Cienka warstwa: zależność kątowa

Przyjmując oznaczenia jak na rysunku 6.4 i uwzględniając tylko energię anizotropii magnetokrystalicznej i pola odmagnesowania, otrzymujemy na-stępujące wyrażenie na całkowitą energię magnetyczną w zależności od kąta obrotu α:

E = −HMScos(ϑ − α) + 2πMS²cos²ϑ (E.18) które minimalizujemy:

∂E

∂ϑ ^{= 0} (E.19)

Uwzględniając, że

cos(ϑ − α) = cos ϑ cos α − sin ϑ sin α ^(E.20) otrzymujemy:

∂E

∂ϑ = −HMS(− sin ϑ cos α − cos ϑ sin α) − 4πMS²cos ϑ sin ϑ = 0 (E.21) skąd po podstawieniu:

cos ϑ = ^M MS

(E.22) możemy obliczyć sin ϑ:

sin ϑ = ^{HMSsin α}

4πMMS − HMScos α (E.23)

Teraz wstawiając cos ϑ i sin ϑ do tożsamości trygonometrycznej:

sin²ϑ = 1 − cos²ϑ (E.24)

otrzymujemy równanie: ( ^{HMSsin α} 4πMMS − HMScos α⁾ 2 = 1 − ^M 2 M2 S (E.25) skąd, po przekształceniu otrzymujemy równanie czwartego stopnia na M:

16π²M⁴− 8πH cos αM³+ (H²− 16π²M_S²)M²+

+ 8πH cos αM_S²M − H²cos αM_S² = 0, (E.26) które wykorzystujemy dalej w rozdziale 6.8 jako równanie 6.41.

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