X-rays are high-energy electromagnetic radiation. They have energies ranging from about eV to 1 MeV, which puts them between γ-rays and ultraviolet (UV) . X-ray Diffraction: A Practical Approach. C. Suryanarayana and M. Grant Norton. Plenum Press, New York and London. xiii + pages. (hardback, $ . teshimaryokan.info: X-Ray Diffraction: A Practical Approach (Artech House Telecommunications) (): C. Suryanarayana, M. Grant Norton: Books.
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XRD Textbook teshimaryokan.info Dewi Kiniasih. X-Ray Diffraction A Practical Approach X-Ray Diffraction A Practical Approach c. Suryanarayana Colorado. Part I presents the basics of x-ray diffraction and explains its use in. watermarked, DRM-free; Included format: PDF; ebooks can be used on all reading devices. X-Ray Diffraction. A Practical Approach Download book PDF Pages PDF · Practical Aspects of X-Ray Diffraction. C. Suryanarayana, M. Grant Norton.
This structure is adopted by diamond one of the crystal- line forms of carbon and the industrially important elemental semicon- ductors silicon Si and germanium Ge. We will show you how this works a little later. For example, you will be able to index diffraction patterns from materials with cubic and hexagonal structures. The structure factor is independent of the shape and size of the unit cell. The vector VI in Fig.
Throughout this text except in Experimental Module 8 we use the nanometer. Gamma-rays and x-rays differ mainly in how they are produced in the atom. As we shall see presently, x-rays are produced by interactions between an external beam of electrons and the electrons in hv v I.. Part of the electromagnetic spectrum. Note that the boundaries between regions are arbitrary.
The usable range of x-ray wavelengths for x-ray diffraction studies is between 0. On the other hand, v-rays are produced by changes within the nudeus of the atom. A part of the electromagnetic spectrum is shown in Fig. Each quantum of electromagnetic radiation, or photon, has an energy, B, which is proportional to its frequency, v: So, using the energies given at the beginning of this section, we can see that x-ray wavelengths vary from about 10 nm to 1 pm.
Notice that the wavelength is shorter for higher energies. The useful range of wavelengths for x-ray diffraction studies is between 0. You may recall that interatomic spacings in crystals are typically about 0.
Electrons are produced by heating a tungsten filament cathode. The cathode is at a high negative potential, and the electrons are accelerated toward the anode, which is normally at ground potential. The electrons, which have a very high velocity, collide with the water-cooled anode. The loss of energy of the electrons due to the impact with the metal anode is manifested as x-rays. A typical x-ray spectrum, in this case for molybdenum, is shown in Fig.
As you can see, the spectrum consists of a range of wavelengths. For each accelerating potential a continuous x-ray spectrum also known as the white spectrum , made up of many different wavelengths, is obtained. The continuous spectrum is due to electrons losing their energy in a series of collisions with the atoms that make up the target, as shown in Fig.
Schematic showing the essential components of a modem x-ray tube. We don't normally use the continuous part of the x-ray spectrum unless we require a number of different wavelengths in an experiment, for example in the Laue method which we will not describe. If an electron loses all its energy in a single collision with a target atom, an x-ray photon with the maximum energy or the shortest wavelength is produced.
When referring to electron energies, we use either eV or keV; but when referring to the accelerating potential applied to the electron, we use V or kV. This process is illustrated in Fig. When this hole is filled by an electron from an outer shell, an x-ray photon with an energy equal to the difference in the electron energy levels is produced.
The energy of the x-ray photon is characteristic of the target metal. The sharp peaks, called characteristic lines, are superimposed on the continuous spectrum, as shown in Fig. It is these characteristic lines that are most useful in x-ray diffraction work, and we deal with these later in the book. X-ray spectrum of molybdenum at different potentials. The potentials refer to those applied between the anode and cathode. The linewidths of the characteristic radiation are not to scale.
Illustration of the origin of continuous radiation in the x-ray spectrum. Each electron. The energy of the emitted photon is equal to the energy lost in the collision. Illustration of the process of inner-shell ionization and the subsequent emission of a characteristic x-ray: If the entire electron energy is converted to that of the x-ray photon, the energy of the x-ray photon is related to the excitation potential V experienced by the electron: The x-ray wavelength is thus he A.
This wavelength corresponds to ASWL; the characteristic lines will have wavelengths longer than ASWL ' The accelerating potentials necessary to produce x-rays having wavelengths comparable to interatomic spacings are therefore about 10 kV. Higher accelerating potentials are normally used to produce a higher-intensity line spectrum characteristic of the target metal. The use of higher accelerating potentials changes the value of ASWL but not the charac- teristic wavelengths. For an applied potential V, the intensity of the K lines shown in Fig.
