Biomedical Engineering Reference
In-Depth Information
E
κ
β
D
−
c
d
p
−
t
t
p
c
d
f
ε
S
T/C
c
s
α
α
C/T
σ
T
−
f
Figure 3.2
The mathematical relationships between the mechanical, electrical, and thermal prop-
erties of a crystal (By permission of Oxford University Press)
The piezoelectric coupled effects are shown on the left of the diagram. The direct
piezoelectric effect is given in differential form by:
d
P
i
=
d
ijk
d
σ
jk
(3.5)
since,
D
i
=
k
0
E
i
+
P
i
,thend
P
i
=d
D
i
−
k
0
d
E
i
. Therefore, if the electric field in the crystal
is held constant, d
P
i
=d
D
i
we may write Equation 3.5 as:
d
D
i
=
d
ijk
d
σ
jk
(E
constant
)
(3.6)
The converse piezoelectric effect is the relationship between the electric field and the
strain (see Figures 3.1 and 3.2). Since these coupled fields relate to a first-rank tensor
(
E
i
or
D
i
) and to a second-rank tensor (
σ
ij
or
ε
ij
), they are themselves given by third-rank
tensors. The coupled effects on the right of the diagram are concerned with
pyroelectricity
.
They all connect a vector (
E
i
or
D
i
) to a scalar (
S
or
T
), and therefore are expressed by
first-rank tensors. The equation for the pyroelectric effect may be written as:
d
P
i
=
p
i
d
T
(3.7)
where
p
i
is the pyroelectric coefficient of the crystal and, assuming a constant electric
field, we have:
d
D
i
=
p
i
d
T(E
constant
)
(3.8)
The pyroelectricity effect is sometimes categorized into
primary
and
secondary pyro-
electricity
. If during the heating a crystal, its shape and size are held fixed (crystal
clamped), it is called the primary effect. On the other hand, the crystal may be released so
that thermal expansion can occur quite freely. In this case, an extra effect can be observed
