Photoelectric Emission from Dust Grains Exposed to Extreme Ultraviolet and X-ray Radiation Joseph C. Weingartner1, B. T. Draine 2,3, and David K. Barr1 ABSTRACT 6 0 0 Photoelectric emission from dust plays an important role in grain charging 2 n and gas heating. To date, detailed models of these processes have focused pri- a marily on grains exposed to soft radiation fields. We provide new estimates of J 3 the photoelectric yield for neutral and charged carbonaceous and silicate grains, 1 for photon energies exceeding 20eV. We include the ejection of electrons from 1 both the band structure of the material and the inner shells of the constituent v atoms, as well as Auger and secondary electron emission. We apply the model to 6 9 estimate gas heating rates in planetary nebulae and grain charges in the outflows 2 1 of broad absorption line quasars. For these applications, secondary emission can 0 be neglected; the combined effect of inner shell and Auger emission is small, 6 0 though not always negligible. Finally, we investigate the survivability of dust / h entrained in quasar outflows. The lack of nuclear reddening in broad absorption p line quasars may be explained by sputtering of grains in the outflows. - o r t s Subject headings: ISM: dust a : v i X 1. Introduction r a Cosmic grains are subjected to electrical charging processes and the resulting non-zero charge can have important astrophysical consequences. In Galactic H I regions, the grain chargeis primarily determined by a balancebetween starlight-induced photoelectric emission and the collisional capture of electrons from the gas. Furthermore, in these regions the gas heating is dominated by photoelectric emission from dust. Thus, there have been numerous detailed studies of UV-induced photoelectric emission (Spitzer 1948; Watson 1972; de Jong 1Department of Physics and Astronomy, George Mason University, MSN 3F3, 4400 University Drive, Fairfax, VA 22030,USA; [email protected], [email protected] 2Princeton University Observatory,Peyton Hall, Princeton, NJ 08544,USA; [email protected] 3 Osservatorio Astrofisico di Arcetri, Largo E Fermi 5, 50125 Firenze, Italy – 2 – 1977; Draine 1978; Tielens & Hollenbach 1985; Bakes & Tielens 1994; Weingartner & Draine 2001b). Photoelectric emission from grains exposed to extreme ultraviolet (EUV) and X-ray radiation has received much less attention. Dwek & Smith (1996) modelled, in detail, the dust and gas heating for neutral grains exposed to high-energy radiation, but did not address the grain charging. Charging by high-energy photons can be very important for grains immersed in hot plasma and exposed to a hard radiation field. The grain potential is limited by the highest photon energy present in the incident radiation; as this increases, the grain can reach higher potentials. The efficiency of the gas heating decreases as the grain charge increases, since the photoelectrons have to climb out of the potential well. Hard radiation fields can be found, e.g., in planetary nebulae and active galactic nuclei (AGNs). If the gas contains dust, then a fraction of the incident radiation will be processed into infrared (IR) radiation, i.e., thermal dust emission. Dwek & Smith (1996) were moti- vated by the prospect of learning about these environments through the analysis of the IR emission. Ferland et al. (2002) stressed that the X-ray and IR spectra in AGNs may be correlated as a result of the interaction between hot grains and gas. Detailed modelling of these regions will only be possible once the grain charging is properly modelled, since the grain and gas heating rates depend on the charging. In a previous paper (Weingartner & Draine 2001b, hereafter WD01), we modelled in detail the photoelectric emission from grains exposed to UV radiation; this is briefly sum- marized in §2. In §§3 through 8, we extend the WD01 model to include EUV and X-ray radiation. Applications to planetary nebulae and quasar outflows are described in §9 and §10, respectively. 