 1.1 Types of Solids
 1.2 Order – Spatial and Dimensional
 1.3 Symmetry in Crystals
 1.3.1 Translation Symmetry
 1.3.2 Crystal Systems in 2D
 1.3.3 Crystal Systems in 3D
 1.3.4 Bravais Lattices in 2D
 1.3.5 Bravais Lattices in 3D
 1.3.6 Lattice + Basis = Crystal Structure
 1.3.7 Miller Planes
 1.4 Xray Diffraction
 1.4.1 Interference of Waves
 1.4.2 Bragg's Law for Diffraction
 1.4.3 Powder Xray Diffraction
 1.4.4 Systematic Absence
 1.4.5 Structure Factor
 1.4.6 von Laue Condition for Xray Diffraction
 1.4.7 Reciprocal Lattice
 1.4.8 Ewald Construction
 1.4.9 Structure Factor in Terms of the Reciprocal Lattice Vector
 1.4.10 Basic Concepts of Xray Structure Solution and Refinement
 1.5 Neutron and Electron Diffraction
 1.5.1 Neutron Diffraction
 1.5.2 Electron Diffraction
 1.6 Common Crystal Structure Motifs
 1.7 Quasicrystals: A Brief Note
 Note added after first publication
1: Solid State Structure

Published:14 Sep 2022

Product Type: Textbooks
Core Concepts for a Course on Materials Chemistry, The Royal Society of Chemistry, 2022, pp. 134.
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Fundamental insight into structure–property or form–function correlations is essential to gain an understanding of materials, design new systems and realize their potential applications. The composition and structure therefore form the rational starting point for studying and analyzing materials, and explaining or predicting their properties and functions. Materials in the form of solids are the most extensively studied and the best understood; they provide the logical foundation and framework for the study of materials. A brief outline of the nature of solids is presented in this chapter, followed by an overview of order in solids. This leads naturally to the study of crystalline structure and the closely related Xray diffraction problem.
1.1 Types of Solids
Study of any field is greatly facilitated by a logical and systematic classification of the objects of interest in the study. This is particularly true when we learn about matter and its various characteristics and manifestations. Solids form a central theme in the study of materials and hence understanding their structure is clearly the starting point for learning about them and their attributes and functions.
Classification of solids can take different routes. Some of the general criteria that can be chosen to group solids into different families or classes are the:

elemental composition of the material

nature of interactions between the constituent atoms, ions or molecules

prominent properties of the materials

functional attributes or applications
The nature of interactions between the building blocks is a commonly used criterion for classifying solids, as it helps in understanding the structure of the material, and the structure–property correlations. Figure 1.1 is a schematic representation of the common types based on such a classification. The general classes of solids are:

ionic: popular examples being NaCl, KCl and MgO

covalent: Si and diamond are the classic systems of this kind

metallic: very familiar in everyday life, Cu and Fe being common examples

molecular (sometimes described also as van der Waals solids): Ar, I_{2} and C_{60} are typical members of this family
1.2 Order – Spatial and Dimensional
Order in a solid implies predictability of atom/molecule position and molecule orientation; translational periodicity (see the schematic diagram in Figure 1.2): if moving along a specific direction, an atom/molecule is repeated at a distance r, then it will be found to repeat along that direction at integral multiples of the same distance (2r, 3r, 4r, …).
Even though order is a commonly used term, it can be technically, a relatively complex concept. For example, it can be visualized at different levels:

Spatial: depending on the extent in space (length scale) up to which order (typically, in a 3dimensional (3D) structure) persists, the following can be identified

crystal (designated also as single crystal, millimeters or higher)

microcrystal (micrometers)

nanocrystal (nanometers)

amorphous solid (none)


Dimensional: depending on the number of dimensions in which, the longrange order persists

crystal (typically 3D)

liquidcrystal (with orientational order, typically along 1 or 2dimensions (1D, 2D)); can be further classified into various types such as nematic, smectic, and cholesteric

liquid (none, isotropic – implying no longrange order along any dimension)

1.3 Symmetry in Crystals
The basic point group symmetry operations are defined based on the condition that at least one point in space is not moved during the operation. These operations, often discussed in the context of molecular symmetry, are:

identity (E)

rotation (C _{ n })

reflection (σ)

rotation–reflection (S _{ n })

