(M_{\rm eq} - M)$ where $M_{\rm eq} = \chi H$ is the equilibrium magnetisation and $\nu \propto u n$. In general it is difficult to deconvolve these various factors. In Ref. it was achieved by independent measurement of $n(T)$ and $\chi(T)$ to reveal $u(T)$. In the present time-dependent experiments we cannot perform such a direct separation, but by studying [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples with different isotopes, it seems reasonable to assume that the susceptibility and starting density are roughly the same, so the variation in mobility (hop rate) will dominate differences between the samples. Inclusion of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} in the comparison gives a further point of reference: the starting monopole densities (see above) and susceptibilities for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} are expected to be comparable to those of [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, while the the tunnel splitting (which controls the intrinsic mobility) is also estimated to be of the same order [@Tomasello] in the appropriate range of internal fields (see Fig. 1 and Ref. , Fig. 5).

![image](figure3.pdf){width="12cm"}

Fig. 3 summarizes results for the relaxation of the magnetization $M(t)$ for the different samples, as well as the value of $M(t = 400~{\rm s})$ and the monopole current $J_{\rm m}(t = 0)$ as a function of wait time, for a 