MSE405 Microstructure Characterization XRD-1 Lab X-ray diffraction in crystallography
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1 X-ray diraction in crystallography I. Goals Crystallography is the science that studies the structure (and structure-derived properties) o crystals. Among its many tools, X-ray diraction (XRD) has had the widest historical impact in the development o crystallographic knowledge and remains one o the most widely use due to its power and versatility. In this lab you will: 1. Learn how to operate the x-ray diractometers.. Index the diraction peaks rom elements with cubic and hexagonal crystal structures. 3. Measure and understand the structure actor o simple crystals. 4. Understand how to use the measured diraction intensities to gain inormation about the structure o the basis o atoms decorating each lattice point. II. Theoretical background A crystal is a solid constituted by a three-dimensional regular arrangement o units. These units can be single, atoms, molecules, or groups o several atoms or ions. The deining property o a crystal is that the surrounding environment is the same or every repeating unit. The mathematical way o describing a crystal structure is with a lattice and a basis. The lattice is a set o points in the space that repeats periodically by translation. A position in space r belongs to the lattice (is a lattice point) i it satisies the condition: r= n a+ m b + l c, (1) where n, m, and l are integers. The vectors a, b, and c (the lattice vectors) are usually speciied by their moduli (the lattice constants) a, b, c and the three angles in between them α, β, γ. The crystal is constructed by repetition o the same unit at each lattice point. This unit (the crystal basis) is described by a list o the atoms or ions that integrate it and their positions, given in terms o the lattice vectors: r = ua+ vb+ wc, 0 u, v, w< 1. () basis For example, all the structures in the cubic group can be described by a lattice o three perpendicular vectors o the same length (a = b = c, α = β = γ = 90 ). In addition to the common lattice, dierent basis give rise to dierent crystal structures. The simple cubic (sc) structure has only one atom at the origin (u, v, w) = (0, 0, 0). The body centered cubic structure (bcc) has a two-atom basis: (0, 0, 0) and (½,, ½, ½). The ace centered cubic (cc) structure has atoms at the cube corners (0, 0, 0) and the periodic repetitions generated by the lattice, and at the ace centers: (½, ½, 0), (½, 0, ½), and (0, ½, ½). Three important structures (NaCl, diamond, and ZnS) are based on the cc structure by repeating a two-atom basis at each point. The basis or NaCl consists o one atom A at (0,0,0) and another atom at (½, ½, ½,). oth diamond and ZnS have the same basis: A at (0, 0, 0) and at (¼, ¼, ¼); in the diamond structure A and are atoms o the same element (e.g. C, Si, or Ge) while they are dierent elements in the ZnS structure. In summary:
2 Cubic structures, lattice constant a Simple cubic ody centered Face centered sc bcc cc Diamond ZnS A-(0, 0,0) A-(½, ½, ½) A-(¼, ¼, ¼) A-(¼, ¾, ¾) A-(¾. ¼, ¾) A-(¾, ¾, ¼) -(¼, ¼, ¼) -(¼, ¾, ¾) -(¾. ¼, ¾) -(¾, ¾, ¼) Another way to think about crystals is as a one-dimensional stacking o two-dimensional planes. As you know crystal structures are three-dimensional arrangements, thus this description is not unique and or each crystal there are several sets o planes that can be used to describe it. This way o thinking is useul to describe x-ray diraction by crystal considering specular relection at the planes, which gives rise to ragg's law ( ) λ= d sin θ. (3) In eq. (3), λ is the wavelength o the x-rays, * d is the spacing between () planes, and θ is ragg's diraction angle. It is not necessary to consider the order o diraction in eq. (3) as this can be accounted or by the plane indexes: (00) is the same as the second-order (001) relection. ragg's law gives the position (angle) o the diracted beams, but gives no inormation regarding the diracted intensity. For a perectly random powder, i.e. a well-mix ine powder in which all grain orientations are present with equal probability, the diracted intensity is proportional to I m F. (4) In eq. (4), m is the multiplicity actor which indicates the number o equivalent planes (e.g. (100), (010), etc.) contributing to diraction at a given angle and F, a complex number, is the structure actor, deined as the sum o the scattering actors n o all the atoms in the unit cell F N πi ( hun+ kvn+ lw n) = e. (5) n= 0 n I the atomic positions can be described in term o a cell basis and an atomic basis (e.g. ZnS is cc + two atom basis), the structure actor can be expressed as the product Ncell Nbasis cell atomic πi ( hun+ kvn+ lw n) πi( hum+ kvm+ lw m) F= F F = e ne n= 0 m= 0. (6) Note that the description o the cell lattice points with a basis involves no real atoms, thus cell calculating F involves no atomic scattering actors. (A useul discussion o the meaning and * In this lab, you will use the Cu K α line, λ = Å.
