Possibility of operating quadrupole mass filter at high resolution

International Journal of Mass Spectrometry 408 (2016) 9–19

Contents lists available at ScienceDirect

International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms

Possibility of operating quadrupole mass filter at high resolution Mikhail Sudakov a,∗ , Eugenij Mamontov a , Fuxing Xu b , Chongsheng Xu b , Chuan-Fan Ding b,∗ a

Ryazan State Radio Engineering University, 390035, Gagarina 57, Ryazan, Russia Shanghai Key Laboratory of Molecular Catalysis and Innovative Materials, Department of Chemistry and Laser Chemistry Institute, Fudan University, 220 Handan Road, Shanghai, China b

a r t i c l e

i n f o

Article history: Received 12 July 2016 Received in revised form 7 September 2016 Accepted 7 September 2016 Available online 9 September 2016 To the memory of Vladimir Vladimirovich TITOV. Keywords: Quadrupole mass filter Stability islands Residence time High resolution Faster mass scan Quadrupole excitation One dimensional filtering Increment of instability Nonlinear field distortions

a b s t r a c t A new possibility of improving the resolving power of quadrupole mass filters has been studied theoretically in this work. The results show that with the use of two AC excitations, in addition to the main RF supply, it is possible to modify the first stability diagram for mass filtering by creating a narrow and long band of stability along the X boundary near the tip of first stability region. These newly developed stability regions (the X-band) are similar to higher stability regions, and offer high mass resolution and fast mass separation features. This approach overcomes the many limitations of the normal operation of quadrupole analyzers, while retaining the advantages of using the first stability region. The new operation mode could achieve up to 10,000 mass resolving power with the ion residence time of only 100 RF cycles. In addition, the ion transmission efficiency with the use of the X-band is not only compromised, but is greater than in the normal operation mode. Furthermore, the new mode features one-dimensional mass filtering (in the X direction only) that is not sensitive to nonlinear field distortions, which are particularly problematic for quadrupole mass filters which built with circular rods. Faster mass separation has been confirmed in simulations and theoretical computations of the exponential increment of the trajectory instability. Due to the location of the X-band near the tip of the first stability region, the new operation mode can still have the benefits of traditional techniques (delayed DC ramp) for overcoming the negative effects of fringe fields and improving the ion transmission efficiency. The theoretical simulations show that the method of improving the performance of quadrupole mass filters does not require any modifications of mechanical structures, and only needs different and a little more sophisticated method of electric applications. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Over half a century has passed since the invention of mass analysis of ions using the quadrupole radio frequency field by W. Paul [1]. Nowadays, the different type of ion traps and quadrupole mass analyzers have been applied in wide fields because of their very strong analytical capabilities and compact sizes. Quadrupole mass filters (QMF or quadrupole analyzers) are routinely used in physics and analytical chemistry as standalone mass spectrometers, a part of tandem mass spectrometry for parent ion selection, ion guides for ion beam transmission, or collisional cells for precursor ion fragmentation. Filtration of ions in a quadrupole mass analyzer system is based on the properties of the Mathieu equation that describes

∗ Corresponding authors. E-mail addresses: [email protected] (M. Sudakov), [email protected] (C.-F. Ding). http://dx.doi.org/10.1016/j.ijms.2016.09.003 1387-3806/© 2016 Elsevier B.V. All rights reserved.

the ion motion in quadrupole fields with a harmonic power supply [2]. Thus, the quadrupole mass filtering technology requires a high precision of machining and assembling of electrodes with micrometer accuracy, and a development of very stable and controllable power supplies for generating high frequency and high voltage signals (RF supply). Although quadrupole mass filtering technology has been well established and investigated both experimentally and theoretically in the past several decades, a search for new and unusual modes of operation is on going. Quadrupole analyzers with a distorted electrode geometry [3,4] and with periodic quadrupole excitations (additional AC power supply with different frequency on top of main RF) [5,6] have been investigated over many years in D. Douglas’ laboratory. In spite of the common knowledge that the efficient ion filtration is only achievable with the use of high quality quadrupole fields [7–10], it was found that quadrupole analyzers with significantly modified electrodes can be successfully used for ion filtration under certain conditions [3]. Moreover, the use of

10

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

additional periodic excitations can overcome the destructive influence of nonlinear electric field distortions and obtain a comparable resolving power and ion transmission efficiency with commercial devices [6]. This paper reports a new possibility of operating a quadrupole analyzer at high mass resolution. The “high mass resolution” means the filtering of ions with the mass resolving power significantly greater than 5000, which is a practical maximum for quadrupole mass filters when operating in a normal mode. In this paper, we will first discuss the principle and properties of a normal operation mode of quadrupole mass filters that leads to some basic limitations and possible ways of overcoming them. We further study the possibilities for the most appropriate mass filtering by modifying the stability diagram of ion motion. These modifications are possible with the use of special periodic excitations that result in new properties of the ion motion. It will show that the separation of ions under the new mode occurs much faster than in a normal operation mode and it allows a high resolution mass separation in spite of a short residence time of ions inside a quadrupole analyzer. The proposed mode of operation is verified by direct simulations of ion motion in a two-dimensional quadrupole field where the resolving power of a mass separation and ion transmission efficiency are compared with a normal operation mode under similar conditions. The result of the faster ion separation rate is explained by greater increments of trajectory instability from both sides of the stability regions, and the increments are computed theoretically. A new mode of operation is not sensitive to nonlinear distortions of the quadrupole field, which is confirmed by simulations of the mass peak shapes in quadrupole analyzers with cylindrical rods. Finally, the implications of the new operational mode for some modern mass filtering techniques are discussed. 2. The normal mode of quadrupole mass filter operation and its limitations 2.1. A quadrupole mass filter and equations of ion motion A conventional quadrupole mass filter is composed of four conducting rods with an ideal hyperbolic cross section [1]. They are arranged parallel to the common Z axis and are equally spaced by a distance r0 from the center. An RF voltage V (t) is connected positively to one pair of the rods (the X rods) and negatively to another pair of the rods (the Y rods), resulting in the quadrupole field among the rods: ˚(x, y) = V (t) ·

x 2 − y2 r02

.

(1)

For a mass analysis, the applied voltage V (t) has both DC (U) and radio-frequency (V ) components: V (t) = U − V · Cos(˝t + ˛).

(2)

Here, ˝ is the angular frequency of the RF power supply and ˛ is its initial phase. An ion motion in such field is usually expressed in terms of dimensionless variables: d2 x + (a + 2qCos2) · x = 0, d 2

(3.a)

d2 y − (a + 2qCos2) · y = 0, d 2

(3.b)

˝t + ˛ 8eU 4eV ,a = ,q= 2 M˝2 r02 M˝2 r02

M is the ion mass and e is its charge.

