A different approach to understanding the reciprocal lattice

The word "crystal" is from the Greek krustallos, "ice," and was first applied to quartz, which seemed to be a kind of permanent ice. After the late middle ages, it came to mean any substance, usually a mineral, showing plane faces in symmetrical relations. Until X-rays allowed us to examine these bodies in microscopic (actually, sub-microscopic) detail after 1912, the faces were the only obvious clue to internal structure. Crystal faces, produced naturally, vary widely in shapes and sizes, even in the same crystal. Niels Stensen (Nicolaus Steno, 1638-1686) recognized in 1669 that the angles between the normals to faces were more fundamental than any accidents of shape, and were the same in all crystals of a given substance. This allowed the classification of the symmetry of crystals, and established *goniometry* as the basis for mathematical crystallography. Many crystalline substances do not occur as crystals with well-defined faces (as *euhedral* crystals), and the individual crystals may be tiny, so their natures remained obscure. All metals, for example, are crystalline. When we say "microscopic," we generally mean "on an atomic scale," not literally microscopic. "Macroscopic" is used when the atomic constitution is not apparent.

Crystalline substances have certain distinguishing characteristics aside from the existence of faces. First, they have definite chemical compositions; second, they melt sharply, unlike glasses which soften and melt over a range of temperature; third, they may exhibit *cleavage*. Calcite is well-known to be easily cleavable into cleavange rhombs, all with the same angles between faces. Some crystals have only one cleavage direction, and many have none. Some transparent crystals (those of low enough symmetry) show optical properties that vary with direction. Again, the *double refraction* in calcite is famous, and quartz is also optically anisotropic. These properties are all evidence of some structural regularity in crystals.

The only properties that can be measured macroscopically are averages over volumes that contain many fundamental units, <Ω> = (1/ΔV)∫Ω(**r**)d^{3}**r**, where Ω is a typical property. In a *homogeneous* substance, these properties are invariant under translation, Ω(**r** + **a**) = Ω(**r**), for any translation **a**. As a typical number, properties averaged over 10*isotropic* substance, the properties are invariant under rotation. Crystals are, in general, anisotropic, but cubic crystals are isotropic. Properties that vary with rotation can be classified by their *tensor* nature, as scalar, vector and so on. Scalars are naturally invariant under rotation. Vectors, and other tensors, transform in definite and particular ways under rotation. The existence of tensor properties in crystals is very important, providing evidence of their internal structure. The optical properties of crystals are macroscopic ones, since the wavelengths of 500 nm or so average over many lattice points, and depend on indices of refraction, which are macroscopic parameters. Crystal symmetries are often reflected in macroscopic properties, of course.

In 1784, René Haüy (1743-1822) made the first step in connecting internal structure with external form by showing that the observed faces of calcite crystals could be produced by stacking cleavage rhombs. If these elementary rhombs were small enough, the faces would appear macroscopically smooth, though stepped microscopically. Robert Hooke (1635-1703) had proposed similar ideas in 1665, as did Christiaan Huygens about the same time in connection with calcite. The rise of the atomic theory after 1800 suggested that the elementary units were atoms or molecules arranged in regular space *lattices*. In 1848, Auguste Bravais (1811-1863) made these suggestions concrete by showing that there were 14 possible space lattices belonging to the 7 systems of crystal symmetry. This model confirmed the empirical Law of Rational Indices established by goniometry, and provided strong evidence for the atomic theory of matter.

Two atoms of the same kind are absolutely identical. This means that crystal lattices are perfect, far more exact than is ever possible in the macroscopic world of peas in a pod. This precision is seen in the beauty and regularity of crystal faces, which are far more exact than in any artificially cut gem. The faces of a cut gem have no connection with the internal structure, though cleavage is often used to rough out the shape of a gem. The perfection of a lattice is disturbed by lattice vibrations (phonons), which have observable effects on microscopic properties, such as X-ray diffraction, as well as by impurities and lattice defects such as dislocations.

In this article, the broad and important subject of crystal symmetry will not be treated in any detail, and details of X-ray scattering, beyond the existence of diffraction maxima, will also not be considered. For these matters, the reader should consult the References. This article deals principally with the utility of the reciprocal lattice, a fundamental concept in the study of the crystalline state, and one with many important applications.

