ASM Metals Handbook - Desk Edition (ASM_ 1998) Episode 14 ppsx

Fig. 4 Planes in three-dimensional crystals defined by their Miller indices. Source: Ref 2 In simple crystal structures (e.g., cubic), planes with various combinations and permutations of the same indices have identical interplanar spacings. These spacings are referred to as types of planes. For example, in a cubic crystal (112), (121), (211), (-112), (-211), etc., all spacings are referred to as planes of the type {112}. A number of different types of information can be determined from XRD experiments. The primary types of analyses and their uses are described in the following sections, along with information about the threshold sensitivity and precision, limitations, sample requirements, and capabilities of related techniques. References cited in this section 1. R. Jenkins and R. Snyder, Introduction to X-Ray Powder Diffractometry, John Wiley, 1996 2. C. Barrett and T. Massalski, Structure of Metals, McGraw-Hill, 1966 Identification of Compounds and Phases Using X-Ray Powder Diffraction Typical Uses X-ray powder diffraction is used to identify the crystalline phases present in a sample. Examples of the types of questions that can be answered using x-ray powder diffraction include: • Is a heat treated steel sample 100% martensite, or does it contain some retained austenite? • What compounds are present in the corrosion product that formed when an aluminum alloy was exposed to sea spray? • What compounds are present in the scale formed on a ingot during high-temperature forging? Solving these types of problems by x-ray powder diffraction is the most common use of XRD in metallurgy. Experimental Approach In x-ray powder diffraction, an x-ray beam of a single known wavelength is used to determine the interplanar spacings of the planes in the sample. The sample is typical polycrystalline, ideally containing a semi-infinite number of randomly oriented crystals. Bragg's law dictates that each set of crystal planes will diffract at its own characteristic angle. Once these angles at which diffraction occurs have been experimentally determined, the interplanar spacings corresponding to each of them can then be calculated by substituting these angles and the known wavelength of radiation used into Bragg's law. The resulting set of interplanar spacings provides a "fingerprint" of the sample that can be compared to the "interplanar spacing fingerprints" of over 100,000 known compounds, thus permitting identification of the compound(s) present in the sample. The results of a typical x-ray powder diffraction analysis are shown in Fig. 5. The plot of diffracted intensity versus diffraction angle exhibits a number of peaks, each corresponding to a particular set of crystallographic planes and its characteristic d-spacing. The most important information is the angle at which diffraction occurs. The relative intensities of the peaks are determined by a number of factors beyond the scope of this article. Suffice it to say that, for a sample made up of randomly oriented crystals of a given metal or compound, the relative intensities of the various diffraction peaks are predictable and reproducible within a few percent. However, these intensities are of secondary importance in solving the powder diffraction pattern. Fig. 5 X-ray powder pattern of Al 2 O 3 . Source: Ref 1 The pattern can also be summarized as a table of d-spacings (each corresponding to a angle) and corresponding intensities, typically expressed as a percentage of the most intense peak, as shown in Tables 1(a), 1(b), and 1(c). Table 1(a) Identification of powder diffraction pattern from Al 2 O 3 using the Hanawalt search method The diffracting angles and intensities (area under each peak) are measured from the unknown sample (in this example from the diffractometer trace in Fig. 5). d-spacings are then calculated for each diffraction peak using Bragg' s law and the known wavelength of radiation used (in this case, Cu K , = 1.54178 ). 