In the equatorial plane of the DCDM electron model, the centrifugal force is balanced by the attractive electrical force between the outer shell and central core and the repulsive electrical force on the outer shell. Along the spin axis, the same electrical forces exit, but there is no counterbalancing centrifugal force. By stretching the sphere slightly along the spin axis, two things happen:

● The radius of curvature of the surface at the pole is increased, thereby increasing the repulsive electrical force on the outer shell at that point;

● The distance between the central core and the outer shell at the pole is increased, thereby reducing the attractive force between the two.

An ellipsoid, or ellipse of revolution, seems like a reasonable approximation for the shape required by the model.

The parameters for an ellipsoid are to be calculated for which the electrical forces on the electron surface at the spin axis pole are balanced. Thus, there will be no force on the surface at the pole. The two forces to be balanced are

● The attractive force on the outer shell by the central core, and

● The repulsive force created by the outer shell’s charge.

The repulsive electric field at the ellipsoid pole is calculated by slicing the ellipsoid into many circle rings, all having axes along the spin axis. The charge on the surface of each ring is first calculated. Then, the electric field along the spin axis at the pole is calculated for each charged ring. Finally, the fields for all rings are added together, and from this sum, the ellipsoid parameters are calculated.

The equation for an ellipse is:

( x A ) 2 + ( y B ) 2 = 1 or y = B 1 − ( x A ) 2

where x = major semi-axis which aligns with the electron spin axis,

y = minor semi-axis which lies in the electron equatorial plane,

A is the intersection of the ellipse (pole) with the spin axis, and

B is the intersection of the ellipse with the equatorial plane.

The ellipsoid is formed by revolving the ellipse about its major axis. To calculate the difference between A and B required to provide a balance of electrical forces at the pole, the ellipsoid is divided into many circular rings, all having axes along the spin axis. The radius of each ring is y, and the location of the ring from the electron center is x. To calculate the charge on each ring, the width Δc of each ring is first calculated.

( Δ c ) 2 = ( Δ x ) 2 + ( Δ y ) 2

where Δx and Δy are the projections of Δc on the x and y axes, respectively.

y − Δ y = B 1 − ( x + Δ x A ) 2

y Δ y B 2 = 1 − ( x A ) 2 − [ 1 − ( x A ) 2 ] [ 1 − ( x + Δ x A ) 2 ]

y Δ c = [ ( y Δ x ) 2 + ( y Δ y ) 2 ] 1 / 2 = [ B 2 ( Δ x ) 2 ( 1 − ( x A ) 2 ) + ( y Δ y ) 2 ] 1 / 2

Δq º ring charge

σº charge surface density

Δ q A 2 = 2 π σ y Δ c A 2

For the following equation expansions,

B = A + Δ A

( B A ) 2 = ( 1 + Δ A A ) 2 = 1 + 2 Δ A A + ( Δ A A ) 2 = 1 + 2 Δ A A

It is assumed that Δ A A ≪ 1 , so terms containing higher orders of Δ A A are deleted. The final evaluation of Δ A A will validate this assumption. The expanded equation for Δ q A 2 is

Δ q A 2 = 2 π σ k 1 [ 1 + ( k 2 + 1 ) Δ A A ]

where k 1 = [ 1 − ( x A ) 2 ] ( Δ x A ) 2 + ( y Δ y B 2 ) 2 and k 2 = ( y Δ y B 2 ) 2 k 1 .

The distance d between the charge on a ring and a polar point A is

d = [ ( A − x ) 2 + y 2 ] 1 / 2 = A 2 ( 1 − x A ) 2 + B 2 [ 1 − ( x A ) 2 ] .

d A = 2 [ 1 − x A + [ 1 − ( x A ) 2 ] Δ A A ] 1 / 2

( 1 − x A ) ( d A ) − 3 = 2 − 3 / 2 ( 1 − x A ) − 1 / 2 [ 1 − 3 2 1 − ( x A ) 2 1 − x A Δ A A ]

The electric field along the spin axis at a pole A due to a single ring is

Δ q d 2 ( A − x d ) = Δ q A 2 ( 1 − x A ) ( d A ) − 3 = π σ 2 k 3 [ 1 + k 4 Δ A A ]

where k 3 = k 1 ( 1 − x A ) − 1 / 2 and k 4 = 1 + k 2 − 3 2 1 − ( x A ) 2 1 − x A .

The following electric fields along the spin axis at pole A are defined:

E e + º repulsive field created by the ellipsoid surface charge q⁺

= the sum of the electric fields due to all of the ellipsoid rings. Δ A ≠ 0 .

