## 1. Introduction

A lot of high-energy physical processes develop within large domains of space along the direction of particle motion (see, for example monographs [1-3] and references in them). In the case of electromagnetic processes the size of these domains can substantially exceed sometimes not only interatomic distances of substance but the size of experimental facility (detectors) as well [1,2,4-13]. Essential in this case is the fact that interaction of particles with atoms and experimental facility situated within such domains and outside them can substantially differ. Such situation arises, for example, when considering long-wave radiation in processes of bremsstrahlung and transition radiation by ultra relativistic electrons. Therefore, it is necessary to know what happens within such regions and what the peculiarities of evolution of such processes in space and time are. The present chapter is dedicated to the consideration of different aspects of this problem, which concern the behavior of high-energy wave packets, which take place in processes of bremsstrahlung and transition radiation.

We begin with consideration of the behavior of localized high-energy wave packets of a scalar massive particle in wave mechanics [2,14], of Gaussian packets of free electromagnetic waves and of packets, which take place in the equivalent photon method [15]. It is shown that in all cases high energies make the stabilizing effect upon the packet motion. Some peculiarities of dispersion of such packets and their reconstruction into the packets of diverging waves are considered as well.

Further we show that the discussed wave packets naturally arise in processes of bremsstrahlung and transition radiation by high-energy electrons. For this purpose, firstly, the process of bremsstrahlung at an ultra relativistic electron instantaneous scattering to a large angle is considered [16,17]. The consideration is made on the basis of classical electrodynamics. In this case the moving electron is considered as a charge with its own coulomb field moving together with it. At the instantaneous scattering the perturbation of this field occurs. This perturbation is treated here as appearance of a packet of free plane electromagnetic waves, which reconstructs then into a packet of diverging waves. For ultra relativistic particles, however, this does not happen at once. The length within which this process develops has a name of the coherence length of the radiation process [1,2]. It is

It is shown further that analogous effects take place in the process of transition radiation by an ultra relativistic electron during its traverse of thin ideally conducting plate as well [16-18]. The picture of evolution in space and time of the electromagnetic field, which arises before and after the electron traverse of thin metallic plate is considered here. The main attention is paid to effects in the process of backward transition radiation. In this case the wave packets of the field reflected from the plate are the packets of free waves, which reconstruct into the field of transition radiation. We show that the structure of these packets is in many respects analogous to the structure of the packets, which take place at instantaneous scattering of the particle to a large angle. This fact explains the presence of analogous effects in transition radiation and bremsstrahlung in the considered cases.

The special attention is drawn to the transition radiation by a scattered electron, which own field is not totally reconstructed after the scattering [17-19]. During a long period of time in this case the electron is in ‘half-bare’ state, which is the state in which some Fourier harmonics in the field around the electron are suppressed compared to the equilibrium coulomb field. The large values of distances, which the electron covers in this state allow us to place the plate within these distances and to consider the transition radiation by such electron on this plate. We show that in this case characteristics of backward transition radiation substantially differ from transition radiation characteristics in the case when the target is situated on large distances from the scattering point. The effect of transition radiation suppression and the effect of oscillatory dependence of transition radiation characteristics on the distance between the plate and the scattering point take place in this case. The causes of such effects are discussed.

## 2. High energy wave packets

### 2.1. Dispersion of relativistic wave packets

The general solution of the wave equation can be presented in the form of a wave packet, which spatially disperses in course of time. In semiclassical approximation such packet does not disperse. It moves according to the laws of classical mechanics (see, for example [2, 20]). It is going beyond the semiclassical approximation that leads to the packet dispersion. The high-energy wave packets are of special interest because the speed of their dispersion decreases with the increase of their energy. Let us pay attention to some peculiarities of dispersion of such packets. Significant here is the fact that characteristic features of this dispersion are similar for all fields. Therefore it is sufficient to consider just scalar field.

