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Nanowire Thermoelectric Devices

Nanowire thermoelectric devices, now under development, are intended to take miniaturization a step beyond the prior state of the art to exploit the potential advantages afforded by shrinking some device features to approximately molecular dimensions (of the order of 10 nm). The development of nanowire-based thermoelectric devices could lead to novel power-generating, cooling, and sensing devices that operate at relatively low currents and high voltages. Recent work on the theory of thermoelectric devices has led to the expectation that the performance of such a device could be enhanced if the diameter of the wires could be reduced to a point where quantum confinement effects increase charge-carrier mobility (thereby increasing the Seebeck coefficient) and reduce thermal conductivity. In addition, even in the absence of these effects, the large aspect ratios (length of the order of tens of microns diameter of the order of tens of nanometers) of nanowires would be conducive to the maintenance of large temperature differences at small heat fluxes. The predicted net effect of reducing diameters to the order of tens of nanometers would be to increase its efficiency by a factor of .3. Nanowires made of thermoelectric materials and devices that comprise arrays of such nanowires can be fabricated by electrochemical growth of the thermoelectric materials in templates that contain suitably dimensioned pores (10 to 100 nm in diameter and 1 to 100 microns long). The nanowires can then be contacted in bundles to form devices that look similar to conventional thermoelectric devices, except that a production version may contain nearly a billion elements (wires) per square centimeter, instead of fewer than a hundred as in a conventional bulk thermoelectric device or fewer than 100,000 as in a microdevice. It is not yet possible to form contacts with individual nanowires. Therefore, in fabricating a nanowire thermoelectric device, one forms contacts on nanowires in bundles of the order of 10-microns wide. The fill factor for the cross-section of a typical bundle is about 1/2. Nanowires have been grown in alumina templates with pore diameters of 100 and 40 nm.

Borshchevsky, Alexander↗

What it Takes to Compute Highly Accurate Rovibrational Line Lists for Use in Astrochemistry

We review the Best Theory + Reliable High-resolution Experiment (BTRHE) strategy for obtaining highly accurate molecular rovibrational line lists with InfraRed (IR) intensities. The need for highly accurate molecular rovibrational line lists is two-fold: a) assignment of the many rovibrational lines for common stable molecules especially those that exhibit a large amplitude motion, such as NH3, or have a high density of states such as SO2; and b) characterization of the atmospheres of exoplanets which will be one of the main areas of research in astronomy in the coming decades. The first motivation arises due to the need to eliminate lines due to common molecules in an astronomical observation in order to identify lines from new molecules, while the second motivation arises due to the need to obtain accurate molecular opacities in order to characterize the atmosphere of an exoplanet. The BTRHE strategy first consists of using high quality ab initio quantum chemical methods to obtain a global potential energy surface (PES) and dipole moment surface (DMS) that contains the proper physics. The global PES is then refined using a subset of the reliable high-resolution experimental data. The refined PES then gives energy level predictions to an accuracy similar to the reproduction accuracy of the experimental data used in the refinement step in the interpolation region (i.e., within the range of the experimental data used in the refinement step). The accuracy of the energy levels will slowly degrade as they are extrapolated to spectral regions beyond the high-resolution experimental data used in the refinement step. However, because the degradation is slow, the predicted energy levels can be used to assign new high-resolution experiments, and the data from these can then be used in a subsequent refinement step. In this way, the global PES eventually can yield highly accurate energy levels for all desired spectral regions including to very high energies and high J values. We show that IR intensities computed with the BTRHE rovibrational wavefunctions and the DMS can be very accurate provided one has minimized the fitting error of the DMS and tested the completeness of the DMS. Some examples of our work on NH3, CO2, and SO2 are given to highlight the usefulness of the BTRHE strategy and to provide ideas on how to further improve its predictive power in the future. In particular, it is shown how successive refinement steps, once new high-resolution data is available, can lead to PESs that yield highly accurate transition energies to larger spectral regions. The importance of including non-adiabatic corrections to reduce the J-dependence of errors for H-containing molecules is shown with work on NH3. Another very important aspect of the BTRHE approach is the consistency across isotopologues, which allows for highly accurate line lists for any isotopologue once one is obtained for the main isotopologue (which has more high-resolution data available for refinement).

