Pulse EPR of Paramagnetic Centers in Solid Phases

Abstract

We present an overview of the most used Electron Spin Echo techniques and their applications to the study of structure and dynamics of paramagnetic centers in solid phases. A short theoretical section presents the tools necessary to understand the experiments. Three sections describe the experiments that are used to get information on the spin and spatial dynamics of the system, on the distribution of paramagnetic centers in the solid matrix, and on their local environment. Many examples of applications to different paramagnetic centers in various research fields are given.

Appendix

The 2p echo decays are all given by a stretched exponential function

$$ {V_d}(2\tau ) = {V_0}\exp {( - 2\tau /{T_M})^x} $$

(2.37)

• 1. In the case of relaxation due to time modulation of anisotropic terms of the spin Hamiltonian, solution of Eq. (2.17) in fast motion régime (Redfield limit) [16], x = 1,

  $$ {{1} \left/ {{{T_M}}} \right.}{ = }{\Delta^{{2}}}\frac{{{\tau_c}}}{{{1 + }{\omega^{{2}}}\tau_c^2}} $$

  (2.38)

  where Δ is the magnetic anisotropy (in frequency units) averaged by the motion and τ _c is the correlation time of the motion.

• 2. In the case of instantaneous diffusion, x = 1 and with $2{\theta_1} = {\theta_2}$ (see Sect. 2.4.1)

  $$ {{1} \left/ {{{T_M}}} \right.} = A + B \cdot {\sin^2}({{{{\theta_2}}} \left/ {2} \right.}) $$

  (2.39)

  where A is the contribution due to any other process, and B is due to instantaneous diffusion and it is proportional to the concentration see Eq. (2.27), and θ₂ is the tilting angle of the π pulse (assuming non selective pulses).

In the case of spin-spin interactions (spectral diffusion) the values of the exponent and of 1/T _M depend on the particular process and model, see Table 2.1.

Table 2.1 Analytical expressions for echo decay in the presence of spectral diffusion for given models and approximations

The general expression found for the echo intensity is obtained from solution of Eq. (2.13) as [60]:

$$ {{{V}}_{{Y}}}\left( {{2}\tau + {{T}}} \right) = {{n}}\left( {{\omega_{{k}}}} \right){\hbar^{{2}}}\gamma {\omega_{{k}}}{/}\left( {{4kT}} \right){{Re}}\left[ {{ \exp }\left[ {{{i}}\int_{{0}}^{{{2}\tau }} {{{s(t^\prime)}}\delta {\omega_{{k}}}{{(t^\prime}}} {{)dt^\prime}}} \right]} \right] $$

(2.40)

where ω_k is the resonance frequency of the k-th spin packet, δω_k its frequency shift due to coupled-spin flips, and s(t’) is a step function (equal to +1,0 or –1 during the different free evolutions).

Evaluation of the integral can be performed on the basis of the type of spin-flip mechanism with different models. The expression is given for the three pulses (stimulated) echo, for the 2p-echo the time T = 0.