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Interaction of Radiation with Matter
« Element of modern x-ray physics »
J. Als-Nielsen et D. McMorrow
« Processus d’interaction entre
photons et atomes »
C. Cohen-Tannoudji,…
• Particles: probes
• Two process of interaction
Absorption
and
scattering
dz
dW
I0
kd
I
ki
l
2q
Characteristics of particles
Three types of particles
Are used in condensed matter physics
•
• Tender and hard X-ray photons: 3-100 keV
• Low or high energy electrons: 150 eV-100 keV
• Hot, thermal or cold neutrons: 120-25-10 meV
• Interference effects:
Wave length of particle must be
smaller
than interatomic distances
2𝑑 sin 𝜃 = 𝑚𝜆
𝜆 ≤ 2𝑑
Characteristics of particles
Description
X Photons
Neutrons
Electrons
Electromagnetic field
Particle
Particle
𝑬 = 𝑬0 𝑒 𝑖(𝒌∙𝒓−𝜔𝑡)
Energy
Momentum
kBT/E
E=hn=hc/l
l(Å)=12398/E(eV)
l=1 Å, E=12.4 keV
n=3.1018 Hz (EHz)
E
p
p=hk=hn/c
300K
3.10-6 << 1
y ~ exp(i k.r)
y ~ exp(i k.r)
E2=p2c2+mn2c4; E=p2/2mn
E=p2/2me
l(Å)=0.286/E0.5(eV)
l(Å)=12.265/E0.5(eV)
l=1 Å, E=81.8 meV
l=1 Å, E=150 eV
vn = 4000 m/s
ve = 7274 km/s
p=hk (=mv)
~
1
p=hk (=mv
~
10-5
Interaction
Charge
sth ~ Z2 barn
Moments magnétiques
sd ~ 10-6 barn
Noyaux (forte)
sd ~ 5 barn
Moments magnétiques
sd ~ 3 barn
Potentiel electrostatique
sd ~ 108 barn
Absorption
4700 barn (Z=28, 1,5 Å)
Typique : 0,1-1 barn
-
Absorption cross section
• After going through matter of width dz,
beam intensity decreases by dI
dz
𝑑𝐼 = −𝐼 𝑧 𝜇𝑑𝑧 ⟹ 𝐼 = 𝐼0 𝑒 −𝜇𝑙
I0
• m attenuation coefficient (cm-1)
• Beer-Lambert law
l
• F0: flux incident particles (s-1/cm2), F = I/S
Number of absorbed particles 𝑑𝑁𝑞 per time unit
𝑑𝑁𝑞 = 𝜙 𝑧 𝑁(𝑑𝑧)𝜎𝑎
•
sa:
absorption cross section, expressed in barn = 10-24 cm2
The cross section depends on the element,
its environnement (RX) and on the particle energy
Ex: 2D lattice
Unit cell 0.3 nm
Surface per atom is
s~10-15 cm2
I
𝑁 𝑑𝑧 = 𝜌𝑎 𝑆𝑑𝑧
𝜇 = 𝜎𝑎 𝜌𝑎
Scattering cross section
• Scattering process
dW
• number of scattered particles
𝒒
𝑑𝜎
𝑑𝑁𝑑 = 𝜙𝑑Ω
𝑑Ω
ki
Scattering differential cross section
• Wave function of the scattered particle
𝑒 𝑖𝑘𝑑 𝑟
𝜓𝑑 𝒓 = −𝑏(𝒒)
𝑟
𝑏(𝒒): scattering length
Neutrons: b independent on
• Differential cross section
𝑑𝜎
𝑑Ω
𝑠
𝑘𝑑
=
𝑏
𝑘𝑖
2
𝑑𝜎
𝑑Ω
= 𝑏
