In the Laboratory
X-ray Diffraction Investigation of Alloys R. A. Butera and D. H. Waldeck Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260 X-ray diffraction study of solid materials and single crystals is a primary structural tool in chemistry, physics, biology, and materials science. This probe of structure is perhaps the most clear cut; however, its use is limited to cases in which crystals can be obtained from the material under investigation. This restriction has limited the application of X-ray studies in fields such as biology. At a fundamental level, the phenomenon of Xray diffraction is a clear demonstration of the wave nature of light and the discrete nature of matter. For these reasons, it is important that undergraduate students be exposed to this structural tool. In this laboratory experiment, students use the Xray diffraction of metallic powders to determine the lattice parameter of a cubic lattice. Specifically, they investigate the metals Ni and Cu. In addition to studying these two pure metals, they prepare a physical mixture of the metals and solid solutions (alloys) of the two metals at three different stoichiometries. They use a coldboat induction furnace to prepare the metal alloys. From the powder diffraction patterns, students are able to determine the lattice parameter for the pure metals and the alloys. The comparison of the diffraction pattern for the physical mixture with the other diffraction patterns provides a dramatic illustration of the distinction between a mixture and a homogeneous solution. This experiment helps students to draw connections between aspects of physical chemistry and materials science, and it provides them with experience in the preparation of materials. Background The form of an X-ray diffraction pattern is determined by the crystal structure, via the Bragg condition n λ = 2d sin θ
(1)
where n is an integer, λ is the wavelength of the radiation, d is the distance between atomic layers, and θ is the diffraction angle. If n = 1, the reflection is said to be first order. If n = 2, it is termed second order, and so on. Note, however, that a second-order reflection for a given spacing d is at the same angle as a first-order reflection from planes of spacing d/2. Thus second-order reflections from (100) planes of a simple cubic crystal should be indistinguishable from first-order reflections from a hypothetical set of (200) planes. It is more convenient to treat the order of a reflection in this alternative way and write the Bragg condition as λ = 2dhkl sin θ
(2)
where dhkl is the distance between the hkl planes of the lattice. The integers h, k, and l are the Miller indices. A more detailed discussion of these issues may be found in standard sources (1–3). Clearly the diffraction pattern depends on the arrangement of atoms in the crystal. The shape of a crystal can be classified by defining its three
axes and the angles between its faces. Any crystal falls into one of seven crystal systems (1–3), and these systems are defined by the length of the axes (a, b, c), the angles between faces (α, β, γ), and the corresponding formula for dhkl. For example, a crystal that is described by three mutually perpendicular axes of the same length is a member of the cubic system—the case studied here. Knowledge of the crystal system is not all that is needed to describe the X-ray diffraction pattern, however. Two other important factors are (i) the local symmetry of the molecules or atoms that constitute the lattice points and (ii) the scattering factor for the different atoms, or molecules, comprising the lattice. The local symmetry of the atoms places restrictions on the values of h, k, and l that are seen in the diffraction pattern. For example, it is found that all integer values of h, k, and l are allowed for a simple cubic lattice. In contrast, for a face centered lattice, only those integer values for which h, k, and l are all even or all odd are allowed. Other restrictions may occur for a unit cell containing a large number of atoms or molecules (2, 3). The Bragg condition determines the angular position of the diffraction lines but the intensities of the lines will depend on the scattering factors of the atoms that comprise the unit cell. The form of the atomic scattering factor f is shown qualitatively in Figure 1. Note that f depends on not only the atomic number, but also on θ and λ. Because the different atoms are in distinct planes of the lattice, the constructive and destructive interference discussed for the simple Bragg picture is not complete. This partial interference creates variations in the diffraction peaks’ intensities and may lead to either the presence or the absence of certain peaks. The intensity of an (hkl) reflection is given as I(hkl) ~ F(hkl), where F(hkl) is the crystal structure factor, such that
F(hkl) = Σ f j exp 2πni hx j + k yj + lz j j
(3)
where n specifies the order of the diffraction and (xj, yj, z j) is the position of atom j in the unit cell,given in units of the lattice parameters: e.g., xj = r j/a where r j is the spatial coordinate of xj. For more exposition of the topics presented here see refs 2 and 3. For the experiment described here, students are provided with a computer program to fit the observed diffraction patterns. This program fits patterns for the different crystal systems, but it does not include the atomic form factors or the local symmetry effects. For this reason the students’ fits may have extra peaks that are not present in the observed data. These latter influences could be implemented in the fitting program if more computational power were available.
