The document discusses the Bohr-Sommerfeld atomic model, which extends Bohr's original atomic theory to explain the fine structures of hydrogen's spectral lines through elliptical electron orbits and relativistic corrections. It highlights the limitations of Bohr’s model, such as its inability to account for spectral line splitting due to electric and magnetic fields, and introduces Sommerfeld's concepts such as azimuthal quantum numbers and modified energy equations. The analysis includes the derivation of allowed electron transitions leading to the hydrogen alpha spectral line and its fine structure, showcasing Sommerfeld's contributions to atomic theory.

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BOHR-
SOMMERFELD
ATOMIC MODEL
https://jayamchemistrylearners.blogspot.com/
ABSTRACT
This PDF document discusses about Sommerfeld
explanation for the hydrogen fine structures,
relativistic correction and hydrogen alpha fine
structures.
By Jayam chemistry learners
PDF notes

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Introduction:
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.
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.
Drawbacks 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 atomic 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.

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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
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 pr and pφ,
respectively. The Wilson-Sommerfeld quantum condition amounted to integrals for each coordinate
over one complete rotation.
∮ 𝑝𝑟 𝑑𝑟 = 𝑛𝑟ℎ
∮ 𝑝𝜑𝑑𝜑 = 𝑘ℎ
Where,
pr = radial momentum
pφ = angular momentum
r = radial coordinate
φ = angular coordinate
nr = radial quantum number
k= angular or azimuthal quantum number
h= Planck’s constant
Sommerfeld was inspired by Kepler’s law and considered the planetary motion of the electron in
elliptical orbits rather than circular. And he thought that it would solve the problem of hydrogen fine
structures. But very soon, he realized that he could not split the energy levels of the atom to account
for the fine structures with the principal quantum number. He searched for a new quantum
condition to break the principal energy level of the hydrogen atom into unequal sub-energy states.

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In this journey, he found that the electron’s energy has a significant contribution to the orbital
motion of the electron. And his quest offered a new quantum entity that deals with the orbital
angular momentum energy distributions. He named it as 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
𝑛 = 𝑛𝑟 + 𝑘
Where,
n= principal quantum number
nr = radial quantum number
k= azimuthal quantum number
Sommerfeld’s relativistic correction:
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.
𝑚 =
𝑚0
√1 −
𝑣2
𝑐2
Where,
m = relativistic mass of the body
m0 = rest mass of the body
v= velocity of the body
c= velocity of light
To explain this concept, he 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.
Sommerfeld explained the velocity of the electron is minimum at the aphelion point. And it is
maximum at the perihelion point.

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Sommerfeld’s modified energy equation:
Sommerfeld derived an expression for the energy of the electron orbiting in the elliptical orbit prior
to the invention of relativistic concept.
𝐸 = −
𝑚𝑍2
𝑒4
8𝜀0
2ℎ2𝑛2
This equation relies only on the principal quantum number without considering the azimuthal
quantum number. Hence, it was unable to explain the splitting of the spectral lines based on the
energy of the orbits.
When the motion of the electron is considered relativistic, there was a considerable variation in the
electron’s velocity on the elliptical orbit that added a new relativistic correction term to the total
energy of the electron. Now, the modified Sommerfeld’s energy equation is below.
𝐸 =
𝑚𝑒4
𝑍2
8𝜀0
2ℎ2𝑛2
−
𝑚𝑒4
𝑍2
𝛼2
8𝜀0
2ℎ2
(
1
𝑘𝑛3
−
3
4𝑛4
)
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.
The electron transitions to the slightly varied energy levels show the spectral line splitting. 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’s angular momentum condition:
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. He revived Bohr's angular
momentum condition and replaced the principal quantum number with the new quantum entity. It
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

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constant. This quantized angular momentum condition proved the discrete orbicular paths'
existence for the spinning electron.
Sommerfeld’s angular momentum condition for the relativistic motion of the electron is shown
below;
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚𝑣𝑟 =
𝑘ℎ
2𝜋
= 𝑘ħ
Eccentricity and its conditions:
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;
(1 − 𝜀2) = (
𝑘
𝑛
)
2
For an ellipse, it can be written as;
(1 − 𝜀2) = (
𝑏
𝑎
)
2
By comparing the above two equations, we get;
(
𝑘
𝑛
) = (
𝑏
𝑎
) = (
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑖𝑛𝑜𝑟 𝑎𝑥𝑖𝑠
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠
)
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.
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.
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.

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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.
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.
Example-2:
For n=2, k has two values, such as k=1 and k=2.
For n=2 and k=1,
𝑘
𝑛
=
𝑏
𝑎
=
1
2
= 0.5

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In this ellipse, the length of minor axis is equal to half the length of the major axis.
For n=2 and k=2,
𝑘
𝑛
=
𝑏
𝑎
=
2
2
= 1
We have, b=a. It is a circle.
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
𝑘
𝑛
=
𝑏
𝑎
=
2
3
= 0.6
𝑏 = 0.6𝑋𝑎
It is an ellipse with minor axis 0.6 times less than the major axis
For n=3 and k=1
𝑘
𝑛
=
𝑏
𝑎
=
1
3
= 0.3
𝑏 = 0.3𝑋𝑎
It is an ellipse with minor axis 0.3 times less than the major axis

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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.
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

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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.
𝛥𝑘 = 𝑘2 − 𝑘1
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.
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

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successfully explained the cause for the appearance of the fine structures of the hydrogen atom with
his relativistic model.