resonance frequency Δf N and the deposition of an additional mass, Δm, rigidly
connected to the quartz crystal surface:
Δf N ¼ ÀN
2f N
2
μ q ρ q
À
Á 1=2 Á
Δm
A
where f N is the unperturbed resonance frequency of the N-th mode, μ q is the
piezoelectric shear strength of the quartz crystal, ρ q is the mass density of the quartz
crystal, and A is the electrode surface. It is important to note that the coefficient of
proportionality depends only on intrinsic characteristics of the quartz crystal
resonators.
2.2 Sauerbrey’s Mass Sensitivity
Assuming that the rigid film is uniformly deposited on the quartz crystal surface, it is
possible to define the sensitivity of the quartz crystal resonators S N , measured in the
CGS unit as g
À1 cm
2 s
À1 , as the ratio between the frequency variation Δf N and the
variation of the surface mass density:
S N ¼
Δf N
Δm=A
ð
Þ
which can be written in the form:
S N ¼ ÀN
2f N
2
μ q ρ q
À
Á¼ À
2f N
2
ρ q v s
For an AT-cut quartz crystal, which is a specific cutting of original crystal stones
characterized by a cut angle of 35
15
0 respect to the crystallographic Z-axis, the
piezoelectric shear strength is μ q ¼ 2.947 Â 10
11 g cm
À1 s
À2 , the mass
density ρ q ¼ 2.648 g cm
À3 , and speed of propagation of the shear wave is
v s ¼ (μ q /ρ q )
1/2
¼ 3.340 Â 10
5 cm s
À1
. Based on the equations above, for a quartz
crystal oscillating at the unperturbed fundamental frequency of 10 MHz, a frequency
shift of 1 Hz is caused by a mass deposited per unit area equal to 4.49 Â 10
À11 g.
The mass sensitivity of the quartz is not uniform over the entire surface of the
quartz but has a maximum in the center and decreases as it approaches the edges of
the electrodes. The experimental results [21] show that the spatial distribution of
mass sensitivity on the quartz surface follows the distribution of the vibration
amplitude. Both the sensitivity and the acceleration follow a Gaussian distribution
and, in particular, the sensitivity is proportional to the square of the radial
displacement.
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
319
connected to the quartz crystal surface:
Δf N ¼ ÀN
2f N
2
μ q ρ q
À
Á 1=2 Á
Δm
A
where f N is the unperturbed resonance frequency of the N-th mode, μ q is the
piezoelectric shear strength of the quartz crystal, ρ q is the mass density of the quartz
crystal, and A is the electrode surface. It is important to note that the coefficient of
proportionality depends only on intrinsic characteristics of the quartz crystal
resonators.
2.2 Sauerbrey’s Mass Sensitivity
Assuming that the rigid film is uniformly deposited on the quartz crystal surface, it is
possible to define the sensitivity of the quartz crystal resonators S N , measured in the
CGS unit as g
À1 cm
2 s
À1 , as the ratio between the frequency variation Δf N and the
variation of the surface mass density:
S N ¼
Δf N
Δm=A
ð
Þ
which can be written in the form:
S N ¼ ÀN
2f N
2
μ q ρ q
À
Á¼ À
2f N
2
ρ q v s
For an AT-cut quartz crystal, which is a specific cutting of original crystal stones
characterized by a cut angle of 35
15
0 respect to the crystallographic Z-axis, the
piezoelectric shear strength is μ q ¼ 2.947 Â 10
11 g cm
À1 s
À2 , the mass
density ρ q ¼ 2.648 g cm
À3 , and speed of propagation of the shear wave is
v s ¼ (μ q /ρ q )
1/2
¼ 3.340 Â 10
5 cm s
À1
. Based on the equations above, for a quartz
crystal oscillating at the unperturbed fundamental frequency of 10 MHz, a frequency
shift of 1 Hz is caused by a mass deposited per unit area equal to 4.49 Â 10
À11 g.
The mass sensitivity of the quartz is not uniform over the entire surface of the
quartz but has a maximum in the center and decreases as it approaches the edges of
the electrodes. The experimental results [21] show that the spatial distribution of
mass sensitivity on the quartz surface follows the distribution of the vibration
amplitude. Both the sensitivity and the acceleration follow a Gaussian distribution
and, in particular, the sensitivity is proportional to the square of the radial
displacement.
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
319