Width Doppler

Generally, the Lorentzian line profile with the natural linewidth 8v , as discussed in Sect. 3.1, cannot be observed without special techniques, because it is completely concealed by other broadening effects. One of the major contributions to the spectral linewidth in gases at low pressures is the Doppler width, which is due to the thermal motion of the absorbing or emitting molecules. 

Consider an excited molecule with a velocity v = vx, Vy, v ] relative to the rest frame of the observer. The central frequency of a molecular emission line that is a o in the coordinate system of the molecule is Doppler shifted to 

Similarly, one can see that the absorption frequency coo of a molecule moving with the velocity v across a plane EM wave E = Eo xp(icot-k-r) is shifted. The wave frequency co in the rest frame appears in the frame of the moving molecule as 

The molecule can only absorb if co coincides with its eigenfrequency coq. The absorption frequency co = coa is then 

As in the emission case, the absorption frequency co is increased for A r 0 (Fig. 3.6b). This happens, for example, if the molecule moves parallel to the wave propagation. It is decreased if A - v 0, e.g., when the molecule moves against the light propagation. If we choose the -hz-direction to coincide with the light propagation, with k = 0, 0, and A = 2ti/X, (3.39a) becomes 

As in the emission case the absorption frequency is increased for k-v 0 (Fig. 3.6b). This happens, for example, if the molecule moves parallel to the 

Consider an excited molecule with a velocity v = v, v, v relative 

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This is best done graphically, using Table 3.3 (a = 0.001). The data are just barely compatible with the lines being on the linear part of the curve of growth, in which case the EW of Di is 200 mA and the column density is 2.0 x 1012cm-2 from Eq. (3.38). In this case, the equivalent width of D2 could be at most about 1 Doppler width,... 

Since the energy of a photon is hv = 3.3 x 10 J, the average number of photons for that mode is 1.7 x 10. This indicates that the number of photons per mode in the visible region for a spectral width of 10 s (in the order of the Doppler width) is a very small nnmber compared to unity. This has important conseqnences, as we will see in Section 2.3. 

The width of the gain profile in a CO2 laser is given as 66 MHz (close to the Doppler width of the emission band of the gas). If the eigenfrequency of the laser resonator is tuned to the center of the laser gain profile, what is the maximum length of resonator for which the laser can oscillate in a single mode ... 

The resolution in these measurements is, in the case of singlemode lasers, limited by the doppler width of the absorption lines and reaches 0.001 cm 4) compared to o.o5 cm with conventional nearinfrared spectrometers. In Section IV it will be shown that even this limitation may be overcome in favorable cases. 

Since the single-mode laser linewidth is small compared to the absorption linewidth, one can probe the absorption profile by tuning the laser line across it, getting more information than by measuring the absorption coefficient averaged over the whole doppler width 6). 

By modulating the electric field and using phase-sensitive detection methods, Uehara et al. 8 ) were able to increase the sensitivity considerably and they could even detect Stark splittings of less than the doppler width of the components. Fig. 3 shows the Stark spectrum of HDCO for different electric field strengths. Because of the Stark modulation technique the absorption lines appear differentiated the zero points represent the center of each line. 

The absorption measurement via observation of the total fluorescence has advantages when the probe cannot be placed inside the laser cavity. It is not necessary to employ any monochromator or spectrograph. The spectral resolution limit, which is set by the finite Doppler width of the absorbing gas and which is already far lower than the resolution of most spectographs, may be drastically reduced by using an atomic or molecular beam perpendicular to the laser beam. 

With this technique the Doppler width could be reduced by two orders of magnitude below the natural linewidth, and spectral structures within the Doppler width could be resolved. Examples are the resolution of hyperfine structure components in an 12-beam using a single-mode argon laser (tunable within a few gigahertz) or the investigation of the upper state hfs-splitting in the atomic... 

If the molecular absorption lines do not overlap within their Doppler width, excitation with single-mode lasers succeeds in populating one single rotational-vibrational level. 

