14.15 Draft: Spin Data

The net in­ter­nal an­gu­lar mo­men­tum of a nu­cleus is called the nu­clear spin. It is an im­por­tant quan­tity for ap­pli­ca­tions such as NMR and MRI, and it is also im­por­tant for what nu­clear de­cays and re­ac­tions oc­cur and at what rate. One pre­vi­ous ex­am­ple was the cat­e­gor­i­cal re­fusal of bis­muth-209 to de­cay at the rate it was sup­posed to in sec­tion 14.11.3.

This sec­tion pro­vides an overview of the ground-state spins of nu­clei. Ac­cord­ing to the rules of quan­tum me­chan­ics, the spin must be in­te­ger if the to­tal num­ber of nu­cle­ons is even, and half-in­te­ger if it is odd. The shell model can do a pretty good job of pre­dict­ing ac­tual val­ues. His­tor­i­cally, this was one of the ma­jor rea­sons for physi­cists to ac­cept the va­lid­ity of the shell model.

14.15.1 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 spin is zero. This pre­dic­tion is fully vin­di­cated by the ex­per­i­men­tal data, fig­ure 14.31. There are no known ex­cep­tions to this rule.

Fig­ure 14.31: Spin of even-even nu­clei. [pdf][con]

14.15.2 Draft: Odd mass num­ber nu­clei

Nu­clei with an odd mass num­ber $A$ have ei­ther an odd num­ber of pro­tons or an odd num­ber of neu­trons. For such nu­clei, the odd-par­ti­cle shell model pre­dicts that the nu­clear spin is the net an­gu­lar mo­men­tum (or­bital plus spin) of the last odd nu­cleon. To find it, the sub­shell that the last par­ti­cle is in must be iden­ti­fied. That can be done by as­sum­ing that the sub­shells fill in the or­der given in sec­tion 14.12.2. This or­der­ing is in­di­cated by the col­ored lines in fig­ures 14.32 and 14.33. Nu­clei for which the re­sult­ing nu­clear spin pre­dic­tion is cor­rect are in­di­cated with a black check mark. (If the check mark is white rather than black, some reser­va­tion was ex­pressed about the mea­sured spin in NUBASE 2003.)

Fig­ure 14.32: Spin of even-odd nu­clei. [pdf][con]

Fig­ure 14.33: Spin of odd-even nu­clei. [pdf][con]

The pre­dic­tion is cor­rect right off the bat for a con­sid­er­able num­ber of nu­clei. That is very much non­triv­ial. Still, there is an even larger num­ber for which the pre­dic­tion is not cor­rect.

One ma­jor rea­son is that many heavy nu­clei are not spher­i­cal in shape. The shell model was de­rived as­sum­ing a spher­i­cal nu­clear shape and sim­ply does not ap­ply for such nu­clei. These are the non­spher­i­cal, ro­ta­tional nu­clei, the ones that are eas­ily ex­cited; the non­s­mall squares in 14.45, the red (or yel­low) big R squares in fig­ure 14.22, the small squares in fig­ure 14.19. Their main re­gions are for neu­tron num­ber $N$ above 132 and in the deep in­te­rior of the $Z$ $\raisebox{.3pt}{$<$}$ 82, $N$ $\raisebox{.3pt}{$>$}$ 82 wedge. For these nu­clei, the shell model sim­ply does not work.

For al­most all re­main­ing nu­clei near the sta­ble line, the spin can be ex­plained in terms of the shell model us­ing var­i­ous rea­son­able ex­cuses, [36, p. 224ff]. Some ex­cuses are marked.

In par­tic­u­lar, if the sub­shell with the odd neu­tron is above a sub­shell of lower spin, a par­ti­cle from the lower sub­shell may be pro­moted to the higher one. This par­ti­cle can then pair up at a higher spin, which is be­lieved to be en­er­get­i­cally fa­vor­able. Since an odd nu­cleon oc­curs now in the lower shell, the spin of the nu­cleus is pre­dicted to be the one of that shell. So the nu­clear spin is low­ered com­pared to the bare shell model.

