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Lecture 14. Atomic Spectroscopy

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Lecture 14. Atomic Spectroscopy
Selection Rules, Coupling, and Terms
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Lecture 14. Quantum Principles: Atomic Spectroscopy
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UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:00:41 Atomic Spectroscopy 0:09:08 Emission Spectra 0:21:41 Coupling Schemes 0:26:17 LS Coupling 0:29:42 Adding Angular Momenta 0:31:47 Vector Model for Coupling 0:38:40 Symmetry Considerations 0:41:48 LS Coupling 0:44:28 Term Symbols 0:50:32 Selection Rules
Potenz <Homöopathie> Übergangszustand Insulin shock therapy Chemical element Complication (medicine) Set (abstract data type) Wursthülle Atomic number Chemistry Man page Deterrence (legal) Clenbuterol Mercury (element) Electron Tool steel Sea level Lithium Dipol <1,3-> Process (computing) Transition metal Separation process Chemical element Area Storm surge Whitewater Action potential Adamantane Protonation Electronegativity Tanning Electronic cigarette Systemic therapy Emission spectrum Spectroscopy Optische Analyse Azo coupling Valence (chemistry) Proton-pump inhibitor Dipol <1,3-> Separator (milk)
Cheminformatics Potenz <Homöopathie> Übergangszustand Chemical element Lightning Wursthülle Chemistry Clenbuterol Electron Ionisationsenergie Emission spectrum Common land Gemstone Sea level Chemical formula Transition metal Chemical element Nitrogen fixation Action potential Gold Tanning Knot Wine tasting descriptors Emission spectrum Sea level Metabolic pathway Optische Analyse Valence (chemistry) Azo coupling Graphiteinlagerungsverbindungen Common land Pharmacy Calculus (medicine)
Sense District Übergangszustand Separation process CHARGE syndrome Wursthülle Clenbuterol Wine tasting descriptors Tiermodell Electron Ionisationsenergie Kernproteine Sea level Atom Chemical formula Transition metal Inositol Elektrostatische Wechselwirkung Protonation Gold Erdrutsch CHARGE syndrome Tanning Wine tasting descriptors Systemic therapy Sodium Sea level Tiermodell Magnetism Blue cheese Optische Analyse Atomic orbital Azo coupling Kernproteine Regression <Geomorphologie> Graphiteinlagerungsverbindungen Common land Calculus (medicine)
Übergangszustand Sense District Bohrium Atomic orbital Wursthülle Materials International Space Station Experiment Deterrence (legal) Tiermodell Electron Sequence alignment Meeting/Interview Shuttle-Vektor Exciter (effect) Kernproteine Process (computing) Pitting corrosion Chemical property Starvation response Walking Adamantane Katalase Protonation Erdrutsch Tanning Wine tasting descriptors Magnetism Blue cheese Optische Analyse Atomic orbital Azo coupling Kernproteine Caffeine Atom
Claus process Man page Clenbuterol Tiermodell Electron Match Colourant Nanoparticle Mineral Process (computing) Beer Set (abstract data type) Chemical property Adamantane Solution Electronegativity Stockfish Erdrutsch Knot Wine tasting descriptors Systemic therapy Stem cell Combine harvester Anaerobic digestion Blue cheese Optische Analyse Cardiac arrest Mixing (process engineering)
Semiotics Übergangszustand Lebensmittelfarbstoff Atomic orbital Wursthülle Sodium Atomic number Reflexionsspektrum Sample (material) Clenbuterol Electron Emission spectrum Exciter (effect) Explosion Nanoparticle Dipol <1,3-> Waterfall Transition metal Flame Chemical element Semiotics Singulettzustand Adamantane Protonation Gold Knot Tanning Emission spectrum Heck-Reaktion Sodium Drop (liquid) Sea level Optische Analyse Dipol <1,3-> Chemical structure Wasserwelle <Haarbehandlung> Atom Stereoselectivity
welcome back to chemistry 131 today we're going to continue our exposition of atomic spectroscopy focusing on selection rules coupling and terms Of course atomic spectroscopy is the basic tool by which everybody learned that the Adam was quantized and so is of a very central importance in in this hole area of endeavor of quantum mechanics applied to small things various sets of transitions are observed here is energy level diagram for the hydrogen atom not to scale because if you do it to scale the levels crowd together too much at the top and it's very hard to see what's going on there is a series called the Lyman series in which an electron jumps from the angle student equals 1 or 3 2 141 and so forth and then there's another series Obama series that's in the visible range in which the electron jumps to the end equals to state and then there are the series and they have names according to who worked on the main thing To keep in mind here is that any change in the quantum number and in the energy is allowed in principle but the only thing the only thing that has to be satisfied is that there has to be conservation of energy and that's taken care of automatically because the photon is emitted will require major new which is exactly equal to the energy losses the so energy is concerned In this diagram the 0 0 reference energy is at the top and all the energies quoted there are negatives being more stable