Both Eqs. (3.a) and (3.b) are linear, second order differential equations with periodic coefficients known in literature as the Mathieu equations [11]. Note that the parameters of Eq. (3.a) are taken for a basis while parameters for the Y motion have an opposite sign. The Mathieu equation appears in physics as a mathematical model for a description of a parametric resonance, but there are some regions of a and q parameters for which the parametric resonance does not occur and which correspond to a stable motion. The graphical description of these regions on a plane of a and q parameters is termed as “a stability diagram”. Ions can be transmitted through a quadrupole analyzer if only both X and Y motions are stable simultaneously. The regions where the stability diagrams of the X and Y motions overlap are called the regions of common stability or “stability zones”. There is an infinite number of such regions. The first zone of the common stability, called “the first stability region” (see Fig. 1 ), is usually used for conventional quadrupole mass filtering. It is evident from Eq. (4) that for ions of different mass-to-charge ratio, a and q parameters appear at the same “operating” line a =  · q, where  = 2U/V . The a and q parameters are inversely proportional to the ion mass, so heavy ions have parameters (“working point”) located closer to the origin of the diagram. By an appropriate selection of the U/V ratio, the operating line can be located so that it crosses the tip of the first stability region, as shown in Fig. 1. The mass filtering method in quadrupole analyzers is evident − only ions that have mass-to-charge ratio within some range [Mmin , Mmax ] will have stable motion inside a quadrupole mass filter. When a beam of ions of different mass-to-charge ratios is injected into a quadrupole analyzer, the heavier ions experience instability in the Y direction and are ejected by the Y rods, while the lighter ions are unstable in the X direction. Only the ions that have working points within a stable zone will be transmitted to the end of the quadrupole analyzer and then detected by an ion detector. The mass selection using a stability triangle at the tip of the first stability region is referred further as “a normal” or “a conventional” mode of the quadrupole analyzer operation. 2.3. The resolving power of a mass selection in a normal mode of operation The mass resolution is defined by the width of a transmitted mass range M = Mmax − Mmin and is usually expressed in Thomson unit which has been introduced by Prof. G.Cooks [12] as an appropriate measure of the mass-to-charge ratio, 1 Thomson equals 1 Da mass unit 1.66053873·10−27 kg divided by the electron charge 1.602176462·10−19 C. The mass resolving power, R, is a dimensionless measure that is equal to the ratio of the nominal mass, Mnom = (Mmin + Mmax ) /2, and the transmitted mass range, M: R=

Mnom . M

(4)

(5)

In practice, quadrupole analyzers, are operated by scanning RF voltage amplitude, while keeping the U/V ratio fixed (when scanning in a wide mass range, the U/V ratio is adjusted slightly in order to keep a similar resolution at each mass) and recording ion current at a detector. The mass assignment is obtained from the RF voltage, V , using a definition of q parameter from Eq. (4): Mnom =

where =

2.2. Stability diagram and mass filtering

4e ˝2 r02

·

V . q1

(6)

Here, q1 = 0.705996 is the coordinate value of the stability tip. In experiments, the mass resolution, M, is defined for ion species of similar mass as the peak width measured at some level of the peak maximum (usually 10%).

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

11

Fig 1. First zone of common stability diagram for a quadrupole mass filter. Region of stable motion is marked by grey colour. Tip of stability region is at (a1 = 0.236994,q1 = 0.705996).

It has been discovered by Von Zhan [13] that the maximum mass resolving power delivered by a quadrupole mass filter is limited by the residence time of ions inside the analyzer: Rmax =

n2 C

(7)

Here, n is the number of the RF cycles that an ion spends in a quadrupole analyzer, and C is a constant. C depends on the anticipated theoretical resolution and the level at which a peak width is measured (e.g. C = 10 ÷ 20 for 10% of maximum at the tip of the first stability region [14,15]). This equation, referred as the “Von Zhan’s rule”, indicates the major practical limitation of obtaining the highest resolution with a normal operational mode. For example, with the residence time of 100 RF cycles, the maximum resolving power is only 500 according to Eq. (7). 2.4. Methods of improving the resolving power in a normal mode of operation There are number of methods to increase the ion residence time inside quadrupole analyzers for improving the mass resolution. For example, to make quadrupole rods longer. A 5.8 m long quadrupole analyzer has even been designed and built to deliver the resolving power of 16000 [16]. In practice, however, this method is limited by an accuracy of the mechanical manufacturing and an assembly of the quadrupole analyzer electrodes. With current state of the art techniques, quadrupole analyzers with 4–6 mm inscribed radii and a length from 100 to 300 mm are typically manufactured and used. M.H. Amad and R. S. Houk have used ion reflections inside a quadrupole analyzer in order to ensure a multi-pass motion and obtained the resolving power over 22000 [17]. Another method is to reduce the longitudinal velocity of the ion beam. For estimations, let us suppose a 200 mm long quadrupole analyzer with an inscribed radius of 5 mm is powered by 1 MHz RF supply, so the residence time of 100 cycles for a singly charged 609 Da ions corresponds to the longitudinal energy of 12 eV, which results in the maximum resolving power of 500 only. The reduction of the beam energy down to 3 eV results in 200 cycles and the resolving power increases to 2000. Unfortunately, many ion sources generate ions with the energy spread of several eV and it is practically difficult to reduce the longitudinal energy of the beam below approximately several eV. The third method is to increase the number of the ion cycles by using higher frequency RF power. But this method also has some practical limitations. According to Eq. (4), to increase the RF frequency by a factor of 2 would require 4 times greater RF voltage in order to bring the same mass-to-charge ratio (m/z) ion at the tip of stability. This decreases the mass range of detected ions

since it is limited by the maximum voltage that can be applied to the rods without any danger of electrical discharge. Furthermore, an increase in the ion residence time correspondingly increases the time that ions spend in the fringing field of a quadrupole analyzer system. At the entrance and exit areas of the quadrupole analyzer, the electric field distributions differ significantly from Eq. (1) which results in an increase of the ion beam lateral energy and a position spread, and reduces the transmission efficiency. The discussion above shows that improving the resolving power of quadrupole mass filters by increasing the residence time of ions is possible, but it is very limited with some present methods. Another way to overcome the Von Zhan’s rule could be implemented by increasing the speed of the mass separation. It is defined by properties of the first stability region and cannot be changed, but it is different at higher stability regions. As it have been discussed above, there is an infinite number of the common stability regions, which lie at higher a and q values on the stability diagram. Konenkov et al. [18] have used the third stability region to perform a residual gas analysis with the resolving power of 5000. Operations at other higher stability regions have been investigated in detail by D.Douglas and co-workers [14,19,20], and up to 14000 resolving power has been demonstrated at the fourth stability region [20]. Unfortunately, these methods have little practical applications because of their extremely low ion transmission efficiency. On the way through the fringing field at the entrance of a quadrupole mass filter, ions pass a wide unstable area from zero to nominal a and q parameters inside the quadrupole analyzer system. An unstable motion in the fringing field results in a substantial increase of the lateral energy spread leading to a massive ion loss during the ion transmission. It has been noticed in a previous work [20] that a sufficient mass separation of ions with the use of higher stability regions can be achieved with the beam energies up to several kiloelectronvolts, which corresponds to the ion residence time of only a few RF cycles. If it works with the use of the first stability region, then the ion transmission is not necessarily sacrificed. We will discuss this possibility and its implementation method in details next.

