RADIATION REACTION
2.1 PHYSICAL DESCRIPTION AND HISTORICAL ORIGIN
A good introduction to the physical significance of the
radiation reaction force is given by Robert March in Physics for Poets:[i]
...when
electric charge is moved, an additional force is required above and beyond that
needed to accelerate an uncharged mass.
This force is a consequence of the interaction of the charge with its own field, regardless of
whether or not other charges are present.
In the process, forces are exerted on, and thus momentum is imparted to
and work is done on, the field itself.
This description alludes to the importance of the idea of
the electromagnetic field, the necessary carrier of the radiation reaction
force as well as the carrier of the outgoing radiation emitted by the
accelerating charge. The radiation reaction problem has been subjected to
investigation for nearly as long as Maxwell’s equations for the electric and
magnetic fields have existed. Thus,
radiation reaction has been studied theoretically for over a hundred years,[ii]
from the work of Larmor and Stewart in the 1870s, and Planck, Lorentz, and
Abraham around the turn of the century, to Dirac in the 1930s, and the work of
Bethe, Feynman, Schwinger, Tomonaga, and Dyson in the late 1940s that resulted
in a renormalizable theory of quantum electrodynamics. “Renormalizable” means, among other things,
that a way was found around the difficulties presented by the infinite
electromagnetic mass of a point electron interacting via radiation reaction
with its own electromagnetic field.
One of the recurrent questions is whether the electron is a
point charge or an extended charge.
Partly because of this open question, there have been many papers written
about the issue of classical radiation reaction since the advent of quantum
electrodynamics. The review article just
cited is a comprehensive guide to research published from the 1870’s to 1961. The Quantum Vacuum: An Introduction to
Quantum Electrodynamics, by Peter W. Milonni,[iii]
provides a more recent discussion of both classical and quantum radiation
reaction research.
2.2 THE ABRAHAM-LORENTZ EQUATION
The most significant early work was done in
the years 1903-1905, when the German physicist Max Abraham and the Dutch
physicist H. A. Lorentz independently developed an equation of motion for a
spherical electron.[iv] Their equation includes a radiation reaction
force.
A derivation of the Abraham-Lorentz equation involves
calculating the force of each part of an accelerating spherical shell of charge
on all the other parts and summing the contributions, giving a net self-force
due to the retardation time or propagation time of the electromagnetic field
across the finite extent of the electron.
As described by Griffiths and Szeto:[v]
Lorentz
showed that the radiation reaction force is attributable to the breakdown of
Newton's Third Law in classical electrodynamics. When an extended charge accelerates, the
force of one part on another is not
equal and opposite to the force of the second part on the first. When these imbalances are integrated over the
entire configuration, the result is a net force of the charge on itself.
Lorentz’s calculation is given in his
book Theory of Electrons,[vi]
and similar calculations appear in several recent texts.[vii] The method uses a series expansion of the vector potential in the retardation
time 2a/c, where a is the electron radius and c
is the speed of light. For a radius of 10-15 meters the
retardation time is on the order of 10-24 seconds, and the electron is considered to
remain approximately stationary during this very short time interval. In this approximation “only terms which are
linear in the particle’s velocity or its time derivatives”[viii]
are kept. Thus, magnetic effects are not
considered, and the calculation is therefore nonrelativisitic. It can be made fully relativistic by
including terms of order (v/c)2
and higher.
The nonrelativistic result for the
self-force, simplified to the case of one-dimensional motion, is given by
Feynman:[ix]
Fself (x,t) = —(e2/6ε0ac2)
d2x/dt2 + (e2/6ε0c3) d3x/dt3 + (ae2/6ε0c4)
d4x/dt4 + …
The first
term is a constant times acceleration, and thus the units of the constant are
mass units (kilograms in the mks system used here). This is the electromagnetic mass term. The electron radius a appears in the denominator of this term, but does not appear in
the second term and then appears linearly in the third term and to succeedingly
higher powers in subsequent terms. Thus
if the electron radius approaches zero, as it does in the limit of a point
charge, the higher order terms go to zero, the second term is unaffected, and
the electromagnetic mass term approaches infinity.
For finite but small values of the
retardation time (the assumption that leads to the series approximation), the
third and higher order terms are negligible. The second term, which shows no
dependence on the electron’s structure, is the radiation reaction term. As described in Chapter 1, the effect of this
term on the subsequent motion of the charged particle is often considered to be
negligible. The radiation reaction term,
however, is necessary to precisely account for energy conservation, as will be
shown in Section 2.3.
Putting the remaining two
self-interaction terms into the equation of motion for an electron with bare
mass m0 subjected to an
external force gives
m0 d2x/dt2 = —(e2/6ε0ac2)
d2x/dt2 + (e2/6ε0c3)
d3x/dt3 + Fext
or
(m0 + e2/6ε0ac2)
d2x/dt2 = (e2/6ε0c3)
d3x/dt3 + Fext
The sum of
mechanical mass and electromagnetic mass on the left-hand side of this equation
is considered to be the observable or experimental mass of the electron, m. Writing the equation in
terms of m gives the one-dimensional
Abraham-Lorentz equation of motion for the electron,
m
d2x/dt2 = (e2/6ε0c3)
d3x/dt3 + Fext
The coefficient of the
d3x/dt3 term is often written
as mτ , where τ is given by
τ = e2/6ε0 mc3
which is of the same order
of magnitude as the retardation time.
Then the Abraham-Lorentz equation is written
m(d2x/dt2 - τd3x/dt3) = Fext
As noted by
Jiménez and Campos[x]
(who use a for acceleration
da/dt term, which involves
an extension of the phase space and the specification of a at some time.” The
consequences of the departure from a standard Newton’s second law structure are
discussed in Section 2.4.
As Erber[xi]
points out, however, the A-L equation despite some of its pathological
solutions has been accepted as valid because it accounts for the dynamical
effects of radiation reaction “through an augmented form of Newton’s second
law” so that one “universal form of [Fself] is sufficient for all cases,” and
because the equation has a “natural place in a relativistic setting.”
For a bound electron oscillating under the action of a
linear restoring force, the equation of
motion including radiation reaction becomes
to be written here soon!
If the
radiation reaction is considered to be small compared to the other terms, the
simple harmonic motion (specified by assuming x(t) = xocos(wt))
allows replacement of
, giving
For an electric field Ex(t) as the external force,
the equation becomes
which is
the classical equation of motion for an electron in an atom interacting with light. (The Appendix discusses the first derivation
of these equations, published in 1897 by Max Planck. Planck was considering the effects of
electromagnetic damping on the entropy of abstract electromagnetic dipole
oscillators, one aspect of the so-called blackbody problem.)
2.3 RADIATION REACTION AND ENERGY CONSERVATION
When an uncharged particle of mass m is accelerated by a net force F, the work done in accelerating the
particle is equal to the change in its kinetic energy. However, an accelerating charged particle, in
addition to experiencing a change in kinetic energy, produces electromagnetic
radiation. As Griffiths and Szeto[xii]
describe it, "radiation carries away energy, which, for a structureless
particle, must come at the expense of kinetic energy." Under this assumption--that translational
kinetic energy alone is converted into electromagnetic energy--radiation
reaction must somehow be included in a work-energy calculation in order to
satisfy energy conservation.
For the case of periodic motion, this
will be accomplished if the rate at which the radiation reaction force does
work on the charge is shown to be equal to the rate of energy loss to
radiation, which is given by the Larmor formula.
The self-force equation, truncated to
two terms, is
Since the
first term is the electromagnetic mass term and is joined with mechanical or
bare mass in the electron equation of motion, it will be included in the rate
of change of kinetic energy of the electron caused by an external force. Multiplying the self-force equation by
shows this and also
shows how the rate of energy loss to radiation comes from the radiation
reaction term. Using
and multiplying the
first term by
gives
showing how
the electromagnetic mass contributes to the electron’s kinetic energy for the
classical charged spherical shell model. The radiation reaction term multiplied
by
is
which can
be written as
The first
term on the right-hand side averages to zero over any integral multiple of the
period of motion. The second term is the
square of the acceleration and thus
does not average to zero over a period of the motion. The result is that the average rate of energy loss due to the radiation reaction
force is identical to the Larmor formula for energy radiated
The
radiation reaction force, although it may be small compared to external forces,
is required for the energy of an accelerating charge to be conserved.
