(Radiation direction / Radiation mechanism / Radiation spectrum)
In this section, Feynman discusses the direction of radiation, radiation
mechanism, and radiation spectrum
pertaining to bremsstrahlung. The
term bremsstrahlung originated from the early days of radiation research
when physicists observed that high-energy electrons, when passing through
matter, appeared to slow down as they emitted radiation, including X-rays. In a
sense, bremsstrahlung is a misnomer (Melrose, 1980) because it does not simply
refer to the process of slowing down, but rather to the emission of bremsstrahlung
radiation caused by the scattering or deflection of electrons.
1. Radiation direction:
“So when very energetic
electrons move through matter they spit radiation in a forward direction. This
is called bremsstrahlung (Feynman et al., 1963, p. 34–6).”
Perhaps there is a little humor when Feynman says
that electrons move through matter and they spit radiation in a forward
direction. In Chapter 28, Feynman
explains: “our formula said that the
field should be the acceleration of the charge projected perpendicular to
the line of sight... So that checks the first rule, that there is no effect
when the charge is moving directly toward us.” Thus, the rule could be succinctly
expressed as “energy is most strongly radiated
perpendicular to the acceleration (Hecht, 2002, p. 60).” For
example, if the electron is accelerating in a straight line,
the energy radiated is strongest in the directions perpendicular to the
acceleration and there is no radiation in the forward direction. It
is a coincidence that the electron in circular motion radiates energy in the forward direction
because the direction of radiation is perpendicular to the centripetal
acceleration.
Feynman could have provided a diagram to show the
direction of bremsstrahlung radiation. Many diagrams of bremsstrahlung are
misleading, e.g., some textbooks and websites seem to suggest only
one photon can be radiated or there is only one direction of radiation (See figure
below). However, these diagrams do not show that electrons move through matter and spit radiation
in a forward direction. Importantly, it is an idealization to consider energy is
radiated only in the forward direction. If the
speed of a circulating electron is increased gradually, the backward lobe
(radiation pattern) will shrink and the forward lobe will elongate in the
direction of motion. That is, the electron moving near the speed of light would
radiate more energy along a narrower lobe (smaller solid angle) in the forward
direction.
2.
Radiation mechanism:
“Suppose
that there are charged particles in a piece of matter and a very fast electron,
say, comes by (Fig. 34–9). Then, because of the electric field around
the atomic nucleus the electron is pulled, accelerated, so that the curve of
its motion has a slight kink or bend in it (Feynman et al., 1963, p. 34–6).”
The bremsstrahlung is due to the deflection or
scattering of electrons by electric fields of nuclei and it results in a
decrease of kinetic energy of electrons. Feynman analyzes the radiation from a
kinematical perspective, i.e., the curve of its motion has a slight kink or
bend in it. It is worth mentioning that the exact path of an individual
electron during the bremsstrahlung phenomenon cannot be directly observed
during the process. Essentially, the bremsstrahlung radiation is emitted in
various directions and high energy electrons could have interacted with each
other, or electric fields of many nuclei and their surrounding electrons. The
collective behavior of a large number of electrons undergoing bremsstrahlung has
been studied by analyzing the radiation and its characteristics, such as energy
distribution and angular distribution.
“Remember
our rule: we take the actual motion, translate it backwards at speed c,
and that gives us a curve whose curvature measures the electric field. It was
coming toward us at the speed v, so we get a backward motion, with the
whole picture compressed into a smaller distance in proportion as c−v
is smaller than c. So, if 1−v/c << 1, there is a very sharp and
rapid curvature at B′, and when we
take the second derivative of that we get a very high field in the direction of
the motion (Feynman et
al., 1963, p. 34–6).”
Feynman explains that there is a very sharp
curvature at B′ by visualizing how the whole picture is compressed into a
smaller distance in proportion as c−v is smaller than c
provided 1−v/c ≪ 1. We can relate this to the classical Doppler
effect for an electron, which has a factor of 1/(1−v/c). If the
electron is moving close to the speed of light, we expect the effect of time
dilation (relativistic Doppler effect) and more intense radiation, such as
x-ray. (Coincidentally, Feynman needs to derive the formula for relativistic
Doppler effect in the next section.) The forward and backward lobe of an electron are
related to the relativistic Doppler effect and dependent
on the frame of reference as shown below. In essence, the blue-shifted
radiation (shorter l) emitted in the forward direction has a
narrower and longer lobe, whereas the red-shifted radiation (longer l) emitted in the backward
direction has a broader and shorter lobe.
