DOI: 10.7795/710.20251208
title: Additional data for the publication: Domain formation and structural stabilities in mixed-species Coulomb crystals induced by sympathetically cooled highly charged ions

description: We investigated the structural stability and the formation of vibrational domains in mixed-species ion Coulomb crystals with highly charged ions using molecular dynamics simulations. The data presented here is the result of the evaluated ion trajectories generated using our in-group simulation code. The data for each figure was compiled in separate CSV files, making it possible to reconstruct each graph of each figure individually.
description: description of the individual files:
 - block: all .csv files in the different subfolders (labeled by figure)
 research object: dynamics of ion Coulomb crystals (single-species and mixed-species)
 method: combination of molecular dynamics simulations and analysis of experimental EMCCD fluorescence images
 format: .csv (comma-separated plain text with header row)
 recommended software: any software capable of reading CSV files (e.g. Python/NumPy, MATLAB, Origin, Excel)
 parameters:
 - Figure 3 – Molecular-dynamics simulation of a four-Be+ ion string (RMS motion and hopping rates)
 Files (in subfolder "Figure 3"):
 * RMSD_4_Be_panel_a_rmsd_vs_T.csv
 Description:
 Data for Fig. 3(a). Normalized root-mean-square (RMS) ion displacement s_ℓ as a function of temperature T
 for different total simulation times, compared to the harmonic approximation and the saturation value for
 continuous ion reordering.
 Columns:
 - T_abs_mK:
 Absolute ion crystal temperature T in millikelvin [mK].
 - T_rescaled:
 Normalized temperature T/τ (dimensionless), where τ is the characteristic temperature scale defined in the paper.
 - rmsd_harm:
 Normalized RMS displacement s_ℓ(T) from the harmonic approximation (no hopping, purely linear response).
 - rmsd_100ms:
 Simulated normalized RMS displacement s_ℓ(T) for a total simulation time t_sim = 100 ms.
 - rmsd_10ms:
 Simulated normalized RMS displacement s_ℓ(T) for a total simulation time t_sim = 10 ms.
 - rmsd_1ms:
 Simulated normalized RMS displacement s_ℓ(T) for a total simulation time t_sim = 1 ms.
 - rmsd_rand:
 Saturation value of the normalized RMS displacement s_ℓ for effectively continuous ion reordering
 (randomized ion permutations; upper dashed line in Fig. 3(a)).

 * RMSD_4_Be_panel_b_hopping_data.csv
 Description:
 Data points for Fig. 3(b). Normalized hopping rates for the individual ion pairs and the total hopping
 rate as a function of temperature.
 Columns:
 - T_abs_mK:
 Absolute ion crystal temperature T in millikelvin [mK].
 - T_rescaled:
 Normalized temperature T/τ (dimensionless).
 - hop_inner_norm:
 Normalized hopping rate of the inner ion pair (ions 2–3), hop / (ω′_z/2π), as defined in the paper.
 - hop_outer_norm:
 Normalized hopping rate of the outer ion pairs (ions 1–2 and 3–4), hop / (ω′_z/2π).
 - hop_total_norm:
 Total normalized hopping rate (sum over all neighboring pairs), hop / (ω′_z/2π).

 * RMSD_4_Be_panel_b_hopping_fits.csv
 Description:
 Arrhenius-type fit curves for the hopping rates shown as dashed lines in Fig. 3(b).
 Columns:
 - T_abs_fit_mK:
 Temperature grid in millikelvin [mK] on which the fitted hopping-rate curves are evaluated.
 - hop_inner_fit_norm:
 Fitted normalized hopping rate for the inner ion pair (ions 2–3), hop / (ω′_z/2π).
 - hop_outer_fit_norm:
 Fitted normalized hopping rate for the outer ion pairs (ions 1–2 and 3–4), hop / (ω′_z/2π).
 - hop_total_fit_norm:
 Fitted total normalized hopping rate (sum of the fitted inner and outer contributions), hop / (ω′_z/2π).

