Anisotropic lattice expansion

Finally, molecular dynamics calculations were performed by Oumi et al. in order to obtain information about the anisotropic lattice expansion experimentally observed for TS-1 [68]. Using a simple two-body interatomic potential, they predicted an equilibrium d i-o of 1.85 A, slightly higher than that observed both experimentally and theoretically. Comparing the effect of Ti substitution on the lattice expansion over the twelve independent T-sites, the authors concluded that T8 is the most probable site for Ti incorporation. 

Under normal conditions tetradymite-type Bi2Tc3 does not undergo any phase transition from 4 to 600 K as follows from X-ray measurements [518]. No anomalous temperature dependence of the (anisotropic) lattice expansion was observed as a function of charge-carrier density or carrier type. 

Fig. 14. Anisotropic thermal expansion of GdCuSn measured by x-ray powder diffraction (Gratz and Lindbaum 1998). The lines represent the extrapolation of the lattice contribution from the paramagnetic range by fitting...

Fig. 28. Anisotropic thermal expansion of the orthorhombic FeB phase of GdCu obtained from neutron diffraction experiments on a bulk polycrystalline sample (Blanco et al. 1999). The lattice parameters (Ip) at 180 K are o = 7.I5 0.01 A, b = 4.527 0.008 A, c = 5.471 0.008 A.

Fig. 38. Anisotropic thermal expansion of the orthorhombic Gr Ni, measured by single crystal x-ray diffraction (the data points have been extracted from Kusz et al. (2000)). The lines represent extrapolations of the lattice...

Displacement parameters of atoms are also expected to be different as the temperature of the powder diffraction experiment varies. Furthermore, it is also feasible that atomic positions may change due to generally anisotropic thermal expansion of crystal lattices. These considerations are in addition to the most obvious cause (different lattice parameters) preventing combined refinement using powder diffraction data collected at different temperatures. In general, material may also be polymorphic but this is not the case here, as was established in Chapter 6, sections 6.10 and 6.11. 

Software that enables the refinement of twinned crystals is now widely available and has been shown to provide satisfactory results in a large number of applications. As data collection is done almost always at low temperatures, anisotropic thermal expansion of the crystal lattice will invariably introduce local strains into twinned crystals. This induced strain may be unevenly distributed within the... 

In further work the same team [82] used angle-dispersive X-ray diffraction in the pressure range 2-6 GPa to study crystalline GeSc2. This confirmed the transition to a three-dimensional crystalline structure above 2 GPa at 698 K. This transition was explained by an anisotropic lattice distortion due to the cooperative tilting of rigid Ge(Sc4) tetrahedra. The authors claimed that a similar transition, including anomalous compressibility and thermal expansion phenomena, could be observed in... 

At 373K < T < 873K, swelling of BeO should be attributed only to lattice defect growth (anisotropic crystal expansion) and microcracking. 

The thermal expansion of gadolinium has been studied several times (for a collection of data see e.g. Touloukian et al. (1976)). Since there is some disagreement mainly concerning the temperature variation of the lattice parameters, we remeasured the anisotropic thermal... 

These expressions show that a deformed polymer network is an extremely anisotropic body and possesses a negative thermal expansivity along the orientation axis of the order of the thermal expansivity of gases, about two orders higher than that of macromolecules incorporated in a crystalline lattice (see 2.2.3). In spite of the large anisotropy of the linear thermal expansivity, the volume coefficient of thermal expansion of a deformed network is the same as of the undeformed one. As one can see from Eqs. (50) and (51) Pn + 2(iL = a. Equation (50) shows also that the thermoelastic inversion of P must occur at Xim (sinv) 1 + (1/3) cxT. It coincides with F for isoenergetic chains [see Eq. (46)]. 

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

This intermolecular potential for ADN ionic crystal has further been developed to describe the lowest phase of ammonium nitrate (phase V) [150]. The intermolecular potential contains similar potential terms as for the ADN crystal. This potential was extended to include intramolecular potential terms for bond stretches, bond bending and torsional motions. The corresponding set of force constants used in the intramolecular part of the potential was parameterized based on the ab initio calculated vibrational frequencies of the isolated ammonium and nitrate ions. The temperature dependence of the structural parameters indicate that experimental unit cell dimensions can be well reproduced, with little translational and rotational disorder of the ions in the crystal over the temperature range 4.2-250 K. Moreover, the anisotropic expansion of the lattice dimensions, predominantly along a and b axes were also found in agreement with experimental data. These were interpreted as being due to the out-of-plane motions of the nitrate ions which are positions perpendicular on both these axes. 

Atomic and molecular displacement under constraint. Thermal expansion and compressibility are large and anisotropic. Sometimes structural data have been extrapolated from the room temperature (RT) down to low temperature (LT) simply by considering changes in lattice dimensions. This has led to disappointing results since, even in the absence of a phase transition, molecular shapes and orientations may change substantially. Similarly, if we find an isostatic pressure at room temperature whose effect is equivalent to a given temperature decrease at ambient pressure for, say, the chain contraction, the equivalence will not usually match for, say, the... 

The thermal expansion of solids depends on their structure symmetry, and may be either isotropic or anisotropic. For example, graphite has a layered structure, and its expansion in the direction perpendicular to the layers is quite different from that in the layers. For isotropic materials, ay w 3 a . However, in anisotropic solid materials the total volume expansion is distributed unequally among the three crystallographic axes and, as a rule, cannot be correctly measured by most dilatometric techniques. The true thermal expansion in such case should be studied using in situ X-ray diffraction (XRD) to determine any temperature dependence of the lattice parameters. 

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