- 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

The derivations include a term due to an effect that was mentioned in the initial 1952 derivation by B. Stech, [44]. 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
, with1, 2, and 3 for , , andrespectively. - 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

(D.26) |

The energy difference can be expressed in terms of the energy

Note that none of my sources includes the commutator in the first
term, not even [16]. (The original 1952 derivation by
[44] 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

To prove the lemma, start with the left hand side

where subscripts

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

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

In that case:

The quickest way to prove this is to take the

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

where

To reduce this, take the factor

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

Plugging these results back into the expression for the matrix
element, renotating

where

and

Here

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,

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

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

where

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

The direction of integration in the expression for

Long photon wave length corresponds to small photon wave number

This is readily integrated to find

and

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

This assumes

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

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

The inner product should normally be of the same order as the one of

The third matrix element to find is the magnetic multipole one

Note that in index notation

where

where

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

Using the same long wave length approximation for

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

The value of

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

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 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

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

Fortunately, it turns out that this makes no difference. For example,
by symmetry the state

Now the first term produces decays to

So as long as the final state

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

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

For these quantum numbers, the initial and final states are

where the square roots come from figure 12.5 in the

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

Next, this gets multiplied by the

The second term in

However, the

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

as seen in table 4.3 or {D.64}. The constants

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

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

More interestingly, the same expression turns out to hold if neither
the initial nor the final angular momentum equals

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.