Ferromagnetic resonance is not under control of Kittel's equation. Nature 'knows' what to do regardless of our models, often simplified. The Kittel's equation assumes uniform magnetization of the sample. This is only possible (in uniform external field) when the sample boundary has shape of second order polynomial, in practice it is an three-axial ellipsoid. An ordinary sphere, infinitely long straight line or just a very thin plate are also possible as limiting cases. In all other cases the demagnetizing tensor has components with varying values from point to point. But there is one more assumption: the overall sample size is much smaller than the wavelength of exciting microwave radiation, say below 3 mm at X-band (~9 GHz). Maybe the small cylinder with aspect ratio exceeding 4 is close enough to infinitely long straight wire; approximates it well enough.

For macroscopic samples (single crystals) it is not true that their anisotropy constants are size dependent. It is the so called magnetocrystaline anisotropy, depending only on elementary cell structure. At micro- and nanoscale one also has to take into account the surface anisotropy. In this sense the overall anisotropy appears as a size-dependent quantity. On the other end, in case of relatively bulk polycrystaline samples we have an effect of disappearing magnetic anisotropy the bigger the sample. This results from unavoidable averaging of randomly oriented easy/hard magnetic axes (but not in grain-oriented steel!).

Your another claim that magnetization is size-dependent is clearly false if you mean saturation magnetization. I'm not sure how about coercivity: is it or isn't it size-dependent? But I am sure that it is orientation-dependent, even in your case of high aspect ratio cylinder.

Dear Marek,

Your sentence:

Your another claim that magnetization is size-dependent is clearly false if you mean saturation magnetization.

is wrong. The magnetization M depends of the number of spins in a given volume, the magnetocrystalline energy depends also of the volume and the balance between the magnetocrystalline energy with respect to the magnetostatic one creates magnetic domains. They depend thus very strongly of the size and also their walls.

Dear Mohammad,

Your question is not true what is assumed. The resonance frequency depends in general on the form of the specimen, and we have a simple result only when the specimen is in the form of a general ellipsoid, and here only when the dimensions of the specimen are small enough in relation to the skin depth and to the wavelength in the material so that the radio frequency field penetrates uniformly and so that the phase of the radio frequency field is substantially the same throughout the specimen

Dear Baldomir,

Aren't you too quick? You have written: The magnetization M depends of the number of spins in a given volume ... I fully agree. But what is the consequence of the sentence above? In other words, the magnetization is nothing else but density of magnetic moments. Why should his quantity be size dependent? If it is not, then also the saturation magnetization has to be size insensitive.

Next: please don't mix magnetocrystalline anisotropy with its density. Magnetocrystalline energy is written as K*V, with K being anisotropy constant, and V- sample volume. Maybe I wasn't clear enough using the name of 'overall anisotropy' for K*V and just 'anisotropy' (not anisotropy density) for K - but this is common practice in contemporary literature.

Last: there are no domains in ferromagnetic resonance experiments, nor in its description as expressed by Kittel's formula. The sample is assumed to be magnetically saturated. On the other hand, many years ago, our group was investigating ferromagnetic resonance in small spherical garnet sample. The sample had no earlier magnetic history and the experiment was carried at fixed microwave frequency and with sweeping external field. During first stage, i.e. when field was increasing from 0 to 0.5 T, we have observed literally a forest of absorption bands, 50 lines or maybe more. Up to some point when this picture suddenly disappeared and no more absorption lines were visible. During second stage, when the field was decreasing, only a single, strong absorption line was detected. We were unable to replicate this result with this sample, just because it became magnetically saturated (single domain) and was small enough to remain in this state, without creating domain structure. I was lucky to witness this phenomenon; it happened only once in my life.

In summary: your comments are generally true, but not tightly related to ferromagnetic resonance (the subject of this thread).

Dear Marek,

I have only answered to your sentence:

Your another claim that magnetization is size-dependent is clearly false if you mean saturation magnetization

This is wrong, if you have magnetic domains or not. Obviously if you have magnetic domains as it corresponds to a big sample, the creation of domains do the sentence even worse.

Dear Marek,

I don't know also how can you write the sentence:

Last: there are no domains in ferromagnetic resonance experiments, nor in its description as expressed by Kittel's formula

There are many experiments of ferromagnetic resonance when the material has more than one domain. Let me to show you

http://iopscience.iop.org/article/10.1143/APEX.4.063006/meta

Article Stripe domains in Permalloy films as observed by ferromagnet...

The formalism is not Kittel but LLG or others. The question made is obviously wrong and confuse in all its terms as I have written in a previous post to Mohammad. It is not sense to restrict you to its constraints.

