MMon. Not. R. Astron. Soc. , 1–12 (2013) Printed 18 November 2018 (MN L A TEX style file v2.2)
A snowflake’s chance in heaven
Mark A. Walker (cid:63)
Manly Astrophysics, 3/22 Cliff St, Manly 2095, Australia
Accepted ................................ Received ................................ In original form ................................
ABSTRACT
We consider the survival of solid H in the diffuse interstellar medium, with applicationto grains which are small enough to qualify as dust. Consideration of only the thermalaspects of this problem leads to the familiar conclusion that such grains sublimaterapidly. Here we show that charging plays a critical role in determining the sublima-tion rate, because an electric field helps to bind molecules to the grain surface. A keyaspect of the charging process is that the conduction band of solid hydrogen lies abovethe vacuum free-electron energy level, so low-energy electrons cannot penetrate thesolid. But they are attracted by the dielectric and by positive ions in the matrix, sothey become trapped in vacuum states just above the surface. This charge-separatedconfiguration suppresses recombination and permits overall neutrality, while support-ing large electric fields at the surface. Charging ceases when the potential energy justoutside the electron layer coincides with the conduction band minimum. By that stagethe heat of sublimation has increased tenfold, effecting a huge reduction in the subli-mation rate. Consequently hydrogen grains may be able to survive indefinitely in thediffuse ISM. There are good prospects for identifying H grains, if they exist, as fully-localised surface electrons should exhibit discrete energy levels, with a correspondingspectral line signature. Key words:
ISM: molecules — dust, extinction — molecular processes
Solid hydrogen is thought to be absent from the diffuse in-terstellar medium (ISM). The reason is simple: typical pres-sures in the diffuse medium are far below the saturatedvapour pressure of H at all accessible temperatures, so sub-limation is very rapid (Greenberg and de Jong 1969; Field1969). This appears to be a compelling argument for thenon-existence of interstellar solid hydrogen because it onlyrelies on knowledge of the phase-diagram of H , and thathas been accurately determined in the laboratory. It is, how-ever, misleading. Laboratory experiments exclude the vari-ous ionising- and charged-particles which pervade interstel-lar space and, as this paper demonstrates, that makes a bigdifference. When surface charging is considered, it appearsthat hydrogen grains may be able to survive indefinitely inthe diffuse ISM.This paper is tightly focused on the thermodynamics ofH surfaces in an interstellar context. As our starting pointwe take macroscopic grains of pure hydrogen, in contrast tocircumstances where H molecules form a monolayer aroundother species. Hydrogen monolayers may be important bothat the nanoparticle level (Duley 1996; Bernstein, Clark and (cid:63) [email protected]
Lynch 2013), and as coatings on other bulk materials (Sand-ford and Allamandola 1993). But the electronic propertiesof these composites differ from those of bulk, solid H andwe do not consider them here.Even if we had no reason to suspect the presence ofsnowflakes, the overwhelming abundance of hydrogen in as-tronomical contexts would still make it worth communicat-ing the results reported herein. But to motivate readers toengage with this material it behoves us to add some contextaround the narrow issue of snowflake survival, as follows.Hydrogen snow cannot be formed in the diffuse ISM — itrequires gas in self-gravitating clouds. The clouds must besufficiently cold and dense that they lie close to the satu-rated vapour-pressure curve of H (see figure 1 of this pa-per). Such clouds would be difficult to detect – i.e. theywould be a form of baryonic dark matter – and current ob-servational constraints are not able to exclude a Galacticpopulation of this type. Various specific models have beenproposed, including a Galactic thin-disk population of mas-sive, fractal clouds (Pfenniger, Combes and Martinet 1994;Pfenniger and Combes 1994), a disk population of spher-ical clouds (Lawrence 2001), and a Galactic halo popula-tion of spherical clouds (e.g. Henriksen and Widrow 1995;Gerhard and Silk 1996; Walker and Wardle 1998; Draine1998; Sciama 2000; Walker 2007). The likely presence of solid c (cid:13) a r X i v : . [ a s t r o - ph . GA ] J un Walker H within such clouds was pointed out by Pfenniger andCombes (1994), and it was argued by Wardle and Walker(1999) that snowflakes can confer thermal stability on theself-gravitating gas.There are two ways in which such clouds can seed thediffuse ISM with hydrogen dust. First, as they move throughthe Galaxy, small amounts of material – including H grains– may be stripped from their surfaces and left behind in thediffuse ISM. Secondly, an individual cloud may be heatedfaster than it can cool by radiation. In this circumstancethe cloud must cool by expanding, causing precipitation of asubstantial fraction of the H . If the total heat supplied suf-fices to unbind the cloud, this solid material will be injectedinto the diffuse ISM. Examples of such disruptive heatingevents include the UV flash from a nearby supernova explo-sion, and a low-speed (a few km s − ) collision between a pairof clouds. We note that the cloud-cloud collision rate may besignificant over the life of a galaxy (Gerhard and Silk 1996;Walker 1999).The other important contextual point relates to thespectral features expected from particles of solid hydrogen.Longward of the far-UV, H has no electric-dipole transi-tions and the pure solid consequently displays only veryweak absorption. Inevitably, though, an initially pure H grain would acquire trace molecular ion species as a resultof ionising particles impinging on it, and the electric-dipoletransitions of these ions provide a signature of the solid. Itis known that the ionisation chemistry of solid H differsgreatly from the gas phase: whereas H +3 is the dominantionisation product in H gas, in the condensed phase H +6 is favoured (Lin, Gilbert and Walker 2011, and referencestherein). This molecular ion was discovered only recentlyand its infrared spectrum has yet to be measured in thelab. But it is a small molecule that has been tackled witha high level of ab initio electronic structure theory, yield-ing accurate wavenumbers for five fundamental vibrationaltransitions (Lin, Gilbert and Walker 2011). And for the twoisotopomers which are expected to dominate in practice (i.e.H +6 and (HD) +3 ), these authors found a striking match to theobserved pattern of mid-infrared spectral bands of the ISM.What those spectral coincidences mean is at present unclear,but to sort out that issue we certainly need a good under-standing of the durability of interstellar solid hydrogen. Thispaper is a step towards that goal.It was pointed out by Lepp (2003, citing Dyson andWilliams 1997), in the context of pre-galactic chemistry, thatfree charges enhance the stability of hydrogen grains, be-cause of the energy associated with polarisation of the H molecules. Here