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Published in "Encyclopedia of Astronomy and Astrophysics" (IOP Publishing and MacMillan), 1266-1273, 2001


INTERSTELLAR GRAINS

B. T. Draine


Princeton University


Table of Contents

OVERVIEW

OBSERVATIONAL EVIDENCE: SUMMARY

EXTINCTION BY INTERSTELLAR DUST GRAINS

EXTINCTION FEATURES

ALIGNMENT OF DUST GRAINS

SCATTERING OF LIGHT BY DUST GRAINS

DUST GRAIN LUMINESCENCE

INFRARED EMISSION FROM DUST GRAINS

MICROWAVE EMISSION FROM DUST GRAINS

INTERSTELLAR DUST GRAINS IN THE SOLAR SYSTEM

GRAPHITE-SILICATE GRAIN MODEL

ULTRASMALL DUST GRAINS

OTHER GRAIN MODELS

DYNAMICS OF INTERSTELLAR GRAINS

EFFECTS OF INTERSTELLAR GRAINS

FORMATION AND DESTRUCTION OF INTERSTELLAR GRAINS

BIBLIOGRAPHY


1. Overview

Submicron solid particles are dispersed through interstellar gas. These interstellar grains absorb and scatter light, thus shielding some regions from ultraviolet radiation, but also limiting our ability to detect photons which have been emitted by astronomical objects. Dust grains reradiate absorbed energy in the infrared, thus contributing to the overall emission spectrum of astronomical systems ranging from dusty disks around stars to ultraluminous starburst galaxies.

A naked-eye view of the sky from a dark site on a clear summer night reveals dramatic dark patches in the Milky Way. These dark regions are not due to a deficiency of stars - they are instead the result of obscuration by dust clouds interposed between the Earth and distant stars. The obscuration tends to be greater at shorter wavelengths; as a result, the light reaching us from distant, obscured stars is "reddened". This reddening by interstellar dust can be understood as arising from scattering and absorption by a population of interstellar submicron dust grains.

The grain population spans a range of sizes, from molecules (1) containing only tens of atoms to particles as large as ~ 0.3 µm, containing ~ 1010 atoms. Most of the grain mass appears to be due to two types of solid, in approximately equal amounts: (1) amorphous silicate mineral, and (2) carbonaceous material. A number of elements - including silicon and iron - are primarily in solid form in the interstellar medium. Approximately 2/3 of the interstellar carbon in diffuse clouds is in solid form (see INTERSTELLAR ABSORPTION LINES).

It is of course important to characterize the wavelength-dependent interstellar extinction so that astronomical observations can be "corrected" for the obscuring effects of dust. In addition, the infrared emission from dust grains provides a valuable probe of dense regions, and the dust grains themselves play important roles in interstellar chemistry (shielding from ultraviolet radiation, and catalyzing the formation of H2), interstellar gas dynamics (radiation pressure forces on dust grains, and coupling of charged dust grains to magnetic fields) and heating and cooling of interstellar gas. Dust grains are central to many problems in modern astrophysics.

2. Observational Evidence: Summary

There are many different astronomical phenomena which both reveal the existence of interstellar dust grains, and provide information allowing us to infer the properties of this dust. Some of the information is quite direct, due to absorption, scattering, or emission of light by the grains:

Although limited, we also have some quite direct evidence concerning interstellar grains:

In addition, there are a number of astrophysical phenomena which provide indirect information about interstellar grains:

3. Extinction by Interstellar Dust Grains

The "extinction curves" shown in Figure 1 represent the wavelength-dependent extinction Alambda at wavelength lambda observed on different kinds of sightlines. Careful studies of the extinction on many lines of sight reveals that the extinction curves can be approximately described by a one parameter family of extinction curves, and the curves shown in Figure 1 are obtained from fitting functions originally developed by Cardelli, Clayton, & Mathis (1989) and others. It is convenient to take RV ident AV / (AB - AV) as the parameter, where V = 550 nm and B = 440 nm. Values of RV as small as 2.75, and as large as 5.5, are observed in different regions.

Grains in diffuse regions tend to have RV approx 3.1, while larger values of RV tend to be seen when studying the extinction by dust in dense clouds. Larger values of RV - corresponding to "greyer" extinction - are indicative of larger grains. The observed tendency for larger values of RV to be found in denser regions strongly suggests that characteristic grain sizes are larger in these regions. It can be shown that the increased values of RV cannot be explained simply by accretion of atoms and molecules from the gas phase - because of the very large amount of surface area contributed by the smaller grains, accretion from the gas phase would result in only a very small increase Deltaa in the radii of all grains, with minimal effect on the extinction at B and V. Instead, small grains must coagulate with larger grains to convert diffuse cloud dust to dense cloud dust.

4. Extinction Features

The strongest spectroscopic feature in the extinction curve is a conspicuous "bump" at 217.5 nm (see Figure 1). This feature has approximately the wavelength and width expected for small (radius a ltapprox 15 nm) particles of graphite. While identification of the feature is still not certain, it appears highly likely that it is due to small carbonaceous particles with sp2 carbon-carbon bonds as in graphite and polycyclic aromatic hydrocarbons. The strength of the 217.5nm feature requires that ~ 15% of the solar abundance of carbon be present in small (a ltapprox 5 nm) particles.

Between 1.32 µm and 400 nm there are many weak diffuse extinction features, known as the "diffuse interstellar bands", or DIBs. Approximately 300 such features have been identified, with full width at half-maximum (FWHM) ranging from ~ 0.05 nm up to ~ 4 nm. The strongest DIB is at 442.9 nm.

The first DIBs were recognized in 1934 by Merrill, yet they remain essentially unidentified to this date. They must be due to either "impurities" within dust grains, or to small free-flying molecules (i.e., ultrasmall dust grains). A few of the weaker features have been identified recently as electronic transitions of C7-, and it seems likely that many other of the DIBs are also produced by small hydrocarbon molecules, either neutral or charged (either positively or negatively).

A broad extinction feature at 3.4 µm is attributed to the C-H stretching mode in aliphatic (chainlike) hydrocarbons. This feature is present in the diffuse interstellar medium.

There are strong infrared extinction features peaking at 9.7 µm and 18 µm which are almost certainly due to amorphous silicates with a composition approximating that of olivine (MgxFe2-xSiO4). In regions where the dust is hot (e.g., circumstellar dust, or the dust near the Trapezium in the Orion Nebula), these features appear in emission. In some circumstellar dust shells and disks (e.g., the dusty disk around the Herbig Ae/Be star HD 100546), sharp features characteristic of crystalline silicates appear in emission, but these features have not been detected in either emission or absorption in the interstellar medium, indicating that the bulk of interstellar silicates are amorphous.

In dark clouds, a number of additional features appear in the infrared extinction, presumably due to growth of molecular ice mantles on the refractory dust grain cores. The strongest feature is at 3.08 µm and is due to amorphous H2O ice. Additional features have been identified as frozen CH3OH (3.53 µm), CO (4.67 µm), CH4 (7.65 µm), and CO2 (15.2 µm).

5. Alignment of Dust Grains

When initially unpolarized starlight passes through the dusty interstellar medium, it acquires both linear and circular polarization. The linear polarization is due to preferential attenuation of one linear polarization over the other, due to a population of nonspherical dust grains which are somehow aligned. The linear polarization peaks near V = 550 nm, with peak polarization P approx 0.03AV. The circular polarization, generally quite small, is due to conversion of linear to circular polarization when the alignment direction of the dust grains undergoes a twist along the line-of-sight to the source.

Spinning dust grains have magnetic moments antiparallel to their angular velocities, and the angular momentum therefore precesses around the local galactic magnetic field. As a result, the observed direction of starlight linear polarization must be either parallel or perpendicular to the projection of the magnetic field on the sky. Theories of grain alignment lead us to expect the grain angular momentum to tend to align with the magnetic field, and the "long" axis of the grains to tend to be perpendicular to the angular momentum. The long axis of the grain therefore tends to be perpendicular to the magnetic field, and as a result light becomes linearly polarized parallel to the projection of the magnetic field on the plane of the sky.

Our understanding of the physics of dust grain alignment is not yet complete, but it appears that the observed alignment in diffuse clouds is produced mainly by a combination of paramagnetic dissipation (originally proposed by Davis and Greenstein in 1951) and radiative torques on irregular dust grains due to anisotropic starlight. In some cases the alignment may result from drift of the dust grains through the gas cloud.

