## 4. Single-Lorentz oscillator modelsActually, in view of the widely assumed formation mechanisms of the
carbonaceous component responsible for the UV bump, the highly
anisotropic dielectric function of planar graphite is unlikely to be a
good model of the purported dielectric function of the bump carrier.
First, the exposure of the carbonaceous interstellar grains to UV
radiation, though conducive to a "graphitization" of the material
through dehydrogenation, is unlikely to produce perfect graphite
sheets as an end-product (Sorrell 1990). There must still be
considerable defects in the structure, the topology of the grains
being a collection of randomly oriented graphitic crystallites in a
sp Virtually all UV bumps observed to date can be reproduced extremely well by a "Drude" profile. This is the profile generated by a sphere whose optical properties are characterized by a single-Lorentz oscillator model. Thus, in this section, we consider a single-Lorentz oscillator model to approximate the dielectric function hypothesized for the interstellar UV feature carrier, and see how shape and clustering can affect the peak position, width, and shape of the UV bump. We assume that variability in the width is attributable to shape and clustering effects, as well as to intrinsic variations in chemical composition along different lines of sight. Variability in the peak position, which is observed to be uncorrelated with width, is attributed to variations in mean chemical composition alone. The dielectric function of a Lorentz oscillator is given by , where , , and , are the plasma frequency, the peak position, and the damping constant, respectively, all in units of the inverse wavelength (). A Drude model is characterized by and usually describes metals (or semi-metals, like graphite). Through these parameters, the model can simulate the variability of the chemical composition of the grains and provide some physical insight into the mechanisms involved. Interstellar grains cannot be expected to be perfectly smooth spheres in all environments. For example, one may expect aggregation of the primary grains, especially in denser regions. Furthermore, the grains could be characterized by surface roughness and/or by porosity, as well as chemical inhomogeneities. An interesting question is whether such shape and clustering effects (neglecting chemical inhomogeneities within a given grain) can conserve the Drude-like profile that is observed, along with the other observational constraints. ## 4.1. Combining a Lorentz oscillator model with shape and clusteringThe effect of shape and clustering on the scattering properties of
chemically homogeneous grains in the Rayleigh limit can be modelled
via a spectral density, , of the geometric
factor Combining a single-Lorentz oscillator model of dielectric function with a model of shape and clustering using , the extinction cross section per unit volume can be written as where denotes the imaginary part. The familiar form of for spheres is obtained from . Note that using this model has the form of the "bump" term in Eq. (1) if . In that case, the maximum occurs close to One advantage of this type of representation is that a statistical
ensemble of grains of various shapes or clustering states can also be
represented by an average . Actually some mean
and mean could even
describe qualitatively a statistical ensemble of grains with varying
shape The irregular grains considered here arise from the agglomeration of spheres (i.e. the spheres can touch their neighbour at one point, but they are not allowed to interpenetrate). In the case of agglomerates of spheres, the spectral representation approach brings about a considerable reduction in computer burden compared to direct computations using a multiple Mie sphere code since the problem has to be solved only once. Results between the two approaches in the case of clusters of spheres using the optical constants of graphite are very similar. To compute the spectral density we use a code kindly provided by Hinsen & Felderhof (1992). This code computes ensemble-averaged electromagnetic interactions between identical spheres in the electrostatic limit. As stated in Stognienko et al. (1995), the spectral density depends on the choice of the maximum order for the multipole expansion. We present the spectral densities computed with multipole expansion order of 9 for the CCA clusters and the compact clusters in Fig. 3. Note that a multipole expansion order of 6 is sufficient to obtain convergence for the results using Eq. 2 with the optical constants of graphite.
For our simple Lorentz model, we assign one of the smallest widths
derived from interstellar extinction curves,
(for HD 93028; Fitzpatrick & Massa 1986), to spheres. This
might not be completely correct, but the sphere is a useful reference
shape. We assume here that broader widths arise For spheres, a pure Drude profile is obtained. Due to shape and clustering effects, however, the synthetic bump computed from may differ from a pure Drude profile. We wish to find out under which conditions such clustering models can produce an increase in width of the bump without appreciably changing its peak position or its Drude-like shape. ## 4.2. Results for CCA clustersFig. 4 shows the extinction cross section per unit volume, , and the real and imaginary part of the dielectric function, and , for various single-Lorentz oscillator models using the computed from CCA clusters. The models were parameterized using and . This choice of gives rise to a peak at for spheres ( in Eq. [ 3]). The damping constant is set to . For spheres, this produces a Drude profile of width .
In order of increasing amplitude, results shown are for and 0.0 (solid lines). Also shown for comparison are the fits using the procedure of Fitzpatrick & Massa (1990; dotted lines). Note the considerable amount of structure in the profiles, except for the weakest model shown, . The corresponding is shown in Fig. 3 (dashed line). A comparison between the two figures indicates that the of models with the smallest ("strong" Lorentz models, i.e. closest to a Drude model) are just a convolution of with a broadened Drude profile. Thus, for "strong" Lorentz models, extra peaks in the profile of will translate into structure in the curves. Conversely, for "weak" Lorentz models ( in Eq. [ 2]) the profile will be close to a "Drude" profile, irrespective of the form of . Table 2 is similar to Table 1, but for the one-Lorentz
oscillator models using the of CCA clusters. In
the present case, and are
meaningless for since the curves are poorly
represented by a "Drude" profile (thus being inconsistent with
observations of actual interstellar extinction curves). Again, note
the significant discrepancies between "PEAK" and
, and between "FWHM" and
because of the spurious fitted linear rise. Only
values of less than 500 are listed,
corresponding to models that are acceptable. Note that, though the
curves are not strictly Drude-like, "PEAK" shifts to
The ratio of carbon locked up in interstellar grains relative to
hydrogen, , is thought be be around
part-per-million (ppm) by number (Cardelli et
al. 1996). Assuming the bump grains have a density of
2 g cm Thus cluster-cluster agglomeration (i.e. fluffy interstellar
grains) ## 4.3. Results for compact clustersWe also computed using Eq. (2) for a corresponding to an ensemble of compact clusters containing spheres. Fig. 5 is similar to Fig. 4, but for an ensemble of compact cluster models. The corresponding is shown in Fig. 3 (solid line). Note that the of compact clusters also peaks at , but that it is more "compact" and contains less structure than the of CCA clusters. This translates into profiles that are closer to a Drude profile, except again for "stronger" Lorentz models (). This emphasizes how tightly constrained the shape and clustering of grains described by the stronger single-Lorentz oscillator models are in terms of their allowable profile. Table 3 lists the same parameters as Table 2 for the various compact clusters using these Lorentz models. Models with satisfy most of the observational constraints. Only models with violate the cosmic carbon abundance constraint. Note again the trend of larger FWHM's usually leading to larger values of "PEAK", as required observationally.
© European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |