The thermal fluctuation of the cantilever position measured experimentally with a DSFM detection unit is calculated from the noise density in analogy to relation 5: For low frequencies, that is, for frequencies well below the resonance frequency G ( ν ) = α$_{th}$ and the total thermal fluctuation within a bandwidth bw is Δ a$_{th}$ ( ν$_{c}$, bw ) ≃ α$_{th}$ √ bw . At the resonance frequency G ( ν ) = α$_{th}$Q and the total thermal fluctuation is as is well known from the literature. While the first relation is usually correct for static SFM applications, the second requires the condition that the measurement bandwidth is smaller than the width of the resonance curve: bw ≃ 1 /τ ≪ ν$_{0}$/Q . We note that in general this assumption may not be correct. In fact, for typical imaging applications (scan speed of 1 second/line, 250 points/line) a minimum bandwidth of 250Hz is required, which implies Q ≪ 4000 for typical cantilevers ( ν$_{0}$ ≃ 100 kHz) in order to fulfill the “small bandwidth” approximation. However, in UHV much higher Q values have been reported[25]. Note that the relation Δ a$_{th}$ ( ν$_{0}$ ) = α$_{th}$Q √ bw overestimates the thermal noise for large bandwidth, in fact for bw > ( π/ 2)( ν$_{0}$/Q ) the total thermal noise according to relation 8 would be larger than $^{√}$kT/c , which is non-physical since it is in contradiction with the Equipartition Theorem (rel. 1). As discussed below, for a sufficiently large bandwidth the total thermal noise is Δ a$_{th}$ = $^{√}$kT/c and is independent of the measuring bandwidth. A simple illustrative approximation for the spectral noise density of the tip-sample system is This approximation shown in figure 2 satisfies as the correct noise density the rela- tions α$_{appr}$ ( ν$_{0}$ ) = Q α$_{appr}$ (0), width of the peak of the order ν$_{0}$/Q , total noise Δ a = $^{(∫}$dν α 2 $_{appr}$( ν ) ) 1 / 2 = $^{√}$kT/c and, for large Q , α (0) = α$_{th}$ . The relation of noise in the “peak”, to that in the “flat” part is Δ a$_{peak}$/ Δ a$_{flat}$ = Q . For high Q factors most of the 8 Δ a$_{th}$ ( ν$_{c}$, bw ) = α$_{th}$$^{√}$$^{∫}$ ν$_{c}$ + bw/ 2 ν$_{c}$ − bw/ 2 dν | G ( ν ) | 2 (7) Δ a$_{th}$ ( ν$_{0}$, bw ) ≃ α$_{th}$Q √ bw = $^{√}$2 kT Q π c ν$_{0}$ √ bw (8) α$_{appr}$ ( ν ) =      √ Q Q $_{+1}$α$_{th}$ for 0 ≤ ν < ν$_{0}$ $^{(}$1 − π 4 Q ) and ν$_{0}$ $^{(}$1 + π 4 Q ) < ν < ν$_{0}$ $^{(}$1 + π 2 Q ) √ Q Q $_{+1}$Q α$_{th}$ for ν$_{0}$ $^{(}$1 − π 4 Q ) < ν < ν$_{0}$ $^{(}$1 + π 4 Q ) 0 for ν ≥ ν$_{0}$ $^{(}$1 + π 2 Q )     