Finding a periodic image inside the WS cell about brillouin.jl HOT 5 CLOSED

@jaemolihm I added this in 933853e with the name reduce_to_wignerseitz, which takes a point and a basis as arguments.
Let me know if you feel another name is more suitable: otherwise I'll probably tag a new version in a day or so.

Here's a quick visualization of the results:

Unreduced points	Reduced points

The implementation itself is quite simple[edit: a little tricky]: first, reduce the input point to the "rectangular" reduced unit cell with coordinates in [-1/2, 1/2); next, check all immediately neighboring point to see if there's a point with smaller norm in cartesian space; [edit: then do that recursively until convergence]; finally, return point with minimum norm.

reduce_to_symmetric_unitcell(x::Real) = mod(x + 1//2, 1) - 1//2
reduce_to_symmetric_unitcell(v::AbstractVector{<:Real}) = reduce_to_symmetric_unitcell.(v)

function reduce_to_wignerseitz(v::StaticVector{D, <:Real}, Vs::BasisLike{D}) where D
    # reduce `v` to "rectangular" unit cell with coordinates in [-1/2, 1/2)
    v₀ = reduce_to_symmetric_unitcell(v)
    v₀ᶜ = cartesianize(v₀, Vs)
    d₀ = norm(v₀ᶜ)
    # check whether `v₀` has the smallest norm of all adjacent equivalent points; if not,
    # we initiate a recursive search the new possible minimum until convergence
    return search_shortest_norm_among_neighbors_recursive(v₀, d₀, Vs)
end

function search_shortest_norm_among_neighbors_recursive(
            v₀::StaticVector{D, <:Real}, d₀::Real, Vs::BasisLike{D}) where D

    for I in CartesianIndices(ntuple(_->-1:1, Val(D)))
        iszero(I) && continue
        v′ = v₀ .+ Tuple(I)
        v′ᶜ = cartesianize(v′ᶜ, Vs)
        d′ = norm(v′ᶜ)
        if d′ < d₀
            return search_shortest_norm_among_neighbors_recursive(v′, d′, Vs)
        end
    end
    return v₀
end

I'm sure there is a faster way, but this doesn't seem likely to occur in performance sensitive contexts.

Edit 2: If a faster version of this is ever needed, I think the best way forward would be to define a function is_reduced_in_wingerseitz that performs an inclusion test using the convex hull structure of a provided Wigner-Seitz cell, e.g. via its facets normal vectors & origins (as here) and then use that to test whether the current candidate is actually already the right one before searching for alternatives.

from brillouin.jl.

Thanks a lot for this feature! The visualization is also very nice 😄 The name reduce_to_wignerseitz seems adequate to me.

from brillouin.jl.

Can't you just convert to reduced coordinates, use modulo and then convert back?

from brillouin.jl.

Hmm, that would just transform the point back to the "cubic" (reduced) unit cell though, not necessarily the associated Wigner-Seitz cell - right?

There is indeed a function in SymmetryReduceBZ: seems it first reduces to the "cubic" (reduced coordinates) unit cell and then searches through the immediately neighboring points for whichever one has the shortest Cartesian norm. That seems sound.

from brillouin.jl.

Closed by 933853e

from brillouin.jl.