FRACEP, J. A. Giannini (7/25/2012)



What Is the Size of the Composite Particles?

May 17, 2007 (revised 9/26/10)


The Standard Model (SM) has the fermions as part of its list of fundamental particles despite considerable evidence to the contrary. The FRACEP Model, Part 1a [Ref. 1] is a new heuristic model. It presents a composite structure for the fermions that is consistent with the SM observations of those particles in mass, spin, electromagnetic charge, and decay components. This paper takes the next step. It shows that those FRACEP built-up composite structures have a classical radius that is consistent with the best maximum estimate by the SM for the observed fermions. This offers further support for the concept of non-fundamental fermions.





2.1. The Macro World Size Description

2.2. The Quantum World Size Description


3.1. Historical Basis of the Fundamental Scales

3.2. The Planck Mass

3.3. The Planck Length

3.4. The Planck Time


4.1. Assumptions Used in the Size Determination

4.2. Sizes of the IBB's and Their Components






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The Standard Model of Particle Physics (SM) is a model of the world of the small (atoms and molecules, and sub-atomic particles). Its theoretical part is based on quantum mechanics; and, its data base includes its fundamental particles (the fermions and bosons). The generally accepted definition of "fundamental" includes the requirement of indivisibility of the particles. However, the observed data for the majority of the fermions (and many of the bosons) indicate that these particles spontaneously decay. This appears to violate a necessary requirement for a fundamental nature in a particle. At the same time, it seems to support the possibility of the picture of fermions as composite particles. In recognition of this state of affairs, the FRACEP Model was developed to address the feasibility of describing fermions as composite particles.


In Part 1[Ref. 1], the structures (required components) for all the fermions and anti-fermions were developed, based on a set of Intermediate Building Blocks (IBB's) that were themselves built up from fractal-like structures of only two fundamental spin-less, charge-less particles (G0p and G0m). The set of IBB's contains: a positive (SGp) and a negative (SGm) spin carrier; a negative (QGp) and a positive (QGm) charge carrier; several mass carrying particles (GXp); and several momentum carrying rings (RXp). The X in the particle designation indicates the fractal level of the particle. The X in the ring structure designation indicates that the ring is made of 6 GXp's particles. The built-up composite FRACEP fermions agree with the SM fundamental fermions in four characteristics: mass, spin, electromagnetic charge, and decay products. But, these four characteristics are only some of what are necessary to validate the FRACEP picture.


In order to further support the FRACEP, it is necessary to consider the sizes of the composite particles by showing that they are consistent with the SM estimates of the fermion sizes. The purpose of this exercise is to provide a best estimate of the expected sizes, defined as classical radius (Rclass), of the particles and components. The size computations are necessarily approximate given the challenge of estimating the size of something that cannot be seen.


The size discussion, in this paper, is presented in several steps. First, there is a discussion of the Concept of Size and Its Different Descriptions. The natural tendency is to perceive size in the quantum sense as the SM sees it because that is where the FRACEP particles lie based on their mass. But, the FRACEP recognizes the extension of the classical description as also valid in the context presented here. So this section discusses the difference in the descriptions between classical and quantum measures of size.


Next, there is a discussion of the Fundamental Scales of Nature because the FRACEP is dealing with a true fundamental particle (G0p) that is profoundly interrelated with the fundamental scales in determining its size.


After that, the Composite Particle Building Block Sizes as defined by the FRACEP are computed.


And finally, the Sizes of the Composite Version of the SM Fermions are computed. In this section, there is also a discussion of the FRACEP composite Planck particle because of its fundamental scales relation.


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On the face of it, size, in general, seems to be a simple concept. However, our experience in describing size is based mostly on our perception of the world we see (the macro world). But when dealing with objects too small to see, such as the fermions, the description of size changes to fit the rules of the quantum world. To put the size question in perspective, it is valuable to briefly discuss what is meant by size in each of the two regimes, recalling that, although the fermions reside in the quantum world (they have wave-like behavior), they are not totally divorced from the macro world (they also have coherent particle-like behavior). Because of this, the FRACEP extends the macro world size description (Rclass which is the classical radius concept) to its composite versions of the SM fundamental particles.


2.1. The Macro World Size Description


From the macro perspective of every day life, size is an easy concept. Objects have a fixed surface or boundary confining them. The size is determined by measuring their dimensions (length, width and height) with a ruler. An example of the every day scales is a human who weighs ~200 lbs (mh ~ 90.0 kg ~ 5x1031 MeV/c2) and is ~ 6 ft (~1.8 m) tall. (This approximately cylindrical person is roughly equivalent in volume to a sphere of radius rclass-h ~ 0.85 m. The equivalent sphere is useful for providing a perspective for every day scales of spherical objects).


As one progresses from the every day scales to the astronomical scales, the concept of size is maintained though the nature of the ruler is adjusted to accommodate the larger scales of interest. The first step of the astronomical scales includes the planets [Ref. 2, chapt. 1-13] that still have discernable, relatively solid surfaces, many surrounded by an atmosphere. For example, earth is roughly spherical with a radius rclass-e ~ 6.4x106 m, and a mass me ~ 6.0x1024 kg (~3.4x1054 MeV/c2). Its atmosphere has numerous layers of gases and charged particles, and is very rarefied with a thickness on the order of rclass-e.


The next step in the astronomical scales is to the very large, such as stars and galaxies [Ref. 2, chapt. 2, p.18-26]. These objects lose their fixed surface, so the boundary defining them is based on large density changes. For stars, the bulk of the mass is found within an approximately spherical volume that contains the very high density, gaseous, energy producing region. This region is then surrounded by a much lower density chromosphere. For example, our sun has a radius rclass-s ~ 7.0x108 m and a mass ms ~ 2.0x1030 kg (~1x1060 MeV/c2). Its chromosphere is thin compared to rclass-s (~ 0.007 x rclass-s).


The galaxies are larger than the stars. They have a general elliptical shape, though some are freeform. Many, like our own Milky Way, have a spiral structure within a flattened elliptical disc around a roughly spherical bulge that forms a core. Many are believed to have a super massive black hole at their center. Our galaxy, the Milky Way, is estimated to have a radius rclass-MW ~ 5x104 lt-yrs (~4.7x1020 m) depending on where the cut-off density boundary is set. (The lt-yr is defined as the distance travelled at the speed of light in one year. That is, ~3x108 m/s x 365.25 days/year x 86,400 seconds/day = 9.5x1015 m.)


