Perspective Chapter: Slowing Down the “Internal Clocks” of Atoms – A Novel Way to Increase Time Resolution in Time-Resolved Experiments through Relativistic Time Dilation

Abstract

Traditional time-resolved studies typically rely on a pump laser beam that triggers a reaction dynamic in an atom or molecule and is subsequently probed by a probe pulse of photons, electrons or neutrons. This traditional method is reliant on advancements in creating ever shorter probe and pump pulses. The shorter the pulses the higher is the time resolution. In this chapter we would like to present a novel idea that has the potential to achieve 2–3 orders of magnitude higher time resolutions than is possible with laser and electron compression technology. The proposed novel method is to slow down the ‘internal clock’ of the sample. This can be achieved by accelerating the sample to relativistic speeds, which can be realized in particle accelerators such as cyclotrons and synchrotrons.

1. Introduction

Until a few decades ago capturing molecular dynamics was in the realm of Gedanken- or thought experiments [1, 2]. Chemists know about the reactants and the final products of chemical reactions, but how the molecules and atoms rearrange themselves to produce the reaction products had always remained in the realm of imagination. This is due to the technical difficulties in making these measurements. In solids, chemical reactions occur at the speed of sound ($\sim$1000 m/s) and atomic bond lengths are on the order of 1 Å, which means that the time resolution required is on the order of femtoseconds [3, 4].

Recent advances in laser technology have made it possible to produce laser pulses that are femtoseconds and even attoseconds in duration [5]. This has enabled rapid developments in the field of ultrafast science. Typically, a short laser pulse initiates a photo-induced reaction dynamic in a molecular sample, which is then probed by a probe pulse. Probe pulses can be short X-ray pulses [6] in X-ray free electron lasers (XFELs) [7] or compressed electron pulses [8] in tabletop experiments [3, 9]. Additionaly, laser pulses can be used in table-top spectroscopy experiments, which temporally probe molecules and atoms but not with the same spatial resolution as X-rays [10, 11]. XFELs are multibillion dollar facilities that are very costly to operate and entail very complex engineering [12]. However, electron beams are generated in table-top experiments. The electrons are usually accelerated via a DC electric field for a short distance in order to avoid rapid expansion due to Coulomb forces between them [13], or they are accelerated via a DC field, then compressed via an RF field [14, 15]. There are also designs where acceleration and compression take place through the same RF field [16]. Furthermore, utilizing relativistic electron sources can also improve brightness and time resolution since they greatly reduce pulse broadening effects [17, 18, 19]. In all cases a short probe pulse is produced. Probe pulses capture molecular dynamics and produce a diffraction pattern. By varying the time delay between pump and probe pulses, the molecular dynamics at different time points can be captured and a’molecular movie’ can be generated. As a result, the time resolution is mainly limited by the technological ability to produce ever-shorter laser and electron pulses [20], both for triggering a photo-induced dynamic rapidly and for imaging it. In other approaches, the probe pulse is dissected to increase time resolution. In the case of electrons, streak cameras [21] that spatially separate a long electron beam into smaller pieces have been developed, enabling higher time resolutions [22]. Another proposed method, known as optical gating [23], uses ultrashort laser pulses to dissect the electron beam and achieve a higher time resolution than was originally possible based on the length and speed of the beam.

For the sake of making this chapter as self-encompassing as possible, we will start with a review of special relativity (SR) and the concepts of time dilation and length contraction. This shall make for a smoother understanding of the core ideas of the proposed novel experiment.