we explore that idea further for the case ofgrains in the diffuse ISM of our Galaxy. We restrict attentionto collisional charging, which occurs because of the diluteplasma in which the grains are immersed. Although photo-electric charging of solid H is potentially important, therehas been no experimental investigation of the photoelectricyield and so the charging rates are currently unknown, evento order of magnitude. We do know that the band-gap ofsolid H (approximately 14.5 eV: Inoue, Kanzaki and Suga1979) is greater than the ionisation energy of atomic hydro-gen. For grains immersed in the typical interstellar radiationfield, that means that the rate of photoexcitations into theconduction band is many orders of magnitude smaller thanfor graphite or silicate grains (Bakes and Tielens 1994; Wein- gartner and Draine 2001). It is also small compared to therate of photodissociations, which we identify in § grains. Section 3 describes theproperties of vacuum electronic states above the surface ofsolid H . We then turn to collisional grain charging in § § phase equilibrium and grain sublimationrates, this time accounting for the influence of macroscopicelectric fields on the heat of sublimation. These calculationsdemonstrate that interstellar hydrogen grains may be long-lived. Survival of H grains in the ISM has been consideredby Greenberg and de Jong (1969) and by Field (1969).E.S. Phinney (1985, unpublished manuscript) gave a thor-ough treatment of the sublimation of H snowballs in a cos-mological context. The treatment given here follows Phin-ney’s development, but adapted to small particles in thediffuse ISM of our own Galaxy. H Suppose we have a gas of H at temperature T and pressure P , in thermodynamic equilibrium with a lump of solid H .The flux of molecules, each of mass m , incident upon thesolid is F in = P √ πmkT , (1)and over the temperature range of interest to us here almostall of the incident molecules are expected to stick to thesurface. In thermodynamic equilibrium the pressure of H must equal the saturated vapour pressure, P sat : P sat = kT (2 πmkT ) / h exp (cid:16) − b o kT − ∆ (cid:17) , (2)where h is Planck’s Constant, b o is the molecular heat ofsublimation, and ∆ ≡ π ( T / Θ) /
5, with Θ (cid:39)
105 K the De-bye temperature of the solid. The value of b o can be obtainedby comparing equation (2) with measurements. For para- H at the Triple Point ( T tp = 13 . P sat = 7 ,
030 Pa, from which we deduce b o /k = 90 . The data refer to H whose nuclear spin statistics equilibratedat T = 20 . ortho- H content is small, but not zero.c (cid:13)000
030 Pa, from which we deduce b o /k = 90 . The data refer to H whose nuclear spin statistics equilibratedat T = 20 . ortho- H content is small, but not zero.c (cid:13)000 , 1–12 nowflakes in heaven Figure 1.
The saturated vapour pressure of H (solid curve) fortemperatures below the triple-point ( T tp = 13 . T = T cmb = 2 .
73 K, and the horizontaldashed line shows the typical pressure in the diffuse ISM, P ism ∼ ,
000 K cm − . The fact that P sat (cid:29) P ism for all T (cid:62) T cmb arguesthat phase equilibrium between solid and gaseous phases of pureH cannot be achieved in the diffuse ISM. In equilibrium there is no net flux of molecules from thesurface, so the sublimation rate, F out , must equal the fluxarriving, F in . We thus obtain F out = kT πmkTh exp (cid:16) − b o kT − ∆ (cid:17) , (3)as the sublimation rate at temperature T .The kinetic energy flux of the molecules arriving at thesurface of the grain is 2 kT F in . On average, each arrivingmolecule releases an amount of heat equal to the heat ofsublimation, b o , so the grain heating rate per unit area dueto the arriving molecules is ( b o +2 kT ) F in . In equilibrium thismust equal the grain cooling rate due to departing molecules,and the sublimation cooling rate per unit area is therefore ( b o + 2 kT ) F out .Because of the exponential factors in equations (2) and(3), both P sat and F out are strong functions of temperature.The form of P sat ( T ) is shown in figure 1, from which wesee that P sat greatly exceeds typical pressures in the dif-fuse ISM ( P ism ∼ ,
000 K cm − ; Jenkins and Tripp 2011),even for T = T cmb = 2 .
73 K, the temperature of the CosmicMicrowave Background (CMB). As the steady-state temper-ature of the ISM must exceed T cmb , this graph leads us toexpect that solid H cannot be in phase equilibrium with thediffuse interstellar gas. Consequently if the diffuse ISM con-tains any hydrogen grains we expect them to be shrinkingby sublimation. H grains Equations (1) and (3) correctly describe the influx and ef-flux of molecules even if the phases are not in equilibrium.So for P ∼ P ism (cid:28) P sat we can neglect F in in comparisonwith F out and treat the sublimation as if it is occuring invacuum. The sublimation rate depends on the grain temper-ature, and that in turn is set by the balance between heat- Phinney (1985) gives b o + 5 kT/ ing and cooling. Sublimation itself contributes to cooling, asdoes thermal photon emission from the grain. On the otherside of the equation there is heating due to absorption fromthe InterStellar Radiation Field (ISRF), including the CMB,and due to impinging material particles, i.e. cosmic-rays andthe thermal particles which constitute the interstellar gas.Material particles have significant energy-densities inthe ISM, and an interaction cross-section that is close tothe geometric cross-section of the grain. However, grains arehighly transparent to cosmic-rays, with each particle de-positing only a tiny fraction of its energy in the grain, socosmic-ray heating is negligible. Thermal particles depositall of their energy in the grain, but they move so slowly thattheir contribution to grain heating is also small comparedto the absorption of starlight.An unusual aspect of solid H is that energy absorbedfrom a radiation field is not necessarily thermalised in thecrystal as a whole, but may remain localised as electronicand/or ro-vibrational excitations on the absorbing centre. Itis for this reason that solid H is used for matrix isolationspectroscopy (e.g. Oka 1993; Anderson et al 2002). There-fore in determining the grain temperature we can neglectheating due to the ISRF except where there is a manifestcoupling to lattice vibrations. This is the case for microwaveradiation, which couples directly to the phonon modes of thecrystal, and for far-UV radiation which can dump mechan-ical energy into the lattice as a result of dissociation of theH molecules. Henceforth we use the term “far-UV” to referspecifically to wavelengths λ such that 912 < λ (˚A) (cid:46) grains.We adopt the analytic description of the ISRF given byDraine (2011), based on the tabulation by Mathis, Mezgerand Panagia (1983). For that description the far-UV energy-density is U fuv = 9 . × − erg cm − , with a mean photonenergy of (cid:104) hν (cid:105) = 12 . σ is U fuv σ c . In this paperwe will consider only spherical grains of radius a (cid:38) . µ m,for which the geometric cross-section, σ = πa , is an ade-quate approximation in the far-UV band. (Such grains alsoexperience only small temperature fluctuations, (cid:46) .
01 K,on absorption of a single far-UV photon.) All H moleculesin the solid are expected to be in the ro-vibrational groundstate, for which the absorption of a far-UV photon by anisolated H molecule leads to a dissociation probability of p dis (cid:39) .
13 (Draine and Bertoldi 1996). We assume that thesame average branching ratio holds for solid H . Subtractingthe energy required for dissociation ( G = 4 .
48 eV; Herzberg1969), we infer a lattice heating rate due to dissociations of˙ E dis = p dis U fuv (cid:18) − G (cid:104) hν (cid:105) (cid:19) πa c (cid:39) . × − (cid:18) a µ m (cid:19) erg s − . (4)Microwave heating/cooling depends on the low-frequency behaviour of the dielectric constant of solid H , (cid:15) = (cid:15) + i(cid:15) . The real part was measured by Constable, Clarkand Gaines (1975) to be (cid:15) = 1 . c (cid:13) , 1–12 Walker perature variation across the range 0 . − . (cid:15) ∝ /λ , at low frequencies, but theconstant of proportionality is uncertain for solid hydrogen.Here we use the same assumption as Wardle and Walker(1999), namely (cid:15) = λ /λ , with λ = 1 µ m. For (cid:15) ∝ /λ and a (cid:28) T − T cmb . For our adopted value of λ , balancingphotodissociation heating plus net microwave heating withsublimation cooling leads to an equilibrium grain tempera-ture of T eq = 2 .
42 K, for a grain of radius a = 1 µ m. Thisis not a true steady state as the grain is shrinking and willeventually vanish.With the grain temperature below that of the CMB,increasing the microwave coupling (larger λ , or larger a )leads to greater heating by the CMB and therefore slightlygreater equilibrium temperatures. However, the dependenceof T eq on a and λ is very weak because sublimation coolingand photodissociation heating are the dominant processes,and sublimation cooling is very temperature sensitive.Later in this paper we will be examining circumstanceswhere the sublimation rate is drastically lowered by surfacecharging of H grains. In that case the sublimation cooling isnegligible and thermal equilibrium is achieved by a balancebetween photodissociation heating and net microwave cool-ing. With our choice of λ , the equilibrium temperature fora grain of 1 µ m radius would then be 7 . ∼ − ) of the starlight in the ISRF. Previous authors have considered composite grainstructures in which a solid H mantle surrounds a core ofgraphite (e.g. Wickramasinghe and Reddish 1968; Hoyle,Wickramasinghe and Reddish 1968; Wickramasinghe andKrishna Swamy 1969). In that case the graphite core wouldthermalise a substantial fraction of the ISRF, so a compositegrain is expected to be warmer than a pure H grain. H grains The sublimation timescale of an interstellar hydrogen grainof radius a is t sub = a ρ ∗ mF out , (5)where ρ ∗ = 0 .
087 g cm − is the density of solid H . For T = 2 .
42 K this evaluates to t sub (cid:39) × ( a/ µ m) s. Inthe absence of a mechanism for supplying dust to the ISMon a similar, or shorter timescale, this calculation suggeststhat the abundance of interstellar H grains ought to benegligible.Any H grain which has a temperature different from T eq = 2 .
42 K will be driven towards that equilibrium valueby the imbalance between heating and cooling. The temper-ature evolution is governed by If our adopted microwave dielectric constant is extrapolated toshorter wavelengths it implies that the matrix will absorb far-IRradiation from the ISRF. Including that as an additional heatsource for the matrix raises the equilibrium temperature of amicron-sized grain from 7 . . C d T d t = ˙ E, (6)where the heat capacity of the grain is (Debye model) C = 16 πa ρ ∗ m k ∆ , (7)and ˙ E is the net heating rate. A fiducial value is ˙ E = ˙ E dis forwhich we can estimate the corresponding heating timescale t heat ∼ ( a/ µ m) s at T = T eq = 2 .
42 K. The heatingtimescale is small compared to the sublimation timescale,so the lifetime of an H grain is not lengthened significantlyif it is initially very cold.The foregoing calculations appear to preclude any pos-sibility of H grains persisting in the diffuse interstellarmedium. However, it is clear that the sublimation rate andthe saturated vapour pressure are both very sensitive to theheat of sublimation, b o , so any effect which significantly in-creases the binding of H molecules to the surface can affectthis conclusion.As noted earlier, the polarisation of H molecules asso-ciated with static electric fields increases their heat of subli-mation. The rest of this paper is devoted to grain chargingand the influence of charges on the longevity of H grains.Before modelling the charging process itself, we first describethe nature of the electronic surface states above solid H . There is an extensive literature on Surface State Electrons(SSE) above liquid helium, solid hydrogen and, to a lesserextent, solid neon (e.g. Cole 1974; Grimes 1978; Edel’manand Faley 1983; Leiderer 1992; Smolyaninov 2001). Boundsurface states arise when there is a surface barrier whichprevents electrons from penetrating the solid. Such a bar-rier exists for condensed noble gases, and for H , because atan atomic/molecular level these species have closed electronshells which strongly repel any additional electrons. Conse-quently the electronic structure of these materials exhibitsa conduction band which lies above the vacuum electronenergy level.For solid H the barrier height calculated using theWigner-Seitz method is V = 3 .
27 eV (Cole 1970). An earlyexperimental investigation of field emission in liquid H sug-gested a low barrier (0 . ± . have yielded re-sults which are in close accord with predictions based on theWigner-Seitz model (Johnson and Onn 1978). We thereforeadopt Cole’s (1970) theoretical value of the barrier height.Whereas the surface barrier prevents electrons from en-tering solid H , the electrical polarisation which they inducenevertheless attracts them to the dielectric. Any positiveions lodged in the matrix likewise attract electrons to thesurface, and as a result there exists an electron potentialwell just outside the solid. That well supports bound elec-tron states. Low energy electrons which are incident on thesurface cannot penetrate the solid, but in the process of re-flection from the surface they excite lattice vibrations, losekinetic energy and may become trapped in bound vacuumstates. Once there is a significant SSE population, incidentelectrons may also lose energy by scattering from that pop-ulation, which increases the capture efficiency (i.e. the frac- c (cid:13) , 1–12 nowflakes in heaven tion of incident electrons which are captured into bound,vacuum states). For the purposes of our charging calcula-tions ( §
4) we assume that the capture efficiency is unity ifthe incident electron has insufficient energy to penetrate thesolid.
