In a study of line broadening mechanisms in low-pressure laser-induced plasmas, Gornushkina et al.10 present the following expression for the half width for Doppler broadening of an atomic line.
where is the wavelength at the center of the emission line, k is Boltzmannโs constant, T is the absolute temperature, M is the atomic mass, and c is the velocity of light. Ingle and Crouch11 present a similar
equation in terms of frequencies.
where is the Doppler half width and 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 4s4p transition for atomic nickel at 361.939 nm (3619.39 ร
) at a temperature of 20,000 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 .
(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
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 4s 4p transition at 385.9911 nm (3859.911 ร
) 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 K that would have the velocity calculated in (e).
(g) Create a spreadsheet to calculate the Doppler half width in nanometers for the nickel and iron lines cited in (b) and (e) from 3000โ10,000 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.
