Radiation feedback

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The radiation feedback is an effect in electrodynamics . It occurs when an electrically charged object does not move at a constant speed in an electromagnetic field . Except for very special field configurations such as a Wien filter , this is always the case.

Radiation feedback arises from the fact that accelerated charged particles themselves emit electromagnetic radiation, which influences the external field and thus also the future development of the particle trajectory. As a rule, the effects of the radiation feedback are ignored, since this is a very small contribution to the equations of motion . Nevertheless, the treatment of this effect leads to fundamental problems in both classical physics and special relativity . Among other things, under certain circumstances, on the one hand, the mass of a particle seems to become infinitely large, on the other hand, its speed either becomes infinitely high or the information about the future movement of the particle is included in the initial conditions of the particle orbit. The first two statements are in blatant contradiction to the reality experienced, the last statement contradicts the principle of causality .

In the context of classical physics, the problem was first examined in 1902 by Max Abraham and in 1903 by Hendrik Lorentz . The special theory of relativity was taken into account in 1938 by Paul Dirac . An approximate solution that does not deal with the problem of infinite speeds was given by Lew Landau and Jewgeni Lifschitz in their textbook on theoretical physics . This solution in turn has the problem of not always taking energy conservation into account.

The formulas that describe the radiation reaction are called the Abraham-Lorentz equation , Abraham-Lorentz-Dirac equation and Landau-Lifschitz equation after their discoverers .

The problem remains unsolved in classical physics to this day. In 2010, David J. Griffiths described the radiation reaction as the "corpse in the basement of classical electrodynamics".

The equations of motion of elementary particles , including the radiation reaction , are completely described in quantum physics by quantum electrodynamics ; nevertheless the problem with infinite mass also arises in this context.

background

The force on a charged object is determined by the electric and magnetic fields in its environment and its speed and is called Lorentz force , where the electric charge of the object and its speed. is the electric field strength and the magnetic flux density of the environment, denotes the vector product . According to Newton's First Law , the force is proportional to the acceleration of an object, i.e. the change in its speed and its mass .

On the other hand, according to Liénard and Wiechert, an accelerated, charged object in turn generates an electromagnetic field in its environment and, according to the Larmor formula, loses energy in the form of electromagnetic radiation.

As a rule, these two questions are separated from each other: Either the movement of an object in the external field and its loss of energy or the field generated by the object itself is considered. The former is the case, for example , when treating synchrotron radiation when a particle is accelerated in a synchrotron on a circular path, the latter for example when considering the Hertzian dipole when calculating the emitted electromagnetic field of a transmitter.

In fact, however, the law of conservation of energy and momentum must not be violated, so that the electromagnetic field radiated by the charged object interferes with the external electromagnetic field and the future movement of the object is thereby influenced. The fact that this does not play a special role in practical everyday life is due to the fact that the effects of the radiation feedback are negligibly small compared to the strength of the original field. They only need to be taken into account if the time intervals in which the external field acts or the observed distances over which the particle moves are very small.

This time interval is called characteristic time and can be estimated by comparing the energy that is lost through radiation with the energy that it receives through acceleration. With the speed of light and the electric field constant , this characteristic time is :

The characteristic length is the characteristic time multiplied by the speed at which an electromagnetic field propagates, the speed of light. Both sizes are therefore inversely proportional to the mass and are greatest for very light objects. The lightest charged particle currently known is the electron . The characteristic time for this is 6.3 · 10 −24 seconds, the characteristic length 10 −15 meters. For a comparison of the order of magnitude, this is less than a hundredth the diameter of a hydrogen atom.

Abraham-Lorentz equation

The Abraham-Lorentz equation for point charges is:

In addition to the designations already introduced, the jerk of the particle is a change in acceleration over time and the index "ext" on the force designates the Lorentz force induced by the externally applied electromagnetic field.

Since the speed itself is defined as the temporal change in the location, the Abraham-Lorentz equation is therefore a third-order differential equation of the location, provided that the external force is location-dependent. This is precisely the case when the electric field is not homogeneous. The property that, in contrast to the usual equations of motion, the jerk occurs explicitly in the equation leads to weighty problems in solving the equation.

Heuristic derivation

The form of the Abraham-Lorentz equation can already be deduced heuristically through dimensional analysis : the only parameter that can be included in the formula if the radiation feedback is to be taken into account is the characteristic time . In addition, two other conditions should be met:

  1. If the acceleration of the particle disappears, then the radiation reaction must also disappear.
  2. The radiation feedback must be proportional to an even power of the charge, since the sign of the charge cannot have any influence on the radiated power.