Laboratory investigations of the properties of SSE have allbeen undertaken in the limit of low charge densities andsmall applied fields. For this circumstance the attraction ofan electron to the surface can be described by the vacuumpotential associated with the image-charge (i.e. dielectricpolarisation). In this section we summarise relevant resultsfrom the review of Cole (1974).An electron near a planar dielectric boundary has po-tential energy − e Φ( z ) = − ξe π(cid:15) o z , ( z > , (8)due to interaction with its image charge. Here z is the dis-tance above the surface, and ξ ≡ (cid:15) − (cid:15) + 1) . (9)For solid H we have (cid:15) = 1 . ξ = 0 . E n = − ξ R n + p ⊥ m e , (10)where R is the Rydberg Constant, n is a positive integer, and p ⊥ is the electron momentum in the plane of the surface. Thevalues of p ⊥ are not quantised, so electrons bound in thesestates can be thought of as a two-dimensional, free-electrongas. For solid H the lowest energy level is E = −
13 meV,which is sufficiently small in magnitude (i.e. | E | (cid:28) V ) thatan infinite barrier is a good approximation.The corresponding wavefunctions are also “hydrogenic”in z , being products of the associated Laguerre polynomialswith an exponential factor. The ground state wavefunctionis ψ = 2 z √ z o exp[ − z/z o ] , (11)where z o = 1 /ξ in atomic units. For solid H z o (cid:39) . a (cid:38) . µ m (cid:29) z o ).Surface electron states around small dielectric particles havebeen considered by Rosenblit and Jortner (1994). (See alsoKhaikin, 1978, who proposed a possible connection with in-terstellar grains.)The solutions given in equations (10) and (11) are ap-propriate to the early stages of grain charging. However, inthis paper we are particularly interested in the late stages,where the confining potential is primarily due to a popula-tion of sub-surface positive ions. We now turn to that case. In the presence of a high surface-density of positive ions,bound in the matrix of the solid, the potential immediatelyabove the surface can be approximated byΦ( z ) = Φ(0) − E z, ( z > , (12)where E is the electric field. We again assume a large sur-face barrier for electrons, approximating this with the limit − e Φ( z < → ∞ . The wavefunctions appropriate to thispotential are Airy Integrals: ψ k ( z ) = 1 √ s Ai (cid:48) ( − ζ k ) Ai (cid:16) zs − ζ k (cid:17) , (13)with energy eigenvalues E k = − e Φ(0) + ζ k e E s + p ⊥ m e , (14)where s ≡ (cid:20) ¯ h m e e E (cid:21) / , (15)and Ai( − ζ k ) = 0. In our subsequent development we re-strict attention to the ground state, k = 1, for which theappropriate eigenvalue is ζ (cid:39) . We will see in § z o , s (cid:28) a , the po-tential variation through the SSE can be approximated bya step function. The amplitude of the step, δ Φ sse , can bedetermined by integrating Poisson’s equation through theSSE, with the volume charge density given by − eN e | ψ | ,where − eN e is the surface charge density in the SSE. If thecharge density in the solid matrix is small, the appropriatewavefunction is that given in equation (11), whereas equa-tion (13) (with k = 1) is a better approximation for largevalues of N p . In either case the result can be expressed as δ Φ sse = µ e(cid:15) o N e , (16)with the appropriate length scale being µ = 1 . z o for theimage-charge confining potential (exact result), and µ (cid:39) . s for a linear potential (approximate result obtainedfrom numerical integration). The characteristic scale s , inthe wavefunctions of equation (13), decreases as the surfacecharge density increases. Those wavefunctions are appropri-ate to high surface charge densities. Thus the potential stepthrough the SSE can be approximated by equation (16),with µ = min(1 . z o , . s ) . (17)We adopt this prescription in our model of hydrogen graincharging in § c (cid:13) , 1–12 Walker
Coulomb. If the surface density of positive charges embed-ded in the solid is N p (which may be positive or negative),then we haveΦ c ( r ) = ea (cid:15) o r ( N p − N e ) , (18)at radii such that r − a (cid:29) µ . The rate at which charged particles impinge upon the sur-face of an uncharged grain can be written in the same formas equation (1). In the case of a charged grain, the flux ofcharged particles is increased or decreased, depending onthe sign of the charge, because of the associated attrac-tion or repulsion. The magnitude of this effect depends on γ = − V /kT ism , the ratio of electrostatic potential energy atthe grain surface to the characteristic thermal energy in thegas. Henceforth we assume for simplicity that in the inter-stellar gas the positive ions are protons and that they havethe same number density as the electrons. Denoting the ion-isation fraction by x e , the resulting fluxes are (Spitzer 1941): F e,p = P ism (cid:112) πkT ism m e,p (cid:16) x e x e (cid:17) F ( γ e,p ) , (19)where F ( γ ) = 1 + γ, ( γ (cid:62)
0) (20)= exp( γ ) , ( γ < . (21)The net number-density of positive charges lodged inthe solid matrix, N p , evolves according tod N p d t = F p ( γ p ) − F e ( γ e ) , (22)i.e. the flux of protons, minus that of electrons. Defining N p ≡ N p /N , with N := kT ism (cid:15) o /e a , and τ ≡ t/t , with t := N /F e (0), this becomesd N p d τ = (cid:18) m e m p (cid:19) / F ( γ p ) − F ( γ e ) . (23)The relationships between the two γ ’s and the net densityof ions on the surface differ markedly between solid H andmore conventional grain materials. The conventional mate-rials are simpler in this respect and we briefly review thatcase before turning to solid hydrogen. For silicates and graphitic grains, any electrons or ions whichreach the grain surface will stick there and then recombineat some rate. Neglecting the polarisation of the grain, whichcan be important in some circumstances (Draine and Sutin1987), the external potential is Coulomb and is completelydetermined by the net charge on the grain. In this case wehave γ e = N p = − γ p , (24)and we can proceed to solve our differential equation (23),subject to the initial condition N p = 0 at τ = 0.Initially the derivative in equation (23) is approximatelyequal to − Figure 2.
The development of net charge on a grain of conven-tional composition (i.e. silicate or graphite), in the case of purecollisional charging. This graph shows the solution to the differ-ential equation (23), starting from an uncharged grain at t = 0. Table 1.
