Section 34–3 Synchrotron radiation

 (Uniform magnetic field / Polarized light / Synchrotron radiation spectrum)

 

In this section, Feynman discusses synchrotron radiation from the perspective of unform magnetic field, polarized light, and synchrotron radiation spectrum. Synchrotron radiation was initially sometimes known as “Schwinger radiation” based on Schwinger’s contribution on the development of synchrotron.

 

1. Uniform magnetic field:

We have very fast electrons moving in circular paths in the synchrotron; they are travelling at very nearly the speed c, and it is possible to see the above radiation as actual light!... In the synchrotron we have electrons which go around in circles in a uniform magnetic field. (Feynman et al., 1963, p. 34–3).”

 

In a synchrotron, it is not true that we have electrons which go around in circles in a uniform magnetic field. The electrons do not move circularly in the synchrotron because of insertion devices, i.e., undulators and wigglers, that generate alternating magnetic fields. These insertion devices are arrays of magnets used to control the electrons to generate more intense radiation. In Chapter 29 of Volume II, Feynman elaborates on the alternating magnetic field: “Let’s return now to the synchrotron guide magnet. We can consider that it consists of an alternating sequence of ‘positive’ and ‘negative’ lenses with a superimposed uniform field...” However, he did not specifically mention undulators or wigglers that are used to laterally deflect the electrons inside a storage ring of a synchrotron.

 

“A particle moving on a circle of radius 3.3 meters, or 20 meters circumference, goes around once in roughly the time it takes light to go 20 meters. So the wavelength that should be emitted by such a particle would be 20 meters—in the shortwave radio region (Feynman et al., 1963, p. 34–4).”

 

Particle physicists may not agree with Feynman’s derivation of wavelength because the magnetic field of a synchrotron is not constant. More important, the electron is moving at a relativistic speed and thus there should be a factor of g multiplies the mass, such that the angular velocity of the electron is expressed as w = qB/mg. On the other hand, the formula w = qB/m is applicable to a classical cyclotron if the magnetic field is constant and the electron is not moving at a relativistic speed. For higher speeds, a synchrocyclotron can be used to accelerate the electron. Currently, complicated mathematical expressions such as Bessel’s functions and Fourier transform are used to explain synchrotron radiation.

 

We should not assume that electrons can maintain stable circular orbits under a uniform magnetic field. During an interview, as Saxon recalled, ‘[Schwinger] and I worked together on a paper on the stability of synchrotron orbits. We both knew exactly the same about the problem, namely zero. That's one of the times I really had the rare privilege of seeing him start from absolute scratch. We struggled the first night. The question was how to do it – we worked all through the night trying to figure it out. The next day I came in and said, “I think I know how to do it!” and Julian said, “So do I.” He had the complete theory. I knew how to do it, but he had done it. The amusing thing about the story is that people thought the orbit was unstable, because they had run it on a computer, actually a differential analyzer, and there were instabilities in the formulation. But Julian and I demonstrated that it was stable (Schwinger, 2000, p. 300).’

 

2. Polarized light:

“The acceleration, which involves a second derivative with respect to time, gets twice the “compression factor” of 8×10⁶ because the time scale is reduced by eight million twice in the neighborhood of the cusp. Thus we might expect the effective wavelength to be much shorter, to the extent of 64 times 10¹² smaller than 20 meters, and that corresponds to the x-ray region (Feynman et al., 1963, p. 34–5).”

 

Perhaps some may prefer Schwinger’s explanation of effective frequency that is equivalent to effective wavelength. In Schwinger’s (2002) words, “the detected electromagnetic field changes rapidly as the narrow beam swings past. Rapid change means high frequency; it works out that the frequency is increased by about a factor given by the value of g for the electron’s speed. But that is not all…… the detected radiation arrives in an interval much shorter than the duration of the emission, because the radiating electron almost keeps pace with its own radiation. That produces a further boost in frequency; it is about equal to the square of g. Taken together these relativistic effects increase the frequency of the detected radiation by a factor that is now equal to about the cube of g (pp. 117-118).” Alternatively, the detected radiation can be explained using relativistic Doppler effect (i.e., classical Doppler effect and time dilation).

 

“Thus, even though a slowly moving electron would have radiated 20-meter radiowaves, the relativistic effect cuts down the wavelength so much that we can see it! Clearly, the light should be polarized, with the electric field perpendicular to the uniform magnetic field (Feynman et al., 1963, p. 34–5).”