2.4 RUNAWAY SOLUTIONS
The
Abraham-Lorentz equation is abnormal as an equation of motion for a single
particle. Although it is linear, it
contains a third time derivative of
position, which requires that an initial value of acceleration in addition to
initial velocity and position. Third
order equations can show up in other single particle situations, for instance
in solving simultaneous second order equations.
One example is the motion of a charged particle in a region with both a
magnetic and an electric field.[xiii] In this case the solutions make sense
physically.
The Abraham-Lorentz equation has been
described as predicting “unphysical behavior because it is higher order in time
differentiation than a mechanical equation of motion should be”[xiv]
and this “involves an extension of the phase space.”[xv] Erber[xvi]
notes that the equation is actually a functional equation rather than a
differential equation, and this “implicit x(t)
dependence of Fself . .
. endows the classical theory with its astonishing wealth of solutions.”
The wealth of solutions includes a
prediction of exponentially increasing acceleration even when no external force
is applied. This is the homogeneous
equation
and solving
for the acceleration gives
which
implies that radiation reaction causes spontaneous acceleration of the
electron. The general solution--i.e.,
when an external force is included--necessarily contains the solution to the
homogenous equation. The general
solution can be written as[xvii]
which
predicts that acceleration increases exponentially no matter what the form the
external force takes.
In Section 4.1, one method of avoiding
the runaway solutions is discussed, but it leads to the problem of
pre-acceleration. In Section 4.2, where
the Abraham-Lorentz equation is rewritten as a delay-differential equation, the
problem of runaways and pre-acceleration is shown to be avoidable by avoiding
the point charge limit and its associated infinite electromagnetic mass. First the concept of electromagnetic mass
needs to be discussed in more detail.
[i]
Robert March, Physics for Poets, 2nd
ed. (McGraw-Hill, New York, 1978), p. 76.
[ii]
Thomas Erber, “The Classical Theories of Radiation Reaction,” Fortschr. Phys. 9, 343-392 (1961).
[iii]
Peter W. Milonni, The Quantum Vacuum: An
Introduction to Quantum Electrodynamics (Academic
Press, San Diego, 1994), Chapter 5.
[iv]
Jackson, p. 786.
[v]
D. J. Griffiths and Ellen Szeto, “Dumbbell model for the classical radiation
reaction” Am. J. Phys. 46, 244-248
(1978).
[vi]
H. A. Lorentz, Theory of Electrons (B. G. Teubner, Leipzig 1909), Note 18. This is a series of lectures Lorentz
delivered at Columbia University in 1906, later reprinted in paperback (Dover
Publications, New York, 1952).
[vii]
See, for instance, Jack Vanderlinde,
Classical Electromagnetic Theory (John Wiley, New York, 1993), Chapter 12.
[viii]
D. H. Sharp, “Radiation Reaction in Nonrelativistic Quantum Theory,” in Foundations of Radiation Theory and Quantum
Electrodynamics, ed. A.O. Barut (Plenum, New York, 1980), p. 130.
[ix]
R. P. Feynman, R. Leighton, and M. Sands, The
Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading MA,
1964), Chapter 28.
[x]
J. L. Jiménez and I. Campos, “A critical examination of the Abraham-Lorentz
equation for a radiating charged particle,” Am. J. Phys. 55, 1017-1023 (1987).
[xi]
Erber, p. 349.
[xii]
Griffiths and Szeto, p. 244.
[xiii]
K.R. Symon, Mechanics, 3rd edition
(Addison-Wesley, Reading MA, 1971), pp. 144-148.
[xiv]
Jackson, p. 798.
[xv]
Jiménez and Campos, p. 1019.
[xvi]
Erber, p. 371.
[xvii]
Milonni, p. 157.
