Decay of Excited Atoms: Probability & Nonlinearity

[Trollfaz]

I learned that the probability of radioactive decay for an atom is always the same. However, is the decay of and excited atom or electron non linear(decay probability varies with time)?

 

[Riotto]

Radioactive decay is governed by the exponential law $$N(t)=N_0e^{-\lambda t}$$ where $N_0$ denotes the number of nuclei at $t=0$ , and $\lambda$ is the decay constant which is different for different nuclei. If you use the word "probability of radioactive decay" for "rate of radioactive decay" it is not same for all nuclei. In case of radioactive decay too, the decay rate $\frac{dN}{dt}$ also varies with time in a nonlinear fashion.

In presence of the environment, the eigenstates of the atomic Hamiltonian are not true eigenstates of nature. Therefore, each atomic level has a certain lifetime $\tau$ which depends upon the interaction it is subjected to. In general, a single atom continues to make transitions between the atomic energy levels.

 

[Trollfaz]

I mean is decay of excited atoms or electrons governed by the exponential law too?

 

[mfb]

The probability that an existing atom will decay within the next second is always the same.

If you follow a speific atom, the probability that it exists (in the original state) goes down over time - because it might have decayed. That means the probability that you observe the decay after X seconds is smaller than the probability that you observe it after Y seconds if X>Y.Every radioactive decay follows an exponential decay.

There are no excited electrons. Excited atoms exist, and if you don't influence them from the outside, they will decay exponentially.

 

[Riotto]

I mean is decay of excited atoms or electrons governed by the exponential law too?

If at any instant of time all the atoms are excited to the higher energy level i.e., a population inversion is achieved, it will eventually relax back to a Boltzmann distribution in presence of a temperature bath.