Magnetic excitations from the linear Heisenberg antiferromagnetic spin trimer system $A_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$ ($A=\mathrm{Ca}$, Sr, and Pb)

Inelastic neutron-scattering experiments were performed on $A_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$ ($A=\mathrm{Ca}$, Sr, and Pb), which consists of one-dimensional array of linear copper spin trimers. It was found that the magnetic excitations are well described by the linear Heisenberg antiferromagnetic spin trimer model, in which the main interaction is the nearest-neighbor intratrimer interaction and the weak next-nearest-neighbor intratrimer and intertrimer interactions are expected. The intratrimer coupling constants are determined to be 9.45(3), 10.04(3), and 9.13(2) meV for $A_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$ with $A=\mathrm{Ca}$, Sr, and Pb, respectively. The other interactions were found to be very small.

Figure 1

(a) Projection of the exchange pathways in $A_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$ ($A=\mathrm{Ca}$, Sr, and Pb). The large and filled circles represent the ${\mathrm{Cu}}^{2+}$ moments. The small and open circles represent the oxygen ions. The black and gray circles represent the Cu(1) and Cu(2), respectively. The thick black and gray lines show the pathways of $J_1$ and $J_2$, respectively. (b) Schematic view of the major magnetic interactions $J_1$, $J_1^{\prime }$, and $J_2$ in the one-dimensional array of copper spin trimers. $J_1^{\prime }$ is the next-nearest-neighbor interaction between edge spins in the trimer.

Figure 2

(Color online) Image plot of the neutron-scattering intensity in $\hslash \omega -Q$ space at 8 K in ${\mathrm{Pb}}_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$.

Figure 3

(Color online) (a) Constant-$Q$ scans at various temperatures in ${\mathrm{Pb}}_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$. The solid lines are the results of fits with Gaussians. Fifty counts are added at each successive temperature so that the scans are compared on one graph. The elastic-scattering intensity originates from both the nuclear incoherent scattering and paramagnetic scattering. (b) $Q$ dependence of neutron-scattering intensity at $\hslash \omega =5\mathrm{meV}$ at $T=60\mathrm{K}$ and at $\hslash \omega =9\mathrm{meV}$ and 13.5 meV at $T=8\mathrm{K}$. For the data at $T=8\mathrm{K}$, background intensity measured at a different energy, at which the magnetic signal is almost negligible, is subtracted from the raw data. For the data at $T=60\mathrm{K}$, background intensity measured at $T=8\mathrm{K}$ is subtracted from the raw data. The lines show the intensity calculated with Eq. (3).

Figure 4

Temperature dependence of the integrated intensities at 4.9, 9, and 13.7 meV in ${\mathrm{Pb}}_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$. The lines are calculated using Eq. (3).

Figure 5

(a) Constant-$Q$ scans at 8 K in ${\mathrm{Ca}}_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$. The solid line is the result of a fit with two Gaussians. (b) $Q$ dependence of neutron-scattering intensity at 9.5 meV and 14.25 meV. Background intensity measured at a different energy is subtracted from the raw data. The lines show the intensity calculated using Eq. (3).

Figure 6

(a) Constant-$Q$ scans at 8 K in ${\mathrm{Sr}}_3{\mathrm{Cu}}_3{\left(\mathrm{P}{\mathrm{O}}_4\right)}_4$. The solid line is the result of a fit with two Gaussians. (b) $Q$ dependence of neutron-scattering intensity at 9.9 meV and 15.15 meV. Background intensity measured at a different energy is subtracted from the raw data. The lines show the intensity calculated using Eq. (3).