Post on 07-Apr-2020
X-Ray Interaction with Matter:Absorption, Scattering
and Diffraction
David Attwood
University of California, Berkeley
(http://www.coe.berkeley.edu/AST/srms)
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
+Ze +Ze
+Ze +Ze
Primaryelectron
Photoelectron(E = ω – EB)
KLL Augerelectron
Photon(ω)
Scatteredprimary electron (Ep)
Secondaryelectron (Es)
(Ep)
′
KL
M
KL
M
(a) Electron collision induced ionization
e–
e–
e–
e–
(b) Photoionization
(c) Fluorescent emission of characteristic radiation (d) Non-radiative Auger process
KL
M
KL
M
e–
ω
Basic Ionization and EmissionProcesses in Isolated Atoms
Ch01_F02VG.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
ApxB_1_47_Jan07_lec2.ai
Electron Binding Energies, in Electron Volts(eV), for the elements in their Natural Forms
www.cxro.LBL.gov
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007Professor David AttwoodUniv. California, Berkeley
K-shellAuger
L3-subshellfluorescence
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00 20 40 60
Atomic number
Fluo
resc
ence
and
Aug
er y
ield
s
80
(Courtesy of M. Krause, Oak Ridge)
100
Fluorescence and Auger Emission Yields
Ch01_F03VG_2005.ai
L3-subshellAuger
K-shellfluorescence
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
1 10 100Energy above Fermi level (eV)
1000
100
10
1
0.1
0
1
2
3
4
4
0
1
2
3
Al
Au
Mea
n fre
e pa
th (n
m)
100 101 102 103 104
(a)
(b)
(c)
Electron Mean Free Paths As a Function of Energy
Ch01_F05VG.ai
Courtesy of: Penn (a & b), Seah and Dench (c)
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
x
x
(a)
(c)
(b)
(d)
(e)
na
Ι0
Ι0
Ι0
Ι(x)ω
ρ
ω
ωΙ(x)
II0
ω
Cu foil
Cu atom
3p
3s4s
3d
2p2s
10 100Photon energy (eV)
σabs
1000
10 100Photon energy (eV)
Distance, x
Inte
nsity1000
100
10
1
0.1
0.01
Mbatom
106
105
104
103
102
cm2
g
Exponentialdecay (e–ρµx)
= e–ρµx
Photoabsorption by Thin Foils and Isolated Atoms
Ch01_F08VG_Aug05.ai
II0
= e–naσabsx
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Atomic Energy Levels and Allowed Transitions in the Bohr Atom
Ch01_Eqs4_5_6_9VG.ai
Equate Coulomb Force Ze2/4π0r2 to the centripetal force mv2/r:
En = (1.4)
(1.5)
(1.6)
(1.9)
mZ2e4
32π2220
1n2
rn = · n24π02
me2Z
me4
32π202
rn = ; a0 = 0.529 Å
13.6 eV
a0n2
Z
1nf
1ni
ω = Ei – Ef = – Z2
2 2
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Quantum Mechanics Basedon a Probabilistic Wave Function, Ψ(r, t)
Ch01_Eqs10_13_15VG.ai
– ∇2Ψ(r, t) + V(r, t)Ψ(r, t) = i
P(r, t)dr = Ψ∗(r, t) Ψ(r, t)dr
quantum numbers: n, , m, ms
selection rules for allowed transitions: ∆ = ± 1 ∆j = 0, ± 1
r = rP(r, t)dr = Ψ∗(r, t)rΨ(r, t)dr
(1.10)
(1.13)
(1.15)
2
2m∂Ψ(r, t)
∂t
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter, Scattering and Diffraction, EE290F, 18 Jan 2007
Lower state
Upper state
1
0Time
Pro
babi
lity
Osc
illat
ion
ampl
itude
Time
Radiative Decay Involves An Atom Oscillating Between Two Stationary States at the Frequency if = (Ei – Ef) /
Ch01_F09VG.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
20
15
10
5
0 0.5 1.5
2s
3s
3p
1s
2pR
adia
l cha
rge
dens
ity d
istri
butio
n
1.0 2.0Normalized radius, r/a0
2.5 3.0 4.03.5
Probabilistic Radial Charge Distribution (e/Å)in the Argon Atom
Ch01_F12VG.ai
Courtesy of Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles.
