the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu alright guys so today is gonna be the last day of Neutron physics as promised we're gonna talk about what happens as a function of time when you perturb the reactor like you all did about a month ago did any of you guys notice the old fashioned analog panel meter that said reactor period when you were doing your power manipulations we're gonna do that today and you're gonna explore that on the homework so I'm arranging for all of your actual power manipulation traces to be sent with to you so each one you'll have your own reactor data you'll be able to describe the reactor period and see how well it fits our sort of infinite medium single group equations which it turns out is not very well but that's okay because you'll get to explain the differences first before we get into transients I wanted to talk a bit about criticality and perturbing it so let's say we had our old single group criticality in relation and I'd like to analyze just intuitively or mentally with you guys a few different situations let's say we're talking about a light water reactor or a thermal reactor like the MIT reactor or pretty much all the reactors we have in this country what sort of things could you do to perturb it and how would that affect criticality like for example let's say you shoved in a control rod let's take a sieve the simplest scenario control rods in what would happen to each of the terms in the criticality condition and then what would happen to K effective so let's just go one by one does nu ever change ever actually yeah it does over time you'll start that that nu right there remember that's a new bar number of neutrons produced per fission as you start to burn you to thir or consume you two or thirty-eight add neutrons and as you guys saw through a complicated chain of events on the exam eventually make plutonium 239 which is a fissile fuel the new for 238 it's actually different than the new for 239 so I don't want to say that new never changes it's just that shoving the control rods into the reactor isn't God not going to change new but it does change slowly over time as you build up plutonium what about Sigma fission if this were a blended homogeneous reactor our reactor in a blender what would happen to Sigma fission as you then shove in an absorbing material does it change you say no and I'm gonna adhere homogeneous so in this case remember if we define the average Sigma fission as a sum I'll add bits to it of each materials volume fraction or let's say atomic fraction times each materials Sigma fission if we throw new materials into the reactor then this homogeneous Sigma fission does change when we put materials in or take materials out so you guys want to revise your idea yes thank you there's only one other choice now the question is by how much if you put in a control rod where let's say the control rods Sigma fission would be equal to zero but volume would be equal to small can't be any more specific than that how much of an effect do you think you'll have on Sigma fission very small so let's say a little down arrow like that what about Sigma absorption the volume is still small but the control rod by definition Sigma absorption equals huge so what do you think it's gonna increase a little or a lot quite a bit now let's look at the diffusion constant and remember that the diffusion constant is 1 over 3 Sigma total minus the average cosine scattering angle Sigma scattering what do you think is going to happen to the neutron diffusion coefficient as you throw in an absorbing material something that's got an enormous absorption cross-section is also going to have an enormous total cross section because Sigma total the Sigma absorption plus Sigma scattering and Sigma scattering doesn't change that much but if Sigma absorption goes up Sigma total goes up if Sigma total goes up then what happens to the diffusion coefficient yep it's got a decrease and how does inserting a control rod change the geometry very very close so it yeah you're right the control rod better not change the geometry but what I do want to let you remind you of is that this buckling term includes let's say this was a one-dimensional infinite slab Cartesian reactor that little hat over there means we have some extrapolation distance remember if we were to draw our infinite reactor with a thickness a and we wanted to draw a flux profile on top of that it would have to be symmetric about the middle and let's say we had our axes of this as X and this is flux flux can't go to zero right at the edge of the reactor because that would mean that no neutrons were literally leaking out so there's going to be some small extrapolation distance equal to about two times the diffusion coefficient so this geometric buckling is actually PI over the reactor geometry plus two times the diffusion coefficient and if the diffusion coefficient goes down but it's also very very small compared to the geometric buckling how much does the buckling change and by how much and in what direction increases very slightly so the buckling might increase very slightly what's the overall net effect on K effective should go down you would hope if you put a control rod in it should make K effective go down because there's a little decrease here things kind of cancel out there but the big one is putting an absorption material like a control rod in should make K effective go down and that's the most intuitive one but you can work out one term at a time what's generally going to happen so let's now look at some other scenarios for the same criticality condition I'll just rewrite it so that we can mess it all up again now we want to go for the case of boil or avoid your coolant and now we're getting into the concept of different feedback mechanisms we've already talked once about how raising the temperature of something tends to increase cross sections in certain ways but now let's say what would happen if you boil your coolant if things got really hot and the water started to boil what do you want to happen to K effective you want it to increase decrease thank you you want it to decrease or else you get a Chernobyl and we'll talk about how that happened in a week or two so now let's reason through each one of these let's assume that nu doesn't change when you boil the coolant what about Sigma fission of the whole reactor you're taking a little bit of material out of the reactor by taking liquid water which is fairly dense and making it gaseous water which is less dense so overall there are more fissile atoms in the reactor proportionally when the coolant is boiled away then when it's not so what happens to Sigma fission the average Sigma fission for the reactor will go up ever so slightly probably not enough to matter what about Sigma absorption if the coolant disappears yeah water is an absorber hydrogen and oxygen but really just hydrogen have some pretty non negligible absorption coefficients and if those go away then you're losing a bit of absorber aren't you oxygen actually it's interesting oxygen is the lowest thermal cross section of any element so we can treat it as pretty much transparent now how about the diffusion coefficient we've got the formula for it up there if all of a sudden your neutrons don't have much to moderate from or there's not much to moderate your neutrons yeah your scattering just like disappears right but some so does some of your total cross-section so chances are those neutrons are going to go farther before they undergo any given collision because there's no water in the way so you'd expect Neutron diffusion to go up and what about geometric buckling the fusion goes up then the geometric buckling I'm just gonna make it really small but the net effect here once again K effective goes down we didn't talk about anything to do with the actual temperature effects on the cross-sections this is just a density thing on the coolant itself so let's now look at that what about if you have some power spike raise fuel temperature I'll write it again so we can mess it up again so let's say you raise the fuel temperature and that's going to cause every cross-section effectively to increase if you're doing this average scenario let's talk a little bit about why it's not it's not as simple as just saying the cross-sections go up so let's say we had two different temperatures cold and hot so this would be your Sigma fission cold and this would be your Sigma fission hot for cold Sigma fission looks something like that and as the temperature goes up these resonances which I'll just label right here resonances being specific energy is where the absorption suddenly goes up suddenly goes down will actually decrease in height but they'll start to spread out more that's about as well as I can draw it very crudely and same thing goes well not just for Sigma fission but for Sigma anything including absorption including total whatever you want and so if your goal is to get your neutrons from the fast region where they're born into the thermal region where you get fission broadening these cross-sections makes it more likely that if the neutron loses any amount of energy it's going to hit one of these big resonance regions and get absorbed or taken away before it gets a chance to go to the fission region so what this is really going to do it's it's kind of funny to say it in terms of a one group criticality relation but your fission cross-sections actually gonna go down one reason is that the fuel physically spreads out and so this just from the density modification you're not going to get as much but then you've also got that effect of increasing fission from these resonance regions spreading out the question is which one is a bigger effect I can't answer that with a simplest statement you'll go over a lot more of that in 2205 when you talk about what actually defines a residence region how do you calculate them and how do they Doppler broaden our broaden with temperature how about Sigma absorption yeah Sigma absorption well it's gonna go down because things spread out but it might also go up because the the cross sections spread out or the residence is spread out what's really gonna happen though is the reactor atoms are effectively like spreading themselves apart the coolants less dense the structural materials in the fuel and everything are still there they're less dense but there's not few of them in the reactor but there is going to be less coolant in the reactor because it has the ability to spar safai or get less dense and kind of squeeze out the inlet and outlet of the reactor so what's really gonna happen here is we know diffusions going to go up which might cause a corresponding change in buckling and the net effect as we would hope k effective would go down and so what we've talked about now here is directly controlling reactivity with control rods what's called a void coefficient where you actually want to have a negative void coefficient so if you boil your coolant too much K effective should go down and that's one of the mechanisms that a light water or thermal reactor can help stabilize itself and you can see that now from just the really simplified one group criticality rate relation and if you raise the fuel temperature let's say the fuel