Fundamentals of Diffraction

Chapter 7 Fundamentals of Diffraction

When X-rays propagate through a substance, the occurrence of the following processes should be considered in the phenomenon of diffraction:

– Coherent scattering (Sect 7.1), which produces beams with the same wavelength as the incident (primary) beam. In other words, the energy of the photons in

a coherently scattered beam remains unchanged when compared to that in the primary beam. – Incoherent (or Compton 1 ) scattering, in which the wavelength of the scattered beam increases due to partial loss of photon energy in collisions with core elec- trons (the Compton effect).

– Absorption of the X-rays, see Sect 8.6.5, in which some photons are dissipated in random directions due to scattering, and some photons lose their energy by ejecting electron(s) from an atom (i.e., ionization) and/or due to the photoelectric effect (i.e., X-ray fluorescence).

Incoherent scattering is not essential when the interaction of X-rays with crys- tal lattices is of concern, and it is generally neglected. When absorption becomes significant, it is usually taken into account as a separate effect. Thus, in the first ap- proximation only coherent scattering results in the diffraction from periodic lattices and is considered in this chapter.

Generally, the interaction of X-rays (or any other type of radiation with the proper wavelength) with a crystal is multifaceted and complex, and there are two different levels of approximation – kinematical and dynamical theories of diffrac- tion. In the kinematical diffraction, a beam scattered once is not allowed to be scat- tered again before it leaves the crystal. Thus, the kinematical theory of diffraction is based on the assumption that the interaction of the diffracted beam with the crystal

1 Arthur Holly Compton (1897–1962). The American physicist, best known for his discovery of the increase of wavelength of X-rays due to scattering of the incident radiation by free electrons –

inelastic scattering of X-ray photons – known today as the Compton effect. With Charles Thomson Rees Wilson, Compton shared the Nobel Prize in physics in 1927 “for his discovery of the effect named after him.” See http://nobelprize.org/nobel prizes/physics/laureates/1927/compton-bio.html for more information.

134 7 Fundamentals of Diffraction is negligibly small. This requires the following postulations: (1) a crystal consists of

individual mosaic blocks – crystallites 2 – which are slightly misaligned with respect to one another; (2) the size of the crystallites is small, and (3) the misalignment of the crystallites is large enough, so that the interaction of X-rays with matter at the length scale exceeding the size of mosaic blocks is negligible.

On the contrary, the theory of the dynamical diffraction accounts for scattering of the diffracted beam and other interactions of waves inside the crystal, and thus the mathematical apparatus of the theory is quite complex. Dynamical effects become significant and the use of the theory of dynamical diffraction is justified only when the crystals are nearly perfect, or when there is an exceptionally strong interaction of the radiation with the material. In the majority of crystalline materials, however, dynamical effects are weak and they are usually noticeable only when precise sin- gle crystal experiments are conducted. Even then, numerous dynamical effects (e.g., primary and/or secondary extinction, simultaneous diffraction, thermal diffuse scat- tering, and others) are usually applied as corrections to the kinematical diffraction model.

The kinematical approach is simple, and adequately and accurately describes the diffraction of X-rays from mosaic crystals. This is especially true for polycrystalline materials where the size of crystallites is relatively small. Hence, the kinematical theory of diffraction is used in this chapter and throughout this book.

7.1 Scattering by Electrons, Atoms and Lattices

It is well-known that when a wave interacts with and is scattered by a point object, the outcome of this interaction is a new wave, which spreads in all directions. If no energy loss occurs, the resultant wave has the same frequency as the incident (primary) wave and this process is known as elastic scattering. In three dimensions, the elastically scattered wave is spherical, with its origin in the point coinciding with the object as shown schematically in Fig. 7.1.

When two or more points are involved, they all produce spherical waves with the same λ, which interfere with each other simply by adding their amplitudes. If the two scattered waves with parallel-propagation vectors are completely in-phase, the resulting wave has its amplitude doubled (Fig. 7.2 top), while the waves, which are completely out-of-phase, extinguish one another as shown in Fig. 7.2 (bottom).

The first case seen in Fig. 7.2 is called constructive interference and the second case is termed as destructive interference. Constructive interference, which occurs on periodic arrays of points, increases the resultant wave amplitude by many or- ders of magnitude and this phenomenon is one of the cornerstones in the theory of diffraction.

