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:

$$H_{21} = \big\langle\psi_{\rm {L}}\big\vert H \big\vert\psi_{\rm {H}}\big\rangle$$

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):

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

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

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

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

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 $\vphantom0\raisebox{1.5pt}{$-$}$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 [31] and [10] 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:

$$H_{21,i} = \big\langle\psi_{\rm {L}}\big\vert H \big\vert\... ...g_i{\widehat S}_{i,y})] \big\vert\psi_{\rm {H}}\big\rangle$$

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:

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

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:

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

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

$$\fbox{$\displaystyle H = \sum_i \left[\frac{1}{2m_i}\lef... ...al B}_i \cdot {\skew 6\widehat{\vec S}}_i \right] + V $} %$$ (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.38}. See the Dirac equation and in particular {D.75} 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 [42]. 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, [31].)

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

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

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

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

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

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, ${\hat\imath}_{\cal B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$${\hat\imath}$.

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

$$\begin{eqnarray*} H_{21,i} & = & \big\langle\psi_{\rm {L}}\big\vert H_i \big\v... ...{\skew 6\widehat{\vec S}}_i \big\vert\psi_{\rm {H}}\big\rangle \end{eqnarray*}$$

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:

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

where

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

$$\fbox{$\displaystyle H_{21,i}^{\rm M\ell} \approx - \f... ...\cal B}}\Big) \big\vert\psi_{\rm{H}}\big\rangle $} \, %$$ (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

$$\vert H_{21,i}^{\rm E\ell}\vert = \frac{\vert q_i\vert{\ca... ...1}(r_{i,{\cal E}}/R) \big\vert\psi_{\rm {H}}\big\rangle \vert$$

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

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

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

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

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 $\vphantom0\raisebox{1.5pt}{$-$}$$\hbar^2\nabla^2$$\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

$$\big\langle\psi_{\rm {L}}\big\vert x_i\big\vert\psi_{\rm {H}}\big\rangle$$

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 $\vphantom0\raisebox{1.5pt}{$-$}$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, ($\vphantom0\raisebox{1.5pt}{$-$}$0 is 0 but say $\vphantom0\raisebox{1.5pt}{$-$}$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 $\vphantom0\raisebox{1.5pt}{$-$}$1 that this state picks up under axes inversion cancels the $\vphantom0\raisebox{1.5pt}{$-$}$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

$$\big\langle\psi_{\rm {L}}\big\vert\L _{i,y}+g_i{\widehat S}_{i.y}\big\vert\psi_{\rm {H}}\big\rangle$$

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

$$\big\langle\psi_{\rm {L}}\big\vert x_i\big\vert\psi_{\rm {H}}\big\rangle$$

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 $m_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$1, respectively 1.

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

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

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

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

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

$$\big\langle\psi_{\rm {L}}\big\vert\L _{i,y}+g_i{\widehat S}_{i.y}\big\vert\psi_{\rm {H}}\big\rangle$$

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:

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

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

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

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

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

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

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},

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

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},

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

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},

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

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 [31] and [[5]], 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:

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

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

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

$$\fbox{$\displaystyle \vert h_{21}^{\rm M\ell}\vert = \su... ...t{\vec S}}_i}{\hbar}\Big) \vert\psi_{\rm{H}}\rangle $} %$$ (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 $g_i$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$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 [31] and [10] 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:

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

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

$$\Theta_{lj}^{m_j}(\theta_i,\phi_i) = c_1 Y_l^{m_j-\frac1... ...row} + c_2 Y_l^{m_j+\frac12} (\theta_i,\phi_i) {\downarrow}$$

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.38}. 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:

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

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,

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

and an angular one,

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

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

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

$$\fbox{$\displaystyle \lambda^{\rm M\ell} = \alpha \omega... ...f^{\rm ang,\ell}_{\rm LH} f^{\rm mom,\ell}_{\rm LH} $} %$$ (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, [31]:

$$\vert l_{\rm H}-l_{\rm L}\vert \mathrel{\raisebox{-.7pt}{$... ...\mathrel{\raisebox{-.7pt}{$\leqslant$}}l_{\rm H}+l_{\rm L} %$$ (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 14.12, 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:

$$\fbox{$\displaystyle \mbox{Weisskopf: } f^{{\rm rad},\... ...} = \left(\frac{1.5}{\ell+3}\right)^2 \mbox{ or } 1 $} %$$ (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, [31, p. 878],

$$\fbox{$\displaystyle f^{\rm ang,\ell}_{\rm LH} = (2j_{\r... ...4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle ]^2 $} %$$ (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 [10, p. 9-178] gives the above factor for electric transitions as

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

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

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

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}:

$$\fbox{$\displaystyle \begin{array}{r@{\;\,}c@{\;\,}l} ... ...2 \end{array} \end{array} \right. \end{array} $} %$$ (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

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

$$\fbox{$\displaystyle \lambda^{\rm M\ell}_{\rm Moszkowski... ...l+2)^2} \left(g_i - \frac{2}{\ell+1}\right)^2\ell^2 $} %$$ (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 $(\ell+3)^2$, [9, p. 9-49], [34, p. 676], [5, p. 242], because, hey, anything for a laugh. Other sources keep the $(\ell+2)^2$ factor just like it is, [10, p. 9-178], [29, 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 [[5]], as reference the new and improved magnetic transition rate may be:

$$\fbox{$\displaystyle \lambda^{{\rm{M}}\ell}_{\rm Weissko... ...(\frac{\hbar}{m c R}\right)^2 \frac{90}{(\ell+3)^2} $} %$$ (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, [9, 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, [9, 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, [10, 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 [31]. (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 [34, 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 [29, 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.