N.36 Draft: Cage-of-Fara­day pro­posal

The gi­gan­tic er­rors in the­o­ret­i­cal half-life pre­dic­tions in sec­tion 14.20.5 are dis­con­cert­ing to say the least. They im­ply that the pre­dicted gamma de­cay rates, (es­sen­tially the in­verse of the half-lifes), are typ­i­cally ei­ther much less than the­ory or much larger than the­ory.

To ex­plain why some of the gamma de­cay rates, like the E2 and high en­ergy E3 ones, are so much faster than ball­park is rel­a­tively straight­for­ward. De­cay much faster than ball­park is only pos­si­ble if not just one pro­ton, but a lot of nu­cle­ons par­tic­i­pate in the tran­si­tion. And since the ef­fect is sys­tem­atic in E2 and high en­ergy E3 tran­si­tions, ap­par­ently it is nor­mal for a lot of nu­cle­ons to par­tic­i­pate in gamma de­cay. Or at least it is for these types of gamma de­cay. And since the the­ory as­sumes that only one pro­ton par­tic­i­pates, the mis­er­able pre­dic­tions of the­ory can be ex­plained.

A much big­ger prob­lem is to ex­plain why other tran­si­tions end up so far be­low ball­park in a cred­i­ble way. Con­sider in par­tic­u­lar the ${\rm {E1}}$ tran­si­tions in fig­ures 14.63 and 14.65. How come that they are not just oc­ca­sion­ally, but typ­i­cally slower than the­ory by four or­ders of mag­ni­tude?

Ba­si­cally, you can give two rea­son­able types of ex­pla­na­tion:

1.

You can as­sume that only one pro­ton par­tic­i­pates in these tran­si­tion, not many as in E2 and E3 tran­si­tions. Then you must as­sume that in ad­di­tion there is a sys­tem­atic very poor over­lap be­tween the ini­tial and fi­nal states in the rel­e­vant in­ner prod­uct. We do not re­ally know the ini­tial and fi­nal states, so, why not? Prob­lem solved. Next ques­tion?

This is es­sen­tially the ex­pla­na­tion that ba­sic nu­clear text­books that the au­thor has seen give. Un­for­tu­nately, there are two big prob­lems with it. First, how come that un­like the E2 and E3 tran­si­tions, sud­denly only one pro­ton par­tic­i­pates in E1, low en­ergy E3, M1 and most M2 tran­si­tions? If there is a very big sys­tem­atic ef­fect, there must be a rea­son. Worse, fig­ure 14.65 seems to ex­clude the pos­si­bil­ity of just one pro­ton par­ti­pat­ing in at the very least M1 tran­si­tions, and surely at least some very slow E1 tran­si­tions.

The sec­ond prob­lem is to ex­plain why the over­lap is sys­tem­at­i­cally ex­tremely bad in some types of tran­si­tions, but ap­par­ently ex­cel­lent in oth­ers. Again, this is a big sys­tem­atic ef­fect, as the fig­ures show. So there must be a rea­son for this too.

Your hands are not enough to wave these prob­lem away. If you want peo­ple to take you se­ri­ously, you should have a be­liev­able and com­pre­hen­sive dis­cus­sion.

2.

Al­ter­na­tively, you can as­sume that in the tran­si­tions that are much slower than the­ory, still many nu­cle­ons par­tic­i­pate. One im­me­di­ate ad­van­tage is, of course, that this would ex­plain why the the­ory per­forms mis­er­ably bad on these tran­si­tions too. And you do not have to ex­plain why in some tran­si­tions only a sin­gle pro­ton par­tic­i­pates while in oth­ers many nu­cle­ons do. In par­tic­u­lar, this re­moves fig­ure 14.65 as an is­sue. It can also ex­plain why for some very light nu­clei, E1 and M1 tran­si­tions are at ball­park or even no­tice­ably faster than ball­park.

But now of course, you face the ap­par­ently daunt­ing prob­lem of ex­plain­ing why some types tran­si­tions can be so ex­tremely slow, even now that there are many nu­cle­ons par­tic­i­pat­ing. In par­tic­u­lar, you need to pro­vide a rea­son­able ex­pla­na­tion why for some types of tran­si­tions, the par­tic­i­pa­tion of many nu­cle­ons ac­tu­ally slows down the emis­sion of elec­tro­mag­netic ra­di­a­tion greatly, rather than in­crease it greatly. And ap­par­ently, this ef­fect re­quires the pres­ence of enough nu­cle­ons, as very light nu­clei do not have the prob­lem.

While stan­dard nu­clear text­books give the first ex­pla­na­tion above, the holes in the ar­gu­ment are wor­ri­some. The mag­ni­tude of the ef­fects pointed at by the text­books just does not seem big enough to ex­plain the data. And it is hard to think up rea­sons why not. There just is a lack of sus­pen­sion of dis­be­lieve for an en­gi­neer think­ing in terms of ball­parks.

There­fore, this book wants to ar­gue that more se­ri­ous con­sid­er­a­tion should be given to the sec­ond ex­pla­na­tion. Its main li­a­bil­ity is to ex­plain why some tran­si­tions get slowed down greatly, rather than sped up, if a lot more nu­cle­ons par­tic­i­pate.

Since nu­clear wave func­tions are poorly un­der­stood, that would be dif­fi­cult to ex­plain from a quan­tum-me­chan­i­cal view­point. So maybe it is again time to do what has been done be­fore for nu­clei; look for macro­scopic mod­els. And surely the macro­scopic model that stands out in killing off elec­tro­mag­netic ef­fects is the cage of Fara­day. (In this case the cage is as­sumed to shield the out­side from the in­side.)

Maybe then the nu­clear sur­face acts as such a cage in some sense. In the liq­uid-drop idea, nu­cle­ons at the sur­face are in a state of in­creased en­ergy. So it may not be such a crazy idea that nu­cle­ons at the sur­face might be­have dif­fer­ently from nu­cle­ons in the in­te­rior.

As­sume now at first, in this macro­scopic model, that the nu­clear sur­face is spher­i­cal and con­duct­ing. Then elec­tric charge changes in the in­te­rior of the nu­cleus would not leak out. That would kill off the ca­pa­bil­ity of tran­si­tions to emit ra­di­a­tion. So the model can pro­vide a macro­scopic ex­pla­na­tion why some elec­tro­mag­netic tran­si­tions can be greatly slowed down. Charges can still move around in­side the nu­cleus, but be­cause the sur­face nu­cle­ons move to com­pen­sate, that does not pro­duce ra­di­a­tion out­side it. So par­tic­i­pa­tion of many nu­cle­ons does in­deed re­duce, rather than in­crease, elec­tro­mag­netic ra­di­a­tion greatly.

Do note that un­like nor­mal cages of Fara­day, a nu­cleus con­tains a net pos­i­tive charge. And if a macro­scopic cage of Fara­day con­tains a net charge, there must al­ways ne a nonzero elec­tric field out­side. (That is due to Maxwell's first law.) But as long as the sur­face re­mains spher­i­cal and con­duct­ing, the out­side elec­tric field will not change if charges in­side the sur­face are moved around. So in that case, there will be an elec­tric field, but still no elec­tro­mag­netic ra­di­a­tion ra­di­ated away. And the rea­son for that is still be­cause many nu­cle­ons are in­volved, rather than a sin­gle pro­ton.

But things change when the nu­clear sur­face changes shape. A con­duct­ing sur­face makes only the elec­tric field tan­gen­tial to the sur­face zero. There­fore there will be vari­a­tions in the elec­tric field out­side the sur­face if it changes shape. So now we have a sit­u­a­tion where ra­di­a­tion is in fact be­ing trans­mit­ted, and again with many nu­cle­ons in­volved in do­ing that.

That opens up the pos­si­bil­ity of ex­plain­ing why some tran­si­tions can be so far from the sin­gle pro­ton ball­park. And why it de­pends on the type of tran­si­tion whether the tran­si­tion turns out to be much slower than ball­park or much faster than ball­park.

At least for rel­a­tively light nu­clei, (but still with enough nu­cle­ons that the macro­scopic pic­ture makes sense), and small ex­ci­ta­tions, sur­face ten­sion would pro­mote a spher­i­cal sur­face. And sur­face rough­ness would not nec­es­sar­ily make much of a dif­fer­ence. That is just like small holes in a macro­scopic cage do not make a dif­fer­ence. The field out­side the nu­cleus is gov­erned by the so-called Laplace equa­tion. This equa­tion is known to kill off short-scale per­tur­ba­tions quickly.

On the other hand, changes in a de­formed nu­clear sur­face shape would def­i­nitely pro­duce non­triv­ial long-range elec­tric field per­tur­ba­tions. Now nu­clei are of­ten mod­eled as spher­oids or el­lip­soids. Changes in such a shape would pro­duce quadru­pole and hexa­de­ca­pole per­tur­ba­tions in the elec­tric field out­side the nu­clei. So they would pro­duce ${\rm {E2}}$ ra­di­a­tion, but not ${\rm {E1}}$ ra­di­a­tion. That is ex­actly what is needed to make some sense out of the elec­tric tran­si­tions.

While macro­scopic cages of Fara­day do not block sta­tic mag­netic fields, they do block changes in mag­netic fields. So con­cep­tu­ally the model could also ex­plain why mag­netic tran­si­tions of low mul­ti­pole or­der are of­ten so slow. Note that there is no net mag­netic charge in­side the cage. Mag­netic monopoles do not ex­ist. So sur­face shape would not nec­es­sar­ily af­fect mag­netic tran­si­tions much.

While this model leaves many ques­tions unan­swered, at least it sug­gests a rea­son­able way to un­der­stand how it is pos­si­ble at all that the one-pro­ton model is not just ex­tremely mis­er­able, but sys­tem­at­i­cally mis­er­able in the ob­served way. For one, it seems to make fig­ure 14.65 far less un­ex­plain­able.