Equations cyclotron motion

Fourier transform mass spectrometry is made possible by the measurement of an AC current produced from the movement of ions within a magnetic field under ultra-high vacuum, commonly referred to as ion cyclotron motion.21 Ion motion, or the frequency of each ion, is recorded to the precision of one thousandth of a Hertz and may last for several seconds, depending on the vacuum conditions. Waveform motion recorded by the mass analyzer is subjected to a Fourier transform to extract ion frequencies that yield the corresponding mass to charge ratios. To a first approximation, motion of a single ion in a magnetic field can be defined by the equation... 

Equations (A9) show that the electron exhibits the expected cyclotron motion in the presence of the magnetic field. However, collisions must also be taken into account. Let N(t) be the number of particles that have not experienced a collision for time t (after some arbitrary beginning time, t = 0). Then it is reasonable to assume that the rate of decrease of N(t) will be given by dN oc —Ndt = —Ndt/1. The solution of this equation is N(t) = N0 exp (—t/ ), where N0 is the total number of particles. It can easily be shown that x is simply the mean time between collisions. The probability of having not experienced a collision in time t is, of course, N(t)/N0 = exp(—t/ ). The... 

ICR-MS [17] is based on the phenomenon of cyclotron motion of the ions in the analyzer cell. This motion is caused by a combination of magnetic (B-) and electric (E-) fields (trapping plates). The ions are forced to circulate as a result of the Lorentz force (equation 1 m, v, q are mass, velocity and charge of the ions). 

When a gaseous ion drifts into or is formed in a strong magnetic field, its motion becomes circular in a plane perpendicular to the direction of the field. The angular frequency of this motion is called the cyclotron frequency, Equation 20-8 can be rearranged and solved lor v/r. which i.s the cyclotron frequency in radians per second. 

Figure 1. The cyclotron resonance principle as applied to mass spectrometers. An alternating electric field whose frequency equals the cyclotron frequency (Equation 1) for a particular ion mass, excites the cyclotron motion of that ion. An oscillator is connected to the plates of a capacitor, whose dimensions define the sample volume, and gives rise to an alternating electric field within the capacitor. If the frequency of the oscillator equals the cyclotron frequency (Equation 1) of an ion located within the capacitor, the radius of the ion s cyclotron orbit will be increased (i.e., the ion cyclotron motion is excited). This phenomenon is called cyclotron resonance. The kinetic energy of the ion increases as the ion follows the spiral path shown, and the presence of cyclotron resonance is detected by measuring the signal that is induced in the plates of the capacitor by the excited ion motion.

In the Fourier transform ion cyclotron resonance (FT-ICR) spectrometer s,10"24 the cyclotron motion of ions of many different masses is excited essentially simultaneously (see Figure 2). The presence of excited cyclotron motion is then detected as the alternating voltage (Equation 2) after the exciting oscillator is turned off. Thus, unlike the conventional ICR spectrometers in which ion excitation and ion detection are simultaneous, ion excitation and ion detection in the FT-ICR spectrometer are temporally distinct. 

The cyclotron equation shows that the frequency at which an ion undergoes cyclotron motion is inversely proportional to its mass-to-charge ratio. Thus, when the cyclotron frequency is measured, m/z may be calculated. 

Residual magnetic fields with a component in the radial plane will couple the radial and axial motions of the ions. The effect of the cyclotron motion on the resonance frequencies co+ can be judged by considering the simpler single-ion situation. Here, the coupled equations of motion read... 

F = mv /r) and the magnetic force, we can obtain the angular velocity (oo ) of the cyclotron motion from Equation 3.30 ... 

The frequency of the cyclic motion of ions,m, within the cell is given by the cyclotron equation ... 

In addition to high mass resolution, another important feature of FTMS is its wide mass range (7,8). From examination of the cyclotron equation (Eq. 1), the mass range of FTMS appears to have no upper limit. However, an instrumental upper limit in excess of 100,000 has been suggested (16), based on a detailed study of ion motion (30). From a practical standpoint, the cyclotron... 

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