CONDENSED MATTER ASTROPHYSICS: A PRESCRIPTION FOR DETERMINING THE SPECIES-SPECIFIC COMPOSITION AND QUANTITY OF INTERSTELLAR DUST USING X-RAYS

In an X-ray-absorption spectroscopy (XAS) experiment, the incident X-ray photon promotes a deep-core electron into an unoccupied state above the Fermi energy. For the iron L edge experiments shown in this paper, that deep core state is a 2p state. As the incident photon energy is varied using a variable line spacing plane grating monochromator, the cross section of this absorption process is monitored by measurement of the decay of a higher-lying electron into the core hole. The fluorescence of a photon and emission of Auger and secondary electrons can be monitored in tandem. Because the mean-free path of the electron is limited to a few nanometers or less, the electron yield measurement is mostly sensitive to the surface of the sample. The fluorescent photon, on the other hand, has a larger mean-free path, and therefore is a better probe of the bulk of the sample.

The samples measured in this paper were fine powders, either purchased as such or prepared by grinding in an agate mortar and pestle. A portion of these fine powders was sprinkled onto conducting carbon tape, another portion was pressed onto a flattened strip of indium metal. The carbon tape and indium strips were affixed to a glass microscope slide for mounting onto the sample holder. The fluorescence measurement is made using a photodiode pointed at the sample. However, since all of our samples were thick compared to the penetration depth and highly concentrated in iron, the fluorescence data were significantly attenuated by the self-absorption effect (see, e.g., Booth & Bridges 2005). All data in this paper were measured in electron yield mode.

To measure the electron yield, an alligator clip was clamped onto the glass slide and electrical contact was made with the otherwise isolated carbon tape and indium strip, and used to carry the photocurrent into a current-to-voltage amplifier. This experimental arrangement provided for redundant measurements of each sample. Some samples proved more amenable to measurement on the carbon tape substrate and others to the indium. For every sample, each measurement was repeated three times. In each case, the measurement yielding the highest quality, most reproducible data was selected for presentation. The three scans for each sample have been calibrated in energy, aligned, and averaged.

Absolute energy calibration was attained by measuring an iron foil and calibrating it to the tabulated energies of the iron L_II and L_III edges (Brennan & Cowen 1992). We had no capacity to prepare the surface of our foil, thus we relied upon the fluorescence measurement. Although the foil measurement is severely attenuated by self-absorption, the edge position can be reliably extracted. Iron-edge scans on other materials were aligned to the iron foil data using reproducible features in the spectrum of the incident intensity.

In an XAS measurement, the absolute scale of the measured cross section is ambiguous. The size of the measured step at an absorption edge energy depends on a wide variety of factors, including the concentration of the absorbing atom in the sample, the electronic gains on the signal chains, and the efficiencies of the detectors. In order to compare measurements on disparate samples under differing experimental conditions, it is standard practice to perform a common normalization of every measured spectrum. In this work, we choose to perform this normalization by matching the measurements to tabulated values for bare-atom cross sections, as described in this section. In this way, we can directly and consistently compare measurements on our samples to one another as well as to astronomical observations.

The normalization of XAS L edge data is more complicated than for K edge data due to the close proximity of the L_II and L_III edges in the soft X-ray regime. In order to properly normalize these spectra, we adopt the MBACK normalization technique of Weng et al. (2005) with modifications. Here, we describe the MBACK technique in brief, and our modification, which we found to be necessary for data with ≳0.2% noise, as typical of each individual cross section measure. We note that the typical practice has been to first co-add all individual cross section measures to increase S/N, before normalization. However, we wished to be more rigorous in our methodology by first renormalizing the individually measured cross sections before averaging to create the final cross section to facilitate fitting purposes. The MBACK technique ensures better normalization for the more complex L-edge cross sections by calculating a smooth normalizing function over the entire measured data range, rather than extrapolating from independently pre- and post-edge linear fits as would be sufficient for the less complex K-edge cross sections. For our data, this normalizing function, μ_norm, is dominated by the Legendre polynomial term (in brackets) of the function

$$\begin{equation} \mu _{\rm norm}(E_i) = \left[\sum _{j=0}^{m}C_{j}(E_i-E_{\rm edge})^j \right] + A\,\cdot {\rm erfc}\left({{E_i-E_{em}}\over \xi }\right), \end{equation} \tag{ 1 }$$