The seven crystal systems are: triclinic (no symmetry axes), monoclinic (one 2-fold axis), orthorhombic (3 2-fold axes), tetragonal (one 4-fold axis), trigonal (one 3-fold axis), hexagonal (one 6-fold axis) and cubic (four 3-fold axes and three 4-fold axes). Axes of other multiplicity, such as 5 or 7, are not found in crystals. Each class includes crystals of the full symmetry, with all other symmetry elements consistent with the class (planes of symmetry, 2-fold axes, centres of symmetry, etc.), as well as those of lesser symmetry consistent with the crystal class.

The three Bravais lattices belonging to the cubic system are shown at the right. These *unit cells* are to be imagined as repeated to fill space. If the "molecules" are spherically symmetric, then each lattice will have the full symmetry of a cube, and crystals with these lattices are called *holohedral*, since all crystal faces are possible. The simple cubic lattice has one lattice point per unit cell, or one lattice point per volume of a^{3}, where a is the side of the unit cell. Note that there are 8 corners, and each corner is shared with 8 unit cells, so it contributes 1/8 of a lattice point to the unit cell. If a is known, then the theoretical density of the crystal can be calculated. This calculation can be performed in reverse to obtain the value of a. It is found that a is on the order of 10^{-10} m, or an Ångstrom unit, more than a thousandth of the wavelength of visible light. Consequently, it is impossible to see the structure of a crystal by visual imaging.

The simple cubic lattice can be defined by the three basis vectors **a**_{1}, **a**_{2} and **a**_{3}. In this case, they are of equal length and directed along the x,y and z axes. The volume of the unit cell is given by V = **a**_{1}·**a**_{2}**x****a**_{3}, the triple scalar product of the three basis vectors, which in this case is a^{3}. Sometimes we may write this volume as (**a**_{1}**a**_{2}**a**_{3}), to make clear which basis vectors are involved. The simple cubic lattice is far too simple and special for deriving general results, but offers clear illustrations. In general, however, we can define any unit cell by three basis vectors, and its volume is given by their triple product. In general, however, the basis vectors will not be the same length and will not be at right angles.

The body-centered cubic lattice has an additional lattice point at the center of the unit cell, so the unit cell contains 2 lattice points. Each lattice point then occupies a volume of a^{3}/2. The face-centered cubic lattice has an additional lattice point at the center of each of the six faces, which is shared with the adjoining unit cell, so that a total of 1 + 3 = 4 lattice points are contained in each unit cell, each occupying a volume of a^{3}/4. A unit cell that contains only one lattice point is called a *primitive* unit cell, and for some purposes it is essential to work with primitive unit cells. The simple cubic unit cell is already primitive, and it is possible to find primitive unit cells for the body-centered and face-centered lattices as well.

One way of doing this for the body-centered lattice is shown at the left. A corner point is joined with three nearest body points to give the basis vectors for a rhombohedral primitive unit cell. If repeated, it fills space just as the body-centered unit cell does, and gives exactly the same lattice. These basis vectors are equal in length, but are clearly not at right angles, and the rhomb-shaped unit cell gives no evident hint of the cubic symmetry. This is the reason for the use of the non-primitive unit cells. Simple cubic crystals are not uncommon (CsCl is the usual example), but elements seem to avoid this structure, since it is not energy-favored. Body- and face-centered cubic lattices are very common and important (most elements and metals crystallize in these lattices). The reader may check that the primitive body-centered unit cell is indeed of half the volume of the cube by finding the triple scalar product of the basis vectors given. Spheres may close-pack in either the face-centered cubic (fcc) or the hexagonal close-packed (hcp) structures, and ideally occupy 74% of the volume.

To specify any position in the lattice, we could choose any lattice point as origin and define three orthonormal unit vectors **i,j,k** or **e**_{1},**e**_{2},**e**_{3} in arbitrarily chosen x,y and z or x_{1},x_{2},x_{3}directions. Note the different ways the three directions might be distinguished. This would be especially appropriate for simple cubic crystals, where the axes could be chosen along the sides of the unit cell. Because the unit vectors satisfy the orthonormality relation **e**_{i}·**e**_{j} = δ_{ij}, it is easy to express any vector in terms of rectangular components: **x** = x_{i}**e**_{i} (summation convention: sum over i from 1 to 3!) or **x** = x**i** + y**j** + z**k**. The scalar product of two vectors is then simply **x**·**y** = x_{i}y_{i}, the sum of the products of corresponding components. It is convenient to use unit vectors to specify directions as well. This system is so convenient that we adopt it wherever it is applicable.

