XRD-Intro

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Figure 1 A modern x-ray diffractometer. The x-ray source with its warning lights is located

on the left, the detector and its liquid nitrogen dewar are on the right, and the sample holder is in the center.

Spreadsheet Applications for Materials Science

Introduction to X-ray Powder Diffraction

Introduction

X-ray powder diffraction is a powerful analytical technique which is widely used in many fields of science and in many industries for applications ranging from basic research to routine quality control. It is used to identify new compounds and crystalline structures, to identify unknown materials in terms of its crystalline structure, and to look for deviations from the perfect structure which may indicate the presence of impurities, strain, the size of the crystals and other fine-scale structural defects.

The principle behind this technique relies on the constructive and destructive interference of x-rays emitted from a sample that had been illuminated by a filtered and focused x-ray beam. If certain conditions are met then the intensity of these x-rays will be strong. These conditions tell us about the spacing between planes of atoms in the crystal structure as well as a host of other details. This tutorial covers the principles of x-ray diffraction with an emphasis on understanding the basics. It starts with the criteria that must be met in order for diffraction to occur, followed by the intensity of the diffracted beams, then secondary issues which alter the diffraction pattern slightly, and finally an example of how x-ray diffraction is used to study materials. When done the student should have a good understanding of what is involved in producing a powder diffraction pattern, how to interpret it, and will have a spreadsheet that can be used in laboratory work.

A Typical X-ray Powder Diffraction System

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Figure 2 A close-up look at an x-ray diffractometer.

Top – soller slits, divergence and scatter slits and $ -filter, Second – solid-state detector, Third – sample and x-ray tube, Bottom – angles and dimensions.

at the sample, a detector that is also pointing at the sample, and two of these three components move so that a range of diffraction angles (22) is scanned. Figure 1 shows a modern computer-controlled diffractometer where the x-ray source (left, moves), detector (right, moves), and a specimen holder (center, does not move), are all enclosed in a cabinet that protects the operator from exposure to x-rays. The top of figure 2 shows the source in more detail, including the warning lights, the $-filter that blocks unwanted x-rays, and a pair of slits that control the divergence of the incident beam. The second image in figure 2 shows a liquid nitrogen cooled x-ray detector which also has a pair of slits (under the cover) that limits the angles from which x-ray can be detected. The third image shows a sample in the center of the goniometer. The x-ray source can be seen in the background, aimed at the sample. The principal dimensions and angles for this system are shown in the bottom image in figure 2. In this system the source and detector are located 250 mm from the center of the goniometer (diffractometer circle). The angle between the source and the horizontal is called T and in this figure is shown at 10 degrees. The detector is also positioned at 2=10 degrees during an experiment both the source and detector can scan angles from 2 to 70 degrees. The sample is located in the center of this circle. Care is taken to ensure that the sample is as close as possible to this center, that it is not tilted, and that it is smooth and flat.

A Typical X-ray Diffraction Analysis

During a typical experiment using the system shown in the figures above, the sample position is fixed while the angles of the source and detector scan a specified range of angles, typically 20 to 90 degrees 22 for the routine analysis of metals and simple compounds, and 10 to 60 degrees 22 for many minerals. In this system the source and detector angles are always equal, T=2, 22=T+2.

The scan rate, the rate at which 22 changes, is chosen to provide the desired quality of results, slower scans providing higher quality than faster, more economical, scans. A quick preliminary scan may

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Figure 3 Screen shot of Scintag’s DMSNT software. The top graph shows the raw data, the

center graph shows plots of the peaks found by the software, and the bottom graph shows the results of the search-match procedure which found the pattern for quartz best matched the data.

take only 5 minutes but the peaks may not be well defined and small peaks may be difficult to distinguish from the background signal. The higher quality scans required for detailed analyses may require 1 or more hours depending on the size and type of sample being analyzed.

Typical results from an x-ray diffraction analysis are shown in figure 3. The top graph shows the raw data, plotted as the intensity of the diffracted x-ray beam as a function of the diffraction angle 22. The peaks indicate angles at which all of the conditions for x-ray diffraction have been met. The middle graph shows which peaks the software found while the bottom graph shows a plot of a diffraction pattern from a database of over 100,000 elements, alloys, and compounds. The software searched this database in only a few seconds, comparing the positions and intensities of the peaks in the measured pattern to those in the database, and found several matches. The pattern which was the closest match, in this case quartz, is shown in this figure.

This analysis illustrates a common application of x-ray diffraction, identifying a substance in terms of its crystalline structure. Many other types of analyses are also possible. Slight shifts of the peaks may indicate the presence of impurities or stress. The shapes of the peaks may be able to tell us about the size of the crystals and the intensities of the peaks can tell us about the concentration of this material if it were a mixture.

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Figure 4 Illustration of the reflection analogy of diffraction.

Figure 5 Fraunhoffer diffraction through a series of

slits. The order of diffraction is determined by which wave fronts form the diffracted beam.

Geometry of Diffraction

Two criteria must be met before a peak in the diffraction pattern can be obtained. The first involves the geometry of diffraction in which the combination of the wavelength of the incident x-ray, the angle between the incident beam and the diffracting plane, and the interplanar spacing of this plane satisfy conditions that will produce constructive interference, or at least not totally destructive interference, of the diffracted beam. Figure 4 illustrates the reflection analogy that is often used to represent this criterion. Rays of the incident beam are in phase when they reach the sample. Some rays “reflect” off the first plane, others off the second and other parallel planes, and will reemerge from the sample. These rays will be in phase if the incident angle and interplanar spacing are such that the path difference 2A traveled by each ray are equal to integral values of the wavelength. The geometric requirements of diffraction are summed up in Bragg’s law

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where n is the order of diffraction, 8 is the wavelength of the x-ray, d is the interplanar spacing and 2 is the incident/diffracted angle.

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The concept of order of diffraction is illustrated in figure 5. Order of diffraction refers to the wave fronts that form the diffracted beam. Note that diffraction can occur at the same angle for 2nd_order

diffraction from slits whose spacing is twice that of the slits that produce 1st_{order diffraction. In}

crystals, the same conditions can be met for 1st_{order diffraction from (200) planes and 2}nd_order

diffraction from (100) planes. In general, the value of n is assumed to be 1 and peaks for the (100) and (200) planes will be found at different angles.

