The Reciprocal Space View

The presence of well-defined reflections in reciprocal space, the quantity accessible to diffraction experiments, is a distinguishing feature of crystalline as compared to amorphous phases. In a periodic crystal these reflections will lie on points of a three-dimensional reciprocal lattice. The observation of additional, so called, satellite reflections in the diffraction pattern of a material is the most unequivocable means possible to characterising an incommensurate phase. The first satellite reflections to be correctly identified as such were in diffraction patterns of NaNO$_2$ in 1961 [25] (although such spots had been observed as long ago as 1936 in studies of calverite). They are so called because they do not lie on the nodes of the three-dimensional reciprocal lattice but rather lie at fractional positions centred around the fundamental reflections. The satellites are well-defined, their sharpness is limited only by the degree of disorder and the size of the crystal, and may appear as sharp as the Bragg reflections of a periodic crystal. This is a reflection of the long range order, demonstrating beyond doubt that microscopic periodicity is not a prerequisite for macroscopic crystallinity. It was to encompass this that the International Union of Crystallography, only as late as 1992 [26], officially sanctioned the redefinition of the term ``crystal'' as being ``any solid having an essentially discrete diffraction diagram''.

The route to understanding the reciprocal space picture of a crystal is thus to start with a representation of the material as a superposition of plane waves:

$$n( {\bf r} ) = \sum_{{\bf k} in L} n_{\bf k} exp( i {\bf k . r} )$$ (4.3)

The quantity $n({\bf r})$ may represent any local physical variable. In the case of x-rays the quantity with which they interact is the electron density. A scattering experiment means sensing the Fourier component for a selected wavevector k, if it is present. Each location in Fourier (reciprocal) space may potentially represent a value of k. The complete description of the structure requires that the full set of wavevectors $L$ be determined. It is the restrictions upon which k components are allowed to go to make up $L$ which determine the nature of the crystalline state, and are the key to an understanding of the reciprocal space image.

The traditional definition of a crystal (a periodic crystal) had the requirement that n(r) should be invariant under any translation of the form ${\bf T} = u_1{\bf a}_1 + u_2{\bf a}_2 + u_3{\bf a}_3$ where ${\bf a}_{(1,2,3)}$ are the crystal axes and $u_{(1,2,3)}$ are integers. And therefore

$$n ( {\bf r} ) = n ( {\bf T} + {\bf r} )$$ (4.4)

This condition for translational invariance means that the only components of n(r) which are allowed are ones which contain reciprocal lattice vectors G, such that G can be expressed in terms of reciprocal lattice axes b$_{(1,2,3)}$: ${\bf G} = v_1{\bf b}_1 + v_2{\bf b}_2 + v_3{\bf b}_3$ where $v_{(1,2,3)}$ are integers. The condition also restricts the symmetry point group of the structure to one of the 14 Bravais lattices and is the basis for the three-dimensional space group classification. The result is, of course, also in accord with the new requirement for a discrete diffraction diagram. The Fourier coefficients $n_k$ will determine the weights of the reciprocal lattice nodes i.e. the intensity of the fundamental reflections.

The same starting point, equation 2.3, is again applicable when considering the new broader crystal definition which includes aperiodic structures. The existence of a discrete diffraction diagram requires merely that a finite number of the wavevectors k in the set $L$ possess significant coefficients $n_k$. The periodic case also necessitated that just three generating vectors be able to express all k wavevectors in the set $L$. But, more generally, without the periodic restriction, the minimum number of generating vectors, ${\bf b}_i$, whose integral linear combinations express all $L$ may vary. The minimum number of ${\bf b}_i$ necessary is known as the rank, $D$, where:

$${\bf k} = \sum_{i=1,D} a_i{\bf b}_i$$ (4.5)

The case when the rank is 3, can be seen to be that of the already described periodic crystal, the three generating vectors being the axes of the reciprocal lattice. When the rank exceeds 3, however, the condition for translational invariance is no longer met, it can no longer be classified by the traditional space group scheme, and a quasiperiodic crystal is the result.

A new crystallographic scheme, advanced by Mermin [27], has been developed to encompass this enlarged crystallographic view. With the concept of periodicity now no longer central to crystallinity, the Mermin scheme instead starts from a more fundamental basis. The constraints on the set $L$ are derived from the requirements for physical stability itself. At the heart of the method is the redefinition of point group symmetry using the concept of indistinguishable, as opposed to identical, densities. Briefly stating Mermin's definition, two densities are indistinguishable if any substructure on any scale that occurs in one occurs in the other with the same frequency. Under indistinguishability, the condition on n(r) in equation 2.4 can now be satisfied for any value of the rank, $D$. The scheme is able to treat both periodic and quasiperiodic materials on an equal footing, with periodicity merely reducing to the simplest case amongst many other possibles.

