Sub­sec­tions


14.20 Draft: Gamma De­cay

Nu­clear re­ac­tions and de­cays of­ten leave the fi­nal nu­cleus in an quan­tum state of el­e­vated en­ergy. Such an ex­cited state may lower its en­ergy by emit­ting a pho­ton of elec­tro­mag­netic ra­di­a­tion. That is called gamma de­cay. It is a com­mon way to evolve to the ground state.

Gamma de­cay is in many re­spect sim­i­lar to al­pha and beta de­cay dis­cussed in ear­lier sec­tions. How­ever, the type of nu­cleus does not change in gamma de­cay. Both the atomic and mass num­ber of the nu­cleus stay the same. (Of course, an ex­cited nu­clear state can suf­fer al­pha or beta de­cay in­stead of gamma de­cay. That how­ever is not the sub­ject of this sec­tion.)

Gamma de­cay of ex­cited nu­clei is the di­rect equiv­a­lent of the de­cay of ex­cited elec­tron states in atoms. The big dif­fer­ence be­tween gamma de­cay and the ra­di­a­tion emit­ted by the elec­trons in atoms is en­ergy. The en­ergy of the pho­tons emit­ted by nu­clei is typ­i­cally even higher than that of the X-ray pho­tons emit­ted by in­ner elec­trons. There­fore the ra­di­a­tion emit­ted by nu­clei is gen­er­ally re­ferred to as “gamma rays.”

Both atomic ra­di­a­tion and nu­clear gamma de­cay were an­a­lyzed in con­sid­er­able de­tail in chap­ter 7.4 through 7.8 and ad­denda {A.20} through {A.25}. There is no point in re­peat­ing all that here. In­stead this sec­tion will merely sum­ma­rize the key points and dis­cuss some ac­tual ob­ser­va­tions.

How­ever, the ex­ist­ing data on gamma de­cay is enor­mous. Con­sider NuDat 2, a stan­dard data base. At the time of writ­ing, it con­tains over 3 100 nu­clei. Al­most every nu­cleus but the deuteron has many ex­cited en­ergy lev­els; there are over 160 000 in NuDat 2. Gamma de­cays can pro­ceed be­tween dif­fer­ent states, and NuDat 2 con­tains over 240 000 of them. There is no way that this book can cover all that data. The cov­er­age given in this sec­tion will there­fore be anec­do­tal or ran­dom rather than com­pre­hen­sive.

How­ever, based on a sim­ple model, at least ball­park tran­si­tion rates will be es­tab­lished. These are called the Weis­skopf units. They are com­monly used as ref­er­ence val­ues, to give some con­text to the mea­sured tran­si­tion rates.

One big lim­i­ta­tion of gamma de­cay is for nu­clear states of zero spin. A state of zero spin can­not tran­si­tion to an­other state of zero spin by emit­ting a pho­ton. As dis­cussed in chap­ter 7.4, this vi­o­lates con­ser­va­tion of an­gu­lar mo­men­tum.

But there are other ways that a nu­cleus can get rid of ex­cess en­ergy be­sides emit­ting an elec­tro­mag­netic pho­ton. One way is by kick­ing an atomic elec­tron out of the sur­round­ing atom. This process is called “in­ter­nal con­ver­sion” be­cause the elec­tron is out­side the nu­cleus. It al­lows tran­si­tions be­tween states of zero spin.

For atoms, two-pho­ton emis­sion is a com­mon way to achieve de­cays be­tween states of zero an­gu­lar mo­men­tum. How­ever, for nu­clei this process is less im­por­tant be­cause in­ter­nal con­ver­sion usu­ally works so well.

In­ter­nal con­ver­sion is also im­por­tant for other tran­si­tions. Gamma de­cay is slow be­tween states that have lit­tle dif­fer­ence in en­ergy and/or a big dif­fer­ence in spin. For such de­cays, in­ter­nal con­ver­sion can pro­vide a faster al­ter­na­tive.

If the ex­ci­ta­tion en­ergy is high, it is also pos­si­ble for the nu­cleus to cre­ate an elec­tron and positron pair from scratch. Since the quan­tum un­cer­tainty in po­si­tion of the pair is far too large for them to be con­fined within the small nu­cleus, this is called “in­ter­nal pair cre­ation.”


14.20.1 Draft: En­er­get­ics

The re­duc­tion in nu­clear en­ergy dur­ing gamma de­cay is called the $Q$-​value. This en­ergy comes out pri­mar­ily as the en­ergy of the pho­ton, though the nu­cleus will also pick up a bit of ki­netic en­ergy, called the re­coil en­ergy.

Re­coil en­ergy will usu­ally be ig­nored, so that $Q$ gives the en­ergy of the pho­ton. The pho­ton en­ergy is re­lated to its mo­men­tum and fre­quency through the rel­a­tivis­tic mass-en­ergy and Planck-Ein­stein re­la­tions:

\begin{displaymath}
Q = E_{\rm N1} - E_{\rm N2} = p c = \hbar \omega
\end{displaymath} (14.65)

Typ­i­cal tab­u­la­tions list nu­clear ex­ci­ta­tion en­er­gies as en­er­gies, rather than as nu­clear masses. Un­for­tu­nately, the en­er­gies are usu­ally in eV in­stead of SI units.

In in­ter­nal con­ver­sion, the nu­cleus does not emit a pho­ton, but kicks an elec­tron out of the sur­round­ing atomic elec­tron cloud. The nu­clear en­ergy re­duc­tion goes into ki­netic en­ergy of the elec­tron, plus the bind­ing en­ergy re­quired to re­move the elec­tron from its or­bit:

\begin{displaymath}
Q = E_{\rm N1} - E_{\rm N2} = T_{\rm e} + E_{\rm B,e}
\end{displaymath} (14.66)


14.20.2 Draft: For­bid­den de­cays

The de­cay rate in gamma de­cay is to a large ex­tent dic­tated by what is al­lowed by con­ser­va­tion of an­gu­lar mo­men­tum and par­ity. The nu­cleus is al­most a math­e­mat­i­cal point com­pared to the wave length of a typ­i­cal pho­ton emit­ted in gamma de­cay. There­fore, it is dif­fi­cult for the nu­cleus to give the pho­ton ad­di­tional or­bital an­gu­lar mo­men­tum. That is much like what hap­pens in al­pha and beta de­cay.

The pho­ton has one unit of spin. If the nu­cleus does not give it ad­di­tional or­bital an­gu­lar mo­men­tum, the to­tal an­gu­lar mo­men­tum that the pho­ton car­ries off is one unit. That means that the nu­clear spin can­not change by more than one unit.

(While this is true, the is­sue is ac­tu­ally some­what more sub­tle than in the de­cay types dis­cussed pre­vi­ously. For a pho­ton, spin and or­bital an­gu­lar mo­men­tum are in­trin­si­cally linked. Be­cause of that, a pho­ton al­ways has some or­bital an­gu­lar mo­men­tum. That was dis­cussed in chap­ter 7.4.3 and in de­tail in var­i­ous ad­denda such as {A.21}. How­ever, the in­her­ent or­bital an­gu­lar mo­men­tum does not re­ally change the story. The bot­tom line re­mains that it is un­likely for the pho­ton to be emit­ted with more than one unit of net an­gu­lar mo­men­tum.)

The nu­clear spin can also stay the same, in­stead of change by one unit, even if a pho­ton with one unit of an­gu­lar mo­men­tum is emit­ted, In clas­si­cal terms the one unit of an­gu­lar mo­men­tum can go into chang­ing the di­rec­tion of the nu­clear spin in­stead of its mag­ni­tude, chap­ter 7.4.2. How­ever, this only works if the nu­clear spin is nonzero.

