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1、BASICS OF X-RAY DIFFRACTION INTRODUCTION TO POWDER/ POLYCRYSTALLINE DIFFRACTION 3 Basics of X-ray Diffraction INTRODUCTION TO POWDER/POLYCRYSTALLINE DIFFRACTION About 95% of all solid materials can be described as crystalline. When x-rays interact with a crystalline substance (Phase), one gets a dif
2、fraction pattern. In 1919 A. W. Hull gave a paper titled, “A New Method of Chemical Analysis.” Here he pointed out that “.every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the other
3、s.” The x-ray diffraction pattern of a pure substance is, therefore, like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases. Today about 50,000 inorganic and 25,000 organic single component, crystall
4、ine phases, diffraction patterns have been collected and stored on magnetic or optical media as standards. The main use of powder diffraction is to identify components in a sample by a search/match procedure. Furthermore, the areas under the peak are related to the amount of each phase present in th
5、e sample. 4 THEORETICAL CONSIDERATIONS In order to better convey an understanding of the fundamental principles and buzz words of x-ray diffraction instruments, let us quickly look at the theory behind these systems. (The theoretical considerations are rather primitive, hopefully they are not too in
6、sulting.) Solid matter can be described as: Amorphous:The atoms are arranged in a random way similar to the disorder we find in a liquid. Glasses are amorphous materials. Crystalline:The atoms are arranged in a regular pattern, and there is as smallest volume element that by repetition in three dime
7、nsions describes the crystal. E.g. we can describe a brick wall by the shape and orientation of a single brick. This smallest volume element is called a unit cell. The dimensions of the unit cell is described by three axes: a, b, c and the angles between them alpha, beta, gamma. About 95% of all sol
8、ids can be described as crystalline. An electron in an alternating electromagnetic field will oscillate with the same frequency as the field. When an x-ray beam hits an atom, the electrons around the atom start to oscillate with the same frequency as the incoming beam. In almost all directions we wi
9、ll have destructive interference, that is, the combining waves are out of phase and there is no resultant energy leaving the solid sample. However the atoms in a crystal are arranged in a regular pattern, and in a very few directions we will have constructive interference. The waves will be in phase
10、 and there will be well defined x-ray beams leaving the sample at various directions. Hence, a diffracted beam may be described as a beam composed of a large number of scattered rays mutually reinforcing one another. This model is complex to handle mathematically, and in day to day work we talk abou
11、t x-ray reflections from a series of parallel planes inside the crystal. The orientation and interplanar spacings of these planes are defined by the three integers h, k, l called indices. A given set of planes with indices h, k , l cut the a-axis of the unit cell in h sections, the b axis in k secti
12、ons and the c axis in l sections. A zero indicates that the planes are parallel to the corresponding axis. E.g. the 2, 2, 0 planes cut the a and the b axes in half, but are parallel to the c axis. 5 If we use the three dimensional diffraction grating as a mathematical model, the three indices h, k,
13、l become the order of diffraction along the unit cell axes a, b and c respectively. It should now be clear that, depending on what mathematical model we have in mind, we use the terms x-ray reflection and x-ray diffraction as synonyms. Let us consider an x-ray beam incident on a pair of parallel pla
14、nes P1 and P2, separated by an interplanar spacing d. The two parallel incident rays 1 and 2 make an angle (THETA) with these planes. A reflected beam of maximum intensity will result if the waves represented by 1 and 2 are in phase. The difference in path length between 1 to 1 and 2 to 2 must then
15、be an integral number of wavelengths, (LAMBDA). We can express this relationship mathematically in Braggs law. 2d * sin = n * The process of reflection is described here in terms of incident and reflected (or diffracted) rays, each making an angle THETA with a fixed crystal plane. Reflections occurs
16、 from planes set at angle THETA with respect to the incident beam and generates a reflected beam at an angle 2-THETA from the incident beam. The possible d-spacing defined by the indices h, k, l are determined by the shape of the unit cell. Rewriting Braggs law we get : sin = = = = =/ 2d Therefore t
17、he possible 2-THETA values where we can have reflections are determined by the unit cell dimensions. However, the intensities of the reflections are determined by the distribution of the electrons in the unit cell. The highest electron density are found around atoms. Therefore, the intensities depen
