Calculating Lifetimes in a Three-Level System with Einstein A & B Coefficients?

In summary, the equations state that the number of atoms in state ##i## is equal to the sum of the atom numbers in states ##A_{ac}+A_{ab}##, where ##A_{ac}## and ##A_{ab}## are the atom numbers in states ##a## and ##b##, respectively. The lifetime is simply the time it takes for the number of atoms in state ##i## to decay by half.
  • #1
bananabandana
113
5

Homework Statement


Screen Shot 2016-06-03 at 17.23.13.png


Homework Equations

The Attempt at a Solution



Very confused by this problem. For one thing, it doesn't specify if there is or isn't any light present to drive the stimulated emission/absorbtion. I guess there's no reason to assume that there is no light - but since the question is asking about lifetimes, that would seem more sensible...[plus introducing an unspecified ##\rho(\omega_{0})## seems odd]

Assuming there is no light involved, then this is just a three level system, with three sets of coupled differential equations describing the behaviour - let ##N_{i}## be the number of atoms in state ##i##:
(1) $$ \frac{dN_{A}}{dt} = -(A_{ac}+A_{ab})N_{A} $$
(2) $$ \frac{dN_{B}}{dt} = A_{ab}(N_{A} -N_{B}) $$
(3) $$ \frac{dN_{C}}{dt} = A_{ac}N_{A}+A_{bc}N_{B} $$
[Though,one of them is made redundant by the fact that total particle number must be constant.]

We can easily solve the first equation:
$$ N_{A} = N_{A0}exp\bigg[ - (A_{ac}+A_{ab})t) \bigg] $$
By substituting this into the equation (2), we can then solve for ##N_{B}##:

$$ \frac{dN_{B}}{dt} +N_{B}A_{ab} = A_{ab}N_{A0}exp\bigg[ - (A_{ac}+A_{ab})t) \bigg] $$

So by using the standard method of integrating factors:
$$ N_{B} = Cexp(-A_{ab}t)-\frac{A_{ab}}{A_{ac}}N_{A0}exp(-A_{ac}+A_{ab}t) $$
But how am I meant to define a lifetime for that?, Assuming ##N_{B}(t=0)=0##, i.e:
$$ N_{B} =\frac{A_{ab}}{A_{ac}}N_{A0}exp(-A_{ab}t)\bigg[1-exp(-A_{ac}t)\bigg]$$
Do we take the dominant exponential to define the lifetime?
 
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  • #2
You are only told to consider the decays a-c and b-c.
The presence of photons is not relevant to the lifetime calculation for a state ... check the definition of "lifetime" in your notes.
 
  • #3
Simon Bridge said:
You are only told to consider the decays a-c and b-c.
The presence of photons is not relevant to the lifetime calculation for a state ... check the definition of "lifetime" in your notes.

Ah,okay, so lifetimes are generally defined to only involve the Einstein ##A## coefficients.
So I can just ignore completely ##\psi_{A} \rightarrow \psi_{B}##? I wasn't sure the question implied that... I guess if it does:

$$ \frac{dN_{A}}{dt} = -A_{ac}N_{A} $$
$$ \frac{dN_{B}}{dt} = -A_{bc} N_{B} $$

Implying that ## \lambda_{a} = A_{ac}## ##\lambda_{b} = A_{bc}##. We know that ## \frac{1}{\tau} = \lambda## where ##\tau ## is the lifetime, so:
$$ \frac{\tau_{A}}{\tau_{C}} = 2 $$

and then the information about the matrix element is completely redundant? It's that simple??
[Edit : I mean, it tells us the same thing]
Thanks!
 
  • #4
Well done.
 
  • #5
Oh dear, that's embarrassing.. Ah well, many thanks for the help, much appreciated!
 

Related to Calculating Lifetimes in a Three-Level System with Einstein A & B Coefficients?

What are Einstein A & B Coefficients?

Einstein A & B coefficients are a set of mathematical values that describe the probability of an atom or molecule absorbing or emitting a photon of a specific frequency. They were developed by Albert Einstein and are used in the study of atomic and molecular processes.

How are Einstein A & B Coefficients calculated?

Einstein A & B coefficients are calculated using the principles of quantum mechanics and the properties of the atom or molecule in question. They take into account factors such as energy levels, transition probabilities, and the strength of the electric dipole moment.

What is the significance of Einstein A & B Coefficients?

Einstein A & B coefficients are important in understanding the behavior of atoms and molecules in the presence of electromagnetic radiation. They are used in a variety of fields, including astrophysics, spectroscopy, and laser technology.

How do Einstein A & B Coefficients relate to the absorption and emission of light?

Einstein A & B coefficients describe the likelihood of an atom or molecule absorbing or emitting a photon of a specific frequency. The A coefficient represents the probability of spontaneous emission, while the B coefficient represents the probability of stimulated emission or absorption.

Can Einstein A & B Coefficients be measured experimentally?

Yes, Einstein A & B coefficients can be measured experimentally using techniques such as spectroscopy. By analyzing the absorption and emission spectra of a substance, scientists can determine the values of these coefficients and use them to further understand the behavior of the atoms and molecules in the sample.

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