gets really hot because there's been some power spike you also want the reactor to shut itself down which you can see that it does let's make things a little trickier let's now talk about a sodium reactor fast reactor this one relies a lot more on fast fission of u-238 so if we were to draw the two cross-sections of 235 Sigma fission and 238 Sigma fission remember uranium-235 looked like the one that we drew before whereas you 238 go something like that with no actual scale given I'm not going to even go there but uranium 238 does not need moderation for the neutrons to induce more fission so let's now write the same criticality reaction which again is a super simplified view of things but that's okay what would happen to each of these terms in a sodium fast reactor if you void the coolant so new won't change what about Sigma fission well if the coolant goes away then on average there's fissile materials contributing more to that cross-section but not that much so do you want to get technical might be the slightest of increases but doesn't matter that much what really matters though is the stuff on the bottom sodium does have a low but non negligible absorption cross-section so if the sodium were to boil away then the absorption would go down by a non negligible amount and then what about diffusion well we've got the formula for it up there if there's not as much coolant in the way then the neutrons of God I got are gonna be able to get further on on average let's say they're not going to be scattering around with as much of the sodium so there might be a small increase in diffusion and corresponding small increase in buckling but this is where the one group kind of fails right what the sodium is actually doing is providing a little bit of moderation so that some of those neutrons when they bounce off of sodium leave the fast fission region and get absorbed and that's part of the balance of the reactor if all of the neutrons are then born fast and don't really slow down and just get absorbed then you might have an overall positive void coefficient so this would tell you that in a fast reactor where you're depending on your coolant not just to cool the reactor but to absorb somewhat and to moderate some what you don't want to boil the coolant in a fast reactor and this is a lot of the reason why most fast reactor coolants tend to have extremely high boiling points sodium is approximately 893 Celsius LED business is approximately 1670 Celsius molten salts about 1,400 Celsius so those all those cool ins except for the sodium one you'll melt the steel that the reactor is made out of before your boil to coolant so boiling the coolant is a bad day in a fast reactor because then things will go from bad to worse because in this case the feedback coefficient can be positive for voiding the coolant that's no good so you want to keep the reactor submerged and that's another reason why a lot of these fast reactors are what's called pool type reactors the reactor is not a vessel with a bunch of piping under it that can break and fail but instead it's designed as a huge pool of liquid sodium and then the core is somewhere in here with a bunch of pumps sending the coolant in and back out or through some heat exchanger or something so there's not really any penetrations on the bottom of this pool and you make sure that you maintain either when you have sodium or lead bismuth eutectic or a liquid lead or some other fast reactor coolant so these are some kind of interesting scenarios to think about I think one of them that I put in the homework was imagine you have the MIT reactor and replace the coolant with molten sodium what's going to happen well let's say you got all the water out first and it wouldn't just blow up right what would actually happen to the criticality relation that's something I want you to think about because one of the big problems on the homework is doing exactly this for scenarios that have happened to the MIT reactor except for the sodium one that's never happened and hopefully never will yeah I can't even imagine but so now let's talk a little bit about when you perturb a reactor by doing something to it putting the control rods in or pulling them out or doing whatever you want you're by definition going to take one of our first assumptions about how the neutron diffusion equation works and throw it out the window so we're now moving into the transient regime so to study what happens in a reactor transient or when something changes as a function of time let's first go from K effective to what we call K infinity the multiplication factor for an infinite medium we're only doing this because it's analytically easier to understand and still gets the point across so we'll say that our K infinity is still a balance between production and destruction the difference is if we have an infinite medium there's no leakage you can't leak out of an infinitely sized reactor should one ever exist and so it just comes out as new Sigma fission over Sigma absorption a much simpler form and so now we can write what would happen to the flux in the reactor as a function of time and in this case is gonna be one over velocity I'm going to make this a very obvious wide V that change in the reactor flux is going to just be proportional to the imbalance now in the number of neutrons produced and destroyed so the number of neutrons produced would be proportional to very sharp new Sigma fission minus the number of neutrons destroyed Sigma absorption times Phi as a function of T you all with me so far so this right here is a change which is proportional to an imbalance between production and destruction