2. Special relativity: a quick review

2.1 A brief history

Towards the end of the nineteenth century there was one major inconsistency plaguing the structure of theoretical physics. Newton’s¹ equations described very well the mechanics of moving objects ranging from tiny objects on earth to the orbits of planets in space. Maxwell² had successfully completed the theoretical framework of electromagnetism in 1865, a monumental task that crowned the gradual understanding of electromagnetism throughout the eighteenth and nineteenth centuries through the work of scientists such as Coulomb³, Ampère⁴ and Faraday⁵. Despite the enormous success of both Newton’s and Maxwell’s frameworks in describing mechanical objects and light, respectively, they were inconsistent with each other. Maxwell’s equations predicted a constant speed of light and Newton’s equations suggest that objects moving at different speeds should measure the speed of light to be different. On a more rigorous level, Newton’s equations are invariant under Galilean⁶ transformations while Maxwell’s equations are invariant under Lorentz⁷ transformations. So, each set of equations suggested a different symmetry present in nature. Moreover, in 1887, on the experimental side, Michelson⁸ and Morley⁹ tried to detect a difference in the speed of light for observers moving at different speeds but their results were decidedly negative. There were some attempts to resolve this consistency but it was Einstein,¹⁰ who, in 1905, successfully presented the correct solution through a radical theory that would change our understanding of nature forever.

To resolve the inconsistency Einstein suggested that:

1. The laws of physics are invariant in all inertial frames of reference.

2. The speed of light in vacuum is constant in all frames of reference.

Simply put, he suggested that Newton’s equations were fundamentally wrong and he replaced them with the equations of SR that were invariant under Lorentz transformations. The implications were dramatic: different observers experience different rates of time, lengths can shrink or elongate and many other peculiar effects take place as objects approach the speed of light. Newton’s equations were only a very good approximation as long as objects moved slowly compared to the speed of light.

2.2 Mathematical framework

There are many ways to approach SR from a mathematical point of view. In this chapter we will present the mathematical framework of SR in a simple manner.

2.2.1 Galilean transformations

Limiting our consideration to one spatial dimension for simplicity, the most general way one can transform between two coordinate systems $O$ and $O^{\prime }$, where $O^{\prime }$ is moving with speed $v$ in the positive x-direction compared to $O$, is the following:

$$\left[\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right]=\left[\begin{array}{cc}a& b\\ e& f\end{array}\right]\left[\begin{array}{c}x\\ t\end{array}\right]$$E1

Assuming that space is homogeneous and noticing that the principle of relativity requires that $O$ moves at speed -v compared to $O^{\prime }$, we are restricted to linear transformations. A deeper mathematical analysis of this is outside the scope of this chapter. Taking all this into account the forward and backward transformations are calculated to be:

$$\left[\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right]=\left[\begin{array}{cc}a& -\mathit{av}\\ \frac{1-a^2}{\mathit{av}}& a\end{array}\right]\left[\begin{array}{c}x\\ t\end{array}\right]$$E2

$$\left[\begin{array}{c}x\\ t\end{array}\right]=\left[\begin{array}{cc}a& \mathit{av}\\ \frac{a^2-1}{\mathit{av}}& a\end{array}\right]\left[\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right]$$E3

The Galilean transformations coincide with our everyday intuition (a = 1). Velocities are additive, acceleration is invariant and time is the same for all observers.

$$x^{\prime }=x-\mathit{vt}$$E4

$$t^{\prime }=t$$E5

2.2.2 Lorentz transformations

Going back to the general form of the transformations, we have:

$$x^{\prime }=\mathit{ax}-\mathit{bt}$$E6

$$x=a{x}^{\prime }+b{t}^{\prime }$$E7

By setting $x^{\prime }=0$, we can calculate the relative velocity of $O^{\prime }$ with respect to $O$

$$V^{\prime }=b/a\equiv v$$E8

and similarly, by setting $x=0$, we calculate

$$V=-b/a=-v$$E9

Assuming the speed of light is constant in all inertial frames of reference, we consider a light signal when the origins coincide ($t^{\prime }=t$, $x^{\prime }=x$). The propagation of the light signal in both frames is:

$$x=\mathit{ct}$$E10

$$x^{\prime }=c{t}^{\prime }$$E11

Substituting Eqs. (10) and (11) into Eqs. (6) and (7) yields:

