Lifetime broadening
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص437-441
2025-12-03
38
Lifetime broadening
It is found that spectroscopic lines from gas-phase samples are not infinitely sharp even when Doppler broadening has been largely eliminated by working at low temperatures. The same is true of the spectra of samples in condensed phases and solution. This residual broadening is due to quantum mechanical effects. Specifically, when the Schrödinger equation is solved for a system that is changing with time, it is found that it is impossible to specify the energy levels exactly. If on average a system survives in a state for a time τ (tau), the lifetime of the state, then its energy levels are blurred to an extent of order δE, where
This expression is reminiscent of the Heisenberg uncertainty principle (eqn 8.40), and consequently this lifetime broadening is often called ‘uncertainty broadening’. When the energy spread is expressed as a wavenumber through δE = hcδv, and the values of the fundamental constants introduced, this relation becomes
No excited state has an infinite lifetime; therefore, all states are subject to some life time broadening and, the shorter the lifetimes of the states involved in a transition, the broader the corresponding spectral lines. Two processes are responsible for the finite lifetimes of excited states. The domin ant one for low frequency transitions is collisional deactivation, which arises from collisions between molecules or with the walls of the container. If the collisional life time, the mean time between collisions, is τcol, the resulting collisional linewidth is δEcol ≈$/τcol. Because τcol = 1/z, where z is the collision frequency, and from the kinetic model of gases (Section 1.3) we know that z is proportional to the pressure, we see that the collisional linewidth is proportional to the pressure. The collisional linewidth can therefore be minimized by working at low pressures.
The rate of spontaneous emission cannot be changed. Hence it is a natural limit to the lifetime of an excited state, and the resulting lifetime broadening is the natural linewidth of the transition. The natural linewidth is an intrinsic property of the transition, and cannot be changed by modifying the conditions. Natural linewidths depend strongly on the transition frequency (they increase with the coefficient of spontaneous emission A and therefore as ν3), so low frequency transitions (such as the microwave transitions of rotational spectroscopy) have very small natural linewidths, and collisional and Doppler line-broadening processes are dominant. The natural lifetimes of electronic transitions are very much shorter than for vibrational and rotational transitions, so the natural linewidths of electronic transitions are much greater than those of vibrational and rotational transitions. For example, a typical electronic excited state natural lifetime is about 10−8 s (10 ns), corresponding to a natural width of about 5 × 10−4 cm−1 (15 MHz). A typical rotational state natural lifetime is about 103s, corresponding to a natural linewidth of only 5 × 10−15 cm−1(of the order of 10−4Hz).
Observations by the Cosmic Background Explorer (COBE) satellite support the long-held hypothesis that the distribution of energy in the current Universe can be modelled by a Planck distribution (eqn. 8.5) with T = 2.726 ± 0.001 K, the bulk of the radiation spanning the microwave region of the spectrum. This cosmic microwave background radiation is the residue of energy released during the Big Bang, the event that brought the Universe into existence. Very small fluctuations in the background temperature are believed to account for the large-scale structure of the Universe. The interstellar space in our galaxy is a little warmer than the cosmic background and consists largely of dust grains and gas clouds. The dust grains are carbon-based compounds and silicates of aluminium, magnesium, and iron, in which are embedded trace amounts of methane, water, and ammonia. Interstellar clouds are significant because it is from them that new stars, and consequently new planets, are formed. The hottest clouds are plasmas with temperatures of up to 106 K and densities of only about 3 × 103 particles m−3. Colder clouds range from 0.1 to 1000 solar masses (1 solar mass = 2 × 1030 kg), have a density of about 5 × 105 particles m−3, consist largely of hydrogen atoms, and have a temperature of about 80 K. There are also colder and denser clouds, some with masses greater than 500 000 solar masses, densities greater than 109 particles m−3, and temperatures that can be lower than 10 K. They are also called molecular clouds, because they are composed primarily of H2 and CO gas in a proportion of about 105 to 1. There are also trace amounts of larger molecules. To place the densities in context, the density of liquid water at 298 K and 1 bar is about 3 ×1028 particles m−3.
It follows from the the Boltzmann distribution and the low temperature of a molec ular cloud that the vast majority of a cloud’s molecules are in their vibrational and electronic ground states. However, rotational excited states are populated at 10–100 K and decay by spontaneous emission. As a result, the spectrum of the cloud in the radiofrequency and microwave regions consists of sharp lines corresponding to rotational transitions (Fig. 13.8). The emitted light is collected by Earth-bound or space borne radio telescopes, telescopes with antennas and detectors optimized for the collection and analysis of radiation in the microwave–radio wave range of the spectrum. Earth-bound radio telescopes are often located at the tops of high mountains, as atmospheric water vapour can reabsorb microwave radiation from space and hence interfere with the measurement.
Over 100 interstellar molecules have been identified by their rotational spectra, often by comparing radio telescope data with spectra obtained in the laboratory or calculated by computational methods. The experiments have revealed the presence of trace amounts (with abundances of less than 10−8 relative to hydrogen) of neutral molecules, ions, and radicals. Examples of neutral molecules include hydrides, oxides (including water), sulfides, halogenated compounds, nitriles, hydrocarbons, aldehydes, alcohols, ethers, ketones, and amides. The largest molecule detected by rotational spectroscopy is the nitrile HC11N. Interstellar space can also be investigated with vibrational spectroscopy by using a combination of telescopes and infrared detectors. The experiments are conducted
Fig. 13.8 Rotational spectrum of the Orion nebula, showing spectral fingerprints of diatomic and polyatomic molecules present in the interstellar cloud. Adapted from G.A. Blake et al., Astrophys. J. 315, 621 (1987).
primarily in space-borne telescopes because the Earth’s atmosphere absorbs a great deal of infrared radiation (see Impact I13.2). In most cases, absorption by an inter stellar species is detected against the background of infrared radiation emitted by a nearby star. The data can detect the presence of gas and solid water, CO, and CO2 in molecular clouds. In certain cases, infrared emission can be detected, but these events are rare because interstellar space is too cold and does not provide enough energy to promote a significant number of molecules to vibrational excited states. However, infrared emissions can be observed if molecules are occasionally excited by high energy photons emitted by hot stars in the vicinity of the cloud. For example, the poly cyclic aromatic hydrocarbons hexabenzocoronene (C48H24) and circumcoronene (C54H18) have been identified from characteristic infrared emissions.
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