BRUKER ADVANCED X-RAY SOLUTIONS

BRUKER ADVANCED X-RAY SOLUTIONS Configuration Measuring circle diameter Angle range 360 (Theta and 2 Theta, without additional equipment) Horizontal or vertical, Theta/2 Theta or Theta/Theta geometry (can be reconfigured on-site) Measuring circle diameter 435, 500, and 600 mm predefined, all other intermediate settings possible Max. usable angular range -110 < 2 Theta 168 (depends on extra equipment) Angle positioning Smallest addressable increment 0.0001 Reproducibility ± 0.0001 Stepper motors with optical encoders Maximum angular speed 30 /s (depends on extra equipment) General space requirements Exterior dimensions 2035 x 1400 x 1260 mm 80.12 x 55.12 x 46.90 inch (h x w x d) Weight (without optional electronics) 550 kg Cooling water supply 3.6 l/min, pressure: 4.0 to 7.5 bar (without optional internal with no back pressure, water chiller) Temperature: 10 to 20 C Power supply Single phase: 208 to 240 V, Three phases: 120 V, 230 V, 240 V; 47 to 63 Hz Maximum power consumption (without controllers for optional equipment) Technical Data 6.5 kva BRUKER AXS GMBH Östliche Rheinbrückenstr. 49 D-76187 KARLSRUHE GERMANY BRUKER AXS INC. 5465 East Cheryl Parkway MADISON, WI 53711 USA TEL. (+49) (0) 721 595-2888 FAX (+49) (0) 721 595-4587 EMAIL info@bruker-axs.de www.bruker-axs.de TEL. (+1) (800) 234-XRAY TEL. (+1) (608) 276-3000 FAX (+1) (608) 276-3006 EMAIL info@bruker-axs.com www.bruker-axs.com D8 and DIFFRAC are registered trademarks of the US Office of Patents and Trademarks. All configurations and specifications are subject to change without notice. Order No. B88-E00014 V1 2006 BRUKER AXS Printed in Germany. AIM.DE 01626

Providing Solutions To Your Diffraction Needs. Chapter 7: Basics of X-ray Diffraction Scintag has prepared this section for use by customers and authorized personnel. The information contained herein is the property of Scintag and shall not be reproduced in whole or in part without Scintag prior written approval. Scintag reserves the right to make changes without notice in the specifications and materials contained herein and shall not be responsible for any damage (including consequential) caused by reliance on the material presented, including but not limited to typographical, arithmetic, or listing error. 10040 Bubb Road Cupertino, CA 95014 U.S.A. www.scintag.com Phone: 408-253-6100 Fax: 408-253-6300 Email: sales@scintag.com Page 7.1 Scintag, Inc. AIII - C

TABLE OF CONTENTS Chapter cover page 7.1 Table of Contents 7.2 Introduction to Powder/Polycrystalline Diffraction 7.3 Theoretical Considerations 7.4 Samples 7.6 Goniometer 7.7 Diffractometer Slit System 7.9 Diffraction Spectra 7.10 ICDD Data base 7.11 Preferred Orientation 7.12 Applications 7.13 Texture Analysis 7.20 End of Chapter 7.25 Page 7.2

INTRODUCTION TO POWDER/POLYCRYSTALLINE DIFFRACTION About 95% of all solid materials can be described as crystalline. When X-rays interact with a crystalline substance (Phase), one gets a diffraction pattern. In 1919 A.W.Hull gave a paper titled, A New Method of Chemical Analysis. Here he pointed out that.every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the others. The X-ray diffraction pattern of a pure substance is, therefore, like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases. Today about 50,000 inorganic and 25,000 organic single component, crystalline phases, diffraction patterns have been collected and stored on magnetic or optical media as standards. The main use of powder diffraction is to identify components in a sample by a search/match procedure. Furthermore, the areas under the peak are related to the amount of each phase present in the sample. Page 7.3

