Radiation by an Accelerated Charge: Scattering by Free and ...attwood/srms/2007/04_new_2007.pdf ·...

asin2ΘΘ

Title_Lec4_Ch02_F00.aiProfessor David AttwoodUniv. California, Berkeley Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007

Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons

Maxwell’s Equations

Wave Equation

Ch02_Eqs_1.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 Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 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– δδ + + ββ


n = 1

n = 1– δ + iβ

(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 Radiation by an Accelerated Charge: Scattering by Free and Bound Electrons, EE290F, 25 Jan 2007

Ch02_WaveEq.ai

The Wave Equation

(2.2)

(D.7)

(2.7)

(2.8)

(2.10)

(3.1)

From Maxwell’s Equations, take the curl of equation 2.2

and use the vector identity from Appendix D.1, pg. 440,

to form the Wave Equation

where

For transverse waves the Wave Equation reduces to

where J(r,t) is the current density in vacuum and ρ is the charge density:J(r,t) = qn(r,t)v(r,t)

∇ ×


Ch02_ConsrvChrg.ai

Conservation of Charge

Vector identity: ∇ (∇ × H) 0

where the current density is

(2.9)

(2.10)

(D.9)

ρ


Ch02_RadPwrPoyntgs.ai

Radiated Power and Poynting’s Theorem

(2.27)

(2.28)

To obtain Poynting’s Theorem, begin with Maxwell’s Equationsand form the difference

To obtain

Integrating over a volume of interest and using Gauss’ Divergence Theoremone obtains the integral form of Poynting’s Theorem

Use B = µ0H and D = 0E, and the vector identity ∇ (A × B) = B (∇ × A) – A (∇ × B), Eq. (D.5), to obtain Poynting’s Theorem:

S

S


Ch02_RelatnshpsPlWv.ai

Relationships Among E, H and Sfor a Plane Wave in Vacuum

(2.29)

(2.31)

(2.37 & 2.39)

Among E and H, take , with B = µ0H in vacuumto obtain

Write E and H, in terms of Fourier representations

and

and

with time average (one cycle)

using the inverse transforms

The ∇ and ∂/∂t operates on the fields become algebraic multipliers on theFourier-Laplace coefficients Ekω and Hkω

E(r, t) = ∫∫Ekω e–i(ωt – k r) dωdk

(2π)4kωH(r, t) = ∫∫Hkω e

–i(ωt – k r) dωdk(2π)4kω


Ch02_RadiatedFlds.ai

Radiated Fields Due to a Current J

To calculate the radiated fields we want to solve the wave equation for Ein terms of J. We introduce Fourier-Laplace transform methods, to algebrizethe and ∂/∂t operators, then integrate in k, ω-space.

In a plane wave representation

We start with the transverse wave equation:

(2.12a)

(2.12b)where the amplitudes are

In k, ω-space, the algebrized form of the wave equation becomes

so that

(ω2 – k2c2)

– c22 E(r, t) = – JT(r, t)

E(r, t) = ∫∫Ekω e–i(ωt – k r)

∂2

∂t2

dωdk(2π)4

1o

∂∂t

kω

Ekω = ∫∫E(r, t)ei(ωt – k r)dr dt r,t

∂/∂t → –iω and → ik


Ch02_NaturModeOscil.ai

Natural Modes of Oscillation

(2.18)

(2.19)

(ω2 – k2c2)

(ω2 – k2c2) Ekω = 0

which has an inverse transform

Returning now to the case of interest with a finite source

with solutions ω = ±kc , or fλ = c (where k = 2π/λ and ω = 2πf)

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

a Green’s function-like solution for E in the presence of a source term J in afrequency-wavenumber (energy-momentum) space. To perform the integrationswe need a specific source function JTkω.

= 0


Ch02_CurrntDensty.ai

Current Density J Associatedwith an Accelerated Electron

For an oscillating point charge (size << λ), e.g., a single electron

To calculate the radiated field, as in eq.(2.19), we need an expression for JTkω.Using the same Fourier-Laplace representation

with transverse component

for the point source

where VT is the component of V perpendicular to the wave’s propagation direction k0 (a unit vector).

