Larmor Formula: radiation from non-relativistic particles Feb. 28, 2011 Larmor Formula: radiation from non-relativistic particles Dipole Approximation Thomson Scattering

The E, B field at point r and time t depends on the retarded position r(ret) and retarded time t(ret) of the charge. Let Field of particle w/ constant velocity Transverse field due to acceleration

Qualitative Picture: transverse “radiation” field propagates at velocity c

Radiation from Non-Relativistic Particles For now, we consider non-relativistic particles, so Then E the RADIATION FIELD is and

Magnitudes of E(rad) and B(rad): Poynting vector is in n direction with magnitude

ASIDE: If Show that Need two identities: So… Now

Energy flows out along direction with energy dω emitted per time per solid angle dΩ cm2 ergs/s/cm2 so

Integrate over all dΩ to get total power LARMOR’S FORMULA emission from a single accelerated charge q

NOTES 1. Power ~ q2 2. Power ~ acceleration 2 3. Dipole pattern: No radiation emitted along the direction of acceleration. Maximum radiation is emitted perpendicular to acceleration. 4. The direction of is determined by If the particle is accelerated along a line, then the radiation is 100% linearly polarized in the plane of

The Dipole Approximation Generally, we will want to derive for a collection of particles with You could just add the ‘s given by the formulae derived previously, but then you would have to keep track of all the tretard(i) and Rretard(i)

One can treat, however, a system of size L with “time scale for changes” tau where so differences between tret(i) within the system are negligible Note: since frequency of radiation then or If This will be true whenever the size of the system is small compared to the wavelength of the radiation.

Can we use our non-relativistic expressions for ? yes. Let l = characteristic scale of particle orbit u = typical velocity tau ~ l/u tau >> L/c  u/c << l/L since l<L, u<c  non-relativistic

Using the non-relativistic expression for E(rad): If Ro = distance from field to system, then we can write L where Dipole Moment

Emitted Power power per solid angle power DIPOLE APPROXIMATION FOR NON-RELATIVISTIC PARTICLES

What is the spectrum for this Erad(t)? Simplify by assuming the dipole moment is always in same direction, let then

Let fourier transform of then

Then Recall from the discussion of the Poynting vector: Integrate over time: (1)

Parseval’s Theorem for Fourier Transforms  (2) Since E(t) is real FT of E  (See Lecture notes for Feb. 16) so Thus (2) 

Substituting into (1)  Thus, the energy per area per frequency  substituting and integrate over solid angle

NOTE: 1. Spectrum ~ frequencies of oscillation of d (dipole moment) 2. This is for non-relativistic particles only.

Thomson Scattering Rybicki & Lightman, Section 3.4

Thomson Scattering EM wave scatters off a free charge. Assume non-relativistic: v<<c. E field e = charge Incoming E field in direction electron Incoming wave: assume linearly polarized. Makes charge oscillate. Wave exerts force: r = position of charge

Dipole moment: so Integrate twice wrt time, t So the wave induces an oscillating dipole moment with amplitude

What is the power? Recall time averaged power / solid angle (see next slide) So

Aside: Time Averages The time average of the signal is denoted by angle brackets , i.e., If x(t) is periodic with period To, then

The total power is obtained by integrating over all solid angle: or

What is the Thomson Cross-section? Incident flux is given by the time-averaged Poynting Vector Define differential cross-section: dσ scattering into solid angle dΩ cross-section per solid angle cm2 /ster Power per solid angle erg /sec /ster Time averaged Poynting Vector erg/sec /cm2

Thus so since (polarized incident light) we get Thomson cross section classical electron radius

Integrate over dΩ to get TOTAL cross-section for Thomson scattering Thomson Cross-section NOTES: 1. Thomson cross-section is independent of frequency. Breaks down when hν >> mc2, can no longer ignore relativistic effects. 2. Scattered wave is linearly polarized in ε-n plane

Electron Scattering for un-polarized radiation Unpolarized beam = superposition of 2 linearly polarized beams with perpendicular axes

Differential Cross-section Average for 2 components Thomson cross-section for unpolarized light