14.16 Draft: Par­ity Data

The par­ity of a nu­cleus is even, or one, if its wave func­tion stays the same if the pos­i­tive di­rec­tion of all three Carte­sian axes is in­verted. That re­places every ${\skew0\vec r}$ in the wave func­tion by $\vphantom{0}\raisebox{1.5pt}{$-$}$${\skew0\vec r}$. The par­ity is odd, or mi­nus one, if the wave func­tion gets mul­ti­plied by $\vphantom{0}\raisebox{1.5pt}{$-$}$1 un­der axes in­ver­sion. Nu­clei have def­i­nite par­ity, (as long as the weak force is not an ac­tive fac­tor), so one of the two must be the case. It is an im­por­tant quan­tity for what nu­clear de­cays and re­ac­tions oc­cur and at what rate.

This sec­tion pro­vides an overview of the ground-state spins of nu­clei. It will be seen that the shell model does a pretty good job of pre­dict­ing them.

14.16.1 Draft: Even-even nu­clei

For nu­clei with both an even num­ber of pro­tons and an even num­ber of neu­trons, the odd-par­ti­cle shell model pre­dicts that the par­ity is even. This pre­dic­tion is fully vin­di­cated by the ex­per­i­men­tal data, fig­ure 14.37. There are no known ex­cep­tions to this rule.

Fig­ure 14.37: Par­ity of even-even nu­clei. [pdf][con]

14.16.2 Draft: Odd mass num­ber nu­clei

For nu­clei with an odd mass num­ber $A$, there is an odd pro­ton or neu­tron. The odd-par­ti­cle shell model says that the par­ity is that of the odd nu­cleon. To find it, the sub­shell that the last par­ti­cle is in must be iden­ti­fied, sec­tion 14.12.2. This can be done with a fair amount of con­fi­dence based on the spin of the nu­clei. Nu­clei for which the par­ity is cor­rectly pre­dicted in this way are shown in green in fig­ures 14.38 and 14.39. Fail­ures are in red. Small grey signs are shell model val­ues if the nu­cle­ons fill the shells in the nor­mal or­der.

Fig­ure 14.38: Par­ity of even-odd nu­clei. [pdf][con]

Fig­ure 14.39: Par­ity of odd-even nu­clei. [pdf][con]

The fail­ures above $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 50 and in­side the $Z$ $\raisebox{.3pt}{$<$}$ 82, $N$ $\raisebox{.3pt}{$>$}$ 82 wedge are ex­pected. The shell model does not ap­ply in these re­gions, be­cause the nu­clei are known to be non­spher­i­cal there. Be­sides that, there are very few fail­ures. Those near the $N$ $\vphantom0\raisebox{1.5pt}{$=$}$ 40 and $N$ $\vphantom0\raisebox{1.5pt}{$=$}$ 60 lines away from the sta­ble line are pre­sum­ably also due to non­spher­i­cal nu­clei. The highly un­sta­ble ni­tro­gen-11 and beryl­lium-11 mir­ror nu­clei were dis­cussed in sec­tion 14.12.6.

14.16.3 Draft: Odd-odd nu­clei

For odd-odd nu­clei, the odd-par­ti­cle shell model pre­dicts that the par­ity is the prod­uct of those of the sur­round­ing even-odd and odd-even nu­clei. The re­sults are shown in fig­ure 14.40. Hits are green, fail­ures red, and un­able-to-tell black. Small grey signs are shell model val­ues for the sur­round­ing even-odd and odd-even nu­clei. How­ever ac­tual even-odd and odd-even val­ues were used in the pre­dic­tion.

Fig­ure 14.40: Par­ity of odd-odd nu­clei. [pdf][con]

Fail­ures for spher­i­cal nu­clei in­di­cate that some­times the odd pro­ton or neu­tron is in a dif­fer­ent shell than in the cor­re­spond­ing odd-mass neigh­bors. A sim­i­lar con­clu­sion can be reached based on the spin data.

Note that the pre­dic­tions also do a fairly good job in the re­gions in which the nu­clei are not spher­i­cal. The rea­son is that the pre­dic­tions make no as­sump­tions about what sort of state, spher­i­cal or non­spher­i­cal, the odd nu­cle­ons are in. It merely as­sumes that they are in the same state as their neigh­bors.

14.16.4 Draft: Par­ity Sum­mary

Fig­ure 14.41 shows a sum­mary of the par­ity of all nu­clei to­gether. To iden­tify the type of nu­cleus more eas­ily, the even-even nu­clei have been shown as green check marks. The odd-odd nu­clei are found on the same ver­ti­cal lines as the check marks. The even-odd nu­clei are on the same hor­i­zon­tal lines as the check marks, and the odd-even ones on the same di­ag­o­nal lines.

Fig­ure 14.41: Par­ity ver­sus the shell model. [pdf][con]

Par­i­ties that the shell model pre­dicts cor­rectly are in green, and those that it pre­dicts in­cor­rectly are in red. The par­i­ties were taken straight from sec­tion 14.12.2 with no tricks. Note that the shell model does get a large num­ber of par­i­ties right straight off the bat. And much of the er­rors can be ex­plained by pro­mo­tion or non­spher­i­cal nu­clei.

For par­i­ties in light green and light red, NUBASE 2003 ex­pressed some reser­va­tion about the cor­rect value. For par­i­ties shown as yel­low crosses, no (unique) value was given.