6. Scattering of Light by Dust Grains

Reflection nebulae such as NGC2023 or the beautiful filamentary structures near the Pleaides show that dust grains scatter starlight. While most conspicuous when a bright star is located near the surface of a dense clouds, reflected starlight also manifests itself as the "diffuse galactic light" - reflected starlight seen in all directions in the sky where there is dust. Measurements of the surface brightness of reflection nebulae, or of the diffuse galactic light, provide constraints on the scattering properties of interstellar dust grains - both the total "albedo" omega = Csca / Cext, and the angular dependence of the scattering, often characterized by g ident <costheta> , where theta is the scattering angle. At visible wavelengths, the albedo omega approx 0.5 and the grains are moderately forward-scattering, g approx 0.5, consistent with current models for interstellar grains which reproduce the wavelength-dependent extinction as well as the infrared emission from dust.

Scattering by dust grains can also be observed at X-ray wavelengths. A grain is essentially transparent to hnu gtapprox 0.5 keV X-ray photons. Since the refractive index of the grain material is very close to 1, the scattering can be calculated in the "Rayleigh-Gans" approximation, and one finds that only small-angle scattering is expected. X-ray halos around compact X-ray sources have been imaged by the Einstein and ROSAT observatories; the observed X-ray halos appear to be approximately consistent with the scattering expected for a grain model developed to account for the optical-UV extinction curve.

7. Dust Grain Luminescence

Observations of reflection nebulae (e.g., NGC 7023) as well as the general interstellar diffuse clouds or "cirrus" appear to show evidence of far-red continuum radiation in excess of what is expected from simple scattering of starlight by dust grains. The far-red emission peaks near ~ 700 nm, and has a spectrum resembling the luminescence from hydrogenated amorphous carbon illuminated by lambda ltapprox 550 nm radiation. This suggests that some of the grain material may resemble hydrogenated amorphous carbon, although ultrasmall silicon grains have also been proposed as the source of the emission. Even if it is assumed that hydrogenated amorphous carbon is responsible for the observed emission, an accurate estimate of the quantity of material required is not possible, since luminescence efficiencies of hydrogenated amorphous carbon in the laboratory depend on the preparation of the sample.

8. Infrared Emission from Dust Grains

Starlight is in part absorbed by dust grains, and the absorbed energy is reradiated by dust grains in the infrared. The observed emission spectrum for interstellar dust is shown in Figure 3. It consists of far-infrared emission peaking at lambda approx 140 µm, plus substantial emission at shorter wavelengths.

The emission at lambda < 12 µm shows conspicuous emission features at 11.3, 8.6, 7.7, 6.2, and 3.3 µm; emission peaks at these same wavelengths have also been observed from reflection nebulae, planetary nebulae, HII regions, and circumstellar dust. The emission features have been identified as characteristic of polycyclic aromatic hydrocarbons (PAHs): C-H stretch (3.3 µm), C-C stretch (6.2 and 7.7 µm), in-plane C-H bend (8.6 µm) and out-of-plane C-H bend (11.3 µm). Variations in the relative strengths and precise wavelengths of these features from one object to another may be due to changes in the PAH mixture, including changes in the fraction which are positively or negatively charged.

The "classical" grains with radii 0.01 - 0.3 µm radiate a spectrum characteristic of thermal emission at the (steady) temperature of the grain, ~ 15-20 K for grains in the diffuse interstellar medium. This accounts for nearly all of the emission at lambda > 60 µm. For very small grains, however, absorption of a single starlight photon can appreciably change the grain temperature. For example, a single photon of energy hnu = 10 eV can heat a 230 atom graphite grain to a peak temperature T approx 300 K. At this temperature the grain can radiate effectively at wavelengths as short as lambda approx 8 µm. The observed emission feature near 7.7 µm could therefore be due to thermal emission from grains with ~ 100 - 300 atoms. The relatively large amount of power radiated at lambda < 10 µm requires that the grains with ltapprox 300 atoms account for an appreciable fraction (gtapprox 15%) of the total absorption of starlight by interstellar grains.

9. Microwave Emission from Dust Grains

Sensitive observations of the cosmic microwave background radiation have revealed 10-60 GHz emission from interstellar matter with intensities greatly exceeding what would be expected from an extrapolation of the thermal far-infrared emission to these lower frequencies. It appears likely that the observed 10-60 GHz emission is largely rotational electric dipole emission from very rapidly rotating ultrasmall grains, although a fraction of the radiation could be thermal emission from grains containing materials that are ferromagnetic (e.g., metallic Fe inclusions) or ferrimagnetic (e.g., magnetite Fe3O4).

