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Astron. Astrophys. 329, 1035-1044 (1998)

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2. Optical behaviour of powders

For our approach to be understood and the experimental spectra to be interpreted, it is necessary to recall here some results of Mie's theory, as applied to ellipsoids of a homogeneous dielectric material. This is made much easier by the availability of the extensive work of Bohren and Huffman (1983), who, in their Chap.12, used precisely SiC as a textbook example, because its lattice vibrations are so accurately described by the single-oscillator model. SiC is an ionic crystal whose dielectric function may be written

[EQUATION]

where [FORMULA] is the constant, high frequency limit of the electronic dielectric function; [FORMULA] (MKSA units), N: valence electron density; [FORMULA]: frequency of the transverse optical mode of resonant vibration of the crystal atoms and [FORMULA]: damping constant of this vibration. Due to damping, the line width of this resonance is [FORMULA] (FWHM). The low-frequency limit of [FORMULA] is [FORMULA]

If [FORMULA] is written as [FORMULA], its real and imaginary parts are linked to those of the refractive index, [FORMULA], by

[EQUATION]

For the bulk material, an absorption coefficient [FORMULA] can be defined as [FORMULA], which, in this case, exhibits a single peak at [FORMULA], with a peak value

[EQUATION]

where [FORMULA] and [FORMULA] is the velocity of light in vacuum. The absorption spectrum of grains of the same material, and size a, is quite different in intensity as well as frequency. Bohren and Huffman (1983) studied the case of small ([FORMULA]) ellipsoidal grains in detail. In this case, the self depolarization of the particle in an electromagnetic field has to be taken into account, and is of course different in the three principal orientations. Three geometrical factors are therefore defined, [FORMULA],with [FORMULA], each of which enters the corresponding expression for the absorption cross-section of the ellipsoid (against an e.m. wave whose electric field is parallel to direction i):

[EQUATION]

[EQUATION]

where v: volume of particle, [FORMULA]: dielectric function of the embedding matrix, [FORMULA],

[EQUATION]

and

[EQUATION]

For pure crystals, [FORMULA] is normally very small, so that Eq. 4 represents a very narrow feature.

The resonant, or Frohlich, frequency, [FORMULA], is distinctly different from the bulk frequency, [FORMULA], and the corresponding extinction cross- [FORMULA] tion is

[EQUATION]

in the principal orientation considered. The global extinction spectrum, therefore, exhibits three peaks of width [FORMULA] each. Outside the interval [FORMULA], there is no resonance and the continuum is made up of the wings of the three resonances.

The three geometrical factors all equal 1/3 for spheres, 0, 1/2, 1/2 for needles and 0, 0, 1 for discs. The sum of the three is always unity. The larger factors correspond to the smaller principal dimensions and vice versa.

A good approximate expression for [FORMULA] is

[EQUATION]

At this frequency,

[EQUATION]

This is applicable as long as [FORMULA], which is not the case for a needle or a disc parallel to the electric field (when L =0).

From (6), [FORMULA] is constrained to fall between the transverse ([FORMULA]) and longitudinal ([FORMULA]) optical modes of the lattice ions, corresponding to [FORMULA] and 1 respectively. Here,

[EQUATION]

From (7) and (5), the extinction cross-section for a given particle volume, v, decreases quickly with increasing L and [FORMULA], but less so in a dielectric matrix. This is understandable since a larger L corresponds to a shorter transverse size of the particle, while a stronger dielectric produces a stronger electric field.

In the degenerate case of the sphere, the three absorption peaks merge at

[EQUATION]

with [FORMULA]. There is no linear relation between [FORMULA] and [FORMULA] of the bulk. For a sphere embedded in a dielectric matrix (e.g. Kbr, CsI...), [FORMULA], hence a red shift of the resonance with respect to vacuum. Note, however, that [FORMULA] is independent of the particle radius; an assembly of small spheres can only exhibit one, narrow, extinction peak at a frequency depending only on the material and the matrix. A wide extinction band, therefore, can only result from a distribution of particles of different shapes.

For real particles, the shapes are likely to be continuously distributed. From (7) and (8), there results a continuous spectrum of extinction cross-section, [FORMULA], which can be deduced by eliminating [FORMULA] Bohren and Huffman (1983) considered in [FORMULA] ail the case of an arbitrary, continuous distribution of joint probabilities of L 's for ellipsoidal shapes. In particular, for a uniform probability distribution, the extinction spectrum extends nearly uniformly in the range [FORMULA]. In such a case, the plateau extinction efficiency is smaller than the peak efficiency for a sphere by a factor [FORMULA].

So much so for for the available knowledge. What can we further infer from it for our present purposes?

First, the most important clue to the identification of an astronomical dust from its optical signature is the interval over which the latter extends, not necessarily the peak [FORMULA] wavelength, which depends on the distribution of shapes.