(2.21)

(2.13b)

(2.20)

δ(r) δ(x) δ(y) δ(z)


Ch02_ElectrcFld.ai

Electric Field Radiated by an Accelerated Charge

or

Then

Previously we determined that

Where now for an accelerated electron

(2.24)

(2.19)

(2.21)

(2.25)

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 integra-tion 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 inte-gration are given in Appendix E. The result of the k-integration is

e+ikr → 0 for large r

ki

krPole atk = –ω/c

Pole atk = ω/c

(2.22)


Ch02_PowrRadiatd.ai

Power Radiated by an Accelerated Point Charge

Combining

For an angle Θ between the direction of acceleration, a, and the observation direction, k0, the instantaneous power per unit area is

one obtains the instantaneous power per unit area radiated by an accelerated electron

and

k0, propagation direction

Noting that S = (dP/dA)k0 and dA = r2 dΩ, one obtains the power per unit solid angle

the well known “donut-shaped” radiation pattern characteristic of a radiator whose sizeis much smaller than the wavelength (“dipole radiation”).

(2.31)(2.25)

(2.32)

(2.33)

(2.34)

a

Θ k0

t0′′

t0′

a

k0

sin2ΘΘ

a(b)(a)


Ch02_TotalPwrRad.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:

where for 0 ≤ Θ ≤ π and 0 ≤ φ ≤ 2π we have dΩ = sin Θ dΘ dφ, thus

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.

Thus the instantaneous power radiated to all angles by an oscillating electron of acceleration a is

For sinusoidal motion, averaging over a full cycle, sin2ωt or cos2ωt, introduces a factor of 1/2

(2.35)

(2.36)

12


Ch02_ScattCrSecs.ai

Scattering Cross-Sections

Measures the ability of an object to remove particles or photonsfrom a directed beam and send them into new directions

λ

Diminishedby bothscattering andabsorption

• Isotropic or anisotropic?• Energy or wavelength dependent?

σ area =Pscattered

S

(power)

(power/area)


Ch02_ScattFreElctrn1.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

For an incident electromagnetic wave of electric field Ei(r,t)

For a free electron the incident field causes an oscillatory motion described by Newton’s second equation of motion, F = ma, whereF is the Lorentz force on the electron

Thus the instantaneous acceleration is

(2.42)

(2.40)

(2.39)

(2.38)

small


Ch02_ScattFreElctrn2.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”

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

One obtains the scttering cross-section for a single free electron

(2.47)

(2.45)

(2.44)


Ch02_ScattBndElctrn.ai

Scattering by a Bound Electron

For an electromagnetic wave incident upon a bound electron of resonantfrequency ωs, the force equation can be written semi-classically as

the harmonic motion will be driven at the same frequency, ω, so that

Following the same procedures used earlier, one obtains the scatteringcross-section for a bound electron of resonant frequency, ωs

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

(2.48)

(2.49)

(2.50)

(2.51)


Ch02_SemiClassScatt.ai

Semi-Classical Scattering Cross-Sectionfor a Bound Electron

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.

(2.51)

(2.52)

Note that below the resonance, for ω2 << ωs2

σ = σT(ωs/γ)2

σ = σT(ω/ωs)4

(γ/ωs) = 10–1

Lorentzian line shape of half width γ/2

100

90

80

70

60

50

2

1

0 0.5 1.0 1.5 2.0 2.5 3.0Frequency ω/ωs

Sca

tterin

g cr

oss-

sect

ion,

σ(ω

)/σT


The sky appears blue because of the strong wavelength dependence of scattering by bound electrons.

Ch02_F09VG.ai

Sun

Settingsun

Earthly observer

More blue thangreen or red

σR – σT~

~

λsλ

4

Atmosphere(O2, N2, . . .)λS – 150 nm

• 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


Radiation by an Accelerated Charge: Scattering by Free and ...attwood/srms/2007/04_new_2007.pdf ·...

Documents

Transcript of Radiation by an Accelerated Charge: Scattering by Free and ...attwood/srms/2007/04_new_2007.pdf ·...