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DOE OSTI2025

SCGSR Final Report

Dual phase noble liquid time projection chambers (TPCs) have an unprecedented sensitivity towards dark matter searches and neutrinos, owing to their ability to detect single electrons. When a particle deposits energy into one of these detectors, electrons are released and drifted into a gaseous region of high electric field. In the gas, the electrons are accelerated to produce extra light into what is called an S2 pulse. This light is far easier to detect than the current induced from a single electron. The two widely used elements for noble liquid detectors are Xe and Ar. Xe has a larger chance of being hit by neutrinos or dark matter, but ends with a lower (i.e. harder to detect) energy when struck. However, Ar light is hard to reflect and detect due to its short wavelength which is absorbed by most materials. While one may use TPB to re-emit this light at a longer wavelength, spatial variations of TPB thickness can smear the amount of re-emitted light, and thus cause a poor S2 energy resolution and spatial resolution. Furthermore, a single electron pulse in Ar is spread over a much longer time than one in Xe, which makes it harder to identify single electrons in Ar. The goal of Xe doping is to combine the best aspects of Ar with the best aspects of Xe. More specifically, this means a detector with a larger yield of photons and electrons produced per energy deposition, an ability to collect more of the light given by those signals, and a narrower pulse from the electron signal. During this project, we developed and operated a detector that can overcome the thermodynamic challenges of Xe doping – namely, preventing Xe from freezing. In the first experimental run during this award period, we found that we can achieve at least 5% of Xe in LAr, and measure the Xe concentration in the gas. In the second run, we installed our detector and saw the first S2 pulses in a dual phase Xe doped LAr TPC. With the data we took, we quantified the effect of Xe doping on the detectability of wavelength-shifted light produced in a Xe doped Ar mixture, as well as the single electron pulse shape. While the Ar light has successfully been wavelength shifted, we learned that much to our surprise, Xe doping actually makes the S2 pulse wider before it makes it narrower.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2000

Innovative Research Program: Supershields for Gamma-Ray Astronomy

The supershield project evaluated the importance of novel shield configurations for suppressing neutron induced background in new classes of gamma-ray detectors such as CZT. The basic concept was to use a two-part shield. The outer shield material heavily moderates the incoming neutron spectrum. This moderated neutron beam is then more easily absorbed by the inner material, which is an efficient neutron absorber. This approach is, in principle, more efficient than that in previous attempts to make neutron shields. These previous attempts involved biatomic, monlithic shields (eg. LiH) in which the shield consisted of a single material but with two types of atoms - one for moderating and one for absorbing. The problem with this type of monolithic shield is that moderating neutrons, without the efficient absorption of them, leads to the leakage into the detector of neutrons with a low energy component (approx. 10-100 KeV). These energy neutrons are particularly problematic for many types of detectors. The project was roughly divided into phases. In the first phase we attempted to carefully define the neutron source function incident on any space instrument. This is essential since the design of any shield depends on the shape of the incident neutron spectrum. We found that approximations commonly used in gamma-ray astronomy for photon background is inadequate. In addition, we found that secondary neutrons produced in any passive shield, and dominated by inelastic neutron scattering, are far more important than background due to neutron activation. The second phase of our work involved design of supershield geometries (one and three dimensional) in order to compare different shield configurations and materials for their effectiveness as neutron shields. Moreover we wanted to compare these supershields with previous neutron shields to confirm the performance differences between the supershield (two material) and monolithic (one material) designs and to understand the physics origins of these differences more clearly. The third phase of the supershield program involved the benchmarking of the supershield designs through direct experimental verification. This required fabricating various supershields and exposing them to beams of neutrons to directly characterize their performance. With explicit verification that our modeling procedures can be used with confidence, we are now in a position to design shields for realistic space geometries. Using the supershield modeling capacity developed as part of this program we are attempting to evaluate their utility for a specific proposed mission--the Energetic X-ray Imaging Survey Telescope (EXIST). It is anticipated that this experiment, which is limited by internal background at high energies, might benefit from a neutron shield.

