Applications of Stochastic Process in the Quadrupole Ion traps

: The Brownian motion or Wiener process, as the physical model of the stochastic procedure, is observed as an indexed collection random variables. Stochastic procedure are quite influential on the confinement potential fluctuation in the quadrupole ion trap (QIT). Such effect is investigated for a high fractional mass resolution spectrometry. A stochastic procedure like the Wiener or Brownian processes are potentially used in quadrupole ion traps (QIT). Issue examined are the stability diagrams for noise coefficient, as well as ion trajectories in real time for noise coefficient, . The simulated results have been obtained with a high precision for the resolution of trapped ions. Furthermore, in the lower mass range, the impulse voltage including the stochastic potential can be considered quite suitable for the quadrupole ion trap with a higher mass resolution.


Introduction
There is the possibility that an ion trap mass spectrometer incorporates such traps as the Penning, 1 Paul 2 or Kingdon 3 traps. In 2005, the Orbitrap was introduced according to the Kingdon trap. 4 The two most popular kinds of ion traps are the Penning and the Paul traps (quadrupole ion trap). [5][6][7][8] Of course, it is also possible that other kinds of mass spectrometers utilize a linear quadrupole ion trap selected as a mass filter. Interestingly, ion trap mass spectrometry has undergone many developmental stages in order to achieve its current condition with relatively high performance level and growing popularity. Paul and Steinwedel 9 invented Quadrupole ion trap (QIT) commonly used in mass spectrometry, 5-8 ion cooling and spectroscopy, 10 frequency standards, 11 quantum computing 12 and others. However, different geometries have also been suggested and utilized for QIT. 13 Main properties of Wiener process A Wiener process 14,15 (notation ) is named in the honor of Prof. Norbert Wiener; other name is the Brownian motion (notation ). Wiener process is Gaussian process. As any Gaussian process, Wiener process is completely described by its expectation and correlation functions 14,15 Main properties of : • • Trajectories of Wiener process are continues functions of (see Figure (1  In can be shown that as , converges in distribution to a stochastic process , termed the Wiener process or Brownian motion. [14][15][16][17] The motions of ion inside quadrupole ion trap with stochastic potential form  2) shows indicatesa the schematic perspectives of a quadrupole ion trap (QIT). The quadrupole ion trap is the ion trap which including hyperbolic geometry and also is composed of involves a ring and two end cap electrodes that facing face each other in the z-axis (see Fig. (2)). Here, can also be considered the distance that begins started from the center of the QIT to the end cap. Also, is regarded as the distance from the outset point in the center of the QIT and extends to the nearest ring surface. Forcing the particles to swing and vibrate in confined space 6,18-20 can be the best approach for trapping charged particles. Such force can be written as follows, Where R is regarded as the distance from the center of the swing to the particle position and k is seen as a constant. Such force generated via the parabolic potential will move the oscillating particle around the equilibrium point. It can be expressed as follows, Here, , . Also, can be considered as Cartesian space components. Furthermore, any possible free space should satisfy the Laplace equation, given as follows, As Eq. (2) is unable to satisfy the Laplace situation, thus for confining the ions in two dimensions, it seems to be necessary to use a complicated potential as follows, For satisfying Eq. (4), in a Laplace situation, , the following equations are required, , . Therefore, This possibility is generated via four hyperbolic electrodes. In order to achieve such type of electrodes, the surfaces can be considered with the same potential and , as follow, and .
These situations make us able to find, and x y z , , , therefore . Consequently, electrodes shaped for the potential (4) can be obtained as follows, Eq. (7) represeznts a hyperbolic equation for this potential. Also, the potential used in hyperbolic electrodes is as follow, (8) thus, the stochastic potential, , can be written as, where W t is a Wiener procedure and is the noise coefficient determining the size of the stochastic term. 14,15 In this regard, the noise coefficient, , explains the amount of fluctuation potentially. For , the deterministic potential or normal potential can be stated as . Here, the parameter is selected, therefore is set to be about 14% as a common fluctuation in a potential. Also, the potential is usually written as follows, field elements in the trap therefore becomes, Where is the gradient. From Eq. (11) we obtain, The equations of motion for a singly charged positive ion in the QIT is represented thusly, The a and q are for the z and r parts and the dimensionless parameter are as follows, where m can be regarded as the ion mass and e as the electronic charge.
Thus, is considered as the drive radio frequency (rf), z 0 as one-half the shortest separation of the end cap electrodes, as the square of ring electrode radius and a z and q z as the trapping parameters. The standard Wiener procedure can be defined by a time step as follows, (15) here, is seen as the standard normal distribution that is the normal distribution including mean and variance and density function given as, In Matlab, the command "randn" was used to add the elements of distribution .

