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Interference Interference Peter Hertel X-ray diffraction Laue Interference conditions Electron and neutron diffraction Peter Hertel Quantum physics University of Osnabr¨uck,Germany Neutron- molecule scattering Lecture presented at APS, Nankai University, China From full to no interference http://www.home.uni-osnabrueck.de/phertel Spring 2012 Interference Crystal Peter Hertel X-ray diffraction Laue conditions • a crystal is a regular array of identical unit cells Electron and neutron diffraction • location xl = l1a1 + l2a2 + l3a3 Quantum • l = −M ; −M + 1;:::;M − 1;M physics 1 1 1 1 1 Neutron- • l2 and l3 likewise molecule scattering • each unit cell serves as an antenna From full to no interference • it is excited by a primary electromagnetic wave • and emits a secondary wave • response is described by a complex number f • which describes the responsiveness and the retardation Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to a1 no interference a3 a2 a Unit cell and primitive cell of NaCl like crystal Interference Source, crystal and detector Peter Hertel X-ray diffraction Laue conditions • the X-ray source is at xs = −Rsnin Electron and neutron diffraction • the detector is at xd = +Rdnout Quantum • what is the distance between source and a unit cell? physics Neutron- • rsl = jxs − xlj = jRsnin + xlj molecule −1 scattering • = Rsjnin + Rs xlj From full to p −1 no interference • = Rs 1 + 2Rs xl · nin + ::: • = Rs + xl · nin + ::: • distance between detector and unit cell likewise • rdl = Rd − xl · nout + ::: Interference Spherical wave Peter Hertel X-ray diffraction Laue conditions • for simplicity, we work with a scalar electric field E 2 Electron and • wave equation ∆E + k E = 0 neutron diffraction • vacuum wave number k = !=c = 2π/λ Quantum physics • spherical solution for primary field Neutron- ikr molecule e scattering Epr = A From full to r no interference • field strength at distance r from source • secondary field emitted by unit cell ikr e E = fEpr(rs ) sd l r • field strength at distance r from unit cell Interference Peter Hertel X-ray diffraction S Laue D conditions Electron and n neutron in diffraction nout Quantum physics Neutron- l molecule scattering From full to no interference X-ray scattering.Primary field emitted at source S. A secondary field is emitted by the unit cell labeled l. It is detected at D. Interference Detected intensity Peter Hertel X-ray diffraction • field strength of secondary wave at detector Laue conditions ikRs ikRd e e iknin · xl −iknout · xl Electron and El = Af e e neutron Rs Rd diffraction • this is the contribution from unit cell l Quantum physics • the entire field strength is the superposition Neutron- i i molecule kRs kRd X e e X −i∆ · xl scattering E = El = Af e Rs Rd From full to l l no interference • wave vector transfer ∆ = k(nout − nin) • detected intensity of secondary waves 2 jAj2jfj2 X −i∆ · x jEj2 = e l R2R2 s d l Interference Peter Hertel • the sum has three factors, Σ = F1F2F3 X-ray • each of the form diffraction M Laue X −i l ∆ · a conditions F = e Electron and l=−M neutron diffraction • with Quantum −i∆ · a physics z = e Neutron- • the factor is molecule M scattering X sin(N∆ · a=2) F = zl = From full to sin(∆ · a=2) no interference l=−M • where N = 2M + 1 (number of unit cells in this dimension) • intensity at detector is proportional to 2 2 2 2 jΣj = jF1j jF2j jF3j Interference Peter Hertel 25 X-ray diffraction Laue 20 conditions Electron and neutron diffraction 15 Quantum physics Neutron- molecule 10 scattering From full to no interference 5 0 0 2 4 6 8 10 12 14 16 18 20 jF j2 versus ∆ · a for N = 5 Interference Laue conditions Peter Hertel X-ray diffraction • Each factor is tiny unless ∆ · ai is an integer multiple of 2π Laue conditions • this must be fulfilled for a1 and a2 and a3 Electron and neutron • Laue conditions diffraction Quantum ∆ · ai = νi 2π physics • sharp refraction peak if Laue conditions are fulfilled Neutron- molecule • scattering recall that ∆ = k(nout − nin) depends on angles and From full to wave number no interference • different kinds of X-ray diffraction experiments • Max von Laue 1912, Nobel prize 1914 • William Henry