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MasterEquations

< Simulation of quadrupole and Zeeman interaction | Index | The master equations for 1/T2 >

The description of the spin relaxation process is just another example of the influence of a small coupling to a thermodynamic reservoir on the quantum dynamics of a simple system (the spin system, in our case).

We will describe this process giving due emphasis to the conceptual steps in the calculation.

As in the case of simpler spectral features, like the unperturbed precession dynamics of isolated spins, we shall calculate the average values of the spin observables ({$O$}) with the density matrix formalism

{$ \langle O \rangle = Tr \rho O $}

Here, however, two distinct averages are needed: that on the quantum states of the spin system and that on the statistical ensemble which describes the reservoir (electrons, in our first example). The second one will be explicitly indicated by an overline:

{$ \overline{\langle O \rangle} = \overline{Tr \rho O} =Tr \overline \rho O $}

where we assume that the two averages commute (i.e. we disregard correlations between, say, the electron and the nuclear dynamics).

The simple example that we work out in details is that of a nucleus in a paramagnetic compound, where nuclei and electrons are coupled by the isotropic hyperfine interaction. This may either directly be the Fermi contact interaction to an on-site unpaired electron spin, as in simple metals or in atomic hydrogen. Most often it is the so-called super-transferred interaction, mediated by the indirect polarization of an orbital in the bond, such as in M-O-P, where M is a paramagnetic species, P the ion of the probe nucleus and O the oxygen ion in the bond.

We now imagine one nucleus, subject to a strong Zeeman interaction with the external field, {$ {\cal H}_0=-\hbar\gamma B I_z$}, and to an additional interaction with one paramagnetic electron, {$ {\cal H_1}(t)=\hbar\gamma \mathbf{B_e}(t) \cdot \mathbf{I} $}, where the electron field {$\mathbf B_e=\frac 2 3 \hbar\gamma_e \mu_0 |\Psi|^2 \mathbf{s}(t)$} is considered a random variable with known statistical behaviour.

Typically the instantaneous value of {$\mathbf{B_e}$} is large, typically even larger than the external field {$B$}, but its average {$\overline {\mathbf{B_e}(t)}$} either vanishes or is much smaller than {$B$}, in which case the parallel component will be simply included in {$B$} and it gives rise to the Knight shift. The other relevant features of {$\mathbf{B_e}$} statistical behaviour will be introduced shortly.

< Simulation of quadrupole and Zeeman interaction | Index | The master equations for 1/T2 >