Lecture 7: Powder diffraction and chemical analysis

Lecture 7: Powder diffraction and chemical analysis Contents 1 Introduction 1 2 Indexing pattern for cubic crystals 2 3 Indexing for non-cubic systems 5 4 Phase diagram determination 9 5 Super lattice structures 12 6 Qualitative analysis by XRD 15 7 Quantitative analysis by XRD 17 7.1 External standard......................... 17 7.2 Direct comparison......................... 18 7.3 Internal standard method.................... 18 8 Chemical analysis - x-ray fluorescence 18 1 Introduction The original reason for doing x-ray diffraction is to identify the structure of the unknown material. Single crystals are favorable but for most metals, ceramics, intermetallics, single crystals are not necessarily available. For such materials powder patterns from polycrystals are used for identifying the crystal structure. Powder patterns give two useful information 1. The shape and size of the unit cell - this is from the position of the diffraction lines (2θ) 1

2. The arrangement of atoms in the unit cell - this is from the relative intensities of the different lines. To give an example, for a cubic system the lattice constant a determines the values of 2θ for the various planes. The arrangement of atoms in the cubic system, whether simple cubic, bcc, or fcc, determines the relative intensities and the absence and presence of some lines. Thus, given a structure it is easy to calculate the diffraction pattern, especially for simple metals and intermetallics. But doing the reverse (which is what is expected from X-ray diffraction) is not easy. This is especially true for following phase transformations in multi component systems where more than one system, with closely spaced diffraction lines, is present. Finding the different phases and their relative amounts becomes challenging. There are three major steps involved in phase determination. 1. From the angular position (2θ) of the lines we get an idea of the shape and size. We start by assigning a crystal system to the material (out of 7) and based on that calculate Miller indices to the various lines. If they do t fit go back and reassign a new crystal system and iterate. 2. From the density, known chemical composition, and shape and size of the unit cell the number of atoms per unit cell are calculated. 3. Finally, from the intensity of the lines the atom positions are calculated. There are some sources of error in this approach. 1. Lack of truly monochromatic source - if the X-ray is not truly monochromatic, K β lines are also present along with the K α line then there will be extra lines in the diffraction pattern. Usually, these can be minimized by using the appropriate filters. Also, the extra lines have a specific angular relation with the lines from the K α radiation which can be calculated and then eliminated. 2. Impurities in the unknown material - any presence of crystalline impurities in the sample will again cause extra lines. This depends on the specimen properties and can be eliminated by processing. 2 Indexing pattern for cubic crystals A cubic crystal gives diffraction lines where the angle (θ) obeys the following relation sin 2 θ h 2 + k 2 + l = sin2 θ = λ2 = constant (1) 2 s 2 4a2 2

Table 1: 2θ values for Al. First 5 diffraction lines Intensity 2θ (in deg) 100 38.37 45.4 44.61 23.0 64.90 22.7 77.97 6.2 82.14 where a is the lattice constant, λ is the x-ray wavelength, and (hkl) refers to the Miller indices of the plane. Equation 1 is obtained using Bragg s law and the fraction is a constant for diffraction lines from a given x-ray source. Since, hkl are integers s is also an integer and can only take certain values. The values that s can take changes for the different cubic systems (sc, bcc, and fcc) and this is based on the structure factor rules. The problem is finding the values of s for the different 2θ values. The s values for the different cubic systems are simple cubic - All (hkl): 1, 2, 3, 4, 5, 6, 8, 9, 10... bcc - (h + k + l) = even: 2, 4, 6, 8, 10, 12... fcc - (hkl) all even or odd: 3, 4, 8, 11, 12... From the diffraction lines it is possible to calculate the various values of s using equation 1. This information is summarized graphically in figure 1 for the different cubic systems. These can be tried against the different sets for the cubic systems. If there is no match then the system is not cubic. Consider the case of Al. The first five diffraction lines for Al, in order of increasing 2θ are given in table 1. The radiation used is Cu Kα with wavelength 1.54 Å. Using equation 1 it is possible to calculate the values of sin2 θ s taking different values of s for simple cubic, fcc, and bcc. These are tabulated in table 2. From table 2 it is clear that only for the fcc system does sin2 θ s become a constant as indicated in equation 1. Thus, Al crystallizes in a fcc structure. The lattice spacing can be calculated using equation 1 where the value of the constant from table 2 is 0.036. sin 2 θ s = λ2 = 0.036 (2) 4a2 From equation 2 the lattice constant a of Al is 4.06 Å. Given that the density of Al (ρ) is 2.7 gcm 3 it is possible to calculate the number of atoms per unit 3

