Physics Tutorial: Elementary Particles

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In this Physics tutorial, you will learn:

  • What are quantum numbers? How many quantum numbers are there?
  • What are atomic orbitals? What do they represent?
  • How to find the electronic distribution in various orbitals?
  • What does Pauli's Exclusion Principle say on electronic configuration in atomic orbitals?
  • What are elementary particles? Where do they come from?
  • What is the phenomenon of "pair annihilation"? When does it occur?
  • What did Yukawa predicted?
  • What are mesons? Why are they called so?
  • How to find the lifespan and energy of elementary particles?

Introduction

What is the smallest particle you know? Do you think it is the smallest particle in the universe? Explain your opinion.

Do you think all electrons in an atom have the same energy regardless their distance from nucleus?

Do you think there exists any form of energy produced when two electrons collide with each other? (Have you ever noticed anything strange when you strike two rocks against each other?)

This tutorial is an introduction on elementary particles - a further penetration on the fascinating mysteries of microscopic world. However, first some properties of electrons (one of elementary particles) that determine their behavior in atomic shells are explained, as a pre-requisite for a full understanding of elementary particles properties.

Background and Introduction to Quantum Numbers and Orbitals

As we discussed in Section 20 (Nuclear Physics), atoms are composed by a nucleus at centre that contains protons and neutrons, as well as electrons that revolve around the nucleus in determined orbits (layers or shells). However, when studying the atom more in detail, we see that a shell may contain a number of subshells, and so on. The more closely we observe the atom, the more specific features we discover in it.

a. Quantum numbers

Quantum numbers represent a method used to define the trajectory and movement of an electron within an atom. There are four quantum numbers for every electron in an atom, which are combined to give a unique configuration. This is like describing a person physically through a number of attributes which provide information to other people who don't know him (for example tall, blonde, slim and smiley). Quantum number in itself is a value used to describe the energy available in atoms or ions. Quantum numbers are nothing more but different shapes of orbitals drawn, depending on the possibility of finding electrons around the nucleus in the space enclosed by such shapes. The four quantum numbers are:

  1. n - the principal quantum number that expresses the energy levels (atomic orbitals). The principal quantum number n is always an integer. Additionally, it is equivalent to the number of electron shells. Hence, its value is at least one and higher. Principal quantum number is never zero or negative (n = 1, 2, 3, 4 ) as any atom has at least one electronic shell. Only four principal quantum numbers are in use so far for known elements, despite theoretically they can be more than four). This is because the number of chemical elements is finite (118 to this date).
  2. - the azimuthal or angular momentum quantum number that describes the subshell. The angular momentum quantum number is also an integer, which represents the value of an electron's orbital. Hence, ℓ is either greater than or equal to zero, and lower or equivalent to n -1 (ℓ = 0, 1, 2, , n - 1).
  3. mℓ (or simply m) - the magnetic quantum number that expresses the orbital of subshells, that is the mathematical functions often employed in order to determine the probability of finding an electron (belonging to an atom) in a specific region around the nucleus of the atom. The magnetic quantum number symbolizes the orientation of the orbital. The integer values of magnetic quantum number are ranging from -ℓ to +ℓ. Thus, for p orbital, where ℓ = 1, the magnetic quantum number m can have values of -1, 0, 1.
  4. ms (or simply s) - the spin quantum number expressing the spin, that is one of two types of angular momentum in quantum mechanics (the other is orbital angular momentum). The spin quantum number has a half-integer value, which is either - 1/2, known as 'spin down' or + 1/2 called 'spin up'. In practice, it describes the intrinsic angular momentum or 'spin' of an electron within an orbital, as the word itself suggests. Moreover, it provides a projection of the spin angular momentum (s) along a particular axis.

b. Atomic orbitals

The "orbit" is defined as the definite path of an electron that moves around the nucleus in an atom. This is similar to a planet which moves around the sun. Atomic orbitals on the other hand, are the space or region around the nucleus where the electron are calculated to be present. So orbits and orbitals have totally different meanings and it is important you remember that key difference..

We understand from lessons in chemistry that the four atomic orbitals in use are s, p, d and f where s can hold 2 electrons at maximum, p can hold 8 electrons, d can hold 18 electrons and f can hold 32 electrons (that is 2n2 where n is the principal quantum number).

Let's describe more in detail the four orbitals and their orientation.

s-orbitals

s-orbitals represent solid spherical shapes around the nucleus. When the principal quantum number n = 1 and azimuthal quantum number ℓ = 0, that is 1s orbital which is closest to the nucleus. When n = 2 and ℓ = 0, (i.e 2s orbital) there is an orbital which contains one node. When n = 3 and ℓ = 0, (i.e. 3s), we have an orbital which contains two nodes. The pictorial representation of these orbitals is shown below:

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

Hence, we can say that s-orbitals have always a spherical shape, regardless of the principal quantum number, size and the number of nodes they contain.

p-orbitals

p-orbitals are dumb-bell shapes containing two lobes, like two identical balloons tied together. The two lobes are apart from each other along the axial line. When n = 1, there are no p-orbitals but only a s-orbital. When n = 2 and ℓ = 1, the possible magnetic quantum numbers are m = -1, 0, +1. Thus, three dumb-bell shape p-orbitals are found pointing towards the three axes x, y and z, which are perpendicular to each other. These three orbitals are named as px , py and pz respectively. The nodal plane is the plane where it is not possible to find any electrons. The nodal planes of px , py , pz are yz , xz and xy respectively.

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

d-orbitals

d-orbitals have different shapes and these are only available when the principal quantum number is n = 3 or more. When n = 3, then ℓ = 2, so the magnetic quantum number m can take the following values: m = -2, -1, 0, +1 and +2. That means five d-orbitals are available in any atom. The directions, names and the shapes of these orbitals are as follows:

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

You don't have to remember all these configurations and symbols; they are just for illustration purpose as such things are extensively explained in chemistry courses.

f-orbitals

f-orbitals also have different shapes and they are only available when the principal quantum number is n = 4 or more. When n = 4, then ℓ = 3, so m = +3, +2, +1, 0, +1, +2 and +3. In this way, seven f-orbitals are available in an atom. The directions, names and the shapes of these orbitals are as follows:

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

c. Electrons distribution in orbitals

s-orbitals

The maximum number of electrons that each s-orbital can hold is two, regardless of the number of principal quantum number (n). Thus, we have 1s2, 2s2, 3s2 etc. The spin of these two electrons must be opposite, as stated earlier.

p-orbitals

Each p-orbital can possess at maximum two electrons which means six electrons in total; two electrons for each of three p-orbitals. We can write that either 2p6 or 2px2 2py2 2pz2. The spin of each of these orbitals must be opposite.

d and f-orbitals

The total number of electrons in d-orbitals and f-orbitals is 10 and 14 respectively. Again, two electrons at maximum can occupy each suborbital of these d or f-orbitals.

On the other hand, the number of orbitals is found through the formula

N = 2l + 1

where is the azimuthal quantum number.

d. Spin quantum number

The spin of every two electrons, in each orbitals, will be always be in the opposite direction. Spin is represented schematically through vertical arrows, where each suborbital (orbital in a given direction) is represented through a square-shaped box. The following image shows the maximum electron distribution in each orbital.

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

The following figure shows what "spin up" and "spin down" mean in the trajectory of an electron.

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

There is an important principle (known as Pauli's Exclusion Principle) to bring in mind when dealing with atomic orbitals. It says that in a single atom no two electrons will have an identical set or the same quantum numbers (n, l, ml and ms). This means each electron in an atom has an unique configuration regarding quantum numbers and their orientation.

Definition of Elementary Particles. Antiparticles

Elementary particles represent the smallest known building blocks of the universe. They are thought to have no internal structure, meaning that they are considered as as zero-dimensional points that occupy no space. Electrons are probably the most familiar elementary particles, but the Standard Model of physics, which describes the interactions of particles and almost all forces, recognizes in total 10 elementary particles. Some of them originate from outside the Solar System, so we can identify a number of elementary particles only through the radiation coming from remote stars by means of cosmic rays, which provide one of our few direct samples of matter from outside the solar system.

Cosmic rays are fluxes of high-energy particles that move through space at high speeds, very close to the speed of light. Most cosmic rays are atomic nuclei stripped of their atoms with protons (hydrogen nuclei) being the most abundant type but nuclei of elements as heavy as lead have been identified as well. In cosmic rays however, we also find other subatomic particles like neutrons, electrons and neutrinos.

In 1931, Paul Dirac - a famous British scientist of 20th century - based on theoretical reasoning, introduced the revolutionary hypothesis that besides the negative electron, there must exist the positive electron as well. This new particle must have a positive charge of +e and a spin of 1/2. (It cannot be proton, as a proton has no spin). Later on (in 1934) the positive electron (in short positron) was identified experimentally in cosmic rays coming from remote sources and immediately after this event, positron was also detected in terrestrial conditions. This elementary particle was identified as a particle produced by radiation emitted by radioactive nuclei of Phosphorus-30.

The related but opposite elementary particles such as electron and positron are known as antiparticles. More precisely, an antiparticle is a subatomic particle having the same mass as a given particle but opposite electric or magnetic properties.

There are many other antiparticles besides the electron-positron pair. As seen in beta decay examples explained in tutorial 20.3 "Radioactivity and Half-Life", electron is symbolically denoted as (e-) or (0/-1e) while positron as (e+) or (0/+1e). In the following paragraph, we will explain more in detail the electron-positron relationship and the interaction between them.

Electron-Positron Pair

Positron does not exist in the structure of common matter; the electron-positron pair appears only during the collision with matter of charged particles or high-energy gamma rays. This process is known as "pair production". It must be noted here that pair production is not an exclusive process of electron-positron only; it applies in all matter-antimatter pairs of elementary particles. During this process, electric charge must be conserved and an amount of energy E = 2 me · c2 sufficient to overcome the rest energy of the two particles. Hence, the minimum energy needed for pair production process must be

Emin = 2me ∙ c2
= 2 ∙ (9.1 × 10-31 kg) ∙ (3 × 108 m/s)2
= 1.64 × 10-13 J

The process of pair production is shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

Schematically, the process of electron-positron pair production is written as:

γ → e- + e+

The reverse process may also occur. Thus, when an electron enters in contact with a positron, a gamma particle is produced. More specifically, if a positron encounters an electron on its way, this electron-positron pair is transformed into a pair of gamma quants according the reaction

e- + e+ = γ + γ

The energy of this pair of gamma quants is not less than 2 me · c2, This radiation propagates in opposite direction to the movement of original particles. Thus, when a particle and an antiparticle collide with each other, they are annihilated, emitting energy. This process is (not so rightfully) called "pair annihilation".

The above fact is a demonstration that besides the mutual transformation of particles from matter to matter (i.e. within the family of material objects), a matter to energy or vice-versa transformation is also possible. More specifically, matter is transformed in EM (gamma) radiation and vice-versa.

We can find identify the positron as elementary particle during the explosion of unstable particles as well. A nucleus that contains too many neutrons emit an electron e- during a beta minus decay to become more stable. In this way, a neutron converts to proton, increasing the atomic number Z by 1 while the number of neutrons N decreases by 1. On the other hand, if an atomic nucleus is abundant in protons but lacks the sufficient number of neutrons to be stable, releases a positron e+, transforming in this way a proton into neutron. Such nuclei cannot be found in nature but they can be produced artificially in nuclear reactors.

Example 1

Calculate the rest mass of positron in MeV/c2 if the minimum energy needed to produce a positron is 1.02 MeV.