2. Photoelectric Emission Induced by Low-Energy Photons As a starting point for this study, we will adopt the WD01photoelectric emission model. Here, we briefly discuss some of the important features of this model and mention two minor modifications. The photolectric emission rate depends on the absorption cross section and the photo- electric yield Y (i.e., the probability that anelectron will be emitted following the absorption of a photon), both of which depend on the photon energy hν. WD01assumed spherical grains, forwhichtheabsorptioncross sectionisQ πa2, where abs – 3 – a is the grain radius and Q is the absorption efficiency factor. They used Mie theory to abs compute Q and adopted dielectric functions from Li & Draine (2001) and Weingartner & abs Draine (2001). Here, we will use Mie theory to compute Q when x ≡ 2πa/λ < 2 × 104 abs (λ is the wavelength of the radiation) and anomalous diffraction theory when x > 2× 104 (Draine 2003). We also will use somewhat newer dielectric functions (Draine 2003). The WD01 prescription for determining the threshold photon energy for photoelectric emission, hν , is discussed in their §§2.2 and 2.3.1 and equations 2 through 7. Here, we will pet adopt a different expression for the minimum energy that an electron must have to escape a negatively-charged grain, E . Instead of equation 7 from WD01, we adopt equation 1 min from van Hoof et al. (2004).1 Thus, hν for photoelectric emission from the band structure pet is taken to be IP (Z,a) , Z ≥ −1 V hν (Z,a) = (1) pet IP (Z,a)+E (Z,a) , Z < −1 (cid:26) V min with the “valence band ionization potential” 1 e2 e20.3˚A IP (Z,a) = W + Z + +(Z +2) (2) V 2 a a a (cid:18) (cid:19) and 0 , Z ≥ −1 E = ; (3) min θ (ν = |Z +1|) 1−0.3(a/10˚A)−0.45|Z +1|−0.26 , Z < −1 ν (cid:26) W is the work function and θν is de(cid:2)fined in equation 2.4 of Draine &(cid:3)Sutin (1987). We retain the WD01 estimates of W = 4.4eV (8eV) for carbonaceous (silicate) grains. WD01adopted the following expression for the photoelectric yield of a grainwithcharge Ze (their equation 12; e is the proton charge): Y(hν,Z,a) = y (hν,Z,a)×min[y (Θ)y (a,hν),1] , (4) 2 0 1 where hν −hν +(Z +1)e2/a , Z ≥ 0 pet Θ = (5) hν −hν , Z < 0 (cid:26) pet (WD01 equation 9). For a bulk solid, Y(hν) = y (Θ = hν−W). WD01 present approxima- 0 tions, derived from laboratory measurements, for y (Θ) for carbonaceous and silicate grains 0 (their equations 16 and 17, respectively). 1Since van Hoof et al. (2004) define Emin as the negative of our Emin, the minus sign in their equation 1 is absent here. – 4 – We adopt the size-dependent yield enhancement factor y used by WD01: 1 β 2 α2 −2α+2−2exp(−α) y (a,hν) = (6) 1 α β2 −2β +2−2exp(−β) (cid:18) (cid:19) where β ≡ a/l and α ≡ a/l +a/l . It depends on the photon attenuation length, l (the a a e a e-folding length for the decrease of radiation intensity as it propagates into the material) and the electron escape length, l (roughly the distance that the electron travels in the material e before losing its energy). The electron escape length depends on the energy of the excited photoelectron; WD01 adopted l = 10˚A in all cases, which is a reasonable approximation e for the low-energy electrons excited by visible and UV radiation. Finally, the factor y (hν,Z,a) accounts for the fact that not all of the electrons that 2 breach the surface barrier have sufficient energy to escape to infinity. WD01 assumed a parabolic electron energy distribution: 6(E −E )(E −E) f0(E) = low high , E ≤ E ≤ E , (7) E (E −E )3 low high high low where f0(E)dE gives the fraction of attempting electrons with energy (with respect to E infinity) between E and E +dE. When Z < 0, E = E and E = E +hν −hν ; low min high min pet when Z ≥ 0, E = −(Z + 1)e2/a and E = hν − hν . The fraction of attempting low high pet electrons that escape to infinity is given by EhighdEf0(E) = E2 (E −3E )/(E −E )3 , Z ≥ 0 y (hν,Z,a) = 0 E high high low high low . (8) 2 1 , Z < 0 (cid:26)R 3. Photoelectric Emission Induced by High-Energy Photons Photoelectric emission induced by high-energy photons differs from that induced by low-energy photons in the following ways: 1. Whereaslow-energy photonscanonlyexcite electronsfromthebandstructure ofthesolid, high-energy photons can also excite inner shell electrons. Except very near the absorption edge (see, e.g., Draine 2003) the absorption cross section from inner shell electrons will be essentially identical to that for isolated atoms. Dwek & Smith (1996) find that the transition between band-like and atomic-like absorption typically occurs for photon energies around 50eV. However, laboratory-measured photoelectron yields are