We report the discovery of weak magnetic fields in three white dwarfs within the local 20pc volume (WD 0816-310, WD 1009-184, and WD 1532+129), and we confirm the magnetic nature of a fourth star (WD 2138-332) in which we had previously detected a field at a 3 sigma level. The spectra of all these white dwarfs are characterised by the presence of metal lines and lack of H and He lines, that is, they belong to the spectral class DZ. The polarisation signal of the Ca II H+K lines of WD 1009-184 is particularly spectacular, with an amplitude of 20% that is due to the presence of a magnetic field with an average line-of-sight component of 40kG. We have thus established that at least 40% of the known DZ white dwarfs with an He-rich atmosphere contained in the 20pc volume have a magnetic field, while further observations are needed to establish whether the remaining DZ white dwarfs in the same volume are magnetic or not. Metal lines in the spectra of DZ white dwarfs are thought to have originated by accretion from rocky debris, and it might be argued that a link exists between metal accretion and higher occurrence of magnetism. However, we are not able to distinguish whether the magnetic field and the presence of a polluted atmosphere have a common origin, or if it is the presence of metal lines that allows us to detect a higher frequency of magnetic fields in cool white dwarfs, which would otherwise have featureless spectra. We argue that the new highly sensitive longitudinal field measurements that we have made in recent years are consistent with the idea that the magnetic field appears more frequently in older than in younger white dwarfs.
We present a set of ultramassive white dwarf models, focused on masses above $1.3\,M_\odot$. Given the uncertainties about the formation and compositions of such objects, we construct parameterized model sequences, guided by evolutionary calculations including both single star and double white dwarf merger formation channels. We demonstrate that the cooling of objects with central densities in excess of $10^9\,\rm g\,cm^{-3}$ is dominated by neutrino cooling via the Urca process in the first $\approx 100$ Myr after formation. Our models indicate that the recently discovered ultramassive white …
We investigate the luminosity and cooling of highly magnetized white dwarfs where cooling occurs by the diffusion of photons. We solve the magnetostatic equilibrium and photon diffusion equations to obtain the temperature and density profiles in the surface layers of these white dwarfs. With increase in field strength, the degenerate core shrinks in volume with a simultaneous increase in the core temperature. For a given white dwarf age and for a fixed interface radius or temperature, the luminosity decreases significantly from $\sim 10^{-6}\, L_{\odot}$ to $10^{-9}\, L_{\odot}$ as the field s…
The evolution of white dwarfs is a simple gravothermal process. This process can be tested in two ways, through the luminosity function of these stars and through the secular variation of the period of pulsation of those stars that are variable. Here we show how the mass of the axion can be constrained using the white dwarf luminosity function.
We present theoretical analysis on old white dwarf stars with \emph{some} hydrogen, that possess a mass of surface hydrogen from $1\times10^{-11}M_{\sun}$ to $1\times10^{-7}M_{\sun}$. The evolution of such objects is complicated by convective mixing from surface convection zone to the underlying helium layer. In this paper, we provide first self-consistent, quantitative investigation on the subject of convective mixing. Numerical cooling curves and chemical evolution curves are obtained as a function of white dwarf mass and hydrogen content. Such results will be applied to the investigation of…
We present an empirical determination of the white dwarf cooling sequence in the globular cluster 47 Tucanae. Using spectral models, we determine temperatures for 887 objects from Wide Field Camera 3 data, as well as 292 objects from data taken with the Advanced Camera for Surveys. We make the assumption that the rate of white dwarf formation in the cluster is constant. Stellar evolution models are then used to determine the rate at which objects are leaving the main sequence, which must be the same as the rate at which objects are arriving on the white dwarf sequence in our field. The result …
An accurate assessment of white dwarf cooling times is paramount to place white dwarf cosmochronology of Galactic populations on more solid grounds. This issue is particularly relevant in view of the enhanced observational capabilities provided by the next generation of Extremely Large Telescopes, that will offer more avenues to employ white dwarfs as probes of Galactic evolution and test-beds of fundamental physics. We estimate for the first time the consistency of results obtained from two independent and widely used evolutionary codes (BaSTI and LPCODE) for white dwarf models with fixed mas…