times the actual flux that you have in some given time so to make the simpler let's multiply everything by V where's my green substitute color multiply everything by V and the only unfortunate situation is we have a Vienna knew next to each other I'm going to try to keep them looking really different those go away and then we end up with if we divide by Phi and those Phi's go away and we have v prime over Phi equals V nu Sigma fission minus V Sigma absorption and now we can start to define things in terms of our K infinity factor and a new quantity I'd like to introduce called the prompt lifetime it's a measure of how long a given Neutron tends to live before something happens to it before it's either absorbed or leaks out well not from our infinite reactor and so we can define this as 1 over their Neutron Vil ASSA tea time Sigma absorption and just to check the unit's here velocities in meters per second macroscopic cross-sections are in one over meters so those cancel out and we're left with a total units of seconds that's nice we would want a mean Neutron lifetime or a prompt lifetime to have units of seconds or time at least yep why the DV DT squared went away okay so that's because we assume we're gonna be analyzing an infinite medium so right here this two real able these terms this would be the total production term that right there represents absorption and that right there represents leakage but if we're analyzing an infinite medium you can't leak out because it takes up the entire universe and Beyond depending on what you believe metaphysically or whatever anyway that's different costs so this right here we can rewrite as 1 over lifetime that makes it easier and this right here if we note that new SiC that's a new I'm gonna be really explicit about that new Sigma fission over Sigma absorption this kind of looks like well this is looking to be like K whoops wrong color like RK infinity over LP so all of a sudden we have a much simpler relation we have v prime over Phi equals K infinity minus 1 over the prompt Neutron lifetime so if we solve this this is just the next financial so we end up with our Phi as a function of T is whatever flux we started at like for your power in amid manipulations it would be whatever the Neutron flux was before you touch the control rod times e to the T or e to the that stuff okay infinity minus one over L P times T which we can rewrite as T over capital T where we're going to define this symbol as what's called the reactor period the reactor period actually measures how what the reactor period actually says is how long before the flux increases by a factor of e and so this is actually what that meter was measuring on the reactor it's the reactor period or the time it would then take for the reactors power to increase by a factor of e because it's an exponential to tell you what these typical reactor periods tend to be for a thermal reactor T is about 0.1 seconds corresponding to an average prompt Neutron lifetime of 10 to the minus 4 seconds seems fast doesn't it like really fast so the question I ask to you guys is why don't reactors just blow up yes there is something we've neglected from here it's like what Sarah said and it deserves its own board there is a fraction of delayed neutrons well good well give that fraction the symbol beta and for a uranium 235 it equals about point Oh Oh 6 4 so there's less than a percent of all the neutrons coming out of a reactor have some delay to them because they're not made directly from fission in the 10 to the minus 14 seconds that we talked about in the timeline but they come out of radioactive decay processes with delayed lifetimes ranging from about 0.2 seconds to about 54 seconds this is the whole reason why reactors don't just blow up so you can actually make a reactor go supercritical but if the K effective is less than 1 plus beta then the reactor is not what we call prompt supercritical and so the reason for that is let's say you raise the reactor power by some amount and the K effective goes up to like one point zero five there's still this fraction point zero six four of the neutrons are not going to be released immediately they're gonna be released not in ten to the minus fourteen seconds but in like ten to the two seconds so you know amis Li 15 orders of magnitude slower meaning that there's actually some ability for this reactor to raise its power level and these delayed neutrons even though that's such a small fraction takes the reactor period from its T infinity value of about 0.1 seconds to about a hundred seconds so the same reactor when you account for the delayed neutrons increases in power by a factor of e and it takes it about a hundred seconds which means this is totally controllable now I have a question for you guys would you guys like me to derive this formula or do you want to go into more of the intuitive implications of it because we can go either way there is a formula that will tell you what the reactor period and time dependence will be and you will hit it in 22:05 probably I can't guarantee it cuz I'm not teaching it or we can talk a little bit more about some of the intuition behind delayed neutrons so a bit of your choose-your-own-adventure math or intuition intuition okay that's fine good so that that was the route that was the derivation I'll post that anyway if you guys want to see says I think in the yep reading it says like okay let's account for the delayed neutrons intuitively we find that the answer ends up being so I'll skip the derivation and it comes out - oh boy find out eetu the beta minus 1 times K minus 1 over L T plus beta Phi naught over