Question

X-ray from a tube with a target $A$ of atomic number $Z$ shows strong $K$ lines for target $A$ and weak $K$ lines for impurities. The wavelength of $K_{\alpha }$ lines is ${\lambda }_z$ for target $A$ and ${\lambda }_1$ and ${\lambda }_2$ for two impurities.
$\frac{{\lambda }_z}{{\lambda }_1}=4and\frac{{\lambda }_z}{{\lambda }_2}=\frac{1}{4}$
Assuming the screening constant of $K_{\alpha }$ lines to be unity, select the correct statement(s)

• A. The atomic number of first impurity is $2z-1$
• B. The atomic number of first impurity is $2z+1$
• C. The atomic number of second impurity is $\frac{(z+1)}{2}$
• D. The atomic number of second impurity is $\frac{z}{2}+1$

Answer 1

Answer: (A), (C)

The correct options are
A The atomic number of first impurity is $2z-1$
C The atomic number of second impurity is $\frac{(z+1)}{2}$

For $K_{\alpha }$ line, $\lambda \propto \frac{1}{(Z-1{)}^2}$.

For first impurity, Let atomic number be $Z_1$

So, $\frac{(Z_1-1{)}^2}{(Z-1{)}^2}=4\Rightarrow Z_1-1=2Z-2\Rightarrow Z_1=2Z-1$

For second impurity, Let atomic number be $Z_2$

So, $\frac{(Z_1-1{)}^2}{(Z-1{)}^2}=1/4\Rightarrow Z_1-1=(Z-1)/2\Rightarrow Z_1=(Z+1)/2$