# Superspace

### From Online Dictionary of Crystallography

Superespace, hyperespace (*Fr*). Superraum (*Ge*). Superspazio (*It*). 超空間 (*Ja*). Superespacio (*Sp*).

## Definition

*Superspace* is a Euclidean vector space with a preferred (real) 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( r)~=~\sum_{ k \in M^*} \hat{\rho}( k \exp (2\pi i k. r ),[/math]

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

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

for a basis **a**_{i}^{*} with the property that [math]\sum_i h_i 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~=~( r,~ 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*^{*}.
The function ρ(**r**) then can be embedded as

[math] \rho_s(r_s)~=~\sum_{k_s\in\Sigma^*} \hat{\rho}(\pi k_s= 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( r)~=~\rho_s( 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).