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20 Light Scattering Spectroscopy Slow Mode

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lasses in polymer Dynamics based
on George Phillies book phenomenology of polymer solution dynamics Cambridge University Press 2011
ends today this lecture E is lecture 20
dynamic structure factor and the slow
I'm professor valleys and today we're going to advance to continue our discussions of light scattering spectroscopy and the dynamic structure factor of polymer solution there our last lecture which for the wider audience was while ago I discussed something of how light scattering spectroscopy works the core issue is you said in a laser beam it's coherent monochromatic light it scares power angle scattering process can be described by the first-order report approximation because at least in usable systems scattering is quite weak as a result the incident beam is the same intensity everywhere along its length the call it's a plane wave to some optics questions which are actually quite tricky are collecting optics just pick up plane waves and with reasonable design can get by also and minimize losses scattered light all of the things that does the scattering is the forest faithful for a component of the index of refraction "quotation mark alternatively the density of scary Park and so we're looking at a spatial sign-waving concentration and the spatial "quotation mark where the case in time to diffuse another few minor questions you'll have your 1st you completely unfamiliar with this the 1st used the word that spatial co-signed with come from did you use some complicated layering technique inject inserted into the system and the answer is no the particles move around rent if they were killed in a raid in a crystal with a few exceptions the spatial Fourier components of the density with all these 0 well except for 1 and you see no scary for all however the particles do move around if we do a description of the concentration of versus position it has spatial foreign components which fluctuate time and then but she it's harmful .period large and other points the small what we do through numerical technique known as the intensity correlation function is to ask and answer the following question suppose song given moment as a result of the ground motion the spatial 48 component of the the concentration is quite large certainly happen randomly there's going to be very small part of the time should be very large part of having found those moments when the fluctuations of quiet words we then ask what happens on the apparatus as time goes on and the answer is the of regression hypothesis is that on the average as time goes on the concentration of particles goes down it goes down to well we're looking at especially for a component goes down to 0 in on the average and the intensity of the scattered life on the average relaxes back toward 0 In fact since we're looking at a concentration camp outside where you stick the co-sign ways and fix long diffusion and outcomes and dying now this technique is a 1960's technique as originally created by the recent shows up at 1st in the 60's and then ballooned in the eighties and 90's is not as a series of technical advances which make them as possible the 1st technical advance was the laser now the issue monopolizes the conclusions was the 1st edition with laser visitors source of code during long life and as a result the issue is here here's our staggering volume the light comes in from the side it's scattered from particles in different sides of the region it heads out toward the observer and you would like interference between whites scared from here and white scared from here well in order to do that like here and the light here have to be consistent in their faces triangles however we have a Ladies European going this way it has a consistency in phases over some distance on at that distance has which is called coherence 1 has to be large relative to the ball and we're looking at what sounds impressive realizes the 1960's commercial leaders had coherence wants it to be easily km however think about this realize sample volume we're talking about not kilometers along its of the focus with laser beam which might be 100 microns and it might be the length of the ski volume given that we have collecting involves a millimeter as long as the coherence length as long by comparison with the millimeter you have coherency if you remember Michaelson interferometry treaty with the mercury arc source as opposed to Malaysia you realize that mm actually quite sure and a classical light source if you filtered back has plenty of coherence .period the experiment this has been confirmed experimentally people who don't cause elastic light scattering spectroscopy using highly filtered sunlight however they didn't problem and the problem they get is that the other advantage of lasers the important thing was the important thing is to always is right In the light laser game which is very narrow you were putting through a great deal of power in May Seiji lasers 21 a lot so what it was what light bulb is almost nothing however I have
