Phonon renormalization of the N\`{e}el transition in KCuF$_{3}$*

Critical magnetic fluctuations in the one-dimensional antiferromagnet KCuF$_3$, in the form of diffuse scattering around the magnetic (001) Bragg peak, have been studied with resonant soft x-ray scattering. Using x-rays near the Cu L$_{3}$ edge to exploit the 2\textit{p} $\to$ 3\textit{d} dipole transition, the (001) was directly observed at temperatures ranging from 23K to above the transition temperature (T$_{N}$ $\approx$ 43K). Notably, the phase transition exhibits hysteresis, with T$_{N}$ sensitive to whether the sample was cooled or heated prior to measurement. This suggests that the phase transition is weakly first order, as might be expected by a transition renormalized by phonons. The temperature dependence of the coherence length and the diffuse scattering, as well as the role played by Jahn-Teller phonons in the transition are discussed.