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Radiation reaction in strong fields from an alternative perspective

Current classical theory of radiation reaction has several deficiencies such as"runaway solutions"and violation of causality. The Landau-Lifshitz approximation to the exact equation introduced by Lorentz, Abraham and Dirac is widely used, though questions remain regarding its domain of validity. This thesis explores an alternative treatment of the motion of a radiating electron, based on an equation first proposed by Ford and O'Connell. A general condition is found for solutions of this equation to deviate from those of Landau-Lifshitz. By exploring radiation reaction effects on a particle colliding with an ultra-intense laser pulse we show that the regime where there is a significant deviation of these two approaches can never be reached with existing or proposed laser facilities. The methods used to explore single particle interaction with an intense laser pulse are extended to describe the interaction of a particle bunch with various realistic laser pulses. We find that the interaction leads to a decrease in average momentum and relative momentum spread. However, the decrease appears to be independent of the length of the pulse and depends only on the energy in the pulse regardless of how it is distributed. Radiation reaction effects occuring during the scattering of an electron by a heavy, highly-charged nucleus are studied. Radiation reaction is seen to affect the particle's motion. We find noticeable differences between the predictions of the Ford-O'Connell and Landau-Lifshitz equations, albeit in regimes where quantum effects would be important.


Thesis overview
An accelerating charged particle emits radiation [1]. With this emission the particle loses energy and momentum, which influences its motion via the recoil force (so called "radiation reaction") in order to fulfill energy and momentum conservation. The question of how the particle interacts with the radiation it produces, however, remains unclear, despite investigations stretching back more than a century; see Refs. [2,3,4].
Until recently, interest has been motivated principally by theoretical curiosity, since the radiation reaction force is in general a negligible correction to the Lorentz force from the external fields. However, with the advent of the modern ultra-intense laser, this question is becoming relevant to experimental investigations.
This will be of particular relevance to research at the Extreme Light Infrastructure (ELI) [5] facility that is currently under construction, which will be one of the leading high power laser facilities in the world. The work in this thesis will be most relevant to the fourth pillar of ELI (location to be decided), where the laser intensities are expected to reach ∼ 10 25 Wcm −2 and the power to exceed that of the other three ELI pillars by at least one order of magnitude.
The work presented in this thesis is focused on exploring the effects of classical radiation reaction in regimes where this effect is believed to be non-negligible. We present an alternative model for classical radiation reaction and compare its predictions with traditional approaches to help establish the limitations of commonly used methods.
In the remaining part of this chapter we will discuss two common theoretical descriptions of radiation reaction: that of Lorentz [6], Abraham [7] and Dirac [8], and that of Landau and Lifshitz [9], respectively. While the former leads to difficulties such as exponentially growing acceleration ("runaway solutions") and violation of causality ("pre-acceleration"), the latter is perturbative, which raises questions on its validity in extremely high fields.
In Chapter 2 we introduce an alternative approach based on the Ford-O'Connell equation [10,11,12], which addresses these issues. The relation of this approach to | 2

Radiation reaction models
other commonly used treatments is explored. We also introduce and discuss a condition allowing us to predict where the Landau-Lifshitz model is not valid.
Chapter 3 is devoted to the study of the interaction of a high-energy electron with an intense laser pulse. By analysing this interaction we find that radiation reaction prevents the particle from accessing the regime where the Landau-Lifshitz approximation is not valid. The results presented in this chapter are summarised in Refs. [13,14,15,16].
In Chapter 4 we present a study of how radiation reaction influences the momentum spread of a bunch of particles propagating through an intense laser pulse.
In Chapter 5 we apply the model developed in Chapter 2 to the scattering of a particle off a heavy nucleus. We present a model that includes the classical radiation reaction corrections, and derive relevant equations of motion. Numerical simulations are presented and relevance of quantum effects during the interaction is discussed.

Radiation reaction models 1.2.1 Lorentz-Abraham-Dirac equation
An equation of motion for a non-relativistic point particle of charge q and mass m in an external electromagnetic field is as follows: where E and B are the external electric and magnetic fields, respectively.
Equation (1.1) can be written covariantly as: where F a b is the electromagnetic field tensor and the dot here denotes a derivative | 3

Radiation reaction models
with respect to proper time 1 . Since F a b is antisymmetric (1.2) preserves the 4-velocity normalisation condition:ẋ aẋ a = −1 .
We use the Einstein summation convention and indices are raised and lowered using the metric tensor η ab = diag (−1, 1, 1, 1).
As mentioned earlier, a particle undergoing acceleration emits radiation, which produces a recoil force. For this to be taken into account, the above equation of motion must be modified to account for energy carried away by the radiation. This can be achieved by including a radiation reaction force in addition to the Lorentz force.
Equation (1.2) then becomes:ẍ where P a is the recoil force or so called radiation reaction force, which accounts for the emitted energy and momentum andĊ a is the so-called Schott term [17,18], which ensures that the normalisation condition is preserved.
The electromagnetic power emited by a relativistic particle can be obtained using the Larmor formula [1]: where τ = q 2 /6πm = 2r e /3c ∼ 10 −23 s for an electron (with r e the classical electron radius).
Differentiating ( (1.7) 1 Here proper time is the time elapsed between two events as seen by the particle in its rest frame. Throughout this thesis, we refer to proper time as s.
It can be seen that the radiation reaction force involves the so-called "jerk " term (time derivative of acceleration) making it a third order differential equation for the position of the particle. To solve this equation it is neccessary to specify initial conditions for position, velocity and acceleration. However the above equations (1.10, 1.11) have a few important pathologies 2 .
2 The term pathologies is widely used to describe the non-physical solutions of the Lorentz-Abraham-Dirac equation.