Par­ity must also be pre­served, chap­ter 7.4. Par­ity is even, or 1, if the wave func­tion stays the same when the pos­i­tive di­rec­tion of all three Carte­sian axes is re­versed. Par­ity is odd, or $\vphantom{0}\raisebox{1.5pt}{$-$}$1, if the wave func­tion changes sign. Par­i­ties of sep­a­rate sources are mul­ti­plied to­gether to com­bine them. That is un­like for an­gu­lar mo­men­tum, where sep­a­rate an­gu­lar mo­menta are added to­gether.

In the nor­mal, or al­lowed, de­cays the pho­ton is emit­ted with odd par­ity. There­fore, the nu­clear par­ity must re­verse dur­ing the tran­si­tion, chap­ter 7.4.2.

(To be picky, the so-called weak force does not pre­serve par­ity. This cre­ates a very small un­cer­tainty in nu­clear par­i­ties. That then al­lows a very small prob­a­bil­ity for tran­si­tions in which the ap­par­ent par­ity is not con­served. But the prob­a­bil­ity for this is so small that it can al­most al­ways be ig­nored.)

Al­lowed tran­si­tions are called elec­tric tran­si­tions be­cause the nu­cleus in­ter­acts mainly with the elec­tric field of the pho­ton. More specif­i­cally, they are called “elec­tric di­pole tran­si­tions” for rea­sons orig­i­nat­ing in clas­si­cal elec­tro­mag­net­ics, chap­ter 7.7.2. For prac­ti­cal pur­poses, a di­pole tran­si­tion is one in which the pho­ton is emit­ted with one unit of net an­gu­lar mo­men­tum.

Tran­si­tions in which the nu­clear spin change is greater than one unit, or in which the nu­clear par­ity does not change, or in which the spin stays zero, are called for­bid­den. De­spite the name, most such de­cays will usu­ally oc­cur given enough time. How­ever they are gen­er­ally much slower.

One way that for­bid­den tran­si­tions can oc­cur is that the nu­cleus in­ter­acts with the mag­netic field in­stead of the elec­tric field. This pro­duces what are called mag­netic tran­si­tions. Mag­netic tran­si­tions tend to be no­tice­ably slower than cor­re­spond­ing elec­tric ones. In mag­netic di­pole tran­si­tions, the pho­ton has one unit of net an­gu­lar mo­men­tum just like in elec­tric di­pole tran­si­tions. How­ever, the pho­ton now has even par­ity. There­fore mag­netic di­pole tran­si­tions al­low the nu­clear par­ity to stay the same.

Tran­si­tions in which the nu­clear spin changes by more than one unit are pos­si­ble through emis­sion of a pho­ton with ad­di­tional or­bital an­gu­lar mo­men­tum. That al­lows a net an­gu­lar mo­men­tum of the pho­ton greater than one. But at a price. Each unit of ad­di­tional net an­gu­lar mo­men­tum slows down the typ­i­cal de­cay rate by roughly 5 or­ders of mag­ni­tude.

The hor­ror story is tan­ta­lum-180m. There are at the time of writ­ing 256 ground state nu­clei that are clas­si­fied as sta­ble. And then there is the ex­cited nu­cleus tan­ta­lum-180m. Sta­ble nu­clei should be in their ground state, be­cause states of higher en­ergy de­cay into lower en­ergy ones. But tan­ta­lum-180m has never been ob­served to de­cay. If it de­cays at all, it has been es­tab­lished that its half life can­not be less than 10$\POW9,{15}$ year. The uni­verse has only ex­isted for roughly 10$\POW9,{10}$ years, and so tan­ta­lum-180m oc­curs nat­u­rally.

Fig­ure 14.56: En­ergy lev­els of tan­ta­lum-180. [pdf]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
% vertical spacing 137...
...\put(13.2,0){\makebox(0,0)[l]{1}}
}
\end{picture}}
\end{picture}
\end{figure}

The tan­ta­lum-180 ground state shows no such id­iocy. It is un­sta­ble as any self-re­spect­ing heavy odd-odd nu­cleus should be. In fact it dis­in­te­grates within about 8 hours through both elec­tron cap­ture and beta-mi­nus de­cay at com­pa­ra­ble rates. But tan­ta­lum-180m is an ex­cited state with a hu­mon­gous spin of 9. Fig­ure 14.56 shows the ex­cited en­ergy lev­els of tan­ta­lum-180; tan­ta­lum-180m is the sec­ond ex­cited en­ergy level. It can only de­cay to the 1$\POW9,{+}$ ground state and to an 2$\POW9,{+}$ ex­cited state. It has very lit­tle en­ergy avail­able to do ei­ther. The de­cay would re­quire the emis­sion of a pho­ton with at least seven units of or­bital an­gu­lar mo­men­tum, and that just does not hap­pen in a thou­sand years. Nor in a petayear.

You might think that tan­ta­lum-180m could just dis­in­te­grate di­rectly through elec­tron cap­ture or beta de­cay. But those processes have the same prob­lem. There is just no way for tan­ta­lum-180m to get rid of all that spin with­out emit­ting par­ti­cles with un­likely large or­bital an­gu­lar mo­men­tum. So tan­ta­lum-180m will live for­ever, spin­ning too fast to reach the sweet obliv­ion of the quick death that waits be­low.

Elec­tric tran­si­tions are of­ten gener­i­cally in­di­cated as ${\rm {E}}\ell$ and mag­netic ones as ${\rm {M}}\ell$. Here $\ell$ in­di­cates the net an­gu­lar mo­men­tum of the pho­ton. That is the max­i­mum nu­clear spin change that the tran­si­tion can achieve. So elec­tric di­pole tran­si­tions are ${\rm {E1}}$, and mag­netic di­pole tran­si­tions are ${\rm {M1}}$. Names have also been given to the higher mul­ti­pole or­ders. For ex­am­ple, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 tran­si­tions are quadru­pole ones, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 3 oc­tu­pole, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 4 hexa­de­ca­pole, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 5 tri­akon­tadi­pole, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 6 hexa­con­tate­tra­pole, etcetera. (If you are won­der­ing, the pre­fixes in these names are pow­ers of two, ex­pressed in a mix­ture of Latin and Greek.)

For elec­tric tran­si­tions, the nu­clear par­ity changes when $\ell$ is odd. For mag­netic tran­si­tions, it changes when $\ell$ is even. The tran­si­tion rules are sum­ma­rized in ta­ble 14.4.


Ta­ble 14.4: Nu­clear spin and par­ity changes in elec­tro­mag­netic mul­ti­pole tran­si­tions.
\begin{table}\begin{displaymath}
\begin{array}{rcc}\hline\hline
& \mbox{maximu...
...0 to spin 0 transitions.}} \\ \hline
\end{array} \end{displaymath}
\end{table}


That leaves tran­si­tions from nu­clear spin 0 to nu­clear spin 0. Such tran­si­tions can­not oc­cur through emis­sion of a pho­ton, pe­riod. For such tran­si­tions, con­ser­va­tion of an­gu­lar mo­men­tum would re­quire that the pho­ton is emit­ted with­out an­gu­lar mo­men­tum. But a pho­ton can­not have zero net an­gu­lar mo­men­tum. You might think that the spin of the pho­ton could be can­celed through one unit of or­bital an­gu­lar mo­men­tum. How­ever, be­cause the spin and or­bital an­gu­lar mo­men­tum of a pho­ton are linked, it turns out that this is not pos­si­ble, {A.21}.

