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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the NPIPE and FFP10 data releases

We present a topological analysis of the temperature fluctuation maps from the Planck 2020 Data Release 4 NPIPE dataset and the Planck 2018 Data Release 3 FFP10 dataset. We performed a multiscale analysis in terms of the homology characteristics of the maps, invoking relative homology to account for the analysis in the presence of masks. We performed our analysis for a range of smoothing scales spanning sub- and super-horizon scales corresponding to a full width at half maximum (FWHM) of 5',10',20',40',80',160',320', and 640', and employed simulations based on the standard model for comparison, which assumes the initial fluctuation field to be an isotropic and homogeneous Gaussian random field. Examining the behavior of topological components, represented by the 0D homology group, we find the observations to be approximately 2σ or less deviant from the simulations for all resolutions and scales for the NPIPE dataset. For the FFP10 dataset, we detect a 2.96σ deviation between the observations and simulations at N = 128, FWHM = 80'. For the topological loops, represented by the first homology group, the simulations and observations are consistent within 2σ for most resolutions and scales for both the datasets. However, for the NPIPE dataset, we observe a high deviation between the observation and simulations in the number of loops at FWHM = 320', but at a low dimensionless threshold ν = –2.5. Under a Gaussian assumption, this would amount to a deviation of ~4σ. However, the distribution in this bin is manifestly non-Gaussian and does not obey Poisson statistics either. In the absence of a true theoretical understanding, we simply note that the significance is higher than what may be resolved by 600 simulations, yielding an empirical p-value of at most 0.0016. Specifically in this case, our tests indicate that the numbers arise from a statistically stable regime, despite being based on small numbers. For the FFP10 dataset, the differences are not as strong as for the NPIPE dataset, indicating a 2.77σ deviation at this resolution and threshold. The Euler characteristic, which is the alternating sum of the ranks of relative homology groups, reflects the deviations in the components and loops. To assess the significance of combined levels for a given scale, we employed the empirical and theoretical versions of the χ 2 test as well as the nonparametric Tukey depth test. Although all statistics exhibit a stable distribution, we favor the empirical version of the χ 2 test in the final interpretation, as it indicates the most conservative differences. For the NPIPE dataset, we find that the components and loops differ at more than 95%, but agree within the 99% confidence level with respect to the base model at N = 32, FWHM = 320'. The Euler characteristic at this resolution displays a per mil deviation. In contrast, the FFP10 dataset shows that the observations are consistent with the base model within the 95% confidence level, at this and smaller scales. This is consistent with the observations of the Planck analysis pipeline via Minkowski functionals. For the largest smoothing scale, N = 16, FWHM = 640', both datasets exhibit an anomalous behavior of the loops, where FFP10 data exhibit a deviation that is larger by an order of magnitude than that of the NPIPE dataset. In contrast, the values for the topological components and the Euler characteristic agree between observations and model to within a confidence level of 99%. However, for the largest scales, the statistics are based on low numbers and may have to be regarded with caution. Even though both datasets exhibit mild to significant discrepancies, they also exhibit contrasting behaviors at various instances. Therefore, we do not find it feasible to convincingly accept or reject the null hypothesis. Disregarding the large-scale anomalies that persist at similar scales in WMAP and Planck, observations of the cosmic microwave background are largely consistent with the standard cosmological model within 2σ.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

BEYONDPLANCK VI. Noise characterization and modeling

We present a Bayesian method for estimating instrumental noise parameters and propagating noise uncertainties within the global BeyondPlanck Gibbs sampling framework, and apply this to Planck Low Frequency Instrument (LFI) time-ordered data. Following previous literature, we initially adopt a 1/ f model for the noise power spectral density (PSD), but find the need for an additional lognormal component in the noise model for the 30 and 44 GHz bands. We implement an optimal Wiener-filter (or constrained realization) gap-filling procedure to account for masked data. We then use this procedure to both estimate the gapless correlated noise in the time-domain, n corr , and to sample the noise PSD parameters, ξ n = {σ 0 , f knee , α, A p }. In contrast to previous Planck analyses, we assume piecewise stationary noise only within each pointing period (PID), not throughout the full mission, but we adopt the LFI Data Processing Center (DPC) results as priors on α and f knee . On average, we find best-fit correlated noise parameters that are mostly consistent with previous results, with a few notable exceptions. However, a detailed inspection of the time-dependent results reveals many important findings. First and foremost, we find strong evidence for statistically significant temporal variations in all noise PSD parameters, many of which are directly correlated with satellite housekeeping data. Second, while the simple 1/ f model appears to be an excellent fit for the LFI 70 GHz channel, there is evidence for additional correlated noise not described by a 1/ f model in the 30 and 44 GHz channels, including within the primary science frequency range of 0.1–1 Hz. In general, most 30 and 44 GHz channels exhibit deviations from 1/ f at the 2–3σ level in each one hour pointing period, motivating the addition of the lognormal noise component for these bands. For some periods of time, we also find evidence of strong common mode noise fluctuations across the entire focal plane. Overall, we conclude that a simple 1/ f profile is not adequate to fully characterize the Planck LFI noise, even when fitted hour-by-hour, and a more general model is required. These findings have important implications for large-scale CMB polarization reconstruction with the Planck LFI data, and the current work is a first attempt at understanding and mitigating these issues.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

BeyondPlanck: VII. Bayesian estimation of gain and absolute calibration for cosmic microwave background experiments

We present a Bayesian calibration algorithm for cosmic microwave background (CMB) observations as implemented within the global end-to-end BEYONDPLANCK framework and applied to the Planck Low Frequency Instrument (LFI) data. Following the most recent Planck analysis, we decomposed the full time-dependent gain into a sum of three nearly orthogonal components: one absolute calibration term, common to all detectors, one time-independent term that can vary between detectors, and one time-dependent component that was allowed to vary between one-hour pointing periods. Each term was then sampled conditionally on all other parameters in the global signal model through Gibbs sampling. The absolute calibration is sampled using only the orbital dipole as a reference source, while the two relative gain components were sampled using the full sky signal, including the orbital and Solar CMB dipoles, CMB fluctuations, and foreground contributions. We discuss various aspects of the data that influence gain estimation, including the dipole-polarization quadrupole degeneracy and processing masks. Comparing our solution to previous pipelines, we find good agreement in general, with relative deviations of -0.67% (-0.84%) for 30 GHz, 0.12% (-0.04%) for 44 GHz and -0.03% (-0.64%) for 70 GHz, compared to Planck PR4 and Planck 2018, respectively. We note that the BEYONDPLANCK calibration was performed globally, which results in better inter-frequency consistency than previous estimates. Additionally, WMAP observations were used actively in the BEYONDPLANCK analysis, which both breaks internal degeneracies in the Planck data set and results in an overall better agreement with WMAP. Finally, we used a Wiener filtering approach to smoothing the gain estimates. We show that this method avoids artifacts in the correlated noise maps as a result of oversmoothing the gain solution, which is difficult to avoid with methods like boxcar smoothing, as Wiener filtering by construction maintains a balance between data fidelity and prior knowledge. Although our presentation and algorithm are currently oriented toward LFI processing, the general procedure is fully generalizable to other experiments, as long as the Solar dipole signal is available to be used for calibration.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Analysis of Needlet Internal Linear Combination performance on B -mode data from sub-orbital experiments

