14.14 Draft: Fis­sion

In spon­ta­neous fis­sion, a very heavy nu­cleus falls apart into big frag­ments. If there are two frag­ments, it is called bi­nary fis­sion. In some cases, there are three frag­ments. That is called ternary fis­sion; the third frag­ment is usu­ally an al­pha par­ti­cle. This sec­tion sum­ma­rizes some of the ba­sic ideas.

14.14.1 Draft: Ba­sic con­cepts

What makes fis­sion en­er­get­i­cally pos­si­ble is that very heavy nu­clei have less bind­ing en­ergy per nu­cleon than those in the nickel/iron range, as shown ear­lier in fig­ure 14.4. The main cul­prit is the Coulomb re­pul­sion be­tween the pro­tons. It has a much longer range than the nu­clear at­trac­tions be­tween nu­cle­ons. There­fore, Coulomb re­pul­sion dis­pro­por­tion­ally in­creases the en­ergy for heavy nu­clei. If a nu­cleus like ura­nium-238 di­vides cleanly into two pal­la­dium-119 nu­clei, the en­ergy lib­er­ated is on the or­der of 200 MeV (200 000 000 eV). That is ob­vi­ously a very large amount of en­ergy. Chem­i­cal re­ac­tions pro­duce maybe a few eV per atom.

The liq­uid drop model pre­dicts that the nu­clear shape will be­come un­sta­ble at $Z^2$$\raisebox{.5pt}{$/$}$​$A$ about equal to 48. How­ever, only the weird­est nu­clei like ${}\fourIdx{293}{118}{}{}{\rm {Ei}}$ come close to that value. Be­low $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 100 the nu­clei that de­cay pri­mar­ily through spon­ta­neous fis­sion are curium-250, with $Z^2$$\raisebox{.5pt}{$/$}$​$A$ equal to 37 and a half life of 8 300 years, cal­i­fornium-254, 38 and two months, and fer­mium-256, 39 and less than 3 hours.

In­deed, while the fis­sion prod­ucts may have lower en­ergy than the orig­i­nal nu­cleus, in tak­ing the nu­cleus apart, the nu­clear bind­ing en­ergy must be pro­vided right up front. On the other hand the Coulomb en­ergy gets re­cov­ered only af­ter the frag­ments have been brought far apart. As a re­sult, there is nor­mally a en­ergy bar­rier that must be crossed for the nu­cleus to come apart. That means that an “ac­ti­va­tion en­ergy” must be pro­vided in nu­clear re­ac­tions, much like in most chem­i­cal re­ac­tions.

For ex­am­ple, ura­nium has an ac­ti­va­tion en­ergy of about 6.5 MeV. By it­self, ura­nium-235 will last a bil­lion years or so. How­ever, it can be made to fis­sion by hit­ting it with a neu­tron that has only a ther­mal amount of en­ergy. (Zero is enough, ac­tu­ally.) When hit, the nu­cleus will fall apart into a cou­ple of big pieces and im­me­di­ately re­lease an av­er­age of 2.4 “prompt neu­trons.” These new neu­trons al­low the process to re­peat for other ura­nium-235 nu­clei, mak­ing a “chain re­ac­tion” pos­si­ble.

In ad­di­tion to prompt neu­trons, fu­sion processes may also emit a small frac­tion of “de­layed neu­trons” neu­trons some­what later. De­spite their small num­ber, they are crit­i­cally im­por­tant for con­trol­ling nu­clear re­ac­tors be­cause of their slower re­sponse. If you tune the re­ac­tor so that the pres­ence of de­layed neu­trons is es­sen­tial to main­tain the re­ac­tion, you can con­trol it me­chan­i­cally on their slower time scale.

Re­turn­ing to spon­ta­neous fis­sion, that is pos­si­ble with­out the need for an ac­ti­va­tion en­ergy through quan­tum me­chan­i­cal tun­nel­ing. Note that this makes spon­ta­neous fis­sion much like al­pha de­cay. How­ever, as sec­tion 14.11.2 showed, there are def­i­nite dif­fer­ences. In par­tic­u­lar, the ba­sic the­ory of al­pha de­cay does not ex­plain why the nu­cleus would want to fall apart into two big pieces, in­stead of one big piece and a small al­pha par­ti­cle. This can only be un­der­stood qual­i­ta­tively in terms of the liq­uid drop model: a charged clas­si­cal liq­uid drop is most un­sta­ble to large-scale de­for­ma­tions, not small scale ones, sub­sec­tion 14.13.1.

14.14.2 Draft: Some ba­sic fea­tures

While fis­sion is qual­i­ta­tively close to al­pha de­cay, its ac­tual me­chan­ics is much more com­pli­cated. It is still an area of much re­search, and be­yond the scope of this book. A very read­able de­scrip­tion is given by [36]. This sub­sec­tion de­scribes some of the ideas.

From a va­ri­ety of ex­per­i­men­tal data and their in­ter­pre­ta­tion, the fol­low­ing qual­i­ta­tive pic­ture emerges. Vi­su­al­ize the nu­cleus be­fore fis­sion as a clas­si­cal liq­uid drop. It may al­ready be de­formed, but the de­formed shape is clas­si­cally sta­ble. To fis­sion, the nu­cleus must de­form more, which means it must tun­nel through more de­formed states. When the nu­cleus has de­formed into a suf­fi­ciently elon­gated shape, it be­comes en­er­get­i­cally more fa­vor­able to re­duce the sur­face area by break­ing the con­nec­tion be­tween the ends of the nu­cleus. The con­nec­tion thins and even­tu­ally breaks, leav­ing two sep­a­rate frag­ments. Dur­ing the messy process in which the thin con­nec­tion breaks an al­pha par­ti­cle could well be ejected. Now typ­i­cal heavy nu­clei con­tain rel­a­tively more neu­trons than lighter ones. So when the sep­a­rated frag­ments take in­ven­tory, they find them­selves overly neu­tron-rich. They may well find it worth­while to eject one or two right away. This does not change the strong mu­tual Coulomb re­pul­sion be­tween the frag­ments, and they are pro­pelled to in­creas­ing speed away from each other.

Con­sider now a very sim­ple model in which a nu­cleus like fer­mium-256 falls cleanly apart into two smaller nu­clear frag­ments. As a first ap­prox­i­ma­tion, ig­nore neu­tron and other en­ergy emis­sion in the process and ig­nore ex­ci­ta­tion of the frag­ments. In that case, the fi­nal ki­netic en­ergy of the frag­ments can be com­puted from the dif­fer­ence be­tween their masses and the mass of the orig­i­nal nu­cleus.

In the fis­sion process, the frag­ments sup­pos­edly pick up this ki­netic en­ergy from the Coulomb re­pul­sion be­tween the sep­a­rated frag­ments. If it is as­sumed that the frag­ments are spher­i­cal through­out this fi­nal phase of the fis­sion process, then its prop­er­ties can be com­puted. In par­tic­u­lar, it can be com­puted at which sep­a­ra­tion be­tween the frag­ments the ki­netic en­ergy was zero. That is im­por­tant be­cause it in­di­cates the end of the tun­nel­ing phase. Putting in the num­bers, it is seen that the sep­a­ra­tion be­tween the frag­ments at the end of tun­nel­ing is at least 15% more than that at which they are touch­ing. So the model is at least rea­son­ably self-con­sis­tent.

Fig­ure 14.30: Sim­pli­fied en­er­get­ics for fis­sion of fer­mium-256. [pdf][con]

Fig­ure 14.30 shows the en­er­get­ics of this model. In­creas­ing red­ness in­di­cates in­creas­ing en­ergy re­lease in the fis­sion. Also, the spac­ing be­tween the squares in­di­cates the spac­ing be­tween the nu­clei at the point where tun­nel­ing ends. Note in par­tic­u­lar the dou­bly magic point of 50 pro­tons and 82 neu­trons. This point is very neu­tron rich, just what is needed for fis­sion frag­ments. And be­cause it is dou­bly magic, nu­clei in this vicin­ity have un­usu­ally high bind­ing en­ergy, as seen from fig­ure 14.9. In­deed, nu­clei with 50 pro­tons are seen to have the high­est fis­sion en­ergy re­lease in fig­ure 14.30. Also, they have the small­est rel­a­tive spac­ing be­tween the nu­clei at the end of tun­nel­ing, so likely the short­est rel­a­tive dis­tance that must be tun­neled through. The con­clu­sion is clear. The log­i­cal thing for fer­mium-256 to do is to come apart into two al­most equal frag­ments with a magic num­ber of 50 pro­tons and about 78 neu­trons, giv­ing the frag­ments a mass num­ber of 128. Less plau­si­bly, one frag­ment could have the magic num­ber of 82 neu­trons, giv­ing frag­ment mass num­bers of 132 and 124. But the most un­sta­ble de­for­ma­tion for the liq­uid drop model is sym­met­ric. And so is a spher­oidal or el­lip­soidal model for the de­formed nu­cleus. It all seems to add up very nicely. The frag­ments must be about the same size, with a mass num­ber of 128.

Ex­cept that that is all wrong.

Fer­mium 258 acts like that, and fer­mium-257 also mostly, but not fer­mium 256. It is rare for fer­mium-256 to come apart into two frag­ments of about equal size. In­stead, the most likely mass num­ber of the large frag­ment is about 140, with only a small prob­a­bil­ity of a mass num­ber 132 or lower. A mass num­ber of 140 clearly does not seem to make much sense based on fig­ure 14.30.

The pre­cise so­lu­tion to this rid­dle is still a mat­ter of cur­rent re­search, but physi­cists have iden­ti­fied quan­tum ef­fects as the pri­mary cause. The po­ten­tial en­ergy bar­rier that the fis­sion­ing nu­cleus must pass through is rel­a­tively low, on the or­der of say 5 MeV. That is cer­tainly small enough to be sig­nif­i­cantly af­fected by quan­tum shell ef­fects. Based on that idea, you would ex­pect that mass asym­me­try would de­crease if the ex­ci­ta­tion en­ergy of the nu­cleus is in­creased, and such an ef­fect is in­deed ob­served. Also, the sep­a­ra­tion of the frag­ments oc­curs at very low en­ergy, and is be­lieved to be slow enough that the frag­ments can de­velop some shell struc­ture. Physi­cists have found that for many fis­sion­ing nu­clei, quan­tum shell ef­fects can cre­ate a rel­a­tively sta­ble in­ter­me­di­ate state in the fis­sion process. Such a state pro­duces res­o­nances in re­sponse to spe­cific ex­ci­ta­tion en­er­gies of the nu­cleus. Shell cor­rec­tions can also lower the en­ergy of asym­met­ric nu­clear fis­sion­ing shapes be­low those of sym­met­ric ones, pro­vid­ing an ex­pla­na­tion for the mass asym­me­try.

Imag­ine then a very dis­torted stage in which a neu­tron-rich, dou­bly magic 50/82 core de­vel­ops along with a smaller nu­clear core, the two be­ing con­nected by a cloud of neu­trons and pro­tons. Each could pick up part of the cloud in the fi­nal sep­a­ra­tion process. That pic­ture would ex­plain why the mass num­ber of the large frag­ment ex­ceeds 132 by a fairly con­stant amount while the mass num­ber of the smaller seg­ment varies with the ini­tial nu­clear mass. Whether or not there is much truth to this pic­ture, at least it is a good mnemonic to re­mem­ber the frag­ment masses for the nu­clei that fis­sion asym­met­ri­cally.