By a direct derivation of the equations of motion for the spins in CsNiF3 we show that the extensively used sine-Gordon equation fails to describe the dynamics of this 1D magnet. Instead of this soliton-bearing model we use the spin-wave theory and, without going to the continuum approximation, calculate the dynamic structure factors of the scattering of neutrons on CsNiF3. Complete analytical solutions for the dynamic structure factors in the frequency domain are obtained both within the classical consideration and with quantum corrections.PACS numbers: 75.10. Jm, 75.10.Hk, 75.30.Ds, 75.40.Gb, 78.70.Nx
IntroductionFor more than three decades the magnetic compound CsNiF 3 has attracted an intense attention of investigators [1]. This is mainly due to a relative simplicity of the Heisenberg model used to describe this magnet above the temperature 2.7 K, where it exhibits essentially 1D behavior, and to a possible existence of nonlinear excitations of spins -integrable (solitonic) or non-integrable [2,3], depending on whether the externally applied magnetic field is directed parallel or normal to the anisotropic axis. In the work by Mikeska [4], the fully integrable sine--Gordon (SG) model was proposed to describe the spin dynamics in CsNiF 3 . He has also calculated the "parallel" dynamic structure factor (DSF) of inelastic scattering of neutrons on kink and antikink solitons of the SG model. This and other results of the soliton theory were soon doubted by several authors [5], first in the work [6]. However, flaws of the derivation of the classical SG equation for CsNiF 3 are, to our knowledge, not directly shown in the literature. In this contribution we thus returned to the original model and obtained equations of motion for the spins. We have found that the soliton theory is a very crude approximation for CsNiF 3 . Especially for large fields and low temperatures the spin-wave (SW) theory, without going to the continuum approximation, is much more substantiated. We have calculated, both in the classical limit and with quantum corrections, the longitudinal (with respect to the applied field oriented perpendicularly to the chain of spins) and transversal DSF of inelastic neutron scattering. As distinct from the previous calculation by Reiter [6], exact analytical expressions in the frequency domain have been obtained for the DSF which explicitly satisfies the detailed balance condition. The analysis shows that the DSF peaks -the central peak and the satellite ones -are always separated. The static structure factor has been also found. We thus propose the most complete solution for the DSF of 1D magnets of the type of CsNiF 3 within the SW theory. The