The Structure Factor for a Crystal

If Bragg s law is not satisfied for this family of planes, it doesn t matter what the distribution of electron density in the unit cell is, since no diffraction from the crystal occurs. Scattered waves from any one unit cell will be out of phase with those from all others, producing destructive interference. 

When Bragg s law is satisfied, the resultant wave produced by this unit cell will be duplicated by every unit cell in the crystal, and they will sum together, or constructively interfere, to yield an observable resultant wave. But what will the wave from an individual unit cell be What will be its amplitude and phase  

The wave produced by all of the atoms in a unit cell, when Bragg s law is satisfied, will simply be the sum of the waves scattered by the individual atoms within the cell. To obtain the diffracted wave produced by the entire crystal (which is what we measure as a diffraction intensity), we need only sum the waves scattered by each atom in one unit cell and then multiply by the total number of unit cells, N, in the crystal. What are the amplitudes and relative phases of these individual waves with respect to one another, or to some other reference wave  

Consider further an atom that lies exactly between two planes. If the difference in phase between two successive planes is 2n, then this atom would scatter with a phase oilit/l = n. In fact any atom in the cell lying an arbitrary distance D from the nearest plane would scatter with a phase of (D/dhu)2jt. In other words, the phase angle of the wave scattered by any atom in the unit cell is simply a function of its distance from a plane of that particular family. Can we determine how far any atom is from an hkl plane and, therefore, its phase We can indeed, so long as we know the positions, the x, y, z coordinates, of the atoms in the unit cell. 

The working vector v having unit length and which is normal to the planes can be drawn. It passes not only through this unit cell but identically through all unit cells in the crystal. We use this unitary vector simply to define a direction. The phase of a wave scattered by an atom with respect to the set of planes will be 0 = 2ji(D/dhia), where D is the atom s distance from the nearest plane. But D is the projection of 3ri onto i , that is, l) = x Ft, so that 0 = 2jt(xi v/dhu)- 

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R is a function of incident angle (6), F d is the structure factor for a specific crystal plane, v is unit cell volume of crystal and temperature factor is e 2M, and p is the multiplicity factor of a crystal, which is the number of crystal planes that have the same plane spacing. For a cubic crystal, the p value of 001 is six and p value of 111 is eight, because there six and eight planes in the plane family, respectively. The temperature factor can be extracted from Figure 2.27. 

If the structure factor for an ideally smooth surface, Fldeai, is written by summing laterally over a single surface unit cell and vertically over all z (including the semiinfinite sum over the bulk crystal, the sum over the surface layer and any fluid layer above the surface) as written in Equation (16), the structure factor for a rough surface can be written ... 

Equation (3.71) is similar to equation (3.64) which represents the structure factor for a centrosymmetric crystal. Equation (3.71) can be interpreted as the structure factor for a fictive squared structure. The squared structure is centrosymmetric. It is made up of atoms with a scattering power [/ ],[/ ], at the positions r - r = (x - x> + (y - y )b + (z - z )c. These coordinates represent the interatomic vectors of the real structure. An atom with the scattering power 2 Umif is found at the origin which represents all the zero vectors. If the... 

The first term for <7 = 0 is not interesting (po can be found by other techniques, e.g. by dilatometry). The product term with pi is a result of the convolution theorem and we already have the two Fourier transforms mentioned, namely, the structure factor of unstructured Hquid, that is Lorentzian (5.39a) and the structure factor of a crystal that is delta-functions, Eq. 5.36 ... 

For small values of a, and I, this is indeed equal to the kinematical result because P = 1, sin cti cti and, from the expression for the structure factor along a crystal truncation rod,... 

Table 5.1 lists some values of the correlation factor for a variety of diffusion mechanisms in some common crystal structure types. 

The summation is taken over all unit cells in the crystal. The overall scattering amphtude J is given by A=F J (where F is the structure factor for the hkl reflection) and the intensity 1 by the square of the amplitude... 

Normally, if the assumed model for a crystal structure has an R value of 0.5 and resists attempts to refine to a lower residual, then the model structure is rejected as false, and a new model is tried until a fit between the observed and calculated structure factors yields an acceptable residual (R < 0.25). (Other models were tried for this complex, but they either gave Fourier maps which were uninterpretable or they converged to the present model). However, the normal crystal structure is solved with data obtained from crystals which have dimensions of the order of 0.1 mm. In the crystals available for this experiment, two of the dimensions were of the order of 0.01 mm. Thus, long exposures were required to give a small number of relatively weak diffraction spots. (Each Weissenberg photograph was exposed for five days with Cuka radiation 50 kv., 20 ma. loading, in a helium atmosphere). 

Suppose we are able to locate a heavy atom in the unit cell of derivative crystals. Recall that Eq. (5.15) gives us the means to calculate the structure factors Fhkl for a known structure. This calculation gives us not just the amplitudes but the complete structure factors, including each of their phases. So we can compute the amplitudes and phases of our simple structure, the heavy atom in the protein unit cell. Now consider a single reflection hkl as it appears in the native and derivative data. Let the structure factor of the native reflection be Fp. Let the structure factor of the corresponding derivative reflection be FHp. Finally, let FH be the structure factor for the heavy atom itself, which we can compute if we can locate the heavy atom. 

In all properties studied with pseudopotenlial theory, the first step is the evaluation of the structure factors. For simplicity, let us consider a metallic crystal with a single ion per primitive cell -either a body-centered or face-centered cubic structure. We must specify the ion positions in the presence of a lattice vibration, as we did in Section 9-D for covalent solids. There, however, we were able to work with the linear force equations and could give displacements in complex form. Here the energy must be computed, and that requires terms quadratic in the displacements. It is easier to keep everything straight if we specify displacements as real. Fora lattice vibration of wave number k, we write the displacement of the ion with equilibrium position r, as... 

