Optimum Radius Size between Cylindrical Ion Trap and Quadrupole Ion Trap
 DOI : 10.5478/MSL.2015.6.3.59
 Author: Chaharborj Sarkhosh Seddighi, Kiai Seyyed Mahmod Sadat, Arifin Norihan Md, Gheisari Yousof
 Publish: Mass Spectrometry Letters Volume 6, Issue3, p59~64, 30 Sep 2015

ABSTRACT
Quadrupole ion trap mass analyzer with a simplified geometry, namely, the cylindrical ion trap (CIT), has been shown to be wellsuited 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/ Δm of the confined ions in the first stability region was analyzed by the fifth order RungeKutta 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.

KEYWORD
Quadrupole ion trap , cylindrical ion trap , optimum radius size , fifth order Runge Kutta method , stability regions , ion trajectory , fractional mass resolution

Introduction
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,211 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 trap16 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.1921
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
Electric field inside CIT
In CIT, the hyperbolic ring electrode,24 as in Paul ion trap, is replaced by a simple cylinder and the two hyperbolic endcap electrodes are replaced by two planar endplate electrodes.25 The potential difference applied to the electrodes2426 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, andz _{1} expresses the distance from the center of the CIT to the end cap andr _{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 motions10,21,24,25 of the ion of mass
m ande can be written asand the following
Where
J_{0} andJ_{1} are the Bessel functions of the first kind of order 0 and order 1, respectively, whereasch is the hyperbolic cosine function,m_{i}r is the roots of equationJ _{0} (m_{i}r ) = 0. To obtainλ_{i}’ s the Maple software was employed to findJ _{0} (λ_{i} ) = 0 roots. Eqs.(5) and (6) are coupled inu andv (respectiver andz ), 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.27The motions of ion inside quadrupole ion trap
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 andr _{0} is the distance from the center of the QIT to the nearest ring surface. In each of the perpendicular directionsr andz , the ion motions of the ion of massm and chargee 5,24,28,29 may be treated independently with the following substitutions:CIT and QIT stability parameters
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
The optimum radius between cylindrical ion trap and quadrupole ion Trap
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} andv' =z/z _{0}. Whereα andχ are the trapping parameters, whichλ_{i} is the root of equationJ _{0} (m_{i}r _{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}, andv =v (0) =v _{4}, were assumed. Here,u (0),v (0),u' (0),v' (0) are the initial values foru,v,u' andv' , respectively. Now, from Eqs.(13) and (14) withd ^{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 andd ^{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} andq_{z} we have,with Eq.(18) gives the optimum value of
z _{1} andz _{0} for CIT and QIT with conditionsc _{1} =c _{3} andc _{2} =c _{4}. After substitutingλ_{i}',s,c _{1} =c _{3} = 0.01 andc _{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} = 1.01978z _{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} = 1.04978z _{0}) is almost comparable with the optimal radius size in Eq.(11)(z _{1} =z _{0}) whenχ =q_{z} 21,24 For the variousu; 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 withc _{1} =c _{3} = 0.005,c _{2} =c _{4} = 0.01 andc _{1} =c _{3} = 0.05,c _{2} =c _{4} = 0.01, we havez _{1} = 1.4976z _{0}. andz _{1} = 1.05024z _{0}.Numerical results
> Stability regions
There are two stability parameters which control the ion motion for each dimension
z (z =u orz =v ) and (z =z orz =r ), anda_{z}, q_{z} in the case of cylindrical and quadrupole ion traps,24 respectively. In the plane (a_{z}, q_{z} ) and for thez axis, the ion stable and unstable motions are determined by comparing the amplitude of the movement to one for various values ofa_{z}, q_{z} .26,30 To compute the accurate elements of the motion equations for the stability diagrams, we have used the fifth order RungeKutta 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} = 1.04978z _{0}, (a): first stability region and (b): second stability region. Figure 1 shows that the apex of the stability parametersa_{z} stayed the same and the apex of the stability parametersq_{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.> Phase space ion trajectory