NGC 6791 is a well studied open cluster1 that it is so close to us that can be imaged down to very faint luminosities. The main sequence turn-off age (~8 Gyr) and the age derived from the termination of the white dwarf cooling sequence (~6 Gyr) are significantly different. One possible explanation is that as white dwarfs cool, one of the ashes of helium burning, 22Ne, sinks in the deep interior of these stars. At lower temperatures, white dwarfs are expected to crystallise and phase separation of the main constituents of the core of a typical white dwarf, 12C and 16O, is expected to occur. Thi…
White dwarfs are stellar embers depleted of nuclear energy sources that cool over billions of years. These stars, which are supported by electron degeneracy pressure, reach densities of 1e7 grams per cubic centimetre in their cores. It has been predicted that a first-order phase transition occurs during white-dwarf cooling, leading to the crystallization of the non-degenerate carbon and oxygen ions in the core, which releases a considerable amount of latent heat and delays the cooling process by about one billion years. However, no direct observational evidence of this effect has been reported…
We present Hubble Space Telescope observations of the upper part (T_eff> 10 000 K) of the white dwarf cooling sequence in the globular cluster 47 Tucanae and measure a luminosity function of hot white dwarfs. Comparison with previous determinations from large scale field surveys indicates that the previously determined plateau at high effective temperatures is likely a selection effect, as no such feature is seen in this sample. Comparison with theoretical models suggests that the current estimates of white dwarf neutrino emission (primarily by the plasmon channel) are accurate, and variations…
We present full evolutionary calculations appropriate for the study of hydrogen-rich DA white dwarfs. This is done by evolving white dwarf progenitors from the zero age main sequence, through the core hydrogen burning phase, the helium burning phase and the thermally pulsing asymptotic giant branch phase to the white dwarf stage. Complete evolutionary sequences are computed for a wide range of stellar masses and for two different metallicities: Z=0.01, which is representative of the solar neighborhood, and Z=0.001, which is appropriate for the study of old stellar systems, like globular cluste…
Using images from the Hubble Space Telescope Advanced Camera for Surveys, we measure the rate of cooling of white dwarfs in the globular cluster 47 Tucanae and compare it to modelled cooling curves. We examine the effects of the outer convective envelope reaching the nearly isothermal degenerate core and the release of latent heat during core crystallisation on the white dwarf cooling rates. For white dwarfs typical of 47 Tuc, the onset of these effects occur at similar times. The latent heat released during crystallisation is a small heat source. In contrast, the heat reservoir of the degener…
We present a new set of cooling models and isochrones for both H- and He-atmosphere white dwarfs, incorporating accurate boundary conditions from detailed model atmosphere calculations, and carbon-oxygen chemical abundance profiles based on updated stellar evolution calculations from the BaSTI stellar evolution archive - a theoretical data center for the Virtual Observatory. We discuss and quantify the uncertainties in the cooling times predicted by the models, arising from the treatment of mixing during the central H- and He-burning phases, number of thermal pulses experienced by the progenit…
Type Ia supernovae (SNe Ia) play a key role in measuring cosmological parameters, in which the Phillips relation is adopted. However, the origin of the relation is still unclear. Several parameters are suggested, e.g. the relative content of carbon to oxygen (C/O) and the central density of the white dwarf (WD) at ignition. These parameters are mainly determined by the WD's initial mass and its cooling time, respectively. Using the progenitor model developed by Meng and Yang, we present the distributions of the initial WD mass and the cooling time. We do not find any correlation between these …
Using deep Hubble Space Telescope imaging, color-magnitude diagrams are constructed for the globular clusters 47 Tuc and NGC 6397. As expected, because of its lower metal abundance, the main sequence of NGC 6397 lies well to the blue of that of 47 Tuc. A comparison of the white dwarf cooling sequences of the two clusters, however, demonstrates that these sequences are indistinguishable over most of their loci - a consequence of the settling out of heavy elements in the dense white dwarf atmosphere and the near equality of their masses. Lower quality data on M4 continues this trend to a third c…
The precise astrometric measurements of the Gaia Data Release 2 have opened the door to detailed tests of the predictions of white dwarf cooling models. Significant discrepancies between theory and observations have been identified, the most striking affecting ultramassive white dwarfs. Cheng et al. (2019) found that a small fraction of white dwarfs on the so-called Q branch must experience an extra cooling delay of $\sim 8\,$Gyr not predicted by current models. $^{22}$Ne phase separation in a crystallizing C/O white dwarf can lead to a distillation process that efficiently transports $^{22}$N…
