The decay of the excited ${}_{}{}^4{}_{}{}^{}S_{\frac{3}{2}}^{}{}_{}{}^{}$ state of ${\mathrm{Er}}^{3+}$ ions in La${\mathrm{F}}_3$ at 295°K has been investigated in detail. The energy is transferred from the thermally populated multiplet ${}_{}{}^2{}_{}{}^{}H_{\frac{11}{2}}^{}{}_{}{}^{}$. The decay curve of the luminescence from ${}_{}{}^4{}_{}{}^{}S_{\frac{3}{2}}^{}{}_{}{}^{}$ after a flash lamp excitation is explained well by taking into account diffusion and cross relaxation simultaneously. The obtained diffusion coefficient $D$ is inversely proportional to the fourth power of the average separation between ${\mathrm{Er}}^{3+}$ ions, indicating that the coupling between the ions responsible for the diffusion is the electric dipole-dipole interaction. The energy-transfer constant $C_{ \mathrm{dd}}$ of the diffusions ${}_{}{}^2{}_{}{}^{}H_{\frac{11}{2}}^{}{}_{}{}^{}\leftrightarrow {}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^4{}_{}{}^{}S_{\frac{3}{2}}^{}{}_{}{}^{}\leftrightarrow {}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}$ and the energy-transfer constant $C$ of the cross relaxations ${}_{}{}^2{}_{}{}^{}H_{\frac{11}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{9}{2}}^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{13}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}H_{\frac{11}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{13}{2}}^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{9}{2}}^{}{}_{}{}^{}$ are $7.0\times {10}^{-40\mathrm{}}$ ${\mathrm{cm}}^6$/sec and $4.5\times {10}^{-41\mathrm{}}$ ${\mathrm{cm}}^6$/sec, respectively. The fact that $C_{\mathrm{dd}}$ is larger than $C$ by more than one order of magnitude indicates that the energy transfer is essentially the diffusion-limited relaxation. The other cross relaxations, ${}_{}{}^4{}_{}{}^{}F_{\frac{7}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{11}{2}}^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{11}{2}}^{}{}_{}{}^{}$, and ${}_{}{}^4{}_{}{}^{}F_{\frac{5}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}F_{\frac{9}{2}}^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{13}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^4{}_{}{}^{}F_{\frac{5}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}I_{\frac{13}{2}}^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}I_{\frac{15}{2}}^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}F_{\frac{9}{2}}^{}{}_{}{}^{}$, have been observed in La${\mathrm{F}}_3$:${\mathrm{Er}}^{3+}$. The ensemble-averaged energy-transfer constants of these cross relaxations calculated by the theory of Dexter for the electric dipole-dipole interaction are in good agreement with the energy-transfer constants obtained by the phenomenological rate-equation model.

• Received 16 December 1974

DOI:https://doi.org/10.1103/PhysRevB.11.5103