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At least 487 records · Page 27

Eccentric loading of microtensile specimens

Ceramic materials have a lower density than most metals and are capable of performing at extremely high temperatures. The utility of these materials is obvious; however, the fracture strength of brittle materials is not easily predicted and often varies greatly. Characteristically, brittle materials lack ductility and do not yield as other materials. Ceramics materials are naturally populated with microscopic cracks due to fabrication techniques. Upon application of a load, stress concentration occurs at the root of these cracks and fracture will eventually occur at some not easily predicted strength. In order to use ceramics in any application some design methodology must exist from which a component can be placed into service. This design methodology is CARES/LIFE (Ceramics Analysis and Reliability Evaluation of Structures) which has been developed and refined at NASA over the last several decades. The CARES/LIFE computer program predicts the probability of failure of a ceramic component over its service life. CARES combines finite element results from a commercial FE (finite element) package such as ANSYS and experimental results to compute the abovementioned probability of failure. Over the course of several tests CARES has had great success in predicting the life of various ceramic components and has been used throughout industry. The latest challenge is to verify that CARES is valid for MEMS (Micro-Electro Mechanical Systems). To investigate a series of microtensile specimens were fractured in the laboratory. From this data, material parameters were determined and used to predict a distribution of strength for other specimens that exhibit a known stress concentration. If the prediction matches the experimental results then these parameters can be applied to a desired component outside of the laboratory. During testing nearly half of the tensile Specimens fractured at a location that was not expected and hence not captured in the FE model. It has been my duty to investigate the nature of this phenomenon in hopes of finding a better correlation between theory and empirical results. To investigate I built complete FE models of all of the tensile specimens using ANSYS. It is suspected that some misalignment naturally occurs during testing and thus additional bending stresses are present in the specimens. I modeled this eccentric loading and ran several FE trials using ANSYS/PDS (a probabilistic design system in ANSYS). My objective this summer has been familiarize myself with the CARES/LIFE program in hopes of using it in conjunction with ANSYS to help verify that CARES is applicable to MEMS-scale (greater that 1 micron, less than 1 millimeter) components.

Trapp, Mark A.↗

Theoretical Treatment of the X(sup 1)Sigma(sup +), A(sup 1)Sigma(sup +), and B(sup 1)Pi states of LiH

Ab initio self-consistent-field plus configuration-interaction calculations are reported for the X(sup 1)Sigma(sup +), A(sup 1)Sigma(sup +), and B(sup 1)Pi states of LiH using a 22(sugma)12(pi)7(delta) function Slater basis set. The resulting dissociation energies, with the experimental values in parentheses, are D(sub e)(X(sup 1)Sigma(sup +)) = 19 972 (20288)/ cm, and D(sub e)(A(sup 1)Sigma(sup +)) = 9042 (8682)/ cm, and D(sup e)(B(sup 1)Pi) = 284 (288) /cm. This is the first ab initio treatment to quantitatively account for the binding in the B(sup 1)Pi state. Calculated dipole moments and electronic transition moments for the X(sup 1)Sigma(sup+)-A(sup 1)Sigma(sup +), X(sup 1)Sigma(sup +)-B((sup 1)Pi, and A(sup 1)Sigma(sup +)- B(sup 1)PI band systems are in excellent agreement with existing theoretical and experimental data. Radiative transition probabilities and lifetimes, including both the bound-bound and bound-free contributions, are computed for all vibrational levels of the A(sup 1)Sigma(sup+) and B(sup 1)Pi states. Comparison with previous results using experimentally based potentials provides insight into the sensitivity of the radiative lifetimes to the detailed nature of the uppermost region of the potentials. Our calculated lifetimes for the lower vibrational levels of the A(sup 1)Sigma(sup +) state are within the experimental uncertainty. Our calculated lifetimes for the three vibrational levels of the B(sup 1)Pi state are in excellent agreement with those of Zemke and Stwalley (values in parentheses), increasing with (upsilon)' from 11.3 (11.3) nsec at (upsilon)' = 0, to 17.0 (17.0) nsec at (upsilon)' = 1, and then to 23.5 (24.0) nsec at (upsilon)' = 2.