Intermetallic Solutions When metal atoms form a solid they usually adopt one of several regular packing structures: namely, body-
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centered cubic (bcc), primitive cubic, face-centered cubic (fcc), and hexagonal–closest packed (hcp). The fcc (atoms adopt a packing sequence of ABCABCABC…) and hcp (atoms adopt a packing sequence of ABABAB…) lattices are the most widely found structures. These primary packing lattices are described in a variety of undergraduate textbooks; for example, see refs 1 and 4. In any packing arrangement of atoms that nearly touch, empty spaces, called interstitial sites, will occur. Two common types of interstitial site are found for closepacked solids. First, tetrahedral sites (see Fig. 2A) are formed if three atoms are “capped” by one atom. Second, the three atoms may be capped by three more atoms (Fig. 2B) to form an octahedral site. The presence of these interstitial sites in the lattice is important in determining the structure of binary (and higher) solids and has an important impact on the properties of solids (e.g., diffusion, reactivity) and the strength of metals. When two different metals (M and M9) are mixed, two types of alloys may be created, either single-phase or twophase. The type of alloy formed is largely determined by the relative sizes of the atoms, but is also influenced by the chemical bonding between atoms. If the atomic radii of M and M9 are nearly the same, the lattice sites can be interchanged with little change in the lattice energy arising from the change in packing. Clearly, however, the lattice energy will change from changes in the strength of the bonding. In the case where the bonding is also similar, atoms can be interchanged at random with little change in the system’s binding energy (i.e., the metals are miscible). This situation corresponds to the case of a substitutional alloy. The atomic radii must be equal to within ca. 15% to achieve this substitution without destabilizing the primary lattice. This criterion is one of the Hume–Rothery Rules (5–7). Nickel’s atomic radius is 1.24 Å and copper’s is 1.28 Å. These atoms form a continuous series of alloys, or solutions, whose percentage composition ranges from pure copper to pure nickel; in other words, the atoms are miscible in all proportions (see Fig. 3). In contrast, if an atom in the alloy mixture is much smaller than the larger primary atoms, these smaller atoms may occupy the interstitial sites of a given type in the larger atom’s lattice. If the smaller atom’s radius is 22–41% of the larger atom’s radius it can occupy the tetrahedral sites; if it is 41–73% of the larger atom’s radius it must occupy the octahedral sites. Sometimes the interaction of two dissimilar metals is so electronically favorable that discrete stoichiometries are obtained (e.g., MgZn2, Cu3Zn, Na5Zn21). Such materials are called intermetallic compounds.
nents. First is the preparation of the sample using the cold boat RF induction furnace. Second is measurement of the powder diffraction pattern of the samples. Typically, students study six samples in two 4-hour laboratory periods: pure copper (Cu), pure nickel (Ni), a physical mixture of nickel and copper (nearly 1:1), an alloy that is 0.25 mole fraction copper, an alloy that is 0.50 mole fraction copper, and an alloy that is 0.75 mole fraction copper. The first three samples do not require use of the induction furnace. Students prepare the three samples first and then begin taking their X-ray diffraction spectra. Each X-ray spectrum takes about 30 min on a Diano XRD-5 X-ray diffractometer. While a diffraction pattern is being obtained the students prepare the alloy samples. SAFETY PRECAUTIONS: The primary safety items in this experiment are the precautions for avoiding exposure to X-rays and the power supply for the cold boat induction furnace, which can be lethal.
Preparation of Pure Metals and Physical Mixtures Students are provided with cylindrical pieces of copper rod and nickel rod. They use a hand file to generate metal filings of the bulk metal specimens for the diffraction measurements. Portions of each of the pure metals are retained as samples. In addition, a fraction of the Ni filings is placed into a vial with Cu filings and the filings are mechanically mixed by shaking. This procedure creates a physical mixture of the two metals. Xray slides are prepared by mounting the metal filings onto a microscope slide using silicone grease (e.g., Dow Corning High Vacuum Grease) as a bonding agent.
Figure 1. The qualitative dependence of the diffraction intensity (i.e., form factor) on the angle and atomic number.
Preparation of Alloys
The Experiment In this experiment, students prepare several alloys of Cu and Ni by the cold boat induction method (8). The phase diagram and structural information for this alloy may be found in ref 6. The phase diagram (see Fig. 3) is relatively simple and one need not be concerned about the formation of intermetallic compounds or eutectic materials. Typically, students are asked to prepare three solutions of differing stoichiometry (specifically, Ni mole fractions near 0.25, 0.50 and 0.75). From the influence of the alloy’s composition on the X-ray diffraction pattern, they are able to probe selected regions of the phase diagram. It should be clear from their data that the alloy formed is a substitutional alloy (single phase), rather than a two-phase alloy. The laboratory procedure has two major compo-
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Figure 2. This figure illustrates the presence of tetrahedral holes (A) and octahedral holes (B) in a body centered cubic cell. The plus sign marks the location of the hole, and the filled circles show its nearest neighbor sites.
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In the Laboratory
First the students cut single pieces of copper and nickel, so that the materials will contain 0.25, 0.50, and 0.75 mole fraction of copper after they are reacted. Typically, the samples are cut from the metal rod using a hacksaw and range in thickness from about 0.1 in. to 0.5 in. The two metal pieces (Cu and Ni) are fused by radio frequency (RF) induction melting using the cold boat technique. In this method the sample is placed in a water-cooled copper vessel fabricated to allow it to act as the secondary coil of a transformer. The transformer is coupled to the RF generator so as to produce a levitating force on the sample. This levitating force allows the sample to be reacted and/or melted without contamination; that is, the reaction occurs with the molten sample floating in free space. It is the RF energy that heats the sample to the melting point. This technique can readily achieve temperatures exceeding 2000 K. Efficient mixing of the components is accomplished by the eddy currents induced in the sample. The entire procedure is performed under an argon atmosphere to prevent the formation of oxides. After the fusion process, metal filings are produced from the metal alloy ingots, and the X-ray studies are performed. This procedure is the same as described for the pure metals.
X-Ray Spectra The students measure powder patterns for all six samples using a Diano XRD-5 X-ray diffractometer, which is interfaced to a Leading Edge (IBM compatible) computer. During the scan, the computer collects the diffracted X-ray intensity at equally spaced divisions of 2θ and records these data, as well as other relevant information, into a file that is named by the user. Figure 4 shows an X-ray scan for an alloy that is approximately Ni3Cu in stoichiometry. A second program, Xrayfit, is provided so that students can plot their spectra and then generate a stick spectrum using the equations for the crystal classes. This program is written in a general manner so students may fit using any of the seven crystal classes; hence they may adjust the three lengths (a, b, c) and the angles (α, β, γ). For the cubic system, which is appropriate for this experiment, students use a literature value for the lattice parameter as a starting value and optimize the fit by adjusting the guessed lattice parameter. Note that this program calculates all of the allowed lines; however, it does not contain any intensity information, so that the students may in fact not observe all the calculated lines.
Figure 3. Phase diagram of the Cu/Ni alloy in the solid, liquid, and biphasic region.
Among other things, students are asked to explain the absence of these lines in the laboratory report. Figure 5 shows a fit to the diffraction pattern of Ni, over an angular range of 40–80°. The best fit lattice parameter is 3.52 Å, in agreement with the literature value (9). Results and Discussion The diffraction patterns shown in Figures 4 (Ni3Cu) and 5 (Ni) are representative of the type observed in this experimental exercise. For the pure metals and the alloys, the diffraction pattern changes between samples by the value of the spacing between the peaks, hence the lattice parameter of the solids. As one proceeds from pure Ni to pure Cu, the diffraction peaks evolve continuously from that of pure Ni to that of pure Cu. The observation that the diffraction patterns have the same overall structure (just different peak positions) is consistent with the fact that Cu and Ni form a homogeneous substitutional alloy. In contrast, if the metals form specific intermetallic compounds or a two-phase alloy, one would expect to observe new spectral features (diffraction peaks) or a superlattice, respectively (7). Figure 6 plots the measured lattice parameter as a function of the mole fraction of Ni. The data shown in this figure correspond to reported literature values (diamonds) and values obtained by students using the above procedure (squares). The linear relationship between the mole fraction and the value of the lattice parameter is evident in the plot. In addition, good agreement between student measurements and literature values is observed. The diffraction pattern of a physical mixture (see Fig. 7) stands in stark contrast to the spectra observed for the pure elements and their