Radiation reaction models
Consider equation (1.11) in the absence of an external force: (1. 13) Assume motion only in one direction: (1.14) Hereẋ 0 = cosh α andẋ 1 = sinh α satisfy the normalisation condition (1.3). An equation of motion for proper acceleration can be produced. It has the form: τα =α , (1.15) which leads to the solution: This type of solution is known as a runaway solution as the particle's acceleration increases exponentially with time, unless α(0) = 0. This solution is clearly unphysical and is one of the main pathologies of the Lorentz-Abraham-Dirac equation.
Another pathology of this equation [25,26,27] can be most readily seen by examining the non-relativistic limit of (1.11): which can be integrated to give: where t 0 is a constant and F ext is given by the right hand side of (1.1). The previously discussed runaway solutions can be eliminated by introducing a mix of initial and final conditions rather than just the initial ones. Demanding that the final acceleration is 0 | 6

Radiation reaction models
once all the forces have finished acting we eliminate the previous pathology.
This corresponds to t 0 → ∞ and a (t 0 ) → 0 in (1.18). Applying the change of variable z = (t − t) /τ , the original equation becomes: This solution uncovers yet another pathology of the Lorentz-Abraham-Dirac equation.
From (1.19) it can be seen that the acceleration at time t depends on the applied force still to come. Therefore removal of the runaway pathology leads to the preacceleration pathology, which is also unphysical. However, in this case, unlike the runaway pathology, the influence of the unphysical part of the solution is very small as it is supressed by the e −z term. The same pathology remains in the relativistic equation of motion [8].

The Landau-Lifshitz equation
In 1962 Landau and Lifshitz [9] proposed a perturbative method for removing runaway and pre-acceleration pathologies by reducing the order of the Lorentz-Abraham-Dirac equation (1.10). Assuming τ 6·10 −24 s is small compared to the timescale over which the Lorentz force varies (as measured in the rest frame of the particle) they considered the radiation reaction force (proportional to τ ) to be a small perturbation about the Lorentz force:ẍ allowing us to approximate the jerk term: If we now substitute (1.21) into the original Lorentz-Abraham-Dirac equation (1.11) and drop terms of O τ 2 we obtain the Landau-Lifshitz equation: It has been shown [28] that (1.22) approximates the non-runaway solutions of (1.10).
As this equation no longer depends on the derivative of the acceleration it is free from the pathologies of the original Lorentz-Abraham-Dirac equation (1.11) and is usually accepted to be the correct classical equation of motion for the relativistic charged point particle.
As noted earlier, the fields in forthcoming ultra-high intensity laser facilities will be sufficiently strong that the forces due to an electron's emission can exceed the Lorentz force of the electron due to the laser pulse, which raises questions regarding the domain of validity of the Landau-Lifshitz approach [29,30,31,32,33].

Alternative radiation reaction models
Given the fundamental flaws of the Lorentz-Abraham-Dirac equation, a number of alternative approaches have been proposed throughout the last century. This section is devoted to a brief introduction of the most common of these to compare attempts to solve this problem. Advantages and limitations of these approaches will be discussed.

Mo-Papas equation
In 1971 an alternative approach was presented by Mo and Papas [34]. Instead of a rigorous derivation from first principles, they heurestically claimed that an equation of motion should balance inertia and radiation forces with the Lorentz force and an additional acceleration-dependent generalization of the Lorentz force. This led to an | 8

Alternative radiation reaction models
equation of the following form: It can be seen that the second term on the left-hand side is responsible for the compensation of the losses due to radiation, while the second term on the right-hand side is the new force. To fulfill the normalisation condition g = −τ q/m was defined.
The Mo-Papas equation has been criticised on a number of occasions. One of the most important criticisms [35] is that for the case of linear motion this equation coincides with the Lorentz force describing the motion with no radiation reaction taken into account. The radiation losses in this case are small, but non-zero.

Sokolov equation
One of the most popular recent endeavours that have gained significant attention is the equation presented by Sokolov [36]. The fundamental assumption of the derivation of this equation is that the 4-momentum does not have to be collinear with the 4-velocity.
To justify this claim Sokolov uses the fact that part of the momentum of a charged particle may be regarded as distributed throughout space in its Coulomb field.
If we take the momentum and velocity to be parallel, p a = mẋ a , the Einstein relation for energy and momentum corresponds to the normalisation condition with respect to proper time: If we now accept Sokolov's assumption and assume that due to acceleration momentum p a and velocityẋ a are no longer parallel only one of (1.24) holds. From the first equation in (1.24) via some additional assumptions we acquire: 1.3. Alternative radiation reaction modelṡ (1.26) If we now substitute p a /m −→ẋ a into (1.26), apart from the field derivative terms it agrees with the Landau-Lifshitz equation. The novel feature of the Sokolov theory is given by (1.25), which describes the non-collinearity ofẋ a and p a .
Using (1.25), the normalisation of velocities can be written as: It can be seen that for sufficiently large fields and/or high energies, it is possible thatẋ aẋ a 0. This would lead to the failure of the notion of proper time which would allow a massive particle to move with the speed of light or faster. This is one fundamental problem with the Sokolov theory indicating that it violates causality in extreme circumstances.

The Ford-O'Connell equation
One of the first attempts to address the original issues related to the Lorentz-Abraham-Dirac equation was undertaken by Dirac's student Eliezer, in 1948 [37]. He noticed that the equation of motion of a nonrelativistic non-pointlike electron of radius R can be written as: where the c n denote coefficients dependent upon the particle structure. If we now adjust the radius and charge density such that the c n give: the equation of motion becomes: (1.30) | 10

Alternative radiation reaction models
The relativistic form of (1.30) can be written as follows:  [11,12] while looking at the classical limit of a quantum theory for non-pointlike electrons.
In their work, they chose a form factor for the electron to be the following: with Ω a cut-off frequency. The pointlike electron arises from Ω → ∞ giving rise to the Abraham-Lorentz equation [11]. However, for Ω = τ −1 we recover (1.30), a pathology-free equation of motion for a non-pointlike particle.

Ford-O'Connell equation in an electromagnetic field
In the presence of an arbitrary external force f a , the Ford-O'Connell equation [11,12] readsẍ where ∆ a b = δ a b +ẋ aẋ b is the projection operator.
Consider the normalisation condition of the 4-velocity: shows thatẋ must be ⊥ toẍ.
Combining (2.3) with (2.1) it can be seen that: The first component 2f ·ẋ = 0 is due to the property of an external force being orthogonal to the 4-velocity and the second component reflects the property of the projection operator ∆ to annihilate theẋ component of a vector, shown below: These coupled together ensure that the normalisation condition (2.2) is preserved.
The dominant forces on a classical charged particle are electromagnetic, so we use the Lorentz force: (2.6) | 13

Ford-O'Connell equation in an electromagnetic field
as the applied force in (2.1), leading to: (2.7) Because F is antisymmetric ∂F b c ∂x dẋbẋ c = 0, so (2.7) can be rearranged as follows: Apart from the term involving derivatives of the fields, this coincides with the Mo-Papas equation [34], which reads: The latter was derived heuristically, rather than either from first principles or as an to act only on vectors that are orthogonal toẋ and will produce vectors orthogonal tȯ x. Choosing V to be: where G a b = q m ∆ a c F c d ∆ d b is the "sandwiched" tensor, representing the magnetic field as seen by the particle.
we need to show that M can be inverted. However, care must be taken in defining the inverse: taken as a matrix acting on all 4-vectors, M annihilatesẋ, and therefore cannot be inverted. However, from (2.11), we only require M to act on (and produce) vectors orthogonal toẋ.
In the particle rest frame, denoted by , the 4-velocity is given byẋ a = δ a 0 and the projection operator is given by: and its spatial components In this section Greek indices run from 1 to 3.
Based on (2.13) and (2.14) the time components of the matrix M in the particle's rest frame are: and, according to (2.15) the spatial components are:

Matrix form of the Ford-O'Connell equation
The definition of the inverse of M in this frame is given by: It follows that: which is true since in the rest frame of the particle, M 0 λ = 0. Also, because both M −1 0 a and M a 0 are 0, the following holds: Therefore, the inverse can be defined as To specify its determinant, consider the 3 × 3 matrix of spatial components M µ ν in the rest frame of the particle (2.17). Its determinant is given by: where ε αβγ is the completely antisymmetric tensor of rank 3 with ε 123 = 1.
This can also be written as

Matrix form of the Ford-O'Connell equation
with ε abcd the completely antisymmetric tensor of rank 4 with ε 0123 = 1.
Due to the antisymmetric property of ε, (2.24) can be written covariantly as The combination ε abcd ε ef gh can be expanded as follows: To produce general expression for the determinant the respective contractions of ε abcd ε ef gh with M 's must be carried out. To simplify the calculation we start with the contraction withẋ dẋ h , which yields: Considering that M is orthogonal toẋ it can be seen that all the terms of ε abcd ε ef ghẋdẋ h involvingẋ will vanish after all the contractions have taken place leaving only terms that contribute to the final result: Based on the result in (2.28) and proceeding with the remaining contractions we obtain: where M = M a a = tr (M ).
Considering the form and properties of M (2.13) all the required terms involving M can be written out in terms of "sandwiched" electromagnetic field tensors G as follows: and allowing the determinant from (2.25) to be specified as: Expressing the electromagnetic field tensor in terms of the electric and magnetic fields | 18 2.3. Inversion of the M matrix as seen by the particle: the property of the projection operator ∆ will lead to the annihilation of theẋ Combined with the Lorentz invariant F ab F ab = 2 B 2 − E 2 , the form of the determinant (2.33) in terms of the fields measured by an observer comoving with the particle can then be found: It follows that M is invertible, and the Ford-O'Connell equation is a viable description for the motion of a charged particle.

Inversion of the M matrix
To consider the Ford-O'Connell equation as an equation of motion written in the following form:ẍ we require an analytical form for the inverse M −1 a b .
To solve (2.22) for the inverse it is convenient to express M −1 as a linear combination of a set of known matrices.
A convenient choice of basis matrices can be constructed using the fields seen by the particle: where λ runs from 1 to 9. In terms of the electromagnetic field tensor, the electric field | 19

Inversion of the M matrix
E, magnetic field B and Poynting vector S are as follows: and The tensor product in (2.37) leads to a combination of 9 linearly independent 1 components: The S c S a matrix in (2.37) can be expanded as: which explains the presence of ∆ c a in (2.41) instead of S c S a .
The matrix inverse can then be described as a sum: where α λ is a set of coefficients we need to determine in order to obtain an expression for the inverse in terms of the electric and magnetic fields.
Taking into account the properties of the inverse (2.22) and the above relation (2.43) it can be straightforwardly seen that: The matrix to be inverted can also be rewritten in terms of electric and magnetic fields: Equation (2.45) can be written explicitly as a linear combination of Γ λ components.
Because ε abcd is antisymmetric the ε abcd B cẋd term can be expanded as: Contracting equation (2.46) with B b leads to the left hand side vanishing due to properties of the completely antisymmetric tensor ε abcd , therefore: which is equivalent to: Because of the linear independence of E, B and S it can be clearly seen that: The analogous contraction of equation ( which is equivalent to: From (2.51) it can be seen that: Equating (2.49) with (2.52) leads to: After substitution of this result into (2.49), we obtain: Recombining the ζ coefficients with (2.46) we obtain an expression for ε a b mn B mẋn in terms of the electric and magnetic fields, and the Poynting vector: This allows us to rewrite (2.45) in the form of: To obtain the inverse we can write out each λ-component of (2.44) separately, giving us the form of the inverse in terms of E, B and S: (2.60) (2.61) (2.63) (2.65) The above combined with (2.42) allows us to rewrite the sum in (2.44) in terms of the corresponding α λ coefficients and basis elements, (Γ λ ) c a :

Inversion of the M matrix
in front of every other element of the basis to be 0. This leads to a set of 9 algebraic equations with 9 unknowns, that are the α λ coefficients we require. These are: To obtain these coefficients we need to solve the above set of equations. From equation (2.68) it can be seen that: Substituting this result into equation (2.67) we obtain: leading to: (2.78) | 25