inversion (i)
In addition to the above point group symmetries, crystals with their periodic structure possess translation symmetry. In fact, this is a defining characteristic of a crystal. In order to understand the point group and translation symmetries, consider a 2D square lattice; note that the lattice extends to infinity along the x and y directions. Figure 1.3 shows the lattice; the rotation (by 90°) and translation (by vectors connecting the nearest points, called unit cell vectors) operations leave the lattice invariant (unchanged); therefore these are typical symmetry operations. The operations can be visualized by coloring some lattice points as shown in the figure.
Combination of rotation and translation (along the rotation axis, by a fraction of the unit cell vector) gives a new symmetry operation, ‘screw rotation’ in crystals. Similarly, combination of reflection and translation (along an axis parallel to the plane of reflection) gives ‘glide reflection’.
1.3.1 Translation Symmetry
Translation symmetry imposes restrictions on the possible rotation symmetries. This can be geometrically illustrated as shown in Figure 1.4; a is the unit cell length (n has to be an integer, positive, negative or zero, due to the translational symmetry) and θ is the rotation angle allowed by symmetry. Table 1.1 shows that rotations of order 1, 2, 3, 4, and 6 only are compatible with the translational symmetry of the crystal lattice.
n  3  2  1  0  − 1 
θ (°)  180  120  90  60  0 
Order of rotation  2  3  4  6  1 
n  3  2  1  0  − 1 
θ (°)  180  120  90  60  0 
Order of rotation  2  3  4  6  1 
1.3.2 Crystal Systems in 2D
Considering all the possible point group symmetries with the restriction imposed by the translational symmetry, the only crystal systems that can be visualized in 2D are the following, where a and b are the unit cell lengths, and γ is the angle between the unit cell axes (shown schematically in Figure 1.5).

square (a = b, γ = 90°)

rectangular (a ≠ b, γ = 90°)

hexagonal (a = b, γ = 120°)

oblique (a ≠ b, γ ≠ 90°)
It is a useful exercise to list the symmetry elements in each case above. It is also an interesting task to verify and confirm that no periodic lattices in 2D are possible with a set of point group symmetries that are different from the four cases above.
1.3.3 Crystal Systems in 3D
Extending the logic used earlier for the 2D systems, it is found that there are seven crystal systems in 3D. They are the following, where a, b, and c are the unit cell lengths, and α, β, and γ are the angles between the unit cell axes b and c , a and c , and a and b respectively (Figure 1.6).

cubic (a = b = c, α = β = γ = 90°)

tetragonal (a = b ≠ c, α = β = γ = 90°)

orthorhombic (a ≠ b ≠ c, α = β = γ = 90°)

monoclinic (a ≠ b ≠ c, α = γ = 90° ≠ β)

triclinic (a ≠ b ≠ c, α ≠ β ≠ γ)

trigonal (a = b = c, α = β = γ < 120°, ≠ 90°)

hexagonal (a = b ≠ c, α = β = 90°, γ = 120°)
1.3.4 Bravais Lattices in 2D
Adding no further point group symmetries, but incorporating any additional translational symmetry operations possible to the crystal systems, one ends up with five Bravais lattices in 2D. They are the following (Figure 1.7(a)).

square

rectangular

centered rectangular

hexagonal

oblique
It may be noted that the rectangular and centered rectangular lattices have exactly the same set of point group symmetries, but the latter has a new translational symmetry; one way to visualize this symmetry is to imagine the lattice translation depicted in Figure 1.7(b). A contrasting exercise is to imagine a centered square lattice, and realize that it is just another simple lattice (Figure 1.7(c)), and therefore not a new Bravais lattice.
1.3.5 Bravais Lattices in 3D
Similar to the case of the Bravais lattices in 2D, these follow from the crystal systems shown in Figure 1.6. The 14 Bravais lattices are listed in Table 1.2, with the standard notations, P (primitive), F (facecentered), I (bodycentered), and C (edgecentered).
Crystal system .  Bravais lattices .  Number . 

Cubic  P, F, I  3 
Tetragonal  P, I  2 
Orthorhombic  P, C, F, I  4 
Monoclinic  P, C  2 
Triclinic  P  1 
Trigonal  P  1 
Hexagonal  P  1 
Total  14 
Crystal system .  Bravais lattices .  Number . 

