**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 *vs*, undergoes a slowing down of its 'internal clock' by a factor of *γ*, where

$$\gamma = \mathbf{1} / \sqrt{\mathbf{1} - \mathbf{v}\_s^2 / c^2} \tag{23}$$

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 *vs*, 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

#### **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.*

*Perspective Chapter: Slowing Down the "Internal Clocks" of Atoms – A Novel Way… DOI: http://dx.doi.org/10.5772/intechopen.102931*

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 *τ*<sup>0</sup> *<sup>d</sup>* according to the sample's clock is

$$
\pi\_d' = \frac{L}{\upsilon\_s \chi}.\tag{24}
$$

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 [25–27] and spectroscopy [28] experiments is � <sup>10</sup><sup>15</sup> cm�3. Proton bunches at the LHC contain 1*:*<sup>15</sup> � 1011 protons with a proton number density of � <sup>10</sup><sup>16</sup> cm�<sup>3</sup> [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 � 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 �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 Pbþ<sup>81</sup> 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 Pbþ<sup>81</sup> 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*þ, rest mass: 3*:*<sup>16</sup> � <sup>10</sup>�<sup>26</sup> kg) accelerated to an energy of 1*:*<sup>8</sup> TeV would experience a slowing down of time with a *γ* factor of 100. Since

$$E = \gamma mc^2,\tag{25}$$

the time resolution scales proportionally to energy, so an energy of 18 TeV would result in an astonishing *γ* ¼ 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�<sup>4</sup> to 10�<sup>8</sup> [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 [40–43] 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.

a. **Lab frame of reference.** The pulse duration in the *x*-direction *τ<sup>x</sup>* is given by

$$
\pi\_{\mathfrak{x}} = \frac{l\_{\mathfrak{x}}}{\upsilon\_p},
\tag{26}
$$

where *lx* indicates the pulse length in the *x*-direction and *vp* indicates the speed of the pump/probe pulse. The pulse duration in the *y*-direction *τ<sup>y</sup>* is given by

$$
\pi\_{\mathcal{V}} = \frac{l\_{\mathcal{V}}}{v\_s},
\tag{27}
$$

where *ly* indicates the pulse length in the *y*-direction and *vs* indicates the speed of the sample. *Vs* will always be close to the speed of light *c* in the proposed experiment.

b. **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

$$l'\_{\mathcal{Y}} = \frac{l\_{\mathcal{Y}}}{\mathcal{Y}}.\tag{28}$$

The pulse duration in the *y*-direction *τ*<sup>0</sup> *<sup>y</sup>* is then given by

$$
\pi\_y' = \frac{l\_y}{\rho v\_s}.\tag{29}
$$

The pulse duration in the *x*-direction *τ*<sup>0</sup> *<sup>x</sup>* is given by

$$
\pi'\_{\mathbf{x}} = \frac{pl\_{\mathbf{x}}}{v\_p} \tag{30}
$$

as *vp* transforms to

$$v\_p' = \frac{v\_p}{\chi} \tag{31}$$

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<sup>∘</sup> does not hit the sample perpendicularly in the sample frame of reference. The closer *vs* 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 *vp* ¼ *c*. Utilizing Eq. (41), the longitudinal component of velocity in the sample frame of reference is given by *c* cos *θ*<sup>0</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>c</sup>* cos *<sup>θ</sup>i*�*<sup>v</sup> c* 1� cos *θ<sup>i</sup> v c* , where *θ<sup>i</sup>* 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 *<sup>θ</sup><sup>i</sup>* <sup>¼</sup> *<sup>π</sup>* <sup>2</sup>, we end up with �*v* as the longitudinal component. The total speed of the

photon probe pulse is then given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *c γ* � �<sup>2</sup> þ �ð Þ*v* 2 r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>c</sup>*<sup>2</sup> <sup>1</sup> � *<sup>v</sup>*<sup>2</sup> *c*2 � � þ �ð Þ*<sup>v</sup>* 2 q ¼ *c*, which is expected as the speed of light is constant in all frames of reference.

