Core Concepts in Radiation Physics, Solar Cells, and Electromagnetism

Posted on Jan 27, 2026 in Physics

Radiation Interaction with Matter

1. Alpha (α) Particles

• Heavy, positively charged particles.
• Interact strongly with matter, leading to high ionization.
• Lose energy quickly and stop within a few centimeters of air.

2. Beta (β) Particles

• Fast-moving electrons.
• Moderate interaction, resulting in moderate ionization.
• Can travel a few meters in air.

3. Gamma (γ) Rays

• Electromagnetic waves (no mass, no charge).
• Weak interaction, causing low ionization.
• Travel long distances and penetrate deeply.

Penetrating Power Comparison

• Alpha → Lowest (stopped by paper or skin)
• Beta → Medium (stopped by plastic or aluminum)
• Gamma → Highest (needs thick lead or concrete)

Ionization Capability

• Alpha → Highest ionization (strong collisions, slow speed)
• Beta → Moderate ionization
• Gamma → Lowest ionization (no charge)

Radiation Shielding Requirements

• Alpha: Paper sheet or skin is sufficient.
• Beta: Thin metal sheet (aluminum) or plastic.
• Gamma: Thick lead, concrete, or heavy materials required.

Summary: Ionization vs. Penetration

• Alpha = Strong ionizer, weak penetration
• Beta = Medium ionizer, medium penetration
• Gamma = Weak ionizer, strong penetration

Radiation Detectors: GM Counter

Working Principle

• Radiation ionizes gas inside the GM tube.
• This creates a big avalanche of charges, resulting in one counting pulse.

Use

• Primarily counts radiation events.

Advantages

• Cheap, simple, and very sensitive.

Radiation Detectors: Scintillation

Working Principle

• Radiation hits a scintillator, producing light flashes.
• A Photomultiplier Tube (PMT) converts light into an electrical pulse (the size of which indicates energy).

Use

• Measures radiation energy.

Advantages

• Fast, high efficiency, and good energy resolution.

Radiation Detectors: Semiconductor

Working Principle

• Radiation creates electron–hole pairs in silicon or germanium.
• These charges generate a pulse directly proportional to the radiation energy.

Use

• Precise X-ray and gamma-ray measurement.

Advantages

• Best energy resolution, compact, and very accurate.

Detector Comparison Summary

• GM Counter → Counts radiation only (poor energy information).
• Scintillation → Counts + gives energy (good resolution).
• Semiconductor → Counts + very accurate energy (best resolution).

10-Step Silicon Solar Cell Fabrication Process

1. Silicon Wafer Preparation

Process:

Start with high-purity monocrystalline Si (Czochralski method).

Principle:

Pure Si leads to fewer recombination centers.

Parameters:

Resistivity 1–10 Ω·cm, thickness approximately 180 μm.

Effect:

Higher purity results in higher efficiency.

2. Surface Cleaning

Process:

Chemical cleaning (RCA clean) removes dust and metal ions.

Principle:

Removes defects, which reduces electron–hole recombination.

Parameters:

NH₄OH, H₂O₂, HCl at 70–80°C.

Effect:

Clean surface ensures better junction quality.

3. Surface Texturing

Process:

Etching pyramids on Si using KOH or NaOH.

Principle:

Traps light, thereby increasing absorption.

Parameters:

KOH 1–2%, 70–80°C, 15–20 min.

Effect:

Less reflection results in more current (J_sc).

4. N-Type Doping (Emitter Formation)

Process:

Diffuse phosphorus (POCl₃) onto p-type Si.

Principle:

Forms the p–n junction, which separates charges.

Parameters:

800–900°C furnace diffusion.

Effect:

A good emitter leads to higher voltage (V_oc).

5. Phosphorus Glass Removal (PGR)

Process:

Remove phosphorus silicate glass formed during diffusion.

Principle:

Ensures a clean junction for electrical contacts.

Parameters:

HF dip for 1–2 min.

Effect:

Reduces contact resistance.

6. Edge Isolation

Process:

Remove short-circuit paths along wafer edges using plasma etching or laser.

Principle:

Prevents p–n junction shunting.

Effect:

Increases voltage and reduces leakage.

7. Anti-Reflective Coating (ARC)

Process:

Deposit SiNx (silicon nitride) layer using PECVD.

Principle:

Minimizes surface reflection and passivates surface defects.

Parameters:

Thickness approximately 70–80 nm.

Effect:

More absorbed light plus lower recombination results in higher efficiency.

8. Front Contact (Metal Grid) Formation

Process:

Screen printing of Ag paste, followed by high-temperature firing.

Principle:

Provides a low-resistance path for electrons.

Parameters:

Line width < 100 μm, firing ~700–800°C.

Effect:

A fine grid means less shading and better current collection.

9. Back Surface Field (BSF) Formation

Process:

Aluminum paste applied to the back, followed by heat treatment to create a p+ region.

Principle:

Pushes minority carriers away, reducing recombination at the back surface.

