The harmonic temperature factor

According to the Fourier convolution theorem, further discussed in section 5.1.3, the Fourier transform of the convolution in expression (2.14) is the product of the Fourier transforms of the individual functions, or 

the temperature factor T(S) is the Fourier transform of the probability distribution P(u) T(S) = F P(u). In the common case that the rigidly vibrating groups are considered to be the individual atoms, T(S) is the Fourier transform of the atomic probability distribution. 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

For an isotropic potential, the three-dimensional probability distribution is 

The integral in Eq. (2.17) is the product of the integrals over each of the Cartesian 

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As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

Compared with the Gram-Charlier temperature factor of Eq. (2.31), the entire series now occurs in the exponent, so, in the cumulant formalism, terms are added to the exponent of the harmonic temperature factor P0(H) = exp — fijkhjhk. ... 

Since the anharmonic term is small relative to the leading harmonic term, the corresponding temperature factor can be written as... 

The great potential of the X-ray data for obtaining motional information has recently led to a molecular dynamics test197 of the standard refinement techniques that assume isotropic and harmonic motion. Since simulations have shown that the atomic fluctuations are highly anisotropic and, in some cases, anharmonic (see Chapt. VI.A.1), it is important to determine the errors introduced in the refinement process by their neglect. A direct experimental estimate of the errors resulting from the assumption of isotropic, harmonic temperature factors is difficult because sufficient data are not yet available for protein crystals. Moreover, any data set includes other errors that would obscure the analysis, and the specific correlation of temperature factors and motion is complicated by the need to account for static disorder in the crystal. As an alternative to an experimental analysis of the errors in the refinement of proteins, a purely theoretical approach has been used.197 The basic idea is to generate X-ray data from a molecular dynamics simulation... 

It is clear that nonconfigurational factors are of great importance in the formation of solid and liquid metal solutions. Leaving aside the problem of magnetic contributions, the vibrational contributions are not understood in such a way that they may be embodied in a statistical treatment of metallic solutions. It would be helpful to have measurements both of ACP and A a. (where a is the thermal expansion coefficient) for the solution process as a function of temperature in order to have an idea of the relative importance of changes in the harmonic and the anharmonic terms in the potential energy of the lattice. 

Fig. A3.1. Some lowest-order diagrams for the temperature GF (A3.6). The dashed and solid lines correspond to the GF for high-frequency and resonance low-frequency vibrations of a molecular planar lattice in the harmonic approximation (see Eq. (A3.9) and (A3.10)). Each vertex is associated with the factor -y/N, the integration and summation being performed over each vertex coordinates r, from 0 to / , and over all internal wave vectors K. At ptiClK 1, the main contribution is provided by a-type diagrams.184...

Thus, in the high-temperature limit, the mean-square displacement of the harmonic oscillator, and therefore the temperature factor B, is proportional to the temperature, and inversely proportional to the frequency of the oscillator, in agreement with Eq. (2.43). At very low temperatures, the second term in Eq. (2.51a) becomes negligible. The mean-square amplitude of vibrations is then a constant, as required by quantum-mechanical theory, and evident in Fig. 2.5. 

The preceding discussion assumed a pure liquid was used for the measurement. Most molecules of interest, however, are not in the liquid state at room temperature. In this case it is common to dissolve the compound in an appropriate solvent and conduct the measurement. Contributions to the second harmonic signal are therefore obtained from both the solvent and solute. Since r and the local field factors that are related to e and n, (the dielectric constant and refractive index respectively) are concentration dependent, the determination of p for mixtures is not straightforward. Singer and Garito (15) have developed methods for obtaining r0, eQ, and nQ, the values of the above quantities at infinite dilution, from which accurate values for p can be obtained in most cases. 

This method requires only a crude structural model as a starting model. In this analysis, the starting model was a homogeneous spherical shell density for the carbon cage. As for the temperature factors of all atoms, an isotropic harmonic model was used an isotropic Gaussian distribution is presumed for a La atom in the starting model. Then, the radius of the C82 sphere was refined as structural parameter in the Rietveld refinement. 

Independent of whether or not a well-defined crossover temperature can be observed in NS data above Tg, it has been well known for a considerable time that on heating a glass from low temperatures a strong decrease of the Debye-Waller factor, respectively Mossbauer-Lamb factor, is observed close to Tg [360,361], and more recent studies have confirmed this observation [147,148,233]. Thus, in addition to contributions from harmonic dynamics, an anomalously strong delocalization of the molecules sets in around Tg due to some very fast precursor of the a-process and increases the mean square displacement. Regarding the free volume as probed by positron annihilation lifetime spectroscopy (PALS), for example, qualitatively similar results were reported [362-364]. 

Fig. 26. Temperature dependence of various properties of myoglobin crystals , frequency of the O-D band maximum (IR) —, dielectric relaxation time of water (schematic) ---—, Lamb-Mossbauer factor,/o, after subtracting the harmonic mode (sche-...

In myoglobin, we find that the anharmonic contribution significantly enhances thermal conduction over that in the harmonic limit, by more than a factor of 2 at 300 K. Moreover, the thermal conductivity rises with temperature for temperatures beyond 300 K as a result of anharmonicity, whereas it appears to saturate around 100 K if we neglect the contribution of anharmonic coupling of vibrational modes. The value for the thermal conductivity of myoglobin at 300 K is about half the value for water. The value for the thermal diffusivity that we calculate for myoglobin is the same as the value for water. Thermal transport coefficients for other proteins will be presented elsewhere. 

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