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| Measuring attenuation of light: Limiting the acceptance angle of the detector | Prev topic | Next topic Fig. 1, Fig. 1a, Fig. 2 |
Hodkinson (1966) suggested that the maximum acceptance half-angle, θ1/2 [radians], of the detector should be less than one tenth of the first angular minimum in the Fraunhofer diffraction pattern (for example, http://en.wikipedia.org/wiki/Fraunhofer_diffraction) of a disk equal to the projected area of the particle in the monodisperse turbid medium, i.e.:
| θ1/2 ≤ 0.122 (λ / D) | (1) |
where λ is the wavelength in the medium and D is the particle diameter, here understood as the equivalent circular diameter.
By extrapolating results of Mie theory one expects that particles much larger than the wavelength of light scatter relatively more light in the forward direction, while the smaller particles scatter relatively more light at the larger angles (Fig. 1 in Mie theory: Angular patterns). If the particle sizes in the sample are known, the detector acceptance half angle should be less than that given in Eq. 1 for the largest known particle.
For a polydispersion with a frequency particle size distribution, n(D), a representative particle size, D, to be used in Eq. 1 is the optically effective diameter, Deff (for example, Mitchell DL 2002, see also Effective optical particle size):
| Deff = <D 3> / <D 2> | (2) |
where <D q> is the q-th normalized moment of the particle size distribution, defined as follows:
| <D q> = ( 1 / Ntot ) ∫0∞ D q n(D)dD | (3) |
and Ntot is the total number concentration of the particles, i.e.:
| Ntot = ∫0∞ n(D)dD | (4) |
The optically effective diameter is a reasonable alternative to the mean diameter, i.e. <D 1> in the present context, because the former is generally larger than the latter. In addition, dispersions that have similar radiative characteristics but different particle size distributions, tend also to have similar Deff values (for example, see also Mitchell DL 2002, Lenoble J 1993 - Section 16.2.3).
Experimental arrangements with the detector acceptance angle ranging from ~10 mrad (Swanson NL et al 1999) down to 3.4 mrad (Zaccanti et al 2003) have been used to measure attenuation coefficients of turbid media with optical thicknesses, τ, ranging from ~10 to ~13, respectively. The limit on the reduction of the acceptance angle is placed by the divergence of the light beam itself. Hence, Piskozub J et al 2004 suggest that the acceptance angle should be large enough to just enclose the beam divergence.
Typically, the acceptance angle of the detector in a transmissometer is limited by using a plano-convex lens and an aperture (pinhole), coaxial with and positioned at the focal plane of the lens (Fig.1, Fig. 1a). Such an arrangement utilizes the Fourier transform action of the convex lens, which converts the direction in the object space of the lens to the radial distance in the lens focal plane. This arrangement is also referred to as the lens-pinhole system (for example, Swanson NL et al 1999, Deepak A and Box 1978a). It is much more compact than, for example, a two-aperture system (Fig. 2) with a comparable entrance aperture area and acceptance angle.
| CITATION: Swanson N. L., Jonasz M. 2007. Measuring attenuation of light: Limiting the acceptance angle of the detector (www.tpdsci.com/Tpc/AtnCfMsAcptAngLim.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 21-Nov-2007 Modified: 20-Jan-2008 Peer-reviewed: 22-Dec-2007 |
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