As you can see in Fig. The different characteristic lines correspond to electron transitions between different energy levels. The characteristic lines are classified as K, L, M, etc. This terminology is related to the Bohr model of the atom in which the electrons are pictured as orbiting the nucleus in spedfic shells. For historical reasons, the innermost shell of electrons is called the K shell, the next innermost one the L shell, the next one the M shell, and so on. If the hole is in the L shell and we fill it with an electron from the M shell we get an La x-ray.
Figure 6 shows schematically the origin of these three different characteristic lines. The situation is complicated by the presence of subshells. For example, we differentiate the Ka x-rays as Ka l and Ka2.
For example, the electron configuration of silicon Si , atomic number 14, is If'2f'2p 6 3f'3p2. The first number is the value of the prindpal quantum number n. The letter s, p, etc.
These values of 1correspond to the 2s and the 2p levels, respectively. All the shells except the K shell have subshells. Let's do an example to illustrate these different transitions for molyb- denum.
The wavelength of the emitted x-rays is related to the energy difference between any two levels by Eq. The energy difference between the Lm and K levels is Using this energy in Eq. It is the presence of these subshells that gives rise to splitting of the characteristic lines in the x-ray spectrum.
This is the wavelength of the Ka l x-rays of Mo. The energy difference between the Lu and K levels is Using Eq. This is the wavelength of the Ka2 x-rays of Mo. Figure 7 shows the x-ray spectrum for Mo at 35 kY. The right-hand- side figure shows the well-resolved Ka doublet on an expanded energy wavelength scale.
However, it is not always possible to resolve separate the Ka l and Ka2 lines in the x-ray spectrum because their wavelengths are so close. If the Ka l and Ka2 lines cannot be resolved, the characteristic line is simply called the Ka line and the wavelength is given by the weighted average of the Ka l and Ka2 lines. Figure 8 shows the complete range of allowed electron transitions in a molybdenum atom. Not all the electron transitions are equally probable.
For example, the Ka transition Le. Weighted Average Sometimes it is not possible to resolve the Ka l and Ka2 lines in the x-ray spectrum. In these cases we take the wavelength of the unresolved Ka line as the weighted average of the wavelengths of its components.
To determine the weighted average, we need to know not only the wavelengths of the resolved lines but also their relative intensities. The Ka l line is twice as strong intense as the Ka2 line, so it is given twice the weight. The wavelength of the unresolved Mo Ka line is thus 1 - 2 x 0. X-ray spectrum of molybdenum at 35 kYo The expanded scale on the right shows the resolved Kal and Ka2 lines.
The important radiations in diffraction work are those corresponding to the filling of the innermost K shell from adjacent shells giving the so-called Kal' Ka2, and Kf3 lines. For copper, molybdenum, and some other commonly used x-ray sources, the characteristic wavelengths to six decimal places are given in Table 2. For most x -ray diffraction studies we want to use a monochromatic beam x-rays of a single wavelength.
The simplest way to obtain this is to filter out the unwanted x-ray lines by using a foil of a suitable metal whose absorption edge for x-rays lies between the Ka and Kf3 components of the spectrum. The absorption edge, or, as it is also known, critical absorption wavelength represents an abrupt change in the absorption characteristics of x-rays of a particular wavelength by a material. The reason for this, and the absence of other transitions, is based on a series of selection rules governing electron transitions.
A detailed description of why these transitions are absent would require us to discuss the Schr6dinger wave equation the famous equation that relates the wavelike properties of an electron to its energy , which is beyond the scope of this book.
Transitions between any shell prindpal quantum number are allowed e. Therefore the LI to K transition is not allowed. However, in most modem x-ray diffractometers a monochro- matic beam is obtained by using a crystal monochromator. For x-ray diffraction studies there is a wide choice of characteristic Ka lines obtained by using different target metals, as shown in Table 2, but, Cu Ka is the most common radiation used.
The Kalines are used because they are more energetic than La and therefore less strongly absorbed by the material we want to examine. The wavelength spread of each line is extremely narrow, and each wavelength is known with very high preci- sion.
You should already be familiar with the term diffrac- N tion" from introductory physics classes. In this section we review some fundamental features of diffraction, particularly as they apply to the use of x-rays for determining crystal structures. First let's consider an individual isolated atom.
