Notes follow and pats taken fom souces in Bibliogaphy At the end of the 19 th centuy, physics was consideed to be almost completely solved by some. As it tuned out, thee wee seveal inconsistencies which eventually led to two evolutions in physics: the theoy of elativity and the quantum theoy. The most popula model of the atom in ealy 0 th centuy was developed by J. J. Thomson and was known as the plum pudding model. In it, the electons wee tiny point paticles inside a continuous distibution of positive chage, like seeds in pudding. One of Thomson s students, Enest Ruthefod, investigated the stuctue of the atom in a seies of expeiments in 1911. Ruthefod fied α paticles (helium nuclei, emitted in some adioactive decays) at a vey thin sheet of gold foil to measue the deflection angle as they went though. Because the sheet was so thin, each α paticle could only encounte a elatively small numbe of atoms, and because each α paticle was about 10,000 times as massive as an electon (the only solid thing to scatte fom, as fa as anyone knew), the α paticles wee expected to pass though almost completely undeflected. Instead of this esult, Ruthefod found that some of the α paticles undewent huge deflections, of sizes that wee not consistent with collisions with electons. Ruthefod postulated that the positive chage holding the electons togethe was concentated into a vey tiny volume (with a adius of about 10-15 m) at the cente of the atom, athe than being spead out all ove it. This pesented seious poblems fo a numbe of easons. Fist, the Coulomb epulsion of the positive chages confined to the tiny volume of the nucleus would be enomous. At the time, thee was no way to explain what othe foce could exist to hold the nucleus togethe. Additionally, something had to explain the peviously detemined size of the atom itself (aound 10-10 m). The electons would be pulled towads the cente, leading to collapse of the atom. If the electons wee envisioned to move aound the nucleus like the planets aound the Sun, with angula momentum peventing the collapse, thee was a diffeent poblem. A paticle can only move in a cicle unde the influence of a foce which will cause a centipetal acceleation. The poblem in the case of the electon is that an acceleated chage (like one moving in a cicle) must adiate enegy. Calculations indicated that electons moving in cicula obits aound the lage positive nucleus would adiate all thei excess enegy away and collapse into the nucleus in much less than one second, which obviously did not happen. Any theoy of the atom would also need to explain the fact that light emitted fom isolated atoms, like those in a elatively thin gas, is only emitted at cetain discete wavelengths athe than acoss a continuum. The Boh model of the atom was able to explain lage pats of these poblems. In the Boh model, the electons would move in cicula obits without emitting adiation and spialing in to the nucleus. The allowed obits wee esticted to those which had 1
discete values of angula momentum, specifically an angula momentum equal to whee n is an intege geate than zeo and is Planck s constant divided by π. n If the centipetal foce holding the electon in the atom is to be supplied by the Coulomb attaction between the electon and the nucleus, we would get m e v k Z e The v is the electon s velocity in its cicula obit, and Z epesents the total numbe of chages (equal in magnitude to that on the electon). We can solve this fo and use the fact that L mv n h π Combining these ideas, we can wite n h 4π mk Z e and then, since the enegy of a cicula obit is well-known to be E 1 mv k Z e we can wite the enegy as E k Z e π h n mk e 4 Z Fo the hydogen atom (with Z 1), we calculate the enegy fo an electon in level n to be E 13.56eV 1 n
Again, the estiction on n as being an intege geate than 0 still applies. When an electon moves fom a highe level n to a lowe one n 1, it will emit one (o sometimes moe than one) photon(s) with a total enegy equal to E E 1. The Uncetainty Pinciple Anothe key diffeence between quantum mechanics and classical mechanics is the idea of deteminism. Deteminism basically says that if you knew the positions and momenta of all objects in the univese exactly, even fo just an instant, you could then calculate thei positions and momenta at any late time. The outcome of evey coin flip, lottey dawing, football game o stock puchase afte that would be known in advance. This was lagely a philosophical agument, since finding the exact position and velocity of even one paticle had not been done when it was poposed. Still, thee was nothing to pevent this in pinciple it would just take bette and bette measuing instuments which could be expected to be developed though technological pogess. In fact, it was late discoveed that this could not be done, even fo one paticle, and even in pinciple. Heisenbeg s Uncetainty Pinciple says that you (o anyone else) cannot know the exact position and momentum of any one paticle at the same time. It futhe esticts the poduct of the uncetainty in momentum and the uncetainty in position as shown below: p x Theefoe, the bette you know a paticle s position (the lowe x), the less you know about its momentum. You can think of this as follows: if we want to measue the position of an electon, how do we do that? At the most fundamental level, we look at it, which means we have to bounce photons off of it. In doing that, though, we tansfe some momentum to the electon. If we want to measue the electon s position to within one nanomete, fo example, we need to use light with a wavelength of less than one nanomete. What momentum will the light have? We can use ou pevious esults to see that 9 h h if λ < 10 m, p so p > 9 λ 10 m Getting a bette measuement of position means using light with a smalle wavelength, but that means each photon has a high momentum. Some (unknown) faction of that momentum will be tansfeed to the electon, so measuing positional infomation causes us to alte, and theefoe lose, momentum infomation. 3
