INFICON Maxtek PM-700 Plating Monitor User Manual
Page 76
PM-700 SERIES PLATING MONITOR
68
crystal vibrating in the thickness shear mode.
ρ
q
= Density of quartz (gm/cm
3
).
f
q
= Resonant frequency of the uncoated crystal.
f = Resonant frequency of the loaded crystal.
TK
f
= Thickness of plated material
ρ
f
= Density of the material (gm/cm
3
).
This equation proved to be adequate in most cases. However, the constant of
proportionality is not actually constant because one of the terms contains the
crystal frequency which of course changes. Because the achievable frequency
change was small the change in scale factor fell within acceptable limits.
In the late 1960's improvements in sensor crystals and oscillator circuits resulted
in a significant increase in achievable frequency shift. At the same time, low cost
integrated digital circuits became available allowing a significant increase in basic
instrument accuracy so that the frequency squared term in the scale factor became
important.
Substituting 1/period for frequency results in the following equation:
where:
τ
= Period of the loaded crystal (sec).
τ
q
= Period of the uncoated crystal (sec).
Note: Units of N
q
is cm/sec.
Note that the constant of proportionality in this equation is constant.
The original assumption that the addition of a foreign material to the surface of
the crystal produced the same effect as that of the addition of an equal mass of
quartz was, of course, questionable.
Crystals heavily loaded with certain materials showed significant and predictable
deviation between the film thickness measured and that predicted by equation 2.
Analysis of the loaded crystal as a one dimensional composite resonator of quartz
and deposited layer led to the equation below.
(
)
TK
N
f
q
q
f
q
=
⋅
−
ρ
ρ
τ τ
TK
N
R
R
f
q
f
q
z
z
q
=
⋅
⋅
⋅
−
ρ
ρ
τ
π
π
τ τ
τ
arctan
tan
(2)
(3)