## 3 Dipolar Contributions to the Mediated Rate
A contribution that is of an identical order of magnitude to that considered in the last example, Equations (33) and (34), is the rate between an electric dipole donor and a magnetic dipole acceptor, mediated by a third-body \(T\). From Equation (21), the rate for an oriented arrangement of the three particles is  
\[\Gamma^{(\mu\mu)^{D}-(mm)^{A}}=\frac{2\pi\rho_{f}}{h}\mu_{i}^{0m}(D)\overline{ \mu}_{i^{\prime}}^{0m}(D)m_{j}^{m0}(A)\overline{m}_{j^{\prime}}^{m0}(A)A_{kl}^ {T}\overline{\alpha}_{k^{\prime}l^{\prime}}^{T}V_{ik}^{R}\overline{V}_{l^{ \prime}k^{\prime}}^{R}U_{jl}^{R^{\prime}}\overline{U}_{j^{\prime}l^{\prime}}^{ R^{\prime}}. \tag{35}\]  
In addition to Equations (27) and (29), the following orientational average over products of magnetic dipole moments associated with center \(\xi\) is necessary to compute the rate applicable to a fluid sample,  
\[<m_{i}^{0m}(\xi)\overline{m}_{i^{\prime}}^{m0}(\xi)>=\frac{1}{3}\delta_{ii^{ \prime}}|\hat{m}^{\rightarrow 0m}_{i^{\prime}}(\xi)|^{2}, \tag{36}\]  
yielding the isotropic rate result  
\[ \begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(mm)^{A}}>=\frac{\pi\rho_{f}}{135\hbar c ^{2}}\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{1}{(R\kappa^{\prime})^{6}}|\hat{ \mu}^{\rightarrow 0m}(D)|^{2}|\hat{m}^{\rightarrow m0}(A)|^{2}\\ \times(\{[3-9(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{2}R^{\prime 2}+k^{4}R^{ \prime 4})+[5-15(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{4}R^{2}{R^{\prime}}^{2}+k^{6 }R^{2}{R^{\prime}}^{4})\\ +[-1+3(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{6}R^{4}{R^{\prime}}^{2}+k^{8}R^{ 4}R^{\prime 4})\}\alpha_{\lambda\lambda}^{T}\overline{\alpha}_{\mu\mu}^{T}\\ +\{[41-3(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{2}{R^{\prime}}^{2}+k^{4}R^{ \prime 4})+[15-5(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{4}{R^{2}{R^{\prime}}^{2}}+k^{6 }{R^{2}}^{R^{\prime}}^{4})\\ +[13+(\hat{R}\cdot\hat{R}^{\prime})^{2}](k^{6}{R^{4}{R^{\prime}}^{2}}+k^{8}R^{ 4}R^{\prime 4})\}\alpha_{\lambda\mu}^{T}\overline{\alpha}_{\lambda\mu}^{T}),\end{array}  \tag{37}\]  
where \(k=k_{m0}\). It is informative to evaluate the asymptotic limits of result Equation (37) corresponding to short and large relative separation distances. In the near zone, \(kR<<1\) and \(kR^{\prime}<<1\), yielding the asymptote  
\[ \begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(mm)^{A}}>_{NZ}\sim\frac{\pi\rho_{f}}{13 8\hbar c^{2}}\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{k^{2}}{R^{8}R^{4}}|\hat{\mu} ^{\rightarrow 0m}(D)|^{2}|\hat{m}^{\rightarrow m0}(A)|^{2}\\ \times([3-9(\hat{R}\cdot\hat{R}^{\prime})^{2}]\alpha_{\lambda\lambda}^{T} \overline{\alpha}_{\mu\mu}^{T}+[41-3(\hat{R}\cdot\hat{R}^{\prime})^{2}]\alpha_ {\lambda\mu}^{T}\overline{\alpha}_{\lambda\mu}^{T}),\end{array}  \tag{38}\]which is also not a true near-zone limit since \(U_{ij}\) is retarded, and with \(k^{2}=k_{m0}^{2}\) appearing explicitly. Interestingly, the short-range rate exhibits Forster-like \(R^{-6}\) dependence due to electric dipole-dipole transfer between \(D\) and \(T\), and \(R^{\prime-4}\) behavior arising from exchange between electric dipole \(T\) and magnetic dipole \(A\). At very large inter-particle displacements, \(k\)R>> 1 and \(kR^{\prime}>>1\), giving rise to the far zone approximated form of the rate  
\[ \begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(mm)^{A}}>_{FZ}\sim\frac{\pi\rho_{f}}{13 5\hbar c^{2}}\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{k^{8}}{R^{2}R^{\prime 2}}| \overrightarrow{\mu}^{0m}(D)|^{2}|\overrightarrow{m}^{m\to 0}(A)|^{2}\\ \times\{[-1+3(\hat{R}\cdot\hat{R^{\prime}})^{2}]\alpha_{\lambda\lambda}^{T} \overline{\alpha}_{\mu\mu}^{T}+[13+(\hat{R}\cdot\hat{R^{\prime}})^{2}]\alpha_{ \lambda\mu}^{T}\overline{\alpha}_{\lambda\mu}^{T}\},\end{array}  \tag{39}\]  
displaying characteristic inverse square behavior on each of \(R\) and \(R^{\prime}\), indicative of sequential real photon emission from \(D\) to \(A\) via \(T\). From the results Equations (35) and (37), it is straightforward to obtain the mediated transfer rate between a magnetic dipole emitter and an electric dipole acceptor; simply interchange \(\hat{R}\) and \(\hat{R^{\prime}}\), and \(R\) and \(R^{\prime}\).