Element Y has atomic number 11. For another element X, the measured frequency ratio is ν[Kβ(X)] ÷ ν[Kα(Y)] = 128 ÷ 27. Assume the K-series screening constant is 1 for both elements and use the simplified shell-transition model. Find the atomic number of X.
Physics · Atomic Structure and X-rays · x-rays and its types
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Question

Element Y has atomic number 11. For another element X, the measured frequency ratio is ν[Kβ(X)] ÷ ν[Kα(Y)] = 128 ÷ 27. Assume the K-series screening constant is 1 for both elements and use the simplified shell-transition model. Find the atomic number of X.

Physics · Atomic Structure and X-rays · x-rays and its types

G1 Physics · Accepted Answer

For Kβ radiation, ν(Kβ) ∝ (Z−1)²×8 ÷ 9. For Kα radiation, ν(Kα) ∝ (Z−1)²×3 ÷ 4. Therefore, ν[Kβ(X)] ÷ ν[Kα(Y)] = (32 ÷ 27)×[(Z(X)−1) ÷ (Z(Y)−1)]². For Y, Z(Y)−1 = 10. Given, 128 ÷ 27 = (32 ÷ 27)×[(Z(X)−1) ÷ 10]². Cancel 32 ÷ 27: 4 = [(Z(X)−1) ÷ 10]². Taking the positive square root: 2 = (Z(X)−1) ÷ 10. Thus, Z(X)−1 = 20, so Z(X) = 21. Hence the answer is 21.