Spin chirality and high-field phase diagram of the kagom\'e lattice antiferromagnet $A$Fe$_{3}$(OH)$_{6}$(SO$_{4}$)$_{2}$

We have used neutron scattering to study spin chirality and high-field spin re-oriention in single crystal and powder samples of the antiferromagnetic kagom\'e lattice compounds, $A$Fe$_{3}$(OH)$_{6}$(SO$_{4}$)$_{2}$ ($A$ = K and Ag). The $\frac{5}{2}$ Fe$^{3+}$ spins on the kagom\'e lattice order three dimensionally for temperatures below the N\'eel temperature, T$_{N}$ = 65 K. Above the N\'eel temperature, we observed the spin chiral order. Neutron measurements reveal critical spin fluctuations above T$_N$, which indicate that the spin-rotational symmetry and the vector chiral symmetry are not broken simultaneously at T$_N$. Below the N\'eel temperature, neutron scattering measurements in high fields show a spin re-orientation transition, which appear to be first-order in nature. Details of the high field phase diagram will be discussed.