By D.S. Sivia
The possibilities for doing scattering experiments at synchrotron and neutron amenities have grown speedily in recent times and are set to proceed to take action into the foreseeable destiny. this article offers a uncomplicated figuring out of the way those recommendations allow the constitution and dynamics of fabrics to be studied on the atomic and molecular point. even if arithmetic can't be kept away from in a theoretical dialogue, the purpose has been to jot down a publication that almost all scientists will nonetheless locate approachable. To this finish, the 1st chapters are dedicated to supplying an educational heritage within the arithmetic and physics which are implicitly assumed in different texts. Thereafter, the philosophy has been one among maintaining issues so simple as attainable.
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Additional resources for Elementary Scattering Theory: For X-ray and Neutron Users
57). A closer examination of Fig. 5 Fourier optics and physical insight 45 Fig. 18 The diffraction pattern from a pair of slits of width w and separation d can be evaluated from a knowledge of the Fourier transforms of a Young’s double slit, of spacing d, and a single slit, of width w, with the convolution theorem. where we have taken z to be the original direction of propagation, from the aperture to the projection screen, and |k| = 2π/λ. With this observation, it seems plausible that the two-dimensional diffraction pattern, I(kx , ky ), from an aperture in the x–y plane, A(x, y), with y coming out of the page in Fig.
50) −∞ where g ⊗ h is read as ‘g convolved with h’, and physically represents a ‘blurring’ of g(x) by h(x). This can be understood from the example of Fig. 9, where g(x) consists of four spikes, or δ-functions, and h(x) is a broad asymmetric function. The convolution is carried out Fig. 9 The convolution of the spiky function g(x) with the broad asymmetric function h(x): f(x) = g(x) ⊗ h(x). 4 Fourier transforms 35 by replacing each of the the sharp peaks in g(x) with scaled copies of h(x) and adding together the four contributions; those from the two closely spaced components in the middle, shown by dotted grey lines, combine to give a resultant function where the constituent doublet is no longer resolved clearly.
The value of the ACF at a distance L away from the origin is calculated by multiplying f(x) with a copy that’s displaced by L relative to it, f(L+x), and integrating the product; its magnitude is a measure of how much structure there is in f(x) separated by a distance of L. This can be understood most easily by considering the ACF of a function that consists of a few sharp peaks, such as that shown in Fig. 11. Basically, two spikes at x1 and x2 in f(x), with amplitudes A1 and A2 , will contribute a symmetric pair of very sharp components at ± (x1 −x2 ), and magnitude A1 A2 , towards the ACF of f(x); they will also add an amount A12 +A22 to the ACF at the origin.