 

It is unclear why Feynman considers the light or synchrotron radiation should be clearly polarized. However, it may not be obvious that an electron moving slowly in a circular orbit will radiate a doughnut-shaped pattern. If the electron increases its speed, the back lobe of the radiation pattern will shrink and the forward lobe will elongate in the direction of motion* as shown below (related to relativistic Doppler effect). It may also not be obvious that the electron moving near the speed of light will radiate more energy along a narrower lobe in the direction of motion (Hecht, 2002). Interestingly, the synchrotron radiation would be polarized in the plane of the circular motion when v = c, but it is an idealized concept because the electron is unable to move at the speed of light.

Source: Woodruff, D. (2021). Synchrotron Radiation: Sources and Applications to the Structural and Electronic Properties of Materials. Cambridge: Cambridge University Press.

*In the Audio Recordings [22 min: 30 sec] of this lecture, Feynman says something like: a man stands here will see a lot of lights from the synchrotron, whereas another man will see no light… This could be explained by relativistic Doppler effect.

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

 

Perhaps it is more appropriate to say that the synchrotron radiation has a high degree of polarization. From an experimental perspective, there are variations in the magnetic field strengths along the electron’s path such that it is not a circular orbit. Furthermore, the electrons in the synchrotron have a spread in their energies and momenta that lead to variations in the radiated wavelengths and a broader angular distribution of the radiation. Besides, the imperfect practical synchrotron devices can introduce unintentional transverse components in the emitted radiation that further deviate it to form an ideal planar polarization. Thus, the predominant direction of the electric vector of the synchrotron radiation lies in the same plane of the orbiting electrons, i.e., in the direction of the acceleration.

 

3. Synchrotron radiation spectrum:

“The electromagnetic radiation emitted by relativistic charged particles circulating in a magnetic field is called synchrotron radiation. It is so named for obvious reasons, but it is not limited specifically to synchrotrons, or even to earthbound laboratories (Feynman et al., 1963, p. 34–5).”

 

Feynman defines synchrotron radiation as the electromagnetic radiation emitted by relativistic charged particles circulating in a magnetic field. As a suggestion, we may explain synchrotron radiation from the perspective of magnetic field, polarization, and spectrum: (1) magnetic field: relativistic charged particles are moving under alternating magnetic fields; (2) polarization: the predominant direction of the electric field of the synchrotron radiation lies mainly in the same plane of the orbiting particles; (3) spectrum: This radiation is extremely intense over a broad range of wavelengths extending from the infrared through the visible and ultraviolet range, and into the x-ray region (or gamma ray) of the electromagnetic spectrum. To summarize, the synchrotron radiation as produced by alternating magnetic fields has a characteristic polarization over a broad range of frequencies.

 

“(Actually, the cusp itself is not the entire determining factor; one must also include a certain region about the cusp. This changes the factor to the 3/2 power instead of the square, but still leaves us above the optical region.)… In short, the sum of the reflections from all the successive wires is as shown in Fig. 34–6(a); it is an electric field which is a series of pulses, and it is very like a sine wave whose wavelength is the distance between the pulses, just as it would be for monochromatic light striking the grating!  (Feynman et al., 1963, p. 34–5).”

Feynman explains how the synchrotron radiation spectrum can be separated into red light, blue light, and so on using a diffraction grating, which is a lot of scattering wires. The 3/2 factor mentioned by Feynman could be related to Schwinger’s (1949) derivation of radiation power in terms of harmonic frequency (n) and angular distribution. (Instead of using Fig. 34-6, the spectrum consists of numerous harmonic frequencies as shown below because the circular motion is periodic.) In Schwinger’s (1949) words, “The discrete spectrum is effectively a continuum if n >> 1 (p. 1924),” i.e., Schwinger’s equations help to explain how the synchrotron radiation appears as a continuous spectrum. It is worth mentioning that synchrotron radiation is a direct proof of the physical unity of the electromagnetic spectrum from radio waves to x-rays and gamma rays (Schwinger, 2002).

Source: https://asd.gsfc.nasa.gov/Volker.Beckmann/school/download/Longair_Radiation2.pdf

Review Questions:

1. Is it true that electrons go around in circles and under a uniform magnetic field in a synchrotron?

2. Do you agree with Feynman that the synchrotron radiation is clearly polarized?

3. How would you define synchrotron radiation?

 

The moral of the lesson: the synchrotron radiation (produced by alternating magnetic fields) is very intense and has a characteristic polarization over a broad range of frequencies from the infrared through the visible and ultraviolet range, as well as the x-ray or gamma ray.

 

References:

1. 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.

2. Hecht, E. (2002). Optics (4th edition). San Francisco: Addison Wesley.

3. Schwinger, J. S. (1949). On the classical radiation of accelerated electrons. Physical Review, 75, 1912.

4. Schwinger, J. S. (2000). A quantum legacy: seminal papers of Julian Schwinger (Vol. 26). Singapore: World Scientific.

5. Schwinger, J. S. (2002). Einstein's legacy: the unity of space and time. Courier Corporation.