Partridge, Harry↗

Studies of Tropical/Mid-Latitude Exchange Using UARS Observations

At the time this proposal was submitted, recent publications had suggested an important role for transport of midlatitude air into the tropical lower stratosphere. Most of these studies had employed data that gave only a time-averaged picture, making it difficult to determine the nature of the transport processes responsible for the observed behavior. We proposed to analyze observations of long-lived trace gases, such as nitric acid, methane, nitrous oxide, and chlorofluorocarbons, made from the Upper Atmosphere Research Satellite, to investigate the seasonal behavior of mixing between the midlatitudes and tropics. We planned to construct probability distributions of the concentrations of these species over small altitude ranges and to compare them to expectations based on modeled mean concentrations and knowledge of instrument precision. Differences from expectation were to be analyzed with respect to meteorological parameters to determine whether wave activity may have induced apparent mixing.

Avallone, Linnea↗

Development of a New Data Tool for Computing Launch and Landing Availability with Respect to Surface Weather

The Marshall Space Flight Center Natural Environments Branch has a long history of expertise in the modeling and computation of statistical launch availabilities with respect to weather conditions. Their existing data analysis product, the Atmospheric Parametric Risk Assessment (APRA) tool, computes launch availability given an input set of vehicle hardware and/or operational weather constraints by calculating the climatological probability of exceeding the specified constraint limits, APRA has been used extensively to provide the Space Shuttle program the ability to estimate impacts that various proposed design modifications would have to overall launch availability. The model accounts for both seasonal and diurnal variability at a single geographic location and provides output probabilities for a single arbitrary launch attempt. Recently, the Shuttle program has shown interest in having additional capabilities added to the APRA model, including analysis of humidity parameters, inclusion of landing site weather to produce landing availability, and concurrent analysis of multiple sites, to assist in operational landing site selection. In addition, the Constellation program has also expressed interest in the APRA tool, and has requested several additional capabilities to address some Constellation-specific issues, both in the specification and verification of design requirements and in the development of operations concepts. The combined scope of the requested capability enhancements suggests an evolution of the model beyond a simple revision process. Development has begun for a new data analysis tool that will satisfy the requests of both programs. This new tool, Probabilities of Atmospheric Conditions and Environmental Risk (PACER), will provide greater flexibility and significantly enhanced functionality compared to the currently existing tool.

Burns, K. Lee↗

Space Radiation Cancer Risks and Uncertainities for Different Mission Time Periods

Space radiation consists of solar particle events (SPEs), comprised largely of medium energy protons (less than several hundred MeV); and galactic cosmic ray (GCR), which includes high energy protons and high charge and energy (HZE) nuclei. For long duration missions, space radiation presents significant health risks including cancer mortality. Probabilistic risk assessment (PRA) is essential for radiation protection of crews on long term space missions outside of the protection of the Earth s magnetic field and for optimization of mission planning and costs. For the assessment of organ dosimetric quantities and cancer risks, the particle spectra at each critical body organs must be characterized. In implementing a PRA approach, a statistical model of SPE fluence was developed, because the individual SPE occurrences themselves are random in nature while the frequency distribution of SPEs depends strongly upon the phase within the solar activity cycle. Spectral variability of SPEs was also examined, because the detailed energy spectra of protons are important especially at high energy levels for assessing the cancer risk associated with energetic particles for large events. An overall cumulative probability of a GCR environment for a specified mission period was estimated for the temporal characterization of the GCR environment represented by the deceleration potential (theta). Finally, this probabilistic approach to space radiation cancer risk was coupled with a model of the radiobiological factors and uncertainties in projecting cancer risks. Probabilities of fatal cancer risk and 95% confidence intervals will be reported for various periods of space missions.

Kim,Myung-Hee Y.↗

On the Probabilistic Analysis of Neural Networks

Neural networks are powerful tools for automated decision-making,seeing increased application in safety-critical domains, such as autonomous driving. Due to their black-box nature and large scale,reasoning about their behavior is challenging. Statistical analysis is often used to infer probabilistic properties of a network, such as its robustness to noise and inaccurate inputs. While scalable, statistical methods can only provide probabilistic guarantees on the quality of their results and may underestimate the impact of low probability inputs leading to undesired behavior of the network.We investigate here the use of symbolic analysis and constraint solution space quantification to precisely quantify probabilistic properties in neural networks. We demonstrate the potential of the proposed technique in a case study involving the analysis of ACAS-Xu, a collision avoidance system for unmanned aircraft control.

