LINEAR QUADRUPOLES IN MASS SPECTROMETRY
D.J. Douglas*
Department of Chemistry, University of British Columbia, 2036 Main Mall,
Vancouver, BC, Canada V6T 1Z1
Received 4 October 2008; received (revised) 14 November 2008; accepted 14 November 2008
Published online 2 June 2009 in Wiley InterScience (www.interscience.wiley.com) DOI 10.1002/mas.20249
The use of linear quadrupoles in mass spectrometry as mass
filters and ion guides is reviewed. Following a tutorial review of
the principles of mass filter operation, methods of mass analysis
are reviewed. Discussed are extensions of quadrupole mass
filters to higher masses, scanning with frequency sweeps of the
quadrupole waveform, operation in higher stability regions,
and operation with rectangular or other periodic waveforms.
Two relatively new methods of mass analysis the use of
‘‘islands of stability’’ and ‘‘mass selective axial ejection’’ are
then reviewed. The optimal electrode geometry for a quadrupole mass filter constructed with round rods is discussed. The
use of collisional cooling in quadrupole ion guides is discussed
along with ion guides that have axial fields. Finally, mass
analysis with quadrupoles that have large distortions to the
geometry and fields is discussed. An Appendix gives a brief
tutorial review of definitions of electrical potentials and
fields, as well as the units used in this article. # 2009 Wiley
Periodicals, Inc., Mass Spec Rev 28:937–960, 2009
Keywords: linear; quadrupole; mass filter; resolution; stability; mass scans; quadrupole excitation; ion guide; axial fields;
distorted fields; multipoles
I. INTRODUCTION
The linear quadrupole, originally developed for mass spectrometry by Paul and co-workers (Paul, Reinhard, & von Zahn, 1958),
is widely used in many areas as a mass filter, an ion guide, or a
linear ion trap. The history of the first 21 years of quadrupole
devices has been given by Dawson (1976, Ch. 1). His classic
book describing theory and applications of quadrupoles, both
two-dimensional and three-dimensional, was published more
than 30 years ago (Dawson, 1976). Since that time, there has been
much progress in the development and use of quadrupoles in new
ways. The book by March and Hughes (1989), while mostly
concerned with three-dimensional ion traps, contains much
useful information about the properties of ion motion in
quadrupole fields that can be directly applied to two-dimensional
linear quadrupoles. The use of quadrupoles as linear ion traps was
reviewed by Douglas, Frank, and Mao (2005).
This article attempts to review the use of linear quadrupoles
as mass analyzers and ion guides. Section II is a review of the
————
Contract grant sponsor: Natural Sciences and Engineering Research
Council of Canada; Contract grant sponsor: MDS-Analytical Technologies.
*Correspondence to: D.J. Douglas, Department of Chemistry,
University of British Columbia, 2036 Main Mall, Vancouver, BC,
Canada V6T 1Z1. E-mail: douglas@chem.ubc.ca
Mass Spectrometry Reviews, 2009, 28, 937– 960
# 2009 by Wiley Periodicals, Inc.
equations of ion motion and stability regions. Section III
describes the use of linear quadrupoles as mass analyzers with
conventional and new methods of operation including the
development of high mass range quadrupoles, scanning with
frequency sweeps, operation in higher stability regions, and
operation with rectangular waveforms. Two new methods of
mass analysis the use of ‘‘islands of stability’’ and axial ejection
are then reviewed. The optimal electrode geometry for a
quadrupole mass filter constructed with round rods is discussed.
The use of collisional cooling in quadrupole ion guides is
discussed along with ion guides that have axial fields. Section IV
discusses mass analysis with quadrupoles that have large
distortions to the geometry and fields. An Appendix gives a
brief review of definitions of electrical potentials and fields, as
well as the units used in this article. Table 1 shows the units used
in this article.
Many areas of quadrupole research could not be reviewed
here. These include special applications of quadrupoles, development of hybrid instruments, novel electrode geometries, and
micro-machined and miniature quadrupoles. Mathematical
methods, such as matrix methods to calculate stability diagrams
and boundaries, properties of solutions of the Mathieu equation,
and mathematical methods for calculating ion motion in distorted
fields are not covered. The patent literature is not generally
included. Relevant publications up to June 2008 have been
considered.
II. THE LINEAR QUADRUPOLE
A. The Quadrupole Potential
The electric potential of the linear quadrupole, f(x, y), is given by
2
x y2
0
ð1Þ
ðx; yÞ ¼
r02
where x and y are Cartesian co-ordinates. This potential is
produced by four parallel electrodes with hyperbolic shapes,
as shown in Figure 1. The point (x, y) ¼ (0, 0) is the center of
the quadrupole. The parameter r0, called the ‘‘field radius,’’ is
the distance from the center to an electrode. Balanced potentials
are normally applied between the x and y electrodes, that is, a
potential þf0 is applied to the two electrodes in the x direction
and a potential f0 is applied to the two electrodes in the y
direction. In this case the potential at the center of the quadrupole
is 0. It is readily verified that the potential of Equation (1) satisfies
Laplace’s equation. In fact, for a two-dimensional potential
where electric fields vary linearly with position (the simplest
variation with position) Equation (1) can be derived from
Laplace’s equation (Dawson, 1976, pp. 9–10).
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DOUGLAS
TABLE 1. SI units used in this article
B. Other Two-Dimensional Potentials
If a quadrupole does not have an ideal field with the potential
given by Equation (1), the mathematical description of the
potential is more complicated. The potential can be described as a
superposition of two-dimensional multipole fields. The quadrupole potential is just one of the many possible solutions to
Laplace’s equation in two dimensions. In general, the spatial part
of any two-dimensional potential F(r, y) can be written in polar
co-ordinates (r, y) as a sum of two-dimensional multipoles as
follows:
1
X
’N ðr; Þ
ð3Þ
Fðr; Þ ¼ ½ðA þ BÞðC ln r þ DÞ þ
N¼1
The terms jN(r, y) are referred to as ‘‘spatial harmonics’’ or
‘‘circular harmonics’’ (Smythe, 1939, p. 62) and are given in the
most general form when N 1 as
’N ðr; Þ ¼ ðAN cos N þ BN sin NÞðCN rN þ DN r N Þ
If unbalanced voltages are applied to the x and y rods a
quadrupole potential is still formed, but there is an axis potential.
Suppose the magnitude of the voltage applied to the x rods is f0x
and to the y rods f0y. The potential can be written
2
0x þ 0y
x y2
0x 0y
ðx; yÞ ¼
ð2Þ
þ
2
2
r02
where ((f0x f0y)/2) is the axis potential.
Round electrodes are often used to provide an approximate
quadrupole field because it is less expensive to manufacture
round rods with high precision. The optimum choice of the ratio
of the radius of a round rod r to the field radius r0 is discussed in
Section IIIJ.
ð4Þ
For the field within a linear quadrupole only the terms in rN
contribute, so all DN are zero and without loss of generality the
spatial harmonics can be taken as
N
r
’N ðr; Þ ¼
cos N
ð5Þ
r0
In Cartesian coordinates, used in this review, a two-dimensional
potential can be expanded in multipoles fN(x, y) as
Fðx; yÞ ¼
þ1
X
AN N ðx; yÞ
ð6Þ
N¼0
where AN is the dimensionless amplitude of the two-dimensional
multipole fN(x, y). Each of the two-dimensional multipoles
is also a solution to Laplace’s equation. The analytical form of a
two-dimensional multipole can be calculated from
N ðx; yÞ ¼ Imððx þ iyÞN Þ
ð7Þ
N ðx; yÞ ¼ Reððx þ iyÞN Þ
ð8Þ
or
where Im(f(z)) is the imaginary part of the complex function f(z),
Re(f(z)) is the real part of the complex function f(z), and i2 ¼ 1
(Smythe, 1939, p. 70; Feynmann, Leighton, & Sands, 1963). The
set Re(x þ iy)N can be used without loss of generality. The term
f0 represents a potential that is independent of position, f1 a
dipole potential, f2 a quadrupole potential (Eq. 1), f3 a hexapole
potential, f4 an octopole potential, and so on.
Analytical forms of the two-dimensional multipoles have
been given by Szylagyi (1988). The analytical forms of the first
five multipoles are
ð9Þ
0 ¼ constant
1 ðx; yÞ ¼
FIGURE 1. The quadrupole electrodes with applied potentials, showing
equipotential lines. Reproduced from Du, Douglas, and Konenkov
(1999a). Copyright the Royal Society of Chemistry.
938
2 ðx; yÞ ¼
x
r0
x2 y2
r02
ð10Þ
ð11Þ
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
3 ðx; yÞ ¼
4 ðx; yÞ ¼
x3 3xy2
r03
x4 6x2 y2 þ y4
r04
ð12Þ
&
independent because the potential does not contain terms of the
form xmyn. As a result, ion motion in the x and y directions can be
considered independently.
ð13Þ
The hexapole potential corresponds to the potential in a region
between six rods, the octopole potential to the potential in the
region between eight rods, and so on. The multipole fN(x, y)
gives the potential from an array of 2N electrodes with shapes
given by
N ðx; yÞ ¼ 1
2. Ion Motion and Stability in the x Direction
The equation of ion motion for the x direction can be written as
m
ð21Þ
Introducing the variables
ð14Þ
The electrode geometries and applied potentials producing these
multipoles are shown by Szabo (1986).
d2 x
2zex
¼ 2 ðU VRF cos OtÞ
dt2
r0
x¼
Ot
;
2
ax ¼
8zeU
;
mr02 O2
qx ¼
4zeVRF
mr02 O2
ð22Þ
Equation (21) becomes
d2 x
þ ðax 2qx cos 2xÞx ¼ 0
dx2
C. Equations of Motion and the Stability Diagram
1. Electric Fields in the x and y Directions
For mass analysis, the potential applied to the quadrupole rods
has both DC and time-varying parts, and is given by
0 ðtÞ ¼ U VRF cos Ot
A typical operating frequency is approximately 1.0 MHz
(O ¼ 2 p 1 106 s1) which is a radio frequency (RF).
Equation (16) has a periodic cosine variation of the potential
with time. In fact, any periodic function may be used (see
below).
Ion motion is determined by Newton’s law
d~
u
~
f ¼m
dt
ð17Þ
where ~
f is the force on an ion, m is the mass of the ion,~
u is the ion
velocity, and t is the time. The force on a positive ion is
~
f ¼ zerFðx; y; tÞ
ð18Þ
where z is the number of charges on the ion. The electric field in
the x direction is
@Fðx; y; tÞ
2x
¼ 2 ðU VRF cos OtÞ
Ex ¼ @x
r0
@Fðx; y; tÞ
2y
¼ 2 ðU VRF cos OtÞ
@y
r0
ð20Þ
The electric field in x depends only on x and the electric field in y
depends only on y. The electric fields in the x and y directions are
Mass Spectrometry Reviews DOI 10.1002/mas
!n ¼ ð2n þ Þ
O
2
ð24Þ
where n ¼ 0, 1, 2,. . . and b is a function of the ax and qx
parameters.
Many of the methods of operating linear quadrupoles can be
understood in terms of the stability of ion trajectories and stability
diagrams. Solutions to the Mathieu equation are classified as
‘‘stable’’ or ‘‘unstable.’’ Ions oscillate in the x direction in a
complex way.
A solution is ‘‘stable’’ if the amplitude of oscillation remains
finite as t ! 1. Conversely, a solution is ‘‘unstable’’ if the
amplitude of oscillation increases exponentially with time. In this
case the ion will eventually have an oscillation amplitude equal to
r0 and will strike a rod. Examples of stable and unstable
trajectories are given by Dawson (1976, p. 17). The stability of
ion motion depends on the parameters ax and qx. Combinations of
ax and qx that give stable ion motion in the x direction are shown
by the shaded regions labeled ‘‘stable x’’ in Figure 2.
3. Ion Motion and Stability in the y Direction
The equation of ion motion for the y direction is
m
ð19Þ
and the electric field in the y direction is
Ey ¼ Equation (23) is the Mathieu equation (Émile Léonard Mathieu,
1835–1890). Ion motion consists of oscillations with combinations of many different frequencies. The angular frequencies of
ion oscillation are given by
ð15Þ
where U is a DC voltage applied pole to ground, and VRF is a zero
to peak alternating voltage applied pole to ground. In this case the
quadrupole potential is
2
x y2
ðU VRF cos OtÞ
ð16Þ
Fðx; y; tÞ ¼
r02
ð23Þ
d2 y
2zey
¼ þ 2 ðU VRF cos OtÞ
dt2
r0
ð25Þ
Equation (25) is similar to Equation (21), but differs in the sign
of the right side (because of the different signs of the potential in
x and y, Eq. 1). Equation (25) can also be written as the Mathieu
equation with the change of variables:
x¼
Ot
;
2
ay ¼ 8zeU
;
mr02 O2
qy ¼ 4zeVRF
mr02 O2
ð26Þ
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DOUGLAS
FIGURE 2. Regions of stability for the x and y motions in a linear quadrupole. Reproduced from Du,
Douglas, and Konenkov (1999a) with permission. Copyright the Royal Society of Chemistry.
With these variables, the equation of motion for y becomes
2
dy
þ ðay 2qy cos 2xÞx ¼ 0
dx2
ð27Þ
The stability of ion motion in the y direction can be
considered. Because Equation (27) is the same as Equation (23),
the y motion will be stable if ay and qy have same values as those
shown by the shaded regions labeled ‘‘stable x’’ in Figure 2 for ax
and qx.
4. Combined Stability for Ion Motion in the
x and y Directions
What combinations of ax,qx and ay,qy make ion motion stable in
both the x and y directions? The key is to note that
ay ¼ ax ;
qy ¼ qx
ð28Þ
What values of ax and qx make the y motion stable? For the y
motion to be stable the values of ay and qy must be in the shaded
regions ‘‘stable x’’ of Figure 2. If ax and qx have the same
magnitudes but with opposite sign, ay and qy will have these
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values. The stability diagram Figure 2 is symmetric about the a
axis; replacing qx by qx makes no difference. However, for the y
motion to be stable when expressed in terms of the x parameters,
the sign of ax for the stable regions must be changed. This gives
the regions in Figure 2 labeled ‘‘stable y.’’ Regions where both the
x and y motions are stable are called ‘‘combined stability
regions’’ or simply ‘‘stability regions.’’ Eight stability regions
can be found in Figure 2. In fact, there are an infinite number of
stability regions. The a and q values of the tips of the first six
regions have been given by Konenkov, Sudakov, and Douglas
(2002). Almost all commercial mass filters use the stability
region labeled ‘‘1’’ in Figure 2. Operation of quadrupole mass
filters in other regions is reviewed in Section IIIC.
5. Hexapole and Octopole Fields
As discussed below, mass analysis takes advantage of the
stability regions. Can stability diagrams be derived for other
multipole rod sets, such as hexapoles or octopoles? The stability
diagram of Figure 2 was derived by considering first the ion
motion in the x direction and then the ion motion in the y
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
direction. Separate stability diagrams for each direction could
be calculated because the ion motions in the x and y directions
are independent. For a higher multipole such as a hexapole or
octopole this is not possible. Consider the octopole. From
Equation (13) the electric field in the x direction is given by
3
4x 12xy2
ð29Þ
Ex ¼ r04
and the electric field in the y direction by
12x2 y þ 4y3
Ey ¼ r04
ð30Þ
The electric field in x depends on y, and the electric field in y
depends on x. Thus, the x and y motions are coupled and cannot be
considered independently. No stability diagram can be derived.
Stability boundaries are not well defined and become diffuse
(Haag & Szabo, 1986; Gerlich, 1992). Thus, higher order
multipole rod sets have not been used as mass filters.
D. Acceptance and Emittance
Not all combinations of initial position and initial radial velocity
are transmitted at a given operating point (a, q), in the stability
diagram. For a given RF phase, combinations of initial position
(x, y) and radial velocity, x_ ¼ dx=dt, y_ ¼ dy=dt, that are
transmitted, fall within ellipses in the x, x_ and the y, y_ planes
(the ‘‘phase’’ planes) (Dawson, 1976, p. 32) as shown schematically in Figure 3. There are separate ellipses for the x and y
directions. The area of an ellipse is the ‘‘acceptance’’ for that
particular phase. The ellipses rotate about the origin of the phase
plane at the RF frequency. The region where the ellipses overlap
for all phases gives the acceptance in x or y for 100% transmission
regardless of RF phase. The acceptance can also be described
FIGURE 3. An acceptance ellipse for the x direction. Reproduced from
Du, Douglas, and Konenkov (1999a) with permission. Copyright the
Royal Society of Chemistry.
Mass Spectrometry Reviews DOI 10.1002/mas
&
for 50% or some other value of transmission. The area of the
acceptance ellipses scales as r02 f for each of the x and y directions,
so that the combined acceptance in x and y is proportional to r04 f 2 .
The positions and radial velocities of ions from a source will
occupy regions in the x, x_ and the y, y_ planes. In general, these
positions will not form an ellipse. The areas of these regions
are called the source emittance in x and y. The transmission
of a quadrupole will be determined by the overlap of the
quadrupole acceptance and source emittance (Dawson, 1990).
As the resolution of a quadrupole increases, the combined
acceptance in x and y decreases with resolution (R) as 1/R
(Dawson, 1980, 1990).
III. MASS ANALYSIS WITH LINEAR QUADRUPOLES
A. Mass Analysis at the Tip of the First Stability Region
Figure 4 shows the first stability region. The boundaries
correspond to the lines bx ¼ 0, bx ¼ 1, by ¼ 0, and by ¼ 1.
Lines of constant bx and by are shown within the stability region
(‘‘iso-beta lines’’). At the tip of the region ax ¼ 0.23699 and
qx ¼ 0.70600 (Dawson, 1976, p. 21). The stability boundaries
when ax ¼ 0 are at qx ¼ 0.000 and qx ¼ 0.908.
Consider the positions on the stability diagram of three ions
with masses (more accurately, mass-to-charge ratios) m1 < m2 <
m3 for fixed values of U and VRF. For mass analysis the values of U
and VRF are set so an ion, m2 in this example, is just within the
stability region at the tip of the stability diagram. For the same
voltages heavier ions m3 have lower a and q values, and lighter
ions m1 have greater a and q values. The ions lie on an ‘‘operating
line’’ or ‘‘scan line.’’ Ions m2 have stable motion within the
quadrupole, and ions m1 and m3 have unstable motion.
Comparison to Figure 2 shows m1 has motion unstable in the x
direction, and m3 has motion unstable in the y direction. Thus,
if an ion source is placed at the entrance of a quadrupole and a
detector or some other device at the exit of the quadrupole, ions of
mass m2 will be transmitted and ions of other masses, m1 and m3
in this example, will become unstable, will collide with the
electrodes of the quadrupole, and will not be transmitted. Thus
the quadrupole acts as a ‘‘mass filter’’ for ions of mass m2.
To transmit another mass the voltages U and VRF are adjusted
to place that mass at the tip of the stability region. If U and VRF are
scanned together with a constant ratio, ions of increasing mass
will reach the tip of the stability region in order of their mass and
will be transmitted sequentially to produce a mass spectrum.
Scans by decreasing U and VRF are also possible. Alternatively,
the voltages U and VRF can be switched so that the quadrupole
transmits predetermined masses to provide ‘‘peak hopping’’
or single or multiple ion monitoring. The scanning is purely
electronic and is amenable to computer control.
The mass resolution of a quadrupole can be changed by
changing the ratio of DC (U) to RF voltage (VRF) to change the
slope of the scan line. Lower ratios provide lower mass
resolution. If a quadrupole is operated with a constant ratio
U/VRF the resolution will be constant across a mass scan. It is
more common to operate a quadrupole so that the peak width
is constant across a spectrum (e.g., to provide unit resolution
941
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DOUGLAS
0.3
m3
0.2
βx
a
0.1
Operating Line
0.8
m2
0.2
m1
0.4
0.6
βy
0.6
0.4
0.8
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
q
FIGURE 4. The first stability region, showing an operating line. Reproduced from Moradian (2007).
Copyright A. Moradian.
across a spectrum). This can be done by changing the ratio U/VRF
with mass (Marchand & Marmet, 1964). The simplest, although
somewhat approximate, method to do this for positive ions is
to apply a constant negative DC voltage between the rods in
addition to the positive DC voltage U which changes with mass.
In principle, the resolution can be increased without limit by
placing ions closer to the tip of the stability region. In practice, at
least two things limit the resolution—the residence time of ions in
the quadrupole field and field imperfections in the quadrupoles.
The effects of field imperfections are complex and not well
understood and are discussed below. The effects of residence
time are introduced here.
If ions do not spend a sufficiently long time in the quadrupole
field they may be transmitted even though they have a,q values
outside the stability region. The stability boundaries correspond
to the amplitude of ion oscillation increasing exponentially and
are for an infinitely long quadrupole (i.e., t ! 1). If an ion which
would otherwise be unstable reaches the end of the quadrupole
before its amplitude of oscillation reaches r0, it will be
transmitted. When the mass resolution is limited by the ion
residence time in the quadrupole, the resolution measured at half
height, R1/2, is given by
R1=2
n2
¼ RF
h
ð31Þ
where nRF is the number of RF cycles an ion spends in the
quadrupole field, and h is a constant. For operation in the first
stability region Paul, Reinhard, and von Zahn (1958) calculated
h ¼ 12.25. Austin, Holme, and Leck (1976) found h ¼ 20. For this
reason quadrupole mass filters, at least when operated in the first
stability region, provide the highest mass resolution with low
energy ions. High energy ions give poor resolution with tails
on peaks, particularly the low mass side. If an ion beam has a
distribution of energies with a high energy tail, the ions with high
energies will not be well resolved and will produce peak
942
broadening. Thus, quadrupoles benefit from ion beams that have
both low energies and low energy spreads.
Equation (31) shows that the resolution of a quadrupole
might be increased by running at a higher RF frequency. This,
however, requires higher RF voltages for a given mass range.
Manufacturers make choices between resolution, sensitivity, and
mass range. Higher frequencies generally give higher resolution
and somewhat higher sensitivity (because of the greater acceptance), but at the expense of lower mass range. Conversely, lower
frequencies give higher mass range but at some expense of
sensitivity and resolution. Where only a limited mass range is
required, such as for inductively coupled plasma mass spectrometry (ICP-MS) (maximum m/z ca. 250), quadrupoles operated
at frequencies of 2.0–3.0 MHz are commonly used. Another
approach to increase nRF is to use a longer quadrupole. Von Zahn
(1962) used a quadrupole with a length of 5.82 m to obtain a
resolution of several thousands with Xeþ ions. Amad and Houk
(1998, 2000) increased nRF by reflecting ions to give multiple
passes through a quadrupole. A resolution of R1/2 & 11,000 was
obtained at m/z ¼ 28.
B. Fringe Fields
The field of a quadrupole mass filter does not stop abruptly
with its full value at the ends of the rods. At each end of the rods
there are regions of length approximately r0 both outside and
inside the quadrupole where the field grows from 0 to its
maximum value within the quadrupole. The ‘‘fringing fields’’ in
this region can be defocusing. Consider an ion approaching the
tip of the stability region of Figure 4 along the scan line. If the
field grows linearly the ion will be in a region of unstable motion
for the y direction and may be rejected before it reaches the mass
filter (Dawson, 1980). The fringe field can change the acceptance
of the quadrupole. The fringe field is not necessarily defocusing.
The transmission depends on the number of RF cycles nf that ions
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
spend in the fringing field. There is an optimum transit time of
nf & 2 cycles which gives the maximum transmission with a
quadrupole operated in the first stability region (Dawson, 1980).
In general, the fringing field is complex. Ion motion is no
longer independent in the x, y, and z directions. Dawson (1980)
proposed a linear model of the potential in the fringing field as
follows:
ðx; y; zÞ ¼
ðx2 y2 Þz
r03
ð32Þ
Hunter and McIntosh (1989) calculated numerically the fringing
field of a quadrupole with an aperture plate at a distance d from
the end of the rods and fit the fields to
ðx; y; zÞ ¼
x2 y2
f ðzÞ
r02
ð33Þ
with
f ðzÞ ¼ 1 expðaz bz2 Þ
ð34Þ
where the distance z is measured from the aperture plate in units
of r0. Values of a and b were given for different distances of the
end plate from the quadrupole. In a second article, McIntosh and
Hunter (1989) calculated the effects of this more realistic fringe
field model on the quadrupole acceptance. The defocusing effect
of the fringing field can be eliminated with a delayed DC ramp,
short RF-only quadrupole, which has some fraction of the main
RF applied but no DC between the rods (Brubaker, 1968). The
short quadrupole is sometimes called a ‘‘prefilter.’’ As an ion
approaches the quadrupole it first experiences an RF field in
which it is stable and then the DC is applied, but in such a way
that the ion retains stable values of a and q. The fringing field
at the exit of a quadrupole can also be defocusing. If ions leaving
the quadrupole enter a detector floated at a high potential the
defocusing effects at the exit are overcome by the strong
attractive field from the detector, and a delayed DC ramp is not
used at the exit of the quadrupole.
C. Quadrupole Mass Filters for High Mass Ions
For mass analysis an ion is placed near the tip of the stability
diagram where a ¼ 0.23699 and q ¼ 0.70600. For a quadrupole
power supply with a maximum RF voltage output of VRFmax,
there will be a maximum value of the mass-to-charge ratio that
can be mass analyzed, given by
ðm=zÞmax ¼
4eVRFmax
0:70600r02 O2
ð35Þ
For given rod size r0, the maximum mass can be increased by
lowering the operating frequency O. This was demonstrated for a
3D trap by Wuerker, Shelton, and Langmuir (1959) who trapped
small (ca. 1 mm) particles of aluminum in a trap operated at 50 Hz.
With a lower operating frequency the number of RF cycles that an
ion spends in a quadrupole mass filter is reduced, so somewhat
lower resolution can be expected. As well, because the acceptance of the quadrupole scales as O2, the sensitivity can be
expected to be somewhat lower for a quadrupole operated at
lower frequencies. Beuhler and Friedman (1982) described a
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quadrupole operated at 292 kHz with a maximum m/z of
approximately 80,000, used to study cluster ions. Labastie and
Doy (1988) described a quadrupole mass filter operated at
560 kHz, with a maximum mass range of 9,000, also used to
study cluster ions. A resolution of approximately 80 was used,
sufficient for the cluster experiments. With the development of
electrospray ionization (ESI) there has been renewed interest in
quadrupole mass filters with high mass ranges. Winger et al.
(1993) described a quadrupole similar to that of Beuhler and
Friedman (1982), operated at 292 kHz to give a maximum m/z of
45,000. The quadrupole was used to examine low charge states
of protein ions and protein–protein complexes (Light-Wahl,
Schwartz, & Smith, 1993, 1994). The resolution was surprisingly
low, ca. 100, and there was a severe loss of sensitivity for
ions with m/z <500. Collings and Douglas (1997) described a
quadrupole operated at 683 kHz with a mass range of 8,585. Unit
resolution was possible at m/z ¼ 5,000, and below m/z 3,000
the sensitivity was comparable to the same system operated at
1.00 MHz with a mass range of 4,000, as expected from the
slightly lower acceptance with the higher mass range quadrupole.
High mass range quadrupoles have also been used with hybrid
quadrupole time-of-flight (TOF) instruments. Sobott et al. (2002)
described a system with a quadrupole with a calculated mass
range of 32,000, used to study protein–protein complexes. A
resolution of ca. 1,000 at m/z 22,226 was demonstrated.
Interestingly, to collisionally cool (see below) ions of large
protein complexes requires a higher pressure in a quadrupole ion
guide than is used for more highly charged protein ions or
peptides (Chernusevich & Thomson, 2004).
D. Scanning with Frequency Sweeps
If the voltages U and VRF are kept constant and the frequency of
the quadrupole power supply is scanned, ions of different massto-charge ratio will reach the tip of the stability region and a mass
spectrum can be produced. This was first demonstrated by Paul
and Raether (1955) who swept the frequency of a quadrupole
from 2.38 to 2.54 MHz to produce a mass spectrum of Rb isotopes
(see also Paul, 1990). A quadrupole power supply capable of
frequency scans was described by Mellor (1971). Marmet and
Proulx (1982) described a frequency swept quadrupole that was
intended to give greater stability than a conventional quadrupole
with the electronics available then. A frequency scan from 500 to
350 kHz was used to produce a spectrum from m/z ¼ 10 to 40 with
better than unit resolution. A spectrum of Xeþ ions was
demonstrated where the isotopic peaks were baseline resolved.
Landais et al. (1998) described a quadrupole with a frequency
that could be scanned from 0.4 to 1.1 MHz. Resolution of
approximately 100–200 was obtained at m/z 79–176. Nie et al.
(2006) described a quadrupole with the frequency scanned from
100 to 500 kHz. The quadrupole was operated in the third
stability region (see below) to improve resolution. The resolution
with ions of cytochrome c was approximately 30–100. There are
two potential difficulties with a frequency-swept quadrupole.
First, at the lower frequency necessary for higher mass ions,
resolution is limited by the number of RF cycles in the quadrupole. Second, because the acceptance of the quadrupole is
proportional to the square of the frequency, as the quadrupole
943
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DOUGLAS
scans to higher masses (i.e., to lower frequency) the sensitivity
might be expected to decrease.
E. Operation in Other Stability Regions
Most commercial quadrupole systems operate in the first stability
region. However, there have been several investigations of
quadrupole operation in other stability regions with higher a and
q values. In general, operation in higher stability regions provides
higher resolution and the ability to obtain unit resolution with
higher kinetic energy ions because the value of h in Equation (31)
is considerably lower than with operation in the first region.
Different authors use different notations for the stability regions.
The notation used here is from Dawson (1976). Earlier reviews
of operation in higher stability regions have been given by
Konenkov and Kratenko (1991) and by Du, Douglas, and
Konenkov (1999a).
1. The Second Stability Region
The second stability region, Figure 5, is centered at q ¼ 7.547
with a tip at a ¼ 0.0295. Values of bx and by vary from 1 to 2
across the region. The width of the stability diagram at the base
gives a resolution q/Dq of 114 for a quadrupole operated in RFonly mode. A quadrupole that requires VRF ¼ 5,000 V to reach
q ¼ 7.7457 requires only U ¼ 20 V to reach the tip of the region.
A scan line that passes through the tip of the second region will
also pass through the first region. When ions are transmitted at
the center of the second region, ions of m/z 8.3 times greater or
more are transmitted in the first region. Practical operation in
the second region will require a second auxiliary mass analyzer to
prevent transmission of ions in the first region. However, only
very low resolution is required from the second analyzer.
Dawson and Bingqi (1984a, b) first studied this region and
showed that considerably higher resolution is possible than
with a quadrupole operated in the first region. The value of h in
Equation (31) is lower in this region. A resolution of several
thousands at m/z 44 and 129 was shown. With 300 eV ions, a
FIGURE 5. The second stability region. Reproduced from Ying and
Douglas (1996) with permission. Copyright John Wiley and Sons, Ltd.
944
resolution of ca. 1,500, sufficient to separate 131Xeþ from C3 Fþ
5,
was demonstrated. Dawson and Bingqi (1984a) calculated that in
theory R ¼ 180n2RF and measured R ¼ 19n2RF (Dawson & Bingqi,
1984b). Titov (1998) calculated R ¼ 24n2RF . Du, Douglas, and
Konenkov (1999a) measured R1=2 ¼ 26n2RF and Konenkov and
co-workers measured R1=2 ¼ 24n2RF (Shagimuratov et al., 1990;
Konenkov & Kratenko, 1991).
Because of the lower value of h compared to the first region,
ions of higher energy can be mass analyzed with unit resolution in
the second stability region. Hiroki, Kanenko, and Murakami
(1995) showed resolution of 48 at m/z ¼ 32 with a quadrupole
operated in the second region with no DC between the poles
(i.e., a ¼ 0) and an ion energy of 3,000 eV. Du, Douglas, and
Konenkov (1999a) showed unit resolution at m/z ¼ 39 with
1,000 eV Kþ ions.
Because of the higher RF voltages required for scanning in
the second region, it may be most useful for mass analysis of
relatively low m/z ions. Ying and Douglas (1996) and Du,
Douglas, and Konenkov (1999a) discussed the possible advantages for ICP-MS. The ability to mass analyze high energy ion
beams may reduce space charge problems, a source of nonspectroscopic interferences in ICP-MS. With low energy beams a
resolution R1/2 & 5,000–9,000 is sufficient in many cases to
separate atomic ions from molecular ion interferences below
m/z ¼ 80. Ying and Douglas (1996) demonstrated a resolution at
half height of R1/2 ¼ 5,000 at m/z ¼ 56, sufficient to separate
ArOþ from Feþ.
Low energy ions limit the maximum scan speed that can be
achieved with a quadrupole. If the quadrupole mass setting
changes before an ion passes through the quadrupole, the ion will
not be transmitted. With high energy ion beams a quadrupole can,
in principle, scan more quickly. For ICP-MS this will increase the
number of elements that can be detected in a transient sample.
Grimm, Clawson, and Short (1997) demonstrated the use of
the second stability region to mass analyze ions with kinetic
energies up to 1,000 eV for fast GC/MS scans. Scan rates of
1,000 scans/sec over 80 mass units were demonstrated.
The transmission with operation in the second region is
expected to be considerably less than in the first region for
two reasons: (1) the acceptance of the quadrupole is lower in
the second region, and (2) the fringe fields are more defocusing.
When a quadrupole is operated in the second region at a
resolution of 10,000 the acceptance is approximately 103 of the
quadrupole operated in the first region at low resolution (Douglas
& Konenkov, 1998). With operation at low resolution (114) in
the second region, the fringing field can decrease the acceptance
by three orders of magnitude if the ions spend too long in the
fringing field (Dawson & Bingqi, 1984a; Douglas & Konenkov,
1998). However, these effects are not prohibitive. Ions can be
tightly focused into the quadrupole so that the decrease in
transmission between the first and second regions is less than
calculated from the differences in acceptance (Douglas &
Konenkov, 1998). The fringing fields do not defocus if the ion
residence time is less than approximately 0.5 RF cycles (Dawson
& Bingqi, 1984a; Douglas & Konenkov, 1998). Titov (1998)
calculated a maximum in the transmission when nf ¼ 0.1 for a
quadrupole with a distorted field. These short times can be
achieved by accelerating the ions in the fringe field and
decelerating the ions in the quadrupole. A comparison between
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
&
an experimentally measured curve of transmission versus
resolution and a curve calculated from the overlap of the source
emittance with the quadrupole acceptance showed good agreement provided the emittance corresponded to the ions being
tightly focused into the quadrupole (Douglas & Konenkov,
1998). Under these conditions the transmission at low resolution
in the second region is approximately 10 times less than in
the first region. Increasing the resolution to 5,000 causes
about another factor of 10 loss of transmission (Ying & Douglas,
1996).
