Cycloidal Magnetic Ordering in Noncentrosymmetric EuIrGe$_3$

Successive magnetic phase transitions at $T_{\text{N}}$=12.2 K, $T_{\text{N}}^{\;\prime}$=7.0 K, and $T_{\text{N}}^{\;*}$=5.0 K in EuIrGe$_3$, an intermetallic compound with a body centered tetragonal lattice belonging to a polar space group $I4mm$, has been investigated by neutron diffraction and resonant X-ray diffraction. It is shown that EuIrGe$_3$ exhibits an incommensurate longitudinal sinusoidal order with $q\sim (0, 0, 0.792)$ and $m_{q} \parallel c\text{-axis}$ in the high temperature phase ($T_{\text{N}}^{\;\prime}< T < T_{\text{N}}$), which changes to a cycloidal order with $q=(\delta', 0, 0.8)$ ($\delta'\sim 0.017$) and $m_{q} \parallel ac\text{-plane}$ in the intermediate phase ($T_{\text{N}}^{\;*} < T < T_{\text{N}}^{\;\prime}$). In the low temperature phase ($T < T_{\text{N}}^{\;*}$), the cycloidal plane rotates by $45^{\circ}$ to have $q=(\delta, \delta, 0.8)$ ($\delta\sim 0.012$). It is also pointed out that the X-ray scattering amplitude from odd-parity magnetic quadrupole due to the polar environment interfere with that from normal even-parity magnetic dipole in the magnetic ordered phase.