The Milky Way mass is mMW ~ 1.7x1071 kg, though higher and lower estimates can be found. Its central core has a radius r class-c-MW ~ 7.5x103 lt-yrs (~7.1x1019 m). It was generally believed that this very dense core region contained most of the galactic mass. However this perception is currently a matter of debate, so core estimates are widely varied. The black hole at the galaxy's center is estimated to have a mass mc-MW ~ 3.6x1045 kg (~2x1066 MeV/c2).


The largest of the large scales is for the entire Universe. Cosmological models using observations of galaxy densities, estimate the entire Universe has a radius on the order of rclass-U ~ 1025 m and a mass about mU ~ 1080 kg (~8x1079 MeV/c2 [Ref. 3, p10; Ref. 4, p65]).


This brings us to our last and exotic scale - the measure of the black hole. A black hole is formed when a massive star has expended its fuel supply and the outward pressure of the energy production can no longer counterbalance the inward pull of gravity. Under this condition, the star collapses to the point of destroying the integrity of matter causing the smallest of particles to tunnel into each other.


At the more conventional scales considered to this point, the measurement was one of determining the extent in 3-D space of visible matter (and possibly including the infamous "dark matter") using optical, microwave, x-ray, etc. frequency data as the rulers. However, since black holes have no visible matter (and no obvious boundary), the question of size is defined, by definition, as the "event horizon" which is computed as the Schwarzschild Radius (RSW = 2Gm/c2 as indicated in Figure1).


Only matter around the black hole that is outside its RSW is visible. Within the RSW, the gravity well of the black hole is so strong that no light can escape, so its presence is observable only by gravitational effects and the tell-tale glow of surrounding matter trying to escape the black hole's pull. (For most objects, the RSW is well within the observable surface.) (In general, the RSW is not so much an independent measure of size, as it is the lowest possible limit of Rclass at any given mass.)


2.2. The Quantum World Size Description


Having covered the macro world from every day scales to the largest and most exotic, it is now appropriate to turn to the small - which is where the atoms and SM particles lie. In the world of the large, though the ruler and boundary definitions changed, Rclass is the only practical measure of size. However, it is here in the world of the small that the concept of size has more than one practical measure.


In the early 1900's, as quantum mechanics was being developed, it was discovered that the photon had a dual nature [Ref. 5, p. 128,189]. When shining a light thru a diffraction grating onto a screen, the photon acted like a particle producing a single spot of light when the grating spacing was large. But, for spacings smaller than the deBroglie wavelength (or alternatively the Compton wavelength, lC = hc/mc2), the photon demonstrated wavelike properties that appeared as interference patterns on the screen. This appears as a series of bright and dark lines on the screen.


Since those earliest results, the experiments have been repeated with larger particles showing dual behavior for electrons and atoms. For example, Brezger, et. al. [Ref. 6] showed wave diffraction behavior for the carbon-70 molecule. This wave nature, only now becoming obvious in larger objects, is the origin of the second measure of size appropriate to the quantum scales. This measure is the Compton wavelength (lC) and it is the point where wave behavior becomes obvious.


In addition to lC, however, there is still the classical radius, Rclass, which is the macro scale concept extended to smaller particles (i.e., an indication of the physical extent of a particle in 3D space). Early in the 1900's, Rutherford, Thomson, and others used scattering experiments to study atoms. They verified the theory of a positively charged core surrounded by a negatively charged electron cloud. The core was shown to be made up of positively charged protons and zero charged neutrons.


Scattering experiments employ models of the interaction forces between probe and target [Ref. 7, chapt 5, Ref. 8]. Using input data (the energy and approach scattering angle), they predict the expected output (energy and retreat scattered angle) of probe and target. By correlating the predictions with the measured results, the charge distribution of the target particle is mapped out. This provides a measurement referred to by the SM as the scattering cross section which represents the approximate inverse of the Rclass of the target.


According to the SM and such experiments, the minimum scattering cross section of the electron is ~1.8/fm. This gives a maximum size, in 3D space, for the radius of a spherical electron, Rclass-electron ~ 1/1.8 = 0.56 fm (5.6x10-16 m). (Note that lC-electron is ~3.86x10-13 m which is larger than its Rclass). However, given the state of technology, the maximum predicted size for leptons in general (including quarks and electron family members) is accepted as <10-18 m with no discernable difference in their Rclass's. For comparison, the proton is Rclass-proton ~ 8.42x10-16 m and a carbon atom nucleus is Rclass-C12 ~2.5x10-15 m.


Thus, in the quantum world, objects have a dual nature with two different ways to describe their size: 1) the lC is the wave-like size description; and 2) the Rclass is the particle-like size description. Even though the SM concentrates on the lC description, the FRACEP is focusing on the Rclass description (i.e., the physical extent in space).


Figure 1 shows a comparison of the two measures of size for a sample of objects from the largest of the macro world to the smallest of the quantum world. For all non-black-hole objects, theory requires the Rclass to be greater than the RSW (the lowest macro size limit) for any given mass. This is demonstrated in the figure which shows the observed sizes (D's) for all objects except the Milky Way black hole which, because of its collapsed nature, has a defined size of its RSW (its event horizon), and the Planck particle (which is hypothesized but not observed) whose size is predicted to be lp*.


The s-shaped curve shows a nonlinear relation (on the log-log scale) of Rclass to mass. The mid region (from about the sun down to C12) is roughly linear with curved tails in the extremes. The dashed curve was obtained from a fit of the form R = F1.M2 + F2.M + F3 where Fi = Ai + Bi tanh[ 0.125 .(M - 34.0)]. The M = log10(mass/2). A and B are 3-element vector, where A = (5.9092337x10-2, -3.9778756, 55.0231368) and B = (0.1103927, -5.0404973, 71.8671272). The predicted R = log10(radius). The mass is in MeV/c2 and the radius is in meters. The plotted values are log10(mass) and log10(radius).)


Note, that given the quantum mechanical definition of discrete space-time (as discussed in the Fundamental Scales section below), no particle can have an Rclass < lp* (the point where the Planck mass's lC and RSW cross). As stated previously, the FRACEP builds up all of its composite fermions from fractal-like groupings of G0p. In the FRACEP, G0p is the smallest possible mass. It is therefore not unreasonable to assume that it also has the smallest physical extent in space. So, the FRACEP defines the Rclass-G0p = lp*, the smallest length scale supported by space, as indicated by the "Z" on the plot.