$$c{t}^{\prime }=\left(\mathit{ac}-b\right)t$$E12

$$\mathit{ct}=\left(\mathit{ac}+b\right)t^{\prime }$$E13

Substituting Eq. (13) into Eq. (12) and using Eq. (8) yields:

$$a=\frac{1}{\sqrt{1-v^2/c^2}}\equiv \gamma$$E14

$$b=\mathit{av}=\mathit{\gamma v}$$E15

Substituting Eqs. (14) and (15) into Eqs. (6) and (7) yields:

$$x^{\prime }=\gamma \left(x-\mathit{vt}\right)$$E16

$$x=\gamma \left(x^{\prime }+v{t}^{\prime }\right)$$E17

By substituting Eq. (16) into Eq. (17) the transformation of time is obtained:

$$t^{\prime }=\gamma \left(t-\frac{\mathit{vx}}{c^2}\right)$$E18

$$t=\gamma \left(t^{\prime }+\frac{v{x}^{\prime }}{c^2}\right)$$E19

2.2.3 Time dilation

Consider an observer in the frame of reference O at the origin, so $x=0$ with a clock that has a period $\mathrm{\Delta }t=t_2-t_1$. For the observer in $O^{\prime }$ that period is much longer, namely

$$\mathrm{\Delta }{t}^{\prime }=\gamma \mathrm{\Delta }t.$$E20

Hence, the clock is ticking slower for the moving observer.

2.2.4 Length contraction

Similar to the peculiar effect of time dilation, a moving observer experiences a contraction in length along the direction of movement. Consider an observer in the frame of reference O that measures a moving ruler to be of length $\mathrm{\Delta }x$. This measurement happens instantly at one point in time such that $t_2=t_1$. This is an important detail as simultaneous times in one frame are not simultaneous in the other one. For the observer in $O^{\prime }$ that length transforms to

$$\mathrm{\Delta }{x}^{\prime }=\gamma \mathrm{\Delta }x.$$E21

Now, by symmetry, i.e., a ruler at rest in $O$, that is measured to be of length $L_o$ in $O$, is measured to be of length

$$L_o^{\prime }=\frac{1}{\gamma }\mathrm{\Delta }{L}_o$$E22

in $O^{\prime }$. Alternatively, we could have considered an object at rest in $O$ at the beginning but this would have added an extra step in the derivation as we would have considered the transformations both in space and time. Either way we arrive at the same result; moving objects experience length contraction along the direction of motion.

With that knowledge in mind, we are now ready to discuss the proposed experiment.

3. The proposed experiment

This section presents a method for studying molecular and atomic dynamics using time-resolved diffraction or spectroscopy studies with greater time resolution without relying on laser or electron beam technology advancements [24]. In this method, instead of shortening the probe pulse, the ‘internal clock’ of the sample (charged molecule or ion) is slowed down. This can be accomplished by accelerating the sample to relativistic speeds, which can be realized in particle accelerators, such as cyclotrons and synchrotrons. A sample, which is accelerated to speed $v_s$, undergoes a slowing down of its ‘internal clock’ by a factor of $\gamma$, where

$$\gamma =1/\sqrt{1-v_s^2/c^2}$$E23

relative to the lab frame irrespective of its velocity direction. As a result, the time resolution becomes a function of the sample’s energy rather than being mainly reliant on pump and probe pulse durations. This can easily enable new time resolutions that have never been unlocked before.

3.1 Experimental considerations

To successfully implement any novel method, there are several barriers and challenges to overcome. We will introduce the experimental setup and discuss the experimental challenges and limitations, as well as the physics involved in the proposed setup.

3.1.1 Setup

We propose accelerating the samples in a cyclotron or synchrotron and studying them at a fixed energy $E$ and, consequently, at a fixed speed $v_s$, which remains constant during data collection. Figure 1 shows a schematic of the experimental setup. During the experiment, samples are accelerated into a chamber in which a pump pulse is directed parallel to a probe pulse and perpendicular to the direction of the sample’s motion. The delay between the pump pulse and probe pulse can be controlled by changing the distance $L$ between the two beams. The resulting time delay ${\tau }_d^{\prime }$ according to the sample’s clock is

Figure 1.

