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^{1}School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada

^{2}Department of Mathematics, Islamic Azad University, Bushehr Branch, Bushehr, Iran

- *Corresponding Author:
- Sarkhosh Seddighi Chaharborj

School of Mathematics and Statistics Carleton

University, Ottawa, K1S 5B6, Canada

**Tel:**+16139816230

**E-mail:**[email protected]

**Received date:** May 14, 2017; **Accepted date:** May 23, 2017; **Published date:** May 29, 2016

**Citation: **Chaharborj SS, Gheisari Y (2017) Mass Resolution Study in the Cylindrical Ion Trap by Using Three-Point One Block Method. Mass Spectrom Purif
Tech 3:117. doi:10.4172/2469-9861.1000117

**Copyright:** © 2017 Chaharborj SS, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

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Motion equations in cylindrical ion trap (CIT) coupled in u and v (respective r and z), and thus, can only be treated as a rough approximation. Hence, studies on cylindrical ion trap (CIT) equations are more complicated and involved. Therefore, a three point one block method (3POBM) of Adams Moulton type is presented to study a cylindrical ion trap (CIT) motion equations. The advantage of the three point one block method (3POBM) is to estimate the approximate solutions directly at three points simultaneously. Numerical results from three point one block method (3POBM) will compare with fifth order Runge-Kutta method (RKM5). The proposed three points one block method has a potential application to solve complicated linear and nonlinear equations of the charge particle confinement in the cylindrical field especially in fine tuning accelerators, and generally speaking, in physics of high energy. The physical properties of the confined ions in the r and z axises are illustrated and the fractional mass resolutions m/Δm of the confined ions in the first stability region was analyzed by the fifth order Runge-Kutta method (RKM5) and three points one block method (3POBM).

Cylindrical ion trap; Three point one block method; Fifth order Runge-Kutta method; Lagrange interpolating polynomial; Ion trajectories

An ion trap mass spectrometer may incorporate a Penning trap [1], Paul trap [2] or the Kingdon trap [3]. The Orbitrap, introduced in 2005, is based on the Kingdon trap [4]. The two most common types of ion traps are the Penning trap and the Paul trap (Quadrupole ion trap) [5-9]. Other types of mass spectrometers may also use a linear Quadrupole ion trap as a selective mass filter.

A Quadrupole ion trap mass analyzer with simplified geometry, the cylindrical ion trap (CIT), was shown to be well-suited to use in miniature mass spectrometers and even mass spectrometer arrays. Experiments with a single miniature CIT showed acceptable resolution and sensitivity, limited by the ion trapping capacity of the miniature device. The CIT has received much attention of a number of research groups because of several merits. The CIT is easier to fabricate than the Paul ion trap which has hyperbolic surfaces [10]. And the relative simplicity and small size of the CIT make it an ideal candidate for miniaturization. With these interests, many groups in, such as Purdue University [11] and Oak Ridge National Laboratory [12] have researched on the applications of the CIT to a miniaturized mass spectrometry.

Numerical solutions for ODEs have great importance in scientific computation as they were widely used to model in representing the real world problems such as Quadrupol ion trap with damping force. Many of these problems are of higher order and cannot be solved analytically and hence the use of numerical methods is advocated. There are several methods that can be used to solve the higher order ODEs numerically [13]. Block method for numerical solution of higher order ODEs directly had been proposed by several researchers [13-17]. The general theory of block method for solving first order ODE is given in Fatunla [15]. A two point block methods for solving higher order ODEs directly are described by Majid [14]. In this paper, we shall consider two point one block method for the numerical solution of solving second order ODEs in the form

(1)

subject to an initial conditions

in the interval [a, b]. The proposed two point block method will approximate the solutions at two points simultaneously.

In this section, we have divided the interval [a, b] into a series of blocks with each block containing three points. Three approximate values that are *y _{n+1}, y_{n+2}* and

The first point of the method will be obtained by integrating Equation (1) once and twice over the interval [*x _{n}, x_{n+1}*]. By integrating Equation (1) once gives:

(2)

Integrating Equation (1) twice gives:

(3)

The second point of the method obtain by integrating (1) over the interval [*x _{n}, x_{n+2}*] once and twice will respectively give,

(4)

(5)

The same process is applied to derive the third point of the method, integrate (1) over the interval [*x _{n}, x_{n+3}*] once and twice will respectively give,

(6)

(7)

We can rewrite the (2), (3), (4), (5), (6) and (7) into r point k step Adams Moulton:

(8)

(9)

**Figure 1** showed the seven mesh points i.e., *(x _{n−3}, f_{n−3}), (x_{n−2}, f_{n−2}), (x_{n−1}, f_{n−1}), (x_{n}, f_{n}), (x_{n+1}, f_{n+1}), (x_{n+2}, f_{n+2})* and

(10)

with r=1, 2, 3. The function *f(x, y, y ^{0})* in (10) will be replaced by Lagrange interpolating polynomial. Lagrange Interpolating is a method of finding an unique polynomial,

(11)

For the general case when there are *n*+1 points, the polynomial, Pn in Equation (11) are *nth*-order. Seven interpolating points involved in three point one block method, we will obtain the lagrange interpolating polynomial, P6 from Equation (11).