Collisional charging characteristics in four ISM phasesPhase T ism x e F e (0) N t (K) (cm − s − ) (cm − ) (s)CNM 10 × − × × WNM 5 × × − × × WIM 10 × × HIM 10 × × Conventional abbreviations are used for the phases of the ISM,viz: CNM, Cold Neutral Medium; WNM, Warm Neutral Medium;WIM, Warm Ionised Medium; HIM, Hot Ionised Medium. The nu-merical values of N and t quoted here are appropriate to grainsof radius a = 1 µ m; these quantities both scale ∝ /a . Estimatesof the ionised fraction, x e , are taken from Draine (2011). grain with N p (cid:39) − τ . Subsequently the net negative chargemeans that the electron (proton) flux decreases (increases),so the rate of charging diminishes. At late times the rate ofcharging asymptotically approaches zero, and N p → −N ∞ ,with(1 + N ∞ ) exp( N ∞ ) = (cid:112) m p /m e . (25)The solution to equation (25) is the familiar result for colli-sional grain charging (Spitzer 1941) N ∞ (cid:39) . N p with τ , and table 1 gives the char-acteristic values of N and t for a grain of a = 1 µ m radiusin various phases of the ISM. In table 1, and throughoutthe rest of this paper, we adopt the usual acronyms for ISMphases: CNM, Cold Neutral Medium; WNM, Warm Neu-tral Medium; WIM, Warm Ionised Medium; and HIM, HotIonised Medium. If, contrary to our assumption, the positive ions in the diffuseISM include a significant fraction of metals (which are more mas-sive than protons), then the right-hand-side of equation (25) mayincrease by up to a factor of a few, with the result that N ∞ mayincrease by a few tens of percent.c (cid:13) , 1–12 nowflakes in heaven H grains The collisional charging of solid hydrogen grains differs fromthe behaviour just described. First, in order to enter thesolid, electrons must reach the surface with sufficient energyto overcome the barrier, V , presented by the band structureof the lattice. Secondly, those electrons which have insuffi-cient energy to penetrate the lattice will become trapped inthe potential well just above the surface. We therefore haveto keep track of the surface density of this two-dimensionalelectron gas, N e , in addition to the net density of positivecharges on the grain itself. Finally, the presence of electronsabove the surface causes the external potential to deviatefrom the Coulomb form; we approximate that deviation bythe potential step δ Φ sse given in equations (16), (17).We can compute the particle fluxes for a general po-tential as follows. Suppose the radial profile of the potentialenergy is V ( r ) for a particular species, and that V → r → ∞ . If the energy-at-infinity is E ∞ , then a general ex-pression describing the cross-section for the particle to reachradius a is σ ( a ) = min a (cid:26) πr (cid:18) − V ( r ) E ∞ (cid:19)(cid:27) , (26)where min a {} denotes the minimum value over all radii r (cid:62) a , and it is understood that σ (cid:62)
0. For the poten-tial profile appropriate to solid H ( § E ∞ . Subsequentaveraging over a thermal distribution then leads to exactlythe same outcome as equations (19–21), with γ being deter-mined by the potential energy evaluated at the location ofthe minimum in equation (26).For our problem there are three relevant cross-sections,with their associated γ values: the cross-section for an elec-tron or a proton to reach the surface of the grain, andthe cross-section for an electron to penetrate into the solid.For an electron reaching the surface, the minimum in equa-tion (26) occurs just outside the SSE, at r = a + , whereΦ( a + ) (cid:39) Φ c ( a ), and the Coulomb potential Φ c is as perequation (18). Defining N e ≡ N e /N we find the resultingvalue of γ to be γ + e = N p − N e . (27)For protons reaching the surface of the grain, the minimumin equation (26) occurs at the surface itself, where the po-tential is Φ( a ) = Φ c ( a ) + δ Φ sse . The resulting value of γ isthus γ p = N e − N p − µa N e . (28)Although µ (cid:28) a , the potential step through the SSE isimportant in circumstances where N e (cid:29) | N p − N e | , and thefinal term in equation (28) cannot be neglected.Any positive ion which reaches the grain surface willpenetrate the solid, binding into the matrix just below thesurface. But electrons reaching the surface will not penetratethe matrix unless they have enough energy to climb into theconduction band of the solid. The corresponding value of γ is γ e = N p − N e + µa N e − V kT ism . (29)Strictly, γ e should be the smaller of γ + e and the value inequation (29). The evolution of the net surface charge within the solidmatrix is then determined by equation (23), with γ p and γ e given by equations (28) and (29), respectively. To cal-culate how N p varies, we must simultaneously solve for theevolution of the SSE density, N e . Here we assume that anyelectron reaching the surface with sufficient energy to pene-trate the solid does so, and that all other electrons reachingthe surface lose some kinetic energy on impact and becometrapped in bound vacuum states. We therefore model theSSE density evolution usingd N e d τ = F ( γ + e ) − F ( γ e ) , (30)with γ + e and γ e given by equations (27) and (29), respec-tively. We can now proceed to solve equations (23) and (30)as coupled differential equations. H grains We have obtained numerical solutions to equations (23) and(30), describing the pure collisional charging of solid H grains, in the four different phases of the ISM characterisedin table 1. These solutions are shown in figure 3 for a grainradius a = 1 µ m.To understand how N e and N p evolve it is helpful to ex-amine the initial conditions. At t = 0 we have N e = N p = 0,so γ + e = γ p = 0, and γ e = − V /kT ism . In the high tempera-ture regime ( kT ism (cid:29) V ) we therefore expect a small initialgrowth rate in the SSE density, and a net negative chargeaccumulating in the matrix itself, with N p (cid:39) − τ for τ (cid:28) N e (cid:39) τ , and a net positive charge gradually accumulatingin the matrix, with N p (cid:39) τ (cid:112) m e /m p at early times. Thisbehaviour is seen in figure 3 in the warm and cold phases ofthe ISM.The early-time solutions just described are valid until τ ∼ t ∼ t o ), at which point either N e or N p becomessignificant and the corresponding growth rate is moderated.This leads to an approach to overall charge neutrality, with N e (cid:39) N p .The steady-state conditions implied by equations (23)and (30), at late times, are easily obtained by setting theirright-hand-sides to zero. In the case of equation (30) thisimmediately yields the result N e → N e ( ∞ ) = (cid:15) o V e µ , (31)as τ → ∞ . This condition can also be written as e δ Φ sse = V , (32)i.e. the potential drop through the SSE is equal to the heightof the surface barrier. Stated in that way it is easy to see whyequation (31) is the limiting charge density: any electronwhich reaches the top of the SSE will also penetrate intothe solid, so the SSE density can grow no further. c (cid:13) , 1–12 Walker
Figure 3.