Stability regions
Two stability parameters monitor the ion motion for each dimension z (z = z or z = r) and a z , q z in the cases of the quadrupole ion trap for deterministic and stochastic cases respectively. The ion's stable and unstable motions, in the plane and for the z axis, can be determined through making comparison between the amplitude of the movement and different values of a z , q z . For calculating the precise elements of the motion equations for the stability diagrams, a numerical approach was used. The fifth order Runge-Kutta approach (using 0.001 stepwise increments) was used via the Matlab software as well as the scanning approach.  for stochastic case when . Here "st" stands for "stochastic".
From a mathematical viewpoint, stochastic as well as theoretical results are closely related. Thus, employing stochastic procedure in quadrupol ion trap potential makes us able to simulate and obtain the numerical outcomes including high accuracy (see Figs. (5)).
values of q z for QIT including and excluding the stochastic potentials for the equivalent points. Thus, two operating points observed in their corresponding stability diagram have the same : . For the computations, the following equations can be used, Here is the mean of .  (1) we see that an increase in the parameter η, will decrease q z for different values of β.
The values of when for the quadrupole ion trap with and without stochastic potential in the first stability region when is presented in Table (2). Xe with rad/s, V, cm in the first stability area when . Table (3) reveals that as increases, will increase too. To obtain the values of Table (3) by using Eq. (15) we presume is the function of m, z 0 , , e and is written as follows, Now, we use Eq. (19) to calculate for 131 Xe with rad/s and cm when as follows, Fig. (6A) shows the behavior of function for when . As increase, the difference will also increase. Fig. (6B) shows and as a function of in a QIT determined for the first stability area as in parts (a) and (b), respectively. To plot Fig. (6B), we have to used Table and Table when . Fig.  (6B.a) shows that when all factors increase, there is an outcome decrease and Fig. (6B.b) shows that with increasing parameter ; the values of increases also. Higher is shown to have better mass separation particularly for the lower ion mass range.

The effect of stochastic potential form on the mass resolution
Generally, the resolution of a quadrupole ion trap mass spectrometry 21 can be regarded as a function of the mechanical precision of the hyperboloid of the QIT , and the stability performances of electronics tools like, variations in voltage amplitude and the rf frequency 21 which tells us how precise the type of voltage signal   ; with initial conditions, and . for 131 Xe with rad/s, V, cm.
For deriving an influential theoretical formula for fractional resolution, we should consider the stability parameters of the impulse excitation for the QIT including and excluding its stochastic potential, respectively as follows, By taking the partial derivatives associated with the variables of the stability parameters q z for Eq. (20) and for Eq. (21), the expression of the resolution of the QIT including and excluding stochastic potential are as follows, Now, in order to find the fractional resolution, we have, here Eq. (24) and Eq. (25) are the fractional resolutions for QIT with and without stochastic potential, respectively. Fig. (7a) indicates the fractional resolution that is a function of the noise coefficient η and Fig. (7b) displays the resolution of ∆m that is a function of ion mass m, where a dash dot line (red line); represents deterministic cases ( ) and dash lines (green line) represent stochastic case ( ) . Regarding the fractional mass resolution, the following uncertainties were used for the voltage, rf frequency and the geometry; , , , for we have assumed arbitrarily the the noise coefficient . The fractional resolutions obtained are 1113;1448 for 0.07;0.14;0.28, respectively. When stochastic potential is applied ( ), the limited voltage of rf increases by a factor of approximately 1.14; thus, the voltage uncertainties were taken as . Once these fractional resolutions were considered for the tritium isotope mass , then, and 0.001959 with and without stochastic potential, the values for and were achieved, respectively. Theoretically, we have, Ω Ω ⁄ ∆ η m ∆ η 0 = η 0.14 = Thus, . This means that, along with increasing the value of will increase. Therefore, the power of resolution will increase because of the reduction in . Experimentally, this means that the width of the mass signal spectra is better separated. ) . The results represented in Fig. (8) indicates that for the same equivalent operating point in the two stability diagrams (having the same ), the associated modulated secular ion frequencies behavior is almost the same for different values of .

Discussion and conclusion
From a mathematical point of view, the results of stochastic process has higher resolution during mass separation. It has been shown that , this means that, with increase in η the value of also increases and therefore the power of resolution increases too due to a reduction in . Empirically, the width of the mass signal spectra seems to be better separated. Anyway, at least in the lower mass range, the impulse voltage including the stochastic potential is clearly quite suitable for the quadrupole ion trap with higher mass resolution. The fractional resolutions obtained are and ; therefore, and this indicates that when higher resolution in mass separation is involved.