## Thursday, March 2, 2023

### Electrons: UNIZOR.COM - Physics4Teens - Atoms - Elementary Particles

Notes to a video lecture on http://www.unizor.com

Electrons

Electron is one of the three main particles present in every atom - protons and neutrons make up a nucleus, and electrons are rotating around a nucleus.

According to current view, electron is an elementary particle. Physicists have not identified any smaller particles that compose an electron.
There are some theories about some smaller particles inside an electron, but so far they have not been generally supported.

This leaves us to discuss the properties and characteristics of electrons, not their inner structure, like in the case of a proton or neutron with quarks and gluons they are composed of.

The first characteristic of an electron is its electric charge. When discussing subatomic particles, the electric charge of an electron is a unit of measurement. Therefore, its electric charge has absolute value of 1.
The electric field has two types of charge - positive and negative. An electron has a type negative, while a proton has type positive. Hence, we say that electric charge of an electron is −1, while a proton's electric charge is +1.

The usual symbol for an electron is e.

Another important characteristic of an electron is its mass.
The experiments show that its mass is very small relatively to a mass of a proton or a neutron. In fact, it's almost 2000 times less than the mass of a proton.

We discussed the structure of an electron configuration of an atom in a lecture "UNIZOR.COM - Physics 4 Teens - Atoms - Electronic Structure of Atoms - Electrons and Shells" of this course. Let's continue this topic and get deeper into an electron configuration.

We know that electrons occupy shells around a nucleus, sequentially numbered 1, 2, 3 etc.
Every shell has certain number of subshells, and the number of subshells within each shell equals to a shell number:
shell #1 - 1 subshell s
shell #2 - 2 subshells s, p
shell #3 - 3 subshells s, p, d
shell #4 - 4 subshells s, p, d, f
etc.
Electrons within the same subshell have the same energy level.

There is another very important characteristic of an electron we have not discussed yet. It's called a spin.
An important property of an electron is that in the magnetic field it behaves like a little magnet similarly to a behavior of a electrically charged object spinning around an axis.

By analogy, physicists called this property of an electron a spin and, consequently, considered an orientation of the axis of this spin as an important characteristic of an electron.

A famous physicist Wolfgang Pauli suggested so called Pauli exclusion principle, according to which no more than two electrons can share a single trajectory (called orbital) within a subshell, and, if two of them do, they must have opposite orientation of the axis of their spins.

The number of orbitals inside a subshell depends on the subshell number. The first subshell s can have 1 orbital, the second subshell p has 3 orbitals, the third subshell d has 5 orbitals, etc. going along odd numbers, so subshell #X has 2X−1 orbitals.

Summarizing,
(a) electrons are moving within shells (#1, #2, #3 etc.)
(b) that are subdivided into subshells (a shell #N has N subshells with letters substituting the subshell numbers, like s for subshell #1, p for subshell #2, d for subshell #3 etc.)
(c) with 2M−1 orbitals inside a subshell #M
(d) where each orbital capable to hold no more than 2 electrons, which must have opposite spin.

We can represent this structure as the following table:
 Shell Subshell Orbits # of e− #1 #1(s) 1 2 Σ(#1) 1 2 #2 #1(s) 1 2 #2 #2(p) 3 6 Σ(#2) 4 8 #3 #1(s) 1 2 #3 #2(p) 3 6 #3 #3(d) 5 10 Σ(#3) 9 18 #4 #1(s) 1 2 #4 #2(p) 3 6 #4 #3(d) 5 10 #4 #4(f) 7 14 Σ(#4) 16 32

Simple calculations can prove that the number of orbits per shell #N equals to and, consequently, the maximum number of electrons that shell #N can hold is 2·N². This is exactly the formula obtained in the lecture "Electrons and Shells" referenced above.

Double Slit Experiment

There are many articles and videos about double slit experiment on the Web.
In particular, there is a lecture by Richard Feynman about it (almost an hour long).
More recently, Jim Al-Khlili did it in a more theatrical way in about 9 minutes, that I like a lot.
You can find both on the Web.

Below is yet another presentation of this topic, we do it for completeness of our story about electrons.

Imagine a box with two parallel slits at the bottom filled with sand. As sand goes down through slits it accumulates at the tray underneath a box forming two parallel hills, corresponding to two slits above the tray.

This is a clear example of how particles (in this case, sand) independently go through slits without interfering.

Consider an experiment with monochromatic light going through two slits. In this case the slits should be really close to each other and very narrow.
On a screen opposite to slits you will see the bright and dark lines - the result of interference between two rays coming through two slits.
Picture below reflects the intensity of light on a screen - a red curve with oscillating amplitude, maximum in the middle and diminishing to both ends.
Flat wave front of monochromatic light are split into two coherent rays. The bright and dark lines on a screen appear because these two rays come to a corresponding point on a screen in phase or out of phase. In the first case the interference between these rays is positive and enhance the brightness, in the second case rays work against each other and the spot is dark.
The lecture "UNIZOR.COM - Physics 4 Teens - Waves - Phenomena of Light - Interference" explains this process of interference in details.

This is an example of how waves (in this case, waves of electromagnetic field) interfere with each other when going through two slits, making a completely different picture on a screen than if we dealt with particles, like sand above.

In these two experiments, we see two different behaviors of particles and waves going through two slits.
Particles going through different slits do not interfere with each other, while waves do.

Let's see how electrons behave in a similar setting.

Instead of flat wave front of monochromatic light we direct a bunch of electrons. Those that go through two slits will hit a screen with some sensitive to electron material, like the one used in old CRT computer screens.
What will we see on this screen?

Strangely enough, contrary to our perception that electrons are just small particles, the picture is as if electrons are waves that, going through two slits, interfere with each other.

Well, maybe, when a lot of electrons are going through slits, there are some forces among them that distribute them in such a pattern that resembles the interference.
Let's change the experiment and randomly send electrons one by one with sufficient time interval between them, so some will go through the first slit and some - through the second. This way they will not interfere with each other.
What will be a picture on a screen?
In the beginning we will see only individual dots randomly positioned on a screen. After sufficiently large number of electrons fall onto a screen, the pattern will be obvious, and it will be identical to the above pattern of interference.

Common sense tells us that, since electrons are sent to slits one at a time and they randomly go through one or another slit, there should be no information transfer from one electron to another and, therefore, no interference. We should just see two parallel lines on a screen, each across a corresponding slit, similarly to how sand goes through two slits. Yet, the picture was obviously like an interference.

Let's try to analyze which slit each electron goes through. Maybe, this will clarify the situation.
We put a detector near one slit that detects the electron passing by, and repeat the experiment with sending electrons to slits one at a time.
Indeed, about have the times our detector reacted on a passing by electron, as we would expect, considering the randomness of shooting electrons.
But to our surprise the picture on a screen now will be exactly as if individual particles hit slits and accumulate exactly opposite to slits on a screen. Indeed, a particle-like behavior.

That is strange. Do electrons see our detection device and change the behavior?
Let's fool the electrons. Since our detector of electrons requires some electricity to work, we will retain it in place, but unplug it from the wall.
Surprisingly, we will see the interference picture again.

Go figure.

That completes this pseudo-detective story about behavior of electrons going through a two slits configuration.

## Monday, February 27, 2023

### Strong Forces: UNIZOR.COM - Physics4Teens - Atoms - Elementary Particles

Notes to a video lecture on http://www.unizor.com

Strong Nuclear Force

Physical concept of a field assumes that there is a characteristic usually called a charge that participates in the creation of a field or its interaction with objects, manifested as a force.

Electricity has two types of charges that we call positive, carried by protons, and negative, carried by electrons.
Strictly speaking, these are just names, they do not imply that electric charges are positive or negative real numbers, but we can measure them using some clever devices and measuring units, resulting in positive or negative numbers, and they act as if we can add them together using the rules of arithmetic.