FIGURE 1. This shows a comparison of the sizes of objects spanning from the quantum to the macro scales. The horizontal dashed lines indicate the log10(mass) of the indicated particles (with log10 (Mass), log10(Rclass) indicated). The diagonal line, lc, is the particle's quantum description (left side of equation 1); the diagonal line, Rsw, is lowest possible limit for the particle's macro description (right side of equation 1). The s-shaped curve, Rclass, is a fit thru the observed classical radius points (D), with the Rclass-G0p indicated (Z).



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The FRACEP describes the composite fermions as composed of what are referred to as intermediate building blocks (IBB's), that are themselves built-up from fractal-like groupings of the FRACEP fundamental particles (G0p and G0m - note these two particles are identical except for the sign of the mass value where the p denotes positive mass, and the m denotes negative mass). So, before we can adequately estimate the physical size (classical radius) of the composite fermions, we must first address the size of G0p.


The characteristics of mass, em charge, and spin were determined, in Part 1[Ref. 1], by applying the constraints imposed by the SM fermion observations. However, there was no adequate way to determine the size of G0p using those data, so it is necessary to look else where - in the fundamental scales of nature. By linking G0p to the fundamental scales, its size (Rclass-G0p) is established, by the FRACEP, as lp* , the smallest length scale.


3.1. Historical Basis of the Fundamental Scales


Efforts to find a fully unified picture of the universe are not new. Newton's theory of gravity [Ref. 9], in the late 1600's, showed that the interaction of bodies on earth, and the interaction of solar system bodies follow the same rules. This was the first success in unifying observed effects that were believed to be different and independent.


Two hundred years later, Maxwell [Ref. 10] successfully unified the electric and the magnetic effects. He showed that both fields could be described in terms of one another. This meant that they were different aspects of one unified electromagnetic (EM) field.


By the early 1900's, two new forces were identified: the weak (nuclear) field that controls radioactive decay; and the strong (nuclear) field that holds the atom together. In 1967, Weinberg [Ref. 11] provided the first unification of electromagnetism with the weak field (called electroweak theory). The work was successfully modified by 'tHofft [Ref. 12] in 1971 by removing the problematic infinities in the theory. In 1973, Quantum Chromo-Dynamics (QCD) entered the scene with its description of quark interactions and the strong field [Ref. 13].


Since then, extensive efforts have been ongoing to unify electroweak theory with the strong field to produce a Grand Unified Theory (GUT); and, further to unify the GUT with gravity to produce a Super-GUT (or the theory of everything). General understanding assumes that full unification to the Super-GUT level is achieved when all of the characteristic fields of nature (electromagnetism, weak, strong and gravity) have approximately equal strength and that a relationship among all the fields to one another is established [Ref. 14, p134; Ref. 15, p83; Ref. 16, p346].


The non-unified fields vary widely in strength. Consider the relative fields for particle masses around m ~ 10-27 kg (on the order of a few hundred MeV/c2), and at nuclear ranges around r ~ 10-15 m. The strong field is important on the quantum scales of the atom and smaller. It is attractive from the smallest distances up to about 1fm (10-15m), then becomes repulsive and quickly goes to zero at larger distances. The strong field interaction is about 8 times stronger than the EM field interaction between em-charges.


The EM field spans the quantum scale thru macro scales of gravity. It is attractive between unlike charges, and falls off as the inverse of the distance squared.


Gravity is an attractive force and is most evident at the macro scales of every day life and greater. Though its effect is not limited to this range, it is very difficult to measure at the very small scales. Like the EM field, it falls off as the inverse of the distance squared; however, it is 39 orders of magnitude weaker than the EM interaction making it the weakest field by far.

Unifying to the GUT level theoretically describes the three SM fields (EM, weak and strong) in terms of a single unified field. There are several versions of GUT models, none of which are totally consistent at this time. They require over 20 free parameters (e.g., the masses and charges of the fundamental particles, plus more), all of which must be experimentally measured before any predictions are possible; however, despite this, their prediction capability is impressive.


The GUTs predict that the energy at which unification should be observed (MGUTc2 ~ 1015 GeV) is the point where the energy-dependent interaction coefficients for the three fields merge to a single value (merging the three fields to a single unified field at the same time). However, at this energy, it is believed that gravity is still insignificant. (A detailed discussion of the GUT unification can be found in Halzen and Martin [Ref. 16, chapt. 15.7]).


The step to Super-GUT has seen numerous approaches (two examples include: super symmetry where each SM fermion has a super heavy boson partner; and string theory where the topology of space, the gravity-based description, supports vibrating quantum strings that represent the SM particles). Like the GUTs, there is not, as yet, a single self-consistent Super-GUT. However, there is general agreement that full unification of all of the fields (i.e., the unified GUT plus gravity) should be seen at the fundamental Planck scales (i.e., the Planck mass, mp; the Planck length, lp; and the Planck time tp).


3.2. The Planck Mass


Recall that at energies up to MGUTc2, gravity is sill negligible; however, it is predicted that as the energy grows beyond that level, the strength of gravity increases rapidly and ultimately becomes comparable to the unified GUT force at the Planck energy (mpc2 ~ 1x1019GeV). According to quantum-gravity models [Ref. 17, p257], the mass of the Planck particle is determined when the deBroglie wavelength (the quantum wave description) equals the Schwarzschild radius (the gravity description), that is,


2p . h-bar 2G . m
(1) ________ = _______
m .
c c2

giving the mass as:

(2) mp = K .(h-bar .c / G)1/2 ~ 1.22x1022 MeV/c2



This is the mass of the predicted Super-GUT unification. In (2), h-bar (1.0545887x10-34 Js) is Planck's constant divided by 2p, c (2.99792458 x108) is the speed of light in a vacuum, and G (6.6726x10-11) is the empirically determined gravitational constant. A factor 5.60958494x1029 MeV/c2/kg converts units from one system to the other giving mp ~ 2.18x10-8 kg. The leading coefficient, K = \/p, has been set to unity for the order of magnitude value that is sufficient for the purposes of most discussions.


Hoyle et. al. [Ref. 3, chapt 17] , however, provide a different way of estimating mp that is derived from the cosmological-gravity point-of-view, rather than the usual quantum-gravity view. They note that an ambiguity in mp arises because the quantum mechanical side of (1) (i.e., the left) comes from a scale-invariant theory; while, the gravitational side (i.e., the right) comes from General Relativity (GR) [Ref. 18] which is not scale invariant.