If a beam of x-rays is incident on the atom, the electrons in the atom then oscillate about their mean positions. Recall from Section 1. This process of absorption and reemission of electromagnetic radiation is known as scattering. Using the concept of a photon, we can say that an x-ray photon is absorbed by the atom and another photon of the same energy is emitted. On the other hand, inelastic scattering involves photon energy loss.
If the atom we choose to consider is anything other than hydrogen, we would have to consider scattering from more than one electron.
Figure 9 shows an atom containing several electrons arranged as points around the nucleus. Although you know from quantum mechanics that this is not a correct representation of atomic structure, it helps our explanation. We are concerned with what happens to two waves that are incident on the atom.
The upper wave is scattered by electron A in the forward direction. The lower wave is scattered in the forward direction by electron B. The two waves scattered in the forward direction are said to be in phase in step or coherent are other terms we use across wavefront XX' since these waves have traveled the same total distance before and after scattering; in other words, there is no path or phase difference.
A wavefront is simply a surface perpendicular to the direction of propaga- tion of the wave. If the two waves are in phase, then the maximum in one wave is aligned with the maximum in the other wave.
If we add these two waves across wavefront XX' Le. The other scattered waves in Fig. If we add these two waves across wavefront yy', we find that the amplitude of the scattered wave is less than the amplitude of the wave scattered by the same electrons in the forward direction.
The resultant wave is the algebraic sum of the various amplitudes at each point.
This is known as the superposition principle. Figure 10 shows three examples of the superposition of two waves: Since the amplitudes of wave 1 and wave 2 are different, there is some resultant amplitude. If the amplitUdes of waves 1 and 2 are equal, then the resultant amplitude is zero and there is no intensity. Illustration of the superposition of waves.
But as eincreases, the waves become more and more out of phase because they travel different path lengths and, therefore, the amplitude, or f, decreases.
The atomic scattering factor also depends on the wavelength of the incident x-rays. For a fixed value of e, f is smaller for shorter-wavelength radiation. The variation of atomic scat- tering factor with scattering angle for copper, aluminum, and oxygen, is shown in Fig.
The curves begin at the atomic number Z , which for copper is 29, and decrease with increasing values of eor decreasing values of A. The scattered waves from each atom interfere. If the waves are in phase, then constructive interference occurs.
If the waves are out of phase, then destructive interference occurs. A diffracted beam may be defined as a beam composed of a large number of superimposed scattered waves. For a measurable diffracted beam complete destructive interference does not occur. To describe diffraction we have introduced three terms: Scattering is the process whereby the incident radiation is absorbed and then reemitted in differ- ent directions.
Interference is the superposition of two or more of these scattered waves, producing a resultant wave that is the sum of the overlapping wave contributions. Diffraction is constructive interference of more than one scattered wave. There is no real physical difference between constructive interference and diffraction. At the time Young may well not have realized that the phenomenon he observed would have application to other forms of electromagnetic radiation, and certainly he was not aware of x-rays.
Young died in , sixty-six years before the discovery of x-rays by Wilhelm Rontgen in In Young's double-slit experiment two coherent Le. The pattern produced on a screen placed beyond the slits consisted of a series of bright and dark lines, as shown schematically in Fig. If we replace the double slits with a grid conSisting of many parallel slits, called a diffraction grating, and shine a line source of electromagnetic radiation on the grid, we also observe a pattern consisting of a series of bright and dark lines.
The fringe pattern produced on a screen in Young's experiment. Waves passing through two slits interfere, and the pattern observed on the screen consists of a series of white max and dark min lines not drawn to scale. If two diffraction gratings are now superimposed with their lines intersecting at right angles like a possible arrangement of the lattice planes in a crystal , a spot pattern is produced in which the distance between the spots is a function of the spacing in the gratings and the wavelength of the radiation for a given pair of diffraction gratings.
For the experiment to work, the dimensions of the slits in the grating must be comparable to the wavelength of the radiation used. Max von Laue, in , realized that if x-rays had a wavelength similar to the spacing of atomic planes in a crystal, then it should be possible to diffract x-rays by a crystal and, hence, to obtain information about the arrangement of atoms in crystals. Single crystal 2. Polycrystalline 3.
Amorphous Simple schematics representing these three categories are shown in Fig.
A crystal is said to possess long-range order because it is composed of atoms arranged in a regular ordered pattern in three dimensions. This periodic arrangement, known as the crystal structure, extends over dis- grain boundaries a b c FIG.