This already defines the form of the Abraham-Lorentz equation so that the additional dimension of “time” introduced by the characteristic time must be compensated for by the introduction of the time derivative of the acceleration, the jerk. Only through these assumptions one already obtains with two unknown parameters , where a natural number must be:

As a further restriction, the validity of the law of conservation of energy can be required: The radiated power according to the Larmor formula must be equal to the energy lost due to the effect of the radiation feedback. The radiated power according to Larmor is calculated according to

and the relationship between the force acting by the radiation and power is

.

From this follows an integration within the limits of two times that determine the time interval of the acting force

,

partial integration was used in the last step . This leads to the Abraham-Lorentz equation with two restrictions: Firstly, at the start and end times, the speed and acceleration must be orthogonal to each other, so it must be. Second, the condition is considerably weaker than because one can not simply divide by the vector . The last point, however, is made plausible by the considerations made above. Under these conditions, this leads directly to the Abraham-Lorentz equation given above, since, provided the integrals are identical for any , the integrands must also be identical:

Resulting problems

The problem with the ad hoc derivation is that the Abraham-Lorentz equation is only valid under the condition . The bigger problem, however, is that the solution of the Abraham-Lorentz equation, even for vanishing external fields, is not trivially zero, but rather

with the initial acceleration . The acceleration grows exponentially and becomes arbitrarily large after a finite time, which is a physically senseless result. Accompanied by this acceleration, the speed and the distance covered also become arbitrarily large. In the heuristic derivation, however, this solution contradicts the assumption made of the orthogonality of acceleration and speed, because it holds

,

what only disappears when is. Since this restriction does not appear in a solid derivation, this is already the first indication of inadequacies in classical electrodynamics. The technical term for this type of solution is runaway solution , outlier solution.

To avoid these pathological solutions, boundary conditions can be applied to the equations of motion. After a long calculation, the solution of the equations of motion results:

This solution is unphysical for another reason: To calculate the acceleration at a certain point in time , the future path of the particle has to be integrated into an infinitely distant future.

Relativistic generalization

The Abraham-Lorentz-Dirac equation is the relativistic generalization of the Abraham-Lorentz equation by Dirac. It is:

Due to the identity , this can alternatively be called

to be written. In addition to the introduction of four-vectors in four-dimensional space - time , the difference to the Abraham-Lorentz equation is the additional occurrence of the term . Obviously, in the non-relativistic approximation in which is, this term falls out of the equation. In the relativistic case, differentiation is understood as differentiation according to the proper time of the object; in the non-relativistic borderline case, the distinction between proper time and coordinate time is irrelevant.

The relativistic version also suffers from the problem that either the path of the particle will be included in the equations of motion in the distant future or the speed of the particle will approach the speed of light (instead of the speed, the rapidity becomes arbitrarily large, the speed of light represents the highest achievable speed ). In addition, however, the relativistic generalization has the advantage that the restricting condition is automatically fulfilled, since velocity and acceleration are always orthogonal to one another in four-dimensional space-time.

Landau-Lifschitz equation

The Landau-Lifschitz equation is an approximation that regards the radiation feedback as a small effect, which it actually is. The idea of ​​Landau and Lifschitz is to differentiate the solution of the equations of motion in the external field without considering the radiation feedback, namely , to differentiate according to time and to use it in the Abraham-Lorentz equation. Then:

The Landau-Lifschitz equation is a good approximation, provided that the external forces do not vary too much over time. This means that high-frequency alternating fields must not be involved, the wavelength of which is in the order of magnitude of the object, and the fields themselves must not be too strong. The second condition, however, is irrelevant when dealing with the radiation feedback, since classical electrodynamics can no longer be used even at lower field strengths.

The relativistic generalization of the Landau-Lifschitz equation is

and is obtained in the same way as the non-relativistic version.

Equation of motion for general spherically symmetric charge distributions

The starting point of the stringent derivation is the division of the total momentum of the system, i.e. the particle and all electromagnetic fields, into a mechanical and an electromagnetic part. Since the Lorentz force acts on the particle in the electromagnetic external field , is

with the electric field , the magnetic flux density , the charge density and the current density . The electric and magnetic quantities involved in this equation are not only the external fields, but also the proportions of the fields that are generated by the charged particle itself. The limits of integration for the volume integral is the volume of the particle itself.

In order to obtain an equation that takes into account radiation feedback and the influence of external forces, an equation for the total momentum must be found that has the shape

owns. As in the above equation, the Lorentz force is, however, limited to the external electromagnetic fields . With these generally valid prerequisites, a rigid and spherically symmetrical charge distribution in its rest system , i.e. in a system in which its current density disappears, after a long calculation applies to the electromagnetic component of the pulse

.

