A.25 Mul­ti­pole tran­si­tions

This ad­den­dum gives a de­scrip­tion of the mul­ti­pole in­ter­ac­tion be­tween atoms or nu­clei and elec­tro­mag­netic fields. In par­tic­u­lar, the spon­ta­neous emis­sion of a pho­ton of elec­tro­mag­netic ra­di­a­tion in an atomic or nu­clear tran­si­tion will be ex­am­ined. But stim­u­lated emis­sion and ab­sorp­tion are only triv­ially dif­fer­ent.

The ba­sic ideas were al­ready worked out in ear­lier ad­denda, es­pe­cially in {A.21} on pho­ton wave func­tions and {A.24} on spon­ta­neous emis­sion. How­ever, these ad­denda left the ac­tual in­ter­ac­tion be­tween the atom or nu­cleus and the field largely un­spec­i­fied. Only a very sim­ple form of the in­ter­ac­tion, called the elec­tric di­pole ap­prox­i­ma­tion, was worked out there.

Many tran­si­tions are not pos­si­ble by the elec­tric di­pole mech­a­nism. This ad­den­dum will de­scribe the more gen­eral mul­ti­pole in­ter­ac­tion mech­a­nisms. That will al­low rough es­ti­mates of how fast var­i­ous pos­si­ble tran­si­tions oc­cur. These will in­clude the Weis­skopf and Moszkowski es­ti­mates for the gamma de­cay of nu­clei. It will also al­low a gen­eral de­scrip­tion ex­actly how the se­lec­tion rules of chap­ter 7.4.4 re­late to nu­clear and pho­ton wave func­tions.

The over­all pic­ture is that be­fore the tran­si­tion, the atom or nu­cleus is in a high en­ergy state $\psi_{\rm {H}}$. Then it tran­si­tions to a lower en­ergy state $\psi_{\rm {L}}$. Dur­ing the tran­si­tion it emits a pho­ton that car­ries away the ex­cess en­ergy. The en­ergy of that pho­ton is re­lated to its fre­quency $\omega$ by the Planck-Ein­stein re­la­tion:

$$E_{\rm {H}}-E_{\rm {L}} = \hbar\omega_0 \approx \hbar\omega$$

Here $\omega_0$ is the nom­i­nal fre­quency of the pho­ton. The ac­tual pho­ton fre­quency $\omega$ might be slightly dif­fer­ent; there can be some slop in en­ergy con­ser­va­tion. How­ever, that will be taken care of by us­ing Fermi’s golden rule, chap­ter 7.6.1.

It is of­ten use­ful to ex­press the pho­ton fre­quency in terms of the so-called wave num­ber $k$:

$$\omega = k c$$

Here $c$ is the speed of light. The wave num­ber is a phys­i­cally im­por­tant quan­tity since it is in­versely pro­por­tional to the wave length of the pho­ton. If the typ­i­cal size of the atom or nu­cleus is $R$, then $kR$ is an nondi­men­sion­al quan­tity. It de­scribes the ra­tio of atom or nu­cleus size to pho­ton wave length. Nor­mally this ra­tio is very small, which al­lows help­ful sim­pli­fi­ca­tions.

It will be as­sumed that only the elec­trons need to be con­sid­ered for atomic tran­si­tions. The nu­cleus is too heavy to move much in such tran­si­tions. For nu­clear tran­si­tions, (in­side the nu­clei), it is usu­ally nec­es­sary to con­sider both types of nu­cle­ons, pro­tons and neu­trons. Pro­tons and neu­trons will be treated as point par­ti­cles, though each is re­ally a com­bi­na­tion of three quarks.

As noted in chap­ter 7.5.3 and 7.6.1, the dri­ving force in a tran­si­tion is the so-called Hamil­ton­ian ma­trix el­e­ment:

$$H_{21} = {\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\vert} H {\left\vert\psi_{\rm {H}}\right\rangle}$$

Here $H$ is the Hamil­ton­ian, which will de­pend on the type of tran­si­tion. In par­tic­u­lar, it de­pends on the prop­er­ties of the emit­ted pho­ton.

If the ma­trix el­e­ment $H_{21}$ is zero, tran­si­tions of that type are not pos­si­ble. The tran­si­tion is for­bid­den. If the ma­trix el­e­ment is very small, they will be very slow. (If the term for­bid­den is used with­out qual­i­fi­ca­tion, it in­di­cates that the elec­tric-di­pole type of tran­si­tion can­not oc­cur,)

A.25.1 Ap­prox­i­mate Hamil­ton­ian

The big ideas in mul­ti­pole tran­si­tions are most clearly seen us­ing a sim­ple model. That model will be ex­plained in this sub­sec­tion. How­ever, the re­sults in this sub­sec­tion will not be quan­ti­ta­tively cor­rect for mul­ti­pole tran­si­tions of higher or­der. Later sub­sec­tions will cor­rect these de­fi­cien­cies. This two-step ap­proach is fol­lowed be­cause oth­er­wise it can be easy to get lost in all the math­e­mat­ics of mul­ti­pole tran­si­tions. Also, the ter­mi­nol­ogy used in mul­ti­pole tran­si­tions re­ally arises from the sim­ple model dis­cussed here. And in any case, the needed cor­rec­tions will turn out to be very sim­ple.

An elec­tro­mag­netic wave con­sists of an elec­tric field $\skew3\vec{\cal E}$ and a mag­netic field $\skew2\vec{\cal B}$. A ba­sic plane wave takes the form, (13.10):

$$\skew3\vec{\cal E}= {\hat\imath}\sqrt{2}{\cal E}_0 \cos\Big... ...{\sqrt{2}{\cal E}_0}{c} \cos\Big(kz - \omega t - \alpha_0\Big)$$

For con­ve­nience the $z$-​axis was taken in the di­rec­tion of prop­a­ga­tion of the wave. Also the $x$-​axis was taken in the di­rec­tion of the elec­tric field. The con­stant $c$ is the speed of light and the con­stant ${\cal E}_0$ is the root-mean-square value of the elec­tric field. (The am­pli­tude of the elec­tric field is then $\sqrt{2}{\cal E}_0$, but the root mean square value is more closely re­lated to what you end up with when the elec­tro­mag­netic field is prop­erly quan­tized.) Fi­nally $\alpha_0$ is some unim­por­tant phase an­gle.

The above waves need to be writ­ten as com­plex ex­po­nen­tials us­ing the Euler for­mula (2.5):

$$\skew3\vec{\cal E}= {\hat\imath}\frac{{\cal E}_0}{\sqrt{2}}... ...-\omega t-\alpha_0)} + e^{-{\rm i}(kz-\omega t-\alpha_0)}\Big)$$

Only one of the two ex­po­nen­tials will turn out to be rel­e­vant to the tran­si­tion process. For ab­sorp­tion that is the first ex­po­nen­tial. But for emis­sion, the case dis­cussed here, the sec­ond ex­po­nen­tial ap­plies.

There are dif­fer­ent ways to see why only one ex­po­nen­tial is rel­e­vant. Chap­ter 7.7 fol­lows a clas­si­cal ap­proach in which the field is given. In that case, the evo­lu­tion equa­tion that gives the tran­si­tion prob­a­bil­ity is, {D.38},

$${\rm i}\hbar \dot {\bar c}_{2} \approx {H}_{\rm {21}} e^{{\rm i}(E_2-E_1)t/\hbar}$$

Here $\vert c_2\vert^2$ is the tran­si­tion prob­a­bil­ity. For emis­sion, the fi­nal state is the low en­ergy state. Then the Planck-Ein­stein re­la­tion gives the ex­po­nen­tial above as $e^{-{\rm i}\omega_0t}$. (By con­ven­tion, fre­quen­cies are taken to be pos­i­tive.) Now the Hamil­ton­ian ma­trix el­e­ment $H_{21}$ will in­volve the elec­tric and mag­netic fields, with their ex­po­nen­tials. The first ex­po­nen­tials, com­bined with the ex­po­nen­tial above, pro­duce a time-de­pen­dent fac­tor $e^{-{\rm i}(\omega_0+\omega)t}$. Since nor­mal pho­ton fre­quen­cies are large, this fac­tor os­cil­lates ex­tremely rapidly in time. Be­cause of these os­cil­la­tions, the cor­re­spond­ing terms never pro­duce a sig­nif­i­cant con­tri­bu­tion to the tran­si­tion prob­a­bil­ity. Op­po­site con­tri­bu­tions av­er­age away against each other. So the first ex­po­nen­tials can be ig­nored. But the sec­ond ex­po­nen­tials pro­duce a time de­pen­dent fac­tor $e^{-{\rm i}(\omega_0-\omega)t}$. That does not os­cil­late rapidly pro­vided that the emit­ted pho­ton has fre­quency $\omega$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ $\omega_0$. So such pho­tons can achieve a sig­nif­i­cant prob­a­bil­ity of be­ing emit­ted.

For ab­sorp­tion, the low en­ergy state is the first one, in­stead of the sec­ond. That makes the ex­po­nen­tial above $e^{+{\rm i}\omega_0t}$, and the en­tire story in­verts.

The bet­ter way to see that the first ex­po­nen­tials in the fields drop out is to quan­tize the elec­tro­mag­netic field. This book cov­ers that only in the ad­denda. In par­tic­u­lar, ad­den­dum {A.24} de­scribed the process. For­tu­nately, quan­ti­za­tion of the elec­tro­mag­netic field is mainly im­por­tant to fig­ure out the right value of the con­stant ${\cal E}_0$ to use, es­pe­cially for spon­ta­neous emis­sion. It does not di­rectly af­fect the ac­tual analy­sis in this ad­den­dum. In par­tic­u­lar the con­clu­sion re­mains that only the sec­ond ex­po­nen­tials sur­vive.

The bot­tom line is that for emis­sion

$$\skew3\vec{\cal E}= {\hat\imath}\frac{{\cal E}_0}{\sqrt{2}}... ...rac{{\cal E}_0}{\sqrt{2}c}e^{-{\rm i}(kz-\omega t-\alpha_0)} %$$ (A.168)

Also, as far as this ad­den­dum is con­cerned, the dif­fer­ence be­tween spon­ta­neous and stim­u­lated emis­sion is only in the value of the con­stant ${\cal E}_0$.

Next the Hamil­ton­ian is needed. For the ma­trix el­e­ment, only the part of the Hamil­ton­ian that de­scribes the in­ter­ac­tion be­tween the atom or nu­cleus and the elec­tro­mag­netic fields is rel­e­vant, {A.24}. (Re­call that the ma­trix el­e­ment dri­ves the tran­si­tion process; no in­ter­ac­tion means no tran­si­tion.) As­sume that the elec­trons in the atom, or the pro­tons and neu­trons in the nu­cleus, are num­bered us­ing an in­dex $i$. Then by ap­prox­i­ma­tion the in­ter­ac­tion Hamil­ton­ian of a sin­gle par­ti­cle $i$ with the elec­tro­mag­netic field is

$$H_i \approx - q_i \skew3\vec{\cal E}_i\cdot{\skew0\vec r}_i... ...2m_i} g_i \skew2\vec{\cal B}_i\cdot{\skew 6\widehat{\vec S}}_i$$

In gen­eral, you will need to sum this over all par­ti­cles $i$. But the dis­cus­sion here will usu­ally look at one par­ti­cle at a time.

The first term in the Hamil­ton­ian above is like the $mgh$ po­ten­tial of grav­ity, with the par­ti­cle charge $q_i$ tak­ing the place of the mass $m$, the elec­tric field that of the ac­cel­er­a­tion of grav­ity $g$, and the par­ti­cle po­si­tion ${\skew0\vec r}_i$ that of the height $h$.

The sec­ond and third terms in the Hamil­ton­ian are due to the fact that a charged par­ti­cle that is go­ing around in cir­cles acts as a lit­tle elec­tro­mag­net. An elec­tro­mag­net wants to align it­self with an am­bi­ent mag­netic field. That is just like a com­pass nee­dle aligns it­self with the mag­netic field of earth.

This ef­fect shows up as soon as there is an­gu­lar mo­men­tum. In­deed, the op­er­a­tor ${\skew 4\widehat{\vec L}}_i$ above is the or­bital an­gu­lar mo­men­tum of the par­ti­cle and ${\skew 6\widehat{\vec S}}_i$ is the spin. The fac­tor $g_i$ is a nondi­men­sion­al num­ber that de­scribes the rel­a­tive ef­fi­ciency of the par­ti­cle spin in cre­at­ing an elec­tro­mag­netic re­sponse. For an elec­tron in an atom, $g_i$ is very close to 2. That is a the­o­ret­i­cal value ex­pected for fun­da­men­tal par­ti­cles, chap­ter 13.4. How­ever, for a pro­ton in a nu­cleus the value is about 5.6, as­sum­ing that the ef­fect of the sur­round­ing pro­tons and neu­trons can be ig­nored. (Ac­tu­ally, it is quite well es­tab­lished that nor­mally the sur­round­ing par­ti­cles can­not be ig­nored. But it is dif­fi­cult to say what value for $g_i$ to use in­stead, ex­cept that it will surely be smaller than 5.6, and greater than 2.)

