Classical Electron: Intro

(This is my master's thesis in physics, Southwest Texas State University, 1998.  SWT is now called Texas State University-San Marcos.  My thesis, which is called The Binding Energy of the Classical Electron, is more history and its interpretation than actual physics.  This blog version is being edited by me as I have time.  At the moment it doesn't have the equations that are supposed to go with it.  I hope to include them after I figure out how to do that!)

INTRODUCTION

          In the 100 years since the discovery of the electron, no one has produced a satisfactory answer to the most elementary questions about this first inhabitant of what has been called[i] the subatomic particle zoo.  Although it has been defined by precise measurements of its charge, charge-to-mass ratio, and magnetic moment, and even though the experimental value of magnetic moment fits very snugly into the calculational confines of quantum electrodynamics, the electron is still a mystery.  Why does it have the charge and mass it has?  Why is the charge precisely the same as the proton’s charge (but opposite in sign), and why do the electron and proton have the charge-to-mass ratios they have?  Until these conceptual questions are answered and the mass and charge of the electron are predicted by a theory, rather than inserted as parameters in quantum electrodynamics as they are now, we cannot say we understand what an electron is.

           The discovery of the electron in 1897 was accomplished by J. J. Thomson’s experimental determination of the charge-to-mass ratio for the formerly unexplained cathode rays, showing that the “rays” are actually streams of charged particles.  Several kinematical or dynamical similarities between charge and mass indicate the charge-to-mass ratio is of fundamental importance in the development of a theory of the electron:   Point charges and point masses produce inverse square fields, charge and mass are both relativistic invariants (scalars)--although mass is often improperly described as varying with velocity,[ii] and charge and mass both resist being accelerated.  Mass resists acceleration because of its mysterious property called inertia.  Charge resists acceleration because it creates a time-dependent electromagnetic field when accelerated and, in conserving momentum, the field reacts back on the charge with a force called the radiation reaction force. 

          The radiation reaction phenomenon has been chosen as the subject of this thesis because the difficulties which arose in classical models of the electron’s radiation reaction have not been solved by quantum electrodynamics. The phenomenon of radiation reaction also represents a partial unification of the ideas of charge and mass, because the radiation reaction force causes an electron to act as if it has an electromagnetic mass in addition to its bare or mechanical mass.  This apparent combination of two kinds of inertial mass--and their apparent inseparability--is a problem “we still don’t understand,” as Pais says.[iii]

           In classical electrodynamics, the Maxwell equations are used to calculate fields when charges and currents are specified, or fields can be given and the distributions of charge and current calculated.  Forces on charge and current distributions can also be calculated from the Lorentz force equation.  As Jackson[iv] notes, however, both regimes usually ignore the radiation reaction effect, partly because there is “negligible error” when acceleration of charges is not extremely rapid, and partly because “a completely satisfactory treatment of the reactive effects of radiation does not exist.”

          Although quantum electrodynamics has been successful in giving precise calculations of radiative effects, it fails to put such calculations on a sound theoretical foundation.  As Jackson[v] says:

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of an elementary particle....One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties....It is one of the triumphs of comparatively recent years (1948-1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment.  From a fundamental point of view, however, the difficulties still remain.

          One very significant domain where accelerations of a charge are rapid enough that the radiation reaction force cannot be neglected is in classical and quantum models of light emission and absorption by atoms.  Because of the oscillatory nature of the electron’s motion in this case, radiation reaction is called radiation damping.  In the classical model of light emission and absorption, the electron is bound to the nucleus by a linear (Hooke’s law) restoring force, and there is a velocity-dependent radiation damping term in the equation of motion of the electron.  However, this velocity-dependent term is just the simple harmonic motion approximation to the more general case where radiation damping is proportional to  , the rate of change of acceleration. 

          The need for a rate of change of acceleration in a single-particle force law is the distinguishing characteristic of radiation reaction. The presence of   in the electron’s equation of motion--the classical Abraham-Lorentz equation--leads to physically senseless solutions.  In one case, exponentially increasing “runaway” solutions are predicted.  Choosing reasonable asymptotic conditions on the electron’s acceleration and avoiding the runaway solutions leads to “pre-acceleration,” the electron accelerating before an external force is applied. These problems are discussed in Chapter 2 of this thesis.

          Another difficulty of both classical and quantum electrodynamic models of radiation reaction is that the electromagnetic mass of a point electron is infinite.  This subject is discussed in Chapter 3, along with the so-called “4/3 problem”--a disagreement between relativistic and purely electrodynamic predictions of the electromagnetic mass of a spherical shell of charge.  In Chapter 4, classical models of the point charge electron and the extended spherical shell electron are discussed, and the spherical shell model is shown to be free of the difficulties of runaway solutions, pre-acceleration, and infinite electromagnetic mass.

          In Chapter 5, Discussion and Conclusion, the 4/3 problem is  considered as logical consequence of the classical electron requiring a relativistic binding energy, just as a nucleus has a binding energy holding together its positively charged protons.  The problem of predicting the observed stability of the electron is also discussed.

          This thesis seeks to put into perspective the classical ideas of radiation reaction and electromagnetic mass, and to discuss how the classical ideas might influence quantum electrodynamics and thereby improve our understanding of the electron as an elementary particle.

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[i] Cindy Schwarz, A Tour of the Subatomic Zoo:  A Guide to Particle Physics , 2nd ed. (American Institute of Physics, New York, 1997).

[ii] Lev B. Okun, “The Concept of Mass,”  Physics Today, June 1989; Carl G. Adler, “Does mass really depend on velocity, Dad?” Am. J. Phys. 55, 739-743 (1987).

[iii] Abraham Pais, ‘Subtle is the Lord...’: The Science and the Life of Albert Einstein  (Oxford University Press, New York, 1982), p.155.

[iv] J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 781.

[v] Ibid.

Classical Electron: Radiation Reaction

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]

F_self (x,t) = —(e²/6ε₀ac²) d²x/dt²  +  (e²/6ε₀c³) d³x/dt³  + (ae²/6ε₀c⁴) 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  =  —(e²/6ε₀ac²) d²x/dt²  + (e²/6ε₀c³) d³x/dt³  + F_ext

or

(m₀ + e²/6ε₀ac²) d²x/dt²  =  (e²/6ε₀c³) d³x/dt³  + F_ext

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  =  (e²/6ε₀c³) d³x/dt³  + F_ext

The coefficient of the d³x/dt³ term is often written as mτ ,  where τ is given by

τ = e²/6ε₀ 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) = F_ext

As noted by Jiménez and Campos[x] (who use a for acceleration ), “the structure of the equation departs from the Newtonian canon because of the 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 [F_self] 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) = x_ocos(wt)) allows replacement of , giving

For an electric field E_x(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 F_self . . . 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.

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[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.