Quadrupole ion trap mass analyzer with a simplified geometry, namely, the cylindrical ion trap (CIT), has been shown to be well-suited using in miniature mass spectrometry and even in mass spectrometer arrays. Computation of stability regions is of particular importance in designing and assembling an ion trap. However, solving CIT equations are rather more difficult and complex than QIT equations, so, analytical and matrix methods have been widely used to calculate the stability regions. In this article we present the results of numerical simulations of the physical properties and the fractional mass resolutions m/ Δ m of the confined ions in the first stability region was analyzed by the fifth order Runge-Kutta method (RKM5) at the optimum radius size for both ion traps. Because of similarity the both results, having determining the optimum radius, we can make much easier to design CIT. Also, the simulated results has been performed a high precision in the resolution of trapped ions at the optimum radius size. Ion trap mass spectrometry has been developed through several stages to its present situation of relatively high performance and increasing popularity. Quadrupole ion trap (QIT), invented by Paul and Steinwedel, 1 has been widely applied to mass spectrometry, 2 - 11 ion cooling and spectroscopy, 12 frequency standards, quantum computing, 13 and so on. However, various geometries has been proposed and used for QIT. 14 An ion trap mass spectrometer may incorporate a Penning trap, 15 Paul trap 16 or the Kingdon trap. 17 The Orbitrap, introduced in 2005, is based on the Kingdon trap. 18 Also, the cylindrical ion trap CIT has received much attention in a number of research groups because of several merits. The CIT is easier to fabricate than the Paul ion trap which has hyperbolic surfaces. In addition, the relatively simple and small sized CIT make it an ideal candidate for miniaturization. Experiments using a single miniature CIT showed acceptable resolution and sensitivity, and limited by the ion trapping capacity of the miniature device. 19 - 21 With these interests, many groups such as Purdue University and Oak Ridge National Laboratory have researched on the applications of the CIT to miniaturize mass spectrometer. 22 , 23 In CIT, the hyperbolic ring electrode, 24 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. 25 The potential difference applied to the electrodes 24 - 26 is: with Where, U _dc is a direct potential, V _ac is the zero to peak amplitude of the RF voltage, Ω is RF angular frequency, and z ₁ expresses the distance from the center of the CIT to the end cap and r ₁ the distance from the center of the CIT to the nearest ring surface. The electric field in a cylindrical coordinate ( r, z,θ ) inside the CIT can be written as follows: here, ▽ is gradient. From Eq.(3) (grad), the following is retrieved: he equation of the motions 10 , 21 , 24 , 25 of the ion of mass m and e can be written as and the following Where J₀ and J₁ are the Bessel functions of the first kind of order 0 and order 1, respectively, whereas ch is the hyperbolic cosine function, m_ir is the roots of equation J ₀ ( m_ir ) = 0. To obtain λ_i’ s the Maple software was employed to find J ₀ ( λ_i ) = 0 roots. Eqs.(5) and (6) are coupled in u and v (respective r and z ), and thus, can only be treated as a rough approximation. 21 , 25 Therefore, studies on CIT equations are more difficult and complex compared to QIT equations. As stated earlier, the optimum radius size between CIT and QIT helps us to study QIT instead of CIT. 27 A hyperbolic geometry for the Paul ion trap was assumed; Here, z ₀ is the distance from the center of the QIT to the end cap and r ₀ is the distance from the center of the QIT to the nearest ring surface. In each of the perpendicular directions r and z , the ion motions of the ion of mass m and charge e 5 , 24 , 28 , 29 may be treated independently with the following substitutions: If the ions the same species are taken into consideration and the same potential amplitude and frequency, the following relations has been obtained: From Eq.