Temperature dependence of lattice constants

At each temperature one can determine the equilibrium lattice constant aQ for the minimum of F. This leads to the thermal expansion of the alloy lattice. At equilibrium the probability f(.p,6=0) of finding an atom away from the reference lattice point is of a Gaussian shape, as shown in Fig. 1. In Fig.2, we present the temperature dependence of lattice constants of pure 2D square and FCC crystals, calculated by the present continuous displacement treatment of CVM. One can see in Fig.2 that the lattice expansion coefficient of 2D lattice is much larger than that of FCC lattice, with the use of the identical Lennard-Lones (LJ) potential. It is understood that the close packing makes thermal expansion smaller. 

Mercury selenide HgSe 28.61 30-380 °C From temperatur dependence of lattice constant a... 

The temperature dependence of lattice constants is well known. On increasing temperature, there is an expansion in the (001) plane, but a corresponding contraction along c so that the net overall volume of the cell decreases. This peculiar behavior can be explained by the edge sharing of Li-and Al,Si-containing tetrahedra. At room temperature, the Li-(Al,Si) distance... 

Mercury selenide Mercury telluride HgSe HgTe Q perpend 18.1(5) 28.61 4.0 30-380°C 77-300K Perpendicular to c axis, trigonal HgS From temperatur dependence of lattice constant a See Fig. 4.1-176 for temperature dependence... 

There have been quite a number of discrepancies in the compression curves of CeAl2. A discontinuous change in the pressure dependence of lattice constant at room temperature was observed by some authors (Bartholin et al., 1980 Croft and Jayaraman, 1979) around 6 GPa. But other authors did not find it below 20 GPa (Barbara et al., 1986 Vedel et al., 1986). The origin of these discrepancies seems to be a complicated issue, but it may originate mainly from the lattice compression at nonhy-drostatic condition by solidification of pressme medium. 

In order to study the vibrational properties of a single Au adatom on Cu faces, one adatom was placed on each face of the slab. Simulations were performed in the range of 300-1000"K to deduce the temperature dependence of the various quantities. The value of the lattice constant was adjusted, at each temperature, so as to result in zero pressure for the bulk system, while the atomic MSB s were determined on a layer by layer basis from equilibrium averages of the atomic density profiles. Furthermore, the phonon DOS of Au adatom was obtained from the Fourier transform of the velocity autocorrelation function. ... 

At high temperatures, the module of the pyroelectric constants of both compounds increases more significantly and reaches an extremum at about 480K. Fig. 113 shows the temperature dependence of the pyroelectric coefficients. This phenomenon could be related to the change in dilatation mechanism that was observed while investigating the temperature dependence of the lattice parameters (see Fig. 102). 

Fig. 38. Temperature dependence of the lattice constants a, b, c and the unit cell volume V of [Fe(2-pic)3]Cl2 CH3OH. The solid lines are calculated using the parameter values of Table 20. According to Ref. [39]...

Our development has assumed temperature independent force constants. In real liquids, however, there is a small temperature dependence of frequencies and force constants due to anharmonicities, lattice expansion, etc. The incorporation of these effects into the theory is treated in later sections. 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

To determine static properties of the SeO radical in KDP and DKDP, the temperature dependence of the hyperfine interaction between unpaired electron and Se (I = 1/2) nucleus was measured [53]. The hyperfine tensor component A, where the direction is along the c-axis, shows an isotope effect, because its value is higher in DKDP than in KDP. Furthermore, its value shows a jump at Tc for DKDP and a considerable temperature dependence in the PE phase of both crystals, approximated by the relation A (T) = A (0) - B coth(ro/T), where To 570 K for both crystals. It is interesting to note that A, similarly to the As NQR frequency and P isotropic chemical shift, should be constant in the PE phase if the two-state order-disorder mechanism of the corresponding tetrahedron holds. However, while the temperature dependencies of the As NQR frequency and P isotropic chemical shift in the PE phase were explained as originating from a six-state order-disorder mechanism [42] and additional displacive mechanism [46], respectively, here it was assumed that excitation of some extra lattice vibration mode with frequency Tq affects the hyperfine tensor components and causes the temperature dependence of A. ... 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

Magnitude of Stress. We suspect that sources besides stress may, in the aggregate, account for as much as half of the observed spread in v3, so that the most highly stressed C02 experiences the equivalent of at least 20 kbar of pressure. Support for the inference of high local stress comes from a survey of the temperature dependences of the bands observed in 24 different reaction site environments. Since a crystal expands as it warms, one can make an analogy between temperature and pressure. When the temperature is raised, the crystal lattice expands, and the average force constant between stressed molecules decreases [74],... 

The temperature dependencies of the ( 172)0/ 1/2 ratio, where ( 1/2)0 is the 1/2 value measured at room temperature, determined for the CHOH - CH2 - O and CH2 - N units of the hydroxylpropyl ether (HPE) sequence (Fig. 92) in the HMDA network [63] are shown in Fig. 97. It is worth noticing that the 1/2 values of these two types of carbons have the same temperature dependence. Up to 60 °C, the 1/2 values are constant and equal to the rigid-lattice values, indicating that the HPE sequence does not undergo any local motion at a frequency equal to or higher than 105 Hz in this temperature range. Above 60 °C, mobility develops, which leads at 100 °C to motions in the tens of kilohertz for the whole HPE sequence. These results are qualitatively confirmed by data on 13C spin-lattice relaxation time in the rotating frame, Tip(13C). 

Thus from the present calculations it is derived sh — sa = 0.14 eV for MgO Cu2+. This figure concurs with the value [21] derived from the analysis of the temperature dependence of spin-lattice relaxation, 7j. The value of the coupling coefficient for SrO Cu2+ is found to be V = 0.47 eV/A. In comparison with V = 1.05 eV/A found for MgO Cu2+ this reduction reflects the increase of R0- Therefore, the big reduction undergone by force constant, K, on going from MgO Cu2+ to SrO Cu2+ is thus the main responsible for Eji/Ua = 1.9 obtained for SrO Cu2+. This value, which is three times bigger than E]T/hojK = 0.65 derived for MgO Cu2+ can explain the observation [22] of static EPR spectra for SrO Cu2+. 

Fig. 2. Temperature dependence of (a) the lattice constants of a La7/8Sri/8Mn03 single crystal (figure taken from Ref. [10]), (b) the resistivity, and (c) the low-field magnetic moment. The cooperative JT transition at Tjt = 269 K results in a doubling of the resistivity and a sharp drop of the magnetic moment (see inset in (c)). Below the temperature TCo = 147 K the crystal goes into a charge-ordered state and the CJT effect vanishes.

Fortunately, they are several species of low-loss dielectric ceramics with tailored temperature coefficient of dielectric constant, which can be made lower than 1 ppm/K for a certain temperature window around room temperature. Physically, this can be accomplished either by intrinsic compensation of the temperature dependence of thermal volume expansion V(T) and lattice polarizability a(T) via the Clausius-Mossotti relation ... 

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. 

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