Research: Science & Education
Quadrupole Mass Spectrometers: An Intuitive Look at the Math James J. Leary* and Rebecca L. Schmidt Department of Chemistry, James Madison University, Harrisonburg, VA 22807 The quadrupole mass filter is the instrument that chemists use most frequently to obtain a mass spectrum. Unfortunately, some aspects of the theory associated with the operation of quadrupoles can be difficult to understand. Miller and Denton (1) examined the operation of quadrupole mass filters by deriving the equations that describe their behavior and by intuitively examining the physics of their operation. The publication of their work in this Journal helped to move the discussion of the workings of quadrupoles out of monographs on mass spectrometry and into instrumental analysis textbooks, but there is more that can be done. By examining the equations that describe the ion trajectories within the quadrupole as certain limiting conditions are approached, simplified forms of these equations are generated that can lead to a better understanding of the workings of this instrument. Furthermore, such an applied mathematics review can help students to better appreciate the close relationship between instrumental analysis and their required calculus courses.
often the parameter of greatest interest in mass spectrometry, and it is for this reason that this ratio is being written explicitly. The derivation of these equations is presented in the Miller–Denton paper (1) and can also be found in most monographs on mass spectrometry (e.g. refs 2–5). Throughout the remainder of this development it will be assumed that all ions possess a single positive charge (e = 1.60 × 10{19 C). Although this assumption is not essential, it is made to facilitate the discussion. The equations describing the acceleration in the x
Limiting Cases Simplify Quadrupole Math The geometry required for a quadrupole mass filter is usually provided by four cylindrical rods, illustrated in Figure 1. This physical arrangement of the quadrupole rods is designed to provide the geometry that best approximates the mathematical model of a hyperbolic field. The hyperbolic approximation is important because it allows the trajectory of the ion to be treated via equations that are independent along each of the coordinate axes (i.e., no xy cross terms). An xyz reference coordinate system has been overlaid on Figure 1 such that the zaxis is parallel to the long axis of the rods. The “Laws of Motion” that describe the components of acceleration in the x, y, and z directions can be written as follows:
Figure 1. Conventional array of cylindrical rods in a quadrupole mass filter.
and y directions (eqs 1 and 2) are nearly identical, with the exception of a difference in sign. For this reason, only eq 1 will be examined. However, because of the similarities, the conclusions that are drawn about an ion’s trajectory in the x-direction can be extended to elucidate its trajectory in the y-direction. The third equation states that the acceleration of the ion in the z-direction is zero. This means that the z-component of the ion’s velocity (as it moves from the source toward the detector) is unaffected by the potentials on the rods.
d 2x = – x U + V cos (ωt) (m / e)r 02 dt 2
(1)
d2y y = U + V cos (ωt) 2 (m / e)r 02 dt
(2)
Consider how eq 1 changes if we specify that the ion enters the quadrupole from the source precisely at x = 0. Equation 1a is obtained
d 2z = 0 dt 2
(3)
d 2x = 0 dt 2
First Limiting Case ( x = 0)
The variables used to describe the ion’s acceleration are the mass of the ion (m), the charge on the ion (e), one half the distance between the rods (r0), the distance from the origin in either the x or y direction (x or y), the dc potential on the rods (U), the maximum ac potential on the rods (V), the frequency of the ac potential (ω), and the elapsed time (t). The mass-to-charge ratio (m/e) is
and shows that the ion will experience no acceleration due to the potential in the x-direction. However, the influence of the ion’s velocity in the x-direction should also be considered. This is accomplished by integrating eq 1a, which shows that the velocity of the ion is a constant (C).
*Corresponding author.
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(1a)
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d 2x = dx = C dt 2 dt
(4)
Research: Science & Education
If an ion enters at x = 0 and its x-velocity component (C) is not equal to zero, it will move away from x = 0 and it will experience an acceleration dependent on its mass and the potential on the x-rods; eq 1 applies again. If an ion enters at x = 0 and its x-velocity component (C) is equal to zero, its trajectory in the x-direction will be completely unaffected by the potential on the rods, regardless of its mass, and it will experience no acceleration in the x-direction. Although an ion is unaffected by potential in the x-direction, it could still experience an acceleration in the y-direction. If the ion enters the quadrupole at x = 0 and y = 0 and has no x or y component of velocity, it will, regardless of its mass, traverse the quadrupole unaffected by any rod potentials. Only a small number of ions will meet these conditions, and these contribute to the inherent noise in the quadrupole.