Intensity of Diffraction

Satisfying the conditions defined by Bragg’s law does not guarantee that a diffracted beam will result. There are a number of other factors that determine the actual intensity of this beam, but the one that determines if a diffracted beam of any intensity at all will result is the structure factor. The structure factor, which is discussed in more detail below, basically sums up the contribution that each atom in the lattice makes to the diffracted beam. It turns out that the value of the structure factor is 0 for all but a relatively few cases. Also the structure factor is independent of the size of the lattice, and therefore the d-spacings.

Diffracted Intensity

The intensity of the diffracted beam is determined by a number of factors, including: • the intensity of the source radiation

• the distance from the source

• the area of the sample that is illuminated by x-rays • the density of the material

• the combined contribution of each atom in the crystal to the diffracted intensity • geometric details of the instrument

• the ability of the material to absorb x-rays.

Each of these factors are discussed below, and are represented in equation 2.

I₀ intensity of the incident radiation m mass of an electron

c speed of light

8 wavelength of the incident radiation A cross-sectional area of the incident beam R diffractometer (goniometer) radius v volume of the unit cell

|Fhkl| structure factor

p multiplicity factor 2 diffraction angle

e-2M _{temperature factor (a function of 2)}

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(3) (4) (5) (6) (7) Structure Factor

The structure factor, which describes the interaction of the incident radiation with every atom in the unit cell, depicts a central aspect of the actual mechanism of diffraction in crystals. Briefly, as an x-ray passes by an atom it excites the atom by disturbing the orbits of its electrons. As a result, each atom emits x-rays of the same wavelength, but in all directions. If the time between the emission from one atom and any other atom is just right then constructive interference of these rays will occur. The structure factor sums the contributions of these two, plus all other atoms in the unit cell, to the intensity of the diffracted beam.

The mathematical definition of the structure factor is a Fourier transform. For a given plane in the crystal the structure factor is:

where h, k and l are the Miller indices of that plane, u, v, and w are the position indices of the atoms and N is the number of atoms in the unit cell.

The atomic scattering factor f_N represents the ability of the atom to scatter x-rays, relative to that of a single electron, and increases with increasing atomic number (number of electrons around the nucleus). It is dependent on the angle 2 as shown in appendix 3. The structure factor is similar in the sense that it represents the amplitude of scattering by a unit cell relative to that of a single electron.

Fourier transforms are often expressed in their trigonometric form

where

and

Fhkl is a complex number which can be difficult to work with so it is convenient to express it as the

absolute value |Fhkl| where

Multiplicity Factor

A typical powder sample consisting of 20 :m particles can contain several hundred thousand particles. Assuming these particles are randomly oriented, the probability that any plane will be oriented for Bragg diffraction is proportional to the total number of planes in its family. For the

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cubic system there are 6 (100) planes in the {100} family, 8 (111) planes in the {111} family, 12 (110) planes in the {110} family, and so on. Therefore it will be twice as likely to get diffraction from members of {110} than {100}, for example, and the intensity of the diffracted beam will be twice as strong for the {110}planes, assuming all other factors are equal. The multiplicity factor is equal to the number of members in the family of planes. In some compounds, however, each member of the family may not be structurally equivalent, and depends on the locations of the different types of atoms on these planes.

Polarization

The interaction of the incident x-ray and the orbital electron results in the scattering of x-rays in all directions. These intensity of scattering, however, is not equal in all directions. The reason for this can be understood from an analysis of the interaction of the electric field of the x-ray beam and the electric field of the electron. The intensity of the beam scattered at angle 2 is derived from an analysis of the plane polarized components of the incident beam. The result is

where r is the distance from the electron. Since the terms to the left of the parentheses are constant during an experiment they are usually ignored and only the term

known as the polarization factor, is used. This can be done because it is the relative intensities, not the absolute intensities, that are of interest.

Lorentz Factor

Figure 4 depicts diffraction occurring only in the plane of the page. What this simple illustration does not show is that it is also possible to satisfy Bragg’s condition by rotating the crystal so that the diffracted rays come out of this plane, towards of away from the reader. In a powder or polycrystalline sample there will be many crystals so oriented and the result is a cone of diffracted rays rather than a single line. While this can make it easier to find the diffracted rays, it also effects the intensity of x-rays at the detector. This is because, at a given distance from the sample, the cone at higher diffraction angles has a larger diameter. Since all of the energy of diffraction is in this cone, the larger cone will have a lower energy density which results in lower measured intensities. This is a purely geometric phenomenon that is given by the Lorentz factor

In general, measured intensities at higher diffraction angles are lower than those at lower angles, in part due to the Lorentz factor.

Temperature Factor

As temperature increases the thermal vibrations of the atoms also increases. As a result there is a decrease in intensity because the atoms are no longer on mathematically perfect planes. Instead,

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(11) 0 500 1000 1500 2000 2500 3000 Liner Absorpt ion Coef ficient , 1/ cm

Hematite Quartz Iron Graphite Corundum MgO 1,274

115

2,787

19 161 130

Figure 6 Linear absorption coefficients for selected materials and for Cu-k" x-rays. Note the very high

absorption coefficients for iron and iron oxide.

atoms can be thought of as lying in planar zones whose thickness varies with temperature and the material’s elastic constants, which in turn are related to the melting point. This effect is also related to the d-spacing and is more pronounced for smaller values of d. The temperature factor

is therefore also a function of 2 since smaller values of d are related to higher values of 2.

Absorption Factor

Both the incident and diffracted x-rays are absorbed by the sample, resulting in a lower intensity of the diffracted beam. The absorption factor for a flat specimen, where the incident and diffracted angles are equal, is given by 1/2:.

The value of the linear absorption coefficient : can be as high as a few tens of cm-1_{to several}

thousand for typical solids, as shown in figure 6. This can have a pronounced effect on measured intensities. For example, in a system that utilizes a copper x-ray source and where maximum intensities for quartz are around 15,000 counts per second, the maximum intensity for iron oxide (Fe₂O₃, hematite) is only a few hundred counts per second. Much of this is due to the very high linear absorption coefficient of hematite.

The absorption of x-rays has two other important effects on the diffraction pattern. The first is the depth of penetration and this may become a factor in the analysis of thin films. The second is that since diffraction occurs at and below the surface of the sample the sample will in effect be below the center of the goniometer. The resulting error is similar to sample displacement which causes peak positions to shift slightly, typically a few hundredths of a degree 22. This may not seem like much, but the target 22 accuracy for owners of many diffractometers is only 0.01 degrees 22.

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Spreadsheet Exercises

The following exercises help you explore the principles of x-ray diffraction by performing calculations using the equations presented above. The exercises below start out with relatively simple problems. For these, step-by-step instructions and screen shots of the completed exercise are also given to help one get a good start in building well-organized and readable spreadsheets. Both of these elements, organization and readability, are as important in these exercises as they are in any laboratory report. Several of these are used by later exercises and projects.

1. Diffraction Angle - d-spacing Conversions

Powder diffraction data is usually published as the intensity of peaks for specific d-spacings, rather than the diffraction angle 22. The reason for this is that not everyone uses the same x-ray source, and converting from the published d-spacing to a diffraction angle using the wavelength of the x-ray used on one’s diffractometer is easier than converting from an original diffraction angle, to the d-spacing, to the final diffraction angle. In this exercise you will convert the d-spacings given in published work to the 22 positions expected on a diffractometer that uses different x-ray sources.

Note – Figure 7 shows a screen shot of the spreadsheet created for this exercise. Note how it was laid out so that it is easy to use and easy for someone else to understand how the calculations were done.

• Set up a spreadsheet using the recommended format, including the header, sections for physical constants and conversion factors, as appropriate.

• Set up cells (label, value, and units) where you can enter the name of the material, type of structure, and the wavelength of the x-ray source.

• Create a table where the Miller indices, d-spacings, and intensities from the diffraction data can be entered. Make sure this table can hold data for up to 21 peaks.

• Enter the wavelength for the x-ray source you are interested in, then enter the diffraction data for iron (See appendix 1).

• In the last column of this table calculate the diffraction angles.

• This spreadsheet provides a good foundation for performing additional useful calculations, such as the value of s=sin(2)/8 that is widely used in x-ray diffraction calculations.

• Experiment with different wavelengths and diffraction patterns. How much does changing the wavelength change the positions of the peaks? Consider the peaks at high and low angles when answering this question. Note the range of angles of the peaks from the other materials listed in appendix 1. If you were planning to perform diffraction measurements on similar types of materials, what range of angles would you need to scan?

Report – Write a brief 1-2 page report describing what you have learned and how this spreadsheet can be used in other diffraction and crystallography exercises. Include a printout of your spreadsheet, scaled to fit on one or two pages.

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2. Interplanar Spacings for Any Crystal System

X-ray diffraction allows one to measure the interplanar spacings of any crystalline structure. In this exercise you will build a spreadsheet that can be used to calculate the interplanar spacing for any plane in any crystal system. The exercise will start with the cubic system, the simplest case, but will then be generalized for use with any crystal system. This spreadsheet will be very useful in other exercises and in your own x-ray diffraction work.

Note – Figure 8 shows a screen shot of the spreadsheet created for this exercise. Note how by performing the calculations using a series of tables one can more easily tackle complex calculations or build on previous calculations. This technique is recommended for many of the exercises in this module.

• Set up a spreadsheet using the recommended format, including the header, sections for physical constants and conversion factors as appropriate.

• Set up a simple table where the lattice constants for crystal structures can be entered. Include the unit cell dimensions, angles, and units. Also, include entries for the name of the material and/or its crystal system. Finally, enter the lattice parameters for a typical cubic structure.

Figure 7 This screen shot of exercise 1 shows how even simple spreadsheets benefit greatly from being well-organized

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• Set up a second table where the Miller indices for at least 9 planes can be entered in separate columns. The individual Miller indices should be listed vertically. Enter the Miller indices of members of the {100}, {110}, and {111} planes.

• Add one more row to the second table and in it calculate the d-spacing for each plane. Use the equation for d-spacings in the cubic system (appendix 2).

• Examine the results to make sure they are correct. Experiment with the lattice parameters and Miller indices.

• Replace the equation for the d-spacing for the cubic system with that for any crystal system (appendix 2). This will require additional rows in table 1 for intermediate results such as S11,

S₂₂, unit cell volume, etc.

• Density measurements are often used as a way to verify the results of structure determination measurements. Calculate the density for this structure and compare it to values listing in your data books, periodic table, or other sources.

• Examine the results to make sure they are correct. You should be able to repeat the earlier calculations for the cubic system. Experiment with the lattice parameters and Miller indices. Now test your spreadsheet with orthorhombic and tetragonal systems.

3. Bragg Diffraction Angles

Bragg’s law provides a simple relationship between the wavelength of the x-ray, the d-spacing, and the diffraction angle. A typical diffraction pattern will contain peaks for many planes and even for different x-rays when the source radiation is not adequately filtered.

• Continuing with the spreadsheet from the previous exercise, set up a third table for calculating the diffraction angle 22 for each plane in table 2 and for several wavelengths. The wavelengths should be entered in the first column and the diffraction angles calculated in the same columns as the d-spacings in table 2.

• Examine the effect of the wavelengths on the diffraction angles. For instance, chromium characteristic radiation is used in systems designed to measure residual stresses in steels. How do the angles for low and high indices (h2_+k2_+l2_{) planes change when compared to systems}

that use copper x-ray sources?

• Experiment with the Miller indices to see how the diffraction angles change. For example, for cubic systems, are the diffraction angles for the (100), (010) and (001) planes the same? Is the diffraction angle for the (400) plane four times larger than that for the (100) plane?

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Suggestion – try using your spreadsheet’s string-handling functions to generate a string of the form (hkl) where h, k and l are values in the cells of your spreadsheet. Use the @string function to convert the values to strings and use the concatenate operator & to join them into a single string and to wrap them in the parentheses. This skill will be useful in later exercises.

Figure 8 This screen shot montage shows how the calculations were performed using a series of simple tables. Using

this technique can make it possible to perform complex calculations and a spreadsheet that is easy to read and understand.