than a proton and electron at rest and infinite separation Our the day of the selection rules and are a little bit more complicated however and the 1st thing is that of photon has 1 unit of angular momentum h bar angular momentum is conserved and so when the electron makes a transition there has to be a change by 1 unit of angular momentum so that means that although we we draw these things coming down to the 1 in fact they're all coming from specific values of Bell it's to Peter 1 as 3 Peter 1 as for Peter 1 s and so forth and always Petey we can have 2 S 2 1 s because then dealt L is 0 and that's not allowed that doesn't conserving the momentum and likewise we can't have 3 D 2 1 st there because then dealt L is equal to 2 and again that doesn't conserve angular momentum now if we have a different kind of transition worth more than 1 photon is emitted that's different because then of course that could be allowed because angular momentum could be conserved in a more complicated kind of process those kind of processes do happen all the time they're usually just much slower than the single photon electric dipole allowed transition and if we have the transitions to to pee in the bombers a series for example they could be from 3 s or 3 D if we have transitions to to the 2 two-hour state and they have to be again from 3 P 4 p and so on and in general Delta elders plus or minus 1 for elected electric dipole allowed transitions and the surge are generally the very strongest transitions that you see in the spectrum not always however if if the item very heavy like mercury as will see In the next lecture and we could have a strong transition that doesn't seem to follow this rule the weaker transitions magnetic dipole transitions and electric quadruple transitions we won't touch on those much but you should keep in mind that those can and do occur and therefore you mustn't just assume that this is an ironclad rule when we use the terms allowed and forbidden inspect roster be what we're talking about really is strong and weak allowed is the start of the light green forbidden it's red but some people run red lights here is a compendium from the nest database just to show you how accurate and fastidious the spectroscopy lists have been in cataloging these energy levels this 1 has a different energy 0 in this case energies are quoted from the 1 s level of the hydrogen atom being 0 and then up from there but the amazing thing about these numbers is that they have 6 7 8 9 digits of accuracy so you can see that they had to take a lot of precautions so that when they were doing me and if you look closely there there multiple entries there's to peak and then there's 2 Western there's just too then there's to toupee again and so forth and we will see in a 2nd what these things mean but you have to keep track of individual sub states in the systems and oftentimes you put on a magnetic field to be able to tell which status which and how they move around to keep track of them and they go all the way up to 6 h so in the periodic table we don't go beyond ass but if we go too excited states of animals Adams it's easy to go very high and very high values of the angular momentum if you look closely you find out that the 1st 2 status quo that is ever so slightly lower in energy than to us which is quite a surprise because in other elements of course to U.S. is lower than to be that's why lithium is to West 1 and not to P 1 for example and we don't fill up the Tupi until we get over the border on but in hydrogen is kind of an anomaly 1 of those 2 states is lower than the 2 s and that was extremely interesting to the irritations how that could possibly be and in studying them in great detail they worked out important clues about actually the
residual electromagnetic field that permeates our entire universe like the 0 . energy of a harmonic oscillator you can think of the electromagnetic field some kind of electromagnetic oscillations and it turns out that even in a vacuum at absolute 0 in this universe apparently we can't have nothing there's still some 0 . energy left over and that can bounce the electron around and then the question is if it bounces at 2 as electron around bounces a toupee electron around what's the difference energy actually follow through the calculation it's it's actually very very beautiful work to follow and you can actually derived that the Tupi should be lower and we can use the measured energy once we have a series of the states we can use the measured energy if we're careful to estimate the ionization potential also called the ionization energy and that's just an extrapolation to what the energy would be at any equals infinity and that's very important to know what that number is that's 1 of the main things that little that's listed and freshman chemistry books in the periodic table to try to account for various trends if an electron is easy to ionized and that means the bad elements much more likely to give an electron up to another Adam then and an element that has an electron this extremely hard to ionize for example let's try practice