El Método para la Reexpresión de los Estados Financieros.

Ultimately it is the microwave H1 field that drives the EPR transitions in the sample. The

accuracy and precision of the quantitative measurements depend on the amplitude and the distribution of the H1 field over the sample, and how precisely this can be reproduced for

the reference standard. It has been suggested that errors related to H1 field distribution

over the sample is one of the largest source of uncertainty in Q-EPR.5

The ε’ leads to redistribution of the E1 and H1 fields within the resonator compared to the

situation when the resonator is empty. The sample’s ε’ introduces a “lens” or “suck in effect” which can concentrate the H1 to the sample depending on the size, shape and the

dielectric constants of the sample material. Also any accessory such as variable temperature (VT) dewar or quartz capillaries have an impact on the H1 field distribution.

For example a VT dewar is known to concentrate the H1 field, increasing the effective filling

factor from that expected solely by the volume ratio between the sample and the cavity.5, 17 The distribution of the H1along the sample axis (z) has been investigated by Yordanov et al.

for several types of resonators, and one of the major conclusions was that the distribution of H1 for a specific cavity has to be known before any quantitative work should be

performed.18 Furthermore, as the effect of the ε’ to the EPR signal intensity can be non- linear along the sample axis, the placement and other properties of the unknown and the

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reference standard, including the thickness of the sample tube walls, must be identical as possible for reliable results.19

The discussion above for Q, H1 and η indicates that the sample and reference should be

made exactly the same in terms of size, shape and placement inside the resonator. Furthermore solvent and matrix composition must match so that ε’ and ε’’ won’t lead to unwanted errors in quantification. In practice for liquid samples the referencing should be done using exactly the same diameter and material capillary, positioned in exactly the same way, while matching the solvent and supporting electrolyte composition for the reference standard and the unknown.

Additional problems are introduced due to the temperature and frequency dependency of

ε’ and ε’’, indicating that the experimental conditions have to be carefully controlled. For example a change in temperature changes the dielectric properties of the sample affecting Q-value, but also affects the critical coupling frequency of the spectrometer and the distribution of H1 field over the sample.

4.5. Magnetic field modulation

Because CW EPR relies on phase sensitive detection through the modulation of the external magnetic field H0, the homogeneity of Hmod over the sample volume is important, as its

variation over the sample volume can introduce significant errors in quantitative measurements.20 The investigation of distribution of Hmod for several resonator types has

concluded that the combined effect of H1 and Hmod inhomogeneities can result up to 150 %

error.18

The relationship between Hmod and CW EPR ΔSpp is shown in Figure 4.2a for Lorenzian and

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after which the signal starts to diminish. The implication of increasing Hmod is shown in

Figure 4.2b, where the ΔHpp is plotted against Hmod. For Lorenzian line, around the

maximum ΔSpp where the Hmod is three times the ΔHpp, the line width is broadened by three

fold also. As a rule of thumb, for a Lorenzian line, as long as the Hmod ˂ 1/3 ΔHpp the line is

less than 3 % broadened.9

It is important to note that although the ΔSpp is crucial for the quantification of

paramagnetic species due to its effect on S:N, it is ultimately the double integrated signal intensity (DI) that will be used as a means of quantification. DI is proportional to the modulation amplitude even when the line is slightly broadened, and thus for quantification purposes the loss of resolution due to over modulation can be justified whenever the S:N requirements demand maximization of the EPR signal.

Also vmod can introduce distortion to the EPR line shapes if the frequency of the modulation

approaches the frequency of the line width in Hz. For 0.03 mT ΔHpp, distortion of the

spectra would be observed at:

Figure 4.2. (a) Normalised peak-to-peak amplitude (ΔSpp) for first derivatives of Lorenzian and Gaussian lines

as a function of modulation amplitude (Hmod). ΔHpp(Hmod → 0) is the peak-to-peak separation of 1st

derivative line as Hmod tends towards 0. (b) Relative 1st derivative line widths at increasing values of the

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𝑣𝑚𝑜𝑑 = (𝑔𝜇𝐵⁄ )∆𝐻ℎ 𝑝𝑝≈ 840 kHz , (4.7)

well above the typical modulation frequencies of 100 kHz. This distortion manifests itself as side bands in the spectrum, as the crystal detector employed is not a linear device, and its output includes the sum and difference of the microwave and modulation frequencies. At 100 kHz modulation frequency the side bands would manifest themselves at:

𝑣𝑚𝑜𝑑⁄ = 3.6𝛾𝑒 −6 T = 0.0036 mT , (4.8)

away from each other, i.e. well within the line width of 0.03 mT.9

In more detail, Barklie et al. have investigated the dependency of ΔSpp in terms of H1 and

Hmod amplitudes over cylindrical and point samples, noting that variation in the sample size,

geometry and placement will introduce significant errors if not accounted for. The distribution of Hmod and H1 were determined by a pick-up coil and a point sample of F+ spins

in MgO, respectively, and a correction factor was introduced to account for the variation of

Hmod from the cavity centre.21

While deriving a correction factor for a TE102 cavity in terms of sample size, shape and

position Nagy et al. observed that purely theoretical calculations for the correction factor yielded unsatisfactory results, and thus a semi-empirical approach was adopted. It was also highlighted that the numerical results obtained should not be used for other cavity resonators, but the correction factor must be obtained individually for each resonator. Also it was highlighted that the correction factor only works for non-lossy solid samples, where the sample or the sample holder do not introduce significant perturbation of the microwave field, a situation not applicable for EC-EPR cells and lossy samples.22

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