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| Fig. 1: Picture of the wake formed by an electron beam (blue) in a plasma, similar to wakes behind a travelling boat (analogous to the white travelling regions of ions). This allows acceleration of charged particles (red) of electrons. (Source: Wikimedia Commons) |
Particle accelerators are important for fundamental physics research, developing medical techniques, and advancing materials characterization. The fundamental limit of conventional accelerators is the electric field they can sustain. Radio frequency (RF) cavities, the workhorse technology since the 1930s, can accelerate particles with an electric field of roughly 100 megavolts per meter before the cavity walls start breaking down. The only way to reach higher energies, therefore, is to build longer machines. This results in longer and longer particle colliders, with the Large Hadron Collider at CERN stretching about 27 kilometers beneath the Swiss-French border.
Looking ahead, the next proposed electron-positron collider could span 50 kilometers and cost tens of billions of dollars. This is not a sustainable trajectory. Physicists are looking at a new approach that exploits nature's own extreme electric fields rather than fighting engineering limits. Plasma wakefield acceleration (PWFA) is one promising promising alternative.
PWFA uses the electric fields sustained by collective electron oscillations in an ionized gas to accelerate charged particles.
We can break this down with an analogy to wakes formed behind a boat. When water hits the bow of a boat, the incompressible liquid curves around the boat and forms the wake that we observe behind the boat. The popular sport of wakesurfing involves a surfer using the artificial wake created behind the boat to surf with the boat.
PWFA adopts a similar technique. Plasma is a liquid of charged electrons and (heavier) ions — and we can draw an analogy to plasma as the water in the sea, and a ions as the boat. As illustrated in Fig 1., when an ion beam is sent through a plasma (white areas) the ions are unaffected in their trajectories due to their heavier weight, while the lighter electrons (tiny blue dots) move around the ions due to forces from the external field (from the oven), as well as attraction to the ions (if protons) and repulsion from other electrons. This eventually results in the electrons moving around the ions like a wake forming behind a boat. The ions and electrons in the plasma form the oscillatory pattern seen in Fig 1., known as electron plasma waves. When a charged particle (the red electrons) faces this wake, the electric field from the oscillations of ions and electrons accelerate the particle — just like a surfer riding the wake of a wake boat. [1,2]
Alternatively, without driving an ion beam through the plasma, a laser can be used to used as the external electromagnetic field to drive the lighter electrons around the heavier ions in a plasma. [3] This similarly creates electron plasma waves, which results in the alternative technique called laser wakefield acceleration (LWFA).
In a fully ionized plasma, the free electrons can sustain collective oscillations. The fundamental frequency of these oscillations is the plasma frequency ωp, given (in S.I. units) by
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(1) |
where n0 is the free electron number density, e is the electron charge, ε0 is the permittivity of free space, and me is the electron mass. [1]
The plasma frequency determines whether an electromagnetic wave can propagate through the medium. A plasma is permeable to an external electromagnetic field only when the electromagnetic field frequency ω exceeds ωp. Below this threshold, the free electrons respond fast enough to each cycle of the external field, allowing the plasma to counteract the field. This means that the external field (e.g. from a laser) must satisfy ω > ωp. This is easily achieved for optical lasers (ω ~ 1015 sec-1) in plasmas of density 1017 - 1018 cm-3 (where ωp ~ 1013 sec-1). Theoretical studies have shown that the laser frequency in LWFA setups should ideally be twice the plasma frequency to most effectively generate a wake. [2]
For PWFA, no such constraint applies since the ion beam is not an electromagnetic wave, and the heavier ions propagates freely through the plasma at any density. This implies that the technique of PWFA might have less constraints on driving plasma wakes as compared to LWFA.
The peak fields currently achievable in PWFA hit approximately hundred of gigavolts per meter [2] - exceeding those of conventional RF cavities by two to three orders of magnitude. To understand the exponential gain in peak electric fields, we can look at the electron trajectory in the plasma wake.
A free electron in a longitudinal electric field E(t) obeys Newton's second law:
| me dv/dt = e E(t) | (2) |
In a plasma wave the electric field oscillates sinusoidally at the plasma frequency ωp.
