Bohr-Sommerfeld model-chemistry learners

 Sommerfeld atomic model, relativistic correction, hydrogen alpha transition

The Universe always mesmerizes us with full of mysteries. It poses exciting questions to humans whenever one solves one question. Even today, it grabs our attention to unsolvable mysteries. And it excites science lovers with complex puzzles. One such curious puzzle is the appearance of fine structures in the hydrogen alpha spectral line of the Balmer series. The Bohr-Sommerfeld model solved it.

Atom is the fundamental constituent of the Universe. Hence, the study of atomic structure has much significance in spectroscopy that solved the complexities of astronomy.

Today's blog article discusses the drawbacks of the Bohr atomic model. And it depicts Sommerfeld's extension to overcome the limitations of the Bohr atomic theory.

The Sommerfeld atomic model

A brief introduction to the Sommerfeld atomic model

Bohr's atomic model interpreted the electronic structure in the atom with stationary energy levels. It solved the enigma of hydrogen atomic spectra with quantized photon emissions. Moreover, he observed a single spectral line for one electron transition. But, the advent of a high-power spectroscope showed a group of fine lines in the hydrogen atomic spectrum. Have a look at the postulates of the Bohr's atomic model.

Also, Stark and Zeeman's research on the influence of electric and magnetic fields on spectral emission lines supported the appearance of hydrogen spectral line splitting. 

Even though Bohr's atomic model succeeded in calculating electron energy and radius of circular orbits, it could not explain the reason for the spectral line splitting.

In 1916, Sommerfeld extended Bohr's atomic model with the assumption of elliptical electron paths to explain the fine splitting of the spectral lines in the hydrogen atom. It is known as the Bohr-Sommerfeld model.

Elliptical orbits illustration of the Bohr-Sommerfeld model

According to Sommerfeld, the nuclear charge of the nucleus influences the electron motion that revolves in a single circular path. Hence, the electron adjusts its rotation in more than one elliptical orbit with varying eccentricity. And the nucleus is fixed in one of the foci of the ellipse.

The ellipse comprises a major (2a) and a minor (2b) axes. When the lengths of major & minor axes are equal, the electron's orbit becomes circular. Hence, the circular electron path is a remarkable case of Sommerfeld's elliptical orbits.

Additional reference:

A short PDF notes on Bohr-Sommerfeld atomic model

What is Zeeman effect?

Drawback's of the Bohr's atomic model

The Bohr-Sommerfeld model tried to solve the following limitations of Bohr's atomic model.

1. It failed to explain the spectral splitting into fine lines.
2. It did not explain the effect of magnetic and electric fields on spectral emissions. So, it could not explain the Zeeman and Stark effect.
3. It is silent about the relative intensities of the emitted spectral lines.
4. It contradicted the De-Broglie dual nature of matter and Heisenberg's uncertainty principle.
5. Last but not least, it succeeded in explaining the spectra of mono-electron species. But it failed to clarify the spectra of multi-electron atoms.

Overview of Bohr-Sommerfeld model

In Bohr's atomic model, the electron revolves around the nucleus in stationary circular orbits. And we all know the radius of the circle is a constant quantity. Hence, the distance of the electron from the central nucleus does not change with the electron's movement. But, the angle of rotation changes with time. 

Bohr circular electron path explanation

In Sommerfeld extension, the path of the electron's rotation is an ellipse with a major and a minor axes of 2a & 2b lengths. Out of the two foci on the major axis, the nucleus locates in only one of them.

With orbiting, both the distance of the electron and the rotation angle will vary. Hence, the permitted elliptical orbits deal with these two varying quantities.

• The change in distance of the electron (r) from the fixed focal nucleus
• The variation in the angular position (φ) of the electron orbiting the nucleus 

Sommerfeld elliptical paths explanation

So, he felt that two polar coordinates are essential to describe the location of the revolving electron in the ellipse. They are radial and angular coordinates corresponding to momenta p_r and p_φ, respectively. The Wilson-Sommerfeld quantum condition amounted to integrals for each coordinate over one complete rotation.

The Wilson-Sommerfeld quantum conditions

Where,

p_r = radial momentum

p_φ = angular momentum

r = radial coordinate

φ = angular coordinate

n_r = radial quantum number

k= angular or azimuthal quantum number

h= Planck’s constant

Bohr expressed the electron's energy with the principal quantum number n. Its value varies from 1 to infinity.