2 Crystallite usually means a tiny single crystal (microcrystal). Each particle in a polycrystalline material usually consists of multiple crystallites that join together in different orientations. A small

powder particle can be a single crystallite as well.

7.1 Scattering by Electrons, Atoms and Lattices 135 Scattered spherical

wave

Incident wave

Fig. 7.1 The illustration of a spherical wave produced as a result of elastic scattering of the incident wave by the point object (filled dot in the center of the dotted circle).

+ Fig. 7.2 The two limiting cases of the interaction between two waves with parallel propagation

vectors (k): the constructive interference of two in-phase waves resulting in a new wave with double the amplitude (top), and the destructive interference of two completely out-of-phase waves in which the resultant wave has zero amplitude, i.e., the two waves extinguish one another (bottom).

Diffraction can be observed only when the wavelength is of the same order of magnitude as the repetitive distance between the scattering objects. Thus, for crys- tals, the wavelength should be in the same range as the shortest interatomic dis- tances, that is, somewhere between ∼0.5 and ∼2.5 ˚A. This condition is fulfilled when using electromagnetic radiation, which within the mentioned range of wave- lengths, are X-rays. It is important to note that X-rays scatter from electrons, so that the active scattering centers are not the nuclei, but the electrons, or more precisely the electron density, periodically distributed in the crystal lattice.

The other two types of radiation that can diffract from crystals are neutron and electron beams. Unlike X-rays, neutrons are scattered on the nuclei, while electrons, which have electric charge, interact with the electrostatic potential. Nuclei, their electronic shells (i.e., core electron density), and electrostatic potentials, are all dis- tributed similarly in the same crystal and their distribution is established by the crys- tal structure of the material. Thus, assuming a constant wavelength, the differences

136 7 Fundamentals of Diffraction in the diffraction patterns when using various kinds of radiation are mainly in the

intensities of the diffracted beams. The latter occurs because various types of ra- diation interact in their own way with different scattering centers. The X-rays are the simplest, most accessible, and by far the most commonly used waves in powder diffraction.

7.1.1 Scattering by Electrons

The origin of the electromagnetic wave elastically scattered by the electron can be better understood by recalling the fact that electrons are charged particles. Thus, an oscillating electric field (see Fig. 6.2) from the incident wave exerts a force on the electric-charge (electron) forcing the electron to oscillate with the same frequency as the electric-field component of the electromagnetic wave. The oscillating electron accelerates and decelerates in concert with the varying amplitude of the electric field vector, and emits electromagnetic radiation, which spreads in all directions. In this respect, the elastically scattered X-ray beam is simply radiated by the oscillating electron; it has the same frequency and wavelength as the incident wave, and this type of scattering is also known as coherent scattering. 3

For the sake of simplicity, we now consider electrons as stationary points and disregard the dependence of the scattered intensity 4 on the scattering angle. 5 Each electron then interacts with the incident X-ray wave producing a spherical elastically scattered wave, as shown in Fig. 7.1. Thus, the scattering of X-rays by a single electron yields an identical scattered intensity in every direction.

3 It is worth noting that coherency of the electromagnetic wave elastically scattered by the electron establishes specific phase relationships between the incident and the scattered wave: their phases

are different by π (i.e., scatterred wave is shifted with respect to the incident wave exactly by λ/2). 4 The scattered (diffracted) X-ray intensity recorded by the detector is proportional to the amplitude

squared. 5 The absolute intensity of the X-ray wave coherently scattered by a single electron, I, is determined

from the Thomson equation:

1 + cos 2 2 θ

I =I

where I 0 is the absolute intensity of the incident beam, K is constant (K = 7.94 × 10 −30 m 2 ), r is the distance from the electron to the detector in m, and θ is the angle between the propagation vector of the incident wave and the direction of the scattered wave. It is worth noting that in a powder diffraction experiment all prefactors in the right hand side of the Thomson’s equation are constant and can be omitted. The only variable part is, therefore, a function of the Bragg angle, θ. It emerges because the incident beam is generally unpolarized but the scattered beam is always partially polarized. This function, therefore, is called the polarization factor.

Thomson equation is named after sir Joseph John (J.J.) Thomson (1856–1940) – the British physicist who has been credited with the discovery of an electron. In 1906 he received the Nobel Prize in Physics “in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases.” See http://nobelprize.org/nobel prizes/ physics/laureates/1906/thomson-bio.html for more information.