neural nets↗

A comparison of spacecraft penetration hazards due to meteoroids and manmade earth-orbiting objects

The ability of a typical double-walled spacecraft structure to protect against penetration by high-velocity incident objects is reviewed. The hazards presented by meteoroids are compared to the current and potential hazards due to manmade orbiting objects. It is shown that the nature of the meteoroid number-mass relationship makes adequate protection for large space facilities a conceptually straightforward structural problem. The present level of manmade orbiting objects (an estimated 10,000 in early 1975) does not pose an unacceptable risk to manned space operations proposed for the near future, but it does produce penetration probabilities in the range of 1-10 percent for a 100-m diameter sphere in orbit for 1,000 days. The number-size distribution of manmade objects is such that adequate protection is difficult to achieve for large permanent space facilities, to the extent that future restrictions on such facilities may result if the growth of orbiting objects continues at its historical rate.

Brooks, D. R.↗

Evidence for a warm wind from the red star in symbiotic binaries

A systematic redshift of the high ionization resonance emission lines with respect to the intercombination lines is found from an examination of the ultraviolet spectra of symbiotic stars obtained with IUE. After consideration of other possibilities, this is most probably explained by photon scattering in an expanding envelope optically thick to the resonance lines. Line formation in a wind, or at the base of a wind is therefore suggested. Reasons are also given indicating line formation of the most ionized species in a region with an electron temperature of the order of 100,000 K, probably around the cool star. The behavior of the emission line width with ionization energy seems to support this model. The cool components of symbiotic stars appear to differ from normal red giants, which do not have winds of this temperature. An explanation in terms of a higher rotation velocity due to the binary nature of these stars is suggested.

Friedjung, M.↗

Primitive material surviving in chondrites - Mineral grains

Besides chondrules and various kinds of polymineralic inclusion, carbonaceous chondrites commonly contain, embedded in their matrices, isolated grains of mafic silicates and metallic iron. Most of the silicate grains probably originated in chondrules, but some appear to predate chondrule formation and may have formed as individual grains in the solar nebula. If that was the case, their compositions suggest some departure from equilibrium condensation from a gas of solar composition. Metal-grain compositions are broadly suggestive of nebular formation but the exact nature of the conditions in which they were formed remains problematical.

Steele, Ian M.↗

Development of Fuzzy Logic and Soft Computing Methodologies

Our earlier research on computing with words (CW) has led to a new direction in fuzzy logic which points to a major enlargement of the role of natural languages in information processing, decision analysis and control. This direction is based on the methodology of computing with words and embodies a new theory which is referred to as the computational theory of perceptions (CTP). An important feature of this theory is that it can be added to any existing theory - especially to probability theory, decision analysis, and control - and enhance the ability of the theory to deal with real-world problems in which the decision-relevant information is a mixture of measurements and perceptions. The new direction is centered on an old concept - the concept of a perception - a concept which plays a central role in human cognition. The ability to reason with perceptions perceptions of time, distance, force, direction, shape, intent, likelihood, truth and other attributes of physical and mental objects - underlies the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Everyday examples of such tasks are parking a car, driving in city traffic, cooking a meal, playing golf and summarizing a story. Perceptions are intrinsically imprecise. Imprecision of perceptions reflects the finite ability of sensory organs and ultimately, the brain, to resolve detail and store information. More concretely, perceptions are both fuzzy and granular, or, for short, f-granular. Perceptions are f-granular in the sense that: (a) the boundaries of perceived classes are not sharply defined; and (b) the elements of classes are grouped into granules, with a granule being a clump of elements drawn together by indistinguishability, similarity. proximity or functionality. F-granularity of perceptions may be viewed as a human way of achieving data compression. In large measure, scientific progress has been, and continues to be, driven by a quest to progress from perceptions to measurements. Pursuit of this aim has led to brilliant successes. But alongside the successes stand problems whose solutions are not in sight. Representative of such problems is the problem of automation of driving in city traffic. In this case, as in many others, what can be done with ease by humans - without any measurements and a computations - is an intractable task for machines.