only available for hν up to ≈ 20eV. 2. A sufficiently energetic photon can excite a photoelectron from atomic shells other than the highest occupied shell (e.g., from the 1s shell in C if hν > 291eV). The hole left by – 5 – the photoelectron can then be filled in a radiationless transition in which a second electron fills the vacancy produced by the photoionization, with the excess binding energy going into kinetic energy of a third electron, which then leaves the atom (the Auger effect). The Auger electron can then possibly leave the grain. This process can give rise to a cascade of secondary Auger transitions that fill holes produced in previous Auger transitions. Thus, more than one electron can be emitted from a grainfollowing the absorption of a high-energy photon. 3. Following the absorption of a high-energy photon, the photolectrons and Auger electrons might have enough energy to excite secondary electrons from the grain. In the following sections, we will discuss the emission of primary photoelectrons, Auger electrons, and secondary electrons. 4. Emission of Primary Photoelectrons 4.1. Bulk Yields 4.1.1. Physical Model In the case of high-energy radiation, we assume atomic-like absorption and estimate the bulk yield of photoelectrons ejected from each shell of each atomic constituent of the grain: y (i,s;Θ ) is the probability that a primary photoelectron excited from shell s of element i 0 i,s escapes the bulk solid following the absorption of a photon of energy hν = Θ +I , where i,s i,s I is the ionization energy of shell s of element i. i,s To estimate the bulk photoelectron yields, we adopt a semi-infinite slab geometry and assume that a photoelectron escapes the solid with probability 0.5 exp(−z/l ), where z is e the perpendicular distance between the surface and the point of excitation. The factor 0.5 accounts for the fact that only half of the photoelectrons will travel towards the surface of the solid, if they are emitted isotropically. For simplicity, we ignore refraction and the fact that the reflectivity varies with the angle of incidence, θ. For an isotropic incident radiation field, π/2 ∞ dx 1 cosθ l l e a y (i,s;Θ ) = n σ l dθsinθcosθ exp −x + = n σ l 1− ln 1+ ; 0 i,s i i,s a i i,s e l l l l l Z0 Z0 a (cid:20) (cid:18) a e (cid:19)(cid:21) (cid:20) a (cid:18) e(cid:19)(cid:21) (9) x is the distance along an incident ray from the surface and n σ l is the probability that i i,s a absorption is by shell s of element i. In evaluating σ l , the photon energy hν = Θ +I ; i,s a i,s i,s in evaluating l , the initial energy of the excited photoelectron E = hν−I = Θ . To find e e i,s i,s – 6 – the bulk yield of photoelectrons ejected from the band structure, y (band,Θ ), we sum 0 band y (i,s) over all of the shells that comprise the band, except that we take E = Θ and 0 e band hν = Θ + W. We take photoionization cross sections from Verner & Yakovlev (1995) band and Verner et al. (1996), making use of the FORTRAN routine phfit2.2 The photon absorption length l = λ/(4πImm), where λ is the wavelength in vacuo a and m(λ) is the complex refractive index, can be approximated by l−1(hν) = n σ (hν) . (10) a i i,s i,s X For consistency with our atomistic approach to photoelectric emission, we will use the ap- proximation when hν > 20eV. 4.1.2. Electron Escape Length WD01 took l = 10˚A for hν . 20eV, in approximate agreement with experiments on e C (Martin et al. 1987) and SiO (McFeely et al. 1990) films. For high initial energies E , the 2 e electron escape length is roughly given by (Draine & Salpeter 1979) −0.85 1.5 ρ E l (E ) ≈ 300˚A e , 300eV < E < 1MeV , (11) e e gcm−3 keV e (cid:18) (cid:19) (cid:18) (cid:19) where ρ is the density of the material. Extending equation (11) below 300eV, we would find l = 10˚A at E = 164eV for carbonaceous grains (assuming the ideal graphite density e e of ρ = 2.24gcm−3) and at E = 211eV for silicates (assuming ρ = 3.5gcm−3). The upturn e in l at ∼ 200eV for C is in rough agreement with the results of Martin et al. (1987), who e considered E up to 1keV. Thus, we adopt e 10˚A , E ≤ 164eV l (E ; carbonaceous) = e (12) e e 4.78×10−3˚A(E /eV)1.5 , E > 164eV (cid:26) e e and 10˚A , E ≤ 211eV l (E ; silicate) = e . (13) e e 3.27×10−3˚A(E /eV)1.5 , E > 211eV (cid:26) e e 2The subroutine phfit2 was written by D. A. Verner and is