Cubic  P, F, I  3 
Tetragonal  P, I  2 
Orthorhombic  P, C, F, I  4 
Monoclinic  P, C  2 
Triclinic  P  1 
Trigonal  P  1 
Hexagonal  P  1 
Total  14 
As an example, consider the three Bravais lattices under the cubic crystal system (Figure 1.8(a)); they all belong to the same point group, O _{h} and hence have the same set of point group symmetries. However, they possess different sets of translational symmetries; for example, the bcc lattice has a characteristic half bodydiagonal translation.
It is an interesting exercise to understand why the facecentered and bodycentered lattices are the same in the tetragonal system, but not in the cubic system. This has to do with the fact that a = b in tetragonal, and the F and I lattices are the same with only a rotation of the a, b axes system, and change of magnitude of the parameter a (Figure 1.8(b)). The orthorhombic system has a new lattice, C; if a similar situation is imagined in the tetragonal system, it would simply be another tetragonal P lattice with a different a = b value (extension of the idea illustrated in Figure 1.7(c)).
The hierarchy of crystal symmetries at this stage can be summarized as shown in Table 1.3.
Point group operations  7 Crystal systems 
Point group operation + translation symmetries  14 Bravais lattices 
Point group operations  7 Crystal systems 
Point group operation + translation symmetries  14 Bravais lattices 
Before proceeding further on the symmetry in crystals, a little more insight into the concept of a Bravais lattice is useful. In 3D, a Bravais lattice can be imagined as arising from the collection of all points represented by the set of vectors,
where n_{1}, n_{2}, n_{3} are integers and , , are primitive vectors that determine the symmetry of the lattice.Primitive vectors for a cubic lattice are,
; ; where a is the unit cell length and , , are the unit vectors along the three Cartesian axes.
Similarly, the primitive vectors for the facecentered cubic (fcc) and bodycentered cubic (bcc) lattices are (these are convenient, but not unique choices):
fcc: ; ;
bcc: ; ;
The primitive vectors define a primitive unit cell; Figure 1.9a shows the fcc cell.
Primitive vectors for the tetragonal and orthorhombic lattices are:
tetragonal: ; ;
orthorhombic: ; ;
As mentioned above, the unit cell can be chosen in multiple ways. A special choice is the Wigner–Seitz cell, constituted by all points that are closer to one lattice point than any other; it is the innermost region enclosed by perpendicular planes (lines in 2D) bisecting the lines connecting a lattice point to all the neighbors (Figure 1.9(b)).
Every lattice point in a Bravais lattice has identical environment; observation of the surrounding, from any point is identical. This aspect can be used to distinguish a nonBravais lattice from a Bravais lattice.

For example, a honeycomb lattice (Figure 1.10) is not a Bravais lattice, as the points ○ (red) and ○ (blue) do not have identical environments; however, a combination of two of these points leads to the hexagonal Bravais lattice in 2D.

A cubic closepacked (ccp or fcc) lattice is a Bravais lattice, but a hexagonal closepacked (hcp) lattice is not; Figure 1.11 illustrates this interesting point.
1.3.6 Lattice + Basis = Crystal Structure
The discussion so far has looked at the symmetries of the lattice; ‘lattice’ is simply a set of points in space described by a set of coordinates, two in 2D or three in 3D. A real crystal is made up of atoms, ions, or molecules; the molecule can be as small as H_{2} or a large protein.

The object or set of objects (with atoms at specific locations with respect to each other) placed on the lattice points, is described technically as the ‘basis’.

A crystal consists of the basis organized on a lattice with a specified symmetry.
The basis can be spherical (perfectly symmetric), e.g., a single atom, or nonspherical (with lower symmetry than a sphere) if it has more than one atom, a molecule, etc.

Symmetry of the unit cell remains unchanged when a spherical basis is added.

However, symmetry of the unit cell is reduced when a nonspherical basis is added; it goes to one of the subgroups of the original point group.

The example below, of the square lattice unit cell (Figure 1.12(a)) illustrates the point.
In a similar way, subgroups of the various crystal systems in 3D (Figure 1.6) can be identified. The different point groups based on the cubic system are shown in Figure 1.12(b).
The final hierarchy of crystal symmetries, building on the classification shown earlier in Table 1.3, can be summarized as shown in Table 1.4.
.  Lattice + spherical basis .  Lattice + nonspherical basis . 

Point group operations  7 crystal systems  32 crystallographic point groups 
Point group operations + translation symmetries  14 Bravais lattices  230 space groups 
.  Lattice + spherical basis .  Lattice + nonspherical basis . 

Point group operations  7 crystal systems  32 crystallographic point groups 
Point group operations + translation symmetries  14 Bravais lattices  230 space groups 
1.3.7 Miller Planes
Miller indices are a convenient way of designating planes in the crystal lattice; this will have a direct bearing on the discussion and analysis of the Xray diffraction process later. We look first, at 2D lattices in which they should strictly be called Miller lines. The examples in Figure 1.13 show the designations and the distance between the lines. Note that the distances decrease with increasing values of the indices.
The logical steps involved in naming the Miller planes can be explained using the example shown in Figure 1.13(c).

Consider the point at an arbitrary origin (0, 0) in the x, y axes framework.

The ‘plane’ intercepts the axes at fractional coordinates 1/2 and 1/3 respectively.