3 origin o the scattering actor can be ound in C. Hammond, The asics o Crystallography and Diraction, chapter 9) The ollowing table lists some o the relections in the cubic system and the corresponding plane spacing (in units o the lattice constant), multiplicity actor, and scattering actors or the bcc and cc lattices. () d / a m F bcc F cc (100) (110) (111) (00) (10) (11) (0) (310) (311) () (31) (400) (330) (331) (40) (33) (4) (431) (511) III. Experimental procedure Note: Measuring diraction patterns takes a lot o time. Use the acquisition time to analyze the data you have already acquired. This will save you time and allow you to detect errors and repeat your measurement i you need to.. 1. Load a Cu powder sample (cc) into the diractometer and collect a diraction spectrum 0 θ 100 at an acquisition speed o /min and step size 0.0. Please reer to the written procedure or the diractometer in your station. While you are collecting the spectrum, complete the preparations or data analysis as described in section IV.. Repeat the measurement using a bcc crystal, Mo. While you are collecting data or Mo, start analyzing the Cu data. 3. Repeat the measurement or a diamond structure crystal, Si. While you are collecting data or Si, analyze the Mo data. 4. Acquire more detailed XRD spectra o the two Si peaks with 85 θ 100. Change the step size to 0.005, decrease the scan speed, and pick adequate start and stop angles or each peak. You want to make sure o scanning the whole peak while minimizing the area
4 around it (which contains no inormation) to maximize the inormation you collect. Take into account your available time when picking the scan speed. 5. Collect a XRD spectrum o ZnS (0 θ 100, at an acquisition speed o /min, 0.0 step size). 6. Collect a XRD spectrum o a hcp material, Ti. (0 θ 100, at an acquisition speed o /min, 0.0 step size. The basis atoms or hcp structure are at 0,0,0 and /3, 1/3, 1/). IV. Data analysis A. Preliminary analysis For each o the ollowing materials Cu, Mo, Si, and Ti: Make a table listing the allowed diractions (), d-spacing, expected diraction angle θ, and θ. For Si and Ti, list also the structure actor due to the two-atom basis. Use the inormation provided in section II and the lattice constants listed in the provided periodic table. For Ti, use the ollowing ormula to calculate the d-spacings or all planes with 0 h, 0 k h, and 0 l h + hk+ k l =. d 3 + (7) a c. Diraction pattern analysis For each material, use your calculated values o the diraction angles to index (identiy) the peaks in your experimental spectra. With your report, turn in a printout o the spectra with () labels indexing the peaks. How do the experimental angles compare to the observed ones? Are there any absent or extra lines in the spectra? For each cubic material (Cu, Mo, Si, ZnS) pick the 4 tallest angle peaks and use your experimental θ values to calculate the lattice constant a or each peak. How big is the dispersion in the a value or each material? The dispersion in a arises rom small errors in the measurement o θ. Dierentiating ormula (3) it is ound that d sin θ ( ) ( ) a = = θ. (8) a d cos θ I.e. small errors in measuring θ, e.g. rom a aulty goniometer calibration, get propagated to errors in a. In order to minimize this error, it is better to determine a rom a set o peaks, rather than or each peak individually. For each cubic material (Cu, Mo, Si, ZnS), plot sin(θ ) vs. (d /a) -1 using the (given) tabulated values o d /a and the experimental θ o the 4 tallest angle peaks. Use a least squares it to determine the lattice constant or each cubic material analyzed. Note that the slope o this plot is not the lattice constant, but related to it. How do these values compare to the ones that you used to calculate θ in the irst palce?
5 C. Characterization o the Cu K α doublet Looking careully at the (more detailed) Si spectrum rom 85 to 100 you will see that the diraction peaks are doubled. The reason is that you are resolving two diraction peaks rom the two Cu-K α x-ray lines (λ 1 = Å and λ = Å). Measure the angular position o the two sub-peaks or each relection and use ragg's law to determine the ratio λ 1 / λ o the wavelengths o the two lines. How does this ratio compare with the theoretical one? Use eq (4) and the experimental sub-peak intensities (maximum values) to estimate the intensity ratio o the two Cu-K α lines? Note that you may need to do a linear background subtraction to get accurate intensity values. D. Structure actor analysis What are the dierences (in terms o relections present or absent) between Si, Cu, and ZnS spectra? Explain qualitatively why some relections are present in Cu but not in Si. Are this relections present in ZnS? Why? (Hint, reer to the structure actors that you calculated in section IV.A). A A + Use the measured intensities o the (0), and () ZnS peaks to estimate the ratio A. Use this value to calculate. Which atom (Zn or S) would you expect to have a higher atomic scattering actor? Why? (A useul discussion o the meaning and origin o the scattering actor can be ound in C. Hammond, The asics o Crystallography and Diraction, chapter 9).
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