available at http://www.pa.uky.edu/∼verner/fortran.html. – 7 – 4.1.3. Carbonaceous Grains We adopt the ideal graphite density of 2.24gcm−3; thus, the C atom number density n = 1.12×1023cm−3. There are strong correlations between the 2s and 2p shells, with no C jump in the photoionization cross section at the 2s threshold (Verner et al. 1996). Thus, we treat (2s + 2p) as a single shell, with ionization energy equal to that of the 2p shell = 11.26eV. (See Table 1 for the ionization energies of relevant elements.) The bulk yield computed using equation (9) is displayed as the short-dashed curve in Figure 1. Since the (2s + 2p) shell produces the band structure, this high-energy yield curve should connect continuously with the low-energy curve from WD01 (their equation 16), which is displayed asthelong-dashedcurve inFigure1. Toenforce thiscontinuity, we adopttheyield computed here when hν > 50eV, the WD01 yield when hν < 20eV, and interpolate between these when 20eV < hν < 50eV. The result is displayed as the solid curve in Figure 1. 4.1.4. Silicate Grains For silicate grains, we adopt a stoichiometry approximating MgFeSiO and a density of 4 3.5gcm−3, intermediate between the values for crystalline forsterite (Mg SiO , 3.21gcm−3) 2 4 and fayalite (Fe SiO , 4.39gcm−3). The atomic number densities are thus n = n = 2 4 Mg Fe n = 1.22×1022cm−3 and n = 4.88×1022cm−3. Si O As with C (2s + 2p), O (2s + 2p), Si (3s + 3p), and Fe (3d + 4s) do not display jumps in the photoionization cross section at the threshold energy of the deeper shell. Thus, we treat each of these pairs as a single shell, with ionization energy equal to that of the lower- energy shell. The O 2s and 2p, Si 3s and 3p, Mg 3s, and Fe 3d and 4s shells are considered to comprise the silicate band structure. As with graphite, we adopt the WD01 yield when hν < 20eV, the yield computed here when hν > 50eV, and interpolate between these when 20eV < hν < 50eV. The resulting yield is displayed in Figure 2. 4.2. Size-Dependent Yield For photoelectric emission from the band structure of carbonaceous and silicate grains, we compute y (hν,a) and y (hν,Z,a) using the WD01 prescription, as modified in §2 above 1 2 (eqs. 6 and 8). We estimate y (Θ ) as discussed in §§4.1.3 and 4.1.4; Θ is computed 0 band band using equation (5). Equation (4) is employed to compute the yield, except that the 1 in the min function is replaced by the probability P that the photon absorption occurs in the band – 8 – band structure, rather than in an inner shell. We estimate band P (hν) ≈ l (hν) n σ (hν) , (14) band a i i,s i,s X where the sum is over all shells that are taken to comprise the band structure (see §§4.1.3 and 4.1.4). The yield for photoelectric emission from the inner shells is computed in exactly the same way, except that the work function is replaced by the appropriate ionization energy when computing the photoelectric threshold energy, hν (which is needed to evaluate Θ ; pet i,s see eq. 5). Also, P is replaced with P = l n σ . band i,s a i i,s Verysmallcarbonaceousgrainsaretakentobepolycyclicaromatichydrocarbons(PAHs). For these, we expect Y → 1 as hν → ∞ for the electrons associated with the “band struc- ture”. This applies by construction for the WD01 yields, but does not apply to the yields derived here. Thus, when a ≤ 6˚A (corresponding to ≈ 100 C atoms; see eq. 1 in WD01), we employ y from WD01. When 6˚A < a ≤ 13˚A, we take 0 13˚A−a a−6˚A y = y (WD01) +y (this work) . (15) 0,band 0,band 7˚A 0,band 7˚A 5. Emission of Auger Electrons Supposeaprimaryphotoelectronisejectedfromshellsofelement i. Thereareanumber of Auger transitions that can then occur; we will denote them with index j. Adopting the model for electron escape from §4.1.1, the bulk yield for the j-th Auger transition is l l e a y (i,s,j;Θ ) = p n σ l 1− ln 1+ , (16) 0,A A;i,s A,i,s,j i i,s e l l (cid:20) a (cid:18) e(cid:19)(cid:21) where p is the average number of electrons ejected from the atom via Auger transition A,i,s,j j, photon energy hν = Θ + I is used in evaluating σ and l , and the energy E A;i,s i,s i,s a A,i,s,j of the Auger electron is used when evaluating l . We adopt the values of p and E e A,i,s,j A,i,s,j given in Tables 4.1 and4.2 of Dwek & Smith (1996). Note that the Auger yield is the average number of electrons emitted by the grain via the given