This is not a poblem in ou macoscopic wold, but it can be one on the micoscopic scale. If an electon moving at 1000 m/s has a momentum of about 10-7 kg m/s, an uncetainty of 10-4 kg m/s is idiculously lage. Of couse, an uncetainty in momentum will evolve into an uncetainty in position if the uncetainty in a paticle s velocity ( p/m) is 10 m/s, its uncetainty in position is gowing by 10 m each second. If we localize a paticle vey well (measue it pecisely so that its x is small), the uncetain amount of momentum tansfeed to it may completely emove it fom ou viewing aea. In this case, about all we can say is that a paticle was at this position ealie. Thee is a simila uncetainty elation fo enegy. An accuate detemination of the enegy of a state depends on how long it is obseved. Basically, we ae allowed to violate consevation of enegy (!) vey biefly and boow some enegy fom the vacuum, as long as we etun it vey quickly. The poduct of the enegy uncetainty and the time uncetainty satisfies the same inequality as above: E t h 4π It s stange and pobably a little distubing to see that one of the things so impotant in physics (consevation of enegy) seems to be disappeaing. Befoe you get too woied, do a sample calculation: if you only want to boow some enegy fo a picosecond (10-1 seconds), how much can you have? Plugging 10-1 seconds in fo t in the fomula above gives 5.3 x 10-3 J, o about 3.3 x 10-4 ev. Not exactly a huge amount of enegy! One of the consequences of the uncetainty pinciple is that cetain featues of classical mechanics ae adically alteed. Fo example, if we imagine a ball bouncing inside a dinking glass, it is vey staightfowad to find out whethe it will escape o not. If the kinetic potential enegy of the ball is less than the potential enegy the ball would have at the im of the glass, it absolutely won t get out. The uncetainty pinciple allows the ball to boow a small amount of enegy fom the empty space aound it and etun that enegy once the ball has gone ove the side. Since the time that enegy can be boowed is invesely popotional to the amount boowed, and Planck s constant is so small, we won t see this happen with a ball & dinking glass. The pobability of penetating a baie dops vey quickly as the baie s thickness inceases. Also, if the baie is vey lage compaed to the paticle s enegy, the pobability of penetation dops apidly. In the smoke detecto example, the pocess can be modeled by assuming the α paticle is tapped inside the nucleus like a paticle in a well. The α paticle bounces back and foth within the nucleus until (by andom chance) it gabs a small amount of enegy fom the vacuum, climbs the side of the well, and appeas outside the confines of the nucleus. Once it s on the outside, the epulsive foce between the two potons of the α paticle and 93 emaining ones in the nucleus cause the α paticle to ocket away and be detected by cicuity in the smoke detecto. 4
Fo this eason, the enegy of the escaping α tends to incease as the Z of the nucleus inceases. Something inteesting to notice is that the enegy boowed eally is etuned to the vacuum. We can calculate the enegy an alpha paticle needs to have to escape its confinement by the nucleus, and we find that the emitted alpha paticle has a lowe enegy than this. If the boowed enegy didn t have to be etuned (and theefoe consevation of enegy could be violated on the lage scale), the emitted alphas would be ejected much moe quickly than they eally ae. This pinciple also explains the eason that atoms ae stable. The electon can t adiate photons and fall into the nucleus because that would make its position moe cetain (since the nucleus is smalle, we d know whee it is to within 10-14 m o so instead of 10-10 m), which would mean its momentum would be moe uncetain, making it potentially lage. Inceased momentum means inceased kinetic enegy. A smalle value of position, on aveage, would give a moe negative potential enegy but a moe positive kinetic enegy. The atom s size is the esult of a balancing act between confining an electon to a tiny volume (which the Coulomb foce would like to do) and keeping its momentum and kinetic enegy fom inceasing wildly. Anothe way to look at this uncetainty involves what you do when you execise and measue you pulse. You can count heatbeats fo 10 seconds and then multiply by 6 to get you pulse ate, but the possible answes ae things like 60, 66, 7, etc. You can t get a pulse ate of 65 this way. You could count fo 30 seconds and multiply by two, but you ll only get even numbes in this case. You will get a bette estimate by measuing longe. You could count fo a full minute, but what if you eal pulse ate is 67.4 beats pe minute? You will still have the potential fo eo in the detemination of the pulse ate as long as you count fo a finite time. The Schödinge Equation The equiement of quantized values of the electon s angula momentum can be aived at by anothe means. If we conside an electon with momentum p to be a wave with a wavelength λ h/p (the idea of light as a paticle gained suppot aound the same time that the electon was postulated by de Boglie to have wave-like chaacteistics) and popose that an integal numbe of wavelengths must fit aound the electon s obit ( π in size), we would get n h p π mv n h π which is the same condition on that was seen befoe. If the electon can be consideed to be a wave, thee must be a wave equation descibing how the electon moves in geneal. This equation is known as the Schödinge equation. The wave, which is 5
expected to be a function of position and time, is theefoe known as the wave function and is usually epesented as (x,t). As a tial wave function, we can daw on what we will see will be many paallels between quantum theoy and optics and ty a plane wave of the fom i ( k x ω t ) ( x, t ) e 0 The equation itself is witten (in one dimension) as m d d ( x, t) x V ( x, t) ( x, t) d ( x, t) i d t Substituting in ou plane wave solution and dividing eveywhee by, we get k m V ω Notice that the fist tem could also be witten as p /(m), which is the classical expession fo the kinetic enegy of a paticle of mass m. If we add the kinetic tem to the potential tem (V), we get the total enegy, which is ω, just as it is fo a photon. The Schödinge equation just expesses KE PE E total. It s woth noticing that, since p /m is the nonelativistic expession fo enegy, the Schödinge equation itself is nonelativistic. At enegies whee an electon s KE is not insignificant compaed to its est mass enegy, a elativistic equation is equied (known as the Diac equation fo the electon). Because the Schödinge equation contains the fist deivative with espect to time, but the second deivative with espect to position, it looks vey simila to a diffusion equation, which specifies how two mateials mix (a dop of milk in a glass of wate, fo example). The weid thing about the Schödinge equation is that it is a diffusion equation in imaginay time (notice the i on the ight side). What is diffusing in this case? The wave function itself, which descibes the electon s position/momentum. Because the absolute squae of the wave function is popotional to the chance of finding the electon in any given egion, we can say that this expesses what we saw ealie with the uncetainty pinciple: afte we locate the electon, its position gadually gets less cetain with time (due to p) as the wave function speads out, o diffuses, though space. 6
