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. For more information on this concept, kindly visit our blog article at; https://jayamchemistrylearners.blogspot.com/2022/04/bohr-sommerfeld-model-chemistrylearners.html

1. Bohr-
Sommerfeld
atomic
model
By
Jayam chemistry learners

2. 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.
 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.

3. Overview:
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.

4. Wilson-Sommerfeld quantum condition
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.

5. Azimuthal quantum number role in fine structures
 Sommerfeld searched for a new quantum condition to break the principal energy level of the
hydrogen atom into unequal sub-energy states. 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
Where,
n= principal quantum number
nr = radial quantum number
k= azimuthal quantum number

6. Sommerfeld’s energy equation
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.
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.

7. Eccentricity and 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;
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.

8. 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.

9. 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.
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.

10. Sommerfeld’s relativistic electron motion
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.
Where,
m = relativistic mass of the body
m0 = rest mass of the body
v= velocity of the body
c= velocity of light

11. 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.

12. Rosette 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.
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 above equation represents the precession of perihelion during one orbit.

13. 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.
Sommerfeld’s elliptical orbits concept proved the existence of stationary electronic orbits of the atom
as proposed by Neil Bohr. And his relativistic electron velocity theory successfully explained the fine
structures of the hydrogen spectrum.

14. For more information on this topic, kindly visit our blog article at;
https://jayamchemistrylearners.blogspot.com/2022/04/bohr-sommerfeld-model-
chemistrylearners.html
Our blog discusses interesting topics of chemistry with regular new post updates. You can have a look
at it and follow it. https://jayamchemistrylearners.blogspot.com/
You can send your valuable suggestions and any doubts regarding chemistry subject through these social
media channels.
Instagram : https://www.instagram.com/chemistrylearners/
Pinterest : https://in.pinterest.com/kameswariservices/_created/
Facebook https://www.facebook.com/jayamchemistrylearners
Thank you