Visit: 
http://smartstudyzone.in
For High-Quality, More Free Study Notes
In the study of quantum mechanics and radiation, we encounter one of the most elegant formulations describing the interaction between matter and electromagnetic radiation: Einstein coefficients. These coefficients provide a quantitative framework to understand how atoms absorb and emit radiation, forming the backbone of modern technologies such as lasers, spectroscopy, and optical communication systems.
👉 Why it matters?
From barcode scanners to fiber optics and medical lasers, this concept is the backbone of modern technology.
📌 By the end of this article, you’ll understand:
- The three Einstein processes
- Mathematical derivation (step-by-step)
- Physical meaning of each coefficient
- Real-world applications
- Exam-focused numericals
2.📜 Historical Background:
In 1917, the world was in the midst of a quantum revolution. Max Planck had already described blackbody radiation, but the mechanism of how atoms actually exchanged energy with a radiation field remained a mystery. Albert Einstein, never satisfied with incomplete pictures, proposed a phenomenological approach.
He realized that the then-current understanding of light (absorption and spontaneous emission) wasn’t enough to achieve thermal equilibrium. He predicted a third process—stimulated emission—which laid the theoretical foundation for the MASER and eventually the LASER (Light Amplification by Stimulated Emission of Radiation).
3. ⚙️ Fundamental Concepts of Atomic Transitions:
To understand the coefficients, we must look at a simple two-level atomic system with energy states E1 (ground state) and E2 (excited state). Let N1 and N2 be the number of atoms per unit volume in these states, respectively.
Suppose the energy difference between the two energy states is given by:
E2 – E1 = hν
where h is Planck’s constant and ν is the frequency of radiation.
There are three primary processes through which transitions occur:
3. 1. Induced Absorption:
Imagine an atom sitting quietly in the ground state E1. If a photon with energy hν = E2 – E1 hits it, the atom “absorbs” the photon and jumps to E2. The rate of this process depends on how many atoms are available in the ground state (N1) and how much light (radiation density u(ν)) is hitting them.
Hence, the rate of induced absorption is given by:
Here, B₁₂ is the Einstein coefficient of absorption (also called the induced absorption coefficient). It tells us how readily atoms in state 1 absorb radiation of density ρ(ν).
3. 2. Spontaneous Emission:
An atom in the excited state E₂ is inherently unstable. Since the lifetime of an atom in the excited state is in the order of 10⁻⁸ s. After this time, it will eventually fall back to E₁ and emit a photon. This happens randomly, with no preferred direction and no fixed phase. The rate at which this occurs depends only on how many atoms are in the excited state.
Hence, the rate of spontaneous emission is:
3. 3. Stimulated Emission:
This is the “Laser Process.” When a photon (incoming or emitted by spontaneous emission) of energy hν encounters an atom already sitting in the excited state E₂, it does not get absorbed—it instead triggers the atom to emit a second photon and come down to the ground state.
Critically, this second photon is completely identical to the first: same frequency, same direction of travel, same polarisation, and same phase. You start with one photon and end up with two. This is the mechanism of optical amplification.
The rate of stimulated emission is:
4. Einstein's Coefficients:
Let us assume the system is in thermal equilibrium at temperature T.
In equilibrium, the number of transitions going “up” must be equal to the number of transitions coming “down.” Therefore:
To find the radiation density u(ν), we rearrange the equation:
Now, let’s divide the numerator and denominator by B21N2:
From Maxwell-Boltzmann Statistics:
$$N_1=N_0e^{-E_1/kT}$$
$$N_2=N_0e^{-E_2/kT}$$
Therefore, the ratio of atoms in the two states is:
Substituting this into equation (4):
We also know from Planck’s Law of Radiation that:
By comparing these two equations (5) and (6), we can conclude two vital points:
First, the probability of absorption must equal the probability of stimulated emission for a non-degenerate system:
Second, the ratio of spontaneous to stimulated emission coefficients is:
This ratio shows that as the frequency ν increases (like moving from infrared to X-rays), spontaneous emission becomes much more dominant, making it very difficult to create lasers at very high frequencies.