and William Lawrence Bragg, father and son, 1914, Nobel prize 1915 • proof that X-rays were not particles, but electromagnetic waves Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Max von der Laue, German physicist Interference Massive particle diffraction Peter Hertel X-ray diffraction Laue conditions Electron and neutron • diffraction In 1937, George P. Thomson discovered that electrons Quantum produced the same refraction pattern as X-rays physics • provided the momentum p was identified with k Neutron- ~ molecule • scattering predicted before by Louis de Broglie From full to • in 1946 the same was discovered for neutrons no interference • today preferred because only nucleons are visible • dedicated nuclear reactors and spallation sources Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference George P.Thomson, British physicist Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Louis de Broglie, French physicist Interference Quantum physics Peter Hertel X-ray diffraction Laue conditions Electron and • neutron normally: equal equations have equal solutions (Feynman) diffraction • now: same solutions must come from same equations Quantum physics • guess the mathematical framework of quantum physic Neutron- molecule • probability scattering From full to • complex probability amplitudes no interference • interference • Hilbert space, linear operators, observables, states, expectation values Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Max Planck, German physicist Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Albert Einstein, German physicist Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference David Hilbert, German mathematician Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Erwin Schr¨odinger,Austrian physicist Interference Peter Hertel X-ray diffraction Laue conditions Electron and neutron diffraction Quantum physics Neutron- molecule scattering From full to no interference Werner Heisenberg, German physicist Interference Differential cross section Peter Hertel X-ray diffraction Laue conditions • the target has n scatterer per unit volume Electron and • beam intensity I = I(x) neutron diffraction • the beam is weakened by scattering Quantum physics dI(x) = −I(x) σ n dx Neutron- molecule • therefore scattering −σnx ( ) = (0) e From full to I x I no interference • cross section is solid angle integral Z σ = dΩ σd(#) • differential cross section depends on scattering angle # • but normally not on azimuth φ Interference Neutron-O2 scattering Peter Hertel X-ray diffraction Laue • oxygen nuclei at ±an=2 conditions Electron and • O2 molecule held together by common electron cloud neutron diffraction • ignored by neutrons Quantum physics • differential cross section is 2 Neutron- −ia∆ · n=2 +ia∆ · n=2 molecule σd = f e + f e scattering From full to • i.e. no interference a∆ · n σ = 4jfj2 cos2 d 2 • average over direction n (randomly oriented molecules) sin(∆a) σ = 2jfj2 1 + d ∆a Interference Constructive, destructive and no interference Peter Hertel X-ray • M refers to the randomly oriented molecule, A to the atom diffraction Laue • differential cross section conditions M sin(∆a) A Electron and σd = 2 1 + σd neutron ∆a diffraction • scattering on an atom is isotropic Quantum physics • cross section for scattering on a randomly oriented Neutron- molecule depends on scattering angle molecule scattering • via From full to # no interference ∆ = kjn − n j = 2k sin out in 2 • for ∆a ! 0 (slow or forward): amplitudes add, cross section for scattering on molecule is four times as large as for atom, full interference • fast neutrons: cross section for molecule is twice that for atom - no interference • intermediate: constructive or destructive interference Interference 4 Peter Hertel 3.5 X-ray diffraction 3 Laue conditions Electron and 2.5 neutron diffraction 2 Quantum physics Neutron- 1.5 molecule scattering 1 From full to no interference 0.5 0 0 0.5 1 1.5 2 2.5 3 The ratio of molecular to atomic differential cross section is plotted vs. scattering angle #. The curves are for ka = 1, 2, 4, and 8, according to decrease at forward direction. Values below 2 mean destructive interference..
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