For crystals, however, there is a problem. The coordinates are not obviously related to the lattice points or unit cells when we use orthonormal unit vectors. For cubic crystals, we can get around this by choosing the unit of distance as a = 1, but this cannot be done in general, since the lengths of basis vectors may not be equal, and they may not be at right angles. We will not be able to make any deductions until our coordinate system is adapted to the crystal lattice, and in that case we must give up many of the conveniences of an orthonormal basis.

Therefore, we use the basis vectors of a unit cell as the basis vectors for our coordinates, and write a general position **r** = m_{i}**a**_{i}. Integral values of the m_{i} correspond to the corner of some unit cell in the lattice. Positions differing by integral values of the m_{i} are equivalent in the lattice, since all properties have the periodicity of the lattice. The range 0 ≤ m_{i} ≤ 1 covers the unit cell. This is the *direct lattice* in ordinary space. If we have a periodic function in one dimension, f(x + nL) = f(x), we know that we can expand the function in a Fourier series of sinusoids of wavelengths L, L/2, L/3, ... knowing only f(x) in the interval x = 0 to x = 1, and that this expansion will be valid everywhere. Crystal properties can be expanded in exactly the same way knowing only values within the unit cell. This is one immediate and huge benefit of the new coordinate system, but there are many others.

If we expressed **r** in terms of an orthonormal basis, it would be easy to find, say, the 1-component by scalar multiplication with **e**_{1}: x_{1} = **e**_{1}·**r**. It is not so simple if we expand in terms of the basis vectors of a unit cell. In this case, we require a vector **b**_{1} such that **b**_{1}·**r** = m_{1}. That is, **b**_{1} must be orthogonal to **a**_{2} and **a**_{3}, and its scalar product with **a**_{1} must be unity. It is not difficult to find such a vector. It must be proportional to **a**_{2}**xa**_{3}, so the required vector is **b**_{1} = [**a**_{2}**xa**_{3}] / (**a**_{1}**a**_{2}**a**_{3}). Its two companions **b**_{2} and **b**_{3} are given by similar expressions obtained by cyclically permuting the indices 1,2,3. Now we have three noncoplanar vectors for finding the direct lattice components m_{i}

By direct calculation, we can show that (**b**_{1}**b**_{2}**b**_{3}) = 1/(**a**_{1}**a**_{2}**a**_{3}), and that **a**_{1} = [**b**_{2}**xb**_{3}] / (**b**_{1}**b**_{2}**b**_{3}), with the other two basis vectors found by cyclic permutation. The triplets {**a**} and {**b**} are, therefore, mutually reciprocal, and the unit cells defined by them fill space. The vectors **r** = m_{i}**a**_{i} for integral values of m_{i} define the *direct lattice*, as we have seen, and the vectors **B** = l_{i}**b**_{i} for integral values of l_{i} in the same way define the *reciprocal lattice*. The two lattices have a fixed relation in space, so that if the crystal is rotated, both lattices rotate at the same time by the same amount. They are, however, reciprocally related in size. As direct lattice spacing increases, the reciprocal lattice shrinks, and vice-versa.

Any vector **B** can be expressed as three numbers in parentheses: (h_{1}h_{2}h_{3}). If the h's are integers, then these are the reciprocal lattice points. The vector **b**_{1} or (100) is normal to the plane of lattice points defined by **a**_{2} and **a**_{3}, and similar statements can be made about the other two reciprocal lattice basis vectors, written (010) and (001). We can now show that any such vectors with integral h's correspond to planes of lattice points in the direct lattice. Any plane that cuts the three direct basis vectors of the unit cell at some fractions 1/h, 1/k, 1/l of their lengths, where h,k,l are integers will sooner or later pass through lattice points, and the plane will contain an infinite number of them in an infinite crystal. Be sure you understand why this is so. The same is true for any multiple of these fractions. If, however, the fractions are irrational, the plane will pass through *no* lattice points at all. If a face does not intersect an axis, the corresponding index is 0.

The plane will intersect the sides of the unit cell in three lines. Any two of these intersections, considered as vectors, will define the plane and their cross product will be parallel to the normal to the plane. Two such vectors are **a**_{2}/k - **a**_{1}/h and **a**_{3}/l - **a**_{1}/h. Their cross product can be written in terms of the **b**'s as (V/hkl)[h**b**_{1} + k**b**_{2} + l**b**_{3}]. The normal to this plane is, then, the reciprocal lattice vector (hkl). Normally, no commas appear and minus signs are written over the index. Every lattice point in the reciprocal lattice corresponds to a plane in the direct lattice that passes through lattice points. By goniometry we measure the angles between face normals, and so can check this prediction of the lattice hypothesis. The agreement is, in fact, precise, and is called the Law of Rational Indices. This is a demonstration of the existence of atoms by using only macroscopic measurements, a remarkable thing (there are, of course, many such demonstrations).

The values of (hkl) for a given face depend on the choice of the unit cell. It is important that there always exists some choice that gives small integers for (hkl). The choice of a reference plane (111) defines the lengths of the basis vectors; any plane can be assigned to be (111), but some choices are much better than others. A particularly bad choice may give inconvenient numbers, but the numbers will always be found to be rational, if sufficiently precise measurements can be made. The numbers (hkl) are called the Miller indices of a face. It is a good idea to practice with a simple cubic lattice if you are not familiar with the relation between faces and Miller indices. The (111) plane is taken as the plane through the ends of the basis vectors.

Another remarkable result is that the reciprocal of the length of **B** is the distance d between the plane and a parallel plane through the origin, or d = 1/|**B**|. The scalar product of the vector **a**_{1}/h, whose length is the distance along the 1-axis cut by the two parallel planes, and the vector **B** in the direction of the normal, is equal to unity, since the 1-component of **B** is h, and h/h = 1. It is also equal to the length of **B** times the projection of **a**_{1}/h on the normal, which is just the distance d. Therefore, 1 = |**B**|d, or d = 1/|**B**|, as was to be proved. The planes (222) are twice as closely spaced as are (111) planes, (333) planes three times, and so on.

The scalar product of two vectors, one referred to the **a**'s and one referred to the **b**'s, is simply the sum of the products of the components, just as in the case of the orthonormal basis. In other connections, these components are called *contravariant* and *covariant*, and it is similarly easy to form the scalar product. Of course, it is possible to form the scalar product of two vectors in the direct lattice or two vectors in the reciprocal lattice, but then it is not so easy, and the scalar products of the basis vectors are involved. Any vector can be expanded in terms of either set of basis vectors, but the components will not, of course, be the same.

We have found that the vector product of two direct lattice vectors can be expressed as a reciprocal lattice vector. Conversely, the vector product of two face normals (hkl) and (h'k'l') can be expressed as a direct lattice vector that is in the direction of the line of intersection of the two faces, or a crystal edge. The result is (hk' - kh')**a**_{1} + (lh' - hl')**a**_{2} + (kl' - lk')**a**_{3} times 1/V, an unimportant factor. The coefficients can be written as 2 x 2 matrices. Therefore, if h,k,l and h',k',l' are integers, then so are the coefficients. We can remove any common factors without changing the direction of the edge vector. The vector can be written [pqr] in square brackets, where p = hk' - kh', and so on.

A set of faces whose intersections are parallel lines is called a *zone*. The zone direction is perpendicular to any normal to the faces of a zone. Therefore, the condition that a face (hkl) belong to zone [pqr] is that their scalar product vanish. Since the vectors are in reciprocal spaces, this condition is hp + kq + lr = 0. We note that zone axes also obey the Law of Rational Indices, and are given by [pqr] with p,q,r (usually) small integers. It is easy to see that the condition that three faces (hkl), (h'k'l') and (h",k",l") belong to the same zone is that the 3 x 3 determinant with these indices as rows (or columns) vanish. Summarizing, (hkl) is a direction in the reciprocal lattice, while [pqr] is a direction in the direct lattice.

A set of faces described by three numbers h,k,l in any order and including negative values is called a *form* and denoted {hkl}. For a cubic unit cell, the form {100} consists of the six faces of a cube: (100), (010), (001), (-100), (0-10), (00-1). The minus signs are not written in the proper places because this is hard to do in HTML. Similarly, the form {111} is an octahedron. Here we distribute 0,1,2 and 3 minus signs in all possible different ways. Phillips shows how to draw forms on clinometric axes, a delightful skill.

