The form factors for star-branched polymers with linear branches or with looping branches are calculated. The effect of chain swelling excluded volume is incorporated through an excluded volume parameter approach. The form factor for ring polymers is also included, since it is nicely derived as a special case. Furthermore, the form factor for dendrimers with excluded volume is also calculated. In order to evaluate the form factor for stars with looping branches, the multivariate Gaussian function is used to close the looping branches. Analytical results are possible in the Gaussian chain case (i.e., with no excluded volume), but the calculations are left in a form involving summations over monomers when the general case incorporates excluded volume.
Keywords: dendrimers; excluded volume effect; looping branches; polymer ring; small-angle scattering form factors; star-branched polymers.