Solving the Schödinge Equation The Schödinge equation is usually given in eithe time dependent o time independent fom. The example above is the time dependent case the potential is a function of position as well as time, and the wave function is also. The time deivative on the ight is the signatue of time dependence. In the cases we will examine, the time independent Schödinge equation will be good enough. This means that the potential will no longe be a function of time, and that the time deivative tem on the ight side of the equation will be eplaced by E, the enegy of the paticle. The enegy is a constant in this case, because time independence is deeply connected to the consevation of enegy. In the same way, tanslational invaiance is deeply connected to the consevation of momentum (if you e inteested in leaning moe about this, it s called Noethe's theoem). We now get the simple fom m d ( x) d x V ( x) ( x) E ( x) known as the Time Independent Schödinge Equation (TISE). If we want to begin to use this to solve poblems in quantum mechanics, we can stat with the poblem of a one-electon atom. Fist, we need to ewite the 1-D equation above in thee dimensions. We also need to come up with an expession fo the potential felt by the electon. Finally, just as you may have done in you fist semeste physics couse if you poved Keple s thid law, we have to account fo the fact that, in a classical situation, the electon and the nucleus both move. The electon will move much moe than the nucleus, but to make the math easie, we can define a educed mass which we will use instead of the electon s mass. By effectively changing coodinates to a system whee the electon does all of the moving, we don t need a tem fo the kinetic enegy of the nucleus. If a mass m 1 and a mass m ae in obit aound one anothe, we can define the educed mass as µ m m 1 1 m m We ll use this µ in place of the m in the time independent Schödinge equation. The potential is just what you would have witten in the fist weeks of the second semeste of physics fo the potential enegy between two chages: 7
( ),, z y x Z k e z y x V whee Z is the numbe of potons in the nucleus, k is the constant fom Coulomb s law, e is the magnitude of the chage on an electon/poton, and is the atomic adius. Finally, we have to ewite the kinetic tem in the Schödinge equation in thee dimensions: ),, ( z y x m While it would be simple to wite this new equation in Catesian coodinates, that would ignoe the natual symmety of the poblem. The Coulomb potential has spheical symmety, so solving it will be much easie if we switch to spheical coodinates. Witing out the full equation (including expanding the Laplacian opeato in the kinetic tem) gives us φ θ θ θ θ θ µ E Z k e sin 1 sin sin 1 1 whee is a function of all thee vaiables. The usual method fo solving this equation (and the eason the spheical coodinate system is pefeed) is that the wave function is sepaable into a poduct of functions, each of which depends only on one coodinate. In othe wods, we can estict ouselves to solutions that satisfy (,θ,φ) R() (θ) (φ). When we substitute this compound fom fo into the TISE, we can look fo solutions to the individual pieces of the wave function, which is a much easie poblem. We can ewite the equation above as ( ) ( ) ( ) Φ Θ Z k e E R sin sin sin sin 1 θ µ φ θ θ θ θ θ φ θ and look at the behavio of the compound fom of in the deivative with espect to φ, fo example: 8
(, θ, φ ) φ R ( ) Θ ( θ ) d Φ d φ ( φ ) whee we have done away with the patial deivative. We can move all tems containing (φ) o its deivatives to one side of the TISE, and all othe tems to the othe side. This means one side contains all of the φ dependence and the othe contains all of the and θ dependence. That means (and this is a fequently used technique in physics) that each side must be equal to a constant (which we call m l by convention). The (φ) containing side can then be witten as 1 Φ ( φ ) d Φ dφ ( φ ) d Φ ( φ ) m l m l The solution to the equation above is clealy an oscillating exponential of the fom dφ Φ ( φ ) Φ ( ) i l φ φ e m The foms of the solutions fo R() and (θ) ae unfotunately not quite so simple. The solution fo (θ) can be witten in tems of special mathematical functions known as Legendé polynomials, which ae polynomials in cos(θ) such as cos(θ) itself (the fist Legende polynomial), ½ (cos θ - 1) (the second), etc. The n th Legende polynomial will have a tem popotional to cos n θ in it. The R() solutions ae even weide things known as hypegeometic functions. Rathe than wok though this, we will look them up in tables. Hydogenic Wave Functions Thee ae thee numbes that will chaacteize the solutions descibed above one fo the (φ) pat (the m l mentioned ealie, known as the magnetic quantum numbe, which is also pat of the solution fo (θ)), one that is used fo (θ) and R() (l, the angula momentum quantum numbe), and one used only fo R() (n, the pincipal quantum numbe). The mathematical popeties of the functions above place estictions on the values of n, l, and m l as follows n 1,, 3, l 0, 1,, n-1 m l -l, -l1, l-1, l 9
