Biomedical Engineering Reference
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where we defined the so-called
normalized frequency shift
(Giessibl, 1997)
k
cant
A
3
/
2
γ
(
D
)
:
=
Δ
f
(
D
)
(2.36)
f
0
which is independent of the oscillation amplitude, but a function of the nearest tip-
sample distance
D
(Holscher et al., 2000). To recover the tip-sample interaction force
from a spectroscopy experiment, we have to measure the frequency shift and the
driving amplitude as a function of the tip-sample distance before we calculate the
normalized frequency shift. After that, we introduce this data into Equation 2.35.
For completeness we would like to mention that the “large amplitude approxima-
tion” used for the deviation of Equation 2.25 is not a prerequisite for the reconstruc-
tion of the tip-sample force. Inversion methods suitable for small amplitudes have
been presented by several researchers (Durig, 2000a; Giessibl, 2001; Pfeiffer, 2004;
Sader & Jarvis, 2004). The method of Sader & Jarvis (2004), for example, can also
be applied to our case and adds two additional terms to our previous formula
)=
−
√
2
∂
∂
D
k
cant
A
3
/
2
f
0
∞
Δ
f
(
z
)
F
SJ
ts
(
D
√
z
−
D
D
d
z
z
√
2
k
cant
A
1
/
2
Δ
f
f
0
−
D
π
+
(
z
−
D
)+
(2.37)
4
The calculation of the energy dissipation Δ
E
is straightforward using Equa-
tion 2.32b
π
k
cant
A
2
(
)
A
D
)
=
Δ
E
(
D
,
A
(
D
)
a
exc
−
(2.38)
Q
It might be interesting to note that this equation follows also from the conservation
of energy (Cleveland et al., 1998; Gotsmann et al., 1999).
Since the quantitative value of excitation amplitude
a
exc
is typically unknown in
an experiment, it is convenient to determine
a
exc
from the freely oscillating cantilever
with the help of Equation 2.19b. As a result, we get
π
k
cant
A
(
D
)
Δ
E
(
D
)=
(
A
0
−
A
(
D
))
(2.39)
Q
where
A
0
is the oscillation amplitude of the freely oscillating cantilever (i.e., with
negligible tip-sample force).
2.5 EXPERIMENTAL COMPARISON OF AM- AND FM-MODE
The setup presented in the previous subsections can be easily realized by combining
a commercial dynamic force microscope (MultiMode AFM with NanoScope IIIa
Controller, Veeco Instruments Inc.) with an additional electronics for the CE mode.
In this way, it is also possible to switch between AM- and FM-mode using the same
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