From: jthorn@mach.thp.univie.ac.at (Jonathan Thornburg)
Subject: Re: Laser Interferometry Accuracies
Date: 3 Feb 2000 11:12:52 -0600
Newsgroups: sci.physics.research
Summary: measure motions << size of an atom by averaging over many atoms
Keywords: atom, measure, Angstrom, nanometer, average, sqrt(N),

In article <871826$jei$1@nnrp1.deja.com>,  <siop@my-deja.com> wrote:
>Dr. Baez has said that the LISA project would measure the distance
>between satellites to the order of .1 Angstrom. This is smaller than a
>hydrogen atom. No mirror or antenna is this smooth.
>    The Earthbound gravity wave detectors claim much smaller
>accuracies.
>    Between which parts of which two atoms are the satellites
>measuring [[...]]
>Matter does not seem smooth enough to support this kind of
>accuracy.     Can someone enlighten me in a general way?

The answer to this paradox is that we're not measuring the motion of
*one* *individual* atom, but rather the overall (average) motion of
a *macroscopic* object (a LISA/LIGO/GEO600/etc mirror weighing some
kilograms) which contains something on the order of 1e27 (a thousand
million million million million) atoms.  What LISA/LIGO/GEO600/etal
are actually measuring is the change in optical path length in an
interferometer, i.e.
	[up to a factor of 2 which doesn't matter here]
the change in the *average* position of all the atoms on the front
surface of the mirror.  Since the mirror is a rigid body, and we're
interested in frequencies much lower than its internal vibrational
periods, all the atoms in the mirror are effectively rigidly connected,
so what we're really measuring here is the *average* position of all
1e27 or so atoms.

It's the averaging over this huge number of atoms that lets us define
the overall mean position to much better than the size -- or even the
thermal motion -- of any individual atom.  Roughly speaking, averaging
over N atoms lowers the position error by about a factor of sqrt(N);
for N = 1e27 that's a reduction of around a factor of 3e13 (30 million
million) or so.


In fact, you don't even need high-tech equipment to make maasurements
to smaller-than-atoms accuracy:  If you work out the motion of the
bones in a human ear when listening to normal conversation, you'll
find that these too are moving much less than an atomic diameter.
The reason we can still easily hear the sound is the same as before
-- our ear bones contain *many* atoms, and its the overall average
position that counts.

-- 
-- Jonathan Thornburg <jthorn@galileo.thp.univie.ac.at>
   http://www.thp.univie.ac.at/~jthorn/home.html
   Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik
   "The first strike in the American Colonies was in 1776 in Philadelphia,
    when [...] carpenters demanded a 72-hour week." -- Anatole Beck