White dwarfs are the final remnants of low- and intermediate-mass stars. Their evolution is essentially a cooling process that lasts for $\sim 10$ Gyr. Their observed properties provide information about the history of the Galaxy, its dark matter content and a host of other interesting astrophysical problems. Examples of these include an independent determination of the past history of the local star formation rate, identification of the objects responsible for the reported microlensing events, constraints on the rate of change of the gravitational constant, and upper limits to the mass of wea…
White dwarf evolution is essentially a gravothermal cooling process, which,for cool white dwarfs, sensitively depends on the treatment of the outer boundary conditions. We provide detailed outer boundary conditions appropriate for computing the evolution of cool white dwarfs employing detailed non-gray model atmospheres for pure H composition. We also explore the impact on the white dwarf cooling times of different assumptions for energy transfer in the atmosphere of cool white dwarfs. Detailed non-gray model atmospheres are computed taken into account non-ideal effects in the gas equation of …
We employ a state-of-the-art asteroseismological model of G117-B15A, the archetype of the H-rich atmosphere (DA) white dwarf pulsators (also known as DAV or ZZ Ceti variables), and use the most recently measured value of the rate of period change for the dominant mode of this pulsating star to derive a new constraint on the mass of axion, the still conjectural non-barionic particle considered as candidate for dark matter of the Universe. Assuming that G117-B15A is truly represented by our asteroseismological model, and in particular, that the period of the dominant mode is associated to a puls…
We present a robust statistical analysis of the white dwarf cooling sequence in 47 Tucanae. We combine HST UV and optical data in the core of the cluster, Modules for Experiments in Stellar Evolution (MESA) white dwarf cooling models, white dwarf atmosphere models, artificial star tests, and a Markov Chain Monte Carlo (MCMC) sampling method to fit white dwarf cooling models to our data directly. We use a technique known as the unbinned maximum likelihood to fit these models to our data without binning. We use these data to constrain neutrino production and the thickness of the hydrogen layer i…
Before reaching their quiescent terminal white-dwarf cooling branch, some low-mass helium-core white dwarf stellar models experience a number of nuclear flashes which greatly reduce their hydrogen envelopes. Just before the occurrence of each flash, stable hydrogen burning may be able to drive global pulsations that could be relevant to shed some light on the internal structure of these stars through asteroseismology. We present a pulsational stability analysis applied to low-mass helium-core stars on their early white-dwarf cooling branches going through CNO flashes in order to study the poss…
It has been shown that the shape of the luminosity function of white dwarfs can be a powerful tool to check for the possible existence of DFSZ-axions. In particular, Isern et al. (2008) showed that, if the axion mass is of the order of a few meV, then the white dwarf luminosity function is sensitive enough to detect their existence. For axion masses of about $m_a > 5$ meV the axion emission can be a primary cooling mechanism for the white dwarf and the feedback of the axion emission into the thermal structure of the white dwarf needs to be considered. Here we present computations of white dwar…
We use Hubble Space Telescope (HST) to reach the end of the white dwarf (WD) cooling sequence (CS) in the solar-metallicity open cluster NGC 6819. Our photometry and completeness tests show a sharp drop in the number of WDs along the CS at magnitudes fainter than mF606W = 26.050+/- 0.075. This implies an age of 2.25+/-0.20 Gyr, consistent with the age of 2.25+/-0.30 Gyr obtained from fits to the main-sequence turn-off. The use of different WD cooling models and initial-final-mass relations have a minor impact the WD age estimate, at the level of ~0.1 Gyr. As an important by-product of this inv…
Since the large amplitude 2007 outburst which heated its accreting, pulsating white dwarf, the dwarf nova system GW Librae has been cooling to its quiescent temperature. Our Hubble Space Telescope ultraviolet spectra combined with ground-based optical coverage during the 3rd and 4th year after outburst show that the fluxes and temperatures are still higher than quiescence (T=19,700K and 17,300K vs 16,000K pre-outburst for a log g=8.7 and d=100 pc). The K{wd} of 7.6+/-0.8 km/s determined from the CI1463 absorption line, as well as the gravitational redshift implies a white dwarf mass of 0.79+/-…
A simple cooling model of white dwarf stars is re-analyzed in Palatini $f(R)$ gravity. Modified gravity affects the white dwarf structures and consequently their ages. We find that the resulting super-Chandrasekhar white dwarfs need more time to cool down than sub-Chandrasekhar ones, or when compared to the Newtonian models.
