What does the following notation mean?

$$A_{20}\left<r^2\right> = -100 \mathrm{K}$$

Where $A_{20}$ is termed a Crystal Electric Field Parameter.

I often see this notation when reading papers on the magnetic properties of materials from first principles. What is a Crystal Electric Field parameter and why is its value given in Kelvin?

An example of this can be found in the following paper: http://dx.doi.org/10.7566/JPSJ.83.043702, which has the title:

First-Principles Study of Magnetocrystalline Anisotropy and Magnetization in NdFe12, NdFe11Ti, and NdFe11TiN

In this paper they discuss the changes to the magnetic characteristics of the NdFe12 crystal structure with the addition of Titanium and Nitrogen. To do so they talk about the aforementioned $A_{20}$ Crystal Electric Field Parameter, the Crystal Electric Field is first mentioned in the following paragraph:

The crystal electric field (CEF) combined with strong spin–orbit coupling causes the anisotropy as follows. The CEF breaks symmetry of the space for the f electrons. It determines stable direction of the orbital magnetization. Then, the spin moment, coupled to the orbital magnetization by strong spin–orbit interaction, is aligned in parallel (in the case of heavy rare-earth element) or anti-parallel (light rare-earth) to the orbital moment. The spin direction is mediated to transition metal sites via R-5d electrons, which fixes the direction of large magnetization of the whole compound. Therefore, the CEF parameters at the R sites are good measure for the strength of magnetocrystalline anisotropy.

After this initial paragraph they define how they calculate the Crystal Electric Field Parameter:

The CEF parameters are defined by expanding the potential $v_{CEF}$ at the $\mathbf{R}$ site by the real spherical harmonics $Z_{lm}$ as $$v_{CEF} = \sum_{l=0}^{\infty}\sum_{m=-l}^lA_{lm}\frac{\left<r^l\right>}{F_{lm}}Z_{lm}(\hat{r})$$ where radial dependence is integrated out using the 4f atomic orbital $\frac{\phi}{r}Y_{lm}(\Omega)$ as $X \equiv \int X\phi^2(r)dr$. Here, $A_{lm}$ is the CEF parameter, and $F_{lm}$’s are numerical factors, whose explicit expressions can be found in, e.g., Ref. 8.

$A_{20}$ is one of these CEF parameters, and is the one of interest because it defines in plane or out of plane magnetisation in this particular crystal structure.

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    $\begingroup$ +1. Out of curiosity, are you familiar with crystal field splitting? Also, it wouldn't be a bad idea to give some context, such as one example of the place where you "often see this notation when reading papers"! $\endgroup$ Commented Oct 25, 2021 at 1:55
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    $\begingroup$ If you can find it and write a self-answer, it would be very appreciated since it would help future users that visit the site! I would also help us narrow down our unanswered queue which has become quite and unmanageable huge lately! $\endgroup$ Commented Jan 31, 2023 at 23:59
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    $\begingroup$ Well it looks like they are the coefficients in a multipolar expansion of the Crystal Electric Field. To be sure I would have to check I understand what is meant by that, and quite exactly what the F_lm "numerical factors" are, but to me at least that looks like the essence of what A_lm are. $\endgroup$
    – Ian Bush
    Commented Feb 1, 2023 at 7:44
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    $\begingroup$ Unfortunately some of the papers I would need to read are behind a paywall (ref 8 for the F_lm terms, maybe others) - I might be able to access them when I am on a machine that has VPN for my institution set up, but don't hold your breath, the answer will require some work and I am busy at the moment. $\endgroup$
    – Ian Bush
    Commented Feb 1, 2023 at 11:14
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    $\begingroup$ @Connor OK, this is going to take a lot more time than I have now to answer this - this is the first time I have come across this stuff and I would need to work carefully through the maths in a few of the papers referenced from the above to work out exactly what need to be said - the two papers in reference 8 are much more careful in defining everything than the paper you reference above and might be a good place to start. $\endgroup$
    – Ian Bush
    Commented Feb 2, 2023 at 10:33

1 Answer 1


This answer is based on fuzzy knowledge, if something seems obviously wrong, it probably is. Comments are welcome, and a better answer will be accepted.

Addressing the issue of units first, please see this question and answer. Kelvin are used in studies like this because temperature is a key factor in the preservation of magnetic state. Thermal fluctuations can readily knock atomic spins out of global alignment and thus remove permanent magnetism. The average amount of energy in a thermal fluctuation is related to temperature by $k_BT$, so energies are given in Kelvin to indicate how well the state is preserved at temperature.

This is where it gets a bit more speculative. Crystal Electric Field parameters are weighting parameters in a combined sum that defines the "crystal electric field" as a combination of underlying functions. The crystal electric field is the electric field produced by the subatomic particles of the crystal. It's called the crystal electric field because the electric field is uniquely associated with the crystal's structure.

I'm not aware of whether the underlying functions are analytical, or empirical. I suspect one could use either, but, hopefully, somebody with more domain knowledge can step in and give a definite answer.


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