Optical density, equation

The classical scheme for dichroism measurements implies measuring absorbances (optical densities) for light electric vector parallel and perpendicular to the orientation of director of a planarly oriented nematic or smectic sample. This approach requires high quality polarizers and planarly oriented samples. The alternative technique [50, 53] utilizes a comparison of the absorbance in the isotropic phase (Dj) with that of a homeotropically oriented smectic phase (Dh). In this case, the apparent order parameter for each vibrational oscillator of interest S (related to a certain molecular fragment) may be calculated as S = l-(Dh/Di) (l/f), where / is the thermal correction factor. The angles of orientation of vibrational oscillators (0) with respect to the normal to the smectic layers may be determined according to the equation... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The right part of equation [4], E = e c d, represents Lambert-Beer s law. E is called the extinction, c is the substance concentration, and d is the thickness of the sample. The E values span from 0 (this is the case when all light is transmitted and no absorption takes place, i.e., 1 = Iq) to inhnity, °o (this is the case of maximal extinction when no incident light is transmitted, i.e., 1 = 0). Realistic E values that can be correctly measured by normal spectrometers range between 0 and 2. Instead of using the E expression for extinction, A for absorbance is often used. E and A are dimensionless values, i.e., numbers without units. Nevertheless, OD, the symbol for optical density, is often added to E and A in order to clarify their meanings. 

At the end of 24 hours of continuous process the system was shut down. The knowledge of flowed buffer volumes and of the optical densities inside and downstream each ultrafiltration stage allowed to estimate product distribution (see appendix for mass-balance equations and the calculation procedure). The content of each cell was recovered and ffeeze-dried in order to be stored and used for subsequent kinetic experiments. A schematic flow-sheet of the whole procedure is illustrated in figure 1. 

In equation (5), T is the turhidity, O.D. is the optical density measvired from the photometer, N is the number density of parti-cles, X is the optical path length and Rext extinction... 

Equations (17) and (22) provide the experimentalist with three knobs (see Fig. 2). First, by varying the optical density of the transmitting medium, one may alter the constant term in [Pg.155]

The experimental findings of the optical density in the absorption region of the C=0 group (1770 cm-1 at a layer thickness of h = 0.87 x 10 4 cm), of the molar extinction coefficients of irradiated and non-irradiated copolymer films, and of the intensities of absorbed light (= 405 nm) made it possible to determine the quantum efficiency of C=0 group consumption using the known equation... 

It is well known23 that ionization ratios / can be determined from spectra using the expression /=( > — DB)/ D — D), where D is the measured UV-vis optical density or NMR chemical shift and DA and Z>B refer to the values for the pure acid form (i.e. BH+) and pure base form (i.e. B), respectively. Recasting this in terms of D gives equation (27) ... 

Some simple rearrangement of Equation 3.1 leads to the concepts of transmission T = Io/1 and absorbance A = — log T, with the quantity s c l called the optical density. The choice of units here for the extinction coefficient (M-1 cm-1) is appropriate for measurement of the absorbance of a solution in the laboratory but not so appropriate for a distance Z of astronomical proportions. The two terms and c are contracted to form the absorption per centimetre, a, or, more conveniently (confusingly) in astronomy, per parsec. The intrinsic ability of a molecule or atom to absorb light is described by the extinction coefficient s, and this can be calculated directly from the wavefunction using quantum mechanics, although the calculation is hard. 

Yeung et al. addressed the question of whether or not pull samples gave as good equations as samples measured on-line.47 The so-called add-back and process-stream samples were compared for statistical robustness in the equations they produced and their performance in predicting values in an on-line reaction. Materials such as biomass (optical density), protein, and RNA were measured by both techniques. 

The optical density is defined as OD = log(/o//), so that according to Equation (1.4) the absorption coefficient is determined by... 

That is, by measuring the optical density and the sample thickness, the absorption coefficient can be determined. According to Equation (1.6) we can now determine the absorption cross section a if the density of centers is known. This means that, by a... 

Eor a typical double-beam spectrophotometer, the sensitivity in terms of the optical density is (OD)min 5x 10. Therefore, using Equations (1.6) and (1.10), the minimum concentration of absorbing centers that can be detected is... 

It is clear from Equation (1.15) that the emitted intensity is linearly dependent on the incident intensity and is proportional to both the quantum efficiency and the optical density (this only for low optical densities). A quantum efficiency of < 1 indicates that a fraction of the absorbed energy is lost by nonradiative processes. Normally, these processes (which are discussed in Chapters 5 and 6) lead to sample heating. The proportionality to OD, which only holds for low optical densities, indicates that the excitation spectra only reproduce the shape of absorption spectra for samples with low concentrations. 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

Notice that the fluorescence intensity is directly proportional to the optical density only over a narrow range. Use of the above equation corrects for this apparent loss in fluorescence intensity. 

According to the Cd 18-90 AOCS ° official method, the ANV is 100 times the optical density measured in a 1 cm cell, at 350 nm, of a solution containing 1.00 g of oil in 100 ml of the test solution. The measured absorbance is due to Schiff bases (167) formed when p-anisidine (166) undergoes condensation reaction with carbonyl compounds, according to equation 55. The carbonyl compounds are secondary oxidation products of lipids, such as a, S-unsaturated aldehydes and ketones derived from the hydroperoxides (see Scheme 1 in Section n.A.2.c), and their presence points to advanced oxidation of the oil. 

Equation for the change in optical density of film dosimeters. 

After detg its optical density at a wave length of 410 millimicrons, a correction is introduced and the % of LSt is calcd from the equation... 

The value of the atom fraction H (of the OH + OD), initially, at equilibrium, and at time t, is denoted by x0, xm, and x, respectively. Allowance was made for the fact that the apparent optical densities of the hydroxyl and corresponding deuteroxyl groups are different by as much as 10%. The excess of D2 was at least 5-fold, so that the variation of the different types of hydroxyls could be evaluated separately (17). From Equation 1 the characteristic slope parameter is obtained (19) ... 

I, the extremely low concentration of polymer in the fractions collected reduced the precision in the ultraviolet optical density values measured, the influence of which is amplified in the figures in the last column. Therefore, the average styrene content was used as the composition of the peak point to calculate the MW of the polymer by Equation 5. The MW of the styrene homopolymer which is present in small amounts was taken as the MW of the first block, as explained above. In this manner the molecular structure of the polymer was found to be S( 15,000) B(61,000)S( 14,000) the figures given in parentheses being the MW s of the successive blocks. 

The conservation of mass equations are used with refractometric optics. The absorption optical system will give absorbance (optical density) which is directly proportional to concentration. Unless conservation of mass equations are used, one can only obtain concentration differences from refractometric optics. The Rayleigh optical system will give information proportional to cr — cTm thus Equation 20 or 23 would have to be used to obtain cTm. Note also that the initial concentration c0 is needed. This must be measured by differential refractometry, by boundary-forming experiments, or from ultraviolet light absorption. 

Equation 7.33 given l y Wonacott (19) is incorrect. At present, spots used for inter-film scaling by AXIS may not have a maximum optical density, including background, greater than 1.7 (see section 1.2). 

---