An often used approach is to use the density matrix formalism. The diagonal terms are related to the populations, and the Fourier transform of the two-times correlation function of the dipole operator gives the emission spectrum. Other approach I am using right now is to insert the thermal distribution (which defines the population of the different levels) in the Fermi Golden Rule for transitions by the dipole operator and then use it to simulate the emission spectrum.

The approach described by Hanz is sufficient for thermodynamic trends of fluorescence such as quantum confinement effect in nanoparticles. Essentially, this would relate to the measurements over the ensembles of particles under cw irradiation. However, fluorescence of individual nanoparticles (as probed by e.g. confocal spectroscopy) generally exhibits a more complex dynamics manifested in fluorescence intermittency or blinking phenomenon. The nature of blinking and its universality is a matter of open debate. It is believed to be related to the non-radiative recombination processes into the non-emmitting states and the lifetimes of such states. The main (good to address) problem here is that the recombination mechanisms seem to be very system specific while the blinking statistics is alleged to exhibit some universality. The discussion on the subject can be found in a recent paper and references therein

http://pubs.acs.org/doi/abs/10.1021/nl3035674

I found expressions for the emission spectrum intensity of optical transitions in polar semiconductor quantum dot (QD). The formulas are obtained starting from the Fermi Golden Rule and consequently the QD eigenvalues and eigenfunctions problem should be solved first. As the longitudinal optical phonon effect is taken into account, the Huang–Rhys factors of the optically active levels are necessary to be computed. Synthetically, I introduce the result obtained as follows:

1. I obtained an expression of the intensity of photoluminescence spectra (IPL) in adiabatic optical transitions (strong quantum confinement of the carriers or strong electron–phonon interaction) in T.O. Cheche, M.C. Chang, and S.H. Lin, Chemical Physics 309, 109 (2005), see in Eq. (7). The QDs are assumed of spherical shape, the eigenproblem is solved within an effective mass model of QD, and the excitonic effect is disregarded.

2. I obtained expressions which approximate IPL in the adiabatic exciton and biexciton-exciton emission spectra in T.O. Cheche, European Physics Letters (EPL) 86, 67011 (2009), see Eqs. (8, 9). The QDs are assumed of cylindrical shape, the eigenproblem is solved within an effective mass model by the configurational interaction approach. More details of derivations can be found in “Influence of Optical Phonons on Optical Transitions in Semiconductor Quantum Dots”, Quantum Dots / Book 1, ISBN 979-953-307-308-7, Ed. Dr. Ameenah Al-Ahmadi, InTech, 2012 – Open access (www.intechopen.com/books/fingerprints-in-the-optical-and-transport-properties-of-quantum-dots/influence-of-optical-phonons-on-optical-transitions-in-semiconductor-quantum-dots).

3. For non-adiabatic optical transitions, I found approximate expressions of IPL in T.O. Cheche, M.C. Chang, Chemical Physics Letters 406 479 (2005). The QDs are assumed of spherical shape, the eigenproblem is solved within an effective mass model of QD, and the excitonic effect is disregarded.

I hope information will be helpful to you.

Mr Cheche Tiberius gives very good examples.

Please follow them.

Thank you for all those interesting answers and discussions. I guess I have to explain more precisely the case I am working on.

In fact I am modeling Up conversion nanocrystals , namely fluorides codoped by Rare earth ions Yb3+ and Er3+ because of their uniquely efficient 4f==>4f transitions. Now my problem comes out when I was fitting the experimental emission intensities simply by generating a solution of a set of rate equations that I established in my model.

The aim is to quantify the energy transfer between Yb (donor) and Er(acceptor) ions. Therefore The complexity of our energy diagram and the corresponding rate equations make it very difficult to solve as any other. Without speaking of the numerical difficulties, the problem came out in fitting two emission bands originating from one unique energy level. Now I am trying to apply the derivation of the intensity given in http://www.sciencedirect.com/science/article/pii/S0925346710000844 because we are working with nonlinear effects induced at low excitation intensities and they derived a formula for it which would permit me to extract several emission intensities from one level population kinetics. However I will try to apply the approaches used for Quantum dots, that all of you proposed, to my system of Lanthanides embedded in nanocrystals.

Thank you again to all of you, I just wanted to emphasize that we deal with nonlinear processes such as energy transfer phenomena in which case our RE ions could undergo different kind of interactions than dipole-dipole ones.

Dear Urs,

I would like to ask you for some sold basis references to investigate more deeply the Non-equilibrium green function's formalism because what you proposed about the distribution of size and shapes and their effects are also of great interest for us for improving samples' quality.

Thank you

Dear Urs ,

Thank you infinitely for those references, I am studying them right now .

And I am, since 6 months, interested in precisely this problem of modeling upconversion phenomena in such materials.

If you have any supplementary informations or relevant references concerning this subject please feel free to send it to me through this page or my personal email :

[email protected]

Thank you again,

Best Regards,

Ziyad

Hi Ziyad,

I would also recommend the reading above suggested by Urs Aeberhard - a lot of effort went into getting this rate equation model right, although it has only be done for NaYF4 doped with 20% Er3+. We have the experimental results for everything from 5% - 75% Er3+ doping under a wide range of monochromatic and broadband power densities, but no-one has attempted to redo the rate-equation model for this yet....! This kind of material is relevant to those of us working on the application of up-conversion to silicon solar cells, and therefore does not include the Yb3+ like you are planning to.

Best regards

Bryce