7.1 Scattering by Electrons, Atoms and Lattices 137

scattered waves

Intensity

a sin 2 ϕ=2π

a −x

a(1 −cos 2θ)

λ Fig. 7.3 Top – five equally spaced points producing five spherical waves as a result of elastic

scattering of the single incident wave. Bottom – the resultant scattered amplitude as a function of the phase angle, ϕ. In this geometry, the phase angle is a function of the spacing between the points, a, the wavelength of the incident beam, λ, and the scattering angle, 2θ. The relationship between the phase ( ϕ) and scattering (θ) angles for the arrangement shown on top is easily derived by considering path difference (a − x) between any pair of neighboring waves, which have parallel propagation vectors.

When more than one point is affected by the same incident wave, the overall scattered amplitude is a result of interference among multiple spherical waves. As established earlier (Fig. 7.2), the amplitude will vary depending on the difference in the phases of multiple waves with parallel propagation vectors but originating from different points.

The phase difference between these waves is also called the phase angle, ϕ. For example, diffraction from a row of five equally spaced points produces a pattern shown schematically in Fig. 7.3, which depicts the intensity of the diffracted beam,

I , as a function of the phase angle, ϕ. The major peaks (or diffraction maxima) in the pattern are caused by the constructive interference, while the multiple smaller peaks are due to the superimposed waves, which have different phases, but are not completely out of phase.

For a one-dimensional periodic structure, the intensity diffracted by the row of N equally spaced points is proportional to the so-called interference function, which is shown in (7.1).

sin 2 N ϕ

I (ϕ) ∝

sin 2

138 7 Fundamentals of Diffraction N=5

ϕ Fig. 7.4 The illustration of the changes in the diffraction pattern from a one-dimensional peri-

odic arrangement of scattering points when the number of points (N) increases from 5 to 20. The horizontal scales are identical, but the vertical scales are normalized for the three plots.

or

I (ϕ) ∝ N 2 , when ϕ = hπ, and h = . . . , −2,−1,0,1,2... The example considered here illustrates scattering from only five points. When

the number of equally spaced points increases, the major constructive peaks become sharper and more pronounced, while the minor peaks turn out to be less and less visible. The gradual change is illustrated in Fig. 7.4, where the resultant intensity from the rows of five, ten and twenty points is modeled as a function of the phase angle using (7.1).

When N approaches infinity, the scattered intensity pattern becomes a periodic delta function, that is, the scattered amplitude is nearly infinite at specific phase an- gles ( ϕ = hπ, h = . . ., −2,−1,0,1,2,...), and is reduced to zero everywhere else. Since crystals contain practically an infinite number of scattering points, which are systematically arranged in three dimensions, they also should produce discrete dif- fraction patterns with sharp diffraction peaks observed only in specific directions. Just as in the one-dimensional case (Fig. 7.2), the directions of diffraction peaks (i.e., diffraction angles, 2 θ) are directly related to the spacing between the diffract- ing points (i.e., lattice points, as established by the periodicity of the crystal) and the wavelength of the used radiation.

7.1.2 Scattering by Atoms and Atomic Scattering Factor

We now consider an atom instead of a stationary electron. The majority of atoms and ions consist of multiple electrons distributed around a nucleus as shown schemati- cally in Fig. 7.5. It is easy to see that no path difference is introduced between the waves for the forward scattered X-rays. Thus, intensity scattered in the direction of

7.1 Scattering by Electrons, Atoms and Lattices 139 k⬙ ≠

Scattered wavefront

k ⬘=

Incident wavefront

1s 2s

δ Fig. 7.5 The schematic of the elastic scattering of X-rays by s electrons illustrating the introduction

nucleus

of a path difference, δ, into the wavefront with a propagation vector k ′′ when it is different from the propagation vector, k, of the incident beam. The distribution of electrons in two s-orbitals is determined from the corresponding wave functions.

the propagation vector of the incident wavefront is proportional to the total number of core electrons, Z, in the atom. For any other angle, 2 θ > 0, that is, when the prop- agation vector of the scattered waves, k ′′ is different from the propagation vector of the incident waves, k, the presence of core electrons results in the introduction of a certain path difference, δ, between the individual waves in the resultant wavefront.

