Measurement of ion acceleration and diffusion in a laser-driven magnetized plasma

Domain: physics
Published: 2026-03-02
Editorial outcome: minor_revision

Abstract
Here we present results from an experiment performed at the GSI Helmholtz Center for Heavy Ion Research. A mono-energetic beam of chromium ions with initial energies of ~450 MeV was fired through a magnetized interaction region formed by the collision of two counter-propagating laser-ablated plasma jets. While laser interferometry revealed the absence of strong fluid-scale turbulence, acceleration and diffusion of the beam ions was driven by wave-particle interactions. A possible mechanism is particle acceleration by electrostatic, short scale length kinetic turbulence, such as the lower-hybrid drift instability.

Introduction
Cosmic rays (CRs) and their extraterrestrial origin were discovered more than a century ago1, with measurements of the CR energy spectrum up to 1020 eV2,3. The exact mechanism that leads to such high-energy particles remains controversial, as do the nature of the sites where this acceleration occurs. Although many different processes may result in CR acceleration, the present-day understanding is that turbulence plays an essential role in energizing both electrons and ions. The original mechanism of CR acceleration by turbulence or plasma waves was proposed by Fermi4, where CRs are produced as charged particles gain energy in random scattering events with magnetized clouds. Like turbulence, Fermi acceleration is expected to be ubiquitous in the universe, but laboratory evidence for the process remains elusive.

In addition, inhomogeneous astrophysical environments are susceptible to various instabilities, such as the lower-hybrid drift instability (LHDI)5 which arises from plasma gradients. These instabilities can accelerate particles, possibly playing an important role in the overall energizing of the thermal background6–9. Space probes have detected lower-hybrid waves (LHWs) in the terrestrial magnetotail current sheet and their role in the energy dissipation of electrons10. A wave-particle model involving LHWs has been used to explain the presence of accelerated auroral electrons11 and ions12, and similarly for the simultaneous acceleration of electrons and ions in solar flares13,14.

Laboratory astrophysics experiments are an invaluable tool in studying complex phenomena such as these acceleration mechanisms15–17, used in complement with simulations while addressing some of their limitations. Compared to in situ measurements, where many competing processes can obfuscate the physics that is happening, laboratory plasmas can be designed to study specific mechanisms of interest. This paper presents experimental results18 from an investigation on ion acceleration and diffusion in a magnetized laser-driven plasma conducted at the GSI Helmholtz Center for Heavy Ion Research. The platform was a modified version of that previously fielded to study the turbulent dynamo19–24, coupled with the heavy ion accelerator at GSI. Plasma conditions were measured and supplemented by results from radiation-magnetohydrodynamic (MHD) FLASH simulations25.

Results
The experimental platform employed for the experiment at the GSI Helmholtz Institut for Heavy Ion research is shown schematically in Fig. 1. Supersonic plasma jets are produced via laser ablation of two opposing target foils, each machined with an embedded grid structure to impose density perturbations. Simultaneously, seed magnetic fields in the azimuthal direction are generated by the Biermann battery mechanism from the cross-product of the density and temperature gradients26–28. The two jets collide at the midpoint between the two foils, defined as the target chamber center (TCC), forming an approximately cylindrical magnetized interaction region. The intention of the design was for the initial density perturbations advected by the jets to interpenetrate, introducing shearing motions which would drive turbulent mixing of the plasma. The scale of these instabilities is, however, predicted to be smaller than the experimental resolution of the diagnostics25. Additionally, by only driving a single target foil, the case of reduced fluctuations, plasma density, and field strength was accessed for comparison. Interferometry and ion deflectrometry were fielded as diagnostics for the experimental plasma conditions.

The monoenergetic ion beam from the UNIversal Linear ACcelerator (UNILAC) was directed through the interaction region, acting as a surrogate for CR particles. The UNILAC supplied a Gaussian pulse train (4 ns width, 28 ns spacing), with only one pulse traversing the interaction region in each shot. Each pulse contains approximately 9×105 ions, a sufficiently low number such that it can be treated as a test beam, with no collective effects expected. The energy and initial charge state of the ion beam were tunable parameters, and we present results obtained using the following: 52Cr14+ with energies of 8.75 MeV/u; 52Cr20+ with energies of 8.661 MeV/u.

The ion beam had a diameter of ~1 mm at the TCC, with a divergence of ~10 mrad. A section of the accelerator beamline, located downstream of the laser interaction chamber, was used to direct the ions toward a diamond detector29,30 situated approximately 12 m from the TCC. The data were time-gated by the laser-pulse duration, which is short compared to the system’s dynamical timescale (τ ≪ Lp/cs ~ 10 ns), where cs is the sound speed and Lp ≃ 2 mm is the observed experimental system size, equivalent to the path lengths of the ion beam and interferometry probe through the interaction region. The evolution of the plasma can be tracked with interferometry by changing the probing time in separate shots.