Effect:

Better carrier lifetime results in higher voltage (V_oc).

10. Back Contact Metallization

Process:

Print full-area Al or Ag on the backside and fire.

Principle:

Forms an ohmic contact for current collection.

Effect:

A good back contact ensures a higher fill factor (FF).

This section explains the solar cell I–V equation, including series and shunt resistance, derives expressions for V_oc, I_sc, and Fill Factor, and shows how parasitic resistances reduce efficiency.

Solar Cell I–V Equation and Parasitic Resistance

The practical solar cell model includes a current source (I_L), a diode, series resistance (R_s), and shunt resistance (R_sh).

The I–V equation is:
I = I_L − I₀ [exp((V + I R_s) / (n V_T)) − 1] − (V + I R_s) / R_sh

R_s appears in the exponential term and reduces current.
R_sh appears as a leakage path and reduces voltage.

Short-Circuit Current (I_sc) Derivation

At V = 0,
I_sc = I_L − I₀ [exp(I_sc R_s / n V_T) − 1] − (I_sc R_s / R_sh)

If R_s is small and R_sh is large (ideal case),
I_sc ≈ I_L

Open-Circuit Voltage (V_oc) Derivation

At I = 0,
0 = I_L − I₀ [exp(V_oc / n V_T) − 1] − V_oc / R_sh

If R_sh approaches infinity,
V_oc = n V_T ln(I_L / I₀ + 1)

V_oc increases when I_L increases and I₀ decreases.

Fill Factor (FF) Calculation

FF = (V_mp × I_mp) / (V_oc × I_sc)

Approximate expression:
FF ≈ [v_oc − ln(v_oc + 0.72)] / (v_oc + 1)

where v_oc = V_oc / (n V_T)

A high FF indicates a good-quality solar cell.

Impact of Series and Shunt Resistance

Series Resistance (R_s):

• Reduces current and the slope near I_sc.
• Reduces I_mp and FF.
• Lowers overall efficiency.

Shunt Resistance (R_sh):

• Provides a leakage path.
• Reduces V_oc.
• Reduces FF.
• Lowers overall efficiency.

This section derives the boundary conditions for electromagnetic fields at the interface between two dielectrics using Gauss’s and Stokes’s theorems, showing how tangential E, tangential H, normal D, and normal B vary across the boundary.

Electromagnetic Boundary Conditions

1. Normal Component of Electric Flux Density (D)

Use Gauss’s law:
∮ D · n dS = ρ_s

By choosing a small pillbox across the boundary, we find:

D_1n − D_2n = ρ_s

If there is no surface charge (ρ_s = 0):
D_1n = D_2n
(→ Normal D is continuous)

Since D = εE:
ε₁ E_1n = ε₂ E_2n

2. Tangential Component of Electric Field (E)

Use Faraday’s law (Stokes’s theorem):
∮ E · dl = − dΦ_B/dt

For a very small rectangular loop at the surface, the magnetic flux change Φ_B ≈ 0.

Therefore:
E_1t = E_2t
(→ Tangential E is always continuous)

3. Normal Component of Magnetic Flux Density (B)

Use Gauss’s law for magnetism:
∮ B · n dS = 0

The pillbox argument yields:
B_1n = B_2n
(→ Normal B is always continuous)

4. Tangential Component of Magnetic Field (H)

Use Ampere’s law (Stokes’s theorem):
∮ H · dl = J_s + dΦ_D/dt

For a small loop, the displacement current change dΦ_D/dt is negligible:

H_1t − H_2t = J_s
(→ Discontinuity depends on surface current density J_s)

If no surface current (J_s = 0):
H_1t = H_2t
(→ Tangential H is continuous)

Summary of Boundary Conditions

1. Tangential Electric Field
   E_1t = E_2t (Always continuous)

2. Normal Electric Flux Density
   D_1n − D_2n = ρ_s
   If ρ_s = 0 → D_1n = D_2n

3. Normal Magnetic Flux Density
   B_1n = B_2n (Always continuous)

4. Tangential Magnetic Field
   H_1t − H_2t = J_s
   If J_s = 0 → H_1t = H_2t

This section derives Poynting’s theorem from Maxwell’s equations, defines the Poynting vector, explains energy conservation, and relates intensity to electric field amplitude.

