Question

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 $2\mathring{\text{A}}$. A similar standing wave is again formed if $d$ is increased to $2.5\mathring{\text{A}}$ but not for any intermediate value of $d$. Find the least value of $d$ for which the standing wave of the type described above can form.

• A. $0.4\mathring{\text{A}}$
• B. $0.5\mathring{\text{A}}$
• C. $2\mathring{\text{A}}$
• D. $1\mathring{\text{A}}$

Answer 1

The correct option is B $0.5\mathring{\text{A}}$


From the figure, it is clear that,

$P\times \left(\lambda /2\right)=2\mathring{\text{A}}---\left(1\right)$
And,
$(P+1)\lambda /2=2.5\mathring{\text{A}}---\left(2\right)$

Substracting eq.$\left(2\right)$ to $\left(1\right)$, we get

$\lambda /2=(2.5-2.0)\mathring{\text{A}}=0.5\mathring{\text{A}}$

$\therefore \lambda =1\mathring{\text{A}}$

The least value of $d$ will be when only one loop is formed


$\therefore d_{min}=\lambda /2=0.5\mathring{\text{A}}$

Hence, option $\left(\text{B}\right)$ is correct.