Monoatomic chain

Consider a monoatomic chain of N unit cells with atoms of masses m, lattice parameter a and longitudinal displacements u( ). The total force acting on atom z is given by 

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Figure 8.5 The dispersion curve for a one-dimensional monoatomic chain of atoms.

For mi = m2, the expression reduces to that obtained for a monoatomic chain (eq. 8.18). When q approaches zero, the amplitudes of the two types of atom become equal and the two types of atom vibrate in phase, as depicted in the upper part of Figure 8.10. Two neighbouring atoms vibrate together without an appreciable variation in their interatomic distance. The waves are termed acoustic vibrations, acoustic vibrational modes or acoustic phonons. When q is increased, the unit cell, which consists of one atom of each type, becomes increasingly deformed. At < max the heavier atoms vibrate in phase while the lighter atoms are stationary. 

Fig.l4. STM constant current topograph (lOx 10 nm, 0.5 mV, 3.8 nA) of Co3Pt(lll) annealed at temperatures in the range 960-1060 K [71]. SRO appears as small areas with (1x2) symmetry in monoatomic chains. The bright spots are Pt sites. 

When plotted against k, these roots comprise two modes, or branches, (i.e.y-superscript now has two values, 1 and 2) see Fig. 4.10. The m. root is proportional to A as - 0 (similar to the solutions for the monoatomic chain). This is the acoustic mode (or branch) since it is analogous to the... 

The theoretical modeling of the vibrations of molecular chains with finite length has been nicely treated by Zbinden [18] and by Snyder and Schachschneider [9, 34]. While the approach by Zbinden is mathematically complete, but it applies to simplified monoatomic chains, quite away from chemical reality, the model by Snyder and Schachschneider is more directly applicable to real molecules and will be... 

Steidtner J, Hernandez F, Baltruschat H (2007) The electrocatalytic reactivity of Pd monolayers and monoatomic chains on Au. J Phys Chem C 111 12320-12327... 

For the monoatomic chain both problems have been overcome by the QCA of CoUins [17] for solitary waves and periodic modes. Here the difference operator is inverted instead of expanded. Rosenau [21] developed a still more general approximation scheme which reduces to the result of Collins in the case of solitary waves. 

Since the QCA overcomes the above-mentioned problems we now apply it to our diatomic model with alternating interactions. However, we use a rederivation of the QCA in Fourier space [18]. This formulation is much simpler than the original one [17] and can easily be generalized from the monoatomic chain to our model. 

In Fig. 21.1 this condition is visualized as the intersection of the straight line cq with the dispersion curve (Om q) = 2cm m ql2). Going back to the original units, cOm can be identified as the dispersion of a monoatomic chain of masses M = + M2 with a linear in-... 

Fortunately, an iterative method [18] has been developed for the monoatomic chain. Here the accuracy of taking into account the discreteness effects is increased systematically. In the case of the Toda lattice the iteration converges to the exact one-soliton solution. 

In this chapter, we start with the classical mechanics of the linear diatomic chain with nearest-neighbour interactions. Using periodic boundary conditions the equations of motion are solved and the dispersion relations are discussed. We also discuss the transition from the diatomic chain to the monoatomic chain. 

The cosine curve duplicates the results and in its middle part represents the upper LO curve in Fig.2.7. We note that the optical vibration at q = 0 of the diatomic chain becomes an acoustic vibration at q = n/d of the mono-atomic chain. The upper part AB of the sine curve in Fig.2.7 can be obtained by folding out the LO branch AC, or equivalently, by translation of the LO branch CD through 2-n/a, that is, by a reciprocal lattice vector t = M. This is called an Umptapp process. The dispersion relation (2.43) can of course be obtained if we start directly with the Hamiltonian of the monoatomic chain which follows easily from (2.3) and then solving the resulting equations of motion by assuming a solution of the form... 

Consider the dispersion o)(q) of the monoatomic chain of Problem 2.3.Id. Plot qualitatively the corresponding density of states. Note that there is a discontinuity at the zone boundary frequency 23 = 2(f/m). ... 

The equations of motion for an anharmonic linear and monoatomic chain with nearest-neighbour interactions are... 

Fig.5.2. a) Linear monoatomic chain at T = 0 and K = 0 with N lattice spacing ao- b) Chain at temperature T under the influence of an external force K with lattice spacing a = a(T,K)... 

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