Question

Use the tunneling idea to explain why the decay rates for diffèerent alpha emitters can vary a large factor even though the energy of the alpha particles differ by only a few MeV's?

Answer 1

The laws of quantum mechanics deal with the probability of a system such as a nucleus or an atom being in any of its possible states or configurations at any given time. A fissionable system (uranium-238, for example) in its ground state (i.e., at its lowest excitation energy and with an elongation small enough that it is confined inside the fission barrier) has a small but finite probability of being in the energetically favoured configuration of two fission fragments. In effect, when this occurs, the system has penetrated the barrier by the process of quantum mechanical tunneling. This process is called spontaneous fission because it does not involve any outside influences. In the case of uranium-238, the process has a very low probability, requiring more than 10¹⁵ years for half of the material to be transformed (its so-called half-life) by this reaction. On the other hand, the probability for spontaneous fission increases dramatically for the heaviest nuclides known and becomes the dominant mode of decay for some—those having half-lives of only fractions of a second. In fact, spontaneous fission becomes the limiting factor that may prevent the formation of still heavier (super-heavy) nuclei.

You may won­der why there is so much er­ror in the the­o­ret­i­cal pre­dic­tions of the half life. Or why the the­ory seems to work so much bet­ter for even-even nu­clei than for oth­ers. A de­vi­a­tion by a fac­tor 2 000 like for bis­muth-209 seems an aw­ful lot, rough as the the­ory may be.

Some of the sources of in­ac­cu­racy are self-ev­i­dent from the the­o­ret­i­cal de­scrip­tion as given. In par­tic­u­lar, there is the al­ready men­tioned ef­fect of the value of $$r_1$$ . It is cer­tainly pos­si­ble to cor­rect for de­vi­a­tions from the Coulomb po­ten­tial near the nu­cleus by a suit­able choice of the value of $$r_1$$ . How­ever, the pre­cise value that kills off the er­ror is un­known, and un­for­tu­nately the re­sults strongly de­pend on that value. To fix this would re­quire an ac­cu­rate eval­u­a­tion of the nu­clear force po­ten­tial, and that is very dif­fi­cult. Also, the po­ten­tial of the elec­trons would have to be in­cluded. The al­pha par­ti­cle does reach a dis­tance of the or­der of a tenth of a Bohr ra­dius from the nu­cleus at the end of tun­nel­ing. The Bohr ra­dius is here taken to be based on the ac­tual nu­clear charge, not the hy­dro­gen one.

Also, the pic­ture of a rel­a­tively com­pact wave packet of the al­pha par­ti­cle rat­tling around as­sumes that that the size of that wave packet is small com­pared to the nu­cleus. That spa­tial lo­cal­iza­tion is as­so­ci­ated with in­creased un­cer­tainty in mo­men­tum, which im­plies in­creased en­ergy. And the ki­netic en­ergy of the al­pha par­ti­cle is not re­ally known any­way, with­out an ac­cu­rate value for the nu­clear force po­ten­tial.

A very ma­jor other prob­lem is the as­sump­tion that the fi­nal al­pha par­ti­cle and nu­cleus end up in their ground states. If ei­ther ends up in an ex­cited state, the en­ergy that the al­pha par­ti­cle has avail­able for es­cape will be cor­re­spond­ingly re­duced. Now the al­pha par­ti­cle will most cer­tainly come out in its ground state; it takes over 20 MeV to ex­cite an al­pha par­ti­cle. But for most nu­clei, the re­main­ing nu­cleus can­not be in its ground state if the mech­a­nism is as de­scribed.

The main rea­son is an­gu­lar mo­men­tum con­ser­va­tion. The al­pha par­ti­cle has no net in­ter­nal an­gu­lar an­gu­lar mo­men­tum. Also, it was as­sumed that the al­pha par­ti­cle comes out ra­di­ally, which means that there is no or­bital an­gu­lar mo­men­tum ei­ther. So the an­gu­lar mo­men­tum of the nu­cleus af­ter emis­sion must be the same as that of the nu­cleus be­fore the emis­sion. That is no prob­lem for even-even nu­clei, be­cause it is the same; even-even nu­clei all have zero in­ter­nal an­gu­lar mo­men­tum in their ground state. So even-even nu­clei do not suf­fer from this prob­lem.

How­ever, al­most all other nu­clei do. All even-odd and odd-even nu­clei and al­most all odd-odd ones have nonzero an­gu­lar mo­men­tum in their ground state. Usu­ally the ini­tial and fi­nal nu­clei have dif­fer­ent val­ues. That means that al­pha de­cay that leaves the fi­nal nu­cleus in its ground state vi­o­lates con­ser­va­tion of an­gu­lar mo­men­tum. The de­cay process is called “for­bid­den.” The fi­nal nu­cleus must be ex­cited if the process is as de­scribed. That en­ergy sub­tracts from that of the al­pha par­ti­cle. There­fore the al­pha par­ti­cle has less en­ergy to tun­nel through, and the true half-life is much longer than com­puted.

Note in the bot­tom half of fig­ure 14.12 how many nu­clei that are not even-even do in­deed have half-lifes that are or­ders of mag­ni­tude larger than pre­dicted by the­ory. Con­sider the ex­am­ple of bis­muth-209, with a half-life 2 000 times longer than pre­dicted. Bis­muth-209 has a spin, i.e. an az­imuthal quan­tum num­ber, of $$\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 9}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$$ . How­ever, the de­cay prod­uct thal­lium-205 has spin $$\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$$ in its ground state. If you check out the ex­cited states of thal­lium-205, there is an ex­cited state with spin $$\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 9}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$$ , but its ex­ci­ta­tion en­ergy would re­duce the en­ergy of the al­pha par­ti­cle from 3.2 MeV to 1.7 MeV, mak­ing the tun­nel­ing process very much slower.

One fi­nal source of er­ror should be men­tioned. Of­ten al­pha de­cay can pro­ceed in a num­ber of ways and to dif­fer­ent fi­nal ex­ci­ta­tion en­er­gies. In that case, the spe­cific de­cay rates must be added to­gether. This ef­fect can make the true half-life shorter than the one com­puted in the pre­vi­ous sub­sec­tion. But clearly, In­deed, while the pre­dicted half-lifes of many nu­clei are way be­low the true value in the fig­ure, few are sig­nif­i­cantly above it.