THEORETICAL CONSIDERATIONS In order to better convey an understanding of the fundamental principles and buzz words of X-ray diffraction instruments, let us quickly look at the theory behind these systems. (the theoretical considerations are rather primitive, hopefully they are not too insulting). Solid matter can be described as : Amorphous : The atoms are arranged in a random way similar to the disorder we find in a liquid. Glasses are amorphous materials. Crystalline : The atoms are arranged in a regular pattern, and there is as smallest volume element that by repetition in three dimensions describes the crystal. E.g. we can describe a brick wall by the shape and orientation of a single brick. This smallest volume element is called a unit cell. The dimensions of the unit cell is described by three axes : a, b, c and the angles between them alpha, beta, gamma. About 95% of all solids can be described as crystalline. An electron in an alternating electromagnetic field will oscillate with the same frequency as the field. When an X-ray beam hits an atom, the electrons around the atom start to oscillate with the same frequency as the incoming beam. In almost all directions we will have destructive interference, that is, the combining waves are out of phase and there is no resultant energy leaving the solid sample. However the atoms in a crystal are arranged in a regular pattern, and in a very few directions we will have constructive interference. The waves will be in phase and there will be well defined X-ray beams leaving the sample at various directions. Hence, a diffracted beam may be described as a beam composed of a large number of scattered rays mutually reinforcing one another. This model is complex to handle mathematically, and in day to day work we talk about X-ray reflections from a series of parallel planes inside the crystal. The orientation and interplanar spacings of these planes are defined by the three integers h, k, l called indices. A given set of planes with indices h, k, l cut the a-axis of the unit cell in h sections, the b axis in k sections and the c axis in l sections. A zero indicates that the planes are parallel to the corresponding axis. E.g. the 2, 2, 0 planes cut the a and the b axes in half, but are parallel to the c axis. Page 7.4

THEORETICAL CONSIDERATIONS If we use the three dimensional diffraction grating as a mathematical model, the three indices h, k, l become the order of diffraction along the unit cell axes a, b and c respectively. It should now be clear that, depending on what mathematical model we have in mind, we use the terms X-ray reflection and X-ray diffraction as synonyms. Let us consider an X-ray beam incident on a pair of parallel planes P1 and P2, separated by an interplanar spacing d. The two parallel incident rays 1 and 2 make an angle (THETA) with these planes. A reflected beam of maximum intensity will result if the waves represented by 1 and 2 are in phase. The difference in path length between 1 to 1 and 2 to 2 must then be an integral number of wavelengths, (LAMBDA). We can express this relationship mathematically in Bragg s law. 2d*sin T = n *? The process of reflection is described here in terms of incident and reflected (or diffracted) rays, each making an angle THETA with a fixed crystal plane. Reflections occurs from planes set at angle THETA with respect to the incident beam and generates a reflected beam at an angle 2-THETA from the incident beam. The possible d-spacing defined by the indices h, k, l are determined by the shape of the unit cell. Rewriting Bragg s law we get : sin T =?/2d Therefore the possible 2-THETA values where we can have reflections are determined by the unit cell dimensions. However, the intensities of the reflections are determined by the distribution of the electrons in the unit cell. The highest electron density are found around atoms. Therefore, the intensities depend on what kind of atoms we have and where in the unit cell they are located. Planes going through areas with high electron density will reflect strongly, planes with low electron density will give weak intensities. Page 7.5

SAMPLES In X-ray diffraction work we normally distinguish between single crystal and polycrystalline or powder applications. The single crystal sample is a perfect (all unit cells aligned in a perfect extended pattern) crystal with a cross section of about 0.3 mm. The single crystal diffractometer and associated computer package is used mainly to elucidate the molecular structure of novel compounds, either natural products or man made molecules. Powder diffraction is mainly used for finger print identification of various solid materials, e.g. asbestos, quartz. In powder or polycrystalline diffraction it is important to have a sample with a smooth plane surface. If possible, we normally grind the sample down to particles of about 0.002 mm to 0.005 mm cross section. The ideal sample is homogeneous and the crystallites are randomly distributed (we will later point out problems which will occur if the specimen deviates from this ideal state). The sample is pressed into a sample holder so that we have a smooth flat surface. Ideally we now have a random distribution of all possible h, k, l planes. Only crystallites having reflecting planes (h, k, l) parallel to the specimen surface will contribute to the reflected intensities. If we have a truly random sample, each possible reflection from a given set of h, k, l planes will have an equal number of crystallites contributing to it. We only have to rock the sample through the glancing angle THETA in order to produce all possible reflections. Page 7.6

GONIOMETER The mechanical assembly that makes up the sample holder, detector arm and associated gearing is referred to as goniometer. The working principle of a Bragg-Brentano parafocusing (if the sample was curved on the focusing circle we would have a focusing system) reflection goniometer is shown below. The distance from the X-ray focal spot to the sample is the same as from the sample to the detector. If we drive the sample holder and the detector in a 1:2 relationship, the reflected (diffracted) beam will stay focused on the circle of constant radius. The detector moves on this circle. For the THETA : 2-THETA goniometer, the X-ray tube is stationary, the sample moves by the angle THETA and the detector simultaneously moves by the angle 2-THETA. At high values of THETA small or loosely packed samples may have a tendency to fall off the sample holder. Page 7.7

