2. 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) $-a$.
(D) $\frac{3}{4} a$.
2. (D).

As shown in the figure, fold along $AC$ to form a dihedral angle. Since $AC \perp OB$, $AC \perp OD$ (in the plane figure), then $AC \perp$ plane $BOD$.

Let $\angle A=60^{\circ}$, then $OB=OD=\frac{a}{2}$. Since $\angle BOD$ is the plane angle of the dihedral angle, when $\angle BOD=60^{\circ}$, $BD=\frac{a}{2}$. Draw $OE \perp BD$, then $OE$ is the distance between the two diagonals, $OE=d=\frac{\sqrt{3}}{4} a$. When $\angle BOD=120^{\circ}$, $BD=\frac{\sqrt{3}}{2} a$, at this time $OE=d=\frac{a}{4}$.

Let $\angle A=120^{\circ}$, then $OB=OD=\frac{\sqrt{3}}{2} a$. When $\angle BOD=60^{\circ}$, $BD=\frac{\sqrt{3}}{2} a, OE=d=\frac{3}{4} a$; when $\angle BOD=120^{\circ}$, $BD=\frac{3}{2} a, OE=d=\frac{\sqrt{3}}{4} a$. Comparing the above $d$ values, the maximum value is $\frac{3}{4} a$.