In particular, equal in magnitude positive and negative charges neutralize each other. For example, an atom of hydrogen has one proton (positively charged) and one electron (negatively charged) that neutralize each other and form an electrically neutral atom.

In quantum field theory the electric field activity is manifested itself in exchange of photons - particles that carry field energy.

Gravity has only one type of charge that we call mass and it's always present, wherever we have a matter. We can measure it using some devices and measuring units, and these charges can be dealt with using their positive numerical values. There is no opposite charge to neutralize gravity.

There is a theory (not yet decisively supported by experiments) that gravitational field's activity is manifested itself in exchange of gravitons - particles that carry gravitational field energy.

In both above cases, electric and gravitation fields, the quantum theory tells that the field interaction is an exchange of particles specific for this field.

Let's consider the strong nuclear forces now.
As we suggested in the previous lecture, following the Standard Model, protons and neutrons are comprised of three quarks each:
+1p = +2/3u + +2/3u + −1/3d
0n = +2/3u + −1/3d + −1/3d
where
p is a proton,
n is a neutron,
u is an UP quark,
d is a DOWN quark and
preceding numbers are the electric charges of corresponding particles.

Earlier in this course we mentioned the existence of strong nuclear forces that hold the nucleus together.
These forces hold triplets of quarks together to form a nucleon (a proton or a neutron), overcoming repelling electric forces between similarly charged quarks (two +2/3u quarks or two −1/3d quarks).
Strong nuclear forces are much stronger than electric forces and prevent protons from separating because of their mutually repelling positive electric charge.

What is the nature of these strong nuclear forces?
As in a case of electric or gravitational fields, there are charges that create the strong nuclear force field and there are particles that carry the energy of strong nuclear forces.

The charges that create a strong force field and interact by exerting the force are called color charges. Below is an explanation of the reasons for calling them using colors.

The particles that carry energy of strong nuclear forces are called gluons because quarks (material particles) are "glued" together into a proton or a neutron by exchanging gluons.
So, gluons are radiation type particles, carriers of strong nuclear forces analogously to how photons carry the force of electromagnetic field.

Color Charge

Strong nuclear forces are manifestations of a force field inside a nucleus that maintains the nucleus integrity. As with other fields, there is a charge that produces it and participates in its interaction.
While electric charges involved in electric forces are of two types (positive and negative) and gravitational charges are of only one type (mass), charges involved in strong nuclear field are of three different types.

Obviously, these three types of strong nuclear field charges have to be named. The types of charges for strong nuclear forces are called color charges. It has nothing to do with colors of visible light, it's just a name used to characterize and differentiate the types of charges participating in the interactions of strong nuclear forces.

According to Standard Model, there are three different types of color charges (or simply colors) participating in strong force field called:
red (R),
green (G) and
blue (B).
Again, these are just the names to differentiate the types of charges, totally unrelated to visible light and its colors.

In the world of electric fields a single charged particle (negatively charged electron or positively charged proton) prefers to find an oppositely charged particle to combine and to neutralize the charge, attaining electric neutrality.
For example, a proton and an electron can combine to form an electrically neutral atom of hydrogen.

Similarly, in the world of strong forces quarks, charged with any one of the main colors, look for stability in terms of combining their type of charge with other quarks of other charges to attain stability. This stability comes when all three types of quarks are together to form a stable particle, like a proton or a neutron.

The color analogy helps in this case because a combination of three lights of three main colors, red, green and blue, produces the white (colorless) light.
R + G + B = 0

In real numbers the opposite number to number X is the number −X that, combined with X, results in zero.
Similarly, in colors the opposite light color (we can call it anti-color) is the one that, combined with original, produces colorless (that is, white) light.
Therefore, the anti-colors are
anti-red R=G+B,
anti-green G=R+B,
anti-blue B=R+G.

These three anti-colors are the colors of anti-charges. Thus, an anti-proton is comprised from three anti-quarks, two anti-up quarks and one anti-down quark. Each of these quarks has a different anti-color, so all three anti-colors are present in the anti-proton particle.

While the term "color" applied to this physical characteristic is just a name, it was not chosen arbitrarily.
There is a similarity between a quantum characteristic of three types of strong nuclear field charges called "color" and three main colors of visible light, combination of which in proper proportion can produce any other color.

That's why we say that
anti-red = cyan (R=G+B),
anti-green = magenta (G=R+B),
anti-blue = yellow (B=R+G).

Analogously, a combination of all three anti-colors produces black (also, colorless, that is neutral, not charged) charge
R + G + B = 0

Quarks can have one of three possible color charges (or just colors) (R, G, B).

Anti-quarks have one of three anti-colors (R, G, B).

Three quarks of three different main colors make up a colorless (that is neutral and observable) proton or neutron.
For example,
p=u+u+d or p=u+d+u,
n=d+d+u or n=u+d+d.

Three anti-quarks of three different anti-colors make up a colorless anti-proton or anti-neutron.
For example,
p=u+u+d or p=d+u+u,
n=d+u+d or n=d+d+u.

We can obtain a colorless combination of a quark of some color and an anti-quark of a corresponding anti-color.
These two quarks, one of the main color and an anti-quark of the corresponding anti-color (like, u+d) make up a particle called meson.

Quarks are usually confined to a combination that has zero color charge, like a triplet of quarks of three main colors in a proton or a duo of quark and anti-quark of a corresponding anti-color in a meson.

Gluons

Gluons, like quarks, are color charged. A very important difference is that gluons carry two different color charges at the same time, while quarks are color charged with only one.
More precisely, gluons have one charge of the main color and another charge of an anti-color.
For example, green (G) and anti-red (R).

Let's see what happens when a quark emits a gluon.

Assume, a green quark emits green/anti-red gluon.
In simple terms, we subtract the green from green quark and subtract anti-red from the result.
Subtracting anti-red is equivalent to adding red. Therefore, a quark becomes a red one.
quark(G) → gluon(G,R) =
=
quark(GGR) =
=
quark(GG + R) =
=
quark(R)

If this gluon is subsequently absorbed by a red quark, we add to its color charge green and anti-red. Adding anti-red is equivalent to subtracting red. The original red color, therefore, disappears, but green will be added, so this quark becomes green.
qluon(G,R) → quark(R) =
quark(G + R + R) =
=
quark(GR + R) =
=
quark(G)

The emitting of a green/anti-red gluon by a green quark and subsequent absorbing it by a red quark results in exchange of colors between these quarks. A green quark becomes red and red one becomes green.

So, quarks can change color by emitting or absorbing gluons.
If this happens inside a nucleon (a proton or a neutron), the color neutrality of this nucleon is preserved. This is a conservation of color during this process of emitting and absorbing of a gluon.

Quarks inside a nucleon constantly exchange gluons, thus establishing a strong force that is responsible for the integrity of a nucleon.

While quarks exchange the color charges, a nucleon as a whole always remains color neutral, stable and observable.

Residual Strong Force

As we stated above, three quarks make up a nucleon, uud for a proton or udd for a neutron.
Inside a nucleon these quarks have different color charges (R, G and B) and exchange gluons to maintain their nucleon's color charge as a whole neutral (R+G+B = 0), exerting strong forces to assure a nucleon is stable and observable.

The question is, what keeps different nucleons together?

The answer, as we understand it today, is related to the same strong force and is the force called residual strong force.

Let's start with a process inside a proton, the particle #1 in an interaction we analyze and use a symbol p1 for it. Its three quarks uud constantly exchange gluons, which constantly exchanges their color charge.