According to Hoyle, the implication of this is that under any conformal transformation of the non-scale-invariant theory, the mass is position dependent. This position dependence, by implication, would in turn affect a particle's associated wavelength when compared to its gravitational radius.


To address this difficulty, they developed a scale-invariant form of the gravity equations that reduces to GR with an appropriate choice of scale, leading to a unique particle (the Planck particle - to avoid confusion, the Planck particle derived from the fully scale-invariant theory is referred to as mp*). The mp* has the property that its Compton wavelength (lc = hc / mc2) equals its gravitational radius; and, Hoyle et. al. hypothesize that it represents the largest mass that can form without disrupting the fabric of space-time. On conversion to conventional (MKS) units, the definition of mp* is identical to (2) but with K = (3 / 4p)1/2. With this new value for K, the Planck mass becomes:


(2*) mp* = K .(h-bar .c / G)1/2 = 5.96610354x1021 MeV/c2



(~1.06x10-5 g) using the same constant values for h-bar, c, and G as in (2). Note that mp* is ~51% smaller than mp.


The concept that mp* is the largest mass that can be supported by the space-time fabric is applied, by the FRACEP, to a composite version of mp* (referred to here as mp* ') based on G0p. Defining it as:


(3) mp* ' = 6R44p = 36x944 G0p particles

= 6.02206465x1021 MeV/c2



gives a mass ~1% larger than the mp* mass computed by the scale-invariant theory in (2*). Although it is not a SM fundamental particle (i.e., a fermion or a boson), incorporating the composite Planck particle into the FRACEP is used as a method of connecting G0p to the fundamental scales of nature. Specifically, it is used to determine the physical size (classical radius) of G0p in the next section.


3.3. The Planck Length


The second fundamental scale is the Planck length, which is defined as:


(4) lp* = h-bar / (mp* .c) = 3.3075190x10-35 m



(This is the same definition as used for the non-scale-invariant version, lp; except that for lp, mp is used rather than mp* - note also that lp* is ~2 times larger than lp).


The significance of the Planck length can be found in the concept of quantized space. In the early 1940s, as described by Jammer [Ref. 19, p240], in order to resolve infinities in the quantum field theory development, a smallest length to space-time was proposed (as opposed to a cut-off frequency).


The concept of discreetness to space-time was further elaborated in the early 1960s by appealing to the Heisenberg uncertainty principle (Dp.Dx ~ h, where Dp is the particle momentum uncertainty, Dx is its wave localization uncertainty, and h is Planck's constant). Wheeler hypothesized the principle forced the fabric of space-time to behave like a "fluctuating foam" where the fluctuations became physically significant on the scale of lp.


It is interesting to note that in string theory, the fundamental entity, the string, has a length of ~ lp. This paper does not address the question of quantum fluctuations, but it does embrace the concept of a smallest length supported by space-time in defining the size of G0p as Rclass-G0p = lp*.


3.4. The Planck Time


The third fundamental scale is the Planck time which is defined as:


(5) tp* = lp* / c = 1.1032696x10-43 s


(This is the same definition as used for the non-scale-invariant version, tp; except that for tp, lp is used rather than lp* - note also that tp* is ~2 times larger than tp).


The Planck time sees its importance to particle physics in "Big Bang" cosmology. In theory, starting at time t = 0 (the bang), there was an explosive, exponential expansion of the universe, due to the extreme compression of space, where the thermal energy of matter was greater than the Planck energy (mpc2). This thermodynamic state prevented particles from forming as isolated entities.


The "inflationary" expansion caused cooling until time t = tp*, when the thermal energy had reduced to the Planck energy. This time, tp*, was the boundary (switch-over) from the inflation period to the classical expansion we see today. (The expansion speed is no longer exponential, but has the form v = H0 .d where H0 is the Hubble constant and d is the separation distance of masses such as stars and galaxies. It is this distance that is increasing due to the expansion of space.)


After tp*, with continuing cooling, the universe saw a series of symmetry-breaking events (times when one-by-one the fundamental fields decoupled from the totally unified super-GUT field, and particle formation of the fermions and bosons proceeded). Details of the process can be found in Povh [Ref. 7, chapt. 19.4], Peacock [Ref. 17, chapt. 9], and Guth [Ref. 14]. The relation of tp* to the cosmology-based particle formation will not be considered here; however, the relation of tp* to fermion and boson instability and decay time is presented in Part 2 [Ref. 20], and, to the dynamic properties of the spin effect (Part 4 [Ref. 21] ) and the charge effect (Part 3 [Ref. 22]).

There is one final point of interest regarding the relation of G0p to the fundamental scales. Comparing the values of mG0p and mp* suggests a possible reciprocal relation. A comparison of 1/mp* with mG0p shows a -2.8% difference. This implies a close reciprocal relationship between the two values. Further, if the value of G is increased from 6.6726x10-11 to 7.07x10-11 (just 6%), the reciprocal mass difference drops to +0.02%. This gives a range for mp* (5.96610354x1021 to 5.79600331x1021). It also gives a range for lp* (3.3075190x10-35 to 3.4045876x10-35), and for tp* (1.1032696x10-43 to 1.1356482x10-43).


It is intriguing that the smallest (mG0p) and largest (mp*) masses supported by space have an apparent reciprocal relationship. It is reminiscent of the super-symmetry pairing of small fermions with their super partners - one difference being that mG0p and mp* (or its equivalent composite form mp* ') have no associated spin or charge as do the fermion-boson pairs. We now proceed to estimate Rclass for the IBB's using Rclass-G0p and the fermions.


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The structure of the intermediate building blocks (IBB's) (required components) was determined in Part 1[Ref. 1], along with a configuration (component arrangement) that was show as a planar projection. Since the particles, whose sizes are being determined, cannot be seen directly with today's technology, a series of assumptions are made which are used in the size determination. This section states those assumptions and then proceeds to estimate the size of the IBB's.


4.1. Assumptions Used in the Size Determination


The assumptions used in the size determination affect, not only, the IBB's, but also the composite fermions. The following assumptions address the spacing between adjacent particles in groupings, and the particle configurations that affect how the groupings are seen from the outside.


4.1.1. The Smallest Length Scale of Space Is lp* - This assumption restates the quantum mechanical picture of discrete space with lp* as the smallest length.


4.1.2. The Classical Radius of G0p Is Rclass-G0p = lp* - This assumption re-states the FRACEP picture that the smallest mass particle has the smallest physical size.


