Dear Claudio,

after my understanding the situation can be described as follows:

The Huang Rhys factor S describes the average number of emitted phonons.

The difference between absorbtion and emission line is delta.

The single phonon energy is hv.

The relationship between these quantities is:

S = 0,5*delta/hv

The factor 0,5 or 2(if in the denominator) arises from the fact that delta is twice the phonon shift. In emission, the emission line is reduced by S*hv and in absorbtion enhanced. The square of delta is a mystery. May be, it arises from a special form of line shape analysis.

With Regard

R. Mitdank

Dear Rudiger,

Thank you for the reply. Please note that delta is the difference in some coordinate space between the position of the two parabola (i.e. it is not a difference in energy). Both of these values are dimensionless.

I think that the square comes from the definition of the potential energy of a harmonic oscillator, which is proportional to the square of the displacement. So if delta is the displacement, then S being the excess energy divided by the phonon energy hv (with or without the factor 1/2), must depend on delta squared.

The factor ½ is merely a matter of convention, intended for conveniently casting the expression for the line-shape into the form of a Gaussian distribution [1]. No problem can arise for so long as one uses the same convention in a given analysis. For a discussion of the Huang-Rhys factor in a modern setting, you may wish to consult [2] (Chapter 8 herein).

[1] https://en.wikipedia.org/wiki/Normal_distribution

[2] S Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

Thank you for the answer. As I understand it the convention has to do with the definition of Delta, as the displacement normalized with respect to the zero-point vibration amplitude, (h_bar/m*omega)^(1/2). How one defines the amplitude is what would produce the factor 1/2.

Is it then correct to say that the amplitude in one case is taken as (h_bar/m*omega)^(1/2), and the result is S=Delta^2/2, while in the other case the amplitude is (h_bar/2*m*omega)^(1/2), and S=Delta^2?

Claudiu Iaru : My pleasure. Regarding the HR factor, as I indicated earlier, the factor ½ is just a matter of convention; you can drop it, or you can replace it by, say, 10. What is relevant is that you should then be consistent with the definition you have adopted throughout. The HR factor makes its first appearance as a dimensionless constant in the expression for the dipole correlation function [1], so that defining it without ½, in the latter expression one will have S/2, instead of S. This then leads to having S in the expression for the line-shape, instead of 2S, resulting in the appearance of this expression deviating from that of the Gaussian distribution function.

[1] See Eq. (8.34), p. 218, of the book by Mukamel (Ref. [2] in my earlier comment on this page).

Behnam Farid Thank you again. I understand what you mean by it being a matter of convention. I was trying simply to determine what the convention was in my sources, to make sure that my calculations are consistent. In my case the factor 1/2 does make some difference, since I need to make sure that the value I extract for it is consistent with the observed spectra. The values I obtain with and without the 1/2 are around 0.6 and 1.2 respectively, which would suggest very different phonon coupling behavior, as a HR factor above 1 means that the 1-phonon line is stronger than the 0-phonon line (in PL experiments). This is in contrast to my observations, so if the value were indeed larger than 1, then my calculations would need to be revised. But thanks to your clarification I've determined that not to be the case. Many thanks once again!

Claudiu Iaru : The factor 1/2 cannot be responsible for the deviation of your observations from the expected results, so long as you use the theoretical expressions that have been deduced according to the same set of conventions. The deviations can very well be attributable to the assumptions underlying the theoretical expressions. In your case, for instance the harmonic approximation for the vibrational modes (phonons) may not be valid (this approximation underlies the HR formalism). The assumption of merely two Born-Oppenheimer surfaces may also not be valid. Even if it is valid, there is a further Born-Oppenheimer approximation in the formalism. There is a also the Condon approximation. You will have to check all of these, instead of attempting to attribute the deviations of your observations from the expected theoretical results to the utterly irrelevant factor 1/2, which, I emphasise once more, is a matter of convention and cannot be of any physical consequence. Please go through the formalism as presented in Chapter 8 of the book by Mukamel and see whether you can experimentally verify the assumptions underlying the RH formalism. Any attempt to hold the factor 1/2 responsible amounts to a total waste of time.

Behnam Farid I understand your point. My issue was a bit simpler than that: I wanted to check which definition of the Huang Rhys factor (with or without the 1/2) is consistent with which definition of the displacement in the phonon coordinate. In my experience, when attempting to fit some data, at the end it is always necessary to make sure that no 'irrelevant' factors were missed, such as this 1/2. Your input has helped me do that. Thank you again!