Series-A01 is for submission by Midnight on the 3rd Sunday in February, if possible. Please select answers, then press the submit button at the bottom of the page. Drawing pictures to help visualize, prowling related literature, searching the web, and discussing with colleagues, are all highly recommended.
This material is from UM-StL Physics & Astronomy by P. Fraundorf © 2000. All Fields are Optional...
1a. Consider a simple square 2D lattice with a 4A (0.4nm) separation between adjacent lattice points. Which of the following unit cells, written in the list form {a,b,gamma}, may also be used to describe this lattice?
A:{4A, 4A, 90deg} B:{4A, 4Sqrt[2]A, 45deg} C:{4A, 4Sqrt[5]A, ArcTan[1/2]deg} D:{4A, 4Sqrt[2]A, 33deg}
Cell A only Cell B only Cells A and B only Cells A, B, and C Cells A and D only
1b. The corresponding reciprocal lattice will be:
hexagonal with spacing 0.5/A square with spacing 0.25/A square with spacing 0.5/A square with spacing 4/A
1c. If we "face-center" the square lattice of 1a above, by adding lattice points at {a/2,b/2} in each unit cell, what happens to the reciprocal lattice?
the reciprocal lattice is unchanged in-between spots are added alternating spots are removed all spots are moved by {a/2,b/2}
2a. For something a bit more challenging, consider an oblique 2D lattice generated from an a-vector with cartesian coordinates {5,0}A, and a b-vector with cartesian coordinates {4,3}A. The unit cell has an area of:
6 A^2 9 A^2 12 A^2 15 A^2 18 A^2
2b. The corresponding reciprocal lattice, when projected onto a screen with a camera constant ("LamdaL") equal to 12[mm-A], will have:
a*={3,-4}mm and b*={0,5}mm a*={5,0}mm and b*={4,-3}mm a*={4,-3}mm and b*={5,0}mm a*={0,5}mm and b*={3,-4}mm
3. What is the Miller (reciprocal lattice) index of g3 in the image to the right? Note: if you can figure out the zone index of this projection in the form [u,v,w], then you can find a more completely labeled version of this figure at:
http://www.umsl.edu/~fraundor/p325/zone_uvw.gif
(1,1,1) (1,1,0) (-1,1,0) (1,-1,0) (0,1,0)
4. In the diagram below, one can take the body-centered-cubic (bcc) unit cell on the right, and by geometric construction derive it's reciprocal lattice on the left (which as you can see has a "face-centered" structure). If this done, then the slanted plane pencilled in on the right is associated with what reciprocal lattice spot?
(111) (011) (222) (101) (0.5,0.5,0.5)
5. Methods to infer "reciprocal space structures" from a lattice in "direct space" include:
(A) geometric construction (e.g. "A set of planes with spacing d suggests a reciprocal lattice vector normal to those planes of spacing 1/d"),
(B) vector algebra (i.e. a* = b x c/Vc, b* = c x a/Vc, and c* = a x b/Vc, where Vc = a . b x c ),
(C) "shape-transform" convolution (e.g. thinning the specimen to thickness t spreads all reciprocal lattice points out over a reciprocal distance of 1/t, in the direction of thinning),
(D) diffraction (e.g. hit the specimen with photons, electrons, or neutrons to get a map of intensity as a function of scattering angle, which in the small angle case has peaks a distance g = LamdaL/d away from the unscattered beam for each interplanar spacing d near it's "Bragg" angle with respect to the incident beam), and
(E) Fourier transformation (e.g. ask the computer).
Which of these provides clues to the intensity distribution around each reciprocal lattice point, but does NOT provide direct information on Fourier phases?
A and B C and D D only E only
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