Simulation

Long Wavelength Acoustic Phonon

This simulation shows the atomic vibrations for a very long wavelength longitudinal acoustic phonon. Note that all the atoms move in the same direction and the spacing between adjacent atoms is unchanged. In this case, the motions of adjacent atoms are positively correlated and there is no stretching of the bonds for this mode – the change in the width of the pair distribution function σ with temperature is zero. In this and the next two clips, the amplitudes of the atomic vibrations of each atom are the same.

Short Wavelength Acoustic Phonon

&lambda approximately 31 atom spacings.

Here we show the vibrations for longitudinal acoustic phonons for a wavelength of approximately 30 atoms. The stretching of the bonds between nearest atoms is very small (small change in σ) and the local atomic motions are positively correlated. Note that the end atoms do have a significant phase difference as a result of the finite wavelength.

Very Short Wavelength Optical Phonon

&lambda = 2 atom spacings.

In this simulation we show the atomic vibrations for a short wavelength longitudinal optical mode. The vibration amplitudes of individual atoms is the same as for the previous index. However there is a large stretching of the bonds between the nearest atoms. The motions of the nearest neighbor atoms are opposite to each other; for this situation the atomic motions are negatively correlated and there is a very large increase in σ.

Polaron (static and motion of)

A polaron is a local lattice distortion that follows a mobile charge.

In this cartoon-like simulation we illustrate how a polaron moves through a 1-dimensional lattice. A polaron is a quasi-particle formed of a charge plus a local distortion that follows it; this distortion may extend several atomic spacings from the charge. The top figure shows the static distortions in the vicinity of a polaron while the movie shows the changes in the local atomic environment as the polaron moves through the lattice. Note that these atomic displacements are electronically driven – they are not thermal phonons. The changes in σ can be quite large.

Negative Thermal Expansion

W is the green atom in the center of the triangle with Zr atoms at the vertices. Zn-W linkage is assumed to be relatively stiff in ZrW₂O₈

A simplified cartoon model of the vibrations of the W and Zr atoms in ZrW₂O₈ inferred from X-ray Absorption Fine Structure (XAFS) measurements. The XAFS results show that the fluctuations of the W-Zr distance is very small – the W-Zr linkage is quite stiff. Consequently as the W atom vibrates vertically to the plane formed by the three Zr atoms (in a (1,1,1) plane) the Zr atoms must move together to keep the W-Zr distance approximately constant. This produces a negative thermal contraction of the lattice (the Zr-Zr distance r_Zr-Zr which decreases, is directly related to the cubic lattice constant a; i.e. r_Zr-Zr = a ⁄ √2).

Negative Thermal Expansion in 3-D

Shows the W vibrations along three (111) directions.

A simplified cartoon model of the vibrations of the W and Zr atoms in ZrW2O8 inferred from X-ray Absorption Fine Structure (XAFS) measurements. The XAFS results show that the fluctuations of the W-Zr distance is very small – the W-Zr linkage is quite stiff. Consequently, as the W atom vibrates vertically to the plane formed by the three Zr atoms (in a (1,1,1) plane) the Zr atoms must move together to keep the W-Zr distance approximately constant. This produces a negative thermal contraction of the lattice (the Zr-Zr distance rZr-Zr which decreases, is directly related to the cubic lattice constant a; i.e. rZr-Zr = a ⁄ √2).

Dimeron Motion

Assumes electrons wave functions extend over two lattice sites and hop slowly through lattie.