problems then and have a look at how we might tackle estimating the ionization practice problem 18 consider the following unassigned means we don't know Anna L. emission lines from a hydrogen are all of which terminated in a common level assignment an estimated the ionization potential or the ionization energy and think of hydrogen while the hydrogen artist is we put a big lightning strike a voltage through and we need excite the Adams and then they emit light and if they terminate in a common level that means they're like the other diagram I drew we have arrows coming down to this to this to this and so forth always to the same level and that means the change in energy has to do with the upper level but not the lower level Of course in actual fact he just get all kinds lines and you have to be very clever to figure out which ones go with which in which which go together the various tricks that were worked out to do that so here are the numbers but they all have 4 5 digits in them on 1 . 8 8 8 7 and so forth and let's just take these for numbers quoted in and let's figure out what the ionization energy should be the 1st thing is we have to figure out what common level they terminated and if we don't do that then we are going to be able to figure out anything and so the way we're going to do that is we 1st right this formula the Delta III is equal to the Redbird constant times 1 over and squared common level minus 1 over and and case square where and carries some higher number up dropping down we don't know what the common level is however and we don't know what the value of viruses because it wasn't given and on an exam if you are given the value of on a problem like this that means you're supposed to figure it out without that so let's try to do that we don't know and ,comma we don't know are we have some values of Delta III and we have to try to figure out if they fit this formula a very bad way is to just start plugging things on and seeing what happens a very good ways to proceed systematically and figure out if we can get a straight line plot chemists love straight lines before computers straight lines were absolutely essential and the reason why is that the human eye can distinguish a straight line from any curves but the human eye cannot distinguish curves that slightly different even if their statistically significantly different it's very hard to tell I'm just by looking and before computers people had a plot things on paper often and look at them in order to tell what's what if you could organize your equation so that it came into a straight line plot then you could tell you could tell us there was a systematic deviations sometimes slight curvature one-way where the other way you could tell us there was scatter noise and so forth and all those are very important to quantify when you tried estimates we can cast our our own the equation into a straight line formed by letting Y equals to Delta III but X equal 1 over and squared and then letting the intercept B equals 1 over but rather are over and ,comma and we can guess values of NK and see which look good so we have to assume a value than ,comma now we guess values of NK all greater than that and we see "quotation mark looks good once the line is straight then we can extract the slope why equals B plus and Max and in this case should be minus are the river constant once the line straight that we can get are now the lowest common level could be an equals 1 so in the absence of any other information we should assume that 1st and then work our way out if the common level is very high really really hot on this would be a disaster because we would take forever but we know that it can't be that high because we see a couple of electron volts and so we know it has to be 1 of those levels further down this it's very high and then all the transitions would be very very small numbers because we'd already be most of the way up to the ionization energy so let's try and common equals 1 and in that case we could have MKT the level B 2 3 4 and so forth if some transitions are missing for some reason we just didn't see that they were
faint we made an error or some other reason then this kind of exercise can get very frustrating because the points Stone appear to fit and if you have to start deleting values then it's very tedious but in any kind of exam problem will never have something like that but in the lab for various reasons it can be that some transitions are dark something happens there's another pathway where were