 - Figure 4 – Doppler-cooling fluorescence and spatial thermometry (four-Be+ ion string)
 Files (in subfolder "Figure 4"):
 * experiment_doppler_temp_data.csv
 Description:
 Experimental data underlying Fig. 4. Fluorescence rate per ion as a function of Doppler laser detuning,
 and corresponding ion temperatures extracted via spatial thermometry.
 Columns:
 - detuning_MHz:
 Doppler cooling laser detuning Δν from resonance in megahertz [MHz]. Negative values correspond to
 red detuning.
 - fluorescence_ms^-1:
 Measured fluorescence rate per ion in units of counts per millisecond [ms⁻¹], as plotted in Fig. 4(a).
 - fluorescence_err_ms^-1:
 One-standard-deviation statistical uncertainty of the measured fluorescence rate [ms⁻¹].
 - temp_mK:
 Ion temperature T obtained from spatial thermometry fits of the EMCCD images, in millikelvin [mK],
 as plotted in Fig. 4(b).
 - temp_mK_err:
 One-standard-deviation uncertainty of the extracted ion temperature [mK].

 * experiment_doppler_temp_panel_a_lorentz_fit.csv
 Description:
 Smooth Lorentzian line-shape fit to the fluorescence spectrum used as the curve in Fig. 4(a).
 Columns:
 - detuning_MHz:
 Laser detuning Δν [MHz] at which the fit is evaluated.
 - fluorescence_fit_ms^-1:
 Fitted fluorescence rate per ion [ms⁻¹] from the Lorentzian model (including saturation broadening).

 * experiment_doppler_temp_panel_b_temp_fit.csv
 Description:
 Phenomenological fit to the dependence of ion temperature on Doppler detuning (approximately ∝ 1/|Δν|)
 as shown as dashed line in Fig. 4(b).
 Columns:
 - detuning_MHz:
 Laser detuning Δν [MHz].
 - temp_fit_mK:
 Fitted ion temperature T(Δν) in millikelvin [mK].

 - Figure 5 – Spatial extents of a four-Be+ ion crystal (benchmark of simulation vs experiment)
 Files (in subfolder "Figure 5"):
 * experiment_4be_extents_experiment_data.csv
 Description:
 Experimental spatial extents of the four Be+ ions along the radial (u) and axial (z) directions as a
 function of temperature, underlying Fig. 5(a) and Fig. 5(b). The “outer” ions correspond to ions 1 and 4,
 the “inner” ions to ions 2 and 3.
 Columns:
 - temp_mK:
 Ion temperature T obtained from spatial thermometry, in millikelvin [mK].
 - temp_mK_err:
 One-standard-deviation uncertainty of the extracted temperature [mK].
 - extent_u_out_um:
 Thermal spot size σ_th,u,i of the outer ions (1, 4) along the radial u direction in micrometers [µm].
 - extent_u_out_um_err:
 One-standard-deviation uncertainty of extent_u_out_um [µm].
 - extent_u_in_um:
 Thermal spot size σ_th,u,i of the inner ions (2, 3) along the radial u direction in micrometers [µm].
 - extent_u_in_um_err:
 One-standard-deviation uncertainty of extent_u_in_um [µm].
 - extent_v_out_um:
 Thermal spot size σ_th,z,i of the outer ions (1, 4) along the axial direction (denoted v ≈ z in the
 imaging plane) in micrometers [µm].
 - extent_v_out_um_err:
 One-standard-deviation uncertainty of extent_v_out_um [µm].
 - extent_v_in_um:
 Thermal spot size σ_th,z,i of the inner ions (2, 3) along the axial direction v in micrometers [µm].
 - extent_v_in_um_err:
 One-standard-deviation uncertainty of extent_v_in_um [µm].

