MENA3100, 14.02.11
MEF3100 Exercises
1.
Illustrate by a ray diagram spherical aberration for a convex lens.
2.
What is chromatic aberration?
3. What is Fraunhofer diffraction?
4. What is plane polarized light?
5. What can we learn about a material by using polarized light in an optical microscope?
6. Write Braggs law and argue why it is the way it is.
7.
Electrons of energy 12 keV hit a Cu target, and thus x-rays are produced. What is the
minimum wavelength of the white radiation (Bremsstrahlung)?
8. Make a three dimensional sketch of the reciprocal lattice of a primitive cubic structure, bcc
structure, fcc structure.
9.
Where does a reciprocal lattice point need to be located relative to the Ewald sphere
(reflecting sphere) for the Bragg condition to be fulfilled for that reciprocal point? b) What is
the maximum wavelength that can be used to achieve Bragg reflection for a planar spacing of
2 Å?
10. Which of the following radiation “see” the atoms virtually as points: x-rays, neutrons or
electrons?
11. We perform x-ray powder diffraction experiments on the perovskite BaTiO3 at two different
temperatures, at 200 degrees C where it is cubic with lattice parameter a = 4.01Å and at
room temperature where it is tetragonal due to a small displacement of the atoms along one
of the a-axes that now becomes the c-axis. In the tetragonal phase the lattice parameters
are: a = b=3.99 Å and c = 4.03 Å. We use Cu Kα1 radiation of wavelength 1.5406 Å. a) For the
cubic structure, determine in the powder diffraction pattern the angles for the five peaks
nearest the forward direction (forward direction means  =0. b) What happens to these
peaks in the tetragonal case?
12. In an fcc metal, say Al, all atoms scatter in phase to all allowed reflections. In NaCl, the Na
atoms and the Cl atoms scatter in phase to some of the reflections and  out of phase to
other reflections. Which? Are these reflections strong or week?
13. We perform x-ray powder diffraction on polycrystalline Al at two different temperatures, at
200 K, and at 400 K where the thermal motion of the atoms is considerably larger. Explain
qualitatively how the powder diffraction pattern differs at the two different temperatures.
14. a) Project the atoms of the following cubic structures onto the (001)-plane and make
drawings of the projected unit cells: the primitive SrTiO3, bcc Fe, the fcc Al and silicon with
MENA3100, 14.02.11
the diamond structure. Their lattice parameters are 3.93 Å, 2.87 Å, 4.04 Å and 5.43 Å,
respectively. b) Based on these figures, draw the corresponding electron diffraction patterns
by indicating the diffraction spots (e.g. the 12 innermost) and index them when the parallel
incident electron beam (selected area electron diffraction) is along the [001] direction. c) We
use 200 keV incident electrons. Their wavelength is 2.5 pm. What is the distance from the
central spot (in the forward direction) to the innermost reflection in the four diffraction
patterns when the camera length is 1 m? d) We rotate these crystals Arctg 0.5 = 26.56
degrees around the row of reflections containing 010 reflection (which is perpendicular to
the incident electron beam). Draw the four diffraction patterns that now show up. Along
which crystal direction is now the incident electron beam? e) We have an electron diffraction
pattern of Si along the [011] direction. Draw the diffraction spots that are present based on
the extinction rules. Indicate the additional spots due to double (or multiple scattering). Hint:
A Bragg beam set up in the crystal can be looked upon as a new incident beam in a direction
different from that of the incident beam.
15. Fcc-metals are close-packed. Along the [111]-direction there are alternating A, B and C
planes. Assume, instead of the … ABCABCAB…, that the stacking along the [111] is
…ABABABA… . The result is then a hexagonal close packed (hcp) structure with the hexagonal
c-axis perpendicular to the A-plane (and B-plane). Draw the structure projected onto the
(111) plane for fcc structure and the corresponding drawing for the hcp structure. Draw also
the two corresponding electron diffraction patterns.
16. We perform TEM experiments on a polycrystalline material, on a thin foil of nearly uniform
thickness. We use a parallel beam of incident electrons (selected area). Explain why we see
different gray shades in the different grains, ranging from almost white to almost black in: a)
Bright-field. b) Dark-field. c) In dark-field, some of the grains are much brighter than others.
Why?