Structure amplitude magnitude

The two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

It should be remembered that when a reference atom is moved, all the atoms related to it by symmetry elements move also in a manner determined by the symmetry elements and the problem is to know, for any particular reflection, the direction in which to move the reference atom so that the contribution of the whole group of related atoms either increases or decreases. This problem is best solved by the use of charts which show at a glance the magnitude of the structure amplitude for such a group of atoms for all coordinates of the reference atom. 

Fig. 11.6 shows the noncoplanar-symmetric differential cross sections at 1200 eV for the Is state and the unresolved n=2 states, normalised to theory for the low-momentum Is points. Here the structure amplitude is calculated from the overlap of a converged configuration-interaction representation of helium (McCarthy and Mitroy, 1986) with the observed helium ion state. The distorted-wave impulse approximation describes the Is momentum profile accurately. The summed n=2 profile does not have the shape expected on the basis of the weak-coupling approximation (long-dashed curve). Its shape and magnitude are given quite well by... 

The structure factor F hkl) is the Fourier transform of the unit cell contents sampled at reciprocal lattice points, hkl. The structure factor amplitude (magnitude) F is the ratio of the amplitude of the radiation scattered in a particular direction by the contents of one unit cell to that scattered by a single electron at the origin of the unit cell under the same conditions (see Chapter 3). The first report of the structure factor expression was given by Arnold Sommerfeld at a Solvay Conference. The structure factor F has both a magnitude F(hkl) and a phase rel-... 

Structure factor The structure factor F(hkl ) is the value, at the reciprocal lattice point hkl, of the Fourier transform of the electron density in the unit cell. When appropriately scaled, it is the coefficient, with indices hkl, of the Fourier series that gives the electron density in the crystal. It has both a magnitude (the structure amplitude) and a phase relative to the chosen origin of the unit cell. 

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... 

As a result of symmetry transformation (Eqs. 2.111 and 2.112), both the magnitude of the structure amplitude and its phase may change. Finite symmetry operations (t, tj and tj are all 0) usually affect the phase angle, while infinite operations, i.e. those which have a non-zero translational component, affect both the magnitude and the phase. 

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... 

The idea with the R free is to omit from the least squares calculations, from the observational equations, a subset of the observed structure amplitudes. This set usually constitutes about 10% of the total data and is chosen to be representative of all resolution and magnitude ranges of the reflections. The least squares procedure is then executed as before, but this time only the reflections in the omitted subset are used in the calculation of R free. Two things are true. If the refinement indeed led to an improved model that is closer to the truth,... 

The main sources of error in charge density studies based on high-resolution X-ray diffraction data are of an experimental nature when special care is taken to minimise them, charge density studies can achieve an accuracy better than 1% in the values of the structure factor amplitudes of the simplest structures [1, 2]. The errors for small molecular crystals, although more difficult to assess, are reckoned to be of the same order of magnitude. 

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