Physicists confuse things considerably by multiplying the reciprocal lattice basis vectors **b** by 2π (as in Kittel). This avoids having to write this factor explicitly in several applications (notably to the Fourier transform), but destroys the mutual reciprocity of the direct and reciprocal lattices. It is not hard to allow for this, but the reader should be alert to which definition is used. The method used here to derive the reciprocal lattice is somewhat different than is normally used by either crystallographers or solid-state physicists, and rests more on the general mathematical properties of dual or reciprocal spaces, as the mention of contravariant and covariant components indicates.

It is instructive to admire the importance of the reciprocal lattice when dealing with crystals (or any periodic structure). We can expand vectors in direct space in terms of the basis vectors of the unit cell, whatever its shape, and the coefficients will be integers. Vectors can also be expanded in reciprocal space in terms of the reciprocal basis vectors. Integral coefficients then correspond to planes containing lattice points in direct space. The scalar product of two such vectors, one of each type, will then be integral (whatever the lengths and angles of the unit cell), and from this flow many remarkable results, as we shall see in the case of X-ray diffraction.

Crystals may be probed with anything with a wavelength comparable to lattice spacings. The usual probes are X-rays, neutrons and electrons. Electrons interact strongly with matter because they are charged, and so are useful only for surface investigations. Neutrons require a strong neutron source and BF_{3} detectors, so the experimental setup is expensive and complex. They interact with nuclei, and with the magnetic moments of electrons. X-rays, interacting with the electron density, are by far the most convenient probe, and can easily be used in any laboratory. X-rays can be detected by photographic emulsions, as well as by ionization chambers. The usual energies are 10-50 keV (the wavelength is 1.24/E nm, where E is in keV). The usual methods are Laue diffraction, rotating-crystal and powder camera, the last being the most popular and easiest to use. The powder camera was introduced by Debye and Scherrer, and by Hull, in 1916. For details, see Kittel. The theory of von Laue, and the experiments of Friedrich and Knipping on ZnS, published on 8 June 1912, were the effective beginning of solid-state physics. W. L. Bragg reported the first crystal structure determination, that of KCl, in 1913, using a crystal spectrometer and ionization-chamber detectors. These were events of the first order in Physics, connecting us directly with the microscopic world of atoms and molecules.

Suppose a crystal is irradiated by an X-ray beam of wave vector **k**. The magnitude of this vector is k = 2π/λ, as indicated in the figure. The amplitude of the incident wave at any point **r** is Ae^{i(k·r - iωt)}, where some point O has been chosen as origin. In general, we shall drop the time factor, which is the same for all quantities to be considered. The conventions are those of physicists, who use k and the angular frequency ω instead of 1/λ and f, and the time factor with a minus sign. We observe the radiation scattered by the crystal at some point P at **R**, which is taken as very distant to avoid having to consider the inconsequential details of a finite distance from the crystal to P, just as in optical diffraction theory. The amplitude at P is the sum of the amplitudes scattered by each volume element of the crystal, and the observed intensity is the absolute value squared of this.

Consider the scattering associated with some lattice point Q at position **ρ** in the direct lattice. The amplitude of the spherical scattered wave will depend on e^{ikr}/r and the integral of the electron density at that point. We shall assume each lattice point contributes the same amount, but exactly what that amount is will be ignored, and we shall consider only the r-dependence. The amplitude at P is, then, proportional to (e^{ik·ρ})(^{ikr}/r). If R >> ρ then r is approximately R - ρ cos φ (to see this, expand r as found from the Law of Cosines in powers of ρ/R). Collecting all factors that do not depend strongly on ρ and considering them to be a constant, the amplitude at P is (Ae^{ikR}/R)e^{i(k·ρ - kρ cos φ)}. If now we introduce **k**' = k(**R**/R), the exponential factor can be written e^{iΔk·ρ} in terms of Δ**k** = **k**' - **k**, the change in wave vector in the scattering observed at P.

This factor is easily interpreted in terms of the reciprocal lattice. In the exponent, we have the scalar product of two vectors. One, **ρ**, is a vector [qrs] in direct space, and the components are integers. If, then, Δ**k**/2π should be a vector (hkl) in reciprocal space, their scalar product is simply hq + kr + ls. If h,k,l are integers, then so is the scalar product, and e^{iΔk·ρ} = e^{2πni} = 1. This means that the contribution from every direct lattice point is the same, so that the total amplitude at P will be NA, if A is the contribution from one lattice point, and N is the number of lattice points in the crystal. That is, all the contributions add in phase, and the intensity will be N^{2}|A|^{2}, enormously greater than the intensity due to a single lattice point. This coherent superposition is like that which produces a spectral line from a grating, except that it is in three dimensions, and N is very much larger. Note how easily the reciprocal lattice, and the Law of Rational Indices, give this result!