Take the inverse values; if not integers, scale them to the lowest set of integers: 2, 3.
The distance between the planes can be calculated using simple geometry. In the case of the square lattice, following the above protocol, a ‘plane’ (h k) will intercept the axes at a/h and a/k. The distance between the planes, d, can be obtained using the geometry of the rightangled triangle as shown in Figure 1.13(e).
For a cubic lattice (3D), the lines in Figure 1.13(c) would extend parallel to the z axis to form planes. Correspondingly,

the intercepts will be a/2, a/3 and ∞

the Miller plane will be designated as (2 3 0)
Some common Miller planes are shown in Figure 1.14. Based on the above discussion, the interplanar distance, d for an (h k l) plane of a cubic lattice of unit cell length a, can be shown to be:
1.4 Xray Diffraction
Xray diffraction is the most popular and efficient technique to determine precisely, the structure of crystals. Xrays are electromagnetic waves with wavelength typically in the range of ∼0.5–10 Ångströms (1 Å = 10^{−10} m). Waves can be represented mathematically as a function of displacement, x and time, t:
where A = amplitude, λ = wavelength, υ = frequency. At any time t, f(x) can be visualized as shown in Figure 1.15(a). Superposition of waves is shown schematically using the cases where they are in phase and out of phase by π/2 or π (Figure 1.15(b)–(d)).1.4.1 Interference of Waves
Waves with different path lengths in a medium and hence relative phases (phase differences) interfere, leading to the enhancement or annihilation as shown in Figure 1.15(e); this follows directly from the illustration in Figure 1.15(b)–(d).
When shifted by integral multiples of λ, the waves add to each other fully, and when shifted by halfinteger multiples of λ, they annihilate each other fully. Shifts of intermediate magnitude will lead to partial destructive interference.
1.4.2 Bragg's Law for Diffraction
Bragg's law is essentially the phenomenon described in Section 1.4.1. in action. Xray (could equally well be electron or neutron) waves reflected from parallel planes (h k l) of a crystal undergo interference, with nonzero intensity emerging only when the path difference between the two is an integral multiple of the wavelength, λ (Figure 1.16); the condition for this is related to the angle of incidence, θ and the interplanar distance, d _{ hkl }.

It is an interesting question to consider what happens if the beam deviates slightly from the Bragg angle, θ; will there be partial interference or complete destructive interference? (This has relevance to line broadening in very small crystals).

Another interesting point to note: the Bragg's law equation shows that a reflection for n = 2 is equivalent to an n = 1 reflection from a Miller plane with spacing of d _{ hkl }/2; therefore, in practice one always works with just first order (n = 1) reflections.
1.4.3 Powder Xray Diffraction
The powder Xray diffraction experiment involves scattering a monochromatic Xray beam from a collection of randomly oriented microcrystals. The diffracted beams satisfying the Bragg condition for each Miller plane will form a cone (due to the random orientation of the microcrystals) at scattering angle, 2θ with respect to the incident beam (Figure 1.17).
A typical example of a powder Xray diffraction pattern is provided in Figure 1.18. The pattern corresponds to NaCl; incidentally, the first crystal structure determination by Xray diffraction (Bragg and Bragg) was that of NaCl.

The basic analysis of the powder Xray diffraction pattern involves fitting the observed peak positions (in terms of the 2θ values) to Miller planes corresponding to a unit cell belonging to a specific crystal system. The process is called indexing.

The different crystal systems, usually starting with the highest symmetry one (cubic), are explored. In each case, there is a relationship between the unit cell parameters, the Miller plane indices (h k l) and the interplanar spacing d _{ hkl }; the equation for the cubic system, (Section 1.3.7) is the simplest.

Indexing involves the determination of a common set of a, b, c, α, β, γ within a chosen system that satisfies the observed diffraction peaks (just the 2θ values), so that at the end of this process, one knows the crystal system and unit cell dimensions of the crystal under investigation. Careful analysis of the indexing can also reveal the relevant Bravais lattice, as will be demonstrated in the following sections.
The essential idea of indexing can be described using the NaCl diffraction data, as shown in Table 1.5.

Using the 2θ values from the experimental diffraction pattern and the Xray wavelength, the corresponding d values are calculated (n in the Bragg equation is taken as 1; see Section 1.4.2).

Various combinations of integral values of h, k, and l (starting with, say, (1 0 0)) are tried for each d value to get the same value of (Section 1.3.7) within acceptable numerical deviations (this is essentially a statistical fitting protocol).

In the NaCl case, this works out fine as the crystal does belong to a cubic system.

If the crystal does not belong to the cubic system, then this fitting would never be satisfactory, and one would have to move on to the next system, say tetragonal, and try a similar fitting, now with two unit cell parameters a and c.

The process can go on until the triclinic system where the six independent unit cell parameters will need to be fitted.

A careful observation reveals that in Table 1.5, all the (h k l) sets contain either all odd or all even numbers; cross combinations (for example, (1 0 0)) are absent. This is not accidental, and is determined by the Bravais lattice as explained below.
2θ(deg.) .  d(Å) .  h .  k .  l .  (h ^{2} + k ^{2} + l ^{2})^{½} .  a(Å) . 

