Structure amplitude complex

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... 

When the general arrangement is known it is then necessary to determine precise atomic coordinates. Sometimes the positions of certain atoms are invariant—they are fixed by symmetry considerations—but in complex crystals most of the atoms are in general5 positions not restricted in any way by symmetry. The variable parameters must be determined by successive approximations here the work of calculating structure amplitudes for postulated atomic positions can be much shortened by the use of graphical methods, to be described later in this chapter. It cannot be denied, however, that the complete determination of a complex structure is a task not to be undertaken lightly the time taken must usually be reckoned in months. 

Many binary salts, oxides, and sulphides are a little more complex, two atoms being associated with each lattice point it is necessary to discover the relative positions of the two atoms. This can be done by mere inspection of the set of structure amplitudes, and confirmed by a very moderate amount of calculation. Two examples will be given— calcium oxide and cuprous chloride. 

When the unit cell contains only one atom, the resulting diffracted intensity is only a function of the scattering ability of this atom (see section 2.5.2). However, when the unit cell contains many atoms and they have different scattering ability, the amplitude of the scattered wave is given by a complex function, which is called the structure amplitude ... 

F(h) is the structure amplitude of a Bragg reflection with indices hkl, which are represented as vector h in three dimensions. The structure amplitude itself is often shown in a vector form since it is a complex number ... 

The sums of cosines and sines in Eq. 2.103 signify the real (A) and imaginary (B) components of a complex number, respectively, which the structure amplitude indeed is. Hence, considering the notations introduced in Eq. 2.87, Eq. 2.103 can be rewritten as ... 

Figure 2.56. The structure amplitude, F(h), shown as a vector representing a complex number with its real, A(h), and imaginary, B(h), components as projections on the real and imaginary axes, respectively, in the non-centrosymmetric (left) and centrosymmetric (right) structures. The imaginary component on the left is shifted from the origin of coordinates for clarity.

When the anomalous scattering is present, the structure amplitude even for a centrosymmetric crystal becomes a complex number. This is shown in Eqs. 2.107 and 2.108. The first (general expression) is easily derived by combining Eqs. 2,101 and 2.103 and rearranging it to group both the real and imaginary components. 

Despite the apparent simplicity with which a crystal structure can be restored by applying Fourier transformation to diffraction data (Eqs. 2.132 to 2.135), the fact that the structure amplitude is a complex quantity creates the so-called phase problem. In the simplest case (Eq. 2.133), both the absolute values of the structure amplitudes and their phases (Eq. 2.105) are needed to locate atoms in the unit cell. The former are relatively easily determined from powder (Eq. 2.65) or single crystal diffraction data but the latter are lost during the experiment. 

As suggested by Patterson in 1934, the complex coefficients in the forward Fourier transformation (Eqs. 2.129 and 2.132) may be substituted by the squares of structure amplitudes, which are real, and therefore, no information about phase angles is required to calculate the distribution of the following density function in the unit cell ... 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

In astronomy, we are interested in the optical effects of the turbulence. A wave with complex amplitude U(x) = exp[ irefractive index, resulting in a random phase structure by the time it reaches the telescope pupil. If the turbulence is weak enough, the effect of the aberrations can be approximated by summing their phase along a path (the weak phase screen approximation), then the covariance of the complex amplitude at the telescope can be shown to be... 

---