Stoner–Wohlfarth model

The Stoner–Wohlfarth model is a widely used model for the magnetization of single-domain ferromagnets. It is a simple example of magnetic hysteresis and is useful for modeling small magnetic particles in magnetic storage, biomagnetism, rock magnetism and paleomagnetism. E = K u V sin 2 ⁡ ( φ − θ ) − μ 0 M s V H cos ⁡ φ , {displaystyle E=K_{u}Vsin ^{2}left(varphi - heta ight)-mu _{0}M_{s}VHcos varphi ,,}     (1) η = E 2 K u V = 1 4 − 1 4 cos ⁡ ( 2 ( φ − θ ) ) − h cos ⁡ φ , {displaystyle eta ={frac {E}{2K_{u}V}}={frac {1}{4}}-{frac {1}{4}}cos left(2left(varphi - heta ight) ight)-hcos varphi ,,}     (2) ∂ η ∂ φ = 1 2 sin ⁡ ( 2 ( φ − θ ) ) + h sin ⁡ φ = 0. {displaystyle {frac {partial eta }{partial varphi }}={frac {1}{2}}sin left(2left(varphi - heta ight) ight)+hsin varphi =0.,}     (3) ∂ 2 η ∂ φ 2 = cos ⁡ ( 2 ( φ − θ ) ) + h cos ⁡ φ > 0. {displaystyle {frac {partial ^{2}eta }{partial varphi ^{2}}}=cos left(2left(varphi - heta ight) ight)+hcos varphi >0.,}     (4) h s = ( 1 − t 2 + t 4 ) 1 / 2 1 + t 2 , {displaystyle h_{s}={frac {left(1-t^{2}+t^{4} ight)^{1/2}}{1+t^{2}}},,}     (5) t = tan 1 / 3 ⁡ θ . {displaystyle t= an ^{1/3} heta .,}     (6) h ∥ 2 / 3 + h ⊥ 2 / 3 = 1. {displaystyle h_{parallel }^{2/3}+h_{perp }^{2/3}=1.,}     (7) M a f ( H ) = M r s − M i r ( H ) M d f ( H ) = M r s − 2 M i r ( H ) . {displaystyle {egin{aligned}M_{af}(H)&=M_{rs}-M_{ir}(H)\M_{df}(H)&=M_{rs}-2M_{ir}(H)end{aligned}}.,}     (8) The Stoner–Wohlfarth model is a widely used model for the magnetization of single-domain ferromagnets. It is a simple example of magnetic hysteresis and is useful for modeling small magnetic particles in magnetic storage, biomagnetism, rock magnetism and paleomagnetism. The Stoner–Wohlfarth model was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth and published in 1948. It included a numerical calculation of the integrated response of randomly oriented magnets. Since this was done before computers were widely available, they resorted to trigonometric tables and hand calculations. In the Stoner–Wohlfarth model, the magnetization does not vary within the ferromagnet and it is represented by a vector M. This vector rotates as the magnetic field H changes. The magnetic field is only varied along a single axis; its scalar value h is positive in one direction and negative in the opposite direction. The ferromagnet is assumed to have a uniaxial magnetic anisotropy with anisotropy parameter Ku. As the magnetic field varies, the magnetization is restricted to the plane containing the magnetic field direction and the easy axis. It can therefore be represented by a single angle φ, the angle between the magnetization and the field (Figure 1). Also specified is the angle θ between the field and the easy axis. The energy of the system is where V is the volume of the magnet, Ms is the saturation magnetization, and μ0 is the vacuum permeability. The first term is the magnetic anisotropy and the second the energy of coupling with the applied field (often called the Zeeman energy). Stoner and Wohlfarth normalized this equation: where h = μ0MsH/2Ku.A given magnetization direction is in mechanical equilibrium if the forces on it are zero. This occurs when the first derivative of the energy with respect to the magnetization direction is zero: This direction is stable against perturbations when it is at an energy minimum, having a positive second derivative: In zero field the magnetic anisotropy term is minimized when the magnetization is aligned with the easy axis. In a large field, the magnetization is pointed towards the field.