Although the new Mermin scheme presents a highly aesthetic argument for its adoption it has, however, still to be applied widely by crystallographers. An alternative and less radical scheme, predating that of Mermin, has become more widely accepted for practical purposes. Known as superspace group symmetry, it requires only an extension to, as opposed to the complete reformulation of, the three-dimensional space groups. Although advantages of the Mermin scheme are apparent in the classification of quasicrystals and five-fold symmetry, the superspace method, albeit of a somewhat more abstract nature, has been found to be wholly satisfactory for the description of modulated and misfit layer type structures which are pertinent here. The superspace method will therefore be utilised here in preference to the Mermin scheme.

In order to recover the translational invariance of a quasiperiodic crystal it is necessary to introduce higher dimensions to the space group description. The observation was first noted by De Wolf [28] that a quasiperiodic crystal of rank $D$ could be considered as the three-dimensional section of a lattice which is periodic in $D$ dimensions. A general scheme of the symmetry groups in higher dimensions, known as superspace groups, has subsequently been formulated by Janner and Jannsen [29].

Within the superspace formalism, the requirement upon the set of wavevectors $L$ is that they now be translationally invariant in $D$ dimensions. It follows, in the same way as it does for three dimensional periodicity, that a concomitant periodic reciprocal lattice must exist in $D$ dimensional reciprocal space. Of course this is an imagined construction and the experimental realisation is still three-dimensional. However, the three-dimensional reciprocal space pattern can now be interpreted as the projection of that $D$ dimensional reciprocal lattice. This is illustrated in Figure 2.4 for the simplest example of a four-dimensional reciprocal space. The position of the reciprocal lattice points are lifted out of the three-dimensional plane, represented by $R^*$, and located in the higher dimension represented by the fourth reciprocal axis ${\bf b}_4$. Because ${\bf b}_{i=1,2,3}$ are identical in both three- and four-dimensional reciprocal space the fundamental reflections coincide with the intersection of ${\bf b}_4$ with $R^*$. The superspace reciprocal lattice points, however, are observed as a projection of the points from ${\bf b}_4$ into $R^*$ and correspond to the satellite reflections.

Figure 2.4: A representation of four-dimensional reciprocal space, indicating the lattice points (solid circles) located in the higher dimension along ${\bf b}_4$ and their corresponding satellite reflections (open circles) observed by projection along ${\bf e}_4$ into three-dimensional reciprocal space $R^*$.

The position of a satellite reflection relative to its fundamental reflection can be described by a wavevector q which is related to the higher dimensional reciprocal space coordinates by a vector ${\bf e}_4$ perpendicular to $R^*$,

$${\bf b}_4 = {\bf q} + {\bf e}_4$$ (4.6)

So that the indexing of the three-dimensional reciprocal space pattern can now be achieved using

$${\bf G} = h{\bf a}^* + k{\bf b}^* + l{\bf c}^* + \sum_{j=1}^d m_j{\bf q}_j$$ (4.7)

where the rank $D = 3 + d$, and each reflection is now represented by $D$ indices. As stated, the $hkl$ values are the same in both three-dimensional and $D$-dimensional reciprocal space, so the positions of fundamental reflections, corresponding to $m=0$, will coincide in both. For reflections with $m \neq 0$, that is the satellites, higher orders will be observed corresponding to $m_j{\bf q}_j$ ( $m_j = \pm 1 \pm 2 ...$) but with intensities rapidly decreasing with $m_j$. Each wavevector ${\bf q}_j$ can be represented by a sum of components along the three-dimensional reciprocal axes, such that,

$${\bf q}_j = \alpha_j{\bf a}^* + \beta_j{\bf b}^* + \gamma_j{\bf c}^*$$ (4.8)

For the wavevector to be incommensurate one or more of $\alpha_j, \beta_j,\gamma_j$ will be irrational.

In the application of this theory to studies of the bismuth based high-T$_c$ cuprates, the concern will be predominantly with displacively modulated crystals. In equation 2.7 the value of $d$ is then known as the dimension of the modulation. In the main, the structures of the bismuth based cuprates are one-dimensionally modulated and can therefore be represented in four-dimensional superspace, and by a single wavevector ${\bf q}$. The wavelength of the modulation will be related by $\lambda = 1/\vert{\bf q}\vert$.