X-Ray Interaction with Matter, Scattering and Diffractionattwood/srms/2007/Lec02.pdf ·...

$: X-Ray Interaction with Matter, Scattering and Diffractionattwood/srms/2007/Lec02.pdf · ApxB_1_47_Jan07_lec2.ai Electron Binding Energies, in Electron Volts (eV), for the elements$
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)


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


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


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) /


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.


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


RADIATION AND SCATTERINGAT EUV AND SOFT

X-RAY WAVELENGTHS

Chapter 2

asin2ΘΘ


(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


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)


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:


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


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


Scattering by Density VariationsWithin a Multilayer Coating

Ch04_F01_Sept05.ai

(T. Nguyen, CXRO/LBNL)

Mo/Si


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


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.

X-Ray Interaction with Matter, Scattering and Diffractionattwood/srms/2007/Lec02.pdf ·...

Documents

Transcript of X-Ray Interaction with Matter, Scattering and Diffractionattwood/srms/2007/Lec02.pdf ·...