4 The reciprocal lattice
4.1 The concept of the reciprocal lattice
reciprocal lattice 78
4.2 Definitions 79
4.3 Non-primitive lattices 82 4.4 The reciprocal lattice as the
Fourier transform of the
crystal lattice 86
4.5 Reciprocal space and the
Brillouin zone 87
4.1 The concept of the reciprocal lattice
The concept of thereciprocal lattice arises from the need to have a formal mathematical representation of the planes of atoms within a crystal. The existence of planes of atoms plays an important role in many aspects of crystalline materials – images of planes of atoms in crystals are given in Fig. 4.1. The shapes of crystals are determined by the relative energies of surfaces defined by the crystal planes. For example, the faces of the calcite crystals shown in Chapter 1 are determined by one particular plane of atoms, as shown in Fig. 4.2. Vibrations of the crystal involve motions of planes of atoms.
Diffraction of beams of radiation involve constructive reflections from planes of atoms. With such a diverse range of applications involving planes of atoms, a consistent and general representation is clearly of value, and the formalism of the reciprocal lattice fulfils this role.
Understanding the reciprocal lattice is perhaps the largest conceptual leap in the study of crystalline materials. However, there is a substantial pay-off in working to understand the reciprocal lattice, because it is one of the most powerful tools we have. Three examples will illustrate this point.
First, the reciprocal lattice facilitates the interpretation of diffraction data, as will be discussed in Chapter 6. Second, the reciprocal lattice provides a method for relatively easy formal calculations involving surfaces or lattice planes. Examples of such calculations are common in calculations of physical properties, as discussed in Chapter 7. Third, the reciprocal lattice facilitates calculations involving waves within crystals, whether of electrons or lattice vibrations, as we will meet in Chapters 5, 8, and 12. In fact, although we may proceed by developing some formal definitions of the reciprocal lattice, the same definitions will arise naturally from many general theories, as we will show later in this book.
The starting point is to recognize that the orientation of a plane is usefully described by a vector normal to the plane. If we can identify two independent lattice vectors lying within a plane, the normal will be the vector cross-product of these vectors, as shown in Fig. 4.3. Formally, we can define two lattice vectors lying in the plane, t1= U1a+ V1b+ W1c and t2= U2a+ V2b+ W2c, where U1, V1, W1and U2, V2, W2are sets of integers (Section 3.2.1). The vector normal to t1and t2is
Fig. 4.1 View of the crystal structure of Cu(OH)2. Even in a structure this compli-cated, several planes of atoms with different orientations can clearly be seen.
t1× t2= (U1V2− U2V1)a× b + (V1W2− V2W1)b× c
+ (W1U2− W2U1)c× a (4.1)
Fig. 4.2 Planes of atoms in calcite associated with the common crystal faces. The common face, which is the face seen in the crystals shown in Fig. 1.1, are marked by the plane at the top. This surface is charge-neutral since the charge of the Ca2+cation is balanced by the negative charge of the CO23−molecular anion.
It is clear from this general equation that the set of vector cross-products, a×b, b× c, and c × a, will form a useful basis for the general description of plane normals. Moreover, since the prefactors U1V2−U2V1, etc. are combinations of integers, they are themselves integers which can take on any values. We assign labels h, k, l to the three integers and write
t1× t2= ha × b + kb × c + lc × a (4.2) The important point to note here is that we have three basis vectors which can be combined in any integral multiples. By taking the complete set of integers h, k, l, we define a new lattice. This new lattice is called thereciprocal lattice.
t1
t1×t2 t2
Fig. 4.3 Representation of a plane normal vector being defined as the cross-product of two vectors lying in the plane.
4.2 Definitions
Nomenclature
It is useful to normalize the vector normals. One standard normalization factor is
2π/(a· b × c) = 2π/Vcell (4.3)
where Vcell = a · b × c is the volume of the unit cell, as given by eqn (3.2).
Sometimes the factor of 2π is not included in the normalization – it actually doesn’t matter, provided that it is always understood whether the factor is included or not. With the factor of 2π included, the normalization gives the three vectors
a∗= 2π
Vcellb× c (4.4)
b∗= 2π
Vcellc× a (4.5)
c∗= 2π
Vcella× b (4.6)
We then have the following products:
a∗· a = 2π a∗· b = 0 a∗· c = 0 b∗· a = 0 b∗· b = 2π b∗· c = 0 c∗· a = 0 c∗· b = 0 c∗· c = 2π
(4.7)
The factor of 1/Vcell clearly ensures that the products a∗· a, etc. are reduced to a common result. When the factor 2π is not included in the normalization, we have a∗· a = 1. This may seem to be somewhat neater, but it can be useful to explicitly include the factor of 2π when considering waves, as in diffraction (Chapter 6) and lattice dynamics (Chapter 8).
We can now formally define thereciprocal lattice. It is the lattice of points formed from the infinite set of vectors
d∗hkl = ha∗+ kb∗+ lc∗ (4.8) where h, k, l are the integers we met above, and which can take any values.
The vectors a∗, b∗, and c∗are the basis vectors of the new lattice, and d∗hklis a vector from the origin to any point in this new lattice in a way that is analogous to the lattice vector we met earlier. This vector d∗hklis called areciprocal lattice vector, and the set of integers h, k, l are called the Miller indices. Usually the plane normal to d∗hklis represented by the notation (hkl) – note that the notation includes the brackets.
Just as the crystal lattice gives the unit cell, the reciprocal lattice gives a correspondingreciprocal unit cell, defined by the three vectors a∗, b∗, and c∗. The lengths of these vectors are analogous to the dimensions of the unit cell of the real crystal lattice. We also have the corresponding angles α∗, β∗, and γ∗. The product a∗· (b∗× c∗)is the volume of the reciprocal unit cell.
4.2.1 Geometry of the reciprocal lattice and its link to the crystal lattice
We can now explore the geometry of the reciprocal lattice vectors. Consider the set of equispaced lattice planes shown in Fig. 4.4, normal to the vector d∗hkl, with separation dhkl = 2π/|d∗hkl|. Suppose rhklis a vector from the origin to any point in the first plane. We can decompose rhklinto a component parallel to dhkl∗ , which we denote as dhkl, and a component lying within the plane, which we denote as phkl. It thus follows that
dhkl
Fig. 4.4 A crystal plane (hkl) viewed edge-on. rhklis a vector to a point in the plane. This vector is decomposed into the sum of dhkl, a vector from the origin to the plane and normal to the plane, and phkl, a vector lying in the plane. As demonstrated in the text, the (hkl) plane intersects the a, b axes at the points a/ h and b/k respectively.
rhkl· dhkl∗ = dhkl· d∗hkl= dhkl× |d∗hkl| = 2π (4.9) We now take the special case rhkl= αa. We can therefore write
αa· d∗hkl = 2π (4.10)
Substituting for d∗hkl, we have
αa· (ha∗+ kb∗+ lc∗)= 2παh = 2π (4.11) This immediately gives α= 1/h, leading to the conclusion that the first lattice plane normal to d∗hkl intersects the a axis at a distance a/ h from the origin.
Similarly, d∗hkl intersects the b and c axes at distances b/k and c/ l from the origin respectively. This point is illustrated in Fig. 4.4.
We now consider the special case where h = 1 and k = l = 0, which is represented as (100). This set of planes is shown in Fig. 4.5. Since the first (hkl)plane intersects the b and c axes at distances b/k and c/ l respectively, the zero values of k and l imply that the planes do not intersect these axes and
Fig. 4.5 (100) and (110) planes in the CsCl structure, shown by the shading.
are therefore parallel to them. Thus the (100) plane that includes the origin also contains the vectors b and c, and d∗100 = a∗is normal to both b and c. This is an important point, because in a crystal in which the three axes a, b, and c are not orthogonal, the reciprocal lattice vectors a∗, b∗, and c∗ will not be parallel to the lattice vectors a, b, and c respectively. This point is illustrated for the example of a monoclinic lattice in Fig. 4.6. The important point is that a∗is perpendicular to b and c, and similarly for the other two reciprocal lattice vectors. Only in cases where the crystal axes are orthogonal will the reciprocal lattice vectors lie along the same directions as the corresponding lattice vectors.
It should be appreciated that only in crystals with orthogonal axes are the reciprocal lattice vectors parallel to the corresponding lattice vectors, and only in those crystals where the volume of the unit cell is equal to abc does a∗= 2π/a, b∗= 2π/b, and c∗= 2π/c. In the monoclinic system the volume of the unit cell
[100]
[001]
a*
c*
Fig. 4.6 The directions of the real and recipro-cal lattice vectors in a monoclinic system. The lattice vector b and reciprocal lattice vector b∗ are both normal to the plane of the diagram.
is equal to abc sin β, so that a∗= 2π/a sin β, b∗= 2π/b, and c∗= 2π/c sin β.
In the monoclinic case, we also have β∗= 180◦−β. In the hexagonal or trigonal case (a = b, α = β = 90◦, γ = 120◦), the volume of the unit cell is equal to abcsin γ =√
3a2b/2, giving a∗= b∗= 2π/a sin γ = 4π/√
3a, c∗= 2π/c, and γ∗= 60◦.
4.2.2 Relationship between real and reciprocal lattice parameters
In the general case, the real and reciprocal lattice parameters are related by the following equations:
a∗= 2π
asin β sin γ∗ = 2π asin β∗sin γ b∗= 2π
bsin γ sin α∗ = 2π bsin γ∗sin α c∗= 2π
csin α sin β∗ = 2π csin α∗sin β cos α∗=cos β cos γ − cos α
sin β sin γ cos α=cos β∗cos γ∗− cos α∗ sin β∗sin γ∗
cos β∗= cos γ cos α− cos β
sin γ sin α cos β=cos γ∗cos α∗− cos β∗ sin γ∗sin α∗ cos γ∗= cos α cos β− cos γ
sin α sin β cos γ = cos α∗cos β∗− cos γ∗ sin α∗sin β∗ These equations are greatly simplified if we have orthogonal axes.
4.2.3 Interplanar spacing and the reciprocal lattice parameters
The interplanar spacing dhklcan be obtained from the inverse of|dhkl∗ |2, which is defined in eqn 4.8. For the general triclinic case,|dhkl∗ |2is given as
1
4π2|dhkl∗ |2= 1
dhkl2 = h2a∗2+ k2b∗2+ l2c∗2+ 2klb∗c∗cos α∗ + 2hla∗c∗cos β∗+ 2hka∗b∗cos γ∗ (4.12) The cosines of the angles of the reciprocal unit cell can be calculated from the angles of the real-space unit cell using the relations given above, and once the angles have been calculated it will then be possible to calculate the values of a∗, b∗, and c∗.
4.2.4 Reciprocal lattice vectors and atomic structure
With the geometric link between the crystal and reciprocal lattice established, we can now embelish this by drawing in the atoms in the unit cells. The formalism is particularly useful when we consider crystal surfaces. Figure 4.7 shows a small crystallite of a two-dimensional monatomic close-packed crystal. The faces are clearly cleaved along well-defined lattice planes, and these can be indexed according to the geometric relationships we have established.