beta minus 1 K minus 1 times 1 minus e to the beta minus 1 K minus 1 over L ok so left as an exercise to the reader yeah that's intuitive so but let's actually talk about how intuitive it is I do want to give you the starting and the ending equation and we will not go through the rest yeah charlie no you should it I'm gonna I'm gonna scan it for you guys so don't bother copying it down let's talk about where it comes from and the answer may astound you because we're gonna bring right back the idea of series radioactive decay so let's say you had you want to relate the change in the number in the neutron flux to a 1 minus I'm gonna take a quick look at the original equation so I don't want to screw that up that's the first page and that's the one we want let's say we had some equation that looked something like this 5 plus 5 naught times beta this is the original differential equation from once it came and the intuitive part that I want you to note is that the jump from changing K effective is moderated by this term right here 1 minus beta so that's the fraction of prompt neutrons that as soon as you pull the control rod out that's your instantaneous feedback by instantaneous I mean on the order of like 10 to the minus 4 seconds or something you can't really control this right here represents the delayed fraction this is as mathy as it's gonna get because you have chosen intuition I think you have chosen wisely it's gonna be a lot more fun so what this represents right here is a kind of instant change because whatever you change K effective to is going to be moderated by the prompt fraction how long the neutrons tend to take to undergo that feedback yeah Sara the average one yes this is the average Neutron lifetime so let's define the average Neutron lifetime as simply 1 minus beta times the prompt Neutron lifetime plus the I'm sorry beta times some delayed Neutron lifetime so what no book I've ever seen actually says this is what's referred to as a Maxwell mixing model it's just the simplest thing to say oh if you want to get the average of some variable take the fraction of one species times its variable plus the fraction of the other species times its variable folks do the same thing with electrical resistivity thermal conductivity or any sort of other material property and it is or isn't good in some situations like if you had a piece of material made out of two different things let's say this had thermal conductivity taewon that had thermal conductivity k - what a Maxwell mixing model be appropriate to describe the flow of heat across this thing probably not but in the case of neutrons where they're flying about like crazy and they're mean free path is much larger than the distance between atoms this works great so we can define this mean neutron lifetime and use that in this equation right here so this term right here describes the instantaneous change you pull the control rods out and fraction 1 minus beta neutrons respond immediately what about that fraction of neutrons those are being produced with a fraction beta depending on what the flux was before because they're still waiting to decay from the old power level does anyone notice anything suspicious Li familiar about the final form of this equation for flux you've seen it before with a couple of constants changed around what about the form of this differential equation it is exactly the same as series radioactive decay so the horrible derivation I was going to do for you guys and we're not anymore is use an integrating factor you solve it in exactly the same way you bring everything to one side of the equation find some factor mu that makes this a product rule do a lot of algebra and you end up with a very suspiciously similar looking equation so it's exactly the same posing and solution as series radioactive decay with the difference being that's the constant in front of everything it's instead of a bunch of lambdas and fluxes so what this says here is that the flux as a function of time this is the prompt feedback right here which says that if let's graph it since we're going intuitive there's no room yeah even those all boards are full ok here we go if we graph time and flux right here what that part right there says is that you're going to get some sort of instantaneous exponential feedback but it's going to be moderated by this 1 minus on top so you're gonna end up with a little bit of prompt feedback this stuff right here and then have to draw longer because it's gonna take forever you'll have some delayed feedback because you've got to wait a hundred or so seconds or whatever that new reactor period is for the delayed neutrons to take effect and that's the whole reason you could pull the control rods out at almost any speed you wanted and the reactor doesn't just explode if you pull the control rods out fast enough such that the change in K effective is greater than beta then the reactor goes prompt supercritical which means you don't have any delayed neutrons slowing down the feedback and you've kind of turned your reactor into a weapon a very poor terrible weapon but a prompt supercritical nuclear device nonetheless did anybody pull out the control rods too fast and the controls took over for you what about you guys in training did you ever do things when you watch the cotta Matic control take over no that's what I mean is is the machine takes over and it will kick you off right and stop responding to you I see but the annoying alarm is to stop you from doing something like that like making the reactor go prompt supercritical okay so that's what I would call the Machine taking over yeah okay so if your