Methoden moderner Röntgenphysik: Streuung und Abbildung

Transcription

1 Methoden moderner Röntgenphysik: Streuung und Abbildung Lecture 8 Location Vorlesung zum Haupt- oder Masterstudiengang Physik, SoSe 018 G. Grübel, A. Philippi-Kobs, O. Seeck, L. Frenzel, F. Lehmkühler, M. Martins, W. Wurth Lecture hall AP, Physics, Jungiusstraße Date Tuesdays 13:00-14:30 (starting 3.4.) Thursdays 8:30-10:00 (until 1.7.) Methoden Moderner Röntgenphysik - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 018, G. Grübel

2 Methoden moderner Röntgenphysik II: Streuung und Abbildung Small Angle Scattering, and Soft Matter Introduction, Form Factor, Structure Factor, Applications,... Anomalous Diffraction Introduction into Anomalous Scattering,... Introduction into Coherence Concept, First Order Coherence,... Coherent Scattering Spatial Coherence, Second Order Coherence,... Applications of Coherent Scattering Imaging and Correlation Spectroscopy,... Methoden Moderner Röntgenphysik - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 018, G. Grübel

3 Resonant Scattering (phasing, magnetism,..) Scattering length of an atom: - r 0 f 0 (Q) f 0 (Q) r 0 atomic form factor (fourier transform of charge distribution) thomson scattering length of single electron in order to include absorption effects (f ) atoms a more elaborate model than the free electron gas is needed. Electrons are bound to atoms Forced oscillator modell with resonant frequency ω s and damping constant Γ include dispersion corrections (f, f ): [note: f (k/4πr 0 ) a ] f(q, ω) f 0 (Q) + f (ω) + i f (ω) [in units of r 0 ] 3

4 Resonant Scattering classical model of an electron bound in an atom in E field E(r,t) ^ x Eo exp{-iωt} equation of motion of the electron. x + Γ x + ωs x e E0 ( ) - exp{-iωt} m Γ damping ω S resonant frequency Solution: x (t) x0 exp{-iωt} x0 - e E0 ( ) m 1 (ωs ω iωγ) (A) radiated field strength at distance R and time t Erad(R,t) e ( 4 ε0 R c ) x (t R/c) (B) acceleration at earlier time (t-r/c) 4

5 inserting Erad (R,t) x (t R/c) ω x0 exp{-iωt} exp{i(ω/c)r} ω e (ωs ω iωγ) 4 ε0 m c ( ) using (A) into (B): Eo exp{-iωt} exp{ikr} ( ) R or Erad(R,t) Ein - r0 ω exp{ikr} ( ) (ωs ω + iωγ) R atomic scattering length f s (in units of r0) for bound electron (C) note: f s 1 (ω >>ω s ) total cross-section: 8π σt ( ) 3 σ T (8π/3) r o (free electron) ω 4 (ω ωs) +(ωγ) for Γ 0 and ω << ω s : σ T (8π/3)r o (ω / ω S ) 4 : Rayleigh Scattering r0 5 Methoden moderner Roentgenphysik- Vorlesung im Haupt/Masterstudiengang Physik, Universitaet Hamburg,

6 fs ω ωs + iωγ + ωs - iωγ (ω ωs + iωγ) 1 + ωs - iωγ (ω ωs + iωγ) (Γ 0.1 ω s ) 1 + ωs (ω ωs + iωγ) with: dispersion correction χ(ω) f s ωs (ω ωs) (ω ωs) +(ωγ) χ(ω) f S + i f S ωs (ω ωs + iωγ) f S ωs ω Γ (ω ωs) +(ωγ) 6

7 Note: since f -(k/4π) σ a (E) (see J. A-N. & D. McM. p. 70) it follows that the absorption cross-section for a single oscillator model is: σa,s(ω) 4 π r0 c ωs Γ (ω ωs) +(ωγ) this function has: - sharp peak at ω ωs - ΔωFWHM Γ thus σ a (E) may be written with help of a delta function: σa,s(ω) 4 π r0 c π δ(ω ωs) (D) 7

8 The experimentally observed absorption cross-section is NOT a single line spectrum as suggested by (D). There is a continuum of free states above an absorption edge that the electron can be excited into. This implies a series of different ω s : 8

9 Absorption cross section for multiple harmonic oscillators: σa(ω) π r0 c Σ S g(ωs) δ(ω ωs) where g(ωs) is the relative weight of each transition The real part of the dispersion becomes: f (ω) Σ g(ωs) f S (ω,ωs) (F) S (F) does not describe e.g. white lines or EXAFS oscillations (see figure) in the absorption cross section arising from the particular environement of the resonantly scattering atom. 9

10 measure absorption cross-section and use (E) to obtain f : f (ω) ω 4 π r0 c - ( ) σa(ω) (E) use Kramers-Kronig relations to obtain f : f (ω) + 1 P f (ω) dω π (ω ω) - π P + 0 ω f (ω) dω (ω ω) f (ω) + 1 P f (ω) - dω - π (ω ω) - ω P π + 0 f (ω) (ω ω) dω P stands for principal value (see also comments J. A-N & D. McM p. 4) 10

11 . Friedel s law and Bijvoet pairs The phase problem in crystallography The MAD method (Resonant) Magnetic Scattering 11