There are different spin wave modes supported by magnetic materials. Energy gap arises, when you have non-overlapping spin wave bands... (for example in magnonic crystals - where you have localized standing spin wave modes...)

Both are related due to spin-orbital coupling. The excitations below the ordering temperature are called spin waves or magnons because they involve many lattice sites (while the single-ion anisotropy is not dispersive). When there is not anisotropy, such as in Gadolinium metal, there is no gap, the excitation spectrum is the simplest to be found among the Rare Earths. However, for the other Rare Earths crystal field and magnetoelastic anisotropies modify the excitation spectrum significantly, inducing an energy gap at long wavelengths. You can find more information in J. Phys.: Condens. Matt. 24 (2012) 213201

In general a spin wave gap arise due to anisotropic terms in the spin Hamiltonian. This usually originates from spin orbit coupling or alternatively a dipole-dipole interaction. Anisotropy can originate from either orbital ordering or structural transformation, when this is brought to spin degrees of freedom through the spin orbit coupling. A change in spin order could also increase a spin gap if it lower the symmetry of the system, then probably to a change in the dipole interaction.

If you consider a system with Heisenberg spins, additional terms which cause anisotropy are usually responsible for the gap. This anisotropy usually arises from spin orbit coupling. A classic reference on the measurement and the size of these gaps is Abragam and Bleaney "Electron Paramagnetic Resonance in Transition Metal Ions."

Quantum Magnetism also affords another mechanism for a gap in the spin wave spectrum. In integer spin systems this is known as a Haldane Gap. Basically the fundamental excitation is a pair of spin 1/2 excitations, known as spinons. Unless there is a uniform coupling between inherently s=1/2 particles, there is a spin gap. Therefore similar phenomena are seen in integer spin chains, even leg spin ladders, and alternating spin chains. The really novel system is then the uniform S=1/2 chain which is gapless. The literature on quantum magnetism is too extended to comment on here.

The simplest way to think about it is to imagine finite ferromagnetic ising chain. In such a chain there is a threshold for creating an excitation corresponding to twisting the chain at a given point by one pi. This is the cause for the gap in an ising system. The next step is also straight forward, imagine the spins having a slight anisotropy instead of being ising like. The anisotropy will give rise to a gap in the excitations, which to the first approximation is proportional to the anisotropy. When the anisotropy goes to infinity, you have the other extreme, the ising chain. Garrets gave you a hint where you can find the backround needed for this simple picture.

I regret much that I have to contradict above explanations of the origin of the magnon gap. One unexplained but important experimental observation is that in isotropic magnets with integer spin there is a sizeable magnon gap but in isotropic magnets with half-integer spin there is virtually no gap. This systematics we have described in Chap. 13 of our book in Springer Series in Material Science #127. As an explanation of this phenomenon one must consider that the dynamics of ordered magnets is not controlled by magnons but by field bosons (Goldstone bosons). Consequently, only field theories can give correct description of the Dynamics. When these bosons have a magnetic moment (a non linear dispersion) they are able to interact with magnons (see Acta Physica Polonica 121 (2012) 1176). This interaction is at q=0 only and manifests as magnon gap. In the spin wave dispersions an analytical change occurs at small q-values. This analytical (!) change cannot be described within one theory. Note that magnons with q=0 (the gap) have different temperature dependence than magnons with finite wave vector. This proves their different origin. Magnons with finite wave vector persist in the paramagnetic phase! The situation is different in Ising magnets. Only in Ising magnets the dynamics is controlled by magnons. Since the field particles are essentially magnetic dipole radiation emitted upon precession of the magnetic moments it becomes evident that in Ising magnets there are no field quanta. The field gets not filled with bosons!

I think that Ulrich's answer is more precise and complete than that given by me in an straightforward way.

Thank all of you for your advice! I have further questions for discussion. Let us take two systems as the examples: MnV2O4 and FeCr2O4.

In MnV2O4, there is paramagnetic(PM)-collinear ferrimagnetic(FI) transition at 57 K (no structural transition), followed by another cubic collinear FI- tetragonal noncollinear FI transition at 53 K. The energy gap only emerges below 53 K. Why is there no energy gap at 57 K when the magnetic transition happens? In my understanding, even there is no structural transition or spin-orbital coupling at 57 K, the paramagnetic(PM)-collinear ferrimagnetic(FI) transition should induce the single-ion anisotropy, which is supposed to lead to the energy gap at 57 K.

In FeCr2O4, the cubic- tetragonal structural transition happens at 135 K (Fe becomes ferro-orbital ordered), followed by tetragonal PM- orthorhombic FI transition at 80 K. The energy gap does not emerge below 135 K and only occurs below 80 K. I also cannot understand why does the ferro-orbital ordering of Fe2+ not induce an energy gap? The ferro-orbital ordering of Fe2+ could induce the anisotropy even there is no spin-orbital coupling.