where E_edge is the energy of the onset of the edge absorption, and m and C_j are, respectively, the polynomial order and coefficient. Weng et al. additionally note that a rapidly decreasing pre-edge shape, due to residual elastic scattering, is often seen in fluorescence experiments using energy discriminating detectors, and therefore include an additional error function component in the normalization as defined by the nonbracketed term of Equation (1); see Figure 1 for the shape of this function as determined from our sample fit to γ-FeO(OH)−lepidocrocite. Here, E_em and ξ are, respectively, the centroid energy of the X-ray emission line of the absorbing element and its width, with A as a scale factor. We note, however, that this function is adopted here solely as an effective way to model similar pre-edge feature shapes originating from experimental artifacts of uncertain origin. (Residual elastic scattering effects do not apply to our experiment, which measured total photocurrent.) In any case, our data do not show a large rapidly decreasing pre-edge slope, so this term has negligible effect on our overall normalization; nevertheless, we include it in our modeling efforts for completeness, since this would allow for a global prescription for the determination of μ_norm that is independent of experimental technique. See Figure 2(a) for an illustration of the components that make up the normalizing function for our sample γ-FeO(OH) compound. In general, the details of the normalization variables would clearly depend on the composition of the condensed matter.

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To determine the best normalization (Figure 2(b)), we initially employ the prescribed minimizing function of Weng et al.

$$\begin{eqnarray} &&\hspace{-18pt} {1\over n_1} \sum _{i=1}^{n_1}[\mu _{bf}(E_i)+\mu _{\rm norm}(E_i)-s\mu _{\rm raw}(E_i)]^2\nonumber\\ && +\,{1\over {n_2}} \sum _{i=N-n_2+1}^{N}[\mu _{bf}(E_i)+\mu _{\rm norm}(E_i)-s\mu _{\rm raw}(E_i)]^2, \end{eqnarray} \tag{ 2 }$$

where N represents the total number of data bins, i is the bin index, and n₁ and n₂ are, respectively, the total number of pre-edge and post-edge data bins, such that N − n₂ + 1 represents index 1 of the post-edge bin. The variable μ_bf is the Brennan & Cowen (1992) tabulated absorption cross sections representing the bound-free nonresonant absorption component of Fe. Note that μ_raw is our measured cross sections, which is scaled by s, and μ_norm is the normalization function of Equation (1). However, in our fitting efforts, we found the scale factor s to be very sensitive to noise, often taking on small values for our data which resulted in badly normalized cross sections. In considering a remedy for this, we tested Equation (2) based on simulated cross sections (μ_bf + μ_norm), where we have added controlled levels of Gaussian-distributed noise, as per the prescription (1 + f G_noise) · (μ_bf + μ_norm), where G_noise is a group of random numbers following the Gaussian distribution and modified by the fractional level f, i.e., f = 0.2% or f = 1%, as in Figure 3. (The Gaussian distribution function is defined as $\phi (x)={e^{-(x-\mu)^2/2\sigma ^2}} / \sigma \sqrt{2\pi }$ where the expectation value μ = 0 and the standard deviation σ = 1.) (The μ_norm parameters of Equation (1) are set to C₀ = 3.0, C₁ = 0.01, C₂ = 0.0002, A = 1.0, E_em = 685 eV, ξ = 5.0, E_edge = 707 eV). We then divide the simulated data by the value of s arbitrarily set to 30, before fitting according to the minimization function of Equation (2) to see if we can recover the same value. In doing so, we find that the Weng et al. method, sans modification, is robust only for data with noise at ≲ 0.2%. Accordingly, we modify Equation (2) by dividing sμ_raw(E_i) into the pre-edge and post-edge minimization terms as

$$\begin{eqnarray} &&\hspace{-18pt} {1\over n_1} \sum _{i=1}^{n_1}\left[{\mu _{{\rm tab}}(E_i)+\mu _{\rm norm}(E_i)-s\mu _{\rm raw}(E_i) \over s\mu _{\rm raw}(E_i)} \right]^2\nonumber\\ && +\,{1\over n_2} \sum _{i=N-n_2+1}^{N}\left[{\mu _{{\rm tab}}(E_i)+\mu _{\rm norm}(E_i)-s\mu _{\rm raw}(E_i) \over s\mu _{\rm raw}(E_i)}\right]^2 \end{eqnarray} \tag{ 3 }$$

and find that our method (Equation (3)) is robust to data of up to 2% noise. In total, the normalization procedure will use m + 5 adjustable parameters (m + 1 polynomial coefficients, two scale factors of s and A as well ξ and E_em) to determine the best normalization for the measured cross sections.

