X Rays and the Ordering of the Elements — Question 10

Question 10

Adjacent elements become harder to separate by fractional K-line energy

Assume $E(Z)=C(Z-\sigma)^2$ with fixed C and $\sigma$. Compare neighboring atomic numbers Z and $Z+1$. A spectrometer resolves two lines if their separation exceeds its full width $\delta E$.

1. Derive the exact energy separation and fractional separation relative to $E(Z)$.

2. For $Z=50$, $\sigma=1$, and $E(Z)=25.0\,\mathrm{keV}$, find the separation and the minimum resolving power $\mathcal R=E(Z)/\delta E$ at equality.

Question 10 - Solution

Adjacent elements become harder to separate by fractional K-line energy

(a) Method and calculation

$$\Delta E=C[(Z+1-\sigma)^2-(Z-\sigma)^2]=\boxed{C[2(Z-\sigma)+1]}.$$$$\boxed{\Delta E/E(Z)=2/(Z-\sigma)+1/(Z-\sigma)^2}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta E=25.0(99/2401)=1.0308\,\mathrm{keV}}.$$$$\delta E=\Delta E\Rightarrow\boxed{\mathcal R_{\min}=2401/99=24.25}.$$

Physical check

Absolute separation increases with Z in this model, while fractional separation decreases. Resolving power expresses a fractional-width requirement; a strict separation criterion requires a value just above the equality limit.