Illustration of the difference between a single crystal, b polycrystalline. The boundary between the grains-the grain boundary-depends on the misori- entation of the two grains and the rotation axis about which the misorientation has occurred. There are two special types of grain boundary, illustrated in Fig. In a tilt boundary the axis of rotation is parallel to the grain-boundary plane.
In a twist boundary, the rotation axis is perpendicular to the grain-boundary plane. In general, the axis of rotation will not be simply oriented with respect to either the grain or the grain-boundary plane. Remember, intera- tomic separations are about 0. In a single crystal this order extends throughout the entire volume of the material. A polycrystalline material consists of many small single-crystal regions called grains separated by grain boundaries.
The grains on either side of the grain boundary are misoriented with respect to each other. The grains in a polycrystalline material can have different shapes and sizes.
In amorphous materials, such as glasses and many polymers, the atoms are not arranged in a regular periodic manner. Amorphous is a Greek word meaning "without definite form. The order only extends to a few of the nearest neigh- bors-distances of less than a nanometer.
This arrangement is called a point lattice. If we take any point in the point lattice it has exactly the same number and arrangement of neighbors Le. This condition should be fairly obvious considering our description of long-range order in Sec. We can also see from Fig. This small repeating unit is known as the unit cell of the lattice and is shown in Fig.
The lengths are measured from one corner of the cell, which is taken as FIG. A point lattice. The light lattice points are those that would not be visible when looking from the front of the lattice. But all the points are equivalent. A unit cell. These lengths and angles are called the lattice parameters of the unit cell, or sometimes the lattice constants of the cell.
But the latter term is not really appropriate because they are not necessarily constants; for example, they can vary with changes in temperature and pressure and with alloying. We use a, b, and c to indicate the axes of the unit cell; a, b, and c for the lattice parameters, and a, b, and c for the vectors lying along the unit-cell axes.
How- ever, one of the requirements of a unit cell is that they can be stacked to fill three-dimensional space. Seven unit-cell shapes meet this requirement and are known as the seven crystal systems. All crystals can be classified into these seven categories. We have listed the seven crystal systems in Table 3 in order of increasing symmetry. The triclinic cell has the lowest symmetry, and the cubic cell has the highest symmetry. Quasicrystals are not included in this classification.
They are a relatively new form of solid matter wherein the atoms are arranged in a three-dimensional pattern that exhibits the traditionally forbidden translational symmetries, e. If we put a lattice point at the corner of each unit cell of the seven crystal systems, we obtain seven different point lattices.
However, other arrangements of points also satisfy the requirement of a point lattice; i. Auguste Bravais, in , demon- strated thatthere are 14 possible point lattices and no more. For the three cubic unit cells the number of lattice points per cell is Primitive cubic cubic P 1 Body-centered cubic cubic I 2 Face-centered cubic cubic F 4 All primitive cells have one lattice point per cell. All nonprimitive cells have more than one lattice point per cell.
These lattices are known interchangeably as Bravais lattices, point lattices, and space lattices. The lattice symbols given to the Bravais lattices in Fig. The 14 Bravais lattices. The A face is the face defined by the b and c axes, the B face is defined by the a and c axes, and the C face is defined by the a and b axes.
In Fig. Some texts list only six crystal systems because rhombohedral crystals can always be described in terms of hexagonal axes, so the rhombohedral system is often considered to be a subdivision of the hexagonal system. So far we have discussed only lattice points. What is the difference between a lattice point and an atom? A lattice point represents equivalent positions in a Bravais lattice.
In a real crystal a lattice point may be occupied by one atom or by a group of atoms. In the latter case the atoms are in a fixed relationship with respect to each lattice point. In both cases, the number, composition, and arrangement of atoms is the same for each lattice point. This arrangement is known as the basis. An important difference between lattice points and atoms is that the lattice points tell us nothing about the chemistry or bonding within the crystal; for that we need to include the identity of the atoms and their positions.
The relationship between Bravais lattices and actual crystal structures involves the basis. In the follOwing sections we group crystal structures in terms of their basis. This approach may be somewhat different from that which is used to describe structures in introductory materials sdence classes, but it will help you when we describe the structure factor in Sec.
One Atom per Lattice Point The simplest crystal structures are those in which the basis is one atom located on each lattice point of a Bravais lattice. Remember, each atom must be of the same kind. There is one atom per cell in the sc structure and this atom is located at the origin; Le. Our choice of the origin of the unit cell is entirely arbitrary, as you will see. The simple cubic structure is uncommon; no important metals have this structure. Now let's consider the body-centered cubic Bravais lattice with a basis consisting of one atom.