Hailey, Charles J.↗

DOE OSTI2024

Statistical modelling and Bayesian inversion for a Compton imaging system: application to radioactive source localization

Abstract This paper presents a statistical forward model for a Compton imaging system, called Compton imager. This system, under development at the University of Illinois Urbana Champaign, is a variant of Compton cameras with a single type of sensors which can simultaneously act as scatterers and absorbers. This imager is convenient for imaging situations requiring a wide field of view. The proposed statistical forward model is then used to solve the inverse problem of estimating the location and energy of point-like sources from observed data. This inverse problem is formulated and solved in a Bayesian framework by using a Metropolis within Gibbs algorithm for the estimation of the location, and an expectation-maximization algorithm for the estimation of the energy. This approach leads to more accurate estimation when compared with the deterministic standard back-projection approach, with the additional benefit of uncertainty quantification in the low photon imaging setting.

Tarpau, Cécilia (ORCID:0000000286539490)↗

DOE OSTI2022

Development of an operando characterization stage for multi-modal synchrotron x-ray experiments

It is widely accepted that micro- and nanoscale inhomogeneities govern the performance of many thin-film solar cell absorbers. These inhomogeneities yield material properties (e.g., composition, structure, and charge collection) that are challenging to correlate across length scales and measurement modalities. The challenge is compounded if a correlation is sought during device operation or in conditions that mimic aging under particular stressors (e.g., heat and electrical bias). Correlative approaches, particularly those based on synchrotron x-ray sources, are powerful since they can access several material properties in different modes (e.g., fluorescence, diffraction, and absorption) with minimal sample preparation. Small-scale laboratory x-ray instruments have begun to offer multi-modality but are typically limited by low x-ray photon flux, low spatial resolution, or specific sample sizes. To overcome these limitations, a characterization stage was developed to enable multi-scale, multi-modal operando measurements of industrially relevant photovoltaic devices. The stage offers compatibility across synchrotron x-ray facilities, enabling correlation between nanoscale x-ray fluorescence microscopy, microscale x-ray diffraction microscopy, and x-ray beam induced current microscopy, among others. Furthermore, the stage can accommodate device sizes up to 25 × 25 mm 2 , offering access to multiple regions of interest and increasing the statistical significance of correlated properties. The stage materials can sustain humid and non-oxidizing atmospheres, and temperature ranges encountered by photovoltaic devices in operational environments (e.g., from 25 to 100 °C). As a case study, we discuss the functionality of the stage by studying Se-alloyed CdTe photovoltaic devices aged in the stage between 25 and 100 °C.

14 SOLAR ENERGY↗

DOE OSTI2020

Evolution from Tunneling to Hopping Mediated Triplet Energy Transfer from Quantum Dots to Molecules

Efficient energy transfer is particularly important for multiexcitonic processes like singlet fission and photon upconversion. Observation of the transition from short-range tunneling to long-range hopping during triplet exciton transfer from CdSe nanocrystals to anthracene is reported here. This is firmly supported by steady- state photon upconversion measurements, a direct proxy for the efficiency of triplet energy transfer (TET), as well as transient absorption measurements. When phenylene bridges are initially inserted between a CdSe nanocrystal donor and anthracene acceptor, the rate of TET decreases exponentially, commensurate with a decrease in the photon upconversion quantum efficiency from 11.6% to 4.51% to 0.284%, as expected from a tunneling mechanism. However, as the rigid bridge is increased in length to 4 and 5 phenylene units, photon upconversion quantum efficiencies increase again to 0.468% and 0.413%, 1.5–1.6 fold higher than that with 3 phenylene units (using the convention where the maximum upconversion quantum efficiency is 100%). This suggests a transition from exciton tunneling to hopping, resulting in relatively efficient and distance-independent TET beyond the traditional 1 nm Dexter distance. Transient absorption spectroscopy is used to confirm triplet energy transfer from CdSe to transmitter, and the formation of a bridge triplet state as an intermediate for the hopping mechanism. Lastly, this first observation of the tunneling-to-hopping transition for long-range triplet energy transfer between nanocrystal light absorbers and molecular acceptors suggests that these hybrid materials should further be explored in the context of artificial photosynthesis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

An Exceptionally Powerful, Radiatively Driven Ultrafast Outflow in the Rapidly Accreting AGN RE J1034+396

