Harrick equation

Simply visualised, the infrared beam penetrates (of the order 0.3-3 pm, dependent on its wavelength) just beyond the ATR crystal-specimen boundary before it is reflected back and makes its way through the crystal to the detector. On this short path (of the evanescent wave) into the sample surface layer, light is absorbed, and the reflected beam carries characteristic spectral information of the sample. The decaying amplitude of the evanescent wave and the depth of penetration dp at which it has decreased to a proportion of 1 /e is defined by the Harrick equation (Equation (2)), where X is the wavelength of the incoming... 

The expressions for the electric field components Ex, Ey, and Ez have been evaluated by Haller and Rice (4), based on a theory developed by Harrick (2). We insert these expressions into equation 3-5, and 6, calculate a plot of < > vs D (Figure 2), and estimate the unknown angle < > for a given monolayer from this plot... 

The depth of IR-beam penetration (dp) can be calculated according to the following equation proposed by Harrick (21) ... 

An illustrative description of the implicit parameters was given by Harrick (1967) on the basis of a low-absorption approximation. This applies well for most organic materials, for materials of stronger absorption the Fresnel equations or the thereupon based layer algorithms mentioned in Sec. 6.4.3 should be applied. The depth from which analytical information can be gathered is conveniently characterized by the penetration depth dp. This is the distance in which the amplitude of an electric field decays to a fraction e of its initial value. For the evanescent field one finds... 

One of the features of ATR spectroscopy is that the depth of penetration d of the radiation incident on the sample is a function of the wavelength according to the following equation (Harrick, 1963) ... 

Quantitative analysis taking sample absorption and thickness into account Recently, the validity of Harrick s weak absorber approximations has been checked by comparison with the general thickness-and absorption-dependent model. It was found that the formalism depicted in the section above may be used for film thicknesses up to 20 nm. Especially if the film is in contact with a third bulk medium, e.g. water, the deviation between accurate and approximate calculation of relative electric field components according to Equation [29] was found to be below 3%, i.e. within the error of most experiments. A comprehensive description of ATR spectroscopy of polymers using the general formalism can be found in the Further reading section. 

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