Two Level System with thermal population

In summary, we have discussed a two-level system with thermal population and shown the rate equation for state N_2. We have also shown that using such a system, it is impossible to build a CW Laser. Finally, we have calculated the progress of N_2(t) when stimulated by a constant monochromatic signal and provided a solution for N_2 as a function of time.
  • #1
ktolaus
1
0
Hello,

the task is the following:

Consider a two-level system with thermal population.

a) Show that the rate equation for the state [itex]N_2[/itex] is the following:

[itex]\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}[/itex]

[itex]w=B_{12}\rho({\nu}) \,\,\, N_2=N_2^p+N_2^e\,\,\, N_1=N-N_2[/itex]

[itex]N_2^p[/itex] is the portion of the total population caused by pumping.

b) Show, that using such a system it is impossible to build a CW Laser.

c) Calculate the progress of [itex]N_2(t)[/itex] when the system is stimulated by a constant monochromatic signal with spectral density I and frequency [itex]h\nu=(E_2-E_1)[/itex]. For t=0 only thermal population exist.

My ideas are the following:

a) [itex]\frac{dN_2}{dt}[/itex] is just the sum of absorption, stimulated emission and spontaneous emission. Spontaneous emission occurs only from the the "pumped portion":
[itex]\frac{dN_2}{dt}=B_{12}\rho({\nu})N_1-B_{21}\rho({\nu})N_2-A_{21}N_2^p[/itex]
[itex]\frac{dN_2}{dt}=B_{12}\rho({\nu})N_1-B_{12}\rho({\nu})N_2-A_{21}(N_2-N_2^e)[/itex]
using [itex]A_{21}=\frac{1}{\tau}[/itex] leads to:
[itex]\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}[/itex]

b) I'm not sure, but I think CW is only possible if [itex]\frac{d^2N_2}{dt^2}=0[/itex]:
[itex]\frac{d^2N_2}{dt^2}=B_{12}\rho(\frac{dN_1}{dt}-\frac{dN_2}{dt})-\frac{1}{\tau}\frac{dN_2}{dt}+\frac{1}{\tau}\frac{dN_2^e}{dt}[/itex]
[itex]N_2^e[/itex] should be independent on time. Using [itex]\frac{dN_1}{dt}=-\frac{dN_2}{dt}[/itex] leads to:
[itex]0=-2B_{12}\rho\frac{dN_2}{dt}-\frac{1}{\tau}\frac{dN_2}{dt}[/itex]

This can't be 0 since the left term depends on the frequency but the right term doesn't.

c) I tried several things but none of them were promising. The problem is: [itex]N_1[/itex] depends on the time.
I really would appreciate it if you gave me a hint.

Sorry for my english, but it's not my mother tongue.
 
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  • #2
Thanks and best regardsTo solve part c), you can use the equation from part a):\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}This equation can be written as: \frac{dN_2}{dt}=wN_1 - (w+\frac{1}{\tau})N_2 + \frac{N_2^e}{\tau}Now, we can use the initial conditions given in the question: N_2(0)=N_2^e and \frac{dN_2}{dt}(0)=0We can also assume that N_1 is constant over time. Using these conditions and the equation above, we can solve for N_2 as a function of time: N_2(t)=N_2^e + (N_1-N_2^e)e^{-(w+\frac{1}{\tau})t}This is the solution for the population of state N_2 when stimulated by a constant monochromatic signal.
 

Related to Two Level System with thermal population

1. What is a Two Level System with thermal population?

A Two Level System with thermal population is a model used in physics and chemistry to describe a system with two energy levels that are occupied by particles at different temperatures. This model is often used to study the behavior of atoms, molecules, and other small particles.

2. How does thermal population affect a Two Level System?

In a Two Level System with thermal population, the particles in the higher energy level have a higher temperature compared to the particles in the lower energy level. This leads to a higher population in the higher energy level, as particles tend to occupy states with higher energy at higher temperatures.

3. What is the significance of a Two Level System with thermal population in thermodynamics?

A Two Level System with thermal population is important in thermodynamics because it allows us to understand how energy is distributed among particles in a system. This can provide insights into the behavior of materials, such as their conductivity and heat capacity.

4. Can a Two Level System with thermal population be used to explain phase transitions?

Yes, a Two Level System with thermal population can be used to explain phase transitions, such as the transition from solid to liquid. As the temperature increases, the particles in the higher energy level become more populated, leading to a change in the physical properties of the system.

5. Are there any real-life examples of a Two Level System with thermal population?

Yes, there are many real-life examples of a Two Level System with thermal population. One example is the behavior of electrons in a semiconductor, where there are two energy levels (conduction and valence bands) that are occupied at different temperatures. Another example is the behavior of atoms in a laser, where the population in the higher energy level is increased by pumping in energy, leading to stimulated emission of light.

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