Frequency width

Figure B2.5.13. Schematic representation of the four different mechanisms of multiphoton excitation (i) direct, (ii) Goeppert-Mayer (iii) quasi-resonant stepwise and (iv) incoherent stepwise. Full lines (right) represent the coupling path between the energy levels and broken arrows the photon energies with angular frequency to (Aco is the frequency width of the excitation light in the case of incoherent excitation), see also [111].

A homonuclear spin-system may be excited with radiofrequency (r.f.) pulses that are so Intense (in the order of p.s), compared to the frequency width of the spectrum, that all resonances are excited essentially uniformly. This is a nonselective excitation. A homonuclear spin-system may also be excited with a relatively weak, r.f. pulse (in the order of ms), in the sense that all components of a given multiplet are inverted at time zero, whereas the other resonances in the spectrum remain essentially unperturbed this is a selective excitation. The r.f. pulse may be single-selective, that is, there is an inversion of one multiplet in the spectrum, or double-selective, triple-selective, and so on, where two, three, or more separate multiplets in the spectrum are inverted simultaneously while the remaining resonances remain unperturbed. 

A light pulse of a center frequency Q impinges on an interface. Raman-active modes of nuclear motion are coherently excited via impulsive stimulated Raman scattering, when the time width of the pulse is shorter than the period of the vibration. The ultrashort light pulse has a finite frequency width related to the Fourier transformation of the time width, according to the energy-time uncertainty relation. 

When the full width at half maximum (fwhm) of a Gaussian pulse is 20 fs, its frequency width is 740 cm as the fwhm. Frequency components Ql and fis are present in the pulse and are used to generate the vibrational coherence, where Ql — iis is equal to the vibration frequency ox... 

Modified CNTs feature various spectral changes depending on the methods and the location of modifications. These changes include variations in band frequencies, width, and intensities. For example, aryldiazonium salts [139] were used to modify individual sodium dodecyl sulphate (SDS) coated SWNTs with aryl group. The Raman spectrum of functionalized (SDS-free) SWNTs shows a disorder mode much higher than pristine SWNT the radial breathing modes are nearly unobservable. 

The new vibrational features that were found in the hydrogen passivated n-type layers are shown in Figs. 9 and 10. The frequencies, widths, and relative strengths are given in Table II. There are two near-lying bands above 1500 cm 1 and a third band at 809 cm 1 for each of the three... 

The ultimate (minimum) linewidth of an optical band is due to the natural or lifetime broadening. This broadening arises from the Heisenberg s uncertainty principle, AvAt < U2jt, Av being the full frequency width at half maximum of the transition and the time available to measure the frequency of the transition (basically, the life-... 

In liquid solution the orientation of the ions is rapidly changing relative to the magnetic field. If the reorientation frequency is greater than the frequency width of the powder absorption line, the spectrum collapses to narrow lines and fits the isotropic spin Hamiltonian... 

In most optical excitations the resolution is determined by the Doppler effect or the finite linewidth of the light source. The Doppler effect gives a typical frequency width of 1 GHz, and the width of the light source can be anywhere from 1 kHz to 30 GHz. We assume that these widths are larger than the radiative width. The photoionization cross sections from the ground states of H, alkali, and the alkaline earth atoms are given in Table 3.3. 20... 

An easy way to decide whether this is the case is to compare the bandwidth of the laser with the frequency width of the absorption spectrum of the molecule, where the latter is determined by (E, n deg i ,) 2. Typically, for the case of direct dissociation, the absorption spectrum extends over a few thousands of reciprocal centimeters (cm-1). In order to have a bandwidth broader than this, s(r) must, by Eq. (1.35), be as short as 1 to 5 femtoseconds (fs). Since most pulses used in real photodissociation experiments are much longer, Eq. (2.83) is not a valid description of many photodissociation experiments. 

Experimentally attaining control via Eq. (3.12) requires a light source containing N frequencies to , (i = l,..., N). Both pulsed excitation with a source whose frequency width encompasses these frequencies, as well as excitation with N continuous wave (cw) lasers of frequencies to = coEi, (i = 1.N) are possible... 

Obviously, the details in the time-profile, 7, and the frequency spectrum, Fp, of the incident X-pulse, depend on the experimental setup. However, if the duration of the pulse is either sufficiently short or sufficiently long compared to the time scale of the nuclear dynamics, 7 may be replaced by either a delta function or a constant on the nuclear time scale. Likewise, if the width of Fp can be neglected (known as the static approximation ), we can obtain simplified expressions for the differential scattering signal. However, as pointed out earlier, the frequency widths of X-ray pulses obtained from, e.g., synchrotron radiation are typically on the order of percent of the carrier frequency. Hence, in order to simulate the finer details of the experimental signal, the actual frequency distribution of the incident X-ray pulse must be taken into account [29],... 

Figure 5.1. A stacked plot of the 2D COSY spectrum of thebaine in CDCl at 400 MHz. The aliphatic region only is shown. The original spectrum had a frequency width of 2100 Hz in f/ and f and, after zero-filling once in f , a final matrix size 1024 x 512 which gave a resolution of 4.1 Hz per point in each direction. A sinebell window function was applied and the spectrum has been symmetrized. 128 scans...

Spectrum of light with long persistent time shows narrow frequency width. 

There are then no possibilities for the occurrence of irreversible radiationless decays in such small-molecule limit triatomics. However, interesting effects, arising from the coherent superposition of many levels, may still appear when (2.14) is violated. The presence of hyperfine structure makes this possibility very likely. For instance, Demtroder has observed nonexponential decays of excited states of NO2 in a molecular beam where the spacing between hyperfine levels is claimed to be sufficient to excite a single hyperfine component with his MHz bandwidth laser. Demtroder then has no recourse but to explain the nonexponential decays in terms of some elusive radiationless decay despite the fact that the conditions (2.2) for the small-molecule limit are obeyed and prohibit irreversible decays. It should, however, be recalled that when traveling along with the molecule in the molecular beam, the molecule encounters a pulse of radiation whose duration is given by the laser spatial extent divided by the molecular velocity. For a laser spot size of 10" cm and a molecular velocity of 10 cms -the pulse duration is 10 s. This yields an effective pulse frequency width of 10 MHz which could yield a coherent superposition of a number of hyperfine levels. The nonexponential decay of such a superposition is discussed in Section II. C. 

In Figure 5c, we observe the same time-domain cosine wave as in Figure 5a, but for only a finite period, T sec. The result is that the frequency spectrum is now broadened from an infinitely sharp spike to a signal whose frequency width is of the order of (1/T) Hz. This result is an example of a classical "uncertainty principle" the product of the time-domain width (T) and the frequency-domain width (1/T) is constant. In other words, the only way to determine the frequency of a time-domain signal with perfect accuracy (i.e., infinite frequency "resolution") is to observe it for an infinite length of time. 

The limitation on high resolution spectroscopy until the advent of the laser was provided by the Doppler widths of the lines. The frequency width Av at half maximum intensity (FWHM) is given by the formula... 

The FEL beam has spatial and time structures that reflect the electron beam structure. The radiation pulse length is proportional to the electron bunch length I g the radiation frequency width is given by Su) = 2 tic/Individual laser modes have line widths which are approximately equal to the inverse of the correlation time. The practical limit to the correlation time, and therefore to the widths of individual lines in the spectrum, is set by mirror microphonics. The laser power Pl =Gjnax RF where Prp is the power given to the... 

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