4.1.3. The Minimum Center-to-Center Spacing for Any Adjacent Particles in a Built-Up Structure Is the Rclass of the Larger Particle - This assumption is valid for groupings making up the mass carrying particles (GXps), the momentum carrying rings (RXp's), the ring and particle clumps (MRXp's and MGXp's), the spin carriers (SGp), and the charge carriers (QGp). In the MRXps and MGXps, the M is the number of RXp rings clumped together or the number of GXp particles clumped together.


The assumption also holds for the IBB's and the components within the composite fermions. Figure 2 shows the effects of this assumption. For this example, Dc-to-c = 5r, where r is the radius of one particle and 2r is the radius of the other. This gives a separation of 2r between the two particles because of the radius of the larger particle.






4.1.4. The Components Within a Built-Up Structure Arrange Themselves in a Roughly Spherical Configuration - This assumption is valid for all structures with two qualifiers. First, single ring elements (RXp's - as opposed to clumps) are roughly planar rather than spherical with a horizontal radius rX and a height of h X (determined below in section 4.2.1). Second, the term "roughly spherical" does not imply a uniform distribution of particles within the spherical volume except for the clumps (MRXp's or MGXp's described in section 4.2.4).


4.2. Sizes of the IBB's and Their Components


The IBB's include the mass carrying particles, the momentum carrying rings, the charge carriers, and the spin carriers - all of which are used to build up the composite fermions [Ref. 1]. The basic (lowest order fractal level) ring element (R0p) and the basic mass element (G1p) are presented separately first because the higher order structures are built on them and their configurations and size clarify the higher order size determinations.


4.2.1. The Basic Ring Element (R0p) Configuration and Size


The R0p is the basic (lowest fractal order) ring element. It is composed of six G0p (fundamental) particles arranged in a planar configuration of radius r0 = 3 (normalized by r0) which can be thought of as being the equatorial plane of a sphere with radius r = r0. (The r0 = lp* is the radius of the G0p particle, and for simplicity is set to unity (r0 = 1) in the normalized system). All points (particle centers) on the ring are separated by an internal angle of 60o; and the cord cutting the circle between any two points has a normalized length of 3 as well (Figure 3).

In this configuration, any two adjacent points form an isosceles triangle (equal sides) with the ring center as the third point. The surface-to-surface normalized distance between any two particles is ds-to-s = 1 (a total of 3 center-to-center). The space occupied by the ring is cylindrical in shape (Assumption 4) with a radius r0 = 4 and a height h0 = 2 (from ring center to the outside surface of the particles).




FIGURE 3. This shows the planar ring configuration for R0p with point coordinates for particle centers. All lengths are normalized by r0 = lp* (the radius of the G0p particles making up the ring).



4.2.2. The Basic Mass Element (G1p) Configuration and Size


The G1p is the basic (lowest fractal order) mass element. It is composed of an R0p with three additional G0p particles arranged at 120o apart in the planar projection. However, the three additional G0p's are located out of the plane and are obtained by sliding the particles along the surface of the sphere of radius r = 3 to points that satisfy the center-to-center separation requirement in Assumption 3 (Figure 4). In this configuration, any two adjacent particles have a surface-to-surface normalized distance of 3. The space occupied by the G1p is roughly spherical in shape (Assumption 4) with a radius r = 4 (from sphere center to the outside surface of the particles). For example, G1p appears to the next level (G2p) as a spherical particle that is used in the size determination of G2p.




FIGURE 4. This shows the G0p particle configuration for G1p with point coordinates for particle centers. All lengths are normalized by r0 = lp* (the radius of the G0p particles making up the ring). Note, the particles 8 and 9 are on the sphere surface and below its equatorial plane. Particle 7 is on the sphere but above the equatorial plane.





4.2.3. The Higher Order GXp's and RXp's Configurations and Sizes

The higher levels of GXp and RXp are fractal-like structures based on the previous levels. The number of particles in each structure is: GXp = 9X .G0p's = 9G(X-1)p's, and RXp = 6GXp's (i.e., R0p = 6G0p's, R1p = 6G1p's, .. ; and G1p = 3G0p's + R0p's, G2p = 3G1p's + R1p's, ..). At each level, the configurations are the same as shown in Figures 3 and 4 except that the distances increase with each level (Figure 5). This says that for RXp, the rRXp = rG(X+1)p, and hRXp = 2 . rGXp; and, for GXp the radius is rGx = 4x . r0 .







4.2.4 Clumps of Higher Order Rings (MRXp) Configuration and Size


Clumps of the MGXp's and MRXp's are aggregate groupings of M mass carrying particles (GXps) and M momentum carrying rings (RXps) respectively. The clumps form spherical bodies (Assumption 4). Their size is determined as follows. Suppose the clump in question is M.GXp where GXp has a radius rGXp and M is the number of GXp's in the clump.


1) Enclose the GXp in a cube of side length S1 = 2(rGXp + rGXp). The rGXp assures that the adjacent GXp's in the clump are separated by at least rGXp (Assumption 3).


2) The small (S1) cubes are then arranged to fill a larger cube of side length S2 = M1/3 S1 where the M1/3 is rounded up to the next larger integer (which represent the number of smaller cubes that make up each side of the large cube.


3) The radius of the sphere that encloses the large (S2) cube is now the diagonal of the S2 cube from cube center to any corner. With a little geometry, it is easy to show, that the diagonal in the horizontal plane is



dh = S2 [ ()2 + ()2 ]1/2.



This then gives the cube diagonal from center to corner as rd = S2 [ dh2 + ()2 ]1/2 = 0.866 .S2. This gives


(6) rd = 0.866 .M1/3 .3rGXp



which is the radius of the spherical volume that is used in the other IBB's and fermion size determinations.


4.2.5 The Spin Carrier (SGp) Configuration and Size


Spin in the FRACEP is a dynamic effect explained more fully in Part 4 [Ref. 21]. This paper addresses only the size of the configuration. The spin carrier (SGp) contains two parts: the spin effect component (S0p) and the spin momentum component (MSp). The SGp includes both of these components. The S0p component is the 16th level of a 5-particle fractal-like structure. Each element at any level will be referred to as EX for the discussion here. At the lowest level, E0 = 2G0p's (= G0p - G0p =). The E1 = 5E0's, E2 = 5E1s, ...., E16 = 5E15s = 516(2G0p's). Figure 6a shows the configuration.


