Radiation frequency width

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... 

It would appear that measurement of the integrated absorption coefficient should furnish an ideal method of quantitative analysis. In practice, however, the absolute measurement of the absorption coefficients of atomic spectral lines is extremely difficult. The natural line width of an atomic spectral line is about 10 5 nm, but owing to the influence of Doppler and pressure effects, the line is broadened to about 0.002 nm at flame temperatures of2000-3000 K. To measure the absorption coefficient of a line thus broadened would require a spectrometer with a resolving power of 500000. This difficulty was overcome by Walsh,41 who used a source of sharp emission lines with a much smaller half width than the absorption line, and the radiation frequency of which is centred on the absorption frequency. In this way, the absorption coefficient at the centre of the line, Kmax, may be measured. If the profile of the absorption line is assumed to be due only to Doppler broadening, then there is a relationship between Kmax and N0. Thus the only requirement of the spectrometer is that it shall be capable of isolating the required resonance line from all other lines emitted by the source. 

Besides the uncertainty broadening just discussed, there are other causes of line broadening which make line widths generally considerably greater than the natural width (3.88). The Doppler effect causes an apparent change in radiation frequency for molecules with a component of velocity cobs in the direction of observation of the radiation. Different molecules have different values of cobs and we get a Doppler broadened line. 

The maximum d is reached in the region of the polaron resonance absorption with radiation frequency ft 4/0, // , where the width of the absorp-... 

The performance of the radiation detectors depends on their intrinsic properties, temperature and external conditions of use. They can be compared by using a factor of merit D, known as the detectivity, equal to the inverse of the NEP for a detector with unit area used with an electrical band-width A/ of 1 Hz and expressed in cm Hz1/2 W-1. When a value of D is indicated for a thermal detector, it is considered to be independent of the radiation frequency and the time modulation frequency is assumed to be adapted to the intrinsic time constant t, of the detector. For a photoconductive detector, D peaks at a radiation frequency very close to the band gap for an intrinsic detector or to the ionization energy of the relevant centre for an extrinsic detector and decreases steadily at lower energies. 

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],... 

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. 

The line width A/ is just the range of radiation frequencies that have a high probability of interacting with the molecule. The time At can be vmderstood as the spontaneous emission lifetime of the transition discussed in Section 3.5. The spontaneous decay has an exponential dependence, and therefore the line shape due to natural line broadening is Lorentzian. The natural line width. 

Molecules such as 3,4 and 5 in Figure 2.6, which have a zero velocity component away from the source, behave uniquely in that they absorb radiation of the same frequency Vj-es whether the radiation is travelling towards or away from R, and this may result in saturation (see Section 2.3.4). If saturation occurs for the set of molecules 3, 4 and 5 while the radiation is travelling towards R, no further absorption takes place as it travels back from R. The result is that a dip in the absorbance curve is observed at Vj-es, as indicated in Figure 2.5. This is known as a Lamb dip, an effect which was predicted by Lamb in 1964. The width of the dip is the natural line width, and observation of the dip results in much greater accuracy of measurement of v es. 

In practice the laser can operate only when n, in Equation (9.2), takes values such that the corresponding resonant frequency v lies within the line width of the transition between the two energy levels involved. If the active medium is a gas this line width may be the Doppler line width (see Section 2.3.2). Figure 9.3 shows a case where there are twelve axial modes within the Doppler profile. The number of modes in the actual laser beam depends on how much radiation is allowed to leak out of the cavity. In the example in Figure 9.3 the output level has been adjusted so that the so-called threshold condition allows six axial modes in the beam. The gain, or the degree of amplification, achieved in the laser is a measure of the intensity. 

Dye lasers, frequency doubled if necessary, provide ideal sources for such experiments. The radiation is very intense, the line width is small ( 1 cm ) and the wavenumber may be tuned to match any absorption band in the visible or near-ultraviolet region. 

The second contribution to the line-width is Doppler broadening. While the transition energy AE may be constant, the frequency and therefore the energy of radiation increases if the molecule is approaching the source and decreases if the molecule is receding from the source. In terms of energy... 

These various consequences parallel closely the analogous ones of the fluctuation and frequency modulation theories. There is, however, one important point of difference between the classical and quantum viewpoints which does not seem to have been emphasized previously, namely that transitions from the lowest level of the ground state can occur to several levels of the upper curve. This means that even at very low temperatures, when all the molecules are initially in this lowest energy level, a band of considerable breadth with frequencies rXH + m>(XH Y) will still persist. The temperature independent residual band width is a direct result of the perturbations of the system (in particular the finite change in the distance rxymin) caused by the absorption of a large quantum of radiation of frequency vXH. The same type of explanation may apply to other vibrational bands which remain of finite width at low temperatures the occurrence of such bands have been the cause of considerable discussion [34]. 

The second factor involves the theory that defines the natural width of the lines. Radiations emitted by atoms are not totally monochromatic. With plasmas in particular, where the collision frequency is high (this greatly reduces the lifetime of the excited states), Heisenberg s uncertainty principle is fully operational (see Fig. 15.4). Moreover, elevated temperatures increase the speed of the atoms, enlarging line widths by the Doppler effect. The natural width of spectral lines at 6000 K is in the order of several picometres. 

Welsh suggested correctly that similar transitions take place even if the molecular pair is not bound. The energy of relative motion of the pair is a continuum. Its width is of the order of the thermal energy, Efree 3kT/2. Radiative transitions between free states occur (marked free-free in the figure) which are quite diffuse, reflecting the short lifetime of the supermolecule. In dense gases, such diffuse collision-induced transitions are often found at the various rotovibrational transition frequencies, or at sums or differences of these, even if these are dipole forbidden in the individual molecules. The dipole that interacts with the radiation field arises primarily by polarization of the collisional partner in the quadrupole field of one molecule the free-free and bound-bound transitions originate from the same basic induction mechanism. 

The waves produce a pulse of radiation with a finite width (represented by the arrow in the figure). We will define the uncertainty At as the half width at half maximum thus we would use half the length of this arrow, or approximately five cycles at the center frequency vo. The pulse is large as long as most of the frequency components constructively interfere, then grows small. From the figure, At is approximately five cycles at the center frequency vo. Each cycle lasts for a time 1/vo. The net result is ... 

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