The first term is exactly the structure factor for tlie undistorted crystal, giving a value of 1 for the lattice wave numbers and 0 for other wave numbers. Wc may immediately see that the structure factor is corrected at tlie lattice wave number. Notice that the second term vanishes for lattice wave numbers because c " is the same at every site, though (5r, oscillates from one site to another, averaging to zero. The third term, on the other liand, does not vanish c is again the same at every site, but (q f)r,) can be seen to average to 2(q u)(q u ) by squaring Hq. (17-13) and summing over i. Let us then indicate the lattice wave numbers of the perfect crystal by q wc have found that, to second order in u. 

Fig. 17 (a) Comparison of the Patterson maps of a 5 ML NiO(lll) film on Au(lll) and of the signal crystal (SC), (b) Comparison between the in-plane experimental and calculated structure factors for a model with Ni- ad 0-terminated octopoles separated by a single step (right) experimental and (left) calculated. 

Because the diffraction experiment involves the average of a very large number of unit cells (of the order of 10 in a crystal used for X-ray diffraction analysis), minor static displacements of atoms closely simulate the effects of vibrations on the scattering power of the average atom. In addition, if an atom moves from one disordered position to another, it will be frozen in time during the X-ray diffraction experiment. This means that atomic motion and spatial disorder are difficult to separate from each other by simple experimental measurements of intensity falloff as a function of sm6/X. For this reason, atomic displacement parameter is considered a more suitable term than the terms that have been used historically, such as temperature factor, thermal parameter, or vibration parameter for each of the correction factors included in the structure factor equation. A displacement parameter may be isotropic (with equal displacements in all directions) or anisotropic (with different values in different directions in the crystal). 

In this equation f is the atomic scattering factor or atomic form factor, h,k,l are the Miller indices of the reflecting plane, and x ,y ,z are the coordinates of the scattering atom in decimal fractions of unit cell parameters, a,b,c, respectively. For simple structures the structure factor indicates what types of Bragg planes in a given kind of structure can produce a diffraction peak, i.e., have non-cancelled, coherent scattering, and also indicates the relative intensity of the allowed peaks (Warren 1969). A few structure factors for simple crystal structures are shown below. 

The function F is called the structure factor for the hkl family of planes in the crystal. Because it is a wave, or vector, it has both a magnitude and a phase. The set of all F for all values of h comprise the set of diffracted rays resulting from all possible families of planes in the crystal and thereby constitutes the diffraction spectrum of the crystal. The magnitude of Fis readily measured as the square root of the observed diffracted intensity, that is, /lh = Fh, but there is no experimental means presently available to directly measure its phase 4 ... 

The summation of exponential terms on the right is a Dirac delta function, a discrete function, which is everywhere zero except when the argument is zero or integral. The summation on the left is a continuous function, which determines the value of the entire transform at those nonzero points. Now d ki is normal to the set of planes of a particular family, and d ki I is the interplanar spacing. In order for dhu s = 1, s must be parallel with dhki and have magnitude 1/ Smreciprocal lattice vector. If s h, then there is destructive interference of the waves diffracted by different unit cells, and the resultant wave from the crystal is zero. The elements of the diffraction spectra, the structure factors, for the crystal can therefore be written as... 

The most important consequence the symmetry elements present in a crystal is that some (hkl) planes have F(hkl) = zero, and so will never give rise to a diffracted beam, irrespective of the atoms present. Such missing diffracted beams are called systematic absences. This can most easily be understood in terms of the vector representation described above. Suppose that a crystal is derived from a body centred lattice. In the simplest case, the motif is one atom per lattice point, and the unit cell contains atoms at 000 and Zz V2 Zz, (Figure 6.13a). [Note the cell can have any symmetry]. The structure factor for each (hkl) set is given by ... 

If a structural reaction is rapidly and uniformly initiated in a crystal, what will the likely effects be on the Laue x-ray intensities The very weak intermolecular interactions characteristic of protein crystals make it implausible that existence of a certain tertiary structure in one molecule would favor a particular tertiary structure in its neighbors. That is, all molecules in the crystal are likely to behave independently of each other, as they do in solution. Their populations will evolve smoothly in time in a manner governed by the reactant concentrations and the rate constants. To preserve the x-ray diffraction pattern, spatial coherence must be maintained between molecules, but temporal coherence need not be. Thus, at any instant the crystal will contain many different conformations, each representing a different structural intermediate. The total structure factor for a particular reflection will be a weighted vector sum of the time-independent structure factors of each conformation, where the weights are the time-dependent fractional occupancies. In favorable cases [26), it may be possible to extract the individual time-independent structure factors from this sum, and hence to obtain directly the structure of each intermediate. 

We re-derive the CTR structure factor for a 3D semi-infinite crystal to show its rodlike character explicitly. We assume an orthogonal bulk crystal structure with lattice parameters a, b, c) where the surface is defined by the a-b plane (Fig. 6A). The lattice vectors are oriented so that Q = [(A, Qy, Qz] = [(2 / ) , (2 / ) , (2 / ) ], where H, , and L are the surface Miller indices. If the structure of the crystal is assumed to be identical for every unit cell in the crystal (including all surface layers), then the sum in Equation (14) can be rewritten so that the structure factor of the whole crystal is expressed as the unit cell structure factor, Fuc, multiplied by the phase factors that translate the unit cell structure factor to each and every unit cell in the crystal, both laterally within the surface plane, and vertically into the crystal. In this case,... 

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