From these two cases, it turns out the simultaneous occurrence of structure and magnetic transition with the spin-orbital coupling is necessary to induce the energy gap. Is it true?

Thanks again for your advice!

I do not know the two magnetic materials in detail but quite generally it seems important to consider the following: there are no Heisenberg interactions between chemically different magnetic atoms or ions. Free exchange of electrons requests a mirror plane between the two magnetic species. This commonly is the case between identical magnetic atoms only. As a result, when there are two types of magnetic atoms I expect two distinct ordering temperatures. This effect is most clearly observed in the perovskites containing 3d and 4f elements. The 3d sytem orders at ~500 K but the RE sublattice at a few K only. Of course a weak interaction (by dipolar forces) always is possible. Moreover, the Mn2+ ion has not a stable spin quantum number. For instance in MnO the effective spin is Seff=2.

U. Köbler: consider an FeCo intermetallic compound. It's a ferromagnet with a single ordering temerature? What would be the nature of interaction(s) between neighboring Fe and Co atoms?

We might also consider an atomic mixed dimer FeCo. How would you view it? I don't get your point above.

I agree with kai Fauth. For example R2Fe17 has only one ordering temperature, but in RMn12 the R and Mn ions has different ordering temperatures, because there is not coupling between different sublattices, owing to local symmetry as Ulrich pointed out

Qiang in the materials you mention the energy gaps are probably related to the orbital order. For more information take a look at arXiv:1403.0269v1 and references therein.

Jesus, we should see what U. Köbler exactly means by "Heisenberg interactions". What exactly is the nature of coupling in the R2Fe17's?

Kai, in R2Fe17 there are three exchange interactions that can be distinguished: the exchange interaction JTT between

the transition metal spins, JRT between the rare earth spin

and transition metal spins, and JRR between rare earth spins.

JTT is dominant and responsible for the Curie temperature,

which should be well above room temperature for application

purposes. A strong R–T interaction is important to

maintain the large magnetocrystalline anisotropy of the R

ions up to the highest possible temperature and plays an

important role in the formation of the magnetic structure.

JRR is the RKKY interaction and the strength is very weak

compared with JTT and JRT , so that in the most cases it can

be neglected

Garrett, I know the paper you mentioned well. I have some reservations on the interpretation in that article. They ascribed the energy gap below 110 K to the Fe2+ orbital order. But it is not true since Fe2+ orbital order emerges at 140 K, where no energy gap is observed.

K. Fauth : In metallic magnets the magnetic interactions can be different from the Heisenberg type. One frequently discussed magnetic interaction is the RKKY interaction. Whether this interaction is - as the Heisenberg inteaction- between chemically identical moments only has to be investigated. It seems to me that this limitation does not exist for the RKKY interaction. Here one has to investigate whether the interaction is invariant with respect to permutation of the interacting magnetic moments.

U. Köbler: I'm still a bit confused. Your statement above is very strict. Now, what is the value of: "equivalent" ? Take atoms next to a defect in a crystal. They'll be non-equivalent to others at some level of precision. Does this all of a sudden switch off an interaction channel?

And I'd like to better understand your statement. What exactly is it you name "Heisenberg interaction"? Any place to read on before asking you these questions further? (-> your book?)

Posting this after a long time (gap) in the hope to get the desired answer.

Are spin-gap and magnon-gap are same ?

I am basically talking about spin gaps in dimerized alternate 1D spins.

Dear Prof. Qiang Zhang

In addition to all interesting answers of this thread, you can check the following reference [1] for the basic theory of spin waves & magnons---using the equation of motion of the magnetic moment--- in ferro & antiferromags.:

• A magnon is a wave of the nonuniform precession of atomic magnetic moments in ferro & antiferromagnets according to [1] ch VII.
• If there is no an external field H in ferromagnets, there is not gap in the magnon spectrum epsilon(k)---when only exchage interaction is taken into account pp. 286, please see eq. 70.3 for spin waves & 70.4 for magnons in reference [1]
• If the oscillations of the intrinsic magnetic moment M are taken into account, those oscillations generate a magnetic field h & it has to be taken into account in the magnon description & then, a gap appears in epsilon(k) eq 70.10 pp. 292 in [1] is calculated for an easy axis of magnetization & an uniaxial ferromagnet.
• If magnetic anisotropic is taken into consideration, then a gap appears in the spectrum (see for the case of ferromag eq. 70.12 pp 293 & for the case of antiferromagnets pp. 313 eq 74.12 [1] (Prof. Ch. Kittel 1951 result).

[1] Statistical Physics Vol 2 by E. Lifshitz & E. Pitaevskii ch VII Magnetism, Pergamon 1980.

CC. Prashanta Kumar Mukharjee

Dear Dr Qiang Zhang . See the following useful link: Article Large Spin-Wave Energy Gap in the Bilayer Iridate Sr3Ir2O7: ...

Also see the following very good link: Article Asymmetric dynamics of edge exchange spin waves in honeycomb...