 * experiment_4be_extents_simulation_data.csv
 Description:
 Simulated spatial extents and corresponding harmonic-approximation predictions for the four-Be+ ion
 crystal, used for the solid and dashed lines in Fig. 5(a) and Fig. 5(b).
 Columns:
 - temp_mK:
 Ion crystal temperature T used in the simulations, in millikelvin [mK].
 - sim_extent_harm_u_out_um:
 Harmonic-approximation prediction for σ_th,u,i of the outer ions (1, 4) along the radial u direction
 in micrometers [µm].
 - sim_extent_harm_u_in_um:
 Harmonic-approximation prediction for σ_th,u,i of the inner ions (2, 3) along the radial u direction
 in micrometers [µm].
 - sim_extent_u_out_um:
 Full molecular-dynamics simulated σ_th,u,i of the outer ions (1, 4) along the radial u direction [µm].
 - sim_extent_u_in_um:
 Full molecular-dynamics simulated σ_th,u,i of the inner ions (2, 3) along the radial u direction [µm].
 - sim_extent_harm_v_out_um:
 Harmonic-approximation prediction for σ_th,z,i of the outer ions (1, 4) along the axial direction v ≈ z
 in micrometers [µm].
 - sim_extent_harm_v_in_um:
 Harmonic-approximation prediction for σ_th,z,i of the inner ions (2, 3) along the axial direction v ≈ z
 in micrometers [µm].
 - sim_extent_v_out_um:
 Full molecular-dynamics simulated σ_th,z,i of the outer ions (1, 4) along the axial direction v [µm].
 - sim_extent_v_in_um:
 Full molecular-dynamics simulated σ_th,z,i of the inner ions (2, 3) along the axial direction v [µm].
 - Figure 6 – Gaussian overlap and spatial thermometry of a four-Be⁺ ion crystal
 Files (in subfolder "Figure 6"):
 * experiment_4be_crystal_main_experiment.csv
 Description:
 Experimental data for Fig. 6(a). Average axial Gaussian overlap D_{i,i+1} between neighboring Be⁺ ions
 as a function of ion temperature. The overlap quantifies how strongly the time-averaged axial spatial
 distributions of neighboring ions overlap (D = 0: well separated; D → 1: strong overlap / reordering).
 Columns:
 - T_mK:
 Ion temperature T in millikelvin [mK], obtained from spatial thermometry.
 - T_err_mK:
 One-standard-deviation uncertainty of the extracted ion temperature [mK].
 - D_exp:
 Measured average axial Gaussian overlap D_{i,i+1} between all neighboring pairs in the four-Be⁺ string
 (dimensionless).
 - D_exp_err:
 One-standard-deviation uncertainty of D_exp (dimensionless).

 * experiment_4be_crystal_main_simulation.csv
 Description:
 Simulation curves used as the solid and dashed lines in Fig. 6(a). The file contains the average axial
 Gaussian overlap from full molecular-dynamics simulations and from the harmonic approximation as a
 function of temperature.
 Columns:
 - T_mK:
 Simulation temperature T in millikelvin [mK].
 - D_sim:
 Average axial Gaussian overlap D_{i,i+1} obtained from full molecular-dynamics simulations
 (dimensionless).
 - D_harm_sim:
 Average axial Gaussian overlap D_{i,i+1} from the harmonic approximation (linearized crystal
 dynamics; dimensionless).

 * experiment_4be_crystal_spatial_thermometry.csv
 Description:
 Per-ion spatial thermometry data underlying the right-hand panels of Fig. 6 (“Spatial thermometry”).
 For several representative temperatures, the file lists the radial (u) and axial (z) spot sizes of each ion
 from simulation (thermal + PSF and PSF-only contributions) and from EMCCD image fits.
 Columns:
 - T_mK:
 Ion temperature T corresponding to this EMCCD data set, in millikelvin [mK].
 - emccd_run:
 Numerical identifier of the EMCCD image data set (run number) used for the fits.
 - ion_index:
 Integer label of the ion position along the z axis; {4, 3, 2, 1} from one end of the crystal to the other
 (as in the figure labels).
 - sigma_u_tot_um:
 Total expected radial spot size σ_{u,i} (thermal motion + imaging PSF) for ion i from simulation,
 in micrometers [µm].
 - sigma_u_psd_um:
 Radial contribution of the imaging point-spread function (PSF) alone, σ_{u,PSF}, in micrometers [µm].
 - sigma_u_emccd_um:
 Radial spot size σ_{u,i} obtained from Gaussian fits to the EMCCD images (data), in micrometers [µm].
 - sigma_u_emccd_err_um:
 One-standard-deviation uncertainty of sigma_u_emccd_um [µm].
 - sigma_z_tot_um:
 Total expected axial spot size σ_{z,i} (thermal motion + PSF) for ion i from simulation,
 in micrometers [µm].
 - sigma_z_psd_um:
 Axial contribution of the imaging PSF alone, σ_{z,PSF}, in micrometers [µm].
 - sigma_z_emccd_um:
 Axial spot size σ_{z,i} obtained from Gaussian fits to the EMCCD images (data), in micrometers [µm].
 - sigma_z_emccd_err_um:
 One-standard-deviation uncertainty of sigma_z_emccd_um [µm].

