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Diffraction occurs when the size of the aperture or obstacle is of the same order of magnitude as the wavelength of the incident wave. If the obstacle is a sphere of diameter d, do we have to choose its diameter as size?

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Assuming you are dealing with the non-reflected light, yes. All that matters is the projected size of the sphere in the plane perpendicular to the direction of the light. The sphere's obscuration is the same as a disc with the same diameter.

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