Du, Douglas, and Konenkov (1999b) observed considerable
structure on the peaks with a quadrupole operated in the second
stability region at resolution of ca. 100–500 and ion energies
of 20–100 eV. Two possible sources of the structure were
considered: nonlinear resonances caused by field imperfections
and ion collection effects. It was shown that the structure was
caused by ion collection effects. Ions entering the quadrupole
near the axis can be focused or defocused at the exit of the
quadrupole. When defocused, the ions are collected less
efficiently and a dip appears on the peak. It was shown that
nonlinear resonances caused by hexapole and octopole fields
cannot cause structure on a peak when the quadrupole is operated
at a resolution greater than 800 because the resonance lines do not
pass through the tip of the stability region. Similar observations
of focusing effects were reported for a quadrupole operated in
the third stability region (Du et al., 2000).
2. The Third Stability Region
The third stability region (Fig. 2) has a somewhat rectangular
shape and is near a ¼ 3, q ¼ 3. The upper and lower tips are shown
in Figure 6a and b, respectively. Values of bx vary from 1 to 2 and
by from 0 to 1. A scan line that passes through the center of the
region gives a resolution of 22 (Dawson, 1974). Mass analysis
is possible with scan lines that pass through the upper tip
(a, q) ¼ (3.164, 3.234) or through the lower tip (a, q) ¼ (2.521,
2.815). Scan lines that pass through these tips do not pass through
the first stability region and so a mass filter operated in the third
region does not require any additional device to reject higher
mass ions.
Operation of a mass filter in the third stability region can
provide higher resolution than operation in the first stability
region. Konenkov et al. (1990) first showed the separation of COþ
from Nþ
2 (m/Dm ¼ 2,500 at m/z 28) (reproduced in the review of
Konenkov and Kratenko, 1991), and separation of 40Arþ2 from
20
Neþ (m/Dm ¼ 177) and D2Oþ (m/Dm ¼ 656). Shagimuratov
et al. (1990) demonstrated baseline separation of isotopic peaks
of Krþþ differing in mass by 0.5 Th. Konenkov, Silakov, and
Mogilchenko (1991) demonstrated separation of 4Heþ from Dþ
2
(m/Dm ¼ 157). Hiroki, Abe, and Murakami (1991) noted the
resolution is higher than with operation in the first region and
that the peaks have much less tailing, allowing the detection of
a minor peak beside a major peak. Separation of Heþ from Dþ
2
was demonstrated (Hiroki, Abe, & Murakami, 1992) and separation of 3Heþ from HDþ (m/Dm ¼ 518) (Hiroki et al., 1994).
Detection of 4He in a large excess (104) of D2 was described
(Hiroki, Abe, & Murakami, 1994a) and applied to leak detection
(Hiroki, Abe & Murakami, 1996). In all these studies operation at
Mass Spectrometry Reviews DOI 10.1002/mas
FIGURE 6. The third stability region showing (a) the upper tip and
(b) the lower tip. Reproduced from Du, Olney, and Douglas (1997).
Copyright the American Society for Mass Spectrometry.
the upper tip of the stability diagram was preferred because peaks
had less tailing on the low mass side. Fringing fields of different
lengths were investigated (Hiroki, Abe, &Murakami, 1994b) as
well as use of a prefilter to improve sensitivity by a factor of 2
(Hiroki et al., 1995).
Du, Olney, and Douglas (1997) investigated the use of the
third stability region for ICP-MS. Resolution of up to 4,000 was
obtained at m/z ¼ 59 (59Coþ) with operation at the upper tip, and
ca. 1,000 with operation at the lower tip. These resolutions are not
generally high enough to separate atomic ions from molecular
ion interferences. However, at a resolution of ca. 600–1,000,
readily obtained with operation at the upper tip, the peaks were
found to be remarkably free of tails on the low and high mass
945
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sides compared to operation in the first stability region. An
abundance sensitivity of greater than 107 less than 0.5 Th from
the peak center was demonstrated at m/z 59. This lack of peak
tails was also seen in later computer simulations of peak shapes
by Turner, Taylor, and Gibson (2005), and Hogan and Taylor
(2008).
Somewhat higher energy ions can be analyzed with
operation in the third region. Hiroki, Abe, and Murakami
(1994a) used peak shapes from computer simulations to calculate
that at the upper tip R ¼ 1:7n2RF and at the lower tip R ¼ 0:69n2RF .
The same simulations showed R ¼ n2RF =10:9 for the first region,
in reasonable agreement with the calculation of Paul, Reinhard,
and von Zahn (1958) which gave R ¼ n2RF =12:25. Konenkov and
co-workers found R ¼ 0:7n2RF (Konenkov et al., 1990; Konenkov
& Kratenko, 1991). Dawson (1974) estimated that approximately
five times fewer cycles were required than in the first region so
R 0:7n2RF . Titov (1998) calculated R ¼ 1:4n2RF . Du, Douglas,
and Konenkov (1999a) measured R1=2 ¼ 1:0n2RF for operation at
the upper tip. These calculations and measurements suggest that
ions of somewhat higher energy, a few tens of eV, might be mass
analyzed with operation in the third region. Unit resolution with
63 eV Coþ ions was shown by Du et al. (2000), and resolution of
275 with 120 eV Coþ ions was demonstrated by Du and Douglas
(1999).
Acceptance ellipses for the third region with operation at the
upper and lower tips have been calculated by Dawson (1974),
Konenkov, Mogilchenko, and Silakov (1992), Konenkov and
Dowell (1997), Konenkov (1997), Titov (1998), and Du,
Douglas, and Konenkov (1999a ). The acceptance in the x
direction is much greater than in the y direction. The combined
acceptances (x and y) at the upper and lower tips are similar. Du,
Douglas, and Konenkov (1999a) showed a calculated transmission versus resolution curve for a realistic emittance and an
ion residence time in the fringe field of one RF cycle. Similar
transmissions were calculated for operation at the upper and
lower tips. In contrast, the experiments of Du, Olney, and
Douglas (1997) showed remarkably different transmissions at
the upper and lower tips. The differences between experiment
and modeling remain unexplained, but may be related to
the complex emittance of the ‘‘hollow’’ beam used in the
experiments.
Du, Olney, and Douglas (1997) showed that peaks have
remarkably little tailing provided ion energies remain low
(<10 eV). With higher energy ions, peaks form tails. Hiroki,
Abe, and Murakami (1994a) used computer simulations of peak
shapes to show that, with operation at the upper tip, peaks tail on
the high mass side but remain relatively sharp on the low mass
side. Conversely, with operation at the lower tip, peaks tail on the
low mass side but remain sharp on the high mass side. Du and
Douglas (1999) demonstrated this with 120 eV Coþ ions. Du
and Douglas (1999) then described a tandem quadrupole mass
analyzer that takes advantage of these peak shapes. Two
quadrupole mass analyzers are operated in tandem, the first at
the upper tip and the second at the lower tip and each at relatively
low resolution. The quadrupoles scan together, with each
eliminating tails on one side of the peak. The resulting peak is
free of tails, even with relatively high energy (100 eV) ions. The
sensitivity was found to be highest with the two quadrupoles
placed ca. 2 mm apart with no intervening lens.
946
Several authors have considered the effects of the fringing
field with operation in the third region. In comparison to the first
stability region, fewer cycles in the fringing field give the
optimum transmission. Konenkov (1993) showed that the
transmission is highest if ions spend 1.2 cycles in the fringing
field. Konenkov (1997) showed that with the optimum number of
cycles in the fringe field, ca.1.2 at the upper and lower tips, the
acceptance increased ca. 3 times over that with no fringing field
so that, at a resolution of 100, the transmission was similar to that
of a quadrupole operated in the first region. Konenkov and
Dowell (1997) showed that the optimum residence times are
about nf ¼ 1.0 with operation at the upper tip and nf ¼ 1.2 with
operation at the lower tip. Titov (1998) calculated the optimum nf
at the upper tip is 1.3, and at the lower tip 1.2.
Hiroki, Abe, and Murakami (1994b) studied experimentally
the effects of the inlet and outlet fringing field lengths by
changing the separation between the quadrupole and a source
and detector, and concluded short fringing fields were optimal.
Hiroki et al. (1995) added a short RF-only quadrupole ‘‘prefilter,’’
with an RF voltage approximately 0.2 of that of the mass
analyzing quadrupole, to a quadrupole operated at the center of
the third region, and gained a factor of 2 in transmission. This was
attributed to the ions remaining more stable in the y direction as
they approached the quadrupole, although x motion was still
unstable in the fringe field.
3. The Fourth Stability Region
The fourth stability region is shown in Figure 7. The q values on
the a ¼ 0 axis range from 21.298631 and 21.303174. At the tip
a ¼ 2.06 103. The values of bx and by vary from 2 to 3. When
ions are transmitted in this region, ions with m/z 2.82 times
greater are simultaneously transmitted in the second region, and
ions with m/z 23.46 times greater or more are transmitted in the
first region. With no DC applied to a quadrupole the width of the
region gives a nominal resolution q/Dq of 4,700.
Chen, Collings, and Douglas (2000) investigated the ion
optical properties of this region. With no DC applied between the
rods a resolution of R1/2 ¼ 13,900 was obtained with 40 eV 39Kþ
ions, as expected from the width of the stability region. When DC
was added between the rods, the resolution did not increase. The
resolution was found to vary approximately as R1=2 ¼ 829n2RF ,
and it was calculated that unit resolution should be possible at m/z
39 with 10,000 eV ions. High resolution is possible with
relatively high energy ions. A resolution of R1/2 ¼ 5,000 was
demonstrated with 750 eV ions and a resolution of R1/2 ¼ 1,400
with 4,000 eV ions. The acceptance of the quadrupole was
calculated for various residence times in the fringing field, and
was found to drop severely for times of more than 0.3 RF cycles.
The acceptance was calculated to be 102 to 104 of the
acceptance of a quadrupole operated in the second region
(depending on resolution). Measurements of the transmission in
the fourth region were compared to measurements of the
transmission in the second region when the quadrupole was
operated at resolutions of 1,500–5,000. The transmission in the
fourth region was two to three times less, because high ion
energies (600–4,000 eV) could be used to increase the transmission in the fourth region.
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
a
&
G
2.9
0.0020
2.1
2.8
2.2
2.7
2.3
0.0015
2.6
2.4
βy
βx
2.5
0.0010
2.5
2.4
0.0005
2.6
2.3
2.7
2.8
2.2
0.0000
2.1
2.9
21.299
21.300
21.301
21.302
21.303
q
FIGURE 7. The fourth stability region. Reproduced from Chen, Collings, and Douglas (2000) with
permission. Copyright the American Chemical Society.
F. Mass Filter Operation with Rectangular or Other
Periodic Waveforms
Quadrupoles need not be operated with the sinusoidal waveform
described by Equation (15). Any periodic waveform may be used.
In this case, the equation of motion becomes
d2 u
þ CðtÞu ¼ 0
dt2
ð36Þ
where u is x or y and C(t) is any periodic function of time. This
equation is known as the Hill equation (Dawson, 1976, p. 14).
Richards, Huey, and Hiller (1973) showed that a quadrupole
mass filter can be operated with a periodic rectangular waveform,
like that shown in Figure 8. They considered the case where
jV1 j ¼ jV2 j. The claimed advantages are that more precise
control of the waveform may be possible and, during the periods
of constant voltage, the solutions to the equation of motion can be
obtained analytically. Matrix methods (Pipes, 1953; Dawson,
1976, p. 83; Konenkov, Sudakov, & Douglas, 2002) can then be
used to calculate the stability diagram, which differs somewhat
from the more familiar diagrams of Figures 2 and 4. Provided the
waveform has a net DC component on average, parameters a and
q, analogous to the Mathieu parameters, can be calculated and
mass analysis is possible by placing an ion at the tip of a stability
region. Richards, Huey, and Hiller showed that if the ratio t/T is
set to 0.390, the stability region shrinks to a small region on the q
axis and so mass analysis is possible with no DC component to
the waveform. They demonstrated a resolution of up to 300 at
m/z ¼ 84 (Krþ). When operated with the waveform of Figure 8a,
Mass Spectrometry Reviews DOI 10.1002/mas
a quadrupole will have constant resolution over its mass scan.
