Particle Issues

It seems appropriate to start this discussion with a basic question: what is a particle? However, any initial attempt to find an answer to what seems to be a fairly basic question may only lead to a somewhat circular definition, e.g. a particle is matter, matter is physical substance, particles have rest mass and substance must occupy space. If we ignore the ambiguity of the idea of ‘physical substance’, we might focus on the concept of rest mass [m=kg] associated with sub-atomic particles as described within the development of the classical model of the atom. Today, the form, shown below, might still be the most common type of image presented of an atom.

However, as early as 1904, Einstein’s energy equation [E=mc2] suggested that mass might be a form of energy. As such, we might forward a relatively uncontroversial idea that this mass-energy must have some form of ‘substance’, which occupies ‘space’ and forward the following definition.

A particle is an energy-density within a localised volume of space.

We might attempt some form of comparative measure of the energy-density of an electron compared with the average energy-density of the universe [8.53*10-10J/m3], as assumed by the cosmological model – see Cosmic Calculator for details. We might then initially estimate the energy-density of an electron based on classical assumptions about its mass [me] and radius [re], although both are probably wrong by orders of magnitude.


On the basis of [1], the electron energy-density would be 126,362 times greater than the average energy-density of the universe, but if the electron radius was revised to have a far smaller value, in the range of 10-18m, then the previous figure would be some 20-billion times larger. However, irrespective of the actual figure, a fundamental particle like the electron represents a potentially huge energy-density differential within the universe. This review has also questioned the particle model on the basis that it offers up no structural description of the energy-density or causal explanation of motion. However, despite the apparent issues outlined, we might still follow the development of the classical model into the quantum model, which is still often described as the particle model. Historically within classical physics, measurements invariably related to objects that had meaning on the macroscopic scale of human experience, such that the concept of ‘physical substance’ was not really questioned. Of course, by the beginning of the 20th century, the development of an atomic model was beginning to include evermore details about the sub-atomic structures within an atom, which was once considered to be indivisible.

Note: Today, it is estimated that the 'substance' or 'mass' of an atom, as a percentage of its volume is essentially negligible, such that an atom might be described as 99.9999999999999% empty space. Therefore, it might also be highlighted that the scale of the Bohr model is always misrepresented in any diagram. For example, if an electron was sized at 1 unit, then the proton nucleus might be estimated at 1836 units on this comparative scale. However, possibly the more interesting comparison is that the electron would be positioned in an orbital some 10,000 units from the proton.

Of course, today, we might recognise an additional complexity, if we assume the components of the nucleus, i.e. protons and neutrons, are composite structures made of yet another conceptual particle called a quark, which are also separated by space. Therefore, we might reasonably question whether all this empty space serves some physical purpose? Today, many introductions to the atomic model might start with the development of the periodic table, which was based on the observations of the chemical properties of the known elements. In this context, the idea of the periodic table has a fairly long history of incremental developments. However, in the modern form of the periodic table, the basic elements are now identified by their atomic number, which corresponds to the number of protons in the atom. In a neutrally charged atom, the protons are matched by an equal number of electrons, where the definition of atomic weight and the atomic mass are outlined below.

Note: Atomic mass is the mass of an atom and relates to both the number of protons and neutrons in an atom, such that each isotope of an atom will have its own atomic mass, where electrons are essentially ignored because of the relatively small mass. In contrast, the atomic weight is the mass of an atom averaged over all its isotopes.

From the perspective of chemistry, an atom might be described as a collective structure of sub-atomic particles that form the basis of all chemical reactions, where the key to understanding the properties of a particular atomic element lies within the electron structure. In 1911, before the development of quantum theory, the Rutherford model of the atom proposed that the outer electrons of an atom might explain the interaction between atoms and the formation of chemical bonds. Later, in 1916, Gilbert Lewis, a physical chemist, forwarded the idea of electron valence and its role in chemical bonding – see covalent bonds and ionic bonds for more details.

Note: In 1916, Lewis also forwarded the model of a Cubical Atom, which was later assumed obsoleted by the quantum model. However, some readers might wish to reviewed Gabriel LaFreniere’s wave model of an atom, which also leads to a cubic structure, although for possibly very different reasons to those assumed by Lewis.

