## 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)

## Wednesday, January 18, 2023

### Rydberg Formula: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks of...

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

Rydberg Formula

Rydberg formula is a perfect example of how science is done.
It all started in 1880's with numerous experimental results of spectral lines of hydrogen, emitted after its atoms are excited by some external energy, like electric field or heat.
These spectral lines had certain wave lengths observed through experiments.

Johann Jacob Balmer attempted to connect the wave lengths of observed spectral lines of hydrogen with some kind of empirical formula and found the one:
λ = B·n²/(n²−2²)
where
λ is a wave length of an observable spectral line,
B=3.6450682·10−7 m is a constant that Balmer has suggested,
n ≥ 3 is a sequence number of a spectral line.

Here are a few first wave lengths λ and colors of Balmer series for different sequence number n
 n λ (nm) Color 3 656.1 Red 4 486.0 Cyan 5 433.9 Violet 6 410.1 Violet 7 396.9 Violet 8 388.8 Violet 9 383.4 Violet

Jumping forward, with introduction of Bohr's model with specific stationary electron orbits with fixed energy levels for each orbit of an atom, it was apparent that Balmer has described electron emitting radiation when jumping from some higher orbit to orbit #2.

When electron jumps from any higher orbit to orbit #1 (the closest to nucleus), the emitted radiation is in ultraviolet part of a spectrum and was not observed by Balmer.

A few years later Johannes Rydberg generalized the Balmer formula and described any jump of an electron in a hydrogen atom from orbit #n to orbit #m:
1/λ = R·(1/m² − 1/n²)
where
n ≥ 2 is an orbit number an electron jumps from,
1 ≤ m ≤ n−1 is an orbit an electron jumps to,
R is Rydberg constant.
For m=2 the Rydberg formula is
1/λ = R·(n² − 2²)/2²·n²)
or
λ = (4/R)·n²/(n² − 2²)
which corresponds to Balmer formula if B=4/R.

All the above formulas are empirical, obtained in the process of analyzing the results of experiments. The theoretical foundation of them would be known only after Bohr introduced his atom model in the beginning of the 20th century and his model would undergo certain improvements based on quantum physics.

## Tuesday, January 10, 2023

### History of Atom Model: UNIZOR.COM - Physics4Teens - Atoms - Building Blo...

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

Early History of Atom Model

Let me mention one important fact about Physics in general.
All our statements about Nature and its inner working only represent our model, the fruit of our mind. This model is based on our experience at a particular time and space and seems to correspond this experience to a high degree of precision.

As time passes, we enrich our experience with more facts and observations, which will necessitate updating our model.
No model, no physical law we came up with are absolute and final. Everything is a subject of constant improvement, update, even complete rewrite.

In this chapter we will present a simplified view on what atoms are, which, to a certain degree, can be confirmed experimentally.
By no means this view is the true picture of what atoms really are. It's an approximation to an as high degree as possible at current level of our knowledge, and it will change, with every iteration being closer and closer to what atoms and their components really are.

A drop of water is water.
If we split a drop of water into two smaller drops, each small drop is water.
How small a drop of water can be to retain the properties of water?
Can we divide a drop infinitely, and the smaller and smaller part of it would still be water?
No, there is a limit.

The smallest part of a drop of water that retains the properties of water exists. It has a name - molecule. It has a size and a weight. We cannot divide it further into smaller parts that still retain the characteristics of water.

How about some other kind of matter, like salt?
The same thing. We can divide a salt into smaller and smaller pieces until we reach a limit - a molecule of salt.
Dividing it further will produce something which is not salt anymore.

Every kind of matter that has certain properties is divisible into smaller pieces until we reach the smallest one that retains the original properties of this kind of matter - a molecule of this particular kind of matter.

A small complication to this picture is dealing with a combination of different kinds of matter.
Consider salt dissolved in water - a salty water.
This substance consists of two kinds of molecules mixed together. If we start dividing it, we will, for awhile, have smaller and smaller amounts of salty water. Eventually, we will reach a point that some small amount of substance we deal with is either a molecule of salt or a molecule of water.
Still, we reach the level of molecules, but in this case two different kinds of molecules have been mixed together. The principle of the smallest possible amount of substance that retains its properties is molecule.

How many different molecules exist? Too many to really count with certainty. Some big, some small, some heavy, some light. With all different properties, solid, liquid, gaseous, red, white, transparent etc.

The fundamental question is: "Are there much smaller particles than molecules of a very limited number of types, from which all the molecules consist, differing only in what kinds of these smaller particle participate in a construction of a molecule, what is their number and configuration, as they are organized in a molecule?"