The free electron density, ne, is retrieved from the raw interferograms via the following process. Firstly, the fringes were traced manually to form phase contour maps (Fig. 2A, B). In regions associated with strong density gradients, for example, adjacent to the target foils, individual fringes cannot be resolved—these regions are excluded from the interpolation. Next, the phase contour maps are interpolated to recover a map of the phase shift at each point31,32. The phase shift is converted into the line-integrated electron density via the well-known relation:

ϕ = 2π F (ne / ncrit) (λprobe / λprobe) (1)

where F is the number of fringe shifts, ncrit is the critical density of the plasma, and λprobe is the vacuum wavelength of the probing laser. The result is a 2D map of line-integrated electron density (Fig. 2C). At t = 10 ns, a clear interaction region has formed with a width of ~2 mm. Given the approximate azimuthal symmetry of the setup, the path length traversed by the probe beam is also 2 mm, resulting in an average volumetric electron density in the interaction region of ne ≃ 3 × 1019 cm−3. Assuming full ionization with the given plasma conditions, consistent with atomic kinetics simulations, the corresponding ion density ni ≃ 1.1 × 1019 cm−3. Similar analysis for data recorded at t = 15 ns gives values of ne ≃ 2.5 × 1019 cm−3 and ni ≃ 9 × 1018 cm−3.

The evolution of the inflow plasma jets was captured by interferometry on shots with only one target foil driven by nhelix, as seen in Fig. 3. The expanding jet can be approximated as isothermal, since the ratio of the temperature equilibration timescale to the experiment duration can be shown to be33 ~ 0.05; equivalently, this implies a small Péclet number (Pe ≲ 0.05). An isothermal expansion has a known density profile34,35 dependent on the plasma sound speed cs. Since cs is related to the plasma temperature36, at a time of t = 5 ns, approximately the time the jets collide, the temperature of the inflow jets from this analysis is ~120 eV. Even accounting for collisional heating, we can expect the interaction region plasma to be in the temperature regime of the low hundreds of eV.

Qualitatively, the interferometry data in Fig. 2 show a relatively laminar interaction region, with no evidence of large-scale density fluctuations associated with strong fluid turbulence. To bound any residual fluctuation, we low-pass filter each traced fringe at a cutoff set by the width of the interaction region, take the result as a slowly varying baseline, and subtract it from the raw trace. The residual displacement is then mapped via Eq. (1) to the corresponding fluctuation in line-integrated electron density.

Next, assuming an isotropic plasma, it can be shown20 with a random-walk treatment that the line-integrated density fluctuation is related to the volumetric density fluctuation δne/ne by:

⟨δneℓn⟩ / ⟨neℓ⟩ ≃ (δne / ne) (ℓn / Lp)1/2 (2)

where ℓn is the characteristic length scale of the density fluctuations (approximated as the grid spacing ℓgrid ~ 60 μm) and Lp is the path length of the interferometry laser through the plasma. The magnitude of δne/ne can be used to infer the turbulent velocity uturb using a continuity argument in subsonic, incompressible flow37:

uturb ≃ cs (δne / ne) (3)

Using the values of cs predicted by MHD simulations25 for the case of colliding plasma flows, which are more than twice those extracted from the single-sided experimental fits, this results in values of uturb ≲ 30 km s−1 and 30 km s−1 at t = 10 and 15 ns, respectively. These values should be interpreted as an upper bound for uturb, as the effect of limited dynamic range, as well as the variability of the manual tracing process, cannot be isolated from actual density fluctuations. This analysis will, in turn, provide upper bounds for the expected magnitude of second-order Fermi acceleration, shown in the later discussion.

Another important parameter relevant to transport properties in the plasma is the magnetic field strength, which was constrained experimentally by fielding an ion deflectrometry diagnostic. A nuclear-track detector (CR-39) was placed in the path of the ion beam ~40 cm after the TCC. This was fielded in lieu of the ToF diamond detector, and data were collected both with and without the laser-ablated plasma. For these measurements, the ion beam was gated to the minimum possible length of 10 μs, approximately 350 pulses, to minimize the number of undeflected pulses incident on the plate.

Spatially resolved ion fluence was recorded with this setup. The distribution of pits demarcating the position of ion impacts consists of a saturated central region containing the bulk of the ion beam and a surrounding ‘halo’ decreasing in pit density with increasing radial distance; Fig. 4A, B shows a section of the data aligned radially outwards from a knife edge positioned ~1 mm from the center of the ion beam, for shots without and with plasma respectively. As the ion beam traverses the plasma, it accumulates a random-walk deflection proportional to the magnetic field strength, which dominates over contributions from the electric fields and Coulomb collisions. The relative contributions from magnetic and electric fields were compared by estimating the plasma jet displacement between subsequent interferograms, and vc the initial ion beam velocity. Since the beam ion gyroradius rL ~ 0.3 m is much larger than the magnetic coherence length ℓB ~ 10−4 m, the ions are unmagnetized, and their deflection accumulates in a series of small-angle scattering events as a random walk process; this manifests as a Gaussian deflection as a consequence of the central limit theorem. The average angle of deflection is38