Inversion of the M matrix
Substituting this result into equation (2.76) we obtain: (2.79) Considering equation (2.71) we can write the following condition: (2.80) Substituting the result from equation (2.78) into (2.80) we obtain: Considering equations (2.69, 2.72) we can write out the equations combining α 4 and α 5 coefficients: Substituting equation (2.79) into (2.83) we obtain: leading to: which can then be substituted into (2.82), allowing us to obtain an equation for the α 4 component: Substituting (2.87) into (2.85) we obtain: Considering equation (2.73) and the results we already have, we can find α 6 : From equations (2.70) and (2.74) an analogous relation between α 7 and α 8 can be produced: From (2.91) it can be seen: substituting this back into (2.90) we obtain: As we already know α 1 and α 2 from equations (2.78) and (2.79) respectively we obtain: Considering (2.91) the α 8 coefficient can be written as: Rearranging equation (2.75) we obtain α 9 : Substituting the appropriate solutions for α 2 and α 7 : Following the above calculation, all of the α λ coefficients have been obtained and after small simplifications can be written out as follows: (2.104) Recombining these coefficients with the basis elements and defining D := 1 + τ 2 q 2 m 2 B 2 , the inverted tensor can be explicitly written as: To be able to use this inverse to assemble an equation of motion in terms of the electromagnetic field tensor we require the inverse to be written in the same terms.
From (2.55): and the only term still required is τ 2 q 2 m 2 D B c B a . If we now consider the square of (2.106):

Condition of divergence between Ford-O'Connell and Landau-Lifshitz approaches
If we now expand the S c S a term in (2.107) using (2.42) we acquire: Combining results yields the inverse of M (2.105) in the form: (2.110)

Landau-Lifshitz approaches
It can be seen that, ignoring terms of O τ 2 , the Ford-O'Connell and Landau-Lifshitz equations agree.
After elimination of the remaining terms of O τ 2 we obtain the Landau-Lifshitz equation: For the Landau-Lifshitz equation to be a good approximation to the Ford-O'Connell equation it is necessary that though this involves only the magnetic field seen by the particle, which does not contribute to the applied force. The condition (2.114) is necessary, but not sufficient.
However, we focus on this scalar condition, as it is more readily applicable than the somewhat vague requirement that M a b is "close" to the unit matrix.

Ford-O'Connell vs. Landau-Lifshitz approaches for the case of linear motion
We now consider linear motion in a field: where ε ab is the antisymmetric tensor:

Ford-O'Connell vs. Landau-Lifshitz approaches for the case of linear motion
Consider the "sandwiched" electromagnetic tensor G ab constructed under the above conditions: During the linear motion in the x 1 direction we can assume x 2 = x 3 = 0, therefore the normalisation condition becomes: If we now combine (2.118) with (2.117) it can be seen that for the case of linear motion: In this Chapter we investigate the effect of radiation reaction on particle motion in the radiation dominated regime using the Ford-O'Connell description, which has been described in the previous Chapter.
We explore the conditions under which the perturbative Landau-Lifshitz approximation breaks down, and whether this can be realised in the foreseeable future. To do that we compare the predictions of the Landau-Lifshitz and Ford-O'Connell methods for extreme cases.

Introduction to the theoretical model
In the absence of radiation reaction, the solution to the Lorentz force equation in a plane electromagnetic wave is well known: particularly lucid accounts may be found in [39,40]. Analytical solutions also exist for the Landau-Lifshitz radiation reaction correction [41,42,43].
Radiation reaction will be most prominent for high energy electrons colliding with ultra-intense laser pulses which experience extremely large acceleration, and hence radiate most strongly according to (1.5). We therefore consider the Ford-O'Connell equations of motion in a laser pulse approximated by a plane wave with arbitrary | 34

Introduction to the theoretical model
shape and polarisation, moving in a given direction.
To keep the model as general as possible we do not specify any particular pulse parameters until we solve the equations of motion numerically. To derive these we introduce n a and m a , which are null basis vectors, and the polarisation vectors , λ such that they satisfy the following conditions: We can define the coordinates as follows: Note that ξ and σ are the transverse space-like coordinates whereas ψ and φ (so called lightfront time) are light-like coordinates.
Using the above specified set of coordinates, x a can be expressed as: The normalisation conditionẋ aẋ a = −1 can be written in terms of these coordinates as follows: n a +ξ a +σλ a = −1 .
The electron's energy, normalised to mc 2 , is then given by with (3.8) allowing us to eliminateψ.
as an external force:ẍ where the inverse M −1 a b is defined as: with G ab the "sandwiched" electromagnetic tensor G ab = q m ∆ ac F cd ∆ db and the determinant det M = 1 + (τ 2 /2)G ab G ab .
We focus on deriving a general form for the equations of motion using the Ford-O'Connell radiation reaction force for the case of a particle interacting with an arbitrarily polarised laser pulse. Although realistic laser pulses have important transverse structure, for electrons co-or counter-propagating approximately through the centre of the pulse these are largely unimportant, unless the particle is deflected out of the pulse in the transverse direction. For simplicity then, we will consider a plane wave of the form: where E 1 and E 2 correspond to electric fields of an arbitrary form in the and λ directions respectively. Note the dependence on φ only to satisfy Maxwell's equations | 36

Introduction to the theoretical model
To obtain the relevant equation of motion we need to substitute (3.12) into (3.10).
Considering the form of the electromagnetic field tensor in equation (3.12), the sandwiched tensor G ab can be written out: c are as follows: where prime represents differentiation with respect to φ and subscript i ∈ 1, 2.
Taking a = b and summing in (3.14) leads to the determinant of the form: Having all the necessary components of the Ford-O'Connell equation we can assemble the general equation of motion for a particle interacting with an arbitrarily polarised plane wave laser pulse, which reads: Introduction to the theoretical model (3.17) Contracting equation (3.17) with the basis vectors and using (3.5) we obtain 4 separate equations of motion, one for each component, which are as follows: | 38

Particle equations of motion in a linearly polarised plane wave
By dropping all the terms of O(τ 2 ) we reduce to the Landau-Lifshitz set of equations, which reads:φ

Particle equations of motion in a linearly polarised plane wave
We consider the simple case where we model the laser pulse by a linearly polarised (in direction) plane wave given by

Particle equations of motion in a linearly polarised plane wave
Applying the linearly polarised plane wave (3.28) to the set of Ford-O'Connell equations of motion, they simplify to:φ .
or when as is clearly borne out by Eq.         From Fig. 3.4, it is clear that the electron almost instantaneously loses most of its energy to radiation. After this, it hardly radiates at all, and its evolution is well described by the Lorentz force alone, at a greatly reduced initial energy.
As shown in Fig. 3.5, as the electron propagates through the laser field, a particle initially at the peak of the field starts with T > 1, and in the very early stages of its motion Ford-O'Connell predictions deviate from those of Landau-Lifshitz, indicating that breakdown of the Landau-Lifshitz approximation is possible if the peak field and peak energy coincide. However, after approximately 1/300 of a cycle, T becomes sufficiently small that the two descriptions are indistinguishable. For a particle initially at a node of the wave, T begins at zero, and never approaches unity, as can be seen in The analysis presented above assumes that the laser pulse can be described by a  harmonic plane wave. It has been shown that the radiation reaction effects within this approximation ensure that it cannot enter a regime where the Landau-Lifshitz description breaks down, even when a priori estimates would suggest otherwise, provided the particle does not start at the peak of the field. It is of interest to explore whether the results remain valid for a pulse with a more realistic structure, which we consider next.

