X Rays and the Ordering of the Elements — Question 4

Question 4

Two calibration lines determine a screening constant and a slope

For a characteristic line family assume $\sqrt{E}=a(Z-\sigma)$, with E in keV. Measurements give $\sqrt{E_1}=2.0\,\mathrm{keV^{1/2}}$ at $Z_1=20$ and $\sqrt{E_2}=2.5\,\mathrm{keV^{1/2}}$ at $Z_2=25$.

1. Find a and $\sigma$.

2. Infer Z for a third target with $E_3=5.76\,\mathrm{keV}$.

Question 4 - Solution

Two calibration lines determine a screening constant and a slope

(a) Method and calculation

$$a=\frac{\sqrt{E_2}-\sqrt{E_1}}{Z_2-Z_1}=\boxed{.100\,\mathrm{keV^{1/2}}}.$$$$\sigma=Z_1-\sqrt{E_1}/a=20-20=\boxed{0}.$$

(b) Interpretation and further calculation

$$\sqrt{E_3}=2.40\,\mathrm{keV^{1/2}},\quad Z=\sigma+\sqrt{E_3}/a.$$$$\boxed{Z=24}.$$

Physical check

This supplied calibration yields zero effective screening; it is a hypothetical line family rather than a claim about actual neutral-atom K lines. The fit must determine the intercept as well as the slope.