- D.43.1 Matrix element for linear momentum modes
- D.43.2 Matrix element for angular momentum modes
- D.43.3 Weisskopf and Moszkowski estimates

D.43 Multipole derivations

This derives the multipole matrix elements corresponding to a single particle in an atom or nucleus. These will normally still need to be summed over all particles.

Both a basis of linear momentum photon wave functions and of angular momentum ones are covered. For the angular momentum wave functions, the long wave length approximation will be made that is small. Here is the photon wave number and the typical size of atom or nucleus.

The derivations include a term due to an effect that was mentioned in the initial 1952 derivation by B. Stech, [43]. This effect is not mentioned in any textbook that the author is aware off. That seems to be unjustified. The term does not appear to be small for nuclei, but at the very least comparable to the usual electric multipole element given.

The rules of engagement are as follows:

- The considered particle will be indicated by a subscript .
- The Cartesian axes are numbered using an index , with 1, 2, and 3 for , , and respectively.
- Also, indicates the coordinate in the direction, , , or .
- Derivatives with respect to a coordinate are indicated by a simple subscript .
- If the quantity being differentiated is a vector, a comma is used to separate the vector index from differentiation ones.
- A bare integral sign is assumed to be an integration over all nuclear coordinates.
- A superscript indicates a complex conjugate.

The convoluted derivations in this note make use of a trick. Since
trick

sounds too tricky, it will be referred to as:

Lemma 1: This lemma allows you to get rid of derivatives on the wave function. The lemma assumes nonrelativistic particles. It is a generalization of a derivation of [16].

The lemma says that if is the number of a particle in the atom or
nucleus, and if is any function of the position of that particle
, then

(D.26) |

The energy difference can be expressed in terms of the energy
of the nominal photon emitted in the transition,

Note that none of my sources includes the commutator in the first term, not even [16]. (The original 1952 derivation by [43] used a relativistic Dirac formulation, in which the term appears in a different place than here. The part in which it appears there is small without the term and is not worked out with it included.) The commutator is zero if the potential only depends on the position coordinates of the particles. However, nuclear potentials include substantial momentum terms.

To prove the lemma, start with the left hand side

where subscripts 1, 2, and 3 indicates the derivatives with respect to the three coordinates of particle . Summation over is to be understood. Average the above expression with what you get from doing an integration by parts:

or differentiating out

Combine the first two integrals

and do another integration by parts (I got this from [16], thanks):

Now note the nonrelativistic eigenvalue problems for the two states

Here the sum is over the other particles in the nucleus. These two eigenvalue problems are used to eliminate the second order derivatives in the integral above. The terms involving the Laplacians with respect to the coordinates of the other particles then drop out. The reason is that is just a constant with respect to those coordinates, and that Laplacians are Hermitian. Assuming that is at least Hermitian, as it should, the terms produce the commutator in the lemma. And the right hand sides give the energy-difference term. The result is the lemma as stated.

D.43.1 Matrix element for linear momentum modes

This requires in addition:

Lemma 2: This lemma allows you to express a certain combination of derivatives in terms of the angular momentum operator. It will be assumed that vector is normal to vector .

In that case:

The quickest way to prove this is to take the -axis in the direction of , and the -axis in the direction of . (The expression above is also true if the two vectors are nor orthogonal. You can see that using index notation. However, that will not be needed.) The final equality is just the definition of the angular momentum operator.

The objective is now to use these lemmas to work out the matrix element

where is the constant wave number vector and is some other constant vector normal to . Also is the position of the considered particle, and is the momentum operator based on these coordinates.

To reduce this, take the factor out of and write the
exponential in a Taylor series:

Take another messy factor out of the inner product:

For brevity, just consider the inner product by itself for now. It can trivially be rewritten as a sum of two terms, ([16], not me):

(Recall that and are orthogonal. Also note that the Laplacian of is of essentially the same form as , just for a different value of .) On the second inner product (2), lemma 2 can be applied.

Plugging these results back into the expression for the matrix
element, renotating into for the first part of (1), into
for the second part, which can then be combined with the
first part, and into for (2), and cleaning up gives the final
result:

where

and

Here is the magnitude of . Also is the component of the position of particle in the direction of motion of the electromagnetic wave. The direction of motion is the direction of . Similarly is the component of in the direction of the electric field. The electric field has the same direction as . Further, is the component of the orbital angular momentum operator of particle in the direction of the magnetic field. The magnetic field is in the same direction as . Finally, the factor is

The approximation applies because normally the energy release in a transition is small compared to the rest mass energy of the particle. (And if it was not, the nonrelativistic electromagnetic interaction used here would not be valid in the first place.) For the emission process covered in {A.25}, the plus sign applies, 1.

The commutator is zero if the potential depends only on position. That is a valid approximation for electrons in atoms, but surely not for nuclei. For these it is a real problem, {N.14}.

For addendum {A.25}, the constant should be taken equal to . Note also that the interaction of the particle spin with the magnetic field still needs to be added to . This interaction is unchanged from the naive approximation.

