- A.25.1 Approximate Hamiltonian
- A.25.2 Approximate multipole matrix elements
- A.25.3 Corrected multipole matrix elements
- A.25.4 Matrix element ballparks
- A.25.5 Selection rules
- A.25.6 Ballpark decay rates
- A.25.7 Wave functions of definite angular momentum
- A.25.8 Weisskopf and Moszkowski estimates
- A.25.9 Errors in other sources

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

Here

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

Here

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:

Here

If the matrix element 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

For convenience the

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

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},

Here

For absorption, the low energy state is the first one, instead of the
second. That makes the exponential above

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

The bottom line is that for emission

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

In general, you will need to sum this over all particles

The first term in the Hamiltonian above is like the

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

A special case needs to be made for the neutrons in a nucleus. Since
the neutron has no charge,

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 [32] 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:

This will normally need to be summed over all electrons

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

In the second equality, the summation index was renotated as

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

The terms with

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

is the magnetic part of the field. The spin

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 [43]. 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

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

The term

That makes the interaction Hamiltonian of a single particle

To find that connection requires considerable manipulation. First the
vector potential

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

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

Here the unit vector

The three unit vectors are orthonormal. Note that for a given direction of wave propagation

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

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:

where

Here

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

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

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

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

The magnitudes of the magnetic contributions can be written as

Recall that angular momentum values are multiples of

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

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

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

Here

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

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

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

Consider first the electric dipole contribution

Why would this be zero? Basically because in the inner product integrals, positive values of

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

So if both 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

The angular momentum operators do nothing under axes inversion. One way to see that is to think of

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

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

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

So it is a sum of two states, both with square angular momentum quantum number

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

Here

(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

To get a nonzero inner product, one of the possible states of net
angular momentum above will need to match the quantum numbers

(And if

So the complete selection rules for electric dipole transitions are

where

For magnetic dipole transitions, the relevant inner product is

Note that it is either

One big limitation is that in either an electric or a magnetic dipole
transition, the net atomic or nuclear angular momentum

It follows that the first multipole term that can be nonzero has

A further limitation applies to orbital angular momentum. The angular
momentum operators will not change the orbital angular momentum
values. And the factors

The final limitation is that

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

That assumes that the entire system is contained in a very large periodic box of volume

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

Here

Ballpark matrix coefficients were given in subsection A.25.4.
However, a more accurate estimate is desirable. The main problem is
the factor

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

The electric inner product contains a further factor

Putting it all together, the estimated decay rates become

is the so-called fine structure constant. with

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

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

Here

The contribution of a particle

But now, for electric transitions

The matrix elements can be approximated assuming that the wave length
of the photon is large compared to the size

The subscript

The approximate contribution of the particle to the

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

This assumes that the entire system is contained inside a very big sphere of radius

It is again convenient to nondimensionalize the matrix elements using
some suitably defined typical atomic or nuclear radius

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:

where

The sum is over the electrons or protons and neutrons, with

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

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

These wave functions are very much like the

The coefficients

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:

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,

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

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

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

Note that the magnetic decay rate uses

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

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, [32]:

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

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

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

These results illustrate the limitations of

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,

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

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

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

More generally, the angular factor is given by, [32, p. 878],

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

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

For magnetic multipole transitions, with

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}:

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

The stated values of the orbital angular momentum

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,

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:

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

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

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

The same Handbook, [10, p. 9-110], but a different
author, uses nondimensional

unit where

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 [32]. (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 [35, p. 676]
is missing a factor 1/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 [30, p. 332].
There is a factor

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.