The Doppler analog of the Elsasser band model

Abstract

An expression has been obtained for the spectral absorption coefficient of a distribution of 2 N + 1 equally intense, equally spaced Doppler broadened spectral lines of equal half-width in the limit of large N, i.e., the Doppler analog of an Elsasser band. The expression is P ω=P 0 exp α 2 4β 2 θ 3 πα 2β 2 πi β 2 = S d θ 3 α 2i − β 2 πi where P o¬ 1 n2 π 1 2 S γ , α¬ 2πi(ω−ω 0) d , β= πγ d 1 n2 π 1 2 and the function θ 3( u| τ) is the third Jacobi theta function. Several interesting properties of the distribution have been obtained, namely, (1) P ω±d=P ω , (2) P ω0±nd=P 0θ 3 0 πi β 2 is a maximum, (3) P ω0±(n+ 1 2 )d =P 0θ 2 0 πi β 2 is a minimum, and (4) the ratio of the minimum to maximum values of the absorption coefficient is k 1 2 , where k is the modulus of the complete elliptic integral of the first kind, K( k). This last result may be used to establish conditions for the validity of limiting cases. For example, for values of k 1 2 < 0·01 (corresponding to γ ⧸ d < 0·18), the band may be considered to consist of isolated lines; conversely, for values of >k 1 2 > 0·99 (corresponding to γ ⧸ d > 0·65) the lines are so highly overlapped that the absorption coefficient may be considered constant with a value P ω=P 0 K(k) 2π 1 2 (1+k 1 2 )= S d K′(k) 2π 1 2 (1+k 1 2 )