Classical spins · 1D chain, square and triangular lattices · |S| = 1
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Structure Factor Lab

The static structure factor of a magnetic configuration is the list of weights of the plane waves that make it up. Here you choose a configuration, see its peaks split into longitudinal (s∥, Sᶻ) and transverse (s⊥, Sˣ, Sʸ) components, then pick the peaks one by one and rebuild the configuration by superposing the waves again.

Lab

FIGURE 1Real space
FIGURE 2Reciprocal space
Click a peak: its ±q pair is added to or removed from the synthesis.
FIGURE 3Fourier synthesis: selected waves and their sum
FIGURE 4Correlation function C(r) along a₁
C(r) = Σq s(q) eiq·r. The inverse transform of the structure factor; peak width shows up here as a decay length.

1. Definition and normalization

The Fourier components of a configuration and its static (equal-time) structure factor are

\[ \mathbf S_{\mathbf q}=\sum_{j}\mathbf S_j\,e^{-i\mathbf q\cdot\mathbf r_j},\qquad s^{\alpha\beta}(\mathbf q)=\frac{1}{N^{2}}\big\langle S^{\alpha}_{\mathbf q}{}^{*}\,S^{\beta}_{\mathbf q}\big\rangle ,\qquad \mathbf q=\tfrac{m_1}{L}\mathbf b_1+\tfrac{m_2}{L}\mathbf b_2 .\]

With respect to the spin axis \(\hat z\), the longitudinal and transverse parts are \(s_\parallel=s^{zz}\) and \(s_\perp=s^{xx}+s^{yy}\). The usual definition is \(S^{\alpha\beta}(\mathbf q)=\frac1N\sum_{ij}\langle S^\alpha_iS^\beta_j\rangle e^{i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)}=N\,s^{\alpha\beta}(\mathbf q)\). We use \(1/N^2\) because then the height of a Bragg peak is directly the square of the ordered moment at that wave vector and does not depend on \(N\), while the diffuse contribution from disorder stays \(O(1/N)\).

Sum rule. Parseval's identity gives

\[ \sum_{\mathbf q}s^{\alpha\alpha}(\mathbf q)=\frac1N\sum_j\big\langle (S^\alpha_j)^2\big\rangle ,\qquad \sum_{\mathbf q}\big[s_\parallel+s_\perp\big]=\big\langle|\mathbf S|^2\big\rangle=1 . \]

Whatever the configuration, the total weight is fixed; the structure factor only tells how this weight is distributed over \(\mathbf q\) space and between the components. Σs∥ and Σs⊥ in the readout row check this rule at every step.

Correlation function. \(C^{\alpha\alpha}(\mathbf r)=\frac1N\sum_j\langle S^\alpha_jS^\alpha_{j+\mathbf r}\rangle=\sum_{\mathbf q}s^{\alpha\alpha}(\mathbf q)\,e^{i\mathbf q\cdot\mathbf r}\) (Wiener–Khinchin). A sharp peak corresponds to long-range correlations, a broadened peak to decaying correlations (Figure 4).

Reality and chirality. Since spins are real, \(\mathbf S_{-\mathbf q}=\mathbf S_{\mathbf q}^{*}\) and \(s(\mathbf q)=s(-\mathbf q)\): peaks always come in ±q pairs. The sense of rotation of a spiral does not appear in \(s\); it appears only in the antisymmetric part:

\[ \chi_z(\mathbf q)=\frac{i}{N^2}\big\langle S^{x}_{\mathbf q}{}^{*}S^{y}_{\mathbf q}-S^{y}_{\mathbf q}{}^{*}S^{x}_{\mathbf q}\big\rangle ,\qquad \chi_z(-\mathbf q)=-\chi_z(\mathbf q). \]

For a spiral rotating counterclockwise in the xy plane, \(\chi_z(\pm\mathbf Q)=\pm\tfrac12\). Experimentally this term is measured with polarized neutrons.