2 , degrees Intensity, 0 to 100 d, Miller indices 25.62 67 3.477 012 35.20 89 2.549 104 37.78 35 2.381 110 41.70 1 2.166 006 43.43 100 2.084 113 46.20 1 1.965 202 52.60 45 1.740 024 57.55 100 1.601 116 59.82 4 1.546 211 61.30 12 1.512 122, 018 66.60 40 1.404 214 68.28 53 1.374 300 70.40 2 1.337 125 74.30 2 1.277 208 77.15 22 1.236 1010, 119 80.80 8 1.189 220 84.50 6 1.146 223 86.50 8 1.125 312, 128 89.08 10 1.099 0210 91.00 12 1.081 0012, 134 95.34 18 1.043 226 98.50 1 1.018 042 Table 1(b) Hanawalt search method The d- spacings corresponding to the most intense peaks from the sample are then compared with a database of known compounds ordered by the d-spacings of their most intense peaks, resulting in a tentative identification of the sample. A set of 25 compounds with similar intense peaks is shown below. Note that the 8 most intense peaks for Al 2 O 3 correspond to intense peaks from the sample pattern given in Table 1(a). 2.12 g 2.55 g 4.89 g 1.50 g 1.63 g 1.10 x 1.43 x 2.99 9 cF56 Li 0.75 Mn 0.25 Ti 2 O 4 40-406 2.12 8 2.55 x 1.98 x 1.27 x 1.24 8 3.15 5 1.34 5 1.19 5 tP10 FeW 2 B 2 21-437 2.11 x 2.55 5 2.79 3 1.37 1 1.09 1 1.98 1 1.51 1 1.16 1 cF* Ce 0.78 Cu 8.76 In 3.88 43-1269 2.11 8 2.55 x 2.44 x 2.29 7 1.50 7 1.34 7 7.31 5 3.20 5 hP22 K 6 MgO 4 27-410 2.10 x 2.55 7 2.61 4 1.45 4 1.29 4 1.80 3 3.88 2 2.49 2 tI10 Pd 2 PrSi 2 32-721 2.10 x 2.55 x 2.43 5 1.39 2 0.85 2 3.71 1 1.50 1 1.17 1 hP8 PmCl 3 33-1085 2.09 8 2.55 x 6.31 8 1.68 8 3.16 6 2.79 6 1.61 6 1.56 6 K 0.72 In 0.72 Sn 0.28 O 2 34-711 2.09 8 2.55 x 2.63 8 1.65 8 1.79 7 2.66 6 1.88 6 3.05 5 mC18.80 IrB 1.35 17-371 2.09 x 2.55 9 1.60 8 3.48 8 1.37 5 1.74 5 2.38 4 1.40 3 hR10 Al 2 O 3 10-173 2.09 x 2.55 x 1.60 x 3.48 7 1.37 6 1.74 5 2.38 4 1.40 4 hR10 Al 2 O 3 43-1484 2.08 x 2.55 8 3.22 8 1.57 6 2.00 4 1.61 4 1.75 3 2.40 2 hP6 EuAl 2 EuSi 2 45-1237 2.08 x 2.55 8 2.16 8 1.18 x 1.17 x 2.02 8 1.16 8 2.33 6 oC20 Cr 2 VC 2 19-334 2.08 8 2.55 x 2.14 4 1.23 x 1.32 8 1.17 8 1.30 6 1.64 4 oC8 HfPt 19-537 2.16 x 2.54 6 2.74 5 2.19 4 2.51 4 1.38 2 1.27 2 1.50 1 hR12 ErFe 3 43-1373 2.16 x 2.54 x 2.33 x 2.12 x 1.42 9 1.54 8 1.38 8 1.32 8 hP12 Cr 2 Hf 15-92 2.16 x 2.54 7 1.38 2 4.14 2 1.46 2 1.27 2 2.07 1 0.93 1 cF24 Co 2 Ho 29-481 2.16 x 2.54 6 1.38 2 2.07 2 1.27 2 1.46 2 4.14 1 0.93 1 cF24 TbNi 2 38-1472 2.14 x 2.54 8 2.35 8 1.36 7 2.16 6 0.85 6 1.86 6 0.88 5 oP8 TiB 5-700 2.12 x 2.54 x 2.33 x 2.16 x 1.42 9 1.54 8 1.38 8 1.32 8 hP12 Cr 2 Hf 15-92 2.11 x 2.54 7 1.49 6 4.87 6 0.86 3 1.62 2 0.94 1 1.22 1 cF* AlVO 3 25-27 2.10 x 2.54 7 2.07 6 1.16 8 1.08 8 1.36 7 1.09 7 1.07 7 cF112 RbZn 13 27-566 2.10 8 2.54 x 1.57 x 1.21 x 1.72 8 1.68 8 1.51 8 1.45 8 tP5 LuB 2 C 2 27-301 2.10 x 2.54 x 1.49 7 4.86 6 1.62 4 2.97 2 2.43 2 1.40 1 cF* Mg 1.5 VO 4 19-778 2.09 9 2.54 x 3.49 8 6.39 7 3.69 5 2.76 4 2.13 4 1.38 3 hP8 SmCl 3 12-789 2.09 x 2.54 x 2.59 8 1.28 8 3.85 5 2.46 5 1.63 5 1.60 5 tI10 LaPd 2 P 2 37-994 Table 1(c) Hanawalt search method A card containing all of the information for the tentatively identified compound is compared with the diffraction information from the sample. The card for Al 2 O 3 is shown below. Note that all of the diffraction peaks in the sample pattern can be accounted for by Al 2 O 3 . If unidentified lines were present, it would indicate either that the tentative identification was incorrect, or that one or more additional compounds were present in the sample along with Al 2 O 3 . The information on the card enables identification of the Miller indices of the planes associated with each diffraction peak, shown in the last column of Table 1(b). Historically, results such as those presented in Tables 1(a), 1(b), and 1(c) were compared with the d-spacing and intensity fingerprints for 100,000 known compounds each tabulated on an index card and organized systematically by the d- spacings of the several most intense peaks. This search and match process is now greatly facilitated by computers that contain all of the information on known compounds in an updatable database. The search is based primarily of the d- spacing information, rather than the intensities, because the assumption that the sample consists of randomly oriented crystals is frequently violated, thus altering the intensities. Samples Containing Multiple Phases or Compounds Identification of metals and compounds by x-ray powder diffraction is relatively straightforward when the sample consists of a single phase or