INFICON PLO-10i Phase Lock Oscillator - Manual (Page 50)

PLO-10 PHASE LOCK OSCILLATOR
THEORY OF OPERATION 8-2
f
q
= Resonant frequency of unloaded crystal in Hz.
f
= Resonant frequency of loaded crystal in Hz.
It is important to note that under these assumptions, the change in frequency is a function
of mass per unit area. Therefore, in theory, the QCM mass sensor does not require
calibration. However, keep in mind that the Sauerbrey equation is only strictly applicable
to uniform, rigid, thin-film deposits 21 . Vacuum and gas phase thin-film depositions
which fail to fulfill any of these conditions actually exhibit more complicated frequency-
mass correlations and often require some calibration to yield accurate results.
8.2 Z-MATCH EQUATION
Sauerbrey’s original assumptions were of course, questionable and indeed work with
crystals heavily loaded with certain materials showed significant and predictable
deviations between the measured mass and that predicted by Equation 3. Lu and Lewis 22
analyzed the loaded crystal as a one-dimensional composite resonator of quartz and the
deposited film which led to the equation shown below which is also referred to as the Z-
Match equation.
Equation 4
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
⋅ ⋅ ⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ ⋅
⋅
= ∆ −
f
f f
R
f R
N
m
q
z
z
q q π
π
ρ
tan tan 1
where:
m
∆ = change in mass per unit area in g/cm 2 ,
N
q
= Frequency Constant for AT-cut quartz crystal = 1.668 x 10 5 Hz x cm =
q
q q
ρ
µ ρ
2
ρ
q
= Density of quartz = 2.648 g/cm 3 .
f
q
= Resonant frequency of unloaded crystal in Hz.
f
= Resonant frequency of loaded crystal in Hz.
R
z
= Z-Factor of film material =
f f
q q
µ ρ
µ ρ
⋅
⋅
= Acoustic Impedance Ratio
f
ρ = Density of material g/cm 3
q
µ = shear modulus of quartz = 2.947×10 11 g · cm -1 · s -2
f
µ = shear modulus of film material in g · cm -1 · s -2 .
This equation introduces another term into the relationship which is the ratio of the
acoustic impedance of quartz to the acoustic impedance of the deposited film. The
acoustic impedance is associated with the transmission of a shear wave in the deposited