# 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 that distance d between the atoms of the array is 2Å . A similar standing wave is again formed if d is increased to 2.5 Å but not for any intermediate value of d. Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can form.

Navjyot Kalra
10 years ago
Hello Student,
As nodes are formed at each of the atomic sites, hence
2Å = n (λ / 2) …(1)
[ ∵ Distance between successive nodes = λ / 2]
and 2.5 Å = (n + 1) λ /2
∴ 2.5 /2 = n + 1 / n , 5/ 4 = n + 1 / n or n = 4
Hence, from equation (1),
2Å = 4 λ / 2 i.e., λ = 1 Å
Now, de Broglie wavelength is given by
λ = h / √ 2mK or K = h2 / λ2 2m
∴ K = (6.63 x 10-34)2 / (1 x 10-10 )2 x 2 x 9.1 x 10-31 x 1.6 x 10-19 eV
= (6.63)2 / 8 x 9. 1 x 1.6 x 102 eV = 151 eV
d will be minimum, when
n = 1, dmin = λ / 2 = 1Å / 2 = 0.5Å
Thanks
Navjot Kalra