3. Quadrpole excitation and its effect on the ion motion 3.1. The use of quadrupole excitations in mass filtering. islands of stability Kozo Miseki was probably the first to realise the importance of quadrupole excitations for mass filtering [21]. U.S. Pat. No. 5,227,629 describes a quadrupole mass filter, where additional AC

12

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

voltage, Vex · Cos(ωex t) (excitation AC), is applied to the electrodes of a quadrupole mass spectrometer in addition to the normal DC and RF voltages. The excitation AC voltage has a frequency, ωex , which differs from main RF frequency, and generates some bands of instability within the first stability region, which effectively removes the peak tails, enhances the mass separation resolution, and improves the reliability of the measurements. A strict mathematical method for computing stability diagrams has been proposed, and the use of the quadrupole excitation has been previously investigated both theoretically and experimentally [22]. It has been shown that the excitation produces a number of instability bands on both sides of the first stability region. As a result, a stability triangle at the tip of the first stability region splits into a number of stable islands, as presented in Fig. 2. Two orders of magnitude improvement of the abundance sensitivity have been observed in experiments while operating a mass filtering at the islands of stability. It certainly shows a faster mass separation and indicates a possible way for improving the performance of quadrupole mass filters. Recently, the technology of additional AC excitations has been used in quadrupole analyzers for space explorations [23], and its reliability for improving QMF performance has been proved [24]. 3.2. Two AC excitations. equations of ion motion and stability diagram As it has been noted in a previous paper [22], the best results could be achieved by using the island C in Fig. 2. When the operating line crosses the C island at the center, the Y motion remains stable near and within the island, thus the separation from both sides of the peak occurs along the X direction only (one dimensional filtering). Unfortunately, in this case the operating line also passes island B, which results in overlapping of two mass spectra. A computation of the stability diagram shows that while using a single AC excitation it was impossible to overcome this problem. Fortunately, the quadrupole excitations provide a sufficient flexibility in modifying the shape of islands by using two or more excitations. With two AC excitations, the overall power supply at a quadrupole analyzer is V (t) = U + VCos˝t + Vex1 Cos(ωex1 t + ˛1 ) + Vex2 Cos(ωex2 t + ˛2 ) (8) Here, ωex1 and ωex2 are the frequencies of two AC excitations, where the first frequency is smaller: ωex1 < ωex2 ; Vex1 and Vex2 are the amplitudes (zero-to-peak and pole-to-ground) of the first and second AC excitations; and ˛1 and ˛2 are the initial phases with respect to the main RF power supply. In terms of dimensionless variables, the equations of the lateral ion motion are given as below: d2 x + [a + 2qCos2 + 2qex1 Cos(21  + ˛1 ) + 2qex2 Cos(22  + ˛2 )] · x = 0 d 2

(9.a)

d2 y − [a + 2qCos2 + 2qex1 Cos(21  + ˛1 ) + 2qex2 Cos(22  + ˛2 )] · y = 0 d 2

(9.b)

Here, the dimensionless parameters are  = ˝t/2 and 1 =

ωex1 ωex2 4eVex1 4eVex2 , qex2 = , 2 = , qex1 = ˝ ˝ M˝2 r02 M˝2 r02

(10)

In order to analyze properties of Eq. (9) and to find a stability of the ion motion, the method from the reference [22] is useful here. In the case where the excitation frequencies are exactly equal to some fraction of the main RF frequency: 1 = K1 /P and 2 = K2 /P, where K1 and K2 are integers with the same integer denominator P, all periodic functions in Eq. (9) have a common period, P. It is convenient to refer the period P as the “base period” and the corresponding dimensionless frequency  = 1/P as the “base frequency”. Equations of motion (9) then belong to the class of the Hill equation [25] (second order linear differential equation with periodic coefficients). For such equations, a strict mathematic theory

for computing stability conditions has been developed in reference [26]. An evaluation of stability requires a computation of two independent solutions of Eq. (9) over one complete period P, which gives the so called “monodromy matrix”. Motion is stable if the absolute value of the trace of the monodromy matrix is less than 2. The stability diagram can be plotted within some preselected region of a and q parameters by computing the monodromy matrix for each pixel of the chart area. If both X and Y motions appear to be stable, then a pixel is marked by a color, otherwise it remains white. In the present paper, the monodromy matrix computations have been carried out using a standard 4th order Runge-Kutta algorithm [27] with a fixed /32 integration step. It has been found that smaller integration steps do not affect the visual shape of the stability regions, but increase the computation time. It needs to be mentioned, that with additional parameters (10), the stability diagram actually has more than two dimensions. When plotting in the (a,q) plane, we actually obtain a section of this multidimensional diagram while other parameters are fixed. Note also, that this approach presents a different view on a problem under investigation. Instead of a complicated vibration system with the main RF and two AC excitations at different frequencies and particular phase relationships, the matrix method considers this system simply as the Hill equation with a periodic parameter of a particular shape given by eq.(8). The combination of the main RF and two AC excitations creates a function of a complicated shape, which is periodic with the base period P, that makes it possible to use the general matrix methods. 3.3. Bands of stability The theoretical simulation showed that the use of two AC excitations results in a new quality of the ion motion and stability diagrams. With an appropriate selection of the excitation frequencies ωex1 , ωex2 and amplitudes Vex1 , Vex2 , the influence of the AC excitations are mutually cancelled either for the X or Y motion. Correspondingly, the bands of instability do not appear on the left or on the right sides of the stability triangle, and the islands from the other side appear in a shape of a long stability band. This fact is illustrated by a calculation of the stability diagrams shown in Figs. 3a and b. These diagrams have been computed using the matrix method for the case when 1 = K1 /P and 2 = K2 /P, where K1 = 1 and K2 = 19 and P = 20, which corresponds to the “base period” of 20. For simplicity, both excitations have zero initial phases in the simulations: ˛1 = ˛2 = 0. The first example in Fig. 3a shows the case when the stability of the Y motion is not affected, while the X boundary splits. As a result, a long and narrow band of stability, further referred as “the X-band”, appears on the right side of the original stability zone. In the second example in Fig. 3b, the AC excitations have the same frequencies as before, but opposite initial phases (note the negative sign of qex2 ). Here, the stability of the X motion is not affected and the stability diagram has a long stability band on the left side, the “Y-band”. The X-band is preferable for the mass filtering, because the operating line crosses the stability band almost orthogonally, consequently small deviations due to instability of the RF/DC power supply cause comparatively smaller changes in the ion stability. Secondly, the instability of the ion motion for unwanted ions increases faster along the scan line due to its relative location with respect to the band of stability. Fig. 3 shows an example of stability bands for particular intensities of excitations qex1 and qex2 . Similar results are obtained if the ratio qex1 /qex2 remains constant, namely 2.94 for the case shown in Fig. 3a. With greater intensities of the excitation, the X-band becomes narrower and is shifted to the right. The frequencies of two excitations can be expressed in terms of the base frequency: 1 =  and 2 = 1 − , where the base frequency is  = 0.05 for the case shown in Fig. 3. With the use of the matrix method, similar

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

13

Fig. 2. Islands of stability (red color) due to influence of single AC excitation at frequency 0.95 of the main RF frequency. First three major islands marked as A, B and C. Boundaries of original first stable region are showed by wide solid lines, operating line with slope  = 0.168 − by thin solid line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Common stability diagram of a mass filter near the tip of first stable region in presence of two quadrupole excitations with frequencies 0.05 and 0.95 of the main RF frequency. Intensities of excitations are provided in the legend. Areas of stable motion are marked by color, boundaries of original first stable region are shown by thick black line, operating line goes through the tip of stability. A. Excitations are in the same initial phase. B. Excitations are in the opposite initial phase.

