Low-temperature approximation

We follow this with a low temperature approximation to the integral over the well s phase space population. The resulting equilibrium constant is... 

In the limit that g —> 1, the equilibrium distributions are more delocalized and the low temperature approximation may not be well justified. 

We once again carry out the integral in the low temperature approximation and find... 

The spectral function of local vibration (low-temperature approximation involving collectivized high-frequency and low-frequency modes)... 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

It is often useful to have an approximate relation for VPIE s, especially when complete vibrational analysis is impossible. The AB approximation serves that purpose, and sometimes gives more physical insight than do detailed, but very complicated calculations using Equation 5.24. It is based on the observation that ordinarily condensed phase vibrations fall in two groups the first containing the high frequencies, m > 1 (most often the internal modes, uj = hcvj/kT), where the zero point (low temperature) approximation is appropriate, and... 

The CBC technology operates at relatively low temperatures [approximately 1600°F (870°C)], thus reducing operation costs. The high turbulence produces a uniform temperature and promotes the complete mixing of the waste material during combustion. The effective mixing and relatively low temperature also reduce emissions of carbon monoxide and nitrogen oxides. 

Nickel shows some activity at low temperatures approximately equivalent to the activity of rhodium, but normal catalytic behavior with a constant activation energy is observed only at substantially higher temperatures similar to those necessary for activity with palladium. 

Line spectrum calculated in the low temperature approximation, valid at both 4 °K and 1.2 °K. There are no free parameters the lack of sharp lines in the observed spectra is attributed to spin relaxation (After Lang and Marshall, Ref. 103))... 

Polymethylsiloxanes can also be obtained by the hydrolytic copolycondensation of methylchlorosilanes of various functionality. Thus, by the hydrolytic copolycondensation of methyltrichlorosilane and dimethyldichlo-rosilane one can obtain thermosetting polymethylsiloxane, which quickly solidifies at a relatively low temperature (approximately 150 °C) without any catalysts. It is used as a binding agent in pressure compositions, as well as as a 5% petrol solution to treat the surface of metal pressure molds to facilitate their splitting when molding polymer materials. 

Eq. (9.4) is a low temperature approximation but is easily modified to include the Fermi distribution of the electrons. 

Low-temperature approximation. Zero-point-energy approximation. High-temperature approximation. 

An alternate approach to the analysis of isotope effects at large values of u is the low temperature approximation. As the temperature tends to zero, the function (Equation 1) approaches... 

Hydrothermal synthesis of a-alumina has been well studied. Since the hydro-thermal reaction of aluminum compound yields boehmite at relatively low temperatures (approximately 200°C), transformation of boehmite was examined and it was reported that more than 10 hours is required for complete conversion into a-alumina, even with a reaction at 445°C in a 0.1 N NaOH solution and in the presence of seed crystals. On the other hand, under glycothermal conditions, a-alumina is formed at 285°C for 4 h. The equilibrium point between diaspore (another polymorph of AlOOH) and a-alumina under the saturated vapor pressure of water was determined to be 360°C. However, near the equilibrium point, the transformation rate is very sluggish, and only a small conversion of diaspore is observed. Therefore complete conversion of diaspore into a-alumina requires a much higher temperature. Since boehmite is slightly less stable than diaspore, the hypothetical equilibrium point between boehmite and a-alumina would be lower than that for diaspore-alumina. However, a-alumina would not be formed by a hydrothermal reaction at such a low temperature as has been achieved in the glycothermal reaction. 

Sulfur ylides containing only alkyl, vinyl or aryl groups are very unstable and must generally be generated and used at low temperatures (approximately -70°C). On the other hand, ylides containing carbonyl, cyano, nitro or sulfonyl... 

The time independent product of exponentials in Eq. (A2.70) becomes the exponential of a sum over all modes. Thus Eq. (A2.57) becomes for the individual normal mode, excited to its n harmonic, in its low temperature approximation. 

The Debye-Waller factor always decreases the observed intensity in neutron spectroscopy and significant experimental efforts are made to reduce its impact. To follow this more closely we retrace our steps and, from Eq. (A2.70), write the full expression for 2W, instead of the low temperature approximation used in Eq.(A2.80). 

An interesting problem is the field evolution in a cavity that was initially in the equilibrium state at a finite temperature, when the initial occupation numbers were given by the Planck distribution v = [exp(pw) — 1] 1. Let us consider two limit cases. The first one corresponds to the low-temperature approximation v = exp (—(] ). Then the occupation number of the mth mode is merely the coefficient at vm in the expansion (61) with u = exp( (5). Using the well-known generating function of the Legendre polynomials Pm(z) [Ref. 269, Eq. 10.10(39)], one can obtain the following expression (for y = 0) ... 

Thus a plot of C /r vs. should yield a straight line in the low-temperature approximation. 

In most cases the low-temperature specific heat (at temperatures much lower than the Debye temperature) can be described by Cp = yT+ T, where yT is the so-called linear term due to the excitations of the conduction electrons, and is the low-temperature approximation of the specific heat of the lattice. For normal conductors, y is of the order of l-10mJ/(molK ), while heavy fermions show y-values up to 1000mJ/(molK ). Details about cerium- and uranium-based heavy-fermion materials have been reviewed by Grewe and Steglich (1991), Sereni (1991), Loewenhaupt and Fischer (1993), Fournier and Gratz (1993), Wachter (1994), Nieuwenhuys (1995), Onuki and Hasegawa (1995) and Arko et al. (1999). 

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