Zadeh, L. A.↗

Carrier Modulation Via Waveform Probability Density Function

Beyond the classic modes of carrier modulation by varying amplitude (AM), phase (PM), or frequency (FM), we extend the modulation domain of an analog carrier signal to include a class of general modulations which are distinguished by their probability density function histogram. Separate waveform states are easily created by varying the pdf of the transmitted waveform. Individual waveform states are assignable as proxies for digital ONEs or ZEROs. At the receiver, these states are easily detected by accumulating sampled waveform statistics and performing periodic pattern matching, correlation, or statistical filtering. No fundamental natural laws are broken in the detection process. We show how a typical modulation scheme would work in the digital domain and suggest how to build an analog version. We propose that clever variations of the modulating waveform (and thus the histogram) can provide simple steganographic encoding.

Williams, Glenn L.↗

Using Cloud-to-Ground Lightning Climatologies to Initialize Gridded Lightning Threat Forecasts for East Central Florida

Each morning, the forecasters at the National Weather Service in Melbourn, FL (NWS MLB) produce an experimental cloud-to-ground (CG) lightning threat index map for their county warning area (CWA) that is posted to their web site (http://www.srh.weather.gov/mlb/ghwo/lightning.shtml) . Given the hazardous nature of lightning in central Florida, especially during the warm season months of May-September, these maps help users factor the threat of lightning, relative to their location, into their daily plans. The maps are color-coded in five levels from Very Low to Extreme, with threat level definitions based on the probability of lightning occurrence and the expected amount of CG activity. On a day in which thunderstorms are expected, there are typically two or more threat levels depicted spatially across the CWA. The locations of relative lightning threat maxima and minima often depend on the position and orientation of the low-level ridge axis, forecast propagation and interaction of sea/lake/outflow boundaries, expected evolution of moisture and stability fields, and other factors that can influence the spatial distribution of thunderstorms over the CWA. The lightning threat index maps are issued for the 24-hour period beginning at 1200 UTC (0700 AM EST) each day with a grid resolution of 5 km x 5 km. Product preparation is performed on the AWIPS Graphical Forecast Editor (GFE), which is the standard NWS platform for graphical editing. Currently, the forecasters create each map manually, starting with a blank map. To improve efficiency of the forecast process, NWS MLB requested that the Applied Meteorology Unit (AMU) create gridded warm season lightning climatologies that could be used as first-guess inputs to initialize lightning threat index maps. The gridded values requested included CG strike densities and frequency of occurrence stratified by synoptic-scale flow regime. The intent is to increase consistency between forecasters while enabling them to focus on the mesoscale detail of the forecast, ultimately benefiting the end-users of the product. Several studies took place at the Florida State University (FSU) and NWS Tallahassee (TAE) for which they created daily flow regimes using Florida 1200 UTC synoptic soundings and CG strike densities from National Lightning Detection Network (NLDN) data. The densities were created on a 2.5 km x 2.5 km grid for every hour of every day during the warm seasons in the years 1989-2004. The grids encompass an area that includes the entire state of Florida and adjacent Atlantic and Gulf of Mexico waters. Personnel at the two organizations provided this data and supporting software for the work performed by the AMU. The densities were first stratified by flow regime, then by time in 1-, 3-, 6-, 12-, and 24-hour increments while maintaining the 2.5 km x 2.5 km grid resolution. A CG frequency of occurrence was calculated for each stratification and grid box by counting the number of days with lightning and dividing by the total number of days in the data set. New CG strike densities were calculated for each stratification and grid box by summing the strike number values over all warm seasons, then normalized by dividing the summed values by the number of lightning days. This makes the densities conditional on whether lightning occurred. The frequency climatology values will be used by forecasters as proxy inputs for lightning prObability, while the density climatology values will be used for CG amount. In addition to the benefits outlined above, these climatologies will provide improved temporal and spatial resolution, expansion of the lightning threat area to include adjacent coastal waters, and potential to extend the forecast to include the day-2 period. This presentation will describe the lightning threat index map, discuss the work done to create the maps initialized with climatological guidance, and show examples of the climatological CG lightning densities and frequencies of occurren based on flow regime.