The observation of primordial B modes in cosmic microwave background (CMB) polarisation data represents the main scientific goal of most of the future CMB experiments. This signal is predicted to be much lower than polarised Galactic emission (foregrounds) in any region of the sky, pointing to the need for effective component separation methods. Aims. Among all the techniques, the blind Needlet Internal Linear Combination (NILC) is of great relevance given our current limited knowledge of the B-mode foregrounds. In this work, we explore the possibility of employing NILC for the analysis of B modes reconstructed from partial-sky data, specifically addressing the complications that such an application yields such as E–B leakage, needlet filtering, and beam convolution. We consider two complementary simulated datasets of future experiments: the balloon-borne Short Wavelength Instrument for the Polarisation Explorer (SWIPE) of the Large Scale Polarisation Explorer, which targets the observation of both reionisation and recombination peaks of the primordial CMB B-mode angular power spectrum, and the ground-based Small Aperture Telescope of Simons Observatory, which, instead, is designed to observe only the recombination bump at ℓ ~ 80. We assessed the performance of the following two alternative techniques to correct for the CMB E–B leakage: the recycling technique and the Zhao-Baskaran method. We find that both techniques reduce the E–B leakage residuals at a negligible level given the sensitivity of the considered experiments, except for the recycling method in the SWIPE footprint at ℓ < 20. Thus, we implemented two extensions of the pipeline, the iterative B decomposition and the diffusive inpainting, which enabled us to recover the input CMB B-mode power for ℓ ≥ 5. For the considered experiments, we demonstrate that needlet filtering and beam convolution do not affect the CMB B-mode reconstruction. Finally, with an appropriate masking strategy, we find that NILC foregrounds subtraction allows one to achieve sensitivities on the tensor-to-scalar ratio in agreement with the targets of the considered CMB experiments.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Studying baryon acoustic oscillations using photometric redshifts from the DESI Legacy Imaging survey DR9

Context. The Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Survey DR9 (DR9 hereafter), with its extensive dataset of galaxy locations and photometric redshifts, presents an opportunity to study baryon acoustic oscillations (BAOs) in the region covered by the ongoing spectroscopic survey with DESI. Aims. We aim to investigate differences between different parts of the DR9 footprint. Furthermore, we want to measure the BAO scale for luminous red galaxies within them. Our selected redshift range of 0.6–0.8 corresponds to the bin in which a tension between DESI Y1 and eBOSS was found. Methods. We calculated the anisotropic two-point correlation function in a modified binning scheme to detect the BAOs in DR9 data. We then used template fits based on simulations to measure the BAO scale in the imaging data. Results. Our analysis reveals the expected correlation function shape in most of the footprint areas, showing a BAO scale consistent with Planck’s observations. Aside from identified mask-related data issues in the southern region of the South Galactic Cap, we find a notable variance between the different footprints. Conclusions. We find that this variance is consistent with the difference between the DESI Y1 and eBOSS data, and it supports the argument that that tension is caused by sample variance. Additionally, we also uncovered systematic biases not previously accounted for in photometric BAO studies. We emphasize the necessity of adjusting for the systematic shift in the BAO scale associated with typical photometric redshift uncertainties to ensure accurate measurements.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Plasma processing of low-k dielectrics

This paper presents an in-depth overview of the present status and novel developments in the field of plasma processing of low dielectric constant (low-k) materials developed for advanced interconnects in ULSI technology. The paper summarizes the major achievements accomplished during the last 10 years. It includes analysis of advanced experimental techniques that have been used, which are most appropriate for low-k patterning and resist strip, selection of chemistries, patterning strategies, masking materials, analytical techniques, and challenges appearing during the integration. Detailed discussions are devoted to the etch mechanisms of low-k materials and their degradation during the plasma processing. The problem of k-value degradation (plasma damage) is a key issue for the integration, and it is becoming more difficult and challenging as the dielectric constant of low-k materials scales down. Results obtained with new experimental methods, like the small gap technique and multi-beams systems with separated sources of ions, vacuum ultraviolet light, and radicals, are discussed in detail. The methods allowing reduction of plasma damage and restoration of dielectric properties of damaged low-k materials are also discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2013

Tungsten oxide proton conducting films for low-voltage transparent oxide-based thin-film transistors

Tungsten oxide (WO{sub x}) electrolyte films deposited by reactive magnetron sputtering showed a high room temperature proton conductivity of 1.38 Multiplication-Sign 10{sup -4} S/cm with a relative humidity of 60%. Low-voltage transparent W-doped indium-zinc-oxide thin-film transistors gated by WO{sub x}-based electrolytes were self-assembled on glass substrates by one mask diffraction method. Enhancement mode operation with a large current on/off ratio of 4.7 Multiplication-Sign 10{sup 6}, a low subthreshold swing of 108 mV/decade, and a high field-effect mobility 42.6 cm{sup 2}/V s was realized. Our results demonstrated that WO{sub x}-based proton conducting films were promising gate dielectric candidates for portable low-voltage oxide-based devices.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2013

Ambient condition laser writing of graphene structures on polycrystalline SiC thin film deposited on Si wafer

We report laser induced local conversion of polycrystalline SiC thin-films grown on Si wafers into multi-layer graphene, a process compatible with the Si based microelectronic technologies. The conversion can be achieved using a 532 nm CW laser with as little as 10 mW power, yielding {approx}1 {mu}m graphene discs without any mask. The conversion conditions are found to vary with the crystallinity of the film. More interestingly, the internal structure of the graphene disc, probed by Raman imaging, can be tuned with varying the film and illumination parameters, resembling either the fundamental or doughnut mode of a laser beam.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2013

Growth mechanism of photoreduced silver nanostructures on periodically proton exchanged lithium niobate: Time and concentration dependence

Photodeposition of metallic nanostructures onto ferroelectric surfaces, which have been chemically patterned using a proton exchange process, has recently been demonstrated. By varying the molar concentration of the AgNO{sub 3} solution and the illumination time, one can determine the initial nucleation sites, control the rate of nucleation and the height of silver nanostructures formed, and study the mechanisms by which these processes occurs. The nanoparticles are found to deposit preferentially in the boundary between ferroelectric and proton exchanged regions, in an area proton exchanged via lateral diffusion under the masking layer used for chemical patterning, consistent with our previous results. Using a short illumination time (3 min), we are able to determine that the initial nucleation of the silver nanostructure, having a width of 0.17 {+-} 0.02 {mu}m and a height of 1.61 {+-} 0.98 nm, occurs near the edge of the reactive ion etched area within this lateral diffusion region. Over longer illumination times (15 min), we find that the silver deposition has spread to a width of 1.29 {+-} 0.06 {mu}m, extending across the entire lateral diffusion region. We report that at a high molar concentration of AgNO{sub 3} (10{sup -2} M), the amount of silver deposition for 5 min UV illumination is greater (2.88 {+-} 0.58 nm) compared to that at low (10{sup -4} M) concentrations (0.78 {+-} 0.35 nm), however, this is not the case for longer time periods. With increasing illumination time (15 min), experiments at 10{sup -4} M had greater overall deposition, 6.90 {+-} 1.52 nm, compared to 4.50 {+-} 0.76 nm at 10{sup -2} M. For longer exposure times (30 min) at 10{sup -2} M, the nanostructure height is 4.72 {+-} 0.59 nm, suggesting a saturation in the nanostructure height. The results are discussed in terms of the electric double layer that forms at the crystal surface. There is an order of magnitude difference between the Debye lengths for 10{sup -2} and 10{sup -4} M solutions, i.e., 3.04 vs. 30.40 nm, which suggests the Debye length plays a role in the availability of Ag{sup +} ions at the surface.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2013

Tuning of the nucleation field in nanowires with perpendicular magnetic anisotropy

We report on domain nucleation in nanowires consisting of Co/Pt multilayers with perpendicular magnetic anisotropy that are patterned by electron-beam lithography, sputter deposition, and lift-off processing. It is found that the nucleation field can be tuned by changing the geometry of the wire ends. A reduction of the nucleation field by up to 60% is achieved when the wire ends are designed as tips. This contrasts with the behavior of wires with in-plane anisotropy where the nucleation field increases when triangular-pointed ends are used. In order to clarify the origin of the reduction of the nucleation field, micromagnetic simulations are employed. The effect cannot be explained by the lateral geometrical variation but is attributable to a local reduction of the perpendicular anisotropy caused by shadowing effects due to the resist mask during sputter deposition of the multilayer.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2013

Design of an ultrahigh vacuum transfer mechanism to interconnect an oxide molecular beam epitaxy growth chamber and an x-ray photoemission spectroscopy analysis system

We designed a mechanism and the accompanying sample holders to transfer between a VEECO 930 oxide molecular beam epitaxy (MBE) and a PHI Versa Probe X-ray photoemission spectroscopy (XPS) chamber within a multiple station growth, processing, and analysis system through ultrahigh vacuum (UHV). The mechanism consists of four parts: (1) a platen compatible with the MBE growth stage, (2) a platen compatible with the XPS analysis stage, (3) a sample coupon that is transferred between the two platens, and (4) the accompanying UHV transfer line. The mechanism offers a robust design that enables transfer back and forth between the growth chamber and the analysis chamber, and yet is flexible enough to allow transfer between standard sample holders for thin film growth and masked sample holders for making electrical contacts and Schottky junctions, all without breaking vacuum. We used this mechanism to transfer a barium strontium titanate thin film into the XPS analysis chamber and performed XPS measurements before and after exposing the sample to the air. After air exposure, a thin overlayer of carbon was found to form and a significant shift ({approx}1 eV) in the core level binding energies was observed.

36 MATERIALS SCIENCE↗
DOE OSTI2013

On phase diagrams of magnetic reconnection

Recently, “phase diagrams” of magnetic reconnection were developed to graphically organize the present knowledge of what type, or phase, of reconnection is dominant in systems with given characteristic plasma parameters. Here, a number of considerations that require caution in using the diagrams are pointed out. First, two known properties of reconnection are omitted from the diagrams: the history dependence of reconnection and the absence of reconnection for small Lundquist number. Second, the phase diagrams mask a number of features. For one, the predicted transition to Hall reconnection should be thought of as an upper bound on the Lundquist number, and it may happen for considerably smaller values. Second, reconnection is never “slow,” it is always “fast” in the sense that the normalized reconnection rate is always at least 0.01. This has important implications for reconnection onset models. Finally, the definition of the relevant Lundquist number is nuanced and may differ greatly from the value based on characteristic scales. These considerations are important for applications of the phase diagrams. This is demonstrated by example for solar flares, where it is argued that it is unlikely that collisional reconnection can occur in the corona.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2013

Quantitative magnetic imaging at the nanometer scale by ballistic electron magnetic microscopy

We demonstrate quantitative ballistic electron magnetic microscopy (BEMM) imaging of simple model Fe(001) nanostructures. We use in situ nanostencil shadow mask resistless patterning combined with molecular beam epitaxy deposition to prepare under ultra-high vacuum conditions nanostructured epitaxial Fe/Au/Fe/GaAs(001) spin-valves. In this epitaxial system, the magnetization of the bottom Fe/GaAs(001) electrode is parallel to the [110] direction, defining accurately the analysis direction for the BEMM experiments. The large hot-electron magnetoresistance of the Fe/Au/Fe/GaAs(001) epitaxial spin-valve allows us to image various stable magnetic configurations on the as-grown Fe(001) microstructures with a high sensitivity, even for small misalignments of both magnetic electrodes. The angular dependence of the hot-electron magnetocurrent is used to convert magnetization maps calculated by micromagnetic simulations into simulated BEMM images. The calculated BEMM images and magnetization rotation profiles show quantitative agreement with experiments and allow us to investigate the magnetic phase diagram of these model Fe(001) microstructures. Finally, magnetic domain reversals are observed under high current density pulses. This opens the way for further BEMM investigations of current-induced magnetization dynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Nucleation of single GaN nanorods with diameters smaller than 35 nm by molecular beam epitaxy

Nucleation mechanism of catalyst-free GaN nanorod grown on Si(111) is investigated by the fabrication of uniform and narrow (<35 nm) nanorods without a pre-defined mask by molecular beam epitaxy. Direct evidences show that the nucleation of GaN nanorods stems from the sidewall of the underlying islands down to the Si(111) substrate, different from commonly reported ones on top of the island directly. Accordingly, the growth and density control of the nanorods is exploited by a “narrow-pass” approach that only narrow nanorod can be grown. The optimal size of surrounding non-nucleation area around single nanorod is estimated as 88 nm.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2013

