Thermal expansion tensor

The RUM model provides a natural explanation for the phenomenon of negative thermal expansion, the property in which materials shrink they are heated. Recently the study of materials with this property has become quite active, particularly since the publication on detailed work on ZrW20g (Mary et al. 1996, Evans 1999), for which the (negative) thermal expansion tensor is isotropic and roughly constant over a very wide range of temperatures. 

The mineral calcite, CaC03, is trigonal (Section 2.8). In order to determine the thermal expansion tensor e (4.74), the Bragg angles for the lines 330 and 00,18 were measured from powder patterns obtained at two different temperatures ... 

THERMAL EXPANSION OF MOLECULAR CRYSTALS. III. THE THERMAL EXPANSION TENSOR OF DIBENZYL. //ENGLISH TRANSLATION OF KRISTALLOGRAFIYA 15 111 326-8.1970.// RYZHENKOVAP... 

FIGURE 83 Components of the thermal expansion tensor (A) and volumetric thermal expansion coefficient (B) versus rare earth cation radius in rare earth gallates. The dashed lines are guides for the eye. 

The volume expansion coefficient is obtained by plotting the expansion tensor - i.e. by the sum of the three primary coefficients -but also, because the plot of a square matrix is invariant in a changed system of coordinates, by the sum of the three diagonal terms in the thermal expansion tensor, which gives us the relations ... 

For aligned polymers, a thermal expansion tensor otxyz that is very anisotropic down to low temperatmes can be established. At very low temperatures, all components converge to the same value a -> 0. The influence of different modes is difficult to distinguish even for aligned polymers. Information... 

In order to reproduce the temperature variation of the lattice constants, the anisotropy of the lattice expansion has to be taken into account. For this purpose, the tensor of thermal expansion ot is introduced instead of the scalar a , and the tensor of deformation due to the HS <- LS transition is employed instead of the dilation (Fh — Fl)/Fl. Each lattice vector x T) can now be... 

Table 20. Tensor elements referred to the principal axes of the tensor of thermal expansion 0 and the deformation tensor E due to the HS<->LS transition in [Fe(2-pic)3]Cl2 CH3OH according to Ref. [39]...

The thermal conductivity tensor may likewise be split into symmetric and antisymmetric parts, with expansions in powers of B as in eqs. (35) and (36). But Z is not necessarily a symmetric tensor at B = 0, and so the expansion of the antisymmetric part of Z in an equation like eq. (36) is not applicable. Instead,... 

P. The latter have to be interpreted in terms of thermal expansion and compressibility, respectively. Both quantities are described by a second-rank deformation tensor U which can be directly computed from the six sets of parameters versus constraint [37,107] ... 

Here, ay is the thermal expansion coefficient which is assumed to be equal in both lattices and 6 is the dilatation coefficient (Khs — ls)/ ls> where Fls and Fhs are the unit cell volumes of the pure LS and HS species at 0 K, respectively. In order to account for the anisotropy of the lattice, the thermal expansion v and dilatation e coefficients must be introduced as tensors instead of scalars. Similarly, an equivalent expression could be defined for pressure-induced spin conversions. 

The following properties are characterized by symmetric tensors of rank 2 magnetic susceptibility (negative eigenvalues for diamagnetic materials) electrical and thermal conductivities (these tensors are symmetrical according to the Onsager principle) thermal expansion. 

The wavelength of the radiation used was A = 1.54051 A (CuK i). Calculate the components of the tensor e. In which direction is the thermal expansion zero ... 

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