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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.

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. 

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
9 years ago
Hello Student,
Please find the answer to your question
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
askIITians Faculty

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