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Acknowledgements, Sunil Sinha Roger Pynn Jacob Tosado. UC San Diego U Indiana U Maryland, Acknowledgements. Outline, 1 Scattering geometry basics Plane waves. and Fourier transforms, 2 Scattering cross sections for neutrons and x rays. 3 Scattering from ensemble of atoms, and diffraction.
Scattering geometry basics,Plane waves and,Fourier transforms. Scattering geometry basics The sinusoidal wave, , sin sin A. cos , cos , A amplitude , , angle 2 2, phase difference. cos sin 2 , A, sin , Scattering geometry basics The wavenumber k. sin , k wavenumber , x position A, wavelength, 2 .
x, , k has SI units of rad m 1, A, sin , Scattering geometry basics The travelling wave. , Wave moves in x direction with time t, , sin A, 0 initial phase angle. phase after time t, angular frequency x, 2 , 0 , A. sin 0 , Scattering geometry basics The plane wave. We define a plane wave , Amplitude in the z direction y k.
Propagates in y and x directions , ky, r direction of propagation. kx, k wavevector , 2 , , , sin 0 x, Scattering geometry basics The traveling plane wave. Plane wave in x direction only Plane wave in xy direction. Animation courtesy of Dr Dan Russell Grad Prog Acoustics Penn State. Scattering geometry basics Complex numbers, Useful to work with exponential Im Argand diagram. over sinusoidal waves, Complex numbers allow us to. simplify wavefunction, equations i , b, r, i imaginary number .
Re, a, i cos i sin , i , sin 0 , , i , Re z a Im z b. Scattering geometry basics The Fourier series, We approximate a periodic structure through a sum of cosines and sines . Let f x be a function expanded by a Fourier series. 0 1 cos 2 cos 2 3 cos 3 Goes to, zero if, 1 sin 2 sin 2 3 sin 3 f x f x . n 1 fundamental harmonic n 3 higher harmonics included. The Fourier coefficients, We write sum more efficiently if we pick the coefficients correctly . Now a definition and not approximation , , We extend the analysis to a non .
periodic function, , The Fourier coefficients become. where continuous functions we call F k , 1, 1 , i d 2 . , 0, The Fourier transform, The limiting case is and 0. We call F k the Fourier transform of f x and vice versa. We can toggle between real space x and reciprocal space k . , 1 i , 1, d i d , 2 2 , , A duck in, Im Argand diagram for real. A duck in real space, reciprocal space, and imaginary.
Re components, Credit Dr David Cowtan , University of York. Fourier optics Young s double slit experiment, kf ksin . and Diffraction Efrain E Rodriguez Chemistry and Biochemistry University of Maryland College Park Acknowledgements Sunil Sinha UC San Diego Roger Pynn U Indiana Jacob Tosado U Maryland Acknowledgements Outline 1 Scattering geometry basics Plane waves and Fourier transforms 3 Scattering from ensemble of atoms and diffraction 2 Scattering cross sections for neutrons and x rays