As discussed above quadrupoles are more often operated at
constant peak width (often unit resolution). Richards (1977)
show that with a rectangular waveform the resolution can be
adjusted during a scan by changing the duty cycle t/T.
Any periodic waveform can be used to operate a quadrupole.
Richards, Huey, and Hiller (1973) also discussed the use of a
trapezoidal waveform. More complex waveforms can be used,
provided they are periodic. The stability diagrams for a variety of
rectangular waveforms were described by Konenkov, Sudakov,
and Douglas (2002). Sheretov (2000) derived properties of the
FIGURE 8. A rectangular waveform for use with a quadrupole mass
filter. In general, V1 6¼ V2 and the ratio t/T can be adjusted.
947
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DOUGLAS
ion motion for a rectangular waveform and a harmonic waveform
of the form f0(t) ¼ U VRF(cos Ot þ k cos jOt), where j is an
integer (usually 2 or 3). Sheretov et al. (2000) calculated stability
diagrams for 3D traps and quadrupole mass filters operated with
these waveforms, and described experimental investigations of
the use of such waveforms with a quadrupole mass filter and
three-dimensional ion traps. A quadrupole mass filter was
operated with a rectangular waveform in two of the higher
stability regions. Resolution matched that expected from the
width of the stability regions and scan lines used. Sheretov and
co-workers (Sheretov, 2002; Sheretov et al., 2002) later showed
methods to calculate the properties of ion motion for any periodic
waveform, including various rectangular and harmonic waveforms. In any real system, the rectangular waveform will have
finite rise times and possibly ringing. Sudakov and Nikolaev
(2002) showed that these distortions of the waveform make
only minor changes to the stability diagram.
stability region along lines with bu values determined from
Equation (42). Islands of stability are formed between the bands
of instability.
Figure 9 shows, for example, the stability islands and bands
of instability near the tip of the first stability region formed
with n ¼ 1/10 and q0 ¼ 0.0050. As the excitation amplitude q0
increases, the width of the lines of instability increases. Mass
analysis can be done with a scan line that passes through the tip
of a stability island. The excitation parameters and scan line must
be chosen so that the scan line does not pass through any other
G. Mass Analysis with Islands of Stability
Another method of mass analysis takes advantages of ‘‘islands’’
of stability formed by quadrupole excitation. An auxiliary
quadrupole excitation voltage with amplitude V0 and frequency
oex is added, so that the quadrupole potential becomes
2
x y2
ðU VRF cos Ot V 0 cosð!ex t gÞÞ
Fðx; y; tÞ ¼
r02
ð37Þ
where g is the phase of the excitation which can be taken as 0. The
excitation is applied at a frequency which is a rational fraction
of the main RF frequency, so that oex ¼ nO where n ¼ Q/P where
Q and P are integers. The equations of motion become
d2 x
þ ða 2q cos½2x 2q0 cos½2nxÞx ¼ 0
dx2
ð38Þ
d2 y
ða 2q cos½2x 2q0 cos½2nxÞy ¼ 0
dx2
ð39Þ
where x, a, and q are as in Equation (22) and
q0 ¼
4ezV 0
V0
¼q
2 2
VRF
mO r0
ð40Þ
The angular frequencies of oscillation in the quadrupole field are
given by Equation (24). With quadrupole excitation, resonances
are excited when
!ex ¼ jl þ u j
O
K
ð41Þ
where u is x or y, K ¼ 1, 2, 3,. . . and l ¼ 0, 1, 2, 3,. . .
(Collings & Douglas, 2000; Sudakov et al., 2000). When n ¼ Q/P,
the b values of the quadrupole resonances are determined by
Q jl þ u j
¼
P
K
ð42Þ
The stability diagram forms P 1 relatively strong resonance
lines for the x and y motions. Bands of instability form in the
948
FIGURE 9. The first stability region with quadrupole excitation with
n ¼ 1/10 and (a) q0 ¼ 0.0025, (b) q0 ¼ 0.0050, and (c) q0 ¼ 0.0100.
Reproduced from Konenkov et al. (2001) with permission. Copyright
Elsevier Science B. V.
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
islands. The upper or lower tip of an island can be used for mass
analysis (e.g., island A in Fig. 9c).
The use of auxiliary excitation to improve peak shapes was
first described by Devant et al. (1989). It was shown that addition
of an auxiliary excitation waveform with frequency approximately 101 of the main quadrupole frequency and amplitude
approximately 102 of the main quadrupole RF voltage could
remove tails from peaks. However, this was not described in
terms of bands of instability or islands of stability. The formation
and use of islands of stability to improve peak shape was
described by Miseki (1993) who again noted that islands can be
used to remove tails on the low and high mass sides of a peak, and
therefore improve the resolution. Konenkov et al. (2001) reported
extensive calculations of island positions and described experiments with an ICP-MS system to map the island positions. The
island positions and boundaries were calculated using a matrix
method. For any values of a and q two ion trajectories, calculated
over one period of the applied waveforms (PT where T ¼ 2p/O)
can be used to determine if the ion motion is stable or unstable
(Konenkov, Sudakov, & Douglas, 2002). Trajectories were
calculated for a large number of a and q values and used to
map the bands of instability and islands of stability. Measured
island positions agreed well with the calculated positions. While
the use of islands for mass analysis did not improve the limiting
resolution of the quadrupole, the abundance sensitivity was
improved from 106 –107 to 109 –1010. Baranov, Konenkov, and
Tanner (2001) described mass analysis with islands of stability
with an ICP-MS system. They showed that use of an island
prevented peak tails from forming when the ion energy in the
quadrupole was increased from approximately 5 to 13–15 eV.
The increased ion energy improved the sensitivity by approximately two times. Glebova and Konenkov (2002) modeled
peak shapes and resolution with operation at the tip of island
‘‘A’’ (notation of Fig. 9). For a quadrupole constructed with
round rods (r/r0 ¼ 1.130), operation with an island with
q0 ¼ 0.012 and n ¼ 9/10 removed tails on the low mass side of
a peak. This effect was not seen with an ideal quadrupole field,
and it was concluded that use of an island of stability can improve
peak shape for weakly distorted quadrupole fields.
Sheretov, Gurov, and Kolotilin (1999) considered the
general case of periodic variations of the quadrupole operating
parameters that produce parametric resonances and hence bands
of instability in the stability diagram. They pointed out that bands
of instability can be formed by modulating both the applied RF
and DC together, or by modulating just the DC or just the RF
amplitudes, and calculated that the number of RF cycles required
in the field for a given resolution is decreased when an island of
stability is used for mass analysis. Glebova and Konenkov (2002)
compared conventional mass analysis with an ideal quadrupole
field and mass analysis using an island of stability at a resolution
of approximately 100, and concluded there were no advantages to
using an island of stability. Later, however, Konenkov, Korolkov,
and Machmudov (2005) used matrix methods to calculate island
positions and boundaries, and ion trajectory calculations to
determine peak shapes and resolution with mass analysis in
islands. The effects of modulating the RF amplitude, and both
the RF and DC amplitudes, were compared. Ideal fields were
compared to results with a round rod set with ratio of rod diameter
to field radius r/r0 ¼ 1.13. With amplitude modulation the
Mass Spectrometry Reviews DOI 10.1002/mas
&
resolution can be adjusted in three ways: (1) by changing the
slope of the scan line at the upper or lower tip of an island, (2) by
setting the scan line to pass through the center of an island and
adjusting the modulation amplitude, or (3) by setting the scan
line to pass through the center of an island and adjusting the
modulation frequency. It was concluded that the preferred (and
simplest) method is to fix the modulation amplitude and
frequency and adjust the slope of the scan line. Modulation of
the RF gave similar results to modulation of the RF and DC.
Because modulation of the RF alone is simpler, it is preferred.
With an ideal quadrupole field, amplitude modulation of the
RF was found to remove peak tails at a resolution of 390 with
100 cycles of the field. This contrasts with the finding of Glebova
and Konenkov (2002) that for an ideal field there are no benefits
from using islands of stability, because the earlier study
considered only low resolution (100). Konenkov, Korolkov, and
Machmudov (2005) also found that use of islands removed peak
tails in a quadrupole constructed with round rods. Finally, it was
concluded that the number of RF cycles in the field required to
reach a resolution of 400 was decreased from approximately 150
without modulation, to 75 with modulation of the RF amplitude.
H. Mass Analysis with Radial Ejection
If a quadrupole is used as a linear ion trap, ions may be ejected in
order of their mass radially through a slot in a rod or rods to
produce a mass spectrum. This can be done by scanning the
trapping RF so that ions reach the stability boundary at a ¼ 0,
q ¼ 0.908, or by applying dipole excitation between one of the
rod pairs to excite ions for ejection (Schwartz, Senko, & Syka,
2002). This is not discussed in detail here because the method is
used exclusively with trapped ions (reviewed by Douglas, Frank,
& Mao, 2005).
I. Mass Analysis with Axial Ejection
A quadrupole mass filter may also be operated in RF-only mode
for mass analysis. Ions of a broad range of mass-to-charge ratios
are transmitted through the quadrupole. A stopping potential is
applied to an electrode or electrodes at the quadrupole exit. Ions
of a selected mass-to-charge ratio are excited in the radial
directions. In the fringing field at the exit of the quadrupole the
x and y motions are mixed with axial motion in the z direction.
Some of the increased energy in the radial direction is converted
to axial kinetic energy so that the excited ions overcome the
potential barrier and are transmitted to a detector. The original
motivation for this work was to overcome the defocusing fringing
fields at the entrance to a quadrupole mass filter operated
conventionally with applied RF/DC voltages, to increase the
acceptance over that of a conventional mass filter, and to
overcome deleterious effects of field imperfections.
Brinkmann (1972) first described this concept. A quadrupole was operated in RF-only mode with a potential barrier at the
exit. The quadrupole was scanned so that ions of increasing m/z
reached the stability limit at q ¼ 0.908, where they gained kinetic
energy in the radial directions. A resolution of more than 1,400
was obtained at m/z ¼ 500, and the transmission was approximately 10 times greater than that of the same quadrupole
949
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operated conventionally. This work was extended by Holme and
co-workers (Holme, 1976; Holme, Sayyid, & Leck, 1978) who
reported that there is less dependence of the performance on
mechanical imperfections, and less defocusing of ions at the
quadrupole entrance. Yang and Leck (1982) extended the mass
range to 600 and reported this mode of operation provided
uniform transmission over the mass range, although with low
resolution. They also reported a relative lack of susceptibility of
the performance of the RF-only mass spectrometer to contamination and construction errors. Yang and Leck (1984) showed
that the performance was only slightly affected by small spurious
potentials applied to the rods, and that therefore an RF-only mass
analyzer might be more suitable for use in a dirty vacuum system.
Ross and Leck (1983) modified the collector assembly and
reported the transmission depended less critically on operating
parameters in comparison to conventional operation of the
quadrupole, and that for a given transmission, the resolution
was higher than with conventional operation. Dawson (1985)
reported a detailed investigation of peak shapes, peak tailing, and
resolution. The quadrupole was operated at various frequencies
up to 3.0 MHz. The major finding was that reasonable peak
shape and resolution were possible provided the ions spent
sufficient time in the RF field. In all these studies there were
difficulties with a continuum background caused by relatively
high-velocity low q-value ions.