Figure 1: Calculated diffraction patterns for the various lattices. Taken from Elements of X-ray diffraction - B.D. Cullity. 4

Table 2: 2θ values for Al with corresponding values of sin2 θ s cubic systems for the various sin 2 θ Intensity 2θ (in deg) d Å sin 2 θ sc s bcc fcc 100 38.37 2.343 0.108 0.108 0.054 0.036 45.4 44.61 2.029 0.144 0.072 0.038 0.036 23.0 64.90 1.435 0.288 0.036 22.7 77.97 1.224 0.396 0.036 6.2 82.14 1.172 0.431 0.036 cell, Z. This is given by the relation ρ = Z. at.wt a 3 N A (3) where N A is Avogadro s number. From equation 3 the value of Z is 4. Thus, Al has a fcc structure with 4 atoms per unit cell. 3 Indexing for non-cubic systems Cubic systems are easy to solve, since they have only one lattice constant and the angle are all 90. Things before more difficult if we have non-cubic systems. Usually, graphical methods are used for solving these systems. Consider a tetragonal system with a = b c and all 3 angle 90. The relation between d spacing and the lattice constants for this system is 1 d 2 = h2 a 2 + k2 b 2 1 d = h2 + k 2 2 a 2 + l2 c 2 + l2 c 2 1 d 2 = 1 a 2 [(h2 + k 2 ) + l2 (c/a) 2 ] (4) Taking logarithm on both sides give 2 log d = 2 log a log[(h 2 + k 2 ) + l2 (c/a) 2 ] (5) 5

If there are 2 planes with spacing d 1 and d 2 and Miller indices (h 1 k 1 l 1 ) and (h 2 k 2 l 2 ) then equation 5 modifies to 2 log d 1 2 log d 2 = log[(h 2 1 + k 2 1) + l2 1 (c/a) 2 ] log[(h2 2 + k 2 2) + l2 2 (c/a) 2 ] (6) Equation 6 shows that the logarithm of the difference between d spacing for 2 planes in the tetragonal system depends only on the logarithm of the (c/a) ratio and the Miller indices. It is possible to make log plots of the second term in the right hand side of equation 5 vs. (c/a) for all possible (hkl). This will give a series of curves. The experimental pattern can be superimposed on these curves and the (c/a) value can be obtained. Such types of curves are called Hull-Davey curves. A partial Hull-Davey curve for the tetragonal system is shown in figure 2. A complete one for the body centered tetragonal system is shown in figure 3. Hull-Davey curves can be constructed for different crystal systems taking into account the relation between the lattice constants and the lattice angles. In the case of the hexagonal system the relation between the d spacing and the lattice constants is 1 d = 4 (h 2 + k 2 + hk) + l2 (7) 2 3 a 2 c 2 After a similar manipulation followed for the tetragonal system, this can be rearranged as 2 log d = 2 log a log[ 4 3 (h2 + k 2 + hk) + l2 (c/a) 2 ] (8) A Hull-Davey chart can be constructed similar to than for the tetragonal system to get the lattice constants. As the number of independent lattice constants of the of the crystal increases (length and angles) it becomes more difficult to use the graphical methods. Now, there are computer programs that are used to index patterns by searching and matching with known databases. The powder diffraction patterns for known materials are stored in the ICDD (International Center for Diffraction Data) database. For unknown systems with more than one type of atom in the unit cell we need the intensities of the lines to find the atom positions. This is done by relating then intensities to the structure factor, F, which is related to the atomic scattering factors, and atom positions. This is usually a trial and error process, where an initial structure is assumed and the diffraction pattern calculated. This is matched with the experimental pattern and refinement is carried out to the trial structure. This process is repeated until there is a match. Sometimes, for complex molecules (e.g. organic), single crystals are needed for structure determination. 6