The development of charge density on grains of solidH , of radius a = 1 µ m, in the case of pure collisional charging.The panels show grains located in different phases of the ISM (topto bottom): HIM; WIM; WNM; and CNM (with characteristicsgiven in table 1). The SSE density is shown in red, whereas thedensity of charge in the solid matrix is shown in black if it ispositive, and blue if it is negative. At high charge densities the SSE is confined primarilyby the electric field of the positive ions in the solid matrix, soit is appropriate to calculate the potential drop δ Φ sse usingthe wavefunctions of § µ (cid:39) . s , and with s as given in equation (15) the limitingcharge density N e ( ∞ ) is determined by noting the near-equality between N e ( ∞ ) and N p ( ∞ ). The result is N e ( ∞ ) = 8 . × cm − , (33)corresponding to a limiting surface field E → E max = 1 . × V m − . (34)Excepting the case of very small grains in the HIM (seelater), these limiting conditions are universal.It is worth noting that the field estimate in equation(34) is comparable to the limiting fields which may be ex-pected to arise from (i) field ion emission, which Draineand Sutin (1987, citing Muller and Tsong 1969) give as ∼ × V m − for silicate/graphite grains, and (ii) thespatial gradient in the conduction band energy at the sur-face, which is ∼ V /el , where l is the nearest neighbourdistance in the solid, with l ρ ∗ ∼ m , so this evaluates to ∼ V m − . We do not attempt to model those processeshere, so we cannot say whether either of them imposes alimit that is smaller than the one given in equation (34),but the possibility is acknowledged.Analogous to equation (31), describing the steady-stateconditions for N e , the asymptotic limit for N p is obtainedby requiring the right-hand-side of equation (23) to vanish.In this way we obtain the limit N p → N e − δ N ∞ , where (cid:16) δ N ∞ − V kT ism (cid:17) exp[ δ N ∞ ] = (cid:112) m p /m e . (35)In the high temperature regime, kT ism (cid:29) V , this conditionis approximately the same as equation (25), so that δ N ∞ (cid:39)N ∞ (cid:39) .
5. Whereas in the opposite, low-temperature regimethe asymptotic difference in charge densities can be approx-imated by δ N ∞ (cid:39) V kT ism . (36)Although this value is large compared to unity, it is nev-ertheless a fraction µ/a (cid:28) N e (cid:39) N p at late times.Consequently we observe near coincidence of N e and N p atlate times in the CNM (figure 3).At late times figure 3 shows that the SSE density ( N e ,red curves) indeed approaches N e ( ∞ ), in warm and hot me-dia. In the cold, neutral medium, however, both N e and N p exhibit very slow growth at late times, and have reachedonly ∼ . × their asymptotic values by t (cid:39) N e ∼ N e ( ∞ ) so it haslittle influence on the result.For H grains of different sizes charging proceeds in afashion which is qualitatively similar to that of the a = 1 µm grains shown in figure 3. The main quantitative change withgrain size is that N e and N p converge at earlier times in thecase of larger grains. That is because grain neutralisation isa response to the Coulomb potential becoming significant,and a given Coulomb potential corresponds to N e ∝ /a . c (cid:13) , 1–12 nowflakes in heaven In this paper we have restricted attention to grainslarger than 0 . µ m. For smaller grains in the HIM the elec-trostatic potential energy associated with an unshieldedcharge density | N p | ∼ N e ( ∞ ) may be small compared to kT ism . In this circumstance hydrogen grains charge up ina manner similar to silicate or graphite grains of the samesize (i.e. as per figure 2), and the SSE is largely irrelevant.The net charge on such grains is negative, and their surfaceelectric fields are stronger than the value given in equation(34).The asymptotic limit in equation (31) is different inform, and may be much larger than that obtained in § µ m is expectedto acquire a (negative) surface charge density ∼ cm − ,almost four orders of magnitude less than the limit for hy-drogen grains. Because of the electric field at the surface of the grain, sub-limating molecules are polarised, with an associated elec-trostatic energy E pol <
0. At high charge densities, where N e (cid:39) N p , | E pol | falls rapidly to zero as a molecule is movedoutward through the SSE. Sublimation from a polarised sur-face therefore requires an additional energy b pol = − E pol tobe supplied to each sublimating molecule. The sublimationrate and saturated vapour pressure are then given by equa-tions (3) and (2), but with the replacement b o → b o + b pol .We now evaluate b pol . The molecular polarisability, α , of H was calculated to highprecision by Ko(cid:32)los and Wolniewicz (1967). For the rovibra-tional ground state they obtain (in atomic units) α (cid:107) = 6 . , α ⊥ = 4 . , (37)parallel and perpendicular, respectively, to the molecularaxis. If there is no preferred orientation of the moleculesone works with the spherical average of these values, (cid:104) α (cid:105) =( α (cid:107) + 2 α ⊥ ) /
3. But in our case the electric fields, E , are suf-ficiently strong that the magnitude of the molecular polari-sation energy | E pol | = 12 (cid:8) α (cid:107) E (cid:107) + α ⊥ E ⊥ (cid:9) , (38)is large compared to the typical thermal energy in the solid,and the surface molecules will preferentially orient them-selves so that E ⊥ (cid:28) E (cid:107) (cid:39) E .In § E max (cid:39) . × V m − . From thiswe can now evaluate the electrostatic contribution to theheat of sublimation, it is b pol = − E pol (cid:39) α (cid:107) E max (cid:39)
84 meV . (39) To convert to SI units (F m ), multiply by 1 . × − . Alignment also alters the van der Waals forces between neigh-bouring molecules, but in this paper we neglect that effect.
Figure 4.