The general solution of the wave equation

for a scalar particle with the mass

where

Let us consider the dispersion of the wave packet, which at the initial moment of time coincides with the Gaussian packet modulated by the plane wave with large value of the momentum

where

We can write the obtained expression for the field

in which

Having made in this expression the variable substitution

In the case of large energies it is possible to make the expansion over

where

The formula (9) can be written in the following form as well:

in which

In the case of

The formulae (13) show that in longitudinal and transverse directions the squares of the widths of the packet

In conclusion let us note that while deriving the formula (9) we neglected the terms proportional to

where

Thus the formula (9) is valid for the time interval

### 2.2. Dispersion of a high-energy packet of electromagnetic waves

Now let us consider high energy packets of free electromagnetic waves. Scalar and vector potentials of such packets are the solutions of the wave equation (1) with

where

Here

The obtained formulae show that the initially Gaussian packet does not disperse in the direction parallel to the

When considering a process of radiation by relativistic electrons it is often necessary to deal with packets, which are constructed of plane waves with wave vectors, which directions are close to the direction of a given vector

where

The coefficients

in which

where

Let us note that the considered case corresponds to the wave packet, which consists of the plane waves the directions of the wave vectors of which have some small scatter around the

The given expression for the wave packet has the same structure as the corresponding expression for the packet (16). If the substitutions

The formula (22) shows that for

and for

For

where

the form of the packet (22) coincides with the form of the packet at

the transformation of the packet (22) to the packet of spherical diverging waves occurs.

Let us note that in the theory of radiation of electromagnetic waves by a moving electron the spatial region in which the formation of spherical diverging waves occurs has a name of the wave zone (see for example [21]). In particular, for nonrelativistic charged particles the wave zone begins on distances from the radiation region, which exceed the length of the radiated wave

which are much larger than the wave length

### 2.3. Wave packets in the equivalent photon method

The problem of dispersion of wave packets naturally arises in the equivalent photon method (or the method of virtual photons) in which at the certain moment of time (

For this purpose we write the scalar potential of the coulomb field of the electron moving along the

in which

Here

In the equivalent photon method it is assumed that for

For

where

Here

The function

where

The main contribution to (31) is made by the values

So on distances

In this case for the evaluation of the integral in (32) over

where

The value

## 3. The bremsstrahlung at an electron instantaneous scattering

### 3.1. The electromagnetic field structure at an electron instantaneous scattering. The ‘half-bare’ electron

The electromagnetic wave packets similar to the ones considered above arise, for example, in the processes of bremsstrahlung by relativistic electron at its instantaneous scattering to a large angle and in the process of transition radiation during an electron traverse of thin metallic plate in vacuum. The present section is dedicated to the analysis of evolution of electromagnetic wave packets and peculiarities of formation of radiation by relativistic electron in the wave and the pre wave zones in the first process, while the next one – to the analogous questions concerning the second one.

Let a relativistic electron move along the

Scalar and vector potentials of the total electromagnetic field, which takes place in such process can be defined from inhomogeneous Maxwell equations

in which

where

In the case of a uniform particle motion with the velocity

which are the coulomb potentials of the moving particle. Here

In order to obtain the solution of (37) for the considered case of the particle instantaneous scattering it is convenient to express the potentials in the form of Fourier-integrals. Let us consider, for example, the vector potential:

In order to obtain the expansion in the form of the retarded potential we should calculate the Fourier-component

Making here the substitution

Substituting into (42)

As integration over

The different and rather interesting situation takes place after the scattering moment (for

The integration over

where

The expression (43) shows that before the scattering moment the total field around the electron coincides with its own coulomb field, which moves with the velocity

The first item in braces in (44) corresponds to the nonequilibrium field, which the scattered electron has already managed to rebuild around itself by the moment of time

The second item in braces in (44) describes the field, which as though `tears away' from the electron at the scattering moment. It is a packet of free electromagnetic waves, which moves in the direction of the initial electron's velocity

The equipotential surfaces of the scalar potential of the field around the electron after its scattering to a large angle are presented on Figure 1.