27.367  3.256  1  1  1  1.732  5.639 
31.704  2.820  2  0  0  2.000  5.640 
45.448  1.994  2  2  0  2.828  5.639 
53.869  1.700  3  1  1  3.317  5.639 
56.473  1.628  2  2  2  3.464  5.639 
66.227  1.410  4  0  0  4.000  5.640 
73.071  1.294  3  3  1  4.359  5.641 
75.293  1.261  4  2  0  4.472  5.639 
83.992  1.151  4  2  2  4.899  5.639 
90.416  1.085  3  3  3  5.196  5.638 
90.416  1.085  5  1  1  5.196  5.638 
2θ(deg.) .  d(Å) .  h .  k .  l .  (h ^{2} + k ^{2} + l ^{2})^{½} .  a(Å) . 

27.367  3.256  1  1  1  1.732  5.639 
31.704  2.820  2  0  0  2.000  5.640 
45.448  1.994  2  2  0  2.828  5.639 
53.869  1.700  3  1  1  3.317  5.639 
56.473  1.628  2  2  2  3.464  5.639 
66.227  1.410  4  0  0  4.000  5.640 
73.071  1.294  3  3  1  4.359  5.641 
75.293  1.261  4  2  0  4.472  5.639 
83.992  1.151  4  2  2  4.899  5.639 
90.416  1.085  3  3  3  5.196  5.638 
90.416  1.085  5  1  1  5.196  5.638 
1.4.4 Systematic Absence
The Xray diffraction patterns observed for crystals with primitive cubic, bodycentered cubic and facecentered cubic lattices (Figure 1.19) demonstrate the idea of systematic absences.

The primitive cube with only the vertices of the cube as lattice points, shows the diffraction from all the possible Miller planes.

The systematic absences in the bcc and fcc lattices can be visualized as arising due to the destructive interference between the Xrays scattered from the lattice points at the vertices and at the body center/face centers respectively.

A deeper look at the interference of waves scattered from specifically related lattice positions is required to understand this phenomenon better.

The idea of structure factor discussed in the following section provides a general framework to understand this.
1.4.5 Structure Factor
A physical approach to construct the structure factor for the Xray scattering from a Miller plane (h k l) involves the path and hence phase difference between the Xrays (satisfying the Bragg condition for that plane) reflected by the different atoms of the basis. The atom positions are welldefined with respect to each other. A different approach will be discussed later in Section 1.4.9. As an illustrative case, we first look at the 1D problem with a unit cell of length a (Figure 1.20) along the xaxis.
d_{100} = a; for plane (h 0 0) that is parallel to (1 0 0),
The reflection from (h 0 0) is equivalent to the hth order reflection from (1 0 0) (Section 1.4.2); this follows from Bragg's law:
Path difference between the waves 1 and 2 (1′ 2′) satisfying the Bragg condition, Δl = δ_{2} = λ.
Using similar triangles, the path difference between the waves 1 and 3 (1′ 3′),
Phase difference for 1′ 3′, [note: Δl = λ ⇒ Δφ = 2π].
For the 3D case, phase difference between the two diffracted Xrays can be worked out in a similar way to give Δφ = 2π (hx + ky + lz). Two waves with the same frequency, scattered from different atomic layers with phases φ _{1} and φ _{2}, and amplitudes A _{1} and A _{2} (for example, if the atoms are not the same) can add up to form a wave 3 as shown in Figure 1.21(a).

Scattered Xray intensity will be due to the sum of the contributions from the different individual waves.