In the case of a common upper level the fluorescence from this level due to excitation by the monochromatic wave will reflect this selective population with molecules having only velocity components inside the range Auz. Thus, observing the fluorescence in z direction on any transition from this upper level yields linewidths much smaller than the normal doppler width 322). In this way lines which are not resolved in normal fluorescence spectroscopy can be separated even if their frequency difference is much less than their doppler width 323). 

The important point is that the Lamb-dip widths for most visible and near infrared transitions at low pressures are several orders of magnitude smaller than the doppler widths and are therefore well suited for high resolution spectroscopy. When probing with a... 

Lamb dip spectroscopy provides a very sensitive tool for studying small frequency shifts and broadening of spectral lines which normally would be undetectable because they may be small compared to the doppler width. These investigations yield information about collisions at low pressures, where the effect of far distant collisions is not suppressed by the more effective close collisions. This allows the potential between the collision partners at large intermolecular distances to be examined. 

Collision widths considerably smaller than the doppler width have been observed by Szoke and Javan 344) when studying the effects of collisions on the saturation behaviour of the 1.15 ju Ne transition. The measurements showed that, in addition to pressure dependent broadening due to hard collisions, there exists an appreciable broadening due to soft collisions, and that the collisions cause an asymmetry in the average frequency response of individual atoms. 

The Doppler width (FWHM) is given by Townes and Schalow, 1955)... 

Fig. 3 Upper traces, apparent peak absorptance vs AxDop/Axres, the Doppler width per unit resolution. Each trace is identified by the actual peak absorptance. Lower traces, percentage error incurred when (Ax op + Axr2es)1/2 is used to approximate Axobs for an absorption line vs A cDop/Ajcres. The curves are labeled with the appropriate equivalent width per unit Doppler width as EQW/DOPW.

Fig. 24 Tunable-diode-laser spectrum of RQ0 of v9 of ethane. Trace (a) is the average of 250,000 scans and exhibits linewidths of 0.0022 cm-1 (the Doppler width is 0.0018 cm-1). Trace (b) results from the deconvolution of the data in trace (a) using a gaussian with a FWHM of 0.0022 cm-1 as a response function. Trace (c) is the Q branch calculated using a model that includes torsional splitting effects Av = 1.95 mk. Trace (c) is calculated for Av = 0.00075 cm-1, which is less than one-half the 300 K Doppler width.

In data-point units, the original infrared peaks were about 34 units wide (full width at half maximum). This corresponds to an actual width of approximately 0.024 cm-1. The impulse response function was about 25 units wide. After inverse filtering and restoration of the Fourier spectrum, the resolved peaks were 11 and 14 units wide, respectively. This is close to the Doppler width of these lines. 

Second, we consider the diffuseness in gas-phase spectra.74- 78 An account of those aspects of gas-phase spectra that relate to electronic relaxation is only possible if the other causes of diffuseness are first made clear. We therefore summarize some of the characteristics of vibration-rotation spectra with special reference to a molecule like naphthalene at a vapor pressure of 1 mm76 The Doppler width of each line is 0.022 cm-1 the rotational line spacing can be as small as 0.0004 cm-1, or 50 lines per Doppler width the length of a sequence is 300 cm 1 and the average sequence spacing is 5 cm-1 for anthracene the average sequence spacing... 

The ultraviolet absorption spectrum of formaldehyde consists of many sharp discrete bands of Doppler width. The isotopic shifts due to C and O atoms are sometimes 5 to 10 cm-1 in the 3000 to 3100 A region [see Moore (715)]. Hence, it is possible to selectively excite a specific carbon or oxygen isotopic species in mixtures of other isotopic species. 

If the isotopic shift of a spectral line in an atom or in a molecule is more Hi... the Doppler width, it is in principle possible to selectively excite a parti, id... isotopic species from isotopic mixtures by monochromatic light of w.u. length in coincidence with the absorption of the particular isotopic spe> < In a typical example, 2°2Hg atoms in natural Hg vapor containing 204. o 201, 200, 199, and 198 isotopes are preferentially excited by the 2b i V resonance line of 202Hg atoms. It has recently been demonstrated tlm 235U atoms are enriched in the photoionization processes of Mi. t... 

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