Friedel pair

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. 

Also the amplitudes from HREM images are modulated by the CTF. It is difficult to compensate for these effects completely, especially near the cross-overs of the CTF. Furthermore, for those projections with p2 symmetry, crystal tilt can not be compensated by imposing the symmetry, since reflections have only their Friedel pairs as symmetry related reflections. For these reasons, we used amplitudes from the ED patterns (Fig. 4) instead of the amplitudes from the HREM images. The amplitudes were calculated as the square root of the intensities extracted from the corresponding ED patterns of an exposure series. 

Expressions for Sq(H) for / < 2 and a subvolume of parallelepipedal shape are given in Table 7.1. Though the shape factor for the dipole moment is imaginary, combination of the Friedel pairs F(H) and F(H) in the summation... 

However, FriedeFs law no longer holds if a compound containing an anomalous scatterer, besides other elements, crystallizes in a non-centrosymmetric space group then the following inequalities between Friedel pairs apply ... 

The determination of absolute configuration (or, more generally, absolute structure see Section 4.2.2.1.1.) is usually based on the small intensity differences of Friedel pairs, e.g., Im vs. 

Based on the inequalities in equation 10 (see Section 4.2.2.1.1.), a comparison is made between the measured intensities of Friedel pairs (Ihkl and and the calculated values of their... 

The Friedel pairs are a subset of the so-called Bijvoet pairs, i.e., pairs of any two reflections which are equivalent by Laue symmetry but not by the point group symmetry of the crystal (see Table 2 a full listing of these reflections is available4 1). All Bijvoet pairs may be used for a comparison, but since the measured intensity differences between general Bijvoet pairs are more likely to be affected by systematic error, the conclusion based on their comparison may be less reliable than in the case of Friedel pairs. 

In Chapter 4, Section HI.G, I mentioned Friedel s law, that lhkl = h k i-It will be helpful for later discussions to look at the vector representations of pairs of structure factors Fhkl and F h k l, which are called Friedel pairs. Even though hkl and l h k l are equal, Fhkl and F h k l are not. The structure factors of Friedel pairs have opposite phases, as shown in Fig. 6.3. 

Figure 6.3 Structure factors of a Friedel pair. F k ( is the reflection of Fhkl in the real axis.

This means that F h k l is the mirror image of Fhkl with the real axis serving as the mirror. Another way to put it is that Friedel pairs are reflections of each other in the real axis. 

So the disparity between intensities of Friedel pairs in the anomalous scattering data set establishes their phases in the nonanomalous scattering data set. The reflection whose phase has been established here corresponds to the vector Fhp in Eq. (6.9). Thus the amplitudes and phases of two of the three vectors in the Eq. (6.9) are known (l)FHp is known from the anomalous scattering computation just shown, and (2) FH is known from calculating the heavy-atom structure factors after locating the heavy atom by Patterson methods. The vector Fp, then, is simply the vector difference establish-... 

Like phases from the MIR method, each anomalous scattering phase can only serve as an initial estimate and must be weighted with some measure of phase probability. The intensity differences between Friedel pairs are very small, so measured intensities must be very accurate if any usable phase information is to be derived. To improve accuracy, the crystallographer collects intensities of... 

Friedel partners under very similar conditions, and always from the same crystal. Diffractometry is ideal for anomalous scattering because of its inherently great accuracy in measuring intensities, and because the diffractometer can be programmed to collect Friedel pairs in succession, thus assuring that the crystal is in the same condition during collection of the two reflections. But diffractometry is slow compared to collecting data with area detectors. 

The structure of atisinium chloride as 5 was confirmed recently by a single-crystal X-ray analysis (SO). The absolute configuration of atisinium chloride was determined as 45, 5S, 8/ , 10/ , 12/ , and 155 by Hamilton s method and confirmed by examination of sensitive Friedel pairs. A recent X-ray crystallographic study of isoatisine confirmed the assigned structure 75. The absolute configuration was established as 45, 55, 8/ , 10/ , 12/ , 155, and 195 for isoatisine. It is worth noting that isoatisine does not exist as a mixture of C-19 epimers. Early work on the chemistry of atisine and isoatisine... 

Reciprocal lattices and therefore, diffraction patterns are generally centrosymmetric regardless of whether the corresponding direct lattices are centrosymmetric or not Thus, pairs of reflections with the opposite signs of indices, hkl) and hkl) -ihe so-called Friedel pairs - usually have equal intensity. Yet, they may be different in the presence of atoms that scatter anomalously (see section 2.11.4) and this phenomenon should be taken into account when multiplicity factors are evaluated comprehensively. Relevant details associated with the effects of anomalous scattering on the multiplicity factor will be considered below in section 2.12.2. 

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).

This is known as Friedel s law and the pairs of related reflections Fh and F b-h-i are called Friedel pairs. 

The pair of Bragg reflections Ti, and F h arise from the same stack of planes but they are the scattering from the opposite sides these are a Friedel pair of reflections. They always exactly overlap in a powder pattern since h = h, but they may not have the same value of F if the structure is noncentrosym-metric and there is significant resonant scattering from some of the atoms. Their average intensity is given by ... 

In the absence AD effects, the intensity of I i,i is the same as the intensity of its inverse, I, . However, with AD effects, and I j,i will be unequal. and are referred to as Friedel pairs. For many crystallographic applications, it is assumed that Friedel pairs are, at least approximately, equal. However, accurate measurement of Friedel pair differences can be used to extract starting phases if the AD effect is large enough. 

Figure 2.6 illustrates how AD from a single heavy atom can cause the intensities of a Friedel pair to be different. In the left-hand diagram (a), the vectors F + or F represent the total scattering by normal atoms without AD effects. The vectors F, or F/j represent the sum of the normal and real AD scattering values... 

Morals (1) When there is the possibility that the space group can be acentric and, especially, if it can be polar (i.e., origin of space group not fixed by symmetry elements), collect intensities for hkl as well as hkl (Friedel pairs). (2) Believe your chemical intuition. 

Such pairs of reflections are called Friedel pairs. Because of this, the intensities can be expressed as ... 

The intensities of Friedel pairs will be equal. This will cause the diffraction pattern from a crystal to appear centrosymmetric even for crystals that lack a centre of symmetry. Diffraction is thus a centrosymmetric physical property, which means that the point symmetry of any diffraction pattern will belong to one of the 11 Laue classes, (see Section 4.7). 

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