Consider a metal used to produced some characteristic X-ray, energy of X-ray are given by `E` and wavelength as represented by `lambda`. Then which of the following is true:

Options

A. `E(K_(alpha)) gt E(K_(beta)) gt E(K_(gamma))`
B. `E(M_(alpha)) gt E(L_(alpha)) gt E(K_(alpha))`
C. `lambda(K_(alpha)) gt lambda(K_(beta)) gt lambda(K_(gamma))`
D. `lambda(M_(alpha)) gt lambda(L_(alpha)) gt lamda(K_(alpha))`

Explanation

To solve the problem regarding the characteristic X-ray energies and wavelengths, we will follow these steps:

### Step 1: Understand the Concept of Characteristic X-rays
Characteristic X-rays are emitted when electrons transition between energy levels in an atom. The energy of the emitted X-ray is equal to the difference in energy between the two levels involved in the transition.

### Step 2: Use the Energy-Wavelength Relationship
The energy (E) of a photon is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
where \(h\) is Planck's constant and \(c\) is the speed of light. This implies that energy is inversely proportional to wavelength.

### Step 3: Identify the Energy Levels
For the K, L, and M shells, the energy levels can be calculated using the formula:
\[ E_n = -\frac{13.6 \text{ eV}}{n^2} \]
where \(n\) is the principal quantum number (1 for K, 2 for L, and 3 for M).

### Step 4: Calculate the Energy Differences for K-series
1. For Kα (transition from L to K):
- \(E_{K} = -13.6 \text{ eV}\) (n=1)
- \(E_{L} = -3.4 \text{ eV}\) (n=2)
- Energy of Kα = \(E_{K} - E_{L} = -13.6 + 3.4 = 10.2 \text{ eV}\)

2. For Kβ (transition from M to K):
- \(E_{M} = -1.51 \text{ eV}\) (n=3)
- Energy of Kβ = \(E_{K} - E_{M} = -13.6 + 1.51 = 12.09 \text{ eV}\)

3. For Kγ (transition from N to K):
- \(E_{N} = -0.85 \text{ eV}\) (n=4)
- Energy of Kγ = \(E_{K} - E_{N} = -13.6 + 0.85 = 12.75 \text{ eV}\)

### Step 5: Compare Energies
From the calculations:
- \(E_{Kα} = 10.2 \text{ eV}\)
- \(E_{Kβ} = 12.09 \text{ eV}\)
- \(E_{Kγ} = 12.75 \text{ eV}\)

Thus, the order of energy is:
\[ E_{Kα} < E_{Kβ} < E_{Kγ} \]

### Step 6: Analyze the Options
Given the relationships derived, we can analyze the options provided in the question. If an option states that \(E_{Kα} > E_{Kβ} > E_{Kγ}\), it is incorrect. The correct relationships should reflect the order derived from the calculations.

### Conclusion
Based on the analysis, the correct statements regarding the energies and wavelengths of the X-rays can be determined from the calculations.