The amplitude of the scattered beam is therefore, a gradually decaying function of the scattered angle and it varies with ϕ and with θ. The intrinsic angular depen- dence of the X-ray amplitude scattered by an atom is called the atomic scattering function (or factor), f , and its behavior is shown in Fig. 7.6 (left) as a function of the phase angle.

Thus, when stationary, periodically arranged electrons are substituted by atoms, their diffraction pattern is the result of a superposition of the two functions, as shown in Fig. 7.6, right. In other words, the amplitude squared of the diffraction pattern from a row of N atoms is a product of the interference function (7.1) and the corre-

sponding atomic scattering function squared, f 2 (ϕ): sin 2 N ϕ

It is worth noting that it is the radial distribution of core electrons in an atom, which is responsible for the reduction of the intensity when the diffraction angle increases. Thus, it is a specific feature observed in X-ray diffraction from ordered arrangements of atoms. If, for example, the diffraction of neutrons is of concern,

140 7 Fundamentals of Diffraction

-3π -2π -π 0 π 2 π 3 π ϕ(-)

ϕ=0 ϕ(+) Fig. 7.6 The schematic showing the dependence of the intensity scattered by an atom, i.e., the

atomic scattering factor, f 2 ∝A 2 , as a function of the phase angle (left), and the resultant decrease of the intensity of the diffraction pattern from the row of five regularly spaced atoms, also as a function of the phase angle (right).

they are scattered by nuclei, which may be considered as points. Hence, neutron- scattering functions (factors) are independent of the diffraction angle and they re- main constant for a given type of nuclei (also see Table 6.2).

7.1.3 Scattering by Lattices

The interference function in (7.2) describes a discontinuous distribution of the scat- tered intensity in the diffraction space. 6 Assuming an infinite number of points in

a one-dimensional periodic structure (N → ∞), the distribution of the scattered in- tensity is a periodic delta-function (as mentioned earlier), and therefore, diffraction peaks occur only in specific points, which establish a one-dimensional lattice in the diffraction space. Hence, diffracted intensity is only significant at certain points, which are determined (also see (7.1), Figs. 7.4 and 7.6) from:

sin 2 h π In three dimensions, a total of three integers (h, k and l ) 7 are required to define

the positions of intensity maxima in the diffraction space:

6 Diffraction space, in which diffraction peaks are arranged into a lattice, is identical to reciprocal space.

7 The integers h, k and l are identical to Miller indices.

7.1 Scattering by Electrons, Atoms and Lattices 141 sin 2 2 N 1 h π sin 2 N 2 k π sin 2 N 3 l π

I (ϕ) ∝ f (ϕ) sin 2 h πϕ sin 2 k π sin 2

(7.4) l π

where N 1 ,N 2 and N 3 are the total numbers of the identical atoms in the correspond- ing directions. On the other hand, when the unit cell contains more than one atom, the individual atomic scattering function f (ϕ) should be replaced with scattering by the whole unit cell, since the latter is now the object that forms a periodic array. The scattering function of one unit cell, F, is called the structure factor or the structure amplitude. It accounts for scattering factors of all atoms in the unit cell, together with other relevant atomic parameters. As a result, a diffraction pattern produced by a crystal lattice may be defined as

2 sin U 2 k π sin 2 U 3 l I π (ϕ) ∝ F (ϕ)

sin 2 U 1 h π

sin 2 l π where U 1 ,U 2 and U 3 are the numbers of the unit cells in the corresponding direc-

tions. The phase angle is a function of lattice spacing (Fig. 7.3), which is a function of

h , k and l. As seen later (Sect. 9.1), the structure factor is also a function of the triplet of Miller indices (hkl). Hence, in general the intensities of discrete points (hkl) in the reciprocal space are given as:

sin 2 U h π sin 2 2 1 U 2 k π sin 2 U 3 l π

I (hkl) ∝ F (hkl)

sin 2 l π The scattered intensity is nearly always measured in relative and not in absolute

sin 2

sin 2 k π

units, which necessarily introduces a proportionality coefficient, C. As we estab- lished earlier, when the phase angle is n π (n is an integer), the corresponding in-

terference functions in (7.6) are reduced to U 2 1 ,U 2 2 and U 2 3 and they become zero everywhere else. Hence, assuming that the volume of a crystalline material pro- ducing a diffraction pattern remains constant (this is always ensured in a properly arranged experiment), the proportionality coefficient C can be substituted by a scale

factor K = CU 2 1 U 2 2 U 2 3 . In addition to the scale factor, intensity scattered by a lattice is also subject to different geometrical effects, 8 G , which are various functions of the diffraction an- gle, θ. All things considered, the intensity scattered by a lattice may be given by the following equation:

(7.7) This is a very general equation for intensity of the individual diffraction (Bragg)

I (hkl) = K × G(θ) × F 2 (hkl)

peaks observed in a diffraction pattern of a crystalline substance, and it is discussed in details in Sect. 8.6, while the geometry of powder diffraction, that is, the direc- tions in which discrete peaks can be observed, is discussed in the following two sections.