18、d on what kind of atoms we have and where in the unit cell they are located. Planes going through areas with high electron density will reflect strongly, planes with low electron density will give weak intensities. 6 SAMPLES In x-ray diffraction work we normally distinguish between single crystal an
19、d polycrystalline or powder applications. The single crystal sample is a perfect (all unit cells aligned in a perfect extended pattern) crystal with a cross section of about 0.3 mm. The single crystal diffractometer and associated computer package is used mainly to elucidate the molecular structure
20、of novel compounds, either natural products or man made molecules. Powder diffraction is mainly used for “finger print identification” of various solid materials, e.g. asbestos, quartz. In powder or polycrystalline diffraction it is important to have a sample with a smooth plane surface. If possible
21、, we normally grind the sample down to particles of about 0.002 mm to 0.005 mm cross section. The ideal sample is homogeneous and the crystallites are randomly distributed (we will later point out problems which will occur if the specimen deviates from this ideal state). The sample is pressed into a
22、 sample holder so that we have a smooth flat surface. Ideally we now have a random distribution of all possible h, k, l planes. Only crystallites having reflecting planes (h, k, l) parallel to the specimen surface will contribute to the reflected intensities. If we have a truly random sample, each p
23、ossible reflection from a given set of h, k, l planes will have an equal number of crystallites contributing to it. We only have to rock the sample through the glancing angle THETA in order to produce all possible reflections. 7 The mechanical assembly that makes up the sample holder, detector arm a
24、nd associated gearing is referred to as goniometer. The working principle of a Bragg-Brentano parafocusing (if the sample was curved on the focusing circle we would have a focusing system) reflection goniometer is shown below. The distance from the x-ray focal spot to the sample is the same as from
25、the sample to the detector. If we drive the sample holder and the detector in a 1:2 relationship, the reflected (diffracted) beam will stay focused on the circle of constant radius. The detector moves on this circle. For the THETA:2-THETA goniometer, the x-ray tube is stationary, the sample moves by
26、 the angle THETA and the detector simultaneously moves by the angle 2-THETA. At high values of THETA small or loosely packed samples may have a tendency to fall off the sample holder. 8 GONIOMETER For the THETA:THETA goniometer, the sample is stationary in the horizontal position, the x-ray tube and
27、 the detector both move simultaneously over the angular range THETA. DIFFRACTOMETER SLIT SYSTEM The focal spot for a standard focus x-ray tube is about 10 mm long and 1 mm wide, with a power capability of 2,000 watt which equals to a power loading of 200 watt/mm2. Power ratings are dependent on the
28、thermal conductivity of the target material. The maximum power loading for an Cu x-ray tube is 463 watt/mm2. This power is achieved by a long fine focus tube with a target size of 12 mm long and 0.4 mm wide. In powder diffraction we normally utilize the line focus or line source of the tube. The lin
29、e source emits radiation in all directions, but in order to enhance the focusing it is necessary to limit the divergence in the direction along the line focus. This is realized by passing the incident beam through a soller slit, which contains a set of closely spaced thin metal plates. In order to m
30、aintain a constant focusing distance it is necessary to keep the sample at an angle THETA (Omega) and the detector at an angle of 2-THETA with respect to the incident beam. For a THETA:THETA goniometer the tube has to be at an angle of THETA (Omega) and the detector at an angle of THETA with respect
31、 to the sample. 9 DIFFRACTION SPECTRA A typical diffraction spectrum consists of a plot of reflected intensities versus the detector angle 2-THETA or THETA depending on the goniometer configuration. The 2-THETA values for the peak depend on the wavelength of the anode material of the x-ray tube. It
32、is therefore customary to reduce a peak position to the interplanar spacing d that corresponds to the h, k, l planes that caused the reflection. The value of the d-spacing depend only on the shape of the unit cell. We get the d-spacing as a function of 2-THETA from Braggs law d = = = = =/ 2 sin Each
33、 reflection is fully defined when we know the d-spacing, the intensity (area under the peak) and the indices h, k, l. If we know the d-spacing and the corresponding indices h, k, l we can calculate the dimension of the unit cell. 10 ICDD DATABASE International Center Diffraction Data (ICDD), formerl
