell you come
up with must fill all space if translated to every lattice point.)
3. Describe the lattice as a rectangular lattice plus basis. (Give lattice vectors in
vector notation and locations of basis atoms using the coordinate system from
#1.)
4. Find the 2D packing fraction of the centered rectangular lattice above.
5. Given the book's primitive translation vectors for an FCC lattice, using them,
find the volume of the primitive cell.
6. For the FCC lattice find the angle between any pair of the primitive lattice
vectors in #7 using vector properties.
7. Using the primitive translation vectors for an FCC lattice as previous, describe
the sodium chloride crystal as a lattice plus basis. Assume a lattice constant of
5Å. Put the Cl- ion at the cube corner and the Na+ ion in the center, etc. (Give
the coordinates of the basis atoms using the FCC lattice.)
8. Find the indices of the lattice planes pictured below.
9. To use a neutron beam for diffraction of a lattice with spacing 5Å, find the
speed of the neutrons in m/s?
10. Find the reciprocal lattice for the rectangular lattice in #3. (Give the lattice
vectors in vector notation.)
11. Sketch the 1st and 2nd Brilllouin Zones of this lattice.
12. Assume the center atom in the centered rectangle in #3 is different from the
corner atom. Assign form factors of fA for the corner atom and fB for the center
atom. Find the structure factor.
13. Sketch the form of the Lennard-Jones potential and discuss the limits
referring to the terms in the potential.
14. Given the quantities in the attached tables, find the cohesive energy per ion
pair in eV for LiF. (Use the calculated value for the lattice energy.) Specifically
find the energy of the crystal vs. separated neutral atoms.
S13
1. Draw an SCC lattice and indicate the (111) plane.
2. Find the indices of the (110) plane in the cubic system using the primitive
translation vectors of the FCC lattice.
3. Describe the FCC lattice as a lattice plus basis using the SCC lattice with
lattice constant, a. Give the primitive translation vectors and basis in vector
notation.
4. Find the angle between the primitive translation vectors of the BCC lattice.
5. Find the 1st and 2nd Brillouin Zones for a 1D lattice with lattice constant a.
Sketch the lattice, the reciprocal lattice, and the BZ's.
6. Given a 1D lattice consisting of identical atoms with alternate distances a/4
and 3a/4 between them:
a. Sketch the lattice.
b. Describe the system as a lattice with a 2-atom basis (give lattice and basis in
vector notation).
c. Find the reciprocal lattice in vector notation.
d. Find the structure factor using f as the atomic form factor.
e. Find the zeroes of the structure factor.
7. Given a lattice with translation vectors of a1 = 5.00 Å i ,and a2 = (4.00 Å i +
3.00 Å j), and a3 = (2.50 Å j + 6.00 Å k) find the volume of the cell.
8. a. Describe the CsCl crystal as an SCC lattice plus basis. Give the primitive
lattice vectors and the basis vectors using the lattice constant of 4.11 Å. Use
vector notation. Assume that the Cl atom is in the center.
b. Find the reciprocal lattice and express it in vector notation.
c. With atomic form factors, fCs and fCl, find the structure factor for the system.
9. Identify the rare gases and the alkali metals on the periodic table.
10. Explain briefly why the lattice energy of rare gas crystals is so low.
11. Calculate the cohesive energy of a KBr crystal, which forms in the NaCl
crystal structure. Consider all relevant energies, including the Madelung energy.
Not Used
b. Find the reciprocal lattice vector of the (110) plane ???
4. Find the packing fraction of the SCC lattice.
5. Find the packing fraction of the FCC lattice.
8. Find the 1st and 2nd BZ's for 2D cubic lattice.
10. 2D structure factor or make one up in 3D ; 2 atom basis ???
12. The Van der Wal's interaction was ignored in calculating the lattice energy
of alkali halide crystals. Is this reasonable? Explain.
14. Find the cohesive energy and/or Ro for BCC Ar.
15. Some sort of 1D calculation with Madelung energy.
6. Find the packing fraction of an SC lattice. (Clearly show/describe your work.)
14. Find the reciprocal lattice for the centered rectangular lattice in #1.
16. Describe the side centered cube as a lattice plus basis (SCC - SC with
additional atoms at midpoints of cube sides (not faces)).
18. Given ..... , find the cohesive energy per atom in eV of Xe in an FCC
structure.
4. Sketch the Wigner-Seitz primitive cell for this (rectangular) lattice. Hint: the
answer should not be the same as #2.
5. Give the 2D packing fraction for a crystal with this (rectangular) lattice.
13. Find if fA = fB, what happens to the structure factor (are there any zeros,
etc.)? Explain.