Structure amplitude normalized

E map A Fourier map (equivalent to an electron-density map) with phases derived by direct methods and structure amplitudes F(hkl) replaced by normalized structure amplitudes E(hkt). Since the E values correspond to sharpened atoms, the peaks on the resulting Fourier map are higher and sharper, and therefore easier to identify, than those in an electron-density map calculated with F values. 

Normalized structure factor The ratio of the value of the structure amplitude I F I to its root-mean-square expectation value. This ratio is denoted E(hkl). 

The F(obs) map this Fourier summation is calculated by use of Equation 9.1.1 in Table 9.1. It can be used to determine the arrangement of the atoms in the crystal structure. The relative phase angles a hkl) used to calculate this map have been derived by one of the methods described in Chapter 8. The coefficients are generally either experimentally determined structure amplitudes, F hkl) o, structure amplitudes calculated from a model, F(hkl) c, or normalized structure factor amplitudes E(hkl). ... 

As a result, the structure amplitude can be represented by its magnitude, lF(h), and phase angle, a(h), which varies between 0 and 27t. When the crystal structure is centrosymmetric (i.e. when it contains the center of inversion), each atom with coordinates (x, y, z) has a sjmimetrically equivalent atom with coordinates (-x, -y, -z). Thus, considering that cos(-y) = cos(y) and sin(-y) = -sin(y) and assuming that all atoms scatter normally, Eq. 2.103 can be simplified to... 

Figure 2.57. The relationships between different components of the structure amplitude in a Friedel pair when all atoms scatter normally (left) and when there are anomalously scattering atoms in the crystal structure (right).

Other types of Fourier transformations may also be calculated when the coefficients in Eq. 2.132 are modified or substituted. For example, when squared observed structure amplitudes IF" (in this case phase angles are not required ) or normalized structure amplitudes are used instead of the resultant maps usually provide means to solve the crystal structure (see next section). Different modifications of may reveal the... 

The phase relationships within a triplet are not strict and their probability depends on the magnitude of the associated structure amplitudes. The latter are scaled and normalized in order to reduce their dependence on the atomic scattering factor and vibrational motion since both reduce the structure amplitude exponentially at high sin0/X, see Eq. 2.91, Figure 2.53 and Figure... 

Finally, the generated phase angles are used in a forward Fourier transformation combined with the normalized structure amplitudes... 

Observed structure amplitudes are scaled and normalized structure amplitudes are calculated using Eq. 2.141. Reflections with large normalized structure amplitudes (a standard cut-off is Em E in =1.2) are selected for phase determination and refinement. 

As established above, individual intensities (or structure amplitudes) of Bragg reflections are needed in order to solve the crystal structure using direct or Patterson methods. In the first case, accurate normalized structure amplitudes are required to generate phase angles and to evaluate their probabilities. In the second case, accurate structure amplitudes result in the higher accuracy and resolution on the Patterson map. 

In the first, reflections with low accuracy in individual intensities (those that are completely or nearly completely overlapped) are simply discarded. This works best when direct methods are used for the structure solution because substantial errors even in some of the normalized structure amplitudes may affect phase angles of many other reflections. 

Once differences in structure amplitudes, AF, have been obtained by comparison of the diffraction data from ligand-saturated and native crystals, a Fourier synthesis can be computed in the normal way with phase angles calculated from the protein structure alone. These phases are not exactly the phases that should be attached to the observed differences forming the Fourier coefficients. The correct phases would be those calculated from the ligand correctly disposed in the crystal unit cell, which is, of course, what is being sought, what is not known. Using phases calculated from the native structure in conjunction with... 

Minimized structures gained from MD simulations are also often basis of normal mode analysis (NMA) [41-43]. NMA assumes that all atoms harmonically oscillate around their equilibrium points. The oscillations deflned by frequency and amplitude (normal mode) are extracted and reflect directions of internal protein motions. Given all its normal modes, the entire protein motion can be expressed as a superposition of modes. The modes vith lo vest frequency correspond to rather delocalized motions in proteins in vhich a large number of atoms oscillate in coordinated motion vith considerable amplitude. Modes vith higher frequency represent more localized motions. Linear combinations of the most relevant normal modes can be employed to depict essential protein motions. Stepwise displacement of atoms of the original structure along the modes can be applied to build up an ensemble of relevant protein conformations [44, 45]. 