10. Interstellar Dust Grains in the Solar System

The solar system was formed out of interstellar gas and dust approximately 4.5 billion years ago. The formation of planetesimals and planets was accompanied in many cases by high temperatures and violent conditions, and most interstellar dust particles were destroyed. However, the class of meteorites known as carbonaceous chondrites (see METEORITES) contain small particles with unusual isotopic ratios (see ISOTOPIC ANOMALIES) which indicate that they did not form in the solar nebula, but rather must have been formed in a region with an anomalous composition (e.g., outflow from an evolved star) long before the formation of the solar system. Therefore these particles must have been part of the interstellar grain population prior to the formation of the solar nebula.

To date, 5 different type of presolar grains have been isolated and identified (see Table 1). The evidence that the grains listed in Table 1 were truly interstellar is compelling, but it is important to realize that they apparently do not include typical interstellar grains, for the simple reason that the procedures used to isolate interstellar grains in meteorites are designed to deliberately destroy silicate material (which comprises the bulk of the carbonaceous chondrite meteorite "matrix"). These laboratory procedures are therefore not going to find interstellar silicate grains even if they are present.

Table 1. Interstellar Grains in Meteorites

Composition diameter(µm) Abund. (2)

C (diamond) 0.002 5 × 10-4
SiC (3) 0.3-20 6 × 10-6
C (graphite) (3) 1-20 1 × 10-6
Al2O3 (corundum) 0.5-3 3 × 10-8
Si3N4 ~ 1 2 × 10-9

Detectors aboard the Ulysses and Galileo probes measured impacts by interplanetary dust particles, but also observed impacts attributed to the flux of interstellar grains expected due to the 20 km s-1 motion of the Sun relative to the local interstellar medium. The detectors are sensitive only to collisions with the large-size end of the interstellar grain distribution, but the observed distribution of impact energies appears to be consistent with both the overall numbers and size distribution expected based on studies of interstellar extinction.

The lines of evidence listed above serve to strongly constrain theoretical models for the interstellar grain population. Unfortunately, interstellar grain researchers have thus far not been able to uniquely determine the grain model from the available observational constraints, so debates continue concerning the details.

Nevertheless, there is broad consensus on a number of properties of interstellar grains:

11. Graphite-Silicate Grain Model

It is not possible to invert the observations to obtain a unique grain model. Instead, one makes some assumptions concerning the grain composition and the form of the size distribution, and then attempts to adjust the model to achieve a good match to the observed interstellar extinction, infrared emission, and other constraints such as observed gas phase abundances.

One grain model which has proven fairly successful in conforming to observations consists of a mixture of carbon grains and silicate grains. The carbon grain material is taken to have the optical constants of crystalline graphite. A good fit to the extinction can be obtained if both graphite and silicate grains have size distributions which are approximately a power-law, dn / da propto a-3.5, truncated at amin approx 5 nm and amax approx 250 nm. The 217.5 nm feature is then reproduced by the a ltapprox 5 nm graphite grains. This grain model - the combination of graphite and silicate grains, and the dn / da propto a-3.5 power-law - was first put forward in 1977 by Mathis, Rumpl, and Nordsieck, and is often referred to as the "MRN" model. The MRN model achieves a good fit to the RV = 3.1 extinction curve for diffuse clouds with essentially all of the Mg, Si, and Fe in the silicate grains, and approximately 2/3 of the C in the graphite grains, in reasonable agreement with observed depletions. The grains are heated to temperatures ~ 18 K by starlight, and the resulting thermal emission is approximately consistent with the observed far-infrared emission at lambda > 60 µm.

12. Ultrasmall Dust Grains

Unfortunately, the original graphite-silicate grain model described above fails to reproduce the lambda < 50 µm infrared emission shown in Figure 3. The model lacks the ultrasmall grain component required to explain the observed 3 ltapprox lambda ltapprox 50 µm infrared emission from interstellar dust. The simplest modification is to allow the carbon grain distribution to extend down to very small sizes, with the smallest grains assumed to have infrared optical properties suitable to explain the emission features at 3.3, 6.2, 7.6, 8.6, and 11.2 µm when heated to the appropriate temperatures by absorption of single starlight photons. In order to have sufficient numbers of ultrasmall grains, while still reproducing the extinction curve, the size distribution of at least the carbonaceous grains can no longer be approximated by a single power-law.

