14.7 Draft: Bind­ing en­ergy

The bind­ing en­ergy of a nu­cleus is the en­ergy that would be needed to take it apart into its in­di­vid­ual pro­tons and neu­trons. Bind­ing en­ergy ex­plains the over­all trends in nu­clear re­ac­tions.

Fig­ure 14.4: Bind­ing en­ergy per nu­cleon. [pdf][con]
...makebox(0,0)[l]{$\fourIdx{62}{28}{}{}{\rm Ni}$}}

As ex­plained in the pre­vi­ous sec­tion, the bind­ing en­ergy $E_{\rm {B}}$ can be found from the mass of the nu­cleus. The spe­cific bind­ing en­ergy is de­fined as the bind­ing en­ergy per nu­cleon, $E_{\rm {B}}$$\raisebox{.5pt}{$/$}$$A$. Fig­ure 14.4 shows the spe­cific bind­ing en­ergy of the nu­clei with known masses. The high­est spe­cific bind­ing en­ergy is 8.8 MeV, and oc­curs for ${}\fourIdx{62}{28}{}{}{\rm {Ni}}$ nickel. Nickel has 28 pro­tons, a magic num­ber. How­ever, non­magic ${}\fourIdx{58}{26}{}{}{\rm {Fe}}$ and ${}\fourIdx{56}{26}{}{}{\rm {Fe}}$ are right on its heels.

Nu­clei can there­fore lower their to­tal en­ergy by evolv­ing to­wards the nickel-iron re­gion. Light nu­clei can “fu­sion” to­gether into heav­ier ones to do so. Heavy nu­clei can emit al­pha par­ti­cles or fis­sion, fall apart in smaller pieces.

Fig­ure 14.4 also shows that the bind­ing en­ergy of most nu­clei is roughly 8 MeV per nu­cleon. How­ever, the very light nu­clei are an ex­cep­tion; they tend to have a quite small bind­ing en­ergy per nu­cleon. In a light nu­cleus, each nu­cleon only ex­pe­ri­ences at­trac­tion from a small num­ber of other nu­cle­ons. For ex­am­ple, deu­terium only has a bind­ing en­ergy of 1.1 MeV per nu­cleon.

The big ex­cep­tion to the ex­cep­tion is the dou­bly magic ${}\fourIdx{4}{2}{}{}{\rm {He}}$ nu­cleus, the al­pha par­ti­cle. It has a stun­ning 7.07 MeV bind­ing en­ergy per nu­cleon, ex­ceed­ing its im­me­di­ate neigh­bors by far.

The ${}\fourIdx{8}{4}{}{}{\rm {Be}}$ beryl­lium nu­cleus is not bad ei­ther, also with 7.07 MeV per nu­cleon, al­most ex­actly as high as ${}\fourIdx{4}{2}{}{}{\rm {He}}$, though ad­mit­tedly that is achieved us­ing eight nu­cle­ons in­stead of only four. But clearly, ${}\fourIdx{8}{4}{}{}{\rm {Be}}$ is a lot more tightly bound than its im­me­di­ate neigh­bors.

It is there­fore ironic that while var­i­ous of those neigh­bors are sta­ble, the much more tightly bound ${}\fourIdx{8}{4}{}{}{\rm {Be}}$ is not. It falls apart in about 67 as (i.e. 67 10$\POW9,{-18}$ s), a tragic con­se­quence of be­ing able to come neatly apart into two al­pha par­ti­cles that are just a tiny bit more tightly bound. It is the only al­pha de­cay among the light nu­clei. It is an ex­cep­tion to the rule that light nu­clei pre­fer to fu­sion into heav­ier ones.

But de­spite its im­mea­sur­ably short half-life, do not think that ${}\fourIdx{8}{4}{}{}{\rm {Be}}$ is not im­por­tant. With­out it there would be no life on earth. Be­cause of the ab­sence of sta­ble in­ter­me­di­aries, the Big Bang pro­duced no el­e­ments heav­ier than beryl­lium, (and only trace amounts of that) in­clud­ing no car­bon. As Hoyle pointed out, the car­bon of life is formed in the in­te­rior of ag­ing stars when ${}\fourIdx{8}{4}{}{}{\rm {Be}}$ cap­tures a third al­pha par­ti­cle, to pro­duce ${}\fourIdx{12}{6}{}{}{\rm {C}}$, which is sta­ble. This is called the “triple al­pha process.” Un­der the ex­treme con­di­tions in the in­te­rior of col­laps­ing stars, given time this process pro­duces sig­nif­i­cant amounts of car­bon de­spite the ex­tremely short half-life of ${}\fourIdx{8}{4}{}{}{\rm {Be}}$. The process is far too slow to have oc­curred in the Big Bang, how­ever.

For ${}\fourIdx{12}{6}{}{6}{\rm {C}}$ car­bon, the su­pe­rior num­ber of nu­cle­ons has be­come big enough to over­come the dou­bly magic ad­van­tage of the three cor­re­spond­ing al­pha par­ti­cles. Car­bon-12’s bind­ing en­ergy is 7.68 MeV per nu­cleon, greater than that of al­pha par­ti­cles.