One-dimensional chain of atoms

Figure 8.4 One-dimensional chain of atoms with interatomic distance a and force constant K.

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

Figure 6.1. A one-dimensional chain of atoms, of length L and lattice spacing a, connected by springs that obey Hooke s law. With the end-atoms fixed, only standing waves of atomic displacement are possible.

Figure 1.2 Scattering of a plane wave by a one dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Fig. 6.21. Schematic of a one-dimensional chain of atoms (with periodic boundary conditions) illustrating the connection between the number of A and B atoms, and the number of AA, AB and BB bonds.

We consider a simple case of an infinite one-dimensional chain of atoms, with only one atom, mass m, per unit cell. They are coimected by springs, of force constant f, and interact only with their nearest neighbours. The atom / is displaced, m/, from its equilibrium position. Then /+i and m/ i are the (different) displacements of the neighbouring... 

The temperature dependence of the conductivity can differentiate between metals (for which G T as T i ) and semiconductors (for which a i as T i). We next discuss a very useful way of looking at the physics of infinite one-dimensional chains of atoms or molecules, namely the Hubbard Hamiltonian. 

To find an approximate form of this modulating function a tight-binding descrip>-tion of a one-dimensional chain of atoms possessing one atom per primitive unit ceU and two orbitals per atom was studied [280]. The sinc ( ii jy) function found was simply extended to the three-dimensional cubic lattice case. Unfortunately, the form of this modulating function depends on the direct lattice of superceUs and its extension to noncubic lattices is nontrivial. 

Applications The HMM has been applied to the study of friction between two-dimensional atomically flat crystal surfaces, dislocation dynamics in the Frenkel-Kontorova model (i.e., considering a one-dimensional chain of atoms in a periodic potential, coupled by linear springs ), and crack propagation in an inhomogeneous medium. ... 

Analogous to the simple chain, a one-dimensional chain of atoms with two different atoms per unit cell can be treated. This leads to an equation of motion for... 

The one-dimensional chain of hydrogen atoms is merely a model. Flowever, compounds do exist to which the same kind of considerations are applicable and have been confirmed experimentally. These include polyene chains such as poly acetylene. The p orbitals of the C atoms take the place of the lx functions of the H atoms they form one bonding and one antibonding n band. Due to the Peierls distortion the polyacetylene chain is only stable with alternate short and long C-C bonds, that is, in the sense of the valence bond formula with alternate single and double bonds ... 

It soon became apparent, once the structure of the yttrium compound was bared, that either or both of two central features might account for superconductivity at those record-high temperatures. One was the puckered, two-dimensional plane of copper and oxygen atoms ilar to the flat plane seen earlier in the structure of another superconductor, made of lanthanum, strontium, and copper oxide, that became superconducting at around 40° K. The other was unexpected the one-dimensional chain of copper and oxygen atoms, a sequence unknown in earlier superconductors. The challenge was fairly clear to both theorists and experimentalists. Were the planes or the chains responsible for superconductivity above 90° K ... 

Fig. 2.10 The same as Figure 2.9, but for a one-dimensional chain of nitrogen atoms spaced at 2.0 A. The labels indicate the types of chemical interaction in terms of the symmetry properties.

Fig. 2.11 Symmetry properties of the 2s and 2pz (a) and 2px (b) bands beionging to the one-dimensional chain of nitrogen atoms with respect to the horizontal mirror plane cr the bands are plotted in the middle of the zone for k = n/2a.

For chemists, it is probably not a big surprise that a one-dimensional chain of hydrogen atoms does not exist and that it will immediately decompose into isolated H2 molecules. The Peierls distortion has important consequences for one-dimensional systems, such as polyacetylene with C-C bond-length alterations (instead of equal C-C distances) [74], infinite molecules with platinum-platinum bonding such as Krogmann s salts K2[Pt(CN)4]Xo,3 3H2O with X = Cl or Br [75], or other one-dimensional systems [76], and it also affects three-dimensional systems, in particular elemental structures (see Section 3.4). From a group-theoretical point of view, Peierls distortions are characterized by a loss of translational symmetry in the above example, the nonequidistant chain of H atoms is less symmetrical (in terms of translational symmetry) than the equidistant one. 