13. Other Grain Models

Other grain models, with different grain geometries and/or compositions, have been proposed to account for the observed interstellar extinction and infrared emission.

Grain models generally tend to have the bulk of the Fe, Mg, Si in dust, approximately 2/3 of the C, and about 20% of the O, resulting in dust-to-gas mass ratios of ~ 0.007.

14. Dynamics of Interstellar Grains

Interstellar grains are acted on by forces due to a number of distinct physical processes, including:

Because the gas and grains are subject to different forces, the dust grains generally have a drift velocity relative to the gas. In typical diffuse clouds, anisotropic starlight can result in drift velocities of order ~ 0.1 km s-1. In regions with strongly anisotropic ultraviolet radiation fields (such as photodissociation regions) the drift velocities can be larger.

The rotational dynamics of interstellar grains are also of great interest, in connection with both the problem of grain alignment (which requires alignment of the grain angular momentum vector with the local magnetic field) and with electric dipole radiation which will be radiated by very rapidly-rotating ultrasmall grains. The grain angular momentum is affected by:

Because the interstellar medium is far from thermodynamic equilibrium, some of these torques can act systematically.

Grains can be driven to rotational kinetic energies much larger than kTgas (where Tgas is the gas temperature) by collisions with gas atoms and molecules (because the grain and gas temperatures differ), by photoelectric emission or by absorption and scattering of starlight (because the starlight radiation field is not in thermodynamic equilibrium with the grain temperature), and by H2 formation (because the H2 abundance and gas and grain temperatures are not in thermodynamic equilibrium).

Gradual alignment of the grain angular momentum with the magnetic field can be produced by paramagnetic dissipation (because the grain rotational kinetic energy is not in thermodynamic equilibrium with the "vibrational" temperature of the grain).

15. Effects of Interstellar Grains

15.1. Photoelectric Heating

Perhaps the most important effect of interstellar dust grains is their role in heating the gas via photoelectric emission. When a dust grain absorbs an ultraviolet photon of energy hnu , there is a probability Y(hnu) - often referred to as the "yield" - that a photoelectron will escape from the grain surface and thermalize with the local gas. When this happens, the kinetic energy of the emitted electron (at "infinity", since the grain it is escaping from may be charged) acts to heat the interstellar medium. On average, grains must capture electrons as rapidly as they are ejected, and the net heating rate then is equal to the difference between the mean energy of the emitted electrons and the captured electrons. Since photoelectric yields Y may be of order ~ 10%, and since the kinetic energy of the electron at infinity may be of order ~ 1 eV (if the grain is not highly positively charged) the photoelectric heating mechanism - at best - converts ~ few % of the absorbed ultraviolet starlight energy into heating of the gas. While this ~ few % conversion efficiency may seem small, the availability of energy in starlight is so great that photoelectric heating is generally the dominant heating mechanism for diffuse interstellar gas.

15.2. Radiative Forces on Dust Grains

In astrophysics, anisotropic radiation fields are the rule, rather than the exception, and these anisotropic radiation fields can have dynamical consequences for dust and gas. Circumstellar grains are of course subject to extremely anisotropic radiation from the star. At a typical point in the interstellar medium, the nonuniform distribution of stars in the galaxy, together with patchy obscuration by interstellar dust, results in starlight with appreciable anisotropy. The dipole component of the starlight anisotropy may be typically ~ 10%.

When the radiation field is anisotropic, scattering and absorption by the grain produces a net force on the grain, commonly referred to as the "radiation pressure" force. If the grain is coupled collisionally to the gas by gas drag and perhaps magnetic fields, then the force exerted on the grains is transmitted to the gas. These radiation pressure forces can dominate the dynamics of gas in certain regions, such as dust-forming winds from cool stars.