Solution 1

From theory, we know that electron and positron have the same rest mass denoted by me and the minimum energy needed for the electron-positron pair to produce is given by

Emin = 2me ∙ c2 = 1.02 MeV

Rearranging to isolate me, we obtain for the rest mass of positron:

me = Emin/2 ∙ c2
= 1.02 MeV/2 ∙ c2
= 0.51 MeV/c2

Physics of Elementary Particles. The Yukawa Theory

As we know, atomic nuclei are made of protons and neutrons. Thus, the hydrogen nucleus contains a proton only, the helium nucleus one proton and one neutron and so on. In previous articles we explained why the structure of nucleus is so stable giving the repelling force existing between like charges (protons). We explained that a new force (we called it "nuclear force" but henceforth we will call it the "strong force" - this is the scientific name of nuclear force as one of the four fundamental forces acting in nature - will appear in order to balance the effect of electric force between protons. This force however is evident only in very small distances (1 - 2 × 10-15 m). It decreases drastically with the increase in distance between particles. This is why the action of strong (or nuclear) force is not observed in macroscopic events.

In classical Physics we have the interaction between charged point objects relying on the concept Coulomb's force. In quantum physics however, this interaction is explain through the exchange (emission or absorption) of photons. When two electrons repel each other, one of them emits a photon and the other electron absorbs it. The following figure shows schematically how this event takes place.

Physics Tutorials: This image provides visual information for the physics tutorial Elementary Particles

From the above figure, it is evident that in this process electrons change their direction as well. Since there is a time interval during which the collision process occurs, it is clear that the collision of electrons at the point A occurs prior to their separation at the point B. In other words, the collision brings a emission of photon by one of electrons at the point A which is absorbed by the other electron at the point B.

In this way, we can say that the interaction of charged particles takes place through the mediation of photons. The question that arises here is: "Where does the energy needed for the generation of photons come from?" The Heisenberg Uncertainty Principle helps in explaining this point. According to this principle, a short-term state has an uncertainty of energy ΔE given by the relation

∆E ∙ ∆t ≥ h/

where Δt is the time interval during which the process occurs. Based on this principle, the generation of photons having the energy ΔE is possible with the condition that the generation time not exceeds the time interval Δt provided in the Heisenberg formula. Such a photon having a lifespan as much as allowed by Heisenberg uncertainty principle is known as "virtual photon." In analogy with the financial system, we can say that the uncertainty principle relation acts like a "bank" in which we can borrow energy and settle it within a given time. Based on the above relation, it is evident that the more energy is borrowed, the soonest it must be settled.

Now, let's discuss a little more about the carriers of strong interaction, which result in the generation of nuclear force. In 1935, the Japanese scientist Hideki Yukawa introduced the idea that the strong interaction between nucleons is made possible through the exchange of certain particles of mass 200-300 times the mass of electron. The reasoning used by Yukawa to draw this conclusion was more or less as follows:

The distance in which nuclear forces can act, depends on the mass of particle that produces the interaction. The lifespan of this particle must be long enough to allow it move within the range of nuclear forces action. On the other hand, nuclear forces act in distances that are smaller than the nucleus dimensions. Based on this fact (and other information obtained during the experiments) it resulted that the action range of nuclear forces is about r0 = 1.5 × 10-15 m (it is a kind of diameter, not radius). Assuming the speed of the unknown particles comparable to the speed of light (as they are a kind of energy), the lifespan Δt of such particles must be

∆t = r0/c
= 1.5 × 10-15 m/3 × 108 m/s
= 5 × 10-24 s

Thus, based on the uncertainty principle relation

∆E ∙ ∆t ≥ h/

we obtain for the minimum uncertainty of energy ΔE during this process:

∆Emin = h/2π ∙ ∆t
= 6.626 × 10-34 J ∙ s/2 ∙ 3.14 ∙ 5 × 10-24 s
= 2.11 × 10-11 J

The equivalent mass of this energy is

∆m = ∆E/c2
= 2.11 × 10-11 J/(3 × 108 m/s)2
= 2.34 × 10-28 kg

This value is about 250 times greater than the mass of electron (9.1 × 10-31 kg). Yukawa proposed the idea of the existence of this particle as a mediator in explaining the operating principle of nuclear forces. This idea was very revolutionary for the time, when no experiment could not confirm yet the existence of this particle.