Fig. 2.19 Semiempirical (extended Huckel theory) band structure, density-of-states, and crystal orbital overlap population for a one-dimensional chain of hydrogen atoms spaced at 2.0 A. Due to a Gaussian smoothing, DOS and COOP plots appear slightly...

In the past dozen years, there has been renewed interest in a variety of polyplatinum oxides. Crystallographic investigation of several of these compounds has revealed one-dimensional chains of platinum atoms in more than one direction. Careful studies are necessary to verify the dimensionality of these materials. Recently, a mercury complex has been reported. It is an additional example of this multidimensional one-dimensionality. The chemical and physical properties of such compounds are reviewed below. 

Our discussion so far has focused on polyacetylene and related examples. The broad results however, are transfciable to many other systems. Algebraically, for example, our discussion applies equally well to the case of a one-dimensional chain of hydrogen atoms bearing l.v orbitals. Recall the one-to-one correspondence between the orbitals of finite molecules and their polyene analogs. With one electron pet-atom, Figure 13.3 indicates a half-filled band foi the geometry 13.34. This will distort (in a Peierls fashion) to a solid composed of H2 molecules (13.35) as chemical... 

Finally, there is the important domain of intermediate density, which is really the only experimentally accessible one lattices with r.>1.3 are unstable relative to the molecular solid, while those with ra<1.0 are likely to remain experimentally inaccessible in the foreseeable future. In this domain, neither the van der Waals nor the electron gas model is appropriate. There are no calculations in this regime to which comparison can be made, although results from calculations on one-dimensional chains of hydrogen atoms [35] suggest that the dimensional scaling values may be somewhat low here. 

We show, in Fig, 5a, a one-dimensional chain of 4 electrons on 4 sites (atoms or molecules) spaced an equal distance (lattice period) b apart. At high temperature, if the on-site Coulomb repulsion U is small to medium, i,e, smaller than the bandwidth (U<4t), then the electrons are paired on two of the 4 sites (1/2-filled band), and the k-space states are occupied up to the Fermi wavevector kp= 7c/2b, and Bragg or Umklapp scattering occurs between the -kp state and the +kp state (a difference of 2kp), Since the energy levels are only filled... 

Problem 1 The band structure of a one-dimensional chain of main group atoms. 

Such a one-dimensional chain of ions where each ion can be selectively excited, can be used as the basis of a quantum computer [1247, 1248]. This can be seen as follows Every computer is based on the two bit states 0 and 1 . In the atomic quantum computer these bit states correspond to an atom in the ground state and in a long living excited state, e.g. a metastable electronic state or an excited hyperfine... 

Fig. 41. Example of crystal wavefunctions with different -vectors. A one-dimensional chain of identical atoms is shown, and we consider the wavefunctions that can be constructed from an s-atomic function on each atom in the chain, (a) Although the atomic functions are identical, their contribution to a wavefunction of the chain may vary, in particular, the phase at each atom need not be the same (we show one phase as open circles and the other as shaded circles). The value of the -vector for a given phase pattern along the chain is related to the distance over which the phase changes, (b) If no reciprocal space sampling is used with the smallest repeat unit, the band structure is effectively ignored and only band center states with uniform phase are considered, (c) and (d) Use of supercell calculations allow specific phase patterns for the chain to be included and so some -points are effectively reintroduced.

Basic idea of the theory will be demonstrated in simplified form for cubic crystal with two atoms in the elementary nnit. Let ns suppose just one-dimensional lattice vibrations (i.e. all atoms in certain lattice plane move together) - either parallel, or perpendicnlar to the wave propagation direction. In such case, we could further simplify our model to one-dimensional chain of two regularly repeated atoms. 

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