Xinchuan Huang↗

Quantum mechanical calculations to chemical accuracy

The accuracy of current molecular-structure calculations is illustrated with examples of quantum mechanical solutions for chemical problems. Two approaches are considered: (1) the coupled-cluster singles and doubles (CCSD) with a perturbational estimate of the contribution of connected triple excitations, or CCDS(T); and (2) the multireference configuration-interaction (MRCI) approach to the correlation problem. The MRCI approach gains greater applicability by means of size-extensive modifications such as the averaged-coupled pair functional approach. The examples of solutions to chemical problems include those for C-H bond energies, the vibrational frequencies of O3, identifying the ground state of Al2 and Si2, and the Lewis-Rayleigh afterglow and the Hermann IR system of N2. Accurate molecular-wave functions can be derived from a combination of basis-set saturation studies and full configuration-interaction calculations.

Bauschlicher, Charles W., Jr.↗

Low dark current photovoltaic multiquantum well long wavelength infrared detectors

The authors have, for the first time, demonstrated photovoltaic detection for an multiple quantum well (MQW) detector. With a blocking layer, the MQW detector exhibits Schottky I-V characteristics with extremely low dark current and excellent ideality factor. The dark current is 5 times 10(exp -14) A for an 100x100 square micron 10 micron detector at 40 K, 8 to 9 orders of magnitude lower than that of a similar 10 micron MQW detector without blocking layer. The ideality factor is about 1.01 to 1.05 at T = 40 to 80 K. The measured barrier height is consistent with the energy difference between first excited states and ground states, or the peak of spectral response. The authors also, for the first time, report the measured effective Richardson constant (A asterisk asterisk) for the GaAs/AlGaAs heterojunction using this blocking layer structure. The A asterisk asterisk is low approx. 2.3 A/sq cm/K(exp 2).

Wu, C. S.↗

High-Performance LWIR Superlattice Detectors and FPA Based on CBIRD Design

We report our recent efforts on advancing of antimonide superlattice based infrared photodetectors and demonstration of Focal Plane Arrays (FPA) based on a complementary barrier infrared detector (CBIRD) design. By optimizing design and growth condition we succeeded to reduce the operational bias of CBIRD single pixel detector without increase of dark current or degradation of quantum efficiency. We demonstrated a 1024x1024 pixel long-wavelength infrared focal plane array utilizing CBIRD design. An 11.5 ?m cutoff FPA without anti-reflection coating has yielded noise equivalent differential temperature of 53 mK at operating temperature of 80 K, with 300 K background and cold-stop. In addition, we demonstrated 320x256 format FPA based on the n-CBIRD design. The resulting FPAs yielded noise equivalent differential temperature of 26 mK at operating temperature of 80 K, with 300 K background and cold-stop. These results advance state-of-the art of superlattice detectors and demonstrated advantages of CBIRD architecture for realization of FPA.

unipolar barrier↗

Scintillating Quantum Dots for Imaging X-rays (SQDIX) for Aircraft Inspection

Scintillation is the process currently employed by conventional x-ray detectors to create x-ray images. Scintillating quantum dots or nano-crystals (StQDs) are a novel, nanometer-scale material that upon excitation by x-rays, re-emit the absorbed energy as visible light. StQDs theoretically have higher output efficiency than conventional scintillating materials and are more environmental friendly. This paper will present the characterization of several critical elements in the use of StQDs that have been performed along a path to the use of this technology in wide spread x-ray imaging. Initial work on the SQDIX system has shown great promise to create state-of-the-art sensors using StQDs as a sensor material. In addition, this work also demonstrates a high degree of promise using StQDs in microstructured fiber optics. Using the microstructured fiber as a light guide could greatly increase the capture efficiency a StQDs based imaging sensor.

