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Magnetic interaction energy of a magnetic dipole is potential energy stored in the magnetic dipole when it is placed in an external magnetic field. It is given as,

$\mathbf{U}\mathbf{=}\mathbf{-}\stackrel{\mathbf{鈫�}}{\mathbf{渭}}\mathbf{.}\stackrel{\mathbf{鈫�}}{\mathbf{B}}$

whereis the interaction energy,$\stackrel{\mathbf{鈫�}}{\mathbf{渭}}$ is magnetic moment of the dipole and$\stackrel{\mathbf{鈫�}}{\mathbf{B}}$is the external magnetic field.

The z-component of the spin magnetic moment${\mathbf{渭}}_{\mathbf{z}}$ is related to${\mathbf{S}}_{\mathbf{z}}$by,

${\mathbf{渭}}_{\mathbf{z}}\mathbf{=}\mathbf{-}\left(2.00232\right)\frac{\mathbf{e}}{\mathbf{2}\mathbf{m}}{\mathbf{S}}_{\mathbf{z}}$

where is spin angular momentum for${\mathbf{m}}_{\mathbf{s}}$ as spin magnetic quantum number.

The 1s state corresponds to orbital quantum number l=0 so, the magnetic quantum number is ${\mathrm{m}}_{\mathrm{l}}=0$.

There is no orbital dipole moment for this state and the interaction with external magnetic field is due to the intrinsic spin of electrons.

The z-component of spin magnetic moment is given as,

$$\begin{gathered} {\mathrm{渭}}_{\mathrm{z}}=-\left(2.00232\right)\frac{\mathrm{e}}{2\mathrm{m}}{\mathrm{m}}_{\mathrm{s}}\mathrm{魔} \\ {\mathrm{渭}}_{\mathrm{z}}=-(11.589脳{10}^{-5}\mathrm{eV}/\mathrm{T})脳{\mathrm{m}}_{\mathrm{s}} \end{gathered}$$

The angle between z-component of spin magnetic moment and z-component of magnetic field is$180掳$ as both are oppositely oriented.

This gives the magnetic interaction energy as,

$$\begin{gathered} \mathrm{U}=-{\mathrm{渭}}_{\mathrm{z}}{\mathrm{B}}_{\mathrm{z}}脳\cos 180掳 \\ \mathrm{U}=-{\mathrm{渭}}_{\mathrm{z}}{\mathrm{B}}_{\mathrm{z}}脳-1 \\ \mathrm{U}=-11.589脳{10}^{-5}\mathrm{eV}/\mathrm{T}脳{\mathrm{m}}_{\mathrm{s}}\mathrm{B} \end{gathered}$$

From above equation it is clear that interaction energy for spin up${\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}$ is lower than that of spin down ${\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}$.

The spin magnetic quantum number$\left({\mathrm{m}}_{\mathrm{s}}\right)$ takes value either$-\frac{1}{2}$ or $+\frac{1}{2}$.

So, the interaction energy separation between two levels corresponding to each value of$\left({\mathrm{m}}_{\mathrm{s}}\right)$ is,

$$\begin{gathered} \mathrm{螖贰}={\mathrm{U}}_{(1/2}-{\mathrm{U}}_{-1/2} \\ \mathrm{螖贰}=\left[2.00232脳5.788脳{10}^{-5}鈥�\mathrm{eV}/\mathrm{T}\left(\frac{1}{2}-\left(\frac{-1}{2}\right)\right)\right]脳1.45鈥�\mathrm{T} \\ \mathrm{螖贰}=1.68脳{10}^{-4}\mathrm{eV} \end{gathered}$$

Therefore, the energy difference between them is $1.68脳{10}^{-4}\mathrm{eV}$. The level$m_s=\frac{1}{2}$ has lower energy.