Environmental Engineering Reference
In-Depth Information
There is no change in the axial susceptibility in the axial ferromag-
net, for which θ = 0, but the higher derivatives are affected by the
modifications κ 2 (0) = κ 2 (1
4
6
35 2 . The correction
to the Callen-Callen theory is proportional to b ,whichisoftheorder
1 /J times the ratio between the anisotropy and the exchange energy
(
7 b )and κ 4 (0) =
B 2 J (2) /J 2
( 0 )), and hence becomes smaller for larger values of J .
This calculation may be extended to higher order and to non-zero tem-
peratures, but the complications are much reduced by the application of
the Holstein-Primakoff transformation which utilizes directly the factor
1 /J in the expansion, as we shall see in the discussion of the spin-wave
theory in Chapter 5.
In the ferromagnetic phase, the ordered moments may distort the
lattice, due to the magnetoelastic couplings, and this gives rise to addi-
tional contributions to F ( θ, φ ). We shall first consider the effects of the
γ -strains by including the magnetoelastic Hamiltonian, incorporating
(1.4.8) and (1.4.11),
H γ =
i
J
2
B γ 2 Q 2 ( J i ) γ 1 + Q 2
( J i ) γ 2
c γ ( γ 1 + γ 2 )
2
(2 . 2 . 23)
B γ 4 Q 4 ( J i ) γ 1
( J i ) γ 2 ,
Q 4
4
retaining only the lowest-rank couplings ( l = 2 and 4 of respectively the
γ 2and γ 4 terms). The equilibrium condition
∂F/∂ γ 1 =
H γ /∂ γ 1 =0 ,
(2 . 2 . 24)
and similarly for γ 2 , leads to the equilibrium strains
γ 1 = B γ 2
Q 4 /c γ
Q 2
+ B γ 4
(2 . 2 . 25)
γ 2 = B γ 2
/c γ .
Q 2
2
Q 4
4
B γ 4
The conventional magnetostriction parameters C and A are introduced
via the equations
2
γ 1 = C sin 2 θ cos 2 φ
A sin 4 θ cos 4 φ
(2 . 2 . 26 a )
γ 2 = C sin 2 θ sin 2 φ + 2
A sin 4 θ sin 4 φ
(Mason 1954). Expressing Q l in terms of O l , and retaining only the
terms with m = 0, we may derive these parameters from (2.2.25), ob-
taining
1
c γ
B γ 2 J (2) I 5 / 2 [ σ ]
C =
(2 . 2 . 26 b )
2
c γ
B γ 4 J (4) I 9 / 2 [ σ ] .
A =
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