ESR Saturation and Double Resonance in Liquids

Abstract

The well-known result from the steady-state (s.s.) solution of the Bloch Equations is that the absorption is given by the y-component of magnetization ${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over M} }}_{\rm{y}}}$ in the rotating frame:

$${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over M} }}_{\rm{y}}} = {{{\rm{\gamma }}{{\rm{H}}_1}{{\rm{T}}_2}} \over {1 + {{\left( {{{\rm{T}}_2}\Delta {\rm{\omega }}} \right)}^2} + {{\rm{\gamma }}^2}{\rm{H}}_1^2{{\rm{T}}_1}{{\rm{T}}_2}}}{{\rm{M}}_{\rm{0}}}$$

((1))

with M₀ the equilibrium magnetization. When we switch to a quantum mechanical description, we can calculate:

$${{\rm{M}}_ \pm } = {{\rm{M}}_{\rm{X}}} \pm {\rm{i}}{{\rm{M}}_{\rm{y}}} = \left( {{{{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over M} }}}_{\rm{x}}}{\rm{i}}{{{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over M} }}}_{\rm{y}}}} \right){{\rm{e}}^{ \pm {\rm{i\omega t}}}}$$

((2))

statistically from its associated quantum mechanical operator

$${{\rm{m}}_ \pm } = \Re {\rm{ }}{{\rm{Y}}_{\rm{e}}}{\rm{S}}{{\rm{ }}_ \pm }$$

((3))

where ℜ is the concentration of electron spins, by taking a trace of the spin density matrix σ(t) with the spin operator S_±:

$${{\rm{M}}_ \pm }\left( {\rm{t}} \right) = \Re {{\rm{Y}}_{\rm{e}}}{\rm{ Tr}}\left[ {{\rm{\sigma }}\left( {\rm{t}} \right){{\rm{S}}_ \pm }} \right]$$

((4))

The trace is invariant to a choice of zero-order basis states. The equation of motion for σ(t) is taken to be the relaxation matrix form given by Eq. VIII-20, and we shall neglect effects of higher order than R⁽²⁾.