 - Figure 9 – Gaussian overlap and spatial thermometry of a mixed Be⁺/HCI Coulomb crystal
 Files (in subfolder "Figure 9"):
 * experiment_hci_crystal_main_experiment.csv
 Description:
 Experimental data for Fig. 9(a). Axial Gaussian overlap D_{34} of the central Be⁺ ion pair surrounding
 the highly charged ion (HCI) as a function of temperature. This overlap quantifies how strongly the axial
 spatial distributions of the two Be⁺ ions enclosing the HCI overlap.
 Columns:
 - T_mK:
 Ion temperature T in millikelvin [mK], obtained from spatial thermometry.
 - T_err_mK:
 One-standard-deviation uncertainty of the extracted ion temperature [mK].
 - D_exp:
 Measured axial Gaussian overlap D_{34} of the Be⁺ pair adjacent to the HCI (dimensionless).
 - D_exp_err:
 One-standard-deviation uncertainty of D_exp (dimensionless).

 * experiment_hci_crystal_main_simulation.csv
 Description:
 Simulation curves shown in Fig. 9(a). The file contains the axial Gaussian overlap D_{34} from full
 molecular-dynamics simulations and from the harmonic approximation as a function of temperature.
 Columns:
 - T_mK:
 Simulation temperature T in millikelvin [mK].
 - D_sim:
 Axial Gaussian overlap D_{34} of the Be⁺ pair obtained from full molecular-dynamics simulations
 (dimensionless).
 - D_harm_sim:
 Axial Gaussian overlap D_{34} of the Be⁺ pair from the harmonic approximation (dimensionless).

