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In the helium-neon laser of laser action occurs between two excited states of the neon atom. However, in many lasers, laser action (lasing) occurs between the ground state and an excited state, as suggested in fig. (a) Consider such a laser that emits at wavelength lambda=550nm. If a population inversion is not generated what is the ratio of the population of atoms in state E_(x) to the populationin the ground state E_(0), with the atoms at room temperature? |
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Answer» SOLUTION :The naturally occurring population ratio `N_(x)//N_(0)` of the two states is due to thermal agitation of the gas atoms `N_(x)//N_(0)=e^(-(E_(x)-E_(0))//kT)` (39-22) To find `N_(x)//N_(0)` with we need to find the energy separation `E_(x)-E_(0)` between the two states. (2) We can obtain `E_(x)-E_(0)` from the given wavelength of 550 nm for the LASING between those two states. The lasing wavelength gives us `E_(x)-E_(0)=hf=(hc)/(lambda)` `=((6.63xx10^(-34)J.s)(3.00xx10^(8)m//s))/((550xx10^(-9)m)(1.60xx10^(-19)J//eV))` `=2.26eV`. We also need the mean energy of thermal agitation kT for an atom at room temperature (assumed to be 300 K), which is `kT=(8.62xx10^(-5)eV//K)(300K)=0.0259eV`, in which k is Boltzmann.s constant. SUBSTITUTING the last two results into give us the population ratio at room temperature: `N_(x)//N_(0)=e^(-(2.26eV)(0.0259eV))` `~~1.3xx10^(-38)` This is an extremely small number. It is not unreasonable, however. Atoms with a mean thermal agitation energy of only 0.0259 eV will not often impart an energy of 2.26 eV to another atom in a collision. (b) For the conditions of (a), what temperature would the ratio `N_(x)//N_(0)` be 1/2? Calculation: Now we want the temperature T such that thermal agiation has bumped ENOUGH neon atoms up to the higher-energy state to give `N_(x)//N_(0)=1//2`. Substituting that ratio into taking the natural logarithm of both sides, and solving for T yield `T=(E_(x)-E_(0))/(k(ln2))=(2.26eV)/((8.62xx10^(-5)eV//K)(ln2))` `=38 000K`. This is much hotter than the surface of the SUN. Thus, it is clear that if we are to invert the population of these two levels, some specific mechanism for bringing this about is needed-that is, we must ..pump.. the atoms. No temperature, however high, will naturally generate a population inversion by thermal agitation. |
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