⟨δθ2⟩ ≃ N ⟨δθ2⟩ (4)

where N is the total number of scattering events and can be approximated as N ~ Lp/ℓB. The RMS deflection angle is thus39

δθB ≃ Zc e Brms Lp ℓB / (Ec) (5)

where Z, m, and E are the charge state, mass and energy of the beam ions. The magnetic coherence length ℓB is the characteristic length over which the stochastic magnetic fields maintain coherence and is essentially a measure of the size of the stochastic field structures. An upper estimate for ℓB is found by taking the grid spacing (60 μm) on the target foils and accounting for the approximated doubling in spatial extent of the plasma: ℓB ≲ 0.12 mm. This is on the order of the outer scale of turbulence, which in previous experiments (on similar setups) has been larger than ℓgrid20. Using the experimental value of δθB ≃ 0.1° obtained from the width of the distribution in Fig. 4D, this gives a minimum field strength of Bmin = 40 kG. An upper bound for the field strength can be defined using a simple geometric argument: assuming all the counts in the distribution of Fig. 4D come from ions at the center of the beam, we obtain δθB,max ≃ 0.6°, which is found by summing the initial ion beam divergence and the measured δθB. This corresponds to a field strength of Bmax = 230 kG. Thus, we estimate 40 ≲ Brms ≲ 230 kG, which has implications for later discussion on acceleration from fluid and short-scale electrostatic turbulence.

The ToF diamond detector fielded allows for an energy measurement of each pulse within the ion beam. Figure 5A, B presents the resulting energy spectra, comparing the pulse of interest (PoI), that is, the ion pulse that coincides with the plasma, with a background averaged over pulses acquired prior to plasma formation. There is significant shot-to-shot variation even for shots with similar plasma conditions probed at the same time, reflecting the inherently stochastic nature of the beam-plasma interaction. To quantitatively compare shots across the campaign, two main metrics were used: the shift in mean energy of the pulse ΔEc = Ec − E0, calculated by integrating the energy profile to find its center-of-mass; and the change in the energy spread of the pulse, defined as twice the standard deviation σm of the pulse around its mean energy.

Figure 6 shows the distribution of the measured acceleration and diffusion from this experiment, revealing a clear distinction between single- and double-sided drive configurations. Single-sided shots cluster near the origin, exhibiting minimal energy shift and broadening, whereas double-sided shots trace a broader curve with significantly larger magnitudes in both metrics. Nevertheless, outliers are present in both cases, reflecting the inherent stochasticity of the beam-plasma interaction. Notably, the double-sided data appear to be divided into two regimes: one characterized by net acceleration with minimal broadening, and another by increased width accompanied by a reduction in mean energy.

In addition, charge transfer processes between the beam and plasma ions introduce a non-trivial correction to Zc as the ion beam traverses the plasma. To account for this effect, a semi-empirical model40 based on the Gus’kov model41 was employed to estimate the mean non-equilibrium charge state of the beam ions. This upshifted value is accounted for in the Zc estimate (see Table 1).

One possible explanation for deceleration is the charged-particle stopping power of the plasma—the magnitude of this effect can be calculated following a generalized Fokker–Planck equation42,43 which accounts for both small and large-angle binary collisions:

dE/dx ≃ − (Zc e)2 ωp2 lnΛ / vc2 (6)

where vc is the initial ion beam velocity, ωp is the plasma frequency, and lnΛ is the Coulomb logarithm. Using the experimentally constrained plasma conditions in Table 1, the electron-ion stopping dominates, with values of −0.37 and −0.58 MeV for Zc = 16.1 and 20.3, respectively, assuming a path length of 2 mm. This magnitude of deceleration cannot account for the observed shifts in mean energy. Similarly, the expected parallel diffusion associated with Coulomb collisions is found to be negligible (≪1 eV) under the experimental conditions, due to the high initial velocity of the ion beam.

The results can be compared with theoretical predictions, considering two candidate mechanisms informed by previous theoretical work25. The first is second-order Fermi acceleration44, originally proposed to account for the origin of high-energy cosmic rays by interactions between particles and mirror fields in the interstellar medium4. This process can be generalized to a diffusive interaction between charged particles and magnetized plasma turbulence. For this experiment, as the ion beam traverses the interaction region, the ions undergo many deflections by magnetic fluctuations, which result in a total energy change of

ΔEFermi ≃ Zc e uturb Brms Lp / ℓB (7)

where c is the speed of light and uturb and Brms are the RMS velocity and magnetic field, respectively. Taking Brms ≃ Bmax (bounded by deflectrometry) and uturb ≃ uturb (bounded by interferometry), the maximum ΔEFermi associated with second-order Fermi acceleration (see Table 1) is much smaller than the initial energy spread of the ion beam (≲0.1% 0.4 MeV) and the energy resolution of the diamond detector. As T and hence uturb are only bounded from below by interferometry, one might consider the effect of higher temperature on the magnitude of ΔEFermi. A conservative back-of-the-envelope scaling (uturb ∝ T1/2) shows that one would need temperatures of order ≳750 keV before the Fermi term could become competitive—clearly not physical for this platform and inconsistent with the observed density and expansion dynamics. Consequently, while this mechanism may contribute, it cannot be responsible for the changes measured—a conclusion that is robust to substantially higher local temperatures.

