First-order perturbative calculation of the frequency-shifts caused by static cylindrically-symmetric electric and magnetic imperfections of a Penning trap

Abstract

The ideal Penning trap consists of a uniform magnetic field and an electrostatic quadrupole potential. Cylindrically-symmetric deviations thereof are parametrized by the coefficients Bη and Cη, respectively. Relativistic mass-increase aside, the three characteristic eigenfrequencies of a charged particle stored in an ideal Penning trap are independent of the three motional amplitudes. This threefold harmonicity is a highly-coveted virtue for precision experiments that rely on the measurement of at least one eigenfrequency in order to determine fundamental properties of the stored particle, such as its mass. However, higher-order contributions to the ideal fields result in amplitude-dependent frequency-shifts. In turn, these frequency-shifts need to be understood for estimating systematic experimental errors, and eventually for correcting them by means of calibrating the imperfections. The problem of calculating the frequency-shifts caused by small imperfections of a near-ideal trap yields nicely to perturbation theory, producing analytic formulas that are easy to evaluate for the relevant parameters of an experiment. In particular, the frequency-shifts can be understood on physical rather than purely mathematical grounds by considering which terms actually drive them. Based on identifying these terms, we derive general formulas for the first-order frequency-shifts caused by any perturbation parameter Bη or Cη

A parts-per-billion measurement of the antiproton magnetic moment

“…Imperfections of the electric quadrupole potential and the residual magnetic field inhomogeneity in the precision trap give rise to small amplitude-dependent frequency shifts of the antiproton's eigenfrequencies and thereby of the determined g-factor. A comprehensive analysis of amplitude-dependent systematic effects can be found in the references 22,30 . In our case, the dominant systematic uncertainties scale all with the residual magnetic bottle where χ(ν,ν z ,Δ ν z ) denotes the dip lineshape function for a constant axial frequency 31,32 ν z + Δ ν z .…”

“…In our case, the dominant systematic uncertainties scale all with the residual magnetic bottle where χ(ν,ν z ,Δ ν z ) denotes the dip lineshape function for a constant axial frequency 31,32 ν z + Δ ν z . ν z denotes the unperturbed axial frequency, T z is the temperature of the axial detection system, C 4 and C 6 characterize potential perturbations in the trap and give rise to an amplitude-dependent axial frequency shift 22,30 Δ ν z (E z ,C 4 ,C 6 ). A variation of the voltage ratio that is applied to the correction electrodes V ce and the ring electrode V ring -the tuning ratio TR = V ce /V ringchanges C 4 and C 6 and thereby the signal-to-noise ratio of the dip.…”

High-precision mass spectrometer for light ions

“…In absence of electric field anharmonicities, shifts in the axial and modified cyclotron frequency occur due to B 2 and special relativity [22,70]:…”

QEC value of the superallowed β emitter Sc42

“…We used the Ramsey excitation pattern, 25 ms (on) -750 ms (off) 1 See for example Ref. [15,16] for a complete description of ion motions in a Penning trap -25 ms (on), to produce TOF-ICR resonance curves for each ion species as shown for 42 Sc in Fig. 1.…”