Nu­clei for which such spin low­er­ing due to pro­mo­tion can ex­plain the ob­served spin are in­di­cated with an L or l in fig­ures 14.32 and 14.33. For the nu­clei marked with L, the odd nu­cleon can­not be in the nor­mal sub­shell be­cause the nu­cleus has the wrong par­ity for that. There­fore, for these nu­clei there is a solid ad­di­tional rea­son be­sides the spin to as­sume that pro­mo­tion has oc­curred.

Pro­mo­tion was only al­lowed for sub­shells im­me­di­ately above one with lower spin in the same ma­jor shell, so the nu­cleon could only be pro­moted a sin­gle sub­shell. The idea is that the gained pair­ing en­ergy should not be big enough to make ma­jor mod­i­fi­ca­tions to the shell model.

For some nu­clei, the ba­sic shell model is valid but the odd-par­ti­cle as­sump­tion fails. The odd par­ti­cle as­sump­tion im­plies that all nu­cle­ons ex­cept the odd one pair up in states of zero spin. But some­times at least three nu­cle­ons com­bine in a state with one unit less spin than the sin­gle odd par­ti­cle would have. This is only pos­si­ble for sub­shells with at least three par­ti­cles and three holes (empty spots for ad­di­tional nu­cle­ons). Nu­clei for which this unit spin re­duc­tion might to be the case are marked with a mi­nus sign. It is ev­i­dent, for ex­am­ple, for odd nu­cleon num­bers 23 and 25.

Flu­o­rine-19 and its mir­ror twin neon-19 are rare out­right fail­ures of the shell model, as al­ready dis­cussed in sec­tion 14.12.6.

Among the re­main­ing fail­ures, no­table are nu­clei with odd pro­ton num­bers just above 50. The 5g$_{7/2}$ and 5d$_{5/2}$ sub­shells are very close to­gether and it de­pends on the de­tails which one gets filled first.

In a few ex­cep­tional cases, like the highly un­sta­ble ni­tro­gen-11 and beryl­lium-11 halo nu­clei, the the­o­ret­i­cal model pre­dicted the right spin, but it was not counted as a hit be­cause the nu­clear par­ity was in­con­sis­tent with the pre­dicted sub­shell. In all cases, it was de­manded that the nu­clear par­ity, if known, did not con­flict with the pro­posed shell model ver­i­fi­ca­tion (or ex­pla­na­tion) of the spin.

For nu­clei marked with a cross, no ex­pla­na­tion for the spin us­ing the above rules could be found. In gen­eral, you might not want to take nu­clei well away from the sta­ble line that se­ri­ous. For yel­low squares, NUBASE 2003 gave ei­ther no value for the spin or more than one pos­si­ble value.

14.15.3 Draft: Odd-odd nu­clei

If both the num­ber of pro­tons and the num­ber of neu­trons is odd, the nu­clear spin be­comes much more dif­fi­cult to pre­dict. Ac­cord­ing to the odd-par­ti­cle shell model, the net nu­clear spin $j_{\rm N}$ comes from com­bin­ing the net an­gu­lar mo­menta $j^{\rm p}$ of the odd pro­ton and $j^{\rm n}$ of the odd neu­tron. Then ac­cord­ing to quan­tum me­chan­ics, the net nu­clear spin $j_{\rm N}$ can be any in­te­ger in the range

$$\vert j^{\rm p}-j^{\rm n}\vert \mathrel{\raisebox{-.7pt}{$\... ...m N}\mathrel{\raisebox{-.7pt}{$\leqslant$}}j^{\rm p}+j^{\rm n}$$ (14.25)

That gives a to­tal of $2\min(j^{\rm p},j^{\rm n})+1$ dif­fer­ent pos­si­bil­i­ties, or at least two.

That is not very sat­is­fac­tory of course. You would like to get a spe­cific pre­dic­tion for the spin, not a range. The so-called Nord­heim rules at­tempt to do so. The un­der­ly­ing idea is that nu­clei like to align the spins of the odd pro­ton and odd neu­tron, just like the deu­terium nu­cleus does.

