Superspace

Superspace

Superespace, Hyperespace (Fr.)

Definition

Superspace is a Euclidean vector space with a preferred subspace V_E that has the dimension of the physical space, usually three, but two for surfaces, and one for line structures.

Applications

Superspace is used to describe quasi-periodic structures (See aperiodic crystal). We denote the dimension of the physical space by m. Then a function ρ(r}) is quasi-periodic if its Fourier transform is given by

[math]\rho({\bf r})~=~\sum_{{\bf k} \in M^*} \hat{\rho}({\bf k} \exp (2\pi i {\bf k}.{\bf r} ),[/math]

where the Fourier module M^* is the set of vectors (Z-module) of the form

[math]{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^* , ~~~(h_i~{\rm integers}),[/math]

for a basis a_i^* with the property that [math]\sum_i h_i{\bf a}_i^*$=0[/math] implies h_i=0 (all i) if the indices h_i are integers. The number n of basis vectors is the rank of the Fourier module.

The vectors r_s have two components: r and r_I. The relation is written as

[math]r_s~=~({\bf r},~{\bf r}_I)[/math]

In the superspace there is a reciprocal basis a_si^* such that a_i^* is the projection of a_si^* on the subspace V_E. The reciprocal lattice Σ in superspace then is projected on the Fourier module M<su>*</sup>. The function ρ(r) then can be embedded as

[math] \rho_s(r_s)~=~\sum_{k_s\in\Sigma^*} \hat{\rho}(\pi k_s={\bf k}) \exp (2\pi i k_s.r_s).[/math]

The function ρ(r) is the restriction of ρ_s(r_s) to the subspace

[math] \rho({\bf r})~=~\rho_s({\bf r},0).[/math]

Because the Fourier components in superspace belong to reciprocal vectors of Σ^*, the function ρ_s(r_s) is lattice periodic. Its direct lattice Σ is the dual of Σ^* and its basis vectors are a_is satisfying a_si.a_sj^* = δ_ij. Therefore, the symmetry group of ρ_s(r_s) is a space-group in the n-dimensional superspace, where the dimension n is equal to the rank of the Fourier module. In the superspace the usual crystallographic notions and techniques may be applied. For point atoms, the function ρ_s(r_s) is concentrated on surfaces of co-dimension equal to the dimension of the physical space (See Figure).

Figure Caption: An example of the embedding of a one-dimensional modulated chain with two modulation wave vectors. It consists of an array of two-dimensional surfaces, with three-dimensional lattice periodicity. The intersection of the surfaces with the physical space (a line) gives the positions of the modulated structure.