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Scattering Theory - General

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. Scattering Theory - General. (Read Roe, section 1.2)

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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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