The second possibility is a wave-particle interaction, such as the lower-hybrid drift instability. LHDI is a kinetic instability driven by strong density and magnetic field gradients5, which can excite lower-hybrid waves (LHWs) that in turn induce an electric field for ion acceleration6. To demonstrate that the experimental data are not inconsistent with the formation of the instability, we infer an upper bound for the LHDI growth rate using experimental values. The maximum linear growth rate of this instability γmax is given by

γmax ≃ (2π / rL)2 ωLH (8)

where rL ~ 0.2 mm is the gyroradius of the plasma ions, ωLH is the lower-hybrid frequency, and ϵn is the inverse gradient scale length, which can be approximated as

ϵn = (1/ne) (∂ne/∂x) ≃ 1 / ℓgrid (9)

Using the magnitude of density fluctuations bounded by interferometry, γmax ≃ 6.2 × 108 s−1 at t = 10 ns, corresponding to an e-folding time of 1.6 ns. Additionally, the gradient scale length Ln = 1/ϵn ~ 0.6 mm satisfies the steepness condition Ln/rL < (mi/me)1/445 required to excite LHDI. Note that this analysis only includes density gradients associated with resolvable fluctuations, limited by the dynamic range of the interferometry diagnostic. Gradients on a smaller scale, closer to rL, would instead excite kinetic instabilities, with higher growth rates on the order of ωLH ~ 1.24 × 1010 rad s−145–47.

In the context of this experiment, LHWs are assumed to interact linearly and non-resonantly with the ion beam in a weakly turbulent plasma48. This can be treated as a diffusion problem, with an associated energy change of

ΔELHDI ≃ κ Zc Ac−1 Brms cs (vc Lp)−1 ne−4 (10)

where κ ≃ 4 × 10−14 C m−4 s−2 is a constant of proportionality that captures the level of lower-hybrid turbulence (see ref. 25 for details) and Ac = 52 u the atomic mass of the beam ions. The theoretically predicted values (Table 1) for LHDI are sufficiently high for measurement and are expected to dominate over the effect of second-order Fermi acceleration.

An alternative avenue for the generation of electrostatic waves near the lower-hybrid frequency is via beam-driven instabilities, such as the modified two-stream instability (MTSI)49. To demonstrate that the ion beam acts as a test beam and does not excite plasma instabilities, we consider the MTSI growth rate: the ion beam density of ~107 cm−3 results in growth rates of the order 105–106 s−1. This is much smaller than the growth rate for the lower-hybrid drift instability, and therefore, we discount the possibility of turbulence being driven by the ion beam.

The observation of bulk energization cannot be explained by the broadening process outlined above alone, and suggests that coherent fields which facilitate particle acceleration are generated in the plasma. A tantalizing possibility consistent with our experiment is the role of short-scale electrostatic turbulence (below the interferometry diagnostic’s resolution) in developing coherent structures within the plasma. In literature, both Langmuir wave50 and LHW51,52 turbulence have been shown to drive the evolution of these wave structures. On the other hand, we can also consider electromagnetic turbulence, which can be driven by plasma instabilities such as the ion Weibel instability and ion filamentation instability (IFI)53. To investigate this possibility, 1D particle-in-cell (PIC) simulations using the OSIRIS PIC code were performed25, showing that the counter-streaming phase is short-lived (3 ns) compared to the e-folding times of the IFI and Weibel instabilities (~10 ns). It is therefore unlikely that these instabilities contribute to field amplification and ion acceleration.

In summary, we present observations of ion acceleration and diffusion in a laser-driven magnetized plasma. Despite the absence of large-scale fluid turbulence as diagnosed by interferometry, evidence for ion acceleration and energy diffusion was observed with a time-of-flight detector, suggesting that a wave-particle interaction was the underlying acceleration mechanism. Analysis showed that the lower-hybrid drift instability was consistent with the measured acceleration and diffusion, although other wave-particle interactions could also be responsible.

Methods
Retrieval of free electron density from interferometry using MAGIC

The free electron density is extracted from the interferometry data with the help of MAGIC, an analysis software (available at https://github.com/jdranczewski/Magic2).

First, in an image-editing software, the interferograms (one captured during the plasma interaction and one reference image taken before the interaction) are cropped to a region containing both the interaction plasma as well as a region containing no plasma—this defines a point with zero fringe shift that allows for the absolute density to be calculated. Next, the position of the fringe intensity minima and/or maxima is manually traced to produce the specially formatted image files required by the MAGIC software as inputs. The fringes in the interaction region can be complicated, and tracing them requires a certain degree of human interpretation and experience. The fringes are automatically scanned by the software, and then we label the fringes in order of increasing or decreasing phase. The software performs a 2D linear triangular interpolation between the fringes to produce phase maps; subtracting the reference from the data produces a map of phase shift, which is converted to line-integrated electron density via Eq. (1). This requires knowledge of the real spatial size corresponding to one pixel width, which is estimated using the known distance between the two target foils.

