Figure 1

Photon-energy dependence of HWHM and scattering intensity of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ with $\overrightarrow{q}=(\frac{1}{2},0,\frac{1}{4})$ and $\sigma$ polarization at 30 K. The intensity was obtained by fitting the $q$ scan at each photon energy with a Lorentzian function and a linear background. Inset: Scattering geometry with $\sigma$ and ${\pi }^{\prime }$ polarizations for incident and scattered photons, respectively. With $\overrightarrow{q}$ near $(\frac{1}{2},0,\frac{1}{4})$, the angle between the scattered x ray and the $a$ axis is $\sim 6°$ and that between the incident x ray and the $c$ axis is $\sim 19°$.Reuse & Permissions

Figure 2

Temperature-dependent $\overrightarrow{q}$ of AFM order of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$: (a),(b) $q_a$ component measured with $\pi$ and $\sigma$ polarizations; (c) $q_c$ component. The vertical size of rectangular symbols indicates the uncertainties of $q_a$ and $q_c$. Intensities normalized to that of the commensurate AFM order at 30 K with $\sigma$ polarization are expressed by means of color. The temperature dependence of $q_c$ is without showing relative intensities. Intensities of incommensurate AFM order with $\overrightarrow{q}=(\frac{1}{2}\pm {\delta }_a,0,\frac{1}{4}+{\delta }_c)$ are denoted as $I_{\pm }$, and of the commensurate ordering as $I_C$. The insets of (a) and (c) are, respectively, scattering intensities of $q_a$ and $q_c$ scans at selected temperatures.Reuse & Permissions

Figure 3

Temperature dependence of AFM order and ferroelectric properties of ${\mathrm{TbMn}}_2{\mathrm{O}}_5$: (a) Comparison of $P$ and intensities $\sum _{\overrightarrow{q}}(I_{\sigma }{I}_{\pi }{)}^{1/2}$; (b) comparison of 1 kHz $\epsilon$ along the $b$ axis and simulation of ${\chi }_b$ with $|S_q{|}^2=(I_{\sigma }+I_{\pi })$. Here ${\chi }_0{g}_2(q_{IC})=0.01$, ${\chi }_0{g}_2(q_C)=-0.01$, $g_1(q_{IC})=10$, and $g_1(q_C)=0$. $⟨|\delta {\overrightarrow{S}}_q{|}^2⟩$ is simulated with a Lorentzian distribution centered at 37.5 K with a HWHM of 0.5 K. Note that $I_{\sigma }$ and $I_{\pi }$ are the average intensities for $\sigma$ and $\pi$ polarizations, respectively, i.e., $I_{\sigma /\pi }\equiv (I_{+}+I_{-})/2$ for incommensurate ordering and $I_{\sigma /\pi }\equiv I_C$ for commensurate ordering. $P$ and $\epsilon$ are reproduced data of zero magnetic field from Ref. 4.Reuse & Permissions