We report the analysis of ∼1 Ms of XMM-Newton observations of the rapidly accreting active galactic nucleus RE J1034+396. The 0.3–9 keV EPIC-pn spectra are well described by a model consisting of steep continuum emission from the corona accompanied by relativistically blurred reflection from a highly ionized accretion disk. The source is known to exhibit strong excess soft X-ray emission, which we show is well represented by thermal disk photons Comptonized by a warm plasma spanning the inner accretion flow. Additionally, the EPIC-pn data provide compelling evidence (ΔC ∼ 60 for four additional parameters) for the presence of an ultrafast outflow (UFO) with a line-of-sight velocity v/c = 0.307$^{+0.001}_{-0.005}$, and an emission signature consistent with reflection of the corona from modestly ionized, outflowing gas. The simultaneous 0.5–2.5 keV RGS spectra show clear absorption lines. Modelling of these data confirms the presence of the UFO and constrains its equivalent hydrogen column density, log N H /(atom cm −2 ) = 21.7$^{+0.1}_{-0.2}$. The RGS data also reveal at least two warm absorber components with a modest outflow velocity (1680$^{+40}_{-50}$ km s -1 ). The measured properties and time evolution of the UFO in RE J1034+396 suggest that it is formed from collisionally ionized plasma, launched from the disk surface and accelerated by radiation pressure. The high terminal velocity and substantial absorbing column density imply that the outflow carries sufficient momentum and energy to transform its environment, being capable of driving out essentially all dust and gas it interacts with along the line of sight, even if the AGN were initially surrounded by a Compton-thick absorber.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Structural Stability of Tin-Lead Halide Perovskite Solar Cells

One of the key features that makes halide perovskite solar cells such an attractive and intensely researched photovoltaic (PV) technology, is the tunability of the bandgap of these halide perovskite materials [1], [2]. Shortly after it was established that the bandgap could be increased to 1.7-1.8 eV, investigations into the application for silicon based tandem solar cells launched. Within a few years, conversion efficiencies of up to 28 % have been reported [3]. Concurrently, the interest in commercial application of this technology emerged and have rapidly increased. Whereas there are advantages to piggy backing on an established PV technology such as silicon, efforts to develop all-perovskite tandem solar cells are of considerable interest. Partly substituting the metal cation in the ABX3 perovskite enables the low bandgap absorbers required for an all-perovskite tandem solar cell. One common approach is to partially substitute the lead with tin, however these materials often suffered from poor stability. Therefore the development of an efficient, stable mixed tin-lead perovskite is key to enabling all-perovskite tandem solar cells.

degradation↗

NASA NTRS1996

3C 279 Multiwavelength Monitoring: The Ground-Based Campaign - 2

The optically violently variable quasar 3C 279 was monitored simultaneously from radio to gamma-ray frequencies in 1992 December - 1993 January. We report a detailed study of the ground-based results from radio to optical wavelengths. These data show that 3C 279 has a typical blazar spectrum, slightly rising at radio frequency and then progressively steeper above a first turnover frequency between 37 and 90 GHz. In the millimeter wavelength region, a simple power law is not an adequate description of the spectrum. We suggest that the millimeter "shoulder" corresponds to an additional emission component, self-absorbed between 150 and 375 GHz, possibly associated with the detachment of a new Very Long Base Interferometry (VLBI) knot and with the start of radio flare. A flux increase of 20% over 20 days was observed at 37 and 90 GHz, while contemporaneously the R-band flux doubled in about two weeks. The lack of strong variability in contemporaneous X-ray light curves (possible X-ray variations are less than 30%) implies no direct (i.e., zero lag) correlation between the optical and X-ray fluxes. If X-rays are produced by inverse-Compton scattering of relativistic electrons on some seed photons, the above results exclude that the observed optical photons are the seeds and/or that the relativistic electrons radiating via synchrotron in the optical band are responsible for the scattering to X-ray energies. We suggest that the X-rays are instead produced through the inverse-Compton process by electrons of lower energy, which radiate via synchrotron in the radio to millimeter wave bands and which scatter either on the synchrotron photons themselves or on external photons.