 * experiment_hci_crystal_spatial_thermometry.csv
 Description:
 Per-ion spatial thermometry data for the mixed-species (Be⁺ + Ar¹¹⁺) crystal underlying the right-hand
 panels of Fig. 9. Since the HCI does not fluoresce, only selected Be⁺ ions are included. For each shown
 temperature, the file lists radial (u) and axial (z) spot sizes from simulation (thermal + PSF and PSF-only)
 and from EMCCD image fits.
 Columns:
 - T_mK:
 Ion temperature T corresponding to this EMCCD data set, in millikelvin [mK].
 - ion_index:
 Integer label of the Be⁺ ions used in the spatial thermometry plots. The indices {4, 3, 1} correspond
 to the outer Be⁺ ion at one end, the Be⁺ next to the HCI, and the outer Be⁺ ion at the opposite end
 of the chain (ion index 2 denotes the non-fluorescing HCI and is not listed).
 - sigma_u_tot_um:
 Total expected radial spot size σ_{u,i} (thermal motion + PSF) for ion i from simulation,
 in micrometers [µm].
 - sigma_u_psd_um:
 Radial PSF-only contribution σ_{u,PSF} in micrometers [µm].
 - sigma_u_emccd_um:
 Radial spot size σ_{u,i} obtained from Gaussian fits to the EMCCD images (data), in micrometers [µm].
 - sigma_u_emccd_err_um:
 One-standard-deviation uncertainty of sigma_u_emccd_um [µm].
 - sigma_z_tot_um:
 Total expected axial spot size σ_{z,i} (thermal motion + PSF) for ion i from simulation,
 in micrometers [µm].
 - sigma_z_psd_um:
 Axial PSF-only contribution σ_{z,PSF} in micrometers [µm].
 - sigma_z_emccd_um:
 Axial spot size σ_{z,i} obtained from Gaussian fits to the EMCCD images (data), in micrometers [µm].
 - sigma_z_emccd_err_um:
 One-standard-deviation uncertainty of sigma_z_emccd_um [µm].
 - Figure 7 – Critical temperature of single-species ion Coulomb crystals
 Files (in subfolder "Figure 7"):
 * single_species_melting_panel_a_alpha_vs_Tc.csv
 Description:
 Data for Fig. 7(a). Critical temperature Tc of the orientational melting transition in single-species
 linear ion Coulomb crystals as a function of the trap anisotropy parameter α, for several ion numbers N.
 The temperatures are expressed in units of the characteristic scale τ, and analytic fits capture the
 approximate scaling Tc/τ ∝ (α − 1)^{1/3}.
 Columns:
 - N:
 Number of ions N in the single-species linear Coulomb crystal (dimensionless).
 - alpha:
 Trap anisotropy parameter α = (ω_x/ω_z)² used in the simulations (dimensionless), where ω_x and ω_z
 are the radial and axial secular trap frequencies.
 - Tc_over_tau:
 Numerically determined critical temperature Tc/τ (dimensionless) at which orientational melting
 occurs for the given (N, α).
 - Tc_over_tau_fit:
 Value of the fitted function Tc/τ evaluated at the same α for the corresponding N (dimensionless),
 used to draw the smooth fit curves in Fig. 7(a).

 * single_species_melting_panel_b_Tc_vs_N.csv
 Description:
 Data for Fig. 7(b). Critical temperature Tc as a function of ion number N for three different strategies
 of adapting the trapping parameters when increasing N:
 (i) “no match”: all trap frequencies are kept constant;
 (ii) “match dist.”: the zigzag-mode frequency ω_zz is kept constant by adjusting the radial confinement;
 (iii) “match both”: both the nearest-neighbour distance of the inner ions d_in and ω_zz are kept
 approximately constant by tuning both radial and axial confinement.
 The corresponding smooth curves are obtained from an empirical exponential-plus-linear fit.
 Columns:
 - N:
 Number of ions N in the single-species linear Coulomb crystal (dimensionless).
 - Tc_no_match_mK:
 Critical temperature Tc in millikelvin [mK] for the case where all trap frequencies are kept fixed
 as N is increased (“no matching”; lower trace in Fig. 7(b)).
 - Tc_match_dist_mK:
 Critical temperature Tc in millikelvin [mK] for the case where the zigzag-mode frequency ω_zz is
 kept constant by adjusting the radial trapping frequency, while the axial frequency is fixed
 (“matching ω_zz”; middle trace in Fig. 7(b)).
 - Tc_match_both_mK:
 Critical temperature Tc in millikelvin [mK] for the case where both the distance of the inner ions
 d_in and the zigzag-mode frequency ω_zz are approximately kept constant by tuning both radial and
 axial confinement (“matching d_in and ω_zz”; upper trace in Fig. 7(b)).
 - Tc_no_match_fit_mK:
 Value of the fitted Tc(N) curve in millikelvin [mK] for the “no match” scenario, evaluated at the
 given N (used to draw the smooth lower curve in Fig. 7(b)).
 - Tc_match_dist_fit_mK:
 Value of the fitted Tc(N) curve in millikelvin [mK] for the “match dist.” scenario (constant ω_zz),
 evaluated at the given N (smooth middle curve).
 - Tc_match_both_fit_mK:
 Value of the fitted Tc(N) curve in millikelvin [mK] for the “match both” scenario (constant d_in and
 ω_zz), evaluated at the given N (smooth upper curve).
 - Figure 10 – Localized melting in mixed-species crystals (RMSD vs temperature)
 Files (in subfolder "Figure 10"):
 * mixed_species_RMSD_3_Be_1_Ar11_grouped_MSD_damping_points.csv
 Description:
 Simulated normalized root-mean-square (RMS) displacements s_l as a function of temperature T for the
 mixed-species configuration with 3 Be⁺ and 1 Ar¹¹⁺ ion. The data are grouped by ion domains (Be⁺ groups
 and the Ar¹¹⁺ ion), corresponding to the different curves shown in Fig. 10(a).
 Columns:
 - series_index:
 Integer index identifying the different ion domains in the plot. The mapping of series_index to
 specific domains matches the legend in Fig. 10(a) (e.g. Be⁺(3,4), Be⁺(1), Ar¹¹⁺(2)).
 - m_num:
 Numerical proportionality factor m of the harmonic theory for this domain, entering the harmonic
 prediction s_l^(harm)(T) ∝ sqrt(m · T).
 - T_mK:
 Ion temperature T in millikelvin [mK] for each simulation point.
 - s_l:
 Normalized RMS displacement s_l of the corresponding ion domain (dimensionless, normalized to the
 mean ion spacing as defined in the main text).
 - s_l_err:
 One-standard-deviation statistical uncertainty of s_l (dimensionless).