Schematic of the proposed experimental setup: Samples are accelerated to a fixed energy. A pump pulse and a probe pulse are directed parallel to each other and perpendicular to the sample direction of motion. The delay between the pump pulse and probe pulse can be controlled by changing the distance between the two beams.

$${\tau }_d^{\prime }=\frac{L}{v_s\gamma }.$$E24

As a result, the signal will reflect the changing dynamics according to the ‘internal clock’ of the sample.

3.1.2 Sample suitability

To begin with, this novel method can only be applied to electrically charged ions or molecules so that they can be accelerated to relativistic speeds, and to molecules that are in the gas phase in order to achieve the required energies. To run the experiment for a given time resolution, the higher the mass of the sample, the more energy is needed, so the best candidates for such studies are light, charged molecules, and ions.

The typical number density in gas phase UED [25, 26, 27] and spectroscopy [28] experiments is $\sim$${10}^{15}$ cm${}^{-3}$. Proton bunches at the LHC contain $1.15$$\times$${10}^{11}$ protons with a proton number density of $\sim$${10}^{16}$ cm${}^{-3}$ [29, 30]. There have been many schemes to reduce bunch length, and in fact an order of magnitude shorter bunch length has been produced for the purpose of accelerating electrons with plasma wakefields of proton bunches [31].

It is also necessary for a sample to be stable when subjected to the accelerator conditions. H-anions with a binding energy of $\sim$ 0.75 eV are accelerated regularly to 520 MeV at TRIUMF [32]. Molecules with covalent bonds typically have binding energies of 1 eV or higher. The typical length scale of a covalent bond is 1 Å. Hence, to break a bond, a force of $\sim$1600 pN is needed. The forces exerted by typical electric fields ($<$10 MV/m) and typical magnetic fields ($<$10 T) are orders of magnitude less than 1600 pN. Furthermore, second ionization energies for atomic ions are typically significantly higher than 1 eV.

In addition to originally being designed to accelerate only protons and positively charged ions, the LHC ring has recently been used to accelerate partially stripped Pb${}^{+81}$ ions with one electron to an energy of 6.5 Q TeV [33] as part of the gamma factory proposal [34] to create a new type of high intensity light source. Although originally designed to accelerate protons or positively charged ions only, the LHC ring has recently accelerated partially stripped Pb${}^{+81}$ ions with one electron to an energy of 6.5 Q TeV [33], where Q is the ion charge number, as part of the gamma factory proposal [34] to create a new type of high intensity light source. Currently, engineering challenges with regard to collimation are being addressed [35].

3.2 Theoretical considerations

3.2.1 Energy considerations

At the moment, the Large Hadron Collider at CERN can accelerate protons to energies on the order of 7 TeV [36] and lead ions to 5 TeV collision energies [37], which is enough to boost the resolution of time measurements significantly. As an example, a hydronium molecule ($H_3{O}^{+}$, rest mass: $3.16\times {10}^{-26}$ kg) accelerated to an energy of $1.8$ TeV would experience a slowing down of time with a $\gamma$ factor of $100$. Since

$$E=\gamma {\mathit{mc}}^2,$$E25

the time resolution scales proportionally to energy, so an energy of $18$ TeV would result in an astonishing $\gamma =1000$. Additionally, time resolution is inversely proportional to mass, so hydrogen ions, for example, would experience an order of magnitude more gain in time resolution than hydronium molecules with the same energy.