Hager (1999) reported operation of a quadrupole in RF-only
mode with a greatly reduced continuum background. Ions formed
by electrospray were collisionally cooled in a 20-cm long RFonly quadrupole operated at 7 mTorr, and then entered a 24-mm
long RF-only quadrupole for mass analysis. Because it is the
fringing fields that cause ions to gain energy in the axial direction
when they are excited at q ¼ 0.908, a very short quadrupole
(24 mm) could be used. A stopping potential, typically 7 Vabove
the rod offset of the quadrupole, was applied to an aperture lens
at the exit of the quadrupole. The RF of the quadrupole was
scanned to bring ions to the stability boundary at q ¼ 0.908. The
continuum background was reduced by applying unbalanced RF
to the rods and 3.0 Vof DC between the rods. It was found that
applying some of the quadrupole RF voltage to the exit lens
improved the transmission by nearly an order of magnitude. It
is possible that the improved performance in this work derives
partially from the collisional cooling of ions before they enter the
quadrupole. This provides an ion source with smaller spatial and
energy spreads than were used in prior studies (see below).
Hager (2002) later extended this work to a study with a
triple-quadrupole mass spectrometer system. Precursor ions,
mass selected in a first quadrupole (Q1), formed fragment ions in
a quadrupole collision cell (Q2). The fragment ions could be
trapped and accumulated in either Q2 (operated at 7 103 Torr)
or a downstream quadrupole (Q3) (3 105 Torr). For axial
ejection ions were excited by auxiliary dipole or quadrupole
fields. The trapping RF voltage was scanned to bring ions of
different m/z values into resonance with the excitation. Higher
scan speed was possible with trapping of ions at the lower
pressure in Q3. With ejection from Q2 a resolution of 6,000 at
m/z 609 was possible with very slow scans (5 Th/sec), and a
resolution of 1,000 was possible at 1,000 Th/sec. With ejection
from Q3, resolution of 6,000 was possible at a scan speed of
100 Th/sec. Higher scan speeds gave lower resolution. The major
950
advantage of this technique is that fragment ions can be
accumulated in Q2 or Q3 operated as a linear trap to dramatically
improve the sensitivity for tandem mass spectrometry. Hager
(2002) reported an improvement in sensitivity of 16 times for
MS/MS of reserpine ions in comparison to a conventional
fragment ion scan with Q3. Hopfgartner, Hussel, and Zell (2003)
reported a sensitivity improvement of 60 times for MS/MS
analysis of drug metabolites.
Londry and Hager (2003) have used analytical and
numerical calculations in a detailed investigation of forces on
ions in the fringing field of a quadrupole used for axial ejection.
The potential in this region consists of three parts: (1) the
quadrupole RF potential which decreases towards the exit lens
(the positive z direction), (2) the potential from the exit lens
which decreases in the quadrupole, and (3) the potential from an
auxiliary dipole excitation field. The decreasing quadrupole
potential causes a net force (averaged over one RF cycle) towards
the exit lens. This force increases in x and y as the square of the
distance from the quadrupole center. The stopping potential on
the exit lens produces a force in the opposite direction. At some
point these two forces balance each other. Ions on the exit lens
side of the balance point are ejected. Ions on the quadrupole side
of the balance point are reflected back into the quadrupole. A map
of the points where ions are reflected gives a cone shaped region
near the quadrupole exit. Excitation of ions moves ions from
regions near the quadrupole center, where they are reflected, to
regions closer to the quadrupole rods where they are ejected.
J. The Ratio r/r0
While hyperbolic electrodes can produce the best approximation
to a quadrupole field (aside from construction errors and the finite
truncation of the electrodes), round rods are often used because
they are easier to manufacture and mount to high precision.
If the quadrupole electrodes have fourfold rotational symmetry,
the symmetry of the potential requires that the amplitudes
of higher multipoles are 0 except A2, A6, A10, A14,. . . (Denison,
1971). Several authors have discussed the ratio of rod radius r to
field radius r0 that is optimum for a mass filter. With round rods,
additional multipoles are added to the potential (Eq. 6). Dayton,
Shoemaker, and Mozley (1954) showed that a ratio r/r0 ¼ 1.148
makes the amplitude of next highest multipole in the expansion,
A6, 0. Lee-Whiting and Yamazaki (1971) used conformal
mapping to calculate the potential from four parallel rods and
showed that A6 ¼ 0 when r/r0 ¼ 1.14511. Denison (1971)
calculated the field of a quadrupole in a cylindrical housing of
radius 3.44r0 and found a ratio r/r0 ¼ 1.1468 makes A6 ¼ 0.
Reuben et al. (1994, 1996) again used conformal mapping to
determine the potential of a quadrupole in a grounded case and
also found the ratio r/r0 ¼ 1.14511 makes A6 ¼ 0. This ratio was
found to be independent of the radius of the case. The same ratio
was calculated by Douglas et al. (1999).
In these studies it was assumed that the ratio r/r0 that makes
A6 ¼ 0 would produce the best mass filter. However, manufacturers did not necessarily use this ratio. It was shown in 1999
that a ratio that balances the effects of A6 and A10 to decrease
the effects of nonlinear resonances is preferred (Schulte,
Shevchenko, & Radchik, 1999). A ratio r/r0 ¼ 1.10 was proposed
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
to minimize losses from nonlinear resonances. Gibson and
Taylor (2000) used trajectory calculations to simulate peak
shapes for quadrupoles with hyperbolic and round rods with a
ratio r/r0 ¼ 1.148 and found that the round rods produced tails on
the peaks, especially on the low mass side. Gibson and Taylor
(2001) subsequently investigated the effects of the ratio r/r0 on
the performance of a quadrupole and proposed that a ratio 1.12r0
to 1.13r0 would give the highest resolution and transmission.
Douglas and Konenkov (2002) used simulations to show that a
ratio r/r0 ¼ 1.128–1.130 gives peak shapes and transmission
similar to those of an ideal quadrupole field. When r/r0 ¼ 1.130,
A6 ¼ 1.00 103 and A10 ¼ 2.44 103. The A6 and A10 terms
have similar magnitudes but opposite signs and compensate for
each other. The A6 term shifts the pass band to low mass. The A10
term shifts the pass band to high mass. When both are present
with the correct amplitude, the pass band corresponds to that of an
ideal field. Thus, a ratio r/r0 that makes A6 ¼ 0 does not produce
the best mass filter. A ratio r/r0 & 1.128 is preferred.
K. Ion Guides
1. Collisional Cooling and Focusing
As described above, to obtain high transmission with a quadrupole mass filter, it is desirable to match the emittance of the ion
source to the acceptance of the quadrupole. As the resolution of
the quadrupole increases, the acceptance decreases (as 1/R) for a
quadrupole operated in the first stability region (Dawson, 1980,
1990). As mass spectrometry has pushed to higher mass for
applications in life sciences, the demands for higher resolution
and higher sensitivity have increased. The shape of the emittance
from the source can be changed with an ion lens (Regenstreif,
1967; Dawson, 1980). For example, a nearly parallel beam with a
spread of x positions but a small spread of radial velocities can be
focused so it has a smaller spread of positions but a greater spread
of radial velocities. However, there is a limit to using this to match
a source to a mass analyzer. Liouville’s theorem (Joseph
Liouville, 1809–1882) states that the area of the beam in phase
space before and after an ion lens remains constant (Regenstreif,
1967; Dawson, 1980). Radial velocities can be traded for large
radial positions, or vice versa, but an ion lens or a quadrupole ion
guide cannot reduce the area in phase space occupied by the
beam.
The use of RF ion guides, and particularly quadrupole ion
guides, has provided a method to overcome this limitation. If a
quadrupole is operated in RF-only mode, that is, with only RF
between the rods, the Mathieu parameter a will be 0, ions of a
broad range of m/z values will lie on the a ¼ 0 axis and have stable
trajectories in the quadrupole. The quadrupole can be used as an
‘‘ion guide’’ to transport ions from a source to an analyzer.
Initially, it was believed that the pressure in the ion guide should
be as low as possible to prevent losses of ions by scattering
(see the review by Thomson, 1998). However, Douglas and
French (1992) reported that the ion transmission through an ion
guide and into a quadrupole mass filter increased as the pressure
was increased from 5 104 to 5 103 Torr (see also Douglas
& French, 1990). Collisions between ions and the gas in the ion
guide cause the ions to move to the center of the ion guide and
Mass Spectrometry Reviews DOI 10.1002/mas
&
also to lose radial energy. Ions are ‘‘focused’’ to the center of the
ion guide with a small spread of positions and small radial
energies. Thus, the emittance of ions at the exit of the ion guide is
less than at the entrance. At the same time the kinetic energy
distributions of ions at the exit of the ion guide narrow to
distributions with energies and energy spreads of approximately
1 eV per charge or less. The beam quality at the exit of the higher
pressure ion guide is greatly improved for transmission into a
downstream mass analyzer, particularly a quadrupole. Collisional focusing requires that the ions be injected into the ion
guide with relatively low energies, ca. 10 eV/charge. At higher
energies collisions cause ions to scatter or fragment. Subsequently, Xu et al. (1993) reported measurements of the
transmission of Oþ
2 ions through a hexapole ion guide with
collisions with He. They noted the ions at the exit of the ion guide
had energies and energy spreads of approximately 0.8 eV.
Calculations of ion trajectories showed the ions moved to a
region of approximately 0.5 mm in diameter at the center of the
guide and therefore also showed collisional focusing. Collisional
cooling of the ion beam in these experiments apparently
overcomes Liouville’s theorem because the region occupied by
the ions in phase space shrinks. However, Liouville’s theorem is
not violated because the phase space of the gas molecules is not
considered.
The use of collisional cooling or collisional focusing in
ion guides was soon extended to ESI orthogonal injection
TOF systems (Krutchinsky et al., 1998a) and vacuum MALDI
orthogonal injection TOF systems (Krutchinsky et al., 1998b).
With MALDI a nearly continuous beam of ions is obtained at the
exit of the ion guide, and the properties of the ions are largely
independent of the source parameters. The improvements for
orthogonal injection TOF, higher sensitivity and higher resolution, have been described by Standing (2000). Ion guides with
collisional cooling have since found widespread application in
many areas such as ESI-FTICR (Jebanathirajah et al., 2005), ICPMS (Tanner, Baranov, & Bandura, 2002), atmospheric pressure
MALDI (Schneider, Lock, & Covey, 2005), and nuclear physics
(see, for example, Nieminen et al., 2001; Kellerbauer et al.,
2002). A complete review is beyond the scope of this article.
Collisional cooling and focusing have also found applications in tandem mass spectrometry. In triple-quadrupole mass
spectrometers, mass selected precursor ions are fragmented by
energetic collisions with gas in a quadrupole ion guide. If the ion
guide is operated near single collision conditions the kinetic
energies of fragment ions Ef are related approximately to the
kinetic energies of the precursor ions Ep by
mf
Ep
ð43Þ
Ef mp
Thus, the fragment ions have mass-dependent kinetic energies.
To obtain high mass resolution in the downstream quadrupole
mass analyzer Q3, it is desirable that the fragment ions have low
kinetic energies within Q3 (Section IIIA). Ions can be slowed by
increasing the rod offset potential of Q3. However, this requires
knowing the kinetic energies of the fragments. The problem is
that Equation (43) is only approximate. If the rod offset of Q3 is
too low, the resolution is degraded; if too high, the ions are not
transmitted. If the kinetic energies of the ions are measured, then
the rod offset of Q3 can be set correctly to provide high
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transmission and resolution (Shushan et al., 1983) but this is not
practical. In many applications, to avoid any loss of signal the rod
offset of Q3 was commonly set equal to the rod offset of Q2. In
this case fragment ions had relatively high kinetic energies in Q3
and the mass resolution was degraded. A solution to this problem
was found when it was realized that collisional cooling could
be applied to the fragment ions (Thomson et al., 1995). If the
collision cell pressure is increased from ca. 5 104 to ca.