The gound state of the Hydogen atom theefoe has n 1, which foces it to have l m l 0. It spends most of its time nea the nucleus (i.e., the cente of the atom). A few wave functions fo vaious values of n, l, m l ae shown below (fom the Eisbeg & Resnick book) nl m l 100 Wave function 1 Z 3 / Z / a0 1 Z 3/ Z Z / a0 00 10 1 1 8 4 4 π π π π a 0 a 3/ 0 e a 0 e 1 Z Z Z / a0 a a 0 a a 0 e e 3/ 1 Z Z Z / a0 0 0 cosθ sin θ e The a 0 epesents the Boh adius, which is about 5.9 x 10-11 m. Notice that the wave function fo the 100 state dops off exponentially as you move away fom the nucleus, meaning the chance of finding it is lagest at 0. In the 10 state (the fist one with angula momentum), the spheical symmety that was pesent in the 100 and 00 wave functions (which had no θ o φ dependence) is now gone. The 10 state has a cosθ dependence in addition to the adial dependence. The 1-1 state featues φ dependence as well. Because the wave function epesents a pobability amplitude, though, we have to multiply it by its complex conjugate (i.e., the same wave function but with all i tems eplaced by i. That will emove the φ dependence. Because the pobability of finding the electon at any infinitesimal point will be zeo, we can only talk about pobability densities. We need to integate * ove a finite volume to get a pobability. As you can see, the dimensions of the wave function ae metes -3/, so the squae of the wave function integated ove a volume will give a dimensionless quantity, which pobability must be. Because the Coulomb potential itself is spheically symmetic, thee can be no enegy dependence on eithe l o m l. Only the pesence of a magnetic o electic field beaks this symmety. iφ 10
The estictions on l and m l explain the aangement of the peiodic table of the elements. No two electons can be in the same state (i.e., no two can have the same quantum numbes). To completely descibe the electon s state equies the intoduction of two moe quantum numbes, howeve. The value of the spin angula momentum quantum numbe s is always ½ fo an electon (fo this eason, it is also known as the electon s intinsic angula momentum, o the angula momentum it has just because it is an electon). The othe quantum numbe is m s, and it has the same elationship to s that m l has to l: it anges fom s to s in intege steps. Of couse, that means thee ae only two choices fo m s, and they ae ± ½. This is the eason the fist ow of the peiodic table contains only two elements H and He. The gound state of the H electon is the 100 state. The He electons ae both in the 100 state, but this is allowed because each electon has a diffeent value of m s. In the second ow, lithium has its thid electon in the n state. Thee ae a total of 8 elements in the second ow, with neon having two electons (one of each spin) in the 00 state, two in the 10 state, two in the 11 state, and two in the 1-1 state. This is the eason that the noble gases (which all have full oute shells) ae so eluctant to combine with othe elements thee ae no emaining states in the full shell, and the atom itself is electically neutal and not attactive to othe electons as a esult. Nuclea Physics The potons and neutons inside the nucleus see a vey diffeent envionment than what the electons can see. The electons moving in the atom ae dominated by the effects of the electomagnetic foce. Inside the nucleus, the electomagnetic epulsion of the potons is vey lage. Using Coulomb s law and a sepaation between potons of 1 Femi, we can calculate a epulsive foce of ove 100 N an incedibly lage foce acting on such a tiny mass. Also, we can calculate an electostatic potential enegy of ove 1 MeV fa lage than the few ev binding a typical oute-shell electon to an atom. Clealy, thee is a much stonge foce binding the potons and neutons to one anothe. This foce is also only impotant ove vey shot distances, having a typical ange of appoximately 1 F. If we assume that foces ae mediated by the exchange of paticles (just as the electomagnetic foce is mediated by the exchange of photons), we can use the uncetainty pinciple to estimate the mass of the messenge paticle. A paticle with a est mass of m will have a minimum enegy of mc, meaning that if it appeas fom the vacuum via the uncetainty pinciple, it must etun to the vacuum within a time t mc With this limit on its lifetime, we can then say that it cannot move faste than the speed of light, so its ange is limited by x c t. Following the same pocess backwads and 11
using 1 fm as ou maximum ange, we pedict a est mass enegy of appoximately 100 MeV. Using this line of easoning, expeimentes seached fo and found the pion (actually, thee diffeent pions the π, π -, and π 0 ). A poton emitting a π will become a neuton, as a poton absobing a π - will also. The π and π - ae antipaticles of each othe, while the π 0 is its own antipaticle. The actual est mass of the π and π - is about 140 MeV/c, and about 135 MeV/c fo the π 0. Within the nucleus, we can imagine a flood of these pions thoughout, stabilizing the collection of potons and neutons. The pions ae scala paticles, meaning they have spin 0. Massive scala paticles have thei own wave equation, known as the Klein-Godon equation. This is a elativistic equation, but we can look at the time-independent case and if we solve it, we get the Yukawa potential: V α e / λ whee the λ epesents the typical ange of the potential and α is the coupling constant that descibes the stength of the inteaction. Notice that we could apply this idea to the photon: if its est mass is zeo, the effective ange of the inteaction is infinite. The eason is that, fo any paticle sepaation, thee is always a photon with a low enough enegy to make the tip between the two paticles. The ange limit fo the exchange of a massive paticle aises because, in that case, thee is a minimum enegy the enegy equied to poduce the paticle. In the photon case, if the ange is infinite, the exponential tem in the potential above will appoach one. The potential then becomes α/, which is in fact just the odinay Coulomb potential. Binding Enegy and Mass Defect Because the potons and neutons ae bound to each othe in the nucleus, they ae in a lowe enegy state than if they wee to be sepaated fom each othe. This means that enegy would have to be added to the nucleus to split it into individual paticles, and since enegy and mass ae connected via Einstein s fomula E p c m c 4, a nucleus of Z potons and N neutons actually has a mass less than Z*m p N*m n. The diffeence between the actual nuclea mass and the mass of the pats that make it up is known as the mass defect. The mass defect can be multiplied by c to give the total binding enegy of the nucleus. If we then divide that by the numbe of nucleons (ZN in the pevious paagaph), we ae left with the binding enegy pe nucleon. This is a measue of the stability of a given nucleus. We can make a cuve of the binding enegy pe nucleon as a function of 1
