MatSE 405: Microstructure Determination

Problem set #3; due Thursday February 7
1. I stated in class that the ratio of the populations of the spin-up and spin-down states of the nucleus in a magnetic field is given by
 NÝ Nß = exp æè DE kBT öø
(1)

 DE = g(h/2p)  B0
(2)
where g is the gyromagnetic ratio of the proton, and B0 is the static magnetic field. g/(2p) = 42.58 MHz/Tesla.
We observe a Larmor frequency of 15 MHz in the relatively small-field NMR experiments in the teaching lab; in other words, gB0/(2p) = 15 MHz. If the total number of protons in the sample volume is Ns, what is NÝ-Nß ? (Comments: You will find that DE << kBT; and recall that the Taylor series expansion of exp(x) » 1 +x.)
2. Draw the arrangements of atoms in a (110) plane of a bcc crystal. a) Draw the centered rectangular unit cell and label the positions of all of the symmetry elements (rotations, mirror lines, glide lines) of this two-dimensional lattice. b) Draw the primitive oblique unit cell for this two-dimensional lattice. What the lengths of the two base vectors of the primitive unit cell? What is the angle between the two base vectors of the primitive unit cell?
3. Exercise 1.9 in the textbook.
4. One of the basic things we have to do in analyzing data is find the position (along some x-axis such as angle, wavenumber, energy or frequency) of a peak in the signal intensity (e.g., the intensity of diffracted x-rays, or the intensity of Raman scattered photons). Use this MatLab script peak_simul_ps3.m to simulate and then fit a Gaussian shaped x-ray diffraction peak for a wide range of values of the variable peak_counts that describes the intensity of the peak. Plot the uncertainty in the fit of the peak position as a function of the the intensity of the peak. (You can use the difference between the upper and lower confidence bounds as a measure of this uncertainty; the confidence bounds will be listed in the command window.) Plot your result on a log-log axes (i.e., plot the data on a log scale; do not plot the log of the data on a linear scale; you can do this is MatLab using loglog(x,y,'o')). What do you conclude from this plot?

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On 01 Feb 2008, 14:37.