Engineering physics , The Emission - Einstein's Co-Efficients Derivation

 Einstein's derivation of the emission and absorption coefficients involves the fundamental principles of quantum mechanics and statistical physics. Let's break down the derivation:




 Basic Setup:




Consider a photon interacting with an atom in a gas. The atom can absorb or emit a photon depending on its energy states. Let the energy levels be represented by two states:

- E₁: Lower energy level (ground state).

- E₂: Higher energy level (excited state).




 Key Assumptions:

 

1. The atom can absorb a photon, transitioning from the lower energy state (E₁) to the higher energy state (E₂).

2. The atom can emit a photon in two ways:

   - **Spontaneous Emission**: The atom transitions from the higher energy state (E₂) to the lower energy state (E₁) without any external influence, emitting a photon.

   - **Stimulated Emission**: The atom transitions from the higher energy state (E₂) to the lower energy state (E₁) due to the interaction with an incoming photon, emitting a photon in the process.




### Einstein Coefficients:

In the context of radiation-matter interaction, Einstein introduced the following coefficients:

- **A₁₂**: The **spontaneous emission coefficient**, which describes the probability per unit time that an atom in the excited state E₂ will spontaneously emit a photon and drop to the ground state E₁.

- **B₁₂**: The **absorption coefficient**, which describes the probability per unit time that an atom in the ground state E₁ will absorb a photon and transition to the excited state E₂.

- **B₂₁**: The **stimulated emission coefficient**, which describes the probability per unit time that an atom in the excited state E₂ will emit a photon (due to the presence of another photon) and drop to the ground state E₁.




### The Einstein Relation:

Einstein derived a relationship between these coefficients, which is essential in understanding how radiation interacts with matter.




The detailed process of the derivation involves the concept of detailed balance. The system must be in thermal equilibrium, meaning the rate of absorption must equal the rate of emission (both spontaneous and stimulated). This is what leads to the famous Einstein relation.




#### Energy Density and Photon Flux:

Let \( \rho(\nu) \) be the energy density of radiation at frequency \( \nu \), and let \( n_1 \) and \( n_2 \) represent the populations of atoms in the ground state (E₁) and excited state (E₂), respectively.




- **Absorption rate**: The rate of absorption depends on the energy density of the radiation and the transition probability \( B_{12} \):

  \[

  \text{Absorption rate} = B_{12} \rho(\nu) n_1

  \]




- **Spontaneous emission rate**: The rate of spontaneous emission depends on the probability of spontaneous emission \( A_{21} \) and the population of atoms in the excited state \( n_2 \):

  \[

  \text{Spontaneous emission rate} = A_{21} n_2

  \]




- **Stimulated emission rate**: The rate of stimulated emission depends on the energy density of the radiation and the probability of stimulated emission \( B_{21} \):

  \[

  \text{Stimulated emission rate} = B_{21} \rho(\nu) n_2

  \]




### Detailed Balance Condition:

At thermal equilibrium, the rate of absorption equals the rate of emission (the sum of spontaneous and stimulated emissions):




\[

B_{12} \rho(\nu) n_1 = A_{21} n_2 + B_{21} \rho(\nu) n_2

\]




Using the Boltzmann distribution for the populations of the two energy states, we have:




\[

\frac{n_2}{n_1} = \exp\left(\frac{-h\nu}{kT}\right)

\]




where \( h \) is Planck's constant, \( \nu \) is the frequency of the radiation, \( k \) is the Boltzmann constant, and \( T \) is the temperature.




### Einstein's Coefficients Relation:

Now, by combining the above equation and applying the equilibrium condition, Einstein derived the following relationship between the coefficients:







\frac{B_{12}}{B_{21}} = \frac{g_2}{g_1} \quad \text{and} \quad A_{21} = \frac{8 \pi h \nu^3}{c^3} B_{21}







where \( g_1 \) and \( g_2 \) are the degeneracy factors (the number of available quantum states for each energy level), and \( c \) is the speed of light.




This relation shows how the absorption and stimulated emission coefficients are related to the spontaneous emission coefficient, and how the overall radiation field behaves in thermal equilibrium.




### Summary:

The Einstein coefficients are derived through a combination of quantum mechanics and statistical mechanics, with the assumption of detailed balance in thermal equilibrium. The relation between the coefficients allows us to understand the absorption, spontaneous emission, and stimulated emission processes in a system of atoms interacting with electromagnetic radiation.