Burke, Eric↗

Computational Study of Field Initiated Surface Reactions for Synthesis of Diamond and Silicon

This project involves using quantum chemistry to simulate surface chemical reactions in the presence of an electric field for nanofabrication of diamond and silicon. A field delivered by a scanning tunneling microscope (STM) to a nanometer scale region of a surface affects chemical reaction potential energy surfaces (PES) to direct atomic scale surface modification to fabricate sub-nanometer structures. Our original hypothesis is that the applied voltage polarizes the charge distribution of the valence electrons and that these distorted molecular orbitals can be manipulated with the STM so as to change the relative stabilities of the electronic configurations over the reaction coordinates and thus the topology of the PES and reaction kinetics. Our objective is to investigate the effect of applied bias on surface reactions and the extent to which STM delivered fields can be used to direct surface chemical reactions on an atomic scale on diamond and silicon. To analyze the fundamentals of field induced chemistry and to investigate the application of this technique for the fabrication of nanostructures, we have employed methods capable of accurately describing molecular electronic structure. The methods we employ are density functional theory (DFT) quantum chemical (QC) methods. To determine the effect of applied bias on surface reactions we have calculated the QC PESs in various applied external fields for various reaction steps for depositing or etching diamond and silicon. We have chosen reactions which are thought to play a role in etching and the chemical vapor deposition growth of Si and diamond. The PESs of the elementary reaction steps involved are then calculated under the applied fields, which we vary in magnitude and configuration. We pay special attention to the change in the reaction barriers, and transition state locations, and search for low energy reaction channels which were inaccessible without the applied bias.

Musgrave, Charles Bruce↗

The ZBOT-NC Experiment - Effects of Non-Condensable Gases on Propellant Tank Pressurization and Pressure Control

Introduction Integral to all phases of NASA’s projected planetary expeditions is affordable and reliable cryogenic fluid storage for use in propellant or life support systems. It is greatly advantageous to develop innovative vent-less pressure control designs based on cooling/mixing of the bulk tank fluid to allow storage of the cryogenic fluid with zero or reduced boil-off. The presence of noncondensable gases, can interfere with the condensation at the interface impacting tank pressure control during subcooled jet mixing especially in microgravity. The Zero-Boil-Off Tank (ZBOT) Experiments are a series of small-scale experiments aboard the International Space Station (ISS) that use a transparent volatile simulant fluid in a transparent sealed tank to delineate various fundamental fluid flow, heat and mass transport, and phase change phenomena associated with storage tank pressurization and pressure control in microgravity. The ZBOT-1 experiment was performed on the ISS 2017-2018 timeframe and collected data to validate a state-of-the-art CFD model for tank pressurization and pressure control for a pure system. The ZBOT-NC Experiment is the second experiment in the series to be performed on the International Space Station (ISS) in 2025. Its goal is to investigate the effects of noncondensable gases on interfacial evaporation and condensation during self -pressurization and jet-mixing pressure control in microgravity for a two-component system. Materials & Methods In this work, we will describe the detailed features of the ZBOT-NC experimental hardware and Diagnostics that includes nonintrusive Quantum Dot Thermometry (QDT) for whole field temperature measurement. All microgravity pressurization and pressure control tests will be performed for both the pure and the two-components systems with Xenon and Neon as the two noncondensable gases spanning small and large molecular weights and sizes. The two-phase CFD model that is developed as part of the project will be also presented and discussed. Results Ground-based pressurization and jet mixing experiments and CFD simulation results are compared to each other to validate both the fidelity of the CFD model predictions and the accuracy of the QDT measurement. Model simulations for noncondensable gas effects will also be compared against large Cryogenic LH2-GHe experiments to indicate the noncondensable gas effects on tank pressure control during jet mixing in 1G. Finally, CFD results will be presented to predict the effects of the noncondensable gas during subcooled jet mixing during the ZBOT-NC in advance of the microgravity experiment in 2015.

Evaporation Condensation↗

The Zero-Boil-Off Tank Experiment: the Effects of Non-Condensable Gases on Pressurization and Pressure Control of Propellant Tanks in Microgravity