(a) For some reason during this process a pair of quark d and anti-quark d are born as a result of these interactions.
p1(uud)p1(uud) + d + d

(b) The next step is a replacement of one quark u inside a proton with quark d born above, which effectively changes quark set from uud (proton) to udd (neutron) and releases quark u.
This released quark u combines with anti-quark d born above making a virtual (very short lived) π-meson (short name is pion) π(ud).
p1(uud) + d + d
n1(udd) + π(ud)

(c) Now a proton (particle #1) has transformed into a neutron and π-meson π(ud) that immediately contacted a neighboring particle #2 which is neutron n2(udd).
An anti-quark d of π-meson annihilates with quark d of this neutron (particle #2) leaving an empty spot in a neutron.
Quark u from π-meson takes an empty spot, effectively converting the whole particle into proton p2(uud).
π(ud) + n2(udd)p2(uud)

Analogous transformation of a particle #1, neutron, transforming into a proton, and, subsequently, a particle #2, proton, transforming into a neutron, is possible as well:
(a)n1(udd)n1(udd) + u + u
(b) n1(udd) + u + u
p1(uud) + π(du)
(c) π(du) + p2(uud)n2(udd)

The above process of transformation of a proton into a neutron and a neighboring neutron into a proton with corresponding exchange of quarks is the manifestation of residual strong forces that keep the nucleons (protons and neutron) together inside a nucleus.

That's why we observe neutrons inside any multiple protons nuclei, they are the results of constant transformation and interaction with protons, carriers of electric properties of an atom.
That's why the number of neutrons is equal (for lighter nuclei) or greater (for heavier nuclei) than the number of protons.

Obviously, residual strong forces on distances comparable with a radius of a proton should be stronger than electric repulsion between protons to keep the nucleus from disintegration.

## Saturday, February 18, 2023

### Quarks: UNIZOR.COM - Physics4Teens - Atoms - Elementary Particles

Notes to a video lecture on http://www.unizor.com

Quarks

Let's recall the steps of progress of our understanding of the structure of matter.

1. Splitting a drop of water or cutting the wire we gradually reached the smallest particle that retains the properties of an original object - molecule.

2. Then we realized that thousands of different molecules consist of different combinations of about a hundred of different atoms that have their own characteristics.

3. Later discoveries led to inner structure of every atom as consisting of a nucleus surrounded by shells filled with moving electrons.

4. Farther experiments proved that nucleus itself is a combination of protons and neutrons. So, three main particles - electrons, protons and neutrons - are the building blocks of all atoms.

5. Thousands of experiments showed the existence of other particles that differ from the main ones mentioned above. More than 200 particles have been discovered by physicists through numerous experiments. The number of these new particles made the picture of the structure of matter that we had in mind significantly less "structural" or, plainly, just messy.

It's time to go deeper into inner structure of all particles.
Are there any small number of smaller than proton or neutron particles, whose combinations make up already discovered particles inasmuch as about a hundred different atoms make up thousands of different molecules by grouping in different combinations?

The first theoretical answer was suggested in 1964 by Murray Gell-Mann and George Zweig. It was partially confirmed by experiments and, as a result, is accepted as a working model.

Their proposal was that protons and neutrons are not elementary particles, but, in turn, are composed of smaller particles called quarks.
The usefulness of such an approach can be demonstrated using an example from the previous lecture about isotopes.

Recall how isotope carbon-14 is formed.
Cosmic radiation hits the atoms in the upper layers of the atmosphere, breaking them into individual particles and producing free neutrons.
A free flying neutron hits an atom of nitrogen in our atmosphere, kicks off one proton, replacing it with itself, and frees up an electron.
Kicked off proton and an electron combine into a hydrogen causing the following reaction
714N + 01n → 614C + 11H

The process of carbon-14 decaying is rather complex, but can be described in an oversimplified form as follows.
A neutron in the nucleus of unstable carbon-14 transforms into a pair proton+electron. This does not change the electric neutrality of the atom, does not change the atomic mass, it remains 14, but atomic number increases by 1, thereby creating an atom of stable nitrogen.

This nuclear reaction can be described (in an oversimplified form) as
614C → 714N + e + ?
where ? signifies additional participants in this transformation that we cannot discuss at this point because it requires knowledge of other elementary particles beyond the main ones (electron, proton and neutron).

While a transformation of an atom of nitrogen into an atom of carbon-14 is physically easy to understand (a neutron kicks off a proton and electron, replacing and taking place of a proton in a nucleus), the reverse transformation of a neutron to proton and electron via decay is less understandable, as it seems to happen without material physical factors.

It would be more understandable if this transformation can be expressed as replacing of something with something else, as in the transformation of nitrogen into carbon-14 by replacing a proton with a neutron.

Developing this idea, physicists came to a model of neutron and proton based on some smaller elementary particles and a transformation of one into another as a replacing of one elementary component with another.

The first obstacle to overcome is electric charge of a proton and electric neutrality of a neutron. If these main particles have smaller elementary components, the electric charge must be distributed among them.
Taking electric charge of a proton as +1, its components must have it as a fraction.

Here is one way to implement it.
Assume, a proton is a combination of two elementary particles, X with an electric charge +½ each, and a neutron is a combination of a particle X with an electric charge +½ and a particle Y with electric charge −½.
Then the total charge of a proton (X+X) is +½+½=+1 and for a neutron (X+Y) it is +½−½=0 as it should be.

Consider a different model.
Particle X has charge +2/3 and particle Y has charge −1/3. Then a proton can be composed of X+X+Y with charge +2/3+2/3−1/3=1, while a neutron can be composed of X+Y+Y with charge +2/3−1/3−1/3=0.

The next characteristic to satisfy using this type of modeling protons and neutrons as consisting of smaller components is mass of a particle.
These components must have masses, sum of which is equal to corresponding larger particles. Actually, almost equal because of mass and energy relationship E=m·c².

What's more difficult is to come up with a model that satisfies not only charges and masses of protons and neutrons, but also other particles observed in experiments (antiprotons, antineutrons, pions, barions, mesons etc.).
Mathematically, it means to solve a system of many linear equations (as many as the number of particles we consider as consisting from the components times the number of parameters we have to match, like electric charge and mass, separately) with as few unknowns as possible.

The latest solution to this problem is so called Standard Model that proposed relatively few elementary (not divisible any more) particles with certain characteristics.

According to Standard Model, six different "flavors" of elementary (not divisible) particles called quarks are the building blocks of about 200 observable particles.
These different quarks are called Up (u), Down (d), Charm (c), Strange (s), Top (t) and Bottom (b).

Each quark has an electric charge (positive, negative or zero), mass and other characteristics.
Here is an electric charge table with the unit of charge 1 being associated with proton (+1) or electron (−1):
Up: +2/3
Down: −1/3
Charm: +2/3
Strange: −1/3
Top: +2/3
Bottom: −1/3

They are combined into different combinations, producing different particles.
For example, proton is made from two Up and one Down quark:
p = uud
Electric charge of proton is +2/3+2/3−1/3=+1, as it should.
Neutron is made from one Up and two Down quarks:
n = udd
Electric charge of a neutron is +2/3−1/3−1/3=0, as it should.

The process of transformation of a neutron into a proton and electron can be imagined as replacing one u quark with a d quark.
Where the new d quark comes from, how electron is created and where a released u quark goes to is a separate issue, which is beyond the scope of this lecture.

But complexity of the Standard Model is, actually, higher than this.
For each quark of the above six types there is an anti-quark.
Its electric charge is opposite to the corresponding quark.

In addition, quarks have a characteristic called color with values red, green and blue. These "colors" have nothing to do with the colors observable by our eyes, they are just used to characterize particular quarks or their combination.

Electron belongs to another group of particles, it's not constructed from other particles, it's an elementary particle. So are muons and tau particles.

Yet another group of elementary particles (not quarks) is responsible for carrying forces between particles (electrostatic, strong, weak etc.) One of the particles in this group is photon that carries electromagnetic force. Gluon is a particle responsible for strong forces inside a nucleus.

In a word, the contemporary model of the structure of particles is very complex and requires deep studying to understand it completely.

This complexity is well represented by the following illustration of the Standard Model authored by Chris Quigg and taken from the Web site https://www.quantamagazine.org

(to view all the details, click the right button and open it in a new tab)

## Sunday, February 12, 2023

### Isotopes: UNIZOR.COM - Physics4Teens - Atoms - Nucleus and Electrons

Notes to a video lecture on http://www.unizor.com

Isotopes

Isotopes are atoms that have the same number of protons (atomic number Z) but different number of neutrons (N).
Since the number of electrons and their distribution among shells and subshells are the same for different isotopes of atoms with the same atomic number, these isotopes have practically the same chemical properties.

