“Powder” Diffraction

“Powder” Diffraction http://www.bruker-axs.de/index.php?id=x_ray_diffraction http://pubs.usgs.gov/info/diffraction/xrd.pdf

P301 Lecture 10“Compton scattering” A. H. Compton Phys. Rev. 21 p483 (1923) Phys. Rev. 22 p409 (1923)

P301 Lecture 16“What are (x-ray) photons?” • As discussed in the lecture before last week’s exam, we have now looked at a number of experiments, and when you compare some of their results, you get confronted with something that appears to be rather troubling: • Diffraction/Interference phenomena (diffraction gratings, interference effects at shadows edges etc.) have been unambiguously identified for light (and diffraction effects have been identified even in the x-ray wavelength range Bragg’s law etc.). => Light must be a wave. • On the other hand: • PE and Compton effects clearly demonstrate that light carries momentum and energy in tiny packages of well-defined size (BB too). => Light must be a particle. • How can both views be correct? • “Consistency is the hobgobiln of little minds”? (R.W. Emerson) • Just like relativity, it is unreasonable to expect models from the classical world to work in situations of an extremely different nature (very small in this case)

P301 Lecture 16“What are (x-ray) photons?” • The key here is that whether light (or indeed ANY collection of energy) mimics waves or particles (note the word mimics), depends on the question you are asking about the system. • In many ways the jury is still out on exactly how to resolve this conundrum (current research under such topics as “measurement theory” the “collapse of the wave function” “interpretations of quantum mechanics”). To paraphrase Feynman: It is not true, as some say, that only a handful of people understand relativity, but “I think I can safely say that nobody understands QM.” (the Character of Physical Law). • The book follows the conventional “Copenhagen Interpretation” of QM, and this is what we will try to give you some comfort level with here. • Most working physicists (like yours truly) simply take that attitude that we have rules that allow us to make real predictions about real experiments, and we’re happy to let the Philosophers try to come up with internally consistent explanations for the weirdness.

P301 Lecture 17“Matter (De Broglie) Waves” l=h/p Electron diffraction using a Transmission Electron Microscope a). Single (“quasi”)-crystal (full sample is orientationally coherent) b). Polycrystalline material (many small crystals in all orientations).What is the wavelength of these electrons?

Neutron Scattering“SNS and LENS” 2 of 4 possible instruments now running 12 of 24 instruments now running

Neutron Scattering“Data Collected last week” 3rd and 4th order reflections from Ge 111 crystal planes using neutrons at LENS.

P301 Lecture 17“Complex waves” exp(iq) = cos(q) + isin(q) Where i2 = -1 F(x,t) = exp(i[kx-wt]) [= cos(kx-wt) + isin(kx-wt)] d2F/dx2 = -k2F(x,t) d2F/dt2 = -w2F(x,t) Hence: v2 (d2F/dx2 )= w2/k2 (d2F/dx2)= -w2F(x,t)= d2F/dt2 Therefore: v2 (d2F/dx2 )= d2F/dt2 And we see this satisfies the classical wave equation, provided v=w/k.

P301 Lecture 17“Fourier Decomposition/analysis” http://demonstrations.wolfram.com/WavepacketForAFreeParticle/ http://www.jhu.edu/signals/fourier2/index.html http://www.optics.rochester.edu/~stroud/animations/

Two-slit experiment with particles http://en.wikipedia.org/wiki/Double-slit_experiment

Schrodinger’s Equation The above is taken from Wikipedia, and here the “Laplacian” operator in the first term on the right hand side is simply a short-hand for (s2= d2/dx2 + d2/dy2 + d2/dz2). We will concentrate (for the most part) upon the version that does not involve time and restrict ourselves to one dimension.

“Powder” Diffraction