A spe­cial case needs to be made for the neu­trons in a nu­cleus. Since the neu­tron has no charge, $q_i$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, you would ex­pect that its con­tri­bu­tion to the Hamil­ton­ian is zero. How­ever, the fi­nal term in the Hamil­ton­ian is not zero. A neu­tron has a mag­netic re­sponse. (A neu­tron con­sists of three charged quarks. The com­bined charge of the three is zero, but the com­bined mag­netic re­sponse is not.) To ac­count for that, in the fi­nal term, you need to use the charge $e$ and mass $m_{\rm p}$ of the pro­ton, and take $g_i$ about $\vphantom{0}\raisebox{1.5pt}{$-$}$3.8. This value of $g_i$ ig­nores again the ef­fects of sur­round­ing pro­tons and neu­trons.

There are ad­di­tional is­sues that are im­por­tant. Of­ten it is as­sumed that in a tran­si­tion only a sin­gle par­ti­cle changes states. If that par­ti­cle is a neu­tron, it might then seem that the first two terms in the Hamil­ton­ian can be ig­nored. But ac­tu­ally, the neu­tron and the rest of the nu­cleus move around their com­mon cen­ter of grav­ity. And the rest of the nu­cleus is charged. So nor­mally the first two terms can­not be ig­nored. This is mainly im­por­tant for the so-called elec­tric di­pole tran­si­tions; for higher mul­ti­pole or­ders, the elec­tro­mag­netic field is very small near the ori­gin, and the mo­tion of the rest of the nu­cleus does not pro­duce much ef­fect. In a tran­si­tion of a sin­gle pro­ton, you may also want to cor­rect the first term for the mo­tion of the rest of the nu­cleus. But also note that the rest of the nu­cleus is not re­ally a point par­ti­cle. That may make a sig­nif­i­cant dif­fer­ence for higher mul­ti­pole or­ders. There­fore sim­ple cor­rec­tions re­main prob­lem­atic. See [33] and [11] for fur­ther dis­cus­sion of these non­triv­ial is­sues.

The given Hamil­ton­ian ig­nores the fact that the elec­tric and mag­netic fields are un­steady and not uni­form. That is the rea­son why the higher mul­ti­poles found in the next sub­sec­tion will not be quite right. They will be good enough to show the ba­sic ideas how­ever. And the quan­ti­ta­tive prob­lems will be cor­rected in later sub­sec­tions.

A.25.2 Ap­prox­i­mate mul­ti­pole ma­trix el­e­ments

The last step is to write down the ma­trix el­e­ment. Sub­sti­tut­ing the ap­prox­i­mate Hamil­ton­ian and fields of the pre­vi­ous sub­sec­tion into the ma­trix el­e­ment of the in­tro­duc­tion gives:

$$H_{21,i} = {\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\... ...i{\widehat S}_{i,y})] {\left\vert\psi_{\rm {H}}\right\rangle}$$

This will nor­mally need to be summed over all elec­trons $i$ in the atom, or all nu­cle­ons $i$ in the nu­cleus. Note that the time de­pen­dent part of the ex­po­nen­tial is of no in­ter­est. It will in fact not even ap­pear when the elec­tro­mag­netic field is prop­erly quan­tized, {A.24}. In a clas­si­cal treat­ment, it drops out ver­sus the $e^{{\rm i}(E_2-E_1)t/\hbar}$ ex­po­nen­tial men­tioned in the pre­vi­ous sub­sec­tion.

To split the above ma­trix el­e­ment into dif­fer­ent mul­ti­pole or­ders, write the ex­po­nen­tial as a Tay­lor se­ries:

$$e^{-{\rm i}kz_i} = \sum_{n=0}^\infty \frac{(-{\rm i}kz_i)^n... ...\sum_{\ell=1}^\infty \frac{(-{\rm i}kz_i)^{\ell-1}}{(\ell-1)!}$$

In the sec­ond equal­ity, the sum­ma­tion in­dex was reno­tated as $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell-1$. The rea­son is that $\ell$ turns out to be what is con­ven­tion­ally de­fined as the mul­ti­pole or­der.

Us­ing this Tay­lor se­ries, the ma­trix el­e­ment gets split into sep­a­rate elec­tric and mag­netic mul­ti­pole con­tri­bu­tions:

$$\begin{eqnarray*} & \displaystyle H_{21,i} = \sum_{\ell=1}^\infty H_{21,i}^{\r... ...y}+g_i{\widehat S}_{i.y}){\left\vert\psi_{\rm {H}}\right\rangle} \end{eqnarray*}$$

The terms with $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 are the di­pole ones, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 the quadru­pole ones, 3 the oc­tu­pole ones, 4 the hexa­de­ca­pole ones, etcetera. Su­per­script ${\rm {E}}$ in­di­cates an elec­tric con­tri­bu­tion, ${\rm {M}}$ a mag­netic one. The first con­tri­bu­tion that is nonzero gives the low­est mul­ti­pole or­der that is al­lowed.

A.25.3 Cor­rected mul­ti­pole ma­trix el­e­ments

The mul­ti­pole ma­trix el­e­ments of the pre­vi­ous sub­sec­tion were rough ap­prox­i­ma­tions. The rea­son was the ap­prox­i­mate Hamil­ton­ian that was used. This sub­sec­tion will de­scribe the cor­rec­tions needed to fix them up. It will still be as­sumed that the atomic or nu­clear par­ti­cles in­volved are non­rel­a­tivis­tic. They usu­ally are.

The cor­rected Hamil­ton­ian is

$$\fbox{$\displaystyle H = \sum_i \left[\frac{1}{2m_i}\left(... ...{\cal B}_i \cdot {\skew 6\widehat{\vec S}}_i \right] + V $} %$$ (A.169)

where the sum is over the in­di­vid­ual elec­trons in the atom or the pro­tons and neu­trons in the nu­cleus. In the sum, $m_i$ is the mass of the par­ti­cle, ${\skew 4\widehat{\skew{-.5}\vec p}}_i$ its mo­men­tum, and $q_i$ its charge. The po­ten­tial $V$ is the usual po­ten­tial that keeps the par­ti­cle in­side the atom or nu­cleus. The re­main­ing parts in the Hamil­ton­ian ex­press the ef­fect of the ad­di­tional ex­ter­nal elec­tro­mag­netic field. In par­tic­u­lar, $\varphi_i$ is the elec­tro­sta­tic po­ten­tial of the field and $\skew3\vec A_i$ the so-called vec­tor po­ten­tial, each eval­u­ated at the par­ti­cle po­si­tion. Fi­nally

$$\skew2\vec{\cal B}=\nabla\times\skew3\vec A$$

is the mag­netic part of the field. The spin ${\skew 6\widehat{\vec S}}_i$ of the par­ti­cle in­ter­acts with this field at the lo­ca­tion of the par­ti­cle, with a rel­a­tive strength given by the nondi­men­sion­al con­stant $g_i$. See chap­ter 1.3.2 for a clas­si­cal jus­ti­fi­ca­tion of this Hamil­ton­ian, or chap­ter 13 for a quan­tum one.

Non­rel­a­tivis­ti­cally, the spin does not in­ter­act with the elec­tric field. That is par­tic­u­larly lim­it­ing for the neu­tron, which has no net charge to in­ter­act with the elec­tric field. In re­al­ity, a rapidly mov­ing par­ti­cle with spin will also in­ter­act with the elec­tric field, {A.39}. See the Dirac equa­tion and in par­tic­u­lar {D.74} for a rel­a­tivis­tic de­scrip­tion of the in­ter­ac­tion of spin with an elec­tro­mag­netic field. That would be too messy to in­clude here, but it can be found in [44]. Note also that since in re­al­ity the neu­tron con­sists of three quarks, that should al­low it to in­ter­act di­rectly with a nonuni­form elec­tric field.

If the field is quan­tized, you will also want to in­clude the Hamil­ton­ian of the field in the to­tal Hamil­ton­ian above. And the field quan­ti­ties be­come op­er­a­tors. That goes the same way as in {A.24}. It makes no real dif­fer­ence for the analy­sis in this ad­den­dum.

It is al­ways pos­si­ble, and a very good idea, to take the un­per­turbed elec­tro­mag­netic po­ten­tials so that

$$\varphi = 0 \qquad \nabla\cdot\skew3\vec A=0$$

See for ex­am­ple the ad­den­dum on pho­ton wave func­tions {A.21} for more on that. That ad­den­dum also gives the po­ten­tials that cor­re­spond to pho­tons of def­i­nite lin­ear, re­spec­tively an­gu­lar mo­men­tum. These will be used in this ad­den­dum.

The square in the above Hamil­ton­ian may be mul­ti­plied out to give

$$H = H_0 + \sum_i \left[-\frac{q_i}{m_i}\skew3\vec A_i\cdot ... ...\skew2\vec{\cal B}_i \cdot {\skew 6\widehat{\vec S}}_i \right]$$

The term $H_0$ is the Hamil­ton­ian of the atom or nu­cleus in the ab­sence of in­ter­ac­tion with the ex­ter­nal elec­tro­mag­netic field. Like in the pre­vi­ous sub­sec­tion, it is not di­rectly rel­e­vant to the in­ter­ac­tion with the elec­tro­mag­netic field. Note fur­ther that ${\skew 4\widehat{\skew{-.5}\vec p}}$ and $\skew3\vec A$ com­mute be­cause $\nabla\cdot\skew3\vec A$ is zero. The term pro­por­tional to $\skew3\vec A^2$ will be ig­nored as it is nor­mally very small. (It gives rise to two-pho­ton emis­sion, [33].)

That makes the in­ter­ac­tion Hamil­ton­ian of a sin­gle par­ti­cle $i$ equal to

$$\fbox{$\displaystyle H_i = -\frac{q_i}{m_i}\skew3\vec A_i\... ...m_i}\skew2\vec{\cal B}_i\cdot{\skew 6\widehat{\vec S}}_i $} %$$ (A.170)

Note that the fi­nal spin term has not changed from the ap­prox­i­mate Hamil­ton­ian writ­ten down ear­lier. How­ever, the first term ap­pears com­pletely dif­fer­ent from be­fore. Still, there must ob­vi­ously be a con­nec­tion.

To find that con­nec­tion re­quires con­sid­er­able ma­nip­u­la­tion. First the vec­tor po­ten­tial $\skew3\vec A$ must be iden­ti­fied in terms of the sim­ple elec­tro­mag­netic wave as writ­ten down ear­lier in (A.168). To do so, note that the vec­tor po­ten­tial must be re­lated to the fields as

$$\skew3\vec{\cal E}= - \frac{\partial\skew3\vec A}{\partial t} \qquad \skew2\vec{\cal B}= \nabla\times\skew3\vec A$$

See, for ex­am­ple, {A.21} for a dis­cus­sion. That al­lows the vec­tor po­ten­tial cor­re­spond­ing to the sim­ple wave (A.168) to be iden­ti­fied as:

$$\skew3\vec A= -{\hat\imath}\frac{{\cal E}_0}{\sqrt{2}{\rm i}\omega} e^{-{\rm i}(kz-\omega t-\alpha_0)}$$

This wave can be gen­er­al­ized to al­low gen­eral di­rec­tions of wave prop­a­ga­tion and fields. That gives:

$$\skew3\vec A= - {\hat\imath}_{\cal E}\frac{{\cal E}_0}{\sqr... ...c} e^{-{\rm i}({\vec k}\cdot{\skew0\vec r}-\omega t-\alpha_0)}$$

Here the unit vec­tor ${\hat\imath}_{\cal E}$ is in the di­rec­tion of the elec­tric field and ${\hat\imath}_{\cal B}$ in the di­rec­tion of the mag­netic field. A unit vec­tor ${\hat\imath}_k$ in the di­rec­tion of wave prop­a­ga­tion can be de­fined as their cross prod­uct. This de­fines the wave num­ber vec­tor as

$${\vec k}\equiv {\hat\imath}_k k \qquad {\hat\imath}_k = {\... ...l B}\qquad {\hat\imath}_{\cal E}\cdot{\hat\imath}_{\cal B}= 0$$

The three unit vec­tors are or­tho­nor­mal. Note that for a given di­rec­tion of wave prop­a­ga­tion ${\hat\imath}_k$, there will be two in­de­pen­dent waves. They dif­fer in the di­rec­tion of the elec­tric field ${\hat\imath}_{\cal E}$. The choice for the di­rec­tion of the elec­tric field for first wave is not unique; the field must merely be or­thog­o­nal to the di­rec­tion of wave prop­a­ga­tion. An ar­bi­trary choice must be made. The elec­tric field of the sec­ond wave needs to be or­thog­o­nal to that of the first wave. The ex­am­ple in the pre­vi­ous sub­sec­tions took the wave prop­a­ga­tion in the $z$-​di­rec­tion, ${\hat\imath}_k$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hat k}$, and the elec­tric field in the $x$-​di­rec­tion, ${\hat\imath}_{\cal E}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hat\imath}$, to give the mag­netic field in the $y$-​di­rec­tion, ${\hat\imath}_{\cal B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hat\jmath}$. In that case the sec­ond in­de­pen­dent wave will have its elec­tric field in the $y$-​di­rec­tion, ${\hat\imath}_{\cal E}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hat\jmath}$, and its mag­netic field in the neg­a­tive $x$-​di­rec­tion, ${\hat\imath}_{\cal B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$${\hat\imath}$.

The sin­gle-par­ti­cle ma­trix el­e­ment is now, drop­ping again the time-​de­pen­dent fac­tors,

$$\begin{eqnarray*} H_{21,i} & = & {\left\langle\psi_{\rm {L}}\hspace{0.3pt}\righ... ...kew 6\widehat{\vec S}}_i {\left\vert\psi_{\rm {H}}\right\rangle} \end{eqnarray*}$$

The first term needs to be cleaned up to make sense out of it. That is an ex­tremely messy ex­er­cise, banned to {D.43}.

How­ever, the re­sult is much like be­fore:

$$H_{21,i} = \sum_{\ell=1}^\infty H_{21,i}^{\rm E\ell} + H_{2... ...m {M1}} + H_{21,i}^{\rm {E2}} + H_{21,i}^{\rm {M2}} + \ldots$$

where

$$\fbox{$\displaystyle H_{21,i}^{\rm E\ell} = - \frac{q_i{\c... ...1}r_{i,{\cal E}} {\left\vert\psi_{\rm{H}}\right\rangle} $} %$$ (A.171)

$$\fbox{$\displaystyle H_{21,i}^{\rm M\ell} \approx - \frac... ...\cal B}}\Big) {\left\vert\psi_{\rm{H}}\right\rangle} $} \, %$$ (A.172)

Here $r_{i,k}$ is the com­po­nent of the po­si­tion of the par­ti­cle in the di­rec­tion of mo­tion. Sim­i­larly, $r_{i,{\cal E}}$ is the com­po­nent of po­si­tion in the di­rec­tion of the elec­tric field, while the an­gu­lar mo­men­tum com­po­nents are in the di­rec­tion of the mag­netic field.

This can now be com­pared to the ear­lier re­sults us­ing the ap­prox­i­mate Hamil­ton­ian. Those ear­lier re­sults as­sumed the spe­cial case that the wave prop­a­ga­tion was in the $z$-​di­rec­tion and had its elec­tric field in the $x$-​di­rec­tion. In that case,

$$\mbox{example:}\qquad r_{i,k} = z_i \qquad r_{i,{\cal E}} ... ..._{i.y} \qquad {\widehat S}_{i,{\cal B}} = {\widehat S}_{i,y} %$$ (A.173)

Not­ing that, it is seen that the cor­rect elec­tric con­tri­bu­tions only dif­fer from the ap­prox­i­mate ones by a sim­ple fac­tor 1/$\ell$. This fac­tor is 1 for elec­tric di­pole con­tri­bu­tions, so these were cor­rect al­ready. Sim­i­larly, the mag­netic con­tri­bu­tion dif­fers only by the ad­di­tional fac­tor 2$\raisebox{.5pt}{$/$}$​$(\ell+1)$ for the or­bital an­gu­lar mo­men­tum from the ap­prox­i­mate re­sult. This fac­tor is 1 for mag­netic di­pole con­tri­bu­tions. So these too were al­ready cor­rect.

How­ever, there is a prob­lem with the elec­tric con­tri­bu­tion in the case of nu­clei. A nu­clear po­ten­tial does not just de­pend on the po­si­tion of the nu­clear par­ti­cles, but also on their mo­men­tum. That in­tro­duces an ad­di­tional term in the elec­tric con­tri­bu­tion, {D.43}. A ball­park for that term shows that this may well make the listed elec­tric con­tri­bu­tion quan­ti­ta­tively in­valid, {N.14}. Un­for­tu­nately, nu­clear po­ten­tials are not known to suf­fi­cient ac­cu­racy to give a solid pre­dic­tion for the con­tri­bu­tion. In the fol­low­ing, this prob­lem will usu­ally sim­ply be ig­nored, like other text­books do.

A.25.4 Ma­trix el­e­ment ball­parks

Re­call that elec­tro­mag­netic tran­si­tions are dri­ven by the ma­trix el­e­ment. The pre­vi­ous sub­sec­tion man­aged to split the ma­trix el­e­ment into sep­a­rate elec­tric and mag­netic mul­ti­pole con­tri­bu­tions. The in­tent in this sub­sec­tion is now to show that nor­mally, the first nonzero mul­ti­pole con­tri­bu­tion is the im­por­tant one. Sub­se­quent mul­ti­pole con­tri­bu­tions are nor­mally small com­pared to the first nonzero one.

To do so, this sub­sec­tion will ball­park the mul­ti­pole con­tri­bu­tions. The ball­parks will show that the mag­ni­tude of the con­tri­bu­tions de­creases rapidly with in­creas­ing mul­ti­pole or­der $\ell$.

But of course ball­parks are only that. If a con­tri­bu­tion is ex­actly zero for some spe­cial rea­son, (usu­ally a sym­me­try), then the ball­park is go­ing to be wrong. That is why it is the first nonzero mul­ti­pole con­tri­bu­tion that is im­por­tant, rather than sim­ply the first one. The next sub­sec­tion will dis­cuss the so-called se­lec­tion rules that de­ter­mine when con­tri­bu­tions are zero.

The ball­parks are for­mu­lated in terms a typ­i­cal size $R$ of the atom or nu­cleus. For the present pur­poses, this size will be taken to be the av­er­age ra­dial po­si­tion of the par­ti­cles away from the cen­ter of atom or nu­cleus. Then the mag­ni­tudes of the elec­tric mul­ti­pole con­tri­bu­tions can be writ­ten as

$$\vert H_{21,i}^{\rm E\ell}\vert = \frac{\vert q_i\vert{\cal... ...r_{i,{\cal E}}/R) {\left\vert\psi_{\rm {H}}\right\rangle}\vert$$

There is no easy way to say ex­actly what the in­ner prod­uct above will be. How­ever, since the po­si­tions in­side it have been scaled with the mean ra­dius $R$, its value is sup­pos­edly some nor­mal fi­nite num­ber. Un­less the in­ner prod­uct hap­pens to be zero for some spe­cial rea­son of course. As­sum­ing that this does not hap­pen, the in­ner prod­uct can be ig­nored for the ball­park. And that then shows that each higher nonzero elec­tric mul­ti­pole con­tri­bu­tion is smaller than the pre­vi­ous one by a fac­tor $kR$. Now $k$ is in­versely pro­por­tional to the wave length of the pho­ton that is emit­ted or ab­sorbed. This wave length is nor­mally very much larger than the size of the atom or nu­cleus $R$. That means that $kR$ is very small. And that then im­plies that a nonzero mul­ti­pole con­tri­bu­tion at a higher value of $\ell$ will be very much less than one at a lower value. So con­tri­bu­tions for val­ues of $\ell$ higher than the first nonzero one can nor­mally be ig­nored.

The mag­ni­tudes of the mag­netic con­tri­bu­tions can be writ­ten as

$$\vert H_{21,i}^{\rm M\ell}\vert \approx - \frac{\vert q_i\... ...rt\psi_{\rm {H}}\right\rangle}\vert \ell \frac{\hbar}{2m_icR}$$

Re­call that an­gu­lar mo­men­tum val­ues are mul­ti­ples of $\hbar$. There­fore the ma­trix el­e­ment can again be ball­parked as some fi­nite num­ber, if nonzero. So once again, the mul­ti­pole con­tri­bu­tions get smaller by a fac­tor $kR$ for each in­crease in or­der. That means that the nonzero mag­netic con­tri­bu­tions too de­crease rapidly with $\ell$.

That leaves the ques­tion how mag­netic con­tri­bu­tions com­pare to elec­tric ones. First com­pare a mag­netic mul­ti­pole term to the elec­tric one of the same mul­ti­pole or­der $\ell$. The above es­ti­mates show that the mag­netic term is mainly dif­fer­ent from the elec­tric one by the fac­tor

$$\frac{\hbar}{2 m_i c R} \approx \left\{ \begin{array}{cc} ... ...ac{\raisebox{-.5ex}{1\mbox{ fm}}}{10 R}} \end{array} \right.$$

Atomic sizes are in the or­der of an Ångstrom, and nu­clear ones in the or­der of a few fem­tome­ters. So ball­park mag­netic con­tri­bu­tions are small com­pared to elec­tric ones of the same or­der $\ell$. And more so for atoms than for nu­clei. (Tran­si­tion rates are pro­por­tional to the square of the first nonzero con­tri­bu­tion. So the ball­park tran­si­tion rate for a mag­netic tran­si­tion is smaller than an elec­tric one of the same or­der by the square of the above fac­tor.)

A some­what more phys­i­cal in­ter­pre­ta­tion of the above fac­tor can be given:

$$\frac{\hbar}{2 m_i c R} = \sqrt{\frac{T_{\rm bp}}{2 m_ic^2}} \qquad T_{\rm bp} \equiv \frac{\hbar^2}{2 m_i R^2}$$

Here $T_{\rm {bp}}$ is a ball­park for the ki­netic en­ergy $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar^2\nabla^2$$\raisebox{.5pt}{$/$}$​$2m_i$ of the par­ti­cle. Note that this ball­park is ex­act for the hy­dro­gen atom ground state if you take the Bohr ra­dius as the av­er­age ra­dius $R$ of the atom. How­ever, for heav­ier atoms and nu­clei, this ball­park may be low: it ig­nores the ex­clu­sion ef­fects of the other par­ti­cles. Fur­ther $m_ic^2$ is the rest mass en­ergy of the par­ti­cle. Now pro­tons and neu­trons in nu­clei, and at least the outer elec­trons in atoms are non­rel­a­tivis­tic; their ki­netic en­ergy is much less than their rest mass en­ergy. It fol­lows again that mag­netic con­tri­bu­tions are nor­mally much smaller than elec­tric ones of the same mul­ti­pole or­der.

Com­pare the mag­netic mul­ti­pole term also to the elec­tric one of the next mul­ti­pole or­der. The trail­ing fac­tor in the mag­netic el­e­ment can for this case be writ­ten as

$$\frac{\hbar}{2 m_i c R} = kR \frac{T_{\rm bp}}{\hbar\omega}$$

The de­nom­i­na­tor in the fi­nal ra­tio is the en­ergy of the emit­ted or ab­sorbed pho­ton. Typ­i­cally, it is sig­nif­i­cantly less than the ball­park ki­netic en­ergy of the par­ti­cle. That then makes mag­netic ma­trix el­e­ments sig­nif­i­cantly larger than elec­tric ones of the next-higher mul­ti­pole or­der. Though smaller than the elec­tric ones of the same or­der.

A.25.5 Se­lec­tion rules

Based on the ball­parks given in the pre­vi­ous sub­sec­tion, the $\rm {E1}$ elec­tric di­pole con­tri­bu­tion should dom­i­nate tran­si­tions. It should be fol­lowed in size by the $\rm {M1}$ mag­netic di­pole one, fol­lowed by the $\rm {E2}$ elec­tric quadru­pole one, etcetera.

But this or­der gets mod­i­fied be­cause ma­trix el­e­ments are very of­ten zero for spe­cial rea­sons. This was ex­plained phys­i­cally in chap­ter 7.4.4 based on the an­gu­lar mo­men­tum prop­er­ties of the emit­ted pho­ton. This sub­sec­tion will in­stead re­late it di­rectly to the ma­trix el­e­ment con­tri­bu­tions as iden­ti­fied in sub­sec­tion A.25.3. To sim­plify the rea­son­ing, it will again be as­sumed that the $z$-​axis is cho­sen in the di­rec­tion of wave mo­tion and the $y$-​axis in the di­rec­tion of the elec­tric field. So (A.173) ap­plies for the mul­ti­pole con­tri­bu­tions (A.171) and (A.172).

Con­sider first the elec­tric di­pole con­tri­bu­tion $H_{21,i}^{\rm {E}1}$. Ac­cord­ing to (A.171) and (A.173) this con­tri­bu­tion con­tains the in­ner prod­uct

$${\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\vert}x_i{\left\vert\psi_{\rm {H}}\right\rangle}$$

Why would this be zero? Ba­si­cally be­cause in the in­ner prod­uct in­te­grals, pos­i­tive val­ues of $x_i$ might ex­actly in­te­grate away against cor­re­spond­ing neg­a­tive val­ues. That can hap­pen be­cause of sym­me­tries in the nu­clear wave func­tions.

One such sym­me­try is par­ity. For all prac­ti­cal pur­poses, atomic and nu­clear states have def­i­nite par­ity. If the pos­i­tive di­rec­tions of the Carte­sian axes are in­verted, atomic and nu­clear states ei­ther stay the same (par­ity 1 or pos­i­tive), or change sign (par­ity $\vphantom{0}\raisebox{1.5pt}{$-$}$1 or neg­a­tive). As­sume, for ex­am­ple, that $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ have both pos­i­tive par­ity. That means that they do not change un­der an in­ver­sion of the axes. But the fac­tor $x_i$ in the in­ner prod­uct above has odd par­ity: axes in­ver­sion re­places $x_i$ by $-x_i$. So the com­plete in­ner prod­uct above changes sign un­der axes in­ver­sion. But in­ner prod­ucts are de­fined in a way that they do not change un­der axes in­ver­sion. (In terms of chap­ter 2.3, the ef­fect of the axes in­ver­sion can be un­done by a fur­ther in­ver­sion of the in­te­gra­tion vari­ables.) Some­thing can only change sign and still stay the same if it is zero, ($\vphantom{0}\raisebox{1.5pt}{$-$}$0 is 0 but say $\vphantom{0}\raisebox{1.5pt}{$-$}$5 is not 5).

So if both $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ have pos­i­tive par­ity, the elec­tric di­pole con­tri­bu­tion is zero. The only way to get a nonzero in­ner prod­uct is if ex­actly one of $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ has neg­a­tive par­ity. Then the fac­tor $\vphantom{0}\raisebox{1.5pt}{$-$}$1 that this state picks up un­der axes in­ver­sion can­cels the $\vphantom{0}\raisebox{1.5pt}{$-$}$1 from $x_i$, leav­ing the in­ner prod­uct un­changed as it should. So the con­clu­sion is that in elec­tric di­pole tran­si­tions $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ must have op­po­site par­i­ties. In other words, the atomic or nu­clear par­ity must flip over in the tran­si­tion. This con­di­tion is called the par­ity “se­lec­tion rule” for an elec­tric di­pole tran­si­tion. If it is not sat­is­fied, the elec­tric di­pole con­tri­bu­tion is zero and a dif­fer­ent con­tri­bu­tion will dom­i­nate. That con­tri­bu­tion will be much smaller than a typ­i­cal nonzero elec­tric di­pole one, so the tran­si­tion will be much slower.

The $H_{21,i}^{\rm {M}1}$ mag­netic di­pole con­tri­bu­tion con­tains the in­ner prod­uct

$${\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\vert}\L _{i,y}+g_i{\widehat S}_{i.y}{\left\vert\psi_{\rm {H}}\right\rangle}$$

The an­gu­lar mo­men­tum op­er­a­tors do noth­ing un­der axes in­ver­sion. One way to see that is to think of $\psi_{\rm {H}}$ as writ­ten in terms of states of def­i­nite $y$-​mo­men­tum. Then the an­gu­lar mo­men­tum op­er­a­tors merely add scalar fac­tors $m\hbar$ to those states. These do not af­fect what hap­pens to the re­main­der of the in­ner prod­uct un­der axes in­ver­sion. Al­ter­na­tively, note that $\L _y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar(z\partial/\partial{x}-x\partial/\partial{z})$$\raisebox{.5pt}{$/$}$​${\rm i}$ and each term has two po­si­tion co­or­di­nates that change sign. And surely spin should be­have the same as or­bital an­gu­lar mo­men­tum.

If the an­gu­lar mo­men­tum op­er­a­tors do noth­ing un­der axes in­ver­sion, the par­i­ties of the ini­tial and fi­nal atomic or nu­clear states will have to be equal. So the par­ity se­lec­tion rule for mag­netic di­pole tran­si­tions is the op­po­site from the one for elec­tric di­pole tran­si­tions. The par­ity has to stay the same in the tran­si­tion.

As­sum­ing again that the wave mo­tion is in the $z$-​di­rec­tion, each higher mul­ti­pole or­der $\ell$ adds a fac­tor $z_i$ to the elec­tric or mag­netic in­ner prod­uct. This fac­tor changes sign un­der axes in­ver­sion. So for in­creas­ing $\ell$, al­ter­nat­ingly the atomic or nu­clear par­ity must flip over or stay the same.

If the par­ity se­lec­tion rule is vi­o­lated for a mul­ti­pole term, the term is zero. How­ever, if it is not vi­o­lated, the term may still be zero for some other rea­son. The most im­por­tant other rea­son is an­gu­lar mo­men­tum. Atomic and nu­clear states have def­i­nite an­gu­lar mo­men­tum. Con­sider again the elec­tric di­pole in­ner prod­uct

$${\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\vert}x_i{\left\vert\psi_{\rm {H}}\right\rangle}$$

States of dif­fer­ent an­gu­lar mo­men­tum are or­thog­o­nal. That is a con­se­quence of the fact that the mo­men­tum op­er­a­tors are Her­mit­ian. What it means is that the in­ner prod­uct above is zero un­less $x_i\psi_{\rm {H}}$ has at least some prob­a­bil­ity of hav­ing the same an­gu­lar mo­men­tum as state $\psi_{\rm {L}}$. Now the fac­tor $x_i$ can be writ­ten in terms of spher­i­cal har­mon­ics us­ing chap­ter 4.2.3, ta­ble 4.3:

$$x_i = \sqrt{\frac{8\pi}{3}} r_i \left(Y_1^{-1} - Y_1^1\right)$$

So it is a sum of two states, both with square an­gu­lar mo­men­tum quan­tum num­ber $l_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, but with $z$ an­gu­lar mo­men­tum quan­tum num­ber $m_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1, re­spec­tively 1.

Now re­call the rules from chap­ter 7.4.2 for com­bin­ing an­gu­lar mo­menta:

$$Y_1^{-1} \psi_{\rm {H}} \qquad \Longrightarrow \qquad j_{\... ...mbox{ or } j_{\rm {H}}+1 \qquad m_{\rm {net}} = m_{\rm {H}}-1$$

Here $j_{\rm {H}}$ is the quan­tum num­ber of the square an­gu­lar mo­men­tum of the atomic or nu­clear state $\psi_{\rm {H}}$. And $m_{\rm {H}}$ is the quan­tum num­ber of the $z$ an­gu­lar mo­men­tum of the state. Sim­i­larly $j_{\rm {net}}$ and $m_{\rm {net}}$ are the pos­si­ble val­ues for the quan­tum num­bers of the com­bined state $Y_1^{-1}\psi_{\rm {H}}$. Note again that $m_x$ and $m_{\rm {H}}$ val­ues sim­ply add to­gether. How­ever, the $j_{\rm {H}}$-value changes by up to $l_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 unit in ei­ther di­rec­tion. (But if $j_{\rm {H}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, the com­bined state can­not have zero an­gu­lar mo­men­tum.)

(It should be noted that you should be care­ful in com­bin­ing these an­gu­lar mo­menta. The nor­mal rules for com­bin­ing an­gu­lar mo­menta ap­ply to dif­fer­ent sources of an­gu­lar mo­men­tum. Here the fac­tor $x_i$ does not de­scribe an ad­di­tional source of an­gu­lar mo­men­tum, but a par­ti­cle that al­ready has been given an an­gu­lar mo­men­tum within the wave func­tion $\psi_{\rm {H}}$. That means in par­tic­u­lar that you should not try to write out $Y_1^{-1}\psi_{\rm {H}}$ us­ing the Cleb­sch-Gor­dan co­ef­fi­cients of chap­ter 12.7, {N.13}. If you do not know what Cleb­sch-Gor­dan co­ef­fi­cients are, you have noth­ing to worry about.)

To get a nonzero in­ner prod­uct, one of the pos­si­ble states of net an­gu­lar mo­men­tum above will need to match the quan­tum num­bers $j_{\rm {L}}$ and $m_{\rm {L}}$ of state $\psi_{\rm {L}}$. So

$$j_{\rm {L}} = j_{\rm {H}}-1,\; j_{\rm {H}}, \mbox{ or } j_{\rm {H}}+1 \qquad m_{\rm {L}} = m_{\rm {H}}-1$$

(And if $j_{\rm {H}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, $j_{\rm {L}}$ can­not be zero.) But re­call that $x_i$ also con­tained a $Y_1^1$ state. That state will al­low $m_{\rm {L}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m_{\rm {H}}+1$. And if you take a wave that has its elec­tric field in the $z$-​di­rec­tion in­stead of the $x$-​di­rec­tion, you also get a $Y_1^0$ state that gives the pos­si­bil­ity $m_{\rm {L}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m_{\rm {H}}$.

So the com­plete se­lec­tion rules for elec­tric di­pole tran­si­tions are

$$j_{\rm {L}} = j_{\rm {H}}-1,\; j_{\rm {H}}, \mbox{ or } j_{... ...x{ or } m_{\rm {H}}+1 \qquad \pi_{\rm {L}} \pi_{\rm {H}} = -1$$

where $\pi$ means the par­ity. In ad­di­tion, at least one of $j_{\rm {L}}$ or $j_{\rm {H}}$ must be nonzero. And as al­ways for these quan­tum num­bers, $j_{\rm {L}}$ $\raisebox{-.5pt}{$\geqslant$}$ 0 and $\vert m_{\rm {L}}\vert$ $\raisebox{-.3pt}{$\leqslant$}$ $j_{\rm {L}}$. Equiv­a­lent se­lec­tion rules were writ­ten down for the hy­dro­gen atom with spin-or­bit in­ter­ac­tion in chap­ter 7.4.4.

For mag­netic di­pole tran­si­tions, the rel­e­vant in­ner prod­uct is

$${\left\langle\psi_{\rm {L}}\hspace{0.3pt}\right\vert}\L _{i,y}+g_i{\widehat S}_{i.y}{\left\vert\psi_{\rm {H}}\right\rangle}$$

Note that it is ei­ther $\L _y$ or ${\widehat S}_y$ that is ap­plied on $\psi_{\rm {H}}$, not both at the same time. It will be as­sumed that $\psi_{\rm {H}}$ is writ­ten in terms of states with def­i­nite an­gu­lar mo­men­tum in the $z$-​di­rec­tion. In those terms, the ef­fect of $\L _y$ or ${\widehat S}_y$ is known to raise or lower the cor­re­spond­ing mag­netic quan­tum num­ber $m$ by one unit, chap­ter 12.11. Which means that the net an­gu­lar mo­men­tum can change by one unit. (Like when op­po­site or­bital an­gu­lar mo­men­tum and spin change into par­al­lel ones. Note also that for the hy­dro­gen atom in the non­rel­a­tivis­tic ap­prox­i­ma­tion of chap­ter 4.3, there is no in­ter­ac­tion be­tween the elec­tron spin and the or­bital mo­tion. In that case, the mag­netic di­pole term can only change the value of $m_l$ or $m_s$ by one unit. Sim­ply put, only the di­rec­tion of the an­gu­lar mo­men­tum changes. That is nor­mally a triv­ial change as empty space has no pre­ferred di­rec­tion.)

One big lim­i­ta­tion is that in ei­ther an elec­tric or a mag­netic di­pole tran­si­tion, the net atomic or nu­clear an­gu­lar mo­men­tum $j$ can change by no more than one unit. Larger changes in an­gu­lar mo­men­tum re­quire higher mul­ti­pole or­ders $\ell$. These add a fac­tor $z_i^{\ell-1}$ to the in­ner prod­ucts. Now it turns out that:

$$z_i^{\ell-1} \sim \frac{(\ell-1)!\sqrt{4\pi(2\ell-1)}}{(2\e... ...d (2\ell-1)!! \equiv \frac{(2\ell-1)!}{2^{\ell-1}(\ell-1)!} %$$ (A.174)

Here the dots stand for spher­i­cal har­mon­ics with lower square an­gu­lar mo­men­tum. (To ver­ify the above re­la­tion, use the Rayleigh for­mula of {A.6}, and ex­pand the Bessel func­tion and the ex­po­nen­tial in it in Tay­lor se­ries.) So the fac­tor $z_i^{\ell-1}$ has a max­i­mum az­imuthal quan­tum num­ber $l$ equal to $\ell-1$. That means that the max­i­mum achiev­able change in atomic or nu­clear an­gu­lar mo­men­tum in­creases by one unit for each unit in­crease in mul­ti­pole or­der $\ell$.

It fol­lows that the first mul­ti­pole term that can be nonzero has $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert j_{\rm {H}}-j_{\rm {L}}\vert$, or $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 if the an­gu­lar mo­menta are equal. At that mul­ti­pole level, ei­ther the elec­tric or the mag­netic term can be nonzero, de­pend­ing on par­ity. Nor­mally this term will then dom­i­nate the tran­si­tion process, as the terms of still higher mul­ti­pole lev­els are ball­parked to be much smaller.

A fur­ther lim­i­ta­tion ap­plies to or­bital an­gu­lar mo­men­tum. The an­gu­lar mo­men­tum op­er­a­tors will not change the or­bital an­gu­lar mo­men­tum val­ues. And the fac­tors $z_i^{\ell-1}$ and $x_i$ can only change it by up to $\ell-1$, re­spec­tively 1 units. So the min­i­mum dif­fer­ence in pos­si­ble or­bital an­gu­lar mo­men­tum val­ues will have to be no larger than that:

$$\fbox{$\displaystyle \mbox{electric:}\quad \vert l_{\rm{H}... ...rm{min}} \mathrel{\raisebox{-.7pt}{$\leqslant$}}\ell - 1 $} %$$ (A.175)

This is mainly im­por­tant for sin­gle-par­ti­cle states of def­i­nite or­bital an­gu­lar mo­men­tum. That in­cludes the hy­dro­gen atom, even with the rel­a­tivis­tic spin-or­bit in­ter­ac­tion. (But it does as­sume the non­rel­a­tivis­tic Hamil­ton­ian in the ac­tual tran­si­tion process.)

The fi­nal lim­i­ta­tion is that $j_{\rm {H}}$ and $j_{\rm {L}}$ can­not both be zero. The rea­son is that if $j_{\rm {H}}$ is zero, the pos­si­ble an­gu­lar mo­men­tum val­ues of $z_i^{\ell-1}x_ij_{\rm {H}}$ are those of $z_i^{\ell-1}x_i$. And those val­ues do not in­clude zero to match $j_{\rm {L}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. (Ac­cord­ing to the rules of quan­tum me­chan­ics, the prob­a­bil­ity of zero an­gu­lar mo­men­tum is given by the in­ner prod­uct with the spher­i­cal har­monic $Y_0^0$ of zero an­gu­lar mo­men­tum. Since $Y_0^0$ is just a con­stant, the in­ner prod­uct is pro­por­tional to the av­er­age of $z_i^{\ell-1}x_i$ on a spher­i­cal sur­face around the ori­gin. That av­er­age will be zero be­cause by sym­me­try pos­i­tive val­ues of $x_i$ will av­er­age away against cor­re­spond­ing neg­a­tive ones.)

A.25.6 Ball­park de­cay rates

It may be in­ter­est­ing to find some ac­tual ball­park val­ues for the spon­ta­neous de­cay rates. More so­phis­ti­cated val­ues, called the Weis­skopf and Moszkowski es­ti­mates, will be de­rived in a later sub­sec­tion. How­ever, they are ball­parks one way or the other.

It will be as­sumed that only a sin­gle par­ti­cle, elec­tron or pro­ton, changes states. It will also be as­sumed that the first mul­ti­pole con­tri­bu­tion al­lowed by an­gu­lar mo­men­tum and par­ity is in­deed nonzero and dom­i­nant. In fact, it will be as­sumed that this con­tri­bu­tion is as big as it can rea­son­ably be.

To get the spon­ta­neous emis­sion rate, first the proper am­pli­tude ${\cal E}_0$ of the elec­tric field to use needs to be iden­ti­fied. The same rel­a­tivis­tic pro­ce­dure as in {A.24} may be fol­lowed to show it should be taken as

$$\mbox{spontaneous emission:} \quad {\cal E}_0 = \sqrt{\frac{\hbar\omega}{\epsilon_0{\cal V}}}$$

That as­sumes that the en­tire sys­tem is con­tained in a very large pe­ri­odic box of vol­ume ${\cal V}$. Also, $\epsilon_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.85 10$\POW9,{-12}$ C$\POW9,{2}$/J m is the per­mit­tiv­ity of space

Next, Fermi’s golden rule of chap­ter 7.6.1 says that the tran­si­tion rate is

$$\lambda_{\rm H\to L} = \overline{\vert H_{21}\vert^2}\frac{2\pi}{\hbar} \frac{{\rm d}N}{{\rm d}E}$$

Here $H_{21}$ is ap­prox­i­mated as the first al­lowed (nonzero) mul­ti­pole con­tri­bu­tion $H_{21,i}^{\rm {E}\ell}$ or $H_{21,i}^{\rm {M}\ell}$. So the ad­di­tional higher or­der nonzero con­tri­bu­tions are ig­nored, The over­line means that this con­tri­bu­tion needs to be suit­ably av­er­aged over all di­rec­tions of the elec­tro­mag­netic wave. Fur­ther ${\rm d}{N}$$\raisebox{.5pt}{$/$}$​${\rm d}{E}$ is the num­ber of pho­ton states in the pe­ri­odic box per unit en­ergy range. This is the den­sity of states as given in chap­ter 6.3 (6.7). Us­ing the Planck-Ein­stein re­la­tion it is:

$$\frac{{\rm d}N}{{\rm d}E} = \frac{\omega^2}{\hbar\pi^2c^3} {\cal V}$$

Ball­park ma­trix co­ef­fi­cients were given in sub­sec­tion A.25.4. How­ever, a more ac­cu­rate es­ti­mate is de­sir­able. The main prob­lem is the fac­tor $r_{i,k}^{\ell-1}$ in the ma­trix el­e­ments (A.171) and (A.172). This fac­tor equals $z_i^{\ell-1}$ if the $z$-​axis is taken to be in the di­rec­tion of wave mo­tion. Ac­cord­ing to the pre­vi­ous sub­sec­tion

$$z_i^{\ell-1} \sim \frac{(\ell-1)!\sqrt{4\pi(2\ell-1)}}{(2\ell-1)!!} r_i^{\ell-1} Y_{\ell-1}^0 + \ldots$$

The dots in­di­cate spher­i­cal har­mon­ics of lower an­gu­lar mo­men­tum that do not do any­thing. Only the shown term is rel­e­vant for the con­tri­bu­tion of low­est mul­ti­pole or­der. So only the shown term should be ball­parked. That can be done by es­ti­mat­ing $r_i$ as $R$, and $Y_{\ell-1}^0$ as 1/$\sqrt{4\pi}$, (which is ex­act for $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1).

The elec­tric in­ner prod­uct con­tains a fur­ther fac­tor $x_i$, tak­ing the $x$-​axis in the di­rec­tion of the elec­tric field. That will be ac­counted for by up­ping the value of $\ell$ one unit in the ex­pres­sion above. The mag­netic in­ner prod­uct con­tains an­gu­lar mo­men­tum op­er­a­tors. Since not much can be said about these eas­ily, they will sim­ply be es­ti­mated as $\hbar$.

Putting it all to­gether, the es­ti­mated de­cay rates be­come

$$\fbox{$\displaystyle \lambda^{\rm E\ell} \sim \alpha \ome... ...2} \left(\frac{\hbar}{2 m_i c R}\right)^2 f^{\rm M\ell} $} %$$ (A.176)

Here

$$\alpha = \frac{e^2}{4\pi\epsilon_0{\hbar}c} \approx \frac{1}{137}$$

is the so-called fine struc­ture con­stant. with $e$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.6 10$\POW9,{-19}$ C the pro­ton or elec­tron charge, $\epsilon_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.85 10$\POW9,{-12}$ C$\POW9,{2}$/J m the per­mit­tiv­ity of space, and $c$ $\vphantom0\raisebox{1.5pt}{$=$}$ 3 10$\POW9,{8}$ m/s the speed of light. This nondi­men­sion­al con­stant gives the strength of the cou­pling be­tween charged par­ti­cles and pho­tons, so it should ob­vi­ously be there. The fac­tor $\omega$ is ex­pected for di­men­sional rea­sons; it gives the de­cay rate units of in­verse time. The nondi­men­sion­al fac­tor $kR$ re­flects the fact that the atom or nu­cleus has dif­fi­culty in­ter­act­ing with the pho­ton be­cause its size is so small com­pared to the pho­ton wave length. That is worse for higher mul­ti­pole or­ders $\ell$, as their pho­tons pro­duce less of a field near the ori­gin. The fac­tors $f^{\rm {E}\ell}$ and $f^{\rm {M}\ell}$ rep­re­sent un­known cor­rec­tions for the er­rors in the ball­parks. These fac­tors are hoped to be 1. (Fat chance.) As far as the re­main­ing nu­mer­i­cal fac­tors are con­cerned, ...

The fi­nal par­en­thet­i­cal fac­tor in the mag­netic de­cay rate was al­ready dis­cussed in sub­sec­tion A.25.4. It nor­mally makes mag­netic de­cays slower than elec­tric ones of the same mul­ti­pole or­der, but faster than elec­tric ones of the next or­der.

These es­ti­mates are roughly sim­i­lar to the Weis­skopf ones. While they tend to be larger, that is largely com­pen­sated for by the fact that in the above es­ti­mates $R$ is the mean ra­dius. In the Weis­skopf es­ti­mates it is the edge of the nu­cleus.

In any case, ac­tual de­cay rates can vary wildly from ei­ther pair of es­ti­mates. For ex­am­ple, nu­clei sat­isfy an ap­prox­i­mate con­ser­va­tion law for a quan­tity called isospin. If the tran­si­tion vi­o­lates an ap­prox­i­mate con­ser­va­tion law like that, the tran­si­tion rate will be un­usu­ally small. Also, it may hap­pen that the ini­tial and fi­nal wave func­tions have lit­tle over­lap. That means that the re­gions where they both have sig­nif­i­cant mag­ni­tude are small. (These re­gions should re­ally be vi­su­al­ized in the high-di­men­sion­al space of all the par­ti­cle co­or­di­nates.) In that case, the tran­si­tion rate can again be un­ex­pect­edly small.

Con­versely, if a lot of par­ti­cles change state in a tran­si­tion, their in­di­vid­ual con­tri­bu­tions to the ma­trix el­e­ment can add up to an un­ex­pect­edly large tran­si­tion rate.

A.25.7 Wave func­tions of def­i­nite an­gu­lar mo­men­tum

The analy­sis so far has rep­re­sented the elec­tro­mag­netic field in terms of pho­ton states of def­i­nite lin­ear mo­men­tum. But it is usu­ally much more con­ve­nient to use states of def­i­nite an­gu­lar mo­men­tum. That al­lows full use of the con­ser­va­tion laws of an­gu­lar mo­men­tum and par­ity.

The states of def­i­nite an­gu­lar mo­men­tum have vec­tor po­ten­tials given by the pho­ton wave func­tions of ad­den­dum {A.21.7}. For elec­tric ${\rm {E}}\ell$ and mag­netic ${\rm {M}}\ell$ mul­ti­pole tran­si­tions re­spec­tively:

$$\skew3\vec A_\gamma^{\rm E} = \frac{A_0}{k} \nabla\times{\s... ...0 {\skew0\vec r}\times\nabla j_\ell(kr) Y_\ell^m(\theta,\phi)$$

Here $j_\ell$ is a spher­i­cal Bessel func­tion, {A.6} and $Y_\ell^m$ a spher­i­cal har­monic, chap­ter 4.2.3. The az­imuthal an­gu­lar mo­men­tum quan­tum num­ber of the pho­ton is $\ell$. Its quan­tum num­ber of an­gu­lar mo­men­tum in the cho­sen $z$-​di­rec­tion is $m$. The elec­tric state has par­ity $(-1)^{\ell}$ and the mag­netic one $(-1)^{\ell-1}$. (That in­cludes the in­trin­sic par­ity, un­like in some other sources). Fur­ther $A_0$ is a con­stant.