Along the timeline being outlined, we might reference Bohr’s model of the atom, despite its known limitations, as the start of a transition between the classical particle model towards the quantum model, which included the idea of a wave-particle duality. In this context, the quantum model supports the idea of an electron, possibly as both a point-particle and as a waveform, within a revised model of atomic orbitals. So, within the semantics of the quantum model, the description of these orbitals may first make reference to some form of standing wave structure surrounding the nucleus of the atom in order to explain quantisation in terms of an integer wavelength within an orbital. However, the quantum model also explains these orbitals in terms of the mathematical probability of locating the electron, as a point-particle, within a specific orbital. The atomic orbital model also introduces a set of quantum numbers, i.e. principle number [n], azimuthal number [l], magnetic number [mi] and spin number [m]. The principle number identifies the energy level of the electron orbital, where [n=1] corresponds to the energy level closest to the nucleus. However, within this model, a visualisation of the atomic orbitals is presented below, which for historical reasons also describes the shape of the orbital sub-structure, denoted by letters, i.e. s,p,d,f.

So, how does the quantum model explain the orbital sub-structures in terms of the quantum numbers?

We might start by highlighting that while the following description might be considered logical, it does not necessarily provide a causal explanation, i.e. how and why. For it might be highlighted that the shapes suggested in the diagram above are not really physical as they only represent a probability region in which an electron, as a point-particle concept, might be found. However, we shall begin by trying to quantify how the quantum numbers relate to one another in terms of the atomic orbitals, as illustrated in the previous diagram. First, a clarification is possibly necessary, the principle number [n] identifies the orbital, as defined by the original Bohr model, which was subsequently extended to include the idea of sub-orbitals often described as ‘sub-shells’, e.g. [s,p,d,f..], as shown in the table below.

n l (0..n-1) subshells ml (-l..+l) number ms
1 0 s 0 1 ↑↓
2 0,1 s,p -1, 0,+1 3 ↑↓
3 0,1,2 s,p,d -2,-1, 0,+1,+2 5 ↑↓
4 0,1,2,3 s,p,d,f -3,-2,-1, 0,+1<+2,+3 7 ↑↓

In the table, we see the basic relationship between the azimuthal number [l=n-1] and the magnetic number [mi=-l..+l] and the principle number [n], where the spin quantum number [ms] only has two values, i.e. spin up or down [↑↓]. However, the build-up of subshells is outlined in a little more detail in the following table limited to just the first 2 principle [n=1,2] orbits, where each subshell can only support 2 electrons, as required by the Pauli Exclusion Principle.

n orbital # sub-orbitals electrons electrons electrons
1 s 1 l=0, ml=0, ms=↑↓ 2 2 2
2 s 1 l=0, ml=0, ms=↑↓ 2 2 8
p 3 l=1, ml=-1, ms=↑↓ 2 6
l=1, ml= 0, ms=↑↓ 2
l=1, ml=+1, ms=↑↓ 2

So, when [n=1], we only have one subshell [s], which is limited to 2 electrons (↑↓). When [n=2], the value of [l] can be [0,1], which leads to 3 values [ml=-1, 0,+1], so when [n=2], there is a single [s] subshell with 2 electrons plus 3 [p] subshells, each with 2 electrons, accounting for 6 electrons within the overall [p] subshell and a total of 8 in the [n=2] orbital. However, having outlined the basic principles by which electrons fit into this model, it might be realised that this method can quickly become a bit tedious, such that we might wish to use some simplified formulations, as shown in [2].


Using [2], we might show the number of subshells associated with each orbital defined by the principal number [n] and the number of electrons in each subshell plus the maximum number of electrons within an atom.

n Subshell 2n-1 2(2n-1) 2n2
1 s 1 2 2
2 p 3 6 8
3 d 5 10 18
4 f 7 14 32

While it is a reasonable assumption that electrons will fill-up the lowest energy orbitals first, i.e. starting with [n=1], in practice, some of the subshell energy levels overlap. So, while these details are not the focus of this review, reference might be made to the Aufbau Principle, which explains the energy ordering of the electrons within the subshells. So, having briefly outlined some of the basic principles used to described the various atomic orbitals and subshells, we might now show the periodic table, where the subshells have been coloured coded. Again, while all the details associated with the periodic table are not the focus of this discussion, it might be highlighted that the atomic number at the top of each element square corresponds to the number of protons, and electrons, in a charge-neutral atom.

