A quadrupole analyzer consists of four parallel hyperbolic or cylindrical electrodes. The opposite electrodes 2r0 apart are interconnected and subjected to the same potential (Figure 1). The adjacent electrodes are applied to potentials of values Ф0 = 2(U – Vcosωt), sum of a direct voltage U and a high frequency alternating current V (ω). A quadrupole electrostatic field is thus created in the region between the four electrodes. By entering the quadrupole, the ions maintain their longitudinal velocity and then progress in the analyzer, and the potential difference between the input and the output of the quadrupole guides the ions in the desired direction. The motion of an ion in the x and y direction is defined by the solution of Mathieu’s equation. Thus, for fixed U/V, only ions of a certain m/z will have a stable trajectories to the detector and will be detected. Figure 1: Image of a quadrupole consisting of four bars carried at continuous and alternative current.

Ion trajectory equation in a quadrupole

Let’s take a closer look at the trajectories of ions in the quadripole. The trajectory of ions in the quadrupole is described by Mathieu’s equation. The two opposite bars are equidistant from 2r0. A positive ion of a certain mass to charge ratio (m/z) enters the quadrupole with a certain kinetic energy in the z direction, it will be attracted by the bar carried a negative potential but before touching the bar, this one could change of sign in time and send the ion to the adjacent bar which in turn carries a negative potential. The change is set so that the amplitude of the movement of the ions along the x or y axis should not exceed r0, if the change is not properly adjusted, the ion would have touched the bar and be lost. Thus the values ​​U, V and ω determine the stability of an ion that enters the quadrupole. This operating principle is described by Paul and Steinwedel in 1953 . The trajectory of an ion is described by the following equations. x and y are the coordinates of an ion along the x and y axis at time t.

These equations are simply obtained by Newton’s second law F = m.a (F: the force, a: acceleration, m: the mass) and the force F is also equal to the multiplied charge with the electrostatic field exerted.

Using the solution of Mathieu’s equation, we can represent the stability zones in the x direction, and y (the z direction is independent of the quadrupole field) as a function of U and V (Figure 2). Figure 2: (a) and (b) stability zone for an ion of m/z given in the x direction, and y, (c) 4 stability zones in the x and y plane, (d) zoom of the A zone stability.

The stability diagram (Figure 2c) shows four zones (A, B, C and D) where the ions are stable in the xy plane. By applying the U and V values found in these areas, the x and y position of the ions at any time is less than r0, so the ion can cross the quadrupole without touching the bars. For practical reasons, we operate in zone A where V and U have low and positive values.

Equations 1a and 1b show that the position of the next x and y ion also depends on m/z, the different m/z ions have their different stability diagrams. Figure 3 shows a superposition of stability diagrams (only the A area of each diagram is presented) of ions with different m/z. Figure 3: (a) and (b) Stability diagrams of ions with different m/z.

Figure 3a shows the superposition of the ion stability diagrams, if the value of U and V are increased gradually, all the ions will cross the quadrupole one by one. The quadrupole is also used as an ion guide, in this case the value of U is zero, and one operates only according to the value of V. Theoretically if V ≈ 0V, all ions can pass, but at V ≈ 0V , the pseudo-potential of all the ions is very weak, the transmission is bad. If V = V1 the ions of low m/z easily pass the quadrupole, on the other hand high m/z ions passes hardly because their pseudo potentials are weak. At V = V3 the ions of low m/z as m/z1 et m/z2 can not cross, at this voltage (V3), we see that the pseudo-potential of the ion m/z4 is the maximum, consequently the transmission of this ion is the best, and the ions whose m/z is close to m/z4 are also optimized. In practice, if one wants to optimize a given mass range, the value of V should corresponds to the value where the transmission is the best for an ion m/z which is in the middle of this mass range. This behavior is very useful in the case of hybrid devices where the quadrupole is used as an ion guide to focus a mass range to be transmitted in other analyzers such as ion trap or FT-ICR. By focusing a mass range, it avoids the overload of the analyzers and possibly the space charge. The quadrupole has a very good focusing power, but its transmission remains low, to transmit a wide range of mass, we can vary the value of V over time, but this practice is not effective because the instance t where the value of V is optimized for ions of low m/z, high m/z ions are poorly focused and vice versa. For better transmission, the hexapole or octopole is used as an ion guide, whose transmission of these multipoles is better, however, their focusing power is lower.

