Factor temperature

The change in the intensity with temperature is calculated with the temperature factor. This change is produced by the crystal lattice vibrations, that is, the scattering atoms or ions vibrate around their standard positions as was previously explained (see Section 1.4) consequently, as the crystal temperature increases, the intensity of the Bragg-reflected beams decreases without affecting the peak positions [25], Debye and Waller were the first to study the effect of thermal vibration on the intensities of the diffraction maxima. They showed that thermal vibrations do not break up the coherent diffraction this effect merely reduces the intensity of the peaks by an exponential correction factor, named the temperature factor, D(0) [2,26], given by 

as T increases, B will increase, because at a high temperature this parameter is directly related to the thermal energy, kT. 

At any temperature higher than absolute zero, certain frequencies in a phonon spectrum of the crystal lattice are excited and as a result, atoms are in a continuous oscillating motion about their equilibrium positions, which are determined by coordinate triplets (x, y, z). To account for these vibrations, the so-called temperature factor is introduced into the general equation (Eq. 2.87) of the structure amplitude. 

Oscillatory motions of atoms may be quite complex and as a result, several different levels of approximations in the expression of the temperature factor can be used. In the simplest form, the temperature factor of they atom is represented as  

As mentioned at the beginning of this section, the temperature factor absorbs other unaccounted or incorrectly accounted effects. The most critical are absorption, porosity and other instrumental or sample effects (see Chapter 3), which in a systematic way modify the diffracted intensity as a function of the Bragg angle. As a result, the B parameters of all atoms may become negative. If this is the case, then the absorption correction should be re-evaluated and re-refined or the experiment should be repeated to minimize the deleterious instrumental influence on the distribution of intensities of Bragg peaks. 

The next level of approximation accounts for the anisotropy of thermal motions in a harmonic approximation and describes atoms as ellipsoids in one of the three following forms, which are, in fact, equivalent to one another  

As follows from Eqs. 2.94 and 2.95, the relationships between By and Uij are identical to that given in Eq. 2.92 and both are measured in A. The P,y parameters in Eq. 2.93 are dimensionless but may be easily converted into By or Uy. Very high quality powder diffraction data are needed to obtain dependable anisotropic displacement parameters and even then, they may be reliable only for those atoms that have large scattering factors (see next section). On the other hand, the refinement of anisotropic displacement parameters is essential for those crystal structures, where strongly scattering atoms are distinctly anisotropic. 

The total diffracted intensity is obtained by integrating over an infinitely thick specimen  

Here Iq, b, and // are constant for all reflections (independent of 6) and we may also regard a as constant. Actually, a varies with 9, but this variation is already taken care of by the cos 9 portion of the Lorentz factor (see Sec. 4-9) and need not concern us here. We conclude that the absorption factor 1 /2 is independent of 9 for a flat specimen making equal angles with the incident and diffracted beams, provided the specimen fills the incident beam at all angles and is effectively of infinite thickness. 

The criterion adopted for infinite thickness depends on the sensitivity of our intensity measurements or on what we regard as negligible diffracted intensity. For example, we might arbitrarily but quite reasonably define infinite thickness as that thickness t which a specimen must have in order that the intensity diffracted by a thin layer on the back side be rsVo of the intensity diffracted by a thin layer on the front side. Then, from Eq. (4-15) we have 

This expression shows that infinite thickness, for a metal specimen, is very small indeed. For example, suppose a specimen of nickel powder is being examined with Cu Ka radiation at 6 values approaching 90°. The density of the powder compact may be taken as about 0.6 the density of bulk nickel, which is 8.9 gm/cm, leading to a value of for the compact of 261 cm . The value of t is therefore 1.32 x 10 cm, or about five thousandths of an inch. 

So far we have considered a crystal as a collection of atoms located at fixed points in the lattice. Actually, the atoms undergo thermal vibration about their mean positions even at the absolute zero of temperature, and the amplitude of this vibration increases as the temperature increases. In aluminum at room temperature, the average displacement of an atom from its mean position is about 0.17 A, which is by no means negligible, being about 6 percent of the distance of closest approach of the mean atom positions in this crystal. 

---

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. 

They compared the PME method with equivalent simulations based on a 9 A residue-based cutoflF and found that for PME the averaged RMS deviations of the nonhydrogen atoms from the X-ray structure were considerably smaller than in the non-PME case. Also, the atomic fluctuations calculated from the PME dynamics simulation were in close agreement with those derived from the crystallographic temperature factors. In the case of DNA, which is highly charged, the application of PME electrostatics leads to more stable dynamics trajectories with geometries closer to experimental data [30]. A theoretical and numerical comparison of various particle mesh routines has been published by Desemo and Holm [31]. 

The relative molecular dynamics fluctuations shown in Figure 7-17 can be compared with the crystallographic B-factors, which are also called temperature factors. The latter name, especially, indicates the information content of these factors they show how well defined within the X-ray structure the position of an atom is. Atoms with high temperature have an increased mobility. In principle, this is the same information as is provided by the molecular dynamics fluctuations. Using Eq. (48), the RMS fluctuation of an atom j can be converted into a B-factor... 

In equation 26, F is the MTD correction factor and, in general, is a function of the flow configuration and the two temperature factors defined ... 

The Bragg peak intensity reduction due to atomic displacements is described by the well-known temperature factors. Assuming that the position can be decomposed into an average position, ,) and an infinitesimal displacement, M = 8R = Ri — (R,) then the X-ray structure factors can be expressed as follows ... 

Traver.se number temperature factor), (a) The peak gas temperature minus mean gas temperature divided by mean temperature rise in nozzle design, (b) The differenee between the highest and the average radial temperature. 

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. 

Monte Carlo simulations have been done on the TV x x cubic lattice (TV = 27) with the lattice spacing h = 0.8 [47,49] for a bulk system. The usual temperature factor k T is set to 1, since it only sets the energy scale. The following periodic boundary conditions are used = [Pg.714]

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. 

Total no. of parameters refined (includes overall scale and temperature factor) 18,695... 

The atomic temperature factors obtained after crystallographic refinement are significantly higher for cys530 than for the other site cysteine residues. This is also true when the Ni ion is compared to the Fe center. This may reflect conformational disorder due to the fact that the crystals are made of a mixture of different Ni states (40% Ni-A, 10% Ni-B, and 50% of an EPR-silent species) (52). 

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... 

Isotropic temperature factors (B) are in A standard deviations are given in parentheses. 

Crystallographic refinement is a procedure which iteratively improves the agreement between structure factors derived from X-ray intensities and those derived from a model structure. For macro molecular refinement, the limited diffraction data have to be complemented by additional information in order to improve the parameter-to-observation ratio. This additional information consists of restraints on bond lengths, bond angles, aromatic planes, chiralities, and temperature factors. 

---