To solve the problem regarding the K lines emitted from a target with atomic number Z and its impurities, we can utilize the relationships between the wavelengths of the emitted X-rays and the atomic numbers of the elements involved. The K lines correspond to transitions of electrons to the K shell, and their wavelengths are inversely related to the atomic number due to the principles of X-ray emission.
Understanding the Relationships
The relationship between the wavelength of the K lines and the atomic number can be expressed using the formula:
λ = k / (Z - σ)
Here, λ is the wavelength, k is a constant, Z is the atomic number of the target, and σ is the screening constant. In this case, we are assuming the screening constant (σ) is unity.
Setting Up the Equations
Given the information:
- For target A (atomic number Z), the wavelength is Vz.
- For impurity 1, the wavelength is V1.
- For impurity 2, the wavelength is V2.
From the problem, we have the ratios:
- Vz / V1 = 4
- Vz / V2 = 1/4
Expressing Wavelengths in Terms of Atomic Numbers
Using the formula for the wavelengths, we can express V1 and V2 in terms of Z:
- For impurity 1: V1 = k / (Z1 - 1)
- For impurity 2: V2 = k / (Z2 - 1)
Substituting the Ratios
From the ratio Vz / V1 = 4, we can substitute:
Vz = 4V1
Substituting the expressions for Vz and V1 gives:
k / (Z - 1) = 4 * (k / (Z1 - 1))
By simplifying, we find:
Z1 - 1 = 4(Z - 1)
This leads to:
Z1 = 4Z - 3
Now, applying the second ratio Vz / V2 = 1/4:
Vz = (1/4)V2
Substituting gives:
k / (Z - 1) = (1/4) * (k / (Z2 - 1))
After simplification, we have:
Z2 - 1 = (1/4)(Z - 1)
Thus, we can express Z2 as:
Z2 = (1/4)(Z - 1) + 1 = (Z - 1)/4 + 1 = (Z + 3)/4
Final Results
In summary, the atomic numbers of the impurities in terms of Z are:
- Atomic number of impurity 1 (Z1): 4Z - 3
- Atomic number of impurity 2 (Z2): (Z + 3)/4
This analysis shows how the wavelengths of emitted X-rays can be used to deduce the atomic numbers of impurities based on their relationships with the primary target's atomic number. Understanding these relationships is crucial in fields like materials science and X-ray spectroscopy.
