Scattering by a Multi-Electron Atom, Atomic Scattering Factors; Wave Propagation and Refractive Index

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$Scattering by a Multi-Electron Atom, Atomic Scattering Factors; Wave Propagation and Refractive Index David Attwood University of California, Berkeley$

1 Scattering by a Multi-Electron Atom, Atomic Scattering Factors; Wave Propagation and Refractive Index David Attwood University of California, Berkeley ( Scattering by a Multi-Electron Atom, Atomic Scattering Factors; Wave Propagation and Refractive Index, EE29F, 3 Jan 27

2 Scattering by a Multi-Electron Atom Semi-classical model of an atom with Z electrons and nucleus of charge +Ze at r =. (2.53) Electron at r s r s 2θ k i k = kk r s = r r s r Observer at r >> r s For each electron (2.58) The acceleration has an additional phase term due to the position, r s, within the atom: The scattered electric field at a distance r summed for all Z electrons, is (2.61) where r s r r s and r s = r s. For r >> r s, r s r k r s (2.62) Ch2_ScattMultiElctrn1_Jan7.ai

3 Scattering by a Multi-Electron Atom (continued) (2.65) k i 2θ k k where the quantity f( k, ω) is the complex atomic scattering factor, which tells us the scattered electric field due to a multi-electron atom, relative to that of a single free electron (eq. 2.43). Note the dependence on frequency ω (photon energy ω), the various resonant frequencies ω s (resonant energies ω s ), and the phase terms due to the various positions of electrons within the atom, k r s. k = k k i k = 2k i sin θ Ch2_ScattMultiElctrn2.ai

4 A General Scattering Diagram J(r,t) = e n(r,t)v(r,t) (2.1) k i Incident wave k d k s Scattered wave k d = 2π/d represents a spatial non-uniformity in the medium, such as atoms of periodicity d, a grating, or a density distribution due to a wave motion. d matching exponents ω s = ω i + ω d k s = k i + k d If the density distribution is stationary ω k i = = c k s = ω c = 2π λ 2π λ the scattering diagram is isosceles k i k s θ k d k i + k d = k s k d /2 sinθ = k i sinθ = λ 2d λ = 2d sinθ (2.62) (Reference: See chapter 4, eqs. 4.1 to 4.6) (Bragg s Law, 1913) Univ. California, Berkeley Scattering by a Multi-Electron Atom, Atomic Scattering Factors; Wave Propagation and Refractive Index, EE29F, 3 Jan 27 Ch2_GenScattDia.ai

5 The Atomic Scattering Factor (2.66) In general the k r s phase terms do not simplify, but in two cases they do. Noting that k = 2k i sinθ = 4π/λ sinθ, and that the radius of the atom is of order the Bohr radius, a, the phase factor is then bounded by (2.7) The atomic scattering factor f( k,ω) simplifies significantly when (long wavelength limit) (forward scattering) (2.71a) (2.71b) In each of these two cases the atomic scattering factor f( k,ω) reduces to where we denote these special cases by the superscript zero. (2.72) Ch2_AtomcScatFac.ai

6 Complex Atomic Scattering Factors (2.72) (2.79) which some write as Ch2_ComplexAtomScatFac.ai

7 Atomic Scattering Cross-Sections Comparing the scattered electric field for a multi-electron atom (2.65) with that for the free electron (2.43), the atomic scattering cross-sections are readily determined by the earlier proceedures to be (2.75) (2.76) where (2.77) and where the super-script zero refers to the special circumstances of long wavelength (λ >> a ) or forward scattering (θ << 1). With the Bohr radius a =.529 Å, the long wavelength condition is easily satisfied for soft x-rays and EUV. Note too that we have introduced the concept of oscillator strengths, g s, associated with each resonance, normalized by the condition (2.73) Ch2_AtomcScatXSec.ai

8 Example: Complex Atomic Scattering Factor for Carbon f 1 f (2.79) Carbon (C) Z = 6 Atomic weight = E (ev) (Henke and Gullikson; www-cxro.lbl.gov) E (ev) 1 Note that for ω >> ω s, f 1 Z. This works here for carbon f 1 6, but note that in general this conflicts with the condition λ >> a. For the case of carbon at 4 Å wavelength (λ >> a ), and thus ω = 3 kev (>> ω s ~ 274 ev), the atomic scattering cross-section (2.76) becomes (2.78c) that is, all Z electrons are scattering cooperatively (in-phase) - the so-called N 2 effect. Ch2_Ex_ComplxAtomc.ai

9 Atomic Scattering Factors for Carbon (Z = 6) σ a (barns/atom) = µ(cm 2 /g) E(keV)µ(cm 2 /g) = f Energy (ev) f 1 f 2 µ (cm 2 /g) E E E+ 5.21E E E E E E E E E E E E E E E E E 2 f Carbon (C) Z = 6 Atomic weight = µ (cm 2 /g) E (ev) E (ev) Edge Energies: K ev 1 f (Henke and Gullikson; www-cxro.lbl.gov) Ch2_F13VG.ai

10 Atomic Scattering Factors for Silicon (Z = 14) σ a (barns/atom) = µ(cm 2 /g) E(keV)µ(cm 2 /g) = f Energy (ev) f 1 f 2 µ(cm 2 /g) µ(cm 2 /g) E E E E E E E E E E E+ 1.62E E E E E E E E E f 1 f Silicon (Si) Z = 14 Atomic weight = Edge Energies: E (ev) (Henke and Gullikson; www-cxro.lbl.gov) 1 K ev L ev L ev 99.2 ev L E (ev) 1 Ch2ApC_Tb1F7_Sept5.ai

