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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):
  2.0886170664176004e-13
 -9.367406756859511e-14
  0.14264604656306187

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}[[0.22631496905320617 -0.46998521394289067 0.8531678811684732; -0.37015179729446895 0.7686891962407021 0.5216364313796519; -0.900982342636148 -0.4338557574331349 1.5961959701980758e-11], [0.9844440722591524 -0.17388650855751042 0.0251664605277719; 0.024782822246807223 -0.004377494372247449 -0.9996832744747229; 0.1739416003010786 0.9847559696111011 4.6591902728364205e-11], [0.34240637624084824 0.33359654929975846 -0.8783343416973429; 0.6291167559405981 0.6129301128184867 0.4780468430970918; 0.6978323444097364 -0.7162611388981055 -6.936955992610385e-11]], StaticArraysCore.SVector{3, Float64}[[0.45262993810641217, -0.9399704278857809, 1.7063357623369462] [1.9688881445183046, -0.3477730171150208, 0.05033292105554392] [0.684812752481696, 0.6671930985995165, -1.7566686833946845]; [-0.7403035945889376, 1.5373783924814033, 1.0432728627593035] [0.049565644493614294, -0.008754988744494872, -1.9993665489494448] [1.2582335118811956, 1.2258602256369728, 0.9560936861941829]; [-1.8019646852722955, -0.8677115148662696, 3.192391940396151e-11] [0.34788320060215716, 1.9695119392222018, 9.318380545672838e-11] [1.3956646888194721, -1.4325222777962103, -1.3873911985220765e-10]], 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), [2.0886170664176004e-13, -9.367406756859511e-14, 0.14264604656306187], 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)]")