Lambert, Winnie↗

Using Flow Regime Lightning and Sounding Climatologies to Initialize Gridded Lightning Threat Forecasts for East Central Florida

Each morning, the forecasters at the National Weather Service in Melbourne, FL (NWS MLB) produce an experimental cloud-to-ground (CG) lightning threat index map for their county warning area (CWA) that is posted to their web site (http://www.srh.weather.gov/mlb/ghwo/lightning.shtml) . Given the hazardous nature of lightning in East Central Florida, especially during the warm season months of May September, these maps help users factor the threat of lightning, relative to their location, into their daily plans. The maps are color-coded in five levels from Very Low to Extreme, with threat level definitions based on the probability of lightning occurrence and the expected amount of CG activity. On a day in which thunderstorms are expected, there are typically two or more threat levels depicted spatially across the CWA. The locations of relative lightning threat maxima and minima often depend on the position and orientation of the low-level ridge axis, forecast propagation and interaction of sea/lake/outflow boundaries, expected evolution of moisture and stability fields, and other factors that can influence the spatial distribution of thunderstorms over the CWA. The lightning threat index maps are issued for the 24-hour period beginning at 1200 UTC each day with a grid resolution of 5 km x 5 km. Product preparation is performed on the AWIPS Graphical Forecast Editor (GFE), which is the standard NWS platform for graphical editing. Currently, the forecasters create each map manually, starting with a blank map. To improve efficiency of the forecast process, NWS MLB requested that the Applied Meteorology Unit (AMU) create gridded warm season lightning climatologies that could be used as first-guess inputs to initialize lightning threat index maps. The gridded values requested included CG strike densities and frequency of occurrence stratified by synoptic-scale flow regime. The intent is to improve consistency between forecasters while allowing them to focus on the mesoscale detail of the forecast, ultimately benefiting the end-users of the product. Several studies took place at the Florida State University (FSU) and NWS Tallahassee (TAE) in which they created daily flow regimes using Florida 1200 UTC synoptic soundings and CG strike densities, or number of strikes per specified area. The soundings used to determine the flow regimes were taken at Miami (MIA), Tampa (TBW), and Jacksonville (JAX), FL, and the lightning data for the strike densities came from the National Lightning Detection Network (NLDN). The densities were created on a 2.5 km x 2.5 km grid for every hour of every day during the warm seasons in the years 1989-2004. The grids encompass an area that includes the entire state of Florida and adjacent Atlantic and Gulf of Mexico waters. Personnel at FSU and NWS TAE provided this data and supporting software for the work performed by the AMU.

Lambert, Winifred↗

Using Flow Regime Lightning and Sounding Climatologies to Initialize Gridded Lightning Threat Forecasts for East Central Florida

Each morning, the forecasters at the National Weather Service in Melbourne, FL (NWS MLB) produce an experimental cloud-to-ground (CG) lightning threat index map for their county warning area (CWA) that is posted to their web site (httl://www.srh.weather.gov/mlb/ghwo/lightning.shtml) . Given the hazardous nature of lightning in East Central Florida, especially during the warm season months of May September, these maps help users factor the threat of lightning, relative to their location, into their daily plans. The maps are color-coded in five levels from Very Low to Extreme, with threat level definitions based on the probability of lightning occurrence and the expected amount of CG activity. On a day in which thunderstorms are expected, there are typically two or more threat levels depicted spatially across the CWA. The locations of relative lightning threat maxima and minima often depend on the position and orientation of the low-level ridge axis, forecast propagation and interaction of sea/lake/outflow boundaries, expected evolution of moisture and stability fields, and other factors that can influence the spatial distribution of thunderstorms over the CWA. The lightning threat index maps are issued for the 24-hour period beginning at 1200 UTC each day with a grid resolution of 5 km x 5 km. Product preparation is performed on the AWIPS Graphical Forecast Editor (GFE), which is the standard NWS platform for graphical editing. Until recently, the forecasters created each map manually, starting with a blank map. To improve efficiency of the forecast process, NWS MLB requested that the Applied Meteorology Unit (AMU) create gridded warm season lightning climatologies that could be used as first-guess inputs to initialize lightning threat index maps. The gridded values requested included CG strike densities and frequency of occurrence stratified by synoptic-scale flow regime. The intent was to improve consistency between forecasters while allowing them to focus on the mesoscale detail of the forecast. Several studies took place at the Florida State University (FSU) and NWS Tallahassee (TAE) in which they created daily flow regimes using Florida 1200 UTC synoptic soundings and CG strike densities, or number of strikes per specified area. The soundings used to determine the flow regimes were taken at Miami (MIA), Tampa (TBW), and Jacksonville (JAX), FL, and the lightning data for the strike densities came from the National Lightning Detection Network (NLDN). The densities were created on a 2.5 km x 2.5 km grid for every hour of every day during the warm seasons in the years 1989-2004. The grids encompass an area that includes the entire state of Florida and adjacent Atlantic and Gulf of Mexico waters. Personnel at FSU and NWS TAE provided this data and supporting software for the work performed by the AMU.