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4. Structure Factor

The structure factor is the central term in equation 2 and the only term which can have a value of zero. Calculating it involves summing the contributions of each atom in the lattice and arriving at a single number that represents the scattering ability of the structure. The procedure is not complicated, but it does require several separate steps – calculating a (see equation 4), then b and then |F|2_{, and repeating this procedure for each plane. (Unfortunately the Fourier routines available}

in many spreadsheets cannot be used here.) In this exercise we concentrate on calculating the structure factor for a number of planes and for any structure. The relationship between the reflection rules and the structure factor will then be explored.

Note – This spreadsheet references cells in the previous exercise’s spreadsheet. Note – This spreadsheet can be expanded to included any number of atoms and planes. It will also be very useful in later exercises.

Note – The instructions given below describe each calculation that needs to be performed as well as how to build the spreadsheet. Unlike the first two exercises, however, a screen shot is not provided. The lessons learned during the previous exercises, where you built an orderly spreadsheet that performs fairly complicated calculations, will be important here.

Hint – One of the lessons of the previous, and this, exercise, is that it is always a good idea to break up a complicated equation into smaller parts and then combine these parts to get the final result.

• Set up a new notebook sheet (spreadsheet) in the same notebook used in exercises 2 and 3. Again, use the recommended format which includes a header and sections for physical constants and conversion factors, as appropriate.

• Copy table 1 from the previous exercise. This table will be used to define the dimensions of the unit cell. Enter the lattice parameters for a simple cubic structure such as bcc-iron or fcc-aluminum.

• Create a table that lists, horizontally, data for up to 10 atoms present in the unit cell. The data required for each atom, listed vertically, are the name and/or symbol, the atomic weight, the position indices u,v,w, and the empirical constants that define the sin(2)/8-dependence of the atomic scattering factor (appendix 3). Finally, in the last row, enter a 0 or 1 to indicate that the atom defined in each column is indeed in this structure.

• Calculate the density of the unit cell and compare it to published results, such as a periodic table of the elements, or the value listed with the PDF data in appendix 1.

• In a second table (This table will contain the principal results.) list, in three columns, the three Miller indices of 9 or more planes. Enter the indices for common low and high indices planes, from (100) to (444), for example. In the next 4 columns calculate h2_+k2_+l2_{, d-spacings, and}

diffraction angles (22) for these planes, as well as the quantity sin(2)/8. Add empty columns that will eventually contain the sums of a and b (equations 4-7) and the structure factor |F|2_.

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• Create another table and calculate 2B(ku+kv+lw) for each atom listed in table 1 and each plane listed in table 2. (With this table we are simply splitting up a complicated equation into smaller parts.)

• In the next table calculate the atomic scattering factor for each atom and each plane. Use the equation given in appendix 3 and multiply it by the value in atoms-data table that indicates the presence or absence of the atom in the lattice.

• In the next two tables calculate the real and imaginary terms in the structure factor, a, and b, for each plane listed in table 2.

• Returning to table 2, sum up the values of a and b for each plane and calculate the structure factor |F|2_.

• Examine the results. Experiment with other simple fcc and bcc structures to see if the structure factor agrees with the well known reflection rules for these structures. Compare the diffraction angles with those in the PDF database, if available. They should match those in the PDF file (appendix 1) to better than 0.01 degrees 22.

Suggestion – Your spreadsheet’s logical functions can help make the interpretation of the structure factor in terms of reflection rules easier. For example, the @ISODD and @ISEVEN functions, coupled with the #AND# and #OR# operators, can be used to indicate if the Miller indices for cases where the structure factor is greater than 0 are all even, all odd, or unmixed.

• Experiment with other structures, such as diamond-cubic, halite (NaCl), and others to see if you can find other reflection rules.

• Does your spreadsheet agree with the reflection rules for bcc and fcc structures?

• This spreadsheet performed calculations for a handful of planes and for unit cells that have up to 10 atoms. How easy would it be to expand this spreadsheet to handle unit cells with 40 atoms and for calculations of 100 planes?

• How many members are there in the {100}? {111}? If you wanted to calculate the structure factor for all planes whose indices are 0 and or n, i.e., (-n -n -n) through (n n n), how many planes would you be dealing with when n=1 and when n=4?

5. Lorentz-Polarization Factor

The Lorentz-Polarization factor combines the effects of the interaction of x-rays with orbital electrons and the geometry of the diffractometer on the intensity of the diffracted beam. In this exercise this factor will be plotted so that one can see how much it can change the intensity and at

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which diffraction angles it has its greatest and least influence.

• Set up a spreadsheet using the recommended format, including the header and sections for physical constants and conversion factors as appropriate.

• Perform all of the calculations in a single table which shows the polarization factor, the Lorentz factor and the Lorentz-Polarization factors for 22 from 0 to 180 degrees.

• Plot the results to make it easier to see the trends in these factors.

• Examine the results. Where do the minimum and maximum values for each factor occur? Which factor has the stronger influence on the intensity of the diffracted beam? How would you describe the combined effect of the Lorentz and Polarization factors?

6. Diffracted Peak Intensities

The previous exercises explored various aspects of the geometry and intensity of x-ray diffraction. In this exercise these factors are brought together to calculate the intensities of diffracted beams. It will utilize the spreadsheets created in the previous exercises and will be useful in later exercises.

Hint – This exercise will reference cells from the previous exercises. The tables you set up here will follow the pattern set in those exercises.

• Set up a new notebook sheet (spreadsheet) using the recommended format, including the header, sections for physical constants and conversion factors as appropriate.

• Go back to the spreadsheet from the second exercise and add new specimen parameters – name, mineral name, and chemistry. Set up an identical parameters section on this spreadsheet and reference the data in the cells from the second exercise. Add two more entries in this specimen parameters section for the surface area of the specimen and the linear absorption coefficient. (Assume an initial value of :=10 cm-1_.)

• Create a second parameters section for two additional instrumental parameters – the intensity of the incident radiation and the goniometer radius. Assume a value of 1x109_{for the intensity}

(arbitrary units) and 250 mm for the goniometer radius. To also display the wavelength of the x-ray source here reference the appropriate cell from the second exercise.

• Convert all of these sample and instrumental parameters to a common set of units, such as nanometers.

• In the next table display the planes, d-spacings, angles and structure factors calculated in the previous spreadsheet exercises. Do not copy these values from the previous spreadsheet. Instead, enter their cell addresses.