The effect of relativistic time dilation on dynamical processes will still be extremely fascinating to observe, even if the particle accelerators cannot be commissioned to perform this experiment in the near future. With current laser technology it is possible to observe changes in differential detection, with and without a perturbation, as small as ${10}^{-4}$ to ${10}^{-8}$ [38, 39] using standard modulation techniques and photon detectors. There has also been major advances in laser based particle accelerators up to field gradients as high as $100$ GeV/m [40, 41, 42, 43] that will soon enable particle kinetic energies up to $10$ GeV range or higher. As compared with particle accelerators, this level of relativistic energy would result in only modest time dilation. It would nonetheless constitute a direct measurement of time retardation, which would prove to be an important test case for the development of laser-based particle acceleration with the goal of controlling the time variable directly, asymptotically approaching’stopping’ time. A control of the time variable could open up new avenues, beyond simple imaging, to driving dynamics that are otherwise too rapid to control.

3.2.2 Pump and probe beam dynamics

As in standard ultrafast studies, pulsed pump and probe beams can be used. The time resolution is largely determined by the pulse duration of the pump and probe pulses. In conventional terms, the pulse duration refers to the time during which the full width at half maximum (FWHM) of the pulse crosses the sample. The pump pulse determines the trigger speed, and the probe pulse determines the imaging time resolution.

Besides velocity mismatch [44, 45], which takes place due to the difference in velocity between pump and probe pulses and their different incidence angles, other factors that affect the time resolution are the time of arrival jitter [25] for RF accelerated electron pulses. By the very nature of the experimental geometry, however, a lower resolution due to velocity mismatch or time of arrival jitter is avoided, as both pulses are parallel, so the delay time from time zero is solely determined by the speed of the sample between the pump and probe pulses. According to our proposed setup, the pulse crosses the sample in two directions, and we will thus consider the pulse duration during which the pulse crosses the sample or vice versa, in both directions. To explain the physics we denote the direction, parallel to the direction of propagation of the sample beam, $y$ and denote the perpendicular direction $x$. For clarification, we will treat the problem from the lab frame of reference as well as from the sample frame of reference.

1. Lab frame of reference. The pulse duration in the $x$-direction ${\tau }_x$ is given by

   $${\tau }_x=\frac{l_x}{v_p},$$E26

   where $l_x$ indicates the pulse length in the $x$-direction and $v_p$ indicates the speed of the pump/probe pulse. The pulse duration in the $y$-direction ${\tau }_y$ is given by

   $${\tau }_y=\frac{l_y}{v_s},$$E27

   where $l_y$ indicates the pulse length in the $y$-direction and $v_s$ indicates the speed of the sample. $V_s$ will always be close to the speed of light $c$ in the proposed experiment.

2. Sample frame of reference. With relativistic speeds approaching the speed of light, the effect of length contraction along the direction of sample propagation $y$ becomes significant. Hence, the length of the pulse in the $y$-direction is contracted to

$$l_y^{\prime }=\frac{l_y}{\gamma }.$$E28

The pulse duration in the $y$-direction ${\tau }_y^{\prime }$ is then given by

$${\tau }_y^{\prime }=\frac{l_y}{\gamma v_s}.$$E29

The pulse duration in the $x$-direction ${\tau }_x^{\prime }$ is given by

$${\tau }_x^{\prime }=\frac{\gamma l_x}{v_p}$$E30

as $v_p$ transforms to

$$v_p^{\prime }=\frac{v_p}{\gamma }$$E31

in the sample frame of reference. This happens because of relativistic angle aberration. In the lab frame of reference a pulse that is emitted at ${90}^{\circ }$ does not hit the sample perpendicularly in the sample frame of reference. The closer $v_s$ is to $c$ the smaller is the incidence angle between the beam and the sample’s line of motion in the sample frame of reference. When discussing the observable signal relativistic angular aberration will be discussed in more detail.