Typical pulses are Gaussian temporally and spatially (*τ<sup>x</sup>* ffi *τy*) and so the time resolution *τres* would be determined by the relativistically shortened duration *τ*<sup>0</sup> *<sup>y</sup>*. It would thus be given by

$$
\tau\_{res} = \frac{1}{\chi} \sqrt{\tau\_{pump}^2 + \tau\_{probe}^2},
\tag{32}
$$

where *τpump* and *τprobe* 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 *ly* ¼ 10 *μ*m and *γ* ¼ 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*.

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 *θ* 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

#### **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.*

*Perspective Chapter: Slowing Down the "Internal Clocks" of Atoms – A Novel Way… DOI: http://dx.doi.org/10.5772/intechopen.102931*

that the sample 'sees', *fs* , is given in terms of the frequency in the lab frame of reference, *fl* , by

$$f\_s = f\_l \eta (1 - \beta \cos \theta) \tag{33}$$

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

$$
\beta\_{\epsilon} = \frac{v\_s}{v\_{\epsilon}},
\tag{34}
$$

where *ve* is the speed of the particles but *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *v*<sup>2</sup> *<sup>s</sup> <sup>=</sup>c*<sup>2</sup> <sup>p</sup> 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

$$f\_s = f\_l \gamma. \tag{35}$$

Depending on the angle, there could either be a redshift or a blueshift. For one critical angle *θ<sup>c</sup>* there is no frequency shift. For *β* (*βe*) ≈ 1, as *γ* becomes larger, *θ<sup>c</sup>* 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 *Ei* with angular frequency *ω<sup>i</sup>* polarized in the z-direction and incident at angle *θ<sup>i</sup>* (angle between photon wave vector *k* ! *<sup>i</sup>* and sample velocity vector *v* ! *<sup>s</sup>*), the field would be Lorentz transformed to the sample frame of reference to *E*<sup>0</sup> *<sup>i</sup>* in the following way: [48, 49].

$$\overrightarrow{E}\_i = \hat{z}E\_i \exp\left[ik\_i(\cos\theta\_l y + \sin\theta\_i \mathbf{x})\right] \exp\left(-i\alpha\_l t\right),\tag{36}$$

$$\overrightarrow{E}\_{i}^{\prime} = \hat{\mathbf{z}}^{\prime} \mathbf{E}\_{i}^{\prime} \exp\left[ik\_{i}^{\prime}(\cos\theta\_{i}^{\prime}\mathbf{y}^{\prime} + \sin\theta\_{i}^{\prime}\mathbf{x}^{\prime})\right] \exp\left(-i\alpha\_{i}^{\prime}t^{\prime}\right),\tag{37}$$

where

$$E\_i' = \chi (1 - \beta \cos \theta\_i) E\_i,\tag{38}$$

$$k'\_i = \frac{\alpha'\_i}{\mathcal{c}},\tag{39}$$

$$
\alpha\_i' = \gamma (1 - \beta \cos \theta\_i) a\_i,\tag{40}
$$

$$\theta\_i' = \cos^{-1}\left[\frac{\cos\theta\_i - \beta}{1 - \cos\theta\_i(\beta)}\right].\tag{41}$$

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

$$
\alpha\_i' = \chi (1 - \beta\_\epsilon \cos \theta\_i) a\_i,\tag{42}
$$

$$\theta\_i' = \tan^{-1} \left[ \frac{\sin \theta\_i}{\chi (\cos \theta\_i - \beta\_\epsilon)} \right]. \tag{43}$$

However, as before *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> with *<sup>β</sup>* <sup>¼</sup> *vs <sup>c</sup>* 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*π* 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 *atransverse* <sup>¼</sup> *Ftransverse <sup>γ</sup><sup>m</sup>* . Forces along the ion beam path would cause acceleration according to *alongitudinal* <sup>¼</sup> *Flongitudinal <sup>γ</sup>*3*<sup>m</sup>* . Hence, acceleration due to transverse forces is reduced by a factor of <sup>1</sup> *<sup>γ</sup>* and acceleration due to longitudinal forces is reduced by a factor of <sup>1</sup> *<sup>γ</sup>*<sup>3</sup> 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.