14

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

Table 1 Combination of AC excitation frequencies that result in X-band appearance and corresponding ratio of excitation intensities. Arranged according with increasing frequency (each next band has one of AC frequencies greater than previous band).

1 = 2 = qex2 /qex1 =

R 20000

I

II

III

IV

V

VI

 1− 2.94

 1+ 3.12

1− 2− 7.41

1− 2+ 9.09

1+ 2− 6.90

1+ 2+ 8.33

II

III

I

IV V

15000

VI

10000

Frequency, KHz Conventional Xband-I

DC

RF

AC-1

AC-2

0 184.531 V 184.554 V

1000 1099.507 V 1099.507 V

50 0 10.590 V

950 0 31.135 V

this figure one can find that in order to obtain a similar resolution, the excitations at higher frequencies require greater level of excitation. In all cases of our investigations, they are small compared to the main RF voltage. For some practical implementations of the RF supply [23], higher frequency excitations (cases III–VI) may be preferable due to the higher level of excitation and better stability control. 3.5. Peak shape simulations

5000 0 0.000

Table 2 Settings of power supply for simulations of quadrupole analyzer with hyperbolic rods in conventional mode and with X band. All voltage amplitudes are provided zero-peak, pole-to-ground.

0.005

0.010

0.015

0.020

qex1

Fig. 4. Dependence of theoretical resolution against intensity of first AC excitation with the use of X-band based on frequency  = 0.05. Different curves are for 6 different cases from Table 1. Intensity of the second AC excitation is set according to the ratio from the same table.

results can be easily reproduced for other rational values of  = K/P. No doubt that for irrational values of , where only approximate methods are available, the same phenomena will take place. An important conclusion is that the ratio qex1 /qex2 , which results in the appearance of the X-band, does not depend on the base frequency  in the range from 0 to 0.1, which is of practical interest. Also, it has to be mentioned that a shape of the first stability diagram depends on the AC initial phases, ˛1 and ˛2 , with respect to the main RF, and also on the phase ˛2 relative to ˛1 . For simplicity, we investigated the case when the initial RF phase and the phase of AC1 are both equal to zero. There are other cases which require further investigations in the future. 3.4. Other possibilities to arrange stability bands The suppression of the Y-motion instability and an appearance of the X-bands can also be obtained by two AC excitations with other selection of frequencies. For example, the frequencies 1 =  and 2 = 1 +  will produce the result which is similar to Fig. 3a with the ratio of qex1 /qex2 = 3.12. Some other examples and the corresponding ratios are listed in Table 1. As mentioned above, these ratios are practically independent of the base frequency  within the range from 0 to 0.1. Fig. 3 shows the location of the operating line a = q that passes through the tip of the first stability region. In a conventional mode of operation, the resolving power of the mass filtering is adjusted by the slope of scan line,  = 2U/V . When using mass filtering in a new operation mode with stability bands, the slope is fixed and passes above the tip of the first stability region, so no ions have stable trajectories without AC excitations. The resolution of the mass filtering with the use of the X-band is only determined by the width of the stability band, which depends on the intensities of AC excitations. The theoretical resolving power is R = qcenter /q, where q = qmax − qmin is a part of the scan line that exists inside the X-band, and qcentre = (qmax + qmin )/2 is its middle value. The dependence of the theoretical resolution against the intensity of the first AC excitation as listed in Table 1 are shown in Fig. 4. From

According to Fig. 4, a very high mass resolving power can be obtained with the use of the X-band. One should keep in mind that this is only a theoretical possibility. In practice, the resolution is also limited first by the residence time of ions inside a quadrupole analyzer as has been noted previously. Simulation of a quadrupole mass filter operation is the most convenient way to test a new operational mode. In the present investigation, we consider a typical quadrupole analyzer with an inscribed radius r0 = 5mm (hyperbolic rods) and a total rod length of 200 mm. The electrical field is pure quadrupole all the way along the rod length. Thus, the fringing fields at the entrance and the exit of the quadrupole analyzer are not taken into account at this stage. The quadrupole mass filter is operated with 1 MHz main RF power and in the normal mode it is set to transmit ions of mass 609 Da with a theoretical resolving power of 10,000. The corresponding power supply settings are shown in Table 2. For simulations of a new mode of operation we select the case I from Table 1. The amplitudes of AC excitations are presented in Table 2 and selected in such a way that the width of the X-band corresponds to a theoretical resolving power of 10,000 qex2 = 0.02). (qex1 = 0.0068, For the peak shape simulations, an ion beam with a random normal distribution of the initial lateral locations (standard deviation 0.1 mm) and the normal distribution of the lateral ion velocities (standard deviation corresponds to r.m.s. energy 0.025 eV for ions of corresponding mass) is injected into the quadrupole analyzer along the Z axis with the same longitudinal energy in order to ensure the same flight times along the quadrupole analyzer. With the AC excitations, the resulting RF voltage has a period of 20 s (20 cycles of the original RF). In the present simualtions, ions are injected into the mass analyzer with initial phases uniformly distributed within the whole period of the resulting RF voltage (a uniform distribution of the ion time-of-birth from 0 to 20 s). A particle tracing has been carried out using AXSIM software [28] as a convinient tool. It allows loading potential distributions on a regular grid in a form of potential array (PA) files of the SIMION software [29]. These distributions are then used to compute the electrical field vector at each grid point and at intermediate locations using a bi-linear interpolation. This allows the computation of the Newton equations of the ion motion with time dependant fields produced by electrodes of any arbitrary shape. A standart 4th order Runge-Kutta algorithm [27] has been used to solve the equations of ion motion. For a harmonic voltage, an integration step has been chosen equal to 1/128 of the RF period (7.8125 ns for 1 MHz). Some researchers have sucsessfully used several times greater integration steps to solve similar tasks in RF ion trap investigations [30]. From computations of the stability diagrams, we have found that

