**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**4.3.1 Displacement Sensors**

Diamondoid parts may contain features, or be positioned, in
picometer (1 pm = 10^{-12} m) steps, much smaller than the unavoidable
RMS thermal vibrations (Section 3.5.6). The nanomanipulator
robot arm described in Section 9.3.1.4 may be moved
in picometer increments, and the RMS thermal longitudinal displacements at the
tip of a 20-nm long, 10-nm wide diamondoid lever is ~1 pm.^{10}
Displacement sensitivity of 10 pm is routinely achieved in STMs. It is claimed
this can be improved down to the ~1 pm level;^{433}
STM resolution of 2 pm has been demonstrated experimentally.^{1260}
A C_{60} molecule used as the active element in an electromechanical
amplifier transmits ~100 times more electrical current when physically compressed
by 100 pm,^{561} suggesting a minimum
deformation detection limit of a few picometers. (A strain gauge and a vibration
sensor using a (17,0) carbon nanotube (~2000 atoms) has been proposed.^{2908})

Thermally-induced positional uncertainty in the displacement
of nanoscale components is approximated^{10}
by the classical value

where T is temperature (K) and k_{s} is the Hooke's
law spring constant, or restoring force stiffness (N/m). At 310 K, this classical
approximation is accurate to within <~10% of the quantum mechanical treatment
for harmonic oscillators with RMS displacements Dx
>~ 10 pm.^{10} Spring stiffness k_{s}
is ~0.1 N/m for nonbonded (noncovalent) interatomic interactions, ~30 N/m for
covalent bond angle bending, ~400 N/m for covalent bond stretching, and ~1000
N/m for solid 1 nm^{3} diamondoid blocks.^{10}
At 310 K, springs of such stiffness produce minimum displacement uncertainties
of 200 pm, 10 pm, 3 pm and 2 pm, respectively.

The log ratio of detection energy to noise energy, the signal/noise
ratio (SNR), for a displacement sensor is derived from the harmonic potential
(1/2) k_{s} x^{2} as

_{}
{Eqn. 4.9}

Assuming a very stiff k_{s} = 600 N/m, a minimal SNR
= 1 gives a minimum detectable displacement of 6 pm, or 10 pm at a more reasonable
SNR = 2 (20 dB). The conservative conclusion is that Dx_{min}
~ 10 pm displacements should be reliably detectable by medical nanosensors in
vivo, in a measurement time t_{meas} ~ 10^{-9} sec (Section
4.3.2). This compares favorably with the stereocilia of the inner ear, which
can only detect 100 pm displacements in 10^{-5} sec.^{446}

Last updated on 17 February 2003