Fig. 3: Magnetic diffraction and origin of the satellites.

a Field-dependence of S(q, ω = 0 + ε) measured on CNCS at T = 0.05 K. The color map shows the quasielastic signal along the (00L) direction. The data are integrated within H = [−0.1; 0.1] (r.l.u.); K = [−0.15; 0.15] (r.l.u.) and E = [30; 70]μeV. Blue, green and red dots show positions of Bragg peaks extracted from the FLEXX (circles) and CNCS (squares) measurements; corresponding dotted lines show π + δ(B), 2δ(B), and π + 3δ(B) splittings extracted from the soft-mode fitting. b The red line shows the bare bandstructure in the fermionic representation for a Heisenberg spin chain in a finite magnetic field, whereas the blue dotted lines display two folded bands caused by h_ic. The green horizontal line shows the position of the chemical potential. Red arrows indicate paths for multiple scattering. Note that the small gap at the Fermi level is not shown. c Representative ENS scans along the (00L) direction measured on FLEXX at T = 0.05 K at different magnetic fields. Red arrows mark extrinsic reflections due to the twin. d S^zz(q, ω = 0) within the spin-density wave (SDW) phase calculated using quantum Monte–Carlo (QMC). e S^zz(q, ω = 0) calculated by density-matrix renormalization group (DMRG) method for different external magnetic fields with (left) and without (right) the incommensurate field, h_ic. One can see that h_ic stabilizes the ordering at the fundamental wavevector 2k_F and produces additional harmonics at 2nk_F. f1–f2 Distorted SDW with square-like modulation calculated by DMRG (f1) compared with a harmonic SDW (f2). The distorted SDW in f1 was calculated using Eq. (3) with h_ic = 0.4J. Note that h_ic exceeds the real internal field and was chosen to demonstrate more clearly the nature of the distortions caused by the multiple scattering processes at Q = nq. The intensity axes in c–e are shown on a logarithmic scale and the curves are shifted vertically for ease of viewing. The vertical error bars in c represent statistical errors of one s.d.