The first, as we know, people who attempted to answer this question positively were Greek philosophers Leuccipus, Democritus and those close to them, who lived about 2400 years ago.
They have proposed that all kinds of matter consist of small indivisible particles. It was Democritus, who named these particles a tomos, literally "indivisible".
From our perspective their concept of atomos would correspond to our current concept of a molecule.

Aristotle and Plato criticized many of the details of the ideas presented by Democritus and his school of thought. Primarily, the ideas that a human soul also has atomic structure was unacceptable for them as too mechanical.

After a long gap in philosophical and scientific development related to a structure of matter, experimental science made its first steps with such famous people as Galileo in 17th century.

In 18th century Sir Isaac Newton published his views on atoms similar to that of Democritus.

In the 19th century John Dalton has formulated his view on matter as consisting of individual indivisible atoms. Atoms are specific for each element, all atoms of a particular element are identical to each other and, according to Dalton, any chemical combinations of atoms preserve the total mass. He also suggested that any molecule, however big and complex, consists of certain (not very big) number of atoms of individual elements.
The latter was very significant, because the multitude of thousands of different chemical compounds was reduced to combinations of relatively small number of elements, each of which consists of identical atoms.

At the end of 19th century it became obvious that atoms are not indivisible. Sophisticated experiments conducted by Thomson allowed him to discover a small negatively charged electron as a particle inside an atom.
Knowing the electric neutrality of the atom, Thomson suggested later on that negatively charged electrons are distributed within a positively charged substance - the "plum pudding" model of an atom with electrons resemble plums inside a positively charged "pudding".

Some experiments with radioactive uranium compounds showed that so-called α-particles emitted by this compound go through a thin metallic foil. That brought Ernest Rutherford to an idea of planetary model of an atom, based on Coulomb's Law and Newton's Law, with central positive nucleus and circling around it negative electrons, thus leaving a lot of free space in-between for α-particle to penetrate the space occupied by atoms.

However, the planetary model had a few very important problems. Electrons moving along a circular trajectory around a nucleus move with centripetal acceleration and, as was explained in the part "Waves" of this course, emit energy in the form of electromagnetic oscillations. That would necessitate the gradual fall of these electrons onto a nucleus, that is a destruction of atom, which was not observed.
Another problem surfaced when physicists examined light emitted by gas between electric contacts. This light always had certain spectral lines specific for a particular gas, regardless of the intensity of electricity used to force the emitting of light.
This needed some theoretical explanation, and the planetary model did not provide it.

A few years late, in the beginning of 20th century Albert Einstein suggested that light delivers its energy in units called photons that we spoke about in the previous part of the course, "Waves".
The amount of energy E carried by one photon is proportional to a frequency f of electromagnetic waves:
E=h·f
where h is Planck's constant.

Ernest Rutherford discovered a positively charged particle inside an atom called protons. The positive protons neutralized negative electrons to make an atom electrically neutral. That was the time when physicists realized an important role of electricity in holding an atom together.

This was further developed by Niels Bohr into an atom model (called Bohr's model) with certain stable (not emitting energy) fixed shells around a nucleus specific for each element and having fixed energy levels. Electrons jumping from one such shell to another emit an energy specific for this particular jump, so each element has only specific energy amounts it can emit, thus specific colors of a spectrum of emitted light.
Jumping from a shell with higher energy level Ehi to a shell with lower energy level Elo, electron emits an energy Ehi−Elo, which causes to emit light of frequency f that satisfies the equation
h·f = Ehi−Elo
that is, with frequency equal to
f = (Ehi−Elo) / h

Next step was accomplished by James Chadwick discovering neutron - another particle in the atom's nucleus composition.

All the discoveries made by that time allowed to build a consistent model of an atom that we will be analyzing in this chapter of the part "Atoms" of the course.

## Saturday, January 7, 2023

### Holography: UNIZOR.COM - Physics4Teens - Waves - Holography

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

Holography

Holography is a technique that allows creation of three-dimensional (3D) images of objects, holograms, by recording and later on displaying all the light reflected from these objects.

Ability to record and display all the light reflected from these objects is what differentiates the hologram from a photograph, which records only the light reflected by an object towards the camera.
That is the main distinction between a holographic 3D image and a flat two-dimensional (2D) photograph.

The following two pictures reflect the above distinction.
The top picture is a schematic representation of the formation of a photographic image with only light beams reflected from an object towards a camera lens get recorded.
The bottom picture represents the main feature of a holographic image with all light beams reflected from an object towards any small area of a recording screen being recorded within this small area, representing a view of an object from this particular viewpoint. When we see a real object with our eyes, we can see it in 3D space because our two eyes see it at different angles and we can move the eyes to look at different parts of an object, every time changing the angle of observation.