CR-39 processing

The CR-39 plates are etched with NaOH solution, forming pits that reveal the location of molecular damage caused by ion impacts. The plates are then scanned with an optical microscope to produce high magnification images.

The two images (one with and one without the ion beam traversing the plasma) are spatially aligned and cropped to a region containing the “knife edge”. This region contains on the order of 104 pits in each of the images. A coordinate system is established where the y direction aligns radially away from the center of the beam, and the pits are manually tagged in an image-editing software, with the coordinates of each pit recorded. As the pit density increases towards the center of the beam, pits can start overlapping, and a degree of interpretation is required.

Time-of-flight data processing

The ToF data has to be preprocessed before any analysis can be performed. Firstly, there is an X-ray “flash” created by the laser drive of the target foils, which is recorded by the diamond detector. To remove this effect, the response of the X-ray flash on the diamond detector was isolated by performing background shots with only laser drive but no ion beam and subtracted from the signal. In addition, there is a slow-varying amplitude modulation of the signal associated with source instability on the accelerator side. To correct for this, two cubic splines are fitted to the signal envelope and used to normalize the signal between 0 and 1.

Next, the normalized signal is transformed into energy profiles for each pulse by performing a change of variables from t to E using the relativistic relation:

E(t) = mc c2 [ (1 − (d / (ct))2)−1/2 − 1 ] (11)