But, Sommerfeld described that a portion of electron energy is associated with its orbital motion. To enumerate the orbital angular momentum, he introduced a new quantum number named azimuthal quantum number. The letter 'k' denotes it. And its value varies from 1 to n, where n is a principal quantum number.

The relationship between the principal and azimuthal quantum number is below

The relationship between the principal and azimuthal quantum numbers

Where,

n= principal quantum number

n_r = radial quantum number

k= azimuthal quantum number

Hence, the total principal quantum number is the sum of the radial and azimuthal quantum numbers of the electron rotating in the elliptical paths.

According to Bohr-Sommerfeld theory, the electron's energy depends primarily on the principal quantum number. But, to some extent, on the azimuthal quantum number also. So, the electron transition from one energy level to the other gives slightly different energies due to the possible values of k in the two transition states. It explains the reason for the occurrence of a group of fine lines in the hydrogen spectrum under a refined spectroscope.

The Lyman alpha spectral line splitting

The energy associated with the orbital angular momentum of the electron gives rise to sub-energy levels in the atom. The value of k gives the electron's location in the sub-energy state. The symbols s, p, d, f, and so on represent them.

For example- the 1s-orbit represents the electron's position in the s-subshell of the first main energy level.

Similarly, the 2p-orbit shows the electron's location in the p-subshell of the second main energy level.

The value of the principal quantum number determines the number of subshells in it. For any given value of n except 1, the azimuthal quantum number k has more than one value.

For example- 

• For n=1, k=1. It shows only one subshell in the first main energy level, such as 1s.
• The n=2 shows two subshells, with k values 1 and 2 representing the 2s and 2p levels.
• The n=3 shows three sub-energy levels 3s, 3p, and 3d with k values 1, 2, 3.

The sub-energy levels explanation with the values of the azimuthal quantum number

From the above table, for any principal quantum number with value n, there exists 'n' number of elliptical orbits. And they are known as sub-energy levels. Among them, one subshell is circular. And the remaining (n-1) subshells are ellipses with varying eccentricities.

Have a look at this Quora answer for additional knowledge;

 Why is the circular orbit an exceptional condition for Bohr-Sommerfeld atomic model?

Sommerfeld's possible elliptical electron orbits explanation

Besides, the electron transitions to main and sub-levels of energy are the reason for the spectral splitting.

Eccentricity is the deviation of the elliptical shape of orbit from circularity. The symbol ‘ε’ denotes it. The relationship between the eccentricity and the azimuthal quantum number is below;

The relationship between the eccentricity and the azimuthal quantum number

The eccentricity of an elliptical orbit is the ratio of the lengths of minor and major axes. Any variation in their values changes the eccentricity of the elliptical orbit.

Additional reference:

What is spectrum?

What do you mean by the hydrogen spectral lines?

Necessary conditions:

Case-1: When k=n, then b=a.

It implies that if the lengths of both major and minor axes are equal, then the orbit must be circular.

A circular electron orbit depiction

Case-2: When k<n then b<a

It is the usual scenario of the ellipse. The minor axis length is always less than the major axis length. An important point to consider here is the smaller the value of k increases the eccentricity of the orbit. In case the k value decreases, the ε value increases.

For example

In the below diagram, the elliptical orbit eccentricity decreases with the k value increase. When k=1 and n=4, the orbit is highly elliptical. And the eccentricity decreases with the change in the k value from 1 to 3. At k=n=4, the path of the electron is circular. 

Sommerfeld elliptical orbits for principal quantum number four

Case-3: When k=0

The k value cannot be equal to zero. k=0 means b=0. So there is no minor axis in the ellipse. The k=0 indicates the linear motion of the electron that passes through the nucleus.

Hence, the k value can never be zero for an ellipse. It is a non-zero positive integer with values ranging from 1 to n.

Additional reference:

How is the hydrogen spectrum formed?

Examples of Bohr-Sommerfeld model

Example-1:

For n=1, k has only one value which is k=1. When both n=k=1. It is a circle with a single subshell in the first main energy level.

The circular orbit explanation for the principal quantum number one

Example-2:

For n=2, k has two values, such as k=1 and k=2.

For n=2 and k=1, 

Formula calculating the equation for k=1 and n=2

In this ellipse, the length of minor axis is equal to half the length of the major axis.

For n=2 and k=2,

Formula calculating the equation for k=2 and n=2

We have, b=a. It is a circle.

Elliptical orbit depiction for the principal quantum number two

Example-3:

For n=3, k has three values such as k=1, k=2, and k=3.

For n=3 and k=3

It is a circle with b=a condition

For n=3 and k=2

Formula calculating the equation for k=2 and n=3

It is an ellipse with minor axis 0.6 times less than the major axis

For n=3 and k=1

Formula calculating the equation for k=1 and n=3

It is an ellipse with minor axis 0.3 times less than the major axis

Elliptical orbit depiction for the principal quantum number three

To conclude,

1. The circular orbit is a special condition for the Sommerfeld model.
2. The eccentricity of the orbit is high for a smaller k value. When the orbit broadens, the length of the minor axis increases. This situation continues till n=k. And at b=a, the orbit becomes circular.

Comparison of Bohr-Sommerfeld angular momenta:

The Sommerfeld atomic model considered the planetary motion of the electron as relativistic in elliptical pathways surrounding the nucleus. His envision has successfully explained the fine structures of the hydrogen spectrum.

Since the spectral emissions relate to the electron orientation in orbicular configurations that are established around the central core. He was bound to contemplate the angular momentum of the electron in the distorted circular electron paths. So, he revived Bohr's angular momentum condition and replaced the principal quantum number with a new quantum entity.

As the electron path is no more circular, he put forward a new angular momentum condition in terms of the azimuthal quantum number to account for eccentricity. And he replicated Bohr's angular momentum condition for circular orbits allowing only those stationary electron paths in which the angular momentum is a whole number multiple of reduced Planck's constant. This quantized angular momentum condition proved the discrete orbicular paths' existence for the spinning electron.

According to Bohr, the electron's angular momentum is an integral multiple of reduced Planck's constant and is quantized.

Bohr angular momentum condition

Where,

n= principal quantum number

h= Planck’s constant

ħ = reduced Planck’s constant

Similarly, Sommerfeld explained the quantization of orbital angular momentum with the reduced Planck's constant. And he addressed it as a quantum condition for the eccentricity.

Sommerfeld angular momentum condition

Where,

k= azimuthal quantum number

h= Planck’s constant

ħ = reduced Planck’s constant

Finally, the quantized orbital angular momentum allows only discrete elliptical orbits. Electrons incur only those allowed orbits in which angular momentum is an integral multiple of reduced Planck’s constant. He called those discrete elliptical orbits “quantization of ellipses.”

Sommerfeld tried to explain the number of allowed transitions for hydrogen alpha spectral lines in the Balmer series. We will discuss allowed electron transitions in the selection rule section at the end of this article. But for now, the concept of quantized orbital angular momentum helped Sommerfeld to calculate the number of fine lines possible for the hydrogen spectrum.

Additional reference:

What was Bohr's justification for angular momentum?

Bohr-Sommerfeld energy calculation:

Neil Bohr calculated the electron's energy in the hydrogen atom, which is in good agreement with the experimental values.

Bohr's energy equation

Sommerfeld derived an expression for the energy of the electron orbiting in the elliptical orbit.

Sommerfeld's energy equation

Where,

m = mass of the electron

Z= atomic number of the atom

e = charge of the electron

ε₀ = permittivity of free space

h= Planck’s constant

n = principal quantum number

The quantities m, Z, e, h, ε0 are constant in the above two equations. Besides, the electron's energy depends on the principal quantum number in both Bohr and Sommerfeld models, irrespective of the nature of the orbits. Not to mention, Sommerfeld's energy equation relies only on the principal quantum number without considering the azimuthal quantum number.

For this reason, the energy of the elliptical and circular orbits remains the same. All the main and sub-energy levels of the atom were degenerate in Sommerfeld’s energy equation. So, the electron transitions to these subshells did not bring any variation in the energy of the emitted photon. Hence, it was unable to explain the splitting of the spectral lines based on the energy of the orbits. Moreover, the Sommerfeld elliptical orbit explanation could not explain the fine structures of the hydrogen atom.

Relativistic correction to Sommerfeld's model

In Bohr’s atomic model, the electron's velocity is much less than the speed of the light. Hence, its motion is non-relativistic. The electron's velocity does not change with its mass at different parts of the circle.

