1.3.3 Dispersion
Dispersion has been alluded to several times in the previous discussions, but a more formal description will be provided here. First, dispersion is the separation of polychromatic radiation by a material that alters the velocity of a wave based upon its frequency. Dispersion can be both an important tool as well as a nuisance. For example, some mirrors are coated with polymers to protect them from corrosive environments. If this coating disperses different wavelengths of light, it can deteriorate the integrity of the light by acting like a prism (and separate wavelengths even further).
The positive aspects of dispersion occur in prisms, where separation of wavelengths of light is desirable. When a change in refractive index occurs at an interface between substances in the path of a beam of radiation the net affect is a bending of the light. This bending results from the fact that the waves travel at different speeds in the two substances. Because the amount of bending depends on the speed, and different wavelengths have different speeds, different wavelengths of radiation (light) bend at different angles at the interface. The net result is a separation of wavelengths.
Devices that separate light, such as monochromators and prisms are “rated” by their ability to separate wavelengths, with a number of parameters to quantify their resolution. If the equation, derived above, that describes the constructive interference of radiation
is differentiated while holding i constant, the angular dispersion of a grating can be obtained
This equation relates the angle of diffraction to the wavelength, referred to as angular dispersion. This can be taken one step further by relating the angular dispersion to linear dispersion (D), a measure of the quality of the monochromator, by
where F is the focal length of the monochromator (the distance between the monochromator grating to the exit slit). In this equation, the analyst is concerned with the variation of wavelength as a function of the distance along the line AB in the monochromator figure (Figure 1-8). A more useful measure of dispersion (and the quality of the monochromator) is the reciprocal linear dispersion (D-1) that is commonly used by the manufactory industry
where D-1 typically has values expressed in nm/mm or Angstroms/mm, and small values represent superior instruments.
Furthermore, substitution of the angular dispersion equation into the above equation yields the reciprocal linear dispersion, another figure of merit, for a grating monochromator
From this equation it is evident that angular dispersion increases as the distance, d, between the parallel grating lines decreases (as the number of grating lines per millimeter increases). Typical numbers of parallel grating lines in monochromators range from tens to hundreds of lines per millimeter for IR wavelengths, to several thousands for UV/visible wavelengths.
Finally, the resolving ability (where R is the resolution) of a monochromator refers to its power to separate adjacent wavelengths, represented mathematically by
Values of R for UV-visible monochromators of interest here range from 1000 to 10 000. It can also be shown that R is related to the total number of illuminated grating lines by
where n is the diffraction order, and N is the number of grating lines illuminated by the radiation of interest. The number of illuminated lines is usually determined by the width of the entrance slit to the monochromator, and the “f” number (f#) of the monochromator.
Decades ago, an analyst could request that a monochromator in an instrument be made to specific specifications based on reciprocal dispersion. However, today the above discussion of figures of merit is only of educational interest, as only a limited number of instrument variations are commercially available due to the more economical mass production of a few instrument designs. In essence, technicians purchase what is available and the price of the instrument is correlated to its dispersion and resolution. As a note, there are manufacturers that will custom make a monochromator with specific blaze density and blaze angle, but these are relatively costly.
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