Modern Physics Laboratory

Modern Physics Laboratory X-Ray Diffraction Experiment Supplementary Instructions REFERENCES Tipler and Llewellyn, Modern Physics, 5th edition, pp. 133-135 Laboratory Instructions: 1) X-Ray Diffraction Apparatus; Debye-Scherrer Camera (abbreviated LI1) 2) X-Ray Powder Diffraction (abbreviated LI2) NOTES ON PROCEDURE --The X-ray apparatus is set to produce x-rays using a copper target. The dominant wavelength is K-alpha line. Characteristic wavelengths for various targets are given in a table on the next-to-last page of LI1. --The handout LI1 refers to two different size Debye-Scherrer cameras. The one we work with is the one with diameter 5.73 cm = 57.3 mm. Note that the total length of the X-ray film is therefore (π) (57.3) = 180 mm. If θ is the Bragg angle, then the deflection of a Bragg-diffracted beam from the original X-ray beam direction is 2θ (see illustration at the bottom left of the second page of LI2). The entire range of deflection angles (from 0 to 180 degrees) occurs over one half the length of the film, or 90 mm (see illustration at the bottom right of the second page of LI2). Since 2θ varies from 0 to 180 degrees, θ varies from 0 to 90 degrees. Hence, when the film is measured, the distance in mm along the film from the original beam direction to a Bragg diffraction ring is exactly equal to the corresponding Bragg angle θ in degrees. ANALYSIS It is assumed that the film measurement procedures outlined on the last page of LI1 have been carried out and observed values of the effective interplanar spacing d have been determined from the expression d = λ/(2sinθ) where λ is the X-ray wavelength and θ is the Bragg angle. These values should be tabulated as on the table on the last page of LI1. The following steps deal with estimating the uncertainty in the measured values of d and obtaining predicted values based on knowledge of the crystal structure of the sample. The case of a face-centered cubic (fcc) structure is described explicitly. 1. From the uncertainty in the measured position of the X-ray diffraction rings on the film, estimate approximately the uncertainty in the observed value of d for each ring. (OVER)

2. Look up the cubic lattice constant, a, for the substance used as a sample. See, for example, the Handbook of Metals or a solid state physics text such as the one by Kittel. 3. Predicted values of the interplanar spacing d may be obtained for a cubic crystal with lattice contstant a from the third equation on the first page of LI2. The quantities h, k, and l the Miller indices which specify the family of planes involved. The allowed values of the Miller indices (hkl) for the fcc structure are given on the last page of LI2 (See Table 1 - Allowed fcc Reflections for Aluminum). Calculate d for all the tabulated values of (hkl). These correspond to the first eleven Bragg angles in ascending order. 4. Identify as many of your observed Bragg diffraction rings as possible by inspecting observed and predicted values of d. Where the observed and predicted values of d agree, you may associate the observed Bragg diffraction with the family of planes corresponding to the Miller indices (hkl) that yield the predicted d 5. For each ring identified in Step 4 above, compare the observed and predicted values of d, taking into account the uncertainty in the observed value.