Applications of Quantitative ESR

Abstract

ESR applications concerned with the measurement of the amount of paramagnetic species are the common theme of the chapter. Procedures to obtain absolute concentrations are outlined. Error sources are discussed and procedures to reduce the uncertainties are reviewed. The principles of ESR-dosimetry are presented. Strategies to increase the dose response by using materials other than L-alanine, by isotopic substitution, metal ion doping, and instrumental developments are briefly described. The measurement of the spatial distribution of radiation dose by the ESR imaging (ESRI) method is discussed. Geological dating by ESR using the additive dose method is applicable for periods up to two million years. Procedures to estimate doses by ESR in contaminated areas and after radiological accidents are described as well as ESR analyses for test of irradiated food and of medical equipment. Methods to analyze the ESR line-shapes are considered in the context of obtaining the integrated spectral intensity.

Appendix

1.1 A9.1 Relaxation Times by CW Microwave Saturation Measurements

The method to obtain relaxation times from CW microwave saturation measurements for an inhomogenously broadened line is based on assumptions given in the literature [77, 78]. The ESR line shape is then expressed as a convolution of a Gauss and a Lorentz function [81]:

$$g(r) \propto \frac{{B_0 \beta }}{{\Delta B_G }}\int\limits_{ - \infty }^\infty {\frac{{e^{ - (ar^{\prime} )^2 } }}{{t^2 + (r - r^{\prime} )^2 }}dr^{\prime}} $$

((9.6))

Here β is the transition probability of the line g(r) centered at field B ₀. The variables r and r ^′ are defined in terms of the corresponding magnetic fields B and B ^′ as:

$$r = \frac{{B - B_0 }}{{\Delta B_L }}\quad r' = \frac{{B' - B_0 }}{{\Delta B_L }}$$

((9.7))

ΔB _L and ΔB _G are the widths of the unsaturated Lorentzian, L(B), and Gaussian, G(B), line shapes :

$$\begin{array}{ll}&L(B) = \frac{1}{{\pi\Delta B_L}}\frac{1}{{1+\left({1+\displaystyle\frac{{B-B_0}}{{\Delta B_L}}}\right)^2}}\\ &G(B) = \frac{1}{{\sqrt\pi\Delta B_G}}e^{-\left({\frac{{B-B_0}}{{\Delta B_G}}}\right)^2}\end{array}$$

((9.8))

The parameters t ² and a affecting the shape of the saturation curve are given by:

$$t^2 = 1 + \gamma ^2 B_1^2 \beta T_1 T_2 = 1 + s^2 $$

((9.9))

$$a = \frac{{\Delta B_L }}{{\Delta B_G }}$$

((9.10))

The saturation factor “s” contains the gyromagnetic ratio γ, the amplitude of the microwave magnetic field B ₁ , and the spin-lattice and spin-spin relaxation times T ₁ and T ₂. The amplitude is related to the input microwave power by an expression of the type:

$$B_1 = k\sqrt {Q_L P} = K\sqrt P $$

((9.11))

The constant K depends on the type of resonator and its quality factor Q _L with the sample inserted. It may often be difficult to estimate its value precisely, therefore the quantity P ₀ is introduced:

$$P_0 = \frac{1}{{K^2 \gamma ^2 \beta T_1 T_2 }}$$

((9.12))

Using the microwave power P as variable and P ₀ as a relaxation dependent parameter one obtains:

$$t = \sqrt {1 + {P/{P_0 }}} $$

((9.13))

The conventional saturation parameter s equals 1 for P = P ₀. P₀ was therefore referred to as the microwave power at saturation in [82].

The line-shape given by (9.6) is a Voigt function that can be evaluated numerically by a standard procedure [81]. For a single inhomogenously broadened line, the transition probability β is set to 1 as in a simple two-level system. The relaxation times are given by:

$$T_2 = \frac{1}{{\gamma \Delta B_L }}$$

((9.14))

$$T_1 = \frac{{\Delta B_L }}{{K^2 \gamma P_0 }}$$

((9.15))

The estimate of the spin-lattice relaxation time T ₁ depends on the factor K to calculate the B ₁ field from the corresponding microwave power according to equation (9.11).

The saturation curves must be recorded under slow passage conditions , that is, conditions such that the time between successive field modulation cycles is sufficiently long for each spin packet to relax between cycles. The spin system is then continually in thermal equilibrium and the true line shape is observed. A convenient formulation of the slow passage condition is given by the expression (9.16) [79]:

$$\omega _m B_m < < {{\Delta B{}_L}/{T_1 }}$$

((9.16))

Here ω _m is the modulation circular frequency and B _m is the modulation amplitude. The equation illustrates intuitively that the modulation rate must be much slower than the relaxation rate (1/T ₁). Typical values may be ΔB _L = 0.01 mT, T ₁ = 10 μs and B _m = 0.1 mT, requiring the modulation frequency ν _m = ω _m /2π being less than 1.5 kHz, a condition which often is difficult to achieve with many modern commercial spectrometers.

The formulae were incorporated in a computer program [82]. The input is an experimental array of the 1st derivative peak-to-peak ESR amplitude against the microwave power that is read from a previously prepared file and initial trial parameter values for the Lorentzian and Gaussian line-widths and the microwave power (P ₀) at saturation that are provided interactively. The corresponding theoretical amplitudes were obtained by numerical differentiation of the Voigt function (1). A non-linear least squares fi t of the calculated saturation curve to the experimental data is performed. Output data consist of a graph of the experimental data and the fitted saturation curve, exemplified in Fig. 9.15 below. The parameters with fitting error estimates are printed to a file of the type shown below.

Fig. 9.15

Experimental (+) and theoretical (–) saturation curves for radicals trapped in an X-irradiated crystalline dosimeter material

Analysis with 45 points, 4 parameters
	Parameter	Error
Intensity:	.1614E+01	.5671E-02
Saturating power(mW):	.1289E+01	.2356E-01
Lorentzian width(G):	.3765E+00	.7793E-02
Gaussian width(G):	.2801E+01
Exp. width(G):	.3000E+01
Computed width(G):	.3000E+01
B1(G) = .4400E-01*sqrt(P/mW)
	Parameter	Error
T1(s):	.7424E-05	.2050E-06
T2(s):	.1743E-06	.3185E-08
Modulation frequency =	.156E+04 Hz
Modulation amplitude =	.250E+00 G

Here, the widths are the peak-to-peak values λ _L and λ _G of the first derivatives and can be expressed in terms of the widths of the absorption lines as:

$$\begin{array}{ll}&\Delta B{}_L=\left(\sqrt{3}/2\right)\lambda_{L}\\ &\Delta B{}_G=\lambda_{G}/\sqrt{2}\end{array}$$

((9.17))

The slow-passage condition (9.16) is approximately fulfilled:

$$0.245 \cdot 10^4 = \omega _m B_m \textrm{(G/s)} < < {\Delta B{}_L}/{T_1 } = 0.621 \cdot 10^5$$