But, in the Sommerfeld model, he assumed that the electron travels at nearly the speed of the light. Hence, its motion is relativistic. Moreover, the velocity of the electron moving in the elliptical orbit is different at the various parts of the ellipse. And it causes a relativistic variation in the electron’s mass. He explained the relativistic variation of the electron’s mass with the below formula.

The formula for the relativistic electron's mass

Where,

m = relativistic mass of the body

m₀ = rest mass of the body

v= velocity of the body

c= velocity of light

Why is the electron's velocity vary in the relativistic Sommerfeld model? 

To explain the relativistic motion of electron, Sommerfeld considered two points on the ellipse, namely aphelion and perihelion. The aphelion point is farther away from the focal nucleus. And the perihelion point is closest to the nucleus.

Relativistic electron motion depiction

Sommerfeld explained the velocity of the electron is minimum at the aphelion point. And it is maximum at the perihelion point. The reason for this variation follows Coulomb’s law.

According to Coulomb’s law, the magnitude of the coulombic force varies inversely with the distance between the two electrically charged bodies.

The Coulomb's law

When the electron approaches close to the nucleus, the electrostatic attraction force among them increases. The electron moves with a higher velocity to counteract it. Hence, the velocity of the electron is maximum at the perihelion point.

Similarly, when the distance between the nucleus and the electron increases, the coulombic attraction force decreases. So, the electron moves with a much less speed at the aphelion.

Since the motion of the electron is relativistic, the mass of the electron also varies at the aphelion and perihelion parts of the ellipse.

Sommerfeld's relativistic energy correction term

The considerable variation in the electron’s velocity on the elliptical orbit added a new relativistic correction term to the total energy of the electron. Now, the modified Sommerfeld’s energy equation is below.

The formula for Sommerfeld's modified energy equation

If you observe this equation, you can understand that the electron's energy not only depends on the principal quantum number but also on the azimuthal quantum number. This correction brought a variation in the energy of the elliptical orbits. Now, the elliptical orbits are non-degenerate. Additionally, the elliptical orbit close to the nucleus has higher energy than the one far away from it.

The increasing order of energies for the sub-energy levels is s<p<d<f.

The electron transitions to energy levels that have a slight difference in energies will show spectral line splitting. The splitting of the spectral lines into two or more components with a mild variation in their wavelengths is known as fine structures. Hence, the electron’s energy dependence on both the principal and azimuthal quantum numbers explained the reason for the appearance of fine structures of the hydrogen atom.

Sommerfeld found that the relativistic effect is higher for elliptical orbits with higher eccentricity. The elliptical orbit with a smaller k value is more eccentric. The relativistic effect is more prominent in those elliptical orbits when they have a huge difference in their n and k values.

Additional reference:

What is the fine structure of a hydrogen atom?

Relativistic effect and the path of the electron:

Sommerfeld’s relativistic explanation of the electron’s motion changed the path of the electron from a simple ellipse to a more complicated rosette structure.

The Rosette path of the orbiting electron

In rosette, the nucleus locates consistently at one focus. The electrons move in elliptical paths with a change in their semi-major axis length. The angle through which the semi-major axis of the ellipse shifts is equal to

The formula for the precession of ellipse.

The above equation represents the precession of perihelion during one orbit. And the precession is a change in the orientation of the rotational axis of the rotating body.

The varying elliptical motion of the electron with different eccentricities is known as a precessing ellipse. And it is a function of the time.

Selection rule;

A selection rule describes whether a particular electron transition from one quantum state to another state is allowed or forbidden.

Bohr’s atomic theory explained the electron transitions in the form of hydrogen spectral series. The names of those six hydrogen spectral series are the Lyman series, Balmer series, Paschen series, Brackett series, Pfund series, and Humphreys series. They are governed by the selection rule that Δn =1, 2, 3, …, etc. Only the electron transitions involving principal quantum states are allowed when their difference is a non-zero positive integer.

But with the advent of high-resolution spectroscopes, the scientists observed additional fine spectral lines in the hydrogen spectrum. Bohr’s theory could not explain these additional fine structures of the hydrogen atom.

Later, Schrodinger’s quantum mechanics put forward some selection rules to account for the additional spectral fine structures of the hydrogen atom. Those selection rules are;

1. Δn = any positive integer such as 1, 2, 3,…, etc.
2. Δk= ±1

Here n is principal quantum number and k is azimuthal quantum number.

The electron transitions that obey the above two selection rules are said to be allowed electron transitions.