5. ⚖️ Comparison: Spontaneous vs. Stimulated Emission:
| S. No. | Property | Spontaneous Emission (A₂₁) | Stimulated Emission (B₂₁) |
|---|---|---|---|
1. |
Trigger |
No external field needed; happens naturally |
Requires an incoming photon of energy hν |
2. |
Coherence |
Incoherent — random phase, random direction |
Coherent—identical phase, direction, frequency |
3. |
Photon count |
1 photon out, no amplification |
2 photons out (1 in → 2 out): amplification |
4. |
Rate dependence |
Depends only on N₂ and A₂₁ |
Depends on N₂, B₂₁, and ρ(ν) |
5. |
Frequency dependence |
Increases rapidly with ν (∝ ν³) |
Relatively independent of ν |
6. ✅ Advantages and Limitations:
✅ Advantages
❌ Limitations
- Explains the laser working principle
- Predicts stimulated emission
- Connects quantum theory with radiation
- Assumes thermal equilibrium
- Does not explain population inversion directly
- Simplified atomic model
7. 🚀 Applications of Einstein's Coefficients in Laser:
- Gain Medium Selection: Engineers choose gain media (e.g., Nd: YAG, CO₂, HeNe) based on the B₂₁ value—a high B₂₁ means more stimulated emission per unit of pump energy, increasing efficiency.
- Population Inversion Threshold: The condition for net gain requires N₂ > N₁ (for equal degeneracies), which directly follows from the B₁₂ = B₂₁ relationship — if the coefficients were unequal, the threshold condition would change completely.
- Optical Fibre Amplifiers (EDFA): Erbium-doped fibre amplifiers used in telecommunications are designed using the A₂₁ and B₂₁ values of the Er³⁺ ion at 1550 nm—the exact wavelength where optical fibre losses are minimum.
- Medical Laser Surgery: Pulsed lasers for LASIK eye surgery and tumour treatment are timed using the spontaneous emission lifetime (1/A₂₁) to control pulse duration and ensure tissue-specific energy deposition.
- Laser Spectroscopy and Atomic Clocks: In atomic clocks (e.g., cesium fountain clocks) and laser spectroscopy, the A₂₁ coefficient determines the natural linewidth of the transition, which sets the ultimate frequency precision of the measurement.
8. ⚡ Quick Answer Section:
What are Einstein coefficients?
Einstein’s coefficients (A and B) are mathematical constants that describe the probability of an atom absorbing or emitting light. A represents spontaneous emission, while B represents induced absorption and stimulated emission.
What are Einstein's A and B coefficients?
Einstein’s A coefficient (A₂₁) is the probability per unit time of spontaneous emission from an excited atom. The B coefficients (B₁₂ and B₂₁) are the probabilities of induced absorption and stimulated emission, respectively, when a radiation field of density ρ(ν) is present.
What is A21 coefficient?
It represents the probability of spontaneous emission per unit time.
What is B12 coefficient?
It represents the probability of absorption transition.
Why B12 = B21?
Because absorption and stimulated emission probabilities are equal under equilibrium conditions.
What is the relationship between Einstein's A and B coefficients?
The key relationships are: B₁₂ = B₂₁ (absorption and stimulated emission have equal probability for non-degenerate levels), and A₂₁ = (8πhν³/c³)·B₂₁. This means A₂₁ grows as the cube of frequency — making spontaneous emission dominant at high frequencies (X-rays) and stimulated emission dominant at low frequencies (microwaves, infrared).
Who derived the Einstein coefficients and when?
Albert Einstein derived the A and B coefficients in 1917 in his paper “On the Quantum Theory of Radiation.” He used the condition that his coefficients must reproduce Planck’s blackbody radiation law at thermal equilibrium.
Why is A₂₁/B₂₁ important in laser design?
The ratio A₂₁/B₂₁ = 8πhν³/c³ tells engineers how much the spontaneous emission “noise” competes with useful stimulated emission “gain.” A smaller ratio (lower frequency transitions) means stimulated emission dominates, making lasing easier. This is why infrared and visible lasers are far more common than X-ray lasers.
What is the SI unit of Einstein's A and B coefficients?
The A coefficient has units of s⁻¹ (per second), since it is a transition rate per unit time. The B coefficient has units of m³/(J·s²) or equivalently m³·kg⁻¹·s⁻², since it multiplies the spectral energy density ρ(ν) in J·m⁻³·Hz⁻¹ to give a rate in s⁻¹.
What is the physical significance of B₁₂ = B₂₁?
The equality B₁₂ = B₂₁ means light interacts with atoms with equal probability, whether it causes absorption or stimulated emission. The reason absorption usually dominates in everyday materials is simply that N₁ > N₂ (more atoms in the ground state). In a laser, we reverse this—achieving population inversion—so stimulated emission wins.
How do Einstein's coefficients relate to population inversion?
For net optical gain, stimulated emission must exceed absorption, which requires B₂₁·N₂ > B₁₂·N₁. Since B₁₂ = B₂₁, this simplifies to N₂ > N₁ — population inversion. Einstein’s coefficient equality is precisely why population inversion (and not some other condition) is the threshold requirement for laser action.