At the same time, since the number of neutrons of different isotopes of atoms with the same atomic number is different, certain physical properties of these atoms might be quite different.

Identification of different isotopes involves specifying their atomic mass.
For example, carbon has 6 protons and 6 electrons, but the number of neutrons can be 6, 7 or 8 in its nucleus with atomic masses of these isotopes, correspondingly,
A=Z+N=6+6=12,
A=Z+N=6+7=13 and
A=Z+N=6+8=14.
These isotopes are named, according to their atomic mass, carbon-12, carbon-13 and carbon-14.

Another example: uranium-235 (symbol 92235U) with 92 protons, 92 electrons (atomic number is Z=92) and 143 neutrons (N=143), which has atomic mass A=235, and uranium-238 (symbol 92238U) with the same number of protons, electrons and the same atomic number Z=92, but with 146 neutrons (N=146) and atomic mass A=238.

In most cases one particular isotope of an atom is sufficiently stable and can be found in nature, while other isotopes of the same atom might be more or less radioactive, that is their atoms break with time into other (smaller) atoms - a process called radioactive decay.

The role of neutrons in a nucleus is to stabilize it. Since protons are positively charged and repel each other, neutrons serve as a buffer between them and, therefore, their number is usually equal or greater than the number of protons.
Obviously, so-called strong forces inside a nucleus that bind together all particles (protons to protons, protons to neutrons, neutrons to neutrons) are the main forces that hold a nucleus as a whole.

Interestingly, most nuclei with even atomic number Z and even number of neutrons N demonstrate more stability than those with either Z or N or both are odd.

As an example of the usage of isotopes, let's describe the determination of age of certain samples found by archeologists using radiocarbon dating.

The most stable and abundant isotope of carbon is the one with 6 protons and 6 neutrons in a nucleus - carbon-12 612C.
This carbon isotope plays extremely important role in organisms, as carbon is one of the main building blocks of all living beings on Earth.

Firstly, let's examine where carbon-14 comes from.
High energy cosmic rays bombard the Earth and break atoms of the upper layer of atmosphere, releasing some elementary particles, electrons, protons and neutrons.

Next, some neutrons (01n) hit the atoms of nitrogen (714N) in the air causing production of carbon-14 (614C) and hydrogen (11H) in the following nuclear reaction:
714N + 01n → 614C + 11H
The reaction above describes the process of a neutron hitting an atom of nitrogen, kicking out a proton and an electron that connect to each other in the form of an atom of hydrogen, while replacing a missed proton with itself, thereby reducing the atomic number by one, while retaining the atomic mass, which transforms it into an isotope carbon-14.

Now carbon-14 is formed in the air in some small quantity and participates in the cycle of life. Any living organism absorbs it the same way it absorbs regular carbon-12 during its life time and, therefore, it's present in some quantity inside this organism in proportion to regular carbon-12 similar to its proportion that exists in nature.

As we know, carbon-14 is not a stable element, it decays with a half-life about 5730 years.

The process of decaying is rather complex, but can be described in an oversimplified form as follows.
A neutron in the nucleus of unstable carbon-14 transforms into a pair proton+electron. This does not change the electric neutrality of the atom, does not change the atomic mass, it remains 14, but atomic number increases by 1, thereby creating an atom of stable nitrogen.

This nuclear reaction can be described (in an oversimplified form) as
614C → 714N + e + ?
where ? signifies additional participants in this transformation that we cannot discuss at this point because it requires knowledge of other elementary particles beyond the main ones (electron, proton and neutron).

While an organism lives, the relative amount of carbon-14 in it is maintained on the same level. Whatever is decayed is replenished from the outside world as part of the organism's existence.
As soon as life stops, absorption of carbon-14 stops and, whatever is present in the organism is no longer renewed, but decays according to the laws of half life.

If the amount of carbon-14 in the dead tree is 1/2 of whatever is normal, the tree died approximately 5730·1=5730 years ago.
If the amount of carbon-14 in the dead tree is 1/4 of whatever is normal, the tree died approximately 5730·2=11460 years ago.
If the amount of carbon-14 in the dead tree is 1/8 of whatever is normal, the tree died approximately 5730·3=17190 years ago.
etc.
In general, if the amount of carbon-14 in the dead tree is 1/N of whatever is normal, the tree died approximately 5730·log2N years ago.

## Saturday, February 11, 2023

### Main particles: UNIZOR.COM - Physics4Teens - Atoms - Nucleus and Electrons

Notes to a video lecture on http://www.unizor.com

Main Particles

Three particles constitute the main building blocks of any atom:
electrons,
protons and
neutrons.
Let's examine their properties and functions in maintaining the integrity of an atom.

The basic atom model assumes that electrons are negatively charged particles flying on some orbits around an atom's nucleus that, in turn, contains certain number of positively charged protons and electrically neutral neutrons.

We discussed in details the distribution of electrons in shells (#1, #2, #3 etc.) and subshells (s, p, d etc.) around a nucleus.
These electrons (primarily, those on higher orbits) get involved in chemical reactions among atoms, facilitating creation of different molecules by either ionic or covalent bonding (see previous lectures on this topic).

The number of protons in a nucleus should be equal to the number of electrons circulating on orbits around this nucleus to maintain atom's neutrality.
This number is called an atomic number Z of an atom.

The electrostatic attraction between positive protons and negative electrons keeps the electrons on their stationary orbits.
But electrostatic forces repel from each other particles charged the same way (positive or negative). So, what keeps the protons in the nucleus and holds the atom's integrity?

Apparently, elementary particles inside a nucleus are held together by other forces, much stronger than electrostatic. They are called (not surprisingly) strong or nuclear forces.
These strong forces act only on a very small distance between particles inside a nucleus and have no noticeable influence on electrons around a nucleus.

Strong forces exist between protons, between protons and neutrons and between neutrons. Since neutrons are electrically neutral, greater number of them in the nucleus keeps the nucleus stronger, preventing electrostatic forces or external forces (like bombarding the nucleus with other particles) to break up an atom.
The total number of protons (Z) and neutrons (N) is called atomic mass of an atom (A=Z+N).

Nucleus of an atom takes a very small amount of space relatively to an atom's size.
For example, a nucleus of an atom of hydrogen, which consists of only one proton, has a diameter of the order of 10−15 meter, while a diameter of an atom of hydrogen is more than 100,000 greater (of the order of 10−10 meter).
Heavier elements that contain hundreds of particles (protons and neutrons, commonly called nucleons) in a nucleus have larger diameter of a nucleus, about 10 times larger than hydrogen, and their atoms are thousands times greater than their nuclei.

At the same time, the mass of an atom is concentrated in its nucleus. The nucleus' share in the mass of an atom is, approximately, 99.9%. The rest of mass is the mass of electrons.
These very approximate numbers do not take into consideration the effect of mass-energy relationship, according to Einstein Theory of Relativity formula E=m·c².

Since chemical properties of an element are defined by its electrons, the atomic number (that is, the number of electrons or equal to it the number of protons) defines chemical properties of an atom.
Different number of neutrons with the same number of protons in a nucleus of an element can occur, but these different compositions of a nucleus do not change the chemical properties of an element.
Atoms with the same number of protons but different number of neutrons are called isotopes.
We will address it in the next lecture.

## Thursday, February 9, 2023

### Valence Electrons: UNIZOR.COM - Physics4Teens - Atoms - Interaction of A...

Notes to a video lecture on http://www.unizor.com

Valence Electrons

Previous lecture was about chemical reaction between elements - bonding between atoms that does not affect their nuclei.
We have seen that the most important role in these reactions is played by electrons in the outer most shell of an atom.