The electron transitions that violate those selection rules are known as forbidden electron transitions.

Only the allowed electron transitions give spectral fine lines in the spectrum. It worked well with the practical observations of the hydrogen spectrum. Hence, it is widely accepted. 

Additional reference:

What are the six series of the hydrogen spectrum?

Explanation for hydrogen alpha fine structures:

It is the first line that occurs in the Balmer series of the hydrogen spectrum. Hence, the initial Greek symbol α denotes it. It is symbolized as H-α.

It is a deep red colored spectral line that occurs in the visible region at 656.28 nm in air. And it is the brightest hydrogen spectral line in the hydrogen visible spectrum. The electron transition from the third stationary orbit to the second energy level of the hydrogen atom gives this hydrogen-alpha spectral line. Due to the small energy difference between these two second and third stationary orbits, hydrogen-alpha spectral lines occur at a longer wavelength with the least energy emission in the Balmer series. So, we can see it at the end of the visible region of the electromagnetic spectrum.

As mentioned previously, the electron transition from the third stationary orbit to the second energy level gives a hydrogen-alpha line. The third stationary orbit’s principal quantum number value is three (n=3). So, the azimuthal quantum number has three values such as k=1, k=2, and k=3. It implies the third stationary orbit splits into three sub-energy levels with a slight variation of energy.

The second stationary orbit has the principal quantum number value (n=2) two. The azimuthal quantum number has two values, such as k=1 and k=2. So, the second main energy level has two sub-energy levels with slightly different energies.

The total number of possible electron transitions between the second and third energy levels is six (3X2=6).

But Sommerfeld found that all these electron transitions are not allowed. The allowed electron transitions can be decided based on the selection rule. According to the selection rule, the integral electron transitions with Δk value equal to ±1 are allowed. The remaining electron transitions are forbidden.

Now, let us write all the six possible electron transitions of the hydrogen alpha line. They are 

3→2

3→1

2→2

2→1

1→2

1→1

The left-hand side value represents the final azimuthal quantum number (k₂) of the electron. Similarly, the right-hand side value represents the initial azimuthal quantum number (k₁) of the electron.

The formula for the change in the azimuthal quantum number

So, let us subtract the azimuthal quantum number values to find the allowed electron transitions.

3-2=1

3-1=2

2-2=0

2-1=1

1-2=-1

1-1=0

The allowed electron transitions are;

3→2

2→1

1→2

The forbidden electron transitions are;

3→1

2→2

1→1

Out of those six electron transitions, three are allowed electron transitions and the remaining three are forbidden electron transitions.

The hydrogen alpha allowed and forbidden transitions interpretation

Hence, it is clear that the hydrogen alpha spectral line splits to give three hydrogen fine structures. It matches closely with the observations under refined microscope. So, the Sommerfeld atomic model successfully explained the cause for the appearance of the fine structures of the hydrogen atom with his relativistic model.

Additional reference:

What is the hydrogen-alpha spectral line?

What was the first discovered series of the hydrogen spectrum?

Achievements of the Sommerfeld’s atomic model:

1. Sommerfeld’s elliptical orbits concept proved the existence of stationary electronic orbits of the atom as proposed by Neil Bohr.
2. His relativistic electron velocity theory successfully explained the fine structures of the hydrogen spectrum.
3. He introduced azimuthal and magnetic quantum numbers to explain the position and the energy of the electron and the shape of the atomic orbitals.
4. He revived the old space quantization concept by quantizing the z-component of the angular momentum. It helped to explain the deviation of elliptical orbits from circularity.
5. He introduced the concept of quantum degeneracy which accounts for the energy levels splitting of the atom with a slight variation in their energies.
6. The last but most important significance of the Sommerfeld atomic model is the introduction of the fine structure constant to understand the amount of splitting of spectral lines. It was widely accepted as a fundamental constant.

Limitations of the Sommerfeld’s atomic model:

1. It could not predict the exact number of fine structures possible for a single spectral line.
2. It did not explain the anomalous Zeeman Effect.
3. It is silent about the intensities of spectral lines of the hydrogen spectrum.
4. It did not explain the spectra of multi-electron atoms.
5. It could not explain the discovery of the electron spin.

The fine structure constant:

It is also renowned as the Sommerfeld constant. So, its name signifies that it is a constant quantity introduced by the German physicist Arnold Sommerfeld in 1916 to determine the size of fine-structure splitting of the hydrogen spectrum.

