Debye cutoff energy

The superconducting transition temperature can be expressed in terms of the electron-phonon coupling potential U and the Debye cutoff energy Ed = h-(OD/2n as ... 

Recall that the interaction form (13.52) was chosen to express the close encounter nature of a molecule-bath interaction needed to affect a transition in which the molecular energy change is much larger than Awd where cfD is the Debye cutoff frequency of the thermal environment. This energy mismatch implies that many bath phonons are generated in such transition, as will be indeed seen below. 

Single-ion nonradiative decay for Ln3+ diluted into transparent host elpa-solite crystals, where the energy gap is greater than the Debye cutoff, is primarily due to multiphonon relaxation (with rate kmp). In some cases, first order selection rules restrict phonon relaxation between states, such as between Tig and T4g, or between T2g and T5g, CF states for MX63- systems. The dependence of the multiphonon relaxation rate, kmp, upon the energy gap to the next-lowest state (AE) has been investigated for other systems and is given by a relation such as [353, 354]... 

Equation (1) is derived from the BCS theory it relates with the e-ph pairing potential, V, of electrons near the Fermi level, with Nf as their density of states (DOS). The Debye phonon cutoff is cod and represents the average energy of the phonons involved in the process of pairing. 

This rate has two remarkable properties First, it does not depend on the temperature and second, it is proportional to the bath density of modes g(ct>) and therefore vanishes when the oscillator frequency is larger than the bath cutoff frequency (Debye frequency). Both features were already encountered (Eq. (9.57)) in a somewhat simpler vibrational relaxation model based on bilinear coupling and the rotating wave approximation. Note that temperature independence is a property of the energy relaxation rate obtained in this model. The inter-level transition rate, Eq. (13.19), satisfies (cf. Eq, (13.26)) k = k (l — and does depend on temperature. 

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

Figure 9.15. Typical trajectories of a Gaussian stochastic process x(t) with zero mean and Gaussian (a) or exponential (i>) correlation function. Circles are crossing points of x = 0. Trajectories were generated by regular sampling in the frequency domain, (c) corresponds to the Debye relaxation spectrum with a cutoff frequency. Reorganization energy of the discarded part of the spectrum is 7% of the total. The sampling pattern was the same as in (b).

Due to intrinsic computational complexity of quantum and even semiclassical calculations only the simplest models of bath relaxation are usually employed, namely, the Debye and Ohmic models, as discussed in Section II.A.2. It should be emphasized that these models fail to reproduce Marcus s energy gap dependence in the inverted region. For example, the Debye model predicts in the Golden Rule limit that fei2 (AG-l- ,) for AG — [21]. The Ohmic model gives minus fifth power. This illustrates the importance of the cutoff function for the spectral density. 

Clearly, the considerations that need to be applied to electrolytes are somewhat different. However, Debye-Hiickel screening for all except the most dilute electrolytes helps to limit the range over which discrete sums of Coulomb terms have to be carried out for amorphous systems, and the cell multipole method can be used to compute the longer range contributions (121). Sometimes it is useful to use cutoffs to propagate the MD simulation, and then use Ewald summations to intermittently evaluate the energy to test that the cutoff procedime is sufficiently accurate. 

We can now solve for the energy by putting Equation 17.11 for the C([Pg.324]

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