I'm wondering if a ball of carbon atoms can cast a shadow? If so, how small does the opaque thing need to be for casting a shadow, no matter how faint it is?
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No. A bucky ball ($\text{C}_{60}$) is too small to cast a shadow. The carbon atoms are quite small, so when the light rays pass through a single bucky ball, the rays would diffract around the ball and the atoms within, which would allow the rays to pass through the buck ball. And thus no shadow will be seen. However, if you take millions of bucky balls and forge them into a solid, then the case would be same as that of any other macroscopic object.
What minimum size should an object be such that it can cast a shado(no matter how faint it would be)?
If the dimensions of your object are far more greater than the wavelength of the light which you are using to cast a shadow, then the shadow will be cast. However if you keep the screen(on which the shadow is supposed to be cast) too far away, then again diffraction effects start dominating (See Fresnel distance {1 and 2} for more information).
Now in the cases where the dimension of your object are comparable to the wavelength of the light being used,the diffraction effects start dominating even at closer distances. So in these cases, the placement of screen also becomes important. And this distance where diffraction effects become significant also depends on the wavelength of the light use to cast the shadow (Again, see Fresnel distance links given above).
So there isn't a universal value of minimum size. But it largely depends on the distance of the screen, wavelength the light and many other factors. Moreover, we are also not considering human eye's limitations, which would make the situation even more unpredictable.