Question

For an emission of wavelength 5000 Angstrom, at what temperatures both spontaneous and stimulated emissions will be same?

Answer 1

Answer on Question #80962, Physics / Optics

For an emission of wavelength 5000 Angstrom, at what temperatures both spontaneous and stimulated emissions will be same?

Solution:

The relative rate at which spontaneous and stimulated processes occur in a system in equilibrium at temperature $T$ is

$$R = \frac {A _ {2 1}}{B _ {2 1} \rho (\nu)} = 1$$

The spontaneous emission rate is $A_{21}$, which is independent of external radiation. $B_{21}$ is called the Einstein coefficient for stimulated emission.

We choose the $\rho(\nu)$ appropriate to a black-body radiation field, since such radiation is always present to interact with an excited atom that is contained within an enclosure at temperature $T$.

$$R = \frac {A _ {2 1}}{B _ {2 1} \rho (\nu)} = e ^ {\frac {h \nu}{k T}} - 1 = 1$$ $$e ^ {\frac {h \nu}{k T}} = 2$$ $$\frac {h \nu}{k T} = \ln 2 = 0.69315$$ $$T = \frac {h \nu}{k \ln (2)} = \frac {h c}{k \lambda \ln (2)} = \frac {6.626 \times 10 ^ {- 3 4} \times 3 \times 10 ^ {8}}{1.38 \times 10 ^ {- 2 3} \times 5 0 0 0 \times 1 0 ^ {- 1 0} \times 0.69315} = 41562 K$$

Answer: 41562 K

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