blood-alcohol level is above beta and you try and let's say increase the reactor reactivity too much it will then take over insert a control rod make a horrible noise and say go home you're drunk something like that okay that makes sense to me so what did your guyses reactor power traces look like did they look something like this where there was an initial rise as you pulled the control rod out and then after you pulled the control rod out the power kept rising just a smidge right and what happened when you put the control rod back in let's say you put the control rod back in you're gonna get another prompt to drop not equal to the same prompt gain that you got because now the reactors at a different flux and then some asymptotic feedback like that and so this is why to those who don't understand Neutron physics reactor feedback is very non-intuitive it's not a linear system you can't just pull the control rod out and change the power accordingly this is why there's animated controls and systems to stop you in case like I said if your blood alcohol content suppose beta which is very low by the way you shouldn't be drinking on the job especially at a nuclear reactor plus you're all under 21 so what am I even saying okay that's right good answer what is alcohol that'll be on the exam yeah yeah cool so that's all I want to go into for the intuitive stuff and it's about 5 of 5 of so I'd like to stop here and see if you guys have any questions on Neutron physics at a-hole noting that we're gonna take Thursday's class and turn into a recitation so I would like all of you guys to look at the problem set because it is posted it is hard trust me this one's a doozy so I want to warn you guys because you got 7 days to work on it but I want you to look at it so that we can start formulating strategies for the problems together on Thursday because there are some tricks to it as usual if you guys know me by now right there's always some sort of a trick like do you have to integrate every energy to get the stopping power no you actually don't have to do any integrals at all but you can if you want and your answer will be more accurate and correct it'll just take longer to get to so there's a lot of diminishing returns on these problem sets if you're willing to take an hour and think about how can I do this simpler and with fewer decimal points you're probably on to something and we'll work on those strategies together yes yeah so I posted it yesterday at around noon or whatever the stellar site says yeah I'll also teach you guys sort of explicitly how to use Janus so we got a comment in from the anonymous feedback saying we have to use a lot of software can we have some sort of tutorial for dummy as well you guys aren't dummies but you still deserve a tutorial so I will show you how to export the data you'll need from this problem set for Janus so you can focus on the intuition in the physics and not get frustrated with getting data out of a computer so any questions on anything from the neutron diffusion equation yeah Luke the prompt neutrons come right out of fission if we looked at that time line of let's say the fission event happens here to fission products are released in about 10 to the minus 14 seconds they move a little further apart and then some of them just boil off neutrons because they're so Neutron heavy after around like I would is at 10 to the minus 13 seconds or so these right here are prompt neutrons coming directly from the immediate decay of Neutron rich fission products some of the delayed neutrons come from radioactive decay but of the much later fission products with much less likely occurrences which is why the fractions very low but also because it's much longer half-life those delayed neutrons take seconds instead of picoseconds to show up that's the whole basis behind easier control and feedback of reactor good question so anything starting from neutron transport to simplifying to Neutron diffusion to getting to this criticality condition making the to group get a criticality condition if you want to let's say have fast and thermal or any of the time-dependent stuff that we intuited today yeah was it on the other border from a different day yes so in this case let me get a finer chalk this blue one would be for low temperature and this red one would be for high temperature so this blue graph there are resonances which have very high values but they're very narrow and because the width of a resonance doesn't matter it doesn't affect the probability that a neutron scatters up here and moves some distance down the energy spectrum thinner resonances get tend to get passed over especially like if your reactors full of hydrogen some of those neutrons will be born and immediately jump into the thermal region where it's easy to tell how much fission they'll undergo as you go up in temperature you undergo what's called Doppler broadening which causes these resonances to spread out and also go down in value so the actual value of the cross-section at these residences is lower but the widths are larger so there's a higher probability that a neutron scattering around and losing energy will hit one of these higher cross-section regions call the resonance at a higher temperature that's the difference there is these two plots show the same cross-section at low and high temperature these plots show the difference between uranium-235 and uranium-238 good question anyone else cool ok well for the first time in history I'll let you out a minute early bring all your questions on Thursday so we'll start off with a janice tutorial and then we'll start attacking this problem set together