Zero-phonon line position and width

To describe the effect of the change of the elastic springs on the optical spectrum of an impurity center, we use the adiabatic approximation. In this approximation, phonons are described by different phonon Hamiltonians in different electronic states. The optical spectrum, which corresponds to a transition between different electronic states is determined by the expression /( ) = const X oj1 1 I(oj) [28], where the — sign corresponds to the absorption spectrum and the + sign stands for the emission spectrum, 

In the case of the Ai-E optical transition one can present H and V as the sum of the independent terms belonging to diflFerent representations. In this case the Fourier transform Fit) is the product of the multipliers belonging to these representations. We are interested in the multiplier, which describes the contribution of the e-vibrations. Here we consider the case of strong Jahn-Teller effect. ZPL in this case is described by the optical transitions for configurational coordinates qi and q2 in the vicinity of the AP minima. This means that one can use equation (4) with V given by equation (3). In this approximation the configurational coordinates q and q2 contribute to F(t) independently. Later we consider a contribution of one of these coordinates and omit the index of the line of the representation. 

We are interested only in the temperature dependence of the position and the width of the ZPL. We suppose that coordinates q in the linear term (vq) in equation (5) are orthogonal to the coordinates contributing to the quadratic term (qwq)/2. In this case the linear term does not contribute to these characteristics of the ZPL. This allows us to exclude this term from the further consideration. 

The ZPL, being the narrowest line of the spectrum is determined by the large time asymptotic of the Fourier transform. In the large t limit pt(a ) — 8(oj). Taking also into account that pt(G) — t/2tt. one gets in the large t limit [5,6] 

Let us separate the T = 0 part and the temperature-dependent part of y and 8. The first part equals 

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