Orbach processes

A resonant Orbach process occurs when the energy of the coupled vibrational modes is equal to the energy A of the first excited level of the paramagnetic center. This leads to the temperature dependence 1/Ti oc (exp(A/ BT) 1) expi- /ksT) when ksT < A. 

The Orbach process is a two-phonon process that takes place via population of an excited electronic state with energy Eq. The temperature dependence of the relaxation rate is given by... 

Fig. 3.3. Lattice and spin transitions are coupled by (A) direct processes, (B) Raman processes, (C) Orbach processes. The proximity of the excited electronic state favors both Orbach and Raman processes. The electronic states are labeled with n, the lattice vibrational states are labeled with V. A and 8 indicate energy separations with excited states coupled to the ground state by spin-orbit coupling.

Arrhenius analysis showed two-phonon Orbach process was dominant in the temperature range 25-40 K in the Tb complex, and 3-12 K in the Dy complex. 

The energy barrier through which the Orbach process occurs was estimated to be 2.6 x 102cm-1 and 3.1 x 101 cm-1 for the Tb and Dy complexes, respectively. These values are close to the energy differences between the lowest and the second lowest sublevels (Fig. 7), supporting the dominance of the Orbach process. 

Orbach process the spin center transfer to another energy level via a real intermediate state involving two phonons through a resonant process. 

The first equation refers to non-Kramer ions the second refers to Kramer ions where the energy between doublets is sufficiently large compared with ksT the third refers to Kramer ions doublets where the energy gap is small compared with ksT. The term in the equations with the coefficient A is the direct process part the terms with coefficients B and C refer to Raman and Orbach process, respectively. 

At low temperature, the processes of spin-lattice relaxation between the triplet substates are slow. With temperature increase, the sir rates increase strongly. For Pt(2-thpy)2, three different processes govern the sir. At a temperature below T = 3 K, the sir rate is exclusively determined by the direct process. Above T = 3 K, the Orbach process and above T = 6 K, the Raman process, become additionally important. For Pd(2-thpy)2, the processes that govern the sir have not been determined yet, but it is suggested that the Raman process is of main importance, since no real electronic state lies in the energy vicinity of the Tj state. (Figs. 19, 21, and Refs. [24,60,62,64,65].)... 

Av is the frequency difference between two anisotropic ESR resonance lines, the resulting spectrum is the superposition of the individual configurations. On the contrary, if r < (2 JtAv), we have an isotropic spectrum the resonance frequency is the average of the anisotropic components of the individual configurations As discussed in detail by Ham , motional narrowing can be produced by three relaxation mechanisms, which are characterized by a different temperature dependence an Arrhenius-type dependence (r" = Voe ) for an Orbach process, and a linear dependence or proportional to T for direct and Raman processes, respectively. Therefore, the temperature dependence of the isotropic spectrum gives information about the relaxation mechanism and consequently on the vibronic level scheme. 

Indeed, both the exponential temperature dependence that characterize the Orbach process and the T behavior associated with the Raman type process have been observed in spin lattice relaxation ... 

In the Orbach process the transfer rate is composed of a temperature-dependent part associated with the phonon-occupation number nq and a temperature independent part corresponding to energy transfer accompanied by spontaneous phonon emission. 

Ti times between 693 [ts (5.5 K) and 0.55 [ts (11.0 K). The temperature dependence of Tx could be fitted equally well to the equations describing Orbach or Raman processes. The unrealistic obtained exponent value in case of the Raman process led the authors to prefer the Orbach process. The Orbach energy gap was derived to be 57 cm. This energy gap corresponds to the energy of the first excited spin state, which allowed choosing between two sets of exchange coupling parameters that fitted the susceptibility curve equally well. 

Fig. 4. Experimental results for the spin-lattice relaxation rate of Nd ions in lanthanum magnesium nitrate (LaMN), magnetic field normal to c-axis. The data are fitted to a combination of an Orbach process, 6.3 x lO expf—47.6/T) and a direct process, 0.3cotanh(h( /2fcr). (After Ruby et al. 1962.)...

Fig. 5. The spin-bath relaxation rates for Ce in LaMN, fitted to an Orbach process, 2.7 X 10 exp( —34/D, combined with R cotanh (ho)/2kT) at lower temperatures, where the lower rate for the higher concentration shows that it is limited by the phonon-bottleneck between spins and bath. (After Ruby et al. 1962.)...

The above discussion has focused upon multiphonon decay (i.e., A is greater than the highest energy phonon). When crystal field energy levels are closer together, nonradiative relaxation from the upper to lower level can occur by direct phonon emission, or by Raman or Orbach processes ([2], pp 228-234). 

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