the electron can go and it's very hard to to see that and is sometimes it takes a lot of patience to be able to work out what's going on so let's plot these I drew up a table I have the Delta E values for wine and then I have the 1 over and take square Stanford to its quarter for 3 years and 9 and so forth as as we go up and if we plot those which I've done here well 1st of all we noticed that the slope is negative Excuse me and that's of course exactly what we expect because the slope should be negative on hours of positive number so that parts OK less plot them now and here I drawn plot now therefore black dots on this plot on slide 370 and if you just draw a line through the 4 black dots by linear regression you will get a pretty good said In other words there is a line that misses them all but is pretty good and so you have to decide if if back ,comma line is going to be good enough for what you're doing In this case it's not nearly good enough and what I've tried to do is emphasized that by 1st drawing a line through the 1st 2 points on the left the red line that has a certain slope and keep a mind that these points have very very very tiny errors although I put these black dots so that you can see them on the slide the points themselves are very very accurately determine we draw that red line and then we compare that with the slope from the last 2 points the blue line we only have 4 points so we have to become a careful and we plot that and when you do it that way what you see is that the blue line has a much shallower negative slope than the red line but it is quite a bit different and so there and if you took the center 2 points it would have a slope in between and what that means is that this is curve it's not an equals and common equals 1 and this is not good enough but you could easily assume at work good enough if you're used to other kinds of data that isn't isn't this well determine In any case were going to discard that and say Well and ,comma Nichols 1 doesn't seem to work let's go up to the next level so this is not close enough if we assume the next level up any equals to for the common level and then we have 3 4 5 and 6 4 4 In cases where there we get 1 9th 116 20 5th 136 we put those values and we get this table that I have shown on slide 371 and using this data we get the plot on the next slide and now the differences it's dead on so I put 1 blue line and boy it goes right through all those points perfectly through them nice and straight so there's quite a bit of difference between the 2 when you view it this way and if you pick I'll let you try it on your own if you pick the common level equals 3 as far as the terminating level and do the same exercise and plotted you will see that it is significantly curved again but not by much it's curve but you have to have a critical eye in order to see it and you have to understand how accurately and this plot should come out if you do a linear regression on this and you take the slope down the slope and is equal to minus 13 . 5 9 8 4 electron volts that's are and that's the ionization energy because if we put any equals 1 and the Formula One and equal sincerity the ionization energy is just ah and they're there for the ionization energy for hydrogen according to this analysis with those 4 points should be 13 . 5 9 8 4 electron volts if you go back to the nest but compendium of data what you'll find out is that we did pretty well that comes out very close to the exact value of the ionization energy for hydrogen and doing that with just 4 points as it's pretty is pretty good because we didn't sign during many of the transition it's especially important to know that it's not enough to fit the data if you don't do a statistical analysis of the 5th if you just get a set it can look great and it can be completely wrong and you have to do a statistical analysis based on what you think the errors in the data points might be and how well it fits if you don't bother to do that and most of the time people don't they just do something that looks good they assume it's right and they "quotation mark you can get really crossed out if you have the systems that are in fact very subtle the curvature subtle and the data is very very highly precise so in these cases if it's not right on then your model is incorrect keep that in mind when you're doing these kinds of problems OK now I want to talk about the coupling schemes and this has to do with why were 2 values of 2 in the mist databases 1 1 and then another 1 the music while on electron can be and to pay it can be into what's the difference is at 2 PM to no it's not that's not the difference the difference is that the electron itself as a magnetic moment and we have to take that into account when we have a non-zero value of their but will kind of in order to talk this through it will have to assume a kind of a classical model of the atom but this is just a year Ristic a way to kind of look at what's going on we don't actually believe that something's orbiting around but anyway fell is not equal to 0 we can think that something is orbiting around in some sense if I'm an electron and I'm orbiting around the Proton or the nucleus of it's a bigger Adam and I go on the electron Simon electron boys at a wild ride but I see