 So far, this discussion has only attempted to outline some of the historic developments, which were initially anchored to a classical understanding of the particle model, but which over the course of the 20th century developed into the atomic orbital model, as outlined. However, while this model might be described as generally compatible with later quantum field theory, its description often appears confused by the semantics of a wave-particle duality and lacks details of the causal mechanisms that create and maintain this model. As a consequence, many textbooks, even when related to the quantum model, still describe the ‘workings’ of an atom in terms of a particle model, as illustrated by the carbon atom right. Within this diagram, we see the idea of a central nucleus comprised of positive protons and neutral neutrons surrounded by orbiting negative electrons.

Note: It is recognised that the use of this type of depiction of the atom is possibly only trying to avoid the inherent complexity of the quantum model. However, it might be suggested that the quantum model simply does not make sense to many people in terms of its use of particle-wave semantics. As such, many descriptions may simply default to the particle description, where orbitals still appear to imply rotational orbits, while providing no indication of the physical scale between the nucleus and electron orbitals, as previously outlined.

While not trying to dismiss the insights provided by the quantum model, we still have to question whether the references to ‘conceptual principles’ is simply a means to side-step the issue of causality, while relying on mathematical abstraction to justify its conclusions. As outlined, we might see how the quantum numbers might be useful, although it is not always obvious whether these numbers are supported by empirical observation or are simply the assumptions of the quantum model. At this point, we may have simply returned to the issue of epistemological knowledge versus ontological causality. Of course, in terms of the totality of post-war developments, it might be argued that the quantum model has provided an explanation of not only the principles underpinning the periodic table, but the ever-expanding model of particle physics. However, doubts over the ambiguity created by the wave-particle duality remains – see the Particle Model  or more details of the diagram below.

Today, the particle model has grown to define over 200 different particles, although the existence of many are only transitory in high-energy collision experiments. In more general terms, most of the particle model can be defined in terms of 17 fundamental particles, i.e. 6 types of quarks, 6 types of leptons, plus 4 types of force-carrying bosons and a Higgs boson. However, all matter particles have an anti-matter counterpart, e.g. the electron counterpart is called a positron.

The semantics of particles can appear particularly confusing, when gauge bosons are described as particles, which are assumed to act as ‘ force carriers’. In the previous diagram, we see a hierarchical model that makes reference to various quantum models, but where everything appears to be defined in terms of particles to the point that even a classical force has to be described as an interaction mediated by the force carrier bosons, shown top right. In this model, there is a specific type of force-carrier associated with each of the four fundamental forces of nature, but now described in terms of interactions. For example, the strong interaction is described as a ‘mechanism’ that binds the quark substructure of both protons and neutrons, but where any causal mechanism can quickly be lost within the abstraction of Quantum Field Theory and a sub-field known as Quantum ChromoDynamics.

Note: Experiments indicate that a neutron will decay into a positive proton and a negative electron plus an anti-neutrino. Clearly, this suggests that a neutron has a substructure, although we might question whether another sub-particle, i.e. the quark, is really providing a causal description or simply another abstraction.

While only alluding to the full complexity of the particle model, we might try to further simplify the scope in the earlier diagrams to those basic particles that make-up atoms, i.e. protons, neutrons and electrons. In the simplified table below, we see that an electron is classed as a ‘lepton that has no known particle sub-structure, but where the ‘hadron’ class is first divided into protons and neutrons and sub-divided into up/down quarks.

Leptons Hadrons
Electron Proton Neutron
Up Down

We might add some detail to this particle model of an atom, i.e. protons, neutrons and electrons, in terms of the estimated energy-mass of each particle, but shown in Mev for numeric simplicity, although often considered in terms of mass [kg].

Particle Symbol Quarks Charge MeV/c2 Lifetime
proton p uud + 1 938.272   Stable
neutron n udd 0 939.565 8.881*102
electron e n/a −1 0.511 Stable
Where 1 MeV = 1.782662E-30 kg = 1.602176565E-13 joules

As illustrated in the table above, an electron has no quark substructure, such that the notation [u,d] only applies to the proton having 2 up-quarks and 1 down-quark and the neutron assumed to have 1 up-quark and 2 down-quarks. While there are 6 types of quarks, the following table is simplified to show only the up-down quarks required by an atom.