Poynting’s Theorem and Energy Flow

1. Derivation of Poynting’s Theorem

Start with Maxwell’s equations:

1. Faraday’s law
   ∇ × E = − ∂B/∂t

2. Ampere–Maxwell law
   ∇ × H = J + ∂D/∂t

Step 1: Form the Dot Products

Take the dot product of H with Faraday’s law:

H · (∇ × E) = − H · ∂B/∂t …(1)

Take the dot product of E with the Ampere–Maxwell law:

E · (∇ × H) = E · J + E · ∂D/∂t …(2)

Subtract (1) from (2):

E · (∇ × H) − H · (∇ × E) = E · J + E · ∂D/∂t + H · ∂B/∂t

Step 2: Use Vector Identity

Vector identity:
∇ · (E × H) = H · (∇ × E) − E · (∇ × H)

Rearranging the previous equation using the identity:

− ∇ · (E × H) = E · J + E · ∂D/∂t + H · ∂B/∂t

Step 3: Use Constitutive Relations

Assuming linear media (D = εE and B = μH):

E · ∂D/∂t = ∂/∂t (½ εE²)
H · ∂B/∂t = ∂/∂t (½ μH²)

Substituting these back:

− ∇ · (E × H) = E · J + ∂/∂t (½ εE² + ½ μH²)

Final Poynting Theorem

∇ · S + ∂u/∂t + E · J = 0

where S = E × H (Poynting Vector) and
u = ½ (εE² + μH²) is the electromagnetic energy density.

This equation expresses energy conservation.

2. The Poynting Vector (S)

S = E × H

• Gives power flow per unit area (W/m²).
• Its direction is the direction of energy propagation.
• Its magnitude is the instantaneous energy flux.

3. Principle of Energy Conservation

Poynting’s theorem states that the sum of:

• Energy leaving a volume (∇·S)
• Increase in stored EM energy (∂u/∂t)
• Work done on charges (E·J)

must equal zero. Total EM energy is conserved at every point in space.

4. Intensity and Electric Field Amplitude

For a time-harmonic plane wave:

Average intensity (I):

I = ½ E₀ H₀

Since H₀ = E₀ / η (where η is the intrinsic impedance),

I = E₀² / (2η)

Thus:

Intensity ∝ (Electric field amplitude)²

A higher electric field amplitude corresponds to a higher energy flow.

This section explains phase velocity and group velocity, derives both from the dispersion relation, shows the v_p × v_g = c² relationship, and explains why group velocity represents the speed of information.

Phase Velocity, Group Velocity, and Dispersion

1. Phase Velocity (v_p)

For a single-frequency wave:
E(x,t) = E₀ cos(ωt − kx)

Phase velocity is the speed at which a single-frequency phase (e.g., a peak) travels.

v_p = ω / k

2. Group Velocity (v_g)

A wave packet is formed by the superposition of many frequencies. Its envelope travels at:

v_g = dω / dk

Group velocity represents the speed of the wave energy or information.

3. Derivation from Dispersion Relation

A medium is characterized by its dispersion relation: ω = ω(k).

Phase velocity is derived as the ratio:
v_p = ω / k

Group velocity is derived as the derivative:
v_g = dω / dk

Both velocities are determined directly from the function ω(k).

4. Relation v_p × v_g = c²

For an electromagnetic wave in a lossless, non-magnetic dielectric, the relationship v_p v_g = c² holds true only in a nondispersive medium (where the refractive index n is constant).

In general, for a dispersive medium:

v_p = c / n
v_g = c / (n + ω dn/dω)

If the medium is nondispersive (dn/dω = 0):

v_p v_g = (c/n) × (c/n) = c² / n²

If n = 1 (vacuum), then v_p v_g = c².

5. Group Velocity as Information Speed

• Phase velocity only moves the peaks of a continuous sinusoidal wave, which carries no unique information.
• A signal or pulse requires modulation, forming a wave packet.
• The envelope of this packet, which contains the energy and modulation, moves at v_g.

Therefore, information, energy, and power flow travel at the group velocity, not the phase velocity.

This section describes rectangular and circular waveguides, derives the TE₁₀ cutoff frequency, and explains the behavior of waves below and above cutoff.

Waveguides: Types and Cutoff Frequency

1. Rectangular and Circular Waveguides

Rectangular Waveguide

• Cross-section: a × b.
• Supports Transverse Electric (TE) and Transverse Magnetic (TM) modes.
• Dominant mode: TE₁₀.
• The electric field varies sinusoidally across the wider dimension (a).

Circular Waveguide

• Circular cross-section (radius = a).
• Supports TE and TM modes.
• Dominant mode: TE₁₁.
• Field variation is described using Bessel functions.

2. Derivation of TE₁₀ Cutoff Frequency

For TE modes (E_z = 0), the cutoff wavenumber (k_c) is given by:

k_c² = (mπ/a)² + (nπ/b)²

For the TE₁₀ mode (the dominant mode):

m = 1, n = 0 →
k_c = π / a

The cutoff frequency (f_c) is related to k_c by f_c = (k_c / 2π) c:

f_c = (π / a) × (c / 2π)

Thus for TE₁₀:

f_c = c / (2a)

3. Waveguide Behavior Below and Above Cutoff

Below Cutoff (f < f_c)

• The propagation constant becomes imaginary.
• Fields decay exponentially along the waveguide axis.
• No power transmission occurs.
• The waveguide acts as a high-pass filter.

Above Cutoff (f > f_c)

• The propagation constant is real.
• The wave propagates with low attenuation.
• Phase velocity (v_p) is greater than c.
• Group velocity (v_g) is less than c (representing the energy speed).