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• Add another column to this table and calculate the Lorentz-polarization factor. • Add another column and enter the multiplicity factors.

• Finally, calculate the intensity of the diffracted beam followed by the relative intensities. • Compare the calculated intensities to those in the PDF database (see appendix 1). The

agreement should be within a few percent.

• Experiment with other structures, including diamond, halite, and others. Appendix 4 lists the unit cell dimensions and atom positions for several common structures. You may wish to change or add new planes to your spreadsheet. If you do, don’t forget to update the multiplicity factors.

• Does your spreadsheet do a satisfactory job of calculating a diffraction pattern? Is it laid out in such a way that it is easy to understand and easy to use? What would you do to improve it? How can this spreadsheet be used when performing x-ray diffraction experiments?

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Spreadsheet Projects

The above exercises explored a number of aspects of x-ray powder diffraction and resulted in a spreadsheet that can be used to calculate to all or part of diffraction patterns for any crystalline material. The following projects extends this by looking into other aspects of x-ray diffraction, calculating a more realistic looking diffraction pattern, comparing this pattern to measured experimental patterns, and finally analyzing experimental data to see if it is possible to identify an unknown substance given only the diffraction pattern.

1. K_"1 - k"2 Peak Splitting

Consider the possibility that filtering the x-rays used in x-ray diffraction does not quite produce monochromatic radiation. While many systems can effectively filter the k_$ and other parasitic radiations (i.e., W-L_" radiation), the wavelength for k_"2 radiation is so close to that of k_"1 that it is not filters and as a result both will produce peaks, slightly offset, in the diffraction pattern. The k_"2 peaks must be removed during post-processing of the data, but to do this one must know where they are. In many cases the peaks may overlap, especially at low 22, while at higher angles the peaks will separate completely. This exercise looks at this peak splitting and asks you to calculate this peak splitting ()22) as a function of 22.

Set up a spreadsheet using the recommended format, and calculate and plot )22 as a function of 22 (2 to 140 degrees) for the case of Cu-k_"1 and Cu-k_"2 radiation. Note the degree of peak splitting expected at low and high angles. What does your plot illustrate regarding the interpretation of diffraction patterns? How does all this change if a different x-ray source had been used?

2. Fixed Slits Intensity Corrections

A diffractometer’s divergence and receiving slits limit the spread (divergence) of the incident beam that illuminates the sample and the angle of acceptance of x-rays reaching the detector. (See figure 2.) On many diffractometers the slit sizes are fixed, meaning that as 22 increases during a scan the area of the sample illuminated decreases. This will lead to a predictable decrease in peak intensity in the diffraction pattern. This is a relatively simple geometric problem that can be solved and then added to the corrections made to experimental diffraction patterns.

Calculate and plot the area of the sample illuminated by x-rays in a system which uses fixed slit geometry. The diffractometer has a 250 mm goniometer radius, the divergence angle is 2 degrees, the x-ray beam is 12 mm wide, and the sample measures 30 mm x 30 mm.

Hint – this spreadsheet will require the use of logical functions to correctly calculate the length of the beam when, at low angles, is longer than the sample.

Apply these findings to the intensity of peaks in exercise 6. Does this have a significant effect on the calculated intensities? What is the lowest angle for which all of the incident beam is on the sample?

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3. Overlay the Measured and Calculated Diffraction Patterns

In a previous exercise the intensities of selected peaks were calculated and found to be reasonably close to the intensities listed in the PDF files. It would be interesting to see how closely these calculations agree with actual measured diffraction patterns. In this exercise you will import data from an experiment and plot it together with results from a calculated pattern.

Import the data in the file Quartz.txt into this spreadsheet and plot the diffraction pattern. This raw diffraction data contains artifacts which are normally removed before further analyses are performed. One of these artifacts is the background level which can be seen as the intensity in areas where there are no peaks. Remove this background by simply subtracting a constant intensity value from each data point.

Go back to the spreadsheets built during previous exercises, enter the unit cell data for quartz, and calculate the positions and intensities of the peaks in quartz’s diffraction pattern. Add this data to this spreadsheet and to the plot, but do so in such a way that the calculated pattern is represented by thin vertical lines.

How well do the measured and calculated patterns agree? Are all of the peaks accounted for in the calculated pattern? Is this an effective way to represent the agreement between measured and calculated patterns?

While the calculated and measured patterns are not expected to fit perfectly, the agreement between them should be good considering the simplicity of the calculations. Many factors have not been considered, such as the numerous instrumental parameters that produce systematic errors, and the counting statistics that are responsible for what looks like “noise” in the measured pattern. While these factors will be responsible for a number of minor errors, they should not detract from the overall agreement of the calculated and measured patterns.

Report – Write a brief 1-2 page report describing what you have learned about the effectiveness of your computer model and how well it agrees with the measured diffraction pattern. Note the principal differences and explain the more prominent ones. Copy/paste your plots into this report and attach a printout of your spreadsheet, scaled to fit on one or two pages.

4. Calculating a More Realistic Looking Diffraction Pattern

This project will make the calculated diffraction patterns look more realistic by adding width and shape to the peaks. Using the results from the previous exercises or the PDF files listed in appendix 1, calculate the diffraction pattern from 20 to 90 degrees 22 using a step size of 0.02 degrees 22. Use the Cauchy profile to define the shape of each peak. The intensity of a peaks having the Cauchy profile is given by

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where F is the width of the peak and 2B is the Bragg angle. Peak width is a function of 22 and, for

the purposes of this exercise, is given by the expression F=50-0.37@22B.

Hint – Calculate and plot the intensity for each peak using equation 13 then add the intensities of all peaks to get the total intensity and plot the diffraction pattern.

Hint – This spreadsheet will be rather large, up to 4000+ rows long and 25 columns wide, but this will be easy to build since in involves mostly copying one column 21 times. Your computer might start taking longer to recalculate this spreadsheet, which is normal. It helps to wait until the tables are done before adding the graphs.

How does this diffraction pattern compare to the measured pattern imported and plotted in the previous exercise? Add the experimental data to this plot to see how good the agreement is. Plot the difference in intensities against 22 to illustrate this. Try using different x-axis scaling settings to get better views of sections of the diffraction pattern. Experiment with the peak width parameters to get a better fit or to see what the pattern might look like if larger slits had been used (lower 22 resolution).

