INFICON PLO-10i Phase Lock Oscillator - Manual (Page 49)

PLO-10 PHASE LOCK OSCILLATOR
THEORY OF OPERATION 8-1
8 THEORY OF OPERATION
Sauerbrey was the first to recognize the ability of the Quartz Crystal Microbalance
(QCM) to measure very small mass changes on the crystal surface. His seemingly simple
equations have been used for many years and in many different applications.
8.1 SAUERBREY EQUATION
Equation 1
m C f
f
∆ × − = ∆
Where:
f
∆ = Frequency change in Hz.
f
C
= Sensitivity factor of the crystal in Hz/ng/cm 2
(0.0566 Hz/ng/cm 2 for a 5 MHz crystal @ 20° C)
(0.0815 Hz/ng/cm 2 for a 6 MHz crystal @ 20° C)
(0.1834 Hz/ng/cm 2 for a 9 MHz crystal @ 20° C)
m
∆ = Change in mass per unit area in g/cm 2 .
The Sauerbrey equations assumed that the additional mass or film deposited on the
crystal has the same acousto-elastic properties as quartz. This assumption resulted in a
sensitivity factor, C
f
, which is a fundamental property of the QCM crystal as shown in
equation 2.
Equation 2
q q
f
f n
C
µ ρ Ч
Ч
=
2 2
Where:
n
= Number of the harmonic at which the crystal is driven.
f
= Resonant frequency of the fundamental mode of the crystal in Hz.
q
ρ = Density of quartz = 2.648 g · cm -3
q
µ = Effective piezoelectrically stiffened shear modulus of quartz = 2.947 · 10 11 g · cm -1 ·
sec -2 .
Solving these equations for ∆m yields
Equation 3
2 2
) (
f n
f f
C
f
m
q q q
f
Ч
Ч −
=
∆ −
= ∆
µ ρ
Where: