Influence of higher-order tunneling processes and a finite cooling rate

3 Influence of higher-order tunneling processes and a finite cooling rate 

In order to take into accoimt the cooling process we used the algorithm explained below. Tbe zero-time point is chosen as the time when the cooling process is started. The function n E,u,t) represents the difference in population of the TS energy states at any time t and no(E,T) means this difference in equilibrium (i- oo) at a temperature T. Analog to Eq. 11 the heat release can be written as  

Note that the population difference n depends on E and u because the relaxation rates, that influence the relaxation of the TS, depend on these variables. As already described above, to obtain the function n we need to solve the differential equation given by Eq. 4. In general this is only possible by numerical methods. 

The time dependence of the temperature during the cooling process performed in this work can be well approximated by a linear function given by 

We assume that the sample is at t = 0 (/a) at T (To) /a is the time needed to cool the sample from T to To. The calculations have been done splitting the function T t)m.N steps of time At (Fig. 4.3). During the time At the temperature remains constant and n(E,u,t) shows an exponential behaviour given by Eq. 5 this is qualitatively shown in Fig. 4.3. To obtain the population difference at the time At cooling from Ti at t = 0 for a given E and u we calculate iteratively the value 

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