Debye zero-point energy

The zero point energy follows from quantum theory, according to which atoms do not cease to vibrate at the absolute zero point. For a Debye solid (that is, a homogeneous body of N equal particles) the zero point energy is... 

However, for a real crystal at zero temperature it is impossible to group all vibrational motions on the lowest single vibrational mode and, if the crystal behaves as a Debye solid (see later), zero-point energy is expressed as ... 

The right-hand side can be separated into five parts. The first part is the enthalpy at 0 K, the second represents the zero point energy, the third is the Debye energy term, the fourth is an approximation for the Cp — C correction while the last part arises from the difference in electronic specific heats. 

As for graphite, its zero-point energy, ZPE = R6 + jR0 , is most conveniently deduced from Debye s theory [197,198] by separating the lattice vibrations into two approximately independent parts, with Debye temperatures (in plane) and 6j (perpendicular). A balanced evaluation gives ZPE 3.68 kcal/mol [199]. 

Indeed, the Debye approximation is more appropriate in any problem where the modes of lowest frequency are important, as they are in thermal properties at low temperatures. In cases were all modes are important, such as in the evaluation of the total zero-point energy, the simpler Einstein model may be preferable. Notice that even within the Debye approximation the frequencies are concentrated near the highest frequency, called the Debye frequency. This is illustrated in Fig. 9-7. 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

Table 2. Computational results obtained for the adducts of Figures 2-3. Energy (a.u), dipole moment (Debye), zero-point vibrational energy (ZPE/kcd mof ), ZPE-corrected relative energy of the adducts (AE /kcal mof ) with stability order in brackets (a), activation energy uncorrected (AE /kcal mof ) and corrected (AE corr/kcal mof ) for ZPE.

Fig. 25 (Color online) The different eontributions to the total binding energy from the method of increments are plotted, for the experimmlal lattiee parameters, and compared to the experimental value. The zero-point energy (ZPI correction is estimated from the Debye temperature. ... 

Vibrational Feshbach resonances (VFRs) in a vibrational Feshbach resonance, the interaction of a slow electron takes the form of a virtual excitation of a vibrational level of the neutral molecule with capture of the electron (ffotop et al. 2003 Dessent et al. 2000). For the zero point vibration, the maximum probability of interaction of the electron with parent molecule occurs at zero energy, if the dipole moment of the neutral molecule exceeds the critical value of approximately 2 Debye, the impinging electron maybe trapped into the diffuse bound state, which provides a much longer timescale for the electron to stay near the molecule (fiotop et al. 2003 Dessent et al. 2000 lllenberger 1992), and VFRs may appear as shown in O Fig. 34-5. 

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).

A further simplification is that we need only to consider those ions which are present in excess in the solution and carry the liquid charge of the double layer, since for the mutual neutralization of all other dons, present in equivalent amounts in each point of the space, the net work is zero. The neglection of the mutual Debye-Hiickel energy of the ions seems justified, because this energy either bears no relation to the surface (for the ions in the bulk of the solution) or is only a second order effect (for the ions in the double layer). 

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