(9) and one can obtain In some papers, 21 , 24 stability parameters have been used to determine the optimum radius size for cylindrical ion trap compared to the radius size for the quadrupole ion trap, as following: Eqs.(11) and (12) are from Refs, 21 , 24 respectively. In this study, Eqs.(5), (6) and Eqs.(7), (8) were used for the same propose to find optimum radius size for cylindrical ion trap compared to the radius size for quadrupole ion trap, as: with u' = r/r ₀ and v' = z/z ₀ . Where α and χ are the trapping parameters, which λ_i is the root of equation J ₀ ( m_ir ₁ )=0. Eqs.(13) and (14) are true when ( α,χ ) and ( a_z,q_z ) vqlues belong to stability regions. In this case, u = u (0) = c ₁ , v = v (0) = v ₂ , u' = u' (0) = c ₃ , and v = v (0) = v ₄ , were assumed. Here, u (0), v (0), u' (0), v' (0) are the initial values for u,v,u' and v' , respectively. Now, from Eqs.(13) and (14) with d ² u / dξ ² = d ² c ₁ / dξ ² = 0, d ² v / dξ ² = d ² c ₂ / dξ ² = 0, d ² u' / dξ ² = d ² c ₃ / dξ ² = 0 and d ² v' / dξ ² = d ² c ₄ / dξ ² = 0, the following can be obtained: In adding Eqs.(15)and (16), we have: After substituting α,χ,α_z and q_z we have, with Eq.(18) gives the optimum value of z ₁ and z ₀ for CIT and QIT with conditions c ₁ = c ₃ and c ₂ = c ₄ . After substituting λ_i',s,c ₁ = c ₃ = 0.01 and c ₂ = c ₄ = 0.01 ; in Eq.(18) and by simplification, we have: Therefore, using the Maple software we will have, In Eq.(20), z ₁ = 1.01978 z ₀ is the optimum radius size between the quadrupole and the cylindrical ion traps. For any initial conditions we can obtain same answer with Eq.(20) almost. This optimal radius size ( z ₁ = 1.04978 z ₀ ) is almost comparable with the optimal radius size in Eq.(11)( z ₁ = z ₀ ) when χ = q_z 21 , 24 For the various u; v; r; z when ( α, χ ) and ( a_z,q_z ) belongs to the stability regions, we found almost the comparable optimum values equivalent to Eq. (20) was found. For example with c ₁ = c ₃ = 0.005, c ₂ = c ₄ = 0.01 and c ₁ = c ₃ = 0.05, c ₂ = c ₄ = 0.01, we have z ₁ = 1.4976 z ₀ . and z ₁ = 1.05024 z ₀ . There are two stability parameters which control the ion motion for each dimension z ( z = u or z = v ) and ( z = z or z = r ), and a_z, q_z in the case of cylindrical and quadrupole ion traps, 24 respectively. In the plane ( a_z, q_z ) and for the z axis, the ion stable and unstable motions are determined by comparing the amplitude of the movement to one for various values of a_z, q_z . 26 , 30 To compute the accurate elements of the motion equations for the stability diagrams, we have used the fifth order Runge-Kutta numerical method with a 0.001 steps increment for Matlab software and scanning method. Figure 1 (a) and (b) shows the calculated first and second stability regions for the quadrupole ion trap and cylindrical ion trap, 31) black line (solid line): QIT and blue line (dash line): CIT with optimum radius size z ₁ = 1.04978 z ₀ , (a): first stability region and (b): second stability region. Figure 1 shows that the apex of the stability parameters a_z stayed the same and the apex of the stability parameters q_z decrease for CIT to compare with QIT. Area of first stability regions for QIT and CIT are almost same, as 0.4136 and 0.4087, respectively. Figure 1 reveals almost a comparable stability diagram two methods. Figure 2 shows evolution of different values of the phase ion trajectory for ξ ₀ with red line : QIT with z ₀ = 0.82 cm and blue line: CIT with the optimum radius size The results illustrated in Figure 2 show that for the same equivalent operating point in two stability diagrams (having the same β_z ), the associated modulated secular ion frequencies behavior are almost same for the quadrupole and cylindrical ion traps with the optimum radius size z ₁ = 1.04985 z ₀ . Table 1 presents the values of for the quadrupole and cylindrical ion traps, when a_z = 0 and α = 0 with the optimum radius size z ₁ = 1.04985 z ₀ , respectively for β_z = 0.3.;0.6;0.9. For the computations presented in Table 1 , the following formulas were used: and for QIT and CIT, respectively. Hence, it is important to know that β_z point are the equivalent points; two operating points located in their corresponding stability diagram have the same β_z . 