34、y known as (JCPDS) Joint Committee on Powder Diffraction Standards, is the organization that maintains the database of inorganic and organic spactras. The database is available from diffraction equipment manufacturers or from ICDD direct. Currently the database is supplied either on magnetic or opti
35、cal media. Two database versions are available the PDF I and the PDF II. The PDF I database contains information on d-spacing, chemical formula, relative intensity, RIR quality information and routing digit. The information is stored in an ASCII format in a file called PDF1.dat. For search/match pur
36、poses most diffraction manufactures are reformatting the file in a more efficient binary format. The PDF II database contains full information on a particular phase including cell parameters. Thermo ARLs newest search/match and look-up software package is using the PDF II format. Optimized database
37、formats, index files and high performance PC-computers make PDF II search times extremely efficient. The database format consists of a set number and a sequence number. The set number is incremented every calendar year and the sequence number starts from 1 for every year. The yearly releases of the
38、database is available in September of each year. 11 PREFERRED ORIENTATION An extreme case of non-random distribution of the crystallites is referred to as preferred orientation. For example Mo O3 crystallizes in thin plates (like sheets of paper) and these crystals will pack with the flat surfaces i
39、n a parallel orientation. Comparing the intensity between a randomly oriented diffraction pattern and a preferred oriented diffraction pattern can look entirely different. Quantitative analysis depend on intensity ratios which are greatly distorted by preferred orientation. Many methods have been de
40、veloped to overcome the problem of preferred orientation. Careful sample preparation is most important. Front loading of a sample holder with crystallites which crystallize in the form of plates is not recommended due to the effect of extreme preferred orientation. This type of material should loade
41、d from the back to minimize to effect of preferred orientation. The following illustrations show the Mo O3 spectras collected by using transmission, Debye-Scherrer capillary and reflection mode. 12 APPLICATIONS Identification:The most common use of powder (polycrystalline) diffraction is chemical an
42、alysis. This can include phase identification (search/match), investigation of high/low temperature phases, solid solutions and determinations of unit cell parameters of new materials. Polymer crystallinity:A polymer can be considered partly crystalline and partly amorphous. The crystalline domains
43、act as a reinforcing grid, like the iron framework in concrete, and improves the performance over a wide range of temperature. However, too much crystallinity causes brittleness. The crystallinity parts give sharp narrow diffraction peaks and the amorphous component gives a very broad peak (halo). T
44、he ratio between these intensities can be used to calculate the amount of crystallinity in the material. Residual stress:Residual stress is the stress that remains in the material after the external force that caused the stress have been removed. Stress is defined as force per unit area. Positive va
45、lues indicate tensile (expansion) stress, negative values indicate a compressive state. The deformation per unit length is called strain. The residual stress can be introduced by any mechanical, chemical or thermal process. E.g. machining, plating and welding. The principals of stress analysis by th
46、e x-ray diffraction is based on measuring angular lattice strain distributions. That is, we choose a reflection at high 2-Theta and measure the change in the d-spacing with different orientations of the sample. Using Hookes law the stress can be calculated from the strain distribution. Texture analy
47、sis:The determination of the preferred orientation of the crystallites in polycrystalline aggregates is referred to as texture analysis, and the term texture is used as a broad synonym for preferred crystallographic orientation in the polycrystalline material, normally a single phase. The preferred
48、orientation is usually described in terms of polefigures. A polefigure is scanned be measuring the diffraction intensity of a given reflection (2-Theta is constant) at a large number of different angular orientations of the sample. A contour map of the intensity is then plotted as a function of angu
49、lar orientation of the specimen. The most common representation of the polefigures are sterographic or equal area projections. The intensity of a given reflection (h, k , l) is proportional to the number of h, k, l planes in reflecting condition (Braggs law). Hence, the polefigure gives the probability of finding a given crystal-plane-normal as function of the specimen orientation. If the crystallites in the sample have a random orientation the recorded intensity will be uniform. We can use the orientation of the unit cell to describ