Embedding in superspace

Janssen, T.

doi:10.1107/97809553602060000637

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 244-245

Section 1.10.1.3. Embedding in superspace

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.10.1.3. Embedding in superspace

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A simple example of a quasiperiodic function is obtained in the following way. Consider a function of n variables which is periodic with period one in each variable. $[f(x_{1},\ldots,x_{n}) = f(x_{1}+1,x_{2},\ldots,x_{n}),\ldots.\eqno(1.10.1.13)]$ Now take n mutually irrational numbers $[\alpha_{i}]$ and define the function [g(x)] with one variable as $[g(x) = f(\alpha_{1}x,\alpha_{2}x,\ldots, \alpha_{n}x).\eqno(1.10.1.14)]$ Because of the irrationality, the function [g(x)] is not periodic. If we consider the Fourier transform of $[f(x_{1},\ldots,x_{n})]$ we get $[f(x_{1},\ldots, x_{n}) = \textstyle\sum\limits_{m_{1}}\ldots \textstyle\sum\limits_{m_{n}} A_{m_{1},\ldots, m_{n}}\exp[2\pi i(m_{1}x_{1} + \ldots + m_{n}x_{n})]\eqno(1.10.1.15)]$ and consequently $[g(x) = \textstyle\sum\limits_{m_{1}\ldots m_{n}}A_{m_{1},\ldots,m_{n}} \exp\left[2\pi i\left(\textstyle\sum\limits_{i = 1}^{n}m_{i}\alpha_{i}\right)x\right],\eqno(1.10.1.16)]$ which proves that the function is quasiperiodic of rank n with n basis vectors $[2\pi\alpha_{i}]$ in one dimension.

The quasiperiodic function [g(x)] is therefore the restriction to the line $[(\alpha_{1}x,\ldots,\alpha_{n}x)]$ in n-dimensional space. This is a general situation. Each quasiperiodic function can be obtained as the restriction of a periodic function in n dimensions to a subspace that can be identified with the physical space. We denote the n-dimensional space in which one finds the lattice periodic structure (the superspace) by $[V_{s}]$ , the physical space by $[V_{E}]$ and the additional space, called internal space, by $[V_{I}]$ , such that $[V_{s}]$ is the direct sum of $[V_{E}]$ and $[V_{I}]$ . In the field of quasicrystals, one often uses the name parallel space for $[V_{E}]$ and perpendicular space for $[V_{I}]$ .

On the other hand, one can embed the quasiperiodic function in superspace, which means that one constructs a lattice periodic function in n dimensions such that its restriction to physical space is the quasiperiodic function. Take as an example the displacively modulated structure of equation (1.10.1.2). Compare this three-dimensional structure with the array of lines $[\left({\bf n} + {\bf r}_{j} + {\bf f}_{j}({\bf Q}\cdot{\bf n} + t), t \right) \quad({\rm real} \;\;t)\eqno(1.10.1.17)]$ in four-dimensional space. The restriction to the three-dimensional hyperplane [t = 0] gives exactly the structure (1.10.1.2). Moreover, the four-dimensional array of lines is lattice periodic. Because $[{\bf f}_{j}]$ is periodic, the array is left invariant if one replaces t by [t+1] , and for every lattice vector $[{\bf m}]$ of the basic structure the array is left invariant if one replaces simultaneously t by $[t-{\bf Q}\cdot{\bf m}]$ . This means that the array is left invariant by all four-dimensional lattice vectors of the lattice $[\Sigma]$ with basis $[{\bf a}_{si} = ({\bf a}_{i}, -{\bf Q}\cdot{\bf a}_{i})\quad (i = 1, 2, 3), \quad{\bf a}_{4} = (0,1). \eqno(1.10.1.18)]$ Indeed the quasiperiodic IC phase is the restriction to $[V_{E}]$ ( [t = 0] ) of the lattice periodic function in four dimensions.