2. Ordered structures: Bragg peaks

A general single-wave-vector (single-Q) structure has the form \(\mathbf S_j=\mathbf m_0+\mathrm{Re}\big[\mathbf m_{\mathbf Q}\,e^{i\mathbf Q\cdot\mathbf r_j}\big]\), where \(\mathbf m_{\mathbf Q}\) is a complex vector. Directly,

\[ s^{\alpha\alpha}(0)=|m_0^\alpha|^2,\qquad s^{\alpha\alpha}(\pm\mathbf Q)=\tfrac14|m^\alpha_{\mathbf Q}|^2 . \]

The component split encodes the geometry of the structure:

A spiral cannot be distinguished from a collinear SDW by \(s_\perp\) alone: an SDW along x with amplitude \(\sqrt2\cos(\mathbf Q\cdot\mathbf r)\) also gives \(s_\perp(\pm\mathbf Q)=\tfrac12\). The distinction comes from the ratio \(s^{xx}\)/\(s^{yy}\), the chiral term, and the higher harmonics produced by the fixed-length constraint.

3. Modulation: amplitude, phase and harmonics

A product of two functions in real space is a convolution of their Fourier components in \(\mathbf q\) space. All the “modulation” behaviour on this page follows from this single principle.

Amplitude modulation (SDW) and squaring-up

For \(S^z_j=A\cos(\mathbf Q\cdot\mathbf r_j+\psi_0)\), \(s_\parallel(\pm\mathbf Q)=A^2/4\) and \(\sum_{\mathbf q}s_\parallel=A^2/2\). For \(A=1\) the total is ½: a sinusoidal SDW is incompatible with the constraint \(|\mathbf S|=1\). In a classical model such a structure appears only as a thermal average (for example just below \(T_N\)) or in itinerant electron systems. As the temperature drops the structure “squares up”; here this is modelled by \(f(u)=\tanh(\beta\cos u)/\tanh\beta\). In the limit \(\beta\to\infty\)

\[ \operatorname{sgn}(\cos u)=\frac4\pi\sum_{n\ \text{odd}}\frac{(-1)^{(n-1)/2}}{n}\cos nu\quad\Longrightarrow\quad s_\parallel(\pm n\mathbf Q)=\frac{4}{\pi^2n^2},\qquad 2\sum_{n\ \text{odd}}\frac{4}{\pi^2n^2}=1 . \]

On the lattice the harmonics fold back to \(n\mathbf Q-\mathbf G\). In the ↑↑↓↓ structure (\(Q=\pi/2\)) the 3rd harmonic lands at \(3\pi/2\equiv-\pi/2\) and coincides with the fundamental peak. This is why ↑↑↓↓ shows only two peaks even though it is a square wave.

Phase modulation: the fan structure

\(\mathbf S=(\sin\phi,0,\cos\phi)\), \(\phi=\phi_m\sin u\). The Jacobi–Anger expansion \(e^{i\phi_m\sin u}=\sum_n J_n(\phi_m)e^{inu}\) gives directly

\[ s_\parallel(\pm2n\mathbf Q)=J_{2n}^2(\phi_m),\qquad s_\perp\big(\pm(2n{+}1)\mathbf Q\big)=J_{2n+1}^2(\phi_m) : \]

even harmonics in the longitudinal component, odd harmonics in the transverse one. Phase modulation in one component transfers weight to the harmonics of the other.

The fixed-length constraint

Multi-Q structures are built as \(\mathbf m=m_0\hat z+\sum_i[\hat z\cos(\mathbf Q_i\cdot\mathbf r)+\hat e_i\sin(\mathbf Q_i\cdot\mathbf r)]\) and then normalized to \(|\mathbf m|=1\). Normalization is a nonlinear operation and produces combination harmonics such as \(\mathbf Q_1+\mathbf Q_2\) and \(2\mathbf Q_1\). When the “|S| = 1 normalization” box is unchecked these peaks disappear and the spin length becomes non-uniform. With the box unchecked the operation is linear: m₀ changes only the q = 0 peak and the fundamental peaks stay in place.

4. Superposition: how the peaks build the configuration

The inverse transform \(\mathbf S_j=\frac1N\sum_{\mathbf q}\mathbf S_{\mathbf q}e^{i\mathbf q\cdot\mathbf r_j}\) turns a ±q pair into a single real plane wave:

\[ S^\alpha_j\Big|_{\pm\mathbf q}=2\sqrt{s^{\alpha\alpha}(\mathbf q)}\;\cos\!\big(\mathbf q\cdot\mathbf r_j+\varphi^\alpha_{\mathbf q}\big),\qquad \varphi^\alpha_{\mathbf q}=\arg S^\alpha_{\mathbf q}. \]

Every peak is a wave. Its amplitude is read from the square root of the peak height; its phase is absent from the structure factor altogether. Figure 3 shows each selected wave and their sum. By Parseval, the error of the synthesis is

\[ \frac1N\sum_j\big|\mathbf S_j-\mathbf S^{\text{synth}}_j\big|^2=\sum_{\mathbf q\notin\text{selection}}s(\mathbf q), \]

that is, the total unselected weight. It appears in the readout row as “captured weight”.

The phase problem

The “Scramble phases” button gives a random phase to every component of every selected ±q pair; each press draws a new set of phases, and “True phases” returns to the original. \(|S_{\mathbf q}|\), and therefore the structure factor, does not change, while the real-space pattern changes completely. The structure factor alone does not determine the configuration. This is the central difficulty in solving magnetic structures from diffraction experiments.

Multi-Q or multi-domain?

On the square lattice, a double-Q skyrmion lattice made of two perpendicular spirals and a mixture of single-Q domains oriented along Q₁ and Q₂ give the same picture in the fundamental peaks: ±Q₁ and ±Q₂. To tell them apart one looks at (i) the \(\mathbf Q_1\pm\mathbf Q_2\) harmonics, present only in a genuine multi-Q structure (they are weak and in experiments can be confused with multiple scattering), (ii) selection of a single domain type by field or strain, (iii) real-space imaging. The topological charge is also a real-space quantity; the readout row gives it as the Berg–Lüscher sum:

\[ Q_{\text{top}}=\frac{1}{4\pi}\sum_{\triangle}\Omega_\triangle,\qquad \tan\frac{\Omega}{2}=\frac{\mathbf S_1\cdot(\mathbf S_2\times\mathbf S_3)}{1+\mathbf S_1\cdot\mathbf S_2+\mathbf S_2\cdot\mathbf S_3+\mathbf S_3\cdot\mathbf S_1}. \]

In the double-Q structure with \(m_0\neq0\), \(Q_\text{top}=\mp1\) per magnetic unit cell; the sign follows m₀. At \(m_0=0\) the field to be normalized vanishes exactly at some sites: there the spin direction and \(Q_\text{top}\) are undefined. This is not an accident: in a continuous texture the topological charge can change only by passing through such a singularity, where \(|\mathbf m|\) vanishes somewhere. The same happens in the triple-Q structure: as m₀ increases, the charge per cell takes the values \(+1\) (\(m_0<1\)), \(-2\) (\(11.5\)), and sites with \(|\mathbf m|=0\) appear at the transitions. Through all these transitions the fundamental peaks change only slightly: topology cannot be read from the fundamental peaks.

Meron–antimeron lattice. On the square lattice \(\mathbf m\propto(\sin Qx,\ \sin Qy,\ 0.5\cos Qx)\) contains two vortices and two antivortices per period; half of the cores point along \(+\hat z\), half along \(-\hat z\). Each core carries roughly \(\pm\tfrac12\) charge (±0.42 on this lattice; it approaches ½ as the cores narrow), and the total charge is zero. The fundamental peaks are again at \(\pm\mathbf Q_1\), \(\pm\mathbf Q_2\): the same peak positions can correspond to three different topologies (skyrmion, meron–antimeron, domain mixture).

5. Disorder and peak shape

Uncorrelated fluctuations. For \(\mathbf S_j=\langle\mathbf S_j\rangle+\delta\mathbf S_j\) with \(\delta\mathbf S\) uncorrelated between sites,

\[ \langle s(\mathbf q)\rangle=\frac{|\langle\mathbf S\rangle_{\mathbf q}|^2}{N^2}+\frac{1}{N}\Big(\langle|\mathbf S|^2\rangle-\overline{|\langle\mathbf S_j\rangle|^2}\Big). \]

The peak does not broaden: the Bragg weight decreases (Debye–Waller-like) and the lost weight spreads over \(\mathbf q\) space as a flat background. Direction noise η and vacancy fraction p show this. For vacancies the Bragg weight is \((1-p)^2\) and the total is \(1-p\).