9. 🧠 Conclusion
Einstein’s coefficients form the foundation of laser physics, linking quantum theory with real-world optical systems. They explain how light interacts with matter and why lasers produce highly directional and coherent beams. From just three rates (A₂₁, B₁₂, and B₂₁) and one equilibrium condition, Einstein revealed the following:
- Stimulated emission must exist as a fundamental quantum process.
- The probability of stimulated emission exactly equals that of absorption (B₁₂ = B₂₁).
- Spontaneous emission dominates at high frequencies via the ν³ factor in A₂₁/B₂₁.
- Population inversion is the necessary and sufficient condition for optical gain.
10. 📚 PYQs & Most Expected Exam Questions
- Define Einstein’s coefficients A₂₁, B₁₂, and B₂₁. State their physical significance.
- Derive the relation between Einstein’s A and B coefficients using Planck’s radiation law.
- Show that B₁₂ = B₂₁ for non-degenerate energy levels. What does this imply for laser design?
- Explain why population inversion is necessary for laser action using Einstein’s coefficient theory.
- The spontaneous emission rate of an atom at frequency 5×10¹⁴ Hz is 3×10⁷ s⁻¹. Calculate the stimulated emission coefficient B₂₁.
- What is the ratio A₂₁/B₂₁? How does it vary with frequency, and what are its implications for X-ray laser development?
- Compare spontaneous and stimulated emission with respect to coherence, directionality, and their roles in laser operation.
- Write short notes on: (a) Einstein’s A coefficient, (b) Einstein’s B coefficient, (c) stimulated emission.
11. ✅ Solved Numerical Examples
11.1. Calculate Einstein's coefficients' ratio:
Question: Calculate the ratio of Einstein’s A and B coefficients for a transition at frequency ν = 6.0 × 10¹⁴ Hz.
ν = 6.0 × 10¹⁴ Hz (visible light, orange-red region)
h = 6.626 × 10⁻³⁴ J·s
c = 3.0 × 10⁸ m/s
A₂₁ / B₂₁ = ?
Using the fundamental Einstein relation derived from Planck’s law:
Substituting the values:
$$\frac{A_{21}}{B_{21}} = \frac{3.596 × 10^{12}}{2.7×10^{25}}$$
11.2. Calculate the spontaneous emission coefficient:
Question: For a gas laser transition at ν = 4.74 × 10¹⁴ Hz, the Einstein B coefficient for stimulated emission is B₂₁ = 2.05 × 10²⁰ m³/(J·s²). Calculate the spontaneous emission coefficient A₂₁ and the natural lifetime of the excited state.
ν = 4.74 × 10¹⁴ Hz
B₂₁ = 2.05 × 10²⁰ m³/(J·s²)
h = 6.626 × 10⁻³⁴ J·s, c = 3.0 × 10⁸ m/s
A₂₁ = ? and natural lifetime τ = 1/A₂₁ = ?
Applying the Einstein relation $$A_{21} =\frac{8\pi h\nu^3}{c^3}\times B_{21}$$
$$A_{21} = 6.98×10^{-14} × 2.05×10^{20} = 1.43×10^7 s⁻¹$$
12. ❓ FAQs (People Also Ask):
Q:Can we have a laser without stimulated emission?
A: No. Stimulated emission is the "amplification" part of a laser. Without it, you just have a standard light source like an LED.
Q: Does temperature affect the A and B coefficients?
A: No, the coefficients themselves are constants for a specific transition. However, temperature affects the population of the states (N1 and N2), which changes the overall rate of the process.
Q: What units are used for these coefficients?
A: A is in s⁻¹ (probability per unit time), while B is typically m3/(J.s2) when used with radiation density.
Q: Can Einstein's B coefficient be negative?
No, B coefficients cannot be negative—they represent probabilities per unit time, which must always be non-negative. However, the net gain coefficient in a laser medium can be written as proportional to (N₂ − N₁), which is negative when N₁ > N₂ (normal thermal equilibrium), leading to net absorption. It becomes positive only under population inversion, enabling amplification.
Q: Do Einstein's coefficients apply to semiconductor lasers ?
Yes, but with modifications. In semiconductor lasers, the discrete atomic energy levels are replaced by energy bands (valence band and conduction band), and the Einstein coefficients become band-to-band transition rates integrated over the joint density of states.
Q: Can Einstein coefficients vary?
Yes, they depend on atomic structure and frequency.