Particle motion in a finite laser pulse
To compare the predictions of Ford-O'Connell and Landau-Lifshitz in a more realistic scenario, we need to specify a finite pulse shape for the profile of the electric field E 1 , though the specific choice does not significantly affect the results. It will be convenient to choose E 1 to have compact support, so the electron can begin and end in vacuum.

(3.40)
This represents an N -cycle pulse of central frequency ω, modulated by a sin 2 -envelope.
a 0 is the usual intensity parameter (sometimes called "normalised vector potential"). To describe the impact of radiation reaction we will again look at the evolution of φ(s), indicating how long the particle experiences the pulse with radiation reaction taken into account, along with the evolution of γ in order to have a quantitive estimate of the energy lost to radiation.
To do this we consider a laser pulse approximated by a plane wave with N = 10 oscillations stretching from φ i = 0 to φ f = 2πN in the lightfront time φ. A particle with initial energy γ in is placed in front of the pulse at φ(0) = −π/2. The transverse   remains strong despite the expected prediction, as for the infinite wave.
Consider the T parameter, which is one possible quantitative measure for the divergence between the two approaches investigated. Because the particle begins in vacuum, initially T = 0, and from Fig. 3.11, it is clear that the electron loses almost all its energy to radiation in the first two cycles, while E 1 ωa 0 . After this, the radiation becomes a small effect, and its evolution is well described by the Lorentz force alone, at a greatly reduced initial energy. As shown in Fig. 3.12, as the electron propagates through the laser pulse, its energy loss occurs at such a rate that T never approaches unity. Thus the Landau-Lifshitz equation remains a good description of radiation reaction phenomena for field strengths and electron energies far exceeding those currently proposed. Comparison with the values of T calculated for a (hypothetical) particle experiencing the Lorentz force alone demonstrates that the validity of the Landau-Lifshitz equation for such high energies is a direct consequence of radiation reaction itself (note the different scales in Fig. 3.12).

Summary
Radiation reaction can have a significant effect on the motion of a charged particle interacting with a laser pulse, even coming to dominate over the applied Lorentz force. Nonetheless, a high energy electron traversing an ultra-intense laser pulse loses most of its energy to radiation in the first few cycles. When it reaches the peak of the field, therefore, its energy is comparatively low. For field strengths and electron energies far exceeding those currently proposed, radiation reaction effects ensure that the instantaneous evolution of the particle's worldline can be accurately described by treating radiation reaction as a small correction, as in the prescription of Landau and Lifshitz.

| 50
In the previous Chapter we considered the interaction of a single particle with an intense laser pulse. However, realistic experiments involve a bunch of particles. For a bunch of particles we need to consider the evolution of bulk properties such as the average momentum and relative momentum spread.  [46], where they showed that the ponderomotive force of an intense laser can cause charge separation, leaving a charged region in a previously neutral plasma.
Particles injected into this region are accelerated, which provides a compact alternative to conventional accelerators. In typical LWFA experiments electron bunches are produced with charge ∼ 10 pC corresponding to 10 8 particles, average energy ∼ 1 GeV [47,48,49], and relative energy spread ∼ 1% [50].
In this Chapter we extend the single particle model presented in Chapter 3 and study how radiation reaction influences the average momentum and momentum spread of a bunch of particles during their propagation through an intense laser pulse. We are particularly interested in how the evolution of the distribution depends on the length of a pulse and its total energy per unit area.

Introduction of the particle distribution function
After considering radiation reaction effects on the motion of a single particle colliding with an intense laser pulse we can extend our study further and look at the behaviour of a bunch of particles colliding with the laser pulse under the influence of radiation reaction.
Because we consider a plane wave, the particles' spatial spread would only define the moment in time when the particular particle enters the pulse, so for simplicity we take all particles to originate from the same point. This is reasonable as we are | 52

Introduction of the particle distribution function
primarily interested in the momentum distribution.
Considering the case with no spread in the transverse directions we are using the initial thermal Maxwellian particle distribution for longitudinal velocities v, which can be written as follows: where θ = kT mc 2 is the thermal momentum spread, with k the Boltzman constant, and N p is the number of particles.
Although such a distribution is usually associated with a non-relativistic thermal momentum spread, whereas we investigate situations which are neither thermal nor non-relativistic it is a convenient distribution to illustrate evolution of the bulk properties of the particle bunch.

The average velocityv
where n a is the particle number current.
We consider the number density, as the a = 0 component of the number current vector: To find the evolution of the distribution f (v, φ) we could solve the Vlasov equation, modified to include radiation reaction [51,52,53,54,55,56]. However for computational efficiency we follow the evolution of a finite number of 500 particles, chosen to represent the distribution (4.1). Because the bunch is moving relativistically we can neglect interparticle interactions over the timescale that the bunch experiences the pulse.
Typically one would sample the velocities of the particle distribution at random which would require a large number of particles to accurately represent the distribution.
Instead, we determine the velocity spacing δv between the particles from the initial | 53 4.2. Numerical simulations. Impact of the pulse length on the particle distribution distribution, by truncating the integral in (4.3) so the particle number increases by 1: leading to particles having initial velocities: , , . . .

(4.5)
Using these initial conditions we then integrate the Landau-Lifshitz equation and using the later spacing between particle velocities apply (4.4) in reverse to reconstruct the distribution, f (v, φ).

