91Ó°ÊÓ

The Zeeman effect is the splitting of atomic energy levels and the associated spectral lines when the atoms are placed in a magnetic field. This effect confirms experimentally the quantization of angular momentum.

Atoms contain charges in motion, so it should not be surprising that magnetic forces cause changes in that motion and in the energy levels. In 1896 the Dutch physicist Pieter Zeeman was the first to show that in the presence of a magnetic field, some spectral lines were split into groups of closely spaced lines. This effect now bears his name.

Bohr magneton is the minimum possible magnetic moment associate with an electron in the atom. It is given as,

$$\begin{gathered} {\mathrm{μ}}_B=\frac{e\sout{h}}{2m} \\ {\mathrm{μ}}_B=5.788×{10}^{-5}eV/T \\ {\mathrm{μ}}_B=9.274×{10}^{-24}J/T \end{gathered}$$

The 5g state corresponds to orbital quantum number $\mathit{l}\mathbf{=}\mathbf{4}$and magnetic quantum number ${\mathit{m}}_{\mathbf{l}}$can have $\mathbf{2}\mathit{l}\mathbf{+}\mathbf{1}$values, i.e.$\mathbf{-}\mathit{l}$, to $\mathbf{+}\mathit{l}$.

So, for $l=4$,$m_l$can be any integer value from $-4$to $+4$.

Total number of values of $m_l=2\left(4\right)+1=9$.

belong to set$\left\{-4,-3,-2,-1,0,+1,+2,+3,+4\right\}$.

Thus, 5g state splits into 9 levels by the interaction of the atom’s orbital magnetic dipole moment with the magnetic field.

The orbital magnetic interaction of an electron with magnetic quantum number${\mathit{m}}_{\mathbf{l}}$in an external magnetic field $\mathit{B}$along $\mathbf{+}\mathit{z}$axis is given by,

$\mathbf{U}\mathbf{=}{\mathit{m}}_{\mathbf{l}}{\mathit{μ}}_{\mathbf{B}}\mathit{B}$

The energy difference between any two $m_l$levels is given by,

$$\begin{gathered} ∆E=U_2-U \\ ∆E=m_2{\mathrm{μ}}_{B}B-m_2{\mathrm{μ}}_{B}B \\ ∆E=\left(m_1-m_2\right){\mathrm{μ}}_{B}B \\ ∆E=∆m_l{\mathrm{μ}}_{B}B \end{gathered}$$

The difference in magnetic quantum number for any two adjacent levels is, $∆m=1$.

So, energy separation between two adjacent levels is,

$$\begin{gathered} ∆E=1×{\mathrm{μ}}_{B}B \\ ∆E=1×\left(5.788×{10}^{-5}eV/T\right)×0.600T \\ ∆E=3.47×{10}^{-5}eV \end{gathered}$$

The energy difference between lowest state and highest state is given by,

$$\begin{gathered} ∆E=U_H-U_L \\ ∆E=m_{lH}{\mathrm{μ}}_{B}B-m_{lL}{\mathrm{μ}}_{B}B \\ ∆E=\left(m_{lH}-m_{lL}\right){\mathrm{μ}}_{B}B \end{gathered}$$

For 5g state, level of lowest energy is for $m_{lL}=-4$and level of highest energy is $m_{lH}=4$.

This will give, $m_{lH},-m_{lL}=4-4\left(-4\right)=8$.

So, the energy separation between lowest energy and highest energy is,

$\begin{array}{ll}∆E=8x5.788x{10}^{-5}& eV/T×0.600T\\ ∆E=2.78×{10}^{-4}eV& \end{array}$

Therefore,

(a) 5g state will split into nine levels in presence of magnetic field.

(b) the energy separation between adjacent levels is $3.47x{10}^{-5}e\mathrm{V}$.

(c) energy separation between the level of lowest energy and the level of highest energy is $2.78×{10}^{-4}eV$