A few years later (in 1945), after processing the experimental data obtained from the cosmic rays, scientist were able to identify a particle of mass 207 times the mass of electron. Since this value of mass is between the mass of nucleons and electrons (nucleons are more than 1800 times heavier than electrons), the new particle discovered was named "μ-meson" ("meson" means medium value in Greek language) or "muon". It was initially identified with the particle predicted by Yukawa but later on, scientists realized that such particles almost do not interact with atomic nuclei. That means they cannot be the carriers of nuclear interaction. The true carriers of strong interaction were discovered in 1947 by Cecil Frank Powell - an English scientist. These particles were the π-mesons (pi mesons) or simply "pions" predicted by Yukawa.

After accurate measurements, the results show that three types of pions exist, these are π +, π - and π 0 and these have electric charges of +e, -e and 0 respectively. As for their masses, we have:

mπ + = mπ - = 273 me and mπ0 = 264 me

All nucleons pass some part of their lifespan by experiencing one of the four transformations shown below:

p ⇄ n + π+
p ⇄ p + π0
n ⇄ p + π-
n ⇄ n + π0

Hence, every nucleon is surrounded by a cloud of π-mesons which form the field of its nuclear force. The exchange of π-mesons between nucleons results in the strong nuclear interaction. For example, one of such interactions can be expressed as

p + n ⇄ n + π+ ⇄ n + p

Summary

Quantum numbers represent a method used to define the trajectory and movement of an electron within an atom. There are four quantum numbers for every electron in an atom, which are combined to give a unique configuration. They are:

n - the principal quantum number that expresses the energy levels (atomic orbitals). The principal quantum number n is always an integer (n = 1, 2, 3, 4).

- the azimuthal or angular momentum quantum number that describes the subshell. The angular momentum quantum number is also an integer, which represents the value of an electron's orbital (ℓ = 0, 1, 2, , n - 1).

mℓ (or simply m) - the magnetic quantum number that symbolizes the orientation of the orbital. The integer values of magnetic quantum number range from -ℓ to +ℓ.

ms (or simply s) - the spin quantum number expressing the spin, that is one of two types of angular momentum in quantum mechanics (the other is orbital angular momentum). The spin quantum number has a half-integer value, which is either - 1/2, known as 'spin down' or + 1/2 called 'spin up'.

Atomic orbitals represent the space or region around the nucleus where the electron are calculated to be present. The four atomic orbitals in use are s, p, d and f where s can hold a maximum of 2 electrons, p can hold 8 electrons, d can hold 18 electrons and f can hold 32 electrons (that is 2n2 where n is the principal quantum number).

In general, the number of orbitals is found through the formula

N = 2l + 1

where is the azimuthal quantum number. This means the number of maximum electrons that each s-orbital can hold is two, regardless of the number of principal quantum number (n). Each p-orbital can possess at maximum two electrons which means six electrons in total; two electrons for each of three p-orbitals. The total number of electrons in d-orbitals and f-orbitals is 10 and 14 respectively.

Spin of every two electrons in each orbitals will be always in opposite direction.

Pauli's Exclusion Principle says that in a single atom no two electrons will have an identical set or the same quantum numbers (n, l, ml and ms). This means each electron in an atom has a unique configuration regarding quantum numbers and their orientation.

Elementary particles represent the smallest known building blocks of the universe. They are thought to have no internal structure, meaning that they are considered as as zero-dimensional points that occupy no space. We can identify a number of elementary particles only through the radiation coming from remote stars by means of cosmic rays, which provide one of our few direct samples of matter from outside the solar system. Some of elementary particles detected in cosmic rays include neutrons, electrons and neutrinos. The positive electron (in short positron) was another elementary particle identified experimentally in cosmic rays but also in terrestrial conditions. It has a positive charge of + e and a spin of 1/2.

The related but opposite elementary particles such as electron and positron are known as antiparticles. More precisely, an antiparticle is a subatomic particle having the same mass as a given particle but opposite electric or magnetic properties.