Dear Daniel and Marek,

Sorry for getting back late, I was trying to digest your valuable knowledge, far behind my understanding.

I did FMR measurements for Co, Fe and Ni nanowires at different aspect ratios as arrays where the porosity, the density of the nanowires, was different too. I took the porosity into account as it is reported in literature.

For some cases, my results are matching the previous literature, of course, following the same trend of the Kittle equations. But it is not the case for all samples...

So, the first thing that popped in my mind is that it may be due to the aspect ratio, in other word, the size of nanowires. That is why the Kittle equation does not correlate the aspect ratio or size of the samples to resonance frequency. It has demagnetization tensor terms, but they don't change significantly in a way to impact the deviation that I see. I am really confused...

Dear Mohammad,

Unfortunatally these calculations are not so simple and the demagnetiziong field depends of many variables. The Kittel ferromagnetic resonance equation depends strongly, in general, on the form of the specimen. But we have a simple result only when the specimen is in the form of a general ellipsoid, and here only when the dimensions of the specimen are small enough in relation to the eddy currents and to the wavelength in the material so that the radio frequency field penetrates uniformly. So that the phase of such a field is substantially the same throughout the specimen. In the case of a plane specimen these requirements might be relaxed, however, for geometrical reasons. But where it is straightforwardly to obtain an expression for the resonance is in an ellipsoide.

There is another usual form to present this information if you have a single crystal. The resonance frequency depends on the angle the magnetization makes with the crystal axes and therefore you can write the Kittel equation in function of the crystalline anistropy and also of the other anisotropies. Thus you have to take into account the forma and microscopic features of the material as the spin-orbit, etc...for a very accurate calculation. In most of the cases you can find their values in the literature.

Dear Baldomir,

You have written: The question made is obviously wrong and confuse in all its terms as I have written in a previous post to Mohammad. It is not sense to restrict you to its constraints. For me the essence of Mohammad's question is: it seems that something is wrong with Kittel's formula for ferromagnetic resonance, it sometimes fails to describe the experimental results. Well, he also presents some ill-funded suppositions which need corrections. This is obvious, at least for me. But why Mohammed presents his problem? Just because he would like to hear some hints, maybe some pointers to the literature being still unknown to him.

Going back to the paper you have mentioned. I have a feeling that you have seen only its headline. Within the body of this article you can see nice domain structure which practically disappears above |H| = 200 Oe and FMR signal observed at H = 1000 Oe, together with estimated HS (saturation field) estimated as 105 Oe. In my humble opinion all this is in contradiction with article's title.

Here is my point of view: Kittel's formula was created long ago, when we were able to observe ferromagnetic resonance in macroscopic and non-conductive samples, treated essentially as continuous bodies, with their anisotropic properties originating (eventually) from their crystal structure. The observations of FMR in conductors was impossible for two reasons: small penetration length and perhaps even more importantly because of loses created by eddy currents, efficiently masking any other absorption signal. Situation has changed when nanoscale samples become available. Surprise: in thin layers more than a single absorption line is observed in good quality samples! This is usually ascribed to SWR, i.e. spin wave resonance, or, perhaps even more correctly, to the standing wave resonance. In this picture the old good Kittel ferromagnetic resonance becomes a member of grater family, corresponding to standing wave with infinite wavelength (mode 0). In the article mentioned we can clearly see three absorption lines (in thickest sample) - one too much to explain the observation via stripe domain structure. Nanowires are probably even more convenient objects than thin layers to investigate this phenomenon, obviously related to the sample size (more precisely: to their length). All is fine and easy to understand when we consider exactly one wire. The presence of a second, third (and more) wire nearby the first one complicates the analysis enormously, even if they are parallel to each other (long distance dipolar interactions enter the game). A single, wide and asymmetric absorption line appears in place of a nice series of isolated peaks.

Dear Marek,

Ferromagnetic resonance is a technique of measuring the magnetization dynamics. In the case of the nanowires it is one of the techtniques to see the influence of the spin transfer torque on a domain wall. here are more than one frequency of resonance as Kittel formula gives.

Yes, I agree with you that Kittel formula is old and in fact a very special case of LLG equations (Landau-Lifshitz-Gilbert). For instance, small angle precession approach,i.e. e linearization in vicinity of equilibrium state, gives Kittel formula for the eigenfrequency of the magnetization precession in a thin film.

What was wrong in the question is that the Kittle formula obviously depends strongly of the geometry of the sample as it ought to be if the demagnetizing field plays an important role. But most of my remarks were devoted to your explanation better than to the question itself.