Fourier reciprocal relationship

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. 

This reciprocal relationship is a general property of Fourier transforms. The -function of the previous paragraph and its transform demonstrate the same reciprocity. To characterise this property more precisely a bracket function, called a scalar or inner (dot) product, is used to define an overlap integral... 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

FIGURE 1.7 In (a) the object, again exposed to a parallel beam of light, is not a continuous object or an arbitrary set of points in space, but is a two-dimensional periodic array of points. That is, the relative x, y positions of the points are not arbitrary they bear the same fixed, repetitive relationship to all others. One need only define a starting point and two translation vectors along the horizontal and vertical directions to generate the entire array. We call such an array a lattice. The periodicity of the points in the lattice is its crucial property, and as a consequence of the periodicity, its transform, or diffraction pattern in (b) is also a periodic array of discrete points (i.e., a lattice). Notice, however, that the spacings between the spots, or intensities, in the diffraction pattern are different than in the object. We will see that there is a reciprocal relationship between distances in object space (which we also call real space), and in diffraction space (which we also call Fourier space, or sometimes, reciprocal space). 

This theorem shows that if a function/(x) is scaled by shrinking its width by a factor a, its Fourier transform is expanded in width by the same factor (while the height is altered by a factor 1/ a ). It clearly illustrates the reciprocity relationship between fix) and F(s). 

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

The majority of samples used in NMR are cylindrical. The echo decay of a cylindrical sample in the presence of a uniform field gradient perpendicular to the cylinder axis represents a Fourier transform of a distribution of chords of a circle which, in turn, is a semi-ellipse. Thus, the echo can be written in terms of a Bessel function as J (t)/t, where t is the time coordinate, because this represents a Fourier transform of a semicircle which differs from a semi-ellipse only by a multiplicative constant. (See, for example, Chapter 19 of Bracewell, Appendix A.) Because of the way in which the two domains are related (as discussed in sections I.D.l. and I.D.2.), there is a reciprocal relationship between the spread of the Larmor frequencies across the cylindrical sample and the extent of the signal with the shape J (t)/t in the time domain. Thus the information on the field gradient experienced by the sample is contained in the echo if the sample dimension is known. 

It is noteworthy that the units of functions related by a Fourier transform are reciprocals of one another. For instance, consider the reciprocal relationship between the period and frequency of a wave. Whereas the former is the time required for a complete wave to pass a fixed point (units of sec), the latter is the number of waves... 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

While for some purposes it may be necessary to have accurate frequency definition, for others good time discrimination is useful. These are opposite requirements. Because of the Fourier relationship between frequency and time, the more precisely the time of a signal is known, the greater bandwidth of frequencies is necessary (there is a close analogy here with Heisenberg s uncertainty principle). Approximately, the time resolution t is the reciprocal of the bandwidth Bw, so that their product Bwr 1. 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

In crystallography, direct and reciprocal spaces are related to one another as forward and reverse Fourier transformations. In three dimensions these relationships ean be represented by the following Fourier integrals ... 

What is the relationship between the lattice of the crystal and the lattice on which the diffraction intensities fall, the lattice of the transform The relationship is that between the real space lattice of the crystal and the reciprocal lattice. The point where a wave diffracted by a particular family of planes hkl appears in the diffraction pattern is related to the origin of the diffraction pattern by the reciprocal lattice vector h = hkl. The direction of the reciprocal lattice vector h is normal to the family of planes, and its length is 1 /reciprocal lattice vector defines a permissible point in diffraction space where a diffraction wave may be observed. That wave, having both amplitude and phase, is the Fourier transform of that particular family of planes hkl. 

An important property of Fourier transforms that we did not emphasize in the previous chapter is that spatial relationships in one space are maintained in the corresponding transform space. That is, specific relationships between the orientations in real space of the members of a set of objects are carried across into reciprocal space. This is particularly important in terms of crystallographic symmetry, and we will encounter it again when we consider the process known as molecular replacement (see Chapter 8). 

What has been said here is true but obscures another fundamental property of the Fourier transform, one that complicates matters a bit but not hopelessly so. The Fourier transform fails to directly carry translational relationships from one space to another, in particular, from real space into reciprocal space. This means that the transform does not discriminate between asymmetric units based on the distances between them. The immediate relevance of this is that a set of asymmetric units related by a screw axis symmetry operator (which has translational components) in real space is transformed into diffraction space as though it simply contained a pure rotation axis. The translational components are lost. If our crystal has a 6i axis, we will see sixfold symmetry in the diffraction pattern. If we have 2i2j2i symmetry in real space, the diffraction pattern will exhibit 222 (or more properly, mmm) symmetry. 

The Fourier transform as a mathematical tool has been mentioned in Chapter 3 when discussing the relationship between real space and reciprocal space. Here it is emphasized as a tool which transfers information between a function in the time (t) domain and its corresponding one in the frequency ( >) domain. 

Equations (17) and (18) further show a relationship between real and reciprocal space. The function F(hkl) is the Fourier transform of the unit cell contents, expressed in the reciprocal space coordinates h, k. and /. Because the. symmetry operation of translation holds for all three spatial directions in crystals, the Fourier transform of the entire crystal is zero, except at reciprocal lattice points. 

---