Line width natural

Line broadening processes are important in themselves, in the information they can provide concerning the important physics of a molecular gas system. They are also important to the experimental spectroscopist in determining the spectroscopic resolution and hence the amount and accuracy of the information which can be derived. We here review briefly the principal effects which determine spectroscopic line widths in different experiments. 

Many of the processes which determine line widths can be removed by appropriately designed experiments, but it is almost impossible to avoid so-called natural line broadening. This arises from the spontaneous emission process (governed by the Einstein A coefficient) described in the previous section. Spontaneous emission terminates the lifetime of the upper state involved in a transition, and the Heisenberg uncertainty principle states that the lifetime of the state (At) and uncertainty in its energy (A E) are related by the expression 

We must therefore determine At and this is straightforward since it is simply equal to the inverse of the Einstein A coefficient, 

Here v and //, the frequency and transition dipole moment, refer to a specific transition, but a given level can usually decay by spontaneous emission to a number of different levels. However if we take the frequency to be 10 GHz, and the dipole matrix element to be 3 x 10 V) Cm, we obtain a natural line width from (6.350) of 10-9 Hz. In the microwave region, therefore, this contribution is negligible, but in the near ultraviolet the natural line width of an excited electronic state is of the order 1 MHz, unless the state is metastable. 

Any atom being in an excited state, for instance after absorption of a photon, will undergo a relaxation process to a lower state within a finite time, even if there is no interaction with other atoms or molecules. Typical lifetimes r for undisturbed excited states are of the order of 10 to 10 s. After this the atom re-emits the photon and relaxes to the lower state, which is the ground state in the case of resonance transitions. According to Heisenberg s uncertainty principle AE At = h, the finite lifetime r causes an uncertainty of  

If the lower state is not the ground state, it will also show an energy uncertainty corresponding to its own lifetime. In this case Aj/ is given by the sum of both contributions. 

This imcertainty in frequency, which is inversely proportional to the lifetime, generates a line profile of Lorentz shape, centered at vq, with a width Using the relation AAn = (A /c) Ai/N the so-called natural line width AAn is obtained and the corresponding wavelength-dependant intensity distribution /n(A) of the area-normalized profile is given by  

The lifetime of an electron in the excited state in the case of the resonance lines used in AAS is in the range of a few nanoseconds, resulting in AAn of about 0.01 pm. This is a small effect compared to the other broadening mechanisms occurring in AAS, and is therefore neglected in the context of this section. 

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High-resolution spectroscopy used to observe hyperfme structure in the spectra of atoms or rotational stnicture in electronic spectra of gaseous molecules connnonly must contend with the widths of the spectral lines and how that compares with the separations between lines. Tln-ee contributions to the linewidth will be mentioned here tlie natural line width due to tlie finite lifetime of the excited state, collisional broadening of lines, and the Doppler effect. 

Another feature of the spectrum shown in Figure 10.19 is the narrow width of the absorption lines, which is a consequence of the fixed difference in energy between the ground and excited states. Natural line widths for atomic absorption, which are governed by the uncertainty principle, are approximately 10 nm. Other contributions to broadening increase this line width to approximately 10 nm. 

Of the four types of broadening that have been discussed, that due to the natural line width is, under normal conditions, much the smallest and it is the removal, or the decrease, of the effects of only Doppler, pressure and power broadening that can be achieved. 

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 principle all the X-ray emission methods can give chemical state information from small shifts and line shape changes (cf, XPS and AES in Chapter 5). Though done for molecular studies to derive electronic structure information, this type of work is rarely done for materials analysis. The reasons are the instrumental resolution of commercial systems is not adequate and the emission lines routinely used for elemental analysis are often not those most useftil for chemical shift meas-ure-ments. The latter generally involve shallower levels (narrower natural line widths), meaning longer wavelength (softer) X-ray emission. 

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. 

In the ultraslow exchange limit, O -4 1/T, the line shape becomes insensitive to the motion because the extra broadening is much smaller than the natural line width. 

