Phys560Notes-14

# Phys560Notes-14 - Neutron Scattering Counts E E E Spectrum...

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Neutron Scattering Spectrum at a fixed scattering geometry: elastic peak caused by defect scattering (or Bragg diffraction) one-phonon emission peaks at lower energies (Stokes peaks) one-phonon absorption peaks at higher energies (anti-Stokes peaks), usually weaker the ratio between the Stokes and anti-Stokes intensities is related to temperature, approaching unity at high T anti-Stokes peaks vanish at T = 0 (no phonons available for absorption) a background from multi-phonon contributions Neutrons: mass n M , 2 2 n p E M , no charge ne interaction is weak, ignored n nucleus interaction: nuclear force; short range 13 8 10 10 a  cm cm; small cross section; multiple scattering ignored; approximated by a delta function Counts E' E' E E     rR R uR Atomic position , assuming one atom per unit cell    3 3 11 2 ii Ve d k e V   kr - - RR R k rr r R  where the integration over k (2nd step) is over entire reciprocal space, and the sum in k (3 rd step) is carried out over all zones. Cross section transition probability (Fermi’s golden rule) 2 2 ff i i f EE M   , where energy of crystal, E energy of neutron Initial states: neutron / 1 i e V pr  , crystal   is n k , electrons ignored Final states: neutron / 1 i e V , crystal   fs n k , electrons ignored / // f if i f i Me e e e   pp rR QrR R rrR  Qp p scattering wave vector (momentum transfer) i f i  QRuR R

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Assuming ideal static lattice ,   0 uR , | i f ii f Me N   QR Q,K R K = reciprocal lattice vector With energy conservation (from golden rule), f f EE   , if i f   Elastic scattering. Q,K  pp Κ Laue condition; Bragg diffraction of neutrons; just like x-rays. Neutron diffraction is an important tool for structural determination (complementary to x- ray diffraction). Particularly useful for light elements (for which x-ray scattering cross section is weak) and magnetic structures using polarized neutron beams (neutrons have a magnetic moment). Including lattice vibrations (3d, 1 atom per unit cell)  2 1 1 2 i f i e e i    QuR RR Qu R  Recall   1 i ss s eq N kR k ke k and      ek e k .     23 1 ... f i s i e q O q O q N    Rk ke k Q 1 st term elastic diffraction by ideal static crystal ( q = 0) 3 rd and higher terms higher-order processes; weaker matrix elements; background & Debye-Waller factor; later 2 nd term i s fs i s i N    kQR Qe k k   f si q k : 2 s s qa a M  kk k k contains one-phonon operators Initial state: is s
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## This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys560Notes-14 - Neutron Scattering Counts E E E Spectrum...

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