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Of relevance to the cross sections presented in this paper, 18 measurements (nine mounted on indium substrate and nine on carbon) of each sample were taken. However, the carbon substrate measurements were consistently nosier, so we exclude those from the final co-averaged cross sections. For each individual indium substrate cross section measure (see Figure 4, top), the data were normalized using Equations (1) and (3) based on the subsequent steps. To compare with our reference spectrum, each measurement is first shifted by 4.1 eV to correct for the calibration offset of the monochromator. The resultant energy span of our measurements is then 684.1–754.1 eV, which corresponds to N = 700 total data bins of 0.1 eV widths. As described, Equations (1) and (3) were then used to determine the best normalized description for the combined data set unique to each sample. Since all data measured spanned the range noted above, the number of pre-edge and post-edge data bins correspond respectively to n₁ = 179 (684.1–702 eV) and n₂ = 141 (740–754.1 eV) for all samples. The edge energy E_edge = 707 eV corresponding to the Fe L iii, and a polynomial of order m = 2 was deemed sufficient, i.e., the goodness of fit showed negligible improvement for higher polynomial orders. Postnormalization, cross sections of the highest quality were co-added to create the final cross sections of Figure 5. For the final cross sections, no fewer than nine (indium substrate) data sets were co-averaged per compound type. Figure 5 shows our measured cross sections post-normalization, compared against metallic (Kortright & Kim 2000), and what we assume for bound-free continuum absorption by iron (Brennan & Cowen 1992; Henke et al. 1993). For future fitting efforts to astrophysical data, we use the structure-less Fe L tabulated values of Brennan & Cowen (1992) to extrapolate the normalized XAFS data beyond the energy span of our measurements to encompass an energy range that extends from 0.4 keV to 10 keV.

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We plan to avail the community of our cross section measurements⁸ for these and other compounds, as absolutely calibrated standards for interstellar dust studies, in the near future. Table 1 lists details for a few of these, but as stated, the data shown here are intended merely as a vehicle to facilitate a presentation of our new techniques.

Like condensed material, atomic transitions to higher quantum levels within an isolated atom will also give rise to multiplet resonant absorption. To give the example for Fe, these lines would consist of all possible discrete transitions to all possible configurations of the 3d-shell, since the electronic configuration of Z = 26 Fe⁺⁰ (or Fe i in the astronomy notation) is 1s²2s²2p⁶3s²3p⁶4s²3d⁶. Therefore, while the bound-free nonresonant transitions can be modeled by a simple step function as, for example, that of Brennan & Cowen (1992) for Fe L, additional discrete resonant features will have to be included to account for the bound–bound resonant component of absorption.

Figure 6 shows our calculations for the resonant transition for various Fe ions from Fe⁺⁰ − Fe⁺⁴, as evaluated based on the Gu et al. (2006; see also Gu 2005) predictions for oscillator strengths, radiative decay rates, and autoionization rates for these lines. We note that because of the electronic structure of Fe, which favors the removal of the s-electrons prior to d-electrons, the cross sections for Fe⁺⁰ ∼ Fe⁺¹ ∼ Fe⁺², as evident in the figure. The relevant cross sections derived for these atomic transitions are then convolved with the appropriate instrument resolution (ΔE ∼ 0.9 eV at 700 eV Fe L for the Chandra Medium Energy Grating) to approximate the ion-specific cross section for the bound–bound transition, σ_bb. Finally, the cross section for describing the total gas-phase absorption for the species of interest, σ_Zgas, will be described by the combination of cross sections from the bound–bound (μ_{bb − atom}) and bound-free (μ_bf) components. See Section 3.2 for more details.

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For complex astrophysical environments, additional considerations related to the ionization state of the plasma being probed need to be weighed. As such, modeling efforts need to consider the ionization state of the environment expected for the absorption. For the three, cold (T < 200 K), warm (T ∼ 8000 K), and hot (T ∼ 5 × 10⁵ K) phases of the ISM, the iron ions that contribute most strongly to the L-edge region are Fe⁺⁰ followed by Fe⁺¹. At T = 8000 K, the peak ion fractions are Fe⁺⁰ ∼ 0.67, Fe⁺¹ ∼ 0.33, Fe⁺² ∼ 2.8 × 10⁻³ and Fe⁺³ ∼ 1.9 × 10⁻⁶, assuming the ISM ionizing spectrum defined by Sternberg et al. (2002). At the much colder temperatures of the cold ISM, only Fe⁺⁰ contributes, and at the much hotter temperatures of the hottest phase of the ISM, it is Fe⁺²⁵ (at 6.7 keV) which contributes the bulk ion fraction.