8 One of these geometrical effects is the polarization factor introduced earlier in the Thomson’s equation; see Sect. 7.1 and the corresponding footnote (No. 5 on page 136).

142 7 Fundamentals of Diffraction

7.2 Geometry of Diffraction by Lattices

Both direct and reciprocal spaces may be used to understand the geometry of dif- fraction by a lattice. Direct space concepts are intuitive, and therefore, we begin our consideration using physical space. Conversely, reciprocal space is extremely useful in the visualization of diffraction patterns in general and from powders in particu- lar. In this section, therefore, we also show the relationships between geometrical concepts of diffraction in physical and reciprocal spaces.

7.2.1 Laue Equations

The geometry of diffraction from a lattice, or in other words the relationships be- tween the directions of the incident and diffracted beams, was first given by Max von Laue in a form of three simultaneous equations, which are commonly known as Laue equations:

a (cos ψ − cosϕ 1 ) = hλ

b (cos ψ − cosϕ 2 ) = kλ

c (cos ψ − cosϕ 3 ) = lλ

Here a, b and c are the dimensions of the unit cell; ψ 1 −3 and ϕ 1 −3 are the angles that the incident and diffracted beams, respectively, form with the parallel rows of atoms in three independent directions; the three integer indices h, k , and l have the same meaning as in (7.6) and (7.7), that is, they are unique for each diffraction peak and define the position of the peak in the reciprocal space (also see Sect. 1.5), and λ is the wavelength of the used radiation. The cosines, cos ψ i and cos ϕ i , are known as the direction cosines of the incident and diffracted beams, respectively. According to the formulation given by Laue, sharp diffraction peaks can only be observed when all three equations in (7.8) are satisfied simultaneously as illustrated in Fig. 7.7.

Laue equations once again indicate that a periodic lattice produces diffraction maxima at specific angles, which are defined by both the lattice repeat distances (a, b, c)and the wavelength (λ). Laue equations give the most general representation of a three-dimensional diffraction pattern and they may be used in the form of (7.8) to describe the geometry of diffraction from a single crystal.

7.2.2 Braggs’ Law

More useful in powder diffraction is the law formulated by W.H. Bragg and W.L. Bragg (see Footnote 11 on page 41). It establishes certain relationships among the diffraction angle (Bragg angle), wavelength, and interplanar spacing.

According to the Braggs, diffraction from a crystalline sample can be explained and visualized by using a simple notion of mirror reflection of the incident X-ray

7.2 Geometry of Diffraction by Lattices 143

ψ 1 a(cos ψ 1 − cos ϕ 1 ) = hλ

a(cos ψ 1 − cos ϕ 1 )=h λ b(cos ψ 2 − cos ϕ 2 ) = kλ

c(cos ψ 3 − cos ϕ 3 ) = lλ

Incident beam Fig. 7.7 Graphical illustration of Laue equations. A cone of diffracted beams, all forming the same

angle φ 1 with a row of atoms, satisfying one Laue equation is shown on top left. Each of the three cones shown on bottom right also satisfies one of the three equations, while the intersecting cones satisfy either two, or all three equations simultaneously as shown by arrows. A sharp diffraction peak is only observed in the direction of a point where three Laue equations are simultaneously satisfied.

θ, 2θ – Bragg angles

wavefront Reflected

hkl sin θ – path difference 2 ∆ = nλ – constructive interference

2 ∆ = 2d

Incident wave front

(hkl ) ∆ d

hkl

(hkl )

2 θ Braggs’ law: n λ= 2d hkl sin θ hkl

Fig. 7.8 Geometrical illustration of the Braggs’ law.

beam from a series of crystallographic planes. As established earlier (Sect. 1.4.1), all planes with identical triplets of Miller indices are parallel to one another and they are equally spaced. Thus, each plane in a set (hkl) may be considered as a separate scattering object. The set is periodic in the direction perpendicular to the planes and the repeat distance in this direction is equal to the interplanar distance

d hkl . Diffraction from a set of equally spaced objects is only possible at specific angle(s) as we already saw in Sect. 7.1. The possible angles, θ, are established from the Braggs’ law, which is derived geometrically in Fig. 7.8.