As figure1, but for the case of a fully charged H sur-face. Note that P sat (cid:28) P ism at all temperatures below the triple-point, suggesting that grains of H may be able to live indefinitelyin the diffuse ISM. Our estimated radiative equilibrium tempera-ture (i.e. neglecting sublimation cooling) for a micron-sized grainof H is 7 . § Hence b pol /k (cid:39)
980 K. This is an order of magnitude largerthan b o , demonstrating that grain charging has a profoundeffect on the thermodynamics of the surface.Figure 4 shows the influence of b pol /k = 980 K on thesaturated vapour pressure of H . Comparing to figure 1 wesee that saturation for the charged surface occurs at pres-sures which are many orders of magnitude below that for theuncharged case. Indeed figure 4 shows that our estimate ofthe saturated vapour pressure for a fully charged surface liesmany orders of magnitude below P ism at all temperaturesbelow the triple-point.We caution that our estimate of E max obtained in § b pol varies as the square of E max so the partic-ular value we have obtained should not be given undue em-phasis. Nevertheless our calculation serves to demonstratethat grain charging plays a critical role in determining thesublimation rate of interstellar solid hydrogen, and opensup the possibility of H grains surviving indefinitely in thediffuse ISM. Due to a lack of information on yields, we neglected photo-electric charging. We now estimate how small the photoelec-tric yield would have to be in order for that approximationto be good. Because the band-gap of solid H is greaterthan the ionisation energy of atomic hydrogen, the number-density of photons in the ISRF which can excite electronsinto the conduction band is very small. Therefore the contri-bution of such photons to the charging rate is limited even iftheir photoelectron yield is high. Far-UV photons are muchmore numerous up to energies of one Rydberg, and someof those photons may yield photoelectrons because the ex-citonic states which they induce in the solid lie above thevacuum free-electron energy — excitons could decay if theydiffuse to the surface of the H crystal. Using the far-UVISRF energy density given in § . × η cm − s − , where η is the yield. Comparingthis with the lowest collisional rate (i.e. that for the CNM; c (cid:13) , 1–12 Walker table 1), we see that we need η (cid:28) × − for photoelec-tric emission to be negligible. We also note, from figure 3,that after t ∼ s the collisional charging rate in the CNMdeclines substantially, and in that regime of the chargingprocess the yield would have to be much smaller still inorder for photoemission to be negligible. Laboratory inves-tigation of the far-UV photoelectric yield of solid H wouldbe valuable.Figure 3 shows that pure collisional charging in theCNM does not permit a micron-sized grain to reach the fullycharged state within the sublimation time-scale (6 × s,see § , which leaves the charged species in placebut decreases the surface area of the grain. Thus even in thecase of a hydrogen grain in the CNM with zero photoemis-sion, the particle will not disappear but will shrink until thecharge density is high enough to suppress the sublimationrate.Our finding that charging has a strong influence on theviability of hydrogen grains motivates improvements in thedescription of charging. Notable deficiencies of the treatmentwe have given include our use of a single-particle descriptionof the density distribution, and approximating the band-structure of the solid by an infinite potential wall. Thesedeficiencies could be addressed by the application of Den-sity Functional Theory. Spatial structure in the plane of thesurface would also be worth investigating, as the confiningelectric field provided by the sub-surface ions is only ap-proximately uniform. The non-uniformities are interestingbecause they imply spatial variations in the heat of sublima-tion of H , which we have not allowed for in our model. Theyare also interesting in respect of the possibility of confine-ment of individual electrons in the vicinity of individual ions— forming a new, and highly unusual sort of “atom”. Finallywe note that the photo-detachment of surface-state electronsand their tunneling recombination with sub-surface ions areprocesses whose rates should be quantified. In this paper wehave implicitly assumed that these rates are negligible incomparison with the collisional charging rates.It would be valuable to have a better estimate of theradiative equilibrium temperature of a hydrogen grain. Akey area of uncertainty in that calculation lies with the cou-pling of the grain to microwave radiation. Our treatment( § λ = 1 µ m, yielding an absorption efficiency (cid:39)
5% that of asilicate grain of the same size). The surface charge structurewhich develops on solid H will enhance the microwave cou-pling of hydrogen grains in two ways. First, we expect stronginhomogeneities in the electric field within the layer of ionsbound inside the solid. Consequently any lattice vibrationsin that layer will induce large, oscillating electric dipoles andwill thus radiate more efficiently than an uncharged grain.Secondly, the surface-state electrons are expected to couplestrongly to the microwave radiation field and will contributeto absorption as a result of excitation into higher quantumstates.Sublimation is not the only process which might limitthe lifetime of interstellar grains. Sputtering, which we havecompletely neglected, could play an important role. Therehave been some laboratory studies of the sputtering of solid H by energetic particles (Pedrys et al 1997; Schou 2002),which could provide a basis for estimating its effect in an in-terstellar context. However, the experimental investigationshave presumably been undertaken under conditions of lowsurface charge density. It is possible that sputtering couldbe strongly suppressed under conditions of maximal charge,so further laboratory studies may be needed. The surface-state electrons described in § surface, even though the bulkmaterial is itself an insulator, and a strong interaction withradio-waves is expected. The volume fraction occupied bydust in most of the ISM is small enough (Purcell 1969) thatwe don’t expect dust to have much influence on radio-wavepropagation in typical regions of interstellar space. However,one can imagine atypical regions containing neutral heliumgas (which is difficult to detect), plus a substantial volumefraction of hydrogen grains, and in such regions H partic-ulates could make a significant contribution to the radio-wave refractive index. Moreover they would do so withoutany contribution to the fluid pressure. Thus hydrogen grainsmay offer an explanation for circumstances where heavyscattering arises from regions of small spatial extent. Threesuch phenomena are known: the Extreme Scattering Events(Fiedler et al 1987, 1994); the intra-hour variations of com-pact radio quasars (Kedziora-Chudczer et al 1997; Dennett-Thorpe and de Bruyn 2000; Bignall et al 2003; Lovell et al2008); and the pulsar parabolic arcs (Stinebring et al 2001;Cordes et al 2006). Indeed hydrogen dust could give rise toapparent source flux variations across a broad range fromradio to X-ray, and we note that panchromatic variability isobserved in some quasars and BL Lac objects (e.g. Wagnerand Witzel 1995).We have already noted the possibility of forming a newtype of “atom” on the surface of solid H , with specific SSEelectrons bound to specific, sub-surface positive ions. Theseelectrons would be localised in all three dimensions andwould thus have a discrete spectrum of eigenstates, ratherthan the continua associated with a two-dimensional elec-tron gas (equations 10 and 14). We make no attempt hereto calculate the eigenstates of any fully-localised electrons.We simply draw attention to the implication that thereshould be a set of spectral lines which are characteristicof the charged surface of any interstellar H grains. Giventhat electrons are separated from positive ions by several˚Angstr¨oms (i.e. at least one crystal plane), the characteris-tic energies should be a fraction of those for the hydrogenatom. But they cannot be arbitrarily small, because at largeelectron-ion separations the field of other ions becomes rele-vant and there will be no unique pairing of a specific electronwith a specific ion. Therefore the spectroscopic signature ofthese “atoms” will be confined to a spectral band whose up-per boundary lies at photon energies of a few eV. The linesshould not be expected to be sharp because there are per-turbations from adjacent ions and from other SSE electrons;tunneling phenomena could also make a significant contri-bution to linewidth. These aspects are reminiscent of theDiffuse Interstellar Bands (Herbig 1995; Sarre 2006).In contrast to models based on silicates and graphites c (cid:13)000