Second Limiting Case ( V = 0) This case examines how the equation describing the motion of an ion transforms if the ac potential is shut off. By setting V = 0, eq 1 simplifies to eq 1b.
d 2x = – Ux (m / e)r 02 dt 2
(1b)
This equation is similar to the one for a harmonic oscillator that is routinely encountered in physics and chemistry courses; it specifies that the second derivative of x with respect to time will be a constant, {U/(m/e)r02, multiplied by x. The solution to this differential equation must be a sine or cosine function depending upon the initial value of x. Thus, at a particular value of U, ions having a mass within a particular range will oscillate in the x-direction as they travel through the quadrupole. If the mass-to-charge ratio of an ion is small, its oscillations as it moves from the source toward the detector will be large, and it will collide with one of the rods (x = r0) and be filtered out. The value of U could be varied to filter out ions of different mass-to-charge ratios; this topic will be reexamined in the sections on problems that deal with how a quadrupole can be used to generate a mass spectrum. In addition, reference 1 provides an easyto-follow nonmathematical examination of the physics associated with this process.
Third Limiting Case ( U = 0) When U is set equal to zero, eq 1 is transformed into eq 1c:
d 2x = – x V cos (ωt) (m / e)r 02 dt 2
(1c)
The goal at this point is to examine how the trajectory of an ion is influenced by the frequency of the ac potential. To accomplish this equation 1c will be examined at the limits ω → 0 and ω → ∞. When ω → 0, cos (ωt) → 1, and eq 1c simplifies to eq 1d, which is very similar to eq 1b (see section on second limiting case):
d 2x = – xV (m / e)r 02 dt 2
(1d)
In the “Problems” section we will prove the relationship between V and U that normally exists in quadrupole instruments used to produce mass spectra. When the equation is examined at the other limit, ω → ∞, the inertia of the ion prevents it from keeping
up with the changing ac potential. This causes the ion to experience only the average ac potential, which appears to the ion to be zero (Vapparent = 0). Substituting V = 0 into eq 1c provides an alternative route to eq 1a, and the discussion of the first limiting case again applies. In reality, a good compromise is obtained if the frequency of the quadrupole is in the radio frequency range (i.e., around 1 MHz). Obtaining a Mass Spectrum with a Quadrupole The three limiting cases examined how the ac potential and the dc potential can effect an ion’s acceleration. Furthermore, we have tried to illustrate that these potentials interact with the ion’s mass, charge, position, and velocity to determine whether or not it will successfully traverse the quadrupole. In order to examine the means by which a quadrupole can be made to scan through a mass range (produce a mass spectrum), a new set of variables (a, q, ξ) is defined based upon eqs 5–7. These definitions can be substituted into either eq 1 or 2 (see refs 1, 2, or 4 for details), thereby producing eq 8, which is a general equation written in terms of the defined variables and the general variable u, which can represent either x or y.
a = ax = – ay =
4U ω 2r02(m / e)
(5)
q = qx = – q y =
2V ω 2r02(m / e)
(6)
ξ = tω 2
(7)
d 2u = ± a + 2q cos (2ξ) u 2 dξ
(8)
Equation 8 is in the standard form of Mathieu’s differential equation, and the shaded area in Figure 2 represents its allowed solutions. Ion sources of mass spectrometers are usually designed and tuned to maximize the probability of ions possessing a single charge. Consider any value of q that falls within the shaded area of Figure 2 (e.g., q = 0.6). There will be many a-values that could produce allowed solutions to eq 8. This in turn means that ions of various masses could successfully traverse the quadrupole even if U, ω, r0, and e are constant (see eq 5). Such a result is unsatisfactory because a unique ion mass would not be available. However, if we choose to operate the quadrupole at the singular point describing the apex of the shaded area (q = 0.706, a = 0.237), the situation changes significantly. In order to elucidate some very fundamental properties of quadrupoles, three quantitative problems will be considered. Problems
Problem 1 What is the ratio of the ac potential to the dc potential at the apex in Figure 2? The ratio of the ac potential to the dc potential is determined by substituting the apex q-value into the definition of q and the apex a-value into the definition of a, and dividing the second equation into the first:
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2.98 = V 2U Thus, the ac-to-dc potential ratio (V/U) must equal 5.96. This ratio can be used to calculate either the ac potential or the dc potential provided the other potential is given. For instance, if the dc potential is 200 V, the acpotential would need to be 1.19 × 103 V.