The behavior of certain Fourier-components of the nonequilibrium field of the electron after its scattering and of the field ‘torn away’ from the electron at its scattering is of special interest. According to (44), the values of

The notion of a ‘half-bare’ electron was introduced in the papers of E.L. Feinberg [33,34] who studied the time evolution of the state vector of the system ‘electron + photon’ after the scattering of a fast electron to a large angle on atom. The classical theory of this effect was given in [2, 27, 35].

Let us note that during the period of time

In the theory of radiation by relativistic electrons the length

The field, which ‘tears away’ from the electron at its scattering (the second item in braces in (44)) has the structure similar to the one, which has the packet of free waves considered above in the equivalent photons method. Therefore the main peculiarities of the reconstruction of the ‘torn away’ field to the field of radiation will be the same as the considered above peculiarities of the wave packets evolution. Let us consider this process in detail.

### 3.2. The problem of measurement of bremsstrahlung characteristics

The results presented above show that for ultra relativistic electrons the radiation formation process develops on large distances along the initial and final directions of the electron motion, which can be of macroscopic size. In this case a detector, which registers the radiation characteristics can be situated both in the wave zone (which means on large distances

Making in the second item in (44) the variable substitution

where

and

In (47) and (48) the square root

In ultra relativistic case (

The equations (37) are presented in Lorentz gauge

If we know

In order to determine the total energy radiated in the direction of a small area

where

Proceeding to the Fourier-expansions of the fields

in which

With the use of the Maxwell equation

Let us note that the formulae (52) and (53) are valid for arbitrary distances from the scattering point. Therefore they can be used for radiation consideration both in the wave and the pre wave zones. In the wave zone (which means in the region

On large distances from the scattering point (

As a result we obtain the following expression for the radiation spectral-angular density for

Let us note that the formula (55) is valid on large distances from the scattering point (

where

In the case of ultrarelativistic particle in the region of characteristic for this process small radiation angles

Using (49) we can derive the electric field Fourier-component orthogonal to

For large distances from the scattering point, namely in the wave zone of the radiation process (

where

For ultra high energies of the radiating particle in the region of characteristic small angles of radiation the expression (57) (and hence the expression (58)) is valid for the description of radiation spectral-angular density on small distances from the scattering point as well (in particular, in the pre-wave zone (

where

which coincides with (57) at small angles between

In the pre-wave zone (

where

In the case

From (61) we can conclude that in the pre-wave zone the radiation is mainly concentrated within angles

By the point detector we mean here the detector of the smaller size

The measurements, however, can be made by the extended detector of the larger size than the characteristic transversal length of the radiation process, so that

where

## 4. Transition radiation by relativistic electron on thin metallic plate

### 4.1. Scalar and vector potentials of transition radiation field

The electromagnetic wave packets of the structure analogous to the one considered in the process of an electron instantaneous scattering take place also in the process of relativistic electron traverse of thin conducting plate. Let us consider a problem about transition radiation that arises during normal traverse of thin ideally conducting plate, situated in the plane

Scalar and vector potentials of the electromagnetic field, which is generated by an electron moving in vacuum are the solutions of inhomogeneous Maxwell equations (37). The equations set (37) in the considered problem should be supplemented by a boundary condition, which corresponds to the fact that on the plate's surface the tangential component of the total electric field equals zero. The general solutions of the equations (37) for the electron, which moves uniformly with the velocity

The first items in (63) are the Fourier-expansion of the electron's Coulomb field, for which

The second items in (63) are the Fourier-expansion of the field of induced surface currents on the plate (we will name it the free field), for which in vacuum

Here

and the relation

for the particle's field in vacuum is valid. In general case such relation between potentials is not valid.