De­cay from an ex­cited state with spin zero to an­other state that also has spin zero is pos­si­ble through in­ter­nal con­ver­sion or in­ter­nal pair pro­duc­tion. In prin­ci­ple, it could also be achieved through two-pho­ton emis­sion, but that is a very slow process that has trou­ble com­pet­ing with the other two.

One other ap­prox­i­mate con­ser­va­tion law might be men­tioned here, isospin. Isospin is con­served by nu­clear forces, and its charge com­po­nent is con­served by elec­tro­mag­netic forces, sec­tion 14.18. It can be shown that to the ex­tent that isospin is con­served, cer­tain ad­di­tional se­lec­tion rules ap­ply. These in­volve the quan­tum num­ber of square isospin $t_T$, which is the isospin equiv­a­lent of the az­imuthal quan­tum num­ber for the spin an­gu­lar mo­men­tum of sys­tems of fermi­ons. War­bur­ton & We­neser [47] give the fol­low­ing rules:

1.
Elec­tro­mag­netic tran­si­tions are for­bid­den un­less $\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, or $\pm$1. (Here $\Delta$ means the dif­fer­ence be­tween the ini­tial and fi­nal nu­clear states).
2.
Cor­re­spond­ing $\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pm$1 tran­si­tions in con­ju­gate nu­clei are iden­ti­cal in all prop­er­ties. (Con­ju­gate means that the two nu­clei have the num­bers of pro­tons and neu­trons swapped. Cor­re­spond­ing tran­si­tions means tran­si­tions be­tween equiv­a­lent lev­els, as dis­cussed in sec­tion 14.18.)
3.
Cor­re­spond­ing ${\rm {E1}}$ tran­si­tions in con­ju­gate nu­clei -- whether $\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 or $\pm$1 -- have equal strengths.
4.
$\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ${\rm {E1}}$ tran­si­tions in self-con­ju­gate nu­clei are for­bid­den. (Self-​con­ju­gate nu­clei have the same num­ber of pro­tons as neu­trons.)
5.
Cor­re­spond­ing $\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ${\rm {M1}}$ tran­si­tions in con­ju­gate nu­clei are ex­pected to be of ap­prox­i­mately equal strength, within, say, a fac­tor of two if the tran­si­tions are of av­er­age strength or stronger.
6.
$\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ${\rm {M1}}$ tran­si­tions in self-con­ju­gate nu­clei are ex­pected to be weaker by a fac­tor of 100 than the av­er­age ${\rm {M1}}$ tran­si­tion strength.
7.
$\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ${\rm {M}}\ell$ tran­si­tions in con­ju­gate nu­clei are ex­pected to be of ap­prox­i­mately equal strength if the tran­si­tions are of av­er­age strength or stronger.
8.
$\Delta{t}_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ${\rm {M}}\ell$ tran­si­tions in self-con­ju­gate nu­clei are ex­pected to be ap­pre­cia­bly weaker than av­er­age.
The last four rules in­volve an ad­di­tional ap­prox­i­ma­tion be­sides the as­sump­tion that isospin is con­served.

In a nut­shell, ex­pect that the tran­si­tions will be un­ex­pect­edly slow if the isospin changes by more than one unit. Ex­pect the same for nu­clei with equal num­bers of pro­tons and neu­trons if the isospin does not change at all and it is an ${\rm {E1}}$ or mag­netic tran­si­tion.

As an ex­am­ple, [30, p. 390], con­sider the de­cay of the 1$\POW9,{-}$ iso­baric ana­log state com­mon to car­bon-14, ni­tro­gen-14, and oxy­gen-14 in fig­ure 14.46. This state has $t_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. For oxy­gen-14, it is the low­est ex­cited state. Its de­cay to the $t_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, 0$\POW9,{+}$, ground state is an ${\rm {E1}}$ tran­si­tion that is al­lowed by the spin, par­ity, and isospin se­lec­tion rules. And in­deed, the 1$\POW9,{-}$ ex­cited state de­cays rapidly to the ground state; the half-life is about 0.000 012 fs (fem­tosec­onds). That is even faster than the Weis­skopf ball­park for a fully al­lowed de­cay, sub­sec­tion 14.20.4, which gives about 0.009 fs here. But for ni­tro­gen-14, the equiv­a­lent tran­si­tion is not al­lowed be­cause of rule 4 above. Ni­tro­gen-14 has 7 pro­tons and 7 neu­trons. And in­deed, the par­tial half life of this tran­si­tion is 2.7 fs. That is very much longer. Based on rule 3 above, it is ex­pected that the de­cay rate of the 1$\POW9,{-}$ state in car­bon-14 is sim­i­lar to the one in oxy­gen-14. Un­for­tu­nately, ex­per­i­men­tally it has only been es­tab­lished that its half-life is less than 7 fs.

Some dis­claimers are ap­pro­pri­ate for this ex­am­ple. As far as the oxy­gen tran­si­tion is con­cerned, the NuDat 2 [[12]] data do not say what the dom­i­nant de­cay process for the oxy­gen-14 state is. Nor what the fi­nal state is. So it might be an­other de­cay process that dom­i­nates. The next higher ex­cited state, with 0.75 MeV more en­ergy, de­cays 100% through pro­ton emis­sion. And two or­ders of mag­ni­tude faster than Weis­skopf does seem a lot, fig­ure 14.63.

As far as the ni­tro­gen tran­si­tion is con­cerned, the de­cay processes are listed in NuDat 2. The de­cay is al­most to­tally due to pro­ton emis­sion, not gamma de­cay. The ac­tual half-life of this state is 0.000 02 fs; the 2.7 fs men­tioned above is com­puted us­ing the given de­cay branch­ing ra­tios. The 2.7 fs is way above the 0.006 fs Weis­skopf es­ti­mate, but that is quite com­mon for ${\rm {E1}}$ tran­si­tions.

It may be more rea­son­able to com­pare the for­bid­den ni­tro­gen 8 MeV to 2.3 MeV tran­si­tion to the al­lowed 8 MeV to ground state, and 8 MeV to 4 MeV tran­si­tions. They are all three ${\rm {E1}}$ tran­si­tions. Cor­rected for the dif­fer­ences in en­ergy re­lease, the for­bid­den tran­si­tion is 20 times slower than the one to the ground state, and 25 times slower than the one to the 4 MeV state. So ap­par­ently, be­ing for­bid­den seems to slow down this tran­si­tion by a fac­tor of roughly 20. It is sig­nif­i­cant, though it is not that big on the scale of fig­ure 14.63.

As an­other ex­am­ple, the ni­tro­gen tran­si­tion from the 1$\POW9,{-}$ 5.7 MeV $t_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 state to the $t_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 ground state is also for­bid­den, while the tran­si­tion to the 0$\POW9,{+}$ $t_T$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 state is now per­mit­ted. And in­deed, the de­cay to the ground state is about ten times slower, when cor­rected for en­ergy re­lease, [30, p. 391].

More com­pre­hen­sive data may be found in [47].


14.20.3 Draft: Iso­mers

An “iso­mer” is a long last­ing ex­cited state of a nu­cleus. Usu­ally, an ex­cited nu­cleus that does not dis­in­te­grate through other means will drop down to lower en­er­gies through the emis­sion of pho­tons in the gamma ray range. It will then end up back in the ground state within a typ­i­cal time in terms of fs, or about 10$\POW9,{-15}$ sec­ond.

