Resolution improvements in dipolar resonant excitation have been examined in a round-rod quadrupolar collision cell for values of the Mathieu characteristic exponent  equal to n/p, where n and m are small integers (prime  values) versus other  values where n and p are not small (ordinary values). The trajectories of ions moving in the time-varying electric fields of a quadrupole with and without buffer-gas molecules were calculated to determine the relationship of prime and ordinary  values to frequency resolution for resonant ion excitation and ejection. For prime  values, the ion trajectory in the hyperbolic quadrupole field will be exactly periodic with a period of at most 4 p/⍀, where ⍀ is the angular frequency of the main drive radio-frequency ( . By scanning the RF voltage, trapped ions may be ejected one m/z ratio at a time out of the confining electric field for detection. While the ions are confined, ion-molecule reactions may be carried out. Additionally, the kinetic energy of the trapped ions can be increased by applying a dipolar AC potential that is resonant with the secular frequency of the motion of ions ( 0 ) of one or more m/z values. This increased kinetic energy can be used to selectively eject the ions from the trapping field, or to add internal energy to the confined ions through collisions with a neutral gas added to the trap. This dipolar resonant excitation has been a common method for performing mass-selective ejection and fragmentation of ions confined in Paul traps and, more recently, in linear ion traps [2][3][4]. The theory of instability space, as it relates to auxiliary AC excitation in a linear and 3D quadrupole trap for quadrupolar resonant excitation, has been addressed [5][6][7]. The trajectories of ions in a linear ion trap with dipolar auxiliary excitation have been examined by Franzen [8]; those for hyperbolic and circular quadrupole rods with and without collision gas by Collings et al. [2].In tandem mass spectrometry (MS/MS), resolution of the product ions is sometimes crucial as many fragments have similar m/z values [9 -15]. Unambiguous product-ion assignment often requires better than unit-mass resolution to resolve "isobaric" interferences, which are particularly acute when the analyte is the minor component. Two factors that degrade m/z resolution after dipolar resonance excitation are increasing collisions with buffer-gas molecules [2] and increasing dipolar voltage amplitude [16]. One method to increase resolution is to excite the ions in such a way to reduce the frequency composition and the periodicity of the ions' motion in the electric field. In the present study, we used simulation software to determine the trajectories of ions [17] moving in a time-varying electric field and colliding with buffer-gas molecules in a linear quadrupole reaction cell with the instrumentally common and practical round-rod geometry.It is a common misconception that an ion trajectory within the quadrupolar field is necessarily aperiodic. As we will show below, provided that the...