GONIOMETER For the THETA:THETA goniometer, the sample is stationary in the horizontal position, the X-ray tube and the detector both move simultaneously over the angular range THETA. Page 7.8

DIFFRACTOMETER SLIT SYSTEM The focal spot for a standard focus X-ray tube is about 10 mm long and 1 mm wide, with a power capability of 2,000 watt which equals to a power loading of 200 watt/mm2. Power ratings are dependent on the thermal conductivity of the target material. The maximum power loading for an Cu X-ray tube is 463 watt/mm2. This power is achieved by a long fine focus tube with a target size of 12 mm long and 0.4 mm wide. In powder diffraction we normally utilize the line focus or line source of the tube. The line source emits radiation in all directions, but in order to enhance the focusing it is necessary to limit the divergens in the direction along the line focus. This is realized by passing the incident beam through a soller slit, which contains a set of closely spaced thin metal plates. In order to maintain a constant focusing distance it is necessary to keep the sample at an angle THETA (Omega) and the detector at an angle of 2-THETA with respect to the incident beam. For an THETA:THETA goniometer the tube has to be at an angle of THETA (Omega) and the detector at an angle of THETA with respect to the sample. Page 7.9

DIFFRACTION SPECTRA A typical diffraction spectrum consists of a plot of reflected intensities versus the detector angle 2-THETA or THETA depending on the goniometer configuration. The 2-THETA values for the peak depend on the wavelength of the anode material of the X-ray tube. It is therefore customary to reduce a peak position to the interplanar spacing d that corresponds to the h, k, l planes that caused the reflection. The value of the d-spacing depend only on the shape of the unit cell. We get the d-spacing as a function of 2-THETA from Bragg s law. d =?/2 sin T Each reflection is fully defined when we know the d-spacing, the intensity (area under the peak) and the indices h, k, l. If we know the d-spacing and the corresponding indices h, k, l we can calculate the dimension of the unit cell. Page 7.10

ICDD DATA BASE International Center Diffraction Data (ICDD) or formerly known as (JCPDS) Joint Committee on Powder Diffraction Standards is the organization that maintains the data base of inorganic and organic spactras. The data base is available from the Diffraction equipment manufacturers or from ICDD direct. Currently the data base is supplied either on magnetic or optical media. Two data base versions are available the PDF I and the PDF II. The PDF I data base contains information on d-spacing, chemical formula, relative intensity, RIR quality information and routing digit. The information is stored in an ASCII format in a file called PDF1.dat. For search/match purposes most diffraction manufactures are reformatting the file in a more efficient binary format. The PDF II data base contains full information on a particular phase including cell parameters. Scintag s newest search/match and look-up software package is using the PDF II format. Optimized data base formats, index files and high performance PC-computers make PDF II search times extremely efficient. The data base format consists of a set number and a sequence number. The set number is incremented every calendar year and the sequence number starts from 1 for every year. The yearly releases of the data base is available in September of each year. Page 7.11

PREFERRED ORIENTATION An extreme case of non-random distribution of the crystallites is referred to as preferred orientation. For example Mo O3 crystallizes in thin plates (like sheets of paper) and these crystals will pack with the flat surfaces in a parallel orientation. Comparing the intensity between a randomly oriented diffraction pattern and a preferred oriented diffraction pattern can look entirely different. Quantitative analysis depend on intensity ratios which are greatly distorted by preferred orientation. Many methods have been developed to overcome the problem of preferred orientation. Careful sample preparation is most important. Front loading of a sample holder with crystallites which crystallize in form of plates is not recommended due to the effect of extreme preferred orientation. This type of material should loaded from the back to minimize to effect of preferred orientation. The following illustrations show the Mo O3 spectra's collected by using transmission, Debye-Scherrer capillary and reflection mode. Page 7.12

