14.3: Natural or Radiation Broadening

Of all the physical processes that can contribute to the frequency dependence of the atomic line absorption coefficient, some are intrinsic to the atom itself. Since the atom must emit or absorb a photon in a finite time, that photon cannot be represented by an infinite sine wave. If the photon wave train is of finite length, it must be represented by waves of frequencies other than the fundamental frequency of the line center \(\nu_0\). This means that any photon can be viewed in terms of a "packet" of frequencies ranging around the fundamental frequency. So the photon will consist of energy occupying a range of wavelengths about the line center. The extent of this range will depend on the length of the photon wave train. The longer the wave-train, the narrower will be the range of frequencies or wavelengths required to represent it.

Since the length of the wave train will be proportional to the time required to emit or absorb it, the characteristic width of the range will be proportional to the transition probability (i.e., the inverse of the transition time) of the atomic transition. This will be a property of the atom alone and is known as the natural width of the transition. It is always present and cannot be removed. Its existence depends only on the finite length of the wave train and so is not just the result of the quantum nature of the physical world. Indeed, there are two effects to estimate: the classical effect relying on the finite nature of the wave train, and the quantum mechanical effect that can be obtained for a specific atom's propensity to emit photons. The former will be independent of the type of atom, while the latter will yield a larger broadening that depends specifically on the type of atom and its specific state.

a. Classical Radiation Damping

The classical approach to the problem of absorption relies on a picture of the atom in which the electron is seen to oscillate in response to the electric field of the passing photon. There is a strong analogy here between the behavior of the electrons in the atom and the free electrons in an antenna. The energy of the passing wave is converted to oscillatory motion of the electron(s), which in the antenna produce a current that is subsequently amplified to signal the presence of the photon. It then makes sense to use classical electromagnetic theory to estimate this effect for the single optical electron of an atom. The oscillation of this electron can then be viewed as a classical oscillating dipole.

Since an oscillating electron represents a continuously accelerating charge, the electron will radiate or absorb energy. In the classical picture, the processes of emission and absorption are interchangeable. The emission simply requires the presence of a driving force, which is the ultimate source of the energy that is emitted, while the energy source for the absorption processes is the passing photon itself. If we let \(\overline{W}\) represent the energy gained or lost over one cycle of the oscillating dipole, then any good book on classical electromagnetism (i.e., W. Panofsky and M. Phillips¹ or J. Slater and N. Frank²) will show that \[\frac{d \bar{W}}{d t}=-\frac{2 e^2}{3 c^3} \overline{\left(\frac{d^2 x}{d t^2}\right)^2}\label{14.2.1}\]

where d²x/dt² is the acceleration of the oscillating charge. Now if we assume that the oscillator is freely oscillating, then the instantaneous acceleration is simply \[\frac{d^2 x}{d t^2}=-\omega_0^2 x\label{14.2.2}\]

This is a good assumption as long as the energy is to be absorbed on a time scale that is long compared to the period of oscillation. Since the driving frequency of the oscillator is that of the line center, this is equivalent to saying that the spread or range of absorbed frequencies is small compared to the frequency of the line center.

Equation \ref{14.2.2} can be used to replace the mean square acceleration of equation \ref{14.2.1} to get \[\frac{d \bar{W}}{d t}=-\frac{2 e^2}{3 c^3} \omega_0^4 \overline{x^2}\label{14.2.3}\]

The mean position of the oscillator \(\overline{x^2}\) can, in turn, be replaced with the mean total energy of the oscillator from \[\bar{W}=\langle T\rangle+\langle\Phi\rangle=m \omega_0^2 \overline{x^2}\label{14.2.4}\]

so that the differential equation for the absorption or emission of radiation from a classical oscillating dipole is \[\frac{d \bar{W}}{d t}=-\frac{2 e^2 \omega_0^2 \bar{W}}{3 m_e c^3}=-\gamma \bar{W}\label{14.2.5}\]

The quantity \(\gamma\) is known as the classical damping constant and is \[\gamma=\frac{2 e^2 \omega_0^2}{3 m_e c^3}\label{14.2.6}\]

The solution of equation \ref{14.2.5} shows that the absorption of the energy of the passing photon will be \[I=I_0 e^{-\gamma t}\label{14.2.7}\]

where \(\mathrm{I_0}\) is the presumably sinusoidally varying energy field of the passing photon. The result is that energy of the absorbed or emitted photon resembles a damped sine wave (see Figure 14.1).

