Question

Use Equation 6.59 to estimate the internal field in hydrogen, and characterize quantitatively a "strong" and "weak" Zeeman field.

Short answer

The value of internal magnetic field is 12 T .

The strong and weak Zeeman field is characterized as:

$B_{ext}\gg B_{int}$strong Zeeman field

$B_{ext}\gg B_{int}$weak Zeeman field

Step-by-step solution

Expression for the internal magnetic field

The expression for the internal magnetic field in the hydrogen atom is given as follows,

$\mathit{B}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\mathbf{.}\frac{\mathbf{e}}{\mathbf{m}{\mathbf{c}}^{\mathbf{2}}{\mathbf{r}}^{\mathbf{3}}}\mathit{L}$

Here,${\epsilon }_0$is the permittivity of the free space with value role="math" localid="1658138846877" $8.9\times {10}^{-10}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2$,e is the charge on electron with value $1.6\times {10}^{-19}C,m$, is the mass of the electron with value $9.1\times {10}^{-31}\mathrm{kg}$,c is the speed of light with value $3\times {10}^8\mathrm{m}/\mathrm{s},\mathrm{r}$is the Bohr’s radius with value $0.53\times {10}^{-10}\mathrm{m}$, and $L=ħ$which is Planck’s constant with value $1.05\times {10}^{-34}\mathrm{J}.\mathrm{s}$.

Determination of the internal magnetic field of the hydrogen atom

Assume r=a that is Bohr’s radius, and $L=ħ$.

Substitute the values in the expression for the internal magnetic field in the hydrogen atom.

$$\begin{gathered} B=\frac{1}{4{\mathrm{\pi \epsilon }}_0}.\frac{e}{m_e{c}^2{a}^3}ħ \\ =\frac{1}{4\mathrm{\pi }\left(8.9\times {10}^{-10}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2\right)\times {\displaystyle \frac{1\mathrm{N}.\mathrm{m}}{1\mathrm{J}}}}.\frac{\left(1.6\times {10}^{-19}C\right)\left(1.05\times {10}^{-34}J.s\right)}{\left(9.1\times {10}^{-31}kg\right){\left(3\times {10}^{8}m/s\right)}^2\left(0.53\times {10}^{-10}m\right)} \\ =12\mathrm{C}.\mathrm{m}/\mathrm{s}\times \frac{1T}{1C.m/s} \\ =12\mathrm{T} \end{gathered}$$

Quantitative characterization of strong and weak Zeeman field

It is known that the strong Zeeman field is $B_{ext}>>10\mathrm{T}$and the weak Zeeman field is $B_{ext}\ll 10T$. So,$B_{ext}\gg B_{int}and{B}_{ext}\ll B_{int}$ .