Brownian Motion in 1D Structures

As mentioned in the last of my posts the main problem to observe truly one dimensional Brownian motion is the fabrication of narrow structures. In organic chemistry one important material is tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ)

Structure of TTF-TCNQ (image downloaded from http://www.intechopen.com/books/nanowires-fundamental-research/nanowires-of-molecule-based-conductors)

The large planar molecules are preferentially located on top of each other, and the one dimensionality of the electronic band structure is enhanced by the directional nature of the highest molecular orbitals. With the experimental technique of NMR (Nuclear Magnetic Resonance) one can monitor the motion of the electronic spin in these one dimensional bonds [Soda et. al. J.Phy. 38,931,(1977)].

In an ideal world without perturbations one would observe a sharp delta function agh the nuclear Larmor frequency. However, perturbations generate random magnetic fields at the sites of the nucleus and lead to a broadening of the resonance line. A source of perturbation is the electronic spins which couple to the nuclear spins through hyperfine interaction and thus generating a fluctuating magnetic field (at the nucleus site) reflecting the dynamics of the electron spin motion.

Assuming the electrons perform a random walk in one dimension it can be shown using the spin-spin correlation function that the spin lattice relaxation rate for this case needs to be proportional to x=1/Sqrt(H), where H is the initial external magnetic field.

In the measurement the spin-lattice relaxation rate is measured which is then plotted versus x. There is a wide range where the relaxation rate is proportional to 1/Sqrt(H) indicating 1D diffusion of electronic spins.