## Posts Tagged ‘Magnetism’

### The case of the curious reference

October 11, 2007

Today, I came across a reference to what is called Landau-Lifshitz-Gilbert equation; a little bit of googling got me the reference to Gilbert’s paper:

T.L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetic field, Phys. Rev. 100 (1955) 1243.

However, when I tried Phys. Rev. site, I got the error message:

Phys. Rev. 100 1243
No data available for this citation

A valid journal, volume, and page or article id are required. Please try again.

So, I thought there probably is something wrong with the page numbers; I thought a Google Scholar search might help me get the correct volume and/or page number. That didn’t help me either:

[CITATION] A Lagrangian formulation of gyromagnetic equation of the magnetization field
TL Gilbert – Phys. Rev, 1955
Cited by 272Related ArticlesWeb Search

Now, that is strange; there is only citation (and, a whopping 272 at that), but, no link to the article itself. Surely, there is no way so many of them got it wrong.

I went back to PROLA and browsed through the volume 100 of Phys. Rev. to find that pp. 1236-1272 are missing; however, the stuff that is published on p.1235 of the issue indicates that the missing pages should contain some abstracts, and probably Gilbert’s paper is one among them.

T.L. Gilbert. Phys. Rev., 100:1243, 1955. [Abstract only; full report, Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956] (unpublished).

Wow! Finally, a couple of researchers who not only took the pains to locate the article (abstract, in this case) but also to note it for those who might be trying to hunt it down.

Of course, I do not understand why Phys. Rev does not host pdf pages of these abstracts in their archive. In any case, a visit to the library helped me get the abstract, which reads as follows:

D6. A Lagrangian formulation of the gyromagnetic equation of the magnetization field. T. L. Gilbert, Armour Research Foundation of Illinois Institute of Technology.–The gyromagnetic equation, $d{\mathbf {M}}/dt = \gamma {\mathbf {M}} \times {\mathcal{G}}$, for the motion of the magnetization field ${\mathbf {M(r)}}$, in a ferromagnetic material can be derived from a variation principle, as first shown by Doering.1 Here ${\mathcal{G}}$ is the effective internal field, including the magnetic field and contributions from exchange, anisotropy, and magnetoelastic effects. Using the variational principle, the equations of motion can be recast into a Lagrangian form. This makes possible a consistent derivation of the equations of motion of the magnetization field and other fields to which it may be coupled (e.g., the displacement field of the lattice and the electromagnetic field). It also permits the introduction of viscous damping effects in a consistent manner using the Rayleigh dissipation function. It is shown that viscous damping of the magnetization fields leads to an equation of motion which reduces to the Landau-Lifshitz equation only when the damping is small. It is also shown that this Lagrangian formalism permits the introduction f damping due to disaccomodation in a consistent and very general way.

1 W. Doering, Z. Naturforsch. 3a, 374 (1948)

So, there are a couple of morals to this story: sometimes, if it is good enough, a paragraph like above can get you hundreds of citations; and, if you find some pages of Phys Rev are missing (before 1955; a note in 1955, Vol. 100, issue 4 notes that they will not be published thenceforth), it probably is an abstract of some meeting, and you can only get it in the hard copy format.

Well, the successful resolution of the mystery calls for a cup of coffee, don’t you think! See you around.

PS: For those of you who are interested in using LLG equation to numerically solve domain evolution in giant magnetostrictive materials (of course, using phase field methods — you knew it was coming, didn’t you?), here is a paper:

Title: Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials

Authors: J.X. Zhang and L.Q. Chen

Abstract
:

A computational model is proposed to predict the stability of magnetic domain structures and their temporal evolution in giant magnetostrictive materials by combining a micromagnetic model with the phase-field microelasticity theory of Khachaturyan. The model includes all the important energetic contributions, including the magnetocrystalline anisotropy energy, exchange energy, magnetostatic energy, external field energy, and elastic energy. While the elastic energy of an arbitrary magnetic domain structure is obtained analytically in Fourier space, the Landau–Liftshitz–Gilbert equation is solved using the efficient Gauss–Seidel projection method. Both Fe81.3Ga18.7 and Terfenol-D are considered as examples. The effects of elastic energy and magnetostatic energy on domain structures are studied. The magnetostriction and associated domain structure evolution under an applied field are modeled under different pre-stress conditions. It is shown that a compressive pre-stress can efficiently increase the overall magnetostrictive effect. The results are compared with existing experiment measurements and observations.

Have fun!

### What is Stoner instability?

September 29, 2007

In my earlier post on multiferroic materials, I mentioned Stoner instability. Here is the explanation of Stoner instability (and, this explanation is based on Chapter 7 (p.63) of this text: Physics of magnetism and magnetic materials (E-book)
K H J Buschow and F R de Boer
Kluwer Academic Publishers, NY (2004)).

Consider a 3d transition metal, in which the 3d electrons give rise to magnetism; since the electrons are itinerant (and delocalised) in the metal, the magnetism stems from 3d electron bands. For simplicity’s sake, let us further assume that the 3d bands are rectangular (which means that we are assuming that the density of electron states is a constant over the entire range spanned by the width of the band). The band itself consists of two sub-bands — one for up-spin electrons and another for down-spin electrons. If there are less than ten 3d-electrons in the system, the 3d-band will be partially filled. Further, if the system fills these bands without discrimination, then both the sub-bands will be equally filled. However, if suppose we can define an interaction energy which indicates a reduction in energy if the electrons from one of the sub-bands, say those corresponding to down-spin can be transferred to the up-spin band, then, under certain circumstances it can be shown that this will lead to an instability as discussed below. However, what prevents such an emptying of one of the sub-bands in favour of another is the resultant increase in the kinetic energy of the electrons. In fact, the total variation in energy in such sub-band transfer of electrons can be shown to be equal to $\Delta E = \frac{n^{2} p^{2}}{N(E_F)} [1 - U_{eff} N(E_F)]$, where, $n$ is the total number of 3d electrons per atom, $p$ is the fraction of atoms that move from down-spin sub-band to up-spin sub-band, $U_{eff}$ is the effective interaction energy, and $N(E_{F})$ is the density of energy states at the Fermi level. Thus, if the quantity in square brackets is positive, the state of lowest energy corresponds to $p = 0$ — or, in other words, the metal is non-magnetic. However, if the quantities in the square bracket is negative, the band is “exchange split” — $p > 0$, and hence the metal is ferromagnetic. This is known as the Stoner instability, or sometimes ferromagnetic instability. From the equation, it is clear that such band splitting is favoured for large exchange interaction energy as well as for large density of states. Since the density of states for s– and p-bands are considerably smaller, which, in turn explains why such band magnetism is restricted to elements with partially filled d-band.

Here are the schematics explaining band magnetism in partially filled d-electron systems (based on Fig. 7.1.1 of the reference above):

Paramagnetic:

Weak ferromagnetism:

Strong ferromagnetism with systems in which the d-electrons per atom are less than five, and great than five respectively.