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 /
