Line shapes

Natural line broadening is usually much smaller than the broadening seen in planetary spectra, but it is of fundamental importance as the ultimate limit of the narrowness of a spectral line. Natural line broadening arises from the finite coherence length of the wave train associated with a transition. Fourier analysis of radiation composed of a wave train of finite time, Ar, shows a frequency spread, A/, that is related to this time by 

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. 

From a practical point of view collision broadening is a more important mechanism. If one molecule in the process of a transition collides with another molecule, the phase continuity of the transition is interrupted, thereby reducing At in 

1) from the natural lifetime of the transition to the mean time between collisions, leading in this case to a larger line width, Av = Wc- Under atmospheric conditions of interest, the time between collisions is usually much shorter than the natural transition lifetime. Collision broadening in the infrared can thus be orders of magnitudes larger than natural line broadening. The line profile generated by collisions is 

The collision broadening coefficient varies among molecules and collision partners, but is usually close to 3 gigahertz (0.1 cm ) at one bar. Under typical conditions found in planetary tropospheres and stratospheres pressure broadening usually dominates other forms of line broadening. 

Theoretically, ESR lines should be infinitely narrow, experimentally, they are broadened by various mechanisms, both intrinsic (determined by sample properties and the physics of the resonance experiment) and extrinsic (dependent on spectrometer operating conditions) Line shapes of ESR signals of peroxy radicals in lignin are often asymmetric because of the solid-state or power spectrum effects of the vanous anisotropic interactions 

The idea of a transition between two energy levels suggests that the transition will occur at only one precise frequency as a sharp spike in the absorption or emission spectrum. This is not the case and, in fact, the transitions have an intrinsic width and shape containing information about the local environment of the atoms. The line profile of an atomic transition has contributions from three effects  

Transitions have a natural linewidth associated with their lifetime (via the uncertainty principle) but this is usually small. 

Pressure broadening allows for the presence of nearby molecules to perturb the positions of the energy levels, leading to a spreading of the transition frequency as a function of collisions and hence pressure. 

Doppler broadening allows for observation that some molecules in a gas cloud may be moving towards the observer and some away from the observer in a line of sight. 

The natural linewidth comes from the lifetime, r, of the upper state of a spontaneous transition, which is related to the Einstein A coefficient so that r = A l faster transitions have shorter lifetimes and vice versa, and similarly an allowed transition will have a short lifetime for the upper state whereas forbidden transitions will have a long lifetime. The lifetime consideration is very important in the laboratory where transitions have to occur on the timescale of the experiment, otherwise they are not observed. Hence in the laboratory allowed transitions are observed and in general (but not specifically) forbidden transitions are not seen. For astronomy this does not matter. So what if a forbidden transition has a lifetime of 30 million years - the Universe is 15 billion years old - if you wait long enough it will happen. The rules of spectroscopy need to be understood but in space anything goes  

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The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

Even in semiconductors, where it might appear that the exciton binding energies would be of interest only for low temperaPire regimes, excitonic effects can strongly alter tlie line shape of excitations away from the band gap. 

Spectral lines are fiirther broadened by collisions. To a first approximation, collisions can be drought of as just reducing the lifetime of the excited state. For example, collisions of molecules will connnonly change the rotational state. That will reduce the lifetime of a given state. Even if die state is not changed, the collision will cause a phase shift in the light wave being absorbed or emitted and that will have a similar effect. The line shapes of collisionally broadened lines are similar to the natural line shape of equation (B1.1.20) with a lifetime related to the mean time between collisions. The details will depend on the nature of the intemrolecular forces. We will not pursue the subject fiirther here. 

For a sample at diennal equilibrium there is a distribution of speeds which depends on the mass of the molecules and on the temperature according to the Boltzmaim distribution. This results in a line shape of the form... 

The actual line shape in a spectrum is a convolution of the natural Lorentzian shape with the Doppler shape. It must be calculated for a given case as there is no simple fomuila for it. It is quite typical in electronic... 

Pump-probe absorption experiments on the femtosecond time scale generally fall into two effective types, depending on the duration and spectral width of the pump pulse. If tlie pump spectrum is significantly narrower in width than the electronic absorption line shape, transient hole-burning spectroscopy [101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112 and 113] can be perfomied. The second type of experiment, dynamic absorption spectroscopy [57, 114. 115. 116. 117. 118. 119. 120. 121 and 122], can be perfomied if the pump and probe pulses are short compared to tlie period of the vibrational modes that are coupled to the electronic transition. 

Harde H, Katzenellenbogen N and Grischkowsky D 1995 Line-shape transition of collision broadened lines Phys. Rev. Lett. 74 1307-10... 

Loring R F, Van Y J and Mukamel S 1987 Time-resolved fluorescence and hole-burning line shapes of solvated molecules longitudinal dielectric relaxation and vibrational dynamics J. Chem. Phys. 87 5840-57... 

Kleier D A and Binsch G 1970 General theory of exchange-broadened NMR line shapes. II. Exploitation of invariance properties J. Magn. Reson. 3 146-60... 

A completely different approach, in particular for fast imimolecular processes, extracts state-resolved kinetic infomiation from molecular spectra without using any fomi of time-dependent observation. This includes conventional line-shape methods, as well as the quantum-dynamical analysis of rovibrational overtone spectra [18, 33, 34 and 35]. 

Figure B2.5.12 shows the energy-level scheme of the fine structure and hyperfme structure levels of iodine. The corresponding absorption spectrum shows six sharp hyperfme structure transitions. The experimental resolution is sufficient to detennine the Doppler line shape associated with the velocity distribution of the I atoms produced in the reaction. In this way, one can detennine either the temperature in an oven—as shown in Figure B2.5.12 —or the primary translational energy distribution of I atoms produced in photolysis, equation B2.5.35.

Even if the homogeneous line shape can be extracted, many other processes can contribute. Every decay process contributes to the finite lifetime of an excited species. A, with an individual decay constant k ... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

Blake N P and Metiu H 1995 Efficient adsorption line shape calculations for an electron coupled to many quantum degrees of freedom, applications to an electron solvated in dry sodalites and halo-sodalites J. Chem. Phys. 103 4455... 

Geva E and Skinner J L 1997 Theory of single-molecule optical line-shape distributions in low-temperature glasses J. Chem. Phys. B 101 8920-32... 

Geva E and Skinner J L 1998 Optical line shapes of single molecules in glasses temperature and scan-time dependence J. Phys. Chem 109 4920-6... 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

Now, it is convention to introduce the so-called "line shape" function I (co) ... 

To summarize, the line shape function I(co) produces the net rate of photon absorption... 

This result, when substituted into the expressions for C(t), yields expressions identieal to those given for the three eases treated above with one modifieation. The translational motion average need no longer be eonsidered in eaeh C(t) instead, the earlier expressions for C(t) must eaeh be multiplied by a faetor exp(- co2t2kT/(2me2)) that embodies the translationally averaged Doppler shift. The speetral line shape funetion I(co) ean then be obtained for eaeh C(t) by simply Fourier transforming ... 

Here I (co AE) is a line shape function such as those described earlier each of which contains a set of frequencies (e.g., co = C0fv,iv+ Ei,f/fe coj = co + AE/h) at which absorption or emission occurs. 

Figure 2.5 shows, for a sample in the gas phase, a typical absorption line with a HWHM (half-width at half-maximum) of Av and a characteristic line shape. The line is not infinitely narrow even if we assume that the instmment used for observation has not imposed any broadening of its own. We shall consider three important factors that may contribute to the line width and shape. 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

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