bunsenmagazin - Deutsche Bunsengesellschaft für Physikalische ...

DEUTSCHE BUNSEN-GESELLSCHAFT
b) The quartz in oscillating electric fields
Consider a quartz disk or plate obtained by cutting the quartz
crystal as illustrated in Fig 1. Typically, thin gold layers are deposited
onto the two opposite plane faces. Application of an ac
voltage to these electrodes will generate a rhythmic deformation.
The exact character of the deformation will depend on the
orientation of the electrode planes in the crystal’s X-Y-Z-space.
If the frequency of the ac voltage corresponds to resonance
of this mechanical oscillation, the acoustic amplitude of the
quartz will have a maximum. Correspondingly, the ac current
and the electric admittance have a maximum.
For mass sensor application it is customary to use the “AT
cut” of the quartz crystal that oscillates in the thickness shear
mode. Figure 2 illustrates the standing transverse wave of the
fundamental resonance of the AT quartz. The nodal plane of
the oscillation is in the middle of the plate, at y = 0. The crystal
planes move parallel to one another in the x-direction, without
deformation. The maximum defl ection (ζ in Fig. 2) occurs at
the antinodes, y =±d/2.
Fig. 2
Shear mode deformation of the AT cut quartz under electric ac polariza-
→
tion. X, Y and Z are the crystallographic axes of the quartz, as in Fig. 1; E
is the electric field vector; see ref. 1
According to the above considerations, the plate thickness d
corresponds exactly to one half wavelength of the shear wave
if the quartz oscillates at its fundamental resonance frequency,
f0. Thus, this resonance frequency is defi ned by
f
0
UNTERRICHT
v 1 �
2
tr Q
[1]
f
� �
0 � �m
� vtr
� �m
� 2 f0
� �m
� � �m
�
�f
� � � � � � � � � � � � � �S�
�
2d 2d
�
d�
� A
2
� 2d � � A � � � � A � � A �
where vtr is the velocity of the wave propagation in the Y-direction
(see Fig. 2), μQ= 2.947x10 10 Nm -2 is the relevant piezoelectrically
stiffened shear modulus of the quartz, and ρQ = 2651 kg
m -3 is the density of quartz. With a value of vtr = 3334 m/s 9 the
resonance frequency of a plate is 1.67mm MHz d -1 .
The AT cut quartzes are obtained by cutting the crystal under
angles between ϑ = 35°10’ and 35°15’ with respect to the
crystallographic Z-axis, parallel to the X-axis. An important reason
for using this cut is the temperature stability of the resonance
frequency. For instance, the resonance frequency of
a 10 MHz quartz cut at ϑ =35°15’ will vary by -2.5 Hz/K at
ambient temperature, as shown in Fig. 3.
Q
Fig. 3
The temperature dependence of the resonance frequency Δf/f0 of AT
quartz resonators cut at the indicated angles ϑ. (see ref 1 )
c) The mass load dependence of the resonance frequency
Consider an AT cut plate large enough that the boundaries will
not affect the thickness shear mode oscillation. The surface
of the plate will constitute an antinode, as illustrated in Fig. 2.
This means, the quartz layers near the surface infl uence the
resonance frequency only by their rigid mass, but not by their
elastic properties. A thin homogeneous layer of any rigid material
deposited onto the surface of the quartz plate will consequently
infl uence the resonance frequency of the plate in the
same way as would a quartz layer of the same mass.
The change of the resonance frequency,∆f, caused by the deposition
of a rigid thin layer of mass ∆m is therefore defi ned by
where the subscript Q points to quartz and A is the surface
area. Combined with equ. 1 one obtains the well known „Sauerbrey
equation“ 1
Q
�f
f
0
�d
�mQ
�m
� � � � � � �
d � Ad � Ad
Q
Q
Note that the mass sensitivity constant S depends on the resonance
frequency squared. For example, for a 14 MHz quartz
(thickness d=0.12 mm) operating at the fundamental resonance
frequency, the sensitivity constant S = 4.43x10 7 Hz kg -1
m 2 , corresponding to 0.443 Hz ng -1 cm 2 . For a 5 MHz quartz S
= 0.0565 Hz ng -1 cm 2 . For more details see Box 1.
d) Measurements of the resonance frequency
The electrical behavior of the considered quartz resonator can
be adequately described by the equivalent circuit represented
in Fig. 4. It has two parallel branches 10,11 . The motional branch
is the series combination of an inductance L0, a resistance R0,
and a capacitance C0. The capacitance of the static branch,
Q
Q
Q
[2]
[3]
163