Calculation of Free Electron Energy for First Bragg Reflection in 1-Dimensional Crystal

Given data:
- Atomic spacing (v atomic space) = 5
- First Bragg reflection occurs at energy = 5.9 eV

Explanation:

1. Understanding Bragg Reflection:
When X-rays or other types of electromagnetic radiation are incident on a crystal, they can be scattered by the atoms in the crystal lattice. This scattering phenomenon is known as Bragg reflection.

2. Bragg's Law:
Bragg's law is given by the equation: nλ = 2d sinθ
- n is the order of the reflection
- λ is the wavelength of the incident radiation
- d is the spacing between atomic planes in the crystal lattice
- θ is the angle of incidence

In the given problem, we have a 1-dimensional crystal, so we can consider the angle of incidence as 90 degrees. Therefore, Bragg's law simplifies to: nλ = 2d

3. Calculation of Wavelength:
We know that the energy of a photon is related to its wavelength by the equation: E = hc/λ, where h is the Planck's constant and c is the speed of light.

Since the energy for the first Bragg reflection is given as 5.9 eV, we can convert it to joules by using the conversion factor: 1 eV = 1.602 x 10^-19 J.

Therefore, the energy in joules is: E = 5.9 x 1.602 x 10^-19 J

We can rearrange the equation to solve for wavelength: λ = hc/E

4. Calculation of Atomic Spacing:
In a 1-dimensional crystal, the atomic spacing is given by the expression: d = v atomic space

5. Calculation of Free Electron Energy:
Now, we can substitute the values of wavelength and atomic spacing in Bragg's law to find the free electron energy.

nλ = 2d
n(hc/E) = 2(v atomic space)
n(hc) = 2(v atomic space)E
E = (2n(hc))/(v atomic space)

Substituting the given values and solving the equation will give us the free electron energy for the first Bragg reflection.

Conclusion:
By following the above steps and calculations, we can find the free electron energy for the first Bragg reflection in a 1-dimensional crystal. The correct answer is 5.9 eV.