Calculated phase angles

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

Interpretation of interatomic vectors. Use of known atomic positions for an initial trial structure (a preliminary postulated model of the atomic structure) can be made, by application of Equations 6.21,4 and 6.21.5 (Chapter 6), to give calculated phase angles. Methods for obtaining such a trial structure include Patterson and heavy-atom methods. Such methods are particularly useful for determining the crystal structures of compounds that contain heavy atoms (e.g., metal complexes) or that have considerable symmetry (e.g., large aromatic molecules in which the molecular formula includes a series of fused hexagons). The Patterson map also contains information on the orientation of molecules, and this may also aid in the derivation of a trial structure. 

Calculated phase angle The phase angle o calculated from the atomic positional and displacement parameters for a model structure. 

As soon as phase angles, a w, are established at least approximately, they can be used in combination with available IF" h i to compute a Fourier map and establish the distribution of electron or nuclear density in the unit cell. Even though the phases may be approximate (i.e. inexact), the values of are much more precise and therefore, the Fourier map is usually more accurate than the model employed to calculate phase angles. Thus, a computed Fourier map can be used to improve and refine the model of the... 

To determine the pipeline potentials, the resultant induced field strengths have to be included in the equations in Section 23.3.2. Such calculations can be carried out with computers that allow detailed subdivision of the sections subject to interference. A high degree of accuracy is thus achieved because in the calculation with complex numbers, the phase angle will be exactly allowed for. Such calculations usually lead to lower field strengths than simplified calculations. Computer programs for these calculations are to be found in Ref. 16. 

The amplitudes and the phases of the diffraction data from the protein crystals are used to calculate an electron-density map of the repeating unit of the crystal. This map then has to be interpreted as a polypeptide chain with a particular amino acid sequence. The interpretation of the electron-density map is complicated by several limitations of the data. First of all, the map itself contains errors, mainly due to errors in the phase angles. In addition, the quality of the map depends on the resolution of the diffraction data, which in turn depends on how well-ordered the crystals are. This directly influences the image that can be produced. The resolution is measured in A... 

Flence, for a sinusoidal input, the steady-state system response may be calculated by substituting. v = )lu into the transfer function and using the laws of complex algebra to calculate the modulus and phase angle. 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

For single exponential fluorescence decay, as is expected for a sample containing just one fluorophore, either the phase shift or the demodulation can be used to calculate the fluorescence lifetime t. When the excitation light is modulated at an angular frequency (o = 2itv, the phase angle f, by which the emission modulation is shifted from the excitation modulation, is related to the fluorescence lifetime by ... 

To make the phase angle plot, we simply use the definition of ZGp(joo). As for the polar (Nyquist) plot, we do a frequency parametric calculation of Gp(jco) and ZGp(joo), or we can simply plot the real part versus the imaginary part of Gptjco).1 To check that a computer program is working properly, we only need to use the high and low frequency asymptotes—the same if we had to do the sketch by hand as in the old days. In the limit of low frequencies,... 

We can also use the Bode plot of G to do phase margin calculations. From the textbook definition, we are supposed to find the phase angle ( > = ZG0L where G0L = 1. If the phase margin is 45°, should be -135°. It appears that we need to know Kc beforehand to calculate GOL, but we do not. 

It is evident that the current still leads the voltage but that the "phase angle, a, will vary from close to 90° at low frequencies to close to 0 at high frequencies. Also, at low frequency Z — 1 /tuC and at high frequency Zf — R. In other words, at low frequencies, the circuit behaves like a pure capacitor but at high frequencies it behaves like a pure resistor. Moreover, by fitting the observed current data as a function of frequency to calculated values of Zj and a, an accurate estimate of both R and C can be made. 

This behaviour, depicted in Fig. 5.11 with the values calculated using Equation 5.3, represents the phase-angle variations exhibited by Ru/Ti electrodes containing 20-40 at.% Ru, which show low Rt values. 

The relation between the CCR rates and the pseudorotation phase angle can be calculated using the general formula for /dipole-dipole by converting the values of the projection angles into P and i//ln [47] according to... 

Fig. 16.7 Theoretical curves of the CCR rate as a function of the ribose pseudorotation phase angle P for three different pucker amplitudes (i/m = 30, 35 and 40°. A rotational correlation time of 1.5 ns and a C-H distance of 1.07 A has been used for the calculation.

From the X-ray photographs the dimensions of the unit cell could be calculated but not the phase angle of the crystal, without which unequivocal interpretation of the structure was impossible. Two very different approaches were therefore explored. The first, which became especially applicable if some or all of the primary sequence was known, was to compute the expected diffraction pattern from a postulated three-dimensional structure, and to compare this to the pattern actually obtained from the crystals. 

Now let us look at the Bode plots of some common transfer functions. We have already calculated the magnitudes and phase angles for most of them in the previous section. The job now is to plot them in this new coordinate system. 

Equation (14.23) looks a little complicated but it is easily programmed on a digital computer. Table 14.1 gives a FORTRAN program that reads input and output time functions from a file, calculates the Fourier transformations of the input and of the output, divides the two to get the transfer function G(to)> and prints out log modulus and phase angle at different values of frequency. 

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