Introduction Integral to all phases of NASA’s projected planetary expeditions is affordable and reliable cryogenic fluid storage for use in propellant or life support systems. It is greatly advantageous to develop innovative vent-less pressure control designs based on cooling/mixing of the bulk tank fluid to allow storage of the cryogenic fluid with zero or reduced boil-off. The presence of noncondensable gases, can interfere with the condensation at the interface impacting tank pressure control during subcooled jet mixing especially in microgravity. The Zero-Boil-Off Tank (ZBOT) Experiments are a series of small-scale experiments aboard the International Space Station (ISS) that use a transparent volatile simulant fluid in a transparent sealed tank to delineate various fundamental fluid flow, heat and mass transport, and phase change phenomena associated with storage tank pressurization and pressure control in microgravity. The ZBOT-1 experiment was performed on the ISS 2017-2018 timeframe and collected data to validate a state-of-the-art CFD model for tank pressurization and pressure control for a pure system. The ZBOT-NC Experiment is the second experiment in the series to be performed on the International Space Station (ISS) in 2025. Its goal is to investigate the effects of noncondensable gases on interfacial evaporation and condensation during self -pressurization and jet-mixing pressure control in microgravity for a two-component system. Materials & Methods In this work, we will describe the detailed features of the ZBOT-NC experimental hardware and Diagnostics that includes nonintrusive Quantum Dot Thermometry (QDT) for whole field temperature measurement. All microgravity pressurization and pressure control tests will be performed for both the pure and the two-components systems with Xenon and Neon as the two noncondensable gases spanning small and large molecular weights and sizes. The two-phase CFD model that is developed as part of the project will be also presented and discussed. Results Ground-based pressurization and jet mixing experiments and CFD simulation results are compared to each other to validate both the fidelity of the CFD model predictions and the accuracy of the QDT measurement. Model simulations for noncondensable gas effects will also be compared against large Cryogenic LH2-GHe experiments to indicate the noncondensable gas effects on tank pressure control during jet mixing in 1G. Finally, CFD results will be presented to predict the effects of the noncondensable gas during subcooled jet mixing during the ZBOT-NC in advance of the microgravity experiment in 2015.

Evaporation Condensation↗

A complex T-matrix derivation of a resonance amplitude

Time-dependent perturbation-theory techniques are used to derive a compact expression, valid to arbitrary order and displaying time dependence explicitly, for a quantum-mechanical transition amplitude applicable to the description of resonances. A solution representing a complex-energy generalization of the usual real-energy (nonresonant) amplitude is obtained and generalized to all orders by introducing a complex-energy T matrix. Applications to physical problems such as the extension of the Fermi golden rules to resonances (Norbury and Deutchman, 1984); pion production in relativistic nucleus-nucleus collisions (Norbury et al., 1985); and the determination of resonant cross sections for nuclear, atomic, or molecular processes involving the formation and decay of intermediate discrete or continuum states are indicated.

Norbury, J. W.↗

Rotational and vibrational effects in the E 1Sigma(+)g-X 1Sigma(+)g two-photon transitions of H2, HD, and D2

This paper reports a theoretical study of the dependence of the E 1Sigma(+)g v(E) = 0, J-X 1Sigma(+)g, v(X),J transitions in H2, HD, and D2 on the initial vibrational and rotational quantum numbers v(X) and J. The results demonstrate the concerted nature of two-photon transitions. The magnitude of the two-photon transition moment increases with v(X) between 0 and 2, then decreases rapidly at higher v(X). The J dependence of the two-photon transition moment is most strongly affected by the changes in vibrational wave functions caused by centrifugal distortion of the vibrational potential. It is shown that the J variation can be used as a probe of resonant tunneling in the double-well potential of the E,F state.

Huo, Winifred M.↗

The Coupling of ClONO2, ClO, and NO2 in the Lower Stratosphere From in Situ Observations Using the NASA ER-2 Aircraft

The first in situ measurements of ClONO2 in the lower stratosphere, acquired using the NASA ER-2 aircraft during the Polar Ozone Loss in the Arctic Region in Summer (POLARIS) mission, are combined with simultaneous measurements of ClO, NO2, temperature, pressure, and the calculated photolysis rate coefficient (J(sub ClONO2)) to examine the balance between production and loss of ClONO2. The observations demonstrate the ClONO2 photochemical steady state measurement, [ClONO2](sup PSS) = k[ClO][No2]/J(sub ClONO2), is in good agreement with the direct measurement, [ClONO2](sup MEAS). For the bulk of the data (80%), where T > 220 K and latitudes > 45 N, [ClONO2](sup PPS) = 1.15 +/- 0.36(1-sigma)[ClONO2](sup MEAS), while for T< 220 K and latitudes < 45 N, the result is somewhat less at 1.01 +/- 0.30. The cause of the temperature and/or latitude trend is unidentified. These results are independent of solar zenith angle and air density, thus there is no evidence in support of a pressure-dependent quantum yield for photodissociation of ClONO2 at wavelengths > 300 nm. These measurements confirm the mechanism by which active nitrogen (NOx = NO + NO2) controls the abundance of active chlorine (Clx = ClO + Cl) in the stratosphere.