𝑠
2
q
kd
2q
Characteristics of particles
Description
X Photons
Neutrons
Electrons
Electromagnetic field
Particle
Particle
𝑬 = 𝑬0 𝑒 𝑖(𝒌∙𝒓−𝜔𝑡)
Energy
Momentum
kBT/E
E=hn=hc/l
l(Å)=12398/E(eV)
l=1 Å, E=12.4 keV
n=3.1018 Hz (EHz)
E
p
p=hk=hn/c
300K
3.10-6 << 1
y ~ exp(i k.r)
y ~ exp(i k.r)
E2=p2c2+mn2c4; E=p2/2mn
E=p2/2me
l(Å)=0.286/E0.5(eV)
l(Å)=12.265/E0.5(eV)
l=1 Å, E=81.8 meV
l=1 Å, E=150 eV
vn = 4000 m/s
ve = 7274 km/s
p=hk (=mv)
~
1
p=hk (=mv
~
10-5
Interaction
Charge
sth ~ Z2 barn
Magnetic moments
sd ~ 10-6 barn
Noyaux (forte)
sd ~ 5 barn
Magnetic moments
sd ~ 3 barn
Electrostatic potential
sd ~ 108 barn
Absorption
4700 barn (Z=28, 1,5 Å)
Typique : 0,1-1 barn
-
Scattering length for particles
Solve the Schrödinger equation of the particle
in an interaction potential 𝑉 𝒓
ℏ2 𝑘 2
Stationary states of energy: 𝐸 =
2𝑀
« Mécanique quantique 2, chap.VIII »
Cohen-Tannoudji, Diu, Laloë
with 𝑉 𝑟 =
ℏ2
𝑈(𝒓)
2𝑀
Born aprox. + 𝑘𝑟 ≫ 1
Scattering length =
FT of potential
ℏ2
ℏ2 𝑘 2
∆+𝑉 𝒓 𝜑 𝒓 =
𝜑 𝒓
2𝑀
2𝑀
∆ + 𝑘 2 − 𝑈(𝒓) 𝜑 𝒓 = 0
𝑖𝑘𝑟
𝑒
𝜑 𝒓 ~𝑒 𝑖𝒌𝑖 ∙𝒓 + 𝑏(𝒒)
𝑟
𝑏 𝒒 =−
𝟏
𝟒𝝅
𝑈(𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓
Scattering length
X-rays: FT of the electron density
𝑏 𝒒 = −𝑟0 𝑓 𝒒 = −𝑟0
𝜌 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Rayons X
Phase shift 𝜋 … r0 = 2,82 10-15 Å
Neutrons: FT of Fermi pseudo-potential. It is a constant because
(𝓥 𝒓 ~ δ 𝒓 )
𝑀
𝑏 𝒒 =𝑏=
𝓥 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓
2
2𝜋ℏ
Phase shift 0 ou 𝜋 …
Electrons: TF of potential 𝑉(𝒓)
𝑀
−𝑖𝒒∙𝒓 𝑑 3 𝒓
𝑏 𝒒 =−
𝑉(𝒓)𝑒
2𝜋ℏ2
Electron
𝑏 𝒒 depends n l’énergie
Phase shift 𝛿(q)
Fadley, Physica Scripta, T17,39,1987
Optical theorem
Mécanique quantique II, p. 940
C. Cohen-Tannoudji, B. Diu, Frank Laloë
𝜎𝑡𝑜𝑡 = 𝜎𝑎 +𝜎𝑑 = −
4𝜋
Im(𝑏
𝑘
0 )
𝑒 𝑖𝑘𝑑 𝑟
𝜓𝑑 𝒓 = −𝑏(𝜃)
𝑟
Shadow:
𝜓𝑖 𝒓 = 𝐴𝑒 𝑖𝑘𝑖 𝑟
Interference
between incident wave
and
scattered wave
Absorption
Origin of neutrons
absorption
• Neutrons weaklly absorbed
• Absorbed through nuclear reactions
3He+n  3H-+p
sa
Ni
Pb
Gd
520 2100 74000 4.6 0.17
6Li
10B
Detectors and shields
Energy dependance:
𝑘0
𝜎𝑎 𝑘 = 𝜎𝑎
𝑘
𝑘0 = 34,947 nm−1
Origin of photons
absorption
Free electron energy
𝐸 2 = 𝑚2 𝑐 4 + 𝑝2 𝑐 2
𝑣 ≪ 𝑐 𝐸 = 𝑝2 /2𝑚
Photon energy
(p,E)
𝐸𝑝ℎ = 𝑝𝑐
E
E
?