But some­times a nu­cleus gets stuck in a metastable state that takes far longer to de­cay. Such a state is called an iso­meric state. Krane [30, p. 174] ball­parks the min­i­mum life­time to be con­sid­ered a true iso­meric state at roughly 10$\POW9,{-9}$ s, Bertu­lani [5, p. 244] gives 10$\POW9,{-15}$ s, and NuDat 2 [[12]] uses 10$\POW9,{-1}$ s with qual­i­fi­ca­tion in their poli­cies and 10$\POW9,{-9}$ s in their glos­sary. Don’t you love stan­dard­iza­tion? In any case, this book will not take iso­mers se­ri­ous un­less they have a life­time com­pa­ra­ble to 10$\POW9,{-9}$ sec­ond. Why would an ex­cited state that can­not sur­vive for a mil­lisec­ond be given the same re­spect as tan­ta­lum-180m, which shows no sign of kick­ing the bucket af­ter 10$\POW9,{15}$ years?

But then, why would any ex­cited state be able to last very much more than the typ­i­cal 10$\POW9,{-15}$ s gamma de­cay time in the first place? The main rea­son is an­gu­lar mo­men­tum con­ser­va­tion. It is very dif­fi­cult for a tiny ob­ject like a nu­cleus to give a pho­ton much an­gu­lar mo­men­tum. There­fore, tran­si­tions be­tween states of very dif­fer­ent an­gu­lar mo­men­tum will be ex­tremely slow, if they oc­cur at all. Such tran­si­tions are highly for­bid­den, or us­ing a bet­ter term, hin­dered.

If an ex­cited state has a very dif­fer­ent spin than the ground state, and there are no states in be­tween the two that are more com­pat­i­ble, then that ex­cited state is stuck. But why would low spin states be right next to high spin states? The main rea­son is found in the shell model, and in par­tic­u­lar fig­ure 14.15. Ac­cord­ing to the shell model, just be­low the magic num­bers of 50, 82, and 126, high spin states are pushed into re­gions of low spin states by the so-called spin-or­bit in­ter­ac­tion. That is a recipe for iso­merism if there ever was one.

There­fore, it should be ex­pected that there will be many iso­mers be­low the magic num­bers of 50, 82, and 126. And that these iso­mers will have the op­po­site par­ity of the ground state, be­cause the high spin states are pushed into low spin states of op­po­site par­ity.

Fig­ure 14.57: Half-life of the longest-lived even-odd iso­mers. [pdf][con]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,56...
...6,0){\makebox(0,0)[b]{uncertain}}
}
\end{picture}}
\end{picture}
\end{figure}

And so it is. Fig­ure 14.57 shows the half-lifes of the longest-last­ing ex­ited states of even $Z$ and odd $N$ nu­clei. The groups of iso­mers be­low the magic neu­tron num­bers are called the “is­lands of iso­merism.” The dif­fer­ence in spin from the ground state is in­di­cated by the color. A dif­fer­ence in par­ity is in­di­cated by a mi­nus sign. Half-lives over 10$\POW9,{14}$ s are shown as full-size squares.

Fig­ure 14.58: Half-life of the longest-lived odd-even iso­mers. [pdf][con]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,56...
...6,0){\makebox(0,0)[b]{uncertain}}
}
\end{picture}}
\end{picture}
\end{figure}

Fig­ure 14.58 shows the is­lands for odd $Z$, even $N$ nu­clei.

Fig­ure 14.59: Half-life of the longest-lived odd-odd iso­mers. [pdf][con]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,56...
...0,0)[r]
{$\fourIdx{180\rm m}{73}{}{}{\rm Ta}$}}
}
\end{picture}
\end{figure}

For odd-odd nu­clei, fig­ure 14.59, the ef­fects of pro­ton and neu­tron magic num­bers get mixed up. Pro­ton and neu­tron ex­ci­ta­tions may com­bine into larger spin changes, pro­vid­ing one pos­si­ble ex­pla­na­tion for the iso­mers of light nu­clei with­out par­ity change.

Be­sides tan­ta­lum-180m, which lives for­ever, also note bis­muth-210m in fig­ure 14.59. Bis­muth-210m has the same spin 9$\POW9,{-}$ as tan­ta­lum-180m, but it does man­age to de­cay af­ter about 3 mil­lion years. But it does so through al­pha-de­cay, rather than gamma-de­cay,

Fig­ure 14.60: Half-life of the longest-lived even-even iso­mers. [pdf][con]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,56...
...6,0){\makebox(0,0)[b]{uncertain}}
}
\end{picture}}
\end{picture}
\end{figure}

For even-even nu­clei, fig­ure 14.60, there is very lit­tle iso­meric ac­tiv­ity.


14.20.4 Draft: Weis­skopf es­ti­mates

Gamma de­cay rates can be ball­parked us­ing the so-called Weis­skopf es­ti­mates:

\begin{displaymath}
\fbox{$\displaystyle
\lambda^{{\rm{E}}\ell} = C_{{\rm{E}}\...
...m{M}}\ell} = C_{{\rm{M}}\ell} A^{(2\ell-2)/3}Q^{2\ell+1}
$} %
\end{displaymath} (14.67)


\begin{displaymath}
\begin{array}{rccccc}\hline\hline
\ell: & 1 & 2 & \quad3\q...
...} & 10 & \,3.3~10^{-6} & \,7.4~10^{-13}
\\ \hline
\end{array}\end{displaymath}

Here the de­cay rates are per sec­ond, $A$ is the mass num­ber, and $Q$ is the en­ergy re­lease of the de­cay in MeV. Also $\ell$ is the max­i­mum nu­clear spin change pos­si­ble for that tran­si­tion. As dis­cussed in sub­sec­tion 14.20.2, elec­tric tran­si­tions re­quire that the nu­clear par­ity flips over when $\ell$ is odd, and mag­netic ones that it flips over when $\ell$ is even. In the op­po­site cases, the nu­clear par­ity must stay the same. If there is more than one de­cay process in­volved, add the in­di­vid­ual de­cay rates.

Fig­ure 14.61: Weis­skopf ball­park half-lifes for elec­tro­mag­netic tran­si­tions ver­sus en­ergy re­lease. Bro­ken lines in­clude ball­parked in­ter­nal con­ver­sion. [pdf]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,55...
...put(335,206){\makebox(0,0)[rb]{227}}
\end{picture}}
\end{picture}
\end{figure}

Fig­ure 14.62: Moszkowski ball­park half-lifes for mag­netic tran­si­tions ver­sus en­ergy re­lease. Bro­ken lines in­clude ball­parked in­ter­nal con­ver­sion. [pdf]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(405,26...
...0,0)[rb]{80}}
\put(335,206){\makebox(0,0)[rb]{227}}
\end{picture}
\end{figure}

The es­ti­mates are plot­ted in fig­ure 14.61. For mag­netic tran­si­tions the bet­ter Moszkowski es­ti­mates are shown in fig­ure 14.62. (In­ter­nal con­ver­sion is dis­cussed in sub­sec­tion 14.20.6, where the ball­parks are given.)

A com­plete de­riva­tion and dis­cus­sion of these es­ti­mates can be found {A.25}. Note that many sources have er­rors in their for­mu­lae and/or graphs or use non-SI units, {A.25.9}. The cor­rect for­mu­lae in SI units are in {A.25.8}.

These es­ti­mates are de­rived un­der the as­sump­tion that only a sin­gle pro­ton changes states in the tran­si­tion. They also as­sume that the mul­ti­pole or­der is the low­est pos­si­ble, given by the change in nu­clear spin. And that the fi­nal state of the pro­ton has an­gu­lar mo­men­tum $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. Some cor­rec­tion fac­tors are avail­able to al­low for dif­fer­ent mul­ti­pole or­ders and dif­fer­ent fi­nal an­gu­lar mo­menta, {A.25.8}. There are also cor­rec­tion fac­tors to al­low for the fact that re­ally the pro­ton and the rest of the nu­cleus move around their com­mon cen­ter of grav­ity. Sim­i­lar cor­rec­tion fac­tors can al­low for the case that a sin­gle neu­tron in­stead of a pro­ton makes the tran­si­tion. See {A.25.8} for more.

The ini­tial and fi­nal pro­ton states as­sumed in the es­ti­mates are fur­ther very sim­ple, {A.25.8}. They are like a shell model state with a sim­pli­fied ra­dial de­pen­dence. Cor­rec­tions ex­ist for that ra­dial de­pen­dence too. But the way the es­ti­mates are mostly used in prac­tice is as a ref­er­ence. The ac­tual de­cay rate in a tran­si­tion is com­pared to the Weis­skopf or Moszkowski es­ti­mates. These es­ti­mates are there­fore used as units to ex­press de­cay rates in.

If there is a big dif­fer­ence, it gives hints about the na­ture of the tran­si­tion process. For ex­am­ple, the ac­tual de­cay rate is of­ten or­ders of mag­ni­tude smaller than the es­ti­mate. That can in­di­cate that the state pro­duced by the de­cay Hamil­ton­ian has only a small prob­a­bil­ity of be­ing the cor­rect fi­nal nu­clear state. In other words, there may be a poor match-up or “lit­tle over­lap” be­tween the ini­tial and fi­nal nu­clear states. It is im­plicit in the sim­ple pro­ton states used in the es­ti­mates that the state pro­duced by the de­cay Hamil­ton­ian has a good chance of be­ing right. But ac­tual ${\rm {E}}1$ tran­si­tions can eas­ily be three or more or­ders of mag­ni­tude slower than es­ti­mate, as shown in the next sub­sec­tion. That is sim­i­lar to what was ob­served for the ball­parks for beta de­cays given in sec­tion 14.19.7. One rea­son may be that some of these tran­si­tions are ap­prox­i­mately for­bid­den by isospin con­ser­va­tion.

Con­versely, the ob­served tran­si­tion rate may be sev­eral or­ders of mag­ni­tude more rapid than the es­ti­mate. That may in­di­cate that a lot of nu­cle­ons are in­volved in the tran­si­tion. Their con­tri­bu­tions can add up. This is fre­quently re­lated to shape changes in non­spher­i­cal nu­clei. For ex­am­ple, ${\rm {E}}2$ tran­si­tions, which are par­tic­u­larly rel­e­vant to de­formed nu­clei, may eas­ily be or­ders of mag­ni­tude faster than es­ti­mate.

In­ter­est­ingly, ${\rm {M}}4$ tran­si­tions tend to be quite close to the mark. Re­call that the shell model puts the high­est spin states of one har­monic os­cil­la­tor shell right among the low­est spin states of the next lower shell, 14.15. Tran­si­tions be­tween these states in­volve a par­ity change and a large change in spin, lead­ing to E3 and M4 tran­si­tions. They re­sem­ble sin­gle-par­ti­cle tran­si­tions as the Weis­skopf and Moszkowski es­ti­mates as­sume. The es­ti­mates tend to work well for them. One pos­si­ble rea­son that they do not end up that much be­low ball­park as E1 tran­si­tions may be that these are heavy nu­clei. For heavy nu­clei the re­stric­tions put on by isospin may be less con­fin­ing.

Fi­nally, what other books do not point out is that there is a prob­lem with elec­tri­cal tran­si­tions in the is­lands of iso­merism. There is se­ri­ous con­cern about the cor­rect­ness of the very Hamil­ton­ian used in such tran­si­tions, {N.14}. This prob­lem does not seem to af­fect mag­netic mul­ti­pole tran­si­tions in the non­rel­a­tivis­tic ap­prox­i­ma­tion.

An­other prob­lem not pointed out in var­i­ous other books is for mag­netic tran­si­tions. Con­sider the shell model states, fig­ure 14.15. They al­low many tran­si­tions in­side the bands that by their unit change in an­gu­lar mo­men­tum and un­changed par­ity are ${\rm {M}}1$ tran­si­tions. How­ever, these states have a change in or­bital an­gu­lar mo­men­tum equal to two units. The sin­gle-par­ti­cle model on which the Weis­skopf and Moszkowski es­ti­mates are based pre­dicts zero tran­si­tion rate for such ${\rm {M}}1$ tran­si­tions. It does not pre­dict the Moszkowski or Weis­skopf val­ues given above and in the fig­ures. In gen­eral, the pre­dicted sin­gle-par­ti­cle tran­si­tion rates are zero un­less the mul­ti­pole or­der $\ell$ sat­is­fies, {A.25.8}

\begin{displaymath}
\vert l_1-l_2\vert \mathrel{\raisebox{-.7pt}{$\leqslant$}}\ell \mathrel{\raisebox{-.7pt}{$\leqslant$}}l_1+l_2
\end{displaymath}

where $l_1$ and $l_2$ are the ini­tial and fi­nal or­bital az­imuthal quan­tum num­bers. For­tu­nately this is only an is­sue for mag­netic tran­si­tions, {A.25.8}.

Note that the sin­gle-par­ti­cle model does give a non­triv­ial pre­dic­tion for say an 2p$_{1/2}$ to 2p$_{3/2}$ ${\rm {M}}1$ tran­si­tion. That is de­spite the fact that the sim­pli­fied Hamil­ton­ian on which it is based would pre­dict zero tran­si­tion rate for the model sys­tem. For say a 4p$_{3/2}$ to 2p tran­si­tion, the Weis­skopf and Moszkowski units also give a non­triv­ial pre­dic­tion. That, how­ever, is due to the in­cor­rect ra­dial es­ti­mate (A.187). The cor­rect sin­gle-par­ti­cle model on which they are based would give this tran­si­tion rate as zero. For­tu­nately, tran­si­tions like 4p$_{3/2}$ to 2p are not likely to be much of a con­cern.


14.20.5 Draft: Com­par­i­son with data

This sub­sec­tion com­pares the the­o­ret­i­cal Weis­skopf and Moszkowski es­ti­mates of the pre­vi­ous sec­tion with ac­tual data. The data are from NuDat 2, [[12]]. The plot­ted val­ues are a broad but fur­ther quite ran­dom se­lec­tion of data of ap­par­ently good qual­ity. A more pre­cise de­scrip­tion of the data se­lec­tion pro­ce­dure is in {N.34}. In­ter­nal con­ver­sion ef­fects, as dis­cussed in sub­sec­tion 14.20.6, have been math­e­mat­i­cally re­moved us­ing the con­ver­sion con­stants given by NuDat 2. Com­puted de­cay rates were checked against the de­cay rates in W.u. as given by NuDat 2.

Fig­ure 14.63: Com­par­i­son of elec­tric gamma de­cay rates with the­ory. [pdf]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(404,56...
...n-odd}}
\put(353,375.8){\makebox(0,0)[bl]{odd-odd}}
\end{picture}
\end{figure}

Fig­ure 14.64: Com­par­i­son of mag­netic gamma de­cay rates with the­ory. [pdf]
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(404,56...
...n-odd}}
\put(353,375.8){\makebox(0,0)[bl]{odd-odd}}
\end{picture}
\end{figure}

Fig­ures 14.63 and 14.64 show the re­sults. What is plot­ted is the half life, scaled to an (har­monic) av­er­age nu­cleus size. In par­tic­u­lar,

\begin{displaymath}
\mbox{E$\ell$:}\quad
\tau_{1/2,\rm red} = \tau_{1/2} \left...
...,\rm red} = \tau_{1/2} \left(\frac{A}{32}\right)^{2(\ell-1)/3}
\end{displaymath}

The hor­i­zon­tal co­or­di­nate in the fig­ures in­di­cates the en­ergy re­lease $Q$.

The mul­ti­pole lev­els are color coded. The Weis­skopf val­ues are shown as bro­ken lines. The solid lines are an at­tempt at a “best guess” based on the sin­gle-par­ti­cle model. For the elec­tric tran­si­tions, they sim­ply use the em­pir­i­cal ra­dial fac­tor {A.25.8} (A.187). For the mag­netic tran­si­tions, the best guess was based on the more ac­cu­rate Moszkowski es­ti­mates. The em­pir­i­cal ra­dial fac­tor was again used. The mo­men­tum fac­tor {A.25.8} (A.189) at min­i­mum mul­ti­pole or­der was av­er­aged be­tween the pro­ton and neu­tron val­ues. The $g$ val­ues in those fac­tors were in turn av­er­aged be­tween their free space and the­o­ret­i­cal val­ues.

Sym­bol size in­di­cates the nu­cleus size. Sym­bol shape in­di­cates whether the num­bers of pro­tons and neu­trons are even or odd. A mi­nus sign in­di­cates that the ini­tial par­ity is odd; oth­er­wise it is even. The fi­nal par­ity fol­lows from the mul­ti­pole or­der and type. An open sym­bol cen­ter in­di­cates that the mul­ti­pole level $\ell$ is higher than needed in the tran­si­tion. More pre­cisely, it in­di­cates that it is higher than the change in nu­clear spin.

Please, your mouth is hang­ing open. It makes you look very goofy. You can al­most pre­tend that the mag­netic data are not re­ally as bad as they look, if you cover up those num­bers along the ver­ti­cal axis with your arm.

There is no doubt that if en­gi­neers got data like that, they would con­clude that some­thing is ter­ri­bly and fun­da­men­tally wrong. Physi­cists how­ever pooh-pooh the prob­lems.

First of all, physics text­books typ­i­cally only present the ${\rm {M4}}$ data graph­i­cally like this. Yes, the ${\rm {M4}}$ tran­si­tions are typ­i­cally only an or­der of mag­ni­tude or so off. Ac­cord­ing to the fig­ures here, this good agree­ment hap­pens only for the ${\rm {M4}}$ data. Have a look at the ${\rm {E1}}$ and ${\rm {E2}}$ data. They end up pretty much in the same hu­mon­gous cloud of scat­tered data. In physics text­books you do not re­ally see it, as these data are pre­sented in sep­a­rate his­tograms. And for some rea­son, in those his­tograms the ${\rm {E2}}$ tran­si­tions are typ­i­cally only half an or­der of mag­ni­tude above es­ti­mate, rather than 2.5 or­ders. (The ${\rm {E1}}$ tran­si­tions in those his­tograms are sim­i­lar to the data pre­sented here.)

Con­sider now a typ­i­cal ba­sic nu­clear text­book for physi­cist. Ac­cord­ing to the book, dis­agree­ments of sev­eral or­ders of mag­ni­tude from the­ory can hap­pen. The dif­fer­ence be­tween “can hap­pen” and are nor­mal is not de­fined. The book fur­ther ex­plains: “In par­tic­u­lar, ex­per­i­men­tal dis­in­te­gra­tion rates smaller than the ones pre­dicted by [the Weis­skopf es­ti­mates] can mean that [the Weis­skopf ra­dial fac­tor {A.25.8} (A.187)] is not very rea­son­able and that the small over­lap of the [ini­tial and fi­nal nu­clear wave func­tions] de­creases the val­ues of $\lambda$.

How­ever, the best guess ${\rm {E1}}$ line in fig­ure 14.63 uses a bet­ter ra­dial es­ti­mate. It is not ex­actly enough to get any­where near the typ­i­cal data.

And poor over­lap is an easy cop-out since nu­clear wave func­tions are not known. For ex­am­ple, it does not ex­plain why some mul­ti­pole or­ders like ${\rm {E1}}$ have a very poor over­lap of wave func­tions, and oth­ers do not.

Tran­si­tion rates many or­ders of mag­ni­tude smaller than the­ory must have a good rea­son. Ran­dom de­vi­a­tions from the­ory are not a rea­son­able ex­pla­na­tion. Hav­ing a tran­si­tion rate typ­i­cally four or­ders of mag­ni­tude smaller than a rea­son­able the­o­ret­i­cal es­ti­mate is like rou­tinely hit­ting the bull’s eye of a 10 cm tar­get to within a mm. There must be some­thing caus­ing this.

But what might that be? Con­ser­va­tion of an­gu­lar mo­men­tum and par­ity are al­ready fully ac­counted for. To be sure, con­ser­va­tion of isospin is not. How­ever, isospin is an ap­prox­i­mate sym­me­try. It is not ac­cu­rate enough to ex­plain re­duc­tions by 4 or 5 or­ders of mag­ni­tude. The two ex­am­ples men­tioned in sub­sec­tion 14.20.2 man­aged just 1 or­der of mag­ni­tude slow down. And not all ${\rm {E1}}$ tran­si­tions are for­bid­den any­way. And light nu­clei, for which isospin con­ser­va­tion is pre­sum­ably more ac­cu­rate, seem no worse than heav­ier ones in fig­ure 14.63. Ac­tu­ally, the two best data are small nu­clei.

To be sure, the above ar­gu­ments im­plic­itly as­sume that a bull’s eye is hit by an in­cred­i­bly ac­cu­rate can­ce­la­tion of op­po­site terms in the so-called ma­trix el­e­ment that de­scribes tran­si­tions. There is an al­ter­nate pos­si­bil­ity. The fi­nal nu­clear wave func­tion could be zero where the ini­tial one is nonzero and vice versa. In that case, the in­te­grand in the ma­trix el­e­ment is every­where zero, and no ac­cu­rate can­cel­la­tions are needed.

Fig­ure 14.65: Com­par­isons of de­cay rates be­tween the same ini­tial and fi­nal states.
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(404,55...
...3,187.7){\makebox(0,0)[bl]{odd-odd}}
\end{picture}}
\end{picture}
\end{figure}

But con­sider now the top half of fig­ure 14.65. Here mixed ${\rm {E1+M2}}$ tran­si­tions are plot­ted. These tran­si­tions take some­times place through the elec­tric di­pole mech­a­nism and some­times through the mag­netic quadru­pole one. Note that there are three very fast ${\rm {M2}}$ tran­si­tions, the first, fourth, and tenth. These tran­si­tions oc­cur at rates of 49, 58, re­spec­tively 21 times faster than the best guess based on the sin­gle-par­ti­cle model. So the ini­tial and fi­nal wave func­tions must un­avoid­ably over­lap very well. But how then to ex­plain that the cor­re­spond­ing elec­tric rates are 31 000, 7 800, and 200 times slower than best guess? The ini­tial and fi­nal wave func­tions are the same. While there is a dif­fer­ent op­er­a­tor sand­wiched in be­tween, {A.25}, the pic­ture still be­comes that one op­er­a­tor ap­par­ently achieves a bull’s eye of per­fect can­cel­la­tion. There should be an ex­pla­na­tion.

The ex­am­ple text­book also notes: “Ex­per­i­men­tal val­ues higher than pre­dicted by [the Weis­skopf es­ti­mates] can mean, on the other hand, that the tran­si­tion in­volves the par­tic­i­pa­tion of more than one nu­cleon or even a col­lec­tive par­tic­i­pa­tion of the whole nu­cleus.” The text­book notes in par­tic­u­lar that the rea­son that most ${\rm {E2}}$ tran­si­tions are faster than the­ory is due to the fact that these tran­si­tions are com­mon among col­lec­tive bands, es­pe­cially ro­ta­tional bands in de­formed nu­clei.

