Question about energy transfer between Rare-Earth ions

[HAYAO]

Dear all,

I have a question about one of the values presented in a paper, which is crucial in calculating energy transfer probability between two Yb³⁺ ions.

This is the paper:
T. Kushida, "Energy Transfer and Cooperative Optical Transitions in Rare-Earth Doped Inorganic Materials. I. Transition Probability Calculation", J. Phys. Soc. Jpn. 1973, 34, 1318-1326. DOI: http://dx.doi.org/10.1143/JPSJ.34.1318

The summary of this paper is that it uses similar approach to Judd-Ofelt theory (spherical harmonics with Coulomb interaction operator and extensive use of reduced matrix elements) to derive equations for dipole-dipole, dipole-quadrupole, quadrupole-quadrupole interactions. For the calculation of the actual values of these rates, one needs reduced matrix elements of the electronic transition.

In line 27 and 36-37 of page 1323, the author presents the reduced matrix element values for ²F_5/2→²F_7/2 transition of Yb³⁺ ions, namely <f^{13 2}F_5/2||U⁽²⁾||f^{13 2}F_7/2> = 6/49, <f^{13 2}F_5/2||U⁽⁴⁾||f^{13 2}F_7/2> = 20/49, and <f^{13 2}F_5/2||U⁽⁶⁾||f^{13 2}F_7/2> = 6/7. However, there is no reference to where these values came from. If it is a theoretical value, it still wouldn't make sense because Judd-Ofelt calculations for Yb³⁺ is impossible considering that it only has one excited state level (Judd-Ofelt analysis requires several to make sense).

How do you get these parameters?

Actually, there is another paper by O.L. Malta:
O. L. Malta, "Mechanisms of non-radiative energy transfer involving lanthanide ions revisited", J. non-Cryst. Solids 2008, 354, 4770-4776. DOI: http://dx.doi.org/10.1016/j.jnoncrysol.2008.04.023

In page 4775, there is a calculation of energy transfer rate, but also provides no values they used for the calculation. They provide ΣΩ_k<f^{13 2}F_5/2||U^(k)||f^{13 2}F_7/2>, but that is only enough to calculate dipole-dipole, and every other multipole calculation requires reduced matrix element of each tensor rank.

Am I missing something in the theoretical or experimental aspect?

Thank you,
HAYAO

 

[DrDu]

Wikipedia cites a slideshow, where the definition of these parameters is given. Basically some combination of Racah parameters.
https://www.slideshare.net/bmwalsh/juddofelt-theory-principles-and-practices-presentation.

 

[HAYAO]

Wikipedia cites a slideshow, where the definition of these parameters is given. Basically some combination of Racah parameters.
https://www.slideshare.net/bmwalsh/juddofelt-theory-principles-and-practices-presentation.

Thank you DrDu, but I am very aware of that and it is precisely the reason why I am confused in the case of Yb³⁺.

Racah parameters for rare Earth ions are derived experimentally. A typical procedure requires absorption spectrum of multiple levels (to derive energy of each states), and a fitting is done with theoretical equation to obtain these three Racah parameters as well as spin-orbit coupling constant (and in some cases several other constants). In principle, this means that Racah parameters cannot be derived for Ce³⁺ and Yb³⁺ ions experimentally because the energy level difference is dependent only on the spin-orbit coupling constant for two-level case and the values for Racah parameters are arbitrary.

Consequently, almost all reference I have seen that provides Racah parameters and Spin-orbit coupling constant for all lanthanide ions, shows only spin-orbit coupling constant for Ce³⁺ and Yb³⁺ and no Racah parameters. Without Racah parameters, one is unable to calculate reduced matrix elements in the intermediate coupling scheme.

There is one, a theoretical work, that derives Racah parameters for all lanthanide ions, but they deviate quite off from the experimental values. Thus I cannot trust the values they provided for Yb³⁺.

That means there must be some other way of obtaining racah parameters, or way of directly obtaining the reduced matrix elements for Yb³⁺ and Ce³⁺, in which the author of the paper I have shown above decided not to mention. I want to know how they did it.

 

[DrDu]

From what I understand, the parameters U depend very little on the environment of the f orbitals and can therefore be calculated analytically. However, I have no access to the article you are citing.

 

[HAYAO]

From what I understand, the parameters U depend very little on the environment of the f orbitals and can therefore be calculated analytically. However, I have no access to the article you are citing.

Yes indeed. U as well as F_(k) are only slightly affected by the environment around f orbitals (they are still affected and there are tons of references out there reporting these parameters for different hosts, except for Yb³⁺ and Ce³⁺).^**

But that have nothing to do with obtaining U^(k) itself. They cannot be obtained analytically because we have 4 unknown parameters (5 if you include F⁽⁰⁾ and 9 if you include other parameters for precision) that are independent. It is entirely done by numerical method, most commonly least square fitting of energies of levels and you definitely need more than 2 levels to do this.^**It should be well noted that it is sometimes a poor choice to be overconfident about this fact because some hosts do deviate from average value. For example, using parameters obtained in LaF₃:Er³⁺ and applying it to Y₂O₂S:E³⁺ may lead to error by 17.5% in U⁽²⁾! However, the common practice is to apply parameters obtained in a certain host to another host, and most of the time it works fine, especially if you re-optimize the F^(k) and spin-orbit parameters.

 

[HAYAO]

I know this is an old thread that I made quite a while ago. I am just stopping by because now I understand, and that anyone with similar problem can now understand by coming across this thread.The case for Yb³⁺ and Ce³⁺ is that they can be considered as only having one 4f-electron (Yb³⁺ has 13 but that is same as saying there is one hole). In this case, we can ignore electrostatic interaction Hamiltonian between multiple electrons, and only consider the spin-orbit coupling Hamiltonian. This means that Racah parameters are not needed. Spin-orbit coupling mixes states of different S (total spin angular momentum) and L (total orbital angular momentum), but of the same J (total angular momentum). In the case of Yb³⁺ and Ce³⁺, since there is only one 4f-electron, there are only two 4f-electronic state, i.e.: ²F_5/2 and ²F_7/2. We can see here that the spin-orbit coupling will not mix these two states, and only separates them in energy by spin-orbit coupling coefficient (the order of these two states in energy is reversed between these two ions).

In the calculation of the reduced matrix elements mentioned in the OP, we can simply calculate the values by typical tensor operator techniques, hence the values: <f^{13 2}F_5/2||U⁽²⁾||f^{13 2}F_7/2> = 6/49, <f^{13 2}F_5/2||U⁽⁴⁾||f^{13 2}F_7/2> = 20/49, and <f^{13 2}F_5/2||U⁽⁶⁾||f^{13 2}F_7/2> = 6/7. As such, in the calculation of the energy transfer rate constant, it does not require the actual knowledge of the Ω_(k) parameter (the Judd-Ofelt parameters) for Yb³⁺ and Ce³⁺ because no wavefunctions are mixed. We can simply obtain the entire sum ΣΩ_k<f^{13 2}F_5/2||U^(k)||f^{13 2}F_7/2> from absorption measurement because the value is directly related to the oscillator strength (more precisely the electric-dipole transition oscillator strength).

So the problem I had in OP was that I was not aware of the fact that Yb3+ and Ce3+ were exceptions out of all the other lanthanide ions. Calculations for these ions are simplified, which is probably why the author of the paper (T. Kushida) chose to use Yb³⁺ as an example.Thank DrDu for your help, though.

HAYAO

As a side note, mixing of different J-states by crystal field (which is small in lanthanides) is ignored in the paper.