In fact, Sommerfeld extended the Bohr atomic model to explain the fine structures of the hydrogen spectrum by introducing the relativistic variation of electron mass with velocity in the elliptical electron orbits. To account for the amount of splitting of spectral lines, he entailed a term that he named fine structure constant. In Sommerfeld's analysis, it was the ratio of the electron's velocity in the ground state of the relativistic Bohr atom to the speed of the light in the vacuum. And he used the Greek letter α (alpha) to symbolize it. 

The Sommerfeld’s interpretation of α is below;

The Sommerfeld interpretation for the fine structure constant

To find the velocity of the electron in the ground state of the Bohr atomic model, consider the electrostatic force of repulsion between two electrons of elementary charge e separated at a distance of d.

Additional reference:

Why is Sommerfeld’s constant known as the fine structure constant?

Derivation for Sommerfeld's fine structure constant

According to the Columbic electrostatic force of repulsion is

The Columbic electrostatic repulsion formula

where,

e= magnitude of charge on the electron

d= distance between the two electrons

The centrifugal force arise from the electron’s rotation in the circular orbit is given by;

The centrifugal force formula

Here v is the velocity of the rotating electron

The Sommerfeld fine structure constant formula derivation method steps

For the electron in the first Bohr orbit, the value of the principal quantum number (n) is equal to 1. The above equation can be written as;

The velocity of the electron

Finally, the value of fine structure constant can be expressed as;

Sommerfeld fine structure constant formula

With its help, he could accurately express the gap in the energy difference between the coarse and fine structures of the spectral lines in the hydrogen atom. Hence, it should quantify the electromagnetic interaction of the electrically charged elementary particles and the photon (light radiation). For that, he conceptualized α as a quantity with no physical dimension having the SI unit of measurement of 1, which we call dimensionless quantity. So, α is purely a number with a value equal to 1/137, independent of the system of units. Moreover, this fundamental physical constant appeared naturally in Sommerfeld's fine lines analysis, which agreed well with the experimental observations. But, it became noteworthy after Paul Dirac gave the exact fine structure formula with his linear relativistic wave equation in 1928.

Relationship with the other physical quantities:

Some equivalent definitions of α in terms of other fundamental physical constants are;

In terms of Coulomb constant

k_e is the Coulomb constant. And its value equal to

Relationship between the fine structure constant and the Coulomb constant

In terms of permittivity of free space

Relationship between the fine structure constant and the permittivity of free space

In terms of Planck's constant

The relationship between the fine structure constant and the Planck's constant

In terms of vacuum impedance (Z₀)

The relationship between the fine structure constant and the vacuum impedance 

Measurement units:

The preferred α measurements methods are;

1. Quantum Hall effect or measurement of electron anomalous magnetic moments
2. Photon recoil in atom interferometry

The quantum electrodynamics theory predicts the relationship between the electron magnetic moment (also referred to as the Lande g-factor) and the fine structure constant. The most precise value of α was obtained experimentally on measurement of Lande g-factor by using quantum cyclotron apparatus that involved Feynman diagrams.

In 2018, CODATA (Committee on Data of the International Science Council) specified the value of the reciprocal of the fine structure constant. 

The value of fine structure constant

Even though α is a dimensionless physical quantity, the non-SI units of the other physical quantities can express it.

The non-SI units:

 Electrostatic CGS units:

The stat coulomb measures the electric charge by assuming the permittivity factor as 1 in electrostatic CGS units.  

The fine structure constant in CGS units

Natural units:

In high-energy physics, the numerical values of the selected physical quantities are exactly 1 when expressed in natural units.

The fine structure constant in natural units

Physical interpretations:

The fine structure constant has many physical explanations. Some of them are below;

The fine structure constant can be expressed as the ratio of two energies.

The energy is required to overcome the electrostatic repulsion between the two electrons separated at a distance d apart.

The energy required to overcome the electrostatic repulsion

The energy of a single photon whose wavelength (λ) is 2π multiplied by the distance (d) of separation of the two electrons.

The energy of a single photon

Here λ is the wavelength of the emitted photon.

