The temperature factor

The most important correction term applied to intensity calculations is the temperature factor. The calculations described above assume that the atoms in a crystal are stationary. This is not so, and in molecular crystals and some inorganic crystals the vibrations can be considerable, even at room temperature. This vibration has a considerable effect upon the intensity of a diffracted beam, as can be seen from the following example. The vibration frequency of an atom in a crystal is often taken to be approximately 1013Hz at room temperature. The frequency of an X-ray beam of wavelength 0.154 nm, (the wavelength of copper Ka radiation, which is 

Although chemical bonds link the atomic vibrations throughout the crystal, the thermal motion of any atom in a crystal is generally assumed to be independent of the vibration of the others. Under this approximation, the atomic scattering factor for a thermally vibrating atom, fth, is given by  

Where fa is the atomic scattering factor defined for stationary atoms, B is the atomic temperature factor, (also called the Debye-Waller 

The atomic temperature factor is related to the magnitude of the vibration of the atom concerned by the equation  

Although the value of U gives a good idea of the overall magnitude of the thermal vibrations of an atom, these are not usually isotropic (the same in all directions). To display this, the atom positions in a crystal structure are not indicated by spheres of a radius proportional to 

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The temperature factor (together with the Cartesian coordinates) is the result of the rcfincincnt procedure as specified by the REMARK 3 record. High values of the temperature factor suggest cither disorder (the corresponding atom occupied different positions in different molecules in the crystal) or thermal motion (vibration). Many visualisation programs (e.g., RasMol [134] and Chime [155]) have a special color scheme designated to show this property. 

The atoms of a protein s structure are usually defined by four parameters, three coordinates that give their position in space and one quantity, B, which is called the temperature factor. For well refined, correct structures these B-values are of the order of 20 or less. High B-values, 40 or above, in a local region can be due to flexibility or slight disorder, but also serve as a warning that the model of this region may be incorrect. 

From the temperature factor A, an activation energy may again be calculated which gives a useful indication of the influence of flow-rate on corrosion rate. 

Products that are subjected to a load have to be analyzed carefully with respect to the type and duration of the load, the temperature conditions under which the load will be active, and the stress created by the load. A load can be defined as continuous when it remains constant for a period of 2 to 6 hours, whereas an intermittent load could be considered of up to two hours duration and is followed by an equal time for stress recovery. The temperature factor requires greater attention than would be the case with metals. The useful range of temperatures for plastic applications is relatively low and is of a magnitude that in metals is viewed as negligible. 

The atomic amplitude functions take account of the atomic F- factor, the temperature factor, the Lorentz factor, and the polarization factor. 

The atomic reflecting power Fn as a function of sin B/l or of dhjcl depends on the structure of the atom and also on the forces exerted on the atom by surrounding atoms, inasmuch as the temperature factor (also a function of dh]c ) is included in the J -curve. Values of F for various atoms have been tabulated by Bragg and West. Nov it is convenient to introduce the concept of the atomic amplitude function An, defined by the equation... 

Observed and calculated intensities of reflections on two oscillation photographs, one of which is reproduced in Fig. 5, are given in Table III. The first number below each set of indices (hkl) is the visually estimated observed intensity, and the second the intensity calculated by the usual Bade-methode formula with the use of the Pauling-Sherman /0-values1), the Lorentz and polarization factors being included and the temperature factor omitted. No correction for position on the film has been made. It is seen that the agreement is satisfactory for most of the... 

The factor n is required by the experimental conditions, under which the amount of incident radiation intercepted by the face of the crystal increases linearly with the order of reflection. The temperature factor corresponds to an estimated characteristic temperature of about 530° The /0-values used are those ofPauling and Sherman1). It is seen that the observed intensity relations (800) (600)... 

However, an error was made in the application of the temperature factor, which resulted in incorrect weights therefore the structure factors and their derivatives were recalculated on the basis of these parameters and a second least-squares treatment was carried out as described below. 

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

CNs, the Fe ion is postulated to be coordinated by one SO and a mixture of CO and CNs. The interpretation is based on the temperature factor refinement and pyrolytic analysis of oxidized sulfur species (33). In addition, the bridging ligand is postulated to be an inorganic sulfur ion (instead of an oxo ligand, as proposed for the D. gigas). This... 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

Thus, in order to improve the behavior of the parameter estimates, Watts (1994) centers the temperature factor about a reference value T0 which was chosen to be the middle temperature of 375°C (648 K). The parameters estimates and their standard errors are given in Table 16.19. 

The importance of the first three of these factors has already been discussed. The temperature factor would include the cost of insulation plus the increase in metal thickness necessary to counteract the poorer structural properties of metals at high temperatures. Zevnik and Buchanan17 have developed curves to obtain the average cost of a unit operation for a given fluid process. They base their method on the production capacity and the calculation of a complexify factor. The complexity factor is based on the maximum temperature (or minimum temperature if the process is a cryogenic one), the maximum pressure (or minimum pressure for vacuum systems) and the material of construction. It is calculated from Equation 2 ... 

A small problem arises when the crystal thickness and temperature factors are refined simultaneously, because these parameters are highly correlated. Raising both the thickness and the temperature factors results in almost the same least-squares sum. This is not an artifact of the calculation method but lies in the behavior of nature. Increasing the Debye-Waller factor of an atom means a less peaked scattering potential, which in turn results in a less sharply peaked interaction with the ncident electron wave. It can be shown that a thickness of 5 nm anc B=2 will give about the same results as a thickness of 10 nm and B=6 A. ... 

After merging of the single zones, data sets of approx. 100-300 independent reflections can be obtained as described in chapter 2.5. In a first step a kinematical structure refinement should be performed using the program SHELXL [13]. The temperature factors for FAPPO were chosen as U = 0.06 for C, N and O as U = 0.10 for H atoms apart from H atoms situated at N with U = 0.12 A (Electron scattering factors [20]). To prevent the molecules from being distorted a refinement, where the whole molecule was kept rigid, was performed. This also improves the usually bad parameter/reflection ratio. In the case of modification I we obtained R-values of 31% (481 unique reflections with I > 2cr) for the 100 kV data and of 25% (385 unique reflections with I > 2a). The sparse 100 kV data of modification II was not analysed quantitatively. From 300 kV data we obtained an R-value of 23% (226 unique reflections with I > 2a). 

For anisotropic vibration the temperature factor is more complex because f now depends on the direction of S. The anisotropic temperature factor is often... 

Fig. 15. Temperature factors (B values) for DNA showtrrg areas of structure indetermirracy are represented in a pseudo-color model. Blue represents the lowest temperature factors and red the highest. Each gyre of the DNA is shown, ventral left, dorsal right. The temperature factor is one means of describing the atomic disorder. Sites indicated in red correlate with the holes shown in the previous figure and could be positions of HMGNl binding.

Thus, 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. 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

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. 

Application of Eq. (2.58) to calculate the temperature factors requires knowledge of the full frequency spectrum of the crystal throughout the Brillouin zone. Such information is only available for relatively simple crystal structures such as Al, Ni, KC1, and NaCl (Willis and Pryor 1975, p. 13ff.). Agreement between theory and experiment for such solids is often quite reasonable. 

Since the scale factor is considered an unknown in the least-squares procedure, its estimate is dependent on the adequacy of the scattering model. Other parameters that correlate with k may be similarly affected. In particular, the temperature factors are positively correlated with k, the correlation being more pronounced the smaller the sin 0//. range of the data set, as for a small range the scale factor k and the temperature factor exp ( —fl sin 02ft.2) affect the structure factors in identical ways. 

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