DOI:             10.7795/710.20261005
resource type:   Dataset / Simulation data
title:           Data for the publication: Shell formation and two-dimensional nanofriction in three-dimensional ion Coulomb crystals
description:     We investigated the formation of shells in three-dimensional ion Coulomb clusters and analysed their relative motion and dynamics under the lens of nanofriction 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. Figures with purely graphical elements or schematic representations are excluded.
creators:
  - given name:  Luca-Amadeus
    family name: Rüffert
    ORCID:       https://orcid.org/0000-0001-8980-9794
    Affiliation: PTB Braunschweig
    PTB FB:      Research Group QUEST FG2 "Quantum Clocks and Complex Systems"

funding:
  funder name: EURAMET; DFG
  funding program: EMPIR (18SIB05 ROCIT, 20FUN01 TSCAC); Excellence Strategy EXC-2123 QuantumFrontiers; CRC 1227 (DQ-mat) A07
  funding nr: 18SIB05; 20FUN01; 390837967; CRC 1227 A07
  project acronym: ROCIT; TSCAC; QuantumFrontiers; DQ-mat
  project long title: (see funding acknowledgement text)
  link to project homepage: empir.npl.co.uk/rocit; ptb.de/empir2021/tscac/home; quantumfrontiers.de; dq-mat.de
  funding acknowledgement text: 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.

license:        Creative Commons Attribution 4.0 International License (CC BY 4.0)
data assignment: molecular-dynamics simulation data and data evaluated from simulated ion trajectories
keywords:       ion Coulomb crystals, trapped ions, molecular dynamics, shell formation, nanofriction, commensurability, stick-slip motion
classification: INSPEC A30 Atomic and molecular physics

directory structure:
  structural_data: evaluated data for Figures 4, 6, 7, and 13-18
  dynamical_data: evaluated trajectory and torque-response data for Figures 8-10 and Table I
  raw_sim_data: independently readable CSV conversion of the ion-configuration files used directly by Figures 13-15, 17, and 18

excluded figures:
  Figures 1, 2, 3, 5, 11, and 12 contain experimental images, graphical elements, or schematic representations and therefore have no separate numerical plot-data files in this dataset.

description of the individual files:
  - figure: Figure 4
    method: shell analysis of simulated ion configurations and evaluation of shell-transition lines
    exported files:
      - structural_data/figure_04_shell_number_heatmap.csv
        description: Shell number and dimensionless linear density on the plotted (N, alpha) grid.
        columns: N_ions; alpha; number_of_shells; dimensionless_linear_density
      - structural_data/figure_04_heatmap_transition_overlays.csv
        description: Dashed empirical transition lines and white linear-density prediction markers plotted over the heatmap.
        columns: curve_type; N_ions; alpha
      - structural_data/figure_04_shell_count_scaling.csv
        description: Transition-slope data with x uncertainties and the fitted power-law curve for panel (b).
        columns: record_type; alpha_over_N; number_of_shells; alpha_over_N_uncertainty

  - figure: Figure 6
    exported file: structural_data/figure_06_energy_vs_rotation_angle.csv
    description: Mean-subtracted potential-energy modulation of the rotating outer shell for each selected system.
    columns: alpha; N_ions; rotation_angle_deg; delta_potential_energy_mK; peak_to_peak_energy_mK; returns_to_initial_branch

  - figure: Figure 7
    exported file: structural_data/figure_07_effective_barrier_vs_ion_number.csv
    description: Effective rotational barrier per ion versus ion number for every trapping aspect ratio.
    columns: alpha; N_ions; effective_barrier_per_ion_mK; returns_to_initial_branch; highlighted_system

  - figure: Figure 8
    exported file: dynamical_data/figure_08_normalized_angular_velocity_vs_time.csv
    description: Time-dependent outer-shell angular velocity for the three driving torques shown in the figure.
    columns: source_file; N_ions; drive_torque_over_tau_prime; time_ms; corrected_torque_over_tau_prime; angular_velocity; maximum_angular_velocity; normalized_angular_velocity

  - figure: Figure 9 and Table I
    exported files:
      - dynamical_data/figure_09_sliding_efficiency_vs_driving_torque.csv
        description: Increasing- and decreasing-torque sweeps of the normalized sliding efficiency.
        columns: source_file; alpha; N_ions; sweep_direction; driving_torque_over_tau_prime; effective_torque_over_tau_prime; mean_angular_velocity; maximum_angular_velocity; sliding_efficiency
      - dynamical_data/table_01_pinning_and_depinning_torques.csv
        description: Pinning- and depinning-torque estimates and the relative half-step uncertainty. Decreasing sweeps yield pinning thresholds; increasing sweeps yield depinning thresholds.
        columns: alpha; N_ions; source_file; sweep_direction; transition_type; transition_torque_over_tau_prime; relative_half_step_uncertainty

