Damping of spin waves and singularity of the longitudinal modes in the dipolar critical regime of the Heisenberg ferromagnet EuS

By inelastic scattering of polarized neutrons near the (200) Bragg reflection, the susceptibilities and line-widths of the spin waves and the longitudinal spin fluctuations, δS_sw(q) and δS_z(q)M_s, respectively, were determined separately. By aligning the momentum transfers q perpendicular to both δS_sw and the spontaneous magnetization M_s, we explored the statics and dynamics of these modes with transverse polarizations with respect to q. In the dipolar critical regime, where the inverse correlation length K_z(T) and q are smaller than the dipolar wave number q_d, we observe that (i) the static susceptibility of δS^l_sw(q) displays the Goldstone divergence while for δS^l_z(q) the Ornstein-Zernicke shape fits the data with a possible indication of a thermal (mass) renormalization at the smallest q values; i.e., we find indications for the predicted 1/q divergence of the longitudinal susceptibility; (ii) the spin-wave dispersion as predicted by the Holstein-Primakoff theory revealing q_d=0.23(1) Å^-1, in good agreement with previous work in the paramagnetic and ferromagnetic regime of EuS; (iii) within experimental error, the (Lorentzian) linewidths of both modes turn out to be identical with respect to the q² variation, the temperature independence, and the absolute magnitude. Due to the linear dispersion of the spin waves, they remain underdamped for q<q_d. These central results differ significantly from the well-known exchange-dominated critical dynamics, but are quantitatively explained in terms of dynamical scaling and existing data for T≥T_c. The available mode-mode coupling theory, which takes the dipolar interactions fully into account, describes the gross features of the linewidths but not all details of the T and q dependences.