Symbol MeV/c2 Charge Decay Lifetime
Up u 1.7 to 3.3 + 23 u → d+W+ ?
Down d 4.1 to 5.8 13 d→u +W- ~900s

The ‘charge’ figures shown for the quarks represent the fractional electron charge [e], which results in a net positive charge for the proton, i.e. [uud], and the neutral charge of the neutron. While we might initially question how the charge combination achieves a stable configuration, there is another ‘force’, i.e. strong nuclear, that is assumed to override any repulsion associated with the electrostatic charge – see Quark Interactions for more details. Having introduced this detail, we might now realise that the neutron decay model referenced above actually corresponds to a quark decay model.


In the second table above, we see the additional information related to the neutron decay process in terms of the decay of a down quark with an estimated lifetime. The [W] boson is associated with the ‘weak force’ and is often described in terms of virtual bosons, which only exist within the time frame allowed by the uncertainty principle. The decay time for proton decay, via [u→ d+W+] process is theoretically estimated to be in excess of 1032 years, which being longer than the age of the universe might be ignored. However, a free neutron decay is in the order of 15 minutes.

Note: What was originally described as the weak force was first proposed by Enrico Fermi, in 1933, also referred to as beta decay, where a neutron decays into a proton and an electron. However, later in 1964, the idea of beta decay was developed into the Electro-Weak Theory, although its mathematical abstractions are beyond the scope of this discussion.

Finally, the last two force carriers in this particle model relate to the photon, as described by Quantum Electrodynamics, and the speculative graviton, as described by Quantum Gravity. However, as already outlined, there is considerable speculation about the physical nature of a photon, other than its energy being quantified by Planck’s equation [E=hf].

Note: At this time, quantum gravity has to be described as a hypothesis, which is in the process of trying to unify general relativity and quantum theory. This theory, if realised, would need to provide a model of spacetime, possibly at the Planck scale, where the gravitational field would also be quantized. However, any empirical testing of quantum gravity must be considered questionable, such that any progress may only be based on the assumptions of yet more mathematical abstraction.

While this outline has tried to provide a basic description of the atomic structure and the various sub-structures within the particle model, little has been said about the actual structure of a particle. If we concede that a particle cannot be a ball-bearing like object without structure, we might first consider the quantum description, where a particle is considered to be an ‘excitation’ or ‘ripple’ within one of the quantum fields associated with each particle type. However, the use of such semantics does not necessarily provide an explanation of the causal mechanisms, which allow an energy-density to propagate in space as a function of time. Despite the quantum model describing a ‘ particle as a ripple in a quantum field’, it also appears to support the general idea that an electron must have some form of wave structure in order to explain the quantisation of the wavelength within an atomic orbital.

Note: If an electron has some form of wave structure, this structure must also be used to explain the characteristics of rest mass and charge within gravitational and electrostatic fields, while also being able to propagate through space with a velocity range [v=0..c], such that it can also have the attribute of kinetic energy. 

While this discussion will not speculate further on the potential nature of any wave structure, partly because it is not really supported by the quantum model, we know that the relativistic energy equation in [4] tells us something about the rest-mass energy and kinetic energy that can be associated with the particle concept.


Based on [4], it is not unreasonable to suggest that whatever structure underpins the concept of a particle, it must reflect a stable energy configuration in the form of the rest energy [E=m0c2] when velocity [v=0] plus the kinetic energy [m0v2] when in motion with velocity [v<<c], which might also be described in terms of the particle momentum [p=m0v]. While realising that this discussion has been little more than a brief outline of the particle model, it has questioned how the quantum model, with its description of a probability wave functions provides any further clarity of the causality at work in the quantum realm. However, we might simply try to highlight an overall issue of concern in terms of the following quote attributed to George Box.

All models are wrong. Some models are useful.

By accepting that all models are a simplification, often predicated on unproven assumptions, we might recognise that factual certainty has to remain open to questioning. In this respect, the most obvious assumption of the particle model is that it only requires particles to explain the fundamental workings of the universe. Again, if particles with mass [kg] do not exist, other than as an energy-density, then some other fundamental description is required to explain how this energy-density is structured.