In particular, chemical reactions are expected between elements that have incomplete outer subshells, but complementary to each other to form complete subshells by either borrowing electrons from each other, forming electrovalent or ionic bonding between positively and negatively charged ions or sharing electrons, forming a covalent bonding between atoms.

Recall the chemical bonding between atoms of sodium and chlorine.
Sodium has atomic number 11, and its electrons are arranged as
Na: 1s2 2s2 2p6 3s1
Chlorine has atomic number 17, and its electrons are arranged as
Cl: 1s2 2s2 2p6 3s2 3p5

The mechanism of bonding can be represented in such a case as follows.
1. Sodium atom releases the only electron from the subshell 3s, becoming a positive ion
Na → Na+ + e
2. Chlorine atom captures this electron in its subshell 3p, becoming negative ion
Cl + e → Cl
3. Now all subshells are filled to capacity and both atoms are electrically charged with opposite charges, which makes them stick to each other, forming a molecule of salt.

That was a typical example of electrovalent or ionic bonding.

Valence electrons are those exact electrons that are capable of participating in bonding with other electrons of other atoms.
In most cases they belong to the outer electronic shell of an atom.

Elements with electron structure ending in 1 electron in the outer subshell s, like sodium (Na), have that 1 electron as the only valence electron. The valency of such elements is 1. Atoms of these elements often bond by giving away this electron to complete other atom's outer subshell, like ionic bonding with chlorine (Cl) that needs exactly one electron to complete its outer p subshell. The valency of the counterpart atom in such a case is also 1.
The structure of a molecule that results in bonding of these two atoms can be symbolically expressed as
Na−Cl
where a single bar between symbols of elements corresponds to the valency of each.
Considering this is an ionic bonding, we might as well use more precise symbolics
Na+−Cl

Elements with electron structure ending in 2 electrons in the outer subshell s, like barium (Ba), have these 2 electrons as the only valence electrons. Atoms of these elements often bond by giving away both of these electrons to complete other atom's outer subshell, like ionic bonding with two atoms of chlorine (Cl), each of which needs one electron to complete its outer p subshell, forming a molecule BaCl2.
The structure of a molecule that results in bonding of these three atoms can be symbolically expressed as
Cl−Ba−Cl
where bars correspond to the valency of each atom, 2 for barium and 1 for chlorine.

This process can be described as follows.
1. Barium atom (atomic number 56) frees two electrons from the outer subshell 6s, becoming a positive ion
Ba → Ba2+ + 2e
2. Two chlorine atoms capture one electron each in their outer subshell 3p, becoming negative ions
2Cl + 2e → 2Cl
3. Now all subshells are filled to capacity and three atoms are electrically charged. One atom of barium is positive, two atoms of chlorine are negative, which makes these two atoms of chorine stick to an atom of barium, forming a molecule of barium chloride BaCl2.

So, more precise representation of this molecule would be
Cl−Ba+−Cl

Let's consider a more complicated case of forming aluminum oxide - a molecule combining atoms of aluminum (Al) and oxygen (O).

The configuration of 13 electrons in the atom of aluminum is:
Al: 1s2 2s2 2p6 3s2 3p1
The configuration of 8 electrons in the atom of oxygen is:
O: 1s2 2s2 2p4

As we see, aluminum has one valence electron in subshell 3p. If it loses this electron, its 3rd subshell would still have two electrons in subshell 3s. While technically complete, it's still exposed to be captured by some other atom.

So, an atom of aluminum can lose either (only under special conditions) 1 electron, the 3p1, or (under normal conditions) 3 electrons, the 3p1 and 3s2.
The valency of aluminum is, therefore, either 1 (only under special conditions) or, typically, 3.

A possible molecular bonding between 2 atoms of aluminum and 1 atom of oxygen Al2O that can be established in special laboratory conditions would look like this:
Al−O−Al
This is a very unstable molecule because the 3rd shell of the atom of aluminum with only one remaining subshell 3s and two electrons in it still remains exposed.

Much more stable molecule is formed, when 2 atoms of aluminum contribute 6 electrons (3 electrons from each atom, emptying subshells 3s and 3p) and 3 atoms of oxygen capture them (2 electrons are captured by each oxygen atom to fill its 2p subshell). The formula of this aluminum oxide molecule is Al2O3 with the structure that can be pictured as
O=Al−O−Al=O
where bars correspond to the valency of each atom, 3 for aluminum and 2 for oxygen.

From the position of an exchange of electrons this can be represented as follows.
1. Two aluminum atoms free 3 electrons each from their outer shell #3 (1 electron from subshell 3p and 2 electrons from subshell 3s), becoming positive ions
2Al → 2Al3+ + 6e
2. Three oxygen atoms capture 2 electrons each in their outer subshell 2p, becoming negative ions
3O + 6e → 3O2−
3. Now all subshells are filled to capacity and all 5 atoms are electrically charged. Two atoms of aluminum are positive, three atoms of oxygen are negative, which makes these three atoms of oxygen stick to two atoms of aluminum, forming a molecule of aluminum oxide Al2O3.

The atoms in this molecule are bonded ionically, so we can represent the molecule's structure as
O2−=Al3+−O2−−Al3+=O2−

As you saw from the previous example, in some cases the valency of an atom can be different under different circumstances. While for a bonding between aluminum and oxygen only one stable configuration is observed, in other cases more than one configuration occurs under normal conditions.

The common property of the examples above is that we analyzed the molecules comprised of a metal and non-metal.
Metals have their outer electrons relatively easier departing from their nuclei, that's why in all cases above the bonding between a metal and non-metal was ionic, when metals gave their electrons to non-metals.

As a final example, consider the formation of a molecule of carbon monoxide CO.
The electron configuration of carbon (atomic number 6) is
C: 1s2 2s2 2p2
The configuration of 8 electrons in the atom of oxygen, as we already discussed, is:
O: 1s2 2s2 2p4

None of the elements in a molecule of carbon monoxide is a metal, so their electrons are not capable, ready and willing to depart from the their nuclei, as in metals. As a result, the electrons are not transferred from atom to atom forming ions, but are shared between the atoms to form a covalent bonding.

Two electrons from 2p subshell of carbon and four electrons from 2p subshell of oxygen, totaling 6 electrons (exactly the maximum number for 2p subshell), become a common property of both atoms, completing their corresponding outer subshells 2p.

The act of sharing electrons between atoms of carbon and oxygen is the glue holding the molecule together in covalent bonding.

## Thursday, February 2, 2023

### Chemical Bonding: UNIZOR.COM - Physics4Teens - Atoms - Interaction of Atoms

Notes to a video lecture on http://www.unizor.com

Chemical Bonding

From the atomic viewpoint, classical chemical bonding between atoms are interactions that do not change the composition of these atoms' nuclei.
Only electrons that surround nuclei interact among themselves, breaking some old connections and establishing new ones.

Chemical bonding can occur between atoms of simple elements, like a single atom of sodium Na and a single atom of chlorine Cl forming their compositions, a molecule of sodium chloride NaCl, which is a molecule of regular salt.

Scientists thought about why some elements bond with each other, if mixed in certain proportion and at some external conditions, while some others are inert, that is refuse to combine with other elements. So called noble gases, like helium, neon, argon or krypton are inert.

Let's examine the electron configuration of above mentioned noble gases.

Helium with atomic number 2 has the following electron configuration
He: 1s2

Neon with atomic number 10 has the following electron configuration
Ne: 1s2 2s2 2p6 =
= [He] + 2s2 2p6

Argon with atomic number 18 has the following electron configuration
Ar: 1s2 2s2 2p6 3s2 3p6 =
= [Ne] + 3s2 3p6

Krypton with atomic number 36 has the following electron configuration
Kr: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 =
= [Ar] + 3d10 4s2 4p6

As you see, all subshells of noble gases are filled to capacity:
max(s) = 2
max(p) = 6
max(d) = 10
and this is very important for stability of atoms.

