A particle of mass $$m$$ moves in a circular orbit in a central potential field $$U(r) = U_0 r^4$$. If Bohr's quantization conditions are applied, radii of possible orbitals $$r_n$$ vary with $$n^{1/\alpha}$$, where $$\alpha$$ is ________.

For a circular orbit in the potential $$U(r) = U_0 r^4$$, the force is $$F = -\frac{dU}{dr} = -4U_0 r^3$$. The magnitude of this centripetal force equals $$\frac{mv^2}{r}$$, so $$4U_0 r^3 = \frac{mv^2}{r}$$, giving $$mv^2 = 4U_0 r^4$$ and thus $$v = 2r^2\sqrt{\frac{U_0}{m}}$$.

Applying Bohr's quantization condition $$mvr = n\hbar$$, we get $$m \cdot 2r^2\sqrt{\frac{U_0}{m}} \cdot r = n\hbar$$, which simplifies to $$2r^3\sqrt{mU_0} = n\hbar$$.

Solving for $$r$$: $$r^3 = \frac{n\hbar}{2\sqrt{mU_0}}$$, so $$r \propto n^{1/3}$$. Since $$r_n \propto n^{1/\alpha}$$, we identify $$\alpha = 3$$.