It is seen in Section 2 that the Kramers-Kronig relation requires > 1 for . For the Milky Way diffuse ISM, the 100 µm-1mm dust emission spectrum obtained by the Diffuse Infrared Background Experiment (DIRBE) instrument on the Cosmic Background Explorer (COBE) satellite is well fitted by the product of a single Planck curve of T 17.5 K and an opacity law characterized by 2 (i.e. abs -2; Boulanger et al. ), although other sets of T and are also able to provide (almost) equally good fits to the observed emission spectrum (e.g. T 19.5 K and 1.7; see Draine ). But we should emphasize here that it is never flatter than 1.65 (Draine ).
Smaller in the submm and mm wavelength range has been reported for cold molecular cores (e.g. Walker et al. : 0.9 - 1.8), circumstellar disks around young stars (e.g. Beckwith & Sargent : - 1-1; Mannings & Emerson : 0.6; Koerner, Chandler, & Sargent : 0.6), and circumstellar envelopes around evolved stars (e.g. Knapp, Sandell, & Robson : 0.9) including the prototypical carbon star IRC+10126 (Campbell et al. : 1).
However, these results are not unique since the dust temperature and density gradients in the clouds, disks or envelopes have not been taken into account in deriving - the exponent was usually estimated by fitting the submm and mm spectral energy distribution by a modified black-body - B(T) under the "optically-thin" assumption, with and T as adjustable parameters. If the dust spatial distribution is not constrained, the very same emission spectrum can be equally well fitted by models with different values.
As a matter of fact, an asymptotic value of 2 (i.e. abs -2) is expected for both dielectric and conducting spherical grains: in the Rayleigh regime (where the wavelength is much larger than the grain size)
where () = 1 + i 2 is the complex dielectric function of the grain at wavelength . For dielectric spheres, abs -1 2 -2 as since 1 approaches a constant (>> 2) while 2 -1; for metallic spheres with a conductivity of , abs -1 2-1 -2 as since 2 = 2 / c and 1 << 2. Even for dielectric grains with such an extreme shape as needle-like prolate spheroids (of semiaxes l along the symmetry axis and a perpendicular to the symmetry axis), in the Rayleigh regime we expect abs -2:
where L|| (a / l)2 ln(l / a) is the depolarization factor parallels to the symmetry axis; since for dielectric needles 1 constant and L||(1 - 1) + 1 >> L|| 2, therefore abs -1 2 -2 (see Li  for a detailed discussion). Only for both conducting and extremely-shaped grains abs can still be large at very long wavelengths. But even for those grains, the Kramers-Kronig relation places an upper limit on the wavelength range over which large abs can be attainable (see Li & Dwek  for details). The Kramers-Kronig relation has also been applied to interstellar dust models to see if the subsolar interstellar abundance problem can be solved by fluffy dust (Li ) and to TiC nanoparticles to relate their UV absorption strength to their quantities in protoplanetary nebulae (Li ).
The inverse-square dependence of abs on wavelength derived above applies to both crystalline and amorphous materials (see Tielens & Allamandola  for a detailed discussion). Exceptions to this are amorphous layered materials and very small amorphous grains in both of which the phonons are limited to two dimensions and their phonon spectrum is thus proportional to the frequency. Therefore, for both amorphous layered materials and very small amorphous grains the far-IR opacity is in inverse proportional to wavelength, i.e. abs -1 (Seki & Yamamoto ; Tielens & Allamandola ). Indeed, the experimentally measured far-IR absorption spectrum of amorphous carbon shows a abs -1 dependence at 5 µm < < 340 µm (Koike, Hasegawa, & Manabe ). If there is some degree of cross-linking between the layers in the amorphous layered grains, we would expect 1 < < 2 (Tielens & Allamandola ). This can explain the experimental far-IR absorption spectra of layer-lattice silicates which were found to have 1.25 < < 1.5 at 50 µm < < 300 µm (Day ). For very small amorphous grains, if the IR absorption due to internal bulk modes (for which the density of states frequency spectrum is proportional to -2) is not negligible compared to that due to surface vibrational modes (for which the frequency spectrum is proportional to -1), we would also expect 1 < < 2 (Seki & Yamamoto ).
If there exists a distribution of grain sizes, ranging from small grains in the Rayleigh regime for which ~ 2 and very large grains in the geometric optics limit for which ~ 0, we would expect to be intermediate between 0 and 2. This can explain the small values of dense regions such as molecular cloud cores, protostellar nebulae and protoplanetary disks where grain growth occurs (e.g. see Miyake & Nakagawa ).
The exponent index is temperature-dependent, as measured by Agladze et al.  for silicates at T = 1.2-30 K and = 700-2900 µm, and by Mennella et al.  for silicates and carbon dust at T = 24-295 K and = 20-2000 µm. Agladze et al.  found that at T = 1.2-30 K, first increases with increasing T, after reaching a maximum at T ~ 10-15 K it starts to decrease with increasing T. Agladze et al.  attributed this to a two-level population effect (Bösch ): because of the temperature-dependence of the two-level density of states (i.e. the variation in temperature results in the population change between the two levels), the exponent index is also temperature dependent. In contrast, Mennella et al.  found that increases by ~ 10%-50% from T = 295 K to 24 K, depending on the grain material (e.g. the variation of with T for crystalline silicates is not as marked as for amorphous silicates). The increase of with decreasing T at T = 24-295 K is due to the weakening of the long wavelength absorption as T decreases because at lower temperatures fewer vibrational modes are activated. Finally, it is noteworthy that the inverse temperature dependence of has been reported by Dupac et al.  for a variety of regions.