5 103 Torr (Ar), all fragments at the cell exit have kinetic
energies of 1 eV or less. Thus improved mass resolution is
possible in Q3 which can be operated with a fixed rod offset
voltage, slightly below that of Q2. At the same time, because
collisional cooling causes all ions to move to the center of Q2
with a reduced emittance, the transmission through Q3 increases
greatly. The improvements to MS/MS spectra are dramatic and
have been described by Thomson et al. (1995) and in more detail
by Morris, Thibault, and Boyd (1994). The benefits of increased
sensitivity and mass resolution are also seen where Q3 is replaced
by a TOF system (Standing, 2000).
2. Ion Guides with Axial Fields
The use of collisional focusing in Q2 of triple quadrupoles
provided an ion beam of greatly improved quality for Q3 but
had one drawback. With pressures in the collision cell of
5 103 Torr or greater, the transit time of the ions was increased
and this caused problems with fast scans in some modes of
operation. Suppose Q3 is set to a fixed product ion mass, and Q1
switches rapidly between two precursor masses, ions of mass m1
which produce a fragment at the mass setting of Q3, and ions of
mass m2 which do not. When Q1 switches from m1 to m2, some of
the fragments formed from m1 are still draining from the collision
cell and appear to have been formed from m2. Thus, ‘‘crosstalk’’
problems were encountered with early versions of the highpressure collision cell. A simple solution was to apply a voltage
pulse to empty the collision cell between measurements.
However, this did not address the slow transit of ions that also
caused problems in precursor and neutral loss scan modes. A
more complete solution was to apply a small axial field of the
order of 1 V/m along the axis of the cell to cause the ions to drift
more rapidly to the exit (Thomson, Jolliffe & Javahery, 1996;
Thomson, 1998). Initially, this was done with a quadrupole with
rods made of many segments. Each segment was given a different
DC bias to provide a net axial field along the quadrupole. Many
other methods of providing an axial field are possible (Thomson,
Jolliffe, & Javahery, 1996; Thomson & Jolliffe, 1998). A system
with conical shaped rods, which is both mechanically and
electrically simple, has been described (Thomson, 1998; Loboda
et al., 2000). The conical rods are arranged so that a first opposing
pair of tapered rods have larger diameters at the entrance, and
a second pair of rods have larger diameters at the exit. A DC
potential difference is applied between the rod pairs. At any point
along the quadrupole, the axis potential is determined by the
larger rods. If a positive DC potential is applied to the first pair,
and a negative DC potential to the second pair, the quadrupole
axis potential will be positive at the entrance, zero at the center,
and negative at the exit, so that an axial field is produced.
Properties of a collision cell with an axial field produced this
way were investigated in detail by Mansoori et al. (1998) who
952
concluded that crosstalk problems were eliminated and at the
same time, the benefits of collisional cooling were retained. An
even simpler version that used cylindrical rods that are tilted with
respect to the axis was eventually implemented in commercial
systems. The DC applied between the rods limits the transmission of Q2 to a band pass. For triple-quadrupole instruments
where Q3 scans this is not a problem because the pass band can be
scanned along with Q3 to transmit the ions of interest. For
instruments where Q3 is replaced by a TOF system, the pass band
becomes more of a problem. Loboda et al. (2000) described a
method of adding an axial field by placing additional electrodes
between the rods of Q2. The same potential is applied to all
electrodes. The field on axis is determined by the spacing of
the electrodes from the quadrupole center which varies along
the length of the quadrupole. The pass band with this method can
be up to 10 times that of the quadrupole with tapered rods. The
use of quadrupole ion guides with axial fields has been extended
to other applications such as the measurement of collision
cross sections (Javahery & Thomson, 1997; Guo et al., 2005),
the fragmentation of ions in an ESI-TOF system (Dodonov
et al., 1997), improved scan speed and controlled ion molecule
chemistry with an ICP-MS system (Bandura, Baranov, &Tanner,
2002), and an ion-molecule reactor with heating of ions (Guo,
Siu, & Baranov, 2005). Most importantly, however, the use of
collisional cooling to couple high-pressure sources to a quadrupole mass analyzer, and to improve the emittance of an ion beam
from a collision cell, has contributed to the development of bench
top mass spectrometers with sensitivity and resolution exceeding
those of earlier much larger systems.
IV. LINEAR QUADRUPOLES WITH ADDED
MULTIPOLE FIELDS
Most quadrupole mass filters are constructed with high precision
to make the closest approximation to the desired field. It has long
been argued that small construction errors can degrade the
performance of a mass filter, although the effects of various errors
have not been well described in the literature. Austin, Holme, and
Leck (1976) concluded that to obtain a resolution of 900 required
mechanical errors in general of less than 103. Titov (1995)
calculated that a mechanical error of 10 mm in the rod diameters
reduced the acceptance and hence the transmission by a factor of
nearly 10. Taylor and Gibson (2008) used trajectory calculations
of peak shapes to show that if the y rod of a quadrupole (the rod
with the negative DC with positive ions) is moved in by 0.005r0
(20 mm for a quadrupole with r0 ¼ 4.0 mm), the transmission
decreases by ca. 2 times and structure is formed on a peak
(nominal resolution 300–400). Dawson (1980) calculated the
limiting resolution with various mechanical distortions, and
stated ‘‘quadrupole rods are said to be manufactured to tolerances
in radius of 105 cm and to parallelism of 104 cm.’’ As
discussed, changing the ratio r/r0 of a quadrupole with round
rods from 1.12 to 1.13 significantly changes the peak shape and
transmission. This corresponds to a change in rod radius of 40 mm
for 4.0 mm radius rod.
The field of a quadrupole with distorted electrode geometries contains additional multipoles. Generally these have
amplitudes AN which are 103 or less of A2. For example, when an
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
&
x rod is moved in 0.005r0, the highest multipole amplitudes are
A1 ¼ þ0.0042, A3 ¼ þ0.0011, and A6 ¼ þ0.0011. (A dipole term
is introduced because the field center moves from the center of the
quadrupole.) Recently, Douglas and co-workers have described
quadrupoles that have much greater amplitudes of added
multipoles; added octopole fields with A4 ¼ 0.02–0.04 (Sudakov
& Douglas, 2003), and added hexapole fields with A3 ¼ 0.040–
0.012, Konenkov et al., 2006). These fields were added to
improve the fragmentation efficiencies of ions for MS/MS when
the quadrupoles are used as linear ion traps (Douglas, Frank, &
Mao, 2005). In view of the literature describing the effects of
small field distortions on mass filter performance, it would not be
expected that these rod sets would be capable of mass analysis at
all. Yet in some cases the transmission and resolution with mass
analysis are surprisingly high. As well, the new methods of
scanning quadrupoles using islands of stability or axial ejection
have been found to at least partially overcome the otherwise
deleterious effects of the much larger multipole amplitudes of
these rod sets. These results are reviewed here.
A. Linear Quadrupoles with Added Octopole Fields
A linear quadrupole with an added octopole field has a potential
of the form:
2
4
x y2
x 6x2 y2 y4
þ A4
Fðx; y; tÞ ¼ A2
r02
r04
ð44Þ
ðU VRF cos OtÞ
where A2 & 1.0 and A4 is the amplitude of the added octopole
field. Sudakov and Douglas (2003) showed that an octopole field
can be added to a linear quadrupole constructed with round rods
by placing the two y rods closer to the central axis than the x rods.
However, this method adds significant amplitudes of other
multipole terms. A preferred method is to make the radius of the y
rods (Ry) greater than the radius of the x rods (Rx). This adds an
octopole term and only low levels of other higher multipoles. An
axis potential is formed, but this can be removed by unbalancing
the RF and DC applied to the rods. The changes in rod radius are
much greater than those caused by manufacturing tolerances. For
example, to make A4 ¼ 0.026 requires Ry/Rx ¼1.130, a difference
of rod diameters of approximately 1.0 mm for a quadrupole with
Rx ¼ 4.0 mm. Figure 10 shows a photograph of a rod set with a
2.0% added octopole field (Rx/Ry ¼ 1.220). A detailed description of trapped ion motion and fragmentation efficiencies of these
rod sets used as linear ion traps has been given by Michaud et al.
(2005). Given the discussion above, it would not be expected that
such rod sets would be capable of conventional mass analysis.
Ding, Konenkov, and Douglas (2003) investigated mass
analysis with quadrupoles with added octopole fields of 2.0–
4.0% (i.e., A4/A2 ¼ 0.020–0.040). Conventional mass analysis
was done with RF and DC voltages applied to the rods. With the
positive DC applied to the y rods (the rods of greater diameter)
so that for positive ions the Mathieu parameter ax < 0, only very
broad peaks with low transmission were seen. The added
octopole field prevents mass analysis. However, when the
positive DC was applied to the x rods, so that ax > 0, the
resolution improved dramatically. Figure 11 shows peak shapes
obtained with a quadrupole with a 2.6% added octopole field
Mass Spectrometry Reviews DOI 10.1002/mas
FIGURE 10. A linear quadrupole with 2.0% added octopole field. The
radius of the smaller rods is 4.50 mm and the radius of the larger rods is
5.49 mm. Reproduced from Ding, Konenkov, and Douglas (2003) with
permission. Copyright John Wiley and Sons, Ltd. [Color figure can be
viewed in the online issue, which is available at www.interscience.
wiley.com.]
(Ry/Rx ¼ 1.300). With the positive DC applied to the larger rods,
Figure 11a, a broad peak with nominal resolution of approximately 150 is produced. With the positive DC applied to the
smaller rods, Figure 11b, the isotopic peaks are resolved and
R1/2 & 5,890. Measurements of transmission versus resolution
showed that at R1/2 & 1,000, the transmission of the quadrupole
with 2.0% octopole field equaled or exceeded that of a
conventional quadrupole. At higher resolution the transmission
of the quadrupole with the octopole field dropped more rapidly
with increasing resolution, so that at R1/2 ¼ 3,000 it was
approximately 10 times less than a conventional quadrupole.
With distorted fields, split peaks might be expected, caused by
nonlinear resonances (Dawson & Whetton, 1969; Dawson, 1976,
1980). The octopole field causes a nonlinear resonance when
bx þ by ¼ 1. This line runs through the tip of the stability region.
However, split peaks are not observed (Fig. 11b) because the
residence time of the ions in the quadrupole is too short to allow
nonlinear resonances to build up. Ding, Konenkov, and Douglas
(2003) used trajectory calculations to determine the stability
boundaries for a quadrupole with a 2.0% added octopole field.
With positive ions, when the positive DC is applied to the x rods
(ax > 0), the boundaries are very similar to those of a pure
quadrupole field, and the tip of the stability diagram remains
sharp and well defined. When the positive DC is applied to the y
rods (ax < 0), the boundaries move out and become diffuse. This
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DOUGLAS
FIGURE 11. Peak shapes with mass analysis with a quadrupole with a
2.6% added octopole field with the positive DC applied to (a) the larger
and (b) the smaller rods. Reproduced from Ding, Konenkov, and Douglas
(2003) with permission. Copyright John Wiley and Sons, Ltd.
is consistent with the experimental observations, but does not
explain why the boundaries become diffuse or sharp. Dawson
(1980) estimated that, for a limiting resolution of 2,000, A4
must be less than 3.5 104. The results of Ding, Konenkov,
and Douglas show that mass analysis is possible with values of
A4 nearly two orders of magnitude greater.