Those elements that do not have this characteristic of its subshells will be easier to get into chemical bonding that facilitates sharing electrons to complete top most subshells of all involved components.

Let's examine the chemical process between elements using the above example of producing sodium chloride from sodium and chlorine
Na + Cl → NaCl

First of all, let's analyze the electronic structure of both substances.
Sodium has atomic number 11, and its electrons are arranged as
Na: 1s2 2s2 2p6 3s1
Chlorine has atomic number 17, and its electrons are arranged as
Cl: 1s2 2s2 2p6 3s2 3p5

As you see, the first subshell s of the shell #1 (that is, subshell 1s), which can hold up to 2 electrons, is completely filled for both elements, and there can be no more subshells within this shell #1.

The first subshell s of the shell #2 (that is, subshell 2s) is also completely filled for both elements, but shell #2 can have two subshells, s and p, and subshell p has capacity for 6 electrons. As we see, both 2p subshells are filled to capacity in both elements.

The first subshell s of the shell #3 (that is, subshell 3s) has only 1 electron in an atom of sodium.
The atom of chlorine, on the other hand, has this subshell filled to capacity of 2 electrons, but the next subshell 3p has only 5 electrons out of maximum 6.

Now the sharing of electrons comes to play.
The top subshell of sodium 3s contains 1 electron out of maximum 2. If it loses this electron, all its subshells will be complete, but losing electron means becoming positively charged.
The top subshell of chlorine 3p contains 5 electrons out of maximum 6. If it captures one electron, all its subshells will be complete, but capturing an electron means becoming negatively charged.

The mechanism of bonding can be represented in such a case as follows.
1. Sodium atom frees the only electron from the subshell 3s, becoming a positive ion
Na → Na+ + e
2. Chlorine atom captures this electron in its subshell 3p, becoming negative ion
Cl + e → Cl
3. Now all subshells are filled to capacity and both atoms are electrically charged with opposite charges, which makes them stick to each other, forming a molecule of salt.

Let's make a few conclusive statements about this process of forming molecules.

I. Atoms like their subshells to be completely filled up to a maximum of 4m−2, where m is the subshell number (subshell #1 is labeled s, subshell #2 is labeled p etc.)

II. Some atoms are ready to accept electrons to fill up their top subshell from other atoms that are willing to give up their electrons.

III. Some other atoms are ready to give up electrons from their incomplete top subshell, leaving with themselves only complete subshells, if there is a recipient atom of these extra electrons that needs them to complete its top subshell.

IV. The necessity for an atom to have its shells filled up is very strong. Atoms with incomplete top subshells are looking for partners to get into chemical combination. If there is a fit between two atoms to transfer electrons from an incomplete top subshell of one of them to incomplete top subshell of another that completes both, electrons are transferred.

V. After the transfer of electrons that completes both atoms, they become oppositely charged ions that are attracted to each other, which establishes a stable combination.

VI. The complex combinations of more than 2 atoms can also be created based on the same principle. The mechanism is a little more complex, but follows the same idea to complete top subshells of all participants.

## Monday, January 30, 2023

### Energy Levels: UNIZOR.COM - Physics4Teens - Atoms - Electronic Structure...

Notes to a video lecture on http://www.unizor.com

Energy Levels

In the previous lecture we have addressed how electrons are positioned around nucleus.
We mentioned that, according to Bohr's model, they are only located on stationary orbits within shells and subshells.

We had a nice formula for the maximum number of electrons that can be located on each subshell, depending on its order number, traditionally designated by a letter (subshell #1 is designated letter s, the next #2 is p, then d, f, g etc.) For subshell #m the maximum number of electrons on it is 4m−2.
The number of subshells within shell #N is N.
Consequently, the maximum number of electrons held in each shell is 2·N².

This mathematically perfect picture gets more complicated, when we consider the energy level associated with each subshell.
The main principle of distribution of energy among subshells, that is, the greater radius of a subshell - the greater energy electrons within this subshell possess, is absolutely true.
The problem is, the further we go from the nucleus - the closer to each other are the shells and, subshells of greater radius of a shell #N might overlap with subshells of smaller radius of shell #N+1.

Let's consider a few elements in a sequence of their atomic numbers, that is the number of protons in their nucleus or electrons within shells on different levels.

Hydrogen has only one proton in the nucleus and one electron within the first subshell (designated by a letter s) of the only shell #1.
So, the electron structure of the atom of hydrogen, where the number specifies the shell number and a letter specifies the subshell, where the superscript at the subshell letter indicates the number of electrons within the same subshell, is:
1s1

Helium has two protons in the nucleus and two electrons within the first subshell (designated by a letter s) of the only shell #1.
The subshell s (the first one) can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of helium is:
1s2

Lithium has three protons in the nucleus and three electrons.
The first (and only) subshell s of the shell #1 can hold only 2 electrons. Therefore, a new shell #2 should exist and one electron should be in its first subshell s.
The subshell s of the second shell can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of lithium is:
1s2 2s1

Let's skip a few elements and consider carbon with atomic number 6.
Its 6 electrons should fill the shell #1 with its only subshell s holding 2 electrons, then the shell #2 with its subshell s holding another 2 electrons, then, considering shell #2 has two subshells, s and p, the second subshell p will hold the remaining 2 electrons.
The subshell p, as the second subshell, can hold maximum of 6 electrons, so we still have room.
The electronic structure of an atom of carbon is, therefore,
1s2 2s2 2p2

Skip a few more elements to silicon with atomic number 14.
We already know that the first shell can hold 2 electrons in one and only subshell s.
The second shell can hold 2 electrons in subshell s and 6 electrons in subshell p.
That totals 10 electrons. To accommodate 4 other electrons the third shell should be used.
Its subshell s will hold 2 electron and subshell p will hold remaining 2 ones.
The electronic structure of silicon is
1s2 2s2 2p6 3s2 3p2

So far, increasing in atomic number is synchronous with sequential filling of shells and subshells within shells.
But let's consider element argon. Its atomic number is 18 and it completely fills one subshell s of the first shell, both subshells of the second shell and two (out of three) subshells of the third shell.
Its electronic structure is
1s2 2s2 2p6 3s2 3p6

The next element is potassium with atomic number 19.
It would be reasonable to expect that, since the second shell's last subshell p is already filled as well as two first subshells of shell #3, and the third shell allows three subshells, the extra electron #19 should be in the third subshell d (that is the third subshell's letter) of the shell #3.
However, the experiments showed that the last electron #19 goes to the fourth shell's first subshell s instead. Why?

Here we face the fact that shells overlap, and higher energy level of the third subshell d of shell #3 exceeds the lower energy level of the first subshell s of shell #4.
That's why the location of electron #19 of potassium is not 3d1 but 4s1.
That is the reason why the electronic structure of potassium is
1s2 2s2 2p6 3s2 3p6 4s1

The order of filling the electronic subshells, as the number of electrons in the atom grows, under normal conditions corresponds to an order of increasing their energy levels. This is known as the Aufbau principle or Madelung or Klechkovsky rule.

According to this principle, and taking into consideration that shells get closer and closer to each other as we move away from a nucleus, while the number of subshells is increasing linearly with the shell number, the order of filling the electronic structure of an atom is not the numerical order of shell/subshell numbers, but as follows: 1s→2s→2p→3s→3p→4s→
→3d→4p→5s→4d→5p→6s→
→4f→5d→6p→7s→5f→
→6d→7p→...

In the lecture "Orbiting Electron" of the chapter "Building Blocks of Matter" of this course we have derived a formula for a total energy of an orbiting electron:
E = −k·e²/(2·r)
where
k is Coulomb constant
e is the electric charge of an electron
r is the radius of an orbit.

From the above formula we see that absolute value of the energy is decreasing with increasing of a radius of an orbit, but, since it's negative, the energy itself is increasing.
As a radius of an orbit of an electron is increasing, its (negative) energy is increasing, getting asymptotically closer to zero. At the same time the number of subshells of each shell is increasing with the shell number.
That explains the overlapping energy characteristic of the shells.