APPLICATIONS Identification : Polymer crystallinity : Residual stress : Texture analysis : The most common use of powder (polycrystalline) diffraction is chemical analysis. This can include phase identification (search/match), investigation of high/low temperature phases, solid solutions and determinations of unit cell parameters of new materials. A polymer can be considered partly crystalline and partly amorphous. The crystalline domains act as a reinforcing grid, like the iron framework in concrete, and improves the performance over a wide range of temperature. However, too much crystallinity causes brittleness. The crystallinity parts give sharp narrow diffraction peaks and the amorphous component gives a very broad peak (halo). The ratio between these intensities can be used to calculate the amount of crystallinity in the material. Residual stress is the stress that remains in the material after the external force that caused the stress have been removed. Stress is defined as force per unit area. Positive values indicate tensile (expansion) stress, negative values indicate a compressive state. The deformation per unit length is called strain. The residual stress can be introduced by any mechanical, chemical or thermal process. E.g. machining, plating and welding. The principals of stress analysis by the X-ray diffraction is based on measuring angular lattice strain distributions. That is, we choose a reflection at high 2-Theta and measure the change in the d-spacing with different orientations of the sample. Using Hooke s law the stress can be calculated from the strain distribution. The determination of the preferred orientation of the crystallites in polycrystalline aggregates is referred to as texture analysis, and the term texture is used as a broad synonym for preferred crystallographic orientation in the polycrystalline material, normally a single phase. The preferred orientation is usually described in terms of polefigures. A polefigure is scanned be measuring the diffraction intensity of a given reflection (2-Theta is constant) at a large number of different angular orientations of the sample. A contour map of the intensity is then plotted as a function of angular orientation of the specimen. The most common representation of the polefigures are sterographic or equal area projections. The intensity of a given reflection (h, k, l) is proportional to the number of h, k, l planes in reflecting condition (Bragg s law). Hence, the polefigure gives the probability of finding a given crystal-plane-normal as function of the specimen orientation. If the crystallites in the sample have a random orientation the recorded intensity will be uniform. We can use the orientation of the unit cell to describe crystallite directions. The inverse polefigure gives the probability of finding a given specimen direction parallel to crystal (unit cell) directions. By collecting data for several reflections and combining several polefigures we can arrive at the complete orientation distribution function (ODF) of the crystallites within a single polycrystalline phase that makes up the material. Considering a coordinate system defined in relation to the specimen, any orientation of the crystal lattice (unit cell) with respect to the specimen coordinate system may be defined by Euler rotation (three angular values) necessary to rotate the crystal Page 7.13

APPLICATIONS coordinate system from a position coincident with the specimen coordinate system to a given position. The ODF is a function of three independent angular variables and gives the probability of finding the corresponding unit cell (lattice) orientation. Polefigure data collection : The systematic change in angular orientation of the sample is normally achieved by utilizing a four-circle diffractometer. We collect the intensity data for various settings of CHI and Phi. Normally we measure all PHI values for a given setting of CHI, we then change CHI and repeat the process. Four circle diffractometer Page 7.14

APPLICATIONS In the polefigure plot below, the PHI values are indicated around the circle. The CHI value changes radically, and are indicated along the vertical bar. The continuous irregular lines in the plot (contour levels) are drawn through values of CHI and PHI that have the same constant value of intensity of the reflection we are measuring. The probability of finding the crystal plane normal for the reflection is proportional to the intensity. 2-D Polefigure display Page 7.15

APPLICATIONS 3-D Polefigure display (1 of 4) Page 7.16

APPLICATIONS 3-D Polefigure display (2 of 4) Page 7.17

APPLICATIONS 3-D Polefigure display (3 of 4) Page 7.18

APPLICATIONS 3-D Polefigure display (4 of 4) Page 7.19

TEXTURE ANALYSIS The determination of the preferred orientation of the crystallites in a polycrystalline aggregate is referred to as texture analysis. The term texture is used as a broad synonym for preferred crystallographic orientation in a polycrystalline material, normally a single phase. The preferred orientation is usually described in terms of pole figures. The Pole Figure : Let us consider the plane (h, k, l) in a given crystallite in a sample. The direction of the plane normal is projected onto the sphere around the crystallite. The point where the plane normal intersects the sphere is defined by two angles, a pole distance a and an azimuth ß. The azimuth angle is measured counter clock wise from the point X. N a = 55 D ß = 210 ( h, k, l) X Let us now assume that we project the plane normals for the plane (h, k, l) from all the crystallites irradiated in the sample onto the sphere. Each plane normal intercepting the sphere represents a point on the sphere. These points in return represent the Poles for the planes (h, k, l) in the crystallites. The number of points per unit area of the sphere represents the pole density. Page 7.20

TEXTURE ANALYSIS We now project the sphere with its pole density onto a plane. This projection is called a pole figure. A pole figure is scanned by measuring the diffraction intensity of a given reflection with constant 2-Theta at a large number of different angular orientations of a sample. A contour map of the intensity is then plotted as a function of the angular orientation of the specimen. The intensity of a given reflection is proportional to the number of hkl planes in reflecting condition. Hence, the polefigure gives the probability of finding a given (h, k, l) plane normal as a function of the specimen orientation. If the crystallites in the sample have random orientation the contour map will have uniform intensity contours. The most common spherical projections are the stereographic projection and the equal area projection. Page 7.21