Phase disorder (1D). A long-wavelength random field is added to the phase of the modulation: \(\langle\Delta\phi^2(r)\rangle=2r/\xi\). For a Gaussian field \(\langle\cos\Delta\phi\rangle=e^{-r/\xi}\), hence \(C(r)\propto\cos(\mathbf Q\cdot\mathbf r)\,e^{-r/\xi}\). In 1D this corresponds to a Lorentzian peak:

\[ s(q)\propto\frac{\sinh(1/\xi)}{\cosh(1/\xi)-\cos(q-Q)},\qquad \text{HWHM}\simeq 1/\xi . \]

The ξ in the readout row is obtained from the half width at half maximum of the peak, assuming a Lorentzian, and agrees with the input ξ to within 10–20%. Compare with direction noise η: η lowers the peak but does not broaden it; ξ broadens it. This is the difference between long- and short-range order. The setting exists only in 1D: in 2D the transform of an exponential correlation is not Lorentzian, and on a finite torus the phase difference saturates, leaving a misleading residual Bragg peak. In 2D, peak broadening is shown by antiphase and orientation domains.

Domains. Domains of average size ℓ broaden peaks by \(\sim2\pi/\ell\) (Laue function). With antiphase domains Q does not shift; only the width increases. With ±M domains the spins inside the walls acquire a transverse component, so \(s_\perp\) rises at small \(q\).

Labyrinth: a ring-shaped structure factor. When all waves on the shell |q| ≈ Q are added with random phases, stripes form with a fixed local wavelength and random orientation. The structure factor is then no longer a set of discrete peaks but a ring of radius Q. A ring means “wavelength known, direction not”; it has the same meaning as the rings seen in powder averages and in SANS. The |S| = 1 normalization couples the waves and moves part of the weight off the ring (into harmonics). As the net moment m₀ grows the stripes break up; in the chiral labyrinth a mixture of isolated skyrmions and worms appears, and the ring remains with a q = 0 peak added. On small lattices the shell contains few points, so the ring looks speckled.

Single sample and ensemble. The snapshot \(s(\mathbf q)\) of a random configuration is noisy (a speckle pattern): the value at each \(\mathbf q\) is exponentially distributed and fluctuates as much as its mean. A smooth Lorentzian or a flat background appears only after ensemble averaging. The synthesis, in contrast, uses the complex amplitudes of a single snapshot, because the averaged \(|S_{\mathbf q}|^2\) has no phase.

6. Critical notes

  1. Equal-time versus elastic. What is computed here is \(S(\mathbf q)=\int S(\mathbf q,\omega)\,\mathrm d\omega/2\pi\). Bragg peaks are the elastic (\(\omega=0\)) part, while part of the diffuse scattering is inelastic. A diffraction measurement without energy analysis equals the equal-time correlation only in the quasi-static approximation (incident energy much larger than the magnetic energies).
  2. The projection measured in experiment. The magnetic neutron cross-section is \(\propto|F(\mathbf q)|^2\sum_{\alpha\beta}(\delta_{\alpha\beta}-\hat q_\alpha\hat q_\beta)S^{\alpha\beta}(\mathbf q)\). Only components perpendicular to \(\mathbf q\) are seen, and the form factor suppresses the signal at large \(q\). The ∥/⊥ split used here is a theoretical split with respect to the spin axis (ẑ); it is not the same as the neutron “longitudinal/transverse”.
  3. Classical versus quantum. For quantum spins the sum rule becomes \(S(S+1)\) and the Bragg weight is \(m^2\le S^2\). Zero-point fluctuations move weight from the Bragg peak to the diffuse background. For the 2D S = ½ Heisenberg antiferromagnet \(m\approx0.307\), so only \(m^2/S(S+1)\approx13\%\) of the total weight is in the Bragg peak.
  4. Finite periodic lattice. Q always lies on the \(2\pi m/L\) grid; there is no true incommensurability. Harmonics fold into the zone, and the peak width cannot be smaller than \(2\pi/L\).
  5. Recipes, not minimization. The configurations are analytic recipes. They are not claimed to be ground states or thermal equilibria of any Hamiltonian. The sinusoidal SDW and the unnormalized multi-Q structures deliberately violate \(|\mathbf S|=1\).
  6. Limits of the component split. The sum \(s_\perp=s^{xx}+s^{yy}\) is invariant under rotations about \(\hat z\). In-plane orientation additionally requires \(s^{xx}\), \(s^{yy}\) and \(\chi_z\).