But, we are interested in the behavior of the absorption with wavelength or frequency, for that is what yields the line profile. Since we are interested in the behavior of an uncorrelated collection of atoms, their combined effect will be proportional to the combined effect of the squares of the electric fields of their emitted photons. Thus, we must calculate the Fourier transform of the time-dependent behavior of the electric field of the photon so that \[E(\omega)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} E(t) e^{i \omega t} d t\label{14.2.8}\]

If we assume that the photon encounters the atom at \(\mathrm{t=0}\) so that \(\mathrm{E(t)=0}\) for \(\mathrm{t<0}\), and that it has a sinusoidal behavior \(\mathrm{E}(\mathrm{t})=\mathrm{E}_0 e^{-i \omega_0 t}\) for \(\mathrm{t \geq 0}\), then the frequency dependence of the photon's electric field will be \[E(\omega)=\frac{E_0}{\sqrt{2 \pi}} \int_0^{\infty} e^{-\left[t\gamma / 2+i\left(\omega-\omega_0\right) t\right]} d t\label{14.2.9}\]

Thus the power spectrum of the energy absorbed or emitted by this classical oscillator will be \[I_\omega \propto E_\omega^2=\text { (const) }\left[\left(\omega-\omega_0\right)^2+\left(\frac{\gamma}{2}\right)^2\right]^{-1}\label{14.2.10}\]

Figure 14.1 is a schematic representation of the effect of radiation damping on the wave train of an emitted (absorbed, if t is replaced with -t) photon. The pure sine wave is assumed to represent the photon without interaction, while the exponential dotted line depicts the effects of radiation damping by the classical oscillator. The solid curve is the combined result in the time domain.

It is customary to normalize this power spectrum so that the integral over all frequencies is unity so that \[I_\omega=\frac{\gamma /(2 \pi)}{\left(\omega-\omega_0\right)^2+(\gamma / 2)^2}\label{14.2.11}\]

This normalized power spectrum occurs frequently and is known as a damping profile or a Lorentz profile. Since the atomic absorption coefficient will be proportional to the energy absorbed, \[S_\omega=\frac{\pi e^2}{m_e c} \frac{\gamma}{\left(\omega-\omega_0\right)^2+(\gamma / 2)^2}\label{14.2.12}\]

Here the constant of proportionality can be derived from dispersion theory³. A plot of \(\boldsymbol{S}_\omega\) shows a hump-shaped curve with very large "wings" characteristic of a damping profile (see Figure 14.2). At some point in the profile, the absorption coefficient drops to one-half of its peak value. If we denote the full width at this half-power point by \(\Delta\lambda_\mathrm{c}\), then \[\Delta \lambda_c=\frac{2 \pi c \gamma}{\omega_0^2}=\frac{4 \pi e^2}{3 m_e c^2} \approx 1.18 \times 10^{-4} \AA\label{14.2.13}\]

This is known as the classical damping width of a spectral line and is independent of the atom or line. It is also very much smaller than the narrowest lines seen in the laboratory, and to see why, we must turn to a quantum mechanical representation of radiation damping.

Figure 14.2 shows the variation of the classical damping coefficient with wavelength. The damping coefficient drops to half of its peak value for wavelength shifts equal to \(\Delta\lambda_\mathrm{c}/2\) on either side of the central wavelength. The overall shape is known as the Lorentz profile.

b. Quantum Mechanical Description of Radiation Damping

The quantum mechanical view of the emission or absorption of a photon is rather different from the classical view since it is intimately connected with the nature of the atom in question. The basic approach involves the Heisenberg uncertainty principle as the basis of the broadening. If we consider an atom to be in a certain state, then the length of time that it can remain in that state is related to the uncertainty of the energy of that state by \[\Delta E \Delta t \simeq \hbar\label{14.2.14}\]

If there are a large number of states to which the atom can make a transition, then the probability of it doing so is great, ∆t is small, and the uncertainty of the energy level is large. A large uncertainty in the energy of a specific state means that a wide range of frequencies can be involved in the transition into or out of that state. Thus any line resulting from such a transition will be unusually broad. Thus any strong line resulting from frequent transitions will also be quite broad.