Consider an incident front of waves with parallel propagation vectors, which form an angle θ with the planes (hkl). In a mirror reflection, the reflected wavefront will also consist of parallel waves, which form the same angle θ with all planes. The path differences introduced between a pair of waves, both before and after they are

144 7 Fundamentals of Diffraction reflected by the neighboring planes, Δ, are determined by the interplanar distance

as Δ=d hkl sin θ. The total path difference is 2Δ, and the constructive interference is observed when 2 Δ = nλ, where n is integer and λ is the wavelength of the incident wavefront. This simple geometrical analysis results in the Braggs’ law:

(7.9) The integer n is known as the order of reflection. Its value is taken as 1 in all

2d hkl sin θ hkl = nλ

calculations, since orders higher than 1 (n > 1) can always be represented by first- order reflections (n = 1) from a set of different crystallographic planes with indices that are multiples of n because:

(7.10) and for any n > 1, (7.9) is simply transformed as follows:

d hkl = nd nh ,nk,nl

2d hkl sin θ hkl =n λ ⇒ 2d nh ,nk,nl sin θ nh ,nk,nl =λ

7.2.3 Reciprocal Lattice and Ewald’s Sphere

The best visual representation of the phenomenon of diffraction has been introduced by P.P. Ewald (see Footnote 11 on page 11). Consider an incident wave with a certain

propagation vector, k 0 , and a wavelength, λ. If the length of k 0 is selected as the inverse of the wavelength

|k 0 | = 1/λ

(7.12) then the entire wave is fully characterized, and it is said that k 0 is its wavevector.

When the primary wave is scattered elastically, the wavelength remains constant. Thus, the scattered wave is characterized by a different wavevector, k 1 , which has the same length as k 0 :

(7.13) The angle between k 0 and k 1 is 2 θ (Fig. 7.9, left). We now overlap these two

|k 1 | = |k 0 | = 1/λ

wavevectors with a reciprocal lattice (Fig. 7.9, right) such that the end of k 0 coin- cides with the origin of the lattice. As shown by Ewald, diffraction in the direction of k 1 occurs only when its end coincides with a point in the reciprocal lattice. Con- sidering that k 0 and k 1 have identical lengths regardless of the direction of k 1 (the direction of k 0 is fixed by the origin of the reciprocal lattice), their ends are equidis- tant from a common point, and therefore, all possible orientations of k 1 delineate a sphere in three dimensions. This sphere is called the Ewald’s sphere, and it is shown schematically in Fig. 7.10. Obviously, the radius of the Ewald’s sphere is the same

as the length of k 0 , in other words, it is equal to 1 /λ. The simple geometrical arrangement of the reciprocal lattice, Ewald’s sphere, and three vectors (k 0 ,k 1 , and d ∗ hkl ) in a straightforward and elegant fashion yields

7.2 Geometry of Diffraction by Lattices 145

b*

d* (13)

Diffracted beam

k 1 k 2 1 2 a*

Incident k 0 beam

k 0 Origin

Reciprocal lattice

Fig. 7.9 The incident (k 0 ) and diffracted (k 1 ) wavevectors originating from a common point (left) and the same two vectors overlapped with the two-dimensional reciprocal lattice, which is based on the unit vectors a ∗ and b ∗ (right). The origin of the reciprocal lattice is chosen at the end of k 0 . When diffraction occurs from a point in the reciprocal lattice, e.g., the point (¯13), the corresponding reciprocal lattice vector d ∗ hkl [e.g., d ∗ (¯13) ] extends between the ends of k 0 and k 1 .

