Sommerfeld atomic model

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 prand 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.

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