How much of the structure of an atom can be seen?
Well, it depends on what you mean by "seen"! Since nobody
can actually see an atom with "naked eyes", then of course
you cannot see the internal structure either. However there
are all sorts of probes, and the original discovery of
the nucleus was made by a rather clever experiment of
Rutherford, where he sent helium nuclei passing through a
thin foil of metal, and noticed that most of the nuclei
passed straight through, but some bounced back, some almost
going straight back the way the came. That indicated that
most of atoms are empty space, but there is a small hard
core in there, and you can calculate the size of it just
by figuring out the fraction of your original nuclei
that did not even notice it, and using the thickness of the
foil. I believe this was done in the 1910's or 20's.
Well, there is also atomic force microscopy (AFM) and
scanning tunneling microscopy (STM). These methods have
been used to resolve the electronic force contours of
surfaces and molecules adsorbed onto surfaces with resolution
on the scale of atomic dimensions (i.e., you see "bumps"
where the atoms are, and the bumps are atom-sized).
Then there is angularly-resolved photoemission of electrons,
which have been used to measure the angular distribution of
electrons emitted from atoms, molecules, and clusters. This
allows scientists to "observe" the shape of the electronic orbitals.
In short, although there is no way to directly "see" an atom,
there are literally hundreds of ways that scientists can probe the
details of atomic and molecular structure. Such scientists are
variously called spectroscopists, atomic/molecular physicists,
and/or physical chemists (like me).
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Update: June 2012