5% that of asilicate grain of the same size). The surface charge structurewhich develops on solid H will enhance the microwave cou-pling of hydrogen grains in two ways. First, we expect stronginhomogeneities in the electric field within the layer of ionsbound inside the solid. Consequently any lattice vibrationsin that layer will induce large, oscillating electric dipoles andwill thus radiate more efficiently than an uncharged grain.Secondly, the surface-state electrons are expected to couplestrongly to the microwave radiation field and will contributeto absorption as a result of excitation into higher quantumstates.Sublimation is not the only process which might limitthe lifetime of interstellar grains. Sputtering, which we havecompletely neglected, could play an important role. Therehave been some laboratory studies of the sputtering of solid H by energetic particles (Pedrys et al 1997; Schou 2002),which could provide a basis for estimating its effect in an in-terstellar context. However, the experimental investigationshave presumably been undertaken under conditions of lowsurface charge density. It is possible that sputtering couldbe strongly suppressed under conditions of maximal charge,so further laboratory studies may be needed. The surface-state electrons described in § surface, even though the bulkmaterial is itself an insulator, and a strong interaction withradio-waves is expected. The volume fraction occupied bydust in most of the ISM is small enough (Purcell 1969) thatwe don’t expect dust to have much influence on radio-wavepropagation in typical regions of interstellar space. However,one can imagine atypical regions containing neutral heliumgas (which is difficult to detect), plus a substantial volumefraction of hydrogen grains, and in such regions H partic-ulates could make a significant contribution to the radio-wave refractive index. Moreover they would do so withoutany contribution to the fluid pressure. Thus hydrogen grainsmay offer an explanation for circumstances where heavyscattering arises from regions of small spatial extent. Threesuch phenomena are known: the Extreme Scattering Events(Fiedler et al 1987, 1994); the intra-hour variations of com-pact radio quasars (Kedziora-Chudczer et al 1997; Dennett-Thorpe and de Bruyn 2000; Bignall et al 2003; Lovell et al2008); and the pulsar parabolic arcs (Stinebring et al 2001;Cordes et al 2006). Indeed hydrogen dust could give rise toapparent source flux variations across a broad range fromradio to X-ray, and we note that panchromatic variability isobserved in some quasars and BL Lac objects (e.g. Wagnerand Witzel 1995).We have already noted the possibility of forming a newtype of “atom” on the surface of solid H , with specific SSEelectrons bound to specific, sub-surface positive ions. Theseelectrons would be localised in all three dimensions andwould thus have a discrete spectrum of eigenstates, ratherthan the continua associated with a two-dimensional elec-tron gas (equations 10 and 14). We make no attempt hereto calculate the eigenstates of any fully-localised electrons.We simply draw attention to the implication that thereshould be a set of spectral lines which are characteristicof the charged surface of any interstellar H grains. Giventhat electrons are separated from positive ions by several˚Angstr¨oms (i.e. at least one crystal plane), the characteris-tic energies should be a fraction of those for the hydrogenatom. But they cannot be arbitrarily small, because at largeelectron-ion separations the field of other ions becomes rele-vant and there will be no unique pairing of a specific electronwith a specific ion. Therefore the spectroscopic signature ofthese “atoms” will be confined to a spectral band whose up-per boundary lies at photon energies of a few eV. The linesshould not be expected to be sharp because there are per-turbations from adjacent ions and from other SSE electrons;tunneling phenomena could also make a significant contri-bution to linewidth. These aspects are reminiscent of theDiffuse Interstellar Bands (Herbig 1995; Sarre 2006).In contrast to models based on silicates and graphites c (cid:13)000 , 1–12 nowflakes in heaven (e.g. Zubko, Dwek and Arendt 2004), there is no difficulty inexplaining the requisite volume fraction of interstellar dust(Purcell 1969) if it is made of hydrogen. Furthermore thesize spectrum required for hydrogen grains to fit the ex-tinction data could be very different to that needed for sil-icate/graphite grains (Mathis, Rumpl and Nordsieck 1977).For example, hydrogen dust in the ISM might be much big-ger than the predominantly ∼ . µ m grains which are re-quired by silicate/graphite models, and might contain muchmore mass in total. If so, hydrogen grains may be able to ex-plain spacecraft and radar observations of interstellar dustparticles streaming through the Solar System (Gr¨un et al1993; Taylor, Baggaley and Steel 1996; Landgraf et al 2000;Meisel, Janches and Matthews 2002; Mann 2010; Musci etal 2012). These data indicate large numbers of large par-ticles ( (cid:38) µ m) amongst the interstellar grains, and are intension with silicate/graphite models both in respect of theabundances of the constituent elements and the shape of theinterstellar extinction curve (Draine 2009). All types of interstellar dust grains are expected to acquirecharge, but the “double layer” charge configuration of solidH is unique amongst candidate grain materials. Electronsbound in vacuum states above the surface shield the positivecharge of ions buried in the solid, so that grains can maintainoverall neutrality and charging can continue to high surface-densities of both species. Interstellar grains of solid H arethus expected to develop large electric fields within a few˚Angstr¨oms of their surfaces. Polarisation of H moleculesenhances their binding to the solid, and for a fully chargedsurface the heat of sublimation is greatly increased. Charg-ing must therefore be taken into account when determiningthe sublimation rate and lifetime of solid H in astrophysicalenvironments. It is a mistake to exclude hydrogen grains onthe basis of the volatility of the pure solid.Electrons bound to the grain surface may be localised inonly one dimension, or in all three. The former case yields atwo-dimensional free-electron gas, giving the grain surface ametallic character. Electrons which are localised in all threedimensions possess a discrete spectrum of eigenstates, im-plying that H grains may be identified spectroscopically. ACKNOWLEDGMENTS
I thank Sterl Phinney for making available a copy of hisunpublished manuscript “Cosmic Snowballs” (1985), StefanBromley for an illuminating conversation on the modelling ofthe SSE, Artem Tuntsov for some good discussions on all therelevant physics, and Oxford Astrophysics for hospitality.
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