Grandi, P.↗

NASA NTRS1993

Iron, Cobalt, and Nickel in SN 1987A

During the first several weeks after the explosion of SN 1987A, Fe/Co/Ni clumps, containing approx. 1% of the mass of the supernova envelope, absorbed most of the energy release by Ni-56 and Co-56 decay. As a result, the clumps expanded relative to the substrate, forming a 'nickel bubble' of low-density Fe/Co/Ni. Later the clumps captured approx. 10% of the radioactive luminosity of gamma rays and positrons. Assuming that these elements are not mixed microscopically with other elements, we find that the clumps must occupy approx. greater than 30% of the volume of the emitting region (radial velocity approx. less than 2500 km/s). The result indicates that the emission at late times is dominated by an extra source of heating and ionization, most likely photoionization by two-photon continuum from metastable helium in the gas surrounding the clumps. The resulting 'frothy' structure, consisting of bubbles of low-density Fe surrounded by higher-density filaments of H, He, and other elements, will persist and may be seen in the spectra and structure of supernova remnants.

Li, Hongwei↗

DOE OSTI2022

DEVELOPMENT OF A QUANTUM ELECTRON BEAM DIAGNOSTIC APPARATUS

Spatial properties of electron beams are the essential parameters needed in particle physics and accelerator research. Beam characterization using optical detection methods through the interaction between photons and electrons such as Compton scattering, or electro-optic sampling of the electric field of electrons have been extensively explored and successfully implemented as non-invasive diagnostics at various accelerator facilities. However, such methods often suffer from inherently low sensitivity. Here we present the study of a new type of electron beam diagnostic device for direct optical imaging of electron beams at various energy levels. The concept is based on the extremely high sensitivity of atoms, prepared in a specific "dark superposition" quantum state, to the external electric or magnetic perturbations induced by the passing charged particle, which allows atoms to change polarization of the probe light or absorb probe light and to fluoresce, enabling direct 3D imaging of the charged particles with high resolution. We report our recent experiment results and the design effort on a compact apparatus intended to be tested with the relativistic electron beams at Jefferson Laboratory.

Zhang, S.↗

NASA NTRS1994

Evidence for X-ray flux and spectral modulation by absorption in NGC 6814. 1: The nature of the most rapid variability

The Seyfert 1 galaxy NGC 6814 was observed using the Japanese X-ray astronomy satellite Ginga in 1990 April and October. The rapid variability characteristically associated with this source was reconfirmed. Specifically, three dips were found in the April observation in which the flux dropped to nearly zero in approximately 300 s. The doubling timescale was approximately 50 s. A similar but separated drop and rise in flux was observed in the October data, different in that the flux did not decrease completely to zero. A detailed analysis of the data around the structures of most rapid variability found spectral variability and lags in flux between different energy bands. Lags were on the order of a few to tens of seconds for the April data, and on the order of tens to a couple of hundred seconds for the October data. The sense of the lags was such that during flux decreases the hard flux lagged, while during flux increases the soft flux lagged. Associated significant apparent hardening of the spectrum at low flux was observed in the April data. Apparent hardening of the spectrum also occurred in the October flux decrease, to a photon index of GAMMA = 0.85; however, the spectrum softened at lowest flux to the index of the predecrease level, GAMMA = 1.54. In the April dips, the line flux was found to decrease significantly. A marginal decrease in line flux was also observed in the October data. The variability of the line flux reconfirmed the result of Kunieda et al. (1990), who found that the line production region must be within approximately 300 light-seconds from the source. To explain the observational results, a variable-absorption model was proposed, in which the column density was assumed to vary as a function of time. Other physical processes, including intrinsic spectral changes and warm absorber models, which could account for the fastest variability, could be ruled out by the results of the data analysis. A primary constraint found for geometrical models is that the material which is doing the absorption must have relatively low ionization (xi less than 100). This result leads to severe constraints on a general orbiting cloud model, requiring high densities (n approximately 10(exp 16)/cu cm) and sheetlike geometry.

Leighly, Karen↗

DOE OSTI2024

Laminography as a tool for imaging large-size samples with high resolution

Despite the increased brilliance of the new generation synchrotron sources, there is still a challenge with high-resolution scanning of very thick and absorbing samples, such as a whole mouse brain stained with heavy elements, and, extending further, brains of primates. Samples are typically cut into smaller parts, to ensure a sufficient X-ray transmission, and scanned separately. Compared with the standard tomography setup where the sample would be cut into many pillars, the laminographic geometry operates with slab-shaped sections significantly reducing the number of sample parts to be prepared, the cutting damage and data stitching problems. In this work, a laminography pipeline for imaging large samples (>1 cm) at micrometre resolution is presented. The implementation includes a low-cost instrument setup installed at the 2-BM micro-CT beamline of the Advanced Photon Source. Additionally, sample mounting, scanning techniques, data stitching procedures, a fast reconstruction algorithm with low computational complexity, and accelerated reconstruction on multi-GPU systems for processing large-scale datasets are presented. The applicability of the whole laminography pipeline was demonstrated by imaging four sequential slabs throughout an entire mouse brain sample stained with osmium, in total generating approximately 12 TB of raw data for reconstruction.