Numerical simulations. Impact of the pulse length on the particle distribution
As in the previous Chapter we consider an N -cycle pulse of central frequency ω, modulated by a sin 2 -envelope. a 0 is the usual intensity parameter (so called "normalised vector potential"). We use units such that ω = 1. Pulse parameters being varied between the simulations are: • number of oscillations N of the pulse; • energy (per unit area) of the pulse ∼ N a 2 0 .
The same pulse parameters are considered for the initial average velocityv of 10 2 , 10 3 and 10 4 .
While evaluating cases with different pulse length, we keep the energy in the pulse per unit area constant. The energy in the pulse is given by E = φ f φ i E 2 dφ, which for the pulse we are considering (4.6) is given by: where N is the number of oscillations in the pulse and a 0 is the peak intensity.
Therefore, keeping N a 0 2 constant ensures the above requirement is met.
We are interested in establishing the impact of the pulse length on the width of the velocity distribution after the interaction. Therefore, we fix the initial distribution width to be 1% of the initial average velocity of the distribution and compare with the spread after all the particles have passed through the pulse. The relative distribution width is calculated as:σ              If we now increase the central velocity of the particle bunch tov = 10 3 we observe a respective decrease in the final distribution width toσ f = 0.22%, as shown in Fig. 4.4,   4.5, 4.6, 4.7, 4.8, 4.9, and these figures also demonstrate that the final distribution width still remains independent of the pulse length. This is accompanied by the decrease in average velocity tov f = 220.5.
Moving to still higher average velocities, (the example ofv = 10 4 is presented in | 63 4.3. Numerical simulations. Impact of the pulse energy on the particle distribution An analytical solution for the velocity evolution of the Landau-Lifshitz equation was presented in [43]. Althoughv is not a solution to Landau-Lifshitz, for a sufficiently narrow distribution it will approximate such a solution. According to [43] the velocity changes as:v where and This allows us to rewrite (4.9) as: (4.12) | 64

Numerical simulations. Impact of the pulse energy on the particle distribution
According to (4.11) I 2 is typically O τ 2 . Considering that terms of O τ 2 in (4.12) are negligible due to τ being small and τ E v i the analytical solution (4.9) presented in [43] can be approximated as: wherev i is the initial average velocity,v f is the final average velocity and E is the energy of the laser pulse.
Analogously, an approximate analytical solution for the final distribution width can be obtained. Consider (4.13) with the replacementv i →v i (1 +σ i ). This results in: Using (4.13) the partial derivative and thev i /v f ratio in (4.15) can be expressed as: Combining (4.16) and (4.17) into (4.15) the final distribution width change can be approximated as:σ wherev i is the initial average velocity,σ i is the initial width of the distribution,σ f is the final distribution width and E is the energy contained in the laser pulse.
Plotting the simulation results for three different initial average velocitiesv = 10 2 ,       Approximate analytical solution and simulation data.

Summary
In order to improve the plots' visibility some of the data points representing "abnormal behaviour" have not been plotted in Fig. 4

Summary
Based on results presented in this Chapter we confirm that, while decreasing the energy of particles, radiation reaction also leads to a reduction in the momentum spread when a relativistic particle bunch passes through an intense laser pulse. Both analytical considerations and simulation results presented in this Chapter indicate that the change in average velocity and momentum spread of the particle distribution depends only on the total energy of the laser pulse, and remains completely independent of the way this energy is distributed.
Both (4.13) and (4.18) indicate that further increase of the initial average velocitȳ v i of the particle bunch leads to a unique final average velocityv f = 1/2τ E with zero velocity spread,σ f = 0. This can be interpreted as an effect of phase-space attractors of the Landau-Lifshitz equation [57].
These results remain valid in the classical theory, however it has recently been demostrated [58] that quantum radiation reaction may lead to a broadening of the distribution width.

| 69
Chapter 5 Scattering of an electron by a heavy nucleus | 70 In the previous Chapters we considered particles interacting with intense laser pulses. An alternative source of an extremely high electromagnetic field is the Coulomb field of an atomic nucleus [59]. The value of the electric field close to the surface of the nucleus can be as large as: for the 235 U nucleus, where e = −q is the charge on the proton.
We consider a setup where the particle is fired at a stationary nucleus from a large (compared with nuclear scales) distance with impact parameter b (see Fig. 5.1).
A similar setup has previously played an important role in modern physics: in 1911, Ernest Rutherford performed an experiment in which he fired a beam of alpha particles at layers of gold leaf only a few atoms thick [60]. He noted that while some of the particles passed through with little deflection a small fraction were deflected by very large angles. This result led Rutherford to postulate the existence of the atomic nucleus.
This Chapter will be devoted to an investigation of effects of radiation reaction on the motion of a high energy particle scattered by a heavy nucleus. We are interested in how radiation reaction during the particle-nucleus interaction impacts its trajectory and energy evolution. Regions of deviation of Ford-O'Connell predictions from the Landau-Lifshitz ones are investigated and the importance of quantum effects during the interaction is discussed.
Previously, work has appeared in the literature on related problems. Eliezer [61] considered head-on collisions between an electron and a nucleus using the Lorentz-

Theoretical model
We investigate the effects of radiation reaction on the motion of a particle scattered by a heavy nucleus during a collision. To achieve this goal we consider the trajectory and energy evolution of a particle including radiation reaction described by Ford-O'Connell and compare the outcome with both Landau-Lifshitz predictions and the case with no radiation reaction.

Theoretical model
The relation between this basis and the one used in the previous Chapters is given by The metric tensor is then as follows: To study the problem of an electron scattering off a nucleus, we consider the physical setup shown in Fig. 5.1 which is described by the following parameters: • b -impact parameter, • θ sc -scattering angle, • φ -velocity angle (measured from ξ-axis), • Longitudinal coordinate: ξ = R sin θ, • Transverse coordinate: σ = R cos θ and derive appropriate equations of motion.
Introduce the coordinates such that, with α = −x a η a , ξ = x a a , σ = x a λ a and ζ = x a κ a , we have x a = αη a + ξ a + σλ a + ζκ a . (5.6) For a single particle the scattering occurs in a plane, so we take: without loss of generality. Note that this would not be true for the Lorentz-Abraham-Dirac equation, which requires three initial conditions, which can't be chosen to lie in a plane without loss of generality.

