Nuclear Magnetic Resonance in Gaussian Stochastic Local Field

Abstract

Anderson–Weiss–Kubo model of magnetic resonance is reconsidered to bridge the existing gaps in its applications for solutions of fundamental problems of spin dynamics and theory of master equations. The model considers the local field fluctuations as one-dimensional normal random process. We refine the conditions of applicability of perturbation theory to calculate the spin depolarization and phase relaxation. A counterexample is considered to show that in the absence of temporal fluctuations of local fields, perturbation theory is not applicable even qualitatively. It is shown that for slow fluctuations, the behavior of the longitudinal magnetization is simply related to the correlation function of the local field. Quasi-adiabatic losses are estimated.

Appendix: Resonance Line Shape and Perturbation Theory for AWK Model

The AWK model produces exact result Eq. (29) for free induction decay $F_{0} (t) = \langle \exp (i\varphi (t,0)\rangle_{n}$ and, correspondingly, for the resonance line shape $g(\Delta )$ Eq. (28). Therefore, it is of interest to check the applicability of the perturbation theory here. According to the results of Sect. 2, we can expect that it is applicable for fast fluctuating local field, when $M_{2} \tau_{c}^{2} \ll 1$ and,

$$F_{0} (|t| > \tau_{c} ) = \exp ( - |t|/T_{2} ),\,g\left( {|\Delta | < \frac{1}{{\tau_{c} }}} \right) = \frac{{T_{2} }}{{\pi (1 + \Delta^{2} T_{2}^{2} )}},\quad T_{2} = (M_{2} \tau_{c} )^{ - 1} , $$

(57)

Indeed, at the scale $t \sim T_{2}$, perturbative free induction decay $F_{0}^{p} (t)$ as well as line shape $g_{p} (\Delta )$ at the scale $\Delta \sim 1/T_{2}$, produced below by Eq. (60), are in reasonable agreement with exact results [Eq. (57)]. However, in reality, there exist important fine properties both for free induction decay and resonance line shape, which are produced erroneously by the perturbation theory [18].

To clarify the problem, we can write the free induction decay Eq. (26) as $F_{0} (t) = \langle \widetilde{F}_{0} (t)\rangle_{n} ,$ $\tilde{F}_{0} (t) = \exp (i\varphi (t,0))$. As a projection operators, simple averaging $Af = \langle f\rangle_{n}$ and $\overline{A} = 1 - A$ can be chosen. Then, standard treatment of the equation of motion:

$$ \frac{\text{d}}{{{\text{d}}t}}\widetilde{F}_{0} (t) = i\omega_{l} (t)\widetilde{F}_{0} (t) $$

(58)

produces similar to transformations Eqs. (22), (23), exact master equation:

$$ \begin{array}{l} \frac{\text{d}}{{{\text{d}}t}}F_{0} (t) = - \int_{0}^{t} {{\text{d}}\tau M^{{ ( {\text{phase)}}}} } (t - \tau )F_{0} (\tau ), \hfill \\ M^{{ ( {\text{phase)}}}} (t - \tau ) = \left\langle {\omega_{l} (t)\overline{A} \exp \left( {i\overline{A} \int_{\tau }^{t} {{\text{d}}s\omega_{l} (s)} \overline{A} } \right)\overline{A} \omega_{l} (\tau )} \right\rangle_{n} . \hfill \\ \end{array} $$

(59)

Keeping in the phase memory function $M^{{ ( {\text{phase)}}}} (\tau )$, the leading order in $M_{2} = \langle \omega_{l}^{2} \rangle$, we have $M_{0}^{{ ( {\text{phase)}}}} (t - \tau ) = \langle \omega_{l} (t)\omega_{l} (\tau )\rangle = M_{2} \kappa (|t - \tau |)$, and the main order master equation takes the form:

$$ \frac{\text{d}}{{{\text{d}}t}}F_{0}^{p} (t) = - M_{2} \int_{0}^{t} {{\text{d}}\tau \kappa (\tau )F_{0}^{p} (t - \tau )} . $$

(60)

Here, the superscript “p” indicates that $F_{0}^{p} (t)$ is produced in main order of the perturbation theory.

Using the Laplace transformation, the solution can be written as follows:

$$ F_{0}^{p} (\lambda ) = \int_{0}^{\infty } {{\text{d}}te^{ - \lambda t} } F_{0}^{p} (t) = (\lambda + M_{2} \kappa (\lambda ))^{ - 1} ,\quad \kappa (\lambda ) = \int_{0}^{\infty } {{\text{d}}te^{ - \lambda t} } \kappa (t). $$

(61)

The long time tail of the correlation function $\kappa \left( t \right)$ Eq. (41) produces nonanalytical dependence:

$$ \kappa (\lambda \to 0) = \tau_{c} - 2\pi^{1/2} \lambda^{1/2} (T_{2T} + \tau_{0} )^{3/2} + O(\lambda ), $$