## Thursday, January 26, 2023

### Electrons and Shells: UNIZOR.COM - Physics4Teens - Atoms - Electronic St...

Notes to a video lecture on http://www.unizor.com

Electrons and Shells

First of all, there are some very complicated theories of the electronic structure of an atom. They are experimentally confirmed and, therefore, are considered as true representation of how electrons are arranged around a nucleus.
These theories are above the level of this course, but certain facts based on them we will mention without any proof.

Recall the Bohr's model of an atom and electrons that are supposed to be on stationary orbits (or, rather, within stationary shells), where they do not emit any energy.

Shells of a larger radius can hold more electrons, and electrons in those higher orbit shells have higher level of energy.

Shells are numbered in order of increasing radius as shell #1, shell #2, ...shell #N.

Any shell can have one or more subshells. The number of subshells in each shell corresponds to this shell's number, that is:
shell #1 has 1 subshell,
shell #2 has 2 subshell,
shell #3 has 3 subshell,
shell #4 has 4 subshell,
shell #5 has 5 subshell,
shell #6 has 6 subshell,
shell #7 has 7 subshell,
etc.

For historical reasons subshells within each shell are not enumerated, but rather assigned a letter. The first 4 subshells are called s, p, d and f, then the letter is assigned in alphabetical order.
So, the first few subshell names are:
s for subshell #1,
p for subshell #2,
d for subshell #3,
f for subshell #4,
g for subshell #5,
h for subshell #6,
i for subshell #7,
etc.

To bring a bit of math into this structure, let's use letter N for a shell's number and letter m for a subshell number within a shell.
Using these symbols, we can state the following:
shell #N has N subshells:
#1, #2,...,#N
.

Each subshells has its own capacity to hold electrons. There are theoretical reasons for this based on Quantum Theory. We just state the result of this theory confirmed by experiments:
subshell #m has 4·m−2 electrons.

So,
subshell #1 (s) has 2 electrons,
subshell #2 (p) has 6 electrons,
subshell #3 (d) has 10 electrons,
subshell #4 (f) has 14 electrons,
subshell #5 (g) has 18 electrons,
subshell #6 (h) has 22 electrons,
subshell #7 (i) has 26 electrons,
etc.

Now let's calculate the maximum number of electrons in each shell.

Shell #1 has 1 subshell #1(s) and, therefore, can hold no more than 2 electrons.

Shell #2 has 2 subshells #1(s), #2(p) and, therefore, can hold no more than 2+6=8 electrons.

Shell #3 has 3 subshells #1(s), #2(p), #3(d) and, therefore, can hold no more than 2+6+10=18 electrons.

These calculations can be generalized in a formula for the maximum number of electrons in shell #N:
Σm∈[1,N](4m−2) = 2·N²

This formula can be easily proven by induction.
Indeed, it's correct for N=1 because
4·1−2 = 2 = 2·1²
Assuming the formula is correct for some number N, let's check it for N+1.
Σm∈[1,N+1](4m−2) =
=
Σm∈[1,N](4m−2)+4(N+1)−2 =
= 2·N² + 4·(N+1)−2 =
= 2·N² + 4·N + 2 =
= 2·(N² + 2·N + 1) =
= 2·(N+1)²

which is the same formula, but for N+1.

## Monday, January 23, 2023

### Nucleus of Atom: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks of...

Notes to a video lecture on http://www.unizor.com

Nucleus of Atom

In 1897 J.J.Thomson, experimenting with cathode ray tubes, came to a conclusion that atoms contain tiny negatively charged particles.
He had demonstrated that cathode rays consist of negatively charged particles - electrons.
He then suggested a plum pudding model of an atom with these tiny negatively charged particles embedded into positively charged "soup".

In 1911 Rutherford experimented with a beam of positively charged particles (called alpha particles) directed toward a thin foil. He found that some particles go through a foil, while some are reflected back.
As a result, he came up with a planetary model of an atom with positively charged nucleus and negatively charged electrons rotating around a nucleus on a substantial (relatively to a size of a nucleus) distance, so an atom is substantially empty.

In 1917 Rutherford proved experimentally that nucleus of hydrogen atom is present in atoms of all substances he dealt with.
Later on the nucleus of hydrogen atom was called proton - another building block of an atom together with electron.

A few years later Rutherford suggested that another type of particle - an electrically neutral combination of tightly coupled together proton and electron, called by him neutron - must be present inside an atom's nucleus. While this hypothesis was not confirmed experimentally, the term "neutron" remained and used later on.

In 1932 James Chadwick discovered electrically neutral particles emitted from beryllium bombarded by alpha particles. These electrically neutral particles, in turn, were used to bombard paraffin wax and liberated hydrogen nuclei - protons.
That was a proof of existence of electrically neutral neutrons.

Right after that, in 1932, Dmitry Ivanenko and Werner Heisenberg proposed a proton-neutron structure of an atom's nucleus. This model of a nucleus together with electrons outside of a nucleus moving within stationary shells, each with a specific energy level, remains as the main atom's model.

The number of protons in a nucleus of any element under normal conditions is equal to the number of electrons around a nucleus to maintain electric neutrality of an atom. The properties of any element very much depend on this number and, actually, characterize the element's properties. This number is called an atomic number of an element.

The atomic number of a hydrogen is 1, its nucleus contains 1 proton and there is 1 electron outside a nucleus.
The atomic number of a helium is 2, its nucleus contains 2 proton and there are 2 electrons outside a nucleus.
The atomic number of a carbon is 6, its nucleus contains 6 proton and there are 6 electrons outside a nucleus.
The atomic number of a gold is 79, its nucleus contains 79 proton and there are 79 electrons outside a nucleus.
The atomic number of a uranium is 92, its nucleus contains 92 proton and there are 92 electrons outside a nucleus.

The number of neutrons inside a nucleus is also very important, but can vary for the same element.
Thus, a nucleus of an atom of hydrogen, besides one proton, can have no neutrons, one neutron and two neutrons. Their atomic numbers are the same. To differentiate them, another characteristic is used - a sum of the number of protons and the number of neutrons called atomic mass.

Different kinds of the same element with different numbers of neutrons (that is, different atomic mass) are called isotopes. So, there are three isotopes of hydrogen - with atomic masses of 1 (1 proton, no neutrons), 2 (1 proton, 1 neutron) and 3 (1 proton, 2 neutrons).

Isotopes of the same element have close but not identical properties. To fully identify an element, including its isotope, the following notation is used.
The element is identified by its abbreviated Latin name.
For example,
He for helium,
Au for gold ("aurum" in Latin),
Fe for iron ("ferrum" in Latin) etc.

To fully identify an element, in-front of this abbreviated name two indices are used: top for atomic mass (sum of the numbers protons and neutrons) and bottom for atomic number (the number protons).
Examples:
24He for helium
(2 protons and 2 neutrons),
92238U for uranium
(92 protons and 146 neutrons),
79197Au for gold
(79 protons and 118 neutrons),
2656Fe for iron
(26 protons and 30 neutrons).

Positively charged nucleus with certain number of protons keeps the same number of negatively charged electrons inside an atom because opposite charges attract.

Why a nucleus is held together, when similarly charged protons are lumped together and repel each other?

Apparently, there other attractive forces acting specifically on very small distances between particles inside a nucleus, which are stronger than repelling electrical forces. These forces are called nuclear or strong forces.
They are short range forces of attraction between any particles inside a nucleus, protons and neutrons (collectively called nucleons), and they are millions of times stronger than electric forces. They keep nucleons together inside a nucleus.

### Bohr's Atom Model: UNIZOR.COM - Physics4Teens - Building Blocks of Matter

Notes to a video lecture on http://www.unizor.com

Bohr's Atom Model

In the previous lecture we briefly mentioned the planetary model of an atom, authored by Ernest Rutherford in 1911 and supported by many physicists at that time. We also mentioned two fundamental problems with this model.

