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Statistical Mechanics
Published in Marc J. Assael, Geoffrey C. Maitland, Thomas Maskow, Urs von Stockar, William A. Wakeham, Stefan Will, Commonly Asked Questions in Thermodynamics, 2022
Marc J. Assael, Geoffrey C. Maitland, Thomas Maskow, Urs von Stockar, William A. Wakeham, Stefan Will
The numerical value of (8π2IkBT/h2) is usually (but not always) large compared to 0.42, as a simple calculation reveals. For the extreme case of hydrogen (8π2IkB/h2) ≈ 0.01 K−1 and at a temperature of 300 K (8π2IkBT/h2) ≈ 3; so that ignoring 0.42 gives rise to a fractional uncertainty of 0.1 and the treatment of hydrogen as a classical rotator fails. However, for iodine at a temperature of 300 K, the fractional error is <10−4 and it is thus an acceptable approximation to assume that rotational motion is classical. For molecules with a molar mass greater than hydrogen and for temperatures on the order of 100 K or greater, Equation 2.27 can be approximated by qR≈8π2IkBTsh2,where s is the symmetry number. The symmetry number is s = 1 for linear molecules with no center of symmetry such as hydrogen fluoride and 2 for molecules with a center of symmetry, for example, oxygen. The difference arises because for symmetrical molecules each distinguishable orientation has been counted twice (Reed and Gubbins 1973).
Finding alike supercells of two crystals
Published in Science and Technology of Advanced Materials: Methods, 2023
Very similar crystals might be refined using very different settings. The space group type of perovskite materials differs significantly depending on how the symmetry is lowered. For instance, the five-atom unit cell of BaTiO3 can have cubic, tetragonal, orthorhombic, or rhombohedral symmetry depending on the temperature [2]. The lowered symmetry when B-site octahedra are tilted differently is given in Refs. [3–5]. The number of formula units in the crystallographic conventional cell varies between 1, 2, 4, 6, and 8 among them (Table 1). Layered lithium-ion battery cathode materials with O3 stacking according to the notation by Delmas et al. [6] are based on the rocksalt structure where Li-rich and transition metal-rich cation layers alternate along the (111) direction. The prototypical compound, LiCoO2, has Rm (number 166) symmetry [7]. Other compounds in this family include Li2MnO3 with C2/m symmetry (number 12) [8] and Li(NixLi1/3-2/3Mn2/3-x/3)O2 that can be refined both as an O3-layered structure (space group Rm) and as a spinel structure (space group Fdm, number 227) [9]. There must be an almost cubic supercell for these compounds, but how to derive such an almost cubic supercell from a primitive or conventional cell is not always obvious.