Electrons accelerated from rest through 900 V produce a first-order diffraction maximum from one crystal at a certain Bragg angle. A second crystal has interplanar spacing 1.5 times that of the first crystal. What accelerating voltage should be used with the second crystal so that the first-order diffraction maximum occurs at the same Bragg angle? If the required voltage is N V, find N.
Physics · Photoelectric effect and dual nature of matter · debroglie duel nature of light
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Question

Electrons accelerated from rest through 900 V produce a first-order diffraction maximum from one crystal at a certain Bragg angle. A second crystal has interplanar spacing 1.5 times that of the first crystal. What accelerating voltage should be used with the second crystal so that the first-order diffraction maximum occurs at the same Bragg angle? If the required voltage is N V, find N.

Physics · Photoelectric effect and dual nature of matter · debroglie duel nature of light

G1 Physics · Accepted Answer

For the same diffraction order and the same Bragg angle, Bragg's law requires λ to be proportional to d. Since the second crystal spacing is 1.5 times larger, λ₂ = 1.5λ₁. For electrons accelerated from rest, λ is inversely proportional to √V. Therefore, λ₂ ÷ λ₁ = √(V₁ ÷ V₂). Thus, 1.5 = √(900 ÷ V₂). Squaring: 2.25 = 900 ÷ V₂. Hence, V₂ = 900 ÷ 2.25 = 400 V. Therefore N = 400.