nucleon numbe, and it looks like the gaph below (fom http://www.lbl.gov/abc/wallchat/chaptes/0/3.html ): Two of the notable featues of the cuve ae 1) the peak at A 4 (epesenting 4 He, having two potons and two neutons) and ) the moe gadual (and highest) peak aound A~60. The pesence of the highe peak is vey impotant fo the pocess of ceation of the elements in the coes of stas, known as stella nucleosynthesis. At the cente of stas such as the Sun, gavitational foces contacting the sta incease the tempeatue within it. Highe tempeatues mean highe enegies fo the paticles in the egion. Because the Sun (and the univese) is mostly hydogen, and the high tempeatues at the coe of a sta ae fa moe than necessay to ionize hydogen, the inteio of a sta is a plasma. The potons moving aound at high speed can collide with enough enegy to ovecome the epulsive Coulomb foce. The distance of closest appoach depends on the kinetic enegy of the potons, which in tun depends on the tempeatue. When the potons get close enough fo the pion exchange foce to be felt, they will undego nuclea fusion, combining in the pocess below: p p H e ν e The emission of the positon is to conseve chage as one of the potons becomes a neuton, and the neutino is emitted because lepton numbe also must be conseved. The known leptons ae the chaged electon, muon, tau paticle, the thee neutal neutinos (identified with each chaged paticle) and the six antipaticles of these. The paticles have lepton numbes of 1 while the antipaticles have lepton numbes of -1. This is just the fist step in the poton-poton chain that tuns 4 potons into the helium-4 nucleus and eleases enegy. The enegy is eleased because the 4 He nucleus 13
has a lage binding enegy, meaning it must get id of a lage amount of enegy to exist. (In lage stas, a simila pocess tuns hydogen into helium, but does so using a cabon nucleus as a catalyst. This is known as the CNO cycle.) The enegy eleased by this fusion is about 7 MeV. Notice that this is appoximately ten million times moe enegy than we would expect to be eleased in a chemical eaction, which essentially shuffles electons aound, because electonic binding enegies ae typically a few ev. This is the eason that hydogen bombs (well unde a ton of active mateial) typically have enegy yields equivalent to millions of tons of the chemical explosive TNT. In a sta, the heat and pessue poduced by fusion keeps gavity fom cushing the sta. When the hydogen is exhausted, the sta will contact and tempeatues will ise (just as you notice when pumping up a bicycle tie compessing a gas makes it hotte). Eventually (if the sta is not too small), the tempeatue will ise to the point that helium nuclei can fuse with one anothe. Thee is a paticulaly enegetically favoable method fo combining them, whee thee 4 He nuclei fom a single nucleus of cabon-1, known as the tiple-alpha pocess. Once the helium is scace, the cooling and gavitational compession esume. In lage enough stas, the tempeatue can incease until 1 C stats to fuse with othe nuclei (emaining 4 He that didn t find othe nuclei to fuse to). Of couse, the tempeatue has to go up fo this to happen because the epulsive foce between a 1 C and a 4 He is much lage than that between two 4 He nuclei. This pocess continues in lage stas, foming lage and lage nuclei. Since helium is so abundant, it is common fo the elements to gow by potons and neutons at a time. Since atomic numbe is equal to the numbe of potons, we can look at a chat of elemental abundances in the sola system and notice an inteesting tend. Of couse, thee is a decease in abundance as atomic numbe inceases since the heavie elements ae hade to make. We might then expect that the abundance of an element with Z potons will always be smalle than one with Z-1 potons. This is tue fo helium and hydogen, cetainly. Howeve, take a look at the plot below: 14
Most of this gaph is above the line whee the log of the abundance atios is zeo. This means that elements with even numbes of potons ae moe abundant than those with odd numbes, and the eason is that most elements ae built up two potons (and two neutons) at a time. Also, notice that the even-z elemental abundance is compaed to the abundance of the lowe odd-z neighbo. The geneal tend is that abundance deceases with Z, so this is stonge evidence of the addition of helium nuclei as basic building blocks. The buildup of elements can t continue foeve. As the nucleus gows, we move past the peak of the binding enegy cuve at aound A ~ 60. In a sta, 56 Fe is the enegetic endpoint. Futhe fusion between helium nuclei and heavie nuclei can still happen, but since it will move the nucleus down the binding enegy cuve, it is enegetically unfavoable. Fusion by the addition of single neutons to a nucleus is still possible the neutons do not feel any electomagnetic epulsive foce, so they ae able to get vey nea the nucleus, until the stong foce gabs them and taps them. When a nucleus has too many neutons, it will stat to convet some of them into potons though the pocess of beta decay, which we will examine soon. Heavy elements up to about 08 Bi can be fomed though this slow method (known as the s-pocess) The heavie elements ae believed to fom when a sta s intenal funace finally uns out of useful fuel (i.e., the coe is mostly ion) and the heat and pessue poduced by fusion stat to decline. Gavity keeps woking, howeve, and the sta s coe begins to collapse inwad. When the electons and potons emaining in the coe ae compacted unde high enough gavitational pessue, they fom neutons. The neuton degeneacy pessue is much lage than the electon degeneacy pessue which holds up white dwaf stas. The oute coe of the collapsing sta will be acing inwad at a easonable faction of the speed of light duing the collapse, and when it hits the had bounday of the neuton coe, it ebounds. 15
This ebound ceates a flood of neutons which will iadiate the sta s mateial vey apidly (known as the -pocess) and poduce elements up to at least 38 U. The subsequent explosion scattes the heavy elements thoughout space, to be incopoated in late geneations of planets and stas. The fusion pocess, if contolled on Eath, would povide an almost unlimited souce of enegy since we would be fusing hydogen, feely available by splitting wate. The poblem is that, while a sta has temendous gavity to hold the high-speed, hightempeatue nuclei in place, we don t. We need something that will confine the fusing nuclei at tempeatues ove one million Kelvin. No physical containe can do this without instantly vapoizing and, at the same time, cooling off the eaction. One of the cuent ideas is to ty to confine the fuel using magnetic fields, known as magnetic bottles. Nuclea Fission At the fa end of the binding cuve, we find the heaviest elements. While it is enegetically unfavoable to ty to get huge collections of potons (which epel each othe vey stongly) close enough to fuse, it tuns out that beaking lage nuclei into smalle (moe tightly bound) ones will also elease enegy. We can cause this beakup to occu in many mateials by fiing a neuton at the nucleus. The poblem with doing this andomly is that a nucleus a) might absob the neuton and keep it, making an isotope of the taget element (that is, a nucleus with the same numbe of potons but a diffeent numbe of neutons) b) the nucleus might split into two oughly-equal pieces (fission), but that would be the end of the eaction since those pieces, which will have a lage positive chage, ae unlikely to be able to beak up any othe nuclei o c) the nucleus might split and elease moe neutons. This is the most desiable outcome, because the neutons eleased have a good chance of causing moe fissions, which will elease moe neutons, etc. This is known as a chain eaction, whee the numbe of fissions gows exponentially. One natually occuing candidate fo fission is 35 U. The poblem with obtaining it is that most uanium is 38 U, which will not keep a chain eaction going. Duing WWII, huge amounts of money and effot wee spent to sepaate the two isotopes. Because they ae both uanium, they can t be sepaated chemically, as could be done fo two diffeent elements in a compound. The sepaation had to be physical, based on the mass diffeence. Of couse, the mass diffeence was about 3/38 ~ 1%, so this was vey difficult as well. Adding to the difficulty, the uanium had to be chemically attached to fluoine in the fom of UF 6 to fom a gas. The added mass of 6 fluoine atoms did not make sepaation easie. The two most common methods at the time involved centifuges, whee the uanium hexafluoide could be spun apidly until the slightly heavie 38 U concentated at the outside of the centifuge. In a similaly involved pocess, the same gas was sent though a seies of membanes in a diffusion plant. The lighte 35 UF 6 would get to the end of the diffusion stage a little moe quickly, on aveage, than the heavie 38 UF 6. Afte many stages, the uanium was elatively efined. 16
It was also discoveed that 39 Pu was a good fission candidate, so the govenment also had teams woking on that kind of bomb. Since plutonium has such a geologically shot half-life, thee ae no deposits of it to mine. It had to be ceated by bombading othe mateials with neutons. Nuclea Decay Radioactive decay is a nuclea pocess athe than an atomic pocess. This decay occus when a nucleus is somehow out of balance; thee ae too many potons, o neutons, o both, o thee is too much enegy in the nucleus. The thee most common types of adioactive decay paticles ae alpha, beta, and gamma. The α paticle consists of two potons and two neutons which ae bound togethe vey tightly. This can also be thought of as the nucleus of a helium atom, o He. The β paticle is most commonly thought of as an electon, although the positon is also consideed a β paticle (it s geneally a good idea to emove all ambiguity by efeing to the electon eithe by name o with the symbols e - o β - and the positon eithe by name o with the symbols e o β ). Finally, the γ is just a high-enegy photon If you examine the peiodic table, you ll notice that elements nea the top tend to have weights that ae about twice thei atomic numbes, meaning the numbe of potons and neutons is equal. This makes a stable nucleus, at least fo small nuclei. It s woth mentioning that the atomic weight is not usually an intege fo a few easons, including mass defect as well as the fact that it is an aveage of A weighted by abundance of the diffeent isotopes. Fo lage elements, the numbe of neutons exceeds the numbe of potons in a stable nucleus. The eason is that the pion exchange foce has a vey shot ange, so the nucleons ae effectively bound due to inteactions with thei immediate neighbos. Adding moe nucleons will incease the binding fo othe nucleons that ween t aleady suounded, but it doesn t make much diffeence fo those that wee. Because the electomagnetic foce has infinite ange, though, evey poton in the nucleus epels evey othe poton, tending to make lage nuclei (with moe potons) less stable. The addition of neutons allows the pion exchange foce to continue to stabilize the nucleus while keeping potons futhe away fom one anothe and theeby educing the epulsive foces. When the atoms ae so big that the electomagnetic epulsion stats to win, it is common fo them to emit an alpha paticle. This gets id of two neutons, which doesn t help much, but it also gets id of two potons, so it s woth it enegetically since it impoves the neuton-poton atio. This is a vey common decay mode fo elements at the bottom of the peiodic table. 17
β decay The emaining kind of nuclea decay fo us to examine is β decay. The thee types of β decay ae known as electon emission, positon emission, and electon captue. Each of these pocesses will change Z by one unit as a poton o neuton becomes a neuton o poton. The value of A does not change duing β decay. If the balance of neutons and potons is displaced fom its ideal value, β decay can bing it back to a moe stable atio. This pocess is mediated by the weak nuclea foce, which we will discuss late. We can descibe this decay by n p e ν e The decay above satisfies consevation of chage, lepton numbe, bayon numbe, and mass-enegy. We ll look at bayons again afte leaning about quaks, but fo now, you can just emembe that neutons and potons both have a bayon numbe of 1 and thei anti-paticles have a bayon numbe of -1. Leptons have a bayon numbe of zeo, just as bayons have a lepton numbe of zeo. This decay is also what happens to an isolated neuton. Oddly enough, the neuton is only stable when it is pat of a nucleus. If left alone, half of a sample of isolated neutons will decay in about 10 minutes. Nuclei can also decay via positon emission, also called β decay. This pocess is simila to the one above, but now we have p n e ν e Notice the changes elative to the fist decay. Since the poton has a positive chage on the left, something on the ight must have a positive chage, which is why we get a positon athe than an electon. Also, since the positon has a lepton numbe of 1, this is balanced by an electon-neutino athe than an antineutino. Unlike in the case of the neuton, this pocess cannot happen outside of a nucleus, since the total mass-enegy on the ight is lage than that on the left and would not be conseved. This is only possible when the poton can boow enegy fom somewhee else, like the nucleus to which it belongs. As fa as anyone has been able to detemine, potons ae stable. If poton decay is even possible, as some unified theoies pedict, expeiments show that the half-life of the poton must be lage than at least 10 3 yeas! Finally, just as β decay attempts to fix the poblem of having too many potons fo the numbe of neutons in a nucleus, electon captue is anothe way of doing the same thing (also known as a competing pocess). Afte positon emission, the atom will 18
have lost an enegy of at least *511,000 ev, o about 1 MeV (half of this goes into the positon that is ceated, and the othe half epesents the electon that the atom will lose). If a nucleus has too many potons but enegy of the poduct nucleus (also called the daughte) that would be poduced by positon emission is less than about 1 MeV below the paent nucleus, electon captue is the only way the poton atio can be adjusted. This is most common in heavy elements such as 83 Rb. In these elements, the oute shell electons tend to compact the inne electon obitals so that the innemost electons spend a lot of time inside the nucleus itself, making captue that much easie. This competing pocess is undesiable fom an imaging standpoint, since the PET scanne is looking fo positons athe than evidence of electon captue (abbeviated as EC). Afte EC has occued, thee will of couse be an inne shell vacancy, so an electon fom a highe shell will quickly move to fill it. γ Decay The othe common decay mode is γ decay. If two nuclei have the same numbe of neutons and potons but diffeent enegies (i.e., one is in an excited state), they ae called isomes and the excited state is usually epesented by putting an m fo metastable afte the A supescipt: Tc 99m 99 Tc γ The neutons and potons in a nucleus can be thought of as occupying enegy levels o shells, just as the electons aound an atom do. The majo diffeence is that, while electonic tansitions may involve enegies of a few ev, nuclea tansitions typically involve enegies of millions of ev. If the nucleus ends up in an excited state (usually as a esult of a pevious α o β decay), it will emit a γ to get back to its gound state. The γ can escape completely o it can inteact with one of the atomic electons on the way out. If it hits one of the electons, it will cetainly ionize it, and this pocess is called intenal convesion. This is analogous to the poduction of an Auge electon when X-ays ae involved. In each case, the photon can eithe leave without inteacting o it can give all its enegy to an electon, ionizing it in the pocess. Fo imaging puposes, giving the enegy to an electon is not good. Ou equipment is designed to catch photons, and any competing pocess will only add to the patient s adiation dose without aiding image quality. Gamma ays ae, in geneal, the most penetating of the thee types of adiation we have seen. Depending on the enegy of the photon, the γ can penetate lage thicknesses (metes, in some cases) of concete o steel. 19
Nuclea Models The two pimay models used to descibe events within the nucleus ae the shell model and the liquid dop model. The liquid dop model teats the nucleus as if it wee a dop of wate. Thee ae competing foces in an actual dop of wate thee is suface tension, which aises because molecules at the suface of the dop (analogous to nucleons at the exteio of the nucleus) only feel attactive foces fom thei neighbos, but feel nothing on thei exposed sides. This semi-empiical fomula (found in the Eisbeg and Resnick book in the bibliogaphy) fo nuclea stability has six tems and calculates the mass of the nucleus. Fom that, the mass defect can quickly be found, and we know that the binding enegy is equal to the mass defect multiplied by c. The fist tem in this fomula is theefoe just the mass of the constituent nucleons. Next, the volume tem that pedicts lage nuclei ae moe stable - it can be witten as -a 1 A whee a 1 is a constant and A is the numbe of nucleons ( N Z). The suface tension tem is a coection to this oveestimate of binding enegy which assumes evey nucleon is suounded by neighbos. This is theefoe a positive tem, educing binding enegy. Since the nuclea adius is popotional to A 1/3, the suface aea is popotional to A /3, so we have a tem that can be witten as a A /3. Thee is also a tem epesenting the Coulomb epulsion of the potons. In geneal, the electostatic epulsion enegy could be witten as q 1 q /. A nucleus with Z potons will then have a epulsive enegy of a 3 Z /A 1/3. The fact that nuclei geneally pefe to have Z N, at least fo Z and N elatively small, is eflected in the next tem which is a 4 (Z A/) /A. The squae means that this tem is always positive, educing nuclea stability as it inceases. If ZN, though, it will disappea. Finally, thee is a tem that epesents the pefeence fo even values of Z and N. This tem is zeo if one of the two (Z o N) is odd and one is even. If both ae even, the tem is negative, inceasing the binding enegy. If both ae odd, the tem is positive, deceasing the binding enegy. It usually fits the data best if it is multiplied by A -1/. This model gives easonably good esults in many cases. The Shell Model The shell model shaes some featues with the standad atomic model which is so successful at explaining the peiodic table it assumes that thee ae enegy levels (shells) which can only hold a cetain numbe of electons, and an atom with a filled oute shell is moe stable chemically than one with an unfilled shell. Similaly, thee ae cetain special numbes (known as magic numbes ) of neutons o potons that make a nucleus less likely to decay o incease its lifetime when compaed to nuclei with non-magic numbes of neutons o potons. A nucleus with a magic 0
