Section 23–4 Resonance in nature

(Mechanical resonance / Nuclear resonance / Resonance particle)

In this section, Feynman discusses mechanical resonance, nuclear resonance, and resonance particle. Specifically, there are six different kinds of resonance, namely, tidal resonance, lattice resonance, nuclear magnetic resonance, nuclear resonance, recoilless nuclear resonance, and resonance particle.

1. Mechanical resonance:

“The first two are from mechanics, the first on a large scale: the atmosphere of the whole earth (Feynman et al., 1963, section 23–4 Resonance in nature).”

Tidal resonance: Feynman says the oscillator (earth’s atmosphere) is driven by the moon, which is effectively revolving about the earth. The resonant frequency corresponding to the rotation of the earth under the moon, which occurs at a period of 12.42 hours - 12 hours due to the earth’s rotation and a little more due to the moon’s rotation. It refers to a tidal resonance in which the largest constituent is the “principal lunar semi-diurnal” tide (M₂ waves). The resonance period is equal to half of a tidal lunar day or the time required for the Earth to rotate once relative to the Moon. This is analogous to the time required for the minute-hand on a watch that starts moving with the hour-hand at 12:00 and they meet again at about 1:05 instead of 1:00.

Feynman explains that we can get the resonant frequency ω₀ and the frequency width γ from the size of the atmospheric tides, and from the phase. He considers it to be an example of poor science if we simply draw a beautiful curve using two numbers instead of measuring something else. However, physicists do not have the liberty to vary the forcing frequencies of the earthquakes. One should recall how Fermi responded to Dyson’s model that uses four arbitrary parameters. Fermi’s reply was “I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk (Dyson, 2015, p. 125).” Nevertheless, we can use the Lorentzian resonance curve or spectral lineshape function (g/2)²/[(w₀-w)² + (g/2)²] to draw the so-called beautiful curve.

Lattice resonance: Feynman discusses another mechanical oscillation that involves a sodium chloride crystal. He suggests that we cannot say whether the resonance width in Fig. 23–7 is natural, or whether it is due to inhomogeneities in the crystal or the finite width of the slit of the spectrometer. To be precise, the absorption of infrared radiation is due to a lattice resonance instead of the so-called mechanical oscillation. Currently, experimental data shows that there is an energy band (or reststrahlen band) in which polar solids such as NaCl absorb and reflect light very strongly. The graph should be more complicated because of the interaction with phonons such that some infrared radiations cannot propagate within a given medium (Fox, 2002).

2. Nuclear resonance:

“Our next example has to do with atomic nuclei. The motions of protons and neutrons in nuclei are oscillatory in certain ways, and we can demonstrate this by the following experiment (Feynman et al., 1963, section 23–4 Resonance in nature).”

Nuclear magnetic resonance: Feynman explains that the nuclear magnetic resonance is about a swinging of atoms that have an angular momentum. Essentially, the frequency of the lateral magnetic field that drives this swinging is kept constant, but it is easier to change the magnetic field strength. One may clarify that the absorption and emission of electromagnetic radiations are related to electrons or atomic nuclei that have spins. The energy of electromagnetic radiations corresponds to the work done against the magnetic fields in turning the “tiny nuclear magnets” from one position to the opposite direction. Nuclear magnetic resonance was developed in 1945 by Felix Bloch and Edward M. Purcell, who were awarded the 1952 Nobel Prize in physics for their research and contributions.

Nuclear resonance: By bombarding a lithium atom with protons, the nuclear reactions produce γ-rays and the graph has a very sharp resonance. Feynman elaborates that the horizontal scale is not a frequency, but it is an energy that is related to the frequency of a wave. This is a nuclear resonance that is caused by the formation of new nuclei (Beryllium) in a particular excited state. On the other hand, the vertical scale corresponds to the intensity of gamma radiations, but it may also be measured in terms of the probability that an incident proton causes an emission of gamma-ray (French, 1971). This probability can be described using the effective target area (or cross section) of a nucleus that is hit by the protons.

Recoilless nuclear resonance: Mössbauer, Feynman’s former colleague, was awarded the Nobel prize in physics for his discovery of recoilless nuclear resonance. Feynman explains that the horizontal scale is velocity and the technique for obtaining different frequencies is related to the Doppler effect (or the relative speed between the source and the absorber). Specifically, a free nucleus (source) recoils after emitting a gamma-ray and the total energy (gamma-ray) absorbed by another nucleus (absorber) is lesser by the recoil energy. Strictly speaking, Mössbauer effect is observed when the nuclei are tightly bound such that the whole crystal recoils after the emission of a gamma-ray. In this case, spectral lines become very sharp because the mass of the crystal is practically infinite and thus, it is an essentially “recoil-less” resonance.

3. Resonance particle:

“We thus determine that there is a resonance at a certain energy for the K⁻ meson (Feynman et al., 1963, section 23–4 Resonance in nature).”

One should be cautioned that Feynman uses two different concepts, resonance states and resonance particles, to explain the resonance in particle physics. According to Feynman, there is a resonance found in a nuclear reaction when a K⁻ meson (kaon) and a proton interact. This results in some kind of a state corresponding to the resonance at a certain energy. In other words, it is a resonance state that corresponds to the energy of a kaon. Experimentally, it appears as a “bump” or “jerk” in a curve, but it could be related to a statistical fluctuation or systematic effect. In essence, resonance states are unstable states that are short-lived and they are similar to atomic energy levels. It is different from the concept of a resonance particle that has an invariant mass.

Feynman elaborates that we do not know whether to call a bump like this a “particle” or simply a resonance. When the resonance is very sharp, it means that it corresponds to a very definite energy as if there were a particle of that energy present in nature. In this view, one may conceptualize the resonance behaves effectively like a particle that has a cross-section and it is able to collide with other particles. In a sense, one may question the existence of resonance particles that have very short lifetimes. However, by using Einstein’s mass-energy relationship, we can deduce the invariant mass of a resonance particle.

Another interesting “bump hunting” is the discovery of a new subatomic particle J/y that is recognized for a Nobel prize in 1976. Lederman gives a nice explanation of the sharpness of resonance that is related to Heisenberg's uncertainty relations: “[t]he shorter the lifetime, the wider the distribution of masses. It is a quantum connection. What we mean by a distribution of masses is that a series of measurements will yield different masses, distributed in a bell-shaped probability curve (Lederman & Teresi, 2006, p. 316).” Lederman was possibly the first person to observe this resonance but he was not awarded for this Nobel prize. In his own words, “I was overjoyed at the breakthrough, a joy tinged, of course, with envy and even just a touch of murderous hatred for the discoverers (Lederman & Teresi, 2006, p. 315).”

Questions for discussion (Feynman’s mistakes?):

1. Should the resonance of sodium chloride crystal be explained as a small scale of mechanical oscillation?

2. Is the nuclear magnetic resonance is really about a swinging of atoms that have an angular momentum? 

3. Should the resonance in particle physics be explained using the concept of resonance states or resonance particles?

The moral of the lesson: the research in resonance has resulted in many Nobel Prizes such as nuclear magnetic resonance in 1952, recoilless nuclear resonance in 1961, and the J/y particle in 1976.

References:

1. Dyson, F. J. (2015). Birds and Frogs: Selected Papers of Freeman Dyson, 1990–2014. Singapore: World Scientific.

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. Fox, M. (2002). Optical properties of solids. New York: Oxford University Press.

4. French, A. P. (1971). Vibrations and Waves. New York: W.W. Norton.

5. Lederman, L., Teresi, D. (2006). God Particle: If the Universe Is the Answer, What Is the Question?. New York: Dell.