The assumption of a rigid charge distribution naturally limits the treatment to non-relativistic velocities due to the Lorentz contraction , because in a relativistic theory there can be no rigid bodies . Classically, this equation is almost exact; the only approximation used to derive it is the restriction to linear terms in the time derivative of the velocity in the series expansion .

For further calculations it makes mathematical sense to switch to the Fourier space by means of a Fourier transformation . For the Fourier transforms, and . While there is no name of its own for the Fourier transform of the velocity, the Fourier transform of the charge density is called the form factor (in contrast to the force without a vector arrow); the charge in the Fourier transform ensures that the form factor becomes dimensionless by definition. Then

This formula looks much like the ordinary equation for acceleration in Fourier space, where the mass parameter is represented by the term

was replaced as effective mass .

Renormalization of the mass

For a point charge is . As a result, the integral in the above formula diverges and the effective mass seems to become infinite. Dirac wrote about this in 1938 in his treatise on radiation feedback:

"If we want a model of the electron, we must suppose that there is an infinite negative mass at its center such that, when subtracted from the infinite positive mass of the surrounding Coulomb field, the difference is well defined and is just equal to m . Such a model is hardly a plausible one according to current physical ideas but […] this is not an objection to the theory provided we have a reasonable mathematical scheme. "

- Paul Dirac : Classical theory of radiating electrons

Such a procedure has been common in physics since the development of quantum field theories in the 1940s, since there were further infinities in the calculations where, according to the physical reality experienced, there should be none. This procedure is called renormalization . The renormalization leads to the fact that the parameter that was originally introduced into the equation of motion can no longer be viewed as the “real” physical mass on which the force acts. Such parameters, which appear in the original equation with a certain function, but have to be renormalized in order to result in meaningful values, are called "bare" parameters.

As Dirac pointed out, a renormalization scheme is necessary to carry out the renormalization. For the radiation feedback it makes sense to indicate the physical mass evaluated as effective mass at the point . Then

.

In order to obtain a finite value for the physical mass , in the case of a charge distribution whose form factor does not decrease more than decreasing, the bare mass must assume a negative infinite value, since the integrand is always positive. In the sense of Dirac's statement, such an idea is absurd at first glance, but since the naked mass can never be measured, it is not unphysical. In terms of the physical quantities, the effective mass becomes

.

This last integral does not diverge because of the denominator itself for point charges.

Classic electron

The classic model of an electron is a billiard ball with a radius , on the surface of which the charge is evenly distributed. The form factor for such a charge distribution is . With this form factor, the integration can be carried out via closed and the effective mass results in

For the point charge , this formula is reduced to . The reverse transformation from Fourier space into coordinate space thus gives exactly the Abraham-Lorentz equation, without the restrictive condition .

In the Fourier space the cause of the unphysical solutions is shown in the (positive complex) zero of the effective mass at . These zeros disappear with the condition that the unphysical solutions do not appear for objects that are larger than their own characteristic length. In particular, the classical electron radius is for a classical electron

.

Back in coordinate space, the equation for an object with a radius , on whose shell the charge is evenly distributed, results as:

In this form, the information about the radiation reaction is contained in the difference in speed of the electron at the point in time under consideration and at an earlier point in time . This time difference is the time it takes for an electromagnetic wave to pass through the electron.

literature

Individual evidence

  1. Max Abraham: Principles of the Dynamics of the Electron . In: Annals of Physics . tape 315 , no. 1 , 1903, p. 105-179 .
  2. Hendrik Lorentz: The theory of electrons and its applications to the phenomena of light and radiant heat . 2nd Edition. Teubner, Leipzig 1916 (English).
  3. ^ A b Paul Dirac: Classical theory of radiating electrons . In: Proceedings of the Royal Society A . tape 167 , no. 929 , 1938, pp. 148-169 (English).
  4. ^ Lev Landau and Evgeny Lifschitz: The Classical Theory of Fields . In: Course of Theoretical Physics . 3. Edition. tape 2 . Pergamon Press, Oxford New York Toronto Sydney Braunschweig 1971 (English).
  5. William Baylis and John Huschilt: Energy balance with the Landau-Lifshitz equation . In: Physics Letters A . tape 301 , no. 1–2 , 2002, pp. 7-12 (English).
  6. a b c David J. Griffiths, Thomas C. Proctor and Darrell F. Schroeter: Abraham-Lorentz versus Landau-Lifshitz . In: American Journal of Physics . tape 78 , no. 4 , 2010, p. 391-402 (English).
  7. Mattew D. Schwartz: Quantum Field Theory and the standard model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
  8. Yurij Yaremko: Exact solution to the Landau-Lifshitz equation in a constant electromagnetic field . In: Journal of Mathematical Physics . tape 54 , no. 9 , 2013, p. 092901-1-092901-19 (English).