Auger transition, rather than the probability of electron emission, and may exceed unity. For all grain sizes and charges, we take the threshold photon energy equal to I , even i,s thoughit should really be somewhat less thanthis, due to thepresence ofthe band structure. We use equation (4) to evaluate the Auger electron yield Y , except that the 1 in the min A,j – 9 – function is replaced with Y = P p . We also adopt somewhat different values A,i,s,j;max i,s A,i,s,j of the input energies than for the case of primary photoelectrons. In evaluating y , we 0,A take Θ = hν − I , regardless of grain size and charge. In computing y (hν,a), we A;i,s i,s 1 take E = E (when evaluating l ). We use equation (8) to compute y (hν,Z,a) with e A,i,s,j e 2 E = E − (Z + 1)e2/a; E = E when Z < 0 and E = −(Z + 1)e2/a when high A,i,s,j low min low Z ≥ 0. For Auger electrons, it can occur that E < 0; in this case, we set y = 0. high 2 6. Emission of Secondary Electrons The secondary electron yield is defined as the average number of secondary electrons emitted per absorbed photon; it may exceed unity. This yield can be expressed as a sum of partial yields (denoted with index k), with a term for each type of process that can excite a secondary electron (namely, primary emission from the band structure and inner shells and Auger emission): Y (hν,Z,a) = Y (hν,Z,a) . (17) sec sec,k k X Since secondary electron energies are generally low, only those excited close to the surface have significant probability of escape. Thus, we take Y = 0 if Y = 0; otherwise, we sec,k k approximate ysec(hν,Z,a)E l−1(E ) Y (hν,Z,a) = Y (hν,Z,a) 2,k e,k e e,k (18) sec,k k y (hν,Z,a) ǫ a−1 +(10˚A)−1 2,k where E = hν − W for primary electrons excited from the band structure, hν − I for e,k i,s primary electrons excited from inner shell (i, s), and E for an Auger electron; ǫ is A,i,s,j the average energy loss by the exciting electron per secondary electron created in the solid. Draine & Salpeter (1979) estimate that ǫ = 117eV for graphite and 155eV for lunar dust; we will adopt these values for carbonaceous and silicate grains. The grain size a is included in the last factor to account for the fact that the exciting electron may escape the grain before producing secondary electrons. For the secondary electron energy distribution, we adopt αE˜−2(E −E ) f0(E) = low (19) E 3/2 1+E˜−2(E −E )2 low h i (Draine & Salpeter 1979) with E˜2 = 8eV2 and normalization factor −1 −1/2 α = 1− 1+E˜−2(E −E )2 . (20) high low (cid:26) (cid:27) h i – 10 – Thus, −1/2 −1/2 α 1+E˜−2E2 − 1+E˜−2(E −E )2 , Z ≥ 0 ysec(hν,Z,a) = low high low 2,k  (cid:26) (cid:27) 1 h i h i , Z < 0  (21) when E > 0; otherwise, ysec = 0. For secondary electrons excited by a primary photoelec- high 2,k tron, we take E and E to be the same as for the primary; likewise for those excited by low high Auger electrons. Consequently, the threshold photon energies for the emission of secondary electrons are equal to those for the electrons that excite them. Figure 3 displays the yields of primary, Auger, and secondary electrons for carbonaceous grains with a = 0.1µm. The primary yield Y reaches unity at hν = 104eV, where both p l and l are very large. However, Q Y is a decreasing function at high photon energy. a e abs p Figures 4 and 5 display the total yield (primary plus Auger plus secondary) for carbonaceous and silicate grains of various sizes. 7. Total Photoelectric Emission Rate and Grain Charging The primary photoelectric emission rate (excluding that due to excess “attached” elec- trons on negatively charged grains) is νmax cu J = πa2 dν νQ Y + Y (22) pe abs p;band p;i,s hν Zνpet(band) i,s ! X (c.f. eq. 25 in WD01); hν is the highest photon energy in the incident radiation field and max Y denotes the yield of primary electrons. For negatively charged grains, the excess electrons p undergo photodetachment at a rate νmax cu ν J = dν σ ; (23) pd pdt hν Zνpdt the photodetachment threshold energy and cross section are taken from equations 18 and 19 in WD01, respectively. The Auger electron emission rate is νmax cu J = πa2 dν νQ Y , (24) A abs A;i,s,j hν ZIis,min/h i,s,j X where I is the lowest ionization energy among those of the inner shells. Of course, for all is,min shells the yields are zero when hν < hν . The emission rate of secondary electrons (excited pet