6. If a sphere of radius $R$ contains four identical smaller spheres of radius $r$, then the maximum possible value of $r$ is $(\quad)$.
(A) $\frac{\sqrt{3}}{2+\sqrt{3}} R$
(B) $\frac{\sqrt{6}}{3+\sqrt{6}} R$
(C) $\frac{1}{1+\sqrt{3}} R$
(D) $\frac{\sqrt{5}}{2+\sqrt{5}} R$