31 For the same 0< β_z <1 we have, Here, 1.35 and 1.23 are maximum values of stability diagrams for QIT and CIT when a_z = 0 and α = 0 respectively. Therefore, for β_z = 0 we have and for β_z = 1 we have for QIT and CIT, respectively. To compute Table 1 , Maple software have been used. he resolution of a quadrupole ion trap 9 and cylindrical ion trap mass spectrometry in general with optimum radius size z ₁ = 1.04985 z ₀ , is a function of the mechanical accuracy of the hyperboloid of the QIT Δ r ₀ , and the cylindrical of the CIT Δ r ₁ , and the stability performances of the electronics device such as, veriations in voltage amplitude Δ V , the rf frequency Δ Ω , 9 which tell us, how accurate is the form of the voltage signal. Table 2 shows the values of q _{z max} and V_{z max} or the quadrupole ion trap and cylindrical ion trap with optimum radius size size z ₁ = 1.04985 z ₀ in the first stability region when a_z = 0, respectively. The value of V_{z max} has been obtained for ¹³¹ Xe with Ω = 2Π × 1.05 × 10 ⁶ rad/s, U = 0 V and z ₀ = 0.82 cm in the first stability region when a_z = 0. To obtain the values of Table 2 we suppose V_{z max} as function of for QIT and CIT with z ₁ = 1.04985 z ₀ , respectively as follows, Now, we use Eqs.(23) and (24) to calculate V_{z maxQIT} and V_{z maxCIT} for ¹³¹ Xe with Ω = 2Π×1.05×10 ⁶ rad/s, z ₀ = 0.82 cm and z ₁ = 1.04985 z ₀ as follows, To derive a useful theoretical formula for the fractional resolution, one has to recall the stability parameters of the impulse excitation for the QIT and CIT with z ₁ = 1.04985 z ₀ , respectively as follows, By taking the partial derivatives with respect to the variables of the stability parameters q_zQIT for Eq.(25) and q_zCIT for Eq.(26), then the expression of the resolution Δ m of the QIT and CIT, respectively are as follows, Now to find the fractional resolution we have, Here Eqs.(29) and (30) are the fractional resolutions for QIT and CIT with optimum radius size z ₁ = 1.04985 z ₀ , respectively. For the fractional mass resolution we have used the following uncertainties for the voltage, rf frequency and the geometry; ∆ V / V = 10 ^-15 , ∆ Ω / Ω = 10 ^-7 , ∆ r ₀ / r ₀ = 3×10 ^-4 . The fractional resolutions obtained are m /∆ m = 1638.806949;1638.398047 for QIT and CIT with optimum radius size z ₁ = 1.04985 z ₀ , respectively. When optimum radius size z ₁ = 1.04978 z ₀ is applied, the rf only limited voltage is increased by the factor of approximately 2.6893, therefore, we have taken the voltage uncertainties as ∆ V_CIT / V_CIT = 2.6893 × 10 ^-5 . From Eqs.(29) and (30) we have ( m /∆ m ) _QIT = 1638.8069 and ( m /∆ m ) _CIT = 1598.6598 for QIT and CIT with optimum radius size z ₁ = 1.04985 z ₀ , recpectively. When these fractional resolutions are considered for the ¹³¹ Xe isotope mass m = 3.18, then, we have ∆ m = 0.001940436 and 0.001994156 for QIT and CIT with optimum radius size z ₁ = 1.04978 z ₀ , respectively. This means that, as the value of m /∆ m is decreased, the resolving power is increased due to increment in ∆ m . Experimentally, this means that the width of the mass signal spectra is better separated. In this study, the behavior of the quadrupole and cylindrical ion traps with the optimum radius has been considered. Also, it is shown that for the same equivalent operating point in two stability diagrams (i.e. having the same β_z = 0.3), the associated modulated secular ion frequencies behavior are almost the same with a suitable optimum radius size z ₁ = 1.04978 z ₀ with This optimal radius size ( z ₁ = 1.04978 z ₀ ) is almost comparable with the optimal radius size in Eq.(11) ( z ₁ = z ₀ ) when x = q_z 22 , 25 Table 1 also indicate that for the same equivalent operating point, almost a comparison physical size between two ion traps are shown; z ₁ = 1.04978 z ₀ = 0.86 cm and z ₀ = 0.82 cm. The CIT has a smaller trapping parameter compared to QIT; for example for β_z = 0.3 we have a difference of 0.0564 higher for the QIT. This difference in trapping parameters indicates that for the same rf and ion mass values, we need more confining voltage for CIT than QIT (see Table 2 ). So, higher fractional resolution can be obtained; higher separation confining voltages, especially for light isotopes 9 , 31 (see Figure 3 ).