Lambert, Winifred↗

Analysis of in Vitro Evolution Reveals the Underlying Distribution of Catalytic Activity Among Random Sequences

The emergence of catalytic RNA is believed to have been a key event during the origin of life. Understanding how catalytic activity is distributed across random sequences is fundamental to estimating the probability that catalytic sequences would emerge. Here, we analyze the in vitro evolution of triphosphorylating ribozymes and translate their fitnesses into absolute estimates of catalytic activity for hundreds of ribozyme families. The analysis efficiently identified highly active ribozymes and estimated catalytic activity with good accuracy. The evolutionary dynamics follow Fisher’s Fundamental Theorem of Natural Selection and a corollary, permitting retrospective inference of the distribution of fitness and activity in the random sequence pool for the first time. The frequency distribution of rate constants appears to be log-normal, with a surprisingly steep dropoff at higher activity, consistent with a mechanism for the emergence of activity as the product of many independent contributions.

Abe Pressman↗

Survey of probable micrometer-sized Earth-orbital debris fragments in the NASA-JSC cosmic dust sample collection

A program to collect extraterrestrial dust samples from the stratosphere using impactors mounted on NASA U-2 and WB-57F aircraft is discussed. Sample collection, preparation, and analysis techniques are described and the particle types found are reviewed. Preliminary examination of the particles indicates that they represent not only extraterrestrial material, but some fraction of terrestrial contamination from both natural and manmade sources. This examination involves a combination of optical microscopy, scanning electron microscopy (SEM) and qualitative bulk elemental analysis using an energy-dispersive X-ray spectrometer (EDS) to characterize each particle. The cosmic dust collection contains samples of at least three classes of material that may present some hazard to extended spacecraft operations in near-Earth orbit: (1) micrometeorites/cosmic dust, (2) aluminum oxide spheres, and (3) alloy spherules and fragments.

Clanton, U. S.↗

The effects of transmitter/receiver clock time-base instability on coherent communication system performance

The purpose of this paper is to present a model and associated analysis of the deleterious effects which both transmitter and receiver data clock time-base instability have on bit-error probability. The types of time-base instability modeled and analyzed include what is commonly referred to as bit jitter (epoch jitter), bit-jitter rate (time interval jitter), and data asymmetry. The bit-error probability is determined as a function of the statistical characteristics of the time-base instability, the data-transition density, the signal-to-noise ratio in the bit-rate bandwidth and the mechanizations of both the data-bit detector and bit synchronizer. The latter is manifested through the bit synchronizer's natural frequency and loop-damping parameters. Numerical results which indicate the sensitivity of the system bit-error rate to various parameters of the model are provided.

Chie, C. M.↗

Trace Element Geochemistry of Martian Iddingsite in the Lafayette Meteorite

The Lafayette meteorite contains abundant iddingsite, a fine-grained intergrowth of smectite clay, ferrihydrite, and ionic salt minerals. Both the meteorite and iddingsite formed on Mars. Samples of iddingsite, olivine, and augite pyroxene were extracted from Lafayette and analyzed for trace elements by instrumental neutron activation. Our results are comparable to independent analyses by electron and ion microbeam methods. Abundances of most elements in the iddingsite do not covary significantly. The iddingsite is extremely rich in Hg, which is probably terrestrial contamination. For the elements Si, Al, Fe, Mn, Ni, Co, and Zn, the composition of the iddingsite is close to a mixture of approximately 50% Lafayette olivine + approximately 40% Lafayette siliceous glass + approximately 1O% water. Concordant behavior among these elements is not compatible with element fractionations between smectite and water, but the hydrous nature and petrographic setting of the iddingsite clearly suggest an aqueous origin. These inferences are both consistent, however, with deposition of the iddingsite originally as a silicate gel, which then crystallized (neoformed) nearly isochemically. The iddingsite contains significantly more magnesium than implied by the model, which may suggest that the altering solutions were rich in Mg(2+).

Treiman, Allan H.↗