Positron does not exist in the structure of common matter; the electron-positron pair appears only during the collision with matter of charged particles or high-energy gamma rays. This process is known as "pair production". The minimum energy needed for pair production process must be

Emin = 2me ∙ c2=1.64 × 10-13 J

Schematically, the process of electron-positron pair production is written as:

γ → e- + e+

The reverse process may also occur. Thus, when an electron enters in contact with a positron, a gamma particle is produced. More specifically, if a positron encounters an electron on its way, this electron-positron pair is transformed into a pair of gamma quants according the reaction

e- + e+ = γ + γ

The energy of this pair of gamma quants is not less than 2 me · c2, This radiation propagates in opposite direction to the movement of original particles. Thus, when a particle and an antiparticle collide with each other, they are annihilated, emitting energy. This process is (not so rightfully) called "pair annihilation".

The "Strong force" - the scientific name of nuclear force (as one of the four fundamental forces acting in nature) - appears inside the nuclei to balance the effect of electric force between protons. This force however is evident only in very small distances (1 - 2 × 10-15 m). It decreases drastically with the increase in distance between particles. Because of this, when two electrons repel each other, one of them emits a photon and the other electron absorbs it.

According to Heisenberg's uncertainty principle, a short-term state has an uncertainty of energy ΔE given by the relation

∆E ∙ ∆t ≥ h/

where Δt is the time interval during which the process occurs. Based on this principle, the generation of photons having the energy ΔE is possible with the condition that the generation time not exceeds the time interval Δt provided in the Heisenberg formula. Such a photon having a lifespan as much as allowed by Heisenberg uncertainty principle is known as "virtual photon."

In 1935, the Japanese scientist Hideki Yukawa introduced the idea that the strong interaction between nucleons is made possible through the exchange of certain particles of mass 200-300 times the mass of electron.

A few years later, scientist were able to identify a particle of mass 207 times the mass of electron. Since this value of mass is between the mass of nucleons and electrons (nucleons are more than 1800 times heavier than electrons), the particle discovered was named "μ-meson" ("meson" means medium value in Greek language) or "muon". It was initially identified with the particle predicted by Yukawa but later on, scientists realized that such particles almost do not interact with atomic nuclei. That means they cannot be the carriers of nuclear interaction. The true carriers of strong interaction were discovered in 1947 by Cecil Frank Powell - an English scientist. These particles were the π-mesons (pi mesons) or simply "pions" predicted by Yukawa.

After accurate measurements, it resulted illustrated that three types of pionsexist: π +, π - and π 0 with these bearing electric charges of +e, -e and 0 respectively. As for their masses, we have:

mπ + = mπ - = 273 me and mπ0 = 264 me

All nucleons pass some part of their lifespan by experiencing one of the four transformations shown below:

p ⇄ n + π+
p ⇄ p + π0
n ⇄ p + π-
n ⇄ n + π0

Hence, every nucleon is surrounded by a cloud of π-mesons which form the field of its nuclear force. The exchange of π-mesons between nucleons results in the strong nuclear interaction.

Elementary Particles Revision Questions

1.What are the maximum orbital numbers (n, l, m and s) possible in a neutral chlorine Cl-17 atom?

  1. n = 3, l = 2, ml = 5, ms = 1
  2. n = 3, l = 3, ml = 5, ms = 2
  3. n = 3, l = 3, ml = 5, ms = 2
  4. n = 3, l = 5, ml = 7, ms = 2

Correct Answer: B

2. Using the Heisenberg's Uncertainty Principle relation, determine the lifespan of particles that have an energy of 600 MeV. (1 eV = 1.6 × 10-19 J; h = 6.626 × 10-34 J · s).

  1. 0.11 ns
  2. 1.099 μs
  3. 600 ms
  4. 0.69 ns

Correct Answer: A

3. What is the energy of elementary particles that have an average lifespan of 2.2 μs? (h = 6.626 × 10-34 J · s; 1 μs = 10-6 s).

  1. 3.52 × 10-19 J
  2. 3.02 × 10-28 J
  3. 2.2 × 10-6 J
  4. 4.78 × 10-29 J

Correct Answer: D

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