ABSTRACT
A foremost strength of bioEPR spectroscopy is its applicability as an analytical chemical instrument for the determination of concentrations of prosthetic groups and of their stoichiometries in systems ranging in complexity from single proteins to whole cells. This particular strength clearly discriminates bioEPR from optical spectroscopy, which suffers not only from the practical problem of opacity of complex biological preparations, but also from the fundamental problem of an a priori undetermined wavelength-dependent intensity (or transition probability). In other words, the extinction coefficients of an optical absorption spectrum have to be determined experimentally, while all “extinction coefficients” of an EPR spectrum are equal to unity by definition: the intensity of an EPR spectrum follows directly from theory (i.e., from an interpretation of the spectrum in terms of its EPR parameters), and concentrations are straightforwardly determinable in terms of the molarity of an external EPR standard of known concentration. In mathematical terms, quantitative optical spectroscopy is based on Beer’s law:
in which the spectral amplitude I is a function of an unknown set of molar extinction coefficients ε, of the concentration of the chromophore c, and of an experimentally adjustable optical path length d, and in which other experimental conditions (light intensity, slit width, photomultiplier response function, etc.) are implicitly assumed to be standardized. Similarly, the amplitude of an EPR spectrum depends linearly on the concentration of the paramagnet, c, the sample tube diameter, d, and on a number of other experimental conditions (temperature, modulation amplitude, etc.) assumed to be standardized (see below). However, the EPR equivalent of ε(λ) is
a known quantity, which has been formulated in the literature in several equivalent forms (Holuj 1966; Pilbrow 1969; Isomoto et al. 1970; Abragam and Bleaney 1970: 136), for example,
or
in which the li’s are the direction cosines that we defined in Equation 5.3 and g is the anisotropic g-value of Equation 5.5 or 5.6. The equivalent Equations 6.2-6.3 are yet other examples of expressions of limited practical value because they are valid for frequency-swept spectra only. For field-swept spectra of effective or real S = 1/2 systems, the intensity has to be divided by g(li) (Aasa and Vänngård 1975), and so, for example, Equation 6.3 becomes
in which g, or g(li), is defined in Equation 5.5. This division by g is informally known as “the Aasa correction factor.” To determine the spin concentration of a paramagnet from its powder spectrum Equation 6.4 should be integrated over space (cf. Section 6.3 to follow), but in practice an approximating average is used (ibidem):
When the (effective or real) g-values can be read from the spectrum (with Equation 2.6) then the factor I is known, and the EPR equivalent of Beer’s law at fixed microwave frequency, ν, has no unknowns except for the concentration c
which can then be determined by comparison with the EPR of any (effective) S = 1/2 compound of known concentration.