47 OTHER INSTRUMENTATION↗

NASA NTRS2016

Discovery of X-Ray Emission from the Galactic Supernova Remnant G32.8-0.1 with Suzaku

We present the first dedicated X-ray study of the supernova remnant (SNR) G32.8-0.1 (Kes 78) with Suzaku. X-ray emission from the whole SNR shell has been detected for the first time. The X-ray morphology is well correlated with the emission from the radio shell, while anti-correlated with the molecular cloud found in the SNR field. The X-ray spectrum shows not only conventional low-temperature (kT approximately 0.6 kiloelectronvolts) thermal emission in a nonequilibrium ionization state, but also a very high-temperature (approximately 3.4 kiloelectronvolts) component with a very low ionization timescale (approximately 2.7 times 10 (sup 9) per cubic centimeter per second), or a hard nonthermal component with a photon index Gamma approximately equal to 2.3. The average density of the low-temperature plasma is rather low, of the order of 10 (sup -3) - 10 (sup -2) per cubic centimeter, implying that this SNR is expanding into a low-density cavity. We discuss the X-ray emission of the SNR, also detected in teraelectronvolts with H.E.S.S. (High Energy Stereoscopic System), together with multi-wavelength studies of the remnant and other gamma-ray emitting SNRs, such as W28 and RCW 86. Analysis of a time-variable source, 2XMM J185114.3-000004, found in the northern part of the SNR, is also reported for the first time. Rapid time variability and a heavily absorbed hard-X-ray spectrum suggest that this source could be a new supergiant fast X-ray transient.

Petre, Robert↗

DOE OSTI2021

Synthesis and Photophysical Properties of Light-Harvesting Gold Nanoclusters Fully Functionalized with Antenna Chromophores

The development of efficient light-harvesting systems is important to understand the key aspects of solar-energy conversion processes and to utilize them in various photonic applications. In this study, atomically well-defined gold nanoclusters are reported as a new platform to fabricate artificial light-harvesting systems. An efficient amide coupling method is developed to synthesize water-soluble Au 22 clusters fully protected with pyrene chromophores by taking advantage of their facile phase-transfer reaction. The synthesized Au 22 clusters with densely packed 18 pyrene chromophores (Au 22 –PyB 18 ) exhibit triple-emission in blue, green, and red wavelength regions arising respectively from pyrene monomer, pyrene excimer, and Au 22 emission, producing bright white light emission together. The photoluminescence of Au 22 is enhanced by more than tenfold, demonstrating that pyrenes at the periphery efficiently channel the absorbed energy to the luminescent Au 22 at the center. A combination of femtosecond transient absorption and anisotropy measurements of Au 22 –PyB 18 explicitly reveals three main decay components of 220 fs, 3.5 ps, and 160 ps that can be assigned to energy migration between pyrenes and energy transfer processes from pyrene monomer and excimer to the central Au 22 , respectively.

14 SOLAR ENERGY↗

DOE OSTI2026

Electrochemical Characterization of Photo-Driven Hole-Scavenging by Cadmium Sulfide Quantum Dot-Nitrogenase Biohybrid Complexes

Light-driven biohybrid systems that couple semiconductor nanocrystals with enzymes offer a promising strategy for solar-to-chemical energy-conserving reduction reactions, yet are often limited by inefficient hole scavenging. Hole scavenging is critical for maintaining charge separation in the light-absorbing electron donor molecule and, thus, for sustaining catalytic turnover, making quantification essential to improving system efficiency. Here, we introduce a photoelectrochemical approach for real-time monitoring of hole scavenging in cadmium sulfide (CdS) quantum dot (QD)-nitrogenase MoFe protein biohybrids that catalyze the reduction of H+ and N2. Using hydroquinone (HQ) as the sacrificial electron donor (SED), oxidation of HQ to benzoquinone (BQ) by photogenerated valence-band holes is coupled to the electrochemical reduction of BQ at an electrode, enabling quantification of the hole-scavenging reaction via chronoamperometry. This approach provides a continuous, real-time readout of charge-transfer dynamics under illumination. Systematic variation of light intensity and donor concentration reveals that hole scavenging scales with photon flux and exhibits a non-linear dependence on SED concentration, with an optimum at intermediate HQ concentrations. These results establish chronoamperometry as a powerful tool for probing photoinduced charge transfer, advancing the quantitative understanding of hole-scavenging in biohybrid systems, and offering a generalizable framework for optimizing solar-driven biocatalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Coupled Charge and Radiation Transport Processes in Thermophotovoltaic and Thermoradiative Cells