Mechanical-plowing-based high-speed patterning on hard material via advanced-control and ultrasonic probe vibration

In this paper, we present a high-speed direct pattern fabrication on hard materials (e.g., a tungsten-coated quartz substrate) via mechanical plowing. Compared to other probe-based nanolithography techniques based on chemical- and/or physical-reactions (e.g., the Dip-pen technique), mechanical plowing is meritorious for its low cost, ease of process control, and capability of working with a wide variety of materials beyond conductive and/or soft materials. However, direct patterning on hard material faces two daunting challenges. First, the patterning throughput is ultimately hindered by the “writing” (plowing) speed, which, in turn, is limited by the adverse effects that can be excited/induced during high-speed, and/or large-range plowing, including the vibrational dynamics of the actuation system (the piezoelectric actuator, the cantilever, and the mechanical fixture connecting the cantilever to the actuator), the dynamic cross-axis coupling between different axes of motion, and the hysteresis and the drift effects related to the piezoelectric actuators. Secondly, it is very challenging to directly pattern on ultra-hard materials via plowing. Even with a diamond probe, the line depth of the pattern via continuous plowing on ultra-hard materials such as tungsten, is still rather small (<0.5 nm), particularly when the “writing” speed becomes high. To overcome these two challenges, we propose to utilize a novel iterative learning control technique to achieve precision tracking of the desired pattern during high-speed, large-range plowing, and introduce ultrasonic vibration of the probe in the normal (vertical) direction during the plowing process to enable direct patterning on ultra hard materials. The proposed approach was implemented to directly fabricate patterns on a mask with tungsten coating and quartz substrate. The experimental results demonstrated that a large-size pattern of four grooves (20 μm in length with 300 nm spacing between lines) can be fabricated at a high speed of ∼5 mm/s, with the line width and the line depth at ∼95 nm and 2 nm, respectively. A fine pattern of the word “NANO” is also fabricated at the speed of ∼5 mm/s.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2013

Binary mixed homopolymer brushes grafted on nanorod particles: A self-consistent field theory study

We employ the self-consistent field theory to study phase structures of brush-rod systems composed of two chemically distinct linear homopolymers. The polymer chains are uniformly grafted on the surface of a nanorod particle of finite length and comparable radius to the polymer radius of gyration. A “masking” technique treating the cylindrical boundary is introduced to solve the modified diffusion equations with an efficient and high-order accurate pseudospectral method involving fast Fourier transform on an orthorhombic cell. A rich variety of structures for the phase separated brushes is predicted. Phase diagrams involving a series of system parameters, such as the aspect ratio of the nanorod, the grafting density, and the chain length are constructed. The results indicate that the phase structure of the mixed brush-rod system can be tailored by varying the grafted chain length and/or the aspect ratio of the rod to benefit the fabrication of polymeric nanocomposites.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Rapid production of large-area deep sub-wavelength hybrid structures by femtosecond laser light-field tailoring

The goal of creation of large-area deep sub-wavelength nanostructures by femtosecond laser irradiation onto various materials is being hindered by the limited coherence length. Here, we report solution of the problem by light field tailoring of the incident beam with a phase mask, which serves generation of wavelets. Direct interference between the wavelets, here the first-order diffracted beams, and interference between a wavelet and its induced waves such as surface plasmon polariton are responsible for creation of microgratings and superimposed nanogratings, respectively. The principle of wavelets interference enables extension of uniformly induced hybrid structures containing deep sub-wavelength nanofeatures to macro-dimension.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2013

Ghost imaging for three-dimensional optical security

Ghost imaging has become increasingly popular in quantum and optical application fields. Here, we report three-dimensional (3D) optical security using ghost imaging. The series of random phase-only masks are sparsified, which are further converted into particle-like distributions placed in 3D space. We show that either an optical or digital approach can be employed for the encoding. The results illustrate that a larger key space can be generated due to the application of 3D space compared with previous works.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