Theoretical model
Considering equation (5.6), the normalisation condition,ẋ aẋ a = −1, requires: Since the time coordinate α always increases, its derivative satisfies the conditionα > 0, leaving us with:α = 1 +ξ 2 +σ 2 . (5.9) We consider the nucleus to be sufficiently massive that it can be treated as stationary. There are then two components of the Coulomb field of the nucleus acting on a particle in the ζ = 0 plane: where (m/q)E i are the and λ electric fields, with i ∈ {1, 2}, which can be further defined as: where K = qq n /4πm = −3Zτ /2, with q n = −Zq being the charge on the nucleus. Note that we use Heaviside-Lorentz units, where 0 = 1.
Based on the definition of the sandwiched tensor, G a b then has the following form: which can be further expanded using the definition ofẋ a from (5.6) as: The form of G a b combined with the normalisation condition (5.8) allows us to rewrite | 74

Theoretical model
the determinant as follows: (5.14) If we now consider a head-on collision, the directions of the velocity and position vectors coincide, leading to the following relation: Substituting (5.11) into (5.14) we obtain the expression for the determinant: For more general collisions, the equation of motion that we wish to solve is: Given the spherical symmetry of the field it is appropriate to use polar coordinates: The field is The derivatives of the field are then: In the polar coordinates defined above, the determinant (5.14) has the form The left-hand side of the general equation of motion (5.17) can be expanded by taking into consideration the form ofẋ a , which is as follows: x a =αη a +ξ a +σλ a . (5.28) Taking into account (5. 19, 5.22) and the normalisation condition −α 2 +ξ 2 +σ 2 = −1, we obtainα in terms of polar coordinates: 29) leading to an explicit form ofẋ a : x a = 1 +Ṙ 2 + R 2θ2 η a +Ṙ sin θ a + Rθ cos θ a +Ṙ cos θλ a − Rθ sin θλ a . x a = RṘθ 2 +ṘR + R 2θθ 1 +Ṙ 2 + R 2θ2 η a + 2Ṙθ cos θ − Rθ 2 sin θ +R sin θ + Rθ cos θ a + −2Ṙθ sin θ − Rθ 2 cos θ +R cos θ − Rθ sin θ λ a . (5.31) To complete the equation of motion we also need to convert the right-hand side of (5.17). Combining (5.24) with (5.10) we obtain the form of the field in polar coordinates: Contracting the above equation withẋ b will give us the external force. If we also consider (5.2, 5.3), the expression for the external force becomes: After some manipulation of the above equations we obtain three final sets of equations in polar coordinates describing particle motion in the field of the nucleus with no radiation reaction taken into account, with radiation reaction taken into account using the Landau-Lifshitz force and with the Ford-O'Connell force. These sets are listed below respectively.
With no radiation reaction:R | 77

Significance of quantum effects
With Landau-Lifshitz radiation reaction correction: With Ford-O'Connell radiation reaction correction:

Significance of quantum effects
When we approach ultrahigh fields quantum effects may become significant, therefore we need to estimate their importance. To do so we compare the fields the particle is interacting with, to the Schwinger limit [65,66], at which electron-positron pair creation leads to nonlinearities in the electromagnetic field. We assume that quantum effects can be ignored provided that: is the electric field as seen by the particle and E S = m 2 /q 1.3 · 10 18 Vm −1 is the Schwinger field. The χ parameter is a recognized measure of the | 78
With the polar coordinates used, this parameter corresponds to: Throughout the simulations we trace the value χ to estimate the importance of quantum effects in the given regime. However, exploring the consequence of quantum effects is beyond the scope of this thesis.

Numerical simulations
As we are interested in cases involving high fields we consider a particle with initial energy γ in shot at a highly charged Uranium nucleus with Z = 92 with an impact parameter b.
Throughout the simulations we study the evolution of the particle energy and particle trajectory. The evolution is tracked for the three different approaches: we consider the cases with no radiation reaction, and with the radiation reaction taken into account considering both Landau-Lifshitz and Ford-O'Connell corrections.
Additionally, for each set of parameters we focus on the evolution of the divergence parameter: which is a quantitative measure of the difference between Ford-O'Connell and Landau-Lifshitz. The quantum parameter, χ (5.41) is also tracked so we have an understanding when classical predictions are still reliable.
Parameters being varied between the simulations are: • initial energy γ in of the particle; • impact parameter b.

Numerical simulations
For the case of relatively high initial energy of the particle (γ in = 10 5 ) incident on the nucleus with a relatively large impact parameter (b = 1Å) we observe negligible impact of radiation reaction on the particle trajectory compared to the no radiation reaction case, as shown in Fig. 5 A shining example here is the case of initial energy γ in = 10 2 and impact parameter b = 10 −3Å where Ford-O'Connell predicts particle scattering, whereas Landau-Lifshitz predicts the electron will be captured by the nucleus, see Fig. 5.19. The initially free particle retains this state if radiation reaction isn't taken into account, since its total energy is conserved. However, radiation losses allow the particle's total energy to become negative, indicating that it becomes bound to the nucleus. Since in the Landau-Lifshitz case the particle spirals into the nucleus the ξ coordinate is no longer single valued, as seen in Fig. 5. 19-5.22. Landau-Lifshitz predicts greater energy loss than Ford-O'Connell allowing the transition from the free state to the bound one, which | 91

Summary
is not the case with Ford-O'Connell. However, as can be seen in Fig. 5.22, in this case we remain in a strongly quantum regime.
If we now consider the ratio between the divergence parameter (5.42) and the quantum parameter (5.41) we get:

Summary
It can be clearly seen that radiation reaction has an effect on particle motion in this physical scenario and one can see noticeable differences between the predictions of Ford-O'Connell and Landau-Lifshitz approaches, even when the divergence parameter T is small. This could lead to radiation reaction corrections to the Rutherford cross section (analogous to Dirac's radiation reaction correction to the Thomson cross section [8]).
For the extreme cases (b = 10 −3Å ) significant differences in the particle's behaviour are observed (such as particle capture) in regions where T is no longer small.
However, we find that in these cases quantum effects appear to be very important and radiation reaction is not found to be significant while quantum effects remain small. In subsequent Chapters we presented a study of the interaction of a high-energy particle with an intense laser pulse. By analysing this interaction we found that radiation reaction can have a significant effect on the particle motion and, in certain regimes, even dominates the applied Lorentz force. However, we have also found that the detailed interplay between the particle dynamics and the shape of the pulse ensures that, even with the most advanced technologies currently proposed, radiation reaction prevents the particle from accessing the regime where the Landau-Lifshitz approximation would break down.
Extending this study we proceeded with an exploration of radiation reaction effects on the momentum spread of a bunch of particles during their propagation through an intense laser pulse. We confirm that, while decreasing the energy of particles, radiation | 94

Summary
reaction also leads to a reduction of the initial momentum spread when a relativistic particle bunch passes through an intense laser pulse. This leads to an improvement in the bunch quality at the expense of its energy. One of the main results of this study indicates that the change in momentum spread of the particle distribution depends only on the total energy of the laser pulse, and remains completely independent of the way this energy is distributed. This behviour is backed up by both analytical considerations and simulation results.
We have also shown that, with the increase of initial momentum, the particle bunch asymptotes to a constant final average momentum with zero momentum spread.
We finally considered an atomic nucleus as a source of high fields, which led us to explore a particle being scattered off a heavy nucleus. We presented an analytical model, which includes classical radiation reaction corrections and derived relevant equations of motion for this case and discussed the relevance of quantum effects during the interaction.
We found that radiation reaction has an effect on the particle's motion and noticeable differences between the predictions of the Ford-O'Connell and Landau-Lifshitz approaches are evident, even when the divergence parameter T is small. An extreme case (b = 10 −3Å ) leads to significantly different predictions in the particle's behaviour (such as particle capture), which is observed in regions where the divergence parameter T is no longer small.
However, one of the main conclusions of this work is that in these cases quantum effects appear to be very important and conversely radiation reaction is not found to be significant while quantum effects remain small. This implies the need to develop a quantum model in order to correctly interpret these cases.

Outlook
The work presented in this thesis has been restricted by certain limitations which could be addressed in future work.
Chapter 3 was devoted to the exploration of the effects of radiation reaction for a single particle interacting with a plane wave. The results we obtained strongly indicated that while in the classical domain we do not anticipate strong differences between the Landau-Lifshitz and Ford-O'Connell approaches. However, realistic laser pulses have important transverse structure, and ponderomotive effects can eject the electron from the pulse. It would therefore be of interest to consider more realistic pulse shapes, which benefit from the transverse structure. This will require moving away from the plane wave approximation and would lead to a different condition for divergence between the Landau-Lifshitz and Ford-O'Connell approaches.
In Chapter 4 the evolution of the distribution function of a bunch of particles interacting with a plane wave was considered. One of the main limitations introduced there was the fact that all the particles of the distribution start at the same point in space, therefore having no spatial spread at all. Future extensions could include spatial as well as momentum spread and an investigation of the effects of radiation reaction also on the spatial spread. Consideration of an initial transverse spread, which is currently considered to be 0, and more realistic pulse structures are additional paths for future work.
As for the work in Chapter 5, we have clearly shown that in cases where Landau-Lifshitz and Ford-O'Connell approaches provide qualitatively different predictions, quantum effects appear to be very important and radiation reaction is not found to be significant while quantum effects remain small. This suggests the need to develop a quantum model in order to correctly analyse these cases.

| 96
Appendix A Maxwell's equations for a plane wave | 97 The aim of this Appendix is to demonstrate that, for the case of a plane wave, in order to satisfy Maxwell's equations the electric field depends on x only through the coordinate φ.
To achieve this goal consider Maxwell's equations written in terms of the electromagnetic field tensor: and Considering the form of the electromagnetic field tensor F ab for the plane wave case: where E 1 and E 2 correspond to electric fields of an arbitrary form in and λ directions respectively, and ∂ a F ab = ∂ a E 1 a n b − ∂ a E 1 b n a + ∂ a E 2 λ a n b − ∂ a E 2 λ b n a . (A.5) Using (A.4) and φ = n a x a , ψ = m a x a , ξ = a x a , σ = λ a x a , the field derivatives in (A.5) can be expanded as: where i ∈ {1, 2}.

| 98
Substituting (A.6) into (A.5): leading to the following relations: If we now analogously consider the second of Maxwell's equations (A.2): ∂ a F bc + ∂ b F ca + ∂ c F ab =∂ a E 1 b n c − ∂ a E 1 c n b + ∂ a E 2 λ b n c − ∂ a E 2 λ c n b + ∂ b E 1 c n a − ∂ b E 1 a n c + ∂ b E 2 λ c n a − ∂ b E 2 λ a n c + ∂ c E 1 a n b − ∂ c E 1 b n a + ∂ c E 2 λ a n b − ∂ c E 2 λ b n a . (A.11) Taking into account (A.9) and (A.10) and using the field derivative expansions (A. 6) we obtain the second Maxwell's equation for the case of the plane wave: ∂ξ λ a b n c − λ a n b c − a λ b n c + a n b λ c + n a λ b c − n a b λ c = 0 . For the general case of the plane wave we further differentiate (A.13) and (A.8): (A.14) If we now add (A.14) and (A.15) together: A laser pulse is characterised by a number of properties such as frequency, pulse duration, energy and intesity. For convenience, in this thesis we have been using the "intensity parameter" a 0 = qE/mωc and the number of cycles N . In this Appendix we show how these relate to physically measurable parameters.
• Frequency Since we are working in units such that ω = 1, the relation between τ used throughout the thesis and the SI value τ si gives us the frequency ω in SI units: where τ = 1.5 · 10 −8 .

(B.2)
• Pulse duration Given the above value of the frequency, the pulse duration can then be expressed as: where N is the number of oscillations in the pulse.
• Energy per unit transverse area The energy per unit transverse area (fluence) in the pulse can be expressed as: | 102 • Intensity The intensity is described by: