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8. Useful physics for SPM
8.1 The cantilever resonance
Simple Harmonic Oscillator curve
The resonance curve of a cantilever can be modeled with a harmonic oscillator
function, as in the following formula. This is used to fit the measured cantilever
resonance curve in the spring constant calibration window in the NanoWizard
software. For light damping (high Q), this reduces to a Lorentz curve.
=
⋅
y
(
f
)
A
−
2
2
(
f
f
)
0
Q-factor of the resonance
The Q-factor of a resonance is a measure of the damping in the oscillating
system. The Q-factor can be calculated as the ratio of the energy stored in an
oscillation to the amount of energy that is lost each cycle. This translates to a
measure of how sharp the resonance curve is.
The larger the Q-factor, the sharper the resonance curve. The larger the Q-
factor, the higher the sensitivity of the probe in intermittent contact mode.
Normal AFM probes have a Q-value of a few hundred in air, but this is reduced
to a much smaller value (typically 1-5) in water. This is because of the much
higher damping from the viscosity of the water compared with air. In water the
effective mass also increases, since the cantilever carries some of the
surrounding liquid with it as it moves. Therefore the resonance curves for the
same cantilever in air and in liquid are very different.
8.2 Thermal noise spring constant calibration
Background information
The thermal noise analysis is becoming the main standard for AFM experiments,
because it is available in liquid, online during the experiment, through a fast,
automated software analysis. There are some difficulties in the theoretical analysis
due to cantilever shape, liquid damping, etc., but the convenience and speed
means it is now very widely used.
The position of the end of the cantilever is constantly fluctuating because of the
thermal vibrations from the environment, this can be thought of as a kind of diffusion
restricted or balanced by the restoring force from the spring constant. The thermal
environment of the cantilever is known, and the deflection of the cantilever can be
measured accurately, so the balance between them can be used to calculate the
spring constant. This method is based on measuring the free fluctuations of the
cantilever, so the main advantages are because it is a passive measurement and
can be made in liquid and actually in-situ during an experiment.
2
f
0
2
f
f
+
2
0
Q
JPK Instruments
®
NanoWizard
Handbook
A: Amplitude
f
: Resonant frequency
0
Q: Q-factor
energy
stored
Q =
energy
lost
per
resonant
frequency
Q ≈
full
width
at
half
Version 2.2a
cycle
ma
. x
41
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