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

32 integration steps are sufficient to obtain the accuracy, which is not changed with a reduction of the integration steps. Thus, the integration with 128 steps in the RF cycle provides an adequate description of the ion motion in the quadrupole field. In each separate simulation, 10,000 ions of the same mass to charge ratio enter the quadrupole analyzer and are traced until they hit the rods and are lost, or transmitted to the end of the quadrupole mass filter. The transmitted ions are counted. The simulation is repeated for different m/z ions within some range with a selected step. A plot of the number of transmitted ions against the ion mass represents the peak shape of the quadrupole mass filter transmission. This is a standard method for peak shape simulations used by many researchers, see for example [10] or [31]. For a new mode of operation with additional AC excitations, no other alterations to the method have been made, and exactly the same procedure of simulations as for a conventional mode have been used. The calculated peak shapes are shown in Fig. 5. It can be seen from Fig. 5 (left) that in the normal operation mode, the actual resolving power is limited by the ion residence time. The ions, injected with the axial energy of 12 eV, have the residence time of 100 RF cycles, and the actual resolving power at 10% of peak maxima is only 575 because of the peak tails. The ions with smaller energy 3 eV (200 cycles) result in R = 2175, and finally 0.75 eV (400 cycles) gives R = 4350. The peak maximum is near 700 counts. Due to the lateral energy and the initial position spread in the present simulation nearly 93% of the ions are lost although they have nominally stable trajectories. In a new mode of operation with the X-band (Fig. 5, right) the peak tails are much smaller and almost disappear at the 3 eV beam energy. The simulation results show that the resolving power is 9178 for 12 eV ion beam and R = 10846 for 3 eV ion beam. The maximum peak transmission is over 900 counts (28% greater than in normal mode). Even with the beam energy of 48 eV (50 RF cycles), the peak has reasonably small tails and the resolving power is nearly 4000 at the level of 50% of the peak maximum. Note that the peak center with the X-band appears not at 609 Da, but at 596.56 Da. This occurs due to the shift of the X-band to greater qvalues. According to stability diagram computations in this case the middle of the X-band appears at q = 0.71982 and a = 0.24167 (for a the scan line with 2U/V = 0.3357). These results are summarized in Fig. 6 where the resolving power is plotted against the square of the residence time. For a normal operational mode, the dependence follows Eq. (5) with C ≈ 20. As a result, even the residence time of 200 RF cycles gives only R = 2000. By comparing this with the use of the X-band, nearly 10,000 resolving power is obtained at the residence time of 100 RF cycles. A linear part of the curve (for the residence time below 100 RF cycles) follows the same rule, but with C ≈ 1. It also clearly indicates that a much faster mass filtering rate is possible when using the X-band.

4. Discussion

15

(10) are defined using the main RF frequency in order to have a direct comparison of the excitation AC voltages with respect to the main RF power. If the main RF frequency in Eq. (10) is replaced by the low frequency ωex1 , then both qex1 and qex2 become much greater (400 times in present example). Instead of qex1 = 0.0068 eff eff and qex2 = 0.02, one has qex1 = 2.72 and qex2 = 8, which is reasonable to refer as “effective” excitation parameters. Such values indicate a very high level of excitation of a normal stability diagram without AC excitation, and correspond to higher regions of stability. Regarding Von Zahn’s rule, for higher stability regions, it is modified due to a faster separation speed. For example, with the use of region II, V.V. Titov has reported a rule R1/2 = 24n2 [32], which corresponds to C = 0.04. As it has been discussed above for the islands of stability, Von Zahn’s rule is modified in a similar way. An actual value of the constant C depends on the width of the Xband, which, in turn, is defined by intensities of the AC excitations, but in any case the constant C is much smaller than that in the normal operational mode. Unlike the usual higher stability regions, which are located far from the origin, stability islands appear near the upper tip of the first stability region. As it has been mentioned above, the investigations of D.Douglas et al. have shown that by using higher regions of stability, only a few cycles of the RF are sufficient for mass filtering [20]. Unfortunately, with conventional higher stability regions, while ions passing the fringe field regions, the ion beam experiences a big lateral energy and position spread. As a result, many ions that have nominally stable motion inside the quadrupole analyzer are lost. In contrast to this, the stability islands are located almost at the same place and the same distance from the origin as the tip of first stability region. The ion beam scattering by the fringe fields in this case is much smaller. Also, like in a normal operation mode, the stability islands can benefit from the Delayed DC Ramp − the technique, which is commonly used in quadrupole analyzers in order to overcome the influence of the fringe fields [33]. With all these advantages, the stability islands feature a fast mass separation, typical for higher regions of stability. In the description above we note “stability islands” because all of these properties appear under the influence of a single AC excitation as well. The X-bands feature the same properties because in fact the X-band is an island of a special type which is formed by two AC excitations. 4.2. Explanation of faster mass separation rate with the use of the X-band A faster separation rate in the X-band can be confirmed by direct calculations of the “exponential increment”, . The number exp( ) shows how many times the amplitude of ion oscillations increases over a single RF cycle in the unstable regions (the rate of amplitude growth). It can be computed from the trace of the monodromy matrix using the following equation:



Cosh( ) = s or␮ = Ln s +



s2 + 1



(11)

4.1. Stability islands as higher regions of stability A key issue for improving the mass resolution with the use of the X-band is a faster mass filtering rate. It certainly requires some explanations. The reason is that this property appears due to the use of the AC excitations. The AC excitations are characterised by the base frequency  = ωex1 /˝, which equals 1/20 of the main RF frequency in the examples above. This means that over 100 RF cycles, there are only 5 cycles of the base frequency. It turns out that this time is sufficient for mass filtering, and the explanation lies in the properties of the stability islands. In fact, the stability islands are similar to higher regions of stability on a normal stability diagram. The dimensionless parameters of excitation in Eq.

where s = 0.5 · trace(M). The computation of the monodromy matrix has been described above. Here, we use this method for computing the exponential increment along the scan line for two cases, the normal operation and a filtering with the use of the Xband. Note that for the X-band case, the matrix is computed over the base period, which is equal to 20 RF cycles. In this case, in order to obtain the rate over one RF cycle, the number obtained from Eq. (11) should be divided by 20. The computed rate of the amplitude growth over one RF cycle, exp( ), is presented in Fig. 7. It is obvious that in the case of the X-band, the rate of increase in the vicinity of the stability zone has considerably greater values than in the case of normal operation

16

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

Normal mode

X-Band 1000

900

900

800

12eV

800

700

3eV

700

600

0.75eV

N, counts

N, counts

1000

500 400

400

200

200

100

100 609.0

609.2

609.4

609.6

3eV

500 300

608.8

12eV

600

300

0 608.6

48eV

0 596.0

596.2

596.4

596.6

596.8

597.0

Mass, Da

Mass, Da

Fig. 5. Peak shape simulations for quadrupole analyzer with pure hyperbolic field in normal mode (left) and in a new mode with X band-I (right) at different beam energy.

Resolving power

12000 10000

X-Band Normal

8000 6000 4000 2000 0 0

10000

20000

30000

40000

50000

60000

n2

Fig. 6. Dependence of actual resolving power at 10% of maximum from square of residence time for normal mode of operation and with the use of X band. Both modes are set for theoretical resolving power of 10000.

Fig. 7. Rate of amplitude increase over one RF cycle along scan line corresponding to theoretical resolving power R = 10000 for two cases: A. Normal mode (left) and B. X-band (right). Increment for X motion is red, for Y motion is blue. Values below 1.0 correspond to stable motion. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

(taking into account the exponential amplitude increase over 100 RF cycles). This explains the faster rate of the X-band mass filtering.

Table 3 Settings of power supply for simulations of quadrupole analyzer with cylindrical rods (R/ro = 1.120) in conventional mode and with X band. All voltage amplitudes are provided zero-peak, pole-to-ground.

4.3. One dimensional separation with X-band It can be noticed from Figs. 1 and 7 (left), that for a normal operational mode, the stability region is bounded by the Y-motion instability from the side of smaller q-values and by the X-motion instability from the other side. In Fig. 7 (right), the stable region is bounded by instability of the X-motion from both sides, while the Y motion remains stable along the scan line at least in the vicinity of the X-band. For this reason, the peak maximum at Fig. 5 for the X-band (900 counts) is 28% higher than that at the normal operation. The transmission of a quadrupole mass filter is characterised by the “acceptance”, which is an area of initial conditions of the ions in a phase space that will be transmitted through the quadrupole analyzer. It is known that the acceptance for a quadrupole mass filter in each direction is described as ˝ · r02 . The full acceptance is a product of the acceptances in both X and Y directions. Close to the boundaries of stability, the acceptances become very small. When a band of stability is limited by boundaries of the X and Y motion from both sides, the full acceptance is a product of two small numbers − the acceptances in the X and Y directions and scales as ˝2 · r04 . While using the X-band, the Y motion remains stable and the acceptance in the Y direction is greater than in the normal operation case. Thus, the full acceptance of a mass filter is proportional to the Xacceptance only and scales as ˝ · r02 . This is another reason why the use of the X-band provides better transmission at high resolving powers compared to the normal mode. 4.4. Insensitivity to nonlinear distortions Manufacturing and assembling the rods with a hyperbolic cross section is a relatively difficult and an expensive task. Some of the commercial quadrupole analyzers are actually made of cylindrical rods. The electrical field inside such quadrupole analyzers differs from purely quadrupole (Eq. (1)), and has nonlinear distortions. Through experimental practice and theoretical investigations [31], the optimum ratio of rod radius, r, and the inscribed radius, r0 , has been found to be within 1.120-1.130. In such a geometry, positive and negative nonlinear distortions partially compensate each other to give the field that most closely approximates the quadrupole field. Unfortunately, this compensation is not complete and the quadrupole analyzers with cylindrical rods typically show lower performance in terms of peak quality and the transmission than the analyzers with hyperbolic electrodes. The experiments of D.Douglas et al. showed [6] that the use of the AC excitations overcomes different types of field distortions. This is also a result of a faster mass separation rate, because there is not sufficient time to build up nonlinear phenomena over a few AC cycles. Also, in contrast to the other stability islands, the X-band is not so sensitive to nonlinear distortions. This fact is illustrated by the peak shape simulations presented in Fig. 8. Here, the parameters of the power supply and the ion beam are similar to the previous simulations with the difference that the field inside the quadrupole analyzer corresponds to a system with cylindrical rods with r/ro = 1.120. The electric field distributions inside the quadrupole analyzer for this simulation have been obtained by the method described in the reference [34]. It is a modification of the boundary element method which allows computing a 2D field for a quadrupole analyzer with cylindrical rods with accuracy better than 10 significant digits up to the electrode surface. The potential distribution is saved as a potential array in a format of the SIMION software [29] with a grid step 0.05 mm, and used for the ion tracing with the finite difference method computation of the electrical

17

Frequency, KHz Conventional Xband-I

DC

RF

AC-1

AC-2

0 184.4155 V 184.3460 V

1000 1098.3197 V 1098.3197 V

50 0 10.590 V

950 0 31.135 V

field at the grid locations and further bi-linear interpolation for the locations between the nodes of the grid. The field analysis [34] shows that the intensity of the quadrupole field component in this electrode geometry is slightly greater than unity, A2 = 1.001081. This should be taken into account in a definition of parameters (4) for calculation of the parameters of the power supply in different modes of operation (see Table 3). For a new mode with the use of the X-band, the operating line is set up to pass through the tip of the firsr stability region. No changes to the amplitudes of the additional AC excitations have been made against the previous simulations for a quadrupole mass filter with hyperbolic rods, so that the anticipated resolving power is 10,000. For a conventional mode of operation, the power supply parameters have been selected so that the transmission at the peak maximum is comparable to the X-mode operation. Interestingly, those parameters correspond to the operating line going above the tip of stability zone, which suggests that the peak shapes in Fig. 8.A are obtained due to nonlinear phenomena in a quadrupole analyzer with cylindrical rods because without nonlinear distortions no ions have stable trajectories inside quadrupole at such scan line locations. In a normal mode of operation, the best result in terms of the resolving power is achieved at the beam energy of 0.75 eV, which corresponds to the residence time of 400 RF cycles. At the beam energy 12 eV, the resolving power at 10% level of the peak maximum is less than 1000. In a new mode of operation with the X-band, the resolving power over 1000 is obtained at the beam energy of 48 eV. Compared to the previous simulations for the quadrupole analyzer with hyperbolic rods (Fig. 5B), the peak maximum is lower due to a nonlinear phenomena, but peak shapes do not have distortions except tails. These tails almost disappear at the beam energy below 12 eV, and the resolving power improves up to the theoretical value of 10,000 at the beam energy 3 eV (see Fig. 8B). The transmission at the peak maximum is approximately 2 times less than for the quadrupole mass filter with hyperbolic rods. 4.5. Fringe fields and other implications A discussion of any new modes of a quadrupole mass filter operation is meaningless without considering the fringe field effects that constitute a big problem for quadrupole analyzers, especially when operated at high resolution modes. As it has been previously mentioned in section 2.4, a high resolving power can be obtained with the use of higher stability regions or by multiple reflections. These methods of obtaing a high resolution do so at the expence of the sensitivity and, as a result of this, are unpractical. This is not the case with the new mode of operation because the X-band can take a benefit from the “Delayed DC ramp” technique commonly used to improve the sensitivity of a mass analysis with quadrupole mass filters. In this method [33], a quadrupole analyzer has a short additional section of the rods (“prerods”) at the entrance. The main rods of the quadrupole analyzer have both RF and DC supply, while the prerods are only powered by the RF. As a result, the ion beam enters the prerods section first where the DC field is absent. In the fringing field of the prerods the RF field gradually increases from zero to a nominal value. Only after entering the main rod section,