TEXTURE ANALYSIS The Orientation Distribution Function (ODF) : By collecting pole figure data for several reflections and combining several pole figures we can arrive at the complete orientation distribution function (ODF) of the crystallites within the single polycrystalline phase that makes up the material. Consider a right handed Cartesian coordinate system defined in relation to the specimen. Z Y X In a rolled metal sheet it is natural to choose the x, y and z directions of the sample coordinate system along the rolling direction, transverse direction and normal direction respectively. We also need to specify the crystal coordinate system x, y and z which specifies the orientation of each crystallite in terms of unit cell directions of the crystallites. The crystal coordinate system consist of the same crystal direction in each crystallite, but for each crystallite in the irradiated volume it has a different orientation with respect to the sample coordinate system. We also choose the crystal coordinate system to be the right handed Cartesian and related to the crystal symmetry. Suitable crystal coordinate systems for the cubic and hexagonal systems are shown below. Page 7.22

TEXTURE ANALYSIS The ODF is a function that gives the probability of finding the orientation of the crystallites relative to the sample coordinate system. The orientation of the crystal coordinate system relative to the sample coordinate system can be expressed with three angular values, the so called Euler rotations. Initially the crystal system is assumed to be in a position coincident with the sample system. In the Bunge notation the crystal system is then rotated successively : 1. About the crystal z axis (at this stage coincident with the sample z axis) through the angle PHI 1 (f 1). 2. Then about the crystal x axis through the angle PHI (F). 3. And last about the crystal z axis through the angle PHI 2 (f 2). The angles PHI 1 (f 1), PHI (F) and PHI 2 (f 2) are the three Euler angles which describe the final orientation of the crystal coordinate system (x, y, z ) with respect to the sample coordinate system (x, y, z). Page 7.23

TEXTURE ANALYSIS In the Roe/Matthis notation the crystal system is initially in a position coincident with the sample system. The crystal system is then rotated successively : 1. About the crystal z axis (at this stage coincident with the sample z axis) through the angle PSI (?). 2. Then about the crystal x axis through the angle Theta (?). 3. And last about the crystal z axis through the angle PHI (F). The relations between the two notations ( Bunge & Roe/Matthis) are given by : f 1 =? - p/2, F =?, f 2 = F - p/2 It should be now clear that : The ODF is a function of three independent angular variables, the Euler angles, and represent the probability of finding the corresponding unit cell (crystal lattice) orientation. E.g. in the Bunge notation the orientation of the crystal system has been described by the three angular parameters. It is convenient to plot these parameters as Cartesian coordinates in a three dimensional space, the Euler space. f 1 F1,F,f 2 F f 2 Page 7.24

End of Basics of X-ray Diffraction Page 7.25

19 Jul 04 X-rayDiff.1 X-RAY DIFFRACTION (DEBYE-SCHERRER METHOD) In this experiment, the diffraction patterns of x-rays of known wavelengths will be analysed to determine the lattice constant for the diffracting crystal (NaCl). Theory: In 1912, Max von Laue, a German physicist, discovered that x-rays could be diffracted, or scattered, in an orderly way by the orderly array of atoms in a crystal. That is, crystals can be used as three-dimensional diffraction gratings for x-rays. The phenomenon of x-ray diffraction from crystals is used both to analyze x-rays of unknown wavelength using a crystal whose atomic structure is known, and to determine, using x-rays of known wavelength, the atomic structure of crystals. As mentioned, it is the second application of x-ray diffraction that will be studied in this experiment. The atomic structure of crystals is deduced from the directions and intensities of the diffracted x-ray beams. A crystal is built of unit cells repeated regularly in three dimensions. The directions of diffracted x-rays depend on the repeat distances of the unit cells. The strengths of the diffracted beams depend on the arrangement of atoms in each unit cell. Figure 1 shows the arrangement of Na + and Cl ions in a unit cell of NaCl. Figure 1 One method of interpreting x-ray diffraction is the Bragg formulation. The x-ray waves are considered as being reflected by sheets of atoms in the crystal. When a beam of monochromatic (uniform wavelength) x-rays strikes a crystal, the wavelets scattered by the atoms in each sheet combine to form a reflected wave. If the path difference for waves reflected by successive sheets

19 Jul 04 X-rayDiff.2 is a whole number of wavelengths, the wave trains will combine to produce a strong reflected beam. Figure 2 From Figure 2 it is seen that if the spacing between reflecting planes is d and the glancing angle of the incident x-ray beam is θ, the path difference for waves reflected by successive planes is 2d sin θ. Hence the condition for diffraction (the Bragg condition) is nλ = 2d sin θ where n is an integer and λ is the x-ray wavelength. The atoms of a given crystal can be arranged in sheets in a number of ways; three of the many possible arrangements of sheets in a crystal of sodium chloride are shown in Figure 3.