Second, note that Eq. 7 and 8 define a one-to-one relationship between [FORMULA] and L, for given material and matrix. It is therefore possible to infer the probability distribution of L 's from a measured extinction spectrum.

Third, the red shift of the peak extinction of small ellipsoids embedded in a matrix (relative to vacuum) can be deduced from (8). It depends on L and, hence, on [FORMULA]: it is not the same all over the spectral extent of the signature; in particular it is nul at [FORMULA] and [FORMULA]! There is, therefore, no reason to "correct" IR transmission measurements of SiC in a dielectric matrix by simply blue-shifting the feature as a whole by an arbitrary amount, as has been done in the past.

In order to quantify this in an approximate but transparent manner, we take advantage of the small relative difference between [FORMULA] and [FORMULA]. Let [FORMULA] and [FORMULA] be resonant frequencies corresponding to a shape factor L, in vacuum and matrix, respectively. Define [FORMULA], [FORMULA] and [FORMULA]. Put [FORMULA] and [FORMULA] and solve (8) for [FORMULA], to order 2 in [FORMULA] 's. Eliminating L, it is found that

[EQUATION]

This has a maximum of [FORMULA] at [FORMULA], which corresponds to [FORMULA]

According to (7), the peak intensity of the resonance for a given L is also reduced, in a dielectric matrix, by a factor

[EQUATION]

Note that this factor depends not only on the matrix but also on the particle shape.

In order for these general considerations to be applicable to the comparison between laboratory and astronomical spectra, we have yet to decide which specific constants are to be used in the formulae above. Now, SiC comes in a wide variety of crystallographic types (Schaffer, 1969) which fall roughly in two main classes:

[FORMULA] -SiC: hexagonal and rhomboedric types, similar to the wurtzite variety of ZnS;

[FORMULA] -SiC: cubic, similar to the zincblend variety of ZnS, the more stable of the two above [FORMULA] 1600 oC (Schaffer,1969; Tougne et al., 1993). While a cubic crystal can be expected to be described by one set of optical constants, uniaxial crystals like [FORMULA] -SiC behave differently towards the ordinary and extraordinary polarizations of incident waves. However, Spitzer et al.(1959) showed that, except for a very small subsidiary resonance, there was no significant difference between the two polarisations, as far as [FORMULA] -SiC is concerned. This was confirmed by Pikhtin et al.(1977).

Spitzer et al. also showed that, near the fundamental band, and for all practical purposes, [FORMULA] - and [FORMULA] -SiC may both be characterized by the same optical constants, as follows:

[EQUATION]

[EQUATION]

Thus, there is no point in attempting to identify different celestial spectra with one of the two crystal classes, as was done previously (e.g. Papoular, 1988).

2.1. Size and agglomeration effects

While celestial grains are not expected to be larger than the Rayleigh limit, this may be the case for laboratory particles. Large sizes are indeed encountered when the grinding time is too short, or when the powder is crammed upon a solid, transparent substrate. As predicted by Mie's theory, important modifications then occur in the transmission spectrum: the feature is reduced in intensity and skewed to the red; its peak is red-shifted and its red wing extends beyond [FORMULA] the underlying continuum increases uniformly at first, then with an increasing slope, rising towards higher wavenumbers, due to optical scattering; This behaviour is illustrated by the measurements of Borghesi et al. (1986), on going from raw to ground (G) and, finally, ground and sedimented (GWS) powder, when the mean grain sizes decreases by 1 or 2 orders of magnitudes, and finally satisfies the Rayleigh-Gans criterion,

[EQUATION]

2.2. Amorphicity effects

Disorder in the crystalline structure of the material increases the damping constant, [FORMULA]. This widens the Frohlich resonance and decreases its intensity in inverse proportions (Eq. 7). One may wonder if this effect could be confused with the effect of a distribution of L 's over a corresponding interval [FORMULA].

Indeed, a thin film of pure amorphous, nearly stoichiometric SiC, produced by rf sputtering (Morimoto et al., 1984) or by ion implantation (Serre et al., 1996) exhibits a Gaussian-like feature, centered at [FORMULA] [FORMULA] t, about 250 cm-1 wide at half maximum and extending from [FORMULA] 400 to [FORMULA] 1000 cm-1. Contrary to shape effects, this amorphicity effect, observed in the bulk material, is highly symmetric about [FORMULA] t. Both effects might coexist in amorphous grains. In that case, the feature profile can be computed from the theory above by increasing [FORMULA] accordingly. The outcome is an extension of the feature well beyond the interval [FORMULA] t, [FORMULA] l on both sides.

To ensure that amorphicity effects are small, it is therefore necessary to test the crystallinity of the material at hand, using X-ray diffractometry, for instance.

Having considered individually, above, the various parameters which determine the position and shape of the fundamental vibrational band of a solid, we can arrange to work in the laboratory under conditions in which shape effects are dominant. This will guide us in the interpretation and comparison of laboratory and astronomical data.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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