However, as a quick check, if we assume that the probe beam consists of photons, then $v_p=c$. Utilizing Eq. (41), the longitudinal component of velocity in the sample frame of reference is given by $ccos{\theta }_i^{\prime }=c\frac{cos{\theta }_i-\frac{v}{c}}{1-cos{\theta }_i\frac{v}{c}}$, where ${\theta }_i$ is the angle in the lab frame of reference and $v$ is the speed of the sample in the lab frame of reference. Plugging in ${\theta }_i=\frac{\pi }{2}$, we end up with $-v$ as the longitudinal component. The total speed of the photon probe pulse is then given by $\sqrt{{\left(\frac{c}{\gamma }\right)}^2+{\left(-v\right)}^2}=\sqrt{c^2\left(1-\frac{v^2}{c^2}\right)+{\left(-v\right)}^2}=c$, which is expected as the speed of light is constant in all frames of reference.

Typical pulses are Gaussian temporally and spatially (${\tau }_x$$\cong$${\tau }_y$) and so the time resolution ${\tau }_{\mathit{res}}$ would be determined by the relativistically shortened duration ${\tau }_y^{\prime }$. It would thus be given by

$${\tau }_{\mathit{res}}=\frac{1}{\gamma }\sqrt{{\tau }_{\mathit{\text{pump}}}^2+{\tau }_{\mathit{\text{probe}}}^2},$$E32

where ${\tau }_{\mathit{\text{pump}}}$ and ${\tau }_{\mathit{\text{probe}}}$ are the transit time durations of the sample beam through the pump and probe pulses in the lab frame, respectively. As an example for pump and probe beams with $l_y=10$$\mu$m and $\gamma =100$, the time resolution would be roughly 470 as. Figure 2 shows a visual representation from both frames of reference along the relevant direction $y$.

Figure 2.

Lab and sample frames of reference: Along the $y$-direction the length of the pulse is contracted in the sample frame of reference relative to the lab frame of reference.

The lab frame of reference is the same as the sample frame of reference in conventional pump-probe experiments. Signals (e.g., diffraction patterns) always reflect interaction time according to the clock rate of the sample, so our proposed setup exploits the involved relativistic effects that result from the differences between two frames of reference.

3.2.3 Doppler effect and frequency shifts

In order to properly conduct the experiment, it is imperative to understand how the frequency of a laser, x-ray or electron pulse is ‘seen’ by the sample in its own rest frame. Changing the frequency of a laser can cause it to be outside the absorption spectrum of the sample, preventing the intended interaction. The spatial resolution of scattering x-rays and electrons would decrease if they undergo significant redshifts, for example. Furthermore, there is the relativistic effect of time dilation in addition to the classical Doppler effect. Even if the source and receiver are not crossing paths, relativity dictates a frequency shift known as the transverse Doppler effect [46].

If we let $\theta$ be the angle between the sample wave vector and the wave vector of the pump/probe particles, as measured in the lab frame of reference, then the frequency that the sample ‘sees’, $f_s$, is given in terms of the frequency in the lab frame of reference, $f_l$, by

$$f_s=f_l\gamma \left(1-\beta cos\theta \right)$$E33

for photons (see derivation in Appendix A) [47]. For other particles, e.g., electrons, one needs to replace $\beta$ with

$${\beta }_e=\frac{v_s}{v_e},$$E34

where $v_e$ is the speed of the particles but $\gamma =1/\sqrt{1-v_s^2/c^2}$ remains the same. In our proposed setup, where the pump and probe beams are perpendicular to the direction of motion of the samples, we have

$$f_s=f_l\gamma .$$E35

Depending on the angle, there could either be a redshift or a blueshift. For one critical angle ${\theta }_c$ there is no frequency shift. For $\beta$ (${\beta }_e$) $\approx$$1$, as $\gamma$ becomes larger, ${\theta }_c$ becomes smaller, meaning that the two wave vectors are more collinear. Although this would eliminate frequency shifts entirely, it would decrease the time resolution significantly.

3.2.4 The observable signal

The interaction between light and matter can take many different forms (e.g., absorption, scattering, etc.). In this section we present a general scheme for calculating the final observable signal in our proposed experiment.

