Lattice coupling

Translation to lattice energy transfer is the dominant aspect of atomic and molecular adsorption, scattering and desorption from surfaces. Dissipation of incident translational energy (principally into the lattice) allows adsorption, i.e., bond formation with the surface, and thermal excitation from the lattice to the translational coordiantes causes desorption and diffusion i.e., bond breaking with the surface. This is also the key ingredient in trapping, the first step in precursor-mediated dissociation of molecules at surfaces. For direct molecular dissociation processes, the implications of Z,X,Y y, coupling in the 

PES is far less well understood (and generally ignored). It may, however, be quite important for direct dissociations of molecules heavier than H2 or D2. 

Because first principles knowledge of the dependence of V is generally poor, simple models of the lattice coupling and its consequences for dynamics have played an important role historically. A particularly good discussion of energy transfer to surfaces based on a simple model of atom/molecules interacting with a ID lattice chain is in Ref. [45]. Results from a few simple models are presented below. 

For the cube model, when Eq E , trapping into the adsorption well occurs. Therefore, there is a critical energy Ec such that the trapping coefficient a = 1 for E EC, while a = 0 and only direct inelastic scattering occurs for En Ec. For the stationary cube, 

Because of the large W, a 1 for reactive atoms. However, the dynamics within the well before thermalization is anything but simple. Because of the strong corrugation, there is considerable scrambling of En and E. When /jl is very small (H and D), energy dissipation to the lattice in the well is slow and hot atoms are formed which have considerable reactivity with other species adsorbed on the surface(see Section 4.1.2). 

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With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

Tucker, E.B. 1966. Spin-lattice coupling of a Kramer s doublet Co2+ in MgO. Physical Review 143 264-274. 

The most striking implication of the electron lattice coupling in ID chains is the appearance of the semiconducting state the equal bond ID lattice (metallic state) is unstable (33) with respect to a lattice distorsion and this so called static Peierls instability is the origin of the opening of the intrinsic band gap at the edge of the B.Z. with an infinite density of states there and the presence of band alternation. 

By extension one may say that the power laws (5-7) which determine the magnitude of the linear and nonlinear optical coefficients are consequences of this strong electron-lattice coupling. We now make the conjecture that the time response of these coefficients is severely affected by the dynamics of the electron-lattice coupling in conjugated chains when two or more resonant chemical structures can coexist this is the case for many of the organic chains of Figure 2. 

The solid state polymerisation of diacetylenes (2) with U.V. radiation, heating or shear force is most indicative of the predominant influence of electron-lattice coupling. The details of the chemical changes that occur during th polymerisation process are crucial (2,40) but the overall description only needs part of this chemical information. The kinetics and thermodynamics of the polymerisation process using an elastic strain approach have been worked out in (41). 

The diode model consists of two segments of nonlinear lattices coupled together by a harmonic spring with constant strength kint (see Fig. 6). Each segment is described by the (dimensionless) Hamiltonian ... 

Consider a spin system whose spin Hamiltonian consists of a time-independent Hamiltonian H0 and a stochastic perturbation Hamiltonian H,(t) due to a small spin-lattice coupling,... 

Substituting Eq. (14) into Eq. (12), and neglecting any correlation between the density matrix and the spin-lattice coupling Hamiltonian, one obtains to first order... 

It is found that the relaxation parameter T p as a function of temperature does not follow an increase with chain length, as the square of the number of methylene carbons. Nor is it linear with N, the number of methylene carbons, which should be true if relaxation to the lattice were rate controlling. Rather, it shows a temperature-induced increase of the minimum value of Tjp with about the 1.6 of N. So, both spin diffusion and spin lattice coupling are reflected. For a spin diffusion coefficient D of approximately 2 x 10 12 cm.2/sec., the mean square distance for diffusion of spin energy in a time t is the ft1 = 200/T A, or about 15A on a Tjp time scale. 

To take the above-mentioned ion-lattice coupling into account, the full ion-plus-lattice system must be considered, so that the static Hamiltonian given by Equation (5.2) must be replaced by... 

The 3d orbitals in TM ions have a relatively large radius and are unshielded by outer shells, so that strong ion-lattice coupling tend to occur in TM ions. As a result, the spectra of TM ions present both broad (S > 0) and sharp (S 0) bands, opposite to the spectra of (RE) + ions, discussed in section 6.2.1, which only showed sharp bands (S 0). 

Equation 13 can be solved numerically for Tc as a function of the proton-lattice coupling. The parameters are chosen so as to fit the experimental value of Tc for KDP. For C = 21 732 K/A and g2ygAyf close to those used for perovskite oxides, Tc Ikdp = 115 K. In Fig. 3 Tc is shown as a function of C with all other parameters fixed. Including the deuteration effects (Table 2), Ter = C Idkdp/C Ikdp 1 2. With this estimate TcIdkdp = 168 K. C itself depends only weakly on /, g2y g4 but a strong dependence on/ is observed, which is the coupling between the PO4 shells and the K" " ions. This, on the other hand, should not be dependent on deuteration. 

It has been shown theoretically that an extra electron or hole added to a one-dimensional (ID) system will always self-trap to become a large polaron [31]. In a simple ID system the spatial extent of the polaron depends only on the intersite transfer integral and the electron-lattice coupling. In a 3D system an excess charge carrier either self-traps to form a severely locahzed small polaron or is not localized at all [31]. In the literature, as in the previous sections, it is frequently assumed for convenience that the wavefunction of an excess carrier in DNA is confined to one side of the duplex. This is, of course, not the case, although it is likely, for example, that the wavefunction of a hole is much larger on G than on the complementary C. In any case, an isolated DNA molecule is truly ID and theory predicts that an excess electron or hole should be in a polaron state. 

The importance of lattice coupling in direct molecular dissociation is at present poorly understood. However, there are at least two ways in which inclusion of the lattice can affect direct dissociative adsorption. First, conversion of Et to Eq competes with translational activation in dissociation. Second, thermal distortion of lattice atoms from their equilibrium positions may affect the PES, e.g., the barriers to dissociation V ( ). These two effects can be most simply thought of as a phonon induced modulation of the barrier along the translational coordinate and in amplitude, respectively. 

The other lattice coupling V is also likely to be a general phenomenon in activated direct dissociation since barriers are generally lower for less coordinated sites, i.e., distortion of the lattice can give lower/higher barriers. However, this barrier... 

The presence of lattice coupling in direct dissociation complicates somewhat the definition of V and how V extracted from measurement should be compared to DFT calculations. There are two well defined limits for DFT calculations of V, for a frozen lattice (Ts = 0) and for the case where the lattice is fully relaxed at the transition state (Ts = oo). Since experiments are performed at finite Ts, the best V description is probably intermediate between these two extremes. 

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