Fourier transforms

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 410-411 | 1 | 2 |

Section 4.2.3.1.1. Fourier transforms

H. Jagodzinski^a and F. Frey^b

^aInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany, and ^bInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany

4.2.3.1.1. Fourier transforms

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(1) Normalized Gaussian function

$[(\pi^{3/2} \alpha \beta \gamma)^{-1} \exp \{-(x / \alpha)^{2} - (y / \beta)^{2} - (z / \gamma)^{2}\}. \eqno(4.2.3.12)]$ This plays an important role in statistics. Its Fourier transform is again a Gaussian: $[\exp \{-\pi^{2} (\alpha^{2} H^{2} + \beta^{2} K^{2} + \gamma^{2} L^{2})\}. \eqno(4.2.3.12a)]$ The three parameters α, β, γ determine the width of the curve. Small values of α, β, γ represent a broad maximum in reciprocal space but a narrow one in real space, and vice versa. The constant has been chosen such that the integral of the Gaussian is unity in real space. The product of two Gaussians in reciprocal space $[\eqalignno{&\exp \{- \pi^{2} (\alpha_{1}^{2} H^{2} + \beta_{1}^{2} K^{2} + \gamma_{1}^{2} L^{2})\} &\cr &\quad \times \exp \{- \pi^{2} (\alpha_{2}^{2} H^{2} + \beta_{2}^{2} K^{2} + \gamma_{2}^{2} L^{2})\} &\cr &\quad \quad = \exp \{- \pi^{2} [(\alpha_{1}^{2} + \alpha_{2}^{2})H^{2} + (\beta_{1}^{2} + \beta_{2}^{2}) K^{2} &\cr &\qquad \quad + (\gamma_{1}^{2} + \gamma_{2}^{2}) L^{2}]\} &(4.2.3.12b)}]$ again represents a Gaussian of the same type, but with a sharper profile. Consequently its Fourier transform given by the convolution of the transforms of the two Gaussians is itself a Gaussian with a broader maximum. It may be concluded from this discussion that the Gaussian with $[\alpha, \beta, \gamma \rightarrow 0]$ is a δ function in real space, and its Fourier transform is unity in reciprocal space. The convolution of two δ functions is again a δ function.
(2) Lattices

Lattices in real and reciprocal space may be described by δ functions $[l({\bf r}) = \textstyle\sum\limits_{{\bf n}} \delta ({\bf r - n})]$ and $[L({\bf H}) = \textstyle\sum\limits_{{\bf h}} \delta ({\bf H - h}),]$ where n, h represent the components of the displacement vectors in real and reciprocal space, respectively. The Fourier transforms of lattices with orthogonal basis vectors of unit length and an infinite number of points in all three dimensions correspond to each other. In the following the relation $[l({\bf r}) \leftrightarrow L({\bf H})]$ is used in this generalized sense.

The Fourier transforms of finite lattices are given by $[{\sin \pi N_{1} H \over \sin \pi H} {\sin \pi N_{2} K \over \sin \pi K} {\sin \pi N_{3} L \over \sin \pi L}, \eqno(4.2.3.13)]$ which is a periodic function in reciprocal space, but, strictly speaking, non-periodic in real space. It should be pointed out that the correspondence of lattices in either space is valid only if the origin coincides with a δ function. This fact may easily be understood by applying the law of displacement given in equation (4.2.3.7).
(3) Box functions

The Fourier transform of a box function b(r) with unit height is: $[b({\bf r}) \leftrightarrow {\sin \pi \alpha H \over \pi H} {\sin \pi \beta K \over \pi K} {\sin \pi \gamma L \over \pi L}. \eqno(4.2.3.14)]$ α, β, γ describe its extension in the three dimensions. This function is real as long as the centre of symmetry is placed at the origin, otherwise the law of displacement has to be used. The convolution of the box function is needed for the calculation of intensities: $[\eqalignno{t({\bf r}) &= b({\bf r}) * b(-{\bf r}) &\cr &\leftrightarrow \left({\sin \pi \alpha H \over \pi H}\right)^{2} \left({\sin \pi \beta K \over \pi K}\right)^{2} \left({\sin \pi \gamma L \over \pi L}\right)^{2}. &(4.2.3.15)}]$ t(r) is a generalized three-dimensional `pyramid' of doubled basal length when compared with the corresponding length of the box function. The top of the pyramid has a height given by the number of unit cells covered by the box function. Obviously, the box function generates a particle size in real space by multiplying the infinite lattice l(r) by b(r). Fourier transformation yields a particle-size effect well known in diffraction. Correspondingly, the termination effect of a Fourier synthesis is caused by multiplication by a box function in reciprocal space, which causes a broadening of maxima in real space.
(4) Convolutions

