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Notation and Units. In SI units, the Gilbert equation reads

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Appendix

Notation and Units. In SI units, the Gilbert equation reads

γ

e−1

m = μ ˙

0

m × H , (3.168) where γ

e

= g q

e

/m

e

is the gyromagnetic ratio, q

e

< 0 is the signed electron charge, m

e

is the electron mass and g > 0 is the Land´ e factor ; μ

0

denotes the magnetic permeability of the vacuum, and

H = −(μ

0

M

s

)

−1

δ

m

Φ , (3.169) is the effective field, with M

s

the saturation magnetization and Φ the Gibbs free energy. The standard free energy is written as

Φ =



Ω

φ , (3.170)

with

φ = A |∇m|

2

− K(e · m)

2

− μ

0

M

s

( 1

2 H

s

+ H

e

) · m ,

where H

s

and H

e

are, respectively, the stray field and the external magnetic field, A is the exchange constant and K is the anisotropy constant.

Our notation is recovered upon defining the dimensionless quantities ψ = (μ

0

M

s2

)

−1

φ ,

β = 2(μ

0

M

s2

)

−1

K , h = M

s−1

H , h

s

= M

s−1

H

s

, h

e

= M

s−1

H

e

,

(3.171)

and

α = 2(μ

0

M

s2

)

−1

A ,

γ = M

s

μ

0

γ

e

(3.172)

which have, respectively, the dimensions of (length)

2

and (time)

−1

.

51

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