where d is the distance from the plasma to the diamond detector. To reverse the distortion introduced by this transformation, the signal is multiplied by the 1D Jacobian dt/dE.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents an important experimental result which shows that, even in the absence of strong fluid-like turbulence, ion acceleration and diffusion can be driven in a magnetized plasma by wave particle interactions. This result is relevant to the general field of plasma turbulence, both in the laboratory and in space, in particular in the context of cosmic ray acceleration. The manuscript presents a thorough quantitative analysis of the different physical mechanisms that might explain the experimental findings. In the last part of the manuscript a possible mechanism is proposed based on the role of short scale electrostatic turbulence. However this hypothesis cannot be substantiated because such a turbulence would occur on scale-lengths below the interferometric resolution of the experiment. I found the manuscript clear and well reasoned and I would suggest to accept it for publication as is even if additional investigations will be needed in order to verify the role of short scale electrostatic turbulence and, in particular, of the lower-hybrid drift instability.", "reviewer_2": "General comments. The authors of the manuscript \"Measurement of ion acceleration and diffusion in a laser-driven magnetized plasma\" did an original study and the results very interesting. The authors show that the used ions beam energization and diffusion could occur experimentally in laser-driven colliding plasma due mainly to the lower hybrid wave mechanism. The results presented are quite new. This work is well supported experimentally and well based on modeling and simulations. It is a well-written and -structured manuscript with clear explanations. However, I have some questions to clarify some parts. In experimental set-up part : The authors present the ion beam characteristics of ion beam from UNILAC, could you give the number of ions in each bunch of 4 ns ? In Fig. 1 caption : I propose to replace 200 um by 200 µm Could state about the choose the target grid spacing ? In results part : Could you give reference to the Figure 2 before the equation [1] ? On the Fig.2 : Could you give the direction of the ion beam ? In the Caption of Fig. 2 : in the sentence : “ the images have been cropped, rotated, and their contrast”, why the image is rotated ? Could the give reference to Fig. 3 perhaps in the following paragraph :”The evolution of the inflow plasma jets was captured by interferometry on shots with only one target foil driven by nhelix. The ...” In sentence “ the ratio of deflection force from electric and magnetic fields ∝ u/v_c ≲ 10^−2 , where u is the characteristic fluid velocity and vc the initial ion beam velocity.” How did you estimate this ratio (u/v_c)? In sentence “Since the beam ion gyroradius rc ∼ 0.3 m is much larger than the magnetic coherence length l_B ∼ 10^−4 m,” ; The authors talk about r_c but in the Eq.[4] you talk about r_g ? What is the magnetic coherence length l_B ? In the sentence : “An upper estimate for the magnetic coherence length is found by taking the grid spacing (60 um) “, I suggest to replace 60 um by 60 µm. After the expression [6] I suggest to recall that v_c is the ion velocity. In the Table I : How did you measure or estimate the magnetic field values: 40-230 kG for 10 ns (double drive) ?", "reviewer_3": "The manuscript reports intriguing experimental results that integrate laser-driven colliding plasmas with large-scale heavy-ion accelerators. The laser-driven colliding plasmas are employed to simulate electromagnetic turbulence, a phenomenon ubiquitous in the universe. Meanwhile, the heavy-ion beams serve as proxies for cosmic rays. Plasma density is measured using laser interferometry, and the density profile is reconstructed through isothermal expansion modeling. The key finding is the significant acceleration and broadening of ion beams during their transport through plasmas, which the authors hypothesize may be linked to the origin of high-energy cosmic rays. While the topic is compelling and the manuscript is well-written, I do not believe it is suitable for Nature Communications in its current form. Below, I outline my concerns. 1) The diagnostic approach for plasma conditions is inadequate, as it relies solely on laser interferometry, which is sensitive only to plasma density. While the plasma expansion can be modeled using isothermal assumptions, the temperature evolution during colliding processes is both complex and critical for instability analysis. The absence of reliable temperature diagnostics constitutes a major weakness of this manuscript. 2) In Fig. 5B, the claimed ion beam acceleration is not clearly discernible. 3) In P. Liu et al., Phys. Rev. Letters 132, 155103 (2024), full kinetic simulations were performed, revealing distinct instability mechanisms and turbulence. The authors should compare their results with this study, particularly since the FLASH code likely cannot capture the colliding plasma dynamics accurately—mutual penetration and filamentation turbulence are inherently kinetic processes. 4) In Ren et al., Nature Communications 11, 5157 (2020), a similar experimental setup demonstrated enhanced stopping power of intense proton beams due to collective beam effects. The authors should also compare their findings with this work, emphasizing the role of heavy-ion beam flux as a key parameter."}, "author_response": "Response to Referees\nNCOMMS-25-69449: “Measurement of ion acceleration and diffusion in a\nlaser-driven magnetized plasma”\nJ. T. Y. Chu et al.\nNovember 24, 2025\nWe thank all three reviewers for a careful reading of our manuscript and for their constructive\ncomments. Reviewer #1 is supportive and recommends acceptance as is, noting the relevance of our\nfindings to plasma turbulence and cosmic-ray acceleration. Reviewer #2 is positive about the nov-\nelty and experimental support, suggesting clarifications and minor edits (figure captions/references,\nnotationconsistency,units,ion-bunchdetails,rationaleforgridspacing,andthederivationofcertain\nestimates). Reviewer #3 finds the topic compelling but raises four concerns: the adequacy of tem-\nperature constraints, the visibility of acceleration in Fig. 5b, and comparisons to both fully kinetic\nsimulations (Liu et al., PRL 2024) and collective stopping in a related geometry (Ren et al., Nat.\nCommun. 