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/
Δ
m
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.
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,
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
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 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
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
The 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 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:
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}
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}
= 1.01978
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}
.
Numerical results
 Stability regions
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 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}
= 1.04978
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.
The first and second stability regions, black line (solid line): QIT with z_{0} = 0.82 cm and blue line (dash line): CIT with optimum radius size z_{1} = 1.04985z_{0} and (a): first stability region and (b): second stability region.
 Phase space ion trajectory
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 evolution of the phase space ion trajectory for different values of the phase ζ_{0} for red line: QIT with z_{0} = 0.82 cm and blue line :CIT with optimum radius size z_{1} = 1.04985z_{0}, and z = r_{1}v.
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:
The values offor the quadrupole ion trap and cylindrical ion trap whenaz= 0 andα= 0 withz0= 0.82 cm and optimum radius sizez1= 1.04985z0andforβz= 0.3,0.6,0.9
The values of for the quadrupole ion trap and cylindrical ion trap when a_{z} = 0 and α = 0 with z_{0} = 0.82 cm and optimum radius size z_{1} = 1.04985z_{0} and for β_{z} = 0.3,0.6,0.9
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.
 The effect of optimum radius size on the mass resolution
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
The values ofqzmaxandVzmaxfor the quadrupole ion trap with optimum radius sizez1= 1.04985z0, respectively in the first stability region whenaz= 0
The values of q_{z}_{max} and V_{z}_{max} for the quadrupole ion trap with optimum radius size z_{1} = 1.04985z_{0}, respectively in the first stability region when a_{z} = 0
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.
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 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
).
The resolution of ∆m as function of ion mass m for ^{131}Xe with Ω = 2Π × 1.05 × 10^{6} rad/s, z_{0} = 0.82 cm and z_{1} = 1.04985z_{0}, dash line: for CIT and dash point line: for QIT.
Paul W.
,
Steinwedel H.
1953
Z. Naturforsch.
A8
448 
March R. E
,
Todd J. F. J.
1995
Modern Mass Spectrometry Series
CRC Press
Vol. 13.
Major F. G.
,
Gheorghe V. N.
,
Werth G.
2009
Chraged particle traps
2
(Springer)
Sadat Kiai S. M.
,
Seddighi Chaharborj S.
,
Abu Bakar M. R.
,
Fudziah I.
2011
J. Anal. At. Spectrom.
26
2247 
DOI : 10.1039/c1ja10170f
Kielpinski D.
,
Meyer V.
,
Rowe M. A.
,
Sackett C. A.
,
Itano W. M.
,
Monroe C.
,
Wineland D. J.
2001
Science
291
1013 
DOI : 10.1126/science.1057357
Douglas D. J.
,
Frank A. J.
,
Mao D. M.
1923
Mass Spectrometry Reviews.
21
408 
Hu Q. Z.
,
Noll R. J.
,
Li H. Y.
,
Makarov A.
,
Hardman M.
,
Cooks R. G.
2005
J. Mass spectrom.
40
430 
DOI : 10.1002/jms.856
Baranov V. I.
2003
J. Am. Soc. Mass Spectrom.
14
818 
Badman E. R.
,
Ph. D. Thesis
2001
Miniature cylindrical ion traps and arrays
Purdue University
Ph. D. Thesis
Kornienko O.
,
Reilly P. T. A.
,
Whitten W. B.
,
Ramsey J. M.
1999
Rev. Sci. Instrum.
70
3907 
DOI : 10.1063/1.1150010
Schowartz J. C.
,
Senko M. W.
,
Syka J. E. P.
2002
JASMS.
13
659 
March R. E.
1997
J. Mass Spectrom.
32
351 
Sadat Kiai S. M.
,
Baradaran M.
,
Adlparvar S.
,
Khalaj M. M. A.
,
Doroudi A.
,
Nouri S.
,
Shojai A. A.
,
Abdollahzadeh M.
,
Abbasi D F.
,
Roshan M. V.
,
BabazadehInt A. R.
2005
J. Mass Spectrom.
247
61 
DOI : 10.1016/j.ijms.2005.09.004