High-temperature spin dynamics in the one-dimensional Heisenberg system ${\left(\mathrm{C}{\mathrm{H}}_3\right)}_4$NMn${\mathrm{Cl}}_3$ (TMMC): Spin diffusion, intra- and interchain cutoff effects

High-temperature spin dynamics in the one-dimensional (1d) Heisenberg system ${\left(\mathrm{C}{\mathrm{H}}_3\right)}_4$NMn${\mathrm{Cl}}_3$ (TMMC), is investigated through EPR and proton spin relaxation-time measurements. When the magnetic field $\overrightarrow{\mathrm{H}}$ is parallel to the chain axis $\overrightarrow{\mathrm{c}}$ the field dependence of both the relaxation rate $T_1^{-1\mathrm{}}$ and the relaxation rate in the rotating frame $T_{1\rho }^{-1\mathrm{}}$ fits well an expression of the form $P{H}^{-\frac{1}{2}}+Q$. The $H^{-\frac{1}{2}}$ term in this expression proves that the spectral density $f^{+}\left(\omega \right)\propto 〈S_i^{+}\mathrm{}\left(t\right)\mathrm{}{S}_i^{-}\mathrm{}\left(0\right)\mathrm{}〉$ behaves according to a 1d diffusive law $f^{+}\left(\omega \right)\propto {\left(D_{\omega }\right)}^{-\frac{1}{2}}$ where $D$ represents the diffusion coefficient of the electronic two-spin correlation functions. When $\overrightarrow{\mathrm{H}}\perp \overrightarrow{\mathrm{c}}$ the $T_1^{-1\mathrm{}}$ and $T_{1\rho }^{-1\mathrm{}}$ data show a quite different behavior, being almost constant or decreasing slightly at low field. Such a behavior is interpreted in terms of cutoff effects which limit the $1d$ diffusive behavior of the spectral density $f^z\left(\omega \right)\propto 〈S_i^z\mathrm{}\left(t\right)\mathrm{}{S}_j^z\mathrm{}\left(0\right)\mathrm{}〉$. A cutoff frequency ${\omega }_c^z$ is defined as the frequency at which the ${\omega }^{-\frac{1}{2}}$ divergence in $f^z\left(\omega \right)$ is truncated. The value, the field dependence, and the orientation dependence of ${\omega }_c^z$ show that the cutoff mechanisms are originated from electronic intrachain dipolar and interchain Heisenberg interactions. A theory is presented taking into account both intra- and interchain interactions. Self-consistent expressions are given for evaluating the cutoff frequencies. When