15.3. H2 Formation on Grains

Interstellar chemistry largely starts with the catalysis of H2 on interstellar grain surfaces (see INTERSTELLAR CHEMISTRY). The inferred production rate of H2 in the interstellar medium requires that an appreciable fraction of the H atoms arriving at interstellar grains surfaces must leave the grains as part of H2 molecules. While the details of the kinetics remain uncertain, the overall picture is broadly as follows: When an H atom in a diffuse cloud arrives at a grain surface, it has a high probability of sticking. The H atom then explores some fraction of the grain surface by either thermal diffusion or quantum tunnelling, until it either finds another H atom with which it can recombine to form H2, or it arrives at a location on the grain surface where the binding by either van der Waals forces ("physisorption") or formation of a chemical bond ("chemisorption") is strong enough that it becomes trapped. If such trapping occurs, then the surface coverage of trapped H atoms builds up until newly-arrived H atoms have an appreciable probability of reacting with a previously-trapped H atom rather than becoming trapped themselves, or the adsorbed H atoms are removed by some other process (e.g., photodesorption or thermal desorption).

15.4. Coupling Magnetic Fields to Neutral Gas

Dynamically important magnetic fields are commonly present in interstellar gas (see INTERSTELLAR MAGNETIC FIELDS). In dense molecular gas with very low fractional ionization, a large fraction of the "free" charge resides on positively and negatively charged dust grains. Under these conditions - which prevail in high density regions in molecular clouds, and presumably in protostellar disks - dust grains dominate the coupling of magnetic fields to the gas, since the neutral gas atoms and molecules themselves are not directly coupled to the magnetic field. The charged dust grains are coupled both to the magnetic field (by Lorentz forces) and to the neutral gas (by collisional drag). As a result, the grains will drift with a velocity which is intermediate between the velocity of the neutral gas and the velocity with which the magnetic field lines (and plasma) "slip" through the neutral gas. Thus the dust grains determine the rate of "ambipolar diffusion" of magnetic field lines in dense molecular regions.

Charged dust grains play a similar role in the dynamics of magnetohydrodynamic shock waves in gas of low fractional ionization but dynamically significant magnetic fields.

16. Formation and Destruction of Interstellar Grains

While interstellar grains are observed to be ubiquitous, it is not obvious that this should be so. In the violent interstellar medium (see INTERSTELLAR MATTER, SUPERNOVA REMNANTS), grains can be destroyed when the gas which they are in is overtaken by a shock wave with shock speed vs gtapprox 200 km s-1, as is expected in supernova blastwaves. Estimates of the rate of occurrence of supernovae, together with models for the resulting blastwaves, lead one to estimate that the mass in interstellar grains will be returned to the gas phase on a timescale of ~ 5 × 108 yr, short compared to the age of the Galaxy and the interstellar medium. We know that grains are injected into the interstellar medium in outflows from cool red giants and supergiants, and even in supernova ejecta, but if there were no conversion of gaseous atoms back to solid form in the interstellar medium, we would expect very little grain material to be present at any time. In particular, even for elements like Fe we would expect most of the atoms to be in the gas phase. The fact that this is not so - that most of the interstellar Fe is in fact missing from the gas phase - requires that there be efficient recondensation of Fe, Si, Ca, and other elements back into solid form in the interstellar medium. The "mineralogy" of interstellar grains must therefore largely reflect the surface chemistry which will occur on the surfaces of interstellar grains.

Bibliography

Dust in the Galactic Environment, by D.C.B. Whittet (London: IOP Publishing; 1992), provides an excellent overall description of interstellar dust grains.

The observed optical properties of dust have been recently reviewed by J.S. Mathis (1990), Ann. Rev. Astr. Astrophys., 28, 37.

The ultrasmall grain population is discussed by Puget, J.L., & Leger, A. (1989), Ann. Rev. Astr. Astrophys., 27, 161.

An overall review of interstellar and circumstellar dust is given by Dorschner, J., & Henning, T. (1995), Astr. Astrophys. Rev., 6, 271.

Microwave emission from dust grains is reviewed by Draine, B.T., & Lazarian, A. (1999), in Microwave Foregrounds, ed. A. de Oliveira-Costa & M. Tegmark (San Francisco: Astron. Soc. Pacific), p. 133



1 Since all grains are "molecules", it is natural to consider small molecules as the small-size end of the overall grain population. Back.

2 Mass fraction in "primitive" carbonaceous chondrite meteorites. Back.

3 SiC and graphite grains sometimes contain very small TiC, ZrC, and MoC inclusions. Back.

4 The particles are of course not spherical. By grain "radius" we refer to the radius of a sphere of equal volume. Back.


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