In order to discuss this relation, and to compare with the results known from the literature, we will consider the different approximations for large and low oscillation amplitudes, as well as for small and large bandwidth. For (very) low bandwidth, and large oscillation amplitude we obtain, using relation 8 which is similar to the result reported in the literature[17, 18, 20] (relation 2). As discussed in the previous section, this (very) “low bandwidth approximation” is usually not valid. On the contrary, we believe that in most applications the bandwidth is larger than the width of the resonance curve. Then, as long as instrumental noise is negligible, the correct approximation would be a “large bandwidth” approximation where all the thermal noise is “seen” by the DSFM - detection system. In this case the corresponding (total) frequency noise is which is Δ ν$_{th}$ ≃ ( ν$_{0}$/Q ) a$_{th}$/ (2 √ 2 a$_{ex}$ ) for large amplitude and for π/ $^{(}$2 √ 3 ) ν$_{0}$/Q ≃ 0 . 9 ν$_{0}$/Q for low amplitude. The characteristic frequency determining the thermal frequency noise is therefore the width ν$_{0}$/Q of the resonance curve, and for low oscillation amplitudes ( a$_{os}$ ≪ a$_{th}$ ) the thermal frequency noise is essentially given by the width of the resonance curve. In particular, this implies that, as demonstrated recently for spectroscopy applications[34], DSFM is possible without external excitation of the tip-sample system. Moreover, we believe that a properly designed DFSM-electronics should be able to lock onto the thermal noise of the cantilever. In order to confirm the validity of the relations just discussed, noise measurements have been made as a function of the bandwidth and the oscillation amplitude. A commercial SFM - system based on optical beam deflection[28] was used to measure cantilever motion and analysis of cantilever oscillation was performed either with the DSFM electronics[28] or with a digital lock-in amplifier[29]. The set-up of the SFM system and the essential features of the DSFM electronics are shown in figure 1[30]. A cantilever with a nominal force constant c ≃ 0 . 4 N/m [31] was used and the tip was kept at a large (1 mm ) distance from the sample. V. EXPERIMENTS Δ ν$_{th}$ = ν$_{0}$ 2 Q$^{√}$ 1 π kT ca 2 ex Q ν$_{0}$ √ bw Δ ν$_{th}$ = ν$_{0}$ 2 Q$^{√}$ 1 3 /π 2 + 2 a 2 $_{ex}$/a 2 th (15) 11