| E(t) = Emax cos(ωp t) | (3) |
Substituting Eqn. (3) into Eqn. (2) and integrating over time
| me dv/dt = e Emax cos(ωp t) | (4) |
| v(t) = ∫ (e Emax / me) cos(ωp t) dt = (e Emax) / (me ωp) sin(ωp t) | (5) |
we get that the the peak electron velocity is
| vmax = e Emax / (me ωp) | (6) |
Yet, the plasma wake breaks when the electrons are moving as fast as the wave itself. Going back to the surfer and wave analogy, a surfer does not gain any boost from the current they are surfing on if they are ahead of the current itself. Thus, the maximum electric field that the plasma wake can give an electron through wakefield acceleration is limited by the wave speed, which travels approximately at the speed of light c. Setting vmax = c in Eqn. (6) and solving for Emax,
| c = e EWB / (me ωp) | (7) |
| EWB = me c ωp / e | (8) |
We find the value of EWB, known as the wave-breaking field. [1,2] EWB is the maximum longitudinal electric field a plasma wave can sustain before electrons get swept up into the wave and destroy its coherent structure - analogous to a surfer or boat breaking a wave if they crash into it. The parameters of Eqs. (1) - (8) are summarized in Table 1.
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| Table 1: Parameters used in Eqs. (1) - (8). |
Confirming the results with dimensional analysis, we can determine the maximum electric field along the longitudinal direction EWB that a charged particle experiences when riding the plasma wake, which was then supported by experimental results. Substituting Eqn. (1) into Eqn. (8) to rewrite EWB in a different form,
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(9) |
We see that the main variable we can tune to increase the field experienced by charged particles is the electron density n0 in the plasma sea. At n0 = 1017 cm-3 charged particles experience approximately 30 GV/m of electric field, compared to ~100 MV/m in conventional RF cavities - a factor of ~300.
One should also note that increasing n0 increases the plasma frequency ωp. Based on the limitation stated above, we see that LWFA setups will eventually need to use higher and higher frequencies of lasers to satisfy the ω > ωp condition, as we increase n0 to provide more acceleration to the charge particles.
There are key challenges that ensure successful wakefield acceleration. First, the plasmas have to stay in line with each other, minimizing stray plasmas diverging out from where the laser or ion beam is headed, also known as low emittance. Secondly, the plasma must also have the same density everywhere. If not, particles get uneven acceleration and the acceleration from wakefield falls apart due to incoherent driving, analogous to pushing a kid on a swing at random frequencies. Finally, the plasma must have a low energy spread, meaning that most of the particles are travelling at roughly the same speed, and can drive coherent wakes.
If wakefield acceleration can be done well, some labs even stage multiple plasma sections ahead of each other, boosting the particle acceleration multiple times over.
The PWFA FACET facility at SLAC demonstrated a 1.6 GeV energy gain per particle in a single plasma stage, with an accelerating gradient of ~4.4 GV/m, an energy spread of 0.7%, and energy-transfer efficiency exceeding 30%. [3] More recently, a three-stage metre-scale PWFA setup demonstrated beams exceeding 20 GeV with sub-percent energy spread and normalised emittance of ~2 mmmrad using a 10 GeV particle drive bunch. [4]
The primary motivation for PWFA is a dramatic reduction in accelerator footprint. A TeV-scale lepton collider utilizing PWFA could in principle fit within a few kilometres, rather than the tens of kilometres required by conventional RF cavity technology. [2] Additional applications include compact, high-brightness X-ray free-electron lasers for structural biology and materials science research, probes of strong-field quantum electrodynamics, and even more compact electron and proton sources for cancer radiotherapy. [2]
The central open problem is ion beam quality preservation across multiple staged plasma sections. Maintaining low emittance and narrow energy spread through staging is the active frontier of research at FACET-II and EuPRAXIA. [4-6] Plasma source uniformity, laser and ion beam stability, and precise alignment of plasma and beams must be controlled to incredible precision levels for collider-quality beams. If these challenges are resolved, plasma wakefield acceleration offers a path to particle physics infrastructure that is both smaller and cheaper than the present generation of large-scale machines.
© Raphael Low. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] E. Esarey, C. B. Schroeder, and W. P. Leemans, "Physics of Laser-Driven Plasma-Based Electron Accelerators," Rev. Mod. Phys. 81, 1229 (2009).
[2] T. Tajima and J. M. Dawson, "Laser Electron Accelerator," Phys. Rev. Lett. 43, 267 (1979).
[3] M. Litos et al., "High-Efficiency Acceleration of an Electron Beam in a Plasma Wakefield Accelerator," Nature 515, 92 (2014).
[4] D. Storey et al., "Acceleration of a Trailing Bunch to 20 GeV in a Three-Stage Laser-Ionized Plasma Wakefield Accelerator," Phys. Rev. Accel. Beams 27, 051302 (2017).
[5] "Technical Design Report for the FACET-II Project at SLAC Accelerator Laboratory", SLAC National Accelerator Laboratory, SLAC-R-1072, August 2016.
[6] P. A. Walker et al., "Horizon 2020 dEuPRAXIA Design Study," J. Phys.: Conf. Ser. 874, 012029 (2017).
