Inelastic neutron scattering studies on the eight-spin zigzag-chain compound ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$: Confirmation of the validity of a data-driven technique based on machine learning

We performed inelastic neutron scattering (INS) experiments on ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ powder and compared the experimental results with those calculated for the spin model (an eight-spin zigzag chain with $S=\frac{1}{2}$) using the data-driven technique based on machine learning. We observed magnetic excitations at approximately 3.0, 4.1, 5.9, and 8.8 meV at 5.5 K and at approximately 3.8 and 5.9 meV at 49 K. The excitations corresponding to 3.0, 4.1, and 8.8 meV were magnetic excitations from the ground state to the first, second, and fourth excited states (2.87, 4.23, and 8.53 meV from the calculations), respectively. The excitations corresponding to 3.8 and 5.9 meV were magnetic excitations from the first excited state to the third and fourth excited states (3.78 and 5.67 meV from the calculations), respectively. An excitation was likely to exist between the first and second excited states at approximately 1.35 meV in the experimental results. The excitation energies obtained from the INS experiments were almost consistent with those calculated from the exchange interaction values via the data-driven technique (data-driven values). The experimental $I\left(Q\right)$ curves could not be reproduced. We found that $I\left(Q\right)$ curves could be changed largely by small changes of exchange-interaction values. Therefore, we expect that exchange-interaction values, which can explain not only the magnetic susceptibility, magnetization curves, and excitation energies but also INS intensity, are in the vicinity of the data-driven values.

Figure 1

Schematic of the spin model (an eight-spin zigzag chain) in ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ [15, 16] drawn using vesta [17]. Using the data-driven technique based on machine learning, the exchange interactions were evaluated as $J_1=-8.54\pm 0.51$ meV (AFM), $J_2=-2.67\pm 1.13$ meV (AFM), $J_3=-3.90\pm 0.15$ meV (AFM), and $J_4=6.24\pm 0.95$ meV (FM).

Figure 2

The histogram of the excitation energy evaluated from $J_1=-8.54\pm 0.51, J_2=-2.67\pm 1.13, J_3=-3.90\pm 0.15$, and $J_4=6.24\pm 0.95$ meV determined by the data-driven technique based on machine learning. We divided the uncertainty range into 20 equal parts for each exchange-interaction value and calculated eigenenergies for 194 481 ($={21}^4$) sets of the exchange interactions. Vertical and horizontal lines show the excitation energies and their uncertainties, respectively, estimated from the INS results.

Figure 3

X-ray diffraction pattern (circles) of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ at room temperature. The line on the measured pattern portrays the Rietveld-refined pattern obtained using the crystal structure with $P\overline{1}$ (No. 2) [16]. The line at the bottom portrays the difference between the measured and Rietveld-refined patterns. The hash marks represent the positions of reflections. The lattice constants are $a=7.4273\left(2\right)$ Å, $b=7.8327\left(2\right)$ Å, $c=9.4573\left(2\right)$ Å, $\alpha =108.285{\left(1\right)}^{\circ }, \beta =112.671{\left(1\right)}^{\circ }$, and $\gamma =92.743{\left(1\right)}^{\circ }$. The reliability indices of the refinements are $R_{\mathrm{p}}=1.82$%, $R_{\mathrm{wp}}=2.43$%, and $R_{\exp }=0.82$%.

Figure 4

Temperature $T$ dependence of the specific heat $C\left(T\right)$ of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ (red circles). The blue circles in panel (b) indicate $C\left(T\right)$ calculated for the spin model with $J_1=-8.54, J_2=-2.67, J_3=-3.90$, and $J_4=6.24$ meV.

Figure 5

The INS intensity $I(Q,\omega )$ maps of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ powder below 7.0 meV at various temperatures. The energy of incident neutrons $E_{\mathrm{i}}$ is 15.3 meV. The vertical key on the right shows the INS intensity in arbitrary units.

Figure 6

(a) The INS intensity $I(Q,\omega )$ map of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ powder below 15.0 meV at 5.5 K. The energy of incident neutrons $E_{\mathrm{i}}$ is 25.4 meV. The vertical key on the right shows the INS intensity in arbitrary units. (b) Map between 7 and 11 meV.

Figure 7

$\omega$ dependence of the INS intensity $I\left(\omega \right)$ of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ powder. The energy of the incident neutrons $E_{\mathrm{i}}$ is 15.3 meV. The pink and blue triangles indicate the excitation energies from the GS and 1ES, respectively. The horizontal bars represent the energy resolution.

Figure 8

$Q$ dependence of the INS intensity $I\left(Q\right)$ of ${\mathrm{KCu}}_4{\mathrm{P}}_3{\mathrm{O}}_{12}$ powder (circles). The energy of incident neutrons $E_{\mathrm{i}}$ is 15.3 meV. The lines indicate $I\left(Q\right)$ calculated for (a) the 0-1 and 1-3 excitations, (b) the 0-2 excitation, (c) the 0-4 excitation, and (d) the 1-4 excitation. The pink lines were obtained from $J_1=-8.54, J_2=-2.67, J_3=-3.90$, and $J_4=6.24$ meV evaluated using the data-driven technique based on machine learning [15]. The blue lines were obtained from $J_1=-8.71, J_2=-3.89, J_3=-4.00$, and $J_4=4.27$ meV evaluated using a steepest descent method.

Figure 9

Comparison plots of the magnetic susceptibility at 0.01 T and the magnetization curves at various temperatures between the experimental (blue circles) and calculated results. The red circles were obtained from $J_1=-8.54, J_2=-2.67, J_3=-3.90$, and $J_4=6.24$ meV evaluated using the data-driven technique based on machine learning [15]. The green triangles were obtained from $J_1=-8.71, J_2=-3.89, J_3=-4.00$, and $J_4=4.27$ meV evaluated using a steepest descent method.