Spin relaxation and resonant phonon trapping in $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$

Results of ac-susceptibility, specific-heat, magnetization, and electron-spin-resonance studies of a $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ ($\text{fum}=\text{fumarate}$, ${\text{C}}_4{\text{H}}_2{\text{O}}_4$) powder samples are reported. The results of these measurements enabled us to identify the studied compound as a three-dimensional $S=7/2$ Heisenberg magnet with $T_N=0.19\text{K}$ and dominant crystal-field anisotropy. The susceptibility studies conducted at audio frequencies and temperatures from 2 to 30 K revealed that the application of static magnetic fields induces a slow spin relaxation. The relaxation is not associated with an anisotropy energy barrier and is explicable assuming resonant phonon trapping. The magnetic field dependence of the relaxation time is consistent with the proposed scenario and suggests that a strong spin-lattice interaction may be the mechanism governing the relaxation properties in the studied system.

Figure 1

(Color online) Schematic view of the crystal structure of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ viewed along the $a$ axis. Hydrogen atoms and crystalline water are omitted for clarity.

Figure 2

Electron-spin-resonance line of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ obtained at 70 GHz and 1.75 K (circles). The solid line represents the result of the numerical simulation. See text for a more detailed discussion.

Figure 3

Temperature dependence of the total specific heat of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ (circles). The lattice contribution is denoted by the solid line. The Schottky contributions to the specific heat calculated for an $S=7/2$ paramagnet with $D/k_B=-0.24\text{K}$ and $D/k_B=-0.1\text{K}$ are represented by dashed and dotted-dashed lines, respectively. Inset: temperature dependence of the magnetic entropy of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ (circles). The solid line represents a theoretical prediction for an $S=7/2$ paramagnet with $\left|D\right|/k_B=0.24\text{K}$. The saturation value of the magnetic entropy for an $S=7/2$ paramagnet is denoted by the dashed line.

Figure 4

Temperature dependence of the (a) real and (b) imaginary part of the ac susceptibility of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ in a magnetic field of 100 mT studied at frequencies from 111 to 9999 Hz.

Figure 5

Cole-Cole diagram studied at temperature 10 K for magnetic fields 0.1 T (squares, $\alpha =0.095$), 0.2 T (down triangles, $\alpha =0.093$), 0.3 T (up triangles, $\alpha =0.112$), and 0.5 T (circles, $\alpha =0.140$). Inset: Cole-Cole diagram studied at magnetic field 0.5 T and various temperatures 5 K (stars, $\alpha =0.279$), 10 K (circles, $\alpha =0.14$), and 20 K (pentagons, $\alpha =0.059$). The solid lines denote the results of fitting the data by the equation derived directly from Eq. (2), the resulting values of $\alpha$ for each temperature and magnetic field are given in parentheses. See text for a more detailed discussion.

Figure 6

Temperature dependence of the median relaxation time of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ studied at magnetic fields of 50 (triangles), 100 (squares), and 500 mT (circles). The solid lines represent corresponding fits using the relation $\tau =a\ast T^b$ yielding $b=-2.7\pm 0.1$ for 50 mT, $b=-2.6\pm 0.1$ for 100 mT, and $b=-2.9\pm 0.1$ for 500 mT. Inset: temperature dependence of the median relaxation time in reduced coordinates. The data obtained at 100 mT were analyzed using the relations $\tau =a\ast T^{-1}$ (dashed line), $\tau =b\ast T^{-7}$ (solid line), and $\tau =c\ast \text{exp}\left(-\Delta /k_{B}T\right)$ (short-dashed line). The dashed-dotted line denotes the result of fitting the data from 10 to 20 K using the relation $\tau =c\ast \text{exp}\left(-\Delta /k_{B}T\right)$. See text for a more detailed discussion.

Figure 7

Magnetic field dependence of the median relaxation time of $\left[{\text{Gd}}_2{\left(\text{fum}\right)}_3{\left({\text{H}}_2\text{O}\right)}_4\right]\cdot 3{\text{H}}_2\text{O}$ studied at 5 (squares), 10 (circles), and 20 K (triangles). The solid lines are guide for the eyes. Inset: energy-level scheme for an $S=7/2$ paramagnet with $g=2.0$ and $D/k_B=-0.1\text{K}$ for $B\parallel z$.