Continuous Wave EPR of Radicals in Solids

Abstract

Continuous wave (CW) EPR and ENDOR methods for studies of the structure, dynamics and reactions of radicals in crystalline and powdered samples are reviewed. Improvements of the standard Schonland procedure to obtain the hyperfine (hfc)- and nuclear quadrupole (nqc)-coupling tensors from single crystal measurements are described, including the resolution by ENDOR measurements of the so-called Schonland ambiguity. An account of the influence of the g-anisotropy on the intensity is included in a brief review of EPR studies of powders. The microwave saturation properties are discussed in the context of quantitative EPR using software for the analysis of CW power saturation curves and for simulations of saturated EPR spectra taking into account the different microwave power dependence of the allowed and forbidden (Δm _I = 1) hyperfine lines. The analysis of powder ENDOR spectra by simulations is described in considerable detail, with special emphasis on the combined influence of hfc and nqc of comparable magnitudes. Studies of the dynamics of free radicals, particularly on surfaces are summarized, including recent results concerning the adsorption/desorption and diffusion of nitrogen oxides. Software for the simulation of EPR and ENDOR powder spectra, for the analysis of internal and slow motion, and for the analysis of single crystal data, are described including addresses for downloading, when available.

Appendix

1.1.1 Single Crystal Analysis by the Original Schonland Method

The Schon_fit program in Table 1.1 is based on the original Schonland method as a simplified replacement for software in obsolete code referred to in our previous work [49]. The variation of the g-factor in a crystal plane was analyzed by a least squares fit to the experimental data using an equation of the type (1.4), by which three tensor elements with estimated uncertainties were obtained. Measurements in three mutually orthogonal crystal planes were employed to obtain the complete tensor. The principal values were computed with uncertainties estimated by the relation (1.55) originally given in different notation in [28]. The uncertainties (1.56) of the principal directions considered in [28, 46] were not computed, however. We refer to the literature [28, 46, 170] for procedures to obtain the statistical uncertainties from the propagating errors.

The variation of the hyperfine couplings in EPR measurements can be analyzed in the same manner as for the g-factor. The influence of the nuclear Zeeman term and of possible g-anisotropy was not taken into account in the software, however. Analysis of single crystal ENDOR data to obtain hfc-tensors can be made assuming that the measurements were obtained at or adjusted to a fixed magnetic field. Procedures for the adjustment, see e.g. [171], were not implemented. With these simplifications the program could be compactly written (ca 50 lines of Matlab code).

Axial symmetry constraints, commonly employed in the analysis of defects in crystals of high symmetry [56], can be taken into account by applying (1.49) below. The number of parameters can then be reduced, and measurements in three crystal planes are not always required as schematically illustrated in Fig. 1.13 for the ³³S hfc of the ${{\dot{S}}}{{\text{O}}_3}^{ - }$ radical in an X-irradiated K₂S₂O₆ crystal.

Fig. 1.13

Experimental (o) and fitted variations of a² (a =³³S hyperfine coupling) for a ${{\dot{S}}}{{\text{O}}_3}^{ - }$ radical in the ac plane of K₂S₂O₆, X-irradiated at room temperature. The values A_|| = 138.9 G, A_⊥ = 113.0 G were obtained from an analysis of Q-band EPR measurements (Data provided by Prof. E. Sagstuen) by assuming axial symmetry about the hexagonal crystal axis <c>

1.1.1.1 Axial Symmetry

An axially symmetric g- or hfc-tensor is commonly specified by the principal values T _∥, T _⊥ and the direction of the symmetry axis. The tensor elements in an arbitrary system (x, y, z) are then given by (1.47):

$$ {T_{\textit{ij}}} = {\delta_{\textit{ij}}}{T_{ \bot }} + \left( {{T_{\parallel }} - {T_{ \bot }}} \right){e_i}{e_j} $$

(1.47)

with i, j = x, y, z.