Figure 2: Partial Hull-Davey curve for the tetragonal system with the experimental pattern superimposed. Taken from Elements of X-ray diffraction - B.D. Cullity. 7

Figure 3: Complete Hull-Davey curve for the body centered tetragonal system. Taken from Elements of X-ray diffraction - B.D. Cullity. 8

4 Phase diagram determination Another area where x-ray diffraction is useful is in phase diagram determination. If we want to construct a phase diagram the classical way to do it is thermal analysis followed by microstructure information. But this does not give structure information of the phases. For this we need diffraction. The diffraction pattern for each phase is independent of the other phases. It is also possible to get quantitative information on the relative amounts of the various phases i.e. phase boundaries can also be constructed using XRD. It is also important how the changing composition of the different phases can affect the diffraction patterns 1. If there is solid solubility then as the concentration increases the d- spacing changes. This is because the lattice constant changes. There can either be an increase or decrease in lattice constant depending on the relative atomic sizes of the constituent elements. This leads to a shift in the position of the lines. 2. If there is a 2 phase region then as the concentration change the relative intensity of the different lines changes but there is no change in line position. The following are the principles for collecting x-ray diffraction patterns for phase diagram determination 1. Each alloy must be at equilibrium at the temperature of interest. For high temperature phases not stable at room temperature there are two options for studying the crystal structure. (a) Quench to room temperature and do diffraction. (b) Use x-ray diffraction with high temperature attachment for direct determination. This option is preferred when available for it eliminates the need for preparing large number of samples. 2. The phase sequence: a horizontal line (constant temperature)must pass through a single phase region and 2 phase region alternatively. A line cannot pass from one 2 phase region to the next without passing through a single phase region, can be a line compound. These principles can be understood by looking at figure 4. If we draw a horizontal line then the phases go from single phase α to a mixture of α + γ and then a line compound γ. From γ we again get a two phase region γ + β and then finally single phase β. Within a single phase region as the 9

Figure 4: Phase diagram and lattice constant of a hypothetical binary phase diagram. Taken from Elements of X-ray diffraction - B.D. Cullity. composition changes then line position changes but in the two phase region the relative line intensities change. This information is captured in the series of diffraction patterns for the phase diagram shown in figure 4 and shown in figure 5. In the single phase region where we have a solid solution, these can be of two types 1. Interstitial - when solute atom is much smaller than the solvent atom e.g. C, N, H, B atoms then we can have interstitial solid solutions. Interstitial solid solutions always lead to an increase in lattice parameters. For non cubic structures not all lattice constant change equally. 2. Substitutional - These are of 3 types - random, ordered, or defect. Random and ordered substitutional solid solutions are more common than defect structures. Depending on the relative sizes of the two atoms the lattice constants can increase or decrease. In defect structures the increase in concentration of atom B is accompanied by creating holes where A atoms are present. This is prevalent in compounds that have partial covalent characteristics. They can affect the peak intensities by affecting the structure factor. An example of a defect structure is in NiAl which has a simple cubic structure with Ni atoms at the corner and Al at the center. The phase exists over a composition range 45-61% Ni. For off stoichiometry compositions there will be Ni or Al vacancies in the lattice i.e. defect structures. 10

Figure 5: XRD patterns for different alloys from the hypothetical binary phase diagram in figure 4. Taken from Elements of X-ray diffraction - B.D. Cullity. 11

Figure 6: Ordered and disordered configuration in AuCu 3. Taken from Elements of X-ray diffraction - B.D. Cullity. 5 Super lattice structures These are also called order-disorder transformations. In this a substitutional solid solution that has atoms located at random positions at high temperature transforms into an ordered structure where the different kinds of atoms are located at specific positions. In x-ray diffraction an order-disorder transformation will not affect the positions of the peaks but relative intensities will change. Sometime new peaks are also formed. Ordered structures are also called super lattice structures. The new lines seen in the diffraction pattern are called super lattice lines. The original lines are called fundamental lines. To understand the formation of super lattice lines in XRD consider the example of AuCu 3. The disordered and ordered structure for this is shown in figure 6. AuCu 3 has an fcc structure with 4 atoms per unit cell. From the formula, there are 3 Cu atoms for 1 Au atom. In the disordered structure, the 4 atoms are randomly located in the unit cell while in the ordered structure, Au atoms are located at the corners and the Cu atoms are located at the face center positions. The order-disorder transition temperature for this system is 390 C. Consider the completely disordered structure. The probability of a site being occupied by Au atom is 1 4 while the probability of occupation by Cu is 3 4. Hence it is possible to define an average atomic factor term, f av that is given by f av = 1 4 f Au + 3 4 f Cu (9) where f Au and f Cu are the atomic scattering factors for Au and Cu. The disordered structure can be considered as a regular fcc structure so that the 12

structure factor F hkl is given by F hkl = f av [1 + exp(iπh + k) + exp(iπk + l) + exp(iπl + h)] (10) The structure factor rules for the disordered structure are also similar to a regular fcc lattice i.e. the structure factor F hkl vanishes when hkl are mixed and is non-zero when they are all even or all odd. The difference arises when we have the ordered structure. Now the Cu and Au atoms are located at specific positions and hence the structure factor is calculated by using these specific positions. This gives the structure factor for the ordered structure as F hkl = f Au + f Cu [exp(iπh + k) + exp(iπk + l) + exp(iπl + h)] (11) Using equation 11 we can see that the structure factor does not vanish for certain (hkl). F = (f Au + 3f Cu ) when (hkl) are all even or odd F = (f Au f Cu ) when (hkl) are mixed Thus, the ordered structure has extra diffraction lines, which are called super lattice lines. This can be seen in the case of powder patterns of CuAu 3 in figure 7 where extra lines are visible. Complete order and complete disorder represent the two extremes. In most cases, it is possible to get a mixture of both. In such cases, it is possible to define a long range order parameter, S, given by S = r A F A 1 F A (12) where r A refers to the fraction of A sites occupied by A atoms and F A refers to the fraction of A atoms in the material. In the case of complete order r A = 1 and hence S = 1. In complete disorder r A = F A and S = 0. It is possible to calculate the long range order parameter by comparing the intensity of the super lattice lines with the expected intensity when there is complete order (S = 1). There are certain cases when the super lattice lines have a low intensity and are not visible in the powder pattern. Consider the case of CuZn. The disordered structure is a bcc unit cell with Cu and Zn atoms randomly located either at the corner locations or the body center. In the ordered structure the Cu atoms are located at the corner and the Zn atoms at the center. This is shown in figure 8. In CuZn the order-disorder transformation takes place at 460 C. The disordered structure behaves like a bcc structure with an 13

Figure 7: Powder patterns of (a) disordered (b) partially ordered and (c) completely ordered AuCu 3. Taken from Elements of X-ray diffraction - B.D. Cullity. Figure 8: Ordered and disordered structures in CuZn. Taken from Elements of X-ray diffraction - B.D. Cullity. 14

average atomic scattering factor defined by the average of f Cu and f Zn. In the ordered structure the scattering factor is given by F hkl = f Cu + f Zn [exp(iπh + k + l)] (13) Using equation 13 it is possible to calculate the structure factors for the various values of (hkl) F = (f Cu + f Zn ) when (h + k + l) is even F undamental line F = (f Cu f Zn ) when (h + k + l) is odd Super lattice line Since the super lattice line is given by the difference of the atomic scattering factors its intensity is very weak compared to the fundamental line. Since intensity is directly proportional to the square of the structure factor I super I fundamental = (f Zn f Cu ) 2 (f Zn + f Cu ) 2 (14) For θ = 0 the atomic scattering factors are equal to the atomic numbers so that f Cu = 29 and f Zn = 30. Substituting in equation 14 this gives the ratio to be 3 10 4. It is thus possible for the super lattice lines to be too weak to be detected. Order-disorder transition is an example of long range ordering. It is harder to detect short range order or clustering using x-ray diffraction. 6 Qualitative analysis by XRD X-ray diffraction pattern for a given material does not depend on whether the material is a pure substance or is in a physical mixture. Both qualitative i.e. identification of the crystal structure of the material and quantitative i.e. percentage of the material in a physical mixture analysis are possible. Diffraction patterns provide information on the state of the material but a pure chemical analysis might not be able to do this. In qualitative analysis we are interested in knowing the structure of the unknown material or materials in the given sample. The standard procedure is to compare the unknown diffraction pattern with standard patterns from known materials until there is a match. The problem is given that there are tens of thousands of diffraction patterns and multiple diffraction lines for each pattern doing a match would be almost impossible for an unknown material. It would be better if there is some idea of the constitutive elements so we can narrow down the parameter space for searching. Even then the number of patterns to search could become large. Consider an example 15

Figure 9: Standard ASTM diffraction data card for Molybdenum Carbide. Taken from Elements of X-ray diffraction - B.D. Cullity. of a compound formed by reaction between Ni and Al at elevated temperature. The resultant mixture could be a single intermetallic compound or mixture of intermetallics or mixture of intermetallics and unreacted material. Thus the diffraction pattern of the product has to be checked against Ni, Al, Al 3 Ni, Al 3 Ni 2, AlNi, Al 9 Ni 2, and other Ni-Al intermetallics to find out the constituent phases. Given that there are tens of thousands of diffraction patterns (and the number can only grow) it is a very had indexing pattern to solve an unknown diffraction pattern. Need a system for classifying patterns so that fast indexing is possible. The practice of classifying known diffraction patterns was initiated by Hanawalt and associates in the mid 1930s. Instead of using 2θ and Intensity he used d-spacing and Intensity. The advantage is that the index is independent of the incident x-ray radiation, unlike 2θ. He arranged the patterns in order of decreasing d (or increasing 2θ) and used this to generate a search procedure. Hanawalt started building a collection of known diffraction patterns. This was taken by the Joint Committee on Powder Diffraction Standards (JCPDS). This was later called the International Center for Diffraction Data (ICDD). A standard diffraction data card is shown in figure 9. Earlier printed cards were used for manually searching through the diffraction patterns. Usually the strongest 3 lines were used to characterize each material. This has be replaced by automatic computer searches. Most commercial x-ray diffraction software are preloaded with standard diffraction 16

patterns or with a license to access ICDD diffraction data and come with automatic search and index features. Usually the constituent elements have to be identified but the program will search all possible combinations of the elements to index the patterns. For a single phase search and index is fairly straightforward but it becomes more difficult for mixture of phases. Also, preferred orientation of the material has to be taken into account especially for vapor deposited of grown films where complete polycrystalline samples are might not be available. 7 Quantitative analysis by XRD For quantitative analysis, the intensity of line of a given material in a mixture must be related to its relative amount. The intensity is usually proportional to the relative concentration but it also depends on the absorption coefficient of the mixture. The relation between intensity of a phase (I α )and concentration (c α ) is given by the equation I α = K 1c α µ m (15) where K 1 is a constant which depends on other factors like the Bragg angle and structure factor and µ m is the absorption coefficient of the mixture. The factor K 1 can be eliminated by measuring the intensity with a known standard. There are two approaches for this 7.1 External standard In the case of an external standard we can compare the intensity of the α phase, I α, with the intensity of the same line for the pure material, (I α ) pure. This can be written using equation 15 as I α (I α ) pure = c α (µ m /µ α ) (16) The absorption coefficient of the mixture is still unknown but for a physical mixture µ m can be written as a linear mixture of the absorption coefficients of the components weighted by their relative concentrations. Thus, µ m can be written as µ m = c α µ α + c β µ β (17) Using equations 17 it is possible to rewrite equation 16 as I α (I α ) pure = c α c α (1 µ β µ α ) + µ β µ α (18) 17

Thus knowing the absorption coefficient of the individual components and measuring the line intensity of the pure standard it is possible to calculate the concentration of the α phase in the mixture. 7.2 Direct comparison In the direct comparison method a lien from another phase in the mixture is used to calculate the concentration. The advantage of this method is that multiple spectra, one from the sample and from the standards is not needed. We can also use this technique when pure standard is not possible, for e.g. precipitate phase in an alloy. In the case of direct comparison two line of intensity I α and I β from the 2 phases are used. The ratio of the intensities is given by I α I β = K α K β c α c β (19) The factors K α and K β are dependent on Bragg angle but on other factors that are independent of θ. For more that 2 component systems the calculation of intensities for the individual phases becomes difficult. 7.3 Internal standard method In this method the intensity of the line from the α phase whose concentration is to be measured is compared with the intensity from a known standard material which is mixed with the sample in a specific proportion. The ratio of the intensity of line from the α phase and the standard is linearly related to the weight fraction. Usually, calibration curves are prepared with varying amounts of the α phase and standard. These calibration curves can be compared with the sample with the unknown concentration to know the fraction of the α phase. The standard material has to chosen carefully so that its X-ray lines do not overlap with the x-ray line of the sample so that line intensities can be measured. 8 Chemical analysis - x-ray fluorescence X-rays can also be used for chemical analysis i.e. identification of the constituents (elements) of the unknown material and also the ratios of the constituents. This is based on the concept of x-ray fluorescence. A basic x-ray fluorescence setup is shown in figure 10. The basic idea is that the characteristic x-rays (K α, K β, L α ) are unique for an element and does not depending 18

Figure 10: X-ray fluorescence setup. Taken from Elements of X-ray diffraction - B.D. Cullity. on the chemical nature of the element in the sample. This is because characteristic x-rays are produced when electrons are removed from an inner shell and electrons from an outer shell fall into the inner shell releasing energy as x-rays. Since inner shells usually do not take part in bonding the energies of the lines are insensitive to the chemical state of the atom. A typical fluorescence spectrum from a stainless steel sample is shown in figure 11. X-ray fluorescence does not need a monochromatic x-ray source since only the secondary radiation from the sample is analyzed. The analysis of the secondary radiation can be done by two ways 1. Wavelength dispersive- the output spectrum is analyzed using a crystal with known d-spacing. For each angle (2θ) a certain wavelength (λ) is diffracted and the intensity is measured. 2. Energy dispersive - the output spectrum is analyzed using a Si(Li) counter (cooled to lq. N 2 ) which produces an output current which is proportional to the incident energy. A multi channel analyzer is used to sort out the various pulse heights. The incident radiation can be of two types 1. X-rays - when the incident radiation is x-rays this is called x-ray fluorescence. 2. Electrons - when the incident beam is electrons (could be from an electron microscope) it is called energy dispersive x-ray analysis (EDAX). 19

Figure 11: X-ray fluorescence spectrum from a stainless steel sample. Taken from Elements of X-ray diffraction - B.D. Cullity. 20

In EDAX the Si(Li) or Ge detector is used, which is an energy dispersive detector. The wavelength range of the incident radiation depends on the accelerating voltage. Typical wavelength range is around 0.2-20 Å. X-ray fluorescence technique is complementary with another technique called Auger electron emission. They will be considered in detail when dealing with electron interaction with a material. It is possible to define a fluorescence yield that depends on the atomic number and is complementary to the Auger yield. For light elements (atomic number less than 17, Cl) Auger emission is more common while for heavier elements fluorescence is more common. This limits the ability of x-ray fluorescence to detect light elements. Also, for the light elements the wavelength of the emitted x-rays are large and can get easily absorbed by the atmosphere. Some of this can be eliminated by using an evacuated tube. For qualitative analysis the fluorescence spectra can be compared with known standards to identify the material. Quantitative analysis can be done similar to x-ray diffraction. The intensity depends on the concentration but also on the absorption by the matrix. Sometime matrix enhancement is also possible if the fluorescent lines from one phase can be absorbed by the other phase to enhance its fluorescence. This has to be taken into account when calculating intensities. 21