MatSE 405: Microstructure Determination |

Problem set #3; due Thursday February 7

- 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
N _{Ý}N_{ß}= exp æ

èDE k_{B}Tö

ø(1)

where g is the gyromagnetic ratio of the proton, and BDE = g( ^{h}/_{2p}) B_{0}(2) _{0}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, gB_{0}/(2p) = 15 MHz. If the total number of protons in the sample volume is N_{s}, what is N_{Ý}-N_{ß}? (Comments: You will find that DE << k_{B}T; and recall that the Taylor series expansion of exp(x) » 1 +x.) - 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?
- Exercise 1.9 in the textbook.
- 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.