Simple cubic structure. It is usual to choose a right-handed coordinate system, Fig. The distance is measured in terms of how many lattice parameters we must move along the a, b, and c axes to get from the origin to the point we are interested in.
The coordinates are written as the three distances, with commas separating the numbers. Several metals exist in the bee structure, including sodium Na , chromium Cr , a-iron a-Fe , molybdenum Mo , and tungsten W. The face-centered cubic fcc structure in Fig. Body-centered cubic structure. Face-centered cubic structure. The fcc structure is exhibited by several metals including caldum Ca , copper Cu , gold Au , nickel Ni , and silver Ag.
When referring to the Bravais lattice we write, for example, face-centered cubic, but when referring to a crystal with the face-centered cubic structure we write fcc. Atoms touch across the face diagonal in the fcc structure. The fraction of the unit cell occupied by atoms can be determined by calculating the atomic packing factor APF from the following equation: Look back at the fcc structure in Fig.
This is the maximum possible value for packing spheres of the same size. Crystal structures with an APF of 0. Of the three cubic structures only the fcc structure is close-packed.
The hcp structure is built on the hexagonal Bravais lattice with a basis consisting of two identical atoms associated with each lattice point. Using the relationship in Eq. Hexagonal close-packed structure.
Now, if a Bravais lattice point is located on each corner of the new cell, then a pair of atoms is associated with each Bravais lattice point. To get a better idea of the hexagonal symmetry of the hcp structure, look at Fig.
Only the hcp and fcc structures are close-packed. The Diamond Cubic Structure Another important structure with two atoms per lattice point is the diamond cubic. This structure is adopted by diamond one of the crystal- line forms of carbon and the industrially important elemental semicon- ductors silicon Si and germanium Ge.
The diamond cubic structure Fig. Diamond cubic structure. Although the atoms at these two sets of positions are chemically identical they are all carbon atoms in diamond or all silicon atoms in a crystal of silicon the lattice positions are not equivalent. If the two lattice positions were equivalent, it would be possible to move from one to another by the same translation vector. The vector VI in Fig. The vector v 2 in Fig. If we consider only the atom at 0,0,0 and use the face-centering translations, we produce an fcc crystal structure.
Illustration of the diamond cubic structure showing the atoms which form two face-centered cubic arrangements. There are no systematic names for crystal structures. The IUPAC system is a systematic way of naming many organic compounds on sight, and the name indicates the structure of the compound.
A similar system is not used for naming crystal structures. However, in the next section we introduce a systematic notation for crystal structures, which can be very useful.
Table 4 lists the names of some common crystal structures. Even though two or more materials may share the same crystal structure they may have different properties.
The properties of a material depend on the types of atoms present, the way the atoms are held together bonding , the crystal structure, and the defects present.
You can also see from the table that ZnS exists in both the zinc blende sphalerite and wurtzite structures. This phenomenon is known as polymorphism and is exhibited by many materials.
It is important to realize that, although the fcc structure and the diamond cubic structure both have the same Bravais lattice, they are very different structures and, consequently, materials with a diamond cubic structure have properties significantly different from those with an fcc TABLE 4.
One significant difference between the two structures is the atomic packing factor. The APF is only 0. You will also see that the structure factors selection rules for the observance of reflections in the x-ray diffraction pattern are different for the two structures. The CsCI structure is shown in Fig.
The unit cell contains two atoms. Cesium chloride structure. In the CsCI structure there is a cesium ion at 0,0,0 and a chlorine ion at H,t forming the basis. The Bravais lattice is primitive cubic cubic P: However, in the bcc crystal structure, which we saw in Sec. So in CsCI, where there are two kinds of atoms, the Bravais lattice cannot be body-centered cubic. The NaCI structure Fig.
The sodium ions are in a face-centered cubic arrangement and if we apply the face-centering translations to the chlorine ion at t,o,o we can see that the chlorine ions are also in a face-centered cubic arrangement. So we can describe the NaCI structure as two interpenetrating face-cen- tered cubic lattices, the origin of one displaced from the other by t,o,O. Sodium chloride structure. Zinc Blende Structure The zinc blende, or sphalerite, structure is shown in Fig.
In this structure the arrangement of atoms is the same as in the diamond cubic structure in Fig. But the zinc blende structure has two types of atoms in the unit cell classically Zn and S. Zinc blende structure. Fluorite structure. More than Two Atoms per Lattice Point In the experimental modules in Part II we have concentrated on crystal structures that have either one or two atoms per lattice point. But you should be aware that there are crystal structures with more than two atoms associated with each lattice point.