compound. When multiple phases or compounds are present, however, the task is more complex, as multiple "fingerprints" are superimposed on one another. Fortunately, software is available to analyze and solve such complex patterns. Solution of complex overlapping patterns can be simplified by providing additional information to the computer regarding what elements are, are not, and may be in the sample. This information is typically obtained by x-ray fluorescence spectroscopy. Once this elemental information has been input, the software searches only compounds that are consistent with it. Once the phases or compounds present in a multi-phase sample have been identified, the percentages of each phase or compound present can be deduced from the relative intensities of their diffraction peaks. A common metallurgical example is the determination of the amount of retained austenite present in heat treated steels. The most precise analyses are based on comparison of results from the unknown with those from a number of calibrated standards. Instrumentation X-ray diffraction analyses have historically been conducted using two types of equipment: the Debye-Scherrer camera and the x-ray diffractometer. The Debye-Scherrer camera is used for powdered samples (Fig. 6). The camera is a light- tight hollow cylinder with a removable cover plate. The powdered sample is placed in a hollow capillary tube at the center of the camera. A strip of photographic film is then placed around its inside perimeter, and the cover plate is applied. The camera is then attached to an x-ray generating tube and the x-rays are directed onto the sample through a collimator. The diffracted x-rays are recorded on the film, which is developed following 1 to 4 h of x-ray exposure (Fig. 7). The angles are measured from the film and converted to d-spacings. Intensities are either estimated by eye or quantified using a densitometer. Fig. 6 Schematic of Debye- Scherrer powder method. (a) Relationship of film to sample, incident beam, and diffracted beams. (b) Appearance of film when developed and laid flat. Source: Ref 3 Fig. 7 Debye-Scherrer films identifying phases in copper-zinc alloys of various compositions. Source: Ref 2 The x-ray diffractometer avoids the use of film and is far better suited to automation. The sample is flat, typically either a polished metal surface or a powder adhered to a flat glass slide. The sample is exposed to the incident x-ray beam, and a counter is scanned over the desired range of angles (Fig. 8). The result is a plot of diffracted intensity versus diffraction angle (Fig. 5). Fig. 8 Schematic of x-ray diffractometer. Typically, the x- ray tube remains stationary while the detector mechanically scans a range of angles. The sample also rotates with the detector such that diffraction is recorded from planes parallel to the sample surface. In recent years, position-sensitive wire detectors and solid state charge coupled device (CCD) detectors (in essence, a high resolution array of solid state light sensors) have permitted all of the diffracted signals to be collected simultaneously, thus overcoming the need to mechanically scan a detector scintillation counter over the range of diffraction angles. This method of collection greatly increases the speed with which diffraction information can be obtained, and it provides a digital electronic format that is amenable to computer-assisted data reduction and analysis. Detection Threshold and Precision • Threshold sensitivity: A phase or compound must typically represent 1 to 2% of the sample to be detected • Precision of interplanar spacing and lattice parameter measurements: 0.3% relative in routine measurements; within 0.003% relative in experiments optimized for this purpose • Precision of quantitative analyses of percentage of individual phases or compounds present in samples containing multiple compou nds: 5 to 10% relative or 1 to 2% absolute, whichever is greater (presumes the use of calibrated standards) Amount of Material Sampled • Powdered samples: Entire sample • Polished bulk materials: Typically 1 cm square area, sampling depth usually in the range 10 to 100 m (increases with decreasing average atomic