18

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

1400

700 12eV

R=812

1200

N, counts

0.75eV

800

R=1218

R=1612

12eV

R=6118

R=2215

3 eV

R=9177

600

500

N, counts

3eV

1000

400 300

400

200

200

100

0 608.0

48eV 600

608.5

609.0

609.5

610.0

0 596.3

Mass, Da

596.4

596.5

596.6

596.7

596.8

Mass, Da

Fig. 8. Peak shape of transmission (by simulations) for quadrupole analyzer with cylindrical rods (R/r0 = 1.120) in normal mode (A left) and with the use of X band-I (B right) at different beam energy. Resolving power at 10% level from peak maximum for each curve is provided in a legend. Note different scales on figures A and B.

ions experience the resolving DC field. As a result, a and q parameters for the ions of interest remain within the stable area of the diagram, thus minimizing the beam scattering by the fringe field. While using higher regions of stabillity, this method is not usefull because within the fringe field a and q parameters appear at unstable regions of the diagram. Unlike conventional stability regions of higher orders, the X-band can take a benefit of the delayed DC ramp because ion parameters remain within a stability region. Only near the very tip of the stability region, the ion parameters go through a narrow unstable band before a and q appear within the stable X-band. This happens at the very end of the fringe field, takes very short time and does not result in significant beam scattering. When a QMF is used as a mass analyzer, the transmitted ions are collected when the RF and DC voltages are scanned by steps. Assuming that for each point of a mass spectrum, the quadrupole mass filter collects ions at least for the time equal to the flight time of ions through the quadrupole analyzer, in an example with the RF frequency of 1 MHz, each point requires 0.1 ms. Scanning of 1 Da mass range with the resolving power of 10,000 will need at least 104 points (or more if peak shape is measured), which requires 1 s for scanning. This example shows that a scanning quadrupole mass filter with a high resolution is not very practical. High resolving quadrupole analyzers will be really appreciated in tandem instruments where QMF performs precursor ion selection, while mass analysis is performed faster and more accurate by other type of mass analyzers. Modern QMF, while operated in the normal mode, can provide the resolving power of ion isolation near 5000. Precursor ion isolation can be done inside the ion traps by means of the notched waveform excitation [35] or by resonance excitations combined with slow scanning techniques [36]. In both cases, the limiting resolving power of isolation is about 5000. With the use of the ICR cells, a very high resolving power of 50 K can be achieved [37,38], but the time required for this manipulation makes it unpractical. B.Collings has developed a method of a precursor ion isolation in a low pressure linear ion trap with the resolving power of more than 20 K [39], which is absolute record result for ion traps. Unfortunately, this method suffers from space charge effects and, thus, does not allow large populations of ions, which makes it considered as a nice academic result rather than a practical tool. In contrast to this, quadrupole mass filters do not suffer from space charge effects. As it is presented in this paper, our new mode can perform ion isolation with the resolving power of 10 K (and possibly greater) without significant loss of the transmission. No doubt that this new operation method by simply using two AC power supplies to modify the first stability region can significantly improve quadrupole mass filtering technology. For example,

quadrupole analyzers can be manufactured shorter (for example 100 mm long instead of 200 mm), and the mass range can be extended by the use of lower RF frequency. Additionally, the application of miniature quadrupole analysers will benefit from using this new mode of operation based on the X-band. 5. Conclusion The theoretical simulation results show that a quadrupole mass filter can obtain much higher mass resolving power and better signal sensitivity when it operates under a new mode. It can be achieved by a modification of the first stability diagram with the use of two AC excitations, by re-shaping stable regions in the most convenient way for the mass filtering. The properties of these regions can overcome many limitations of the normal operation mode and have the following advantages: 1. A faster mass separation is obtained. A few cycles of the base AC excitation frequency, , is sufficient to eject unstable ions. It is possible to obtain ∼10,000 mass resolution with the ion residence time of only 100 RF cycles; 2. Mass separation of ions occurs in only one direction, which increases the ion acceptance of the mass filter; 3. The instability bands, which are used for mass separation appear only in the vicinity of the upper tip of the first stability region, so it is possible to use the delayed DC ramp technique to improve the signal sensitivity; 4. The operation with the use of additional AC excitations is not sensitive to many nonlinear field distortions. So, the new method might be applied to quadrupole analyzers with lower mechanical accuracy for improving their mass resolving power. All examples and calculations, illustrated in this paper, have the base frequency  = 0.05, which is a good practical compromise − 100 RF cycles contain 5 cycles of the base frequency. The similar results can be obtained with other base frequency, provided that it is sufficiently small. It should be noted that all simulation results in this paper are given for the ions with mass 609 Da. Experts may joke that most mass spectrometers are build to resolve Reserpine ions only. The actual reason for this is that just like a conventional mass separation in a normal mode at the tip of the first stability region, the separation with the use of the X-bands can be used for ions of any arbitrary mass. At the same time, the beam properties for 4 Da ions and 4000 Da, in practice, are dramatically different. For heavy ions, the collisional cooling in an ion guides upstream to

M. Sudakov et al. / International Journal of Mass Spectrometry 408 (2016) 9–19

a quadrupole analyzer can be used in order to reduce the beam angular and position spreads for improved transmission. For ions of small masses like 4 Da, such a method cannot be used, at least with the Nitrogen buffer gas. Thus, any discussions of a possible resolving power and the transmission for ions of different masses are meaningless without a consideration of the properties of an ion source and downstream ion optics. Bounds of this publication do not allow us to present data of this kind, but only to show a theoretical possibility of achieving a high resolution with quadrupole mass filters. This paper has shown the theoretical possibility of improving QMF performance. Authors may not envisage many problems of practical implementation. Actual resolving power can be limited by the stability of modern power supplies and overall thermal stability of the instrument. The new mode shows a greater tollerance to the field distortions caused by cylindrical rods, but there is a variety of other distortions, such as rod misallighnment, curvature, etc. This may impose basic practical limitations on the resolving power of mass analysis using quadrupole analyzers. As long as the method does not require any modifications to mechanical parts of quadrupole analyzers, but only changes to the method of power supply application, it seems very possible to be implemented with modern technology, which can provide high stability requirements for this purpose. Acknowledgement M.S. is grateful for the financial support from Fudan University (Shanghai, China) under visiting scientist program during preparation of this publication. References [1] W. Paul, H. Steinwedel, Ein neues massenspektrometer ohne magnetfeld, Zeitschrift für Naturforschung A8 (7) (1953) 448–450. [2] P.H. Dawson, Quadrupole Mass Spectrometry and Its Applications, American Institute of Physics, Woodbury, New York, 1995. [3] C.-F. Ding, N.V. Konenkov, D.J. Douglas, Quadrupole mass filters with octopole fields, Rapid Commun. Mass Spectrom. 17 (2003) 2495–2502. [4] N.V. Konenkov, F. Londry, C.-F. Ding, D.J. Douglas, Linear quadrupoles with added hexapole fields, J. Am. Soc. Mass Spectrom. 17 (2006) 1063–1073. [5] M. Moradian, D.J. Douglas, Experimental investigation of mass analysis using an island of stability with a quadrupole with 2% added octopole field, Rapid Commun. Mass Spectrom. 21 (2007) 3306–3310. [6] X. Zhao, Z. Xiao, D.J. Douglas, Overcoming field imperfections of quadrupole mass filters with mass analysis in islands of stability, Anal. Chem. 81 (2009) 5806–5811. [7] W.E. Austin, A.E. Holme, J.H. Leck, Chapter VI. The mass filter: design and performance, in: P.H. Dawson (Ed.), Quadrupole Mass Spectrometry and Its Applications, 125–129, Elsevier, Amsterdam, The Netherlands, 1976. [8] P.H. Dawson, Ion optical properties of quadrupole mass filters, Adv. Electron. Electron Opt. 53 (1980) 153–208. [9] V.V. Titov, Ion separation in the imperfect fields of the quadrupole mass analyzer: part II, Int. J. Mass Spectrom. Ion Processes 141 (1995) 27–35. [10] S. Taylor, J.R. Gibson, Prediction of the effects of imperfect construction of a QMS filter, J. Mass Spectrom. 43 (2008) 609–616. [11] N.W. McLachlan, Theory and Applications of Mathieu Functions, Oxford University Press, Oxford, 1974. [12] R.G. Cooks, A.L. Rockwood, The ‘Thomson’: a suggested unit for mass spectroscopy, Rapid Commun. Mass Spectrom. 5 (1991) 92–93.

19

[13] W. Paul, H.P. Reinghard, U. Von Zahn, Das elektrische massenfilter als massenspektrometer und isotopentrenner, Z. Phys. 152 (1958) 143–153. [14] D.J. Zhaohui Du Douglas, N.V. Konenkov, Elemental analysis with quadrupole mass filters operated in higher stability regions, J. Anal. At. Spectrom. 14 (1999) 1111–1119. [15] U. Sarfaraz, A.H. Syed, T. Hogan, J. Gibson, S. Taylor, Factors influencing the QMF resolution for operation in stability zones 1 and 3, J. Am. Soc. Mass Spectrom. 23 (2012) 988–995. [16] U. Von Zahn, Prazisions-massenbestimmungen mit dem elekrischen massenfilter, Z. Phys. 168 (1962) 129–142. [17] M.H. Amad, R.S. Houk, High-Resolution mass spectrometry with a multiple pass quadrupole mass analyzer, Anal. Chem. 70 (23) (1998) 4855–4889. [18] N.V. Konenkov, V.I. Kratenko, Int J. Mass Spectrom. Ion Process. 108 (1991) 115–136. [19] T.N. Zhaohui Du Olney, D.J. Douglas, Inductively coupled plasma mass spectrometry with a quadrupole mass filter operated in the third stability region, J. Am. Soc. Mass Spectrom. 8 (1997) 1230–1236. [20] W. Chen, B.A. Collings, D.J. Douglas, High-Resolution mass spectrometry with a quadrupole operated in the fourth stability region, Anal. Chem. 72 (2000) 540–545. [21] K. Miseki, Development of quadrupole mass spectrometers and ion optical devices, J. Mass Spectrom. Soc. Jpn. 51 (2009) 3–29. [22] N.V. Konenkov, L.M. Cousins, V.I. Baranov, M.Yu. Sudakov, Quadrupole mass filter operation with auxiliary quadrupolar excitation: theory and experiment, Int. J. Mass Spectrom. 208 (2001) 17–27. [23] D.J. Gershman, B.P. Block, M. Rubin, M. Benna, P.R. Mahaffy, T.H. Zurbuchen, Higher order parametric excitation modes for spaceborne quadrupole mass spectrometers, Rev. Sci. Instrum. 82 (125109) (2011). [24] D.J. Gershman, B.P. Block, M. Rubin, M. Benna, P.R. Mahaffy, T.H. Zurbuchen, Comparing the performance of hyperbolic and circular rod quadrupole mass spectrometers with applied higher order auxiliary excitation, Int. J. Mass Spectrom. 319–320 (2012) 17–24. [25] S. Magnus, W. Winkler, Hill’s Equation, John Wiley & Sons Inc, New York, 1966. [26] N.V. Konenkov, M.Y. Sudakov, D.J. Douglas, Matrix methods for the calculation of stability diagrams in quadrupole mass spectrometry, J. Am. Soc. Mass Spectrom. 13 (2002) 597–613. [27] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition, Cambridge University Press, NY, 2007. [28] M. Sudakov, AXSIM −new software for simulations of mass spectrometry instrumentation, in: Proceedings of the Seventh International Conference of Charged Particle Optics, Cambridge UK 24–28 July, 2006. [29] D.A. Dahl, SIMION 3D v.70 Users Manual, Idaho National Eng. Envir. Lab, 2000. [30] P.M. Remes, J.-E.P. Syka, V.V. Kovtun, J.C. Schwartz, Re-print of insight into the resonance ejection process during mass analysis trough simulations for improved linear quadrupole ion trap mass spectrometer performance, Int. J. Mass Spectrom. 377 (2015) 368–384. [31] D.J. Douglas, N.V. Konenkov, Influence of the 6th and 10th spatial harmonics on the peak shape of a quadrupole mass filter with round rods, Rapid Comm. Mass Spectrom. 16 (2002) 1425–1431. [32] V.V. Titov, Ion separation in the imperfect fields of the quadrupole mass analyzer: part IV, Int. J. Mass Spectrom. Ion Process. 141 (1995) 45–56. [33] W.M. Brubaker, MULTIPOLE MASS FILTER. US Patent 3,410,997 (1968). [34] D.J. Douglas, T. Glebova, N. Konenkov, M.Y. Sudakov, Spatial harmonics of the field in a quadrupole mass filter with circular electrodes, Tech. Phys. 44 (1999) 1215–1219. [35] M.E. Bier, J.C. Schwartz, R. Cole, Electrospray-Ionization quadrupole ion-Trap mass spectromettry, in: Electrospray Ionization Mass Spectrometry: Fundamentals, Instrumentation, and Applications, John Wiley & Sons, NY, 1997, pp. 235–289. [36] M. Ding Li Sudakov, F.L. Brancia, R. Giles, S. Kumashiro, A digital ion trap mass spectrometer coupled with atmospheric pressure ion sources, J. Mass Spectrom. 39 (2004) 471–484. [37] L.J. de Koning, R.H. Fokkens, F.A. Pinske, N.M.M. Nibbering, High front- and back-end resolution MS/MS in Fourier transform ion cyclotron resonance mass spectrometry, Int. J. Mass Spectrom. Ion Process. 77 (1987) 95–105. [38] L.J. de Koning, N.M.M. Nibbering, S.L. van Orden, F.H. Laukin, Mass selection of ions in a Fourier transform ion cyclotron resonance trap using correlated harmonic excitation fields (CHEF), Int. J. Mass Spectrom. Ion Process. 165/166 (1997) 209–219. [39] B.A. Collings, High-resolution excitation of ions in a low-pressure linear ion trap, Rapid Commun. Mass Spectrom. 25 (2011) 66–74.