Radiation by an Accelerated Charge:
Scattering by Free and Bound Electrons
a
Θ
Professor David Attwood
Univ. California, Berkeley
sin2Θ
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Title_Lec4_Ch02_F00.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
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_Eqs_1.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–
1– δδ++iβ
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
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ScatRefrReflc.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
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_Maxwls_WavEqs.ai
The Wave Equation
From Maxwell’s Equations, take the curl of equation 2.2
∇×
(2.2)
and use the vector identity from Appendix D.1, pg. 440,
(D.7)
to form the Wave Equation
(2.7)
where
(2.8)
where J(r,t) is the current density in vacuum and ρ is the charge density:
(2.10)
J(r,t) = qn(r,t)v(r,t)
For transverse waves the Wave Equation reduces to
(3.1)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_WaveEq.ai
Conservation of Charge
ρ
Vector identity:
∇  (∇ × H)  0
(D.9)
(2.9)
where the current density is
(2.10)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ConsrvChrg.ai
Radiated Power and Poynting’s Theorem
To obtain Poynting’s Theorem, begin with Maxwell’s Equations
and form the difference
To obtain
Use B = µ0H and D = 0E, and the vector identity
∇  (A × B) = B  (∇ × A) – A  (∇ × B), Eq. (D.5), to obtain Poynting’s Theorem:
(2.27)
S
Integrating over a volume of interest and using Gauss’ Divergence Theorem
one obtains the integral form of Poynting’s Theorem
S
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
(2.28)
Ch02_RadPwrPoyntgs.ai
Relationships Among E, H and S
for a Plane Wave in Vacuum
Among E and H, take
to obtain
, with B = µ0H in vacuum
Write E and H, in terms of Fourier representations
E(r, t) = ∫∫Ekω e
–i(ωt – k  r)
kω
dωdk
(2π)4
and H(r, t) = ∫∫Hkω e
–i(ωt – k  r)
kω
dωdk
(2π)4
The ∇ and ∂/∂t operates on the fields become algebraic multipliers on the
Fourier-Laplace coefficients Ekω and Hkω
using the inverse transforms
(2.29)
and
(2.31)
with time average (one cycle)
Professor David Attwood
Univ. California, Berkeley
(2.37 & 2.39)
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_RelatnshpsPlWv.ai
Radiated Fields Due to a Current J
To calculate the radiated fields we want to solve the wave equation for E
in terms of J. We introduce Fourier-Laplace transform methods, to algebrize
the  and ∂/∂t operators, then integrate in k, ω-space.
We start with the transverse wave equation:
2
 ∂ – c22E(r, t) = – 1 ∂ J (r, t)
 ∂t2
o ∂t T

In a plane wave representation
E(r, t) = ∫∫Ekω e
–i(ωt – k  r)
kω
where the amplitudes are
so that
Ekω = ∫∫E(r, t)ei(ωt – k
r,t
∂/∂t → –iω
and
 r)
dωdk
(2π)4
dr dt
(2.12a)
(2.12b)
 → ik
In k, ω-space, the algebrized form of the wave equation becomes
(ω2 – k2c2)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_RadiatedFlds.ai
Natural Modes of Oscillation
(ω2 – k2c2)
The electromagnetic waves that naturally propagate in this medium (vacuum)
correspond to finite fields Ekω in the absence of a source, i.e., when JTkω = 0.
In this case
(ω2 – k2c2) Ekω = 0
=0
with solutions ω = ±kc , or fλ = c (where k = 2π/λ and ω = 2πf)
Returning now to the case of interest with a finite source
(2.18)
which has an inverse transform
(2.19)
a Green’s function-like solution for E in the presence of a source term J in a
frequency-wavenumber (energy-momentum) space. To perform the integrations
we need a specific source function JTkω.
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_NaturModeOscil.ai
Current Density J Associated
with an Accelerated Electron
For an oscillating point charge (size << λ), e.g., a single electron
δ(r)  δ(x) δ(y) δ(z)
(2.20)
To calculate the radiated field, as in eq.(2.19), we need an expression for JTkω.
Using the same Fourier-Laplace representation
(2.13b)
for the point source
with transverse component
(2.21)
where VT is the component of V perpendicular to the wave’s propagation
direction k0 (a unit vector).
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_CurrntDensty.ai
Electric Field Radiated by an Accelerated Charge
Previously we determined that
(2.19)
Where now for an accelerated electron
(2.21)
Then
(2.22)
The k-integration can be performed in the complex k-plane using the
Cauchy integral formula. The integrand is seen to have two poles, at
k = ±ω/c, representing incoming and outgoing waves. The integration path can be closed in the upper half plane with a semi-circle
of infinite radius, which makes no contribution to the integral. The
closed path then encloses a single pole, slightly displaced from the
real axis, and the k-integral is readily evaluated. Details of the integration are given in Appendix E. The result of the k-integration is
r
+ ik
e
→0
for large r
ki
Pole at
k = ω/c
kr
Pole at
k = –ω/c
(2.24)
(2.25)
or
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ElectrcFld.ai
Power Radiated by an Accelerated Point Charge
Combining
(2.25)
(2.31)
and
one obtains the instantaneous power per unit area radiated by an accelerated electron
(2.32)
k0, propagation direction
a
For an angle Θ between the direction of acceleration, a, and the
observation direction, k0, the instantaneous power per unit area is
t 0′
k0
Θ
(2.33)
t 0′′
Noting that S = (dP/dA)k0 and dA = r2 dΩ, one obtains the power per unit solid angle
(a)
(2.34)
(b)
a
Θ
a
sin 2 Θ
k0
the well known “donut-shaped” radiation pattern characteristic of a radiator whose size
is much smaller than the wavelength (“dipole radiation”).
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_PowrRadiatd.ai
Total Power Radiated by an Accelerated Point Charge
The total power radiated, P, is determined by integrating S over the area of a distant sphere:
(2.35)
where for 0 ≤ Θ ≤ π and 0 ≤ φ ≤ 2π we have dΩ = sin Θ dΘ dφ, thus
Thus the instantaneous power radiated to all angles by an oscillating electron of acceleration a is
(2.36)
For sinusoidal motion, averaging over a full cycle, sin2ωt or cos2ωt, introduces a factor of 1/2
1

2
The same factor 1/2 appears in the expressions for S (eq. 2.33) and dP/dΩ (eq.2.34) for
sinusoidal motion averaged over one or many cycles.
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_TotalPwrRad.ai
Scattering Cross-Sections
Measures the ability of an object to remove particles or photons
from a directed beam and send them into new directions
Diminished
by both
scattering and
absorption
λ
S



σ
Pscattered
(power)

area =

(power/area) 
• Isotropic or anisotropic?
• Energy or wavelength dependent?
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ScattCrSecs.ai
Scattering by a Free Electron
Define the cross-section as the average power radiated to all angles,
divided by the average incident power per unit area
(2.38)
For an incident electromagnetic wave of electric field Ei(r,t)
(2.39)
For a free electron the incident field causes an oscillatory motion
described by Newton’s second equation of motion, F = ma, where
F is the Lorentz force on the electron
small
(2.40)
Thus the instantaneous acceleration is
(2.42)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ScattFreElctrn1.ai
Scattering by a Free Electron (continued)
The average power scattered by an oscillating electron is
The scattering cross-section is
Introducing the “classical electron radius”
(2.44)
One obtains the scttering cross-section for a single free electron
(2.45)
which we observe is independent of wavelength. This is referred to as the Thomson
cross-section (for a free electron), after J.J. Thomson. Numerically re = 2.82  10–13 cm
and σe = 6.65  10–25 cm2. The differential Thomson scattering cross-section is
(2.47)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ScattFreElctrn2.ai
Scattering by a Bound Electron
For an electromagnetic wave incident upon a bound electron of resonant
frequency ωs, the force equation can be written semi-classically as
(2.48)
with an acceleration term (ma), a damping term, a restoring force term,
and the Lorentz force exerted by the fields. For an incident electric field
the harmonic motion will be driven at the same frequency, ω, so that
(2.49)
(2.50)
Following the same procedures used earlier, one obtains the scattering
cross-section for a bound electron of resonant frequency, ωs
(2.51)
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_ScattBndElctrn.ai
Semi-Classical Scattering Cross-Section
for a Bound Electron
(2.51)
σ = σT(ωs/γ)2
Scattering cross-section, σ(ω)/σT
100
90
80
70
60
Lorentzian line shape
of half width γ/2
50
2
1
0
Note that below the resonance, for ω2 << ωs2
(γ/ωs) = 10–1
(ω/ωs)4
0.5
σ = σT
1.0 1.5 2.0
Frequency ω/ωs
2.5
3.0
(2.52)
This is the Rayleigh scattering cross-section (1899) for a bound electron,
with ω/ωs << 1, which displays a very strong λ–4 wavelength dependence.
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_SemiClassScatt.ai
The sky appears blue because of the strong wave
length dependence of scattering by bound electrons.
Sun
4
σR ~
– σT λs
λ
More blue than
green or red
Atmosphere
(O2, N2, . . .)
λS ~
– 150 nm
Setting
sun
Earthly observer
•
•
•
•
•
UV resonances in O2 and N2, at 8.6 and 8.2 eV
Red (1.8 eV, 700 nm), green (2.3 eV, 530 nm), and blue light (3.3 eV, 380 nm)
Density fluctuations essential
Long path at sunset, color of clouds
Photon energy and wavelength effects. Volcanic eruptions
Professor David Attwood
Univ. California, Berkeley
Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007
Ch02_F09VG.ai