number) Limitations • Noncrystalline samples produce no diffraction peaks • Results represent average of many grains or crystals in the sample, not an individual particle. Fine beams (down to 100 m diameter) can be used to characterize some individual particles. Sample Requirements • Powders: 10 mg is typically enough • Flat metal samples: Diffractometers can usually accommodate samples with lateral dimensions up to 5 cm and thicknesses up to 5 mm. Id eally, the surface should be free of deformation in order to get sharp diffraction peaks. Chemical or electropolishing can be used to remove the last vestiges of deformation in mechanically ground and/or polished samples. References cited in this section 1. R. Jenkins and R. Snyder, Introduction to X-Ray Powder Diffractometry, John Wiley, 1996 2. C. Barrett and T. Massalski, Structure of Metals, McGraw-Hill, 1966 3. Materials Characterization, Vol 10, ASM Handbook, ASM International, 1986, p 335 Measurement of Lattice Parameter Changes due to Alloying or Temperature The addition of alloying elements, heating or cooling, and residual stresses cause slight changes in interplanar spacings, which cause slight shifts in the angles at which diffraction occurs. XRD can be used to quantify these changes. Changes resulting from solid solution alloying can be determined using a diffractometer, as described earlier. Similarly, temperature effects can be characterized using a diffractometer equipped with a furnace to heat the sample to the desired temperature. The systematic errors inherent in XRD measurements decrease with increasing diffraction angle. As a result, lattice parameter measurements are typically based on information obtained at high diffraction angles. The most precise lattice parameter measurements make use of curve-fitting procedures to extrapolate information to the limiting value of 90°. Precision • 0.03% relative with moderate care • 0.003% relative with the greatest care Sample Requirements • Identical to those indicated above for routine diffractometer examination Measurement of Residual Stresses X-ray residual stress measurement is substantially more complex, but its key principles are not difficult to understand. Residual stresses are most often introduced during heat treating or welding and are caused by differential thermal contraction associated with temperature gradients in the material. These stresses cause elastic strains in the material, which manifest themselves as slight departures from the material's normal (unstressed) lattice parameters or interplanar spacings. XRD measurement of these changes in interplanar spacings, then, provides a direct measure of elastic strain, which can be used to calculate residual stress magnitude using the known elastic constants of the material. Because residual stresses must be calculated from small stress-induced changes in interplanar spacing, it is critical to know precisely the interplanar spacings in an unstressed sample of the material in question. The planes that happen to be parallel to and very close to the free surface provide an internal calibration in this regard, because no stresses can be supported perpendicular to a free surface. The problem then becomes one of comparing the interplanar spacings of planes parallel to the free surface with those inclined at various angles to the surface, and using the pertinent equations of elasticity to calculate from these differences the magnitudes of the stresses in various directions parallel to the surface. From this, the principal in-plane stresses directions can be calculated. Instrumentation Residual stresses are typically measured using an x-ray diffractometer equipped with a special specimen holder designed to facilitate measurement of diffraction from planes inclined at various angles to the surface of the sample. (General purpose diffractometers obtain diffraction information only from planes that are parallel to the sample surface. See Fig. 8.) Portable x-ray systems are also frequently used to make field stress measurements on structural components. Residual stresses vary with position in the component, so measurements are frequently made at numerous locations. The spatial resolution of these measurements is defined by the diameter of the x-ray beam used. The diameter is typically in the vicinity of 1 cm, but it can be as small as 30 m. Precision • 5% relative or 5 MPa absolute, whichever is higher (in ideal laboratory conditions) • Less precise information is obtained when portable x-ray equipment used fo r residual stress measurement of non-ideal structural components Limitations • Because XRD obtains information from the near- surface region of the sample, it only provides surface stress information and is not capable of measuring stresses in the interiors of components. Capabilities of Related Techniques Neutron diffraction residual stress measurement is based on the same principles as XRD. Because neutrons penetrate metals to far greater depths, it is possible to measure stresses in the interior or samples, rather than only surface stresses. However, a neutron source is required, so such measurements cannot be made in the field. Characterization of Crystal Size and Defect Density from Peak Width and Shape When Bragg's law was discussed earlier, it was noted that the angular range over which significant diffracted intensity is obtained depends on the number of adjacent planes from which diffracted beams are summed. In the extreme case of only two adjacent planes, diffracted intensity would be maximum at the angles satisfying Bragg's law (where perfectly constructive interference occurs). However, it would vary continuously with diffraction angle, only being zero at the angles where perfectly destructive interference occurs, which would result in very broad diffraction peaks. On the other hand, when the beams scattered by a semi-infinite number of adjacent planes are summed, constructive interference and significant diffracted intensity are only obtained at angles that exactly satisfy Bragg's law. In this case, very narrow diffracted beams are obtained. Intermediate between these extremes, the width of diffraction peaks increases with decreasing crystal size. Peak broadening becomes significant as crystal size decreases below 0.5 m. Most cast or wrought metals have sufficiently large grain sizes to justify the assumption that summing occurs over a semi- infinite number of adjacent planes, hence, narrow diffraction peaks are obtained. Phases formed by low-temperature deposition processes and solid state transformations, however, frequently have much finer grains whose sizes can be estimated from the widths of their diffraction peaks (Fig. 9). Fig. 9 Effect of crystallite size on peak width. Source: Ref 1 Crystal defects, such as dislocations and stacking faults, also interrupt the long-range periodicity of the crystal lattice, thus resulting in relaxation of the conditions for diffraction and broadening of diffraction peaks (Fig. 10). Analysis of peak width and shape (the details of the intensity versus diffraction angle data) can be used to obtain information on the densities of such defects in the sample. Fig. 10 Effect of dislocations introduced by cold working and removed by annealing on width of diffraction peaks in brass. Source: Ref 1 Peak width and shape measurements are typically made from the output of an x-ray diffractometer. The results of such analyses are useful as semiquantitative indicators of crystallite size and defect density, but typically lack quantitative precision. Certain alloys exist as random solid solutions (atoms A and B randomly substituting for one another in identical lattice positions) at high temperature, but ordered compounds (atoms A and B arranged in specific nonrandom patterns) at lower temperature (Fig. 11). Such ordering can give rise to additional diffracted beams, thus enabling ordering to be detected and characterized by XRD. [...]