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Scattering Theory - General

This comprehensive guide delves into the core concepts of scattering theory, including definitions of flux, scattering cross-section, and intensity. Explore the principles of interference and diffraction in relation to X-rays scattered by electrons, along with detailed explanations of energy and scattering amplitude calculations. Gain insights into atomic scattering factors and scattering amplitudes for various scenarios involving multiple scatterers or atomic distributions. Relevant for students and researchers in physics and related fields.

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Scattering Theory - General

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  1. Scattering Theory - General (Read Roe, section 1.2) Definitions of flux, scattering cross section, intensity Flux J - for plane wave, energy/unit area/sec - no. photons/unit area/sec

  2. Scattering Theory - General (Read Roe, section 1.2) Definitions of flux, scattering cross section, intensity Flux J - for plane wave, energy/unit area/sec - no. photons/unit area/sec J = A*A = | A | 2 where A is the amplitude of the wave

  3. Scattering Theory - General (Read Roe, section 1.2) Definitions of flux, scattering cross section, intensity Flux J - for plane wave, energy/unit area/sec - no. photons/unit area/sec J = A*A = | A | 2 where A is the amplitude of the wave For a spherical wave scattered by a point: J is energy/unit solid angle/sec

  4. Scattering Theory - General For a spherical wave scattered by a point: J is energy/unit solid angle/sec Jois flux incident on point scatterer Then we are interested in J/Joand J/Jo = d/d scattered into unit solid angle/sec incident beam flux = differential scattering cross section = intensity I

  5. Scattering Theory - General Interference and diffraction X-rays scattered by electrons Consider 2 identical points scattering wave so in the direction of s Phase difference  betwn scattered waves is (2π/)(so r - sr) s r so Diffraction vector S =(s - so)/ 

  6. Scattering Theory - General Interference and diffraction X-rays scattered by electrons Consider 2 identical points scattering wave so in the direction of s Phase difference  betwn scattered waves is (2π/)(so r - sr) | S | = (2 sin )/  s r so Diffraction vector S =(s - so)/ 

  7. Scattering Theory - General Interference and diffraction Spherical waves scattered by each pt: A1 = Ao b exp (2πi(t-x/)) A2 = A1 exp(iAo b (exp (2πi(t-x/))) exp (-2πi Sr) b = ‘scattering length’ s r so

  8. Scattering Theory - General Interference and diffraction Spherical waves scattered by each pt: A1 = Ao b exp (2πi(t-x/)) A2 = A1 exp(iAo b (exp (2πi(t-x/))) exp (-2πi Sr) and A = A1 + A2 = Ao b (exp (2πi(t-x/))) (1 + exp (-2πi Sr)) b = ‘scattering length’ s r so

  9. Scattering Theory - General Interference and diffraction A = A1 + A2 = Ao b (exp (2πi(t-x/))) (1 + exp (-2πi Sr)) J = A*A = Ao2b2 (1 + exp (2πi Sr)) (1 + exp (-2πi Sr)) So: A(S) = Ao b (1 + exp (-2πi Sr))

  10. Scattering Theory - General Interference and diffraction A = A1 + A2 = Ao b (exp (2πi(t-x/))) (1 + exp (-2πi Sr)) J = A*A = Ao2b2 (1 + exp (2πi Sr)) (1 + exp (-2πi Sr)) So: A(S) = Ao b (1 + exp (-2πi Sr)) If n identical scatterers: A(S) = Ao b Σ exp (-2πi Srj)) n j = 1

  11. Scattering Theory - General Interference and diffraction If n identical scatterers: A(S) = Ao b Σ exp (-2πi Srj)) For a contsassembly of identical scatterers: A(S) = Ao b∫n(r) exp (-2πi Srj)) dr where n(r) = # scatterers indr V

  12. Scattering Theory - General Scattering by One Electron (Thomson) Electrons scatter x-rays z Eo 2 x x y E Ez = Eoz e2/mc2R Ey = (Eoy e2/mc2R) cos 2

  13. Scattering Theory - General Scattering by One Electron (Thomson) Electrons scatter x-rays z Eo 2 x x y E Ez = Eoz e2/mc2R Ey = (Eoy e2/mc2R) cos 2 For an unpolarized beam: Eo2 = 〈 Eoy2〉 + 〈 Eoz2〉 and 〈 Eoy2〉 = 〈 Eoz2〉 = 1/2 Eo2 = Jo/2

  14. Scattering Theory - General Scattering by One Electron (Thomson) Ez = Eoz e2/mc2R Ey = (Eoy e2/mc2R) cos 2 For an unpolarized beam: Eo2 = 〈 Eoy2〉 + 〈 Eoz2〉 and 〈 Eoy2〉 = 〈 Eoz2〉 = 1/2 Eo2 = Jo/2 Thus, the scattered flux is: Jo (e2/mc2R)2(1 + cos22)/2

  15. Scattering Theory - General Scattering by One Electron (Thomson) Ez = Eoz e2/mc2R Ey = (Eoy e2/mc2R) cos 2 For an unpolarized beam: Eo2 = 〈 Eoy2〉 + 〈 Eoz2〉 and 〈 Eoy2〉 = 〈 Eoz2〉 = 1/2 Eo2 = Jo/2 Thus, the scattered flux is: Jo (e2/mc2R)2 (1 + cos22)/2 polarization factor for unpolarized beam

  16. Scattering Theory - General Atomic Scattering Factor f(S) ≡ scattering length and f(S) = ∫n(r) exp (-2πi Sr) dr where n(r) = average electron density distribution for an atom k

  17. Scattering Theory - General Atomic Scattering Factor If electrons are grouped into atoms: f(S) ≡ atomic scattering factor and fk(S) = ∫n(r) exp (-2πi Sr) dr where n(r) = average electron density distribution for an atom k f sin /

  18. Scattering Theory - General Scattering Amplitude A(S) Finally, for N atoms: A(S) = AobeΣ fk(S)exp (-2πi Srk))

  19. Scattering Theory - General Scattering Amplitude A(S) Finally, for N atoms: A(S) = AobeΣ fk(S)exp (-2πi Srk)) Or, for a conts distribution of identical atoms whose centers are represented by nat(r): A(S) = Aobe fk(S)∫nat(r) exp (-2πi Srk)) dr

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