To de­scribe the rules, for­get about quan­tum me­chan­ics for now. Just think of the spin and or­bital an­gu­lar mo­menta in­volved as sim­ple vec­tors that can ei­ther point up or down. And take up to be the di­rec­tion that the aligned pro­ton and neu­tron spin vec­tors $s^{\rm p}$ and $s^{\rm n}$ point. Now sup­pose first that for both neu­tron and pro­ton the or­bital an­gu­lar mo­menta $l^{\rm p}$ and $l^{\rm n}$ also point up­wards (or are zero). Then for both pro­ton and neu­tron, the or­bital and spin an­gu­lar mo­menta add up to a to­tal mo­men­tum $j^{\rm p}=l^{\rm p}+s^{\rm p}$ re­spec­tively $j^{\rm n}=l^{\rm n}+s^{\rm n}$ that also point up­wards. So the sum of the two, the to­tal nu­clear spin $j_{\rm N}$ points up­wards and has mag­ni­tude $j_{\rm N}=j^{\rm p}+j^{\rm n}$.

Con­versely, if both or­bital mo­menta point down­wards (and are nonzero), then spin and or­bital an­gu­lar mo­menta are in op­po­site di­rec­tions and sub­tract. Then pro­ton and neu­tron have net an­gu­lar mo­menta of mag­ni­tude $j^{\rm p}=l^{\rm p}-s^{\rm p}$ re­spec­tively $j^{\rm n}=l^{\rm n}-s^{\rm n}$ that point down­wards. (Re­call from quan­tum me­chan­ics that $l$ is at least 1 if nonzero, so is big­ger than the $s=\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ point­ing up­wards.) The com­bined nu­clear an­gu­lar mo­men­tum is then again $j_{\rm N}=j^{\rm p}+j^{\rm n}$ (point­ing down­wards).

How­ever, if for one nu­cleon the or­bital an­gu­lar mo­men­tum is zero or points in the di­rec­tion of the spin and the other is nonzero and points in the di­rec­tion op­po­site to the spin, then one of $j^{\rm p}$ and $j^{\rm n}$ points up­wards and the other down­wards. That then means that now they are in op­po­site di­rec­tions and sub­tract; $j_{\rm N}=\vert j^{\rm p}-j^{\rm n}\vert$.

That then gives the Nord­heim rules as, [36, p. 239]:

1.

If for both pro­ton and neu­tron, $j=l+s$, or for both $j=l-s$, then the an­gu­lar mo­menta of the two add up and $j_{\rm N}=j^{\rm p}+j^{\rm n}$.

2.

Oth­er­wise they sub­tract and $j_{\rm N}=\vert j^{\rm p}-j^{\rm n}\vert$.

3.

New and im­proved ver­sion: if num­ber 1 above fails, as­sume that the two an­gu­lar mo­menta are op­po­site any­way and the spin is $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert j^{\rm p}-j^{\rm n}\vert$ like in num­ber 2.

Of course, the real quan­tum rules for an­gu­lar mo­men­tum are a lot more com­pli­cated than the sim­pli­fied pic­ture above. Note in par­tic­u­lar from the Cleb­sch-Gor­dan co­ef­fi­cients in fig­ure 12.5 that if $j=l-s$, then that nu­cleon can­not be just in the spin-up state. The two nu­cle­ons can­not fully align then.

Fig­ure 14.34: Spin of odd-odd nu­clei. [pdf][con]

To check those rules is not triv­ial, be­cause it re­quires the val­ues of $l$ for the odd pro­ton and neu­tron. Who will say in what shells the odd pro­ton and odd neu­tron re­ally are? The sim­plest so­lu­tion is to sim­ply take the shells to be the ones that the shell model pre­dicts, as­sum­ing the sub­shell or­der­ing from sec­tion 14.12.2. The nu­clei that sat­isfy the Nord­heim rules un­der that as­sump­tion are in­di­cated with a check mark in fig­ure 14.34. A blue check mark means that the new and im­proved ver­sion has been used. (Yel­low is used if there is un­cer­tainty about the mea­sure­ment.) It is seen that the rules get a num­ber of nu­clei right.