The previous lecture discussed the method of creating the holographic image of a point light. This principle is used to record and recreate an image of any object.

As was explained in the previous lecture, a point light holography is based on interference of two lights - one monochromatic light from a point light itself (all rays are emitted in phase) and another one, as a set of parallel rays of light of the same wave length and in phase with one central ray emitted by a point light.
That was sufficient to create an interference picture on a screen that contained all information needed to recreate an image of a point light, when it's no longer there, by the same set of parallel rays.

Consider any point on a surface of any object. If some monochromatic light, called illumination beam, is directed on it, it will reflect this light in all directions similar to a monochromatic point of light we talked about before.
Therefore, with the help of a set of parallel lights of the same wavelength, called reference beam, we can record the location of this point on a screen similar to the one we used for a point light hologram.

Since the illumination beam illuminates all surface points of an object, all of them will be represented on a screen as an interference picture between two beams.
Next we process the screen surface to develop this interference picture.
To recreate all these points, we will use the same reference beam directed towards a screen, and interference between this beam's rays and rays reflected by a screen.

To produce both beams, illuminating and reference, that should be coherent (same wavelength and phase) we can use partially reflecting mirror that splits the same beam into two separate ones, one for illumination and another as a reference.
Schematically, it can be arranged as on a picture below. (you can click the right mouse button and open this picture in another tab for better view)

Since every point of an object, reflecting an illumination beam, can be viewed as an independent point light, its holographic image will be created on a screen exactly the same way as described in the previous lecture about point light hologram.
This will be sufficient to recreate each such point by the same reference beam at exactly the same point in space, when the object is no longer there.

The above description of holography is very basic, without any technical details and is intended only as a an explanation of a main concept behind this technology.
Issues like color and moving objects are outside the scope of it and require much more technical details addressed by professionals.

### Point Light Hologram: UNIZOR.COM - Physics4Teens - Waves - Holography

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

Point Light Hologram

This lecture is about an idea that is fundamental to understand the holography.
We will explain how a position of a point light (a point in space that emits radial rays of light) can be recorded and, later on, presented, when it's no longer there.

What is discussed in this lecture is a kind of a "thought experiment", but, properly modified, it can be actually implemented, which the progress of holography has achieved.

Assume, we have a point source of monochromatic light of wave length λ.
Our first task is somehow record its position in space relatively to some recording media (a screen).
The second task, when the source of light is no longer there, is to recreate an image of this point light, like a bright point in space, positioned relatively to a screen in the same spot as the original source of light.

The picture below proposes a solution to the first task.
The point source of monochromatic light (orange star S) sheds light (radial orange rays SA, SB and others) onto some flat screen AB, which will be our recording medium.
Assume, SAAB.
The distance between a point light S and a screen AB is SA=d.

We assume that all the rays issued by a source S have the same wave length λ and are issued in phase, that is, coming from the source S, they are coherent.

At the same time a set of parallel monochromatic light rays of the same wave length λ (red rays on the picture), also in phase, are directed perpendicularly to screen AB.
NOTE: the radial light rays are orange and parallel are red only to differentiate them on a picture below, their real color depends on the wave length that is the same for all of them. (click right mouse and open this picture in another tab to see it in full format)

What we will observe on screen AB is the interference picture between two sets of light rays - radial and parallel.

Let's analyze what kind of interference picture we will see on the screen AB.

To simplify this analysis without jeopardizing the logic, assume that both radial and parallel rays that come to point A are in phase and, therefore, constructively interfere with each other, making point A a bright spot on the screen.

Consider any point B on a screen on a distance x from A. Point B will be bright if the radial ray SB and the corresponding parallel ray coming to point B are in phase.
Since all parallel rays are in phase among themselves, and a parallel ray coming to point A is in phase with corresponding radial ray SA, we will have a bright spot at point B if rays SB and SA are in phase, that is
SB − SA = n·λ
where
n is any non-negative integer number and
λ is the wave length (the same for radial and parallel rays).

Incidentally, for any fixed n all bright points B on a flat screen such that SB−SA=n·λ form a circle centered at point A. So, for different n we will see different concentric bright circles, all centered at point A. They are called Newton's rings.
Below is a sample of a real interference picture that can be observed on a screen in a setting similar to the one described above. From the properties of the right triangle ΔSAB, considering SA=d and AB=x, follows
SB² = d² + x²

Therefore,
SB − SA = n·λ = √d² + x² − d
From this follows:
(n·λ + d)² = d² + x²
x² = (n·λ+d)² − d²

The above is a condition for point B on the distance x from the midpoint of a screen A to be a bright spot.