It is often very useful to elucidate the convolution given in equations (4.2.3.4) by introducing the corresponding pictures in real or reciprocal space. Since $[1\cdot f({\bf r}) = f({\bf r}), \delta ({\bf H}) * F({\bf H}) = F({\bf H})]$ the convolution with a δ function must result in an identical picture of the second function, although the function is used as $[f(-{\bf r})]$ in the integrals of equations (4.2.3.4), $[f({\bf r} - {\bf r}')]$ with r′ as variable in the integral of convolution. The convolution with $[f(-{\bf r})]$ brings the integral into the form $[\textstyle\int f({\bf r}')\;f({\bf r}' - {\bf r}) \;\hbox{d}{\bf r}', \eqno(4.2.3.16)]$ which is known as the Patterson function (or self- or auto-convolution) and represents the generalized Patterson function including anomalous scattering [cf. equation (4.2.3.10)].

The change of the variable in the convolution integral may sometimes lead to confusion if certain operations are applied to the arguments of the functions entering the integral. Hence, it seems to be useful to mention the invariance of the convolution integral with respect to a change of sign, or a displacement, respectively, if applied to r′ in both functions. Consequently, the convolution with the inverted function $[a({\bf r}) * b(-{\bf r})]$ may be determined as follows: $[\eqalignno{b'({\bf r}) &= b(-{\bf r}) &\cr a({\bf r}) * b(-{\bf r}) &= a({\bf r}) * b'({\bf r}) = \textstyle\int a({\bf r}') b' ({\bf r} -{\bf r}') \;\hbox{d}{\bf r}' &\cr &= \textstyle\int a({\bf r}') b({\bf r}' -{\bf r}) \;\hbox{d}{\bf r}' = P'({\bf r}). &(4.2.3.17)}]$ This equation means that the second function is displaced into the positive direction by r, then multiplied by the first function and integrated. In the original meaning of the convolution the operation represents a displacement of the second function into the positive direction and an inversion at the displaced origin before multiplication and subsequent integration. On comparing both operations it may be concluded that $[P'({\bf r}) \neq P'(-{\bf r})]$ if the second function is acentric. For real functions both have to be acentric. In a similar way it may be shown that the convolution of $[\eqalignno{a({\bf r - m}) * b({\bf r} - {\bf m}') &= \textstyle\int\limits_{{\bf r}'} a({\bf r}' - {\bf m}) b({\bf r} - {\bf m}' - {\bf r}') \;\hbox{d}{\bf r}' &\cr &= \textstyle\int\limits_{{\bf r}''} a({\bf r}'') b({\bf r} - {\bf m}' - {\bf m} - {\bf r}'') \;\hbox{d}{\bf r}''. &\cr&&(4.2.3.18)}]$ Equation (4.2.3.18) indicates a displacement by $[{\bf m}' + {\bf m}]$ with respect to the convolution of the undisplaced functions. Consequently $[\delta ({\bf r - m}) * \delta ({\bf r} - {\bf m}') = \delta ({\bf r} - {\bf m} - {\bf m}'). \eqno(4.2.3.19)]$ Obviously, the commutative law of convolution is obeyed; on the other hand, the convolution with the inverted function yields $[\delta ({\bf r - m' + m}),]$ indicating that the commutative law (interchange of m and m′) is violated because of the different signs of m and m′.

The effectiveness of the method outlined above may be greatly improved by introducing further Fourier transforms of useful functions in real and reciprocal space (Patterson, 1959).

References

Patterson, A. L. (1959). Fourier theory. In International tables for X-ray crystallography, Vol. II, ch. 2.5. Birmingham: Kynoch Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 410-411