X Rays and the Ordering of the Elements — Question 3

Question 3

Moseley scaling identifies an element through its nuclear charge

Use the approximate $K_\alpha$ law $E_\alpha=C(Z-1)^2$, where C is the same for two neighboring targets. For target A, $Z_A=25$ and $E_A=6.00\,\mathrm{keV}$. Target B has $E_B=6.5104167\,\mathrm{keV}$ within the precision of this model.

1. Infer $Z_B$.

2. Find C and predict the line energy for $Z=27$.

Question 3 - Solution

Moseley scaling identifies an element through its nuclear charge

(a) Method and calculation

$$\sqrt{E_B/E_A}=\frac{Z_B-1}{Z_A-1}\Rightarrow Z_B=1+24\sqrt{6.5104167/6.00}.$$$$\boxed{Z_B\simeq26}.$$

(b) Interpretation and further calculation

$$\boxed{C=6.00/24^2=.0104167\,\mathrm{keV}}.$$$$\boxed{E_\alpha(27)=C(26)^2=7.04167\,\mathrm{keV}}.$$

Physical check

The model orders elements by atomic number rather than mean atomic mass. Screening and relativistic corrections limit its accuracy; inferred integer Z should be judged against those model uncertainties.