Debye cutoff frequency

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. 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

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).

It is usual to express the cutoff frequency cOp with the characteristic Debye temperature 6d ... 

Debye s theory of the specific heats of solids depends on the existence of a high number of standing, high frequency, elastic waves that are associated with thermal lattice vibrations. Central to his approach is the proposal that in a solid the phonon spectral density (p) increases continuously, and with a direct dependence on the square of the frequency (cutoff frequency (Q, at about 10 Hz) above which the phonon density vanishes for a solid continuum containing N atoms in a sample of volume V, the proportionality constant is 6 V/v, where V is the velocity of propagation. At a typical nuclear... 

Debye dodges this little problem by using the dispersion relation for a homogeneous solid which has no cutoff frequency and in which k = (ojvo and dk/do) = 1 /vq. 

To be more precise, 3/r o is sometimes written as l/z + 2/ to distinguish between the different velocities associated with the longitudinal and transverse modes. Debye then proceeds to determine a cutoff frequency by integrating the density of states (Equation 17.11) to a cutoff frequency wd that produces the required number of modes, i.e.. 

Let us examine the validity of the primary assumption of the Debye model, i.e., that the dispersion relation of a chain of atoms can be represented by a linear relationship corresponding to that of a homogeneous medium. Clearly this assumption is valid in the region where the wavelength is long compared to the atomic spacing, as was shown in Chapter 16, but what about the behavior near the cutoff frequency The results are shown in Figure 17.1. 

Dispersion relation for a continuous medium assumed by Debye (dashed line) compared with that of a chain of identical atoms (solid line). The frequency cutoff frequency for the chain of atoms < o = (4)S/m) which occurs at k= nla. 

It is common practice to approximate the true density of states F(co) with the Debye model density of states Fd(co), which has an upper cutoff frequency 0)d... 

Difficulties associated with disparity of timescales may still be encountered even within this approach, in cases where the frequency of the impurity molecule is much larger than the cutoff (Debye) frequency, md, of the host. Note that the rate (13.26) is given by the Fourier transform of the force autocorrelation function, taken at the frequency of the impurity oscillator. The time dependence of this correlation function reflects the time dependence of... 

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

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