 * mixed_species_RMSD_3_Be_1_Ar11_grouped_MSD_damping_theory.csv
 Description:
 Harmonic-theory reference curves for the same ion domains as in
 mixed_species_RMSD_3_Be_1_Ar11_grouped_MSD_damping_points.csv. These are plotted as dashed lines in
 Fig. 10(a).
 Columns:
 - series_index:
 Integer index of the ion domain, identical to the series_index used in the corresponding points file.
 - m_num:
 Proportionality factor m of the harmonic model for this ion domain.
 - T_mK:
 Temperature grid in millikelvin [mK] on which the harmonic curves are evaluated.
 - s_l_theory:
 Harmonic prediction of the normalized RMS displacement s_l^(harm)(T) for this domain (dimensionless).

 * mixed_species_RMSD_6_Be_2_Ar11_grouped_MSD_damping_points.csv
 Description:
 Simulated normalized RMS displacements s_l as a function of temperature T for the mixed-species
 configuration with 6 Be⁺ and 2 Ar¹¹⁺ ions. The data are grouped by ion domains (inner Be⁺ pairs, outer
 Be⁺ ions, and Ar¹¹⁺ ions), corresponding to the different curves in Fig. 10(b).
 Columns:
 - series_index:
 Integer index labelling the different ion domains in the plot. The mapping of series_index to domains
 matches the legend in Fig. 10(b) (e.g. Be⁺(4,5), Be⁺(1,2,7,8), Ar¹¹⁺(3,6)).
 - m_num:
 Proportionality factor m of the harmonic model for this ion domain.
 - T_mK:
 Ion temperature T in millikelvin [mK] for each simulation point.
 - s_l:
 Normalized RMS displacement s_l of the corresponding ion domain (dimensionless).
 - s_l_err:
 One-standard-deviation statistical uncertainty of s_l (dimensionless).

 * mixed_species_RMSD_6_Be_2_Ar11_grouped_MSD_damping_theory.csv
 Description:
 Harmonic-theory reference curves for the ion domains of the 6 Be⁺ + 2 Ar¹¹⁺ configuration, shown as
 dashed lines in Fig. 10(b).
 Columns:
 - series_index:
 Integer index of the ion domain, identical zu dem in der zugehörigen points-Datei.
 - m_num:
 Proportionality factor m of the harmonic model for this ion domain.
 - T_mK:
 Temperature grid in millikelvin [mK] auf der die harmonischen Kurven berechnet werden.
 - s_l_theory:
 Harmonische Vorhersage der normalisierten RMS-Auslenkung s_l^(harm)(T) (dimensionslos).