I see the proton going around me because I'm the fixed Diana protons whizzing around me like that the other direction but the Proton has a charge and so what I see is I see a charge whizzing around me In some kind of a pattern like a circle or something and that to to me as the electron looks like a current loop it looks like electromagnet but as electron I have an intrinsic bar magnet the spin that we discover From the start under lock experiment and therefore my own bar magnet can either be aligned with this magnetic field that I appear to seal or it can be the other way and those 2 could have slightly different energy because there's some magnetic energy we didn't take that into account the Hambletonian week we when we wrote it down but in fact we know that there is and if we were more sophisticated we could take it into account and therefore those 2 orientations there but pretty much as is the resolution of why there's too 2 P E levels they're very close because this magnetic orientation of the electron either being aligned or against this field that the that the proton appears to induce is very tiny compared to the electrostatic interactions with the potential and so it's it's out in several digits out but as will see the world would do a calculation probably with sodium I think this magnetic field but there's nucleus appears to generate at the electron as measured by the difference in the energy it is huge and it's bigger than almost any amendment we can make a Milan so it has a very big effect because things are moving so fast and it's close by so it has a very large effect on the electron compared to the kinds of magnetic fields that we can generate a Milan so quite a bit bigger than 1 Tesla 1 Tesla would be the kind of magnetic field that if went in for an MRI in a whole-body scanner and you have a state-of-the art annular magnet superconducting magnets you might have 1 Tesla In there and it's very much bigger of course than any kind of normal magnet that you would ever find so this kind of interaction between the spin magnetic moment and the orbital motion is called spin or better coupling because it's a magnetic interaction between the apparent magnetic field of the nucleus and intrinsic
magnetic field of despair whenever there is more than 1 electron the electrons themselves can interact magnetically they're little bar magnets they can aligned with each other or they can not a lot if it turns out will see now when we get to it that if they're in the same or but all they have to be but if they aren't the same orbital as there is an excited electron there's another 1 here then they can have any kind of orientation needed for against and if we have an open shall Adam with more than 1 electron then the spends can add up and we 1st add up all the intrinsic bar magnets to give what's called the total spin angular momentum and if there is electrons with non-zero battle and there's orbital angular momentum if we have more than 1 of those we had all those up and I'll explain how in a 2nd together the total orbital angular momentum did and then L and again magnetically interact together the total electronic angular momentum for the Adam which is given the assembled J and S added together if you take at 1 stage further sometimes the nucleus for example protons has a magnetic moment too and it's been one-half like the electron and in that case it can either be aligned or non-aligned and we have to add its angular momentum its magnetic interaction with Jay In that case we have to invent a new letter and that new letters steps we usually put J. but keep in mind that nuclear I do have magnetic properties as well as their spin is not always one-half like the electrons Deuteronomy spend 1 chlorine and spend 3 halves and if you look very closely at electronic transitions you actually have to be very careful to take into account the properties of the nucleus to explain what you see if you try to leave that out than they're missing pieces of the puzzle and you don't appear to get the right answer here on slide 376 is an image adapted from Wikipedia which shows how you can take these 2 columns a big cone for L Fellowes pointing somewhere in another column for S and you can combine them to get a specific value of J and then there are 2 plus 1 sub states magnetic some states of J which are their projections onto this the axis of the total angular momentum How do we had these things up to get bigger well I think there is you you have In order to really do this correctly you have to do a little bit of work with the theory of angular momentum but as I will explain as we go along it makes sense that if I've got 2 electrons let's say and they have orbital angular momentum L 1 and L 2 that I could add them up they could be parallel and that would be 0 1 plus L 2 that would be the maximum as if everything were call linear everything worked the wind blowing from behind and then they could be out of alignment and because things quantized then the next values 1 plus to minus 1 and the worst cases when they're completely miss missile line but the total angular momentum L has to be a positive number because it refers to L. square and so we terminate the series at the absolute value of L 1 minor sell to we say absolute value because we don't know whether I was bigger than L 2 or less than L 2 and so we just put the absolute value there to terminate the series it terminates at a positive number which could be 0 because no 1 and all to could both be be 1 for example this series is called the clutch Gordon series and the same algebra exactly applies to the total spin angular momentum the yes the gas can be S 1 process to or as 1 closest to minus 1 blow blow blow down to the absolute value of S 1 minus as and we can see this in vector picture keeping in mind that the interaction between these guys is magnetic in nature and so it's speed magnetic little bar magnets that are doing the talking here and then there are adding up in certain ways to give these overall angular momentum that we observed a let's for an example let's just take 2 electrons and let's couple the to spend 1 have to get the 2 bar magnets here let's couple them together they're interact and how could they be
