Barcelona, juliol de
B. Introducció General
B.1. Metabolisme del glicogen
B.1.2. Síntesi i degradació del glicogen
Anisotropy is a contribution of energy to the orientation of the atoms within the internal structure of the material. Anisotropy contributions eventually come to equilibrium in order for the material to achieve the lowest possible energy state at any given time. There are several different types of anisotropy that may contribute in magnetic materials. For the majority of cases it is a combination of several of these effects whether it is magnetic, crystallographic, shape or physical deformation (external stress) of such a material. In this section only the effect of anisotropy in ferromagnetic materials is assessed. Magnetocrystalline anisotropy is an intrinsic property of a ferromagnetic crystal and is independent of the crystal size and shape. This property can be most easily seen by measuring magnetization curves along different crystal directions; these materials arrange themselves so that a hard and an easy axis of magnetisation can be experimentally observed, hence there is a preferred crystallographic and magnetic orientation which are not necessarily the same.
Figure 2.2.1.1 shows a Galfenol atom of body centred cubic (BCC or A2) structure which is typical for compositions of Galfenol from 0 to 15 at. % Ga with each atomic position shown as in an [x,y,z] co-ordinate system with respect to an origin of [000], here the Ga atom is positioned in the middle of the structure, but in the A2 disordered lattice the Ga can substitute for any one of Fe atoms in the lattice, hence the term disordered. The easy axes for the Galfenol BCC structure are the <001> family of directions and the hard axes are the <111> directions.
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Figure 2.2.1.1 - A Galfenol Atom of body centred cubic (BCC) ordered structure. [2.2.1.1] (Diagram Created Using CrystalMaker Software, then Adapted using CorelDraw)
The magnetocrystalline anisotropy energy is the energy necessary to deflect the magnetic moment in a sample from the easy to the hard direction. These easy and hard orientations arise from the interaction of the spin magnetic moment coupling to the crystal lattice; this is sometimes called spin-orbit lattice coupling.
In cubic crystals, like Galfenol, the magnetocrystalline anisotropy energy is calculated by a mathematical series in the terms of the angle between the direction of the applied magnetisation field and the axes of the cubic structure as shown by Akulov in 1929 [2.2.1.2]
It is widely accepted that the anisotropy energy in any arbitrary direction can be described by just the first two variable terms in this series expansion, as the successive terms become more and more negligible. For a cubic crystal the anisotropy energy density is expressed as:
Ek = K0 + K1 (α12 α22 +α22 a32 + α12 a32) + K2 (α12 α22 α32) + . . . . (2.2.1.1) [2.2.1.2] Where K0, K1 and K2 are known as the anisotropy constants, K0 is ignored as it is independent of the angle between the magnetisation and the easy axes. α1, α2 and α3 are the direction cosines between the magnetisation and the crystal axes. For the case of Galfenol, K1 is
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positive and K2 is negative (and K1 is of greater magnitude by a factor of ~50) meaning the total anisotropy energy Ek is substantially higher for the <111> directions as opposed to the <100> directions, hence the direction with the lowest associated anisotropy energy is termed the easy axes.
In addition to magnetocrystalline anisotropy, there is another contributing effect that is directly related to the long range spin-orbit lattice coupling. This is called magnetostriction, which manifests itself as stored magnetoelastic energy. This effect is produced from the strain dependence of the anisotropy constants. When the material experiences the magnetisation force from the applied external field, a virgin or unmagnetised crystal experiences a resultant strain that can be measured as a function of applied field along a chosen crystallographic axis. A ferromagnetic material physically changes its dimensions when undergoing magnetisation i.e. magnetostriction. The inverse effect with an applied stress also takes place. A uniaxial stress can produce an easy axis of magnetisation but only if the applied stress is sufficient to overpower all of the other combined anisotropy forces such as the shape and magnetic anisotropies. The magnitude of the stress anisotropy is described by two empirical constants known as the magnetostriction constants, one in the easy direction, λ100and one in the hard direction λ111.These constants are determined by using the magnetoelastic energy equation; which is the coupling between all of the mechanical and magnetic terms.
Within a cubic crystal system using the directions (λ100 and λ111) as previously mentioned, but additionally, the magnitude (σ) and direction cosines (β1s, β2s, β3s) of the applied stress as
well as the orientation of the internal magnetic moments termed (α1, α2, α3) gives:
magnetoelastic σ - λ α β s α β s α β s - λ σ α α β sβ s α α β sβ s α α β sβ s
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Here γσ is a scaling term in order to scale the magnetoelastic energy so that it is relative to the
other anisotropy contributions. A physical basis for this term has not as yet been established; but it is best described as an empirical scaling factor that allows for the effect of magnetoelastic energy within the system. This allows for an accurate calculation of the effect of applying an external stress on the material and observing the resultant magnetomechanical behaviour or applying a field to the sample and observing the resultant deformation of the lattice. [2.2.1.3]
Another type of anisotropy is due to the physical shape or dimensions of the material. A magnetised object produces external magnetic poles on the surface of the material. This surface charge distribution is itself another source of a magnetic field known as the demagnetising field. This field acts in opposition to the externally applied magnetic field that produces it. For a long thin object such as a ribbon or thin film, the demagnetising field along the long axis is weak because the poles are well separated. For magnetisation induced across a relatively thin object the demagnetising field is large because the induced poles are much closer together. The demagnetising field causes a shearing effect or distortion on the initial magnetisation curve [2.2.1.5].
To calculate the demagnetising field within a ferromagnetic material, the pole density on the surface of that object must be determined. If the particular material has a susceptibility , the demagnetising field is represented by the demagnetisation factor, Nd. The only currently known analytical solution for Nd relates to solid ellipsoid shapes; theoretical solutions have been formulated for cylinders and other shaped objects but there seems to be no standardised solutions as yet for hollow objects and shapes, therefore these must be modelled individually to acquire the correct demagnetisation factor [2.2.1.4]. A visual example of the demagnetising field and how it directly opposes the externally applied field can be seen in
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figure 2.2.1.2; where the image also shows how the demagnetising field affects the initial part of the B-H magnetisation curve.
Figure 2.2.1.2 - The Demagnetising Field and the Impact on the B-H Curve (True curve is solid line) Demagnetisation affected curve is the sheared curve (Represented by the dashed line). [2.2.1.5]
In order to quantify this shape anisotropy effect, it is considered to directly oppose the force to the externally applied magnetic field. Equation 2.2.1.3 is revisited and recalculated to include the demagnetising field, Hd.
Quantitively we start with the total magnetic induction equation:
μ
( )
(2.2.1.3)For a non-closed magnetic circuit such as for a flat thin film or long ribbon, this can be expanded to give:
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The demagnetising field within the material is defined as:
d
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d (2.2.1.5)Including Hd and rearranging:
-
-
d(2.2.1.6)
Now utilising Equation 2.1.1.9 and substituting, we can rearrange to give:
(2.2.1.7)
Here
χ
eff is the effective susceptibility of the material [2.2.1.5]. This is distinct from thestandard material susceptibility that would be measured in the case where there were no demagnetising fields. (This effective susceptibility is a combined function of bothand Nd.)