2020). We address each point carefully below in a point-by-point response and have made\ntargeted revisions to improve clarity and completeness.\nAllchangesfromthepreviousversionofthemanuscripthavebeenhighlightedinredinthecurrent\nversion. Figures 2 and 6 have also been modified slightly: the ion beam direction was added to Fig.\n2A, and a mistake in the y-axis label in Fig. 6 was rectified.\nReviewer 1\nReviewer 1 does not suggest any substantial changes to the manuscript – we thank them again for\ntheir careful review.\nReviewer 2\nResponse to minor comments 1, 3, 5, 8, 9:\nComments 1, 3, 5, 8, 9 by Reviewer 2 deal with minor edits, including typographic improvements,\nclarification of variables, and adding references to figures within the text. We thank the reviewer for\nmaking these suggestions and agree that they improve the readability of the paper – all suggestions\nhave been incorporated into the manuscript.\nResponse to more substantial comments:\nComment 2:\nCould state about the choose the target grid spacing ?\nResponse:\nAsmentionedinthemanuscript,theourintentionwastodrivefluidturbulenceastheablationplasma\nfrom the two foils collide. The grid spacing was informed from numerical simulation (Moczulski et\nal., Phys. Plasmas 31, 122105 (2024). However, the experiment showed that the level of fluid tur-\nbulence achieved was insufficient to explain the observed acceleration and energy broadening of the\nionbeam. Thismotivatedus tolookforalternativeexplanations, suchas thoserelatedtoshort-scale\nelectrostatic turbulence.\nIn the Caption of Fig. 2 : in the sentence : “ the images have been cropped, rotated, and their\ncontrast”, why the image is rotated ?\nResponse:\nThere is a small random tilt in the interferometry imaging of ≲ 5◦. A rotation is applied to correct\nfor this tilt, such that the target foils are parallel to the x-axis of the images. We have added this\ninformation in the caption of Fig. 2.\nComment 6:\nIn sentence “ the ratio of deflection force from electric and magnetic fields ∝ u/v ≲ 10−2 , where\nc\nu is the characteristic fluid velocity and v the initial ion beam velocity.” How did you estimate this\nc\nratio (u/v )?\nc\nResponse:\nAn upper limit for the fluid velocity u ≲ 200km/s is found by estimating the supersonic plasma jet\nvelocityfrominterferometrydata. Thisisdonebylookingathowmuchtheplasmajetmovesbetween\nsubsequent timesteps. v =4×104km/s is a known quantity given to us by the UNILAC operators.\nc\nThis has now been clarified in the manuscript.\nComment 7:\nIn sentence “Since the beam ion gyroradius r ∼ 0.3 m is much larger than the magnetic coherence\nc\nlength ℓ ∼10−4 m,” ; The authors talk about r but in the Eq.[4] you talk about r ? What is the\nB c g\nmagnetic coherence length ℓ ?\nB\nResponse:\nWe appreciate the reviewer catching this inconsistency, eq. (4) has been amended to use r as well.\nc\nℓ is the characteristic length over which the stochastic magnetic fields maintain coherency and is\nB\nessentially a measure of the size of the stochastic field structures. We can estimate an upper bound\nfor this as the size of the initial density perturbations, i.e. the grid spacing on the target foils. We\nhave also added this definition in the manuscript for clarity.\nComment 10:\nIn the Table I : How did you measure or estimate the magnetic field values: 40-230 kG for 10 ns\n(double drive) ?\nResponse:\nThese values were measured using the CR-39 detector. The method is outlined in the text after eq.\n(5).\nReviewer 3\nComment 1:\nThe diagnostic approach for plasma conditions is inadequate, as it relies solely on laser interferom-\netry, which is sensitive only to plasma density. While the plasma expansion can be modeled using\nisothermal assumptions, the temperature evolution during colliding processes is both complex and\n2\ncritical for instability analysis. The absence of reliable temperature diagnostics constitutes a major\nweakness of this manuscript.\nResponse:\nWe agree that a direct measurement of temperature is important, but believe that it is sufficiently\nconstrained by our current analysis. The interferometry-based isothermal expansion fit provides a\nlower bound on the inflow temperature of ∼ 120eV at the time the jets meet, and independent\nFLASH simulations place the interaction region in the few-hundred-eV regime, consistent with and\nabove that bound. Thus, the temperature is bracketed by experiment and modeling rather than left\nunconstrained.\nThe only place where a higher temperature could impact our conclusions is through the upper\nbound on the turbulent velocity used to limit second-order Fermi acceleration. In the manuscript we\nestimate\n(cid:112)\nu ≲ c δn /n , c ∝T1/2,\nturb s e e s\nsou –andhence∆E ∝u B –dependsonlyweakly onT (square-rootscaling). Usingthe\nturb Fermi turb rms\nmeasured δn /n and the magnetic–deflectometry bounds on B , our predicted ∆E remains\ne e rms Fermi\nfar below the observed energisation across the entire plausible temperature range. A conservative\nback-of-the-envelope scaling shows that one would need temperatures of order ≳750keV before the\nFermi term could become competitive – clearly not physical for this platform and inconsistent with\nthe observed density and expansion dynamics.\nBycontrast,thewave-particlepathwaywediscuss(e.g.LHDI-mediatedenergisation)isnot weak-\nened by higher T; the relevant scalings are either insensitive to T or become more favorable as c\ns\nincreases, so the hierarchy between mechanisms is preserved. To make this explicit, we have added a\nshortened version of the above discussion into the manuscript text: (i) stating in the results section\nthat interferometry provides “a conservative lower bound on temperature” and (ii) that the conclu-\nsions are “robust to substantially higher local temperatures”.\nComment 2:\nIn Fig. 5B, the claimed ion beam acceleration is not clearly discernible.\nResponse:\nWhile the acceleration might not be visually discernible in Fig. 5B, we can quantify the level of\nacceleration and energy broadening using the two metrics outlined in the paper: “the shift in mean\nenergy of the pulse Em−E , calculated by integrating the energy profile to find its center-of-mass;\nc c\nand the change in the energy spread of the pulse, defined as twice the standard deviation σm of the\nc\npulsearounditsmeanenergy.”Thesemetricsshowsthatthereisastatisticallysignificantmeanshift\nin energy of Em−E =−4.76 MeV and a broadening of 2(σm−σ )=4.01 MeV.