As stated in Section 1, the cross sections we measure in the laboratory of astrophysically likely dust candidates (Section 2) will be applied to X-ray spectra showing structure near photoelectric edges to determine dust composition and quantity. We propose the following prescription to do so:

$$\begin{equation} F_{\rm final} = F_0\,T = F_0\, \rm exp\,[-\tau _{\rm LOS(Z^{\prime })} - \tau _{\rm ionized-gas} - \tau _{(Zgas)} - \tau _{(Zdust)}], \end{equation} \tag{ 4 }$$

whereby the absorption contributions from gas (cold and ionized) and dust can be separately accounted for by each of the exponential terms. Here, F_final and F₀ are respectively the detected and incident flux, and T, represented by the exponential terms, is the transmission. The τ_ionized-gas is determined through photoionization modeling to account for any additional lines arising from the plasma of the source, and should be treated on a source-by-source basis. However, in the Fe L iii and L ii edge regions, we note that additional line contributions would have to come from high-transition (low oscillator strength) lines of He-like O vii (O⁺⁶) 1s² − 1snp for n ⩾ 4, and Fe i–Fe xi (i.e., Fe⁺⁰ − Fe⁺¹⁰). Any significant contribution from oxygen would have to come from very ionized optically thick plasma; for the iron, moderate temperature plasma would have to be present in large quantities to have any appreciable effect in this spectral region. Therefore, for many astrophysical situations, neither ionized oxygen nor iron from the source plasma is expected to contribute strongly (if at all) to the Fe L iii and Fe L ii spectral region. Nevertheless, as stated, these can be modeled out through photoionization studies, the discussion of which is beyond the scope of this paper. For the total line-of-sight (LOS) (gas + dust) ISM absorption from all heavy elements, excluding the species of interest,

$$\begin{equation} \tau _{\rm LOS(Z^{\prime })} = \sum _{Z^{\prime }} \sigma ^{^{\prime }}_{Z^{\prime }} \, N_{\rm H} \, A_{\rm solar}, \end{equation} \tag{ 5 }$$

where $\sigma _{Z^{\prime }}$, A_solar, and N_H are respectively defined to be the cross section, abundance, and line of sight equivalent Hydrogen column summed over all heavy elements present in the broadband X-ray spectrum, with the exception of the element we are trying to measure. One reason for removing the species of interest from the broadband fitting is that we are interested in decomposing the gas and dust contribution specific to that species. For this, we include two additional exponential terms to describe the absorption by the gas and dust for the species of interest whereby

$$\begin{equation} \tau _{\rm Zgas} = {(\sigma _{bf}+\sigma _{bb-\rm gas}) N_{\rm H} xA_Z} = {\sigma _{Z{\rm gas}} N_{\rm H} xA_Z} \end{equation} \tag{ 6 }$$

and

$$\begin{equation} \tau _{\rm Zdust} = (\sigma _{bf}+\sigma _{bb-{\rm dust}}) N_{\rm H} (1-x) A_Z = \sigma _{Z{\rm dust}} N_{\rm H} (1-x) A_Z. \end{equation} \tag{ 7 }$$

In these equations, σ_Zgas and σ_Zdust refer to the cross section of the gas and dust, respectively, for the species of interest. The determination of σ_Zgas has been discussed in Section 3.1. The σ_Zdust and our efforts to measure this for astrophysically viable dust candidates have been discussed at length in Section 2. Given enough S/N, the species-specific abundance A_Z can be fitted for, with the condition that the total abundance should be a combination of both gas and dust, hence the additional multiplicative factor to A_Z of x and (1 − x). Figure 7 shows these components as separated out in a fit to the X-ray binary Cygnus X-1. Details about the dust in Cygnus X-1 will be separately discussed in J. C. Lee et al. (2009, in preparation).

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The path length along the line of sight to our illuminating sources, be they X-ray binaries in our own Galaxy or black hole systems in other galaxies, is complex. Consider, for example, the simplified scenario of only two components distributed in some manner along the path. This problem then has two parts: the identification of the chemical species of those two components and the determination of their distribution along the path length. Assuming we can identify the two species, can we distinguish a heterogeneous arrangement (dust of composition a in location A plus dust of composition b found in location B) from a homogeneous arrangement (dust of mixed composition AB distributed along the path)?

