Wavelength of `K_alpha` line of an element is `lambda_0`. Find wavelength of `K_beta` - line for the same elemetn.

Explanation

To find the wavelength of the K-beta line (\( \lambda_{K\beta} \)) for an element given the wavelength of the K-alpha line (\( \lambda_{K\alpha} = \lambda_0 \)), we can use the following steps:

### Step 1: Write the formula for \( \lambda_{K\alpha} \)
The wavelength of the K-alpha line can be expressed using the formula:
\[ \frac{1}{\lambda_{K\alpha}} = R \cdot (Z - 1)^2 \cdot \left(1 - \frac{1}{2^2}\right) \]
Where \( R \) is a constant, and \( Z \) is the atomic number of the element. Since \( \lambda_{K\alpha} = \lambda_0 \), we can rewrite this as:
\[ \frac{1}{\lambda_0} = R \cdot (Z - 1)^2 \cdot \left(1 - \frac{1}{4}\right) \]
This simplifies to:
\[ \frac{1}{\lambda_0} = R \cdot (Z - 1)^2 \cdot \frac{3}{4} \]
This is our **Equation 1**.

### Step 2: Write the formula for \( \lambda_{K\beta} \)
Similarly, the wavelength of the K-beta line can be expressed as:
\[ \frac{1}{\lambda_{K\beta}} = R \cdot (Z - 1)^2 \cdot \left(1 - \frac{1}{3^2}\right) \]
This simplifies to:
\[ \frac{1}{\lambda_{K\beta}} = R \cdot (Z - 1)^2 \cdot \left(1 - \frac{1}{9}\right) = R \cdot (Z - 1)^2 \cdot \frac{8}{9} \]
This is our **Equation 2**.

### Step 3: Divide Equation 1 by Equation 2
Now we can find the ratio of the two wavelengths:
\[ \frac{\lambda_{K\beta}}{\lambda_{K\alpha}} = \frac{R \cdot (Z - 1)^2 \cdot \frac{8}{9}}{R \cdot (Z - 1)^2 \cdot \frac{3}{4}} \]
The \( R \) and \( (Z - 1)^2 \) terms cancel out:
\[ \frac{\lambda_{K\beta}}{\lambda_{K\alpha}} = \frac{\frac{8}{9}}{\frac{3}{4}} = \frac{8}{9} \cdot \frac{4}{3} = \frac{32}{27} \]

### Step 4: Solve for \( \lambda_{K\beta} \)
Now, we can express \( \lambda_{K\beta} \) in terms of \( \lambda_{K\alpha} \):
\[ \lambda_{K\beta} = \frac{32}{27} \lambda_{K\alpha} \]
Substituting \( \lambda_{K\alpha} = \lambda_0 \):
\[ \lambda_{K\beta} = \frac{32}{27} \lambda_0 \]

### Final Answer
Thus, the wavelength of the K-beta line is:
\[ \lambda_{K\beta} = \frac{32}{27} \lambda_0 \]