... degrees under carefully controlled conditions References 1 2 3 4 R Jenkins and R Snyder, Introduction to X-Ray Powder Diffractometry, John Wiley, 1996 C Barrett and T Massalski, Structure of Metals, McGraw-Hill, 1966 Materials Characterization, Vol 10, ASM Handbook, ASM International, 1986, p 335 Taylor, X-Ray Metallography, John Wiley, 1961, p 444 Microstructural Analysis K.H Eckelmeyer, Microstructural... in this section 3 Materials Characterization, Vol 10, ASM Handbook, ASM International, 1986 References 1 J Goldstein et al., Scanning Electron Microscopy and Microanalysis, 2nd ed., Plenum, 1992, p 85 2 B Gabriel, SEM: A User's Manual for Materials Science, American Society for Metals, 1985, p 110 3 Materials Characterization, Vol 10, ASM Handbook, ASM International, 1986 Surface Analysis K.H Eckelmeyer,... spectroscopy in the article "Bulk Elemental Analysis" and illustrated in Fig 4-6 , wavelength dispersive x-ray analysis has several advantages over energy dispersive analysis: • • • Much better signal-to-noise ratio, which facilitates qualitative analysis and the generation of much higher quality x-ray maps Better separation of x-rays with similar energies (or wavelengths), which enables elements that are... incident x-ray beam In effect, then, the scanning electron microscope provides a mini-x-ray spectrometer with a very fine incident beam that can be used to probe the chemistries of very small operator-selected portions of the sample Many scanning electron microscopes are equipped with an energy dispersive x-ray detector The operation and characteristics of EDS detectors are described in the section on x-ray... features of interest (Fig 9) Fig 9 Backscattered electron image of 82%Au-15.5%Ni-1.75%V-0.75%Mo active braze alloy joined to MoAl2O3 cermet Accompanying x-ray spectrum obtained from area 1 shows the phase forming at the interface to be rich in nickel and vanadium Courtesy of Bonnie McKenzie, Sandia National Laboratories If desired, a single x-ray energy can be selected corresponding to an element of interest,... number of electrons that can be collected in a line-of-sight fashion Backscattered electron images display portions of the sample surface with relatively high average atomic number as bright; portions of the sample surface with lower average atomic number appear darker (Fig 8) Fig 8 Backscattered electron image showing atomic number contrast in as-cast Cu-Al-Mg alloy Brightest areas represent phases containing... electronics provide a histogram of the x-ray energies emitted from the sample As with bulk x-ray spectroscopy, the characteristic x-ray energies observed tell which elements are present, and the relative intensities of the various characteristic x-ray peaks provide information on the relative concentrations of the elements present, as illustrated in Fig 3 of that same article X-ray spectra can be collected either... levels involved Transitions involving inner shell electrons often result in the generation of x-ray photons This is identical to the atomic process described in the section on x-ray fluorescence spectroscopy in the article "Bulk Elemental Analysis." As in x-ray fluorescence spectroscopy, the energies of these x-rays can be compared to the known characteristic energies of each element, enabling the atoms... x-ray maps can provide exceptionally useful information on how various elements are distributed in the sample, particularly when taken in conjunction with secondary electron or backscattered electron images However, because energy dispersive x-ray detectors are characterized by a moderate level of background noise, the x-ray maps they generate are not as high in quality as ones generated when the x-rays... wavelength dispersive crystal spectrometers (See Fig 4-6 in the section "X-Ray Fluorescence Spectroscopy" of the article "Bulk Elemental Analysis" for discussion of the strengths and weaknesses of these different types of x-ray detectors.) Scanning electron microscopes are occasionally equipped with a wavelength dispersive detector to facilitate x-ray mapping, but more often such mapping is done using . Introduction to X-Ray Powder Diffractometry, John Wiley, 1996 2. C. Barrett and T. Massalski, Structure of Metals, McGraw-Hill, 1966 3. Materials Characterization, Vol 10, ASM Handbook, ASM International,. Introduction to X-Ray Powder Diffractometry, John Wiley, 1996 2. C. Barrett and T. Massalski, Structure of Metals, McGraw-Hill, 1966 3. Materials Characterization, Vol 10, ASM Handbook, ASM International,. Cr 2 Hf 1 5-9 2 2.16 x 2.54 7 1.38 2 4 .14 2 1.46 2 1.27 2 2.07 1 0.93 1 cF24 Co 2 Ho 2 9-4 81 2.16 x 2.54 6 1.38 2 2.07 2 1.27 2 1.46 2 4 .14 1