Nanotechnology/AFM/Overview of properties of various cantilevers

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Examples of AFM cantilevers[edit]

For a BS-75kHz[edit]

L=225 w=28 t=3 h=17+3/2 (is actually trapezoidal)}

k_{N}=\frac{1}{4}Yw\left(  \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left(160000\right)  28\left(  \frac{3}{225}\right)  ^{3}=2.\,654\,8

f[Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}}
=\frac{\left(  3\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left(
225\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  }
{3\ast2330}}=79318.Hz

2\left(  \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{28\ast18.5}{3\ast
225}\right)  ^{2}=1.\,177\,8 so k_{lat}~k_{tor}.

k_{tor}=k_{N}\frac{1}{2}\left(  \frac{L}{h}\right)  ^{2}=2.65\frac{1}
{2}\left(  \frac{225}{18.5}\right)  ^{2}=195.\,99

k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left(  160000\right)
3\left(  \frac{28}{225}\right)  ^{3}=231.\,26

For a MPP311[edit]

Specs: 13 kHz, 0.45 N/m : L=440 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

k_{N}=\frac{1}{4}Yw\left(  \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left(
160000\right)  30\left(  \frac{4}{440}\right)  ^{3}=0.901\,58

f [Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}}
=\frac{\left(  4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left(
440\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  }
{3\ast2330}}=27655.Hz

2\left(  \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast
440}\right)  ^{2}=0.220\,96 so k_{lat}<k_{tor}.

k_{tor}=k_{N}\frac{1}{2}\left(  \frac{L}{h}\right)  ^{2}=0.9\frac{1}
{2}\left(  \frac{440}{19.5}\right)  ^{2}=229.\,11

k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left(  160000\right)
4\left(  \frac{30}{440}\right)  ^{3}=50.\,714

For a MPP211[edit]

Specs: 50 kHz, 1.5 N/m : L=215 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

k_{N}=\frac{1}{4}Yw\left(  \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left(
160000\right)  30\left(  \frac{4}{215}\right)  ^{3}=7.\,
727\,6

f [Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}}
=\frac{\left(  4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left(
215\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  }
{3\ast2330}}=1.\,158\,2\times10^{5}Hz

2\left(  \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast
215}\right)  ^{2}=0.925\,43

so k_{lat}~k_{tor}.

k_{tor}=k_{N}\frac{1}{2}\left(  \frac{L}{h}\right)  ^{2}=7.7\frac{1}
{2}\left(  \frac{215}{19.5}\right)  ^{2}=468.\,02

k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left(  160000\right)
4\left(  \frac{30}{215}\right)  ^{3}=434.\,68

For a MPP111[edit]

Specs: 200 kHz, 20 N/m : L=115 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

k_{N}=\frac{1}{4}Yw\left(  \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left(
160000\right)  30\left(  \frac{4}{115}\right)  ^{3}=
50.\,497

f [Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}}
=\frac{\left(  4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left(
115\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  }
{3\ast2330}}=4.\,048\,4\times10^{5}Hz

2\left(  \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast
115}\right)  ^{2}=3.\,234\,6 so k_{lat}>k_{tor}.

k_{tor}=k_{N}\frac{1}{2}\left(  \frac{L}{h}\right)  ^{2}=50.4\frac{1}
{2}\left(  \frac{115}{19.5}\right)  ^{2}=876.\,45

k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left(  160000\right)
4\left(  \frac{30}{115}\right)  ^{3}=2840.\,5