Fig. 6 Structural details obtained by lineshape analysis, (a) In KcsA, labeling at any residue position renders a tetramer with potentially four spin labels, (i) Tandem dimer construct (ii) with cys residues in both protomers (control used to evaluate the effects of the intersubunit linker) and (iii) with only one of the protomers containing a cys (used in the analysis), (b) Rigid-limit X-band EPR spectra obtained at pH 7 (thick line, closed state) and at pH 4 (thin line, open state). Right panel, absorption spectra obtained from integration and relative fits obtained with convolution superimposed, (c) Simulated spin- and amplitude-normalized spectra for the two interspin distances in the figure (100% spin labeling efficiency), (d) Helical wheel representation of residues 100-119. Both closed (top) and open (bottom) states are represented as pairs of helical wheel...

A very sensitive piezoelectric test (2) was negative and the space group PI assumed. Data were collected at room temperature with Zr filtered MoKa radiation on a Nonius integrating Weissenberg camera. The intensities of approximately 1100 reflections were estimated visually and reduced in the normal manner to structure amplitudes. The crystal was carefully gonio-metered and its dimensions measured and an accurate correction for absorption applied. 

From Equation (35), the normalized structure amplitudes l l were obtained. These provided the basis for direct phase determination. 

While most stationary machine components move during normal operation, they are not always resonant. Some degree of flexing or movement is common in most stationary machine-trains and structural members. The amount of movement depends on the spring constant or stiffness of the member. However, when an energy source coincides and couples with the natural frequency of a structure, excessive and extremely destructive vibration amplitudes result. 

Surface force apparatus has been applied successfully over the past years for measuring normal surface forces as a function of surface gap or film thickness. The results reveal, for example, that the normal forces acting on confined liquid composed of linear-chain molecules exhibit a periodic oscillation between the attractive and repulsive interactions as one surface continuously approaches to another, which is schematically shown in Fig. 19. The period of the oscillation corresponds precisely to the thickness of a molecular chain, and the oscillation amplitude increases exponentially as the film thickness decreases. This oscillatory solvation force originates from the formation of the layering structure in thin liquid films and the change of the ordered structure with the film thickness. The result provides a convincing example that the SFA can be an effective experimental tool to detect fundamental interactions between the surfaces when the gap decreases to nanometre scale. 

Another remarkable feature of thin film rheology to be discussed here is the quantized" property of molecularly thin films. It has been reported [8,24] that measured normal forces between two mica surfaces across molecularly thin films exhibit oscillations between attraction and repulsion with an amplitude in exponential growth and a periodicity approximately equal to the dimension of the confined molecules. Thus, the normal force is quantized, depending on the thickness of the confined films. The quantized property in normal force results from an ordering structure of the confined liquid, known as the layering, that molecules are packed in thin films layer by layer, as revealed by computer simulations (see Fig. 12 in Section 3.4). The quantized property appears also in friction measurements. Friction forces between smooth mica surfaces separated by three layers of the liquid octamethylcyclotetrasiloxane (OMCTS), for example, were measured as a function of time [24]. Results show that friction increased to higher values in a quantized way when the number of layers falls from n = 3 to n = 2 and then to M = 1. 

For Comparison Notions of Normal Scattering. As the electron density is assumed to be a real quantity, it directly follows the central symmetry of scattering patterns known by the name Friedel s law [244], Friedel pairs are Bragg reflections hkl and hkl that are related by central symmetry. Concerning their scattering amplitudes, Friedel pairs have equal amplitude Aha = A-g and opposite phase (phki = -scattering intensity the phase information on the structure factor is lost. 

Although metals are generally good conductors of electricity, there is still some resistance to electrical flow, which is known as the resistivity of the metal. At normal temperatures, the resistivity is caused by the flow of electrons being impeded because of the motion of atoms that results from vibration about mean lattice positions. When the temperature is raised, the vibration of atoms about their mean lattice positions increases in amplitude, which further impedes the flow of electrons. Therefore, the resistivity of metals increases as the temperature increases. In a metal, electrons move throughout the structure. There are usually a small number of electrons from each atom that are considered, and because in most structures (fee and hep) each atom has 12 nearest neighbors, there is no possibility for the formation of the usual bonds that require two electrons for each. As a result, individual bonds are usually weaker than those of ionic or covalent character. Because of the overall number of bonds, the cohesion in metals is quite high. 

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