7. Table of configurations

ConfigurationPeakss∥s⊥What it teaches
FM, tilted by θq = 0cos²θsin²θSingle peak; the component split gives the orientation
Néel / stripeQ = (π,π) / (π,0)10Q ≡ −Q on the zone boundary
Spiral (xy)±Q0½ + ½Chirality only in χz
Spiral (Bloch/Néel)±Q¼ + ¼¼ + ¼Rotation plane in the ∥/⊥ ratio
Conical, α0, ±Qsin²α½cos²α + ½cos²αFM and spiral in different components
SDW (sinusoidal)±Q¼ + ¼0Σ = ½: fixed length violated
Squared-up SDW±nQ, n odd4/(π²n²)0Harmonics, folding
Fan, φₘ0, ±nQJ²₀, J²₂, …J²₁, J²₃, …Phase modulation, Bessel weights
Double-/triple-Q skyrmion±Qᵢ, Qᵢ ± Qⱼ, 0depends on normalizationCombination harmonics, Qtop
Meron–antimeron (square)±Q₁, ±Q₂similar to the skyrmion latticeSame peaks, Q_top = 0, ≈ ±½ per core
Labyrinth (collinear / chiral)ring at |q| ≈ Qring; ∥ and ⊥ equal when chiralFixed wavelength, random direction
Orientation domains±Qᵢ (broadened)same fundamental peaks as double-QMulti-Q / multi-domain ambiguity
Short-range spiral (1D)±Q (Lorentzian)0HWHM ≈ 1/ξPeak width ↔ correlation length
Paramagnetnone≈ 1/(3N)≈ 2/(3N)Flat background, speckle noise

8. Numerical checks

The core code was compared with analytic results (single sample, no disorder).

CaseQuantityAnalyticComputed
Spiral xy, N = 96, Q = π/6s⊥(±Q), χz(±Q)0.5, ±0.50.50000, ±0.50000
Conical, α = 35°s∥(0) / s⊥(Q)0.32899 / 0.335510.32899 / 0.33551
Square-wave SDW, N = 240, period 48s∥(nQ), n = 1, 3, 50.4053, 0.0450, 0.01620.4059, 0.0456, 0.0168
↑↑↓↓s∥(±π/2)0.50.50000
Fan, φₘ = 90°J²₀, J²₁, J²₂, J²₃0.2228, 0.3213, 0.0624, 0.00480.2228, 0.3213, 0.0624, 0.0048
120° (triangular, L = 36)s⊥(K), s⊥(K′)0.5, 0.50.50000, 0.50000
Double-Q skyrmion, L = 32, 16 cellsQtop (m₀ = ±0.5)∓16∓16.000
Triple-Q skyrmion, L = 36, 9 cellsQtop (m₀ = 0.5)99.000
Meron–antimeron, L = 32Qtop, core charges, min |m| before normalization0, ±½ (ideal), > 00.000, ±0.42, 0.50
Triple-Q, L = 36, m₀ = 1.2 / 2Qtop per cell−2 / 0−2.000 / 0.000
Phase disorder 1D, N = 240, 64 samplesξ (from HWHM)4, 8, 164.8, 9.2, 16.8

The small deviation for the square-wave SDW is the difference between the continuous square wave and the square wave sampled on the lattice. The deviation in ξ comes from the phase difference growing as \(r(N-r)/N\) on a periodic ring, and from finite-sample noise.

9. Technical note