In a study of line broadening mechanisms in low-pressure laser-induced plasmas, Gornushkina et al." present the following expression for the half width for Doppler broadening $\Delta \lambda_{\mathrm{D}}$ of an atomic line.
$$\Delta \lambda_{\mathrm{D}}(T)=\lambda_{0} \sqrt{\frac{8 k T \ln 2}{\mathscr{M} c^{2}}}$$
where $\lambda_{0}$ is the wavelength at the center of the emission line, $k$ is Boltzmann's constant, $T$ is the absolute temperature, $\mathcal{M}$ is the atomic mass, and $c$ is the velocity of light. Ingle and Crouch $^{11}$ present a similar equation in terms of frequencies.
$$\Delta \nu_{\mathrm{D}}=2\left[\frac{2(\ln 2) k T}{\mathscr{N}}\right]^{1 / 2} \frac{\nu_{\mathrm{m}}}{c}$$
where $\Delta \nu_{\mathrm{D}}$ is the Doppler half width and $\nu_{m}$ is the frequency at the line maximum.
(a) Show that the two expressions are equivalent.
(b) Calculate the half width in nanometers for Doppler broadening of the $4 s \rightarrow 4 p$ transition for atomic nickel at $361.939 \mathrm{nm}(3619.39 \AA)$ at a temperature of $20,000 \mathrm{K}$ in both wavelength and frequency units.
(c) Estimate the natural line width for the transition in (b) assuming that the lifetime of the excited state is $5 \times 10^{-8} \mathrm{s}$
(d) The expression for the Doppler shift given in the chapter and in Problem $8-8$ is an approximation that works at relatively low speeds. The relativistic expression for the Doppler shift is
$$\frac{\Delta \lambda}{\lambda}=\frac{1}{\sqrt{\frac{c-\nu}{c+\nu}}}-1$$
Show that the relativistic expression is consistent with the equation given in the chapter for low atomic speeds.
(e) Calculate the speed that an iron atom undergoing the $4 s \rightarrow 4 p$ transition at $385.9911 \mathrm{nm}$ (3859.911 $\AA$ ) would have if the resulting line appeared at the rest wavelength for the same transition in nickel.
(f) Compute the fraction of a sample of iron atoms at $10,000 \mathrm{K}$ that would have the velocity calculated in (e).
(g) Create a spreadsheet to calculate the Doppler half width $\Delta \lambda_{\mathrm{D}}$ in nanometers for the nickel and iron lines cited in (b) and (e) from $3000-10,000 \mathrm{K}$.
(h) Consult the paper by Gornushkin et al. (note 10 ) and list the four sources of pressure broadening that they describe. Explain in detail how two of these sources originate in sample atoms.