D.43.2 Matrix element for angular momentum modes

This subsection works out the details of the matrix element when angular momentum modes are used for the photon wave function.

The first matrix element to find is

where, {A.21.7},

is the electric multipole vector potential at the location of particle . This uses the short hand

where is the multipole order or photon angular momentum, the photon wave number, a spherical Bessel function, and a spherical harmonic.

Note that the electric multipole vector potential is closely related
to the magnetic one:

The expression for the electric potential can be simplified for
long photon wave lengths. Note first that

where the second equality applied because the vector potentials are solenoidal and the standard vector identity (D.1), while the third equality is the energy eigenvalue problem, {A.21}. It follows that the electric vector potential is of the form

because vector calculus says that if the curl of something is zero, it is the gradient of some scalar function . Here

The direction of integration in the expression for does not make a difference, so the simplest is to integrate radially outwards. The expression for was given in {D.36.2}. That gives

Long photon wave length corresponds to small photon wave number
. All terms above can then be ignored and in addition
the following approximation for the Bessel function applies,
{A.6},

This is readily integrated to find

and is the gradient.

That allows lemma 1 to be used to find the electric matrix element.

This assumes is indeed the lower-energy state. The value of (as defined here) to use in addendum {A.25} is .

The commutator is again negligible for atoms, but a big problem for nuclei, {N.14}.

There is also a term due to the interaction of the spin with the
magnetic field, given by the curl of
as already found above,

Using the property of the scalar triple product that the factors can be interchanged if a minus sign is added, the matrix element becomes

(Note that only acts on the ; is a function, not a differential operator.) In the long wave length approximation of the Bessel function, that becomes

The inner product should normally be of the same order as the one of . However, the second fraction above is normally small; usually the photon energy is small compared to the rest mass energy of the particles. (And if it was not, the nonrelativistic electromagnetic interaction used here would not be valid in the first place.) So this second term will be ignored in addendum {A.25}.

The third matrix element to find is the magnetic multipole one

Note that in index notation

where follows in the cyclic sequence and precedes . By a trivial renotation of the summation indices,

where is the orbital angular momentum operator. Note that the parenthetical term commutes with this operator, something not mentioned in [32, p. 874].

It follows that

or in the long wave length approximation

There is also a term due to the interaction of the spin with the
magnetic field, given by the curl of ,
which equals ,

Using the same long wave length approximation for as before, that becomes

The orbital and spin matrix elements may be combined into one as

The value of to use in addendum {A.25} is .

D.43.3 Weisskopf and Moszkowski estimates

This subsection explains where the radial, angular, and momentum factors in the Weisskopf and Moszkowski estimates come from. These factors represent the nondimensionalized matrix elements.

The electric matrix element is simplest. It is, written out in
spherical coordinates using the assumed wave functions,

The Weisskopf and Moszkowski estimates assume that the radial parts of
wave functions equal a constant until the nuclear edge and are
zero outside the nucleus. To perform the radial integral is then
straightforward:

The first equality is true because the integral in the denominator is 1 on account of the normalization condition of wave functions. The second inequality follows from integrating.

The angular integral above is more tricky to ballpark. First of all, it will be assumed that the matrix element of interest is the lowest multipole order allowed by angular momentum conservation. That seems reasonable, given that normally higher multipole transitions will be very much slower. It follows that . (The possibility that the initial and final angular momenta are equal will be ignored.)

The change in orbital angular momenta could in principle be up to one unit different from the change in net angular momenta because of the spins. But parity conservation allows only .

To simplify even further, assume the following specific angular
states:

which have

If these states are substituted into the angular integral, the product of the spin states is 1 because spin states are orthonormal. What is left is

Now 1 which is just a constant that can be taken out of the integral. There it cancels the corresponding square root in the definition of the matrix element. Then it is seen that the transition can only create a photon for which . The reason is that spherical harmonics are orthonormal; the inner product is only nonzero if the two spherical harmonics are equal, and then it is 1. So the conclusion is that for the given states

The angular integral is 1. That makes the decay rate exactly 1 Weisskopf unit.

One glaring deficiency in the above analysis was the assumption that the initial proton state was a one. It would certainly be reasonable to have an initial nuclear state that has orbital angular momentum and total angular momentum . But a bunch of these nuclei would surely each be oriented in its own random direction. So they would have different magnetic quantum numbers . They would not all have .

Fortunately, it turns out that this makes no difference. For example,
by symmetry the state decays just as happily to
as does to . For other
values of it is a bit more nuanced. They produce an
initial state of the form:

Now the first term produces decays to by the emission of a photon with . However, because of the factor the number of such decays that occur per second is a factor less than the Weisskopf unit. But the second term produces decays to by the emission of a photon with . This decay rate is a factor less than the Weisskopf unit. Since 1, (the normalization condition of the state), the total decay rate is still 1 Weisskopf unit.