The interpretation of the X-ray spectrum passing along this line of sight bears a strong similarity to the well-established technique of XAS as performed at synchrotron facilities. The use of XAS to identify chemical species in a chemically heterogeneous sample is common practice in terrestrial laboratories (Lengke et al. 2006 is but one example among thousands in the XAS literature). The common strategy is to interpret the spectrum from the mixture of species as a linear combination of the spectra from its component species, i.e., ∑ⁿ_{i = 0}c_iα_i, where α_i represent the different species and c_i are their respective fractional contributions as per Lengke et al. (2006). The best determined relative percentages for the linear combination are derived by fits based on minimal χ², which we recast for our purposes to be

$$\begin{equation} \sum _{j=1}^m\left[{\sum _{i=1}^n c_i\, \alpha _{i,j} - \beta _j \over \sqrt{\sum _{i=1}^n (c_i\, \Delta \alpha _{i,j})^2 + (\Delta \beta _j)^2}} \right]^2, \end{equation} \tag{ 8 }$$

where β_j and α_i,j are the normalized cross sections in the jth bin of respectively the mixture and pure forms of the individual compounds, Δβ_j and Δα_i,j are their respective uncertainties, and n is the associated number of compounds and jth bins to sum over; c_i has the condition that ∑ⁿ_{i = 1}c_i = 1. With careful sample preparation, the fractional content of the component species in the spectrum is directly indicative of their fractional content in the physical sample.

To demonstrate this using iron-bearing species of relevance to the ISM, we performed the following experiment in the controlled laboratory of the synchrotron facility. We prepared two samples containing the common terrestrial iron compounds hematite (α-Fe₂O₃), ferrosilicate (Fe₂SiO₄), and lepidocrocite (γ-FeOOH). The first sample contained equal parts by weight of hematite and ferrosilicate and the second contained equal parts by weight of all three materials. Samples for measurement by electron yield at beamline 6.3.1 were prepared as described above and the XAS spectra were measured on both mixtures. The hematite and lepidocrocite were very fine-grained, commercially produced powders. The ferrosilicate was commercially produced, but was ground by agate mortar and pestle from a chunk several millimeters across. Consequently, the ferrosilicate component of each sample was much larger grained than the other two components.

Because equal parts by weight were mixed in our two samples, we might expect the measured spectra to be interpreted by equally weighted fractions of the spectra from the three pure materials. In fact, the nature of the interaction between the X-rays and the morphology of the sample must be considered. Because the ferrosilicate component of each sample was much larger grained than the other components, its surface-area-to-volume ratio was considerably smaller. Because the electron yield measurement is surface sensitive, a much smaller fraction of the iron atoms in the ferrosilicate contributed to the actual measurement than of the other two components.

The data on the two mixtures along with the results of the linear combination analysis are shown in Figure 8. The binary mixture proved to be 16 ± 2% ferrosilicate and 84 ± 2% hematite. As expected from particle size, the ferrosilicate is underrepresented in the measured spectrum. The trinary mixture was 8 ± 2% ferrosilicate, 51 ± 7% hematite, and 41 ± 7% lepidocrocite. The two components of similar particle size are evenly represented, while the ferrosilicate is underrepresented.

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This careful consideration of sample morphology in the synchrotron measurement is directly relevant to the interpretation of the astrophysical X-ray spectra. The astrophysical measurement is akin to a transmission measurement at the synchrotron, while the examples shown in Figure 8 were measured in electron yield. Still, the correct interpretation of the XAS spectra requires consideration of the nature of the interaction of the X-rays with the material through which they pass. A ray interacting with a large particle in the ISM will be lost to the satellite measurement by virtue of being completely absorbed by the particle. Consequently, the satellite measurement will be dominated by the small particles in the ISM as those are the only particles that allow passage of photons for eventual measurement at the satellite.

Finally, we must address the topic of distinguishing the distribution of the chemical species along the path length. By itself, a transmission XAS measurement is incapable of distinguishing between a layered and a homogenized sample. In transmission, an XAS measurement on two powders will be the same whether you prepare the powders separately and stack them in the beam path or mix the powders thoroughly before making the measurement. Similarly, the X-ray satellite measurement by itself cannot address the distribution of the species it is able to identify, if co-located at similar redshifts. However, since an XAFS contribution to a photoelectric edge is scaled according to the optical depth of that particular species, we can reasonably conclude that the compound that best fits the astrospectra is the compound where we are seeing the bulk contributions from, even if there might be lesser (e.g., 10%) contributions from other compounds along the line of sight. Therefore, in combination with other astrophysical studies using photon wavelengths that interact differently with the ISM (e.g., IR), the X-ray measurement can play an important role in the full determination of the composition of the ISM.