We have exploited the very precise parallaxes, proper motions and photometry of \textit{Gaia}\, Data Release~2 to study white dwarf members of the Hyades star cluster. Gaia photometry and parallaxes for the eight DA white dwarfs confirmed members have been then used to compute absolute magnitudes and colours. These were compared to three independent sets of white dwarf evolutionary tracks, to derive cooling times and white dwarf (final) masses. All sets of models provide the same mass values, with only small differences in the cooling ages. The precision in the derived masses and cooling ages …
We investigate the impact of gravity portal to properties of white dwarfs, such as equation-of-state as well as cooling time. We find that the interaction between dark matter spin-zero bosons and electrons in the mean field approximation softens the equation-of-state, and the object coolers down slower compared to the usual case.
We determine the distribution of cooling ages of massive Gaia EDR3 white dwarfs identified with over 90% probability within 200 pc and with mass in the range 0.95-1.25 $M_\odot$. Using three sets of publicly available models, we consider sub-samples of these white dwarfs sorted into three equally spaced mass bins. Under the assumption of a constant white dwarf formation rate, we find an excess of white dwarfs both along the Q branch and below it, corresponding respectively to stars that are in the process of freezing and those that are completely frozen. We compare the cooling age distribution…
We use 10 orbits of Advanced Camera for Surveys observations to reach the end of the white dwarf cooling sequence in the solar-metallicity open cluster NGC 2158. Our photometry and completeness tests show that the end falls at magnitude m_F606W = 27.5 +/- 0.15, which implies an age between ~1.8 and ~2.0 Gyr, consistent with the age of 1.9 +/- 0.2 Gyr obtained from fits to the main-sequence turn-off. The faintest white dwarfs show a clear turn toward bluer colors, as predicted by theoretical isochrones.
We present new diagnostics of white dwarf (WD) cooling sequences and luminosity functions (LFs) in the near-infrared (NIR) bands that will exploit the sensitivity and resolution of future extremely large telescopes. The collision-induced absorption (CIA) of molecular hydrogen causes a clearly defined blue turn-off along the WD (WDBTO) cooling sequences and a bright secondary maximum in the WD LFs. These features are independent of age over a broad age range and are minimally affected by metal abundance. This means that the NIR magnitudes of the WDBTO are very promising distance indicators. The…
Do white dwarfs have inner cores made of iron? Neutron rich nuclei like $^{56}$Fe experience a net gravitational force and sediment toward the core. Using new phase diagrams and molecular dynamics simulations, we show that $^{56}$Fe should separate into mesoscopic Fe-rich crystallites due to its large charge relative to the background. At solar abundances, these crystallites rapidly precipitate and form an inner core of order 100 km and $10^{-3} M_\odot$ that may be detectable with asteroseismology. Associated cooling delays could be up to a Gyr for low mass white dwarfs but are only $\sim$0.1…
We report our study of features at the observed red end of the white dwarf cooling sequences for three Galactic globular clusters: NGC\,6397, 47\,Tucanae and M\,4. We use deep colour-magnitude diagrams constructed from archival Hubble Space Telescope (ACS) to systematically investigate the blue turn at faint magnitudes and the age determinations for each cluster. We find that the age difference between NGC\,6397 and 47\,Tuc is 1.98$^{+0.44}_{-0.26}$\,Gyr, consistent with the picture that metal-rich halo clusters were formed later than metal-poor halo clusters. We self-consistently include the …
We present the initial-final mass relation derived from 10 white dwarfs in wide binaries that consist of a main sequence star and a white dwarf. The temperature and gravity of each white dwarf was measured by fitting theoretical model atmospheres to the observed spectrum using a $χ^{2}$ fitting algorithm. The cooling time and mass was obtained using theoretical cooling tracks. The total age of each binary was estimated from the chromospheric activity of its main sequence component to an uncertainty of about 0.17 dex in log \textit{t} The difference between the total age and white dwarf cooling…
NGC 6791 is a well-studied, metal-rich open cluster that is so close to us that can be imaged down to luminosities fainter than that of the termination of its white dwarf cooling sequence, thus allowing for an in-depth study of its white dwarf population. We use a Monte Carlo simulator that employs up-to-date evolutionary cooling sequences for white dwarfs with hydrogen-rich and hydrogen-deficient atmospheres, with carbon-oxygen and helium cores. The cooling sequences for carbon-oxygen cores account for the delays introduced by both Ne^22 sedimentation in the liquid phase and by carbon-oxygen …