The coupling parameters (T _∥, T _⊥) and the polar angles (θ, ϕ) for the unit vector ${\text{e}} = \left( {\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta } \right)$ along the symmetry axis completely determine the tensor. An alternative specification, obtained by eliminating the direction cosines of the symmetry axis in (1.47) is based on (1.48).

$$ \left( {{T_{\textit{ii}}} - {T_{ \bot }}} \right)\left( {{T_{\textit{jj}}} - T_{\bot}} \right) = T_{\textit{ij}}^2 $$

(1.48)

The first alternative might be suitable for a least squares fitting to the data by a non-linear procedure, while the second can be used with the Schonland method. Assume for instance that the tensor components T _xx , T _yy and T _xy had been obtained from a least squares fit to the data in the xy-plane, in which case equation (1.49) applies:

$$ {T_{ \bot }} = \frac{{{T_{{xx}}} + {T_{{yy}}}}}{2}\pm \sqrt {{{{\left( {\frac{{{T_{{xx}}} - {T_{{yy}}}}}{2}} \right)}^2} + T_{{xy}}^2}} $$

(1.49)

The + and – signs correspond to T_∥<T_⊥ and T_∥≥T_⊥, respectively. An additional measurement along the z axis to obtain T _zz is required to calculate T _∥ = trace(T) – 2T _⊥. The sign ambiguity in (1.49) would be resolved by measurements in two crystal planes, both expected to show the same maximum or minimum g-value or hyperfine splitting associated with T _⊥. Expressions applicable to the yz and zx planes are obtained by cyclic permutations of the indices in Eq. (1.49).

1.1.1.2 Propagation of Errors

The probable errors in the determination of the principal values and the directions of the principal axes are usually estimated from the uncertainties of the tensor elements [28, 46, 170]. The propagation of errors derived by a first order perturbation method is reproduced below in matrix notation.

The principal values t _q and principal vectors c _q are obtained from the eigenvalue equation:

$$ {\mathbf{T}}{{\mathbf{c}}_q} = {t_q}{{\mathbf{c}}_q} $$

(1.50)

Errors in the elements of the tensor T give rise to corresponding uncertainties in t _q and c _q.

$$ \left( {{\mathbf{T}} + \Delta {\mathbf{T}}} \right)\left( {{{\mathbf{c}}_q} + \Delta {{\mathbf{c}}_q}} \right) = \left( {{t_q} + \Delta {t_q}} \right)\left( {{{\mathbf{c}}_q} + \Delta {{\mathbf{c}}_q}} \right) $$

(1.51)

The errors Δc _q are expressed as a linear combination (1.52) of the principal vectors.

$$ \Delta {{\mathbf{c}}_q} = \sum\limits_{{s \ne q}} {{d_s}} {{\mathbf{c}}_s} $$

(1.52)

Equation (1.53) is obtained by combining (1.50), (1.51) and (1.52), neglecting 2nd order terms:

$$ \sum\limits_{{r \ne q}} {{d_r}} {t_r}{{\mathbf{c}}_r} + \Delta {\mathbf{T}}{{\mathbf{c}}_q} = \Delta {t_q}{{\mathbf{c}}_q} + {t_q}\sum\limits_{{r \ne q}} {{d_r}} {{\mathbf{c}}_r} $$

(1.53)

Equation (1.54) is derived after multiplication with c _s, taking into account that the principal vectors are orthonormal.

$$ {d_s}{t_s} + {{\mathbf{c}}_s}\Delta {\mathbf{T}}{{\mathbf{c}}_q} = \Delta {t_q}{\delta_{{sq}}} + {t_q}{d_s} $$

(1.54)

The errors are accordingly estimated by the expressions (1.55) and (1.56):

$$ \Delta {t_q} = {\mathbf{c}}{}_q{\mathbf{\Delta T}}{{\mathbf{c}}_q} $$

(1.55)

$$ \Delta {{\mathbf{c}}_q} = \sum\limits_{{s \ne q}} {\frac{{{\mathbf{c}}{}_s\Delta {\mathbf{T}}{{\mathbf{c}}_q}}}{{{t_q} - {t_s}}}} {{\mathbf{c}}_s} $$

(1.56)