Problem 2 Assuming that the quadrupole is operated at the apex in Figure 2, that it has been designed such that ro is 1.00 cm, that it is operated at a frequency of 1.00 MHz, and that at a particular instant the dc potential is 200 V, what is the mass of the singly charged positive ion that would be transmitted by the quadrupole? In problem 1 the relationship between the ac and dc potentials was proven to be fixed at the apex, and thus the definition of either a or q could be used to solve this problem. Because information about the dc potential is given, the answer will be extracted by working with eq 5, which is rewritten and solved for the mass of the ion (m). Remember that in this calculation the frequency must be converted to angular frequency and that volts times coulombs equals joules (kg m2/s2). 4(1.60 × 10 C) (200V) m = 4eU = ω 2r 02a (2π × 1.00 × 10 6 s –1) 2(1.00 × 10 –2m) 20.237 –19
= 1.37 × 10
–25
kg
By converting from kilograms to grams and multiplying by Avogadro’s number, the mass of the only singly charged ions that could traverse the quadrupole under these conditions is found to correspond to 82.4 g/mol.
Problem 3 What is the mass of the ion that will successfully traverse the quadrupole if all of the conditions specified in problem 2 still apply, except that the dc potential is doubled (U = 400 V)? The process outlined in problem 2 can be used to determine the mass of the ions that can successfully traverse the quadrupole. At the apex the numerical value of a is fixed and likewise the values of e, ω, and r0 are fixed; thus if U doubles, the mass of the ion (m) must also double. This means that only ions at 165 g/mole
could reach the detector. By examining the three problems and their solutions, the means of obtaining a mass spectrum is easily understood. Problem 1 was used to establish that the relationship between U and V is a constant when the quadrupole is operated at the apex. Problem 2 demonstrated that either q or a could be used to determine the mass of the singly charged ions that could reach the detector. Problem 3 was used to elucidate the direct proportionality between either the ac or the dc potential and the mass of the ion. Thus, by increasing the dc potential linearly with time and simultaneously adjusting the ac potential to keep the condition V = 5.96 U satisfied, the quadrupole is made to scan through a known mass range. The plot of detector response vs. the mass-tocharge ratio is the mass spectrum. When designing a quadrupole, consideration must be given to the ion throughput, which is directly related to the intensity of the signal at the detector. Problems 1, 2, and 3 utilized the assumption that the quadrupole is operated exactly at the apex in Figure 2. However, if this were the case the range of ions with this exact massto-charge ratio would approach zero, and the number of ions traversing the quadrupole would be infinitesimal; the sensitivity of the instrument would be seriously impaired. To circumvent this problem, instruments are designed to operate at a point just below the apex. Unfortunately, an increase in ion throughput necessitates that ions with a greater range of mass-to-charge ratios will be able to pass through the quadrupole. A compromise must be made between good sensitivity and high resolution in order to obtain acceptable mass spectra. Usually, this compromise involves working sufficiently close to the apex that singly charged ions differing in mass by an amount proportional to 1 g/mol can be separated throughout the scan range of the instrument. Conclusion Quadrupole-based instruments currently dominate routine mass spectral applications. Furthermore, combinations of quadrupoles (e.g., triple quadrupoles or quadrupoles as part of other MS-MS configurations) are among the most sophisticated instruments for structure elucidation. For these reasons, it is imperative that greater emphasis be devoted to the theory associated with the design and operation of quadrupole-based instruments in instrumental analysis courses. Acknowledgments The authors gratefully acknowledge H. K. Arthur, who participated in a very preliminary examination of this material, and T. N. Gallaher (JMU: Chemistry Deptartment) and W. A. Mattson (Randolph-Macon Women’s College: Chemistry Deptartment), who provided many helpful suggestions during the preparation of the manuscript. Literature Cited 1. Miller, P. E.; Denton, M. B. J. Chem. Educ. 1986, 63, 617. 2. White, F. A.; Wood, G. M. Mass Spectrometry: Applications in Science and Engineering; Wiley- Interscience: New York, 1986. 3. Duckworth, H. E.; Barber, R. C.; Venkatasubramanian, V. S. Mass Spectroscopy, 2nd ed.; Cambridge: New York, 1986. 4. March, R. E.; Hughes, R. J. Quadrupole Storage Mass Spectrometry; WileyInterscience: New York, 1989. 5. Constantin, E.; Schnell, A. Mass Spectrometry; Ellis Horwood: New York, 1990.
Figure 2. Mathieu stability diagram for a quadrupole mass filter.
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