Using Fourier expansion (63) it is possible to derive the potential

For

The values

Thus, taking into account all requirements mentioned above we can write the scalar potential of the free field in the following form:

where

Let us note that the value of

Using (70) and the Lorentz gauge (50) in which the equations (37) are presented we can derive the vector potential:

Making in (70) the substitution

where

Deriving (72), we performed in (70) integration over azimuth angle between

The corresponding expressions for vector potential

### 4.2. The structure of transition radiation field

Let us discuss the structure of the fields that arise during the electron traverse of thin ideally conducting plate. Firstly, let us consider the structure of this field along the

As a result of rather simple calculations (see Appendix) we obtain the following expression for the field produced by the plate on the

The total field produced by the electron and the plate can be obtained by addition of the expression (75) and the electron’s own coulomb field on the

The obtained results show that for

For

The integrals in (73) and (74) can be analytically calculated as well. After rather long calculations we finally obtain the following expression for

The electron’s own coulomb field has the following form:

The structure of the expressions for scalar potential is the same as the structure of these expressions for

In the right half-space the total field equals zero for

After electron's traverse of the plate, which means for

In the right half-space, where the electron is situated after the traverse of the plate, the total field has the following form:

Thus for

The analogous expressions can be obtained for vector potential as well. Namely, for

The expression in square brackets in (80) differs from the same expression for scalar potential (78) only by the sign of second item. The reason of this can be understood from the following reasoning. For

In the region

The obtained results are valid for arbitrary electron velocities. The case of an ultra relativistic particle is of special interest because for such particles the reconstruction of the total field, created by the plate and the electron after its traverse of the plate, into the field of radiation occurs on large distances. The results obtained in this case are illustrated by Figure 4. Here the equipotential surfaces of the scalar potential of the field reflected to the left half-space and the field around the electron on the right of the plate are presented for

In order to understand what occurs with the electric field on the surface of the sphere of radius
* -* sphere) let us consider the structure of the force lines of the total field for

While building the force lines it is necessary to take into account that they should originate or end either on charges or in the infinity and not to cross each other. In the case of an infinite plate all the field lines originate and end either on the surface charges of the plate or on the electron, which traverses it.

Thus each force line of the total field (78), which originates on a surface charge of the plate somewhere in the area
* -* sphere, should be refracted and stretch further along the surface of the sphere, ending on another surface charge of the plate at

*sphere are the force lines of the transition radiation field. Indeed, the field on the*-

*sphere propagates in the radial direction with the speed of light and is perpendicular to this direction. Moreover, as will be shown in further discussion, this field decreases with the distance as*-

It is necessary to note that the given picture of the force lines indicates the necessity of the existence of the field (78) outside the
* -* sphere along with the radiation field on it. It is only in this case that the force lines of the radiation field, which originate (end) on the surface charges of the plate and stretch along the

*sphere can proceed to the region*-

*sphere.*-

### 4.3. The transition radiation field and its intensity

In ultra relativistic case (

Let us consider the radiation, which arises during an electron normal traverse of thin ideally conducting plate [16-18]. In this case by radiation we mean the part of the electromagnetic energy, which belongs to the frequency interval

In further discussion we will mainly concentrate our attention on the peculiarities of the radiation formation process in the left half-space, which is the region of negative

Let us show that in the region of large distances from the target (

The scalar

where

The Fourier-component of this expression has the following form:

Let us note that the relation

Substituting (85) into (55) we obtain the well known expression for spectral-angular density of backward transition radiation [1, 4, 5]:

The presented derivation of the formula (86) for the transition radiation spectral-angular density is based on the analysis of the electron’s field reflected from the plate on large distances

For ultrarelativistic electrons the transition radiation is mainly concentrated inside a narrow cone with the opening angle

For large

Firstly, let us consider on the base of this method the transition radiation on large distances from the plate

The first item in the braces gives the stationary phase point

where the ratio

We took into account the fact that in spherical coordinate system

With the use of (56) the expression

In the point of stationary phase for

Substituting the asymptotic (91) for the potential

For ultra relativistic particles characteristic values of the radiation angles

In this region of radiation angles the characteristic values of the variable

In other words, it is required that the distance between the target and the detector should substantially exceed the coherence length of the radiation process, which means the radiation should be considered in the wave zone.

