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SW15 - Ba₃NbFe₃Si₂O₁₄

This is a Sunny port of SpinW Tutorial 15, originally authored by Sandor Toth. It calculates the linear spin wave theory spectrum of Ba₃NbFe₃Si₂O₁₄. The ground state is an incommensurate spiral, which can be directly studied using the functions minimize_spiral_energy! and SpinWaveTheorySpiral.

Load packages

using Sunny, GLMakie

Specify the Ba₃NbFe₃Si₂O₁₄ Crystal cell following Marty et al., Phys. Rev. Lett. 101, 247201 (2008).

units = Units(:meV, :angstrom)
a = b = 8.539 # (Å)
c = 5.2414
latvecs = lattice_vectors(a, b, c, 90, 90, 120)
types = ["Fe", "Nb", "Ba", "Si", "O", "O", "O"]
positions = [[0.24964,0,0.5], [0,0,0], [0.56598,0,0], [2/3,1/3,0.5220],
             [2/3,1/3,0.2162], [0.5259,0.7024,0.3536], [0.7840,0.9002,0.7760]]
langasite = Crystal(latvecs, positions, 150; types)
cryst = subcrystal(langasite, "Fe")
view_crystal(cryst)

Create a System and set exchange interactions as parametrized in Loire et al., Phys. Rev. Lett. 106, 207201 (2011).

sys = System(cryst, [1 => Moment(s=5/2, g=2)], :dipole)
J₁ = 0.85
J₂ = 0.24
J₃ = 0.053
J₄ = 0.017
J₅ = 0.24
set_exchange!(sys, J₁, Bond(3, 2, [1,1,0]))
set_exchange!(sys, J₄, Bond(1, 1, [0,0,1]))
set_exchange!(sys, J₂, Bond(1, 3, [0,0,0]))

The final two exchanges are set according to the desired chirality $ϵ_T$ of the magnetic structure.

ϵT = -1
if ϵT == -1
    set_exchange!(sys, J₃, Bond(2, 3, [-1,-1,1]))
    set_exchange!(sys, J₅, Bond(3, 2, [1,1,1]))
elseif ϵT == 1
    set_exchange!(sys, J₅, Bond(2, 3, [-1,-1,1]))
    set_exchange!(sys, J₃, Bond(3, 2, [1,1,1]))
else
    error("Chirality must be ±1")
end

This compound is known to have a spiral order with approximate propagation wavevector $𝐤 ≈ [0, 0, 1/7]$. Search for this magnetic order with minimize_spiral_energy!. Due to reflection symmetry, one of two possible propagation wavevectors may appear, $𝐤 = ± [0, 0, 0.1426…]$. Note that $k_z = 0.1426…$ is very close to $1/7 = 0.1428…$.

axis = [0, 0, 1]
randomize_spins!(sys)
k = minimize_spiral_energy!(sys, axis)

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  3.0868734163007545e-12
 -2.9996828345249424e-12
  0.14264604656428487

We can visualize the full magnetic cell using repeat_periodically_as_spiral, which includes 7 rotated copies of the chemical cell.

sys_enlarged = repeat_periodically_as_spiral(sys, (1, 1, 7); k, axis)
plot_spins(sys_enlarged; color=[S[1] for S in sys_enlarged.dipoles])

One could perform a spin wave calculation using either SpinWaveTheory on sys_enlarged, or SpinWaveTheorySpiral on the original sys. The latter has some restrictions on the interactions, but allows for our slightly incommensurate wavevector $𝐤$.

measure = ssf_perp(sys)
swt = SpinWaveTheorySpiral(sys; measure, k, axis)

SpinWaveTheorySpiral(SpinWaveTheory(System([Dipole mode], Supercell (1×1×1)×3, Energy per site 9.601), Sunny.SWTDataDipole(StaticArraysCore.SMatrix{3, 3, Float64, 9}[[0.919424094146979 -0.31821097997188036 0.23108679609043625; -0.2183775633144133 0.07558007123708131 0.9729331388500739; -0.32706356407106363 -0.9450023412971703 1.0656681665242993e-10], [0.5611861001746362 0.3955767180413888 0.7270414163679996; 0.5942460488339039 0.4188804776540448 -0.6865936053340529; -0.57614390078875 0.8173482767975492 -1.281635336393285e-11], [-0.002747611904372585 -0.2863263506665687 -0.9581282124761739; 0.009193855980013304 0.9580841009822714 -0.28633953352132835; 0.9999539607607151 -0.009595642706889188 -4.993777041529958e-11]], StaticArraysCore.SVector{3, Float64}[[1.8388481882939576, -0.6364219599437606, 0.4621735921808726] [1.1223722003492722, 0.7911534360827774, 1.4540828327359991] [-0.005495223808745165, -0.5726527013331368, -1.9162564249523462]; [-0.4367551266288262, 0.15116014247416248, 1.945866277700147] [1.188492097667807, 0.837760955308089, -1.3731872106681051] [0.018387711960026597, 1.9161682019645416, -0.5726790670426565]; [-0.6541271281421271, -1.8900046825943402, 2.131336333048598e-10] [-1.1522878015774998, 1.634696553595098, -2.5632706727865694e-11] [1.9999079215214293, -0.019191285413778368, -9.987554083059912e-11]], Sunny.StevensExpansion[Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0)], [1.5811388300841898, 1.5811388300841898, 1.5811388300841898]), MeasureSpec, 1.0e-8), [3.0868734163007545e-12, -2.9996828345249424e-12, 0.14264604656428487], 3, [0.0, 0.0, 1.0], Matrix{StaticArraysCore.SMatrix{3, 3, ComplexF64, 9}}[[[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 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Calculate broadened intensities for a path $[0, 1, L]$ through reciprocal space

qs = [[0, 1, -1], [0, 1, -1+1], [0, 1, -1+2], [0, 1, -1+3]]
path = q_space_path(cryst, qs, 400)
energies = range(0, 6, 400)
res = intensities(swt, path; energies, kernel=gaussian(fwhm=0.25))
plot_intensities(res; units, saturation=0.7, colormap=:jet, title="Scattering intensities")

Use ssf_custom_bm to calculate the imaginary part of $\mathcal{S}^{2, 3}(𝐪, ω) - \mathcal{S}^{3, 2}(𝐪, ω)$. In polarized neutron scattering, it is conventional to express the 3×3 structure factor matrix $\mathcal{S}^{α, β}(𝐪, ω)$ in the Blume-Maleev polarization axis system. Specify the scattering plane $[0, K, L]$ via the spanning vectors $𝐮 = [0, 1, 0]$ and $𝐯 = [0, 0, 1]$.

measure = ssf_custom_bm(sys; u=[0, 1, 0], v=[0, 0, 1]) do q, ssf
    imag(ssf[2,3] - ssf[3,2])
end
swt = SpinWaveTheorySpiral(sys; measure, k, axis)
res = intensities(swt, path; energies, kernel=gaussian(fwhm=0.25))
plot_intensities(res; units, saturation=0.8, allpositive=false,
                 title="Im[S²³(q, ω) - S³²(q, ω)]")