A wave can be represented as a vector in complex space (Figure 1.21(b)), written as A(cosφ + isinφ) = Ae^{ iφ }
A measure of the scattering amplitude of the Xray from an atom is the atomic scattering factor, defined as:
This is clearly related to the electron density on the atom. Xray wave scattered from an atom, n (scattering factor, f _{ n }), located at a position with coordinates x _{ n }, y _{ n }, z _{ n }, will contribute to the structure factor for the Miller plane (h k l) with phase 2π(hx _{ n } + ky _{ n } + lz _{ n }) and amplitude f _{ n }. The structure factor can now be written as:
where n represents the atoms in the unit cell. In this expression, information on the atom type comes from f _{ n } and the atom position, from the coordinates x _{ n }, y _{ n }, z _{ n }. The physical significance of the structure factor is that the intensity of the scattered Xray from the (h k l) plane is:I _{ hkl }∝S $ hkl * $·S _{ hkl } (its utilization will be discussed further in Section 1.4.10)
The relevance of the structure factor can be illustrated by using it to understand the systematic absence in a bcc lattice as follows:
Imagine the bcc lattice unit cell to be a simple (primitive) cubic cell with two atoms in the basis, A at position (0 0 0) and B at ; these are the fractional coordinates expressing the position with respect to the unit cell axis.
As atoms at A and B are the same in a bcc lattice, f_{A} = f_{A} = f , and therefore,
S_{hkl} = f (1 + e^{πi (h + k + l)})
As e^{nπi} = ±1 for n even/odd, the equation for the structure factor shows that
S_{hkl} = 2f when h + k + l is even
S_{hkl} = 0 when h + k + l is odd
This explains the absence of Xray scattering from those Miller planes which satisfy h + k + l = odd in a bcc lattice.
1.4.6 von Laue Condition for Xray Diffraction
The Bragg's law for Xray diffraction (Section 1.4.2) can be recast in the language of wave vectors and translations in momentum space (reciprocal lattice space) obtained by the Fourier transform of the periodic real lattice space. The meaning of the latter will become clearer through the following discussion.
A wave can be described by a wave vector, defined as , where λ is the wavelength and is the unit vector along the direction of propagation of the wave. Note that represents a momentum vector, since . Consider the Xray scattering in a manner very similar to that described in Figure 1.16, but now represented using the incident wave vector, and scattered wave vector, (from a pair of atoms at positions related by a lattice vector, ); this is illustrated in Figure 1.22.
Projection of the vector on the vector (direction of which is represented by the unit vector ) is given by ; similarly on the vector is given by .
The total path difference of the second wave with respect to the first one is given by .
Since and , .
For constructive interference, Δl = mλ where m is an integer.
Therefore, .
If represents a general translation vector in the Bravais lattice, the condition for constructive interference can be generalized as: .
Equivalently: , as e^{2πmi} = [cos(2mπ) + i sin(2mπ)] = 1.
Introducing the reciprocal lattice: Similar to the Bravais lattice vector for a 3D lattice, (Section 1.3.5), a reciprocal lattice vector, can be defined. The primitive vectors in reciprocal space are a set of orthonormal vectors for the primitive (real) lattice vectors, , , , defined as follows:
This definition ensures that , where m is an integer (this is easily proved; try).
Comparing this to the previous equation, , we get the final von Laue condition for constructive interference:

The physical meaning of this relation is that constructive interference occurs when the change in the Xray wave vector is equal to one of the reciprocal lattice vectors.