19 Jul 04 X-rayDiff.3 Figure 3 In these structural diagrams, the crystal axes are denoted by the letters 0A, 0B, 0C, these being the intervals at which the crystal pattern repeats. Since NaCl is a cubic crystal, 0A = 0B = 0C = a = the length of the edge of the unit cell, also called the lattice constant. The equation for reflection (Bragg condition) can be satisfied for any set of planes whose spacing is greater than half the wavelength of the x-rays used (if d < λ/2, then sin θ > 1, which is impossible). This condition sets a limit on how many orders of diffracted waves can be obtained

19 Jul 04 X-rayDiff.4 from a given crystal using a x-ray beam of a given wavelength. Since the crystal pattern repeats in three dimensions, forming a three-dimensional diffraction grating, three integers, denoted h, k, l are required to describe the order of the diffracted waves. These three integers, the Miller indices used in crystallography, denote the orientation of the reflecting sheets with respect to the unit cell and the path difference in units of wavelength between identical reflecting sheets. The wavelength path difference, m, between identical sheets is the greatest common divisor of h, k, l. The sheet orientation is the plane containing the three points found by moving along the 0A axis h/m units, along the 0B axis k/m units, and along the 0C axis l/m units, where one unit, a, is the length of the edge of the unit cell (lattice constant). If any of h, k, l = 0, the sheet is parallel to the corresponding axis. Note that for NaCl the edge of the unit cell (distance, along crystal axes, between identical atoms) is twice the Na Cl separation. The sheets in Figure 3 a) are denoted (200) = 2(100), indicating a path difference of 2λ between the sheet containing 0 and the sheet containing A. This is the first reflection to appear from the planes perpendicular to 0A because of the presence of the intermediate sheet located halfway between 0 and A. If the path difference between the sheet containing 0 and the sheet containing A is (2j + 1)λ, j an integer, destructive interference occurs due to the immediate sheet and no reflection is obtained. In fact for any set of sheets, each sheet of which contains both sodium and chlorine atoms, the Miller indices must be all even for reflection to occur. The other possible type of sheet orientation is to have sheets alternately occupied by sodium and chlorine atoms (see Figure 3 c)). In this case it is possible to obtain a reflection even for path differences of odd numbers of wavelengths between identical sheets (for example those containing only chlorine atoms). The reason for this is that although the reflections from a chlorine sheet and the neighbouring sodium sheet are exactly out of phase for an identical sheet path difference of odd numbers of wavelengths, the strength of the reflection from a chlorine plane is stronger than for a sodium plane, so only partial cancellation occurs. The net result is that for NaCl, reflections occur for h, k, l all even, or all odd, with the odd order reflections being weaker compared to the even orders. From geometric considerations, the separation of consecutive reflecting sheets, d, is given by d = a /2 2 2 h k + m + m m l 2 where, as mentioned before, a = length of edge of unit cell (= twice Na Cl separation) and m = wavelength path difference between identical sheets = greatest common divisor of h, k, l. d = 2 h ma 2 2 2 + k + l

19 Jul 04 X-rayDiff.5 From the Bragg condition, nλ = 2d sin θ where n is the wavelength path difference between consecutive reflecting sheets. The structure of NaCl is such that because of the symmetric spacing of the Na and Cl atoms, the separation between identical sheets is simply twice the separation between consecutive sheets. Therefore m = 2n. d = h na 2 2 2 + k + l and the Bragg condition is λ = h 2a sinθ 2 2 + k + l 2 Note that because of the unequal scattering strengths for Na and Cl, the Bragg condition is less restrictive for NaCl than it is in general, allowing consecutive sheet path length differences of an integral number of half wavelengths for odd Miller indices (h, k, l). The following table contains the first few allowed sets of h, k, l values: Apparatus: h k l 2 2 2 h + k + l h + k + l 1 1 1 3 1.732 2 0 0 4 2.000 2 2 0 8 2.828 3 1 1 11 3.317 2 2 2 12 3.464 4 0 0 16 4.000 3 3 1 19 4.359 4 2 0 20 4.472 4 2 2 24 4.899 3 3 3 27 5.196 etc. 2 2 2 The experiment consists of analysing two x-ray films exposed in a powder diffraction camera. In the powder (Debye-Scherrer) method, the x-rays fall on a mass of tiny crystals in all orientations, and the diffracted beams of each order h, k, l form a cone. Arcs of the cones are intercepted by a film surrounding the specimen. Figure 4 shows a schematic diagram of the

