The energies of the 3 lowest states of an atom are E 0 = −14 eV, E 1 = −9 eV and E 2 = −7 eV. The Einstein coefficients are A 10 = 3 × 10 8 s −1 , A 20 = 1.2 × 10 8 s −1 and A 21 = 8 × 10 7 s −1 . If a large number of atoms are in the energy level E 2 , the mean radiative lifetime of this excited state is
Mean Radiative Lifetime Explained
The mean radiative lifetime of an excited atomic state represents the average time an atom spends in that state before it transitions to a lower energy state by emitting a photon. This lifetime is determined by the total probability rate of spontaneous emission from that excited state to all possible lower energy states.
Einstein Coefficients and Transition Rates
Einstein coefficients, specifically the Einstein A coefficients ($A_{ij}$), quantify the spontaneous emission probability rate for a transition from a higher energy level $i$ to a lower energy level $j$. If an atom is in an excited state, say state 2, it can transition to any lower energy state, say state 1 or state 0, by spontaneous emission. The rate of transition from state 2 to state 1 is $A_{21}$, and the rate of transition from state 2 to state 0 is $A_{20}$.
The total transition probability rate from state 2 is the sum of the rates of all possible downward transitions:
Calculating Mean Radiative Lifetime
The mean radiative lifetime, denoted by $\tau$, of an excited state is the reciprocal of the total transition probability rate from that state. For state 2, the mean radiative lifetime $\tau_2$ is given by:
Applying Given Values
We are given the following Einstein coefficients for transitions from state 2:
- $A_{20} = 1.2 \times 10^8 \text{ s}^{-1}$ (Transition from state 2 to state 0)
- $A_{21} = 8 \times 10^7 \text{ s}^{-1}$ (Transition from state 2 to state 1)
Note that the energies of the states are given as $E_0 = -14$ eV, $E_1 = -9$ eV, and $E_2 = -7$ eV. Since $E_2 > E_1 > E_0$, transitions from E2 can occur to both E1 and E0.
Now, we can calculate the total transition rate from state 2:
Let's express both coefficients with the same power of 10:
Now, we calculate the mean radiative lifetime $\tau_2$:
Conclusion
The calculated mean radiative lifetime of the excited state E2 is $0.5 \times 10^{-8}$ s. This value represents the average time an atom will remain in the E2 state before spontaneously emitting a photon and transitioning to either the E1 or E0 state.
The final answer is $\boxed{0.5 \times 10^{-8} \text{ s}}$.
Important Questions from Radioactivity
The nuclei of 137 Cs decay by the emission of β - particles with a half life of 30.08 years. The activity (in units of disintegrations per second or Bq) of a 1 mg source of 137 Cs, prepared on January 1, 1980, as measured on January 1, 2021 is closest to
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Radioactivity is the characteristic of which of the following?
Particles which can be added to the nucleus of an atom without changing its chemical properties are
A free neutron decays spontaneously into a proton, an electron and ___________ .