The con­tri­bu­tion of a par­ti­cle $i$ to the ma­trix el­e­ment is as be­fore

$$H_{21,i} = -\frac{q_i}{m_i} {\left\langle\psi_{\rm {L}}\hs... ...e} \qquad \skew2\vec{\cal B}_i = \nabla_i\times\skew3\vec A_i$$

But now, for elec­tric tran­si­tions $\skew3\vec A_i$ needs to be taken as the com­plex con­ju­gate of the pho­ton wave func­tion $\skew3\vec A_\gamma^{\rm {E}}$ above, eval­u­ated at the po­si­tion of par­ti­cle $i$. For mag­netic tran­si­tions $\skew3\vec A_i$ needs to be taken as the com­plex con­ju­gate of $\skew3\vec A_\gamma^{\rm {M}}$. The com­plex con­ju­gates are a re­sult of the quan­ti­za­tion of ra­di­a­tion, {A.24}. And they would not be there for ab­sorp­tion. (The clas­si­cal rea­sons are much like the story for plane elec­tro­mag­netic waves given ear­lier. But here the non­quan­tized waves are too messy to even bother about, in this au­thor’s opin­ion.)

The ma­trix el­e­ments can be ap­prox­i­mated as­sum­ing that the wave length of the pho­ton is large com­pared to the size $R$ of the atom or nu­cleus. The ap­prox­i­mate con­tri­bu­tion of the par­ti­cle to the ${\rm {E}}\ell$ elec­tric ma­trix el­e­ment is then, {D.43.2},

$$H_{21,i}^{\rm E\ell} \approx - {\rm i}q_i c A_0 \frac{(\ell... ... {L}}\vert r_i^\ell Y_{\ell i}^{m*} \vert\psi_{\rm {H}}\rangle$$

The sub­script $i$ on the spher­i­cal har­monic means that its ar­gu­ments are the co­or­di­nates of par­ti­cle $i$. For nu­clei, the above re­sult is again sus­pect for the rea­sons dis­cussed in {N.14}.

The ap­prox­i­mate con­tri­bu­tion of the par­ti­cle to the ${\rm {M}}\ell$ mag­netic ma­trix el­e­ment is {D.43.2},

$$H_{21,i}^{\rm M\ell} \approx \frac{q_i}{2 m_i} A_0 \frac{(... ..._i{\skew 6\widehat{\vec S}}_i\Big) \vert\psi_{\rm {H}}\rangle$$

In gen­eral these ma­trix el­e­ments will need to be summed over all par­ti­cles.

The above ma­trix el­e­ments can be an­a­lyzed sim­i­lar to the ear­lier lin­ear mo­men­tum ones. How­ever, the above ma­trix el­e­ments al­low you to keep the atom or nu­cleus in a fixed ori­en­ta­tion. For the lin­ear mo­men­tum ones, the nu­clear ori­en­ta­tion must be changed if the di­rec­tion of the wave is to be held fixed. And in any cases, lin­ear mo­men­tum ma­trix el­e­ments must be av­er­aged over all di­rec­tions of wave prop­a­ga­tion. That makes the above ma­trix el­e­ments much more con­ve­nient in most cases.

Fi­nally the ma­trix el­e­ments can be con­verted into spon­ta­neous de­cay rates us­ing Fermi’s golden rule of chap­ter 7.6.1. In do­ing so, the needed value of the con­stant $A_0$ and cor­re­spond­ing den­sity of states are, fol­low­ing {A.21.7} and {A.24},

$$A_0 = -\frac{1}{{\rm i}c} \sqrt{\frac{\hbar\omega}{\ell(\e... ...{\rm d}N}{{\rm d}E} \approx \frac{1}{\hbar\pi c} r_{\rm {max}}$$

This as­sumes that the en­tire sys­tem is con­tained in­side a very big sphere of ra­dius $r_{\rm {max}}$. This ra­dius $r_{\rm {max}}$ dis­ap­pear in the fi­nal an­swer, and the fi­nal de­cay rates will be the ones in in­fi­nite space. (De­spite the ab­sence of $r_{\rm {max}}$ they do not ap­ply to a fi­nite sphere, be­cause the den­sity of states above is an ap­prox­i­ma­tion for large $r_{\rm {max}}$.)

It is again con­ve­nient to nondi­men­sion­al­ize the ma­trix el­e­ments us­ing some suit­ably de­fined typ­i­cal atomic or nu­clear ra­dius $R$. Re­cent au­tho­r­a­tive sources, like [33] and [[4]], take the nu­clear ra­dius equal to

$$R = 1.2 A^{1/3}\mbox{ fm} %$$ (A.177)

Here $A$ is the num­ber of pro­tons and neu­trons in the nu­cleus and a fm is 10$\POW9,{-15}$ m.

The fi­nal de­cay rates are much like the ones (A.176) found ear­lier for lin­ear mo­men­tum modes. In fact, lin­ear mo­men­tum modes should give the same an­swer as the an­gu­lar ones, if cor­rectly av­er­aged over all di­rec­tions of the lin­ear mo­men­tum. The de­cay rates in terms of an­gu­lar mo­men­tum modes are:

$$\fbox{$\displaystyle \lambda^{\rm E\ell} = \alpha \omega (... ...ac{2(l+1)}{l(2l+1)!!^2} \vert h_{21}^{\rm E\ell}\vert^2 $} %$$ (A.178)

$$\fbox{$\displaystyle \lambda^{\rm M\ell} = \alpha \omega (... ...\hbar}{2 m c R}\right)^2 \vert h_{21}^{\rm M\ell}\vert^2 $} %$$ (A.179)

where $\alpha$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ 1/137 is again the fine struc­ture con­stant. The nondi­men­sion­al ma­trix el­e­ments in these ex­pres­sions are

$$\fbox{$\displaystyle \vert h_{21}^{\rm E\ell}\vert = \sum_... ..._i^\ell Y_{\ell i}^{m*}/R^\ell \vert\psi_{\rm{H}}\rangle $} %$$ (A.180)

$$\fbox{$\displaystyle \vert h_{21}^{\rm M\ell}\vert = \sum_... ...dehat{\vec S}}_i}{\hbar}\Big) \vert\psi_{\rm{H}}\rangle $} %$$ (A.181)

The sum is over the elec­trons or pro­tons and neu­trons, with $q_i$ their charge and $m_i$ their mass. The ref­er­ence mass $m$ would nor­mally be taken to be the mass of an elec­tron for atoms and of a pro­ton for nu­clei. That means that for the elec­tron or pro­ton the charge and mass ra­tios can be set equal to 1. For an elec­tron $g_i$ is about 2, while for a pro­ton, $g_i$ would be about 5.6 if the ef­fect of the neigh­bor­ing pro­tons and neu­trons is ig­nored. For the neu­tron, the (net) charge $q_i$ is zero. There­fore the elec­tric ma­trix el­e­ment is zero, and so is the first term in the mag­netic one. In the sec­ond term, how­ever, the charge and mass of the pro­ton need to be used, along with $g_i$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$3.8, as­sum­ing again that the ef­fect of the neigh­bor­ing pro­tons and neu­trons is ig­nored.

A.25.8 Weis­skopf and Moszkowski es­ti­mates

The Weis­skopf and Moszkowski es­ti­mates are ball­park spon­ta­neous de­cay rates. They are found by ball­park­ing the nondi­men­sion­al ma­trix el­e­ments (A.180) and (A.181) given in the pre­vi­ous sub­sec­tion. The es­ti­mates are pri­mar­ily in­tended for nu­clei. How­ever, they can eas­ily be adopted to the hy­dro­gen atom with a few straight­for­ward changes.

It is as­sumed that a sin­gle pro­ton num­bered $i$ makes the tran­si­tion. The rest of the nu­cleus stays un­changed and can there­fore be ig­nored in the analy­sis. Note that this does not take into ac­count that the pro­ton and the rest of the nu­cleus should move around their com­mon cen­ter of grav­ity. Cor­rec­tion fac­tors for that can be ap­plied, see [33] and [11] for more. In a sim­i­lar way, the case that a sin­gle neu­tron makes the tran­si­tion can be ac­counted for.

It is fur­ther as­sumed that the ini­tial and fi­nal wave func­tions of the pro­ton are of a rel­a­tively sim­ple form, In spher­i­cal co­or­di­nates:

$$\psi_{\rm {H}} = R_{\rm {H}}(r_i) \Theta_{l_{\rm {H}}j_{\r... ...Theta_{l_{\rm {L}}j_{\rm {L}}}^{m_{j\rm {L}}}(\theta_i,\phi_i)$$

These wave func­tions are very much like the $R_{nl}(r_i)Y_l^{m_l}(\theta_i,\phi_i){\updownarrow}$ wave func­tions for the elec­tron in the hy­dro­gen atom, chap­ter 4.3. How­ever, for nu­clei, it turns out that you want to com­bine the or­bital and spin states into states with def­i­nite net an­gu­lar mo­men­tum $j$ and def­i­nite net an­gu­lar mo­men­tum $m_j$ in the cho­sen $z$-​di­rec­tion. Such com­bi­na­tions take the form

$$\Theta_{lj}^{m_j}(\theta_i,\phi_i) = c_1 Y_l^{m_j-\frac12}... ...arrow} + c_2 Y_l^{m_j+\frac12} (\theta_i,\phi_i) {\downarrow}$$

The co­ef­fi­cients $c_1$ and $c_2$ are of no in­ter­est here, but you can find them in chap­ter 12.8 2 if needed.

In fact even for the hy­dro­gen atom you re­ally want to take the ini­tial and fi­nal states of the elec­tron of the above form. That is due to a small rel­a­tivis­tic ef­fect called “spin-or­bit in­ter­ac­tion,” {A.39}. It just so hap­pens that for nu­clei, the spin-or­bit ef­fect is much larger. Note how­ever that the elec­tric ma­trix el­e­ment ig­nores the spin-or­bit ef­fect. That is a sig­nif­i­cant prob­lem, {N.14}. It will make the ball­parked elec­tric de­cay rate for nu­clei sus­pect. But there is no ob­vi­ous way to fix it.

The nondi­men­sion­al elec­tric ma­trix el­e­ment (A.180) can be writ­ten as an in­te­gral over the spher­i­cal co­or­di­nates of the pro­ton. It then falls apart into a ra­dial in­te­gral and an an­gu­lar one:

$$\vert h_{21}^{\rm E\ell}\vert \approx \int R_{\rm {L}}(r_i... ...}i}^{m_{j\rm {H}}} \sin^2\theta_i{\rm d}\theta_i{\rm d}\phi_i$$

Note that in the an­gu­lar in­te­gral the prod­uct of the an­gu­lar wave func­tions im­plic­itly in­volves in­ner prod­ucts be­tween the spin states. Spin states are or­tho­nor­mal, so their prod­uct is 0 if the spins are dif­fer­ent and 1 if they are the same.

The bot­tom line is that the square elec­tric ma­trix el­e­ment can be writ­ten as a prod­uct of a ra­dial fac­tor,

$$f^{\rm rad,\ell}_{\rm LH} \equiv \left[\int R_{\rm {L}}^*(r_i) (r_i/R)^\ell R_{\rm {H}}(r_i) r_i^2 { \rm d}r_i\right]^2 %$$ (A.182)

and an an­gu­lar one,

$$f^{\rm ang,\ell}_{\rm LH} \equiv \left[ \sqrt{4\pi}\! \int... ...m {H}}} \sin^2\theta_i{\rm d}\theta_i{\rm d}\phi_i\right]^2 %$$ (A.183)

As a re­sult, the elec­tric mul­ti­pole de­cay rate (A.180) be­comes

$$\fbox{$\displaystyle \lambda^{\rm E\ell} = \alpha \omega (... ...2} f^{\rm rad,\ell}_{\rm LH} f^{\rm ang,\ell}_{\rm LH} $} %$$ (A.184)

Here the trail­ing fac­tors rep­re­sent the square ma­trix el­e­ment.

A sim­i­lar ex­pres­sion can be writ­ten for the nondi­men­sion­al mag­netic ma­trix el­e­ment, {D.43.3}: It gives the de­cay rate (A.181) as

$$\fbox{$\displaystyle \lambda^{\rm M\ell} = \alpha \omega (... ...H} f^{\rm ang,\ell}_{\rm LH} f^{\rm mom,\ell}_{\rm LH} $} %$$ (A.185)

In this case, there is an third fac­tor re­lated to the spin and or­bital an­gu­lar mo­men­tum op­er­a­tors that ap­pear in the mag­netic ma­trix el­e­ment. Also, the in­te­grand in the ra­dial fac­tor is one or­der of $r$ lower than in the elec­tric el­e­ment. That is due to the nabla op­er­a­tor $\nabla$ in the mag­netic el­e­ment. It means that in terms of the ra­dial elec­tric fac­tor as de­fined above, the value of $\ell$ to use is one unit be­low the ac­tual mul­ti­pole or­der.

Con­sider now the val­ues of these fac­tors. The ra­dial fac­tor (A.182) is the sim­plest one. The Weis­skopf and Moszkowski es­ti­mates use a very crude ap­prox­i­ma­tion for this fac­tor. They as­sume that the ra­dial wave func­tions are equal to some con­stant up to the nu­clear ra­dius $R$ and zero be­yond it. (This as­sump­tion is not com­pletely il­log­i­cal for nu­clei, as nu­clear den­si­ties are fairly con­stant un­til the nu­clear edge.) That gives, {D.43.3},

$$f^{\rm rad,\ell}_{\rm LH} = \left(\frac{3}{\ell+3}\right)^2$$

Note that the mag­netic de­cay rate uses $\ell+2$ in the de­nom­i­na­tor in­stead of $\ell+3$ be­cause of the lower power of $r_i$.

Ta­ble A.1: Ra­dial in­te­gral cor­rec­tion fac­tors for hy­dro­gen atom wave func­tions.

More rea­son­able as­sump­tions for the ra­dial wave func­tions are pos­si­ble. For a hy­dro­gen atom in­stead of a nu­cleus, the ob­vi­ous thing to do is to use the ac­tual ra­dial wave func­tions $R_{nl}$ from chap­ter 4.3. That gives the ra­dial fac­tors listed in ta­ble A.1. These take $R$ equal to the Bohr ra­dius. That ex­plains why some val­ues are so large: the av­er­age ra­dial po­si­tion of the elec­tron can be much larger than the Bohr ra­dius in var­i­ous ex­cited states. In the ta­ble, $n$ is the prin­ci­pal quan­tum num­ber that gives the en­ergy of the state. Fur­ther $l$ is the az­imuthal quan­tum num­ber of or­bital an­gu­lar mo­men­tum. The two pairs of $nl$ val­ues cor­re­spond to those of the ini­tial and fi­nal states; in what or­der does not make a dif­fer­ence. There are two ra­dial fac­tors listed for each pair of states. The first value ap­plies to elec­tric and mul­ti­pole tran­si­tions at the low­est pos­si­ble mul­ti­pole or­der. That is usu­ally the im­por­tant one, be­cause nor­mally tran­si­tion rates de­crease rapidly with mul­ti­pole or­der.

To un­der­stand the given val­ues more clearly, first con­sider the re­la­tion be­tween mul­ti­pole or­der and or­bital an­gu­lar mo­men­tum. The de­rived ma­trix el­e­ments im­plic­itly as­sume that the po­ten­tial of the pro­ton or elec­tron only de­pends on its po­si­tion, not its spin. So spin does not re­ally af­fect the or­bital mo­tion. That means that the mul­ti­pole or­der for non­triv­ial tran­si­tions is con­strained by or­bital an­gu­lar mo­men­tum con­ser­va­tion, [33]:

$$\vert l_{\rm H}-l_{\rm L}\vert \mathrel{\raisebox{-.7pt}{$\... ...l \mathrel{\raisebox{-.7pt}{$\leqslant$}}l_{\rm H}+l_{\rm L} %$$ (A.186)

Note that this is a con­se­quence of (A.175) within the sin­gle-par­ti­cle model. It is just like for the non­rel­a­tivis­tic hy­dro­gen atom, (7.17). (${\rm {M}}1$ tran­si­tions that merely change the di­rec­tion of the spin, like a $Y_0^0{\uparrow}$ to $Y_0^0{\downarrow}$ one, are ir­rel­e­vant since they do not change the en­ergy. Fermi’s golden rule makes the tran­si­tion rate for tran­si­tions with no en­ergy change the­o­ret­i­cally zero, chap­ter 7.6.1.)

The min­i­mum mul­ti­pole or­der im­plied by the left-hand con­straint above cor­re­sponds to an elec­tric tran­si­tion be­cause of par­ity. How­ever, this tran­si­tion may be im­pos­si­ble be­cause of net an­gu­lar mo­men­tum con­ser­va­tion or be­cause $\ell$ must be at least 1. That will make the tran­si­tion of low­est mul­ti­pole or­der a mag­netic one. The mag­netic tran­si­tion still uses the same value for the ra­dial fac­tor though. The sec­ond ra­dial fac­tor in the ta­ble is pro­vided since the next-higher elec­tric mul­ti­pole or­der might rea­son­ably com­pete with the mag­netic one.

More re­al­is­tic ra­dial fac­tors for nu­clei can be for­mu­lated along sim­i­lar lines. The sim­plest phys­i­cally rea­son­able as­sump­tion is that the pro­tons and neu­trons are con­tained within an im­pen­e­tra­ble sphere of ra­dius $R$. A hy­dro­gen-like num­ber­ing sys­tem of the quan­tum states can again be used, fig­ure 14.14, with one dif­fer­ence. For hy­dro­gen, a given en­ergy level $n$ al­lows all or­bital mo­men­tum quan­tum num­bers $l$ up to $n-1$. For nu­clei, $l$ must be even if $n$ is odd and vice-versa, chap­ter 14.12.1. Also, while for the (non­rel­a­tivis­tic) hy­dro­gen atom the en­ergy does not de­pend on $l$, for nu­clei that is only a rough ap­prox­i­ma­tion. (It as­sumes that the nu­clear po­ten­tial is like an har­monic os­cil­la­tor one, and that is re­ally crude.)

Ta­ble A.2: More re­al­is­tic ra­dial in­te­gral cor­rec­tion fac­tors for nu­clei.

Ra­dial fac­tors for the im­pen­e­tra­ble-sphere model us­ing this num­ber­ing sys­tem are given in ta­ble A.2.

These re­sults il­lus­trate the lim­i­ta­tions of ${\rm {M}}1$ tran­si­tions in the sin­gle-par­ti­cle model. Be­cause of the con­di­tion (A.186) above and par­ity, the or­bital quan­tum num­ber $l$ can­not change in ${\rm {M}}1$ tran­si­tions. A glance at the ta­ble then shows that the ra­dial fac­tor is zero un­less the ini­tial and fi­nal ra­dial states are iden­ti­cal. (That is a con­se­quence of the or­tho­nor­mal­ity of the en­ergy states.) So ${\rm {M}}1$ tran­si­tions can­not change the ra­dial state. All they can do is change the di­rec­tion of the or­bital an­gu­lar mo­men­tum or spin of a given state. Ob­vi­ously that is ho-hum, though with a spin-or­bit term it may still do some­thing. With­out a spin-or­bit term, there would be no en­ergy change, and Fermi’s golden rule would make the the­o­ret­i­cal tran­si­tion rate then zero. That is sim­i­lar to the lim­i­ta­tion of ${\rm {M}}1$ tran­si­tions for the non­rel­a­tivis­tic hy­dro­gen atom in chap­ter 7.4.4.

It may be in­struc­tive to use the more re­al­is­tic ra­dial fac­tors of ta­ble A.2 to get a rough idea of the er­rors in the Weis­skopf ones. The ini­tial com­par­i­son will be re­stricted to changes in the prin­ci­pal quan­tum num­ber of no more than one unit. That means that tran­si­tions be­tween widely sep­a­rated shells will be ig­nored. Also, only the low­est pos­si­ble mul­ti­pole level will be con­sid­ered. That cor­re­sponds to the first of each pair of val­ues in the ta­ble. As­sum­ing an elec­tric tran­si­tion, $\ell$ is the dif­fer­ence be­tween the $l$ val­ues in the ta­ble. Con­sider now the fol­low­ing two sim­ple ap­prox­i­ma­tions of the ra­dial fac­tor:

$$\fbox{$\displaystyle \mbox{Weisskopf: } f^{{\rm rad},\ell... ...m LH} = \left(\frac{1.5}{\ell+3}\right)^2 \mbox{ or } 1 $} %$$ (A.187)

The co­ef­fi­cient 1.5 comes from a least square ap­prox­i­ma­tion of the data. For ${\rm {M}}1$ tran­si­tions, the ex­act value 1 should be used.

For the given data, it turns out that the Weis­skopf es­ti­mate is on av­er­age too large by a fac­tor 5. In the worst case, the Weis­skopf es­ti­mate is too large by a fac­tor 18. The em­pir­i­cal for­mula is on av­er­age off by a fac­tor 2, and in the worst case by a fac­tor 4.

If any ar­bi­trary change in prin­ci­pal quan­tum num­ber is al­lowed, the pos­si­ble er­rors are much larger. In that case the Weis­skopf es­ti­mates are off by av­er­age fac­tor of 20, and a max­i­mum fac­tor of 4 000. The em­pir­i­cal es­ti­mates are off by an av­er­age fac­tor of 8, and a max­i­mum one of 1 000. In­clud­ing the next num­ber in ta­ble A.2 does not make much of a dif­fer­ence here.

These er­rors do de­pend on the change in prin­ci­pal quan­tum num­bers. For changes in prin­ci­pal quan­tum num­ber no larger than 2 units, the em­pir­i­cal es­ti­mate is off by a fac­tor no greater that 10. For 3 or 4 unit changes, the es­ti­mate is off by a fac­tor no greater than about 100. The ab­solute max­i­mum er­ror fac­tor of 1 000 oc­curs for a 5 unit change in the prin­ci­pal quan­tum num­ber. For the Weiskopf es­ti­mate, mul­ti­ply these max­i­mum fac­tors by 4.

These data ex­clude the ${\rm {M}}1$ tran­si­tions men­tioned ear­lier, for which the ra­dial fac­tor is ei­ther 0 or 1 ex­actly. The value 0 im­plies an in­fi­nite er­ror fac­tor for a Weis­skopf-type es­ti­mate of the ra­dial fac­tor. But that re­quires an ${\rm {M}}1$ tran­si­tion with at least a two unit change in the prin­ci­pal quan­tum num­ber. In other words, it re­quires an ${\rm {M}}1$ tran­si­tion with a huge en­ergy change.

Con­sider now the an­gu­lar fac­tor in the de­cay rates (A.184) and (A.185). It arises from in­te­grat­ing the spher­i­cal har­mon­ics, (A.183). But the ac­tual an­gu­lar fac­tor re­ally used in the tran­si­tion rates (A.184) and (A.185) also in­volves an av­er­ag­ing over the pos­si­ble an­gu­lar ori­en­ta­tions of the ini­tial atom. (This ori­en­ta­tion is re­flected in its mag­netic quan­tum num­ber $m_{j{\rm {H}}}$.) And it in­volves a sum­ma­tion over the dif­fer­ent an­gu­lar ori­en­ta­tions of the fi­nal nu­cleus that can be de­cayed to. The rea­son is that ex­per­i­men­tally, there is usu­ally no con­trol over the ori­en­ta­tion of the ini­tial and fi­nal nu­clei. An av­er­age ini­tial nu­cleus will have an av­er­age ori­en­ta­tion. But each fi­nal ori­en­ta­tion that can be de­cayed to is a sep­a­rate de­cay process, and the de­cay rates add up. (The av­er­ag­ing over the ini­tial ori­en­ta­tions does not re­ally make a dif­fer­ence; all ori­en­ta­tions de­cay at the same rate, since space has no pre­ferred di­rec­tion. The sum­ma­tion over the fi­nal ori­en­ta­tions is crit­i­cal.)

Ta­ble A.3: An­gu­lar in­te­gral cor­rec­tion fac­tors $f^{{\rm{ang}},\vert\Delta{j}\vert}_{\rm{LH}}$ and $f^{{\rm{ang}},\vert\Delta{j}\vert+1}_{\rm{LH}}$ for the Weis­skopf elec­tric unit and the Moszkowski mag­netic one. The cor­rec­tion for the Weis­skopf mag­netic unit is to cross it out and write in the Moszkowski unit.

Val­ues for the an­gu­lar fac­tor are in ta­ble A.3. For the first and sec­ond num­ber of each pair re­spec­tively:

$$\ell=\vert j_{\rm {H}}-j_{\rm {L}}\vert \qquad \ell=\vert j_{\rm {H}}-j_{\rm {L}}\vert + 1$$

More gen­er­ally, the an­gu­lar fac­tor is given by, [33, p. 878],

$$\fbox{$\displaystyle f^{\rm ang,\ell}_{\rm LH} = (2j_{\rm{... ...56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}]^2 $} %$$ (A.188)

Here the quan­tity in square brack­ets is called a Cleb­sch-Gor­dan co­ef­fi­cient. For small an­gu­lar mo­menta, val­ues can be found in fig­ure 12.5. For larger val­ues, re­fer to {N.13}. The lead­ing fac­tor is the rea­son that the val­ues in the ta­ble are not the same if you swap the ini­tial and fi­nal states. When the fi­nal state has the higher an­gu­lar mo­men­tum, there are more nu­clear ori­en­ta­tions that an atom can de­cay to.

