A laser actuated cantilever to search for deviations from gravity in the nanometre length scale
Helena Schmidt, Lars Andresen, Helmut Wolff, Gerhard Heinzel* and Ludger Koenders

(*Albert-Einstein-Institut Hannover)

Newton's Law of Gravity: $$ V(r) = -G\frac{m\cdot M}{r}$$
$$ V(r) = -G\frac{m\cdot M}{r}\left(1 + \alpha \mathrm{e}^{-r/\lambda}\right)$$
Frequency shift of a FM-AFM:
$$ \frac{\Delta\omega}{\omega_\mathrm{res}} = -\frac{1}{\pi ak}\int_{-1}^1F(z+a(1+u))\frac{u}{\sqrt{1-u^2}}\mathrm{d}u$$
$a$: Amplitude
$k$: Stiffness
$F()$: Force
$\frac{\Delta\omega}{\omega_\mathrm{res}}$: Frequency shift
$z$: Distance between cantilever and probe

Giessibl, F. J. (1997). Forces and frequency shifts in atomic-resolution dynamic-force microscopy. Phys. Rev. B, 56(24):16010–16015.

Electrostatic ($\sim \frac{1}{d}$):
d [nm] V [V] $\Delta f$ [Hz]
3001 0.01
2001 0.21
1001 0.85
501 3.4
3004 1.52
2004 3.42
1004 13.67
504 54.69
Casimir($\sim \frac{1}{d^3}$):
d [nm] $\Delta f$ [Hz]
3000.0003
2000.0015
1000.024
500.38
2013.7
10193.46
Yukawa:
d [nm] $\lambda$ [$\mu$m] $\alpha$ $\Delta f$ [Hz]
300 1 $10^{10}$ 0.0083
50 1 $10^{10}$ 0.0107
20 1 $10^{10}$ 0.011
300 0.01 $10^{16}$ $10^{-13}$
50 0.01 $10^{16}$ 0.01
20 0.01 $10^{16}$ 0.20
Electrostatic distance calibration
Measured electrostatic force over distance
Current state of measurements