Ewald’s

b*

sphere

Diffracted beam

Reciprocal lattice

Fig. 7.10 The visualization of diffraction using the Ewald’s sphere with radius 1 /λ and the two- dimensional reciprocal lattice with unit vectors a ∗ and b ∗ . The origin of the reciprocal lattice

is located on the surface of the sphere at the end of k 0 . Diffraction can only be observed when a reciprocal lattice point, other than the origin, intersects with the surface of the Ewald’s sphere [e.g., the point (¯13)]. The incident and the diffracted beam wavevectors, k 0 and k 1 , respectively, have common origin in the center of the Ewald’s sphere. The two wavevectors are identical in length, which is the radius of the sphere. The unit cell of the reciprocal lattice is shown using double lines.

146 7 Fundamentals of Diffraction Braggs’ equation. From both Figs. 7.9 and 7.10, it is clear that vector k 1 is a sum

of two vectors, k 0 and d ∗ hkl :

(7.14) Its length is known (1/λ) 9 and its orientation with respect to the incident

k 1 =k 0 +d ∗ hkl

wavevector, that is, angle θ, is found from simple geometry after recalling that |d ∗ | = 1/d:

|k 1 |sinθ = |k 0 |sinθ = |d ∗ 2 | ⇒ 2d sinθ = λ (7.15) The Ewald’s sphere and the reciprocal lattice are essential tools in the visual-

ization of the three-dimensional diffraction patterns from single crystals, as illus- trated in the next few paragraphs. They are also invaluable in the understanding of the geometry of diffraction from polycrystalline (powder) specimens, which is explained in Chap. 8.

Consider a stationary single crystal, in which the orientation of basis vectors of the reciprocal lattice is established by the orientation of the corresponding crystal- lographic directions with respect to the external shape of the crystal, as shown in Fig. 7.11. Thus, when a randomly oriented single crystal is irradiated by monochro-

c*

Ewald’s Reciprocal sphere

lattice

b To area

a*

detector

Incident beam

c Single crystal

b*

Fig. 7.11 The illustration of a single crystal showing the orientations of the basis vectors corre- sponding to both the direct (a, b and c) and reciprocal (a ∗ ,b ∗ and c ∗ ) lattices and the Ewald’s sphere. The reciprocal lattice is infinite in all directions but only one octant (where h > 0, k > 0 and l > 0) is shown for clarity.

9 The lengths of the propagation vectors k 0 and k 1 may also be defined in terms of their wavenum- bers: |k 0 | = |k 1 | = 2π/λ. Equation (7.15) may then be rewritten as |d ∗ | = |Q| = 4πsinθ/λ, thus

defining the so-called Q-vector, which is often used to represent diffraction data in synchrotron radiation experiments. It is also worth noting that since d ∗ is a vector in reciprocal lattice, the value of sin θ/λ is independent of the wavelength.

7.2 Geometry of Diffraction by Lattices 147 f(c*, l, λ)

Fig. 7.12 The two-dimensional diffraction patterns from stationary (left) 10 and rotating (right) 11 single crystals recorded using a CCD detector. The incident wavevector is perpendicular to both the detector and the plane of the figure. The dash-dotted line on the right shows the rotation axis, which is collinear with c ∗ .

matic X-rays, only a few, if any (also see Fig. 7.10), points of the reciprocal lattice will coincide with the surface of the Ewald’s sphere. 12 This occurs because first, the sphere has a constant radius determined by the wavelength, and second, the distrib- ution of the reciprocal lattice points in three dimensions is fixed by both the lattice parameters and the orientation of the crystal. The resultant diffraction pattern may reveal just a few Bragg peaks, as shown in Fig. 7.12 (left).

Many more reciprocal lattice points are placed on the surface of the Ewald’s sphere when the crystal is set in motion, for example, when it is rotated around an axis. The rotation of the crystal changes the orientation of the reciprocal lattice but the origin of the latter remains aligned with the end of the incident wavevector. Hence, all reciprocal lattice points with |d ∗ | ≤ 2/λ will coincide with the surface of the Ewald’s sphere at different angular positions of the crystal. When the rotation axis is collinear with one of the crystallographic axes and is perpendicular to the in- cident beam, the reciprocal lattice points form planar intersections with the Ewald’s

10 The single crystal is triclinic Pb 3 F 5 (NO 3 ): space group P¯1, a = 7.3796(6), b = 12.1470(9), c = 16.855(1) ˚ A, α = 100.460(2), β = 90.076(1), γ = 95.517(1) ◦ . [D.T. Tran, P.Y. Zavalij and

S.R.J. Oliver, A cationic layered material for anion-exchange, J. Am. Chem. Soc. 124, 3966 (2002)] 11 The single crystal is orthorhombic, FePO 4 ·2H 2 O: space group Pbca, a = 9.867(1), b =

10 .097(1), c = 8.705(1) ˚ A. [Y. Song, P.Y. Zavalij, M. Suzuki, and M.S. Whittingham, New iron(III) phosphate phases: Crystal structure, electrochemical and magnetic properties, Inorg. Chem. 41, 5778 (2002)].