alloys. This figure shows a superposition of the two line patterns associated with the pure metals. The independence of the two lattices (Ni and Cu) is clearly evident in these data. To test this independence the students can prepare physical mixtures whose stoichiometry varies. They will observe that the relative line intensities change, but not the line positions. These observations provide direct evidence of the difference between a single-phase alloy and a mixture. Variations on the Theme This experiment may be developed in a number of interesting ways. First, an alternative approach to the experiment described here, which requires less expensive equipment, is outlined. Second, richer experiments, in which intermetallic compounds and more involved phase diagrams could be studied, are proposed. For the type of diffraction patterns studied here (cubic systems) a Tel-X-emeter TYPE 540M diffractometer (Central Scientific) would suffice. Clearly the data reduction would be more tedious, since the data are recorded on photographic film. The RF induction furnace allows high-melting-point systems, such as Cu and Ni, to be easily investigated. However, one could use normal furnace/crucible methods to prepare the Cu/Ni alloy. An additional system is Sn and Pb, which forms a eutectic at 16.1% Pb. The Sn and Pb alloy (6) can be prepared at a temperature of ca. 300 °C, which is readily attainable in a muffle furnace or tube furnace. The phase diagram of this system is available (6). For a system of this sort, one would always observe a mixture of the diffraction patterns for each elemental component. Another variation on this theme is the study of a system in which a well defined intermetallic compound with a well defined stoichiometry is formed. For example, the Mg and Cu system forms two distinct intermetallic
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compounds: MgCu2, which is fcc, and Mg2Cu, which is fcc orthorhombic (6). A system of this sort would produce a diffraction pattern showing the existence of two phases, the pure component and the intermetallic component. For stoichiometries less than ca. 8 at. % Mg, a solid solution of Mg in Cu is formed. In this region, one should see the Cu diffraction pattern with a lattice parameter that varies with Mg composition. Between 8 at. % and 33.3 at. %, the observed diffraction pattern
would consist of a mixture of the solid solution and MgCu2 compound. From 33.3 at. % to 66.6 at. %, the diffraction pattern will be a mixture of the MgCu2 and Mg2Cu patterns. From 66.6 at. % Mg to 100 at. % Mg one would observe a diffraction pattern that is a mixture of the MgCu2 and pure Mg patterns. For a system of this sort, one would always observe a mixture of the diffraction patterns for each elemental component, except at the exact composition of the intermetallic compounds. A similar experiment could be performed using the Cu–Zn system, which forms a rich set of compounds and is commercially important. The intensity information in the experiments described above is qualitative. By spinning the sample, one can remove preferred orientational effects on the diffraction intensity, thereby permitting quantitative use of the intensity information. The relative intensities in the components of a mixture may then be used to quantify the relative amounts present.
Figure 6. Plot of the lattice parameter of the alloy vs. mole fraction of Ni. Diamond symbols are literature values (7) and square symbols are values obtained using the method reported here. Figure 4. X-ray diffraction scans for Ni3Cu over two angular ranges: 40–80° (top) and 90–100° (bottom).
Figure 5. A fit to the powder diffraction pattern of Ni.
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Figure 7. A powder diffraction pattern for a physical mixture of the two metals Cu and Ni. Note the doubling of the number of peaks, as compared to the diffraction pattern in Figure 4.
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Conclusions This work describes an undergraduate laboratory experiment that integrates aspects of physical chemistry, inorganic chemistry, and materials science. Its focus is the synthesis and structural characterization of metal alloys and intermetallic compounds. Acknowledgments The authors thank the University of Pittsburgh for financial support during this lab development. We also thank the teaching assistants and students enrolled in the Physical Chemistry 1430 course during the 1994 and 1995 academic years for their input. Finally, we thank T. D. Burleigh (Materials Science Department, University of Pittsburgh) for advice. Literature Cited 1. Atkins, P. W. Physical Chemistry, 5th ed.; Freeman: New York, 1994. 2. Sproull, W. T. X-Rays in Practice; McGraw Hill: New York, 1946. 3. Stout, W. T.; Jensen, L. H. X-Ray Structure Determination, 2nd ed.; Wiley: New York, 1989. 4. Shriver, D. F.; Atkins, P. W.; Lanford, C. H. Inorganic Chemistry; Freeman: New York, 1991; pp 108–115. 5. Reed-Hill, R. E. Physical Metallurgy Principles; van Nostrand: New York, 1973. 6. Hansen, M. Constitution of Binary Alloys, 2nd ed.; McGraw Hill: New York, 1958. 7. (a) Darken, L.; Gurry, R. Physical Chemistry of Metals; McGraw Hill: New York, 1953; (b) Mott, N. F.; Jones, H. The Theory of the Properties of Metals and Alloys; Dover: New York, 1958. 8. Coles, B. R. Proc. Phys. Soc. 1952, 65B, 221. 9. Greedan, J. E. AIP Conf. Proc. 1971, 5(Part 2), 1425.
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