won a lot like called it gives off light animal direction the laser beam comes out every laser in a straight line you can treat it as a point source but the the the notional point source is many kilometers to the rear of the laser I want lot like Paul kilometers away you don't see the lasers plenty bright enough to damage human eye if you were not respectful of laser light the major advantage of Malaysia and brightness this by focusing lenses 1st so important alliance brightness important can you just give great from water as will make clear a few moments light scattering Tennessee actually 2 photon process here comparatively intensity now and the intensity of a later moments in order to do that need to get a least 1 photon here and at least 1 vote on him as a result the signal not just the signal the signal-to-noise ratio goes up with the brightness of the ways that further work the 2nd reason we're working at a 19 sixties 19 seventies technique is digital electronics now the digital electronics shows up in 2 different ways are the less obvious way is that it shows up that you were running the experiment and because you have a computer and the lab which was extremely exotic when I started as a graduate student in 1960 because you have a computer and what have we had our 1st round 7 years so you can the measure the spectrum do the data analysis in a couple of minutes see immediately she everything is doing well past something has to be adjusted there's funny background noise the sample was no good and because you can do the data in Alsace immediately you can correct on the wall when I started out you measure the spectrum 2 hours you digitize the spectrum to while you type the spectrum and computer cards and you took the computer cards to the computer center and the next day he said at discovering what you meant this was incredibly slow and inefficient and ineffective no 2 ways about it just did not work as a highly effective way for doing large-scale as opposed to look at the new experiments with the demonstration computers laughing stock August 2nd issue which is just as important is that well I suppose you I gave you a laser and were in 1950 with 45 you could still done experiment still have photon counting electronics and you couldn't go the experiment by using a device known as a wave analyzer that measured the frequency spectrum the frequency spectrum although was measured by analog means sets were awaiting analyzes was even if you work very hard if you propose you could get a signal-to-noise ratio in the experiment 100 that you were being wildly optimistic a hundred is abysmally bad to engage with digital electronics kind of so let us consider how we actually measure of correlation function only remember what it is worth it for measuring intensity of time t intensity time he'd lost out and that was taken after now the best you can do is always to do if you're looking at life to vote on I saw a photo on showed up at this moment in time and that is the best you can do if you say Well we've got lots of like and let's do analog signal analysis you were throwing away vast amounts of information however In in practice what news say I haven't detected it gives me an electronic pulses that corresponds to the fact that we detected the photo on the the digitally altered also has some shape and pushed the Shaffer involved signal processors and the signal processors do very nice job uniform advising and then do what is called meeting in the faction that is developed looks say Where's the middle of signal to look at where the signals 1st ramping up and you can actually measure Leading quite accurately pot in in position within nanoseconds or slightly lower systems can have a 2nd 50 years the pulses wired you bet and say it will take time and system bridge the dates and the photons show up at different times and we count the number of times In each time now photon shows up entitled the boundaries were very close unique electronics that are designed so they always treated the same way and always voted show the year into Blondie and or the other .period but not all of the were Niger and that's 1 of the circuit design things that has long since sold this is a list of numbers in the cold and than 2 and 3 and 4 and it is a cow of harmony photons came in each time these numbers comes of photons in time intervals of sun with our intensity any then form the correlation function think the correlation functions as a function of it has number of photons in Beian the number of photons Indian I lost she had indeed to vote to Britain's separated in time by Chadians to get what they need is go and we saw all the time he was 1 of the 1st out to some large number how large a number of well a reasonable dimwits might be a Microsoft or less the you wrote these
early for a reasonable number of minutes so you might actually have a 10th of the minor samples that were work over and 10 samples going into you measure if you have tell the 10 samples of year veteran the statistical error and this can be made quite small look get the the other issue is that there is a kind of way here there was continuous time image as a number of billions but bins have weighed down and we don't How he shit down so we even here intensity here in intensity 1 being later you measuring this correlation function at a time delay which is something we want to help now I have to put in some qualifications the 1st qualification is that this process does not work 318 0 if you do the process for G-8 equals 0 you do not measure the intensity correlation function at times 0 what is the complicated lots of problems we look at the J. Walter is easy to draw the 1st filed on game in some places time between here and there the 2nd vote on pain some places and times between them here and so we have a and 1 which was best 1 and the end to was which was within 2 but the 2 photons who in right on top of each other in which case the sample CEO of Tao for some very short time it could also be that 1 photo and came here and 1 photon there and we sampled sea of power virtually instant sometimes they're very early to galvanize part the result is the what we look at 1st correlated all of these channels saying with is a way to average in which we measure of power excessive power has been an average with waiting for 1 of whom was like this so we put the most weight in the middle and lastly the edges may then asked Well how much there is there in measured in Poland your measurements correlation function the and the answer is correlation function a straight line courses there no error in all its success this correlation function is very slightly higher which is realistic there is an error but Arab instead the outcome of this measurement is the value of