Cyclotron orbit

The energy imparted to the ions depends on the energy of the rf pulse and the duration of the pulse. The energy does not have to be raised in one event but may be absorbed by the ion in small portions. A technique called sustained off-resonance excitation (SORT) (82) uses a low-amplitude rf pulse that is off-resonance to the ion cyclotron frequency. The difference of the cyclotron frequency and the excitation frequency (-500 Hz) causes the ion to experience in- and out-of-phase excitation that has the effect of a repeated expansion and shrinkage of the cyclotron orbit. In this process, the ion undergoes a large number of low-energy collisions and the Ecom slowly increases until the ion dissociates. 

It is clear that it is only possible to observe these resonances if their recurrence times are shorter than the coherence time of the exciting laser. With this point in mind, Main et al.23 made a five fold improvement in their spectral resolution and were able to see new resonances in the Fourier transform spectrum with longer recurrence times, as shown in Fig. 9.8. Tc is the recurrence time for a cyclotron orbit. The spectrum vs tuning energy is not shown since it is composed of... 

The numerical results are consistent with the result of an experiment in which the ion cyclotron orbit sizes of a methane (CHi, ) and benzene (C6H6 ) mixture of ions were varied. In the control experiment, the two ions were excited by low amplitude consecutive RF burst pulses of varied time. The signal ratio was essentially constant over the range of orbits for which signals were detectable. In contrast, for a chirp from 10 kHz to 2 Mhz at 2.094 kHz/usee of varied amplitude, the abundance ratio of CH t/C6H6t decreased from about 90 to about 10 as the orbit size was increased, indicating loss of the lighter ion. 

We have developed a model to explain the time dependent change in sensitivity for ions during excitation and detection. The first step is to describe the image charge displacement amplitude, S(Ap, Az), as a function of cyclotron and z-mode amplitudes. The displacement amplitude was derived using an approximate description of the antenna fields in a cubic cell. The second step in developing the model is to derive a relationship to describe the cyclotron orbit as a function of time for an rf burst. An energy conservation... 

An example of the complexity of the frequency variations in the cubic cell is given in the extreme by the splitting of a peak into a doublet for ions excited to very large orbits. A similar phenomenon was noted by Marshall (47). In the example reported here, the ions were excited by a 0.385 volt peak-to-base RF burst in a 0.0254 m cubic cell with a 1 volt trap and a magnetic field of 1.2 T. Two maxima are discernible for ion-cyclotron-orbit sizes larger than the noptimaln orbit size at about 760 psec excitation time. Local centroids are measurable one increases by ca. 50 Hz and the other... 

Ions are formed by an electron beam offset at 1/2 r because ions cannot be excited at the center of the cell. This is one also to allow the largest possible cyclotron orbit between the z-axis and the ring electrode. The cyclotron mode of the ions is excited across the ring and end caps, and image currents are detected by using a balanced bridge circuit (5.2). 

For low ion populations, a first estimate of achievable ejection resolution might be obtained from the cyclotron frequency spread that occurs over the range of cyclotron orbit radii through which the ion must pass to be ejected. This is based on the notion that an ejection waveform that is just adequate to eject one ion must have a frequency spectral peak that is at least as wide as the above spread of frequencies. Such a waveform would then excite, at least to some extent, all ions with frequencies falling within the width of the peak, thus limiting the ejection resolution. For ions with low z-mode amplitudes, we can use Dunbar s (46) approximate expression for the average radial field strength,... 

Once the ion population is formed and stable and the neutral population is again low, the ions to be activated are accelerated to a cyclotron orbit slightly larger than the disk. Up to this point, the supporting screen and disk have been at the same trapping... 

