Fine structures of hydrogen and sodium, Larmor's precession

Fine Structure of sodium [ Sodium D-lines ]

All the optical properties of dodium atom is determined by electrones in it's valence shell (i.e electron is 2s orbital). When electron in sodium atoms jumps from 3p level to 3s levels then two types of photons are emitted.

For p-state : l= 1, s= 1/2

\(\therefore\;\;\;\;=l\pm s=1+\frac 12= \frac 32, \frac12\)

Multiplicity= 2s+1=2\(\times \frac 12+1=2\)

Spectral notation: \(^2P_{\frac 32}\) and \(^2P_{\frac 12}\)

Hence p-state is splitted into two levels due to spin orbit interaction.

For S-state :

L=0, s=1/2,

j=\(l\pm S=0\pm \frac 12 = +\frac 12(0nly)\)

Multiplicity= 2s+1= 2\(\times \frac 12+1=2\)

Spectral notaiton= \(^2S_{\frac 12}(single)\)

The transition \(^2P_{\frac 12}\) to \(^2S_{\frac 12}\) is known as sodium \(D_1\)-line and has wavelength 5896\(A^\circ\) and the transition from\(^2P_{\frac 32}\) to\(^2S_{\frac 12}\) is known as sodium\(D_2\)-line and has wavelength \(5890^\circ\) .

Fine structure of \(H_\alpha\)-line according to vectro atom model:

Structure of\(H_\alpha\) line is due to transition of electron from n=3 to n=2 level of H- atom.

From Bohr's theory $$\frac{1}{\lambda}=R\biggl(\frac{1}{2^2}-\frac{1}{3^2}\biggr)$$

$$or\;\; \frac{1}{\lambda}=\frac{36}{5}R$$

\(\frac{1}{\lambda}=\frac{36}{5}R=single\;wavelength\)

According to Vector atom model:

n=2, l=0, s=1/2

For L= 0, J=0\(\pm \frac 12= \frac 12 (+ve\;only\Rightarrow^2S_{\frac 12}\))

For L=1,J=1\(\pm\frac 12=\frac 32, \frac 12\Rightarrow ^2P_{\frac 32}, ^2P_{12}\)

For n=3, L=0,1,2

For L=0, s=1/2, J=1/2\(\Rightarrow ^2S_{\frac 12}\)

For L=1, s=\(\frac 12, J=\frac 32,\frac 12\Rightarrow ^2P_{\frac 32}, ^2P_{\frac 12}\)

For L=2, S=\(\frac 12, J=\frac 52, \frac 32\Rightarrow ^2P_{\frac 52}, ^2P_{\frac 32}\)

\(\Rightarrow\;\ n=3 level splits in 5-levels.

There are several possible transition allowed by selection rule \(\Delta L=\pm 1 \;and\; \Delta J=0,\pm 1\).

Larmor's precession:

Statement: The effect of uniform magnetic field on the orbital motion of electron is to superimpose an additional precesion motion about an axis along the direction of applied magnetic field with an anglar frequency (\(\omega =\frac {eB}{2m}\))

Consider an electron of mass m charge e in an atom revolving in an eliptical orbit. The electron having orbital angular momentum experience torque inside magnetic field.

i.e torque\((\tau)= \overrightarrow{\mu}\times\overrightarrow{B}\)

Where \(\mu\)= oorbital magnetic moment

\(\therefore\;\; \mu= \frac {-e}{2m}\overrightarrow L\dotsm(2)\)

Where \(\overrightarrow L\)= orbital angular momentum.

\(\tau= \frac {-e}{2m} (\overrightarrow L\times\overrightarrow B)=\frac {-e}{2m}LBsin\theta\hat n\)

Taking magnitude only,

\(\tau= \frac {eB}{2m}\cdot Lsing\theta\dotsm(2)\)

Since torque can also be defined as rate of change of angular momentum.

\(\tau= \frac {dL}{dt}\)

\(=Lsin\theta\cdot\frac{d\phi}{dt}\)

Where, \(\frac {d\phi}{dt}\)=\(\omega\)=angular frequency due to magnetic field

\(\tau= Lsin\theta\cdot\omega\dotsm(4)\)

From (3) and (4)

\(L sin\theta\cdot\omega=\frac{eB}{2m}Lsin\theta\)

\(\;\; \omega= \frac{eB}{2m}\dotsm(5)\)

Since \(\omega=2\pi f=\frac{eB}{2m}\)

or, f=\(\frac{eB}{4\pi m}\)= precessional frequency

Change in kinetic energy of electrone due to precessional motion.

Let, we be the initial angular frequency of electron in the absence of magnetic field, (\(\omega_\circ+\omega\)) be new frequency in the presence of magnetic field.

\(\therefore K.E= \frac 12 mr^2(\omega_\circ+\omega)^2-\frac 12 mr^2+\omega_{\circ}^2\)

\(=\;\;\frac 12 mr^2[(\omega_{\circ}^2+2\omega\omega_\circ+\omega^2]\)

\(=\;\;\frac 12 mr^2(2\omega\omega_\circ+\omega^2\)

Since, \(\omega^2<<\omega_{\circ}^2\)

The 2nd term can be naglected,

Change in K.E= \(\frac 12 mr^2\cdot 2\omega_\circ\cdot\omega\)

\(\;\;=\frac 12 mr^2\cdot 2\omega_\circ\cdot\biggl(\frac{eB}{2m}\biggr)\)

\(\;\;=(mr^2\omega_\circ)\cdot\frac{eB}{2m}\)

Change in K.E= \(L\frac{eB}{2m}=L\omega\)

Change in K.E is propertional to orbital angular momentum and the strenth of applied magnnetic field.

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.