Accurate modeling of charge transport and both thermal and luminescent radiation is crucial to the understanding and design of radiative thermal energy converters. Charge-carrier dynamics in semiconductors are well-described by the Poisson-drift-diffusion equations, and thermal radiation in emitter-absorber structures can be computed using multilayer fluctuational electrodynamics. These two types of energy flows interact through radiation absorption or luminescence and charge-carrier generation or recombination. However, past research has typically only assumed limited interaction, with thermal radiation absorption as an input for charge-carrier models to predict device performance. To examine this assumption, we develop a fully coupled iterative model of charge and radiation transport in semiconductor devices, and we use our model to analyze near-field and far-field GaSb thermophotovoltaic and thermoradiative systems. By comparing our results with past methods that do not consider cross-influences between charge and radiation transport, we find that a fully coupled approach is necessary to accurately model photon recycling and near-field enhancement of external luminescence. As these effects can substantially alter device performance, our modeling approach can aid in the design of efficient thermophotovoltaic and thermoradiative systems.

30 DIRECT ENERGY CONVERSION↗

NASA NTRS2010

An Overview of High-Resolution, Non-Dispersive, Imaging Spectrometers for High-Energy Photons

High-resolution x-ray spectroscopy has become a powerful tool for studying the evolving universe. The grating spectrometers on the XMM and Chandra satellites initiated a new era in x-ray astronomy. Despite their successes, there is still need for instrumentation that can provide higher spectral resolution with high throughput in the Fe-K band and for extended sources. What is needed is a non-dispersive imaging spectrometer - essentially a 14-bit x-ray color camera. And a requirement for a nondispersive spectrometer designed to provide eV-scale spectral resolution is a temperature below 0.1 K. The required spectral resolution and the constraints of thermodynamics and engineering dictate the temperature regime nearly independently of the details of the sensor or the read-out technology. Low-temperature spectrometers can be divided into two classes - - equilibrium and non-equilibrium. In the equilibrium devices, or calorimeters, the energy is deposited in an isolated thermal mass and the resulting increase in temperature is measured. In the non-equilibrium devices, the absorbed energy produces quantized excitations that are counted to determine the energy. The two approaches have different strong points, and within each class a variety of optimizations have been pursued. I will present the basic fundamentals of operation and the details of the most successful device designs to date. I will also discuss how the measurement priorities (resolution, energy band, count rate) influence the optimal choice of detector technology.

Kilbourne, Caroline↗

NASA NTRS1990

An efficient and accurate technique to compute the absorption, emission, and transmission of radiation by the Martian atmosphere

CO2 comprises 95 pct. of the composition of the Martian atmosphere. However, the Martian atmosphere also has a high aerosol content. Dust particles vary from less than 0.2 to greater than 3.0. CO2 is an active absorber and emitter in near IR and IR wavelengths; the near IR absorption bands of CO2 provide significant heating of the atmosphere, and the 15 micron band provides rapid cooling. Including both CO2 and aerosol radiative transfer simultaneously in a model is difficult. Aerosol radiative transfer requires a multiple scattering code, while CO2 radiative transfer must deal with complex wavelength structure. As an alternative to the pure atmosphere treatment in most models which causes inaccuracies, a treatment was developed called the exponential sum or k distribution approximation. The chief advantage of the exponential sum approach is that the integration over k space of f(k) can be computed more quickly than the integration of k sub upsilon over frequency. The exponential sum approach is superior to the photon path distribution and emissivity techniques for dusty conditions. This study was the first application of the exponential sum approach to Martian conditions.

Lindner, Bernhard Lee↗