\nc c c c\nComment 3:\nInP.Liuetal.,Phys. Rev. Letters132,155103(2024),fullkineticsimulationswereperformed,reveal-\ning distinct instability mechanisms and turbulence. The authors should compare their results with\nthis study, particularly since the FLASH code likely cannot capture the colliding plasma dynamics\naccurately—mutual penetration and filamentation turbulence are inherently kinetic processes.\nResponse:\nWe appreciate the reviewer for bringing this paper to our attention and agree that the possibility of\nelectromagneticturbulenceplayingarolewarrantscarefulconsiderationgiventhesimilaritiesbetween\nthesetupinLiuetal. 2024andours. Asthereviewerpointsout, FLASHisaMHDcodeandcannot\ncapture the kinetic effects that lead to the development of ion filamentation and Weibel instabilities.\nThus, we refer to 1D OSIRIS particle-in-cell simulations performed using initial conditions relevant\nto our experimental setup (see K. Moczulski et al., Phys. Plasmas 31 (12), 122105 (2024)). These\n3\nsimulations show that the counter-streaming phase of the interaction between the two plasma jets is\nshort-lived: it lasts for 3 ns after the two jets start to merge. Since these 1D simulations result in\na longer counter-streaming time than the 3D case, this gives an upper-bound for the duration of the\ncounter-streaming phase.\nWecanthencomparethiscounter-streamingdurationtotheexpectedgrowthtimesoftherelevant\ninstabilities. For the ion Weibel instability, K. Moczulski et al. (2024) calculates an e-folding time\nof 6.7ns. As this is longer than the duration of the counter-streaming phase, it is unlikely for this\ninstability to contribute to field amplification and driving turbulence. Similarly, we can estimate\nthe growth time of the ion filamentation instability (IFI). In P. Liu et al. (2024), they calculate a\ngrowing-up timescale for the IFI to be ∼ 100ps, using a similar temperature (100eV) and electron\ndensity (4×1019cm−3). The main difference between their work and our experiment is the initial\nflow velocity of the plasma jets v : 2000km/s and ∼ 200km/s, respectively. From Liu et al., the\n0\ngrowth rate Γ scales approximately linearly with the anisotropy of the jets A , given by\nIFI i\nA = 2v 0 2+v t 2 i,∥ −1∼2 v 0 2 ,\ni v2 v2\nti,⊥ ti,⊥\nwhere v and v are the ion thermal velocity parallel and perpendicular to the flow velocity,\nti,∥ ti,⊥\nrespectively. Applyingthisflowvelocityscalingleadstoagrowthtimeof∼10nsforourexperiment.\nAs this is significantly longer than the counter-streaming time, we do not expect our plasma to be\nunstable to the IFI.\nWe believe that these findings are both interesting and strengthen our stated conclusions in the\npaper. As such we have included discussion of the possibility of IFI and Weibel instabilities driving\nEM turbulence to the results section, after the contrasting discussion of electrostatic turbulence on\npage6. Weconsider“electromagneticturbulence,whichcanbedrivenbyplasmainstabilitiessuchas\nthe ion Weibel instability and ion filamentation instability”, and show that “the counter-streaming\nphase is short-lived (3 ns) compared to the e-folding times of the IFI and Weibel instabilities (∼ 10\nns)”, concluding that it is“unlikely that these instabilities contribute to field amplification and ion\nacceleration”.\nComment 4:\nIn Ren et al., Nature Communications 11, 5157 (2020), a similar experimental setup demonstrated\nenhanced stopping power of intense proton beams due to collective beam effects. The authors should\nalso compare their findings with this work, emphasizing the role of heavy-ion beam flux as a key\nparameter.\nResponse:\nWe thank the reviewer for bringing this paper to our attention as it presents an interesting result\nthat we had not previously considered. In Ren et al. (2020), PIC simulations and experimental\nmeasurementrevealtheestablishmentofalongitudinalelectricfieldinducedbyabeam-drivenreturn\ncurrent, enhancing the stopping-power beyond that expected with just binary collisions. However,\nthis requires a high beam current (3 × 107A/cm2), compared to our chromium beam current of\n∼ 10−2A/cm2. Similar simulations performed by Ren et al. with a lower current of 3×102A/cm2\nindicate an absence of this collective effect, with good agreement with the stopping power expected\nfrom binary collisions. As our current is safely below this threshold, we can treat our ion beam as a\ntest beam and do not expect an enhanced stopping power due to collective effects. Nevertheless, we\nrecognizethatthisisanimportantpieceofinformationthatiscurrentlymissingfromthemanuscript.\nIn the description of the ion beam in the experimental setup section, we have elaborated that ”each\npulse contains approximately 9×105 ions, a sufficiently low number such that it can be treated as a\ntest beam, with no collective effects expected”.\nThiscommenthasalsomotivatedustorevisitthepossibilityofelectrostaticturbulencedrivenby\ntheionbeamitself. Weconsidertheinstabilityrelevanttoourexperimentalsetup, themodifiedtwo-\nstream instability (MTSI), which can also excite electrostatic waves at the lower-hybrid frequency.\nThe MTSI growth rate scales as\n(cid:114)n m\nγ ∝ c c,\nMTSI n m\np i\n4\nwhere n and m are the beam density and beam ion mass, respectively; n and m are the plasma\nc c p i\ndensity and plasma ion mass, respectively. Given a low beam density of ∼ 107 cm-3, we expect a\nMTSI growth rate on the order of 105 – 106 s-1, much lower than the expected LHDI growth rate.\nFor the MTSI to be competitive with the LHDI would require an ion beam density approximately 6\nto 8 orders of magnitude higher, thus we can safely discount this possibility. We believe this analysis\nfurther strengthens our conclusions, and have included the following discussion in the results section:\n“An alternative avenue for the generation of electrostatic waves near the lower-hybrid frequency is\nvia beam-driven instabilities, such as the modified two-stream instability (MTSI). To demonstrate\nthattheionbeamactsasatestbeamanddoesnotexciteplasmainstabilities, weconsidertheMTSI\ngrowth rate: the ion beam density of ∼ 107 cm-3 results in growth rates of the order 105 – 106 s-1.\nThis is much smaller than the growth rate for the lower-hybrid drift instability and therefore we\ndiscount the possibility of turbulence being driven by the ion beam.”\n5"}, {"round": 2, "reviewer_comments": {"reviewer_2": "For my side the authors have answer well at all my comments and questions. Then I suggest to accept the manuscript for publication.", "reviewer_3": "The authors had well addressed my comments. I believe now the paper can be accepted."}, "author_response": ""}]

Decision letter