Section 31–2 The field due to the material

(Oscillator model / Equation of motion / Deriving dispersion equation)

 

In this section, the three interesting concepts are a simplified Lorentz oscillator model, an equation of motion of electrons (or oscillator equation), and the derivation of a dispersion equation. Based on the simplified version of Lorentz oscillator model, Feynman forms an equation of motion of electrons and then derives the dispersion equation. Thus, the title of this section could be “Simplified version of Lorentz oscillator model.”

 

1. Oscillator model:

“To find what motion we expect for the electrons, we will assume that the atoms are little oscillators, that is, that the electrons are fastened elastically to the atoms, which means that if a force is applied to an electron its displacement from its normal position will be proportional to the force (Feynman et al., 1963, p. 31–4).”

 

To be precise, electrons are connected to the nuclei because electrons are a part of the atoms. On the other hand, instead of using the phrase normal position, Lorentz (1916) writes: “[i]n the first place we shall conceive a certain elastic force by which an electron is pulled back towards its position of equilibrium after having been displaced from it (p. 136).” It is possible that some may argue whether the equilibrium position of the electron is located at the center of the nucleus. However, we can assume the negative charge forms an electron cloud (radius » 10^-10 m) surrounding the nucleus. Furthermore, we may idealize the core electrons to be tightly bound and thus only the valence electrons are oscillating due to the light waves.

 

“You may think that this is a funny model of an atom if you have heard about electrons whirling around in orbits. But that is just an oversimplified picture. The correct picture of an atom, which is given by the theory of wave mechanics, says that, so far as problems involving light are concerned, the electrons behave as though they were held by springs (Feynman et al., 1963, p. 31–4).”

 

Although this model of an atom is described as funny, it is a simplified version of Lorentz oscillator model. Perhaps Lorentz could be recognized for using Maxwell’s equations to explain the refractive index of dense materials. In his Nobel lecture titled The Theory of Electrons and the Propagation of Light, Lorentz mentions: “if attention is focused on the influence of the greater or smaller number of particles in a certain space an equation can be found which puts us in a position to give the approximate change in the refractive index with increasing or decreasing density of the body…” Historically, Lorentz has formulated the refractive index equation that is also known as the Clausius-Mossotti equation. In Volume II (Chapter 32), Feynman discusses the derivation of Clausius-Mossotti equation 3(n²−1)/(n²+2) = Nα (32.32).

 

2. Equation of motion:

“We have already studied such oscillators, and we know that the equation of their motion is written this way: m(d²x/dt²+ω₀²x) = F, (31.11) where F is the driving force. (Feynman et al., 1963, p. 31–4).”

 

Feynman writes the equation of motion of atomic oscillators as m(d²x/dt²+ω₀²x) = F, (31.11) where F is the driving force. However, the equation may also be written as F_driving + F_damping + F_restoring = m(d²x/dt²), i.e., m(d²x/dt²) is placed at the right-hand side as the resultant force. In Volume II, Feynman clarifies: “we did not include the possibility of a damping force in the atomic oscillators... Such a force corresponds to a resistance to the motion, that is, to a force proportional to the velocity of the electron (Feynman et al., 1964).” Specifically, the source of the damping force could be explained as random collisions with other atoms and radiation emitted by the electrons.

 

“For our problem, the driving force comes from the electric field of the wave from the source, so we should use F = q_eE_s = q_eE₀e^iωt,(31.12), where q_e is the electric charge on the electron and for E_s we use the expression E_s = E₀e^iωt from (31.10) (Feynman et al., 1963, p. 31–5).”

 

Feynman says that the driving force is due to the electric field of the electromagnetic wave from the source and we should use F = q_eE₀e^iωt where q_e is the electric charge on the electron. In Volume II, he elaborates: “[w]e assumed that the forces on the charges in the atoms came just from the incoming wave, whereas, in fact, their oscillations are driven not only by the incoming wave but also by the radiated waves of all the other atoms (Feynman et al., 1964).” In essence, the original field is combined with the induced field and results in a new field with a phase shift as compared to the original field. Perhaps Feynman should explain that it is traditional to define q_e as positive although q_e refers to the electron’s charge. However, we can obtain the equation (31.19) n = 1+Nq_e²/2ϵ₀m(ω₀²−ω²) by multiplying another q_e of the same sign from (30.18).