well therefore possibilities they could be like that like this like that your like that because each electron can just be upper down so I "quotation mark that in a notation the which you may see in a more advanced courses and these are called cats but firm for our purposes we don't need to know what they're called they just have the arrows in them there you 2 up but down down up or down down those of the 4 possibilities and we have to see how these can give an overall spend their guests that's not going to be here 1 half like that either suspended the electrons because the clutch Gordon 6 series as well as as big as could be 1 process to that 1 or it could be it's 1 of us to minus 1 that's 0 and that's the absolute value of S 1 minus says to sensible spin 1 half so we could only have big sp 1 or 0 1 we couple dispense together so let's look at these them From a more pictorial view in a vector 10 diagram the here's a very nice picture on slide 379 1 side In Blue is called the single you see why in a 2nd the other side in red is called the trip the same has won state the trip has 3 states that part of that makes perfect sense and we can see I've written these cats next to each orientation for the trip where they could be
both parallel also they could have 1 up 1 down but pointing the same direction like that OK so this this would be a S equals 1 s equals 0 and then they could both be pointing like this that's as equals minus 1 that M sub s Excuse me but those are the 3 values of the M quantum number that would go with an S equals 1 state and stem-cell best equals plus 1 0 or minus 1 and then the single has just and Sebastopol 0 Kazakhstan 0 so that's can be the only value becomes a and for that 1 we dropped slightly differently we drive instead of drawing of like theirs we dropped like that so we've got 1 out in the other pointing the other way they add a 0 but there come like the 2 guys in the stock trek Time Tunnel just stuck battling each other in this state absolutely antisymmetric instead of calling 1 of them up down and the other 1 down up you may notice that I have this route to over 2 down up plus up down for the triple and route to over 2 I'm top-down minus stand up for the for the single why is that that we have to have them well yeah we know of course then since the electron can go through both slits and it's perfectly feasible for quantum systems to decide that they want to exist In a superposition of what we might think of as the most concrete things namely well it's either down operates up down while not on a quantum system it could be 50 per cent of that and 50 per cent of the other 1 together that
could be the solution we see them we've got 4 states and they break apart into 3 and once and that's a common pattern because the odd numbers add up to a perfect square and so if we have a certain angular momentum choice three-year and 3 here we get 9 we can that right that as a sum of 1 3 and 5 and that's basically what the club scored a series says it turns out that neither of the state's up down or down up will do and the reasons why is because these guys either in the trip or the singly or another systems they have to have the same stripes they have to have the same characteristics with respect to swapping the particles the particles are identical so which 1 we call 1 and 2 but is unimportant but if we have 2 2 of them and we swap of we get the same thing if we have to have them down it we swap we get the same thing but if we then say Well I think the middle 1 then for the trip would is up down and we swap we get a new guy we get the other 1 which wasn't part of that part of the mix and we can't have that we have to when we spot but we have to get the same thing because it's part of the same series it has to have the same cemetery as the other ones otherwise it doesn't it doesn't work and therefore we have to take a combination between up down and up down up such that when we swapped over we get the same thing as the M equals 0 stated the triplets and its unless you've seen how to do this is not not quite so easy to figure out how you might do it but let's explore this cemetery by artificially coloring the arrows red and blue that way will keep track of them well we swap we will see what we get in terms of whether or not we get the same then not so at the top here on slide 381 I have shown a picture With the