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 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.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 electrodes24-26 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 motions10,21,24,25 of the ion of mass m and e can be written as
and the following
Where J_{0} and J_{1} are the Bessel functions of the first kind of order 0 and order 1, respectively, whereas ch is the hyperbolic cosine function, m_{i}r is the roots of equation J_{0} (m_{i}r) = 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 e5,24,28,29 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_{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}, 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} = 1.01978z_{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}) 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} = 1.4976z_{0}. and z_{1} = 1.05024z_{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} = 1.04978z_{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} = 1.04985z_{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} = 1.04985z_{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 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} and V_{z}_{max} or the quadrupole ion trap and cylindrical ion trap with optimum radius size size z_{1} = 1.04985z_{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} = 1.04985z_{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} = 1.04985z_{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) and q_{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 are m/∆m = 1638.806949;1638.398047 for QIT and CIT with optimum radius size z_{1} = 1.04985z_{0}, respectively. When optimum radius size z_{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 size z_{1} = 1.04985z_{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} = 1.04978z_{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} = 1.04978z_{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}) 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} = 1.04978z_{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 isotopes9,31 (see Figure 3).