It may be noted that [11, p. 9-178] gives the above fac­tor for elec­tric tran­si­tions as

$$f^{\rm ang,\ell}_{\rm LH} = (2j_{\rm L}+1)(2\ell+1)(2l_{\r... ... L}&\frac12\ j_{\rm H}&l_{\rm H}&\ell \end{array} \right\}^2$$

Here the ar­ray in paren­the­ses is the so-called Wigner 3j sym­bol and the one in curly brack­ets is the Wigner 6j sym­bol, {N.13}. The idea is that this ex­pres­sion will take care of the se­lec­tion rules au­to­mat­i­cally. And so it does, if you as­sume that the mul­ti­ply-de­fined $l$ is $\ell$, as the au­thor seems to say. Of course, se­lec­tion rules might be a lot eas­ier to eval­u­ate than 3j and 6j sym­bols.

For mag­netic mul­ti­pole tran­si­tions, with $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert j_{\rm {H}}-j_{\rm {L}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert l_{\rm {H}}-l_{\rm {L}}\vert+1$, the same source comes up with

$$\begin{eqnarray*} f^{\rm ang,\ell}_{\rm LH} & = &\displaystyle (2j_{\rm L}+1)\... ...c12&\frac12&1\ j_{\rm H}&j_{\rm L}&\ell \end{array} \right\}^2 \end{eqnarray*}$$

Here the fi­nal ar­ray in curly brack­ets is the Wigner 9j sym­bol. The bad news is that the 6j sym­bol does not al­low any tran­si­tions of low­est mul­ti­pole or­der to oc­cur! Some­one fa­mil­iar with 6j sym­bols can im­me­di­ately see that from the so-called tri­an­gle in­equal­i­ties that the co­ef­fi­cients of 6j sym­bols must sat­isfy, {N.13}. For­tu­nately, it turns out that if you sim­ply leave out the 6j sym­bol, you do seem to get the right val­ues and se­lec­tion rules.

The mag­netic mul­ti­pole ma­trix el­e­ment also in­volves an an­gu­lar mo­men­tum fac­tor. This fac­tor turns out to be rel­a­tively sim­ple, {D.43.3}:

$$\fbox{$\displaystyle \begin{array}{r@{\; }c@{\; }l} f^{... ...ax}+\frac12 \end{array} \end{array} \right. \end{array} $} %$$ (A.189)

Here min” and “max re­fer to what­ever is the smaller, re­spec­tively larger, one of the ini­tial and fi­nal val­ues.

The stated val­ues of the or­bital an­gu­lar mo­men­tum $l$ are the only ones al­lowed by par­ity and the or­bital an­gu­lar mo­men­tum con­ser­va­tion con­di­tion (A.186). In par­tic­u­lar, con­sider the first ex­pres­sion above, for the min­i­mum mul­ti­pole or­der $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert\Delta{j}\vert$. Ac­cord­ing to this ex­pres­sion, the change in or­bital an­gu­lar mo­men­tum can­not ex­ceed the change in net an­gu­lar mo­men­tum. That for­bids a lot of mag­netic tran­si­tions in a shell model set­ting, tran­si­tions that seem per­fectly fine if you only look at net an­gu­lar mo­men­tum and par­ity. Add to that the ear­lier ob­ser­va­tion that ${\rm {M}}1$ tran­si­tions can­not change the ra­dial state at all. Mag­netic tran­si­tions are quite hand­i­capped ac­cord­ing to the sin­gle-par­ti­cle model used here.

Of course, a sin­gle-par­ti­cle model is not ex­act for mul­ti­ple-par­ti­cle sys­tems. In a more gen­eral set­ting, tran­si­tions that in the ideal model would vi­o­late the or­bital an­gu­lar mo­men­tum con­di­tion can oc­cur. For ex­am­ple, con­sider the pos­si­bil­ity that the true state picks up some un­cer­tainty in or­bital an­gu­lar mo­men­tum.

Pre­sum­ably such tran­si­tions would be un­ex­pect­edly slow com­pared to tran­si­tions that do not vi­o­late any ap­prox­i­mate or­bital an­gu­lar mo­men­tum con­di­tions. That makes es­ti­mat­ing the mag­netic tran­si­tion rates much more tricky. Af­ter all, for nu­clei the net an­gu­lar mo­men­tum is usu­ally known with some con­fi­dence, but the or­bital an­gu­lar mo­men­tum of in­di­vid­ual nu­cle­ons is not.

For­tu­nately, for elec­tric tran­si­tions or­bital an­gu­lar mo­men­tum con­ser­va­tion does not pro­vide ad­di­tional lim­i­ta­tions. Here the or­bital re­quire­ments are al­ready sat­is­fied if net an­gu­lar mo­men­tum and par­ity are con­served.

The de­rived de­cay es­ti­mates are now used to de­fine stan­dard de­cay rates. It is as­sumed that the mul­ti­pole or­der is min­i­mal, $\ell$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert\Delta{j}\vert$, and that the fi­nal an­gu­lar mo­men­tum is $\frac12$. As ta­ble A.3 shows, that makes the an­gu­lar fac­tor equal to 1. The stan­dard elec­tric de­cay rate is then

$$\fbox{$\displaystyle \lambda^{{\rm{E}}\ell}_{\rm Weisskopf... ...frac{2(\ell+1)}{\ell(2\ell+1)!!^2} \frac{9}{(\ell+3)^2} $} %$$ (A.190)

This de­cay rate is called the “Weis­skopf unit” for elec­tric mul­ti­pole tran­si­tions. It is com­monly in­di­cated by W.u. Mea­sured ac­tual de­cay rates are com­pared to this unit to get an idea whether they are un­usu­ally high or low.

Note that the de­cay rates are typ­i­cally or­ders of mag­ni­tude off the mark. That is due to ef­fects that can­not be ac­counted for. Nu­cle­ons are not in­de­pen­dent par­ti­cles by far. And even if they were, their ra­dial wave func­tions would not be con­stant. The used ex­pres­sion for the elec­tric ma­trix el­e­ment is prob­a­bly no good, {N.14}. And es­pe­cially higher mul­ti­pole or­ders de­pend very sen­si­tively on the nu­clear ra­dius, which is im­pre­cisely de­fined.

The stan­dard mag­netic mul­ti­pole de­cay rate be­comes un­der the same as­sump­tions:

$$\fbox{$\displaystyle \lambda^{\rm M\ell}_{\rm Moszkowski} ... ...(\ell+2)^2} \left(g_i - \frac{2}{\ell+1}\right)^2\ell^2 $} %$$ (A.191)

This de­cay rate is called the “Moszkowski unit” for mag­netic mul­ti­pole tran­si­tions.

Fi­nally, it should be men­tioned that it is cus­tom­ary to ball­park the fi­nal mo­men­tum fac­tor in the Moszkowski unit by 40. That is be­cause Je­sus spent 40 days in the desert. Also, the fac­tor $(\ell+2)^2$ is cus­tom­ar­ily re­placed by $(\ell+3)^2$, [10, p. 9-49], [36, p. 676], [5, p. 242], be­cause, hey, any­thing for a laugh. Other sources keep the $(\ell+2)^2$ fac­tor just like it is, [11, p. 9-178], [31, p. 332], be­cause, hey, why not? Note that the Hand­book of Physics does both, de­pend­ing on the au­thor you look at. Tak­ing the most re­cent of the cited sources, as well as [[4]], as ref­er­ence the new and im­proved mag­netic tran­si­tion rate may be:

$$\fbox{$\displaystyle \lambda^{{\rm{M}}\ell}_{\rm Weisskopf... ...left(\frac{\hbar}{m c R}\right)^2 \frac{90}{(\ell+3)^2} $} %$$ (A.192)

This is called the Weis­skopf mag­netic unit. Note that the hu­mor fac­tor has been greatly in­creased. Whether there is a 2 or 3 in the fi­nal frac­tion does not make a dif­fer­ence. All analy­sis is rel­a­tive to the per­cep­tion of the ob­server. Where one per­ceives a 2 an­other sees a 3. Ever­thing is rel­a­tive, as Ein­stein sup­pos­edly said, and oth­er­wise quan­tum me­chan­ics def­i­nitely did.

Note that the Weis­skopf mag­netic unit looks ex­actly like the elec­tric one, ex­cept for the ad­di­tion of a zero and the ad­di­tional frac­tion be­tween paren­the­ses. That makes it eas­ier to re­mem­ber, es­pe­cially for those who can re­mem­ber the elec­tric unit. For them the sav­ings in time is tremen­dous, be­cause they do not have to look up the cor­rect ex­pres­sion. That can save a lot of time be­cause many stan­dard ref­er­ences have the for­mu­lae wrong or in some weird sys­tem of units. All that time is much bet­ter spend try­ing to guess whether your source, or your ed­i­tor, uses a 2 or a 3.

A.25.9 Er­rors in other sources

There is a no­table amount of er­rors in de­scrip­tions of the Weis­skopf and Moszkowski es­ti­mates found else­where. That does not even in­clude not men­tion­ing that the elec­tric mul­ti­pole rate is likely no good, {N.14}. Or ran­domly us­ing $\ell+2$ or $\ell+3$ in the Weis­skopf mag­netic unit.

These er­rors are more ba­sic. The first edi­tion of the Hand­book of Physics, [10, p. 9-49], gives both Weis­skopf units wrong. Squares are miss­ing on the $\ell+3$, and so is the fine struc­ture con­stant. The other nu­mer­i­cal fac­tors are con­sis­tent be­tween the two units, but not right. Prob­a­bly a strangely nor­mal­ized ma­trix el­e­ment is given, rather than the stated de­cay rates $\lambda$, and in ad­di­tion the square was for­got­ten.

The same Hand­book, [10, p. 9-110], but a dif­fer­ent au­thor, uses $g_i$$\raisebox{.5pt}{$/$}$​2 in­stead of $g_i$ in the Moszkowski es­ti­mate. (Even physi­cists them­selves can get con­fused if some­times you de­fine $g_{\rm {p}}$ to be 5.6 and some­times 2.8, which also hap­pens to be the mag­netic mo­ment $\mu_{\rm {p}}$ in nu­clear mag­ne­tons, which is of­ten used as a nondi­men­sion­al unit where $g_{\rm {p}}$ is re­ally needed, etcetera.) More se­ri­ously, this er­ror is car­ried over to the given plot of the Moszkowski unit, which is there­fore wrong. Which is in ad­di­tion to the fact that the nu­clear ra­dius used in it is too large by mod­ern stan­dards, us­ing 1.4 rather than 1.2 in (A.177).

The er­ror is cor­rected in the sec­ond edi­tion, [11, p. 9-178], but the Moszkowski plot has dis­ap­peared. In fa­vor of the Weis­skopf mag­netic unit, of course. Think of the sci­en­tific way in which the Weis­skopf unit has been de­duced! This same ref­er­ence also gives the er­ro­neous an­gu­lar fac­tor for mag­netic tran­si­tions men­tioned in the pre­vi­ous sub­sec­tion. Of course an ad­di­tional 6j sym­bol that sneaks in is eas­ily over­looked.

No se­ri­ous er­rors were ob­served in [33]. (There is a read­ily-fixed er­ror in the con­ver­sion for­mula for when the ini­tial and fi­nal states are swapped.) This source does not list the Weis­skopf mag­netic unit. (Which is cer­tainly de­fen­si­ble in view of its non­sen­si­cal as­sump­tions.) Un­for­tu­nately non-SI units are used.

The elec­tric di­pole ma­trix el­e­ment in [36, p. 676] is miss­ing a fac­tor 1/$2c$. The claim that this el­e­ment can be found by straight­for­ward cal­cu­la­tion is lu­di­crous. Not only is the math­e­mat­ics con­vo­luted, it also in­volves the ma­jor as­sump­tion that the po­ten­tials de­pend only on po­si­tion. A square is miss­ing in the Moszkowski unit, and the ta­ble of cor­re­spond­ing widths are in eV in­stead of the stated 1/s.

All three units are given in­cor­rectly in [31, p. 332]. There is a fac­tor $4\pi$ in them that should not be there. And the mag­netic rate is miss­ing a fac­tor $\ell^2$. The con­stant in the nu­mer­i­cal ex­pres­sion for ${\rm {M}}3$ tran­si­tions should be 15, not 16. Of course, the dif­fer­ence is neg­li­gi­ble com­pared to re­plac­ing the par­en­thet­i­cal ex­pres­sion by 40, or com­pared to the or­ders of mag­ni­tude that the es­ti­mate is com­monly off any­way.

The Weis­skopf units are listed cor­rectly in [5, p. 242]. Un­for­tu­nately non-SI units are used. The Moszkowski unit is not men­tioned. The non­sen­si­cal na­ture of the Weis­skopf mag­netic unit is not pointed out. In­stead it is claimed that it is found by a sim­i­lar cal­cu­la­tion as the elec­tric unit.