Lifetime Broadening

It is found that spectroscopic lines from gas-phase samples are not initially sharp.The same is true for the spectra from solid state and solutions. There are several reasons for the spectral line broadening. One them is due to quantum mechanical effects. Particularly, if the quantum mechanical system is changing with time it is impossible to specify the energy levels exactly.If the system survives in a quantum state for a time $\tau$, the energy of the level in principle cannot be known with accuracy better than spontaneous decay spontaneous decay

$$\delta E \approx \frac{\hbar}{\tau}$$ (22)

This is fundamental uncertainty relation for energy. In principle, no excited state has infinite lifetime, thus all excited states are subject of the lifetime broadening and the shorter the lifetimes of the states involved in a transition, the broader the corresponding spectral lines. These are two main processes which are responsible to the finite lifetime of excited states.

Firstly, there is the spontaneous decay which is proportional to the corresponding Einstein coefficient $A$. The intensity the spontaneous decay is proportional to the square of the matrix element of interaction with electromagnetic modes of vacuum and in principle cannot be changed. However, for some particular pairs of the energy states these matrix elements can be very small, the corresponding states are called metastable states Of cause, the transition intensity from these states is very small.

Secondly, there is interaction between the molecular quantum states and other particles. In the gas phase this interaction is just inelastic collisions with surrounding particles. In the condensed matter there can be interaction with phonons (vibration of the surrounding lattice). The collision frequency in the gas phase can be reduced by decreasing the number of molecules in the, or by decreasing their velocity (temperature).

In both cases the finite lifetime results in a specific spectral line shape which is called Lorentz line shape:

$$I(\nu) \propto \frac{1}{\pi}\,\frac{\gamma/2}{(\nu-\nu_0)^2 + \gamma^2/4},$$ (23)

where $\gamma$ is the excited state decay rate, $\gamma=1/2\pi\tau$ and $\nu_0$ is the line center.

The Lorentz line is a bell-shaped with characteristic linewidth of $\Delta \nu_L = \gamma$. The natural lifetimes of electronic transitions are very much shorter than that for vibrational and rotational transitions. For example, a typical electronic excited state at the energy of about several $eV$ has a lifetime of about $10^{-8} s$, corresponding to a natural linewidth of about $15 MHz$ ( $5\cdot 10^{-4}cm^{-1}$). A typical rotational state lifetime is about $10^{3}$ s, corresponding to a natural linewidth of $10^{-4}$ Hz ( $5\cdot 10^{-15} cm^{-1}$).

This broadening of the line due to collisions and the respective width is called the homogenous line width because the collisions are isotropic. Under atmospheric conditions, collisions occur within intervals of nanoseconds and we observe widths of a few GHz ($10^{9}Hz$). This value decrease linearly with the gas pressure. In liquids the collisions are much more often resulting in collision times in the picosecond time scale. Therefore, the width of a single transition in liquid is usually about several THz, ($10^{12} Hz$) corresponding linewidth of a few nanometers for transitions in the ultraviolet spectral range of about 300 nm.

In contrast, the line width is said to be inhomogeneous if the anisotropic collisions or interactions occur. Particularly, inhomogeneous line broadening is typical for certain atoms in the lattice of a solid medium.

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