As it was pointed out, the length

In the region of small radiation angles for which

In this case

where

Let us note that during the derivation of the expression (97) we only took into account the fact that the consideration of the radiation process was made in the region of small angles of radiation. Therefore the formula (97) is valid both for large (

Such modification of the radiation angular distribution for

The obtained results show that the effects analogous to the ones which take place in the process of bremsstrahlung at the instantaneous scattering of the electron to a large angle (the broadening of the radiation angular distribution and its dependence on the frequency of the registered photon in the measurements performed by a point detector on small distances from the scattering point) take place also for backward transition radiation in the process of an electron traverse of metallic plate. For ultra relativistic particles, according to (94) and (98), the radiation is mainly concentrated in the region of small angles

## 5. Transition radiation by ‘half-bare’ electron

The wave packets, which arise at ultra relativistic electron instantaneous scattering to a large angle, reconstruct into radiation field on distances along the initial and final directions of the electron’s velocity, which are of the order of the coherence length of the radiation process. For large energies of the electron and low frequencies of the radiated waves, as was stated above, this length can be of macroscopic size. In this case the possibility of investigating of the evolution of such wave packets in space and time by macroscopic devices appears. In [15] one of such possibilities, which concerns the reflection of wave packets from an ideally conducting plate situated on different distances from the scattering point, was discussed. In this case if the plate is situated in the direction of motion of the scattered electron perpendicular to its velocity the reflected field is the backward transition radiation. However, the peculiarity of this process lies in the fact that unlike the ordinary backward transition radiation the considered one is the radiation by the particle with noneqiulibrium field. The plate in this case can be considered as an element of the radiation detector. Let us obtain the formulae, which describe the given process and discuss some of its peculiarities on their basis [17].

Let the ideally conducting plate be situated in the plane

consists of two parts, the first of which describes the equilibrium coulomb field of the electron, which moves with the velocity

From (100) it follows that the rebuilding of the field around the electron occurs in such way that each Fourier-harmonic of frequency

The total field of the electron-plate system consists of the field of 'half-bare' electron

we can find the expression for the Fourier-harmonic of the field of induced surface currents:

where

The expression (101) can be simplified for

and using (57) for spectral-angular density of transition radiation by 'half-bare' electron we obtain:

The expression (103) differs from the corresponding expression for transition radiation by electron with equilibrium field by the interference factor inside the braces and the coefficient two in front of them. As we can see from (103), when the distance

Due to the nonzero frequency resolution

Also due to the nonzero size and, therefore, angular resolution of the detector the oscillations can be observed only inside the region

For large distances

## 6. Conclusion

The behavior of localized high-energy electromagnetic wave packets, which take place in processes of transition radiation and bremsstrahlung by relativistic electrons has been considered. It was shown that with the increase of the energy the stabilization of characteristics of motion of such packets takes place, which consists in substantial decrease of the speed of their dispersion. Essential here is the fact that at high energies the lengths, on which the reconstruction of the form of such packets into packets of diverging waves takes place, can reach macroscopic size, which can exceed the size of experimental facility. In this case both the size of the used detector and its position relative to the region of the wave packet formation become essential for measurements.

Such situation takes place, for example, after the sharp scattering of an electron to a large angle. It was shown that as a result of such scattering the electron’s own coulomb field tears away from it and turns into a localized packet of free electromagnetic waves, which transforms into a packet of diverging waves on large distance from the scattering point. For ultra relativistic electrons such transformation of certain Fourier-harmonics of the packet field takes place within the coherence length of the radiation process, which substantially exceeds the length of the considered wave of radiation. In the case of low-frequency radiation this coherence length can be macroscopic. It gives birth to the problem of bremsstrahlung characteristics measurement by different detectors, which consists in the dependence of the results of measurement on the detector’s size and its position relative to the scattering point.

In the final direction of the electron motion (after the scattering) certain Fourier-components of the field around it do not appear at once. The regeneration of these Fourier-components occurs within the coherence length of the radiation process

The transition radiation formation process also develops within distances of the order of the coherence length of the radiation process,

## Appendix

According to (72) the potential

In order to calculate

where

Integration over

Taking into account that for

we obtain the following expression for

Thus the appearance of the step function

The calculation of

Substituting the obtained expressions (110) and (111) for

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