Simulation of ion behavior in an open three-dimensional Paul trap using a power series method

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean’s stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.

Domain-partition power series in vibration analysis of variable-cross-section rods

A radial dipolar excitation has been applied to a hyperbolic Paul trap with a segmented ring electrode. In addition, this configuration allowed the generation of an elliptical Paul trap by superimposing the RF trapping field with an electrostatic azimuthally quadrupolar field. The changes in the radial ion motions with respect to the conventional Paul trap are discussed and investigated experimentally. As expected, the formerly degenerated radial eigenfrequencies $\nu_{r}$ of the standard Paul trap split into two separate resonances at $\nu_{x}$ and $\nu_{y}$ , while the axial eigenfrequency $\nu_{z}$ remains unaffected. The new trap design adds flexibility and extends the range of applications of Paul traps.

A theoretical method was proposed in this work to study space charge effects in quadrupole ion traps, including ion trapping, ion motion frequency shift, and nonlinear effects on ion trajectories. The spatial distributions of ion clouds within quadrupole ion traps were first modeled for both 3D and linear ion traps. It is found that the electric field generated by space charge can be expressed as a summation of even-order fields, such as quadrupole field, octopole field, etc. Ion trajectories were then solved using the harmonic balance method. Similar to high-order field effects, space charge will result in an "ocean wave" shape nonlinear resonance curve for an ion under a dipolar excitation. However, the nonlinear resonance curve will be totally shifted to lower frequencies and bend towards ion secular frequency as ion motion amplitude increases, which is just the opposite effect of any even-order field. Based on theoretical derivations, methods to reduce space charge effects were proposed.

Methods to reduce mass shifts caused by space charge with mass-selective axial ejection from a linear quadrupole ion trap are investigated. For axial ejection, dipole excitation is applied to excite ions at q ≈ 0.85. The trapping radiofrequency (rf) voltage is scanned to bring ions of different m/z values into resonance for excitation. In the fringing field at the quadrupole exit, excited ions gain axial kinetic energy, overcoming the trapping potential, and are ejected from the trap. Space charge causes the frequencies of ion oscillation to decrease. Thus, greater rf voltages are required to bring ions into resonance for excitation and ejection, and the ions shift to higher apparent masses in a mass spectrum. At the same time, the peaks broaden, lowering resolution. The effects of injection q value, ejection q value, excitation amplitude, quadrupole dc voltages applied to the electrodes, applying an rf voltage to the exit lens, and scan speed, on mass shifts have been studied experimentally. Most experiments were done with only ions of protonated reserpine (m/z 609.3 and its isotopic peaks) in the trap. Some experiments were done with ions of protonated reserpine and ions of m/z 622 in the trap. In general, the mass shifts are reduced with higher ejection q values, higher excitation amplitudes, with quadrupole dc applied, and at higher scan speeds. The application of quadrupole dc appears to increase the ion cloud temperature, which lowers mass shifts. Thus, a proper choice of operating conditions can reduce, but not eliminate, mass shifts caused by space charge.

The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales $R_k$ in the power term $(t/R_k)^k$ is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

The LPCTrap experiment uses an open Paul trap which was built to enable precision measurements in the beta decay of radioactive ions. The initial goal was the precise measurement of the beta-neutrino angular correlation coefficient in the decay of 6He. Its geometry results from a careful optimization of the harmonic potential created by cylindrical electrodes. It supersedes previously considered geometries that presented a smaller detection solid angle to the beta particle and the recoiling ion. We describe here the methods which were used for the potential optimization, and we present the measured performances in terms of trapping time, cloud size and temperature, and space charge related limits. The properties of the ion cloud at equilibrium are well reproduced by a simple numerical simulation using hard sphere collisions, which additionally gives insights on the trapping loss mechanism. The interpretation for the observed trapping liftetimes is further corroborated by a model recently developed for ion clouds in Paul traps. The open trap shall serve other projects. It is currently used for commissioning purpose in the TRAPSENSOR experiment and is also considered in tests of the Standard Model involving the beta decay of polarized $^{23}$Mg and $^{39}$Ca ion in the frame of the MORA experiment. The latter tests require in-trap polarization of the ions and further optimization of the trapping and detection setup. Based on the results of the simulations and of their interpretations given by the model, different improvements of the trapping setup are discussed.

In the present paper, $P-$distributional convergence which is defined by power series method has been introduced. We give equivalent expressions for $P-$distributional convergence of spliced sequences. Moreover, convergence of a bounded $\infty$-spliced sequence via power series method is represented in terms of Bochner integral in Banach spaces.

The aim of this paper is to present Korovkin theorems for positive linear operators of two variables from $H_{\omega }\left( K\right) $ into $%C_{B}\left( K\right) $ via the power series method. In addition, we give an example that our new approximation result works but its classical case does not work. Furthermore, we obtain the rate of convergence of these operators.

In this work, we obtain a Korovkin-type approximation theorem for double sequences of real-valued functions by using the power series method in two-dimensional weighted spaces. We also study the rate of convergence by using the weighted modulus of continuity, and in the last section, we present an application that satisfies our new Korovkin-type approximation theorem but does not satisfy classical one.

This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Pad ´ e (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.

Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.

Mapping the Stability Diagram of a Digital Ion Trap (DIT) Mass Spectrometer Varying the Duty Cycle of the Trapping Rectangular Waveform

The current study utilises Monte Carlo simulation and Mie scattering theory to estimate the reflectance spectra of fabric coated with titanium dioxide nanopigments of various diameters and concentrations. Image processing was carried out and experimental data were gathered to evaluate the performance of Monte Carlo simulation. The distribution and location of the nanopigments on the surface of fabric were determined using the Monte Carlo method. Reflection of the fabric was calculated based on Monte Carlo simulation with the partitive mixing method and Mie theory. According to the experimental and simulation results, the reflectance of coated samples was increased by increasing the concentration and number of titanium dioxide nanoparticles. There was a good match between the results obtained by Monte Carlo simulation and the experimental results. For coated samples (dTiO2: 500 nm), the root mean square error between measured and predicted reflectance by the Monte Carlo and partitive mixing method and by Monte Carlo and Mie theory was 0.022 and 0.0078, respectively. The results indicate that the performance of the Monte Carlo and Mie method was better than that of the Monte Carlo and partitive mixing method. According to t‐test analysis, there was no statistically significant difference between the experimental data and Monte Carlo simulation.

The problem of reflection and refraction of a planar acoustic wave by an inhomogeneous elastic layer whose material possesses general-type anisotropy is considered. The equations of motion of the elastic layer are reduced to a system of ordinary differential equations. The boundary-value problem for this system is solved by two methods: by reduction to problems with initial conditions and by the method of power series. Analytical expressions that describe acoustic fields outside the layer are obtained. Calculation results of the transmission factor for transversely isotropic layers inhomogeneous in thickness are presented.