81 D. If folded along $B D$, let the midpoint of $A C$ be $F$, and the midpoint of $B D$ be $E$, then $A E=\frac{\sqrt{3}}{2} a$,
$$
E F=A E \cos \frac{\theta}{2} \leqslant \frac{\sqrt{3}}{2} a \cos \frac{\pi}{6}=\frac{3}{4} a .
$$

If folded along $A C$, let the midpoint of $A C$ be $F$, and the midpoint of $B D$ be $E$, then $B D=\frac{1}{2} a, E F=B D \cos \frac{\theta}{2} \leqslant$
$$
\frac{\sqrt{3} a}{4} .
$$
81 Given a rhombus $A B C D$ with side length $a$ and $\angle A=\frac{\pi}{3}$, the rhombus $A B C D$ is folded along the diagonal to form a dihedral angle $\theta$, where $\theta \in\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$. Then the maximum distance between the two diagonals is
A. $\frac{3}{2} a$
B. $\frac{\sqrt{3}}{4} a$
C. $\frac{\sqrt{3}}{2} a$
D. $\frac{3}{4} a$