The fine structure constant interpretation in terms of two energies

Here, ħ is the reduced Planck’s constant. And it can be expressed as;

The relationship between the Planck and the reduced Planck constant

Where,

h= Planck’s constant

ħ = reduced Planck’s constant

The fine structure constant expressed in terms of electron's potential energy

The square of the fine structure constant is the ratio of the electron's potential energy in the first circular orbit of the Bohr model of the atom to the energy equivalent to the electron's mass. 

The fine structure constant expressed in terms of potential energy of the electron

The virial theorem gives an equation that relates to the total kinetic energy's average overtime of the stable system of discrete particles bound by potential forces with that of the system's total potential energy.

By using the Virial theorem in the Bohr’s atomic model, we get;

Virial equation

Where,

U_el = potential energy of the electron in the first Bohr orbit.

U_kin = the average electron’s kinetic energy over time

The electron's kinetic energy average equation over time

m_e = mass of the electron

v_e = velocity of the electron

Substituting these values in the Virial equation, we get;

The derivation of the fine structure constant in terms of electron's potential energy

The fine structure constant interpreted in terms of Bohr radius and the classical electron radius

The square of the fine structure constant is the ratio of the classical electron radius to the Bohr radius.

Relationship between the fine structure constant and Bohr radius

Where,

r_e = classical electron radius

a₀ = Bohr radius

Classical electron radius defines a length scale for the electron that interacts with electromagnetic radiation.

The classical electron radius equation

Bohr radius is a physical constant that determines the mean distance between the nucleus and the ground state electron in a hydrogen atom with n value equal to 1.

Bohr radius equation

Dividing the above two equations, we get;

Method of derivation for the fine structure constant in terms of Bohr radius

The fine structure constant expressed in terms of impedance of free space

In electrical engineering, the fine structure constant is one-fourth the product of the impedance of free space and the conductance quantum.

The relationship between the fine structure constant and the impedance of free space

Z₀ = Impedance of free space. Its value is 

The impedance of free space equation

G₀ = Conductance quantum. Its value is

The conductance quantum equation

Derivation:

The value of fine structure constant is 

 

The Sommerfeld fine structure constant formula

The value of permittivity of free space is given by the following equation

The permittivity of free space equation

Substituting ε₀ and ħ values in the above equation, we get;

Derivation of fine structure constant in terms of impedance of free space

Fine structure constant interpreting the positive charge on the nucleus

In Bohr's atomic model, the fine structure constant gives the maximum positive charge to the nucleus, which will allow a stable electron orbit around it.

For an electron orbiting an atomic nucleus with atomic number Z, we have;

The condition of the electron orbiting the nucleus

The momentum/position uncertainty relationship for a relativistic electron motion by following Heisenberg uncertainty principle is

Heisenberg angular momentum equation

For relativistic conditions, v=c and the limiting value for Z can be considered as;

Interpretation of fine structure constant in terms of nuclear charge

Fine structure constant expressed in terms of Planck charge

The fine structure constant is the square of the ratio of the elementary charge to the Planck charge.

Fine structure constant expressed in terms of Planck charge

q_p = Planck charge. And its value is;

Derivation of fine structure constant by using Planck charge

The fine structure constant variation with the energy scale;

With the development of quantum electrodynamics, the fine structure constant exposure has grown from spectroscopic explanations to coupling constant for the electromagnetic field. We now consider α as the coupling constant for the electromagnetic field, similar to the other two fundamental forces, i.e., the weak nuclear force and the strong nuclear force of the standard model of particle physics.

Many physicists thought that α is not constant with the change of space or time. Measurements of hydrogen and deuterium spectral lines showed that the fine structure constant varies negligibly by altering time or location in the Universe.

But, beyond their expectations, it was found that the fine structure constant is a function of energy. 

Is Sommerfeld's constant really a constant physical quantity?

As a matter of fact, α characterizes the coupling strength of the elementary charged particles with the electromagnetic field. So, the quantum electrodynamics shows the logarithmic growth of electromagnetic interaction with the relevant energy scale. Additionally, the electron-positron annihilation proceeds with the production of photons at low energies. It affects the strength of the electrostatic force. Hence, the value of α is approximately equal to 1/137 for lightest charged particles (like electrons and positrons) at low energies.

Conversely, at higher energies, there are particle and anti-particle contributions in addition to electron-positron pairs, which increases the value of α. Hence, for heavier particles like W, Z, Higgs Boson, and a top quark, the value of α is 1/128.

Hence, it was found that the fine structure constant is not a constant quantity. And it varies with energy strength.