Mathematically, the reciprocal lattice is the Fourier transform of the Bravais lattice. The latter is referred to as the real lattice as it represents the real spatial arrangement of the atoms in a crystal; reciprocal lattice can be visualized as the lattice in momentum space, and the Laue condition as the conservation of crystal momentum.
The meaning and construction of reciprocal lattice are discussed briefly in the next section, before returning to the Xray diffraction problem.
1.4.7 Reciprocal Lattice
As noted above, the reciprocal lattice vectors form a set of orthonormal vectors for the real lattice vectors. This can be shown geometrically using first, a 2D Bravais lattice.
In the oblique lattice shown in Figure 1.23(a), and are the real lattice vectors; the unit cell defined by these vectors is shown as the green parallelogram.
The perpendiculars to and give the axial directions of and respectively; this ensures the orthogonality criteria, and .
Clearly, is perpendicular to the (1 0) plane (this is to be imagined, as the reciprocal vector is in inverse space whereas the Miller planes are in real space), and is perpendicular to the (0 1) plane. This shows that each reciprocal lattice vector is associated uniquely to a Miller plane.
The magnitude of is chosen to be and of to be ; this ensures that (Figure 1.23(b) shows that , the spacing for the (1 0) planes). Similarly,
Based on the above definitions, it is clear that the reciprocal lattice vector corresponding to a Miller plane with larger spacing will be shorter and vice versa.
Using the reciprocal lattice vectors, and we construct the reciprocal lattice (blue) shown in Figure 1.23(c); each reciprocal lattice point can be labeled by the coordinates (h, k) corresponding to the Miller plane (h k) associated with it. The point (1, 1) in the reciprocal lattice shows that the vector () is perpendicular to the (1 1) plane in the real lattice.
Similarly one can construct geometrically, the reciprocal lattice for 3D lattices, as follows:
Figure 1.24 shows a triclinic lattice cell with primitive vectors .
From the definition of the reciprocal vector , it is clear that it is perpendicular to the plane defined by and ; the magnitude of this vector (in reciprocal length, of course) is given by:
Similarly the reciprocal vectors, and can be visualized.
The reciprocal lattice vectors can be derived analytically. Using the definition given in Section 1.4.6 (and shown geometrically in this section), the primitive reciprocal lattice vectors of some typical 3D Bravais lattices (cubic lattices) can be derived as follows.
Primitive cube: ; ;
Similarly, and
Clearly, the primitive reciprocal lattice vectors, , , of a simple cubic Bravais lattice form a lattice with cubic symmetry in inverse space.
For a simple cubic lattice of unit cell length, a Å, the reciprocal lattice is cubic with unit cells of length Å^{−1}.
Reciprocal lattice of the fcc lattice can be shown to be a lattice with bcc symmetry!
• ; ;
•
• Similarly, , and
The reverse is also true; the reciprocal lattice of the bcc lattice has fcc symmetry; this is illustrated in Figure 1.25. Note that the reciprocal lattice points correspond to the allowed Miller indices of the bcc lattice, i.e., h + k + l = odd are absent.
; ;
; ;
1.4.8 Ewald Construction
Recall the condition for constructive interference of scattered Xray waves, (Section 1.4.6). This leads directly to the geometric construction illustrating the relation between the incident and scattered Xray waves, the relevant diffraction angle, and the reciprocal lattice.
is any reciprocal lattice vector; it is uniquely associated with the (h k l) Miller plane in the real lattice, and (Section 1.4.7).
The von Laue condition can be visualized geometrically using the Ewald construction in reciprocal space (Figure 1.26).
Placing the incident Xray wave vector, on a lattice point in reciprocal space (chosen as (0, 0, 0) in the figure), and using it as the radius, a circle (sphere in 3D) is drawn. If the circle passes through any other reciprocal lattice point, then it satisfies the condition, , i.e., (note that the scattering is elastic; the wavelength and hence the magnitude of the wave vector are the same).
Combining the diffraction picture in the real lattice (Figure 1.26), one can see that this is consistent with Bragg's law
as every observed d can be related to an integral multiple of a d _{ hkl }.
1.4.9 Structure Factor in Terms of the Reciprocal Lattice Vector
The structure factor can be derived based on the fact that there is interference between the Xrays scattered by the different atoms in the basis (Section 1.4.5). In the derivation of the von Laue condition (Section 1.4.6) it was seen that the path length difference is given by . This translates to a phase difference of (if the path difference is λ, the phase difference is 2π). The contribution of the phase factor to the wave can be written as .
Summing up the contributions from all the atoms in the basis, the structure factor can be written as: , where f _{ n } and d _{ n } are the atomic scattering factor and position vector of atom n in the unit cell respectively.
The above equation is applied below, to determine the structure factor and the systematic absence in the bcc lattice.
As in Section 1.4.5, consider the bcc lattice unit cell to be a simple cubic cell with two atoms in the basis, 1 at position (0 0 0) and 2 at . Therefore, and ; a is the length of the unit cell.
The reciprocal lattice vector corresponding to an (h k l) plane of the cubic lattice considered above, is
If the two atoms have the same scattering factor, f, ; this is the same relation as derived in Section 1.4.5 and the systematic absence conditions follow as discussed there.
1.4.10 Basic Concepts of Xray Structure Solution and Refinement
The essential concepts and steps involved in the actual Xray diffraction data analysis and determination of the crystal structure can be summarized as follows:
The intensity of the Xrays scattered from an (h k l) plane, .
From the intensities measured from different planes in a single crystal Xray diffraction experiment (corrected for experimental factors such as absorption, polarization, reduction in intensity due to lattice vibration and incoherent scattering, etc.), the structure factors are determined. This involves the tricky issue of deciding the phase factor, the famous ‘phase problem’, discussed in specialized text books on crystallography (only S_{hkl} is measured, but since S_{hkl} = S_{hkl}e^{iθhkl}, the phase, θ_{hkl} is also required to determine the atom positions).
As the atomic scattering factor is related to the electron density, one can use the picture of the spatial electron density distribution in the crystal, and replace the summation in the equation for the structure factor by the integral so that,
Fourier transform leads to an electron density map called the Fourier map:
The Fourier map provides the initial structure solution; the total electron densities at specific points determine the type of atom present there.
Using the initial solution, structure factors are calculated for each (h k l) plane; this gives the calculated structure factor list, S_{hkl}(calc). From the experiment, one already has the experimental structure factor list, S_{hkl}(expt).
The least squares method is used to carry out regression of S_{hkl}(calc) against S_{hkl}(expt). Quality of refinement is represented by the socalled ‘R factor’.
The model is revised iteratively to obtain decreasing R factor values. The final model used for the best S_{hkl}(calc) that gives the lowest ‘R factor’ is the refined structure.
1.5 Neutron and Electron Diffraction
Thanks to the wave–particle duality of neutrons and electrons, they can also be used to carry out diffraction experiments to analyze the structure of crystals. The basic principles governing the diffraction process are the same as those discussed for Xray diffraction. From the de Broglie equation (p = momentum, λ = wavelength, h = Planck constant), it is clear that the wavelengths of neutron or electron waves are determined by their momentum, and hence the velocity to which they are accelerated. Scattering of neutron or electron waves with wavelength of the order of Ångströms is used to carry out the diffraction study of single crystals; in high energy electron diffraction experiments much smaller wavelengths are extensively used.
1.5.1 Neutron Diffraction
A fundamental difference between Xray and neutron scattering is that the former is determined by the electron density distribution around the atom nuclei, whereas the latter is directly by the nuclei and the magnetic moment at the atomic site. Because of the weak scattering from light atoms, determination of their positions in the crystal using Xrays is difficult.

