Section 32–3 Radiation damping

 (Oscillator’s Q-factor / Classical electron radius / Atoms’ resonance width)

 

In this section, Feynman discusses an oscillator’s Q-factor, classical electron radius, and atoms’ resonance width that are related to radiation damping or radiation resistance of the oscillator.

 

1. Oscillator’s Q:

“What is the Q of such an oscillator, caused by the electromagnetic effects, the so-called radiation resistance or radiation damping of the oscillator? The Q of any oscillating system is the total energy content of the oscillator at any time divided by the energy loss per radian: Q = −W/(dW/dϕ) (Feynman et al., 1963, p. 31–4).”

 

In the Audio Recordings* [16 min: 10 sec] of this lecture, Feynman actually asks: “What is the Q of such an oscillator, [not caused by the resistance of the motion of the charge in goo, but merely] caused by the electromagnetic effects, the so-called radiation resistance or radiation damping of the oscillator?” (goo is a viscid substance.) In a sense, we can define the Q-factor of an antenna as the power stored in the antenna divided by its radiated power. The term per radian in Feynman’s definition of Q-factor means that there is a cycle (or 2p) in the definition because an oscillatory motion can be viewed as the projection of uniform circular motion onto one axis. That is, we may also define Q as 2p(max stored energy per cycle)/(energy lost per cycle), or Q = (total stored energy)/(energy lost per radian) in which 2p is hidden in per radian. However, Feynman’s explanation of Q in terms of 2p [16 min: 35 sec]* is omitted in the edited lecture.

 

*The Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html

 

“Or (another way to write it), since dW/dϕ = (dW/dt)/(dϕ/dt) = (dW/dt)/[ω(dW/dt)/ω], (32.8) Q = −ωW/(dW/dt). If for a given Q this tells us how the energy of the oscillation dies out, dW/dt =−(ω/Q)W, which has the solution W=W₀e^−ωt/Q if W₀ is the initial energy (at t = 0). (Feynman et al., 1963, p. 31–4).”

 

In the Audio Recordings [16 min: 55 sec]* of this lecture, Feynman says: “no trick, it is just the same expression as here,” but it may involve some non-trivial tricks. There are possibly three tricks or hidden concepts: (1) Decay law or exponential function, (2) Chain rule, and (3) working backward that is related to a damping oscillator. Firstly, dW/dt = −(ω/Q)W is similar to the radioactive decay law dN/dt = −lN in which l is the decay constant. Secondly, dW/dϕ = (dW/dt)/(dϕ/dt) is based on a chain rule to introduce ω = dϕ/dt. Thirdly, (dW/dt)/(dϕ/dt) = (dW/dt)/[ω(dW/dt)/ω] is a “working backward” trick because we have the end result in mind and it justifies the use of ω or 2p in the final formula. Some may need to revise the damping oscillation equation md²x/dt²+ γmdx/dt + mω₀²x = 0 that contains γ as a dissipative constant (See chapter 23).

 

2. Classical electron radius:

“It has been given a name, the classical electron radius, because the early atomic models, which were invented to explain the radiation resistance on the basis of the force of one part of the electron acting on the other parts, all needed to have an electron whose dimensions were of this general order of magnitude (Feynman et al., 1963, p. 31–4).”

 

The classical electron radius r₀ is defined by equating the electrostatic potential energy of a sphere of charge (U = e²/4pe₀r_e) with the mass-energy of the electron, E = m_ec². Perhaps Feynman could have explained why he did not consider another electron’s radius such as Bohr radius (a_B ≈ 5.3 × 10⁻¹¹ m) or Compton wavelength of the electron λ_c. (Note: r_e = αλ_c = α²a_B; α ≈ 1/137 is the fine structure constant.) In section 32.1, Feynman mentions that an electron is not a “little ball,” i.e., the size of an electron is dependent on the nature of the experiment or theoretical model. We may expect a different effective electron size in an antenna due to the radiation damping depending on the antenna frequency.

 

In Chapter 36, Feynman adds: “the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither (Feynman et al., 1963, p. 36-1).” For example, Bender et al. (1984) found from electron scattering experiments at 29 GeV that the upper limit of the size of the electron is about 10^-18 m. Thus, theoretical physicists consider an electron to be a point particle because there is no measurable structure down to 10^-18 m. However, Wilczek (2013) explains: “[t]here is tension between these two observations, that the electron is a simple point-particle, and that it contains the world (p. 31).” In other words, the electron may appear to spread over a region of space that is limited by the size of the universe.

3. Atoms’ resonance width:

“The effective resistance term γ in the resistance law for the oscillator can be found from the relation 1/Q = γ/ω₀, and we remember that the size of γ determines how wide the resonance curve is (Fig. 23–2). Thus we have just computed the widths of spectral lines for freely radiating atoms! (Feynman et al., 1963, p. 31–5).”

 

Instead of writing the relation 1/Q = γ/ω₀, one may prefer to use the symbol Q₀ and state another definition of Q-factor. That is, Q₀ is a ratio of the resonant frequency ω₀ to the full width at half-maximum bandwidth Δω of the resonance curve, i.e., Q₀ = ω₀/Δω. In section 23-2, Feynman elaborates: “we ask for the width Δω of the curve, at one half the maximum height, the full width at half the maximum height of the curve is Δω = γ, supposing that γ is small (Feynman et al., 1963).” It is worthwhile mentioning that the “energy loss” definition Q = −W/(dW/dϕ) is applicable to any ω, whereas the bandwidth definition Q₀ = ω₀/Δω is dependent on ω₀. In essence, these two definitions give different numerical results, but they are approximately equivalent for larger Q values (i.e., lower γ or damping).

 

“This is valid only for atoms which are in empty space, not being disturbed in any way. If the electron is in a solid and it has to hit other atoms or other electrons, then there are additional resistances and different damping (Feynman et al., 1963, p. 31–5).”

Feynman’s computation of the widths of spectral lines for freely radiating atoms in empty space is based on the classical electron radius. Historically, the early atomic models were invented to explain the radiation damping and all that are needed to have an electron whose dimensions were of this general order of magnitude. Currently, the classical electron radius does not have any physical significance in the sense that whether the electron really has such a radius. We may conceptualize the electron to have a different size depending on the solid (e.g., antenna) involved. Perhaps Feynman could have discussed the Q of an antenna empirically and deduced the effective size of the electron in the antenna.

 

Review Questions:

1. Would you define the Q of an oscillator in terms of 2p, e.g., Q = 2p(max stored energy per cycle)/(energy lost per cycle)?

2. How would you explain the physical significance of classical electron radius?

3. Would you compute the widths of spectral lines using classical electron radius that does not have any physical significance?

 

The moral of the lesson: The widths of spectral lines for freely radiating atoms based on Feynman’s computation are valid only for atoms that are in empty space and are not being disturbed in any way (except radiation damping).

 

References:

1. Bender, D., Derrick, M., Fernandez, E., Gieraltowski, G., Hyman, L., Jaeger, K., ... & Va'Vra, J. (1984). Tests of QED at 29 GeV center-of-mass energy. Physical Review D, 30(3), 515-527.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Wilczek, F. (2013). The enigmatic electron. Nature, 498(7452), 31-32.