EO=511 keV
EO=511 keV
EO-EL
Dp.Dr  
p
Free electron:
no absorption
p
Bound electron
absorption
X-ray absorption
Hard X-rays
UV
VUV
XUV
Soft X-rays
• Absorption
At energies smaller than 1000 keV
Gamma
LEAD Z=82
Photoelectric effect
X-ray absorption
For E < 1000 keV
photoelectric effect is dominant
• Photoelectric effect
• Photon is absorbed if hn > EI (EI binding energy of e-)
• Excitation: Photoelectron is emitted ( E=hn - EI -F )
F: work function ~1 eV
• De-excitation: fluorescence photon (hn = EI -EII )
Auger electron ( E= EI -EII -EIII)
Fluorescence photon
Photoelectron
Continuum
Fermi level
Auger electron
-EF
M
(2p3/2)4
L (2p1/2)2
(2s)2
hn
Ka
Core levels
-EII
Kb
K (1s)2
-EI
Excitation
De-excitation
Absorption of photons
Emission of photons and electrons
Order of magnitude
X-rays: l = 1.542 Å
sa
Ni
Pb
Li B
Gd
5,7 36 78300 4760 79800
Neutrons: 1.8 Å
sa
Ni
Pb
Gd
520 2100 74000 4.6 0.17
6Li
10B
Electrons
mean free paths
Distance between
two inelastic collisions with
• Plasmons
• Valence electrons
From A. Zangwill, ‘Physics at Surfaces’, Cambridge Univ. Press.
After this distance (attenuation length), electrons loose their coherence.
Low energy electron diffraction (LEED) is a surface technique
Only surface photoelectrons and Auger electrons escape from the sample
Importance in X-Ray Absorption (XAS)…
Scattering
Scattering: atome-particle system changes of state
Initial state,
ei
Final state,
ef
Elastic scattering:
Does not change the nature or the internal state
of the particles and the target
Photon scattering
• Rayleigh scattering:
• Low energy elastic scattering
• hn << EI , EI -EII ; Fi = Ff ; light scattering, blue sky
• Raman/Brillouin scattering:
• Low energy inelastic scattering
• hn << EI ; Fi  Ff ; scattering on optical/acoustical phonons
• Thomson scattering:
• High energy elastic scattering
• hn >> EI ; Fi = Ff ; X-ray scattering
• Compton scattering:
• High energy inelastic scattering
• hn >> EI ; Fi  Ff ; X-ray scattering
Photons scattering
(pi ,Ei )
(pf ,Ef )
E
𝑝2
𝐸𝑒 =
𝑚
EO
E
EO
EO-EL
p
Free electron (e- mass m)
Compton scattering
𝑝2
𝐸𝑒 =
𝑀
p
Bound electron (atom, crystal mass M»m)
Thomson scattering
Compton scattering
Refraction
𝑅
𝑟
𝑆
A consequence of scattering
𝐷
Wave travelling through a plate of width Δ
Phase shift: 𝑛𝑘Δ-𝑘Δ
𝑅0
Δ
𝑅2 = 𝑅0 2 + 𝑟 2
𝑅𝑑𝑅 = 𝑟𝑑𝑟
𝑖 𝑛−1 𝑘Δ
𝜓
𝑃
=
𝜓
𝑃
𝑒
0
𝑃
= 𝜓0 𝑃 (1 + 𝑖 𝑛 − 1 𝑘Δ)
∞
𝑖𝑘𝑅
𝑒