E s + º repulsive field created by a spherical shell having a radius A and a surface charge q⁺

= E e + for Δ A = 0 = q + A 2

E s − º attractive field created by a charge q^- at the center of the sphere of radius A

= | q − | A 2

For the DCDM electron model with a spherical shape, the attractive and repulsive forces on the outer shell at the pole A are unbalanced. The ratio of the two fields at A is

E s + E s − = q + A 2 | q − | A 2 = q + | q − | and E s − = | q − | q + E s +

The sphere is stretched along the spin axis x to an ellipsoidal shape to balance the attractive and repulsive electrical fields, so that

E e + E s − = 1 and E s − = E e + = | q − | q + E s +

The following equation was solved numerically for Δ A with the ellipsoid sliced into 200 rings:

E e + E s + = | q − | q + = ∑ x = − A x = A E e + ( Δ A ≠ 0 ) ∑ x = − A x = A E e + ( Δ A = 0 ) = ∑ x = − A x = A k 3 ( 1 + k 4 Δ A ) ∑ x = − A x = A k 3

The solution to the above equation for Δ A A is:

Δ A A = − ∑ x = − A x = A k 3 ∑ x = − A x = A k 4 ( | q − | q + − 1 ) = − 0.0119

Note that Δ A A is negative, so the ellipsoid type is prolate, i.e., the distance from the center to the intersection of the electron spin axis with its surface (pole A) is 1.20% longer than the distance from the center to the intersection of the equatorial plane with the surface (R_q).

The mysterious Fine Structure Constant [3] appears in many instances in physics, for example [4] [5] [6], although it is not completely understood why. It is a unitless constant, and is the ratio of other constants having the same unit. It has been suggested [7] that it is one of the most, and maybe the most, fundamental constant in physics.

Table 2 above lists the ratios of attributes of the electron model’s internal components, as calculated according the dual-charge dual-mass model. The table also includes the value of the inverse of the Fine Structure Constant. There is a striking similarity between the attribute ratios and the Constant. In the worst case, the attribute ratio differs from the Constant by less than 15%. Possibly, the attribute ratio should actually equal the Constant, or at least have a value very much closer to it.

Attribute ratios are listed in the table for both ring and spherical electron model shapes. The attribute ratios for a ring shape are less than the Constant, while those for the spherical shape are greater. It has been assumed in the electron model that the charge density on the outer shell surface is uniform. But maybe it isn’t. The fact that the Constant value lies between the ring and sphere attribute ratio values suggests that the charge density might actually be greater near the equator than elsewhere. By changing the surface charge profile from uniform to one of increasing charge density toward the equator, it should be possible to achieve attribute ratios that are within ±0.4% of the Constant value.

For a perfectly spherical outer shell charge, the central core charge exerts a radial force on each charge increment. There is no tangential component to the force. The outer shell charge may be mobile or fixed. It makes no difference. If mobile, the mutually repulsive forces between the charge increments will cause them to spread out uniformly across the shell surface. However, if the outer shell is not perfectly spherical and in fact is a prolate ellipsoid, there will be a tangential force exerted by the central core on each charge increment, which tends to pull it toward the equator. (The exception is at exactly the equator and the poles.) This force is further increased by the lesser distance between the outer shell increment and the central core for increments closer to the equator than farther away. Therefore, for a prolate ellipsoidal shape and a mobile surface charge, the surface charge density will increase from the pole to the equator. The greater the ellipticity, the greater the concentration of charge near the equator.

In reality, the ellipsoid’s charge density near the equator would smoothly taper down toward the poles. A rough approximation to a realistic charge profile is an overlay of a ring charge on a sphere at the equator. The relation between the ring charge value and the sphere’s charge value is calculated in the following. Of course, the values calculated for this composite are approximate, but provide a general idea of how the electron’s internal attribute ratios could be brought closer to the fine structure constant.

A ring-shaped outer shell is superimposed on the spherical outer shell at the equator. The two shells have the same radius r. The charge on the ring is k q + , and ( k − 1 ) q + on the sphere. The scaling factor k is calculated as follows:

The magnetic moments for the ring and sphere are

M ( ring ) = − k q + 2 r , and M ( sphere ) = − ( k − 1 ) q + 3 r . [1]

The composite magnetic moment is

M = M ( ring ) + M ( sphere ) = − ( k + 2 ) q + 6 r ,

where the radius r is

r = − ( m − m ) q 4 m c 3 S [1]

Assume that

− q + q = − m − m = 1 f i n e   s t r u c t u r e   c o n s t a n t = 137

and therefore

q − q = m + m = 1 f i n e   s t r u c t u r e   c o n s t a n t + 1 = 137 + 1 .

By merging and rearranging the above equations,

k = 6 M m c 3 S 137 2 q 5 − 2 = 0.60

r i n g   c h a r g e = 0 .60 q +

s p h e r e   c h a r g e = 0.40 q +

The radius for a ring-shaped outer shell in the dual-charge dual-mass model can be calculated from

R ( ring ) = 2 M m S ( q c ) 3 [1]

The radius for the ring and sphere charge composite can be calculated by replacing “2M” with 6 k + 2 M in the above equation.

R ( ring + sphere ) = 3.255 × 10 − 13 = 1.15 R

Therefore, by allowing the outer shell charge to be mobile on an ellipsoidal surface and assuming the outer shell charge and mass ratios to be equal to the inverse of the Fine Structure Constant, the resulting electron radius is only 15% greater than the Classical Electron Radius.