**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

**4.6.1 Minimum Detectable
Temperature Change**

A sensor system consisting of N atoms has (3N-6) internal
coordinates and thus ~(3N-6) oscillators, each with an average energy of ~kT
assuming no modes are high enough to be frozen out. However, the standard deviation
of the energy of the oscillators around the mean energy is ~kT, and these energy
fluctuations are uncorrelated (ignoring coupling between the oscillators), so
(3N-6) of them will have an average fluctuation of kT (3N-6)^{1/2}.
(By comparison, the energy in one 10-micron infrared photon is ~5 kT.) As a
result, an instantaneous measurement of the total energy of all N atoms gives
a minimum detectable temperature change DT_{min}
of

for a set of N_{meas} independent temperature measurements
using a sensor of volume V_{sensor} constructed with material of atomic
number density n_{d}, which might be maximized using diamond (n_{d}
= 1.76 x 10^{29} carbon atoms/m^{3}).

Thermal equilibration time t_{EQ} ~ L^{2}
C_{V} / K_{t} (Eqn. 10.24)
for a sensor of size L, of heat capacity C_{V} (1.8 x 10^{6}
joules/m^{3}-K for diamond) and of thermal conductivity K_{t}
(2000 watts/m-K for diamond^{460}).
Thus, t_{EQ} ~ 10^{-13} sec for a sensor of size L = 10 nm,
10^{-11} sec for L = 100 nm, and 10^{-9} sec for L = 1 micron,
so thermal sensors up to 1 micron in size should easily achieve thermal equilibration
within a typical measurement cycle time Dt_{min}
~ 10^{-9} sec. Sensor measurement time t_{meas} = N_{meas}
Dt_{min}.

A (57 nm)^{3} sensor can detect DT_{min}
/ T = 10^{-4} (~31 millikelvins at 310 K) in a single measurement (N_{meas}
= 1), with measurement time t_{meas} ~ 1 nanosec. A sensitivity of DT_{min}
/ T = 10^{-6} (~310 microkelvins at 310 K) may be achieved using either
a ~1 micron^{3} sensor with a single measurement (N_{meas} =
1, t_{meas} = 1 nanosec) or a (124 nm)^{3} sensor with N_{meas}
= 1000 independent measurement cycles and t_{meas} = 1 microsec. Finally,
a 1 micron^{3} sensor can detect DT_{min}
/ T = 3 x 10^{-9} (~1 microkelvin at 310 K) with N_{meas} =
100,000 independent sensor cycles, giving a measurement time t_{meas}
~ 100 microsec.

These figures are confirmed by experimental estimates of detector
noise temperature, such as DT_{min} / T =
(k / L^{3} C_{V})^{1/2} ~ 10^{-6} for a silicon
nitride detector^{678} with C_{V}
= 5.2 x 10^{6} joules/m^{3}-K and L ~1 micron. It has been calculated
that sensitivity of ~1 microkelvin is theoretically possible using quartz electronic
microresonators as precision thermometers.^{462,1699}
Thermocouple probes with 100 nm tips have already shown ~100 microkelvin sensitivity,^{463}
tunneling thermometers capable of measuring thermoelectric potential localized
to atomic-scale dimensions have been proposed,^{464}
and experiments leading toward "yoctocalorimetry" were being pursued in 1998.^{2928}

For comparison, heat sensors in human skin have DT_{min}
/ T ~ 3 x 10^{-4 }(~90 millikelvins), and the infrared sensor pit of
the rattlesnake is sensitive to an energy intensity of ~0.8 pJ/micron^{2}
in a measurement time t_{meas} ~35 millisec and achieves DT_{min}
/ T ~ 3 x 10^{-6};^{701,826}
mosquitos register DT_{min} / T ~ 6 x 10^{-6}
at a distance of ~1 cm.

Last updated on 17 February 2003