## Basics of PFG NMR

According to the classical concept of nuclear magnetismus, the phenomenon of NMR is associated with the occurrence of precessional motion for nuclear spins placed in a magnetic field. The precession (Larmor) frequency $\omega_{0}$ is directly proportional to the strength $B_0$ of the magnetic field, $\omega_{0}=\gamma B_{0}$, where $\gamma$ is the gyromagnetic ratio of the nuclear spin. Different nuclei possess different $\gamma$. This allows frequency-based signal separation and provides sensitivity to the chemistry of the molecular species. The spin system can be brought into coherence by a resonant radio-frequency (r.-f.) field in the form of a short pulse with a well-defined duration (the so-called 90$^{\circ}$-pulse) and by letting the system evolve for a time period $\tau$, the spins will acquire a phase $\gamma B_{0} \tau$. The effect of a subsequently applied second r.-f. pulse (or a series of two 90$^{\circ}$-pulses as shown in the Figure 1) is equivalent to inverting the precessional direction so that, at time $\tau$ after the inversion, the initial coherence will again be restored. The thus formed signal is called the spin-echo.

For the diffusion measurements, the series of r.-f. pulses are combined with those of the magnetic field gradients (the so-called 'field gradient pulses', shown by the blue and green rectangles), i.e. the constant magnetic field is superimposed over two short time intervals $\delta$ by an inhomogeneous field. They are typically linear in space with the linearity constant $g=dB/dz$, where the $z$-axis is conventionally chosen to align in the direction of the external magnetic field $B_{0}$. The effect of the two gradient pulses is to encode and to decode positionally the nuclear spins according to their Larmor frequencies. This is essentially done by the phase differences $\gamma \delta gz$ acquired during $\delta$. Thus, if the spins are hypothetically immobilized in space, the effects of the two pulses would compensate each other and the spin-echo signal intensity would remain unchanged in comparison with the situation without gradient pulses. If, however, due to diffusion the spins interchange their positions by $\Delta z=z-z_{0}$, their contribution to the signal will be attenuated by the factor $\cos(\gamma \delta g \Delta z)$.

This is demonstrated by the animation showing the runners running for identical lengths of time $\tau$ forward and backward. The lanes have different covers, determining the runners speeds (different Larmor frequencies). If the runners are not allowed to interchange their lanes (no interlane diffusion or diffusion along the $z$-axis in the respective NMR experiments), at time instant $2\tau$ they all will meet in the starting block. If, however, the runners are let to change their initial lanes (either after time $\tau$ as shown in the animation or during their race), they will cross the starting block at different times. Thus, in the former case the athlets density (coherence) in the strating block will be recovered, but lost for the latter case.

The overall signal $S$ appears as the average over all spins \begin{equation}

S = S(g=0) \int P(\Delta z, t) \cos(\gamma \delta g \Delta z) dz,

\label{NMR_S}

\end{equation}

where $P(\Delta z, t)$, referred to as the mean propagator, stands for the probability (density) that an arbitrarily selected molecule (strictly: nuclear spin) within the sample will be shifted, during $t$, over a distance $\Delta z$ in the direction of the magnetic field gradient. For processes undergoing normal diffusion the mean propagator is given by a Gaussian and eq~(~\ref{NMR_S}) readily simplifies to

\begin{equation}

S = S(q=0) e^{-\frac{1}{2}q^{2} \langle z^{2} \rangle} = S(q=0) e^{-q^{2} D t} ,

\label{NMR_S1}

\end{equation}

where the notion $q=\gamma \delta g$ has been used. The right hand side of eq~(\ref{NMR_S1}) is obtained using the Einstein equation $\langle z^{2} \rangle = 2 D t$ . The possibility to vary the separation time between the magnetic field gradients in the pulse sequence in a range between a few milliseconds to a few seconds allows tracing the mean square displacements as a function of the observation time $t$ in this time window. This time window, depending on the mobility of the species under consideration, may correspond to length scale probed by these species from about hundred nanometers to several micrometers.