23. Photon Radiation
Here we will discuss how a single photon is radiated. This is an important section as it really gives a different view of what is occurring at the atomic level. It reveals that within an atom, an electron is actually an electrostatic ring, that remains stationary about the nucleus except when radiating. From earlier hints of possible electron models, it is further indicated that this annular form is also the form of a free electron.
While we have discussed the electric field required for a photon, we have not determined how such a field could be produced. For a point charge which is accelerated for a short time the (radiative) electric field developed1 is given by:
(1)
where a is the acceleration and q is the angle between the direction of acceleration and the angle of the position vector R. This equation can be used as a Green's function to determine the radiation envelope created by the acceleration (for a short time) of any distribution of charge. For a very short period of acceleration, the difference between the retarded position of the charge and the present position of the charge can be neglected. Imagine a ring of charge which is accelerating radially outward, thus increasing its radius as in the following diagram.
Fig.11: Radially accelerating ring of charge.
It is seen from the symmetry that the electric field generated will be in the r-direction and z-direction. The field in the f-direction is canceled out. It is also seen that these values will be independent of f. The contribution to these fields by a charge element is thus
(2)
and
(3)
where q is the angle off the plane of the ring. This can be expressed by:
(4a)
(4b)
The charge element is
(5)
and the distance to the field point is
(6)
Thus, the total electric field is
(7)
and
(8)
For points distant from the ring, we can use the binomial expansion. This yields (when dropping higher order terms)
(9)
and
(10)
which leads to
(11)
and
(12)
Note that the field in r falls off as 1/R2 at the least. It is not a radiation field. This was expected in analogy to a dipole field where a 1/R2 is canceled and replaced by a 1/R3 field. Here we canceled a 1/R field. This charge accelerating creates an electric field in r that does not propagate. However, what field there is in the z-direction does propagate away leaving no field in that direction, without which the electric field that is left must thus propagate at c, perpendicular to the ring2.
Now imagine that the ring of charge starts at one radius, accelerates radially to constantly changing radius and then decelerates to a new stationary position. During this process, as the acceleration creates the radial field, it propagates away. We have a pulse, a field of finite extent in the direction of propagation. Since the acceleration reversed, we have opposite radial electric fields in sequence which fall off radially as 1/R2 or greater. This is like the field of the photon as previously described, except the intensity along z is not ideally sinusoidal as described in Section 15. It can be concluded thus that for photons to exist, there must exist rings of charge to create the photons.
The implications here are pervasive, affecting the fundamental views of atomic structure. Apparently the reason that charge does not radiate by its motion in the atom is not because classical electromagnetism ceases to function, but rather because the charge simply is not accelerating.
Since hydrogen emits photons, it can be concluded that the associated electron has this annular form and performs the motions described above when a photon is emitted. Thus we can imagine a hydrogen atom as a positive charged nucleus at the center of an annular charge field3. Note that this forms an electrostatic quadrupole.
In Section 3, there was an issue concerning the particle's geometry where the angular propagation of the energy was not completely cyclic. However, with this annular model there does exist an avenue by which the energy can propagate to the back of the particle, either along the outside or the inside of the ring. Thus it seems that this electrostatic ring would be the correct form even when the electron is not bound within the atom.
1. Griffiths1 428 or Jackson2 658.
2. This is still not entirely satisfactory as some of the energy is radiated away (rather than remaining within the photon). The idea is that the photon will take all the energy and then the process can occur in reverse where a photon strikes a ring causing it to change its radius. The photon then would be absorbed.
3. This is obviously not the whole story. There is a magnetism associated with the ring, but this part has yet to be properly established. However, the symmetry of the ring by itself is not sufficient to satisfy the conditions of stability of the charge field (see Section 2). Perhaps in concert with the positive field of the nucleus and/or a particular magnetic field on the ring, these conditions can be met. The stability of the atom as a whole is probably not an issue since it has been seen that Earnshaws' theorem does not necessarily apply (see Section 11). It should be emphasized here that this form is not guessed as a possibly stable formation, but is rather demanded by the nature of the photon itself. It is not a question as to whether this model is stable. We already know that it is. The question is why.
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