1. Tardigrade
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  4. Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one-dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed, if the distance d between the atoms of the array is 2 mathringA. A similar standing wave is again formed, if d is increased to 2.5 mathringA but not for any intermediate value of d. Find wavelength of the beam.

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Q. Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one-dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed, if the distance $d$ between the atoms of the array is $2 \,\mathring{A}$. A similar standing wave is again formed, if $d$ is increased to $2.5 \,\mathring{A}$ but not for any intermediate value of $d$.
Find wavelength of the beam.

Dual Nature of Radiation and Matter

Solution:

From the figure, it is clear that
image
$p \cdot(\lambda / 2)=2 \,\mathring{A}$
$ \Rightarrow (p+1) \cdot \lambda / 2=2.5 \,\mathring{A}$
$\therefore \lambda / 2=(2.5-2.0)\,\mathring{A}=0.5 \,\mathring{A}$
or $ \lambda=1 \,\mathring{A}$