19 Jul 04 X-rayDiff.6 apparatus. The x-ray beam passes through a collimator and strikes the sample (powdered NaCl crystals). A film strip placed along the cylindrical camera wall will be exposed by the scattered radiation and display, after sufficient time and development, the diffraction pattern. The unscattered radiation leaves the camera via the exit port. Figure 4 The Debye-Scherrer method is explained as follows: Figure 5

19 Jul 04 X-rayDiff.7 Consider first a single crystal. Consider one set of reflecting sheets of separation d i. Only one angle, θ, exists for which reflection of the nth order occurs. (Figure 5 a)). Now the crystal is rotated about an axis along the incoming beam direction. The diffracted beam sweeps out a circle as shown in Figure 5 b). This assumes however, that the angle was chosen correctly to get a reflection. To ensure this is so, at least during part of the experiment, the crystal is rotated about an axis perpendicular to the incoming beam, (Figure 5 c)). Thus there will be a ring for each set of sheets in the crystal and each order of reflection from each set of sheets. Finally, there may have been sheets oriented parallel to the paper (Figure 5 d)) and in order to get reflections from them, the crystal is rotated about a third axis perpendicular to the x-ray beam and parallel to the paper. The three rotations result in circular patterns for every sheet separation d i and every order of reflection. Since the three rotations result in the crystal being oriented in every possible position in space, it is equivalent to use powdered crystals which, because of their random orientation, are already oriented in every possible direction. To further ensure randomisation, the powder sample mounting is rotated about axis 2 by an external motor. Thus, for monochromatic x-rays, the diffraction pattern from a powdered sample consists of one circle for each order of each sheet separation d i. The x-rays are produced by bombarding a copper target with electrons. The x-ray spectrum consists of a continuous distribution on which are superimposed the characteristic line spectrum of copper (mainly K α, λ = 0.154 nm, and K β, λ = 0.139 nm). Using a Ni filter, all but the K α radiation is absorbed, providing a reasonably monochromatic x-ray beam. Since the continuous spectrum is of much lower intensity than the characteristic lines, it can be considered low-level background. Procedure and Experiment: Two previously developed x-ray powder diffraction films are provided. These were produced by allowing x-rays from a copper target to strike a NaCl powder sample. One diffraction pattern was obtained using a Ni filter (hence λ = 0.154 nm), the other diffraction pattern was obtained using no filter (hence λ = 0.154 nm and 0.139 nm). The glancing angles corresponding to the rings on the films can be calculated by measuring the ring diameters (s) and knowing the camera radius (R = 2.877 cm) (see Figure 6). Figure 6

19 Jul 04 X-rayDiff.8 Using the calculated θ value and the appropriate λ, and substituting values for a few sets of Miller indices into the equation λ = h 2a sinθ 2 2 + k + l 2 a number of values of a, the lattice constant, are obtained for each diffraction ring. Since a is a constant for a given sample type, the correct indices for a ring are those that yield an a value which agrees with a values obtained for other rings (i.e. one value for a should recur for all the rings when the correct indices for each ring are used in the calculation.) 2 2 2 Note that the correct values of h + k + l will increase with ring diameter for a given λ, and the first ring should have h, k, l = 1,1,1, second ring 2,0,0, third ring 2,2,0, and so on. However, since some of the rings may be of too low an intensity to measure, it is best to use the method described (substitute a few sets of h, k, l and choose the one yielding a constant a), rather than simply assigning indices in order. 1. Measure the diameters of as many diffraction rings as possible for each of the two films. For each ring, also record the relative intensity (faint, light, medium, dark, etc.) 2. Determine the average lattice constant, a, for rings produced by the λ = 0.154 nm x-rays. 3. Determine the average lattice constant a, for rings produced by the λ = 0.139 nm x-rays. 4. Calculate the theoretical value for a, given that a unit cell of NaCl contains 4 NaCl molecules, the molecular weight of NaCl is 58.44 g, and the density of NaCl is 2.165 g/cm 3. (Avogadro s number = 6.022 x 10 23 ). 5. Compare the overall average experimental value and the theoretical value for the lattice constant,a. 6. Comment on the relative intensities of rings corresponding to odd and even indices for a given x-ray wavelength. References: Fretter, Introduction to Experimental Physics, QC 41 Halliday and Resnick, Physics, Part II Harnwell, Experimental Atomic Physics, QC 173 Nuffield, X-Ray Diffraction Methods, QD 945 Woolfson, X-Ray Crystallography, QD 945

