A donor atom in a semiconductor has a loosely bound electron. The… - Vidyadip

MCQ

A donor atom in a semiconductor has a loosely bound electron. The orbit of this electron is considerly affected by the semiconductor material but behaves in many ways like an electron orbiting a hydrogen nucleus. Given that the electron has an effective mass of $0.07 \,m_e$, where $m_e$ is mass of the free electron and the space in which it moves has a permittivity $13 \,\varepsilon_0$, then the radius of the electron's lowermost energy orbit will be close to ................. $\mathring A$ (take, the Bohr radius of the hydrogen atom is $0.53 \mathring A$ )

A
$0.53$
B
$243$
C
$10$
✓
$100$

✓

Answer

Correct option: D.

$100$

d
(d)

From Bohr's atomic model, radius of Bohr's orbit is given by

$r_n=\frac{\varepsilon_g h^2}{\pi m e^2} \cdot \frac{n^2}{Z}$

So, in a medium of relative permittivity $\varepsilon_r$, radius of $n$th orbit of mass $0.07\,m_e$ is

$r_n=\frac{\varepsilon_0 \varepsilon_r h^2}{0.07 \pi m_e e^2} \cdot \frac{n^2}{Z}=\frac{\varepsilon_r h^2}{0.07 Z}\left(\frac{\varepsilon_0 h^2}{\pi m_e e^2}\right)$

where, $\frac{\varepsilon_{0t} h^2}{\pi m_e e^2}=r_0=$ Bohr's radius of H-atom $=0.53 \mathring A$.

$\therefore \quad r_n=\frac{\varepsilon_r h^2}{0.07 Z} \cdot(0.53) \mathring A$

Here, $\quad n=1, \varepsilon_r=13$ and $Z=1$

$\therefore$ Radius of electron's lowermost energy level is

$r=\frac{13 \times 0.53}{0.07}=98.42 \mathring A \approx 100 \mathring A$

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