Therefore, every reciprocal lattice point corresponds to a possible diffraction maximum. A maximum will be observed only if there is a nonzero contribution from each lattice point. The intensity of each maximum depends on the details of the scattering from the electron density around each direct lattice point. The strength and angles of these maxima are measured, and from them the structure of the crystal can be deduced. The Ewald Construction shows clearly the conditions under which a diffracted beam can be observed. In the figure, if the wave vector **k** is used, the reciprocal lattice is the physicist's, expanded by a factor 2π. The construction is to draw **k** ending at any lattice point, and the sphere partly shown with center at the beginning. Then, a maximum is possible if the sphere passes through a reciprocal lattice point. This shows just how difficult it is to satisfy the conditions for a diffraction maximum.

In terms of the reciprocal lattice, let **n** be a unit vector in the direction of the incident radiation, and **n**' a unit vector in the direction of the diffracted beam. A diffracted beam is observed if (**n**' - **n**)/λ is some reciprocal lattice vector **B** with integral components. This assumes elastic scattering of the X-rays, which does not change the wavelength λ. Incidentally, the vector **p** = (h/λ)**n**, where h is Planck's constant, is the momentum of the X-ray photon. This scattering condition implies that the change of momentum is h times a reciprocal lattice vector. This momentum is absorbed by the whole crystal, which does not recoil because of its large mass, and so the scattering is elastic. Interactions with *phonons*, which are unimportant with X-rays, lead to inelastic scattering.

The most familiar description of X-ray scattering by crystals is by Bragg's Law, shown in the figure. An X-ray beam is shown "reflected" from two crystal planes at a glancing angle θ. The path difference ab is seen to be 2d sin θ, and if this is equal to a wavelength the waves from these two planes will interfere constructively, and so on for any number of planes. The integer n is the "order" of the diffraction. Though always presented in elementary discussions, it is certainly not a satisfying derivation of the diffraction condition, since it depends on arbitrary and unlikely assumptions. No "reflection" actually occurs in any sense. We see that the beam is scattered through an angle of 2θ. The Ewald construction gives 2(1/λ)sin θ = |**B**|, or λ = 2d sin θ, which is just the Bragg condition. For a given spacing d, the wavelength and angle must be connected by this relation. If θ is fixed (but perhaps different for different planes), as in a Laue experiment, then an X-ray beam containing a continuum of wavelengths will search out those spacings d for which the equation is satisfied, producing scattered beams only at certain angles. If the X-ray beam is monochromatic and the crystal is rotated, then a diffracted beam will be observed for a given d when the angle is correct. These beams are very sharp, because they result from the interference of a large number of components. Consequently, the measurements can be very precise. Computer processing has reduced the great tedium of analyzing the results.

The integral for the diffraction amplitude looks very much like a three-dimensional Fourier transform of the electron density from the direct space to the reciprocal space. If we knew the amplitudes, the transform could be inverted to give the electron density. What we know, however are the intensities, whose square roots only give the amplitudes up to a phase. This makes the inversion difficult, but progress can be made nevertheless. Since the transform is discrete and not continuous, the inversion is essentially a summation of plane waves, analogous to a Fourier series in one dimension.

These results apply not only to X-ray diffraction in crystals, but to the general problem of diffraction by a three-dimensional lattice. We note, in particular, the property of such diffracting systems of selecting wavelengths. A two-dimensional grating will make diffraction maxima for any incident wavelength, but a three-dimensional grating will not do so, selecting a particular wavelength from a white incident beam, as in white-light holograms or Laue photographs. The concept of the reciprocal lattice makes the explanation of these effects simple and straightforward, as we have seen.

Let's consider first a primitive orthorhombic lattice, in which the direct basis vectors are a**i**, b**j**, c**k**, in terms of the usual orthogonal unit vectors. The volume of the unit cell is V = abc. It is easy to see that the reciprocal basis vectors are (1/a)**i**, (1/b)**j**, (1/c)**k**. The name "reciprocal" is clearly well-chosen. The reciprocal lattice is also orthorhombic, and the volume of its unit cell is 1/abc. the vector (hkl) is **B** = (h/a)**i** + (k/b)**j** + (l/c)**k**. The length of this vector is B = √[(h/a)^{2} + (k/b)^{2} + (l/c)^{2}]. Therefore, the spacing of the (hkl) planes is d = 1/B. These results may be specialized for a primitive tetragonal lattice, a = b ≠ c, or for a cubic lattice, a = b = c. The reciprocal lattice of a primitive hexagonal lattice is also a hexagonal lattice, but with a rotation. The hexagonal unit cell is a prism with angles 120° and 60° between the sides.