5. Multiple Peaks from Unfiltered X-rays

In most calculations Bragg’s law the wavelength is a single number, such as 0.154056 nm for the k"1 characteristic radiation of copper. In practice, it is not possible to obtain such highly

monochromatic radiation. Even with filtering, other characteristic radiation plus a portion of the continuous spectrum, will contribute to the diffraction pattern. In this exercise you will calculate the (101) peaks in the diffraction pattern for quartz for all characteristic radiations that may be present in a real experiment. The results will demonstrate what happens when x-rays are not sufficiently filtered and will help in later exercises that involve real diffraction patterns.

Calculate and plot the (101) peaks in the diffraction pattern of quartz using the source radiations listed in the table below. In the plot show each of the four peaks as well as the sum. These calculations should span a 22 range from 23 to 27 degrees and use a step size of 0.02 degrees 22. Use the Cauchy profile for each peak

Characteristic

Radiation Relative Intensity(%)

Cu- k"1 100

Cu-k"2 45

Cu-k$ 30

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Can you see how one Bragg reflection can result in four, or more, peaks in a diffraction pattern? Would you be able to recognize these peaks in real diffraction patterns? Experiment with the peak width formula to see what happens when the peaks become thinner or broader.

Why would tungsten radiation be present in a system which uses a copper x-ray source? (Hint, the intensity of the tungsten peak increases as the x-ray tube ages.)

6. Five-Fingers of Quartz

An interesting feature in the diffraction pattern for quartz is called the “five-fingers of quartz”. This feature consists of five closely spaced peaks which instrument manufacturers often use to demonstrate the 22 resolution capabilities of their diffractometer. This feature is also curious in that five peaks were produced by three Bragg reflections. In this exercise you will investigate this feature and in the process learn more about an important aspect of x-ray diffraction.

Refer to the PDF data in appendix 1 and/or use the spreadsheet from previous exercises, to find the peak positions and relative intensities for the (212), (203) and (301) peaks of quartz for Cu-k_"1 and Cu-k_"2 radiation. Calculate and plot the diffraction pattern over a 22 range from 67 to 69 degrees using a step size of 0.02 degrees 22. Use the Cauchy profile for each peak. (See the previous exercise.) The plot should show the profile for each peak as well as the sum of all peaks.

It should now be clear why there are only five peaks, not three, and not six, in this region of the diffraction pattern. What lesson does this teach that will be important when planning diffraction measurements and analyzing the results?

Report – Write a brief 1-2 page report describing what you have learned about the five fingers of quartz and how this may apply to future diffraction work. Copy/paste your plots into this report and include a printout of your spreadsheet, scaled to fit on one or two pages.

7. Identifying Unknown Materials

The most common application of x-ray diffraction is identifying materials in terms of their crystalline structure. While the procedures employed often utilize the search-match capabilities of a computerized database, simple patterns can be analyzed manually. During this analysis one can determine the Miller indices of each peak, the lattice parameters, and finally the crystal structure. With the lattice parameter and crystal structure is can be relatively easy to identify the unknown material. In this exercise you will import data from three x-ray diffraction scans, locate and determine the peak positions, and then analyze these results to identify the material.

Import and plot the first data file (Unknown1.txt) into this spreadsheet. Determine the maximum intensity in the pattern and scale the data so that the highest peak is the 100% relative intensity peak. Locate each peak and note the 22 position. Compile this data in a table and analyze it to determine the indices for each peak, the crystal structure, and the lattice parameters. Search your text books, data books, and periodic table to find a match, then repeat this procedure for the remaining two data

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files, Unknown2.txt and Unknown3.txt.

Hint – The three data files used in this project are all for pure elements which have cubic crystal structures.

You should have been able to identify all three materials in terms of their crystal structure and lattice parameters. This exercise worked because the reflection rules were easy to identify. In lower symmetry and more complicated structures this task can be very difficult.

Report – Write a brief 1-2 page report describing what you have learned and how this spreadsheet can be used in your laboratory work. Include a printout of your spreadsheet, scaled to fit on one or two pages.

8. X-ray Diffraction of Non-crystalline Materials

Everything in this module so far has dealt with crystalline materials. But what do the diffraction patterns for non-crystalline materials look like? Do they have sharply defined peaks like the patterns for crystalline materials do? Are the no peaks at all since there are no clearly defined planes, and therefore no characteristics interplanar spacings? Import and plot the file GC.txt to find the answer.

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Appendix 1. Diffraction Data from Selected PDF Files

The International Centre for Diffraction Data has been collecting, evaluating, and publishing data from powder diffraction measurements for over 40 years. The database they have put together, called the Powder Diffraction File (PDF), currently contains over 120,000 entries for elements, metals, ceramics, polymers, pharmaceuticals, minerals, virtually every known crystalline material, natural and synthetic. Most entries are from diffraction experiments but recently it has also included calculated patterns.

The data offered in this appendix are from several of ICDD’s PDF files. Only the data pertinent to the exercises in this module are included here. Only peaks whose relative intensity is 1% or greater and a total of up to 18 peaks are listed. The original PDF files also contain fields describing the quality and source of the data, additional crystallographic information, comments on the measurements and preparation of the sample, color, and other data.

PDF # 04-0831

Name: Zinc

Crystal System: Hexagonal

Lattice Parameters: a=0.2665 nm b= c=0.4947 nm

"= $= (=

Density: 7.136 g/cm3

Diffraction Data

d, nm _IntensityRelative (hkl) d, nm _IntensityRelative (hkl) d, nm _IntensityRelative (hkl)

0.2473 53 (002) 0.11729 23 (112) 0.09064 11 (114) 0.2308 40 (100) 0.11538 5 (200) 0.08722 5 (210) 0.2091 100 (101) 0.11236 17 (201) 0.08589 9 (211) 0.1687 28 (102) 0.10901 3 (104) 0.08437 2 (204) 0.13420 25 (103) 0.10456 5 (202) 0.08245 1 (006) 0.13320 21 (110) 0.09454 8 (203) 0.08225 9 (212) 0.12370 2 (004) 0.09093 6 (105)

