Hi Tim, Colin, Bernhard, Ethan,
This thread reminded me of the case of 'reflection from a hard boundary', ie which involves a 180 degree phase change. (various instructive simulations available to view via google). Is that what explains the 180 degrees phase change in X-ray reflection/re-scattering from an electron?
In reply to an earlier thread, referred to in this thread re QM, a useful source For one's imagination I find is George Gamow's "Mr Tomkins" book eg 'what if Planck's constant had a value of 1?' ...'it would make the game of snooker.....'.
Greetings,
John
John R Helliwell
On 6 Nov 2015, at 07:59, Tim Gruene <[log in to unmask]> wrote:
> Hi Colin, Hi Ethan,
>
> I thought that the 180 degree shift is explained with the negative charge of
> the electron, i.e. I expect the wave scattered by the nucleus to be in phase
> (one would need really high resolution data to measure this, though).
>
> The phase flip that Ethan describes should only happen at high resolution,
> where f' becomes greater than f0. At 2theta = 0, usually |f0|>|f'|, so that
> there is no phase shift due to the negative f', right?
>
> The differential equation for the damped oscillator results in the same
> equation we use to describe anomalous scattering. Maybe that's why someone has
> thought of explaining one with the other. But as with all models there are
> limits, and when comparing phenomena at nuclear level with a mechanistic
> model, that limit is reached rather quickly.
>
> Best,
> Tim
>
> On Thursday, November 05, 2015 11:32:09 PM Colin Nave wrote:
>> For scattering from a single electron isn't there a 180 degree phase change
>> between the incident and scattered wave? The spring demo shows this when
>> the driver frequency is higher than the resonant frequency. I think the
>> strong resonance in the spring demo corresponds to a strong white line. For
>> the real component of the dispersive correction there will then be a change
>> of sign across the white line as in the demo.
>>
>> The damped driven oscillator is a common method used to describe x-ray
>> scattering around resonance. However, I doubt whether the demo corresponds
>> in detail to this. I am also unsure about the degree of damping in the demo
>> (need another knob on the control box for this) but assume that the
>> driving force, amplitude of oscillation and damping all balance out to give
>> a steady state. As you imply, more frequency points would be useful.
>>
>> Colin
>>
>>
>> From: Ethan Merritt [mailto:[log in to unmask]]
>> Sent: 05 November 2015 17:33
>> To: Nave, Colin (DLSLtd,RAL,LSCI)
>> Cc: ccp4bb
>> Subject: Re: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Diffraction as a
>> Single-Photon Process; was RE: [ccp4bb] Twinning Question
>> On Thursday, 05 November 2015 11:51:49 AM
> [log in to unmask]<mailto:[log in to unmask]> wrote:
>>> Ethan
>>>
>>> My understanding is that one would have to have separate springs for each
>>> electron in the atom. Only some would be at resonance for a particular
>>> driving frequency. One would apply some sum for the total scattering of
>>> the atom.
>> Sure. The problem is that the phase in the physical demonstration does not
>> match up
>>
>> with the phase seen for anomalous scattering even when considered one
>> electron at a time.
>>
>>
>>
>> In anomalous scattering the f' term is maximum negative exactly at the
>> resonance point.
>>
>> So far as I can see, that strong negative component "flips the phase" so
>> that the
>>
>> 180° phase shift is seen at (or very near) to the resonance frequency. As
>> the
>>
>> frequency increases from there, the f' term returns to near 0 and the f"
>> term
>>
>> reaches its maximum. This corresponds more or less to a 90° phase shift as
>> the
>>
>> imaginary component dominates over the real component. At even higher
>> frequency
>>
>> the f" term also decays toward zero and the phase gradually returns to the
>> original
>>
>> value. Thus the sequence of phase values is just plain different in the
>> anomalous
>>
>> scattering case and the motor-driven-spring oscillator case.
>>
>>
>>
>> So as the frequency increases, the physical demo highlights an induced phase
>>
>> below -> edge -> above -> high
>>
>> 0 -> 90 -> 180
>>
>> Whereas anomalous scattering shows
>>
>> 0 -> 180 -> 90 -> 0
>>
>>
>>
>> If I were to show this video while teaching, and a student asked me to
>> explain
>>
>> in more detail how it relates to anomalous scattering, I'd be flummoxed.
>>
>>
>>
>> I am inclined to think the narration in the video is simply wrong, or at
>> least
>>
>> misleads the viewer to an incorrect conclusion. The caption on YouTube
>> doesn't help.
>>
>> It's not that the phase is locked at 0 below and 180 above the resonance
>> point;
>>
>> it's just that far from resonance point the input and output phases are
>> decoupled.
>>
>> The camera happened to catch times at which the driver and the suspended
>> object
>>
>> were "in phase" or "out of phase" and the narrator pointed that out, but
>> neither
>>
>> state is a general phenomenon. I think. But maybe I'm confused.
>>
>>
>>
>> Ethan
>>
>>> Of course this is trying to give some physical description for the
>>> electromagnetic field when I was complaining about a similar thing for
>>> quantum mechanics. A nice article by Freeman Dyson illustrates the
>>> difficulty of doing this for both approaches.
>>>
>>> http://www.damtp.cam.ac.uk/user/tong/em/dyson.pdf
>>>
>>>
>>>
>>> Colin
>>>
>>> -----Original Message-----
>>>
>>> From: Ethan A Merritt [mailto:[log in to unmask]]
>>>
>>> Sent: 04 November 2015 21:59
>>>
>>> To: Nave, Colin (DLSLtd,RAL,LSCI)
>>>
>>> Cc: ccp4bb
>>>
>>> Subject: Re: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Diffraction as a
>>> Single-Photon Process; was RE: [ccp4bb] Twinning Question>
>>> On Wednesday, 04 November, 2015 09:48:13 Colin Nave wrote:
>>>> Domenico
>>>>
>>>> Thanks for the kind words!
>>>>
>>>>
>>>>
>>>> I still don't like descriptions such as " Therefore, the anomalous
>>>> scattered photon will still be able to resonate with another anomalous
>>>> scatterer within the crystal" This is an attempt to describe what
>>>> happens to a photon before it has been observed and is therefore an
>>>> attempt to interpret Quantum Mechanics. As Feynman said about his
>>>> formulation - it is ""merely a mathematical description, not an attempt
>>>> to describe a real process that we can measure. Niels Bohr "brainwashed
>>>> an entire generation of physicist into believing that the whole job was
>>>> done 50 years ago" as Murray Gell-Mann said. This might be a bit unfair
>>>> but most physicists accepted the Copenhagen interpretation and
>>>> concentrated on carrying out the necessary calculations from which we
>>>> have all benefitted. Quantum Mechanics works but treat the physical
>>>> descriptions of the processes with scepticism.
>>>>
>>>>
>>>>
>>>> Regarding anomalous scattering I like the classical analogy in terms
>>>>
>>>> of a damped driven oscillator. There is a good video of this sort of
>>>> thing at https://www.youtube.com/watch?v=aZNnwQ8HJHU , for a non damped
>>>> case showing the phase changes near resonance.>
>>> I like the video, but it leaves me scratching my head a bit.
>>>
>>> One comes away from it expecting that there will be a 180° change in the
>>> phase of every Bragg reflection just from choosing a "long" or "short"
>>> wavelength x-ray source. [Or to be more precise a 180° change in the
>>> contribution of the anomalous scattering atoms to every Bragg
>>> reflection].
>>>
>>>
>>>
>>> I realize that if the phase were to flip for all atoms then by Babinet's
>>> principle the same underlying structure should be recoverable with either
>>> choice of phases, but does this really happen? Anyhow, that would not
>>> apply when only a subset of atoms in the structure have an absorption
>>> edge that is spanned to the two wavelengths in question. So does this
>>> demo really match up to what happens in an X-ray experiment?
>>>
>>>
>>>
>>> Ethan
>>>
>>>> A bit of damping is probably apparent but if anyone knows of a better
>>>> example for a damped oscillator I would be interested.
>>>>
>>>>
>>>>
>>>> Colin
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> -----Original Message-----
>>>>
>>>> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of
>>>>
>>>> Dom Bellini
>>>>
>>>> Sent: 03 November 2015 18:05
>>>>
>>>> To: ccp4bb
>>>>
>>>> Subject: Re: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Diffraction as a
>>>>
>>>> Single-Photon Process; was RE: [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> Dear All,
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Sorry for bringing back this old topic but I think I might have an
>>>> explanation to satisfy the original query, which I believe was not
>>>> conclusively put to rest in the end.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> I think the problem that me and the original poster, Jacob, were having
>>>> was that we were confusing energy with amplitude (at least I did).
>>>> I.e., anomalous scattering affects/reduces the amplitude of the atomic
>>>> form factor (or structure factor in case of a crystal), but not the
>>>> energy (or wavelength) of the scattered photon, which is the same as
>>>> that of the incident photon. Therefore, the anomalous scattered photon
>>>> will still be able to resonate with another anomalous scatterer within
>>>> the crystal, without breaking any conservation of energy theory. Since
>>>> anomalous scattering is an elastic effect, if one accepts the
>>>> explanation model of "photon interfering with itself" and "mini-waves"
>>>> in the case without resonators, then this model could be equally valid
>>>> even in the presence of more than one anomalous scatterer.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> I would like to thank Colin Nave to make me realize that I was mixing up
>>>> anomalous scattering with inelastic scattering. I am pretty sure I had
>>>> it clear while doing my PhD many moons ago.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> I hope I understood correctly the original question and that this
>>>> explanation to the query might make some sense to other people as well,
>>>> rather than just me :-).
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Best,
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> and sorry again for bringing this back,
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> D
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> ________________________________
>>>>
>>>> From: CCP4 bulletin board [[log in to unmask]] on behalf of
>>>>
>>>> [log in to unmask]<mailto:[log in to unmask]>
>>>> [[log in to unmask]]
>>>>
>>>> Sent: 31 August 2015 14:12
>>>>
>>>> To: ccp4bb
>>>>
>>>> Subject: [ccp4bb] AW: [ccp4bb] AW: [ccp4bb] Diffraction as a
>>>>
>>>> Single-Photon Process; was RE: [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> Dear Jacob,
>>>>
>>>>
>>>>
>>>> You are not the only one who does not believe in quantum mechanics.
>>>> Albert Einstein was probably the most famous non-believer.
>>>>
>>>>
>>>>
>>>> I agree with you that since we observe interference and diffraction
>>>> patterns, there must occur interference somewhere. Although Niels Bohr
>>>> claimed that you cannot say anything about a quantum system between two
>>>> measurements, my strong feeling is that we see interference between the
>>>> different superimposed quantum states. This is for me the truly spooky
>>>> part of quantum mechanics: instead of a single foton, as long as we do
>>>> not measure, there can be hundreds of fotons haunting our crystal.
>>>> However, the moment we switch on the light, we find only one. The
>>>> position of this foton will have been influenced by all other spooky
>>>> fotons.
>>>>
>>>>
>>>>
>>>> I do not see how quantum mechanics would not conserve energy, but would
>>>> be interested to learn.
>>>>
>>>>
>>>>
>>>> HS
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Von: Keller, Jacob [mailto:[log in to unmask]]
>>>>
>>>> Gesendet: Montag, 31. August 2015 13:06
>>>>
>>>> An: Schreuder, Herman R&D/DE;
>>>> [log in to unmask]<mailto:[log in to unmask]>
>>>>
>>>> Betreff: RE: [ccp4bb] AW: [ccp4bb] Diffraction as a Single-Photon
>>>>
>>>> Process; was RE: [ccp4bb] Twinning Question
>>>>
>>>>> This means that when a foton at the same moment interacts with 100
>>>>> scatterers (or resonators), there are 100 or more different states and
>>>>> in each state the foton interacts with a different scatterer. In each
>>>>> state, one foton interacts with only one scatterer. The moment the
>>>>> measurement is made, we find only one discrete foton, corresponding to
>>>>> one of these states. The distribution of the states, and therefore the
>>>>> possible outcomes, depend on the presence of all scatters/resonators
>>>>> within coherent range.> >
>>>> Then I don't see how interference or diffraction patterns can occur
>>>> without resorting to what others have said, which I don't understand
>>>> really: that interference is not really happening at all, but something
>>>> else with spooky probability distributions which don't need to
>>>> subscribe to conservation of energy.
