The intensity of the peaks depends on the atomic scattering factor of each atom and their arrangement within the unit cell (structure factor).

Intensities of Diffracted Beams (Structure Factor $F_{hkl}$):

$I \propto |F_{hkl}|^2$.
$F_{hkl} = \sum_{j=1}^N f_j e^{2\pi i (hu_j + kv_j + lw_j)}$, where $f_j$ is the atomic scattering factor for atom $j$ at fractional coordinates $(u_j, v_j, w_j)$.
Certain reflections are systematically absent for specific crystal structures (e.g., for FCC, $h,k,l$ must be all odd or all even; for BCC, $h+k+l$ must be even). This helps in identifying the crystal system.

X-ray Diffractometer (XRD Equipment):

X-ray Tube: Generates X-rays (e.g., Cu anode, K$\alpha$ radiation).
Goniometer: Precision instrument that rotates the sample and detector, controlling the incident and scattering angles ($2\theta$).
Sample Holder: Mounts the material to be analyzed.
Detector: Measures the intensity of the diffracted X-rays at various $2\theta$ angles.

Crystal Structure Determination:

Obtaining an XRD pattern (intensity vs. $2\theta$).
Identifying peak positions and intensities.
Indexing: Assigning (hkl) Miller indices to each peak by matching $d_{hkl}$ values calculated from Bragg's Law to possible interplanar spacings for different crystal structures.
Refinement of lattice parameters.

Applications of XRD:

Electron & Neutron Diffractions:

Electron Diffraction: Uses electrons (de Broglie wavelength $\lambda = h/p$).
- Advantages: Stronger interaction with matter (good for thin films, surfaces), high spatial resolution (TEM).
- Disadvantages: High vacuum required, strong interaction makes bulk analysis difficult.
Neutron Diffraction: Uses neutrons.
- Advantages: Penetrates deeply, sensitive to light elements (H, D), sensitive to magnetic moments.
- Disadvantages: Requires a nuclear reactor or spallation source, expensive.

Introduction: Materials exhibiting two key phenomena below a critical temperature ($T_c$):
1. Zero Electrical Resistance: Current flows indefinitely without energy loss.
2. Meissner Effect: Complete expulsion of magnetic flux from the interior, making them perfect diamagnets.
Critical Parameters: Superconductivity is destroyed if temperature ($T$), magnetic field ($H$), or current density ($J$) exceeds critical values ($T_c, H_c, J_c$).
Types of Superconductors:
- Type I Superconductors:
  - Pure metals (e.g., Al, Pb, Hg).
  - Exhibit a sharp transition to the superconducting state.
  - Below $H_c$, they are perfect diamagnets (complete Meissner effect). Above $H_c$, normal state.
  - Limited practical applications due to low $T_c$ and $H_c$.