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PDF # 44-1294

Name: Titanium

"= $= (=

0.2555 25 (100) 0.12776 1 (200) 0.09886 2 (203) 0.2341 30 (002) 0.12481 9 (112) 0.09658 1 (210) 0.2243 100 (101) 0.12324 6 (201) 0.09459 4 (211) 0.17262 13 (102) 0.11707 1 (004) 0.09170 3 (114) 0.14753 11 (110) 0.11215 1 (202) 0.08928 1 (212) 0.13320 11 (103) 0.10643 1 (104) PDF # 06-0696 Name: Iron

Crystal System: Cubic

Lattice Parameters: a=0.28664 nm b= c=

"= $= (=

0.20268 100 (110) 0.11702 30 (211) 0.09064 12 (310)

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PDF # 06-0675

Name: Carbon (Diamond) Crystal System: Cubic

"= $= (=

0.206 100 (111) 0.10754 16 (311) 0.08182 16 (331)

0.1261 25 (220) 0.08916 8 (400)

PDF # 05-0628

Name: Soduim Chloride (NaCl, Halite) Crystal System: Cubic

"= $= (=

0.326 13 (111) 0.1294 1 (331) 0.09491 3 (600) 0.2821 100 (200) 0.1261 11 (420) 0.08917 4 (620) 0.1994 55 (220) 0.1151 7 (422) 0.08601 1 (533) 0.1701 2 (311) 0.1085 1 (511) 0.08503 3 (622) 0.1628 15 (222) 0.09969 2 (440) 0.08141 2 (444) 0.1410 6 (400) 0.09533 1 (531)

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PDF # 46-1045

Name: Silicon Oxide (SiO2, Quartz)

Lattice Parameters: a=0.491344 nm b= c= 0.540524 nm

"= $= (=

0.42549 16 (100) 0.18179 13 (112) 0.13718 5 (301) 0.33434 100 (101) 0.16717 4 (202) 0.12879 2 (104) 0.24568 9 (110) 0.16591 2 (103) 0.12559 3 (302) 0.22814 8 (102) 0.15415 9 (211) 0.12283 1 (220) 0.22361 4 (111) 0.14528 2 (113) 0.11998 2 (213) 0.21277 6 (200) 0.13821 6 (212) 0.11839 2 (114) 0.19798 4 (201) 0.13749 7 (203) 0.11801 2 (310) PDF # 46-1212

Name: Aluminum Oxide (Al2O3, Corundum)

Crystal System: Rhombohedral

"= $= (=

0.34797 45 (012) 0.16015 89 (116) 0.12755 2 (208) 0.25508 100 (104) 0.15466 1 (211) 0.12391 29 (1 0 10) 0.23794 21 (110) 0.15150 2 (122) 0.12343 12 (119) 0.21654 2 (006) 0.15110 14 (016) 0.11931 1 (217) 0.20853 66 (113) 0.14045 23 (214) 0.11897 2 (220) 0.19643 1 (202) 0.13737 27 (300) 0.11600 1 (306) 0.17400 34 (024) 0.13359 1 (125) 0.11472 3 (223)

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PDF # 21-1272

Name: Titanium Oxide (TiO2, Anatase)

Crystal System: Tetragonal

Lattice Parameters: a=0.37852 nm b= c= 0.95139 nm

"= $= (=

0.352 100 (101) 0.14930 4 (213) 0.11894 <2 (008) 0.2431 10 (103) 0.14808 14 (204) 0.11725 2 (303) 0.2378 20 (004) 0.13641 6 (116) 0.11664 6 (224) 0.2332 10 (112) 0.13378 6 (220) 0.11608 4 (312) 0.18920 35 (200) 0.12795 <2 (107) 0.10600 2 (217) 0.16999 20 (105) 0.12649 10 (215) 0.10517 4 (305) 0.16665 20 (211) 0.12509 4 (301) 0.10436 4 (321) PDF # 21-1276

Name: Titanium Oxide (TiO2, Rutile)

Crystal System: Tetragonal

"= $= (=

0.3247 100 (110) 0.14797 10 (002) 0.12006 2 (212) 0.2487 50 (101 0.14528 10 (310) 0.11702 6 (321) 0.2297 8 (200) 0.14243 2 (221) 0.11483 4 (400) 0.2188 25 (111) 0.13598 20 (301) 0.11143 2 (410) 0.20540 10 (210) 0.13465 12 (112) 0.10936 8 (222) 0.16874 60 (211) 0.13041 2 (311) 0.10827 4 (330) 0.16237 20 (220) 0.12441 4 (202) 0.10425 6 (411)

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Appendix 2. Useful Equations

Interplanar Spacing

Cubic:

Triclinic:

Unit Cell Volume

Cubic:

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Appendix 3. Atomic Scattering Factors

The atomic scattering factor is a function of the number of electrons orbiting the nucleus and and decreases with the value of s which is given by

For the purposes of this module the s-dependence is represented by the equation

in which the empirical parameters A0 through A5 were determined by fitting the values listed in

Appendix 10 of B.D. Cullity’s book “Elements of X-ray Diffraction” (Addison -Wesley, Reading, MA, 1967). The table on the following page lists the values of these parameters and the figure below shows the trends and the fit of this equation for selected elements.