correlation function many ways the error is proportional to the Delta house and as long as the correlation function doesn't change much during 1 being with which is the only way anyone right probably equipment the here is small not navigate because life becomes much more complicated than it used to be reserved different weights would be beyond this cost tell you why you're doing this other than to say that if I a lot correlation of hers what got but have called relaxation the looks like this and how you're changing the correlation function may take a 2nd in here to 20 microseconds yes notices difference well splitting listens to 1 might for 2nd attempt by the 2nd pieces makes sense because it's changing microseconds scale but if you were out here and seeing something changing on a 2nd scale splitting it into microseconds pieces is wasteful and their costs the so you actually do this and if you do this the only way the time becomes much more complicated and the answer is a fraction of these it's this time what happened this summer was this time the set exactly the same formula works all of the channels are the same way all this is approximate this is there is some this is approximately this assistance offered this is it's because of that huge less time is with almost all unfortunately there has crept into literature is an extremely severe error noticed that half channel correct and the have channel correction which is probably all of this is shown by a simple and direct analysis is kind to publish review scientific instruments says that if we have 2 channels like this these 2 channels are not separated by the 1 billion people walk by 1 billion minus 1 half of this leads or half of last completely role perhaps channel approximation of erroneous the correct formula comes fairly close to if this way is much larger than so that is the sort of thing we do when we're doing light scattering spectroscopy OK so we actually measure the correlation function the function that is of interest and discussed and passed lectures is not see itself by the time time-dependent part of sea Towery actually use and that's in previous lecture so we won't over over again instead of pushing and ask what happens when you might get respect from holders in Britain as well the shortest form answer is you can will hold series of different measurements on polymer solutions and ask different sorts of questions and what what possibly eligible audiences do you have a light scattering spectrum there's a non Longoria linear plot like this and 1 thing we can do is measure the 1st human on were initially slow also it turns out that's actually fairly accurate as a method of characterizing spectrum the closest exponential you actually do cumulative analysis which we discussed earlier and out of the the 1st humility even further and diffusion coefficient which is unambiguous
reproducible and describes the average relaxation going on in 6th another experiment it is to make the Sierra Nevada March you make the scattering factor March by going to larger angle by using violent light red red light the opposite is you go very well scattering angles becomes smaller meaning and looking at distances over larger distances but you try to make you worry the was RG it's the size of policy last year refused .period meanwhile here is all the more and we're sensitive to concentrations fluctuations will like this way meaning that light scattered from here on the polymer unlike scared from their own all have different phases and interfere and we can see the pall OK what else can we do Wells we will test unity and ask how prosecuting the patterns on the way vector and time and comes trade there's another stop you remember I was discussing Fulton correlation that and it's really hard with classical means you could get out of something like 15 nanoseconds but you have to do some tricks with the the following multiplier to detection to avoid all lawyer always and with modern technology you can get down to a few nanoseconds and a few nanoseconds no corresponds to all 100 matter there is an alternative scheme known as categories overall to the alternatives this interferometry plot in which you measure the frequency spectrum of the light using well fancier than using prison any matter what's going on and it hurts timescale now the gigahertz will not show large-scale over motion on at the time gigahertz frequencies meaning times much shorter than in 2nd Pollard has moved and so you aren't seeing whole polymer dynamics baby can see solid dynamics and will talk about this later on OK what else can you do well some people would say measured using old and from this a dynamic scaling England on the idea of dynamic scaling length comes out of critical phenomena were where like this works very well for describing how things move in binary liquid you approach a consolation .period you approach to temperature and concentration which a binary mixtures face separated since different labels all however if you try to take the same formula applies college infusions discusses backwards they say proteins you discover that this formula was completely what is completely wrong and simple sense the Yukon measured diffusion coefficient you can measure or calculated distance over which oracle motions correlated and if you say take the protein and increase their interactions by charging them off changing age reducing salt concentration you discover that is the you increase the repulsive interactions with fusion becomes faster and the correlation length becomes longer not shorter this fall stressed OK so the sorts of experiments 1 looking wait added from going back many years but the there were people said Well let's start out and look at Diamond followers after all that's already a complicated problem and this was done by the for series papers what he said was following where it's always factor and we see scattering and we see scattering complaint as part of the change in that part change and because we have from 2 laser to these points if light scared from here on the phone changes we're on all change those 2 light waves have different faces and can fear and so has the Polish changes shape these points around the amount of interference change so in order to understand the spectrum we have to understand the molds of motion of Paul weakens and 1 motion was the center of mass the future status you can view that there is a center of mass it does move ended most effusively and that contributes to the however there is also a series of loans and which the Parliament while oxen with respect to each other and when he was able to show was 1 of those loans is and it remains a relaxation of this where most of half so there is the other modes of mean you have any particles in the change there should be 3 and 3 of 6 anyhow current however most of them don't scare a lot of my analyze the 1st 2 months you can predict with the scary expect question What is it that the all the other malls are rearrangements of particle positions in English alone let's see how we describe the all the he didn't scare .period was proximity the mullet that scares we want Oracle's offer on this point imagine we decompose the internal and in into stations Fourier now they're also loads that would save Polish some stuff here and some of the sums of here last year and that was scary life too but I don't get nearly as much of that Palmer doesn't do nearly as much that and describing the polymer moments at a fairly low level of approximation is half of what is a fairly low level approximation well the good news is that Turkey's ties with his victory in the direction of the workers Reisman said here this year a whole has a center of mass location in the center of we can describe it since it has amassed distribution on moments of inertia actually and although oil dollars the motor involved these gasses component which is worth orthogonal everything else which can be described as rotation around an axis doesn't have to be a symmetry axis there after all we're talking about effusive emotional also this is flexible and then we have worked home translation there was rotational and then we have workers who would described this fluctuations language he learned things that small where Paul was actually changing shape and that there an are
arbitrary number of different ways of describing shape I had choosing modes as long as you have just completed complete orthogonal warrants work but the net result is what do the calculation there only 2 loans with interest to see that he thought about it that way .period ourselves this namely the behavior of how this 1 all over town it's a practice some of diffusion coefficient times Hughes where there is the faster the diffusion is slower the most of the shortest-lived motives yes well he describes the motion home the approximate size 1 yes however you want from diffusion coefficient using words size of using object this is a very approximate ended meaning is proportional here and therefore this 1 house for the use where it's actually portion you here now to step up your analysis of what most of what you can do better than the heavyweights on the blackboard menu that gets a few the piano for some power which is sort of like tree what is the implicate always also always calculate intensity remember I said the other intensities were small Bloomberg also struggled with a small translation is known as margin humor this small becomes dominant in between there's average in which both terms Contreras if we step forward not to force President personalized time Ellis and what Dallas did was to use modern digital correlated technology measure all spectrum very high signal-to-noise over a very large range and Ellis was actually able to measure the two-mode relaxation rates separate break spectrum molds and all and look at the Q and the various dependency and Alice was actually the CIA's director however if we go back to 1970 this was not an option with 1970 technology you couldn't do the Moby decomposition with measurements accurately enough you didn't see the relay the offer relaxation spectrum over a wide enough range all of times and therefore what you actually measure was where station 1st funeral and Gamal 1 and gamble 1 was the light scattering intensity Havret weighted average these 2 relaxation so small right mode and relaxation rate was like this on March 2 this was the bright mode and because of this term this that this was the dominant German relaxation rate went like that so I used to say I suppose I take the 1st few minutes which was then sometimes called him on his some favors political capital Omega which it's not a frequency and so security you're seeing there but that was the symbol was used in part to virtually all over the world over the things you what happened as you well assault Caroline is proportional to to use where and divided by you you you get what you large these levels out goes to 0 now you could just as well as table 1 itself and came on itself gone is used wear a small here and then rolled over to you March Q they're just different ways to plot next to their bodies of people who moved on these experiments and the generals sort of consistent outcome is yes you do measurements and things work will come to some issues with things work on time there is a historical issues because you can make predictions as to what is high the new theoretical calculations predict and the theoretical models of the climate to restorative very recess getting high well having set the within 30 % getting high right you can either say my cup is half full or my covers half empty and if you read the papers at the time there was a certain emphasis on half-empty was being negative about the quality of the experiments were very is just there was an emphasis that to theory and experiment for you should look at the number of approximation is you have to making the theory to get this all sort of like the tap dancing look at the issue was not as an elephant tap well the issue is can the elephants have working prefer go out on roller skates and all I have actually seen roller skating elephant the answers they can't of course if you look currently
