 By C. J. Ball and G. E. Bacon (Auth.)

ISBN-10: 0080157866

ISBN-13: 9780080157863

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Sample text

5 will therefore contribute a complex amplitude dA at Ρ such that ia dA = ds e where eis = ^7 ^ n2ny dy Two-dimensional Diffraction 45 and α = 2nk(r'l + r2 — r1—r2). The origin of phase is chosen such that α = 0 for the scattered wave from O. The radius of curvature of the corresponding part of the amplitudephase diagram is D _ ds _ a0af(6)ny da r\r'2k{dr\ dy + dr2) ' S Ρ F I G . 5. Geometry of scattering by a plane o f a t o m s at normal incidence. Now 2 V — r' 2 2 —r = 2r[ dr[ = lydy R ηα0α/(Θ) f k(r[+r 2) 2 — r' — r 2 2r2dr2, 46 Theory of Diffraction The amplitude-phase diagram therefore approximates to a circle, as before.

Yi 0 = ^ m [ J sin (2ny(m/d + A: sin 0)) rfy 4- J sin (2ny(m/d — k sin 0)) dy] A + ¥ m [ J cos (2ny(m/d+k — J cos (2ny(m/d—k sin 0)) d> s'm 0)) rfy]. Consider now one of these integrals, J cos (2ny(m/d—k sin 0)) cfy. If the integral is taken over a distance that is an exact multiple of _ 1 (mid— k sin 0 ) , then the mean value of the integrand, and hence the value of the integral, will be zero. This will not be possible, of course, if k sin 0 = m/d, when the integrand is not periodic but has the value unity.

OS, ON and OA m are coplanar. 4b. Reflection In addition to a scattered wave in the direction of the incident wave there will be a reflected wave. At the point P' in Fig. 6, which is the mirror image of P in the atom plane, for any point 0' in the plane the distances O'P and O'P' will be equal. Therefore the phases at P and P' of the wave scattered from 0' will be the same, and the geometry of the 48 Theory of Diffraction half-period zones will be identical with the case just considered. 4c. Other scattered beams If the atoms were continuously distributed on the plane, or if their separation were less than half the wavelength of the incident radiation, then the forward scattered and reflected waves would be the only scattered waves of significant amplitude.