The evolution of white dwarfs is a simple gravothermal process of cooling. Since the shape of their luminosity function is sensitive to the characteristic cooling time, it is possible to use its slope to test the existence of additional sources or sinks of energy, such as those predicted by alternative physical theories. The aim of this paper is to study if the changes in the slope of the white dwarf luminosity function around bolometric magnitudes ranging from 8 to 10 and previously attributed to axion emission are, effectively, a consequence of the existence of axions and not an artifact int…
We provide a fine and homogeneous grid of evolutionary sequences for He-core white dwarfs with masses 0.15-0.45 Msun, including the mass range for ELM white dwarfs (<0.20Msun). The grid is appropriate for mass and age determination, and to study their pulsational properties. White dwarf sequences have been computed by performing full evolutionary calculations that consider the main energy sources and processes of chemical abundance changes during white dwarf evolution. Initial models for the evolving white dwarfs have been obtained by computing the non-conservative evolution of a binary system…
(Abridged) We present the results of the first unbiased survey for metal pollution among H-atmosphere (DA) white dwarfs with cooling ages of 20-200 Myr and 17000K < Teff < 27000K, using HST COS in the far UV between 1130 and 1435 A. The atmospheric parameters and element abundances are determined using theoretical models, which include the effects of element stratification due to gravitational settling and radiative levitation. We find 48 of the 85 DA white dwarfs studied, or 56% show traces of metals. In 25 stars, the elements can be explained by radiative levitation alone, although we argue …
White Dwarf Merger Remnants with Cooling Delays on the Q Branch Lack Strong Magnetism
Boston University（波士顿大学）；Institute for Advanced Study（普林斯顿高等研究院）；Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma（俄克拉荷马大学霍默·L·多德物理与天文系）；Université de Montréal（蒙特利尔大学）；University of Victoria（维多利亚大学）
We present Hubble Space Telescope data of the low-reddening Sagittarius window in the Galactic bulge. The Sagittarius Window Eclipsing Extrasolar Planet Search field (3'x3'), together with three more Advanced Camera for Surveys and eight Wide Field Camera 3 fields, were observed in the F606W and F814W filters, approximately every two weeks for two years, with the principal aim of detecting a hidden population of isolated black holes and neutron stars through astrometric microlensing. Proper motions were measured with an accuracy of ~0.1 mas/yr (~4 km/s) at F606W~25.5 mag, and better than ~0.5 …
We present the first radiation magnetohydrodynamics simulations of the atmosphere of white dwarf stars. We demonstrate that convective energy transfer is seriously impeded by magnetic fields when the plasma-beta parameter, the thermal to magnetic pressure ratio, becomes smaller than unity. The critical field strength that inhibits convection in the photosphere of white dwarfs is in the range B = 1-50 kG, which is much smaller than the typical 1-1000 MG field strengths observed in magnetic white dwarfs, implying that these objects have radiative atmospheres. We have then employed evolutionary m…
With the Keck I Low-Resolution Imaging Spectrometer we have observed nine white dwarf candidates in the very rich open cluster NGC 2099 (M37). The spectroscopy shows seven to be DA white dwarfs, one to be a DB white dwarf, and one to be a DZ white dwarf. Three of these DA white dwarfs are consistent with singly evolved cluster membership: an ultramassive (1.28$^{+0.05}_{-0.08}$ M$_\odot$) and two intermediate-mass (0.70 and 0.75 M$_\odot$) white dwarfs. Analysis of their cooling ages allows us to calculate their progenitor masses and establish new constraints on the initial-final mass relation…
We present optical spectroscopy and near-infrared photometry of 57 faint ($g= 19-22$) high proper motion white dwarfs identified through repeat imaging of $\approx3100$ square degrees of the Sloan Digital Sky Survey footprint by Munn et al. (2014). We use $ugriz$ and $JH$ photometry to perform a model atmosphere analysis, and identify ten ultracool white dwarfs with $T_{\rm eff}<4000$ K, including the coolest pure H atmosphere white dwarf currently known, J1657+2638, with $T_{\rm eff}= 3550 \pm 100$K. The majority of the objects with cooling ages larger than 9 Gyr display thick disc kinematics…