The use of islands of stability with quadrupoles with added
octopole fields has also been studied for the cases ax > 0 and
954
ax < 0. Konenkov et al. (2007) used trajectory calculations to
determine island positions and peak shapes for quadrupoles that
have added octopole fields of 2.0–4.0%. Rod sets that have only
an added octopole field, and round rod sets with added octopole
fields and multipoles up to N ¼ 10 were modeled. Operation in
the upper stability island at the tip with the greatest jaj gave
resolution peak shape and transmission comparable to conventional operation with a > 0. In this case there is no advantage
to using the island of stability. With the polarity of the applied
DC reversed so a < 0, only low resolution and transmission
are possible with conventional analysis. However, with a < 0 and
use of the uppermost island operated at the tip with the lesser jaj,
the simulations showed that peak shapes and resolution similar to
those with conventional operation with a > 0 are possible. These
simulations were confirmed in experiments by Moradian and
Douglas (2007). An island of stability was formed by using
q0 ¼0.020 and n ¼ 9/10. These values are not necessarily optimum
but were chosen to match the calculations of Konenkov et al.
(2007). With a quadrupole with 2.0% added octopole field,
measured positions of the island tips agreed well with the
calculations of Konenkov et al. With ax < 0 at least unit resolution
was obtained at m/z ¼ 609, sufficient to separate the isotopic
peaks of ions of reserpine. Without the use of the island, under the
same conditions, the peak was ca. 6 Th wide, and isotopic peaks
could not be resolved. In this case use of the island overcomes the
adverse effects of the added octopole field.
Mass analysis with mass selective axial ejection is also
possible with quadrupoles with added octopole fields and was
investigated by Moradian and Douglas (2008). Axial ejection at
q ¼ 0.908 with a quadrupole with 2.6% added octopole field gave
only broad peaks with low resolution, both with ions flowing
continuously through the quadrupole and with trapped ions. The
low resolution was attributed to changes in the x and y stability
boundaries. When a ¼ 0, the x stability boundary is at q ¼ 0.908,
but the y boundary moves out to q & 0.931. It was proposed that
between these q values, ions excited in x lead to excitation in y
where the motion is stable, possibly delaying ion ejection.
Axial ejection of trapped ions was investigated with a
quadrupole with 2.0% added octopole field, with ions excited by
dipole excitation in the x or y directions, quadrupole excitation, or
simultaneous dipole excitation in the x and y directions. With a
linear quadrupole with added multipole fields, the resolution and
scan speed possible depend on the scan direction, as with 3D traps
(Makarov, 1996). Best resolution and sensitivity were obtained
with forward scans from low to high mass, dipole excitation
applied to the smaller rods, and ion excitation and ejection at high
q (0.80). Ejection efficiencies were similar to those obtained with
a conventional quadrupole and decreased from approximately
60% at a scan speed of 15 Th/sec to approximately 1% at
4,000 Th/sec.
B. Linear Quadrupoles with Added Hexapole Fields
A linear quadrupole may also be constructed with an added
hexapole field with the potential given by
2
3
x y2
x 3xy2
þ
A
Fðx; y; tÞ ¼ A2
3
r02
r03
ð45Þ
ðU VRF cos OtÞ
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
x y
r02
2
þ A3
3
x 3xy
r03
2
0.8
0.6
0.4
0.2
0.0
606
608
610
612
614
Apparent Mass (m/z)
¼ c
ð46Þ
where c is a constant. With a quadrupole with a 2% added
hexapole field, and no other higher multipoles, a resolution of ca.
1,130 can be obtained with positive ions. The peak shape and
transmission are similar to those with a pure quadrupole field,
provided the positive DC is applied to the x rods so the Mathieu
parameter a > 0. If the polarity of the DC is reversed so a < 0,
only very low resolution and poor peak shape are obtained.
Increasing the amplitude of the hexapole field causes the peak to
broaden because of changes to the stability diagram, although
resolution of several hundreds could be obtained with quadrupoles with 8% and 12% added hexapole fields.
Mass analysis with quadrupoles with added hexapole fields
of 2–12%, constructed with round rods, was then modeled.
Multipoles up to N ¼ 10 were included. Poor resolution and
peaks with considerable structure were obtained, showing that
the higher multipoles strongly affected the transmission and
resolution. It was shown that the octopole term that is added is
largely responsible for the poor performance. The octopole term
can be removed by making both x rods greater in diameter than
the y rods so that A4 ¼ 0. Modeling the transmission and
resolution with these rod sets showed greatly improved transmission and resolution. Resolution of R1/2 & 700–800 was
calculated for quadrupoles with 8–12% added hexapole field,
and resolution of 1,400 for a quadrupole with 6% added
hexapole. At low and high resolution, peaks were free of
structure. However, at intermediate resolution peaks showed
some structure.
The stability diagram was calculated for a quadrupole with
2% hexapole and no other multipoles, and round-rod quadrupoles
with 2–12% added hexapole fields. With a > 0, the boundaries
remained sharp. Increasing A3 causes the x stability boundary
to move out towards larger q values. With a < 0, only a diffuse
region of stability with low transmission was seen, consistent
with the low resolution and transmission seen in simulations of
mass analysis when a < 0.
Preliminary experimental tests of mass analysis with
quadrupoles with added hexapole fields have appeared (Zhao
et al., 2007; Xiao, Zhao, & Douglas, 2008). Conventional mass
analysis, mass analysis with islands of stability, and mass
analysis with axial ejection have been tested. Figure 12 shows
Mass Spectrometry Reviews DOI 10.1002/mas
b
1.0
Relative Transmission
A2
2
1.0
0.8
0.6
0.4
0.2
0.0
606
608
610
612
614
Apparent Mass (m/z)
2
c
7.5x10
Relative Transmission
a
Relative Transmission
where A3 is the amplitude of the hexapole field. The properties of
quadrupoles with added hexapole fields have been modeled in
detail by Konenkov et al. (2006). It was shown that a hexapole
field can be added to a quadrupole constructed with round rods by
rotating the two y rods through an angle y towards an x rod. With
round rods, other multipoles are added to the potential, especially
an octopole term. Geometries that produce nominal hexapole
fields of 4%, 8%, and 12% were described. Rotating the y rods
towards an x rod moves the field center closer to the x rod.
Mass analysis was modeled by running many trajectories
with a thermal distribution of speeds (300 K) and a Gaussian
distribution of positions in x and y with a standard deviation
s ¼ 0.002r0. Quadrupoles with added hexapole fields and no
other multipoles were modeled first. These can be constructed
with electrodes that have the shapes given by
&
6.0x10
2
2
4.5x10
2
3.0x10
2
1.5x10
0.0
613
614
615
616
617
618
619
620
621
Apparent Mass (m/z)
FIGURE 12. Mass analysis with a quadrupole with a 4.0% added
hexapole field and unequal diameter rods R y ¼ r 0 ¼ 4.500 mm,
Rx ¼ 5.279 mm. a: Conventional mass analysis with the positive DC
applied to the x rods, R1/2 ¼ 2,070, relative sensitivity 9 104 ions/sec.
b: Mass analysis with an island of stability with n ¼ 1/20 and q0 ¼ 0.002,
R1/2 ¼ 2,170, relative sensitivity 2.8 105 ions/sec. c: Mass selective
axial ejection with dipole excitation between the x rods at q ¼ 0.72,
R1/2 ¼ 2,250, ejection efficiency 2%.
955
&
DOUGLAS
peak shapes obtained with a rod set with 4% added hexapole field
and rod diameters chosen to make A4 ¼ 0. With conventional
mass analysis, Figure 12a, the resolution is R1/2 & 2,000, but
there are tails on the peaks. With an island of stability and at
similar resolution, Figure 12b, the tails of the peaks are removed.
The transmission is approximately five times lower than that of a
conventional quadrupole operated at the same resolution. With
axial ejection of trapped ions, Figure 12c, peak shapes and
resolution comparable to those obtained with axial ejection from
a conventional round-rod quadrupole are possible.
These results show that mass analysis with rod sets with
large geometric and field distortions is sometimes possible. The
use of islands of stability can in some cases overcome the effects
of both small and large field distortions. It appears that our
understanding of the effects of field distortions is incomplete.
Dawson (1976) concluded that ‘‘There remains, therefore, a
great deal of scope for further advance in design, performance
and applications, building upon the established technological
foundation . . ..’’ This statement, subsequently proven correct,
would seem to be equally valid in 2009.
ACKNOWLEDGMENTS
This work was supported by the Natural Sciences and Engineering Research Council of Canada and MDS-Analytical Technologies through an Industrial Research Chair.
It follows that the components of the electric field are given by
~
f ðx; y; zÞ ¼ Q ~
Eðx; y; zÞ
ðA:1Þ
where Q is the charge of the ion and ~
Eðx; y; zÞ is the electric field
at that point. The electric field at a point is the force per unit
charge. Force and electric field are vectors; they have both
magnitude and direction. In a Cartesian co-ordinate system they
have components along the x, y, and z axes, fx, fy, fz, and Ex, Ey, and
Ez, respectively.
If a charged particle is moved from a position s1 to a position
s2 through an electric field, the mechanical work done on the
particle is
Z s2
~
w¼
f ~
dl
ðA:2Þ
s1
The work done per unit charge is the change in electrical potential
DF(x, y, z) between the points s1 and s2 so that
Z s2
~
DF ¼ E ~
dl
ðA:3Þ
s1
956
@Fðx; y; zÞ
@x
ðA:4Þ
Ey ¼ @Fðx; y; zÞ
@y
ðA:5Þ
Ez ¼ @Fðx; y; zÞ
@z
ðA:6Þ
Equations (4)–(6) can be written as
~
E ¼ rFðx; y; zÞ
ðA:7Þ
where the gradient operator, r, is given by
r¼
@^ @^ @^
iþ jþ k
@x
@y
@z
ðA:8Þ
where ^i, ^j, and ^k are unit vectors in the x, y, and z directions,
respectively. In free space where there are no charges, such as
the region between the electrodes of a quadrupole mass filter, ~
E
must satisfy
r~
E¼0
ðA:9Þ
@Ex @Ey @Ez
þ
þ
¼0
@x
@y
@z
ðA:10Þ
or equivalently
V. APPENDIX: ELECTRIC POTENTIALS AND
FIELDS: UNITS
This appendix gives a brief review of definitions of electrical
potentials and fields, as well as the units used in this article. More
detail can be found in text books on electromagnetic theory
(Smythe, 1939; Kip, 1969). In linear quadrupoles, as well as
many other devices for manipulating ions, electric potentials are
applied to electrodes to produce forces on ions. The force ~
f on an
ion at a point in space with Cartesian co-ordinates (x, y, z) is given
by
Ex ¼ It follows from Equations (A.10) and (A.7) that:
r2 Fðx; y; zÞ ¼ 0
ðA:11Þ
or equivalently
@ 2 Fðx; y; zÞ @ 2 Fðx; y; zÞ @ 2 Fðx; y; zÞ
þ
þ
¼0
@x2
@y2
@z2
ðA:12Þ
Equation (A.12) is known as Laplace’s equation (Pierre Simon
Laplace, 1749–1824).
A. Units
This article uses SI units. Mass is in kilograms, distance in
meters, time in seconds, force in newtons. The unit of electrical
potential is the volt (the practical unit). Because electric field is
force per unit charge, it has units of newton per coulomb.
However, because electric field is also derivative of potential with
position, it also has units of volts per meter. These units are
summarized in Table 1.
In these units, if a charge of 1 C is moved through a potential
difference of 1 V, the change in potential energy of the charge is
1 J. If a charge of 1 C is at a point where the field is 1 V/m, the
force on the charge is 1 N. If a charge on an electron or proton
of 1 1019 C is moved through a potential difference of 1 V,
the change in potential energy is QDV ¼ 1.6 1019 J. This unit
of energy is called the electron volt.
Mass Spectrometry Reviews DOI 10.1002/mas
LINEAR QUADRUPOLES IN MS
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