Methoden moderner Röntgenphysik II: Streuung und Abbildung

Transcription

1 Methoden moderner Röntgenphysik II: Streuung und Abbildung Lecture 4 Location Vorlesung zum Haupt- oder Masterstudiengang Physik, SoSe 2015 G. Grübel, M. Martins, E. Weckert Lecture hall AP, Physics, Jungiusstraße Date Tuesday 12:45-14:15 Thursday 8:30-10:00 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

2 Methoden moderner Röntgenphysik II: Streuung und Abbildung Introduction Overview, Introduction to X-ray Scattering X-ray Scattering Primer Elements of X-ray Scattering Sources of X-rays, Synchrotron Radiation Laboratory Sources, Accelerator Bases Sources Reflection and Refraction from Interfaces Snell s Law, Fresnel Equations Kinematical Diffraction (I) Diffraction from an Atom, a Molecule, from Liquids, Glasses, Kinematical Diffraction (II) Diffraction from a Crystal, Reciprocal Lattice, Structure Factor, 2 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

3 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,... 3 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

4 Reflection and Refraction from Interfaces Rays of light propagating in air change direction when entering glass, water or another transparent material. Governed by Snell s law: Note: spherical wave e ik r k = nk = n ω = ω c v with v = n phase velocity c (v > c for n < 1; but group velocity d dk c) cosα cosα n = n ω = n (refractive index) 1.2 < n < 2 visible light n < 1 X rays (α < α) n = 1 δ δ 10 5 Total external reflection: for α < α c (critical angle) 4 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

5 Refractive Index Refractive picture: Consider plane wave impinging on a slab with thickness Δ and refractive index n. Evaluate amplitude at observation point P (compared to the situation without slab). no slab: e ik slab: e ink phase difference: e i(nk k) Amplitude: ψ P tot ψ P 0 = eink e ik e iα = cos α + i sin α = e i(nk k) α small 1 + iα ψ P tot ψp [1 + i k n 1 Δ] ($) 0 5 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

6 Refractive Index Scattering picture: R = R x 2 = R 0 2 (1 + x2 R 0 2 ) R x2 R x4 4R 0 2 = R 0 (1 + x2 2R 0 2) 2 = R 0 (1 + x2 2R 0 2) phase difference 2kR between direct rays and rays following path R; 2kx 2 2R 0 = kx2 R 0 Include y direction: iφ x,y e = i x 2 +y 2 k e R 0 Amplitude at P: dψ s P ikr 0 ikr 0 e R 0 ρ dxdy b e R 0 e iφ(x,y) incident wave number of scatters in volume element ρdxdy scattered wave from 1 scatterer phase factor 6 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

7 Refractive Index ψ s P = dψ s P = ρb ( exp i2kr 0 R 0 2 ) exp(iφ x, y )dxdy Amplitude at P without slab: ψ s P = ( exp i2kr 0 2R 0 ) ψ P tot i πr 0 k = = ψ P 0 [1 ( i2πρb ) k ($) ψ P i n 1 k n = 1 2πρb k 2 = 1 δ [Ref. 1] [1] [2] If a homogenous electron density ρ is replaced by a plate composed of atoms: ρ = ρ a f 0 (0) number density x atomic scattering factor δ = 2πρ af 0 0 r 0 k Total external reflection α = 0 for α = α c : cos α = n cos α cos α c = 1 α c 2 2 α c = 2δ = 4πρr 0 k 2 k = 2π λ = 4A 1, b = r 0 = A, ρ = 1e A 3 : δ 10 5 [Ref. 1: Als-Nielsen and McMorrow, p. 66] 7 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

8 Critical Angle for Si α c = 2δ = 4πρr 0 k 2 Silicon: ρ = 0.699e A 3, λ = 1A α c = 4π e 5 1 2π 2 = rad Q c = ( 4π λ ) sin α c = 0.032A 1 8 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

9 Refraction Including Absorption n = 1 δ + i β Wave propagating in a medium: e i nkz = e i(1 δ)kz e βkz Attenuation of amplitude: e μz 2 (when intensity drops according to e μz ) β = μ 2k 9 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

10 Snell s Law and the Fresnel Equations 10 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

11 Snell s Law and the Fresnel Equations k = k I = k R : a I k cos α + a R k cos α = a T nk cos α (B ) : a I a R k sin α = a T nk sin α (B ) α, α small: cos α = cos α (cos z = 1 z2 2 ) a 2 = α 2 + 2δ 2iβ (B +A) = α 2 + α c 2 2iβ (C) Assume that the wave and its derivative is continuous at the interface: a I + a R = a T a I k I + a R k R = a T k T (A) (B) a I a R sin α = n α a I +a R sin α α Fresnel equations: r = a R = α α a I α + α ; t = a T a I = 2α α + α r: reflectivity t: transmittivity (B +A) 11 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

12 Snell s Law and the Fresnel Equations (2) Note: α is a complex number α = Re α + i Im(α ) Consider z-component of transmitted wave: = a T e ik sin α z a T e ikα z = a T e ikre α z e k Im α z exponential damping intensity fall-off: e 2k Im α z Use wavevector notation: Q 2 k sin α 2 k α Q c 2 k sin α c 2 k α c use dimensionless units: q Q Q c 2k Q c α ; sin α = q Q Q c Q 2 k 2k Q c α 1/e penetration depth Λ: z 2k Im α = 1 (z = Λ) 1 Λ = 2k Im(α ) q 2 = q i b μ b μ = 2k 2 4k β =( 2 Q c Q2 ) μ = 2k c 2k Q2 μ c Q c = 2kα c = 2k 2δ (D) 12 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

13 Snell s Law and the Fresnel Equations (3) Use table to extract μ, ρ, f yielding Q c and calculate b μ (b μ 1): b μ = 2kμ Q c 2 Use (D): Get: q 2 = q i b μ r q = q q q + q t q = 2q q + q 1 Λ q = Q c Im(q ) 13 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

14 Snell s Law and the Fresnel Equations (4) Fresnel equations: q 1: R(Q)~ 1 q 4, Λ μ 1, T 1, no phase shift q 1: R 1, Λ 1 q c small, T very small, π phase shift q = 1: T q = 1 4a I 14 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

15 Examples 15 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,

16 Methoden moderner Röntgenphysik II: Streuung und Abbildung Next Lecture Introduction Overview, Introduction to X-ray Scattering X-ray Scattering Primer Elements of X-ray Scattering Sources of X-rays, Synchrotron Radiation Laboratory Sources, Accelerator Bases Sources Reflection and Refraction from Interfaces Snell s Law, Fresnel Equations Kinematical Diffraction (I) Diffraction from an Atom, a Molecule, from Liquids, Glasses, Kinematical Diffraction (II) Diffraction from a Crystal, Reciprocal Lattice, Structure Factor, 16 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg,