Linear entropy: Difference between revisions

Latest revision as of 16:50, 9 October 2013

The inverse magnetostrictive effect (also known as Villari effect) is the name given to the change of the magnetic susceptibility of a material when subjected to a mechanical stress.

Explanation

Whereas magnetostriction characterizes the shape change of a ferromagnetic material during magnetization, the inverse magnetostrictive effect characterizes the change of domain magnetization when a stress is applied to a material. This magnetostriction can be positive (magnetization increased by tension) like in pure iron, or negative (magnetization decreased by tension) like in nickel. In the case of a single stress $σ$ applied on a single magnetic domain, the magnetic strain energy density $E_{σ}$ can be expressed as:^[1]

$E_{σ} = \frac{3}{2} λ_{s} σ \sin^{2} (θ)$

where $λ_{s}$ is the magnetostrictive expansion at saturation, and $θ$ the angle between the saturation magnetization and the stressed direction. When $λ_{s}$ and $σ$ are both positive (like in iron under tension), the energy is minimum for $θ$ = 0, i.e. when tension is aligned with the saturation magnetization. Consequently, the magnetization is increased by tension.

In fact, magnetostriction is more complex and depends on the direction of the crystal axes. In iron, the [100] axes are the directions of easy magnetization, while there is little magnetization along the [111] directions (unless the magnetization becomes close to the saturation magnetization, leading to the change of the domain orientation from [111] to [100]). This magnetic anisotropy pushed authors to define two independent longitudinal magnetostrictions $λ_{100}$ and $λ_{111}$ .

In cubic materials, the magnetostriction along any axis can be defined by a known linear combination of these two constants. For instance, the elongation along [110] is a linear combination of $λ_{100}$ and $λ_{111}$ .
Under assumptions of isotropic magnetostriction (i.e. domain magnetization is the same in any crystallographic directions), then $λ_{100} = λ_{111} = λ$ and the linear dependence between the elastic energy and the stress is conserved, $E_{σ} = \frac{3}{2} λ σ (α_{1} γ_{1} + α_{2} γ_{2} + α_{3} γ_{3})^{2}$ . Here, $α_{1}$ , $α_{2}$ and $α_{3}$ are the direction cosines of the domain magnetization, and $γ_{1}$ , $γ_{2}$ , $γ_{3}$ those of the bond directions, towards the crystallographic directions.

References

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+The inverse magnetostrictive effect (also known as '''Villari effect''') is the name given to the change of the [[magnetic susceptibility]] of a material when subjected to a mechanical stress.
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+== Explanation ==
+Whereas [[magnetostriction]] characterizes the shape change of a ferromagnetic material during magnetization, the inverse magnetostrictive effect characterizes the change of domain magnetization when a stress is applied to a material. This magnetostriction can be positive (magnetization increased by tension) like in pure [[iron]], or negative (magnetization decreased by tension) like in [[nickel]]. In the case of a single stress <math>\sigma</math> applied on a single magnetic domain, the magnetic strain energy density <math>E_\sigma</math> can be expressed as:<ref>{{cite book
+  | last = Bozorth  |  first = R.
+  | title = Ferromagnetism  | publisher = Van Nostrand  |  year = 1951
+}}</ref>
+<math>E_\sigma = \frac{3}{2} \lambda_s \sigma \sin^2(\theta)</math>
+where <math>\lambda_s</math> is the magnetostrictive expansion at saturation, and <math>\theta</math> the angle between the saturation magnetization and the stressed direction.
+When <math>\lambda_s</math> and <math>\sigma</math> are both positive (like in iron under tension), the energy is minimum for <math>\theta</math> = 0, i.e. when tension is aligned with the saturation magnetization. Consequently, the magnetization is increased by tension.
+In fact, magnetostriction is more complex and depends on the direction of the crystal axes. In [[iron]], the [100] axes are the directions of easy magnetization, while there is little magnetization along the [111] directions (unless the magnetization becomes close to the saturation magnetization, leading to the change of the domain orientation from [111] to [100]). This [[magnetic anisotropy]] pushed authors to define two independent longitudinal magnetostrictions <math>\lambda_{100}</math> and <math>\lambda_{111}</math>.
+* In [[Cubic crystal system|cubic]] materials, the magnetostriction along any axis can be defined by a known linear combination of these two constants. For instance, the elongation along [110] is a linear combination of <math>\lambda_{100}</math> and <math>\lambda_{111}</math>.
+* Under assumptions of [[isotropic]] magnetostriction (i.e. [[Magnetic domain|domain]] magnetization is the same in any crystallographic directions), then <math>\lambda_{100} = \lambda_{111} = \lambda</math> and the linear dependence between the elastic energy and the stress is conserved, <math>E_\sigma = \frac{3}{2} \lambda \sigma (\alpha_1 \gamma_1 +\alpha_2 \gamma_2 + \alpha_3 \gamma_3)^2</math>. Here, <math> \alpha_1 </math>, <math>  \alpha_2 </math> and <math> \alpha_3 </math>  are the direction cosines of the domain magnetization, and <math> \gamma_1 </math>, <math> \gamma_2 </math>,<math> \gamma_3 </math> those of the bond directions, towards the crystallographic directions.
+== References ==
+{{reflist}}
+== See also ==
+* [[Magnetostriction]]
+* [[Magnetocrystalline anisotropy]]
+[[Category:Magnetic ordering]]
+{{condensedmatter-stub}}

Linear entropy: Difference between revisions

Latest revision as of 16:50, 9 October 2013

Explanation

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