Stimpfle, R. E.↗

Homodyning and heterodyning the quantum phase

The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.

Dariano, Giacomo M.↗

Copper-Multiwall Carbon Nanotubes and Copper-Diamond Composites for Advanced Rocket Engines

This paper reports on the research effort to improve the thermal conductivity of the copper-based alloy NARloy-Z (Cu-3 wt.%Ag-0.5 wt.% Zr), the state-of-the-art alloy used to make combustion chamber liners in regeneratively-cooled liquid rocket engines, using nanotechnology. The approach was to embed high thermal conductivity multiwall carbon nanotubes (MWCNTs) and diamond (D) particles in the NARloy-Z matrix using powder metallurgy techniques. The thermal conductivity of MWCNTs and D have been reported to be 5 to 10 times that of NARloy-Z. Hence, 10 to 20 vol. % MWCNT finely dispersed in NARloy-Z matrix could nearly double the thermal conductivity, provided there is a good thermal bond between MWCNTs and copper matrix. Quantum mechanics-based modeling showed that zirconium (Zr) in NARloy-Z should form ZrC at the MWCNT-Cu interface and provide a good thermal bond. In this study, NARloy-Z powder was blended with MWCNTs in a ball mill, and the resulting mixture was consolidated under high pressure and temperature using Field Assisted Sintering Technology (FAST). Microstructural analysis showed that the MWCNTs, which were provided as tangles of MWCNTs by the manufacturer, did not detangle well during blending and formed clumps at the prior particle boundaries. The composites made form these powders showed lower thermal conductivity than the base NARloy-Z. To eliminate the observed physical agglomeration, tangled multiwall MWCNTs were separated by acid treatment and electroless plated with a thin layer of chromium to keep them separated during further processing. Separately, the thermal conductivities of MWCNTs used in this work were measured, and the results showed very low values, a major factor in the low thermal conductivity of the composite. On the other hand, D particles embedded in NARloy-Z matrix showed much improved thermal conductivity. Elemental analysis showed migration of Zr to the NARloy-Z-D interface to form ZrC, which appeared to provide a low contact thermal resistance. These results are consistent with the quantum mechanics-based model predictions. NARloy-Z-D composites have relatively high thermal conductivities and are promising for further development.

Bhat, Biliyar N.↗

Copper Multiwall Carbon Nanotubes and Copper-Diamond Composites for Advanced Rocket Engines

This paper reports on the research effort to improve the thermal conductivity of the copper-based alloy NARloy-Z (Cu-3 wt.%Ag-0.5 wt.% Zr), the state-of-the-art alloy used to make combustion chamber liners in regeneratively-cooled liquid rocket engines, using nanotechnology. The approach was to embed high thermal conductivity multiwall carbon nanotubes (MWCNTs) and diamond (D) particles in the NARloy-Z matrix using powder metallurgy techniques. The thermal conductivity of MWCNTs and D have been reported to be 5 to 10 times that of NARloy-Z. Hence, 10 to 20 vol. % MWCNT finely dispersed in NARloy-Z matrix could nearly double the thermal conductivity, provided there is a good thermal bond between MWCNTs and copper matrix. Quantum mechanics-based modeling showed that zirconium (Zr) in NARloy-Z should form ZrC at the MWCNT-Cu interface and provide a good thermal bond. In this study, NARloy-Z powder was blended with MWCNTs in a ball mill, and the resulting mixture was consolidated under high pressure and temperature using Field Assisted Sintering Technology (FAST). Microstructural analysis showed that the MWCNTs, which were provided as tangles of MWCNTs by the manufacturer, did not detangle well during blending and formed clumps at the prior particle boundaries. The composites made form these powders showed lower thermal conductivity than the base NARloy-Z. To eliminate the observed physical agglomeration, tangled multiwall MWCNTs were separated by acid treatment and electroless plated with a thin layer of chromium to keep them separated during further processing. Separately, the thermal conductivities of MWCNTs used in this work were measured, and the results showed very low values, a major factor in the low thermal conductivity of the composite. On the other hand, D particles embedded in NARloy-Z matrix showed much improved thermal conductivity. Elemental analysis showed migration of Zr to the NARloy-Z-D interface to form ZrC, which appeared to provide a low contact thermal resistance. These results are consistent with the quantum mechanics-based model predictions. NARloy-Z-D composites have relatively high thermal conductivities and are promising for further development.