47 Tucanae is one of the most interesting and well observed and theoretically studied globular clusters. This allows us to study the reliability of our understanding of white dwarf cooling sequences, to confront different methods to determine its age, and to assess other important characteristics, like its star formation history. Here we present a population synthesis study of the cooling sequence of the globular cluster 47 Tucanae. In particular, we study the distribution of effective temperatures, the shape of the color-magnitude diagram, and the corresponding magnitude and color distributio…
We investigate the luminosity and cooling of highly magnetized white dwarfs with electron-degenerate cores and non-degenerate surface layers where cooling occurs by diffusion of photons. We find the temperature and density profiles in the surface layers or envelope of white dwarfs by solving the magnetostatic equilibrium and photon diffusion equations in a Newtonian framework. We also obtain the properties of white dwarfs at the core-envelope interface, when the core is assumed to be practically isothermal. With the increase in magnetic field, the interface temperature increases whereas the in…
We present new cooling models for carbon-oxygen white dwarfs with both H- and He-atmospheres, covering the whole relevant mass range, to extend our updated BaSTI (a Bag of Stellar Tracks and Isochrones) stellar evolution archive. They have been computed using core chemical stratifications obtained from new progenitor calculations, adopting a semiempirical initial-final mass relation. The physics inputs have been updated compared to our previous BaSTI calculations: ^{22}Ne diffusion in the core is now included, together with an updated CO phase diagram, and updated electron conduction opacities…
The first solids that form as a white dwarf (WD) starts to crystallize are expected to be greatly enriched in actinides. Previously [PRL 126, 1311010] we found that these solids might support a nuclear fission chain reaction that could ignite carbon burning and provide a new Type Ia supernova (SN Ia) mechanism involving an {\it isolated} WD. Here we explore this fission mechanism in more detail and calculate the final temperature and density after the chain reaction and discuss a number of open physics questions.
We investigate the luminosity suppression and its effect on the mass-radius relation as well as cooling evolution of highly magnetised white dwarfs. Based on the effect of magnetic field relative to gravitational energy, we suitably modify our treatment of the radiative opacity, magnetostatic equilibrium and degenerate core equation of state to obtain the structural properties of these stars. Although the Chandrasekhar mass limit is retained in the absence of magnetic field and irrespective of the luminosity, strong central fields of about $10^{14}\, {\rm G}$ can yield super-Chandrasekhar whit…
We present a set of full evolutionary sequences for white dwarfs with hydrogen-deficient atmospheres. We take into account the evolutionary history of the progenitor stars, all the relevant energy sources involved in the cooling, element diffusion in the very outer layers, and outer boundary conditions provided by new and detailed non-gray white dwarf model atmospheres for pure helium composition. These model atmospheres are based on the most up-to-date physical inputs. Our calculations extend down to very low effective temperatures, of $\sim 2\,500$~K, provide a homogeneous set of evolutionar…
The first solids that form as a cooling white dwarf (WD) starts to crystallize are expected to be greatly enriched in actinides. This is because the melting points of WD matter scale as $Z^{5/3}$ and actinides have the largest charge $Z$. We estimate that the solids may be so enriched in actinides that they could support a fission chain reaction. This reaction could ignite carbon burning and lead to the explosion of an isolated WD in a thermonuclear supernova (SN Ia). Our mechanism could potentially explain SN Ia with sub-Chandrasekhar ejecta masses and short delay times.
Recent observations of Galactic white dwarfs (WDs) with Gaia suggest there is a population of massive crystallizing WDs exhibiting anomalous cooling -- the Q branch. While single-particle $^{22}$Ne sedimentation has long been considered a possible heat source, recent work suggests that $^{22}$Ne must separate into clusters, enhancing diffusion, in order for sedimentation to provide heating on the observed timescale. We show definitively that $^{22}$Ne cannot separate to form clusters in C/O WDs using molecular dynamics simulations, and we further present a general C/O/Ne phase diagram showing …