11 Atomic Scattering Factors for Molybdenum (Z = 42) σ a (barns/atom) = µ(cm 2 /g) E(keV)µ(cm 2 /g) = f Energy (ev) f 1 f 2 µ(cm 2 /g) µ(cm 2 /g) E E E E E E E E E E E E E E E E E E E E E (ev) Edge Energies: (Henke and Gullikson; www-cxro.lbl.gov) 1 f 1 f E (ev) K ev L ev M ev N ev L ev M ev N ev L ev M ev N ev M ev ev M 5 Molybdenum (Mo) Z = 42 Atomic weight = AST 21/EECS 213 Univ. California, Berkeley Ch2ApC_Tb1F12_June28.ai

12 Wave Propagation and Refractive Index at X-Ray Wavelengths n = 1 δ + iβ n = 1 k φ k k Ch3_F_Jan7.ai

13 The Wave Equation and Refractive Index The transverse wave equation is (3.1) For the special case of forward scattering the positions of the electrons within the atom ( k r s ) are irrelevant, as are the positions of the atoms themselves, n(r, t). The contributing current density is then (3.2) where n a is the average density of atoms, and Ch3_WavEq_RefrcIndx1.ai

14 The Wave Equation and Refractive Index (Continued) The oscillating electron velocities are driven by the incident field E (3.2) such that the contributing current density is (3.4) Substituting this into the transverse wave equation (3.1), one has Combining terms with similar operators (3.5) Ch3_WavEq_RefrcIndx2.ai

15 Refractive Index in the Soft X-Ray and EUV Spectral Region Written in the standard form of the wave equation as (3.6) The frequency dependent refractive index n(ω) is identified as (3.7) For EUV/SXR radiation ω 22 is very large compared to the quantity e 2 n a / m, so that to a high degree of accuracy the index of refraction can be written as (3.8) which displays both positive and negative dispersion, depending on whether ω is less or greater than ω s. Note that this will allow the refractive index to be more or less than unity, and thus the phase velocity to be less or greater than c. Ch3_RefracIndex1.ai

16 Refractive Index in the Soft X-Ray and EUV Spectral Region (continued) (3.8) Noting that and that for forward scattering where this has complex components The refractive index can then be written as (3.9) which we write in the simplified form (3.12) Ch3_RefracIndex2.ai

17 Refractive Index from the IR to X-Ray Spectral Region (3.12) (3.13a) (3.13b) Refractive index, n 1 Infrared Visible Ultraviolet X-ray ω IR ω UV ω K,L,M λ 2 behavior δ & β << 1 δ-crossover Ultraviolet Ch3_RefrcIndxIR_XR.ai

18 Phase Velocity and Refractive Index The wave equation can be written as (3.1) The two bracketed operators represent left and right-running waves V φ = c n V φ = c n E z E z Left-running wave Right-running wave where the phase velocity, the speed with which crests of fixed phase move, is not equal to c as in vacuum, but rather is (3.11) Ch3_PhasVelo_Refrc1.ai

19 Phase Velocity and Refractive Index (continued) Recall the wave equation (3.1) Examining one of these factors, for a space-time dependence E T = E exp[ i(ωt kz)] Solving for ω/k we have the phase velocity V φ = Ch3_PhasVelo_Refrc2.ai

20 Phase Variation and Absorption of Propagating Waves For a plane wave (3.14) in a material of refractive index n, the complex dispersion relation is (3.15) Solving for k (3.16) Substituting this into (3.14), in the propagation direction defined by k r = kr or (3.17) where the first exponential factor represents the phase advance had the wave been propagating in vacuum, the second factor (containing 2πδr/λ) represents the modified phase shift due to the medium, and the factor containing 2πβr/λ represents decay of the wave amplitude. Ch3_PhaseVarAbsrb.ai

21 Intensity and Absorption in a Material of Complex Refractive Index For complex refractive index n (3.18) The average intensity, in units of power per unit area, is or Recalling that (3.19) (3.2) (3.17) or the wave decays with an exponential decay length (3.21) (3.22) Ch3_IntenstyAbsrp.ai

22 Absorption Lengths (3.22) Recalling that β = n a r e λ 2 f 2 (ω)/2π (3.23) In Chapter 1 we considered experimentally observed absorption in thin foils, writing (3.24) where ρ is the mass density, µ is the absorption coefficient, r is the foil thickness, and thus l abs = 1/ρµ. Comparing absorption lengths, the macroscopic and atomic descriptions are related by (3.26) where ρ = m a n a = Am u n a, m u is the atomic mass unit, and A is the number of atomic mass units Ch3_AbsorpLngths.ai

23 Phase Shift Relative to Vacuum Propagation For a wave propagating in a medium of refractive index n = 1 δ + iβ (3.23) the phase shift φ relative to vacuum, due to propagation through a thickness r is (3.29) Flat mirrors at short wavelengths Transmissive, flat beamsplitters Bonse and Hart interferometer Diffractive optics for SXR/EUV Reference wave BS r Object M Film or CCD M Object wave BS Ch3_PhaseShift.ai

Multilayer Interference Coating, Scattering, Diffraction, Reflectivity

$Multilayer Interference Coating, Scattering, Diffraction, Reflectivity$ Multilayer Interference Coating, Scattering, Diffraction, Reflectivity mλ = 2d sin θ (W/C, T. Nguyen) Normal incidence reflectivity 1..5 1 nm MgF 2 /Al Si C Pt, Au 1 ev 1 ev Wavelength 1 nm 1 nm.1 nm Multilayer