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This chapter presents some very basic results about the geometry of diffraction from crystals. This is developed in much greater detail in many textbooks but a concise statement of the basic concepts greatly facilitates the understanding of the advanced later chapters so we reproduce it here for the convenience of the reader. Since the results are so basic, we do not make any attempt to reference the original sources. The bibliography at the end of the chapter lists a selection of some of our favorite introductory books on powder diffraction.

X-rays are electromagnetic (em) waves with a much shorter wavelength than visible light, typically on the order of 1 Å (= 1 × 10−10 m). The physics of em-waves is well understood and excellent introductions to the subject are found in every textbook on optics. Here we briefly review the results most important for understanding the geometry of diffraction from crystals. Classical em-waves can be described by a sine wave that repeats periodically every 2π radians. The spatial length of each period is the wavelength λ. If two identical waves are not coincident, they are said to have a “phase shift” with respect to each other (Figure 1.1). This is either measured as a linear shift, Δ on a length scale, in the units of the wavelength, or equivalently as a phase shift, δϕ on an angular scale, such that:

formula
Equation 1

The detected intensity, I, is the square of the amplitude, A, of the sine wave. With two waves present, the resulting amplitude is not just the sum of the individual amplitudes but depends on the phase shift δφ. The two extremes occur when δφ = 0 (constructive interference), where I = (A1 + A2)2, and δφ = π (destructive interference), where I = (A1 − A2)2. In general, I = [A1 +A2 exp (iδφ)]2. When more than two waves are present, this equation becomes:

formula
Equation 2

where the sum is over all the sine-waves present and the phases, ϕj are measured with respect to some origin.

Figure 1.1

Graphical illustration of the phase shift between two sine waves of equal amplitude.

Figure 1.1

Graphical illustration of the phase shift between two sine waves of equal amplitude.

Close modal

X-ray diffraction involves the measurement of the intensity of X-rays scattered from electrons bound to atoms. Waves scattered at atoms at different positions arrive at the detector with a relative phase shift. Therefore, the measured intensities yield information about the relative atomic positions (Figure 1.2).

Figure 1.2

Scattering of a plane wave by a one-dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Figure 1.2

Scattering of a plane wave by a one-dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Close modal

In the case of X-ray diffraction, the Fraunhofer approximation is used to calculate the detected intensities. This is a far-field approximation, where the distance, L1, from the source to the place where scattering occurs (the sample), and then on to the detector, L2, is much larger than the separation, D, of the scatterers. This is an excellent approximation, since in this case D/L1 ≈ D/L2 ≈ 10−10. The Fraunhofer approximation greatly simplifies the mathematics. The incident X-rays form a wave such that the constant phase wave front is a plane wave. X-rays scattered by single electrons are outgoing spherical waves that again appear as plane waves in the far-field. This allows us to express the intensity of diffracted X-rays using Equation (2).

The phases φj introduced in Equation (2), and therefore the measured intensity I, depend on the position of the atoms, j, and the directions of the incoming and the scattered plane waves (Figure 1.2). Since the wave-vectors of the incident and scattered waves are known, we can infer the relative atomic positions from the detected intensities.

From optics we know that diffraction only occurs if the wavelength is comparable to the separation of the scatterers. In 1912, Friedrich, Knipping and Max von Laue performed the first X-ray diffraction experiment using single crystals of copper sulfate and zinc sulfite, proving the hypothesis that X-rays are em-waves of very short wavelength, on the order of the separation of the atoms in a crystalline lattice. Four years later (1916), Debye and Scherrer reported the first powder diffraction pattern with a procedure that is named after them.

The easiest access to the structural information in powder diffraction is via the well-known Bragg equation (W. L. Bragg, 1912), which describes the principle of X-ray diffraction in terms of a reflection of X-rays by sets of lattice planes. Lattice planes are crystallographic planes, characterized by the index triplet hkl, the so-called Miller indices. Parallel planes have the same indices and are equally spaced, separated by the distance dhkl. Bragg analysis treats X-rays like visible light being reflected by the surface of a mirror, with the X-rays being specularly reflected at the lattice planes. In contrast to the lower energy visible light, the X-rays penetrate deep inside the material where additional reflections occur at thousands of consecutive parallel planes. Since all X-rays are reflected in the same direction, superposition of the scattered rays occurs. From Figure 1.3 it follows that the second wave travels a longer distance PN before and NQ after reflection occurs. Constructive interference occurs only if Δ = PN + NQ is a multiple n = 0, ±1, ±2, … of the wavelength λ:

formula
Equation 3

In all other cases, destructive interference results since it is always possible to find a deeper plane, p, for which the relation pΔ =  with n = ±1/2, ± 3/2, …. (i.e., perfect destructive interference) exactly holds. Thus, sharp intensity maxima emerge from the sample only at the special angles where Equation (3) holds, with no intensity in between. As can be easily seen from Figure 1.3, geometrically:

formula
Equation 4

where d is the interplanar spacing of parallel lattice planes and 2θ is the diffraction angle, the angle between the incoming and outgoing X-ray beams. Combining Equations (3) and (4) we get:

formula
Equation 5

the famous Bragg equation.

Figure 1.3

Illustration of the geometry used for the simplified derivation of Bragg's law.

Figure 1.3

Illustration of the geometry used for the simplified derivation of Bragg's law.

Close modal

This simplified derivation of the Bragg equation, although leading to the correct solution, has a serious drawback. In reality the X-rays are not reflected by planes but are scattered by electrons bound to the atoms. Crystal planes are not like shiny optical mirrors, but contain discrete atoms separated by regions of much lower electron intensity, and, in general, atoms in one plane will not lie exactly above atoms in the plane below. How is it then that the simplified picture shown in Figure 1.3 results in the correct result? A more general description (Bloss, 1971) shows that Equation (5) is also valid, if the atom of the lower lattice plane in Figure 1.3 is shifted by an arbitrary amount parallel to the plane (Figure 1.4).

Figure 1.4

Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes.

Figure 1.4

Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes.

Close modal

The phase shift can immediately be deduced from Figure 1.4 as:

formula
Equation 6

From any textbook on trigonometry we know that:

formula
Equation 7

Therefore Equation (7) becomes:

formula
Equation 8

with:

formula
Equation 9

from which the already known Bragg equation follows:

formula
Equation 10

Another equivalent, and highly useful, expression of the Bragg equation is:

formula
Equation 11

with the energy E of the X-rays in keV.

To derive the Bragg equation, we used an assumption of specular reflection, which is borne out by experiment. For crystalline materials, destructive interference completely destroys intensity in all directions except where Equation (5) holds. This is no longer true for disordered materials where diffracted intensity can be observed in all directions away from reciprocal lattice points, known as diffuse scattering, as discussed in Chapter 16.

As a prerequisite, the so-called reciprocal lattice needs to be introduced. Notably, it is not the intention of this book to reproduce basic crystallographic knowledge but, for completeness, some important formalism that recurs throughout the book is briefly presented.

The reciprocal lattice was invented by crystallographers as a simple and convenient representation of the physics of diffraction by a crystal. It is an extremely useful tool for describing all kinds of diffraction phenomena occurring in powder diffraction (Figure 1.5).

Figure 1.5

Two-dimensional monoclinic lattice and its corresponding reciprocal lattice.

Figure 1.5

Two-dimensional monoclinic lattice and its corresponding reciprocal lattice.

Close modal

Imagine that besides the “normal” crystal lattice with the lattice parameters a, b, c, α, β, γ, and the volume V of the unit cell, a second lattice with lattice parameters of a*, b*, c*, α*, β*, γ*, and the volume V*, and with the same origin, exists such that:

formula
Equation 12

This is known as the reciprocal lattice, which exists in so-called reciprocal space. As we will see, it turns out that the points in the reciprocal lattice are related to the vectors defining the crystallographic planes. There is one point in the reciprocal lattice for each crystallographic plane, hkl. For now, just consider h, k and l to be integers that index a point in the reciprocal lattice. A reciprocal lattice vector hhkl is the vector from the origin of reciprocal space to the (hkl) reciprocal lattice point:

formula
Equation 13

The length of the reciprocal base vectors is defined according to:

formula
Equation 14

where the scale factor x can easily be deduced using Equation (12) as:

formula
Equation 15

leading to:

formula
Equation 16

and vice versa:

formula
Equation 17

The relationship between the reciprocal and the real lattice parameters is:

formula
Equation 18

Equation (18) simplifies considerably for higher symmetry crystal systems.

We now rederive Bragg's law using vector notation. The wave vectors of the incoming and outgoing beams are given by s0 and s, respectively (Figure 1.6). They point in the direction of propagation of the wave and their length depends on λ. For elastic scattering (no change in wavelength on scattering), s0 and s have the same length.

Figure 1.6

Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering.

Figure 1.6

Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering.

Close modal

We define the scattering vector as:

formula
Equation 19

which for a specular reflection is always perpendicular to the scattering plane. The length of h is given by:

formula
Equation 20

Comparison with the formula for the Bragg equation [Equation (5)]:

formula
Equation 21

we get:

formula
Equation 22

Setting the magnitude of s to 1/λ, we get the Bragg equation in terms of the magnitude of the scattering vector h:

formula
Equation 23

This shows that diffraction occurs when the magnitude of the scattering vector h is an integral number of reciprocal lattice spacings 1/d. We define a vector d* perpendicular to the lattice planes with length 1/d. Since h is perpendicular the scattering plane, this leads to:

formula
Equation 24

Diffraction can occur at different scattering angles 2θ for the same crystallographic plane, giving the different orders n of diffraction. For simplicity, the number n will be incorporated in the indexing of the lattice planes, where:

formula
Equation 25

e.g., d222* = 2d111* and we get an alternative expression for Bragg's equation:

formula
Equation 26

The vector dhkl* points in a direction perpendicular to a real space lattice plane. We would like to express this vector in terms of reciprocal space basis vectors a*, b*, c*.

First we define dhkl in terms of real space basis vectors a, b, c. Referring to Figure 1.7, we can define that:

formula
Equation 27

with h, k, and l being integers as required by the periodicity of the lattice. These three integers are the Miller indices that provide a unique definition for the set of parallel planes.

Figure 1.7

Geometrical description of a lattice plane in terms of real space basis vectors.

Figure 1.7

Geometrical description of a lattice plane in terms of real space basis vectors.

Close modal

The plane perpendicular vector dhkl originates on one plane and terminates on the next parallel plane. Therefore, OA·d = (OA)dcosα. From Figure 1.7 we see that, geometrically, (OA)cosα = d. Substituting, we get OA·d = d2. Combining with Equation (27) leads to:

formula
Equation 28

and consequently:

formula
Equation 29

By definition, h, k, and l are divided by their largest common integer to be Miller indices. The vector dhkl*, from Bragg's Equation (26) points in the plane normal direction parallel to d but with length 1/d. We can now write dhkl* in terms of the vector d:

formula
Equation 30

which gives:

formula
Equation 31

or written in terms of the reciprocal basis:

formula
Equation 32

which was obtained using:

formula
Equation 33

Comparing Equation (32) with Equation (13) proves the identity of dhkl* and the reciprocal lattice vectorh hhkl. Bragg's equation, Equation (26), can be restated as:

formula
Equation 34

In other words, diffraction occurs whenever the scattering vector h equals a reciprocal lattice vector hhkl. This powerful result is visualized in the useful Ewald construction that is described below.

Useful equivalent variations of the Bragg equation are:

formula
Equation 35

and:

formula
Equation 36

The vector Q is the physicist's equivalent of the crystallographer's h. The physical meaning of Q is the momentum transfer on scattering and differs from the scattering vector h by a factor of 2π.

The Bragg equation shows that diffraction occurs when the scattering vector equals a reciprocal lattice vector. The scattering vector depends on the geometry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald's construction combines these two concepts in an intuitive way. A sphere of radius 1/λ is constructed and positioned in such a way that the Bragg equation is satisfied, and diffraction occurs, whenever a reciprocal lattice point coincides with the surface of the sphere (Figure 1.8).

Figure 1.8

Geometrical construction of the Ewald circle. The “0” marks the origin of reciprocal space. The vectors are defined in the text.

Figure 1.8

Geometrical construction of the Ewald circle. The “0” marks the origin of reciprocal space. The vectors are defined in the text.

Close modal

The recipe for constructing Ewald's sphere§ is as follows (Figure 1.8):

  1. Draw the incident wave vector s0. This points in the direction of the incident beam with length 1/λ.

  2. Draw a sphere centered on the tail of this vector with radius 1/λ. The incident wave vector s0 defines the radius of the sphere. The scattered wave vector s, also of length 1/λ, points in a direction from the sample to the detector. This vector is drawn also starting from the center of the sphere and also terminates at a point on the surface. The scattering vector h = s – s0 completes the triangle from the tip of s to the tip of s0, both lying on the surface of the sphere.

  3. Draw the reciprocal lattice with the origin lying at the tip of s0.

  4. Find all the places on the surface of the sphere, where reciprocal lattice points lie.

This construction places a reciprocal lattice point at one end of h. By definition, the other end of h lies on the surface of the sphere. Thus, Bragg's law is only satisfied, when another reciprocal lattice point coincides with the surface of the sphere. Diffraction is emanating from the sample in these directions. To detect the diffracted intensity, one simply moves the detector to the right position. Any vector between two reciprocal lattice points has the potential to produce a Bragg peak. The Ewald sphere construction additionally indicates which of these possible reflections satisfy experimental constraints and are therefore experimentally accessible.

Changing the orientation of the crystal reorients the reciprocal lattice bringing different reciprocal lattice points on to the surface of the Ewald sphere. An ideal powder contains individual crystallites in all possible orientations with equal probability. In the Ewald construction, every reciprocal lattice point is smeared out onto the surface of a sphere centered on the origin of reciprocal space. This is illustrated in Figure 1.9. The orientation of the dhkl* vector is lost and the three-dimensional vector space is reduced to one dimension of the modulus of the vector dhkl*.

Figure 1.9

Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right). A real powder consists of so many grains that the dots of the reciprocal lattice form into continuous lines.

Figure 1.9

Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right). A real powder consists of so many grains that the dots of the reciprocal lattice form into continuous lines.

Close modal

These spherical shells intersect the surface of the Ewald sphere in circles. Figure 1.10 shows a two-dimensional projection. Diffracted beams emanate from the sample in the directions where the thin circles from the smeared reciprocal lattice intersect the thick circle of the Ewald sphere. A few representative reflected beams are indicated by the broken lines.

Figure 1.10

Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

Figure 1.10

Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

Close modal

The lowest d-spacing reflections accessible in the experiment are determined by the diameter of the Ewald sphere 2/λ. To increase the number of detectable reflections one must decrease the incident wavelength. In the case of an energy dispersive experiment such as time-of-flight neutron powder diffraction, which makes use of a continuous distribution of wavelengths from λmin to λmax at fixed angle, all smeared out cones between the two limiting Ewald spheres can be detected.

In three dimensions, the circular intersection of the smeared reciprocal lattice with the Ewald sphere results in the diffracted X-rays of the reflection hkl forming coaxial cones, the so-called Debye–Scherrer cones (Figure 1.11).

Figure 1.11

Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye–Scherrer cones are drawn.

Figure 1.11

Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye–Scherrer cones are drawn.

Close modal

The smearing of reciprocal space in a powder experiment makes the measurement easier but results in a loss of information. Reflections overlap from lattice planes whose vectors lie in different directions but which have the same d-spacing. These cannot be resolved in the measurement. Some of these overlaps are dictated by symmetry (systematic overlaps) and others are accidental. Systematic overlaps are less serious for equivalent reflections [e.g., the six Bragg peaks (100), (− 100), (010),… from the faces of a cube] since the multiplicity is known from the symmetry. For highly crystalline samples, accidental overlaps can be reduced by making measurements with higher resolution, or by taking data at different temperatures in an attempt to remove the overlap by differential thermal expansion of different cell parameters.

To obtain the maximum amount of information, a spherical shell detector would be desirable, though currently impractical. Often, a flat two-dimensional detector, either film, image plate, or CCD is placed perpendicular to the direct beam. In this case, the Debye–Scherrer cones appear as circles as shown in Figure 1.12a.

Figure 1.12

Debye–Scherrer rings from an ideal fine grained (a, left) and a grainy (b, right) powder sample.

Figure 1.12

Debye–Scherrer rings from an ideal fine grained (a, left) and a grainy (b, right) powder sample.

Close modal

For an ideal powder, the intensity around the rings is isotropic. Conventional powder diffraction measurements, e.g., Bragg–Brentano geometry, take one-dimensional cuts through the rings, either horizontally or vertically depending on the geometry of the diffractometer. If the full rings, or fractions of them, are detected with two-dimensional detectors the counting statistics can be improved by integrating around the rings. If the powder is non-ideal, the ring intensity is no longer uniform, as illustrated in Figure 1.12b, giving arbitrary intensities for the reflections in a one-dimensional scan. To improve powder statistics, powder samples are generally rotated during measurement. However, the intensity variation around the rings can give important information about the sample such as preferred orientation of the crystallites or texture.

Several important relationships in crystallography directly follow from a derivative of the Bragg equation [Equation (5)]. First we rewrite Bragg's law making the d-spacing the subject of the equation:

formula
Equation 37

The uncertainty of the measured lattice spacing is given by the total derivative dd, which can be written according to the chain rule as:

formula
Equation 38

leading to:

formula
Equation 39

and finally:

formula
Equation 40

This equation allows us to discuss several physically important phenomena.

When a crystal is strained, the d-spacings vary. A macroscopic strain changes the interplanar spacing by Δdhkl, giving rise to a shift in the average position of the diffraction peak of Δθ, while microscopic strains give a distribution of d-spacings Δdhkl which broaden the peak by δθ. This is discussed in detail in Chapters 12 and 13.

A constant angular offset due to misalignment of the diffractometer gives rise to a nonlinear error in our determination of dhkl, disproportionately affecting low angle reflections (Figure 1.13). Similarly, our ability to resolve two partially overlapping reflections separated by Δdhkl is limited by the finite angular resolution Δθ of the diffractometer.

Figure 1.13

Percentage error in measured d-spacing as a function of scattering angle arising from a constant angular misalignment of ΔΘ for a well aligned (0.001°), a typically aligned, (0.01°) and a poorly aligned (0.05°) diffractometer.

Figure 1.13

Percentage error in measured d-spacing as a function of scattering angle arising from a constant angular misalignment of ΔΘ for a well aligned (0.001°), a typically aligned, (0.01°) and a poorly aligned (0.05°) diffractometer.

Close modal

There are many geometrical contributions to the angular resolution (e.g., angular width of the receiving slit in front of the detector). Another contribution comes from finite wavelength spread of the incident beam Δλ. From Equation (40) we get the angular dispersion to be:

formula
Equation 41

This is plotted in Figure 1.14, which shows that the resolution due to a finite spread in λ is decreasing at higher angles. In a real experiment the angle dependence of the resolution function can be complicated. In traditional modeling programs the Bragg-peak line shapes are modeled using empirical line-shape functions. More recently, approaches have been developed that explicitly account for the different physical processes that result in the line shapes. This is called the fundamental parameters approach and is described in Chapters 5, 6 and 13.

Figure 1.14

Angular dependence of the intrinsic peak width (resolution function) of the diffractometer due to the wavelength spread between Cu-Kα1 and Cu-Kα2 (about 12 eV).

Figure 1.14

Angular dependence of the intrinsic peak width (resolution function) of the diffractometer due to the wavelength spread between Cu-Kα1 and Cu-Kα2 (about 12 eV).

Close modal

Assuming an infinite stack of lattice planes, Bragg's equation gives the position of delta-function Bragg peaks. Finite size crystallites give rise to Bragg peaks of finite width. This size broadening is described by the Scherrer equation. We now reproduce the simple derivation following Klug and Alexander (1974; see Bibliography).

Figure 1.15 shows the path length difference versus the depth of the lattice plane. When the angle between the incoming beam and the lattice plane Θ is different by an amount ε from the Bragg condition, it is always possible to find a lattice plane inside the crystal where the extra path is Δ = λ/2 producing destructive interference. For a thick crystal this is true for arbitrarily small ε, which explains the sharp Bragg reflections. For a crystal with finite dimensions, for small ε the plane for which holds will not be reached. In this case there is not perfect cancellation of the intensity away from the Bragg condition, thus leading to an intensity distribution over some small angular range. We can use this idea to estimate the broadening of a Bragg reflection due to size effects.

Figure 1.15

Path length difference of the scattered ray versus the depth of the lattice plane in the crystal.

Figure 1.15

Path length difference of the scattered ray versus the depth of the lattice plane in the crystal.

Close modal

The thickness of a crystallite in the direction perpendicular to p (hkl) planes of separation dhkl (Figure 1.15) is:

formula
Equation 42

The additional beam path between consecutive lattice planes at the angle θ + ε is:

formula
Equation 43

The corresponding phase difference is then:

formula
Equation 44

The phase difference between the top and the bottom layer, p is then:

formula
Equation 45

Rearranging Equation (45) leads to:

formula
Equation 46

which gives an expression for the misalignment angle in terms of the crystallite size Lhkl and the phase difference δφ between the reflections between the top and the bottom plane. Clearly, the scattered intensity is at a maximum for δφ = 0 (ε = 0). With increasing ε the intensity decreases giving rise to a peak of finite width. Perfect cancellation of the top and bottom waves occurs a phase difference of δφ = ±π at which point ε = ± λ/(4Lhkl cosθ). On a measured 2θ-scale the measured angular width between these points is:

formula
Equation 47

giving us some measure of the peak width in radians due to the finite particle size. A full treatment taking into account the correct form for the intensity distribution gives:

formula
Equation 48

with a scale factor of K = 0.89 for perfect spheres. In general K depends on the shape of the grains (e.g., K is 0.94 for cubic shaped grains) but is always close to unity. This equation is not valid for crystallites that are too large or too small. With large crystallites the peak width is governed by the coherence of the incident beam and not by particle size. For nano-scale crystallites, Bragg's law fails and needs to be replaced by the Debye equation (see Chapter 16).

Vectors are in bold.

The reciprocal lattice is a commonly used construct in solid state physics, but with a different normalization: a·a*=2π.

§

For practical reasons, plots of the Ewald “sphere” are circular cuts through the sphere and the corresponding slice of reciprocal space.

Strictly speaking, the term crystallite size here refers to the dimension of a coherently scattering domain. Only in a perfect crystal, is this the grain size.

1.
V. K.
Pecharsky
and
P. Y.
Zavalij
,
Fundamentals of Powder Diffraction and Structural Characterization of Materials
,
Kluwer Academic Publishers
,
Boston
,
2003
2.
D. L. BishJ. E. Post (Eds.),
Modern Powder Diffraction
, Reviews in Mineralogy Volume 20,
Mineralogical Society of America
,
Chantilly VA
,
1989
3.
H. P.
Klug
and
L. E.
Alexander
,
X-ray Diffraction Procedures
, 2nd edn.,
John Wiley & Sons
,
New York
,
1974

Figures & Tables

Figure 1.1

Graphical illustration of the phase shift between two sine waves of equal amplitude.

Figure 1.1

Graphical illustration of the phase shift between two sine waves of equal amplitude.

Close modal
Figure 1.2

Scattering of a plane wave by a one-dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Figure 1.2

Scattering of a plane wave by a one-dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Close modal
Figure 1.3

Illustration of the geometry used for the simplified derivation of Bragg's law.

Figure 1.3

Illustration of the geometry used for the simplified derivation of Bragg's law.

Close modal
Figure 1.4

Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes.

Figure 1.4

Illustration of the geometry in the general case where scattering takes place at the position of atoms in consecutive planes.

Close modal
Figure 1.5

Two-dimensional monoclinic lattice and its corresponding reciprocal lattice.

Figure 1.5

Two-dimensional monoclinic lattice and its corresponding reciprocal lattice.

Close modal
Figure 1.6

Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering.

Figure 1.6

Illustration of the important wave and scattering vectors in the case of elastic Bragg scattering.

Close modal
Figure 1.7

Geometrical description of a lattice plane in terms of real space basis vectors.

Figure 1.7

Geometrical description of a lattice plane in terms of real space basis vectors.

Close modal
Figure 1.8

Geometrical construction of the Ewald circle. The “0” marks the origin of reciprocal space. The vectors are defined in the text.

Figure 1.8

Geometrical construction of the Ewald circle. The “0” marks the origin of reciprocal space. The vectors are defined in the text.

Close modal
Figure 1.9

Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right). A real powder consists of so many grains that the dots of the reciprocal lattice form into continuous lines.

Figure 1.9

Illustration of the reciprocal lattice associated with a single-crystal lattice (left) and a large number of randomly oriented crystallites (right). A real powder consists of so many grains that the dots of the reciprocal lattice form into continuous lines.

Close modal
Figure 1.10

Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

Figure 1.10

Illustration of the region of reciprocal space that is accessible in a powder measurement. The smaller circle represents the Ewald sphere. As shown in Figure 1.9, in a powder the reciprocal lattice is rotated to sample all orientations. An equivalent operation is to rotate the Ewald sphere in all possible orientations around the origin of reciprocal space. The volume swept out (area in the figure) is the region of reciprocal space accessible in the experiment.

Close modal
Figure 1.11

Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye–Scherrer cones are drawn.

Figure 1.11

Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye–Scherrer cones are drawn.

Close modal
Figure 1.12

Debye–Scherrer rings from an ideal fine grained (a, left) and a grainy (b, right) powder sample.

Figure 1.12

Debye–Scherrer rings from an ideal fine grained (a, left) and a grainy (b, right) powder sample.

Close modal
Figure 1.13

Percentage error in measured d-spacing as a function of scattering angle arising from a constant angular misalignment of ΔΘ for a well aligned (0.001°), a typically aligned, (0.01°) and a poorly aligned (0.05°) diffractometer.

Figure 1.13

Percentage error in measured d-spacing as a function of scattering angle arising from a constant angular misalignment of ΔΘ for a well aligned (0.001°), a typically aligned, (0.01°) and a poorly aligned (0.05°) diffractometer.

Close modal
Figure 1.14

Angular dependence of the intrinsic peak width (resolution function) of the diffractometer due to the wavelength spread between Cu-Kα1 and Cu-Kα2 (about 12 eV).

Figure 1.14

Angular dependence of the intrinsic peak width (resolution function) of the diffractometer due to the wavelength spread between Cu-Kα1 and Cu-Kα2 (about 12 eV).

Close modal
Figure 1.15

Path length difference of the scattered ray versus the depth of the lattice plane in the crystal.

Figure 1.15

Path length difference of the scattered ray versus the depth of the lattice plane in the crystal.

Close modal

Contents

References

1.
V. K.
Pecharsky
and
P. Y.
Zavalij
,
Fundamentals of Powder Diffraction and Structural Characterization of Materials
,
Kluwer Academic Publishers
,
Boston
,
2003
2.
D. L. BishJ. E. Post (Eds.),
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