The reciprocal basis for (1.10.1.18) consists of the basis vectors $[{\bf a}_{si}^{*} = ({\bf a}_{i}^{*}, 0) \quad (i = 1, 2, 3), \quad {\bf a}_{4}^{*} = ({\bf Q},1). \eqno(1.10.1.19)]$ These span the reciprocal lattice $[\Sigma^{*}]$ . The projection of this basis on $[V_{E}]$ consists of the four vectors $[{\bf a}_{i}^{*}]$ [(i = 1,2,3)] and $[{\bf Q}]$ , and these form the basis for the Fourier module of the quasiperiodic structure.

This is a well known situation. From the theory of Fourier transformation one knows that the projection of the Fourier transform of a function in n dimensions on a d-dimensional subspace is the Fourier transform of the restriction of that n-dimensional function to the same d-dimensional subspace. This gives a way to embed the quasiperiodic structure in a space with as many dimensions as the rank of the Fourier module. One considers the basis of the Fourier module as the projection of n linearly independent vectors in n-dimensional space. This means that for every vector of the Fourier module one has exactly one reciprocal-lattice vector in $[V_{s}]$ . Suppose the quasiperiodic structure is given by some function, for example the density $[\rho ({\bf r})]$ . Then $[\rho ({\bf r}) = \textstyle\sum\limits_{{\bf H}\in M^*} \hat{\rho}({\bf H}) \exp (i{\bf H}\cdot{\bf r}).\eqno(1.10.1.20)]$ One may define a function in n-dimensional space by $[\rho_{s}({\bf r}_{s}) = \textstyle\sum\limits_{{\bf H}_{s}\in \Sigma^*} \hat{\rho}({\bf H}) \exp (i{\bf H}_{s}\cdot{\bf r}_{s}),\eqno(1.10.1.21)]$ where $[{\bf H}_{s}]$ is the unique reciprocal-lattice vector that is projected on the Fourier module vector $[{\bf H}]$ . It is immediately clear that the restriction of $[\rho_{s}]$ to physical space is exactly $[\rho]$ . Moreover, the function $[\rho_{s}]$ is lattice periodic with lattice $[\Sigma]$ , for which $[\Sigma^{*}]$ is the reciprocal lattice.

This construction can be performed in the following equivalent way. Consider a point $[{\bf r}]$ in physical space, where one has the quasiperiodic function $[\rho ({\bf r})]$ . The Fourier module of this function is the projection on physical space of the n-dimensional reciprocal lattice $[\Sigma^{*}]$ with basis vectors $[{\bf a}_{si}^{*}]$ ( $[i = 1, 2,\ldots, n]$ ). The reciprocal lattice $[\Sigma^{*}]$ corresponds to the direct lattice $[\Sigma]$ . A point r in $[V_{E}]$ can also be considered as an element (r, 0) in n-dimensional space. By the translations of $[\Sigma]$ , this point is equivalent with a point $[{\bf r}_{s}]$ with lattice coordinates $[\xi_{i} = {\rm Frac}\left({\bf a}_{si}^{*}.({\bf r},0)\right) = {\rm Frac}({\bf a}_{i}^{*}\cdot{\bf r})\eqno(1.10.1.22)]$ in the unit cell of $[\Sigma]$ , where Frac(x) is x minus the largest integer smaller than x. If one puts $[\rho_{s}({\bf r}_{s}) = \rho ({\bf r})]$ , the function $[\rho]$ determines the function $[\rho_{s}]$ in the unit cell, and consequently in the whole n-dimensional space $[V_{s}]$ . This means that all the information about the structure in $[V_{E}]$ is mapped onto the information inside the n-dimensional unit cell. The information in three dimensions is exactly the same as that in superspace. Only the presentation is different.

In the case in which the crystal consists of point atoms, the corresponding points in d-dimensional physical space $[V_{E}]$ are the intersection of ( [n-d] )-dimensional hypersurfaces with $[V_{E}]$ . For displacively modulated IC phases in three dimensions with one modulation wavevector, one has [n = 4] , [d = 3] and the hypersurfaces are just lines in superspace, as we have seen. For more independent modulation vectors the dimension of the hypersurfaces is larger than one. In this case, as often in the case of composite structures, the ( [n-d] )-dimensional surfaces do not have borders. This in contrast to quasicrystals, where they are bounded. All these hypersurfaces for which the intersection with physical space gives the atomic positions are called atomic surfaces.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 244-245