## 2. Optical behaviour of powdersFor 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 where is the constant, high frequency limit of the electronic dielectric function; (MKSA units), N: valence electron density; : frequency of the transverse optical mode of resonant vibration of the crystal atoms and : damping constant of this vibration. Due to damping, the line width of this resonance is (FWHM). The low-frequency limit of is If is written as , its real and imaginary parts are linked to those of the refractive index, , by For the bulk material, an absorption coefficient can be defined as , which, in this case, exhibits a single peak at , with a peak value where and is the
velocity of light in vacuum. The absorption spectrum of grains of the
same material, and size where and For pure crystals, is normally very small, so that Eq. 4 represents a very narrow feature. The resonant, or Frohlich, frequency, , is distinctly different from the bulk frequency, , and the corresponding extinction cross- tion is in the principal orientation considered. The global extinction spectrum, therefore, exhibits three peaks of width each. Outside the interval , 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 is At this frequency, This is applicable as long as , which is not
the case for a needle or a disc parallel to the electric field (when
From (6), is constrained to fall between the transverse () and longitudinal () optical modes of the lattice ions, corresponding to and 1 respectively. Here, From (7) and (5), the extinction cross-section for a given particle
volume, v, decreases quickly with increasing In the degenerate case of the sphere, the three absorption peaks merge at with . There is no linear relation between
and of the bulk. For a
sphere embedded in a dielectric matrix (e.g. Kbr, CsI...),
, hence a red shift of the resonance with
respect to vacuum. Note, however, that For real particles, the shapes are likely to be continuously
distributed. From (7) and (8), there results a continuous spectrum of
extinction cross-section, , which can be deduced
by eliminating Bohren and Huffman (1983)
considered in ail the case of an arbitrary,
continuous distribution of joint probabilities of 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 wavelength, which depends on the distribution of shapes. Second, note that Eq. 7 and 8 define a one-to-one relationship
between and 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 In order to quantify this in an approximate but transparent manner,
we take advantage of the small relative difference between
and . Let
and be resonant
frequencies corresponding to a shape factor This has a maximum of at , which corresponds to According to (7), the peak intensity of the resonance for a given
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: -SiC: hexagonal and rhomboedric types, similar to the wurtzite variety of ZnS; -SiC: cubic, similar to the zincblend variety
of ZnS, the more stable of the two above 1600
Spitzer et al. also showed that, Thus, ## 2.1. Size and agglomeration effectsWhile 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 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, ## 2.2. Amorphicity effectsDisorder in the crystalline structure of the material increases the
damping constant, . 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 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
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. © European Southern Observatory (ESO) 1998 Online publication: December 16, 1997 |