12 When a stationary single crystal is irradiated by white, polychromatic X-rays, a single Ewald’s sphere shown in Fig. 7.11 becomes a continuum of spheres. Different points of reciprocal lattice

will then rest on surfaces of different Ewald’s spheres, thus producing a much richer diffraction pattern. Thus technique is known as Laue technique, and it is most often employed for examination of symmetry and orientation of single crystals.

148 7 Fundamentals of Diffraction sphere (Fig. 7.11, dash-dotted lines). The planes are mutually parallel and equidis-

tant, and the resultant diffraction pattern 13 is similar to that illustrated in Fig. 7.12 (right).

7.3 Additional Reading

1. International Tables for Crystallography, vol. A, Fifth Revised Edition, Theo Hahn, Ed. (2002); vol. B, Third edition, U. Shmueli, Ed. (2008); vol. C, Third edition, E. Prince, Ed. (2004). All volumes are published jointly with the International Union of Crystallography (IUCr) by Springer. Complete set of the International Tables for Crystallography, Vol. A-G, H. Fuess, T. Hahn, H. Wondratschek, U. M¨uller, U. Shmueli, E. Prince, A. Authier, V. Kopsk´y, D.B. Litvin, M.G. Rossmann, E. Arnold, S. Hall, and B. McMahon, Eds., is available online as eReference at http://www.springeronline.com.

2. R.B. Neder and Th. Proffen, Teaching diffraction with the aid of computer simulations, J. Appl. Cryst. 29, 727 (1996); also see Th. Proffen and R.B. Neder. Interactive tutorial about diffraction on the Web at http://www.lks.physik.uni-erlangen.de/diffraction/.

3. P. A. Heiney, High resolution X-ray diffraction. Physics department and laboratory for re- search on the structure of matter. University of Pennsylvania. http://dept.physics.upenn.edu/ ∼ heiney/talks/hires/hires.html

4. Electron diffraction techniques. Vol. 1, 2. J. Cowley, Ed., Oxford University Press. Oxford (1992). 5. R. Jenkins and R.L. Snyder, Introduction to X-ray powder diffractometry. Wiley, New York (1996). 6. Modern powder diffraction. D.L Bish and J.E. Post, Eds. Reviews in Mineralogy, Vol. 20. Min- eralogical Society of America, Washington, DC (1989).

7.4 Problems

1. A student prepares a sample and collects a powder diffraction pattern on an in- strument that is available in the laboratory overseen by his major professor. The student then takes the same sample to a different laboratory on campus and col- lects a second set of powder diffraction data. When he comes back to his office, he plots both patterns. The result is shown in Fig. 7.13. Analyze possible sources of the observed differences.

2. The following is the list of five longest interplanar distances possible in a crystal lattice of some material: 4.967, 3.215, 2.483, 2.212, and 1.607 ˚

A. Calculate Bragg angles (2 θ) at which Bragg reflections may be observed when using Cr Kα 1 or Cu K α 1 radiation.

13 This type of the diffraction pattern enables one to determine the lattice parameter of the crystal in the direction along the axis of rotation. It is based on the following geometrical consideration:

the distance between the planar cross-sections of the Ewald’s sphere in Fig. 7.11 equals c ∗; the corresponding diffraction peaks are grouped into lines (see Fig. 7.12, right), and the distance be- tween the neighboring lines is a function of c ∗; the distance from the crystal to the detector, l, and the wavelength, λ.

200 Intensity, Y (arb. units)

Bragg angle, 2θ (deg.)

Fig. 7.13 Two powder diffraction patterns collected by a student using the same sample but two different powder diffractometers.

3. Researcher finished collecting a powder diffraction pattern of an unknown crys- talline substance. She used Cu K α radiation, λ = 1.54178 ˚

A. The first Bragg peak is observed at 2 θ = 9.76 ◦ . Based on this information she makes certain conclusions regarding the length of at least one of the three unit cell edges. What are these con- clusions?