Stark effect and line broadening in three-dimensional stochastic fields

A theory is given for (i) the linear Stark effect of the substates of the first excited level of the hydrogen atom and (ii) the linear Stark broadening of the Lyman α line, in a three-dimensional stochastic electric field. The latter is represented in the form of a three-dimensional Fourier series with different random amplitudes and random phases for each vector component of the wave modes, which simulates a perturbing vector field varying with respect to intensity and direction in a random manner during an atomic transition. The line-shape function, which is a moment of the distribution function of the random amplitudes and random phases of the stochastic field, is evaluated by statistical methods. It is shown that the Lyman α line consists of a sequence of discrete spectra (Fourier analysis) at the frequencies ω=nωp, n=0,±1,±2, etc., where ωp is the plasma frequency. The line intensity decreases exponentially with the square of the order number n. The half-width of the spectral line is proportional to the square root of the mean square stochastic electric field. In the case of electron acoustic turbulence, the half-width is related to the unperturbed electron pressure. These results permit a quantitative determination of the intensity of the stochastic electric field and the average electron pressure in turbulent plasmas.