Applications of the fractional calculus to study the physical theory of ion motion in a quadrupole ion trap

In this article, fractional calculus has been applied to study the motion of ions in a three-dimensional radio frequency quadrupole ion trap; we have called this arrangement a fractional quadrupole ion trap. The main purpose of the article is to show that by controlling the fractional parameter of a trapped ion, one can gain a more efficient mass separation. In what follows, we will see that with decreasing the fractional parameter, we can achieve a smaller first stability region. Note that a small stability diagram will result in a good and acceptable mass separation. Various methods can be proposed to obtain a desired ion acceleration with a sufficient accuracy for good mass separation, which is similar to the one obtained by a fractional ion trap. Some of these methods are using the effects of a damping force, a magnetic field or both on the confinement of particles in the quadrupole ion trap. The first stability regions are plotted for all of the aforementioned methods, and simulation results are provided to compare them with those for the fractional case.