Relaxation times of spin states of all ranks and orders of quadrupolar nuclei estimated from NMR z-spectra: Markov chain Monte Carlo analysis applied to 7Li+ and 23Na+ in stretched hydrogels.

The NMR z-spectra of 7Li+ and 23Na+ in stretched hydrogels contain five minima, or critical values, with a sharp "dagger" on the central dip. The mathematical representation of such z-spectra from spin-3/2 nuclei contains nine distinct (the total is 15 but there is redundancy of the ±order-numbers) relaxation rate constants that are unique for each of the spin states, up to rank 3, order 3. We present an approach to multiple-parameter-value estimation that exploits the high level of separability of the effects of each of the relaxation rate constants on the features of the z-spectrum. The Markov chain Monte Carlo (MCMC) method is computationally demanding but it yielded statistically robust estimates (low coefficients of variation) of the parameter values. We describe the implementation of the MCMC analysis (in the present context) and posit that it can obviate the need for using multiple-quantum filtered RF-pulse sequences to estimate all relaxation rate constants/times under experimentally favorable, but readily achievable, circumstances.