Figure 1.29 The effect of increased digital resolution (DR) on the appearance of the NMR spectrum, (a) The spectrum of odichlorobenzene recorded at a digital resolution of 0.1 Hz per point, allowing the sj>ectral lines to be seen at their natural line width, (b) The spectrum of the same molecule recorded at a digital resolution of 0.4 Hz per point.

Another consideration is the natural line width and satellite structure of the x-ray line used. Titanium (TiKa=4510.9 eV) has seen limited use (12) for non-destructive depth profiling, but the observed spectra are complicated by the TiKa satellite structure and the large natural line width of 2.0 eV (13). 

Natural Line Width and Spectral Line Shape... 

Fig. 2.2 Intensity distribution /( ) for the emission of y-rays with mean transition energy Eq. The Heisenberg natural line width of the distribution, F = S/t, is determined by the mean lifetime T of the excited state (e)...

Hence, nuclear resonance absorption of y-photons (the Mbssbauer effect) is not possible between free atoms (at rest) because of the energy loss by recoil. The deficiency in y-energy is two times the recoil energy, 2Er, which in the case of Fe is about 10 times larger than the natural line width F of the nuclear levels involved (Fig. 2.4). 

Fig. 2.8 (a) Fractional absorption of a Mossbauer absorption line as function of the effective absorber thickness t. (b) The depth of the spectrum is determined by fs. The width for thin absorbers, t 1, is twice the natural line width F of the separate emission and absorption lines (see (2.30)). AE is the shift of the absorption line relative to the emission line due to chemical influence... 

Thus, one may state as a rule of thumb that an absorber should have about 21 pg cm of Fe (0.37 pmol cm ) or ca. 1 mg cm of natural iron (17.9 pmol cm ) for a symmetric quadrupole doublet with natural line width. 

In addition, Coulomb excitation can be used to populate the Mossbauer levels of I77,i78,i80j j [165-167]. The experimental line width using these sources is only slightly larger than the natural line width (e.g., the thickness corrected line width of Hf in a tantalum foil is in good agreement with the natural line width Texp = 1.90 0.07 mm s T at = 1-99 0.04 mm s [168]). 

It is a matter of historical interest that Mossbauer spectroscopy has its deepest root in the 129.4 keV transition line of lr, for which R.L. Mossbauer established recoilless nuclear resonance absorption for the first time while he was working on his thesis under Prof. Maier-Leibnitz at Heidelberg [267]. But this nuclear transition is, by far, not the easiest one among the four iridium Mossbauer transitions to use for solid-state applications the 129 keV excited state is rather short-lived (fi/2 = 90 ps) and consequently the line width is very broad. The 73 keV transition line of lr with the lowest transition energy and the narrowest natural line width (0.60 mm s ) fulfills best the practical requirements and therefore is, of all four iridium transitions, most often (in about 90% of all reports published on Ir Mossbauer spectroscopy) used in studying electronic stractures, bond properties, and magnetism. 

MeV a-particles and used the Au/Ir source after annealing without any further chemical or physical treatment. Commercially available sources are produced via Pt(p, n) Au. The most popular source matrix into which Au is diffused is platinum metal although it has the disadvantage of being a resonant matrix - natural platinum contains 33.6% of Pt. Using copper and iridium foils as host matrices for the Au parent nuclide, Buym et al. [327] observed natural line widths and reasonable resonance absorption of a few percent at 4.2 K. 

The first Mossbauer measurements involving mercury isotopes were reported by Carlson and Temperley [481], in 1969. They observed the resonance absorption of the 32.2 keV y-transition in (Fig. 7.87). The experiment was performed with zero velocity by comparing the detector counts at 70 K with those registered at 300 K. The short half-life of the excited state (0.2 ns) leads to a natural line width of 43 mm s Furthermore, the internal conversion coefficient is very large (cc = 39) and the oi pj precursor populates the 32 keV Mossbauer level very inefficiently ( 10%). 

Less than the natural line width materials are magnetic and exhibit a six-line spectrum at low... 

The resolution of XPS is determined by the line width of the X-ray source, the broadening due to the analyzer, and the natural line width of the level under study. The three factors are related as follows ... 

The natural line width is determined by Heisenberg s uncertainty relation... 

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