## 3. Clustering effects on the graphite UV featureIn this section we investigate whether clustering of grains composed of graphite can lead to an increase of the width of the UV feature without appreciably changing its peak position. Draine (1988) and Draine & Malhotra (1993) have confirmed the good agreement between the so called "1/3-2/3" approximation and DDA computations taking explicitly the anisotropy of graphite into account. In this approximation extinction cross sections and are obtained separately for isotropic particles having the components of the dielectric tensor parallel and perpendicular to the c-axis of graphite, respectively. They are then combined by taking 1/3 of the first contribution, and 2/3 of the second, respectively. This procedure is exact in the electrostatic limit for spheres, and is an excellent approximation in the Rayleigh limit (, where is the radius of the equal-volume sphere). It is also expected to be a good approximation for more complicated shapes in the Rayleigh limit provided one assumes that the c-axis of graphite is randomly oriented with respect to any symmetry axis that might be present within the particles. There is no consistent report of the interstellar UV feature being polarized (i.e. the particles are not partially oriented elongated particles). Therefore, in this paper, we assume that the particles are randomly oriented. It has been confirmed (Witt 1989; Witt et al. 1992; Calzetti et al. 1995) that the extinction in the vicinity of the interstellar UV feature is consistent with pure absorption (i.e., carrier with size in the Rayleigh limit). This fact is exploited to simplify the orientational average by taking only three mutually perpendicular orientations of the incident electric field with respect to the particles. This averaging procedure is also exact in the electrostatic limit and is a good approximation in the Rayleigh limit. An added advantage of the Rayleigh limit is that one needs not worry about size distributions of grains since extinction is independent of size (except perhaps indirectly through grains of different sizes possibly having different shapes). It is of interest to see how the width, shape, and peak position of the graphite UV feature change in the case of ensembles of agglomerated spheres to simulate clustered or irregularly shaped interstellar grains. Such a configuration can be handled by a multiple Mie sphere code (Mackowski 1991; Rouleau 1996), provided one assumes that the spheres, though touching, do no interpenetrate each other, and are isotropic. Therefore, we have performed multiple Mie sphere computations to calculate extinction cross sections per unit volume, , for various simple arrangements of spheres and plausible clustering models. The two types of clusters considered are a cluster-cluster
agglomerate (CCA), and a compact cluster. A cluster-cluster
agglomerate is constructed from hierarchically joining together
clusters containing the same number of spheres, The clusters are assumed to be in the Rayleigh limit. The optical constants of Draine (1985) are used for the computations. Qualitatively similar results are obtained for other choices of optical constants of graphite. A multipole expansion order up to 6 was necessary to obtain convergence for the results with the multiple Mie sphere calculations. There are three distinct ways to deal with the anisotropy of graphite in the multiple Mie sphere code. The first way is to use the "1/3-2/3" rule, i.e. to average the extinction cross sections of the agglomerates with either or . The second way is to randomly assign or with probabilities 1/3 and 2/3 to the individual spheres forming the aggregates. The third way is to average the components of the dielectric tensor and to use the resulting dielectric function in the multiple Mie sphere computations. These different approaches should be good approximations if we assume that there is no correlation between the orientations of the anisotropic spheres within the clusters and the direction of their individual c-axes. This is true for the regions outside of the resonance. However, it turns out that the results of the three methods differ from each other markedly in the bump region (Fig. 1). E.g. for the CCA cluster the averaged dielectric function (third way) leads to a much broader feature () at smaller wavenumbers () compared to the results obtained with the "1/3-2/3" rule (results for other arrangements of spheres are discussed in detail below) and the random assignment (second way). Furthermore, the feature obtained with the "1/3-2/3" rule shows a small shoulder in the long-wavelength wing. The appearance of the shoulder is explained by the rather long linear chains of the strong "oscillators". In case of the random assignment the chains are interspersed by the weak "oscillators" which makes the feature less sensitive to the detailed cluster structure (see Sect. 4.2). The averaged dielectric function (third way) is in any case also more or less moderate compared to regardless of the used averaging procedure. The feature position and width, however, certainly depend on the averaging procedure (the numbers given above are for the Bruggeman mixing rule; Bohren & Huffman 1983). Which way and which averaging procedure have to be used to get the best approximation of the of clusters of graphite spheres may be tested with extensive DDA calculations in the future, but this subject goes beyond the goals of this paper. In any case, one should note that more detailed information about the arrangement of basic structured units (in respect to their individual c-axes) from laboratory data are required to get a more reliable basis for the calculations (see Rouzaud & Oberlin 1989). The remaining discussion in this section is restricted to the results obtained with the "1/3-2/3" rule for the various sphere configurations.
Fig. 2 shows for a single sphere, a
randomly oriented bisphere, a randomly oriented linear chain of three
touching spheres, a CCA cluster containing 16 spheres, and a compact
cluster also containing 16 spheres. Other random realizations of these
clusters, also containing 16 spheres, gave very similar results. In
particular the shoulder observed around is
reproduced in all CCA cluster realizations. The appearance of the UV
feature looks very similar if only is plotted.
Note that in going from a single sphere to chains of increasing
Table 1 lists the peak position ("PEAK"), full-width at half-maximum ("FWHM") of the computed , as well as and derived from a fit similar in form to Eq. (1), with fitting coefficients and , using a slightly modified version of the routine given in Fitzpatrick & Massa (1990). To indicate the degree of quality of the fit, the value (in units of ) is also given, which is defined in the usual way with the weight for each data point (sampled at uniform intervals in wavenumber) set to the (arbitrary) value 1 except for the bump region - where the weight is set to 2. Apart from the CDE case (which turns out to be a bad clustering model for the bump), the parameter is quite stable ( relative variation), whereas is substantially increased (by ) in going from a single sphere to clusters. These features satisfy qualitatively the observational constraints, except for the fact that the peak position is wrong ( - compared to ). Note that there is a significant difference between "PEAK" and , and between "FWHM" and . This is due to the fact that a Drude profile fit is applied to a profile that is not purely Drude-like, thus introducing a spurious linear background (coefficients , ) which modifies both the peak position and the width of the feature. The fit in the case of the compact cluster is reasonable, which is indicated by the small value in Table 1, whereas the fit is somewhat poorer in the case of the CCA cluster (due to the presence of the shoulder), yielding a larger .
The peak position is at a larger wavenumber than that observed for
the interstellar UV feature. This is why one has to resort to either
shape or size effects to shift the feature at the position of the
interstellar UV feature (see, e.g., Draine 1988; Mathis 1994).
Discounting the disagreement in peak position and concentrating only
on the qualitative aspect of the feature, one can surmise that some
irregularly shaped or clustered particles © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |