Diffraction catastrophes

The next field of applications of elementary catastrophe theory are optical and quantum diffraction phenomena. In the description of short wave phenomena, such as propagation of electromagnetic waves, water waves, collisions of atoms and molecules or molecular photodissociation, a number of physical quantities occurring in a theoretical formulation of the phenomenon may be represented, using the principle of superposition, by the integral 

The meaning of variables x = (x1.xn), c = (ct,ck) will be exemplified by an electromagnetic wave propagating in a medium from one two-dimensional surface to another (this is the case n = 2, k = 2). The vector x = (xt, x2) describes the position of a source of signal on the first surface and the vector c = (ct, c2) is the position of a point of observation on the second surface. Then, an intensity of the signal at the point (clt c2) is equal to the square of the absolute value of the function (3.24) for n = 2, k = 2, a(x c) being the amplitude of a signal transmitted at the point (xl5 x2) on the first surface and observed at the point (c C ,) on the second surface, while the function /(x c) is related to optical properties of the medium (to the refractive index). 

A function is known to vary most slowly in the vicinity of a minimum or 

The relation with elementary catastrophe theory is now apparent the function /(x c) will play the role of a potential function, whereas critical points of this function determine, as will be shown in Section 3.4.2, the diffraction pattern (just as in the case of a classical potential function its critical points determine the stationary states of a system). 

In many cases the integral (3.24) may be expressed in terms of oscillatory integrals of a simpler type 

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Schrodinger equation and the corresponding diffraction catastrophes occurring at collisions of molecules are important from the viewpoint of the description of chemical reactions taking place upon contact (collision) of molecules. As shown in Section 1.3, recurrent equations appear from a description of the kinetics of chemical reactions. 

In the end, we will describe very briefly the application of catastrophe theory to a description of chemical reactions in terms of individual molecules participating in the reaction. The approach will be based on some methods of quantum chemistry employing solutions to the Schrodinger equation. The description of chemical reactions involving a simultaneous application of the microscopic description, the Schrodinger equation, and of elementary catastrophe theory is notionally similar to the description of diffraction catastrophes for the Schrodinger equation, see Section 3.4.3. 

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

In a later passage of the chapter we shall consider the short-wave limit for the Helmholtz equation, describing optical phenomena, and for the Schrodinger equation, describing quantum phenomena. The relation between the obtained oscillation integrals and the elementary catastrophe functions will be revealed and straightforward examples of application of the canonical diffraction integrals (3.25) discussed. 

For example, the diffraction integral (3.25) containing the potential function of a cusp catastrophe (zf3), F(x c) = x4 + ax2 + bx, describes ligth scattering on a two-dimensional diffraction grating, see Fig. 48. A function defined by such an integral is called the Pearcey function. 

Fig. 48. Diffraction cusp catastrophe M3) (a) diffraction grating (b) oscillation integral Ai.

Fig. 49. Diffraction elliptic umbilic catastrophe (D4 ) diffraction integral D4 in the b, c, a = 0 plane.

Many of the considered problems, such as the problem of stability of soap films, the liquid-vapour phase transition, the diffraction phenomena, descriptions of the heartbeat or the nerve impulse transmission, catastrophes described by non-linear recurrent equations have a close relation to chemical problems. 

The stability of thin films and the catastrophes of film systems may play a crucial role in the case of chemical reactions proceeding at the boundary of a liquid phase and another phase. Phase transitions are of a great significance in physical chemistry. The diffraction phenomena for the... 

In Chapters 5, 6 we will deal with the relation of catastrophes of a diffraction type for the Schrodinger equation (see Section 3.4) and catastrophes occurring in non-linear sequential systems (see Section 3.6) to the catastrophes taking place in chemical systems. 

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