𝜓 𝑃 = 𝜓0 𝑃 +𝜓0 (𝑆)𝑒 𝑖𝑘𝐷
−𝑏
(2𝜋𝑟𝑑𝑟Δ)𝜌𝑑
𝑅
0
∞
2𝜋𝑏Δ𝜌𝑑
𝑖𝑘𝐷
𝑖𝑘𝑅
= 𝜓0 𝑃 −𝜓0 𝑆 2𝜋𝑏Δ𝜌𝑑 𝑒
𝑒 𝑑𝑅 = 𝜓0 𝑃 (1 − 𝑖
)
𝑘
𝑅0
2𝜋𝑏𝜌𝑑
𝑛 =1−
𝑘2
Absorption
𝑅Δ
−1
𝜌𝑑 ~1𝑒Å−3 , 𝑏~ 𝑍 3. ∞
10−5
k~4
Å
−𝜇Å,
𝑒 𝑅0 𝑒 𝑖𝑘𝑅 𝑑𝑅
−5
𝛿 ~ 10
𝑅0
Refraction
a
n
ki
Refraction index
kr
kt
𝑛 = 𝑛𝑟 + 𝑖𝛽
a’
Phase shift and absorption
𝑒 𝑖𝑛𝑘𝑧 = 𝑒 𝑖𝑛𝑟𝑘𝑧 𝑒 −𝛽𝑘𝑧
For X-rays and neutrons
2𝜋𝑏𝜌𝑑
𝜇
𝑛 =1−
+ 𝑖𝛽 = 1 − 𝛿 + 𝑖
2
𝑘
2𝑘
Snell’s law
𝑛𝑟 cos 𝛼′ = cos 𝛼
Existence of a critical angle
above which total reflection
𝛼𝑐 = 2𝛿
ac
ki
kr
Stationnary wave
Measure of the sign of b (holography)
Experimental
techniques
EMISSION :
X-ray
EMISSION
(par rayons X) :
•Fluorescence
(Chemical analysis)
R
• ayons X
Electron Spectroscopy
Fluorescence
•
(Analyse chimique)
Electrons
•
Photoelectrons,
Auger
electrons
(analysis)
Photo-électrons,
•
électrons
Auger
(Spectrométrie, analyse)
Photoelectron
diffraction (structure
(local structure)
Diffraction
•
de photo-électrons
locale)
Photo-émission
•
(Structure
de
bande,
surface
Fermi)
de
Photoemission (band structure)
WAVES/PARTICLES
X-Rays
Neutrons
Electrons
Crystal
Liquid, liquid crystal
Polymer
Surface
REFRACTION :
X-ray, neutrons
Reflectrometry (surfaces, interfaces)
Stationnary waves (surfaces)
ABSORPTION :
X-ray
XAS, EXAFS, XANES (local order)
Dichroism (Magnetism, surfaces)
SCATTERING
X-rays
DIFFUSION
:Diffuse scattering (Disorder, liquids, soft matter)
•
Diffraction (Structures);
R
• ayons X
Compton scattering (electronic
structure)
Diffraction
•
(Etude
des structures)
Diffusion
•
diffuse
(Etude
du désordre
dans nano-particles,
les cristaux, liquides,proteins)
cristaux liquides)
Small angle scattering (Polymer, liquid
crystal,
D
• iffusion
Compton
(Structure électronique)
Magnetic, inelastic,
surface,
coherent
diffraction
Diffusion
• aux petits
angles (Polymères,
cristaux(synchrotrons)
liquides, agrégats, grandes mailles)
Diffusion
•
magnétique,
inélastique,
cohérente…
(synchrotrons)
Neutrons
Neutrons
•
Diffraction, Diffuse
scattering
(Structures,
Hydrogen,
Diffraction,
•
Diffusion
diffuse (Structures,
Hydrogène,contrast)
contraste différent)
Inélastique
•
(Excitations
élémentaires,
phonons, dynamique)
Inelastic scattering
(phonons,
dynamics,
excitations)
Magnétique
•
(Structures magnétique, magnons)
Magnetic (magnetism,
magnons)
Electrons
•
Diffraction,
•
LEED, RHEED (Etude des surfaces)
Electrons
Low- or high-energy electron diffraction (surfaces, thin samples)