The size of S0p is determined much like the higher level momentum rings. It is a cylindrical structure, which when carried to higher levels has a radius roughly rX = 4X .(5/2 r0) and a height hx = 2 .rX-1 . However, because rX and hX are sufficiently close, the spin effect component will be treated as spherical with radius rS0P ~ 416 .(5/2 r0). The spin momentum component MSp is composed of four G16p particles which surround the S0p. The radius of the full spin carrier (SGp) is approximated as half of the longest distance across the group, assuming the surface-to-surface separation between the S0p and any of the four G16p particles is rS0p, that is:



(7) rSGp = [2rG16p + rS0p + 2rS0p + rS0p + 2rG16p]

= 2rs0p + 2rG16p



With r0 = 3.3x10-35 m, then rS0p = 3.45x10-25 m, and rG16p = 1.42x10-25 m, the radius of the spin carrier is approximated as rSGP = 9.92x10-25 m.


4.2.6 The Charge Carrier (QGp) Configuration and Size


Charge in the FRACEP is a dynamic effect explained more fully in Part 4 [Ref. 22]. In this paper, only the size of the configuration is addressed. The charge carrier (QGp) contains two parts: the charge effect component (Q0p) and the charge momentum component (MQp). The QGp includes both of these components. The Q0p component contains two charge effect chains (each are Q0p) where each element of the chains is a pair of G0p's (- G0p = G0p -).




Figure 6. This shows the configurations for spin carrier and charge carrier. In (a) the fractal pattern of the spin carrier shown is for some sub-level E where in the total spin carrier, E = 16. In (b) the total charge carrier has two chains. Each chain has 2x1019 G0p pairs.




There are 4x1019 elements in each chain where each element has the required surface-to-surface separation giving it a length of 6r0 , which makes the total length of a single chain as rQ0p = 6r0 . 419 = 5.5x10-23 m. Figure 6a shows the configuration.


The momentum carrying component is MQp = G19p + 2G13p + 48R13p + 121R16p. Assuming a configuration as presented in Part 1[Ref. 1], and assuming the component separations are the size of the larger of any two components, the rMQp equals half of the maximum extent in any direction:



(8) rMQp = [2rG19p + rG19p + 2r48R13p + r121R16p + 2rR16p]

= [3x9.07x10-24 + 2x6.64x10-26 + 5.45x10-24 + 2x5.89x10-24]



giving the radius of the momentum part of the charge carrier as approximately rMQP = 2.25x10-23 m, and rQGP = 7.75x10-23 m.


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With the sizes of all of the components of the FRACEP composite fermions determined, this section proceeds with the size determination of the fermions themselves. Also, because of its importance to the fundamental scales, the size of the FRACEP composite version of the Planck particle is discussed. Note that in the SM, the Planck particle is considered fundamental in nature though it is not one of the fermions (or bosons, which are discussed in Part 5 [Ref. 23] ).


The procedure for size determination also uses the same philosophy as was used for the components: i.e., 1) the surface-to-surface separation is the radius of the larger of any two adjacent particles; and, 2) the fermion components arrange themselves to occupy a spherical volume (though not necessarily uniformly).


Because of this second assumption and the fact that the size is intended as a best estimate, the radius of the fermion is taken as one-half the maximum extent in any direction for the structure based on the configurations given in Part 1[Ref. 1]. Note that the configuration used is just one possible configuration where configuration is taken to mean an arrangement of components.


Table 1 shows the full structure for each of the fermions where structure refers to the collection of required components to satisfy the SM data. The components used in the size determination are indicated in bold. In the following paragraphs (5.1 thru 5.13), the rclass for each fermion is computed by showing the sums of the rclass's for each component (indicated Table 1), with the separation indicated as a sized unit.


The sizes of the relevant GXp's, RXp's and MRXp's were determined using the expressions and assumptions presented in section 4 above. And finally, the particles with negative mass components and anti-particles (e.g., ne+ or QGm) are assumed to have the same radius as their counter particles (ne- or QGp respectively). Note that in the maximum extent calculation, the particle size is twice the radius; and only the components in the maximum extent are shown.




TABLE 1. This shows the total structure, mass and estimated size of the FRACEP composite fermions. For the large particles, the core mass is the total mass without those momentum that are not required for the decay components. The components in bold (and larger type) represent the longest extent across the structure that are used in the size determination. The size is given in meters, and the mass is in MeV/c2. The maximum expected fermion size estimated by the SM is <10-18m for all fermions without any distinction.

FRACEP Fermion


FRACEP Components

(with maximum extent components larger and in bold)

Estimated Rclass(m)







ne- + G22p




2(QGp + ne-+ G22p) + (QGp + ne-+ G22p)


up (u+)


2(QGm + ne+ + G22p) + 3R22p + nm-


down (d-)


u+ + [ R22p + (ne+ + e-) ]


mu (m-)


(nm- + 2R22p) + 3R22p + 93R22p

+ 3R22p + [2 R22p + (ne+ + e-) ]




nm- + R22p + G24p + G22p


charm (c+) core [core = 123.16

tot. = 1250.39]

2(2R22p + u+ + nm+ ) + 3R22p +

(13R24p + 53R22p -- not part of core) + R22p

(u- + nm- + 2R22p) + 3R22p + (R22p + R24p + nt-)


strange (s-)


c+ core + [ R22p + (ne+ + e-) ]



tau (t-)

[core = 327.27

tot. = 1777.05]

(2R22p + u+ + nm+) + 5R22p + (2R22p + ne+ + e-)

+ 27R22p + 319R19p + 120R16p + 93R22p

(u- + nm- + 2R22p) +5R22p + (R22p + R24p + nt-)

+ (17R24p + 45R22p +343R19p -- not part of core)


top (t+) core

[core = 4397.68

tot. = 1.82e+5]

2(2R24p + c+ + nt+) + 3R22p +

(25R26p + 57R24p + 12R22p -- not part of core) + R22p

(c- + nt-+ 2R24p) + 3R22p + (R22p + R24p + nt-)


bottom (b-)


t+ core + [ R22p + (ne+ + e-) ]






5.1. The Electron Neutrino (ne-) - is the spin carrier computed in section 4.2.5.

rne- = rSQp = 9.92x10-25 m.


5.2. The Muon Neutrino (nm-) - is the ne- plus a mass carrier.

rnm- = [2rne- + (separation + 2rG22p)]

= [2x9.92x10-25 + 3x5.81x10-22 ] = 8.72x10-22 m.


5.3. The Electron (e-) - is triangular with 3 charge/spin chains (QGp + ne-+ G22p), and a maximum extent along any side.

re- = {2x [(2rQGp + sep.) + 2rne- + (sep. + 2rG22p)] + sep}

= 3x7.75x10-23 + 2x9.92x10-25 + 3.5x5.81x10-22 = 2.27x10-21 m.


5.4. The Up-Quark (u+) - is a central chain with 2 charge/spin chains at one end and nm- at the other.

ru+ = [(2rQGm+ sep) + 2rne+ + (sep + 2rG22p) + 3(sep + 2rR22p) + sep + 2rnm-]

= [3x7.75x10-23 + 2x9.92x10-25 + 3x5.81x10-22 + 10x2.32x10-21 + 2x8.72x10-22] = 1.36x10-20 m.

5.5. The Down-Quark (d-) - is an u+ connected to an (e-, ne-) pair by a momentum ring.

rd- = [2ru+ + (sep + 2rR22p)]

= [2x1.36x10-20 + 3x2.32x10-21] = 1.71x10-20 m.


5.6. The Muon - is a central pair of R22p chains connecting a ring clump at one end, and, component groups off the other end of each chain. Note, the muon is more massive and larger that the tau neutrino that follows because this list retains the order (based on number of decay paths) presented in Part 1[Ref. 1].

rm- = [2rnm- + 5(sep + 2rR22p) + sep + 2r93R22p]

= [2x8.72x10-22 + 16x2.23x10-21 + 2x2.49x10-20] = 4.36x10-20 m.


5.7. The Tau Neutrino (nt-) - is the nm- connected to a mass carrier by a momentum ring.

rnt- = [(2rG24p + sep) + (2rR22p + sep) + 2rnm-

= [3x9.29x10-21 + 3x2.32x10-21 + 2x8.72x10-22] = 1.83x10-20 m.


5.8. The Charm-Quark (u+)- is a central pair of R22p chains that terminate at both ends of each chain by neutrinos, and, with up quarks and momentum ring clumps attached to the R22p chains.

rc+ = [2rnm- + 6(sep + 2rR22p) + (sep. + 2rR24p + sep.) + 2 rnt-

= [2x1.83x10-20 + 6x3x2.32x10-21 + 4x3.70x10-20 +2x8.72x10-22]

= 1.14x10-19 m.


5.9. The Strange-Quark (s-) - is an c+ connected to an (e-, ne-) pair by a momentum ring.

rs- = [2rc+ + (sep + rR22p)]

= [2x1.14x10-19 + 3x2.32x10-21 = 1.17x10-19 m.


5.10. The Tau (t-)- is a central pair of R22p chains that terminate at both ends of each chain by neutrinos, and, with up-quarks and momentum ring clumps attached to the R22p chains.

rt- = [2rnm- + 8(sep + 2rR22p) + (sep. + 2rR24p + sep.) + 2 rnt-

= [2x1.83x10-20 + 8x3x2.32x10-21 + 4x3.70x10-20 + 2x8.72x10-20]

= 1.20x10-19 m.


5.11. The Top-Quark (t+)- is a central pair of R22p chains that terminate at both ends of each chain by tau-neutrinos, and, with charm quarks and momentum ring clumps attached to the R22p chains.

rc+ = [2rnt- + (2(sep + 2rR24p) + sep.) + (2(2rR22p + sep.) + 2rR22p)

+ (2(sep + 2rR24p ) + sep.) + 2 rnt-


= [2x8.72x10-20 + 7x3.70x10-20 + 8x2.32x10-21 + 7x3.70x10-20

+ 2x8.72x10-20] = 4.43x10-19 m.


5.12. The Bottom-Quark

b- = t+ + R22p

rb- = [2rt+ + (sep. + rR22p)]

= [2x4.43x10-19 + 3x2.32x10-21] = 4.46x10-19 m.


5.13. The Planck Particle


Finally, although the Planck particle is not a fermion, and it is a hypothesized particle in the SM, because of its relevance to the fundamental scales discussed above, a discussion of its size is made for completeness. As stated in paragraph "a" of the Fundamental Scales section above, the Planck particle is hypothesized as the largest possible mass supported by the fabric of space-time as described in Hoyle [Ref. 3]. (Hoyle, of course does no describe the Planck particle as a composite as does the FRACEP.) As such, it can be assumed to form in the FRACEP as a composite particle.


In determining its size, the surface-to-surface separations are specified as in sections 4.1.3 and 4.1.4 above. It is assumed to have the configuration specified in the Fundamental Scales section (i.e., 6R44p). The size of such a particle would have a radius rP = 2.13x10-7 m. However, it is hypothesized that once formed, the Planck particle would collapse (just as a black hole collapses of its own weight once energy production within ceases). It is assumed that, in the collapsing process, the final state occurs when the G0p's in the Planck particle are close packed but not degenerate. That means that the ring structure is collapsed due to the large mass but the G0p's are not tunneling into each other as one models fundamental matter in a fully collapsed black hole.


At this point the collapsed radius is rP-collapsed = 1.08x10-20 m. The final tunneling by the G0p's (down to the rP-final = lp*, the Planck length) is considered here as destroying the integrity and nature of the fundamental particles (the G0p's) which would represent disruption of the space-time fabric. This would resulting in a creation event - the beginning a new cycle in the build-up of previously non-existing matter - analogous to Hoyle's creation event.


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One of the important aspects of a composite structure in fermions is size. In Part 1 [Ref. 1], a composite structure was determined for the fermions that agreed with the SM data in mass, spin, em charge, and decay components. This paper shows that the FRACEP composite structure (with the stated assumptions) is consistent with yet another characteristic of the SM, that is, the maximum expected size of the fermions.


There is a difference between the SM and the FRACEP on the question of size. The SM estimates the Rclass of the fermions to be <10-18 m; but, because the fermions are considered fundamental, the SM is unable to differentiate the sizes of the many particles except in a quantum mechanical sense, the lC.


The FRACEP, on the other hand, shows a variation in Rclass with mass as would be expected, because it builds up the particles based on their mass and structure. This size-mass variation is seen in nature at the macro scales (e.g., larger mass galaxies are larger in size that smaller mass stars). It is also evident at the atomic scales (uranium is larger than carbon is larger than hydrogen). So it is to be expected that the Rclass of a neutrino should be smaller than that of the larger mass quarks. The FRACEP shows sizes that range from 9.92x10-25 m for the smallest, the electron neutrino, to 4.46x10-19 m for the largest, the bottom quark; and, all of the composite FRACEP fermions are less that the stated SM estimate.

In addition to the fermions, the Rclass of G0p (the FRACEP fundamental particle with mass mG0p = +1.72x10-22 +1.04x10-31 MeV/c2, where the limits of variability in mG0p that still allow it to satisfy the SM observations within their measurement uncertainty) was established by linking it to the fundamental scales of nature. The natural extension of the mass-size variation to this smallest possible particle, and, showing the larger structures built up from it satisfy the SM observation on maximum size, lends validity to the choice of Rclass-G0p = lp* = 3.3x10-35 m (the smallest possible length scale).

One final observation can be made about the FRACEP construction and a possible explanation of the dual nature of particles in the quantum world. The dual nature (both particle-like and wave-like behavior) of particles on the quantum scales was discussed in section 2.


The particle nature on these scales has long been assumed as a natural extension from the world we see and understand. But, very early on in the development of quantum mechanics, it became obvious that these same particles also acted like waves. Because the SM treats fermions as fundamental particles (i.e., homogeneous, uniform and indivisible), there is not practical way to explain the observed diffraction that is characteristic of waves. The FRACEP, on the other hand, may offer an explanation


DeBroglie reflected on the particle-wave duality at the quantum scales, noting that even a simple water wave is granular at the atomic level. By that he meant that water waves are composed of the coordinated motion of a horde of water molecules. We can consider this situation further as we try to understand the duality issue.


On the macro scale, there are two behaviors noticeable in water waves. The usual wave considered by deBroglie can be pictured as a collection of water molecules moving together for some distance. But over time the molecules begin to separate and spread out the wave structure until the height of the wave is finally leveled. This spreading is due to dispersion.


However, there is another kind of wave that is observed. It is referred to as a soliton. In that kind of wave, the motion of the molecules (in a perfect medium) remains coordinated. This coordinated motion continues because the dispersion is counteracted by the nonlinear propertied of the medium [Ref. 24]. In this condition, the soliton wave takes on particle-like properties. This same behavior is observed on the macro scale in light [Ref. 25, -27].


It is possible that this same dual behavior is exhibited in the fermions because their composite structure. As modeled in the FRACEP, the fermions are composed of an extremely large number of G0p particles that make up their components. In one mode, you would get dispersive wave behavior indicated by interference patterns. In the second mode, you would get the soliton particle-like behavior. However, the details of this possible explanation are a line of inquiry left to future work.


The successful validation, to date, in the comparison of the FRACEP with the SM fermion observations is encouraging, though additional work to further support the composite picture is underway, including: the explanation of the charge effect [Ref. 22], the explanation of the spin effect [Ref. 21], and the description of the potential and the decay properties of the composite structures [Ref. 20].


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1 J.A. Giannini, "The FRACEP Model, Part 1a: A Look Inside The Elementary Particles of the Standard Model", in preparation

2 P. Moore, The Data Book of Astronomy, Institute Of Physics Publishing, Midsomer Norton,UK


3 F. Hoyle, G. Burbidge, J.V. Narlikai, A Different Approach to Cosmology, Cambridge U.

Press, Cambridge, UK (2000).

4 S. Majid, ed., On Space and Time, Cambridge Univ. Press, UK (2008).

5 P. Tipler, Foundations of Modern Physics, Worth Pub. Inc., NY (1969).

6 Brezger, et. al., Phys. Rev Letters, 88, 100404, 2002

7 B. Povh, K. Rith, C. Scholz, F. Zetsche, Particles and Nuclei, ed. 2, Springer, NY (1999).

8 T.K. Lim, J.A. Giannini, "Separable Expansion Method for Potential Scattering and the Off-Shell T-Matrix," Phys. Rev. A, 18, No. 2 (1978) 517.

9 I. Newton, Philosophiae Naturalis Principia Mathematica (1686), Principia, University of California Press, London, England (1999).

10 J.C. Maxwell, A Treatise on Electricity and Magnetism, Vol I & II, 3rd ed. (1891), Dover Pub. Inc., Mineola, NY (1954).

11 S.Weinberg, "A Model Of Leptons", Phys. Rev Letters, V19, pp. 1261-6, 1967

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13 D.J. Gross and F. Wilczek, "Ultraviolet Behavior of Non Abelian Gauge Theories", Phys. Re. Lett., vol. 30, pp 1343-6 (1973), and "Asymptotically Free Gauge Theories, I" Phys. Rev., vol. D8, pp 3633-52 (1973); Ref. 14 ,chapt. 7; Ref. 16, chapt 10.

14 A. Guth, Inflationary Universe, Perseus Books, Reading, Mass. (1997).

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16 F. Halzen, A.D. Martin, Quarks & Leptons, John Wiley & Sons Inc., NY (1984).

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18 G. Rindler, Essential Relativity, VanNostrand Reinhold Co., NY (1969).

19 M. Jammer, Concepts Of Space, Dover Publications Inc., Mineola, NY (1993).

20 J.A. Giannini, "The FRACEP Model, Part 2: Why Is There No Evidence Of Internal Structure In Electrons?", in progress.

21 J.A. Giannini, "The FRACEP Model, Part 4: What Is the Spin Effect?", in progress

22 J.A. Giannini, "The FRACEP Model, Part 3: What Is the Charge Effect?", in progress

23 J.A. Giannini, "The FRACEP Model, Part 5: What Is The Structure Of The Bosons And Field Mechanisms?", in progress.

24 J.A. Giannini, J.S. Hansen, L.W. Hart, "Experimental Measurements of Temporal Phase Shifts During Soliton Wave-Wave Interactions," presentation at 35th Annual Meeting Am. Phys. Soc. - Div. Fluid Dyn., New Brunswick, NJ, November 1982.

25 J.A. Giannini and R.I. Joseph, "The Propagation of Bright and Dark Solitons in Lossy Optical Fibers," IEEE J. Quantum Elec., 26, (1990) 2109.

26 J.A. Giannini, "The Propagation of Bright and Dark, Spatial and Temporal Solitons in Nonlinear Optical Materials," Doctoral Dissertation, Johns Hopkins University, (1991).

27 J.A. Giannini and R.I. Joseph, "Propagation in Cylindrically Symmetric Two-Dimensional Nonlinear Media," Phys. Lett. A., 160 (1991) 363.


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