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A beam of light of frquency omega, equal to the resonant frequency of transition of atoms of the gas, passes through that gas heated to temperature T. In this case ħ omega gt gt kT. Taking into account induced radiation, demonstrate that the absorption coefficient of the gas x varies as x=x_(0)(1-e^(-ħ omega//kT)), where x_(0) is the absorption cofficient for T rarr 0. |
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Answer» Solution :Because of the RESONANT nature of the process we can ignore nonresonant processes. We also ignore SPONTANEOUS emission since it does not contribute to the absorption coefficient and is a SMALL term if the beam is intense enough. Suppose `I` is the intensity of the beam at some point. The decrease in the value of this intensity on passing through the layers of the substance of thickness `d x` is equal to `dI=XIdx=(N_(1)B_(12)-N_(2)B_(21))((I)/(c ))ħ omega dx` Here `N_(1)=` No, of atoms in lower level `N_(2)=`No of atoms in the upper level per unit volume. `B_(12), B_(21)` are Einstein coefficients and `I_(c )=` energy density in the beam, `c=` velocity of light. A factor `ħ omega` arises beacuse each trnsition result in a loss or gain of energy `ħ oemga` Hence `x(ħ omega)/(c )N_(1)B_(12)(1-(N_(2)B_(21))/(N_(1)B_(12)))` But `g_(1)B_(12)=g_(2)B_(21)`so `(1-(g_(1))/(g_(2))(N_(2))/(N_(1)))` By Boltzman factor `(N_(2))/(N_(1))=(g_(2))/(g_(1))e^(-^(-H omega//kT)` When ` ħ omega gt gt kT` we can PUT `N_(1)=N_(0)` the total number of atoms per unit volume. Then `x=x_(0)(1-e^(-^h omega//kT))` where `x_(0)=( ħ omega)/(c )N_(0)B_(12)` is the absorption coefficient for `T rarr 0` |
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