This is a well es­tab­lished ar­gu­ment. In prin­ci­ple 50 pro­tons tran­si­tion­ing could in­deed ex­plain why many ${\rm {E2}}$ tran­si­tions in fig­ure 14.63 end up on the or­der of a rough fac­tor 50$\POW9,{2}$ faster than the­ory.

But there are again some prob­lems. For one, the mixed ${\rm {E1+M2}}$ tran­si­tions dis­cussed ear­lier are not among states in the same ro­ta­tional bands. The nu­clear par­ity flips over in them. But the men­tioned ex­am­ples showed that sev­eral ${\rm {M2}}$ tran­si­tions were also much faster than sin­gle-par­ti­cle the­ory. While that might still be due to col­lec­tive mo­tion, it does not ex­plain why the ${\rm {E1}}$ tran­si­tions then were so slow.

Con­sider also the bot­tom of fig­ure 14.65. Here mixed ${\rm {M1+E2}}$ tran­si­tions are plot­ted. Note that the ${\rm {E2}}$ tran­si­tions are again much faster than the­ory, with few ex­cep­tions. But how then to ex­plain that the ${\rm {M1}}$ tran­si­tions be­tween the same ini­tial and fi­nal states are much slower than the­ory? More of these mirac­u­lously ac­cu­rate can­cel­la­tions? There are quite a few tran­si­tions at the higher en­er­gies where the ${\rm {M1}}$ tran­si­tion pro­ceeds slower than the ${\rm {E2}}$ one, de­spite the dif­fer­ence in mul­ti­pole or­der.

It is true that the or­bital ef­fect is rel­a­tively mi­nor in mag­netic tran­si­tions of min­i­mal mul­ti­pole or­der, {A.25.8}. But the tran­si­tions given by black sym­bols with open cen­ters in the bot­tom of fig­ure 14.65 are not of min­i­mal mul­ti­pole or­der. And in any case, rel­a­tively mi­nor gets nowhere near to ex­plain­ing dif­fer­ences of four or five or­ders of mag­ni­tude in rel­a­tive de­cay rates.

The same prob­lem ex­ists for the idea that the spe­cial na­ture of tran­si­tions within ro­ta­tional bands might some­how be re­spon­si­ble. Surely the idea of ro­ta­tional bands is not be far ac­cu­rate enough to ex­plain the hu­mon­gous dif­fer­ences? And some of the worst of­fend­ers in fig­ure 14.65 are def­i­nitely not be­tween states in the same ro­ta­tional band. Those are again the ones where the black sym­bol has an open cen­ter; the nu­clear spin does not change in those tran­si­tions.

There are 65 ran­domly cho­sen ${\rm {E1}}$ tran­si­tions plot­ted in fig­ure 14.63. Out of these 65, only one man­ages to achieve the best guess the­o­ret­i­cal tran­si­tion rate. That is a boron-10 tran­si­tion. (You may have to strain your eyes to see it, it is such a small nu­cleus. It is right on top of the best-guess line, just be­fore 1 MeV.) On the other hand, one tran­si­tion is slower than best guess by more than 8 or­ders of mag­ni­tude, and an­other three are slower by more than 7 or­ders of mag­ni­tude.

Com­pare that with the 67 ${\rm {E2}}$ tran­si­tions. Only one man­ages the three or­ders of mag­ni­tude slower than best guess that is so ho-hum for ${\rm {E1}}$ tran­si­tions. That tran­si­tion is a 2 340 keV $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 29}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\kern-.05em\rule{0pt}{8pt}^{+}$ to 2 063 keV $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 25}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\kern-.05em\rule{0pt}{8pt}^{+}$ ${}\fourIdx{205}{85}{}{}{\rm {At}}$ one. The amount of spin that is in­volved here is not ex­actly run-of-the mill. Note also that the three run­ners-up for be­ing far above the ${\rm {E2}}$ line are ${\rm {E1}}$ tran­si­tions, rather than ${\rm {E2}}$ ones...

Also, why do ${\rm {E3}}$ tran­si­tions act much like ${\rm {E1}}$ tran­si­tions in the first half of the en­ergy range and like ${\rm {E2}}$ ones over the sec­ond half? It seems weird.

The ex­am­ple text­book con­cludes: “In fig­ure [...] one notes very good agree­ment be­tween the­o­ret­i­cal val­ues and ex­per­i­men­tal ones for ${\rm {M4}}$. This be­hav­ior is typ­i­cal for tran­si­tions of high mul­ti­po­lar­ity.” Based on fig­ures 14.63 and 14.64, it seems very op­ti­mistic to call ${\rm {M4}}$ tran­si­tions typ­i­cal for high mul­ti­po­lar­ity.

Need­less to say, then, the au­thor of this book finds the dis­cus­sion of mea­sured gamma de­cay ver­sus the­ory in stan­dard nu­clear text books grossly in­ad­e­quate, and highly un­con­vinc­ing where it is given at all. If you feel the same way, see note {N.36} for one al­ter­na­tive idea.


14.20.6 Draft: In­ter­nal con­ver­sion

In in­ter­nal con­ver­sion, a nu­cleus gets rid of ex­ci­ta­tion en­ergy by kick­ing an atomic elec­tron out of the atom. This is most im­por­tant for tran­si­tions be­tween states of zero spin. For such tran­si­tions, the nor­mal gamma de­cay process of emit­ting a pho­ton is not pos­si­ble since a pho­ton can­not be emit­ted with zero an­gu­lar mo­men­tum. How­ever, the ejected elec­tron, called the “con­ver­sion elec­tron,” can keep what­ever an­gu­lar mo­men­tum it has. (For prac­ti­cal pur­poses, that is zero. Elec­trons that are not in s states have neg­li­gi­ble prob­a­bilil­ity of be­ing found in­side the nu­cleus.) Tran­si­tions in which no an­gu­lar mo­men­tum is emit­ted by the nu­cleus are called ${\rm {E}}0$ tran­si­tions.

A ball­park de­cay rate for ${\rm {E}}0$ in­ter­nal con­ver­sion can be found in Blatt & Weis­skopf [7,8, p. 621]. Where else. Con­verted to look sim­i­lar to the gamma de­cay Weis­skopf es­ti­mate (A.190), it reads

\begin{displaymath}
\fbox{$\displaystyle
\lambda^{{\rm{E}}0}_{\rm Blatt \& Wei...
...mega\equiv\frac{Q}{\hbar} \quad k\equiv\frac{Q}{\hbar c}
$} %
\end{displaymath} (14.68)

Here $\alpha$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^2$$\raisebox{.5pt}{$/$}$$4\pi\epsilon_0{\hbar}c$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ 1/137 is the fine struc­ture con­stant and $Q$ is the en­ergy re­lease. Fur­ther ${m_{\rm e}}c^2$ is the rest mass en­ergy of the elec­tron, which is about half an MeV. The ini­tial and fi­nal par­i­ties need to be the same.

Note that the first three fac­tors in the ex­pres­sion above look much like an ${\rm {E}}2$ elec­tric tran­si­tion. How­ever, the next three fac­tors are very small, though less so for heavy nu­clei. On the other hand the fi­nal fac­tor can be very large if the en­ergy re­lease $Q$ is much less than an MeV. So ${\rm {E}}0$ in­ter­nal con­ver­sion is rel­a­tively speak­ing most ef­fec­tive for low-en­ergy tran­si­tions in heavy nu­clei.

Putting in the num­bers gives the equiv­a­lent of (14.67) as

\begin{displaymath}
\fbox{$\displaystyle
\lambda^{{\rm{E}}0} = 3.8 Z^3 A^{4/3} Q^{1/2}
$} %
\end{displaymath} (14.69)

Once again, the en­ergy re­lease should be in MeV and then the de­cay rate will be per sec­ond. Note that ab­solutely speak­ing the de­cay rate does in fact in­crease with the en­ergy re­lease, but very weakly.