(12)

Evaluate these integrals in (12) using MAPLE. The value of the coefficients, *β _{i}* and

(13)

with r=1, 2, 3. Where the coefficient *β _{i}* and

r |
i |
0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|---|

1 |
βi |
|||||||

γi |
||||||||

2 |
βi |
|||||||

γi |
||||||||

3 |
βi |
|||||||

γi |

**Table 1:** The coefficients for the corrector formulae of three point one block method.

Apply the same process to find the predictor formulae of the two point direct method. The method is the combination of predictor of one order less than the corrector. The Euler method will be used only once at the beginning of the code to find the additional points for the starting initial points. The Euler method also will solve the problem directly. Then, the predictor and corrector direct method can be applied until the end of the interval. The sequence of computation involved was PECE where P and C indicate the application of the predictor and corrector formula respectively and E indicate the evaluation of the function f. The convergence test involved the corrector formulae and it will be iterated to convergence. The iterates are said to have converged when an iterate satisfies as follows:

(14)

where is come from predictor formula and “TOL” mean Tolerance. For numerical results has been used *Tol =*10^{−10}.

In cylindrical ion trap (CIT), the hyperbolic ring electrode [18], as in Paul ion trap, is replaced by a simple cylinder and the two hyperbolic end-cap electrodes are replaced by two planar end-plate electrodes [19]. If opposite pairs of the electrodes have steady potentials Ψ_{0} and −Ψ_{0} [20], the potential difference applied to the electrodes [18,21] is:

(15)

With

(16)

Where *U _{dc}* is a direct potential,

(17)

here, ∇ is gradient. From Equation (17), the following is retrieved:

(18)

The equation of the motions [8,18,19,22-24] of the ion of mass m and charge e can be written as

d^{2}uJ_{1}(λ_{i}u) ch(λ_{i}v)

(19)

(20)

and the following

Where *J _{0}* and

In this section implementation of block method has been done for cylindrical ion trap motion equations. From Equation (13) with Equation (19) and Equation (20), respectively we have, the corrector formulae of the three point one block method as follow:

(21)

(22)

(23)

(24)

(25)

(26)

here y=u or y=v with

and

**Figures 2a** and **2b** show the ion trajectories in real time for cylindrical ion trap with α=0 and *χ*=0.4, *z _{1}*=0.05 and red color (solid line): RKM5 method using

Numerical examples has been tested for t ∈ [0,60] and TOL=10^{−10}. The results obtained from three point one block method and fifth order Runge-Kutta method are comparable for *h*=0.05 and *h*=0.01, respectively. The running steps taken by three-point one block method using *h*=0.05 and RKM5 using *h*=0.01 are 601 and 6000, respectively. Therefore, to obtain the comparable results from both method the total number of running steps for RKM5 is around 10 times more than three-point one block method. Hence, the execution time taken by three-point one block method using *h*=0.05 is less than RKM5 method using h=0.01.

The resolution of a cylindrical ion trap mass spectrometry [22] in general, is a function of the mechanical accuracy of the hyperboloid of the CIT Δr_{1}, and the stability performances of the electronics device such as, variations in voltage amplitude ΔV, the rf frequency ΔΩ [22,28], which tell us, how accurate is the form of the voltage signal. Here we study the resolution of a cylindrical ion trap with RKM5 and three-point one block methods.

**Table 2** shows the values of *χ _{max}* and

Methods ↓ | χ_{max} |
V_{max} |
---|---|---|

RKM5 | 0.64 | 5982.9425 |

3POBM | 0.63 | 6077.9099 |

**Table 2:** The values of* χ _{max} *and

(27)

(28)

Here “RKM5” and “3POBM” mean fifth order Runge-Kutta method and three-point one block method, respectively. By taking the partial derivatives with respect to the variables of the stability parameters *χ _{RKM5}* for Equation (27) and

(29)

(30)

Now to find the fractional resolution we have,

(31)

(32)

here Equation (31) and Equation (32) are the fractional resolutions for CIT with RKM5 and 3POBM, respectively. For the fractional mass resolution we have used the following uncertainties for the voltage, rf frequency and the geometry; *ΔV/V*=10−5, ΔΩ/ Ω=10^{−7}, *Δr _{1}*/r

We have demonstrated the three points one block method to solve some complicated differential equations concerned the cylindrical ion trap. We can observe to obtain the comparable results from both methods the total number of running steps for RKM5 is around ten times more than three point one block method. When these fractional resolutions are considered for the tritium isotope mass m=3.20, then, we have Δ*m*=0.001952640 and 0.1953148 for RKM5 and three point one block methods with using 0.01 and 0.05 steps, respectively. This indicates that as the value of m/Δm decreased, the power of resolution is increased due to increment in Δ*m*. However, it is clear that, at least in the lower mass range the impulse voltage with using the three points one block method is quit suitable than using the RKM5 for the CIT with higher in mass resolution. Apart from numerical result, the method showed a potential application to solve complicated linear and nonlinear equations of the charged particle confinement in the cylindrical field especially in fine tuning accelerators, and generally speaking, in physics of high energy.

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

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