>>>>
>>>>
>>>>
>>>> JPK
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Von: CCP4 bulletin board [mailto:[log in to unmask]] Im Auftrag von
>>>>
>>>> Keller, Jacob
>>>>
>>>> Gesendet: Donnerstag, 20. August 2015 20:42
>>>>
>>>> An:
>>>> [log in to unmask]<mailto:[log in to unmask]<mailto:CCP4BB@JISCMA
>>>> IL.AC.UK%3cmailto:[log in to unmask]>>
>>>>
>>>> Betreff: Re: [ccp4bb] Diffraction as a Single-Photon Process; was RE:
>>>>
>>>> [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> What I don't understand is how a single photon, which I thought by
>>>> definition was an indivisible quantum of energy, can be split up
>>>> arbitrarily amongst a number of scatterers into these "mini-waves."
>>>> Doesn't that self-contradict QM's concept of quanta?
>>>>
>>>>
>>>>
>>>> One might say that somehow there are two energy-related characteristics
>>>> to the photon:
>>>>
>>>>
>>>>
>>>> 1. the actual amount of total energy in the photon, and then
>>>>
>>>>
>>>>
>>>> 2. the "type" or "color" or "frequency" of the photon's energy.
>>>>
>>>>
>>>>
>>>> If you will allow me this dichotomy, then I can understand how it can be
>>>> distributed to different atoms-small portions of energy of the same
>>>> "color" are distributed to all of the resonators. One would also have
>>>> to presuppose another thing, that the resonators themselves are able to
>>>> accept packets of energy of size 1/n, as long as it's of a certain
>>>> color. The problem is, however, that allowing photons and resonators to
>>>> do these things violates what I thought was the central tenet of QM,
>>>> that there are indivisibles known as quanta.
>>>>
>>>>
>>>>
>>>> Maybe, then, one can just drop the bit about there being quanta, or at
>>>> least put a star by it?
>>>>
>>>>
>>>>
>>>> JPK
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> From:
>>>> [log in to unmask]<mailto:[log in to unmask]<mailto:hofkris
>>>> [log in to unmask]:[log in to unmask]>>
>>>>
>>>> [mailto:[log in to unmask]]
>>>>
>>>> Sent: Thursday, August 20, 2015 2:03 PM
>>>>
>>>> To: Keller, Jacob;
>>>> [log in to unmask]<mailto:[log in to unmask]<mailto:CCP4BB@JISCMA
>>>> IL.AC.UK%3cmailto:[log in to unmask]>>
>>>>
>>>> Subject: RE: [ccp4bb] Diffraction as a Single-Photon Process; was RE:
>>>>
>>>> [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> Valid questions.
>>>>
>>>>
>>>>
>>>> The phenomenon of resonance needs some explanation here, in terms we can
>>>> imagine:
>>>>
>>>>
>>>>
>>>> Take first the normal case: let all the n resonating electrons gain 1/n
>>>> in energy from the disappearing photon. These n resonating electrons
>>>> emit partial waves or whatever you want to call them totaling n*1/n in
>>>> energy, which recombines into the new photon. But what happens when the
>>>> phases lead to extinction? Where does the energy go? Well, it just does
>>>> not happen, it won't get scattered in THAT direction. So in the
>>>> probabilistic picture again, IF a photon does gets elastically
>>>> scattered, then it WILL appear again. WHERE it might appear, is given
>>>> by its probability distribution, aka the Fs. No contradiction here,
>>>> although I fully admit that the mini-wave picture results from the need
>>>> to explain, in experience-accessible terms, a non-experienceable
>>>> process. That is the QM conundrum, but not a contradiction.