So as long as the final state has zero orbital angular momentum, the decay is at 1 Weisskopf unit. The orientation of the initial state makes no difference. That is reflected in table A.3. This table lists the angular factors to be applied to the Weisskopf unit to get the actual decay rate. The first row shows that, indeed, when the final angular momentum is , as occurs for zero angular momentum, and the initial angular momentum is , then no correction is needed. The correction factor is 1.

More interesting is the possibility that the two states are swapped. Then the initial state is the one with zero orbital angular momentum. It might at first seem that that will not make a difference either. After all, decay rates between specific states are exactly the same.

But there is in fact a difference. Previously, each initial nucleus had only two states to decay to: the spin-up and the spin-down version of the final state. Now however, each initial nucleus has , i.e. final states it can decay to, corresponding to the possible values of the final magnetic quantum number . That will increase the total decay rate correspondingly. In fact, suppose that the initial nuclei come in spin-up and spin-down pairs. Then each pair will decay at a rate of one Weisskopf unit to each possible final state. That is because this picture is the exact reverse of the decay of the final state. So the pairs would decay at a rate faster than the Weisskopf unit. So by symmetry each nucleus of the pair decays times faster than the Weisskopf unit. That is reflected in the first column of table A.3. (Recall that is the difference in the values.)

If neither the initial nor final state has zero orbital angular momentum, it gets more messy. Figuring out the correction factor in that case is something for those who love abstract mathematics.

Next consider magnetic multipole transitions. They are much messier
to ballpark. It will again be assumed that the multipole order is the
smallest possible. Unfortunately, now the final orbital angular
momentum cannot be zero. Because of parity, that would require that
the initial orbital angular momentum would be . But
that is too large because of the limitation (A.175) on the
orbital angular momentum change in magnetic transitions. Therefore
the simplest possible initial and final states have

For these quantum numbers, the initial and final states are

where the square roots come from figure 12.5 in the tabulation.

Now consider the form of the magnetic matrix element (A.181). First note, {D.43.2}, that the angular momentum and gradient factors commute. That helps because then the angular momentum operators, being Hermitian, can be applied on the easier state .

The -component part of the dot product in the matrix element is then the easiest. The components of the angular momentum operators leave the state essentially unchanged. They merely multiply the two terms by the eigenvalue respectively .

Next, this gets multiplied by the -component of the gradient. But multiplying by the gradient cannot change the spin. So the spin-down first term in stays spin-down. That cannot match the spin-up of . So the first term does not produce a contribution.

The second term in has the right spin. Since spin states are orthonormal, their inner product produces 1. But now there is a problem of matching the magnetic quantum number of . In particular, consider the harmonic polynomial in the gradient. The gradient reduces it to a combination of harmonic polynomials of one degree less, in other words, to polynomials. That limits to a value no larger than , and since the second term in has magnetic quantum number 0, the value in cannot be matched. The bottom line is that the -component terms in the inner product of the matrix element do not produce a contribution.

However, the - and -component terms are another story. The angular momentum operators in these directions change the corresponding magnetic quantum numbers, chapter 12.11. In general, their application produces a mixture of and states. In particular, the and components of spin will produce a spin-up version of the first term in . That now matches the spin in and a nonzero contribution results. Similarly, the orbital angular momentum operators will produce an 1 version of the second term in . Combined with the units from the gradient, that is enough to match the magnetic quantum number of . So there is a total of four nonzero contributions to the matrix element.

Now it is just a matter of working out the details to get the complete
matrix element. The information in chapter 12.11 can be used
to find the exact states produced from
by the and
angular momentum operators. Each state is a multiple of the
state. As far as the gradient term is concerned, the
harmonic polynomials are of the general form

as seen in table 4.3 or {D.65}. The constants are of no importance here. The and derivatives of the first harmonic polynomial will give the needed harmonic. (For values of greater than 1, the third harmonic could also make a contribution. However, it turns out that here the and contributions cancel each other.) The effect of the -derivative on the first harmonic is simply to add a factor to it. Similarly, the -derivative simply adds a factor . Now if you look up in table 4.3, you see it is a multiple of . So the product with the gradient term produces a simple multiple of . The inner product with then produces that multiple (which still depends on of course.) Identifying and adding the four multiples produces

The remaining radial integral may be ballparked exactly the same as for the electric case. The only difference is that the power of is one unit smaller.

A similar analysis shows that the given initial state cannot decay to the version of the final state with negative magnetic quantum number .

And of course, if the initial and final states are swapped, there is again a factor increase in decay rate.

More interestingly, the same expression turns out to hold if neither the initial nor the final angular momentum equals , using the correction factor of table A.3. But the obtained magnetic multipole decay rate is more limited than the electric one. It does require that and that

The momentum factors (A.189) were identified using a computer program. This program crunched out the complete matrix elements using procedures exactly like the ones above. This program was also used to create table A.3 of angular factors. This guards against typos and provides an independent check on the Clebsch-Gordan values.