realize health service the weight on the roller skaters such bearings completely seize up and the elephant was simply walking in verifying shaped shoes but was the I saw myself on any of them having said all you see this you can do those experiments and you sort of get the the right result and that she would probably improve that's both by doing much better experiments and by doing more theory but that's not what's happened yet OK let's from Mexico a lot of this is 1 thing to say we will look at Mumbai's experiments he says we will actually take the concentration of power and there's some question there were several people who worked on this at the the same time it appears to me having looked at the literature and yes papers out of Ph.D. theses show up Showalter late in the period between the credit for the 1st reactor experiment for those who felt Switzerland by the name pulled although there were also a number of French troops that on the work of art certainly about the same time on but he tends to be forgotten so I'm bringing them up so he gets the credit he deserves and what was too don't to Doncaster measure for example the 1st human as a function of concentration on you warrants if you were small due to fewer working at small here all the question of information alone those not a lot of very much because they become invisible you actually have to work piece of leverage to see the of the people on and what both demonstrators was measured with diffusion coefficient of the function of concentration there is international slow along with which he discusses a small this fall is about the same in different weight vectors there was a period early on when there was a feeling that if you measured diffusion coefficient should depend on vectors and he people will try that is distance over which particles small but there was a very long time in which people try experiments and they would measure to dependence that they see absolutely nothing of any interest but in order to see the pall over internal loads need quite very high molecular weight for words and you on the condition of very high molecular weight polymers in the office ability go quite low scattering reliance this all trace all right so here but also other people saw the same thing I will mention very briefly some experiments by the state of the race to the end of the world by hot the measure the future home efficient bruises concentrations but they also did stack might skin that is they measure the intensity scale like otherwise interesting measures the intensity of the scare about well the really interesting measured intensities there was this is the intensity of the scattered light is determined by by how easy it is to create density fluctuations if it's easy to produce density concentration fluctuations they're beginning to get a lot .period scan this what needs to be critical unless it is harder to produce concentration fluctuations of particles push away from each other and can cram galleries of scattered bodies the intensity is determined by the US model compressibility hires the onslaught of procedures used of to Universal the morally Oslo pressure changes to change concentration of harder the particles of telling each other and then universe will weaken the light what time it was stated that if you'd told me it was the worst of the fight at the University of course have all this is the general you don't Weinsteins the way that correct correctly described in light scattering them across the city the actual practices my were designed he said to for until I published the rollout of people involved in the particle interactions did not change the at all because the right the various components hear a reference to people corrections all other also some Johnson's be putting because there's something interesting in this issues
is over the net result of is that Koch all measures this the elevation correctly they the scattering intensities losses firms and derivative directly and therefore they they were able to isolate them and if you go into the book if you go into the year also admitted to place on you can find lots to this day hello television after 4 hours of concentration and you can find where I figure with a slightly specialized function and if it's a white on another however start with that wonderful another then that this is work by carousel the now you have to cooperate with Pittsburgh and what Harrison did was to many the usual coefficient of concentrate what he found was the reason nation region which is quite linear and there is clearly a roll over in the goes down again 5 years there is no need to see the expert sensitive I tell you I have power lost slow someplace in here I even fit is hand Patterson demonstrates an extremely important data quality issues where you would see get artifacts the artifact is fault suppose instead thought and diffusion coefficient the against the lifelong along the against losses the logarithm function has an interesting feature takes this area very close to 0 concentration and stretches it out to minus infinity and so on if you have a lot of some it this I singer function log see little here from stretched out indefinitely flat and if I don't look too far off and concentration I can get something that looks on what what plot like a straight line what Harrison emphasizes is that this original sees this 0 region even phrase used purely an artifact of how you chose plot data there is no season 0 region unless they immediately to clear until the simple linear plot reasons for Patterson's however ,comma support reason this society there's just year there's no season 0 really is that there were several competing theories polymer dynamics to which claims that policies were not interacting therefore fusion coefficient is constant and they were not interacting with concentration was high enough of all boils