Kittel shows that the lattice reciprocal to the body-centered cubic is face-centered cubic, and vice-versa. The first *Brillouin zone* is a specially-chosen reciprocal unit cell. From the origin, vectors are drawn to the nearest lattice points in the reciprocal lattice, and planes drawn perpendicular to these vectors at their midpoints. The smallest volume enclosed by these planes is the Brillouin zone. It is a unit cell, and fills space when it is repeated. Brillouin zones are useful in studying wave motion in periodic structures, especially electron waves. The only allowed waves are those with wave vectors that fall in the Brillouin zone. The first Brillouin zone of a body-centered cubic lattice is a regular rhombic dodecahedron. That of a face-centered cubic lattice is a truncated octahedron.

If the direct basis vectors are given the dimension of length, then the reciprocal basis vectors will have the dimension of (length)^{-1}. This suggests that it may be useful to consider the direct and reciprocal spaces as different in nature. In this case, direct vectors represent distances, while reciprocal vectors represent *spatial frequencies*. Exponents of the form **B**·**r** will then be dimensionless, and suitable for the exponential factors in Fourier transforms. Indeed, we have already made use of this in X-ray diffraction. In quantum mechanics, momentum is related to wavelength by p = h/λ = (h/2π)**k**, so with proper normalization the reciprocal space turns out to be momentum space. This was suggested in connection with the Ewald construction, which follows from momentum conservation.

If we consider the first Brillouin zone as a region in momentum space, then any point within it corresponds to the wave vector of a certain plane wave whose frequency is a function of position. That is, we can specify the *dispersion relation* f = f(**n**/λ) or ω = ω(**k**) for wave vectors in this zone. [Spectroscopists write 1/λ = σ, and call it a *wave number*. Physicists generally multiply things by 2π for some reason, and also write ν instead of f for frequency.] Only wave vectors in this zone are necessary to specify all possible plane waves. As the discrete structure is imagined to become continuous by a decrease in the lattice spacing, the Brillouin zone expands to fill all space, and a wave may have any wave vector. This formalism can also be used to describe electron states, and in expressing the energy-momentum relation E = E(**p**). The first Brillouin zone should not be confused with the Fermi surface in this application. The Fermi surface is also in reciprocal space, but gives those momenta for which E = E_{F}, the Fermi energy of the electron gas. The Wigner-Seitz zone is constructed like the Brillouin zone, but in direct space, and is an alternative unit cell.

The internal energy of an insulating crystal is in the form of lattice vibrations, which can be described by wave vectors in reciprocal space. Physicists usually speak of wave vectors **k** = (2π/λ)**n** in a reciprocal space expanded by a factor 2π. All possible modes have wave vectors in the first Brillouin zone. In the case of X-rays, the wave vectors were large, much greater than the reciprocal lattice spacing. Here, the wave vectors are small. Wave vectors close to the origin represent ordinary sonic and ultrasonic waves in the crystal. These vibrations are, of course, quantized. Quantum effects are not evident for the small wave vectors, but become more prominent as the wave vectors approach the boundary of the Brillouin zone. At a temperature T, the oscillators representing the vibrational modes are excited according to Bose-Einstein statistics. In the familiar Debye approximation, the modes are assumed to be those of a continuum, extending to a maximum frequency f_{D} usually specified by the Debye temperature Θ, where hf_{D} = kΘ.

We can think of the lattice energy as resident in *phonons* of frequency f and wave vector **k**. Curiously, phonons of non-zero **k** do not carry actual momentum, though **k** is conserved in their interactions as if proportional to momentum in the usual way: **p** = (h/2π)**k**. The zero-frequency mode is the only one carrying real momentum, and this is momentum of the lattice as a whole. If the lattice oscillators are perfectly harmonic, then phonons do not interact with one another. If this were true in a real crystal, the lattice could never come to thermal equilibrium. Energy put into phonons at one location would simply move to another location without resistance, and thermal conductivity could not be expressed as dependent on a temperature gradient; it would, in effect, be infinite. We actually see this in the case of ordinary sound waves, which certainly do not diffuse like heat.