 - Figure 12 – Hopping rates in mixed-species Coulomb crystals
 Files (in subfolder "Figure 12"):
 * mixed_species_hopping_3_Be_1_Ar11_hopping_rates_damping_data.csv
 Description:
 Simulated hopping rates for the relevant processes in the mixed-species configuration with 3 Be⁺ and
 1 Ar¹¹⁺ ion, as a function of temperature. The rates are normalized to ω′_z/2π and correspond to the
 symbols in Fig. 12(a) (different hopping channels).
 Columns:
 - series_id:
 Integer index labelling the different hopping channels. The mapping corresponds to the legend in
 Fig. 12(a), e.g. Be–Be hopping (γ_Be), HCI hopping towards the crystal center (γ_Ar→in), and HCI
 hopping towards the edge (γ_Ar→out). The same series_id is used in the corresponding fit file.
 - T_mK:
 Ion temperature T in millikelvin [mK] for each simulated hopping rate.
 - gamma_over_omegaZprime:
 Simulated hopping rate γ_hop normalized to ω′_z/2π, i.e. γ_hop / (ω′_z/2π) (dimensionless).
 - gamma_err_over_omegaZprime:
 One-standard-deviation statistical uncertainty of γ_hop / (ω′_z/2π) (dimensionless).

 * mixed_species_hopping_3_Be_1_Ar11_hopping_rates_damping_fits.csv
 Description:
 Arrhenius-type fit curves for the hopping channels in the 3 Be⁺ + 1 Ar¹¹⁺ configuration, corresponding
 to the smooth lines in Fig. 12(a).
 Columns:
 - series_id:
 Integer index of the hopping channel, identical to the series_id used in the corresponding data file.
 - T_fit_mK:
 Temperature grid in millikelvin [mK] on which the Arrhenius fits are evaluated.
 - gamma_fit_over_omegaZprime:
 Fitted hopping rate γ_hop,fit / (ω′_z/2π) (dimensionless) for the respective hopping channel.

 * mixed_species_hopping_6_Be_2_Ar11_hopping_rates_damping_data.csv
 Description:
 Simulated hopping rates for selected ion groups in the mixed-species configuration with 6 Be⁺ and
 2 Ar¹¹⁺ ions, as a function of temperature, corresponding to the symbols in Fig. 12(b). The different
 series correspond to inner Be⁺ pairs, outer Be⁺ groups, and HCI hopping processes.
 Columns:
 - series_id:
 Integer index labelling the different hopping channels in Fig. 12(b). The mapping to specific channels
 (e.g. inner Be⁺ pair, combined outer Be⁺ groups, HCI hops towards center/edge) is identical in this
 file and in the associated fit file and can be read off from the figure legend.
 - T_mK:
 Ion temperature T in millikelvin [mK] for each simulated hopping rate.
 - gamma_over_omegaZprime:
 Simulated hopping rate γ_hop / (ω′_z/2π) for the given channel (dimensionless).
 - gamma_err_over_omegaZprime:
 One-standard-deviation statistical uncertainty of γ_hop / (ω′_z/2π) (dimensionless).

