**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

**10.1.2.3 Acoustic Transmission
Line Oscillators**

Drexler^{10} describes
a diamondoid acoustic transmission line in which a ~2 nN force pulse is initiated
at one end, travels to the far end at v_{sound} ~ 17,300 m/sec, then
is received by a mechanical displacement probe, possibly with significant energy
recovery. Such lines may also be used to generate precisely delayed acoustic
signal sets suitable for clocking applications.

Consider a "starter" pulse applied to a short feeder line
that symmetrically bifurcates into a set of n_{line} = 10 lines each
of length l_{n} = (1730 n) nm (n = 1, 2, ..., n_{line}), requiring
a total length L_{line} = (1730) n_{line} (n_{line}
+ 1) / 2 = 95,150 nm of transmission line to obtain a set of lines having all
n_{line} time delays. The bifurcated pulses arrive at the end of each
line in 0.1 nanosec (n = 1), 0.2 nanosec (n = 2), ..., 1.0 nanosec (n = 10);
any of these pulses may be drawn off and used for diverse clocking purposes,
or may be fed back into the initiation mechanism and used either to trigger
the next starter pulse or to achieve more robust error correction. If acoustic
lines have a cross-sectional area of ~30 nm^{2}, then the total volume
of all ten lines is ~3,000,000 nm^{3}, which may be coiled into ~0.3%
of the volume of a 1 micron^{3} nanorobot. Such diamondoid acoustic
power transmission lines are essentially lossless (Section
7.2.5.3). If a complete set of lines containing all n_{line} delays
is not required, any desired single delay time may be obtained by connecting
up to n_{line} segments end to end, or by bouncing a pulse off of the
ends of a single line up to n_{line} times.

Aside from the many potential frequency instabilities inherent in pulse detection, signal re-initiation, and acoustic interferometry mechanisms, two fundamental sources of frequency instability include:

1. changes in the velocity of sound due to thermal variations, and;

2. changes in acoustic path length due to elastic longitudinal displacements of thermally excited transmission rods.

First, the speed of transverse sound waves in an isotropic
elastic medium having Poisson's ratio c_{Poisson} (~0.1 for diamond)
is given by:^{10}

where Young's modulus E = 1.05 x 10^{12} N/m^{2}
and density r = 3510 kg/m^{3} for diamond.
In addition to the thermal dependency of E, r varies
as (1 + b_{thermal} T)^{-1 }where
T is temperature, because volume changes according to the volume coefficient
of thermal expansion b_{thermal} = 3.5 x
10^{-6} K^{-1} for diamond, 1.56 x 10^{-5} K^{-1}
for sapphire. In an uncorrected oscillator system, Dn
/ n ~ Dv_{sound}
/ v_{sound} ~ (1/2) b_{thermal} DT
~ 10^{-5} for diamondoid transmission lines, assuming that DT
~ 6 K temperature variations are typically encountered inside the human body
(Table
8.11). Correcting oscillator timing using independent temperature sensors
accurate to DT_{min} / T ~ 10^{-6 }(Section
4.6), and ignoring other possible sources of frequency instability (which
may be significant), could reduce measurement DT
to ~310 microkelvins at T ~ 310 K, thus improving Dn
/ n significantly for this source of frequency instability.

Second, longitudinal displacements DL
/ L_{rod} ~ 10^{-4} - 10^{-5} at 300 K for rods of length
L_{rod} = 1.73-17.3 microns and cross-section A_{rod} ~ 30 nm^{2}
(Figure 5.8 in Drexler^{10}); J. Soreff
observes that DL / L_{rod} ~ (kT / E L_{rod}
A_{rod})^{1/2 }~ 10^{-6} for ~0.01 micron^{3}-volume
systems. For the zeroth longitudinal vibrational mode,^{10}
these displacements occur on a timescale n_{osc}^{-1}
= 4 L_{rod} (r / E)^{1/2} = 0.4-4
nanosec, comparable to signal transit times, taking r
and E for diamond as above and L_{rod} = (1730 n) nm with n = 1, 2,
..., n_{line}. This may restrict Dn / n
to ~ 10^{-6} unless transmission lines can be further rigidized by end-clamping,
sheathing, or latticed bracing at intervals, all of which, in effect, enlarge
A_{rod}.

Last updated on 23 February 2003