91Ó°ÊÓ

The orbital magnetic interaction of an electron with a magnetic quantum number in an external magnetic field B along the +z-axis is given by

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

Where,

$$\begin{gathered} {\mathbf{μ}}_{\mathbf{B}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{788}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{eV}\mathbf{/}\mathbf{T} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{274}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{24}}\mathbf{J}\mathbf{/}\mathbf{T} \end{gathered}$$

The difference between the two adjacent levels is given by

$∆\mathrm{E}={\mathrm{U}}_{{\mathrm{m}}_{\mathrm{l}}+1}-{\mathrm{U}}_{{\mathrm{m}}_{\mathrm{l}}}$

Input the expression

$$\begin{gathered} ∆\mathrm{E}=\left({\mathrm{m}}_{\mathrm{l}}+1\right){\mathrm{μ}}_{\mathrm{B}}\mathrm{B}-\left({\mathrm{m}}_{\mathrm{l}}\right){\mathrm{μ}}_{\mathrm{B}}\mathrm{B} \\ ∆\mathrm{E}={\mathrm{μ}}_{\mathrm{B}}\mathrm{B} \end{gathered}$$

Plug in the values we get

$$\begin{gathered} ∆\mathrm{E}=\left(5.788×{10}^{-5}\mathrm{eV}/\mathrm{T}\right)×\left(0.800\mathrm{T}\right) \\ ∆\mathrm{E}=4.63×{10}^{-5}\mathrm{eV} \end{gathered}$$

The energy is given by

role="math" localid="1664016212100" $$\begin{gathered} ∆\mathrm{E}=4.63×{10}^{-5}\mathrm{eV}\left(\frac{1\mathrm{μ\pm ð³Õ}}{{10}^{-6}\mathrm{eV}}\right) \\ =46.3\mathrm{μ\pm ð³Õ} \end{gathered}$$

The d state corresponds to the orbital quantum number l=3,

And we know that ${\mathrm{m}}_{\mathrm{l}}$can have 2l+1 values from -l and +l,

${\mathrm{m}}_{\mathrm{l}}=-2,-1,0,1,2$

Therefore, the state splits into 5 levels when it is put in the external magnetic field.

The energy diagram is given below

We see that${\mathrm{m}}_{\mathrm{l}}=-2$level will have the lowest energy. The energy level diagram shows no splitting in case of the absence of the magnetic field and shows splitting into 5 states, one in the middle with no defect in energy, two with increased energy, and two with decreased energy.