Spectral line widths and atomic decay rates. The atoms in a...

Spectral line widths and atomic decay rates. The atoms in a gas discharge tube are excited by a very brief burst of electrons at $t=0$. The atoms subsequently fall back to the ground state, emitting visible light belonging to a single spectral line at $5500 \hat{A}$. The intensity of this light falls off with time according to the law $I(t)=I_n e^{-B t}$ with $\beta=5 \times 10^7 \mathrm{sec}^{-1}$. Deduce the spread of wavelength of the spectral line, if the "natural" line width is the only source of broadening.

Key Concepts

Frequency-Wavelength Conversion

Since spectral lines can be described in terms of their frequency or their wavelength, a conversion between these representations is necessary. The relation between frequency and wavelength is given by the speed of light, and differential changes connect the width in frequency to the width in wavelength. This conversion is critical when one needs to express natural line broadening, originally derived in the frequency domain, in terms of wavelength.

Exponential Decay and Fourier Transform Relationship

An exponential decay in the time domain corresponds to a Lorentzian profile in the frequency domain. This is a result of the Fourier transform, which shows that the decay constant of the excited state determines the spread (width) in the frequency components of the emitted radiation. This mathematical relationship is key in understanding how decay rates affect the observed spectral line shape.

Lifetime Broadening

Lifetime broadening refers to the intrinsic spread in energy (or frequency) of a spectral line that arises because of the finite lifetime of an excited state. A shorter lifetime, described by an exponential decay, results in a wider natural line width according to the time-energy uncertainty principle.

Time?Energy Uncertainty Principle

This principle states that there is a fundamental limit to the precision with which energy and time can be simultaneously known. In the context of atomic transitions, a shorter lifetime of an excited state (i.e., a rapid decay) implies a greater uncertainty in its energy, which directly leads to a natural broadening of the spectral line observed in emission or absorption spectra.

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