Radiation pressure on a dust grain is the result of absorption and scattering of photons of different energies. The observed spectra of late-type stars are filled with a lot of molecular features but for the sake of simplicity we approximate the spectral energy distribution by the Planck function with the effective temperature of the star. Note that such an approach may be not exact, in particular for the stars with chromospheres. Furthermore, in the outer parts of the optically thick envelopes, the peak of the energy distribution should shift to longer wavelengths (see, e.g., Netzer & Elitzur 1993; Bagnulo et al. 1995). Nevertheless, in general our skipping the details of stellar spectra does not strongly affect the results of the calculations as radiation pressure on the particles of different shape is mainly compared.
The effective temperatures of late-type giants and supergiants
lie in the range of 
(Pégourié 1987;
Lorenz-Martins & Lefèvre 1994).
Below, the value of 
K is used in calculations.
Some effects of stellar temperature variations are discussed in Sect. 4.4.
Amorphous carbon and amorphous silicates are commonly considered as the major sources of dust emission observed in spectra of carbon-rich and oxygen-rich stars, respectively. The evidence in favor of these materials came from both theoretical modelling (Gail & Sedlmayr 1984) and laboratory analysis (Jäger et al. 1994).
It is a hard task to determine a concrete type of the carbonaceous or silicate material. Moreover, it is evident that a mixture of different dusty components should coexist in the envelopes. For example, silicon carbide, silicate sulfide and even silicate features have been detected in spectra of some carbon-rich stars (Baron et al. 1987; Goebel & Moseley 1985; Little-Marenin 1986). Some emission bands found in oxygen-rich spectra have been tentatively identified with crystalline forms of silicates (Waters et al. 1996). At last, iron and oxidic particles can also condense in the circumstellar shells irrespective of the C/O ratio.
Amorphous carbon and astronomical silicate have been chosen as the basic species in our modelling. The optical constants of these species were taken from the papers of Rouleau & Martin (1991; their AC1 table) and Laor & Draine (1993). We have been also considered other materials: iron (optical constants from Ordal et al. 1988 and Pollack et al. 1994) and magnetite (Huffman & Stapp 1973; Steyer 1974) as examples of highly absorbing materials and artificial dirty silicate proposed by Ossenkopf et al. (1992; OHM-silicate) and clean glassy pyroxene with the ratio Fe/Mg = 0.2 (Dorschner et al. 1995) as examples of different types of silicates. Note that the latter two species cover almost completely the range of possible dirtiness of circumstellar silicates (Jones & Merrill 1976; David & Pégourié 1995).
So far only the formation of spherical grains in the envelopes of late-type stars has been considered (e.g., Draine 1981; Gail & Sedlmayr 1985; Fadeyev 1987; Fleischer et al. 1992; Cadwell et al. 1994 and references therein). The theory of nucleation and growth of non-spherical particles is still in the period of infancy.
We suppose that the circumstellar grains are prolate or oblate homogeneous spheroids with the aspect ratio a/b (a and b are the major and minor semiaxes of a spheroid, respectively). Note that variations of a/b allow to cover a wide set of the particle's shapes from disks to needles.
Dust grains condense and grow in the envelopes of late-type stars.
The size spectrum of the grains extends from tiny
particles to those with radii up to 
or more
(see discussion in Lafon & Berruyer 1991).
The upper cut-off of the size distribution is the subject of the controversy.
However, the models developed to fit available observational data show
that on average the size of grains in oxygen-rich giants should be
larger than that in carbon-rich stars (Jura 1994, 1996;
Bagnulo et al. 1995).
To compare the particles of the same volume but of different shapes,
it is suitable to characterize their size by the radius rV
of the sphere whose volume is equal to that of a spheroid.
The major semiaxis of the spheroid is connected with rV
as follows:
for prolate spheroids and
for oblate ones.
In our calculations, the particles with the size
are considered.
In principle, the grains growing in the envelopes may be fluffy
(or porous). We use a standard approach to model this effect
(see, e.g., Mukai et al. 1992).
The Bruggemann mixing rule
(Bohren & Huffman 1983) is applied
to construct a mean, effective dielectric function 
of an aggregate composed of n materials with dielectric functions
where fi is the volume fraction occupied by the constituent
material of the type i.
The radiation pressure force is calculated for compact
particles with the mean function .
We consider spheroids consisting of vacuum 
and
silicate or carbon with the vacuum filling factor f from
0 up to 0.9.
Collisions with the surrounding gas atoms and molecules cause
rotation of dust grains with high angular velocities
. Interstellar grains are believed to rotate
around their axes of maximum moment of inertia, and on average
the angular velocity vectors of the grains are parallel to the
magnetic fields (Spitzer 1978). Circumstellar dust grains may be aligned
under the action of the anisotropic fluxes of radiation or gas
(Dolginov et al. 1979). Apparently, the radial motion of the grains
should lead to their rotation in the planes containing the radius-vector.
However, non-radial gaseous fluxes or spiral
circumstellar magnetic fields arising as a result of stellar rotation
(e.g., Woitke et al. 1993) may lead to other orientations of grains.
In our modelling, two types of grain alignment are considered:
grains arbitrarily oriented in space (3D-orientaion) and
in a plane (2D-orientaion or perfect rotational orientation).
In the last case, the major axis of a rotating spheroid
remains in the same plane.
The angle between the angular velocity of the grain and
the wave-vector of incident radiation 
is an input parameter 
.