HW01

(1) What is depth of focus and depth of field, respectively?

(2) Assuming the resolution of your naked eyes is C, and the focal length of the following lens is f (the refractive index is n, wavelength is ); calculate the depth of focus and depth of field of the lens (as a function of the lens radius R).

2. About crystal structure:

(1) What are the 7 crystal systems and what are the lattice constants for each of them?

(2) Describe the similarity and difference of CPC and HCP;

give several examples of CPC (FCC) metals and HCP metals, respectively; search online and describe the structure of diamond;

(3) How many atoms are in a unit cell of FCC, HCP, and diamond, and what are the coordinates of the atoms?

(4) Calculate the structure factors of an FCC metal, HCP metal, and diamond. (5) Why does face-centered tetragonal NOT exist?

3. About reciprocal lattice in 2D space. You have a reference Cartesian coordinate system X-Y. Assuming you have a 2D crystal, the lattice vectors of which are

ax3yand bxy. Similarly to Miller index (hkl) for a 3D crystal, we need

only 2 numbers (hk) to index a plane of this 2D crystal. (1) Draw the lattice in the X-Y coordinates.

(2) Calculate a*, b* (the lattice vectors of the reciprocal lattice) and draw it in the same X-Y coordinates.

(3) Draw (12) plane in the real lattice.

(4) Draw the corresponding point of (12) plane in the reciprocal lattice.

4. About reciprocal lattice in 3D space (Require for graduate students but optional for undergraduates).

(1) What are the lattice vectorsof the primitive cell of a BCC and an FCC unit cell, respectively? Calculate the reciprocal latticevectors of a BCC and an FCC unit cell?

(2) Derive Bragg equation from Ewald sphere.

5. About rhombohedral, trigonal and hexagonal (optional for all).

(1) Assuming you have a crystal with rhombohedral structure a=b=c=5 A, ===53o. Calculate its corresponding hexagonal lattice constants. (cos 53o= 0.6) Well, you can use MATLAB or Maple or Mathematica.

(2) What is the difference between trigonal and hexagonal. Hints: you need to discuss hexagonal here is one of whether crystal systems or lattice systems.