Asymptotic solution of a class of boundary-value problems

Resonance Ejection from the Paul Trap: A Theoretical Treatment Incorporating a Weak Octapole Field

“…As the frequency is increased beyond B, it cannot follow the curve from B to D in Figure . Instead, a sudden quenching of the oscillations occurs, and amplitude falls from a B down to that in point C …”

“…As will be shown below, the main features of resonance excitation in nonlinear ion traps start to show up at fairly large amplitudes of axial oscillations 15 after this stabilization already occurred and r / z ≪ 1. At this stage, the role of r - and z -coupling becomes small relative to the z 3 term and does not affect the general trend of excitation …”

“…By letting all transition rates in eq 17 tend to zero, i.e., d a /dτ → 0, dψ/dτ → 0, b → 0, the dependence of amplitude a on the excitation frequency ν can be obtained as shown in Figure , curve MABDCN (the resonance curve). The peculiar shape of the resonance curve is reminiscent of an ocean wave breaking over, in clear contrast with the symmetric shape of the analogous curve for a linear system. , This “turning-over” is caused solely by introduction of the nonlinearity into eq 8. If the damping in the system is sufficiently low, typical one-to-one correspondence of frequency and amplitude disappears in the frequency range from ν D to ν C .…”

“…Detailed analysis of eq 8 by asymptotic methods of the theory of vibrations was first performed by Mitropol'skii . The solution of eq 9 in a first approximation is given by where dimensionless amplitude a and phase ψ should be determined from the following differential equations (variables as defined above): Equations 17 are best analyzed in two stages …”

“…The introduction of nonlinearity changes dramatically the physical picture of ion confinement. Analysis of this picture is based essentially on the results of the theory of vibrations and, perhaps unexpectedly, relates the phenomenon of ultrahigh mass resolution to the quenching of oscillations in radio frequency (rf) resonators or the instability of crankshafts . These results are applied to ion trap mass spectrometry in pursuit of simple but informative theories.…”