numbe of both neutons and potons is known as doubly magic and is geneally vey stable. The fist example of this is the helium nucleus, which has neutons and potons. As in the electonic shell case, is a magic numbe, and helium is so stable that it is common fo an out-of-balance nucleus to eject a whole helium nucleus (alpha decay) athe than just pat of one. In geneal, even numbes of neutons and even numbes of potons ae moe stable than odd numbes (this would help explain the ealie gaph of elemental abundances). The magic numbes ae, 8, 0, 8, 50, 8, and 16. Examples of doubly-magic isotopes would be 4 He, 16 O, 40 Ca, 56 Ni, etc. The shell model is based on the stong inteaction between obital and intinsic angula momentum in the nucleus (sometimes known as L S coupling). Quaks, etc. A moe complete explanation of the nuclea inteio was fomulated in the 1960 s- 1970 s. The nucleons wee no longe believed to be fundamental paticles, but wee thought to be composed of smalle paticles, known as quaks. Thee ae thee families of quaks, with each family containing a pai of quaks. These families ae u d c s t b The uppe quak in each family has a chage of e/3 and the lowe quak has a chage of e/3. Notice that the fundamental unit of electic chage, which has always been e, is now appaently e/3. As it tuns out, these factional chages ae neve obseved. The quaks ae always found eithe in goups of thee (as in the neuton and poton) o in quak-antiquak pais known as mesons (the diffeent π paticles mentioned ealie ae mesons). When thee quaks ae bound togethe, we call the esulting paticle a bayon. Potons and neutons ae bayons, with the poton being composed of two up quaks and a down quak while the neuton is a pai of down quaks and an up quak. Consevation of bayon numbe is one of the things that is equied in nuclea decays, such as the tansfomation of a neuton to a poton in β decay. The foce that holds the quaks togethe as mesons o bayons is known as the stong nuclea foce, and it is the stongest of the fou fundamental foces. It is sometimes also called the colo foce. The eason is that quaks, in addition to having one of the six flavos above, will also have one of thee colos (typically chosen to be ed, geen, and blue). Of couse, these things which ae fa smalle than the wavelengths of visible light don t have colo in the sense we e used to thinking of. The eason this popety is called colo is because you can conside all obsevable paticles to be cololess, o white. It takes ed, geen, and blue togethe to make white, so each quak in a bayon will have a diffeent colo. In the mesons, the quak will have one colo (ed, fo 1
example), and the antiquak will have the opposite colo (antied), leaving a cololess meson. Just as we thought of neutons and potons changing identities when they exchange pions, we can think of quaks changing colos when they exchange gluons. These ae the caies of the colo foce, just as photons ae the caies of the electomagnetic foce. Thee ae quite a few vaieties of gluon, since they can be things like edantigeen, blue-antied, geen-antiblue, etc. The nuclea foce that we have been talking about up until now is eally a kind of leftove of the colo foce holding individual nucleons togethe. It s simila to the way that van de Waals foces hold some molecules togethe, when they ae eally only the esiduals of the electomagnetic foce holding the individual atoms togethe. The colo foce has a shot ange, but it also exhibits something called asymptotic feedom. This woks sot of like the way a leash woks fo a dog: ove a cetain ange, it exets (almost) no foce on the dog, but at the limit of the leash, the foce suddenly gets vey lage. This is one of the poblems involved in a quantum theoy of the colo foce, known as quantum chomodynamics. In quantum electodynamics, which descibes how the photon mediates the foce between chages, the foce dies off with distance, which means a powe expansion is useful since each highe-ode tem should be less impotant than the pevious one. If the foce effectively inceases with distance, the highe-ode tems won t disappea. In an effot to clea up the vaious names used fo the paticles so fa, we can make the chat below:
Anti-bayons (with bayon numbe -1) and anti-mesons would just eplace quaks with antiquaks. The anti-poton, theefoe, is made of two anti-up quaks and one anti-down quak. The antipaticles of mesons ae just othe mesons, since the neutal pion (π 0 ) is its own antipaticle and the π and π - ae each othe s antipaticle. Leptons The leptons ae (in geneal) the lighte matte paticles. Again, thee ae thee families of lepton, and they ae shown below: e ν e µ ν τ µ ν τ Each of the uppe leptons has a chage of e and each of the lowe leptons (the neutinos) is electically neutal. All of these have a lepton numbe of 1. Each family has lage masses than the family to its left. Fo this eason, we can think of the µ and τ paticles as heavy electons. The neutinos, though, ae vey odd paticles. They have only the tiniest chance of inteacting with matte, and almost evey neutino poduced by the Sun and aimed at the Eath will pass though the whole planet without inteacting. Afte many yeas of caeful measuement, it has been detemined that neutinos have est masses well unde 1 ev (the electon, the smallest non-neutino matte paticle, has a mass of 511,000 ev). All quaks and leptons have antipaticles, and the neutinos ae no exception. Just as antibayons have a bayon numbe of -1, antileptons have a lepton numbe of -1. The leptons don t feel the colo foce, but they do feel anothe nuclea foce, known as the weak nuclea foce. This foce is esponsible fo beta decay, among othe things. The messenge paticles fo this foce (like the gluon fo the colo foce and the photon fo the EM foce) ae the W, W -, and Z 0. Notice that the W paticles ae chaged, meaning they will also paticipate in the EM inteaction. In fact, physics now consides the fou fundamental foces (stong, weak, EM, and gavity) to have been patially unified into the stong, electoweak, and gavitational foces. One of the ultimate goals of physics is to continue this pocess of unification of the foces until all foces can be undestood as elated aspects of one foce, just as electicity and magnetism ae no longe consideed to be sepaate foces but ae now undestood to be pats of the electomagnetic foce. We can now take a close look at the β decay of a neuton. Instead of just n p e ν e 3