Environmental Engineering Reference
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which are related to the Cartesian strains as follows:
α
1
=
11
+
22
+
33
α
2
=
3
(2
33
−
11
−
22
)
1
2
γ
1
=
(
11
−
22
)
(1
.
4
.
7)
γ
2
=
12
ε
1
=
13
ε
2
=
23
,
where we have adopted the conventional notation of designating the
Cartesian axes (
ξ, η, ζ
)by(1
,
2
,
3). The
α
-strains are thus symmetry-
conserving dilatations, the
γ
-strains distort the hexagonal symmetry of
the basal plane, and the
ε
-strains shear the
c
-axis. The elastic energy
may then be written
H
el
=
N
2
c
α
1
α
1
+
c
α
3
α
1
α
2
+
2
c
α
2
α
2
(1
.
4
.
8)
c
ε
(
ε
1
+
ε
2
)
,
+
2
c
γ
(
γ
1
+
γ
2
)+
2
where we have defined irreducible elastic stiffness constants per ion, re-
lated to the five independent Cartesian constants by
1
9
c
α
1
=
(2
c
11
+2
c
12
+4
c
13
+
c
33
)
V/N
1
2
c
α
2
=
(
c
11
+
c
12
−
4
c
13
+2
c
33
)
V/N
1
3
(1
.
4
.
9)
c
α
3
=
(
−
c
11
−
c
12
+
c
13
+
c
33
)
V/N
c
γ
=2(
c
11
−
c
12
)
V/N
c
ε
=4
c
44
V/N.
The contributions to the single-ion magnetoelastic Hamiltonian,
corresponding to the different irreducible strains, are
B
l
α
1
α
1
+
B
l
α
2
α
2
O
l
(
J
i
)
+
B
6
α
1
α
1
+
B
6
α
2
α
2
O
6
(
J
i
)
α
H
me
=
−
i
l
=2
,
4
,
6
(1
.
4
.
10)
B
γ
2
O
l
(
J
i
)
γ
1
+
O
−
2
(
J
i
)
γ
2
γ
H
me
=
−
l
i
l
=2
,
4
,
6
(
J
i
)
γ
2
(1
.
4
.
11)
+
l
=4
,
6
B
γ
4
O
l
(
J
i
)
γ
1
−
O
−
4
l
B
ε
1
O
l
(
J
i
)
ε
1
+
O
−
1
(
J
i
)
ε
2
ε
me
=
H
−
l
i
l
=2
,
4
,
6
(
J
i
)
ε
2
.
+
B
ε
5
O
6
(
J
i
)
ε
1
−
O
−
5
6
(1
.
4
.
12)
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