Bhat, Biliyar N.↗

Effects of angular-momentum-changing collisions on dielectronic satellite spectra

A generalization of the low-density corona-model theory of the intensities of satellite lines of the resonance lines emitted by multiply charged helium-like ions in the X-ray spectra of high-temperature plasmas is developed in order to take account of electron collisional transitions between close autoionizing levels. A formalism for determining the population densities of the autoionizing levels and the satellite line intensities is presented, and the quantum mechanical calculation of autoionization, radiative decay and electron-induced collisional transitions is discussed. Generalized emission rate coefficients corresponding to dielectronic recombination and inner-shell-electron collisional excitation are tabulated as functions of electron density and temperature. The procedure is then used to illustrate the effects of angular-momentum-changing electron collisions on the intensities of the satellites of the 1s2p 1P to 1s2 1S resonance line of the ions Al XII and Si XIII. It is concluded that the collisional transitions have their most significant effect on the intensities of satellite transitions from metastable autoionizing states, which may be useful in determining plasma densities.

Jacobs, V. L.↗

Examination of Mars2020 Shock-Layer Conditions via Infrared Laser Absorption Spectroscopy of CO2 and CO

Amid-infrared laser absorption diagnostic was deployed to study a simulated Mars2020 shock layer in the Electric Arc Shock Tube (EAST) facility at NASA’s Ames Research Center. Rapid RF-diplexing techniques enabled quantitative temperature and number density measurements of CO2 and CO with μs-resolution over an incident shock velocity range of 1.39 – 3.75 km/s. Two interband cascade lasers were utilized at 4.17 and 4.19 μm to resolve eight CO2 asymmetric stretch fundamental band (𝜈3) transitions from two different vibrational levels: 0000 (ground state) and 0110 (first excited bending mode). The probed rotational levels span across J” = 58 to J” = 140. Results are compared to DPLR simulations of the shock layer using kinetic mechanisms of Johnston et al. and Cruden et al. At shock velocities below 3.1 km/s, the agreement between the measurements and the Johnston mechanism is typically within 5% for temperature and within 10% for number density. At shock velocities above 3.1 km/s, the CO2 measurement becomes sensitive to a thin boundary layer and corrections of this effect are presented. On test cases with enough energy to dissociate CO2, a quantum cascade laser scanned the P(2, 20), P(0, 31), and P(3, 14) transitions of the CO fundamental band at 4.98 μm. CO formation rate is measured to be close to the Johnston kinetic mechanism at low velocities, and then trending towards the Cruden kinetic mechanism at high velocities. On a few low velocity test cases, rovibrational relaxation of the Martian atmosphere is probed with μs resolution.

Laser Absorption↗

Numerical Studies of Properties of Confined Helium

We carry out state of the art simulations of properties of confined liquid helium near the superfluid transition to a degree of accuracy which allows to make predictions for the outcome of fundamental physics experiments in microgravity. First we report our results for the finite-size scaling behavior of heat capacity of superfluids for cubic and parallel-plate geometry. This allows us to study the crossover from zero and two dimensions to three dimensions. Our calculated scaling functions are in good agreement with recently measured specific heat scaling functions for the above mentioned geometries. We also present our results of a quantum simulation of submonolayer of molecular hydrogen deposited on an ideal graphite substrate using path-integral quantum Monte Carlo simulation. We find that the monolayer phase diagram is rich and very similar to that of helium monolayer. We are able to uncover the main features of the complex monolayer phase diagram, such as the commensurate solid phases and the commensurate to incommensurate transition, in agreement with the experiments and to find some features which are missing from the experimental analysis.

Manousakis, Efstratios↗