>>>>
>>>>
>>>>
>>>> Now the anomalous (n.b.: not inelastic!) case: In this case the net
>>>> effect is a change in the fs and thus Fs, and again all it does is
>>>> change the probability distribution accordingly and above picture
>>>> holds. But wait - where does the X-ray fluorescence come from, and if
>>>> the photon uses all its energy to kick a photoelectron out, how can it
>>>> reappear? It does not. The unlucky photon that generated the
>>>> photoelectron is DEAD, otherwise we violate energy conservation. That
>>>> photoelectron then causes either fluorescence via outer to core
>>>> transitions or can be directly measured in case it manages to escape,
>>>> or make Auger electrons, whatever satisfies energy conservation. The
>>>> lucky photons, passing close to absorption energy, experience only the
>>>> change in scattering factor. If you look at the theoretical QM
>>>> calculations for absorption spectra (Cromer Lieberman etc.), you see
>>>> that the dispersion curves actually show a singularity at precisely the
>>>> orbital excitation energy. That absorption curve is again simply a
>>>> probability function for photon death at a given energy. In solids,
>>>> this curve can be more complicated and have more detail, but still the
>>>> same.
>>>>
>>>>
>>>>
>>>> So, you cannot simultaneously measure diffraction and fluorescence of
>>>> one and the same photon. The fluorescence scan does not come from the
>>>> anomalously but elastically scattered photons. It comes from the
>>>> absorbed dead ones. There is no difference between the normal and
>>>> anomalous 'miniwave' picture other than a change in fs and Fs.
>>>>
>>>>
>>>>
>>>> Radiation damage, btw, is just a process cascade caused by that photon
>>>> death.
>>>>
>>>>
>>>>
>>>> I abstain from digressing into inelastic/incoherent processes.
>>>>
>>>>
>>>>
>>>> Best, BR
>>>>
>>>>
>>>>
>>>> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of
>>>>
>>>> Keller, Jacob
>>>>
>>>> Sent: Thursday, August 20, 2015 9:48 AM
>>>>
>>>> To:
>>>> [log in to unmask]<mailto:[log in to unmask]<mailto:CCP4BB@JISCMA
>>>> IL.AC.UK%3cmailto:[log in to unmask]>>
>>>>
>>>> Subject: [ccp4bb] Diffraction as a Single-Photon Process; was RE:
>>>>
>>>> [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> Do you have any explanation of how a single photon, which contains x
>>>> amount of energy, can cause multiple electrons (at least 1000's!) in
>>>> anomalously-scattering atoms to resonate at that energy?
>>>>
>>>>
>>>>
>>>> We don't find that the presence of different numbers of resonant
>>>> scatterers requires x-rays of different energy; so why, if the energy
>>>> is being divided into different numbers of resonators, does the same
>>>> energy of x-rays work?
>>>>
>>>>
>>>>
>>>> I believe that BR's book says that the photon disappears or annihilates
>>>> briefly, then re-emerges. This must be true, then, across thousands of
>>>> electrons at once, both normal and anomalous?
