The function that I derived above turned out to be a linear combination of the Brillouin function and its first and second derivatives. Unfortunately, however, I've "hidden my light under a bushel" for years by publishing this function in places where most interested parties might not be able to find it. Finally, I'm taking advantage of the Google search engine to lead you to it.
My original publication was relatively short and appeared in an appropriately topical IEEE conference proceedings:
"Particle size effects in the FMR spectra of fine-grained precipitates in glass," D.L. Griscom, IEEE Trans. Magnetics MAG-17 (1981) 2718-2720.After that I became so involved in many other things that I never found an opportunity to explain the derivation and usage of this function ...until one day I found myself writing up a paper for publication in a completely off-topic conference proceedings. So here is the odd place where you will need to go for these details:
"Electron spin resonance of 65-million-year-old glasses and rocks from the Cretaceous-Tertiary boundary," D.L. Griscom, V. Beltrán-López, C.I. Merzbacher, and E. Bolden, in Selected papers from the 18th International Congress on Glass, San Francisco, CA, July 10-15, 1998, R.A. Weeks, Ed., J. Non-Cryst. Solids 253 (1999) 1-22.If you can't find this reference in your library, I'll be happy to send you a paper reprint by snail mail. Please feel free to contact me at email@example.com .
Note that the application of this function to FMR most commonly involves spherical, single-domain ferro- or ferri-magnetic particles precipitated in glasses. Apropos, see, e.g.:
"Microwave resonance thermomagnetic analysis: A new method of characterizing fine grained ferromagnetic constituents in lunar materials," D.L. Griscom, C.L. Marquardt, and E.J. Friebele, J. Geophys. Res. 80 (1975) 2935-2946.The figure below displays several curves (each normalized to 1.0 at T = 0 K) calculated using my function, including non-interacting paramagnetic species characterized by spin S=1/2 and S=5/2 and small non-interacting spherical particles of magnetite of different diameters d.
"Ferromagnetic resonance of precipitated phases in natural glasses," D.L. Griscom, J. Non-Cryst. Solids 67 (1984) 81-118.
Below is a figure from the second-from-the-top publication showing temperature dependence data for the integrated ESR spectra of glasses and rocks from the Cretaceous-Tertiary (K-T) boundary associated with the impact of an asteroid 65 million years ago, creating a 180-km diameter crater at the tip of the Yucatan Peninsula and bringing about the extinction of the dinosaurs. The samples were core drilled from the crater floor (circles), K-T boundary clays from Caravaca, Spain (squares and triangles), and tektite glasses from Beloc, Haiti (diamonds). All spectra are deduced to arise from ferrite nano particles of the sizes entered into GJ(x) to achieve the illustrated fits:
The pair of slides presented below are figures taken from:
"Electron Spin Resonance Study of Fe3+ and Mn2+ Ion Clusters in 17-Year-Old Nuclear-Waste-Glass Simulants Containing PuO2 with Different Degrees of 238Pu Substitution," D.L. Griscom* and W.J. Weber, Journal of Non-Crystalline Solids (in press).The base glass glass studied contained 4.14 mole % Fe2O3, 2.79 mole %, MnO, and 1.77 mole % NiO. Fe3+ and Mn2+ are both S=5/2 ions. And since the ESR intensity of an S=1 Ni2+ ion is calculated via the GJ(x) function to be ~1/3 that of an S=5/2 ion, its contribution to the spectrum is neglected in the analysis to be described.
The following figure shows X-band ESR spectra recorded digitally for the sample without Pu-238 studied as a function of temperature from 78 K down to 4.2 K.
For each temperature, measurements were made of (a) the g=2.06 “broad-line” amplitude, (b) the g=4.3 “sharp line” amplitude, and (c) the all-inclusive numerically integrated intensity. Each of these three quantities was then divided by the non-interacting S=5/2-state ESR-intensity calculation, GJ(x) with J=5/2, pertinent to the corresponding measurement temperature and the (fixed) microwave frequency, and the results were plotted as shown in the figure below (where all three curves have been arbitrarily normalized to unity at 4.2 K).
By construction, these data points reveal the ratio of the experimental temperature dependence to the calculated temperature dependence of non-interacting S=5/2 states – which in the limit of perfect agreement would be a horizontal line in each case.
The gray shaded area in this figure is proportional to the numbers of (presumed S=5/2) paramagnetic states extant at high temperature that are progressively pair-wise removed by (amophous) antiferromagnetic ordering as the temperature is lowered. (The particular gray shaded area shown here is referenced to the open squares and assumes that the asymptote appropriate to these particular data lies at or above 6.5 on the y axis.)
Not surprisingly, the overall integrated intensity plotted in this way (small solid squares) falls between the curves for the amplitudes of the two spectral components that these integrals fully subsume. Within the accuracy of using the g=4.3 amplitude to represent its spin concentration, this well-known-to-be weakly-interacting-Fe3+ signal in glasses  is seen in this figure (solid circles) to obey the non-interacting S=5/2 theory rather well between 10 K and 78 K (i.e., a horizontal line lying between ~1.4 and 1.5 on the y axis). By contrast, both the broad-line amplitude and the total integrated intensity fall away from the asymptotic non-interacting S=5/2 behavior in a concave downward fashion as the temperature is lowered from 78 K toward 4.2 K. Therefore it is reasonably concluded that the “broad line” recorded for this is behaving speromagnetically (i.e., as an amorphous anti-ferromagnetic material).
 T. Castner, G.S. Newell, W.C. Holton, C.P. Slichter, J. Chem. Phys. 32 (1960) 668.
Thank you for visiting this site.
Please let me know if you have questions or if you find new uses for my ESR-intensity temperature-dependence function.
Best wishes, Dave