 

“Each of the electrons in the atoms of the plate will feel this electric field and will be driven up and down (we assume the direction of E₀ is vertical) by the electric force qE (Feynman et al., 1963, p. 31–4).”

“So we shall suppose that the electrons have a linear restoring force which, together with their mass m, makes them behave like little oscillators, with a resonant frequency ω₀ (Feynman et al., 1963, p. 31–4).”

 

In Volume II, Feynman explains restoring force as follows: “[w]e are assuming an isotropic oscillator whose restoring force is the same in all directions. Also, we are taking, for the moment, a linearly polarized wave, so that E doesn’t change direction (Feynman et al., 1964).” Firstly, the restoring force can be expressed as F = -kr by idealizing electrons as isotropic oscillators whereby this force is electric in nature. It is a linear force because the oscillation amplitude is small enough such that high order terms in the Taylor expansion of the electric force are negligible. Next, the source S is idealized (Fig. 31–1) to be infinitely far and thus the electric fields of oscillating electrons are in the same direction. This is known as the plane wave approximation as the direction of electric fields is perpendicular to the motion of the plane wave.

 

3. Deriving dispersion equation:

“Substituting NΔz for η and cancelling the Δz, we get our main result, a formula for the index of refraction in terms of the properties of the atoms of the material—and of the frequency of the light: n = 1+Nq_e²/2ϵ₀m(ω₀²−ω²) (31.19). This equation gives the ‘explanation’ of the index of refraction that we wished to obtain (Feynman et al., 1963, p. 31–5).”

 

Although Feynman claims that the equation (31.19) provides the explanation of the refractive index, it is based on a differential equation (equation of motion) of Newton’s second law of motion. In Volume II, Feynman says: “[t]his method obscures the physical origin of the index (as coming from the re-radiated waves interfering with the original waves), but it makes the theory for dense materials much simpler (Feynman et al., 1964).” In her book titled How the Laws of Physics Lie, Cartwright (1983) cites the above statement of Feynman and writes: “How does Feynman’s study of light in Volume I ‘make clear’ the physical principles that produce refraction? I do not have an answer (p. 162).” This is related to the “funny model of an atom” that involves many idealizations and approximations.

Interestingly, Cartwright (1983) elaborates: “I can tell you what Feynman does in Volume I, and it will be obvious that he succeeds in extracting a causal account from his model for low-density materials. But I do not have a philosophic theory about how it is done (p. 162).” In a sense, one may use the term toy model because this model is simpler than Lorentz oscillator model as it does not include a damping force. However, the phrase “thin gas plate” could be used instead of “glass plate” in the previous section. Perhaps Feynman could have clarified that the total phase shift is dependent on the number of gas atoms per unit volume in the so-called thin plate. Better still, there could be discussions on the accuracy of the toy model by comparing it with the experimental data.

 

To understand the model and derivation of refractive index, one should understand how the velocity of the charges in the equation (31.17) is retarded in time by z/c just like in the equation (30.19) such that it can be eliminated. However, some may disagree with Feynman’s method of approximation in (30.19), e.g., the formula is deduced by assuming a plane of oscillating charges that is infinitely large and without thickness. Furthermore, some may not feel comfortable with his explanation of letting e^−i∞ equal to zero due to the physical situations. He also claims that the formula (30.19) is correct at any distance z and substitutes NΔz for η to relate reflective index to the number of atoms. In summary, the velocity of the charges in the equation (31.17) requires not only many assumptions, such as the plate should be very thin, but this is essentially a toy model.

 

Review Questions:

1. How would you describe the simplified version of Lorentz oscillator model?

2. What are the idealizations and approximations needed to formulate the oscillator equation (or equation of motion)?

3. How would you explain the velocity of the charges in the equation (31.17) can be eliminated to obtain the dispersion equation?

 

The moral of the lesson: The explanation of reflective index is based on the simplified Lorentz oscillator model involving idealizations (isotropic oscillators, linear electric force, infinitely far source) and approximations (far field, plane wave, thin plate…).

 

References:

1. Cartwright, N. (1983). How the Laws of Physics Lie. New York: Oxford University Press.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Lorentz, H. A. (1916). The theory of electrons and its applications to the phenomena of light and radiant heat (2nd Ed.). New York: G. E. Stechert & Co.