Of course, the structures are still based on 1 of the 14 Bravais lattices shown in Fig. In this section we present just three examples of materials having crystal structures more complex than those already considered. The first example is cubic zirconium oxide Zr0 2 , or cubic zirconia as it is commonly known. Part of the spinel structure. There are 32 octahedral sites and 64 tetrahedral sites per unit cell.
Structure of solid C The structure is built on the face-centered cubic Bravais lattice with a basis of three atoms two oxygen and one zirconium.
The second example is the spinel structure. The mineral spinel is MgAl 2 0 4, but many other materials exist in its structure. One example is the magnetic oxide Fe 3 0 4 known by the common name magnetite. The structure of magnetite is shown in Fig.
The Bravais lattice is face-cen- tered cubic once again, but this time there are 14 atoms assodated with each lattice point. There are 56 atoms per unit cell. The last example is the structure of solid C6O' The C60 molecule is quite familiar to many people, even though it was only discovered in , because its shape is characteristic of a geodesic dome and is called a "buckyball.
At each lattice point there is one C60 molecule, so there are 60 atoms associated with each lattice point, or atoms per unit cell! However, a systematic notation for describing crystal structures has been developed by W. This system classifies structures according to crystal system, Bravais lattice, and number of atoms per unit cell. The symbols give successively the crystal system, the Bravais lattice symbol, and the number of atoms per unit cell according to the designation in Table 5.
For the crystal structures in Sec. Even though you can determine the crystal system, the Bravais lattice, and the number of atoms from this notation, you will not be able to differentiate different structures with the same notation. However, this appears to be the best possible notation for the moment and is extensively used by the International Centre for Diffraction Data when referring to crystal struc- tures, as you will see in Experimental Module 8.
Miller indices provide us with such a way. They are determined as follows: Identify the points at which the plane intersects the a, b, and c axes. The intercept is measured in terms of fractions or multiples of the lattice parameters. If the plane moves through the origin, then the origin must be moved. We will show you how this works a little later. Take redprocals of the intercepts. This is done because if the plane never intersects one of the axes the intercept will be at We don't want to have 00 in the indices so we take redprocals.
Clear the fractions. We only want whole numbers in the indices, but don't reduce the numbers to the lowest integers. Enclose the resulting numbers in parentheses. Negative num- bers are written with a bar over the number 1 , and pronounced, for example, as bar one.
These steps give us the Miller indices of any plane in any crystal system, whether cubic, orthorhombic, or triclinic. Also note that the interaxial angles are not important in this case-the A face has indices in all crystal systems, and the C face has indices in all crystal systems.
The actual values of the lattice parameters a, b, and c are also unimpor- tant. Let's determine the Miller indices of the planes A and B in Fig. Detennination of Miller indices of planes.
For plane B: Let's now look at what happens if the plane goes through the origin, as does plane C in Fig. To index this plane we need to move the origin. Remember that the origin of the unit cell is arbitrarily chosen and can be taken to be any point.
So let's move the origin to the point that originally had coordinates 0,1,0. Now we follow the same procedure as before, only using our new origin. For plane c: If, however, you want to compare the Miller indices of planes in a unit cell, the origin must be the same for each plane.
Detennination of Miller indices of a plane when it passes through the origin. Look again at the cubic unit cell in Fig. Notice that plane C is a face of the cube. Hence, the face planes of a cube are all symmetrically related.
There are six different face planes, but they are all equivalent. When a set of planes is equivalent, they are known as a family of planes. These planes all have the same spacing. We can define the mUltiplicity factor, which is the number of planes in a family that have the same spacing. The multiplicity factors are different for different planes, and these values for the cubic and hexagonal systems are listed in Appendix 6.
FAQ Policy. About this book In this, the only book available to combine both theoretical and practical aspects of x-ray diffraction, the authors emphasize a "hands on" approach through experiments and examples based on actual laboratory data. Show all. Pages Hexagonal Structures. Precise Lattice Parameter Measurements. Phase Diagram Determination. Detection of Long-Range Ordering. Determination of Crystallite Size and Lattice Strain.
Quantitative Analysis of Powder Mixtures. Identification of an Unknown Specimen. Back Matter Pages About this book Introduction In this, the only book available to combine both theoretical and practical aspects of x-ray diffraction, the authors emphasize a "hands on" approach through experiments and examples based on actual laboratory data.
Part I presents the basics of x-ray diffraction and explains its use in obtaining structural and chemical information.