An L” or “l in­di­cates that it has been as­sumed that the spin of at least one odd nu­cleon has been low­ered due to pro­mo­tion. The rules are the same as in the pre­vi­ous sub­sec­tion. In case of L, the Nord­heim rules were re­ally ver­i­fied. More specif­i­cally, for these nu­clei there was no pos­si­bil­ity con­sis­tent with nu­clear spin and par­ity to vi­o­late the rules. For nu­clei with an l there was, and the case that sat­is­fied the Nord­heim rules was cherry-picked among other oth­er­wise valid pos­si­bil­i­ties that did not.

A fur­ther weak­en­ing of stan­dards ap­plies to nu­clei marked with N” or “n. For those, one or two sub­shells of the odd nu­cle­ons were taken based on the spins of the im­me­di­ately neigh­bor­ing nu­clei of odd mass num­ber. For nu­clei marked with N the Nord­heim rules were again re­ally ver­i­fied, with no pos­si­bil­ity of vi­o­la­tion within the now larger con­text. For nu­clei marked n, other pos­si­bil­i­ties vi­o­lated the rules; ob­vi­ously, for these nu­clei the stan­dards have be­come mis­er­ably low. Note how many cor­rect pre­dic­tions there are in the re­gions of non­spher­i­cal nu­clei in which the shell model is quite mean­ing­less.

Fig­ure 14.35: Odd-odd spins pre­dicted us­ing the neigh­bors. [pdf][con]

Pre­ston and Bhaduri [36, p. 239] sug­gest that the pro­ton and neu­tron an­gu­lar mo­menta be taken from the neigh­bor­ing pairs of nu­clei of odd mass num­ber. Fig­ure 14.35 shows re­sults ac­cord­ing to that ap­proach. To min­i­mize fail­ures due to other causes than the Nord­heim rules, it was de­manded that both spin and par­ity of the odd-odd nu­cleus were solidly es­tab­lished. For the two pairs of odd mass nu­clei, it was de­manded that both spin and par­ity were known, and that the two mem­bers of each pair agreed on the val­ues. It was also de­manded that the or­bital mo­menta of the pairs could be con­fi­dently pre­dicted from the spins and par­i­ties. Cor­rect pre­dic­tions for these su­per­clean cases are in­di­cated by check marks in fig­ure 14.35, in­cor­rect ones by an E or cross. Light check marks in­di­cate cases in which the spin of a pair of odd mass nu­clei is not the spin of the odd nu­cleon.

Pre­ston and Bhaduri [36, p. 239] write: “When con­fronted with ex­per­i­men­tal data, Nord­heim’s rules are found to work quite well, most of the ex­cep­tions be­ing for light nu­clei.” So be it. The re­sults are def­i­nitely bet­ter than chance. Be­low $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 50, the rules get 43 right out of 71. It may be noted that if you sim­ply take the shells di­rectly from the­ory with no pro­mo­tion, like in fig­ure 14.36, you get only 41 right, so us­ing the spins of the neigh­bors seems to help. The “Nu­clear Data Sheets” poli­cies as­sume that the (unim­proved) Nord­heim rules may be help­ful if there is sup­port­ing ev­i­dence.

Fig­ure 14.36: Odd-odd spins pre­dicted from the­ory. [pdf][con]

The nu­clei marked with E in fig­ure 14.35 are par­tic­u­larly in­ter­est­ing. For these nu­clei spin or par­ity show that it is im­pos­si­ble for the odd pro­ton and neu­tron to be in the same shells as their neigh­bors. In four cases, the dis­crep­ancy is in par­ity, which is par­tic­u­larly clear. It shows that for an odd pro­ton, hav­ing an odd neu­tron is not nec­es­sar­ily in­ter­me­di­ate be­tween hav­ing no odd neu­tron and hav­ing one ad­di­tional neu­tron be­sides the odd one. Or vice-versa for an odd neu­tron. Pro­ton and neu­tron shells in­ter­act non­triv­ially.

It may be noted that the un­mod­i­fied Nord­heim rules im­ply that there can­not be any odd-odd nu­clei with 0$\POW9,{+}$ or 1$\POW9,{-}$ ground states. How­ever, some do ex­ist, as is seen in fig­ure 14.34 from the nu­clei with spin zero (grey) and blue check marks.