3. Radiation spectrum:
“As a matter of fact, the
synchrotron is used, not so much to make high-energy electrons (actually if we
could get them out of the machine more conveniently we would not say this) as
to make very energetic photons—gamma rays—by passing the energetic electrons
through a solid tungsten ‘target,’ and letting them radiate photons from this
bremsstrahlung effect (Feynman et
al., 1963, p. 34–7).”
It is unclear why Feynman suggests the use of a
synchrotron and passing of energetic electrons through a solid
tungsten target to make very energetic photons—gamma rays. Firstly, the gamma
rays can be directly produced in a synchrotron due to the deflection of high energy
electrons under alternating magnetic fields (without the use of a tungsten
target). Secondly, the collisions of high energy electrons and some photons may
undergo the inverse Compton effect, also resulting in the production of
gamma rays. On the other hand, gamma rays are commonly identified as more
energetic radiation emitted from radioactive materials or nuclear processes. Alternatively,
some may explain how the interaction of high energy electrons (without the use
of a synchrotron) with the electrons of tungsten nuclei can create X-rays
instead of gamma rays.
The continuous distribution of X-rays, which forms
the base for the two sharp peaks is called bremsstrahlung.
That is, a synchrotron can generate a bremsstrahlung spectrum, which represents
a continuous range of electromagnetic radiation emitted by high-energy
electrons when they are scattered. The use of a tungsten target can modify the
bremsstrahlung spectrum by introducing characteristic spikes in addition to the
continuous spectrum (See figure below). Feynman’s sentence on the use of a
tungsten target could be revised as follows: “A synchrotron is capable of
generating a bremsstrahlung spectrum, which represents a continuous range of radiation emitted by
high-energy electrons. When a solid tungsten target is employed, the
bremsstrahlung spectrum can exhibit characteristic spikes superimposed on the
continuous spectrum.”
Note:
In Chapter 2, Feynman says “[t]hese two terms, x-rays and gamma rays, are used almost synonymously.
Usually electromagnetic rays coming from nuclei are called gamma rays, while
those of high energy from atoms are called x-rays, but at the same frequency
they are indistinguishable physically, no matter what their source (p. 2-5).”
The physicist who provided a
better explanation of bremsstrahlung is Hans Bethe, Feynman’s immediate boss in
Los Alamos. Bethe developed a comprehensive theory of bremsstrahlung emission with
Heitler, known as Bethe-Heitler theory of bremsstrahlung. In Bethe-Heitler
(1934) words, “[the] stopping power of matter for fast particles
is at present believed to be due to three different processes: (1) the
ionization; (2) the nuclear scattering; (3) the emission of radiation under the
influence of the electric field of a nucleus (p. 83).” There is a screening
effect whereby the presence of other electrons in the atoms can modify the
effective interaction between the high speed electron and the nucleus. Schwinger suggested an additional effect, the interaction
of electrons back on the field after reading Bethe-Heitler theory (Mehra & Milton,
2000). Bethe dismissed the idea by pointing out that the interaction
operator was non-Hermitian and thus unphysical.
Review
Questions:
1.
Do you agree with Feynman’s explanation that electrons moving through matter would spit
radiation in a forward direction?
2.
Do you agree with Feynman that the exact path of an individual electron during the bremsstrahlung phenomenon
usually has a slight kink or bend in it?
3. Do you agree with
Feynman that a synchrotron (tungsten filament?) and tungsten target should be
used to generate gamma rays (X-rays?)?
The
moral of the lesson: Bremsstrahlung is characterized
by a continuous range of spectrum (secondary radiation), which is
most strongly radiated perpendicular to its acceleration, due to the scattering of electrons (primary radiation) by electric fields of
nuclei.
References:
1. Bethe, H., & Heitler, W. (1934).
On the stopping of fast particles and on the creation of positive
electrons. Proceedings of the Royal Society of London. Series A,
Containing Papers of a Mathematical and Physical Character, 146(856),
83-112.
2. Eberhardt, W. (2015).
Synchrotron radiation: A continuing revolution in X-ray science—Diffraction
limited storage rings and beyond. Journal of Electron Spectroscopy and
Related Phenomena, 200, 31-39.
3. 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.
4. Hecht, E. (2002). Optics (4th
edition). San Francisco: Addison Wesley.
5. Mehra, J.
& Milton, K. A. (2000). Climbing the Mountain: The Scientific Biography
of Julian Schwinger. Oxford:
Oxford University Press.
6. Melrose, D. B. (1980). Plasma
astrohysics. Nonthermal processes in diffuse magnetized plasmas - Vol. 1: The
emission, absorption and transfer of waves in plasmas; Vol. 2: Astrophysical
applications. New York: Gordon and Breach.
7. Walker, J., Resnick, R., & Halliday, D. (2014). Halliday and Resnick fundamentals of physics. New Jersey: Wiley.
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