Cramer–Rao bound and unbiased estimator of $\sigma$

Let $x_1,x_2,\dots,x_n$ be a random sample from $N(\mu,\sigma^{2})$ distribution, the parameter $\mu$ is known. Which of the following statements is/are correct?

I. The Cramer–Rao lower bound to the variance of an unbiased estimator of $\sigma$ is $\dfrac{2\theta^{2}}{n}$, where $\theta=\sigma^{2}$.

II. An unbiased estimator of $\sigma$ is $\dfrac{\sum_{i=1}^{n}|x_i-\mu|}{n}$.