>>>>
>>>>
>>>>
>>>> Jacob
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> From: Edwin Pozharski [mailto:[log in to unmask]]
>>>>
>>>> Sent: Thursday, August 20, 2015 9:36 AM
>>>>
>>>> To: Keller, Jacob
>>>>
>>>> Subject: Re: [ccp4bb] Twinning Question
>>>>
>>>>
>>>>
>>>> typo indeed. The point, of course, stands - with older sources there are
>>>> *no* photons inside the crystal for over 99% of the time. (Notice that
>>>> diffraction pattern is still present, Bragg's law satisfied, etc))
>>>> X-ray diffraction is, for all intents and purposes, a single photon
>>>> experiment. Even with the brightest and most coherent sources, when you
>>>> could have multiple photons within a large crystal, these are still
>>>> separated in space by a distance that is at least 100x the coherence
>>>> length. Thus, X-ray photons do not interact with each other (and even
>>>> if they would, it's still does not make them a wave, just good ole
>>>> photons that due to their high spatial density would have detectable
>>>> probability to engage in multi-photon events).> >
>>>> On Wed, Aug 19, 2015 at 5:13 PM, Keller, Jacob
> <[log in to unmask]<mailto:[log in to unmask]<mailto:[log in to unmask]:[log in to unmask]>>>
> wrote:
>>>>> Also, if your X-ray source is not exactly the brightest synchrotron,
>>>>> you are probably looking at ~10^9 photons/sec at best (I am estimating
>>>>> here that it would take at least 15-20 minutes of data collection
>>>>> using early 2000s "RAxisIV" in-house system to get diffraction image
>>>>> of intensity similar to 0.5s exposure at 12-2). That is one photon
>>>>> every nanosecond. Let's continue to ignore the fact that most photons
>>>>> just fly through. A photon zips through a 1mm crystal in about 3fs.
>>>>> Think about this - at a moderate intensity home source (and I can go
>>>>> to sealed tubes), the process of crystal illumination by X-rays is
>>>>> more like single photons flying through with about 300x long pauses
>>>>> between events. To scale this, imagine that a single photon spends a
>>>>> whole second inside a crystal, probing it's electron density. You
>>>>> would then have to wait five minutes for the next photon to arrive.> >
>>>> I was re-thinking through this, and I think one of these numbers is
>>>> wrong, viz, "A photon zips through a 1mm crystal in about 3fs." The
>>>> speed of light is 3x10^8 m/s, so this leads to ~3.3 ps for a 1 mm path,
>>>> and not 3 fs, a difference of ~10^6. Maybe it was just a typo? Anyway,
>>>> it may not make a huge difference, since this would still make for an
>>>> average of ~1 photon in the crystal at a time, assuming a high flux of
>>>> 10^12 photons per second. But of course there would be some Poisson
>>>> statistics involved, and there would be several photons a significant
>>>> part of the time.
>>>>
>>>> Also, I wonder about relativistic effects: in the famous train-in-tunnel
>>>> thought experiment, a large train can fit in a short tunnel if it's
>>>> going close to the speed of light. Is this applicable here, such that
>>>> many photons are in some sense in the crystal at once? Or maybe this is
>>>> a red herring.
>>>>
>>>> But, to change topics a bit: part of the reason I am wondering about
>>>> this is anomalous scattering. Since the resonance energy of an atom is
>>>> a fixed amount, how can one photon provide that energy simultaneously
>>>> to the requisite number (at least thousands, I would think) of resonant
>>>> scatterers? Something's very funny here.
>>>>
>>>> Or, come to think of it, perhaps resonant scattering is no worse than
>>>> normal scattering: if the energy is divided up between the all the
>>>> normally-scattering electrons, you even have a problem with the
>>>> one-photon picture, since the emerging radiation is still of the same
>>>> energy. You want to have everything being scattered with a certain
>>>> energy, but you also want all the scatterers to scatter. The concept of
>>>> "energy" seems to get strange. Does one then need two terms, in which
>>>> "energy" is just a characteristic of radiation, like a color, and then
>>>> there is some other attribute like "probabilistic intensity," which
>>>> describes how much "photon" is there?
>>>>
>>>> It is striking to me how much depth these everyday occurrences really
>>>> have when one starts wondering about them.
>>>>
>>>> Jacob
>>>
>>> --
>>>
>>> Ethan A Merritt
>>>
>>> Biomolecular Structure Center, K-428 Health Sciences Bldg
>>>
>>> MS 357742, University of Washington, Seattle 98195-7742
>>
>> --
>>
>> mail: Biomolecular Structure Center, K-428 Health Sciences Bldg
>>
>> MS 357742, University of Washington, Seattle 98195-7742
>>
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> --
> --
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> Dr. Tim Gruene
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