  - figure: Figure 10
    exported file: dynamical_data/figure_10_outer_shell_angular_velocity_profile.csv
    description: Normalized mean angular velocity of each retained outer-shell ion as a function of its mean axial position.
    columns: alpha; N_ions; z_um; normalized_mean_angular_velocity

  - figure: Figure 13
    exported files:
      - structural_data/figure_13_deltahedron_vertices.csv
        description: Ion coordinates for the eight plotted ground-state deltahedron configurations.
        columns: shape_name; N_ions; ion_index; x_um; y_um; z_um
      - structural_data/figure_13_deltahedron_edges.csv
        description: Unique convex-hull edges connecting the plotted vertices.
        columns: shape_name; N_ions; ion_index_1; ion_index_2

  - figure: Figure 14
    exported files:
      - structural_data/figure_14_shell_histogram.csv
        description: Raw and smoothed counts versus the shell-analysis iteration index.
        columns: shell_index_i; raw_count; smoothed_count; shell_region
      - structural_data/figure_14_shell_structure_coordinates.csv
        description: Coordinates, normalized ellipsoidal radius, shell index, and plotted shell-region assignment of every ion.
        columns: ion_index; x_um; y_um; z_um; normalized_ellipsoidal_radius; shell_index_i; shell_region

  - figure: Figure 15
    exported file: structural_data/figure_15_inner_shell_coordinates_alpha_4.09_N_73.csv
    description: Original and displayed (80 degree rotated) coordinates of the innermost non-empty shell for the selected configuration.
    columns: shell_ion_index; global_ion_index; x_original_um; y_original_um; z_original_um; x_rotated_um; y_rotated_um; z_rotated_um

  - figure: Figure 16
    exported files:
      - structural_data/figure_16_second_energy_difference_residuals.csv
        description: Matched second-energy-difference and effective-barrier series, local trends, residuals, and plotted z scores.
        columns: alpha; N_ions; second_energy_difference_mK; effective_barrier_per_ion_mK; returns_to_initial_branch; second_energy_difference_trend_mK; effective_barrier_trend_mK; second_energy_difference_residual_mK; effective_barrier_residual_mK; z_second_energy_difference_residual; z_effective_barrier_residual
      - structural_data/figure_16_residual_correlations.csv
        description: Pearson and Spearman correlations for all matched points and for the continuous-branch subset.
        columns: alpha; selection; N_min; n_points; pearson_r; pearson_p; spearman_rho; spearman_p

  - figure: Figure 17
    exported files:
      - structural_data/figure_17_energy_decomposition_curves.csv
        description: Plotted changes of the inner-outer, outer-outer, trap, and total energy terms along each relaxed rotational path.
        columns: alpha; N_ions; rotation_angle_deg; delta_U_in_out_mK; delta_U_out_out_mK; delta_U_trap_total_mK; delta_U_trap_outer_mK; delta_U_total_mK
      - structural_data/figure_17_energy_decomposition_summary.csv
        description: Peak-to-peak amplitudes and shell populations for the selected systems.
        columns: source_file; alpha; N; N_inner; N_outer; ptp_U_in_out; ptp_U_out_out; ptp_U_trap_total; ptp_U_trap_outer; ptp_U_total; ptp_U_relevant

  - figure: Figure 18
    exported files:
      - structural_data/figure_18_dynamic_commensurability_curves.csv
        description: Plotted changes of the geometric commensurability score and inner-outer interaction energy along the relaxed rotational path.
        columns: alpha; N_ions; rotation_angle_deg; delta_geometric_commensurability; delta_inner_outer_energy_mK
      - structural_data/figure_18_dynamic_commensurability_summary.csv
        description: Geometric commensurability summary values and the branch-continuity flag for each selected system.
        columns: alpha; N_ions; kappa_geom; C_mean; C_min; C_max; sigma_um; same_config

  - figures: Figures 13-15, 17, and 18
    method: lossless structural conversion of the named ionsimulator text fields; unnamed trailing values are preserved without interpretation
    exported files:
      - raw_sim_data/source_manifest.csv
        columns: source_file; converted_csv; used_for; N_ions; trajectory_rows
      - raw_sim_data/simulation_parameters.csv
        columns: source_file; parameter; value
      - raw_sim_data/timestep_auxiliary_values.csv
        columns: source_file; trajectory_index; unnamed_source_value_1 ... unnamed_source_value_n
      - raw_sim_data/ion_configuration_trajectories/*.csv
        description: One long-format file per source configuration, preserving every saved ion state.
        columns: trajectory_index; ion_index; x_um; y_um; z_um; vx_raw; vy_raw; vz_raw

format and software:
  format: .csv (comma-separated UTF-8 plain text with one header row per file)
  missing values: empty field or NaN where a quantity is not defined for a record type
  recommended software: any software capable of reading CSV files (for example Python/NumPy/pandas, MATLAB, Origin, or Excel)
  raw velocity fields: vx_raw, vy_raw, and vz_raw preserve the native simulation-output values because the source text format does not encode a unit label for these columns.