Ta­ble 14.5: Half lifes for E0 tran­si­tions.
\begin{table}\begin{displaymath}
\renewedcommand{arraystretch}{1.3}
\setlength{...
... 0&.&000003 & 0.18 \\
\hline\hline
\end{array} \end{displaymath}
\end{table}


Ta­ble 14.5 shows how the es­ti­mate stands up to scrutiny. The listed ${\rm {E}}0$ tran­si­tions are all those for which NuDat 2, [[12]], gives use­ful and un­am­bigu­ous data. All these turn out to be 0$\POW9,{+}$ to 0$\POW9,{+}$ tran­si­tions.

The sec­ond-last col­umn in the ta­ble shows a scaled half life. It is scaled to some (har­monic) av­er­age nu­cleus size $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 16, $A$ $\vphantom0\raisebox{1.5pt}{$=$}$ 32. In par­tic­u­lar

\begin{displaymath}
\tau_{1/2,\rm red} \equiv
\tau_{1/2} \frac{Z^3}{16^3} \frac{A^{4/3}}{32^{4/3}}
\end{displaymath}

The fi­nal col­umn shows what the scaled half life should be ac­cord­ing to the the­o­ret­i­cal es­ti­mate above. Note that, ex­clud­ing the fi­nal three nu­clei, the agree­ment is not too bad, as they come. What is an or­der of mag­ni­tude or so be­tween friends? Af­ter the pre­vi­ous sub­sec­tions every­thing would look ac­cu­rate.

How­ever, the fi­nal three nu­clei de­cay much more rapidly than in­ter­nal con­ver­sion pre­dicts. A sec­ond de­cay process oc­curs here: elec­tron-positron pair cre­ation. This re­quires that the nu­clear en­ergy re­lease $Q$ is at least large enough to pro­vide the rest mass en­ergy of the elec­tron and positron. That is a bit over a MeV. How­ever, as the ta­ble sug­gests, to get a sig­nif­i­cant ef­fect, more en­ergy is needed. There should be enough ad­di­tional en­ergy to give the elec­tron and positron rel­a­tivis­ti­cally non­triv­ial ki­netic en­er­gies.

In tran­si­tions other than be­tween states of zero spin, nor­mal gamma de­cay is pos­si­ble. But even in those de­cays, in­ter­nal con­ver­sion and pair pro­duc­tion may com­pete with gamma de­cay. They are es­pe­cially im­por­tant in highly for­bid­den gamma de­cays.

The so-called “in­ter­nal con­ver­sion co­ef­fi­cient” $\alpha_\ell$ gives the in­ter­nal con­ver­sion rate of a tran­si­tion as a frac­tion of its gamma de­cay rate:

\begin{displaymath}
\fbox{$\displaystyle
\alpha_\ell = \frac{\lambda_{\rm IC}}{\lambda_\gamma}
$} %
\end{displaymath} (14.70)

The fol­low­ing ball­park val­ues for the in­ter­nal con­ver­sion co­ef­fi­cient in elec­tric and mag­netic tran­si­tions can be de­rived ig­nor­ing rel­a­tivis­tic ef­fects and elec­tron bind­ing en­ergy:

\begin{displaymath}
\fbox{$\displaystyle
\alpha_{{\rm{E}}\ell} = \frac{1}{n^3}...
...pha Z)^3
\left(\frac{2 m_{\rm e}c^2}{Q}\right)^{\ell+3/2}
$}
\end{displaymath} (14.71)

Here, once again, $\ell$ is the mul­ti­pole or­der of the de­cay, $Q$ the nu­clear en­ergy re­lease, and $\alpha$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ 1/137 the fine struc­ture con­stant. Fur­ther $n$ is the prin­ci­pal quan­tum num­ber of the atomic shell that the con­ver­sion elec­tron comes from, Note the bril­liance of us­ing the same sym­bol for the in­ter­nal con­ver­sion co­ef­fi­cients as for the fine struc­ture con­stant. This book will use sub­scripts to keep them apart.

The above es­ti­mates are very rough. They are rou­tinely off by a cou­ple of or­ders of mag­ni­tude. How­ever, they do pre­dict a few cor­rect trends. In­ter­nal con­ver­sion is rel­a­tively more im­por­tant com­pared to gamma de­cay if the en­ergy re­lease $Q$ of the de­cay is low, if the mul­ti­po­lar­ity $\ell$ is high, and if the nu­cleus is heavy. Ejec­tion from the $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 K shell tends to dom­i­nate ejec­tion from the other shells, but not to a dra­matic amount.

(You might won­der why the ear­lier ball­park for ${\rm {E}}0$ tran­si­tions looks math­e­mat­i­cally like an $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 rate, in­stead of some $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 one. The rea­son is that ${\rm {E}}0$ tran­si­tions do not cre­ate an elec­tro­mag­netic field out­side the nu­cleus, com­pare for ex­am­ple chap­ter 7.4.3. So the in­ter­ac­tion with the elec­tron is lim­ited to the in­te­rior of the nu­cleus. That re­duces the mag­ni­tude of the in­ter­ac­tion greatly.)

In­ter­nal con­ver­sion is es­pe­cially use­ful for in­ves­ti­gat­ing nu­clei be­cause the con­ver­sion co­ef­fi­cients are dif­fer­ent for elec­tric and mag­netic tran­si­tions. There­fore, de­tailed de­cay mea­sure­ments can shed light on the ques­tion whether a given tran­si­tion is an elec­tric or a mag­netic one. Since they also de­pend strongly on the mul­ti­pole or­der, they also help es­tab­lish that. To be sure, the es­ti­mates above are not by far ac­cu­rate enough to do these things. But much more ac­cu­rate val­ues have been com­puted us­ing rel­a­tivis­tic the­o­ries and tab­u­lated.

In­ter­nal pair for­ma­tion sup­ple­ments in­ter­nal con­ver­sion, [7,8, p. 622]. The pair for­ma­tion rate is largest where the in­ter­nal con­ver­sion rate is small­est. That is in the re­gion of low atomic num­ber and high tran­si­tion en­er­gies.

One very old ref­er­ence in­cor­rectly states that in­ter­nal con­ver­sion hap­pens when a gamma ray emit­ted by the nu­cleus knocks a sur­round­ing elec­tron out of the atom. Such a process, the pho­to­elec­tric ef­fect, is in prin­ci­ple pos­si­ble, but its prob­a­bil­ity would be neg­li­gi­bly small. Note in par­tic­u­lar that in many de­cays, al­most no gamma rays are emit­ted but lots of con­ver­sion elec­trons. (While the in­ter­ac­tion be­tween the nu­cleus and the con­ver­sion elec­tron is of course caused by pho­tons, these are vir­tual pho­tons. They would not come out of the nu­cleus even if you stripped away the atomic elec­trons.)

It may be noted that in­ter­nal con­ver­sion is not unique to nu­clei. En­er­getic atomic elec­tron tran­si­tions can also get rid of their en­ergy by ejec­tion of an­other elec­tron. The ejected elec­trons are called “Auger elec­trons.” They are named af­ter the physi­cist Auger, who was the first man to dis­cover the process. (Some un­scrupu­lous woman, Lise Meit­ner, had dis­cov­ered and pub­lished it ear­lier, self­ishly at­tempt­ing to steal Auger’s credit, {N.35}). In fact, in­ter­nal con­ver­sion can give rise to ad­di­tional Auger elec­trons as other elec­trons rush in to fill the in­ter­nal con­ve­rion elec­tron hole. And so can elec­tron cap­ture.