Linear Diatomic Chain

Fig. 4.10 Optic (ojf) and acoustic ( .) branches for a diatomic linear chain.

This text of the two-volume treatment contains most of the theoretical background necessary to understand experiments in the field of phonons. This background is presented in four basic chapters. Chapter 2 starts with the diatomic linear chain. In the classical theory we discuss the periodic boundary conditions, equation of motion, dynamical matrix, eigenvalues and eigenvectors, acoustic and optic branches and normal coordinates. The transition to quantum mechanics is achieved by introducing the Sohpddingev equation of the vibrating chain. This is followed by the occupation number representation and a detailed discussion of the concept of phonons. The chapter ends with a discussion of the specific heat and the density of states. 

Fig.2.1. Diatomic linear chain with masses rni and m2 and nearest-neighbour interactions with force constants f and g. u(j ) is the displacement of atom K in unit cell i from its equilibrium position x( )...

Now, we show in Appendix A that for the diatomic linear chain, the eigenvalues and the atomic displacements are periodic functions of t = Z-rrm/a. The extension of the results to the general three-dimensional case is straightforward and gives... 

For the diatomic linear chain, we have shown that there are singularities in g(co) at certain critical points (c.p.) where the group velocity vanishes (Fig.2.11a). Critical points also exist in two and three-dimensional lattices. The character of these singularities can be discussed by expanding the frequency in a Taylor s series about the frequency co of the c.p. If v. = 0,... 

Fig.3,9. Phonon dispersion and density of state for a crystal with two atoms in the primitive unit cell, a) Qualitative general behaviour, b) Einstein approximation, c) Debye approximation, and d) Hybride Einstein-Debye model. The corresponding situation for the diatomic linear chain is shown in Fig. 2.11...

The eigenvalues A, are identical to those of the diatomic linear chain... 

The optical spectral region consists of internal vibrations (discussed in Section 1.13) and lattice vibrations (external). The fundamental modes of vibration that show infrared and/or Raman activities are located in the center Brillouin zone where k = 0, and for a diatomic linear lattice, are the longwave limit. The lattice (external) modes are weak in energy and are found at lower frequencies (far infrared region). These modes are further classified as translations and rotations (or librations), and occur in ionic or molecular crystals. Acoustical and optical modes are often termed phonon modes because they involve wave motions in a crystal lattice chain (as demonstrated in Fig. l-38b) that are quantized in energy. 

The H-atom chain is to solid-state structures as the diatomic molecule is to polyatomic molecules, e.g., clusters. The geometric-structure problems for H2 and Hoc are so simple that one can focus on the electronic structure problem exclusively. However, real solid-state structures, e.g., a solid with linked clusters or even bulk elemental Al, are not found in the form of a linear chain, square sheet or simple cubic structures so we need a way to treat solids with more complex structures, i.e., define a repeat unit that is more than a single atom. 

FIGURE 16.9 Holstein model with a linear chain of diatomic molecules (N ) with one added electron. 

A heteronuclear diatomic molecule A-B crystallizes end to end to form a linear chain . .. A-B A-B A-B. The intemuclear distance between molecules is significantly longer than that within molecules. The monomer has a bonding valence a MO and a higher energy a MO ... 

The LCPAs are nonprotein components (>3.5 kDa) that have been found encapsulated in the silica matrix of the diatom, and isolated through a series of H F extractions [59]. Unlike silafHns, which have a peptide backbone, these polyamines consist of linear chains of C-N-hnked PEI units that are bonded to a putrescine or putrescine derivative backbone. The chemical composition of these LCPAs is unique for each diatom species, with variable repetitions and degrees of methylation. 

In Chap.5, anharmonic effects are considered. After an illustration of anharmonicity with the help of the diatomic molecule, we derive the free energy of the anharmonic linear chain and discuss the equation of state and the specific heat. The quasi-harmonic approximation" worked out in detail for the linear chain is then applied to three-dimensional crystals to obtain the equation of state and thermal expansion. The self-consistent harmonic approximation" is the basis for treating the effects of strong anharmonicity. At the end of this chapter we give a qualitative discussion of the response... 

I have tried to present both aspects of the subject, descriptive and analytical. The power of simple models to illustrate basic concepts should not be underestimated. Simple models such as simple three-dimensional lattices, the linear chain and even the diatomic molecule are studied at various places in this book. Many figures are included to illustrate both theoretical and experimental results. The interested reader will find all lengthy derivations in an Appendix the basic physical ideas can be understood without the Appendix, but for a deeper understanding of many aspects its content will be helpful. Each chapter contains a number of problems with hints and results. They not only help the reader to exercise newly acquired skills but also contain additional information not contained in the text. It is therefore recommended that readers examine the problems, even if they do not intend to solve them. At the end of each chapter, references to existing literature appear. Despite the inclusion of over 200 references, it is easily possible that I have omitted important papers. If this is the case, it is unintentional and apologies are sincerely offered. 

F1g.2.7. Transition from the diatomic to the monoatomic linear chain. De-... 

As an introduction, the chapter begins with the anharmonic diatomic molecule. Then we study the thermal properties (free energy, equation of state, thermal expansion and specific heat) of the classical anharnx)nic linear chain. Two important concepts are introduced the Gvuneisen pavametev and the quasiharmonic approximation. In this approximation, the temperature dependence of the force constants and phonon frequencies is only due to the... 

The linear O-Au-0 chain that appears systematically in TAA complexes (see the progression of that linear unit in the structures 2b —s- 3a 4a —> 5a —> 6a —> 7a), was identified experimentally as a stable free neutral molecule , formed probably by adding atomic oxygen to the AuO diatomic molecule. We find that the free linear (0-Au-0) anion is more stable than the MA AUO2 anion by 0.23 eV/atom. The angle Au-0-0 in the free MA anion is 121.5, to be compared with the angle ai given in Table 4 for MA complexes. 

The complex morphology of the nanopatterned siUca diatom cell walls has been found to be related to species-specific sets of polycationic peptides, so-called silaffins, which were isolated from diatom cell walls [82], The morphologies of precipitated sihca can be controlled by changing the chain lengths of the polyamines as well as by a synergistic action of long-chain polyamines and silaffins [83,84]. It has been proposed that the delicate pattern formation in diatom shells can be explained by phase separation of silica solutions in the presence of these polyamines [85]. Various linear synthetic analogs of the natural active polyamines in biosilica formation can accelerate the silicic acid condensation even more than the above mentioned... 

For the diatomic chain with non-linear nearest-neighhour interactions a standard decoupling technique in the continuum limit [19] can be used and yields acoustic pulse-type solitary waves and optical envelope-type solitary excitations [20]. These calculations are rather involved, even more for our model which has not only alternating masses but also alternating interactions. We prefer not to apply this method because there are two general problems ... 

A plot of frequency v versus k is known as a dispersion curve. The first period of this function represents the first Brillouin zone. For a linear diatomic chain in which all bonds are identical and the atoms are equally spaced, but have different masses of Mi and Mj, the result is ... 

---