Cyclotron frequency, equation

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.

An interesting feature of this equation is that all ions of a certain m/z have the same cyclotron frequency, independent of their velocity. Hence, energy focusing is not essential for precise determination of m/z. 

Prom these two equations it can easily be seen that the electrons are moving on an orbit in k space which is given by a constant energy surface perpendicular to B. The angular frequency with which the so-called cyclotron orbit is traced is given by the cyclotron frequency Wc = eB/rric, where the cyclotron mass is defined by... 

From Equation (47) it can be seen that while the ion cyclotron frequency (cu) of an ion is a function of its mass, charge, and the magnetic field, it is independent of the ion s initial velocity. 

Equation 4 is the cyclotron equation, coc corresponding to the cyclotron frequency (ICR frequency). The ICR frequencies are between a few kHz and several MHz. Equation 4 indicates that all ions with the same ratio of ion charge to ion mass possess the same ion cyclotron frequency. In contrast to other mass analyzers (magnetic/electric sector field, time-of-flight (TOF), quadrupol), the ion velocity has no direct influence on the relation between the measured value and the ion mass (ICR frequencies toc in equation 4 sector field radius of deflection r for magnetic selection in equation 5 TOF flight time t). 

The formation of ion packages induces alternating currents called picture current. The frequencies of the alternating potentials correspond to the cyclotron frequencies of the ions in the ICR cell. Using equation 6 the corresponding ion masses can be calculated. 

According to Equation 1, an ensemble of ions of differing masses will have a spectrum of cyclotron frequencies which is characteristic of that ensemble. For a magnetic field strength of 1 tesla and a mass range of 15 amu to 1500 amu, the cyclotron frequency spectrun (Equation 1) extends from 10 kHz to 1 MHz and thus falls in the radiofrequency region of the electromagnetic spectrum. 

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. 

Rearranging Eqnation 5.6 gives a quadratic equation in o) that can be solved to give the reduced cyclotron frequency... 

The FT-ICR analyzer is the most complex and difficult to operate, but has by far the highest resolution, mass accuracy, and sensitivity. The operating principle is that ions in a magnetic field will orbit at a frequency that is related to the ion s mass m), charge z), and the strength of the magnetic field (B). This is called the cyclotron frequency fc). The relationship can be described by the following equation ... 

Another form of mass analyzer is Fourier Transform Ion Cyclotron Resonance (FTICR-MS) [6]. This separates ions according to their cyclotron frequency, /, in a fixed magnetic field, according to the following equation. 

This quadratic equation has two solutions, the first of them representing the reduced cyclotron frequency co+... 

Let the magnetic field be in the z-direction and the right-hand circularly polarized radiation be given by Ex = Eoe and Ey = —iEge K The solution to Equation 19.22 can be written as and Vy = —ivoe . Inserting these quantities into Equation 19.22 and identifying the cyclotron frequency as tOc = cBzjm, the Vq can be written as... 

The resonant or cyclotron frequency is detected by absorption of power from the micro-wave beam which becomes a maximum when effective electron mass can then be determined from m = eB z/ c- The effective mass of holes can be found in a similar manner using left-hand polarized microwave radiation. 

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... 

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

Accurate measurements of the electron affinity of simple Ge and Sn radicals have been obtained by threshold photodetachment experiments carried out in ion cyclotron resonance experiments183. In these experiments, measurement of the threshold frequency for removing the electron from the anion yields an upper limit for the electron affinity of the species, as shown for GeH3- in equation 28. 

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... 

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