Scaling behavior of the proton spin-lattice relaxation rate in antiferromagnetic molecular rings

Abstract

We present new and refined data for the magnetic field $(H)$ and temperature $(T)$ dependence of the proton spin-lattice relaxation rate $(1∕{T}_{1})$ in antiferromagnetic molecular rings as well as a new explicit scaling formula that accurately reproduces our data. The key ingredients of our formulation are (1) a reduced relaxation rate, $R(H,T)=(1∕{T}_{1})∕(T\ensuremath{\chi}(T))$, given by $R(H,T)=A{\ensuremath{\omega}}_{c}(T)∕({\ensuremath{\omega}}_{c}^{2}(T)+{\ensuremath{\omega}}_{N}^{2})$, where $\ensuremath{\chi}={(\ensuremath{\partial}M∕\ensuremath{\partial}H)}_{T}$ is the differential susceptibility, $A$ is a fitting constant, and ${\ensuremath{\omega}}_{N}$ is the proton Larmor frequency, and (2) a temperature-dependent correlation frequency ${\ensuremath{\omega}}_{c}(T)$ which at low $T$ is given by ${\ensuremath{\omega}}_{c}(T)\ensuremath{\propto}{T}^{\ensuremath{\alpha}}$, that we identify as a lifetime broadening of the energy levels of the exchange-coupled paramagnetic spins due to spin-acoustic phonon coupling. The main consequences are (1) $R(H,T)$ has a local maximum for fixed $H$ and variable $T$ that is proportional to $1∕H$; the maximum occurs at the temperature ${T}_{0}(H)$ for which ${\ensuremath{\omega}}_{c}(T)={\ensuremath{\omega}}_{N}$; (2) for low $T$ a scaling formula applies, $R(H,T)∕R(H,{T}_{0}(H))=2{t}^{\ensuremath{\alpha}}∕(1+{t}^{2\ensuremath{\alpha}})$, where $t\ensuremath{\equiv}T∕{T}_{0}(H)$. Both results are confirmed by our experimental data for the choice $\ensuremath{\alpha}=3.5\ifmmode\pm\else\textpm\fi{}0.5$.