Lattice distortions

There are several X-ray techniques for finding the strain in a material surface. In general, interplanar spacings are measured as a function of orientation with respect to the surface, and the stress is calculated using the equation 

In the two-exposure method, interplanar distances are measured for two planes, one perpendicular and one at an angle to the surface. This method is rapid but does 

In the sine-squared- / method, lattice spacings are measured at five or more f angles. The governing equation is then least-squares fit to find the best value of O. Observation of the behavior of the quantity di - d d as a function of sin f allows instrument misalignment and preferred orientation to be detected. This method is slower than the two-exposure method, but is more statistically significant. 

A third method is the position-sensitive detector system. This method is actually a modification of detector design, which allows more rapid determination of the peak position at each value of ( ) and /. In standard X-ray diffraction, a single detector is scanned through a set of diffraction angles to find the angle at which the radiation is most intense (i.e., the peak position). The difference for a position sensitive detector system is that an array of detectors is used, allowing accumulation of data from several diffraction angles simultaneously. 

Most techniques for the nondestructive evaluation of ceramic materials fall into two categories high-energy penetrating radiation (for example. X-ray) and high-frequency elastic waves (ultrasonics). Some NDE techniques have been developed for specialized applications, such as optical birefringence for transparent specimens and shearography for laminar composites. 

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Fig. 3. Crystal structure and lattice distortion of the BaTiO unit ceU showiag the direction of spontaneous polarization, and resultant dielectric constant S vs temperature. The subscripts a and c relate to orientations parallel and perpendicular to the tetragonal axis, respectively. The Curie poiat, T, is also shown.

Additional x-ray studies iadicate some degree of lattice distortion ia coatiags prepared from chloride-containing coatiag solutioas. This correlates with an analysis of 3—5% chloride ia the coatiag, which is reduced to aear zero if the coatiag is heated to 800°C. 

Eig. 2. Lattice distortions associated with the neutral, polaron, and bipolaron states in poly(p-phenylene). 

A brief review is given on electronic properties of carbon nanotubes, in particular those in magnetic fields, mainly from a theoretical point of view. The topics include a giant Aharonov-Bohm effect on the band gap and optical absorption spectra, a magnetic-field induced lattice distortion and a magnetisation and susceptibility of ensembles, calculated based on a k p scheme. 

It is known that a metallic ID system is unstable against lattice distortion and turns into an insulator. In CNTs instabilities associated two kinds of distortions are possible, in-plane and out-of-plane distortions as shown in Fig. 8. The inplane or Kekuld distortion has the form that the hexagon network has alternating short and long bonds (-u and 2u, respectively) like in the classical benzene molecule [8,9,10]. Due to the distortion the first Brillouin zone reduees to one-third of the original one and both K and K points are folded onto the F point in a new Brillouin zone. For an out-of-plane distortion the sites A and B are displaced up and down ( 2) with respect to the cylindrical surface [11]. Because of a finite curvature of a CNT the mirror symmetry about its surface are broken and thus the energy of sites A and B shift in the opposite direction. 

Fig. 8. Lattice distortions in a graphite sheet. For an in-plane distortion (left), the bond denoted by a thin line becomes shorter and that denoted by a thick line becomes longer, leading to a unit cell three times as large as the original. For an out-of-plane distortion (right), an atom denoted by a black dot is shifted down and that denoted by a white circle moves up.

In the presence of lattice distortions, the k p equation is given by the 4x4 matrix equation given by... 

The gap parameter or lattice distortion vanishes in the critical AB flux (jt which opens the gap as large as that due to the distortion. For <]) = 0, the gap decreases exponentially as a funetion of the cireumferenee Lla. Table 2 gives some examples for an in-plane Kekul distortion. 

Fig. 9. An example of calculated in-plane lattice distortions induced by a high magnetic field (left) and the dependence of the maximum gap due to in-plane lattice distortions on a magnetic field (right).

Electronic properties of CNTs, in particular, electronic states, optical spectra, lattice instabilities, and magnetic properties, have been discussed theoretically based on a k p scheme. The motion of electrons in CNTs is described by Weyl s equation for a massless neutrino, which turns into the Dirac equation for a massive electron in the presence of lattice distortions. This leads to interesting properties of CNTs in the presence of a magnetic field including various kinds of Aharonov-Bohm effects and field-induced lattice distortions. 

In conclusion we propose ASR as an efficient computational scheme to study electronic structure of random alloys which allows us to take into account the coherent scattering from more than one site. Consequently ASR can treat effects such as SRO and essential off-diagonal disorder due to lattice distortion arising out of size mismatch of the constituents. 

Martensitic phase transformations are discussed for the last hundred years without loss of actuality. A concise definition of these structural phase transformations has been given by G.B. Olson stating that martensite is a diffusionless, lattice distortive, shear dominant transformation by nucleation and growth . In this work we present ab initio zero temperature calculations for two model systems, FeaNi and CuZn close in concentration to the martensitic region. Iron-nickel is a typical representative of the ferrous alloys with fee bet transition whereas the copper-zink alloy undergoes a transformation from the open to close packed structure. ... 

The above technique has the practical inconvenience of requiring as many different sets of Tchebyschev coefficients as the unit cell non equivalent sublattices. Furthermore, for non cubic systems, these coefficients depend on the lattice distortion ratios. Namely, for tetragonal lattices different sets of coefficients are required for each value of c/a. This situation has made difficult the implementation of KKR and KKR-CPA calculations for complex lattice structures as, for example, curates. 

Figure 1 A dilute alloy system, showing a substitutional impurity, an interstitial impurity and an electromigration defect, and its reference system, the unperturbed host system. Some charge transfer effects are shown. Lattice distortion effects are omitted.

In spite of its simplicity this approach, supplemented with Blatt s correction [.3] for lattice distortion, was applied successfully for decades [4, 5] in studies of systematics in the residual resistivity. Its power was the exact treatment of the scattering and the use of the Friedel sum rule [1] as a self-consistency condition ensuring a correct valency difference between impurity and host atom. 

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

In Fig. 6 the arrangement of the manganese atoms is shown in a projection along the a-axis. The unit cells are marked by the shaded regions. It can easily be seen, that no lattice distortion is necessary to form... 

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