On Thu, 2010-03-18 at 12:51 -0500, Jacob Keller wrote:
> Does anybody have a good way to understand this? 

Sure, it just depends on what would one consider a "good" way to
understand.  For a pure empiricist, it's good enough to see one of those
two-dimensional phase swap pictures.  For a "mathematically inclined"
there is nothing better than the song that Fourier transform sings to
them.  Based on the fact that you find the simple statement that
"Fourier synthesis emphasizes phases" "pretty unsatisfying" I would
allow myself to guess that you are a person trying to grasp concepts in
physics without learning its language, which is math.  This is not meant
in a bad sense - it is beyond doubt that true understanding of physics
comes from explaining things with words and analogies, not by writing
down a bunch of equations.

Let's try a couple of things.

1.  Why are some reflections stronger than others?  This is easy to
understand by invoking the Richard Feynman's picture of every scatterer
contributing a little arrow to the final result.  All arrows are the
same length, but are rotated depending on how long does it take for a
photon to fly to the detector.  (Helps if you draw them as we go along).

Every reflection is produced by an imaginary set of Bragg planes.  When
atoms are randomly distributed in the direction perpendicular to the
planes, their corresponding arrows assume all possible orientations and
the resulting "big arrow" is likely close to zero.  Now if atoms tend to
cluster near the Bragg planes, majority of the arrows will all be
pointing the same way and the resulting arrow gets longer.

Other words, the amplitude of the reflection increases when atoms are
arranged in space with periodicity matching the distance between Bragg
planes.  This is how Patterson map gives you the set of interatomic
distances.  Unfortunately, there are so many atoms in proteins that
their motions combined with experimental uncertainties turn it into
incomprehensible mess.

So what is the phase?  It is the orientation of the big arrow, and we
have so far only addressed its length.  If you prevent Bragg planes from
sliding (i.e. one of the planes must pass through the fixed origin),
then the arrow orientation will tell you where exactly in space the
"clustering" of atoms is located - halfway between the planes, or three
quarters away from the origin, etc.  Hopefully you see how this
information is crucial in determining the structure.  Actual structure
determination is done by combining information from different
reflections, and without phase information the corresponding "atom
clusters" or "density packs" can slide all around the place, producing
great deal of uncertainty.

2.  Sports analogies are always popular.  Let's remember that
intensities/amplitudes tell you how many photons have arrived, while
phases tell you when they did.  I'll stick to B-sports.

Baseball.  A blind and deaf catcher only knows how many balls he
received (amplitude).  This can tell him how many pitches were made, but
not how far in the field they landed.   A catcher who is only blind
hears when bat hits the ball and can time how long it took for the field
players to return the ball to him (phase).  He can then delineate how
far balls usually fly, thus determining a 1D structure.  If field
players record when they received the ball (multiple reflections), the
exact place where ball landed can be figured out (actually,you need to
split every ball and send a copy to every field player for triangulation
to work, but sports analogies do not have to be perfect :)

Basketball.  A team always plays a primitive game, where ball is thrown
in and followed by jumpshot.  They are so great, however, that they
never miss.  If you only count the score (amplitude), you will only know
how many possessions they had.  But if you record the time between the
whistle and scoring (phases), and take into account that pass is
horizontal and shot has a significant vertical component thus its
horizontal speed is lower, you can figure out from what distance they
shoot more often.  Again, to pinpoint exact location of all four players
(who never move), you'll need to allow two/three/four/etc passes
(multiple reflections).  But sports analogies do not have to be
perfect :)

Biathlon.  If you only count the shots at all targets of shooting range
(amplitude), you only know how many are running the race.  If you record
the time when shots are made (phases), you know who is running first,
second, etc.  Structure here is inherently 1D, but sports analogies do
not have to be perfect :)

HTH,

Ed.


-- 
Edwin Pozharski, PhD, Assistant Professor
University of Maryland, Baltimore
----------------------------------------------
When the Way is forgotten duty and justice appear;
Then knowledge and wisdom are born along with hypocrisy.
When harmonious relationships dissolve then respect and devotion arise;
When a nation falls to chaos then loyalty and patriotism are born.
------------------------------   / Lao Tse /