Neutron diffraction allows precise determination of the position of light atoms including H, even in the presence of heavier atoms.

Isotopes can be distinguished in a neutron diffraction experiment.

Since neutrons carry a magnetic moment, an ion present in a crystal with different magnetic moments at different sites, would scatter them differently; this allows the differentiation of such ions in the lattice. Neutron diffraction can therefore be used to reveal the magnetic structure of materials (see Section 5.4.5).
1.5.2 Electron Diffraction
Electron diffraction is often carried out together with electron microscopy. For example, a transmission electron microscope can be used to image nanomaterials and nanostructures, and also record electron diffraction from selected areas or features of the sample.

A 100 kV electron beam would have a wavelength of ∼0.04 Å. The accelerating voltage is applied across the electron source (thermal or field emission) and an anode; the schematic diagram in Figure 1.27 shows the electron beam path in a transmission electron microscope.

The diffraction follows Bragg's law and a Fourier transform of the electron diffraction pattern provides a map of the corresponding Miller planes of the lattice structure. Since the electron beam wavelengths are often much smaller than those of typical Xrays used in diffraction studies, the Ewald sphere (Section 1.4.8) in electron diffraction experiments are much larger, sampling more reflections. As the curvature of the Ewald sphere is very small, Miller indices corresponding to the reciprocal lattice points lying on the Ewald sphere can be read out directly using appropriate instrumentation.

Using appropriate apertures, the diffraction pattern can be collected in the back focal plane as shown in Figure 1.27. The image is collected at the image plane as shown.

The lenses are electromagnets which bend the electron beams; collection of electrons scattered from different positions produces the image, whereas electrons scattered at the same angle give the diffraction pattern.
1.6 Common Crystal Structure Motifs
It is important to be familiar with specific types of crystal structures which occur rather frequently or have unique and characteristic features. The structural motif has a direct bearing on the material's properties and functions. A few examples of general crystal structural classes selected at random are listed below with examples of specific materials that possess them.

Rock salt : NaCl, FeO

Fluorite : CaF_{2}, Li_{2}O

Perovskite (ABO_{3}) : BaTiO_{3}

Spinel (AB_{2}O_{4}) : MgAl_{2}O_{4} (inverse spinel : Fe_{3}O_{4})

Rutile : TiO_{2}, NbO_{2}

Diamond: C, Si

Graphite : C

Zinc blende : ZnS, GaAs

Wurtzite : ZnS, AgI
It should be stressed that there are several more structural classes; it is worthwhile to consult textbooks on solid state chemistry, crystallography, etc., for further examples.
1.7 Quasicrystals: A Brief Note
Quasicrystals are a fascinating phenomenon that has been unraveled over the last few decades. As shown in Section 1.3.1, translational periodicity ensures that rotational symmetry occurs only with orders 1, 2, 3, 4, and 6, leading eventually to the 230 space groups. Clearly, a crystal cannot have rotational symmetry of order 5, 7, 9, 10, etc. However, in 1982, Daniel Shechtman (Nobel Prize, 2011) found that electron diffraction of a rapidly cooled Al–Mn alloy produced a diffraction pattern that had 10fold symmetry!
The term ‘quasicrystal’ refers to a structure that is ordered, but not periodic in the observable dimensions like 1, 2, or 3D.

The meaning of ‘ordered’ but not ‘periodic’ can be understood from figures like the famous Penrose tiling (Figure 1.28(a)). It is clearly an ordered structure with a 5fold symmetry; obviously there cannot be, and there is no translational symmetry.

The meaning of quasiperiodicity in a lower dimension mapped on to a periodic structure in a higher dimension can be understood from Figure 1.28(b).

The square lattice (periodic in 2D) has a unit cell shaded deep yellow.

The points within a strip (light yellow) are projected on to an external space (V _{e} axis).

If the V _{e} axis has an irrational slope, the projection gives a quasiperiodic sequence (green (G) and red (R) segments in the ratio equal to the slope of the V_{e} axis (Figure 1.28(c)); for example, if the slope = , the sequence is a Fibonacci chain (1D quasiperiodic structure).


Quasicrystals are such manifestations which are quasiperiodic in 2 or 3D, but periodic in higher dimensions.
Note added after first publication
This chapter replaces the version published September 2022, which contained errors in Figure 1.15.