Figure 2 shows evolution of different values of the phase ion trajectory for
ξ _{0} with red line : QIT withz _{0} = 0.82 cm and blue line: CIT with the optimum radius sizeThe 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 sizez _{1} = 1.04985z _{0}. Table 1 presents the values of for the quadrupole and cylindrical ion traps, whena_{z} = 0 andα = 0 with the optimum radius sizez _{1} = 1.04985z _{0}, respectively forβ_{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 sameβ_{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 whena_{z} = 0 andα = 0 respectively. Therefore, forβ_{z} = 0 we have and forβ_{z} = 1 we have for QIT and CIT, respectively. To compute Table 1, Maple software have been used.> The effect of optimum radius size on the mass resolution
he resolution of a quadrupole ion trap9 and cylindrical ion trap mass spectrometry in general with optimum radius size
z _{1} = 1.04985z _{0}, is a function of the mechanical accuracy of the hyperboloid of the QIT Δr _{0}, and the cylindrical of the CIT Δr _{1}, and the stability performances of the electronics device such as, veriations in voltage amplitude ΔV , the rf frequency ΔΩ ,9 which tell us, how accurate is the form of the voltage signal.Table 2 shows the values of
q _{z} _{max} andV_{z} _{max} or the quadrupole ion trap and cylindrical ion trap with optimum radius size sizez _{1} = 1.04985z _{0} in the first stability regionwhen
a_{z} = 0, respectively. The value ofV_{z} _{max} has been obtained for ^{131}Xe withΩ = 2Π × 1.05 × 10^{6} rad/s,U = 0 V andz _{0} = 0.82 cm in the first stability region whena_{z} = 0.To obtain the values of Table 2 we suppose
V_{z} _{max} as function of for QIT and CIT withz _{1} = 1.04985z _{0}, respectively as follows,Now, we use Eqs.(23) and (24) to calculate
V_{z} _{maxQIT} andV_{z} _{maxCIT} for ^{131}Xe withΩ = 2Π×1.05×10^{6} rad/s,z _{0} = 0.82 cm andz _{1} = 1.04985z _{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} = 1.04985z _{0}, respectively as follows,By taking the partial derivatives with respect to the variables of the stability parameters
q_{zQIT} for Eq.(25) andq_{zCIT} for Eq.(26), then the expression of the resolution Δm 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} = 1.04985z _{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 arem /∆m = 1638.806949;1638.398047 for QIT and CIT with optimum radius sizez _{1} = 1.04985z _{0}, respectively. When optimum radius sizez _{1} = 1.04978z _{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 sizez _{1} = 1.04985z _{0}, recpectively. When these fractional resolutions are considered for the ^{131}Xe isotope massm = 3.18, then, we have ∆m = 0.001940436 and 0.001994156 for QIT and CIT with optimum radius sizez _{1} = 1.04978z _{0}, respectively. This means that, as the value ofm /∆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.Discussion and conclusion
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 sizez _{1} = 1.04978z _{0} with This optimal radius size (z _{1} = 1.04978z _{0}) is almost comparable with the optimal radius size in Eq.(11) (z _{1} =z _{0}) whenx =q_{z} 22,25Table 1 also indicate that for the same equivalent operating point, almost a comparison physical size between two ion traps are shown;
z _{1} = 1.04978z _{0} = 0.86 cm andz _{0} = 0.82 cm. The CIT has a smaller trapping parameter compared to QIT; for example forβ_{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 isotopes9,31 (see Figure 3).

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[Figure 1.] The first and second stability regions, black line (solid line): QIT with z0 = 0.82 cm and blue line (dash line): CIT with optimum radius size z1 = 1.04985z0 and (a): first stability region and (b): second stability region.

[Figure 2.] The evolution of the phase space ion trajectory for different values of the phase ζ0 for red line: QIT with z0 = 0.82 cm and blue line :CIT with optimum radius size z1 = 1.04985z0, and z = r1v.

[Table 1.] The values of for the quadrupole ion trap and cylindrical ion trap when az = 0 and α = 0 with z0 = 0.82 cm and optimum radius size z1 = 1.04985z0 and for βz = 0.3,0.6,0.9

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[Table 2.] The values of qzmax and Vzmax for the quadrupole ion trap with optimum radius size z1 = 1.04985z0, respectively in the first stability region when az = 0

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[Figure 3.] The resolution of ？m as function of ion mass m for 131Xe with Ω = 2Π × 1.05 × 106 rad/s, z0 = 0.82 cm and z1 = 1.04985z0, dash line: for CIT and dash point line: for QIT.