Assigning an integer variable n different values allows to find distances xn of bright spots (more precisely, radii of bright concentric circles) on a screen by resolving the above equation for x.
This results in the following expression for xn as a function of distance d between a point light S and a screen and non-negative integer number n:
xn = √(n·λ+d)² − d²

For n=0 the distance of the bright spot from the midpoint A will be
x0 = 0, that is, it will coincide with midpoint A.
For n=1 the bright spot will have a distance from the midpoint A
x1 = √(λ+d)² − d²
For n=2 the bright spot will have a distance from the midpoint A
x2 = √(2λ+d)² − d²
etc.

An interesting math exercise would be to find how close to each other will be bright circles on a screen.
In other words, how xn+1−xn behaves as index n increasing.

It can be shown that this distance is monotonically diminishes from its maximum value
x1 = √(λ+d)²−d² = √λ²+2λd
(that is greater than λ) to the wave length λ, as index n increases.
That is,
limn→∞(xn+1−xn) = λ

As an example, let's calculate the radii x1, x2 and x3 of the first three bright circles around midpoint A for the monochromatic green light of 500 nanometers wave length and distance d between a source of light S and screen AB of 1 meter.
Let's use micrometers, also known as microns (μm) as units of distance, with 1000nm=1μm and 1,000,000μm=1m.

x1 ≅ 31.6 μm
x2 ≅ 44.7 μm
x3 ≅ 54.8 μm

Therefore, the distance between x0=0 and x1 is 31.6 microns, from x1 to x2 it's 13.1 microns, from x2 to x3 it's 10.1 microns.

To see the diminishing distance between concentric bright circles, let's calculate x97, x98 and x99.

x97 ≅ 315.2 μm
x98 ≅ 316.9 μm
x99 ≅ 318.5 μm

The distance between x97 and x98 is 1.7 microns, from x98 to x99 it's 1.6 microns.

As you see, the bright circles are getting closer to each other as we observe them at a longer distance from the midpoint A.
As n→∞, this distance will get closer and closer to 0.5 microns - the wave length of the monochromatic green light we used in this experiment.

So, we have a pretty good idea of the interference picture developed on a screen.
This interference picture is very specific to a position of our point light S. So, if we will be able to record this interference picture on a screen, there is a possibility to recreate the position of point light S.

Assume, our screen is made of transparent glass. Now, knowing locations of all bright circles on a screen, let's paint in black all the dark areas on a screen (they are circular too), the bright circles remain transparent.
Now our screen represents the recording of the interference picture, which is very specific for a location of the point light S.
Our first task to record the position of a point light is completed. We can take away now both sources of monochromatic light (radial and parallel rays on a picture above).

The second task is to recreate the point light at a location where it was relative to a screen using the information recorded on a screen (concentric black and transparent areas).

Let's take the source of parallel monochromatic rays used before and transfer it behind the screen, so it shines parallel rays of light perpendicularly to a surface of a screen from behind, as presented on a picture below. (click right mouse and open this picture in another tab to see it in full format)
Obviously, only the rays that hit the transparent circles at radii x0, x1, x2, x3 etc. from a screen's midpoint, where the bright spots used to be, will go through a screen. These rays are presented as red and have arrows at the end.
The others (brown without arrows) will be stopped by a screen.

According to Huygens principle, each ray of light that goes through transparent areas of a screen is a source of secondary waves emitted radially at an exit from a screen, as presented on a picture above.
All these secondary light rays will interfere with each other. But, if we consider only the rays going from each transparent area of a screen to a point S, where the point light used to be, they all will be in phase and, therefore, constructively interfere with each other.

Indeed, the difference in distance from the midpoint A to point S and from a circle of radius x1 to point S is exactly the wave length λ.
The difference in length from the midpoint A to point S and from a circle of radius x2 to point S is two wave lengths .
Therefore, the point S will be bright, since rays from each transparent circle will come to it in phase and intensify each other at this point.

What's important is that all rays that go through a screen will constructively or destructively interfere with each other in all other points of space as well, but point S will definitely stand out as the brightest spot because significantly more light rays (actually, as many as the number of transparent circle) will come to it in phase.

If we understand the main principle of recreating a point source of monochromatic light, let's try to make this principle more practical.
Of course, we cannot paint black circles on a screen with a precision required for this experiment to actually implement the process.
What we can do, however, is to cover a screen with photosensitive layer containing silver compound that, being exposed to light at the bright spots on an interference image, makes only these spots reflective.

To create an image of a point light at location S, instead of directing the parallel light rays from behind a screen, we will direct them from the front, they will shine in the same direction as before. Reflected by the reflective circular silver areas on a screen, they will interfere with each, recreating a bright spot exactly at point S, where the original point light used to be.
That's how our second task, recreating an image of a point light, can be accomplished.