The main steps are: (1) transforming the incident field from the lab frame of reference to the sample frame of reference by applying the Lorentz transformations; (2) calculating the resultant signal in the sample frame of reference; (3) transforming the signal from the sample frame of reference to the lab frame of reference by applying the Lorentz transformations one more time.

Without loss of generality, for an incident electric field $E_i$ with angular frequency ${\omega }_i$ polarized in the z-direction and incident at angle ${\theta }_i$ (angle between photon wave vector ${\overrightarrow{k}}_i$ and sample velocity vector ${\overrightarrow{v}}_s$), the field would be Lorentz transformed to the sample frame of reference to $E_i^{\text{'}}$ in the following way: [48, 49].

$${\overrightarrow{E}}_i=\widehat{z}{E}_{i}exp\left[{\mathit{ik}}_i\left(cos{\theta }_{i}y+\mathit{sin}{\theta }_{i}x\right)\right]exp\left(-i{\omega }_{i}t\right),$$E36

$${\overrightarrow{E}}_i^{\prime }={{\widehat{z}}^{\prime }{E}^{\prime }}_{i}exp\left[i{k}_i^{\prime }\left(cos{\theta }_i^{\prime }{y}^{\prime }+\mathit{sin}{\theta }_i^{\prime }{x}^{\prime }\right)\right]exp\left(-i{\omega }_i^{\prime }{t}^{\prime }\right),$$E37

where

$$E_i^{\prime }=\gamma \left(1-\beta cos{\theta }_i\right)E_i,$$E38

$$k_i^{\prime }=\frac{{\omega }_i^{\prime }}{c},$$E39

$${\omega }_i^{\prime }=\gamma \left(1-\beta cos{\theta }_i\right){\omega }_i,$$E40

$${\theta }_i^{\prime }={cos}^{-1}\left[\frac{cos{\theta }_i-\beta }{1-cos{\theta }_i\left(\beta \right)}\right].$$E41

For particles other than photons moving with speed $v_e$ the angular frequency and angle transform in the following way:

$${\omega }_i^{\prime }=\gamma \left(1-{\beta }_{e}cos{\theta }_i\right){\omega }_i,$$E42

$${\theta }_i^{\prime }={tan}^{-1}\left[\frac{sin{\theta }_i}{\gamma \left(cos{\theta }_i-{\beta }_e\right)}\right].$$E43

However, as before $\gamma =1/\sqrt{1-{\beta }^2}$ with $\beta =\frac{v_s}{c}$ remains the same.

Due to the nature of the angular transformations (see derivation in Appendix B), we expect scattering and diffraction angles to be wider than for the static case, and hence, we recommend detectors that cover as much of the 4$\pi$ sr solid angle as possible.

Momentum transfer due to light radiation pressure would have a more negligible effect on changing the ion beam path than in conventional ultrafast gas electron diffraction. Forces perpendicular to the ion beam path would cause acceleration according to $a_{\mathit{\text{transverse}}}=\frac{F_{\mathit{\text{transverse}}}}{\mathit{\gamma m}}$. Forces along the ion beam path would cause acceleration according to $a_{\mathit{\text{longitudinal}}}=\frac{F_{\mathit{\text{longitudinal}}}}{{\gamma }^{3}m}$. Hence, acceleration due to transverse forces is reduced by a factor of $\frac{1}{\gamma }$ and acceleration due to longitudinal forces is reduced by a factor of $\frac{1}{{\gamma }^3}$ as compared to conventional non-relativistic ultrafast gas electron diffraction. This is due to the well-known concept of transverse and longitudinal masses in special relativity.

4. Conclusions

In this chapter, we proposed a novel method for time-resolved studies that relies on taking advantage of relativistic effects rather than on advances in laser technology or electron beam compression. It was shown that by using currently available technology, this method could improve time resolution by $2$ or $3$ orders of magnitude. This has the potential of opening up a whole new domain of ultrafast dynamics that was previously unattainable. With ever more powerful accelerators being proposed, there will be great potential to achieve truly remarkable time resolutions that far exceed the status quo. We hope this would lead to new avenues of collaboration between the particle physics community and the ultrafast science community, which will maximize the research potential of particle accelerator facilities worldwide.

Acknowledgments

Hazem Daoud would like to thank Prof. R.J. Dwayne Miller for his support, and Prof. Pierre Savaria and Prof. David Bailey for fruitful discussions.

Appendix A. Relativistic frequency shifts

We will derive the relativistic frequency shift formula in two steps. First we will consider a classical example to derive the classical Doppler shift then we will consider the relativistic effects involved in the second step.

A.1 Classical Doppler shift

Consider a resting source emitting a photon with frequency $f_0$ and period $T_0$ collinear with an observer moving with speed $u$ in the same direction as the emitted photon. We denote the frequency observed as $f_S$. The observed wavelength ${\lambda }_S$ is given by

$${\lambda }_S=\left(c-u\right)T_0.$$E44

Hence,

$$\frac{c}{f_S}=\frac{c-u}{f_0},$$E45

and so,

$$f_0=\left(1-\beta \right)f_S,$$E46

where $\beta =\frac{u}{c}$.

A.2 Relativistic Doppler shift

1. The time interval between consecutive crests in the source frame of reference is $T_0$.

2. Due to relativistic time dilation, the time interval between consecutive crests in the observer frame of reference is $\gamma T_0$.

3. The time interval between the reception of consecutive crests in the source frame of reference is $T_0\left(1-\beta cos\theta \right)$, where $\theta$ is the angle between the observer’s wave vector and the wave vector of the photon.

4. The time interval between the reception of consecutive crests in the observer frame of reference is $\gamma T_0\left(1-\beta cos\theta \right)$. A calculation analogous to the classical Doppler shift yields $f_0=\gamma \left(1-\beta cos\theta \right)f_S$

Appendix B. Relativistic angular aberration

Consider a probe beam emitted from a source at an angle $\theta$ and speed $u$ with respect to the wave vector of an observer moving with speed $v$ in the frame of reference of the source. Considering a 2-d plane containing the wave vectors of the probe beam and the observer, we can define the components as follows:

$$u_x=ucos\theta$$E47

$$u_y=usin\theta$$E48

In the observer frame of reference (indicated with a prime), using the relativistic velocity transformations, yields:

$$u_x^{\prime }=\frac{ucos\theta -v}{1-\frac{v}{c^2}ucos\theta }$$E49

$$u_y^{\prime }=\frac{\sqrt{1-\frac{v^2}{c^2}}usin\theta }{1-\frac{v}{c^2}ucos\theta }$$E50

Hence,

$$tan{\theta }^{\prime }=\frac{u_x^{\prime }}{u_y^{\prime }}=\frac{\sqrt{1-\frac{v^2}{c^2}}usin\theta }{ucos\theta -v}.$$E51

The probe beam speed in the observer frame of reference is

$$u^{\prime }=\sqrt{{u^{\prime }}_x^2+{u^{\prime }}_y^2}=\frac{{\left(u^2-\frac{u^2{v}^{2}sin{\theta }^2}{c^2}-2ucos\mathit{\theta v}+v^2\right)}^{1/2}}{1-\frac{v}{c^2}ucos\theta }.$$E52

So,

$$cos{\theta }^{\prime }=\frac{u_x^{\prime }}{u^{\prime }}=\frac{ucos\theta -v}{{\left(u^2-\frac{u^2{v}^{2}sin{\theta }^2}{c^2}-2ucos\mathit{\theta v}+v^2\right)}^{1/2}}$$E53

If the probe beam consists of photons, then $u=c$ and this simplifies further to

$$cos{\theta }^{\prime }=\frac{cos\theta -\frac{v}{c}}{1-\frac{v}{c}cos\theta }.$$E54