(62)

while exact solution

$$ F_{0}^{{}} (\lambda ) = \int_{0}^{\infty } {{\text{d}}te^{ - \lambda t} } F_{0}^{{}} (t) = \int_{0}^{\infty } {{\text{d}}te^{ - \lambda t} } \exp \left( { - M_{2} \int_{0}^{t} {{\text{d}}\tau (t - \tau )\kappa (\tau )} } \right) $$

(63)

has analytical expansion for small $\lambda$. As a result

$$ F_{0}^{p} (t \to \infty ) \sim - t^{ - 3/2} $$

(64)

contrary to exact,

$$ F_{0} (t \to \infty ) \sim + \exp ( - M_{2} \tau_{c} t). $$

(65)

The line shape

$$ g(\Delta ) = \frac{1}{\pi }\text{Re} F_{0} (\lambda = \varepsilon - i\Delta ) = \frac{X(\Delta )}{{\pi \left[ {(\Delta - Y(\Delta ))^{2} + X^{2} (\Delta )} \right]}}, $$

(66)

where $\varepsilon \to + 0$ and $X(\Delta ) = \text{Re} M^{{ ( {\text{phase)}}}} (\lambda = \varepsilon - i\Delta ),$ while $Y(\Delta ) = \text{Im} M^{{ ( {\text{phase)}}}} (\lambda = \varepsilon - i\Delta )$. Similar relation takes place for perturbative line shape, as well:

$$ g_{p} (\Delta ) = \frac{{X_{0} (\Delta )}}{{\pi [(\Delta - Y_{0} (\Delta ))^{2} + X_{0}^{2} (\Delta )]}},\,\,M_{0}^{{ ( {\text{phase)}}}} (\varepsilon - i\Delta ) = X_{0} (\Delta ) + iY_{0} (\Delta ). $$

(67)

At small $\Delta$, as a consequence of Eq. (62)

$$ X_{0} (\Delta ) = M_{2} \left( {\tau_{c} - (2\pi |\Delta |)^{1/2} (T_{2T} + \tau_{0} )^{3/2} } \right),Y_{0} (\Delta ) = M_{2} (2\pi |\Delta |)^{1/2} {\text{sign}}(\Delta )(T_{2T} + \tau_{0} )^{3/2} , $$

and

$$ g_{p} (\Delta \to 0) = (\pi X_{0} (\Delta ))^{ - 1} = \frac{{T_{2} }}{\pi }\left( {1 + \frac{{(T_{2T} + \tau_{0} )^{3/2} }}{{\tau_{c} }}(2\pi |\Delta |)^{1/2} } \right), $$

(68)

while it is evident that for exact line shape, we have

$$ g(\Delta \to 0) = \frac{1}{\pi }\int_{0}^{\infty } {{\text{d}}t\cos (\Delta t)} \exp \left( { - M_{2} \int_{0}^{t} {{\text{d}}\tau (t - \tau )\kappa (\tau )} } \right) \approx \frac{{T_{2} }}{\pi }(1 - (\Delta T_{2} )^{2} ). $$

(69)

We see here that the exact solution has smooth maximum at $\Delta = 0$, while perturbative result shows sharp local minimum at the same point.

The high-frequency asymptotics of exact relation:

$$ g(\Delta \to \infty ) = \int_{ - \infty }^{\infty } {\frac{{{\text{d}}t}}{2\pi }} \exp \left( {i\Delta t - M_{2} \int_{0}^{t} {d\tau (t - \tau )\kappa (\tau )} } \right) = \int_{ - \infty }^{\infty } {\frac{{{\text{d}}t}}{2\pi }} \exp ( - S(t)), $$

(70)

should be defined by the nearest to $t = 0$ complex singularity of $S(t)$ or saddle point $t_{0}$ of the integration contour, which is a solution of the equation $S^{\prime } (t_{0} ) = M_{2} \int_{0}^{{t_{0} }} {{\text{d}}\tau \kappa (\tau )} - i\Delta = 0$ and depends on $\Delta$, $M_{2}$ and $\kappa (t)$. For the sake of simplicity, we apply here $\kappa (t) = \exp ( - \mu^{2} t^{2} /2)$ with $\tau_{c} \sim \mu^{ - 1} .$ Then, for perturbative result Eq. (67), we have

$$ g_{p} (\Delta \to \infty ) \sim \Delta^{ - 2} X_{0} (\Delta ) \sim \Delta^{ - 2} \exp \left( { - \frac{{\Delta^{2} }}{{2\mu^{2} }}} \right), $$

(71)

while for exact line shape, as in [19],

$$ g(\Delta \to \infty ) \sim \exp \left( { - \frac{|\Delta |}{\mu }\left( {2\ln \frac{|\Delta |\mu }{{M_{2} }}} \right)^{1/2} } \right). $$

(72)

We see that exact relations Eqs. (65), (69), and (72) are in drastic differences with perturbative results Eqs. (64), (68), and (71). The discrepancy cannot be removed in any finite order of the perturbation theory; it is immanent property of the theory, because the theory is applied to infinite time region. It is the reason why we prefer to use the AWK model in primary phase relaxation problems [2,3,4,5,6, 8, 9] instead of application of master equations at this level.