Rank 0, 1, 2 and 3 magnetic and non-magnetic physical-property tensors

Abstract

still be increasing at 97.5% TD. This suggests that discontinuous grain growth may be taking place. However, the large uncertainties in the scattering volume fraction, g,, and the effective radii, Ree~(0) and Re~(OO), preclude drawing firm conclusions.Although the final-stage samples were originally intended for single-scattering SANS measurements, it was found that even the 97.5% TD sample was within the limit of applicability of the MSANS technique. There was significant wavelength-dependent beam broadening and even the smallest ~, value (~, = 40) is well within the MSANS formalism. However, the fact that q~ is so small means that the change in rc as a function of h is small (see Fig. 4). This nearly fiat slope makes the refinement procedure difficult, resulting in large uncertainties in the fitted Re~ (0) and tp values. To improve the determination of Re~(0) during final-stage sintering, measurements of the higher-density samples are planned with a high-resolution small-angle X-ray scattering instrument (Long, Jemian, Weertman, Black, Burdette & Spal, 1991

Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server

“…The analysis of the restrictions that space-time symmetry produces in the tensors describing the properties of magnetically ordered crystals has been a research subject for more than 50 years (Birss, 1963(Birss, , 1964Cracknell, 1973;Eremenko et al, 1992;Grimmer, 1993Grimmer, , 1994Grimmer, , 2017Kleiner, 1966;Kopsky, 2015;Landau & Lifshitz, 1960;Litvin & Litvin, 1991;Shtrikman & Thomas, 1965;Sirotin & Shaskolskaya, 1982). The topic is interesting because the tensor form determines whether a particular physical phenomenon can take place in the crystal, and also regulates the anisotropy of the different material properties.…”

Classification of second harmonic generation effect in magnetic materials

“…The SHG tensors are usually divided into T -even (itype, χ even ijk ) and T -odd (c-type, χ odd ijk ) parts [24,25]. According to Neumann principle [24,[27][28][29], the symmetries of the SHG tensors are constrained by the magnetic point groups (MPGs) M 0 . Generally, the symmetries of SHG under all MPGs can be obtained directly by numerical calculations [24,27,28].…”

“…According to Neumann principle [24,[27][28][29], the symmetries of the SHG tensors are constrained by the magnetic point groups (MPGs) M 0 . Generally, the symmetries of SHG under all MPGs can be obtained directly by numerical calculations [24,27,28]. However, the relationships between MPGs and SHG tensors and the rules of SHG tensors symmetries under all MPGs cannot be concluded clearly within the numerical method.…”

On the Symmetry Analysis of Surface Induced Effects

“…The bulk symmetry is given in the left-hand-side column and the surface symmetry at the intersection of the bulk symmetry line and the surface orientation column. physical property tensors invariant under non-magnetic and magnetic point groups with various internal symmetries have been derived and are available in print [4][5][6][7][8][9][10] or in computerized tabulations [11][12][13][14]. Nye [15] has shown how to simultaneously present the matrix form of multiple tensor properties: A single 10 by 10 extended matrix of a material's elastic coefficients, piezoelectric coefficients, thermal expansion coefficients, permittivities, pyroelectric coefficients, and heat capacity/absolute temperature was constructed and the form of this extended matrix invariant under one point group of each of the 32 non-magnetic point group types was tabulated.…”