This view of absorption and emission was quantified by Victor Weisskopf and Eugene Wigner^4,5 in 1930. They noted that the probability of finding an atom with a wave function \(\Psi_j\) in an excited state \(j\) after a transition from a state \(i\) is \[P_j(t)=\Psi_j^* \Psi_j e^{-\Gamma t}\label{14.2.15}\]

where \(\Gamma\) is the Einstein coefficient of spontaneous emission \(A_{ji}\). The exponential behavior of \(P_j(t)\) ensures that the power spectrum of emission will have the same form as the classical result, namely, \[I(\omega)=\frac{\Gamma /(2 \pi)}{\left(\omega-\omega_0\right)^2+(\Gamma / 2)^2}\label{14.2.16}\]

If the transition takes place between two excited levels, which can be labeled \(u\) and \(l\), the broadening of which can be characterized by transitions from those levels, then the value of gamma for each level will have the form \[\Gamma_u=\sum_{i<u} A_{u i} \quad \Gamma_l=\sum_{i<l} A_{l i}\label{14.2.17}\]

The power spectrum of the transition between them will then have the form of equation \ref{14.2.16}, but with the value of gamma determined by the width of the two levels so that \[\Gamma=\Gamma_l+\Gamma_u\label{14.2.18}\]

c. Ladenburg f-value

Since the power spectrum from the quantum mechanical view of absorption has the same form as that of the classical oscillator, it is common to write the form of the atomic absorption coefficient as similar to equation \ref{14.2.12} so that \[S_\omega=\frac{\pi e^2}{m_e c} f_{i k} \frac{\Gamma_{i k}}{\left(\omega-\omega_0\right)^2+\left(\Gamma_{i k} / 2\right)^2}\label{14.2.19}\]

The quantity \(f_{ik}\) is then the equivalent number of classical oscillators that the transition from \(i\rightarrow k\) can be viewed as representing. If you like, it is the number that brings the quantum mechanical calculation into line with the classical representation of radiation damping. If the energy levels are broad, then the transition is much more likely to occur than one would expect from classical theory, the absorption coefficient will be correspondingly larger, and \(f_{ik}>1\). The quantity \(f_{ik}\) is known as the Ladenburg f value or the oscillator strength. However, the line profile will continue to have the characteristic Lorentzian shape that we found for the classical oscillator.

Since the \(f\) value characterizes the entire transition, we expect it to be related to other parameters that specify the transition. Thus, the \(f\) value and the Einstein coefficient of absorption are not independent quantities. We may quantify this relation by integrating equation \ref{14.2.19} over all frequencies and by using equation \ref{14.1.4}, substituting into equation \ref{14.1.3} to get \[\begin{aligned}
\frac{\pi e^2 f_{i k}}{m_e c} \frac{1}{2 \pi} \int_0^{\infty} \frac{\Gamma_{i k} d \omega}{\left(\omega-\omega_0\right)^2+\left(\Gamma_{i k} / 2\right)^2} & =\frac{e^2 f_{i k}}{m_e c} \int_{-\omega_0}^{\infty} \frac{a d x}{a^2+x^2} \\
& =B_{i k} \frac{h \nu_0}{4 \pi}
\end{aligned}\label{14.2.20}\]

where \(a=\Gamma_{ik}/2\), and \(\nu_0\) is the frequency of the line center. If we make the assumption that the line frequency width is small compared to the line frequency, then \(\Gamma_{i k} / \omega_0 \ll 1\) and equation \ref{14.2.20} becomes \[f_{i k}=\frac{m_e c}{\pi e^2} \frac{h \nu_0 B_{i k}}{4 \pi}\label{14.2.21}\]

Thus the classical atom can be viewed as radiating or absorbing a damped sine wave whose Fourier transform contains many frequencies in the neighborhood of the line center. These frequencies are arranged in a symmetrical pattern known as a Lorentz or damping profile characterized by a specific width. The quantum mechanical view changes very little of this except that the transition can be viewed as being made up of a number of classical oscillators determined by the Einstein coefficient of the transition. In addition, the classical damping constant is replaced by a damping constant that depends on all possible transitions in and out of the levels involved in the transition of interest. The term that describes this form of broadening is radiation damping and it is derived from the damped form of the absorbed or emitted photon wave train, as is evident from the classical description.

The broadening of spectral lines by this process is independent of the environment of the atom and is a result primarily of the probabilistic behavior of the atom itself. In cases where external forms of broadening are small or absent, radiation damping may be the dominant form of broadening that effectively determines the shape of the spectral line. When this is the case, little about the nature of the environment can be learned from the line shape. However, for normal stellar atmospheres and most lines, perturbations caused by the surrounding medium cause changes in the energy levels that far outweigh the natural broadening of the uncertainty principle. We now consider these forms.