Since heat, which after all is just lattice waves, does come into thermal equilibrium, and crystals show normal heat conduction proportional to a temperature gradient, there must be some mechanism for a system of high-frequency phonons to approach thermal equilibrium. If the lattice oscillators are anharmonic (their potential energy containing terms like ex^{3} + fx^{4}, for example), we know that if vibrations of frequency f_{1} and f_{2} are simultaneously applied, the resulting motion will contain vibrations of frequency f_{3} = f_{1} ± f_{2}. This implies that for phonons, we may observe processes like **k**_{1} + **k**_{2} → **k**_{3}, and that these processes will conserve energy (f_{3} = f_{1} + f_{2}) as well as wave vector. Such phonon "collisions" may be a way to approach thermal equilibrium. A closer consideration of these "N" processes, however, shows that they cannot by themselves result in thermal equilibrium.

To see why, consider the case of translational energy in a gas. This energy equilibrates rapidly; any molecule of larger than normal velocity is quite rapidly slowed down by losing energy in collisions, and the direction of its momentum is quickly randomized. The thermal conductivity K, defined by Q = K(dT/dx), is approximately Cvs/3, where C is the translational specific heat, v the average velocity, and s the mean free path between collisions. In an N process, the wave vector after the collision is in the same direction as the wave vectors before the collision. The hard collisions that produced equilibrium in the case of translation are not available.

However, suppose that the initial wave vectors are large enough that **k**_{3} lies outside the Brillouin zone. To bring it back within the zone, it is necessary to add a lattice vector **B**. As in Bragg diffraction, the momentum difference is absorbed by the lattice as a whole while carrying away a negligible amount of energy, since the lattice is massive. The possible vectors **B** are the smallest reciprocal lattice vectors. The process **k**_{1} + **k**_{2} → **k**_{3} + **B** is called an *umklapp* or "U" process. It is easily seen in a vector diagram that **k**_{3} may be reversed in direction relative to **k**_{1} or **k**_{2}. This supplies the role of the necessary momentum-reversing collisions. The mean free path s in the expression for thermal conductivity is the mean free path between U processes.

The initial wave vectors in a U-process must be large enough that their sum will go outside the Brillouin zone, so U processes will be important only at high enough temperatures for such wave vectors to be available. At low temperatures, s may increase to such an extent that only collisions with the boundaries of the crystal will be effective (these, too, can be "hard" and reverse momenta). This "shape effect" has been observed. The wave vectors of sound waves are much too small for U processes, so they do not equilibrate. These matters were brilliantly explained by R. Peierls, and are another example of the power of the reciprocal lattice.

The structures of face-centered cubic KCl (a = 0.629 nm) and NaCl (a = 0.563 nm) were those first deciphered by W. L. Bragg. He showed conclusively that the crystal was composed of alternating positive and negative ions, not of KCl molecules. The (100) planes are a checkerboard of positive and negative ions, and all these planes scatter equally. The planes are effectively spaced at half the lattice constant. Using the K_{α1} line of Cu (0.1541 nm), the Bragg Law gives 2(0.563/2)sin θ = (0.1541), or θ = 15.89°. This is actually a (200) reflection. The (111) planes are of two types, planes of K^{+} ions halfway between planes of Cl^{-} ions. Scattering from the two planes is opposite in phase, and of nearly equal intensity, so the (111) maxima are very weak (but are there). The (222) maxima are strong, as can be expected. Incidentally, the Mo K_{α1} line at 0.0709 nm is also useful in crystal studies.

F. C. Phillips, *An Introduction to Crystallography*, 3rd ed. (London: Longmans, 1963). An excellent introduction to classical crystallography.

W. H. Zachariasen, *Theory of X-Ray Diffraction in Crystals* (New York: Dover, 1967). Reciprocal lattice as used by crystallographers.

C. Kittel, *Introduction to Solid State Physics*, 3rd ed. (New York: John Wiley & Sons, 1966). Chapter II, pp. 34-76. Reciprocal lattice as used by solid-state physicists.

L. Brillouin, *Wave Propagation in Periodic Structures*, 2nd ed. (New York: Dover, 1946. Especially Chapter VII, but the whole book is worth reading.

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Composed by J. B. Calvert

Created 1 April 2004

Last revised 4 April 2004