22 Sep 04 X-rayDiff.9 PHOTOGRAPHIC DARKROOM TECHNIQUES This experiment is intended to introduce basic film processing and develop good habits in the use of a photographic darkroom. The experiment consists of developing sheet film which has been exposed to a laser diffraction pattern. Since the experiment is concerned only with the experimental technique, and no measurements are made, no report is required. Attach your developed film to a page of your notebook, and submit it with your X-ray Diffraction report. Apparatus: WARNING: LASER LIGHT IS POTENTIALLY DANGEROUS TO YOUR EYES. TO AVOID EYE DAMAGE DO NOT LOOK INTO THE BEAM, AND AVOID VIEWING THE BEAM S DIRECT REFLECTION. The experimental apparatus consists of a He-Ne laser light source, a diffraction slide, and a film holder stand. A cardboard cover with an aperture for the laser beam is provided. Also a black cloth should be placed over the back of the laser source to reduce stray light from its on/off indicator light. A sheet of black cardboard is available to be used as a shutter for the laser light. The laser light source is turned on by a key switch at the back. If the supervisor has not already done so, the laser should be turned on, and the relative positions of the laser, diffraction slide, and film holder stand adjusted to obtain the sharpest possible pattern. A piece of white cardboard placed in the film holder stand can be used as a temporary screen for better viewing of the diffraction pattern. The film holder stand should be roughly 50 cm from the diffraction slide. When the apparatus is correctly positioned, the laser should be switched off. ** All negative film, exposed or not, is to be handled in complete darkness no safelight is permitted with panchromatic film at any time. Before beginning any work in the dark, lock the door! Also, room lights should be switched off using the darkroom controls, rather than the wall switch, since they turn on a warning light in the hallway when the darkroom lights are off. The plastic film holder ( camera ) has two sliding shutters, one on each side, so that two sheets of film may be loaded at once. The film holder must be loaded in TOTAL DARKNESS. The film is placed in the holder with the emulsion toward the object to be photographed. The emulsion side of the film is up when the film is held in the right hand, with the corner notches under the index finger. In the darkroom, keep one side Dry no chemicals or wet hands to be allowed in this area. Use this side for cutting, or loading and unloading film. The other side, or Wet side, will be the sink side where all developing and printing is done.

22 Sep 04 X-rayDiff.10 Set out four trays in order, starting furthest from the sink. These will be: 1. Developer tray 2. Stop Bath tray 3. Fixer tray 4. Hypo-Clearing Solution tray To avoid contamination, always use the same tray for the same solution each time. Use about 1 to 2 cm solution in each tray. Use D-76 developer, indicator stop bath, and fixer solutions direct from stock. The hypo-clear solution is mixed one part stock to 4 parts water. Procedure: - each member of the group is to develop one sheet of film. 1. Set the timer for eight (8) minutes. Place a large tray in the sink under the temperaturecontrolled tap. Turn on tap so that a steady stream flows into the tray. 2. In TOTAL DARKNESS, load the film holder if this has not already been done. Once the holder is loaded, and the film box is closed, the lights may be turned on. 3. Place the holder in the stand. 4. Place the black cardboard shutter over the laser aperture, and turn on the laser. 5. In TOTAL DARKNESS, open the sliding shutter on the film holder. 6. Remove the black cardboard shutter from the laser for an exposure time of 6 seconds, then replace. 7. Turn off the laser. 8. The film is now ready to be developed. Remove the exposed film from the holder and immerse, emulsion side up, in the developer solution. Start timer. Agitate slowly but continuously by gently rocking the tray. 9. When time has expired, remove the film from the developer tray, immerse briefly in water, then place in stop bath solution. Count off 30 seconds, agitating tray twice during this period, then remove film, briefly rinse, and place in the fixer tray. 10. The film is left in the fixer for 2 minutes with agitation every 30 seconds. After the film has been in the fixer for 30 seconds, the lights may be turned on. 11. Remove film and wash in running water at 20 C for one minute, then immerse film in hypoclearing solution. 12. Let film remain in hypo-clearing solution for 2 minutes, agitating every 30 seconds, then remove and wash in running water for 5 minutes. 13. Hang developed film to dry. At the end of the lab, discard all used chemicals in the container that is provided (ask the lab instructor which container to use). Always leave the darkroom the way you would like to find it CLEAN. Ensure all spills are wiped up, counters washed and dried, all used utensils washed and dried, and all stock chemicals properly stored. The developed film is to be submitted, suitably mounted, in your lab notebook.