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
n
N
M
L
K
44...4
NVII 4f7/2..NIV 4d3/2..NI 4s
MV 3d5/2MIV 3d3/2MIII 3p3/2MII 3p1/2MI 3s
LIII 2p3/2LII 2p1/2LI 2s
K 1s
33...0
7/25/2...1/2
33333
22110
5/23/23/21/21/2
222
110
3/21/21/2
1 0 1/2
j
Kα1
Cu Kα1 = 8,048 eV (1.541Å) Cu Lα1
= 930 eVCu Kα2
= 8,028 eV (1.544Å) Cu Lα2 = 930 eV
Cu Kβ1 = 8,905 eV Cu Lβ1
= 950 eV
Absorption edgesfor copper (Z = 29):
EN1, abs = 7.7 eV
.
.EM3, abs = 75 eV.EM1, abs = 123 eV
EL3, abs = 933 eVEL2, abs = 952 eVEL1, abs = 1,097 eV
EK, abs = 8,979 eV (1.381Å)
Kα2
Lα1
Mα1
Lβ2Lα2
Kβ1Kβ3
Kγ3
Energy Levels, Quantum Numbers, andAllowed Transitions for the Copper Atom
Ch01_F11VG_Jan07.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
λ
λλ
λ
λ
(a) Isotropic scattering from a point object (b) Non-isotropic scattering from a partially ordered system
(c) Diffraction by an ordered array of atoms, as in a crystal
(d) Diffraction from a well-defined geometric structure, such as a pinhole
D θnull
(e) Refraction at an interface
n = 1 n = 1–δ+iβ
(f) Total external reflection
1.22λd
n = 1–δ+iβ
θ<θc
λ
θ
mλ = 2d sinθ
θnull =
θ
d
Scattering, Diffraction, and Refraction
Ch01_F13VG.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
RADIATION AND SCATTERINGAT EUV AND SOFT
X-RAY WAVELENGTHS
Chapter 2
asin2ΘΘ
Ch02_F00VG.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
(2.5)
(2.6)
(3.1)
(2.1)
(2.2)
(2.3)
(2.4)
Maxwell’s equations:
The wave equation:
Maxwell’s Equations and the Wave Equation
Ch02_Maxwls_WavEqs.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Scattering, Refraction, and Reflection
Ch02_ScatRefrReflc.ai
Single scatterer,electron or atom,in vacuum.(Chapter 2)
Many atoms, eachwith many electrons,constituting a “material”.(Chapter 3)
• How are scattering, refraction, and reflection related?• How do these differ for amorphous and ordered (crystalline) materials?• What is the role of forward scattering?
λ
λ
n = 1
n = 1– n = 1– δδ + + ββn = 1– δ + β
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Maxwell’s Equations
Wave Equation
Ch02_Eqs_1VG.ai
Radiation by a single electron (“dipole radiation”)Scattering cross-sections
Scattering by a free electron (“Thomson scattering”)
Scattering by a single bound electron (“Rayleigh scattering”)Scattering by a multi-electron atom
Atomic “scattering factors”, f0 and f0′ ′′
Refractive index with many atoms presentRole of forward scatteringContributions to refractive index by bound electronsRefractive index for soft x-rays and EUVn = 1 – δ + iβ (δ, β << 1)
Determining f0 and f0 ; measurements and Kramers-KronigTotal external reflectionReflectivity vs. angleBrewster’s angle
′
′
′′
′′f0 f0
(in a material)(Chapter 3)
(in vacuum)
(Chapter 2)
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Atomic Scattering Factorsfor Silicon (Z = 14)
(Henke and Gullikson; www-cxro.LBL.gov)
Ch02ApC_Tb1F07_9.05.ai
Energy (eV) f1 f2 µ(cm2/g) 30 3.799 3.734E–01 1.865E+04 70 2.448 5.701E–01 1.220E+04 100 –5.657 4.580E+00 6.862E+04 300 12.00 6.439E+00 3.216E+04 700 13.31 1.951E+00 4.175E+03 1000 13.00 1.070E+00 1.602E+03 3000 14.23 1.961E+00 9.792E+02 7000 14.33 4.240E–01 9.075E+01 10000 14.28 2.135E–01 3.199E+01 30000 14.02 2.285E–02 1.141E+00
15
10
5
0
–5
–10
101
100
10–1
10–2
f1
f2
µ(cm
2 /g)
10 100 1000E (eV)
10000
10 100 1000 10000
10 100 1000E (eV)
10000
107
105
103
101
10–1
σa(barns/atom) = µ(cm2/g) × 46.64E(keV)µ(cm2/g) = f2 × 1498.220
0 0
0
0
Silicon (Si)Z = 14
Atomic weight = 28.086
K 1838.9 eV L1 149.7 eV L2 99.8 eV L3 99.2 eV
Edge Energies:
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
WAVE PROPAGATION AND REFRACTIVE INDEXAT EUV AND SOFT X-RAY WAVELENGTHS
Chapter 3
n = 1 – δ + iβn = 1 φ
k
k′
k′′
Ch03_F00VG.ai
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch04_MltlyrMirBragg1.ai
d
neMoSiMoSiMo
MoSi
Si
λ
θ
mλ = 2d sinθ
For normal incidence, θ = π/2, first order (m = 1) reflection λ = 2d d = λ/2if the two layers are approximately equal ∆t λ/4a quarter-wave plate coating.
Multilayer Mirrors Satisfy the Bragg Condition
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Scattering by Density VariationsWithin a Multilayer Coating
Ch04_F01_Sept05.ai
(T. Nguyen, CXRO/LBNL)
Mo/Si
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch04_MltlyrMirReflc.ai
Multilayer Mirrors Have Achieved 70% Reflectivity
Extreme Ultraviolet Lithography
Courtesy of Sasa Bajt, LLNL.
12.0 12.5 13.0 13.5 14.0 14.50.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8R
efle
ctiv
ity
Wavelength (nm)
Mo/B4C/Si70% at 13.5 nm
FWHM = 0.55 nm50 bilayers
ˇ
Professor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
The Derivation of Bragg’s Law
DerivationBraggsLaw.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
d
The angle θ is measured from the crystal plane, and the distance between planes is referred to as the “d-spacing”.
The path difference ofradiation “reflecting”off sequential planesmust be equal to aninterger number ofwavelengths.
mλ
From A.H. Compton and S.K. Allison, X-Rays in Theory and Experiment (D.Van Nostrand, New York, 1926), p.29.Also see M. Siegbahn, The Spectroscopy of X-Rays (Oxford University Press, London, 1925), p.16.
mλ = 2d sinθ
θ
θ
θ
θ
θ
d sinθ
d sinθ
θ d
θ
Bragg Scattering, or Diffraction, Seen as a Reflection from Crystal Planes
BraggScattDiffrac.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Constructive interference occurs when the additional path length is equal to an integral number of wavelengths:
(Bragg’s Law)(m = 1, 2, . . . )mλ = 2d sinθ
R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), section 12.4.
X-Rays are Refracted Entering a Crystal
XRsRefracEnterCrystal.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Refraction of x-rays at a crystal surface requiresa small correction to the Bragg condition:
mλ = 2d sinθ (1 – )4δd2
m2λ2
θθ
λ
λ
θ θd
R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), p. 456.
Face-Centered Cubic Crystal Structure
FaceCenterCubicrCrystal.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter, Scattering and Diffraction, EE290F, 18 Jan 2007
Nearest neighbordistance d = a/2
Primitive vectors:a = ai, b = aj, c = ak
Coordinates of atomswithin unit cell:1: (0,0,0)2: (0,0,a/2)3: (a/2,0,a/2)4: (a/2,0,a)5: (a/2,a/2,0)6: (a/2,a/2,a/2)7: (0,a/2,a/2)8: (0,a/2,a)
Crystal a(Å)Rocksalt Na Cl 5.64Sylvine K Cl 6.28 Ag Cl 5.54 Mg O 4.20Galena Pb S 5.97 Pb Se 6.14 Pb Te 6.34
x
z
yd
a
84
2
6
1
5
73
From R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), section 12.4.
Diffraction of Polychromatic X-Rays from the Various Bragg Planes of a Given Crystal
DiffracPolychromXRs.aiProfessor David AttwoodUniv. California, Berkeley X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Each reflection results in monochromatic x-rays in thegiven direction – the basics for a crystal monochromator.
F.K. Richtmyer, E.H. Kennard, and T. Lauritsen Introduction to Modern Physics (McGraw-Hill, New York, 1955), chapter 8.