Firstly, a purely theoretical problem with planetary model of an atom was related to the fact that an electron rotating around a nucleus, that is going with centripetal acceleration, should produce oscillations of an electromagnetic field and, therefore, is supposed to lose energy. This would cause its falling onto a nucleus, which destroys the fundamental structure of matter.

Secondly, the spectrum of radiation produced by an electron falling onto a nucleus had to be continuous, which contradicted experimental results that showed discrete spectrum.
Experiments showed that gases emit light when exposed to intensive electric field. The electric field supplies energy to electrons, and they increase the radius of their orbits. Then they spontaneously release this energy as visible light (electromagnetic field oscillations in visible spectrum of frequencies) and lower their orbits. This light, going through a prism, produces distinct spectral lines of monochromatic light specific for each gas and independent of an intensity of the electric field applied to it, gas temperature or density.
This consistency of spectral lines had to be explained and planetary model failed to do it.

In 1900 Max Plank, based on his experiments with radiation caused by heat, has suggested that radiation is carried in chunks, and each chunk has an amount of energy E that depends only on the frequency f of this radiation:
E = h·f, where
h=6.62607015·10−34 m²·kg/s is Planck's constant

At the same time, analyzing the process of photoelectric emission, Albert Einstein used the idea of quantum character of the electromagnetic oscillations to introduce a photon as an indivisible unit of absorbed or emitted electromagnetic energy.

Combining the theoretical knowledge and results of experiments, Niels Bohr suggested a new atom model that seemed to be capable of explaining all the experimental results on a new theoretical foundation.

The Bohr's atom model was an enhancement of the Rutherford's planetary model and repeats its geometrical configuration of a central positively charged nucleus and orbiting around it negatively charged electrons.
The main modification to that model was using a quantum character of energy carried by electromagnetic field oscillations.

The development of quantum concepts of electromagnetic energy was a collective effort of famous physicists Planck, Lorentz, Einstein, Haas, Nicholson and others. Bohr used these concepts, applying them to formulate his model of an atom.

Bohr's model of an atom is based on these main principles:

I. For each type of an atom there are certain stable electron orbits (or shells), called stationary, where electron, as long as it stays on such an orbit, emits no energy.
This proposition contradicts the classical theory of electromagnetism, which states that accelerated electron must emit energy.

II. Each stationary orbit is associated with certain level of energy. The larger the radius of an orbit of an electron - the higher energy it possesses.
Energy is absorbed by an electron, when it jumps from a lower energy shell to a higher energy one.
Energy is emitted by an electron, when it jumps from a higher energy shell to a lower energy one.
The amount of energy absorbed or emitted by an electron when it jumps from one shell to another equals exactly the difference in energy levels of these shells.

III. When electron jumps from an orbit of higher energy level Ehi to an orbit of lower energy level Elo, it emits electromagnetic radiation of frequency f, determined by an equation
Ehi − Elo = h·f
where h is Planck's constant.
Obviously, to jump from an orbit of lower energy level Elo to an orbit of higher energy level Ehi, electron absorbs this amount of energy from outside.

IV. The next principle is more complicated and was formulated by Bohr as follows.
The angular momentum of an electron rotating on a stationary orbit equals to an integer multiple of reduced Planck constant ħ=h/(2π) (Latin letter h with a horizontal stroke).
This principle is known as quantization of angular momentum.
As we know, a momentum of a body of mass m moving along a straight line with speed v equal to m·v.
An angular momentum L of a body of mass m uniformly rotating with linear speed v along a circular trajectory of radius r equals to m·v·r.
So, this principle of Bohr's model can be expressed in an equation
L = m·v·r = n·ħ
where n is a positive integer number and ħ is a reduced Planck's constant.

This was Bohr's hypothesis given based on some experimental facts and certain theoretical derivations from them.
At the same time it corresponded to experimentally obtained formula for radii of electron orbits of an atom of hydrogen suggested by Rydberg (see previous lecture on Rydberg Formula).

The theoretical explanation of this last Bohr's principle that quantizes the angular momentum was suggested later on by de Broglie in 1924.

Contemporary explanation, based on duality of a particle and a wave, can be shortened to the following.

The full energy E of a particle of mass m, using the Theory of Relativity, can be expressed as
E = m·c²
where c is the speed of light.

From the Quantum Theory the energy E of a quantum of light (photon) of frequency f and wave length λ is
E = h·f = h·c/λ
where h is Planck's constant.
Therefore,
m·c² = h·c/λ
m·c = h/λ

Expression p=m·c is a momentum of a particle of mass m moving with speed c.
Therefore,
p = h/λ

When an object of mass m rotates with linear speed v along an orbit of radius r. its angular momentum is
L = p·r = h·r/λ

Electron, rotating around a nucleus, from the wave theory, is analogous to a string fixed at both ends, like on a guitar. The wave length of a sound this string produces must fill the length of a string integer number of times, otherwise it will interfere with itself.
Using this principle, the wave length of an electron λ and a radius of its orbit must be in a relation
2π·r = n·λ
where n any positive integer number.

Using the above, we obtain
L = h·r/λ =
= h·r·n/(2π·r) =
= h·n/(2π) = n·ħ

where
n is any positive integer number and
ħ is a reduced Planck constant.

### Orbiting Electrons: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks...

Notes to a video lecture on http://www.unizor.com

Orbiting Electron

Let's analyze the dynamics of an electron rotating around a nucleus of a hydrogen atom on a circular orbit.

Considering the strength of electric forces significantly exceeds the strength of gravitational forces, we will ignore the gravity between a nucleus and an electron.

Assume, the mass of an electron is m (it's a known constant), its negative electric charge is −e (also a known constant) and a radius of its orbit is r (variable).

An atom of hydrogen has only one negatively charged electron. Therefore, to maintain electric neutrality, its nucleus has to have positive electric charge equal in magnitude to a charge of an electron, that is, e.

The Coulomb's Law states that the magnitude of the force of attraction between a nucleus of an atom of hydrogen and its electron equals to
F = k·e·e/r² = k·e²/
where k is a Coulomb's constant.

On the other hand, according to Rotational Kinematics, that same force gives an electron a centripetal acceleration
a = v²/r
where v is a linear speed of an electron circulating around a nucleus.

Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of an electron, its charge and mass:
F = m·a = m·v²/r = k·e²/
or
m·v²·r = k·e²

The direct consequence of this equation is an expression for a kinetic energy of an electron, as a function of a radius of its orbit:
Ekin = m·v²/2 = k·e²/(2·r)

Potential energy of a negative charge e in the centrally symmetrical electric field of a nucleus carrying a positive charge e of the same magnitude (that is, work needed to bring a charge −e from infinity to a distance r from a central charge +e) is
Epot = −k·e²/r

Total energy of an electron is, therefore,
E = Ekin + Epot = −k·e²/(2·r)

### Satellite Speed: UNIZOR.COM - Physics4Teens - Mechanics - Gravity, Weight

Notes to a video lecture on http://www.unizor.com

Satellite Speed

Our task is to find out a linear speed V of a satellite that freely rotates around a planet of mass M on a circular orbit of radius R.

The gravity force, acting on a satellite of mass m and keeping it on a circular orbit with constant angular and linear speed, according to the Law of Universal Gravitation, equals to
F = G·M·m /
where G=6.67·10−11(N·m²/kg²) is a Universal Gravitational Constant.

On the other hand, according to Rotational Kinematics, that same force gives a satellite a centripetal acceleration
a = V²/R

Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of a satellite and mass of a planet:
F = m·a = G·M·m /

Using the expression of centripetal acceleration above, this results in the following:
m·V² /R = G·M·m /

Notice that mass of a satellite m cancels out and the resulting expression for a satellite linear speed on an orbit is
V² = G·M /R
V = √G·M/R

The above formula allows to calculate the period T of rotation of a satellite - the time required to make a complete circle around a planet:
T = 2πR/V = 2π√R³/(G·M)