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Element A0 A1 A2 A3 A4 A5 Mean Error B+3 _1.50 _-0.150 _1.10 _0.25 _0.70 _0.11 _0.02 B 5.10 1.45 0.00 1.00 0.70 0.15 0.03 C 6.30 1.50 1.00 1.00 0.90 0.14 0.03 O-2 _10.00 _2.10 _5.00 _0.90 _1.26 _0.10 _0.07 Na+ _10.00 _0.00 _3.60 _1.30 _1.60 _0.20 _0.03 Mg+2 _10.00 _0.00 _4.10 _1.25 _1.80 _0.35 _0.01 Mg 12.00 0.00 4.30 1.25 1.80 0.20 0.04 Al+3 _10.00 _0.00 _4.95 _1.15 _2.00 _0.25 _0.00 Al 13.00 0.00 4.60 1.40 2.00 0.30 0.04 Si+4 _10.00 _0.00 _5.50 _1.10 _2.30 _0.30 _0.00 Si 14.00 0.00 5.60 1.25 2.30 0.35 0.05 Cl- _19.27 _2.00 _1.89 _2.93 _3.35 _0.67 _0.01 K+ _18.30 _2.40 _8.50 _1.70 _4.20 _0.50 _0.03 Ti+4 _20.00 _2.50 _8.00 _1.90 _5.05 _0.55 _0.02 Ti 22.00 2.00 7.30 2.50 4.60 0.60 0.04 V 23.00 0.00 7.50 2.80 4.90 0.65 0.04 Cr 24.00 0.00 7.60 3.00 5.10 0.72 0.03 Mn 25.00 0.00 8.00 3.10 5.40 0.75 0.03 Fe 26.00 0.00 8.10 3.40 5.70 0.80 0.04 Co 27.00 0.00 8.10 3.50 5.70 0.80 0.04 Ni 28.00 0.00 8.20 3.75 6.30 0.95 0.03 Cu 29.00 0.00 8.30 3.80 6.60 0.98 0.03 Zn 30.00 0.00 8.40 3.90 6.90 0.90 0.05 Ga 31.00 0.00 8.60 4.30 7.30 0.95 0.04 Ge 32.00 0.00 9.00 4.40 7.60 1.00 0.03 As 33.00 0.00 9.40 4.50 7.90 1.05 0.03 Zr 40.00 3.00 14.00 5.10 10.20 1.40 0.02 Mo 42.00 0.00 16.40 4.80 10.90 1.40 0.02 Ag 47.00 0.00 19.40 5.00 12.70 1.60 0.02 Sn 50.00 0.00 21.00 5.50 13.70 1.80 0.02 La 57.00 0.00 25.00 6.00 16.40 2.00 0.01 W 74.00 0.00 34.00 7.80 24.00 2.20 0.02 Pt 78.00 0.00 37.00 7.90 25.00 2.40 0.01 Au 79.00 0.00 38.00 8.00 25.10 2.45 0.01 Pb 82.00 0.00 40.00 8.00 26.70 2.60 0.01 Bi 83.00 0.00 42.00 8.20 27.20 2.65 0.01 U 92.00 0.00 45.00 9.30 31.00 2.90 0.01 Pu 94.00 0.00 46.00 9.40 32.00 3.00 0.01

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Appendix 4. Unit Cell Parameters for Selected Materials

Unit Cell Parameters for Quartz

System: Hexagonal

Density: 2.649 Mg/m3

a=0.491344 nm "=90 degrees

b=0.491344 nm $=90 degrees

c=0.540524 nm (=120 degrees

Atoms in the Unit Cell

Element u v w Si+4 _0.46987 ₀ _0.6667 Si+4 ₀ _0.46987 _0.3333 Si+4 _0.53013 _0.53013 ₀ O-2 _0.4141 _0.2681 _0.7855 O-2 _0.73190 _0.14600 _0.45217 O-2 _0.85400 _0.58590 _0.11883 O-2 _0.2681 _0.4141 _0.21450 O-2 _0.14600 _0.73190 _0.54783 O-2 _0.58590 _0.85400 _0.88117

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Unit Cell Parameters for Corundum (Al2O3) System: Rhombohedral Density: 3.987 Mg/m3 a=0.47587 nm "=90 degrees b=0.47587 nm $=90 degrees c=1.29929 nm (=120 degrees

Element u v w Al+3 ₀ ₀ _0.35500 Al+3 ₀ ₀ _0.64500 Al+3 ₀ ₀ _0.85500 Al+3 ₀ ₀ _0.14500 Al+3 _0.33333 _0.66667 _0.02167 Al+3 _0.33333 _0.66667 _0.52174 Al+3 _0.33333 _0.66667 _0.31167 Al+3 _0.33333 _0.66667 _0.81167 Al+3 _0.66667 _0.33333 _0.68833 Al+3 _0.66667 _0.33333 _0.18833 Al+3 _0.66667 _0.33333 _0.97833 Al+3 _0.66667 _0.33333 _0.47833 O-2 _0.30300 ₀ _0.25000 O-2 _0.69700 ₀ _0.75000 O-2 ₀ _0.30300 _0.25000 O-2 _0.69700 _0.67900 _0.25000 O-2 ₀ _0.69700 _0.75000 O-2 _0.30300 _0.30300 _0.75000 O-2 _0.63636 _0.66667 _0.91667 O-2 _0.33333 _0.96967 _0.91667 O-2 _0.03033 _0.36363 _0.91667

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O-2 _0.33333 _0.36363 _0.41667 O-2 _0.03033 _0.66667 _0.41667 O-2 _0.63636 _0.96967 _0.41667 O-2 _0.96967 _0.33333 _0.58333 O-2 _0.66667 _0.63636 _0.58333 O-2 _0.36363 _0.03033 _0.58333 O-2 _0.66667 _0.03033 _0.08333 O-2 _0.36363 _0.33333 _0.08333 O-2 _0.96967 _0.63636 _0.08333

Unit Cell Parameters for Rutile (TiO₂)

System: Tetragonal

Density: 4.250 Mg/m3

a=0.45940 nm "=90 degrees

b=0.45940 nm $=90 degrees

c=0.29580 nm (=90 degrees

Element u v w Ti+4 ₀ ₀ ₀ Ti+4 _1/2 _1/2 _1/2 O-2 _3/10 _3/10 ₀ O-2 _4/5 _1/5 _1/2 O-2 _7/10 _7/10 ₀ O-2 _1/5 _4/5 _1/2

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Unit Cell Parameters for Anatase (TiO2) System: Tetragonal Density: 3.893 Mg/m3 a=0.37850 nm "=90 degrees b=0.37850 nm $=90 degrees c=0.95140 nm (=90 degrees

Element u v w Ti+4 ₀ _3/4 _1/8 Ti+4 ₀ _1/4 _7/8 Ti+4 _1/2 _3/4 _3/8 Ti+4 _1/2 _1/4 _5/8 O-2 ₀ _1/4 _0.08160 O-2 ₀ _3/4 _0.91840 O-2 ₀ _3/4 _0.33160 O-2 _1/2 _3/4 _0.58160 O-2 _1/2 _1/4 _0.83160 O-2 _1/2 _3/4 _0.16840 O-2 _1/2 _1/4 _0.41840 O-2 _1/2 _1/4 _0.66840