coloring we start with which we know we can have 1 of them because it switches to the other so we know we've got both of them and we know we've got to have 50 per cent of each West forget about the route to over 2 that's the normalization constant just to keep the probability of being in some states equal to 1 analyst just take the combination so that in the let's take up In red down blue clause down in Red Hot and Blue mn let's walk the positions Of the 2 arrows will we get them in the 1st 1 were up and ran down blue if we swap will we get down and blew up in red now in actual fact the red and blue don't matter but lets us keep track of it so we get down up the other 1 which was down red up and blue we get up and blew down right and then if we just change the border we find we get up in blew down and Red Plus down and blew up in red that's what we started with except the colors the swap but the colors don't matter because the electrons are identical particles were just collaring them so that we can keep track of what we're doing otherwise we can't tell what we swap and the key here that we get plus 1 we get the same thing the Eigen value for this swapping His plus 1 that's the kind of a parity value plus 1 for swapping this way plus 1 for this 1 plus 1 for swapping that way they all go together all 3 of them are birds of a feather if 1 wants then the other 1 to be orthogonal can be plus as well and it therefore it has to be minus and if we take that combination which as shown on the 2nd line here and we take up down miners down up and we swap of we get down miners up down and that's equal to minus 1 times what we had before because of the fact that the opposite and what that means then is that the same good has different cemetery then the triplets and so although they both have them equal 0 states states are different because 1 is symmetric under exchange and the others antisymmetric under exchange a more complex systems you have to look at the cemetery of each of the states and decide if it fits in a not and there well documented procedures for doing exactly that if you after and so whether you're symmetric or antisymmetric under exchanges of very important property is not just a trivia OK suppose then we have some these values we've seen we can take S to us 1 have been one-half particles we can get as equals 1 as equals 0 we do the same thing with we could follow that through the same way now got bigger on bigger How do we know what values of Jr well result the Galla something could be too the gas something could be 1 of the 3 halves then particles we have and we want to figure out what values the total angular momentum of the Adams Jr conveyed while the answer is is pretty much the same thing again because any momentum always ads in exactly the same way it doesn't really matter what it is both should they opt Allen as happened to align that's the maximum value of J hello plus and then if they don't then we Decker meant by 1 Kwan and we have l process minus 1 and so forth and we go down to again the absolute value of Bell mind-sets because we don't know whether L arrests is bigger but we know that Jay referring to the square of angular momentum over the square root of the square the angular momentum as like has to be a positive thing so we can have anything go negative and therefore we get another clutch Gordon series helpless S as minus 1 and so forth and therefore what this means is that if you have a certain configuration electronic configuration of an Adam that's open shell it has several electron depending on the details of how these magnetic moments are interacting and since these energies the small then if I have something like a blast like an and electric aka something like that they're all going to be there there is it's not like 1 of one's way higher compared to the others if there's enough energy to populate the excited states there's
probably enough energy to populate all the possibilities randomly statistically just depending on how many possibilities there are for each state and therefore we need some way to keep track of Of these things in order to figure out what kinds of emission lines were to In the spectrum and assignments and the way we do that is with something called the term symbol the term symbol summarizes not only what state the atomization but tells you what Cal yes and J. R and the term symbol has the duty structure 2 West plus 1 on the left big L which is like a letter like P and so forth just like for the electron itself little less little This is bigger the total angular momentum orbital angular momentum and then J. as a right subscript to tell you which which way S & L a adding up To give but the result for the total angular momentum Of the Adam in order to figure out what kinds of transitions you're going to see you have to know what the term symbol that's by far the best and most concise way to look at things and therefore you have to know what these are 2 plus 1 is called the multiplicity and if passes 0 it's called singlet there's 1 line 1 state there like we saw with the S equals 0 for the 2 electrons as equals 1 is called Triple S equals 1 half if we have that that's called Dublin and so on we can have all all the way up and let's try an example with this them and see how we might use this simple to figure out so so let's take here's practice problem 19 let's look at the sodium emission spectrum the of the ground status sodium is gonna 3 s electron outside a closed shop the 1st excited status three-peat because in the sodium electron 3 PM 3 s are quite a bit too far apart and so that's turns out to be a transition that has visible light it's not a very tiny transition energy like radio waves so we can see that easily In the sodium spectrum the question is if we had the electron in the three-piece state outside a closed shell what terms can erupt In other words 1 of the possibilities for these terms samples which way can L S & J well for the P state little Ellis 1 there is 1 electron and therefore Big Ellis 1 because the clutch Gordon series terminates the big Ellis little well in this case because there's just 1 electron why don't we include all the other electrons and sodium the inside a closed shell and inside close shall the angular orbital angular momentum is 0 and the spin angular momentum is 0 and so inside a closed shop just forget about it and throw it away you don't include all those years spend the rest of your life of playing around and there is well what could heck excuse me 1 electron so this is 1 half and therefore the 2 possible values of J Our 1 plus a half and 1 minus ahead the absolute value that said 2 possible values of J R 3 halves and 1 half and therefore the 2 terms are to s plus 1 or 2 times a half plus 1 is to establish P 3 have and don't let P 1 half those of the 2 terms that can arise so as I said closed shells ignored they have a total angular momentum 0 the 2 terms there have closed but not identical energy both of them it turns out will see can emit a photon to the ground state 3 s and therefore we get too closely spaced lines which are called the sodium the lines which are quite famous because they were they were studied very early in fact even before the electron it was figured out that the electron existed as a as a particle by Thompson the year before but Zaman was looking at sodium and putting on magnetic fields and looking at how these lines changed and working out a ton of information from these yellow lines which were very easy to excite for example with sodium vapor with the flame and that's of course the sodium the flame test to see what kind of element to this report flame on a new look at the collar and you can tell if it's yellow it's sodium these are also the the characteristic yellow color of the fog lamps you see down toward the beach in places like San Diego or at least you used to they might not be so energy energy-efficient now but the yellow color was thought to be much better at when you had fog because if you have quite light sometimes what you see with very strong bright light is is you just see the fall you don't see through the fog you instead light up the little droplets and you get a lot of reflection from the fog so you don't see the pedestrian worse if you have the yellow color and without the white which is why use the sodium because you have these 2 lines then you just see the pedestrian and then you put on the brakes the term symbol gives us a really quick way to tell whether a transition is "quotation mark allow or forbid according to the electric dipole selection In light Adams and it are still pretty good quantum numbers the reason why they are perfect quantum numbers is that Al refers to orbital angular momentum that came from hydrogen in hydrogen everything is spherically symmetric because there's that 1 proton and 1 electron but in other Adams there other electrons and they muck things up and what that means is that Al is not quite so good because things are wobbling a little and so although yeah that's going around but there's a little bit wobble and the there's ass in there and these magnetic interactions with more than 1 particle and so there 90 per cent good but not 100 per cent good anymore and that means that you can get some forbidden transitions and if you look closely you'll see some weak things that you can explain by that but what 1 thing we know for sure if we're considering it's the electric parts Of the light wave that's moving the electron around the electorate part of the light can change the magnetic bar of the electron and therefore best can change
in the transition so dealt big has to be 0 and because the photon has 1 unit of angular momentum Delta Big L has to be plus or minus 1 and putting those together we can get the Delta J is 0 or plus or minus 1 however if starts at 0 we can't terminated Jake was 0 because then it's impossible to satisfy the Delta L selection rules as well so in that case we get shut down so those are the electric dipole selection rules and 2 terms of the term symbol dealt A 0 dealt L plus or minus 1 and Delta J 0 plus or minus 1 will explore the the next lecture how we can use this notation to analyze some of the spectra