over became so this was an actual theoretical predictions in which sort of fine people claiming they see the effect they also lost like this what Patterson show was that there is no season 0 regions and the reason is now seeking 0 region is Paul Missouri interacting even at very low concentration OK and can actually see this year face here at the level of course to be 2 so there is a very the results of hackers OK yeah I think we're now going to come up with she what which we are now going to go along with well let's see OK so they can hold their region do anything with it and the answer is yes noted for example that the response papers like and mentioned 1 paper plant costs themselves they measure the members the various international slow pace of the state of nature and I did this for several different polymers they did this for different under different figures on different ways so that multiple measurements but Haiti is something you can hold your life and therefore you can do comparisons of the calculations with the theory and the theories are extensions of current replacement series of shareholders to buy him out of the extension of the current will rise and here's her rise from there's 1 ball and chain In translates it has some answers about which rotates the example ,comma Shane after dynamic interactions as 1 of these moving the float around it must also be moving and the fluid flow has a happened in which Drexel the beads that's hydrodynamic react with young :colon was saying how he have all oils and of the government's fault also and create a hydrodynamic along the way that acts on the use of the other boys and therefore from this calculation you can
calculate what case of the sea and you get a reasonable rate let us now praise Alaska In previous threats to get the ones that we talked about the versus human dilute solution and that was theory by on or originally their experiments by Dr. jostling upon a series of other people not which seems to work and that was diluted solution be pretty darn solution divas here trim there were also people small here at the wave connected exceeding its fall you at the concentration and share have 2 different on sorts of things you can look at 1 the world the same time you could do all measurement the whole picture when you do this for a series of concentration and what people chose plot was the relaxation the a 1st thing to do what do you expect to see and fact is what you see is that you are achieved last longer than followers of often watch out here someplace you get into the region where the kernel is dominated by a huge Q it therefore camera worse use where you you I told you so where the proportion human 1st and I want you to be the 1st Street now this experiment has competed for a series samples of different parts of race and when you 1st find the reference book yeah comes along with the higher concentration and against closely here while and sort of way and then you get up to quite high concentration and at quite high concentration it show on the ruling of postal going to see what would happen a however the diffusion is faster and the the Asian region where Gallup Poll use like you 0 initially increased concentration reaching its brought there's no reason to suppose that this so I had to go various experiments and that is about the best OK the measure all the interesting experiments except the 1 experiment I don't think there is something the stuff we talked about Spanish .period when we discussed status light-scattering along we know that if you experiment with Chief of this fall and put it in the date that forms on the little things wired the invisible the index of refraction match solve so concentration fluctuations not changed index fracture solution and is at the Annex refraction doesn't change at the end of the season of the matrix on year-round because no one's here and now we can look at the diffusion of a matrix chain and March humor through polymeric back unfortunately this experiment has not been done by very many people and is not going race the health of persons in March and Martinsville of problem solving I was quietly amended that I synthetic organic chemists trade standards moreover polystyrene plus 110 kill adult polyvinyl level and he was on the set comment on 40 per cent by weight if you have a there are lots of them because 40 per cent of all is pretty much these polymers and the space
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Herbst
Molekülmasse
Gasdichte
Mark <Maßeinheit>
Wickeln
Weltall
Druckkraft
Sturmgewehr
Modellbauer
Ersatzteil
Papier
Messung
Drehen
Parallelschaltung
Kugelblitz
Linearmotor
Ladungstransport
Energieniveau
Leistungssteuerung
Förderleistung
Regelstrecke
Kette <Zugmittel>
Fernsehempfänger
Multiplizität
Klangeffekt
Tagesanbruch
Stunde
Kaltumformen
Konzentrator <Nachrichtentechnik>
Hydraulischer Aufzug
Tag
Rauschzahl
Patterson-Analyse
Angeregtes Atom
Kernfusion
Satzspiegel
Jahreszeit
Jahr
Intensitätsverteilung
Luftstrom
Papier
Gewicht
Messung
Parallelschaltung
Relaxationszeit
Energieniveau
Kompendium <Photographie>
Satz <Drucktechnik>
Lichtbrechung
Umlaufzeit
Kette <Zugmittel>
Brechzahl
Passat
Herbst
Konzentrator <Nachrichtentechnik>
Mark <Maßeinheit>
Standardzelle
Diffusion
Übungsmunition
Jahreszeit
Mikrowelle
Matrize <Drucktechnik>
Ersatzteil
Absolute Datierung
Quantenfluktuation
Leistungsanpassung

Metadaten

Formale Metadaten

Titel 20 Light Scattering Spectroscopy Slow Mode
Serientitel Classes in Polymer Dynamics
Teil 20
Anzahl der Teile 30 (Lecture 2 missing)
Autor Phillies, George
Lizenz CC-Namensnennung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/17110
Herausgeber Phillies, George
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik

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