By J.M. Cowley

The 1st variation of this hugely profitable ebook seemed in 1975 and advanced from lecture notes for sessions in actual optics, diffraction physics and electron microscopy given to complex undergraduate and graduate scholars. The e-book offers with electron diffraction and diffraction from disordered or imperfect crystals and hired an strategy utilizing the Fourier remodel from the start rather than as an extension of a Fourier sequence treatment.This 3rd revised version is a significantly rewritten and up to date model which now comprises all vital advancements that have taken position in recent times.

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**Example text**

The wave equation (5) may be written [V 2 + 4~ 2k~ + ~q) (t) ] ~ = O, (16) where k 0 represents the wave number for the incident wave in free space and ~ is a parameter which specifies the strength of the interaction with the potential field. As an alternative, most useful for scattering theory, we may write the equivalent integral equation by making use of the Green's function G(r, /). For scattering radiation from a potential field, G(r, /) represents the amplitude at a point of observation r, due to a point of unit scattering strength at / in the field, (see Fig.

It is necessary, therefore, to use c o n s i d e r a b l e caution in m a k i n g use of F o u r i e r t r a n s f o r m e x p r e s s i o n s d e r i v e d from various sources. For more than one d i m e n s i o n we m a y use the v e c t o r form of (12) F(u) = [ f(r) exp{ 2Ei,-r}dr. d (14) The vector u may be c o n s i d e r e d as a vector in " F o u r i e r t r a n s f o r m space". For the t h r e e - d i m e n s i o n a l case, for example, the vector r m a y be c o n s i d e r e d to have c o o r d i n a t e s x,y,z and u m a y be c o n s i d e r e d to have c o o r d i n a t e s u,v,w.

1975]. A number of general relationships may be written for any function f(x), real or complex, thus; Real space Fourier transform space f(x) F(u) (21) f (-x) F (-u) (22) F* (-u) (23) 1F (u/a) (24) (x) f(ax) a f(x) + g(x) F(u) + G(u) f(x- exp a) {2~iau} F(u) (25) (26) df(x)/dx (-2~iu) F (u) (27) dnf(x)/ci~ ( - 2 ~ i u ) n F (u) . (28) These relationships may be readily proved by writing out the relevant integrals. For (24)" f f(ax) exp{2Kiux}dx 1 f f(X) exp { 2~iuX/a }dX a = 1F(u/a) a . 2 For FOURIER TRANSFORMS" GENERAL 33 (26)- S f(x - a) exp { 2Kiux} dx --oo = ~ f(x) = F(u) For (27) : d exp{2Kiua}.