Average deviation or standard deviation for volatility?

To gauge the spread $\sigma$ of i.i.d. Normal returns you compare two estimators. One is built from the average absolute deviation, $d = \frac{1}{n}\sum_i |X_i - \bar X|$, rescaled to target $\sigma$ (since $\mathbb{E}|X - \mu| = \sigma\sqrt{2/\pi}$, use $\hat\sigma_{\text{abs}} = d\sqrt{\pi/2}$). The other is based on the usual root-mean-square deviation, essentially the sample standard deviation.

Which is more efficient under normality, and roughly what is the asymptotic relative efficiency of the absolute-deviation estimator?