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
Basic Ionization and Emission
Processes in Isolated Atoms
(b) Photoionization
(a) Electron collision induced ionization
Primary
electron
(Ep)
Scattered
primary
′)
electron (Ep
e–
e–
+Ze
e–
K
Secondary
electron (Es)
L
Photon
(ω)
Photoelectron
(E = ω – EB)
e–
+Ze
K
L
M
M
(c) Fluorescent emission of characteristic radiation
(d) Non-radiative Auger process
e–
KLL Auger
electron
ω
+Ze
K
+Ze
L
M
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
K
L
M
Ch01_F02VG.ai
Electron Binding Energies, in Electron Volts
(eV), for the elements in their Natural Forms
www.cxro.LBL.gov
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
ApxB_1_47_Jan07_lec2.ai
Fluorescence and Auger Emission Yields
Fluorescence and Auger yields
1.0
0.9
0.8
K-shell
Auger
0.7
0.6
K-shell
fluorescence
L3-subshell
Auger
0.5
0.4
0.3
L3-subshell
fluorescence
0.2
0.1
0
0
20
40
60
80
Atomic number
100
(Courtesy of M. Krause, Oak Ridge)
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F03VG_2005.ai
Electron Mean Free Paths As a Function of Energy
(a)
(c)
4
Al
(b)
Mean free path (nm)
3
100
2
1
0
4
10
Au
3
1
2
1
0 0
10
101
102
103
0.1
104
1
10
Energy above Fermi level (eV)
100
1000
Courtesy of: Penn (a & b), Seah and Dench (c)
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F05VG.ai
Photoabsorption by Thin Foils and Isolated Atoms
(a)
(b)
106
ρ
I = e–ρµx
I0
 cm2 105
 g 
ω
104
ω
(e)
103
x
10
(c)
(d)
na
ω
Ι0
10
100
Cu atom
 Mb 10
atom
ω
Professor David Attwood
Univ. California, Berkeley
0.01
Exponential
decay (e–ρµx)
Distance, x
3p 3d
1
0.1
x
100
1000
Photon energy (eV)
σabs
Ι(x)
Ι0
2
Intensity
Ι(x)
Ι0
Cu foil
4s
10
3s
2p
2s
I = e–naσabsx
I0
100
1000
Photon energy (eV)
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F08VG_Aug05.ai
Atomic Energy Levels and
Allowed Transitions in the Bohr Atom
Equate Coulomb Force Ze2/4π0r2 to the centripetal force mv2/r:
1
mZ 2 e 4
En =
32π 2 20 2 n 2
rn =
4π0  2
me 2 Z
· n2
(1.4)
(1.5)
(1.6)





me 4
 1 – 1  Z2
ω = Ei – Ef =
32π 2 0  2  n f2
n 2i 
13.6 eV
a0n2
rn =
Z
Professor David Attwood
Univ. California, Berkeley
;
a0 = 0.529 Å
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
(1.9)
Ch01_Eqs4_5_6_9VG.ai
Quantum Mechanics Based
on a Probabilistic Wave Function, Ψ(r, t)
∂Ψ(r, t)
2 2
–
∇ Ψ(r, t) + V(r, t)Ψ(r, t) = i
∂t
2m
(1.10)
P(r, t)dr = Ψ ∗ (r, t) Ψ(r, t)dr
(1.13)
r=
rP(r, t)dr = 
Ψ ∗ (r, t)rΨ(r, t)dr
(1.15)
quantum numbers: n, , m, ms
selection rules for allowed transitions: ∆ = ± 1
∆j = 0, ± 1
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_Eqs10_13_15VG.ai
Professor David Attwood
Univ. California, Berkeley
Time
Probability
Oscillation amplitude
Radiative Decay Involves An Atom
Oscillating Between Two Stationary States
at the Frequency if = (Ei – Ef) / 
1
0
Lower state
Upper state
Time
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F09VG.ai
Probabilistic Radial Charge Distribution (e/Å)
in the Argon Atom
Radial charge density distribution
20
1s
15
2p
10
5
2s
3p
3s
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Normalized radius, r/a0
Courtesy of Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F12VG.ai
Energy Levels, Quantum Numbers, and
Allowed Transitions for the Copper Atom
n  j
4
4.
..
4
3
3.
..
0
7/2
5/2
.
..
N
. VII 4f7/2
.
N
.. IV 4d3/2
N
1/2
Absorption edges
for copper (Z = 29):
NI
4s
EN1, abs = 7.7 eV
MV
MIV
MIII
MII
MI
3d5/2
3d3/2
3p3/2
3p1/2
3s
.
.
EM3, abs = 75 eV
.
EM1, abs = 123 eV
M α1
3
3
3
3
3
2
2
1
1
0
5/2
3/2
3/2 M
1/2
1/2
Lα1 Lα2 Lβ2
2 1 3/2
2 1 1/2 L
2 0 1/2
Kα1 Kα2
EL3, abs = 933 eV
EL2, abs = 952 eV
EL1, abs = 1,097 eV
K
EK, abs = 8,979 eV
(1.381Å)
Kβ1 Kβ3 Kγ
3
1 0 1/2 K
Cu Kα1 = 8,048 eV (1.541Å)
Cu Kα2 = 8,028 eV (1.544Å)
Cu Kβ1 = 8,905 eV
Professor David Attwood
Univ. California, Berkeley
LIII 2p3/2
LII 2p1/2
LI 2s
Cu Lα1 = 930 eV
Cu Lα2 = 930 eV
Cu Lβ1 = 950 eV
1s
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F11VG_Jan07.ai
Scattering, Diffraction, and Refraction
(a) Isotropic scattering from a point object
(b) Non-isotropic scattering from a partially
ordered system
λ
λ
(c) Diffraction by an ordered array of atoms,
as in a crystal
(d) Diffraction from a well-defined geometric
structure, such as a pinhole
λ
λ
D
θ
θ null
d
θ
θ null = 1.22λ
d
mλ = 2d sinθ
(e) Refraction at an interface
λ
(f) Total external reflection
λ
θ<θ c
n = 1–δ+iβ
n=1
Professor David Attwood
Univ. California, Berkeley
n = 1–δ+iβ
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch01_F13VG.ai
Chapter 2
RADIATION AND SCATTERING
AT EUV AND SOFT
X-RAY WAVELENGTHS
a
Θ
Professor David Attwood
Univ. California, Berkeley
sin2Θ
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch02_F00VG.ai
Maxwell’s Equations and the Wave Equation
Maxwell’s equations:
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
The wave equation:
(3.1)
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch02_Maxwls_WavEqs.ai
Scattering, Refraction, and Reflection
Single scatterer,
electron or atom,
in vacuum.
(Chapter 2)
Many atoms, each
with many electrons,
constituting a “material”.
(Chapter 3)
n = 1– δ + β
n=1
λ
λ
• How are scattering, refraction, and reflection related?
• How do these differ for amorphous and ordered (crystalline) materials?
• What is the role of forward scattering?
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch02_ScatRefrReflc.ai
Maxwell’s Equations
Wave Equation
)
um
u
ac r 2)
v
(in apte
h
(C
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”,
Professor David Attwood
Univ. California, Berkeley
f0′ and f0′′
(in
(C
am
ha
pte
ate
r3
ria
)
l)
Refractive index with many atoms
present
Role of forward scattering
Contributions to refractive index by
bound electrons
Refractive index for soft x-rays and EUV
n = 1 – δ + iβ (δ, β << 1)
f0′
f0′′
Determining f0′ and f0′′ ; measurements
and Kramers-Kronig
Total external reflection
Reflectivity vs. angle
Brewster’s angle
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch02_Eqs_1VG.ai
Atomic Scattering Factors
for Silicon (Z = 14)
σa(barns/atom) = µ(cm2/g) × 46.64
E(keV)µ(cm2/g) = f20 × 1498.22
Energy (eV)
f10
f20
µ(cm2/g)
30
70
100
300
700
1000
3000
7000
10000
30000
3.799
2.448
–5.657
12.00
13.31
13.00
14.23
14.33
14.28
14.02
3.734E–01
5.701E–01
4.580E+00
6.439E+00
1.951E+00
1.070E+00
1.961E+00
4.240E–01
2.135E–01
2.285E–02
1.865E+04
1.220E+04
6.862E+04
3.216E+04
4.175E+03
1.602E+03
9.792E+02
9.075E+01
3.199E+01
1.141E+00
10
Silicon (Si)
Z = 14
Atomic weight = 28.086
15
10
f10 5
0
–5
–10
10
7
µ(cm2/g)
1000
10000
100
1000
E (eV)
10000
101
105
f20
103
10
100
100
10–1
1
10–1
10
100
Edge Energies: K
1000
E (eV)
1838.9 eV
10000
10–2
10
L1 149.7 eV
L2 99.8 eV
L3 99.2 eV
(Henke and Gullikson; www-cxro.LBL.gov)
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch02ApC_Tb1F07_9.05.ai
Chapter 3
WAVE PROPAGATION AND REFRACTIVE INDEX
AT EUV AND SOFT X-RAY WAVELENGTHS
k′
n = 1 – δ + iβ
n=1
φ
k
Professor David Attwood
Univ. California, Berkeley
k′′
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch03_F00VG.ai
Multilayer Mirrors Satisfy the Bragg Condition
mλ = 2d sinθ
Mo
Si
Mo
d Si
Mo
Si
Mo
Si
ne
θ
λ
For normal incidence, θ = π/2, first order (m = 1) reflection
λ = 2d
d = λ/2
if the two layers are approximately equal
∆t  λ/4
a quarter-wave plate coating.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch04_MltlyrMirBragg1.ai
Scattering by Density Variations
Within a Multilayer Coating
Mo/Si
(T. Nguyen, CXRO/LBNL)
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch04_F01_Sept05.ai
Multilayer Mirrors Have Achieved 70% Reflectivity
0.8
Mo/B4C/Si
70% at 13.5 nm
FWHM = 0.55 nm
50 bilayers
0.7
Reflectivity
0.6
0.5
0.4
0.3
0.2
0.1
0.0
12.0
12.5
13.0
13.5
Wavelength (nm)
14.0
14.5
ˇ Bajt, LLNL.
Courtesy of Sasa
Extreme Ultraviolet Lithography
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
Ch04_MltlyrMirReflc.ai
The Derivation of Bragg’s Law
mλ = 2d sinθ
mλ
θ
θ
θ
d
The path difference of
radiation “reflecting”
off sequential planes
must be equal to an
interger number of
wavelengths.
The angle θ is measured from the crystal plane, and the
distance between planes is referred to as the “d-spacing”.
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.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
DerivationBraggsLaw.ai
Bragg Scattering, or Diffraction,
Seen as a Reflection from Crystal Planes
θ
θ
inθ
ds
inθ
θ
d
ds
θ
Constructive interference occurs when the additional path length is equal
to an integral number of wavelengths:
(Bragg’s Law)
mλ = 2d sinθ
(m = 1, 2, . . . )
R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), section 12.4.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
BraggScattDiffrac.ai
X-Rays are Refracted Entering a Crystal
Refraction of x-rays at a crystal surface requires
a small correction to the Bragg condition:
2
4δd
mλ = 2d sinθ (1 – 2 2 )
m λ
λ
θ
θ
d
λ
θ
θ
R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), p. 456.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
XRsRefracEnterCrystal.ai
Face-Centered Cubic Crystal Structure
Primitive vectors:
a = ai, b = aj, c = ak
z
8
4
Crystal
Rocksalt Na
Sylvine K
Ag
Mg
Galena
Pb
Pb
Pb
Cl
Cl
Cl
O
S
Se
Te
a(Å)
5.64
6.28
5.54
4.20
5.97
6.14
6.34
3
a
d
x
2
6
7
1
Coordinates of atoms
within 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)
y
5
Nearest neighbor
distance d = a/2
From R.B. Leighton, Principles of Modern Physics (McGraw-Hill, New York, 1959), section 12.4.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter, Scattering and Diffraction, EE290F, 18 Jan 2007
FaceCenterCubicrCrystal.ai
Diffraction of Polychromatic X-Rays from the
Various Bragg Planes of a Given Crystal
Each reflection results in
monochromatic x-rays in the
given 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.
Professor David Attwood
Univ. California, Berkeley
X-Ray Interaction with Matter: Absorption, Scattering and Diffraction, EE290F, 18 Jan 2007
DiffracPolychromXRs.ai