The resolution

The quadrupole is a low resolution analyzer, the resolution can be improuved by gradually increasing the value of U and V as a straight line that just passes the top of the stability diagram of each ion (Figure 3b), such as the d1 line. On the other hand, the transmission of the ions is less good, for a better transmission, one can lower the line such as the case d3, but the resolution become bad, at this resolution we can see that the ions m/z4 and m/z5 are overlapped. For optimal use and depending on the sample, it is up to the user to balance between resolution and transmission.

The quadrupole can be coupled with any mass analyzer, and often it is necessary because in general it is necessary to transmit the ions of a high pressure region to another region of low pressure, for the analyzers using an external ionization source. In addition, the focusing power of the quadrupole serves to eliminate ions that are not interesting, and avoids overloading, as well as charge spacing. As it can be coupled with any analyzer, it can very well couple with one or more quadrupoles, this is the case of triple quadrupole mass spectrometers (Figure 4). Figure 4: Diagram of a triple quadrupole analyzer, Q1 and Q3 are quadrupoles to scan ions, and q2 is a collision cell.

It should be noted that the third quadrupole can be replaced by another analyzer such as QTOF or FT-ICR at Bruker.

The triple quadrupole allows to work in several modes:

Fragment ions mode

Figure 5) This mode is used to determine the fragment ions. This mode is usually used, to do this, the Q1 is set to values of U and V so that a single ion population of a certain m/z can cross the quadrupole (Q1), the ions are transferred in the collision cell (q2) to be fragmented. Then the quadrupole (Q3) scans the fragment ions by gradually increasing the values of U and V. Figure 5: “Fragment ions” mode, U and V are fixed in the Q1 so that a certain ions at m1/z can pass, then they will be cleavaged in the collision cell. The Q3 scans fragment ions by gradually increasing U and V.

Precursor ions mode

This mode allows to find all the precursor ions which are capable of giving the fragment ion of a certain m/z (FIG. 6), to do this, the voltages (U and V) of the third quadrupole are fixed in such a way that a single population of ions of a certain m/z can pass. While the first quadrupole works in scan mode, all ions can pass one by one. The set of three quadrupoles works this way:

First, the voltages of the first quadrupole gradually increase, all the ions cross the first quadrupole one by one and they will be fragmented by collision in the collision cell (q2). On the other hand, all the ions are stopped before the third quadrupole (Q3), except for the ion of m/z predefined by the voltages of the Q3 which can pass through this later and arrive until detector. The operation of this mode is quite disturbing, because the ions that actually arrive at the detector are the son ions of m/z given, but the mass spectrum displays the parent ions. Indeed, it is the computer system that calculates the moment when the detector receives the fragment ion and voltages of the first quadrupole to display the mass of ions corresponding to the voltages of the latter. In general, we can consider that Q3 is a part of the detector. Figure 6: the operation of the triple quadrupole in “precursor ions” mode Uf and Vf are voltages allowing a single ion population that can pass the Q3.

Neutral loss mode

This mode allows to find all the ions capable of losing a neutral of a given mass (FIG. 7). First, the Q1 scans the ions by gradually increasing the voltages, allowing all the ions to reach the collision cell and be fragmented. At the same time the Q3 scans the ions but with a voltage delay compared to Q1, this delay corresponds to the mass of the neutral. For example, the ion m crosses the Q1 and breaks up in the collision cell, it loses a neutral of mass n, if n corresponds to the defined mass, the fragment ion resulting from the ion m crosses Q3 and will be detected the ion is then displayed on the mass spectrum, otherwise no ion arrives at the detector the ion a will not be in the mass spectrum. In this mode of operation, it can also be considered that the Q3 is a part of the detector, but a dynamic detector. Figure 7: The operation of the triple quadrupole spectrometer in “neutral loss” mode

 Paul W. et Steinwedel H.S., Z. Naturforsch., 8a, 448 (1953)