F2 F4 - 2F and F3 F4 - Fi, correspond to the forbidden orbits. From a quantum-mechanical point of view there is no semiclassical closed orbit to explain these frequencies. However, they can be understood in the frame of the quantum interference (QI) model [10] as two-arms Stark interferometers [11]. Within the QI model [10] the temperature damping of the oscillation amplitude is given by the energy derivative of the phase difference ((pi -cpj ) between two different routes i and j of a two-arms interferometer. This model states that 5(cpi - cpj) / de = ( /eB) <3Sk / de, where Sk is the reciprocal space area bounded between two arms. Since 3(difference between the effective masses of the two arms of the interferometer, the associated effective mass is given by m = mj - mj, where nij and mj are the partial effective masses of the routes i and j. In our case an interferometer connected with the frequency F3 consists of two routes, abcdaf and abef and another interferometer, connected with the frequency F2, includes two cyclotron orbits, abcdaf and abebef (see Fig 5). 

According to Falicov and Stahowiak [12] the contribution of every segment of the cyclotron orbit to the cyclotron mass parameter is proportional to the subtended angle of this segment, and the total cyclotron mass parameter equals the sum of the partial cyclotron mass parameters. We have estimated the effective mass parameters from the temperature dependence of the SdH oscillation amplitude using the standard formula... 

Thus by measuring the periods of oscillation AH for various applied field directions A , one can map the Fermi surface. The dHvA and SdH oscillations can be seen if the magnetic fields are large enough, so the circumference of the cyclotron orbit is smaller than the mean free path. Therefore dHvA and SdH experiments are usually performed at national or international high-field facilities. 

Fig. 1. An illustration of the use of a rotating magnetic field to induce coherent (and therefore detectable) ion cyclotron orbital motion in an ion packet within an ICR mass spectrometer [12]...

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... 

The Dingle reduction factor, Rd, describes the broadening of the otherwise sharp Landau levels due to scattering of the conduction electrons. The usual parameter which describes this scattering is the relaxation time r averaged over one cyclotron orbit [252]. This effect leads to a reduction factor similar to (3.8) for finite temperatures. As a useful parameter the so-called Dingle temperature... 

The origin for this apparent field dependence of rric is not fully understood. However, the nearly perfect two-dimensionality of -(ET)2l3 discussed in Sect. 2.3.3 seems to be the principal reason for the observed strange temperature and field dependence of the magnetic quantum oscillations. It was suggested that in this extremely 2D system quasiparticles with fractional statistics [365] may occur if the cyclotron orbits lie within an individual 2D conducting plane [363]. Since these quasiparticles do not obey Fermi statistics they should not contribute to the quantum oscillations observed. Hence, the effective cyclotron mass determined by the 3D Lifshitz-Kosevich formula could be underestimated. Further experimental verification for this suggestion is lacking. 

The cyclotron orbits of thermal energy ions when they first enter the ICR cell are both too small and incoherent to be detected. However, if an excitation pulse is applied at the cyclotron frequency, the resonant ions will absorb energy and be brought into phase with the excitation pulse. They will have a larger orbital radius and the ion packets will orbit coherently. The ions may then be detected as an image current induced in the receiver plates. Additionally, this excitation pulse increases the kinetic energy of the trapped ions to the extent that fragmentation can be collisionally induced by ion—molecule reactions. Alternatively, the excitation pulse may be used to increase the cyclotron radius so that ions are ejected from the ICR cell. 

The FT-ICR mass analyzer determines the m/z values of ions based on monitoring of the cyclotron frequency of the ions in a uniform magnetic field. After the charged ions pass from the ionization source into the magnetic field, all ions in the analyzer cell are excited into a higher cyclotron orbit by a pulsed radio frequency (rf) (Fig. 9). The image currents (the flow of electrons in the external... 

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.

FT-ICR mass spectrometry offers ultrahigh resolution. This feature is a result of the large numbers of cyclotron orbits during detection and the fact that cyclotron frequency is independent of ion velocity. Performance is not limited by the initial position, direction or speed of the ions, unlike mass spectrometers such as time-of-flight or sector instruments. 

When some of the cyclotron orbits are not closed but form open orbits, the magnetoresistance increases quadratically and depends on the current direction as... 

To proceed further with the solution of the linearized Boltzmann equation one assumes the magnetic field to be sufficiently weak so that the electrons undergo collisions long before they can complete a cyclotron orbit. Then one can regard the solution for the case B = 0 as slightly perturbed by the addition of the magnetic field and hence write down a power-series expansion of (P — M) in the form... 

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