 * mixed_species_hopping_6_Be_2_Ar11_hopping_rates_damping_fits.csv
 Description:
 Arrhenius-type fit curves for the hopping channels in the 6 Be⁺ + 2 Ar¹¹⁺ configuration, corresponding
 to the smooth lines in Fig. 12(b).
 Columns:
 - series_id:
 Integer index of the hopping channel, identical to the series_id used in the corresponding data file.
 - T_fit_mK:
 Temperature grid in millikelvin [mK] on which the Arrhenius fits are evaluated.
 - gamma_fit_over_omegaZprime:
 Fitted hopping rate γ_hop,fit / (ω′_z/2π) (dimensionless) for the respective hopping channel.
 - Figure 13 – Lindemann criterion at the critical temperature
 Files (in subfolder "Figure 13"):
 * heatmap_tcrit_lindemann_avg_data.csv
 Description:
 Data for Fig. 13. Average Lindemann parameter of single-species linear Coulomb crystals evaluated at the
 critical temperature Tc (determined from the hopping rate), as a function of trap anisotropy α and ion
 number N. The plotted quantity is the mean squared displacement of the ions divided by the squared local
 equilibrium spacing, ⟨(r_i − r_i^0)²⟩ / d_i², averaged over ions, at T = Tc.
 Columns:
 - n_ions:
 Number of ions N in the single-species linear Coulomb crystal (dimensionless).
 - eps:
 Radial-to-axial confinement parameter ε used in the simulations (dimensionless). It parametrizes the
 trap anisotropy via α = ε² − 1/2.
 - alpha:
 Trap anisotropy parameter α = (ω_x/ω_z)² (dimensionless) used on the horizontal axis of Fig. 13.
 - Lindemann_avg:
 Average Lindemann parameter at the critical temperature Tc, i.e. the mean of
 ⟨(r_i − r_i^0)²⟩ / d_i² over all ions in the crystal (dimensionless). This is the quantity plotted on the
 vertical axis of Fig. 13. Values around ≈ 0.14 indicate the onset of orientational melting as discussed
 in the main text.
 - Figure 14 – Pair-correlation functions in mixed-species Coulomb crystals
 Files (in subfolder "Figure 14"):
 * pair_correlation_gr_curves.csv
 Description:
 Data for Fig. 14. One-dimensional pair-correlation functions g(r) of the ion crystal, calculated from
 molecular-dynamics trajectories for several temperatures and corresponding Coulomb-coupling parameters
 Γ. For each (T, Γ), the file contains the smoothed curve g(r) as a function of inter-ion separation r,
 as plotted in Fig. 14.
 Columns:
 - T_K:
 Ion temperature T in kelvin [K] for the respective g(r) curve.
 - T_mK:
 Same ion temperature T expressed in millikelvin [mK] for convenience.
 - gamma:
 Coulomb-coupling parameter Γ (dimensionless), defined as the ratio of characteristic Coulomb energy
 to thermal energy for the given state point.
 - r_um:
 Inter-ion separation r along the considered 1D direction in micrometers [µm]; horizontal axis of
 Fig. 14.
 - g_of_r:
 Pair-correlation function g(r) evaluated at r_um for the given (T, Γ). Values g(r) &gt; 1 indicate enhanced
 probability of finding another ion at distance r compared to a uniform distribution, while g(r) ≈ 1
 corresponds to an uncorrelated distribution.
 - Figure 15 – Global melting map of single-species ion Coulomb crystals
 Files (in subfolder "Figure 15"):
 * single_species_melting_heatmap_data.csv
 Description:
 Data for Fig. 15. Two-dimensional map (heatmap) of the critical temperature at which orientational
 melting occurs in single-species linear Coulomb crystals, as a function of ion number N and trap
 anisotropy α. The critical temperatures are expressed in units of the characteristic temperature τ.
 In the plot, cells with Tm_over_tau = -1 are treated as “no data” (regions where no stable crystal or
 no reliable melting point was obtained).
 Columns:
 - N:
 Number of ions N in the single-species linear Coulomb crystal (dimensionless); vertical axis of
 the heatmap.
 - alpha:
 Trap anisotropy parameter α = (ω_x/ω_z)² (dimensionless), where ω_x and ω_z are the radial and
 axial secular trap frequencies; horizontal axis of the heatmap.
 - Tm_over_tau # Tm_over_tau = -1 markiert Zellen ohne Wert:
 Critical temperature of the orientational melting transition T_m normalized to τ, i.e.
 T_m / τ (dimensionless). This value is encoded by the color scale in Fig. 15. Entries with
 T_m / τ = -1 indicate grid points where no melting temperature is defined or no simulation
 result is available; these cells are effectively masked in the figure.
description: We acknowledge support by the projects 18SIB05 ROCIT and 20FUN01 TSCAC. These projects have received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. This project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers–390837967 and through CRC 1227 (DQ-mat), project A07. This work has been supported by the Max Planck Society; the Max-Planck–Riken–PTB-Center for Time, Constants and Fundamental Symmetries; and the German Federal Ministry of Education and Research (BMBF) through program grant No. 13N15973 (Projekt VAUQSI).