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We get:
<E>= E _{total} /N _{tot}
= ћω [(0× N _{0} )/N ^{0} _{0} + (1× N _{1} ) /N ^{0} _{0} + (2× N _{1} ) /N ^{0} _{0} + (3× N _{3} ) /N ^{0} _{0} +…..]/[ (N _{0} ) /N ^{0} _{0} +(N _{1} ) /N ^{0} _{0} + (N _{2} ) /N ^{0} _{0} + (N _{3} ) /N ^{0} _{0} +..]
assuming N _{N} / N ^{0} _{0} = x ^{N} = Exp[- N ћω/kT] where x= Exp[- ћω/kT]and substituting in Eq.(1.143), we get:
<E>= ћω[0+x+2x ^{2} +3x ^{3} +…]/[1+x+x ^{2} +x ^{3} +…] …………………………………………………….. 1.144
But [1+x+x ^{2} +x ^{3} +…] is a Geometric Progression Series where x« 1 therefore the sum of this G.P. series is 1/(1-x)
Hence denominator of Eq.(1.144) is 1/(1-x)…………………………………………… 1.145
Numerator = [x+2x ^{2} +3x ^{3} +…]
= x[1+x+x ^{2} +x ^{3} + x ^{4} + x ^{5} + x ^{6} +……….
+x+ x ^{2} + x ^{3} + x ^{4} + x ^{5} + x ^{6} +………..
x ^{2} + x ^{3} + x ^{4} + x ^{5} + x ^{6} +……….
x ^{3} + x ^{4} + x ^{5} + x ^{6} +…………
=x[(1/(1-x)+x(1/(1-x))+x ^{2} (1/(1-x))+x ^{3} +………]
Numerator =x(1/(1-x)) [1/(1-x)] ……………………………………………………………….. 1.146
Substituting Eq.(1.145) and Eq.(1.146) in Eq.(1.144)
<E>= ћω {x(1/(1-x)) [1/(1-x)]}/ 1/(1-x)
= ћω Exp[- ћω/kT]/[1- Exp[- ћω/kT]]
<E>= ћω/ [ Exp[ ћω/kT]-1]………………………………………………………………….. 1.147
At ћω, average energy =<E>=<n>ћω………………………………………………………… 1.148
Comparing Eq.(1.147) and Eq.(1.148):
<n>= average number of oscillators at ћω =1/[ Exp[ ћω/kT]-1]…………………………… 1.149
Dividing numerator and denominator by Exp[ ћω/kT]
<n>= Exp[- ћω/kT]/[1- Exp[- ћω/kT]………………………………………………………. 1.150
or <n>+1=1/[1-Exp[- ћω/kT] …………………………………………………………………….. 1.151
Therefore dividing Eq.(1.150) by Eq.(1.151)
<n>/[<n>+1] = Exp[- ћω/kT]………………………………………………………… 1.152
From Statistical Mechanics under energy equilibrium:
N _{e} /N _{g} = Exp[-∆E/kT]= Exp[- ћω/kT] ………………………………………………………….. 1.153
Where N _{e} = number of atoms in excited state,
N _{g} = number of atoms in ground state ;
In energy equilibrium number of photons emitted is equal to number of photons absorbed.
By comparing Eq.(1.152) with Eq(1.153) we obtain:
< n >/[< n >+1] = N _{e} /N _{g}
Therefore N g <n>= N e [<n>+1]…………………………………………………… 1.154
If we assume that the probability of emission of photons is a ^{2} when there are no photons in the cavity then
N _{g} <n>a ^{2} = N _{e} [<n>+1] a ^{2} ……………………………………………… 1.155
At a given temperature T Kelvin, < n > is the average number of oscillators at a circular frequency ω.
Since ground state atoms will absorb photons and excited state atoms will emit photons hence:
N _{g} <n>a ^{2} = rate of absorption
N _{e} [<n>+1] a ^{2} = rate of emission…………………………………………………………. 1.156
The two rates are exactly balanced as is evident from Eq.(1.155).
Eq.(1.156) tells us that if < n > number of oscillators are at the same frequency ω
then probability of absorption is _{} < n > a ^{2} and
probability of emission is [ < n >+ 1] a ^{2} .= [ < n > a ^{2} + a ^{2} ]
Probability of emission = Probability of stimulated emission + Probability of spontaneous emission;
Here stimulated emission is induced emission. A passing photon induces an excited atom to emit a photon of the same frequency and same phase and in the process settle to ground state. The photon emitted by this process of stimulated emission adds to the existing photon. This is known as L ight A mplification by S timulated E mission R adiation. The acronym of this process is LASER action.
Spontaneous Emission takes place when an excited atom settles down to ground state by itself according to Life-Time Law.
Einstein believed that stimulated emission is proportional to the intensity of light causing stimulation and intensity of light is given by the number of photons involved per second.
Therefore stimulated emission multiplicative factor
= absorption multiplicative factor=<n>a ^{2}
spontaneous emission multiplicative factor= a ^{2} ……………………………………………… 1.157.
This bosonian property of stimulated emission is utilized in LASER, the acronym for L ight A mplification by S timulated E mission R adiation. In Figure 1.79, the physics of spontaneous emission is shown.
Figure 1.79. Radiation by Spontaneous Emission.
When an atom relaxes from excited state to ground state in a natural way then we refer to it as spontaneous emission. If an excited solid experiences relaxation it gives out an incoherent beam of light.
A solid can similarly be brought to excited state by irradiation by an incoherent beam of light with a frequency greater then its threshold frequency. The physics of irradiation and subsequent excited state is shown in Figure 1.80.
Figure 1.80. An atom absorbs light of the correct frequency and gets raised to an excited state.
Figure 1.81. Through light irradiation RUBY is being brought to excited state. The spontaneous emission kick starts the LASER Action. The photon emitted by spontaneous emission causes stimulated emission. By stimulated emission a photon of same frequency and phase is emitted. Now there are two electrons. Two electrons are in phase and of same frequency and they constitute a coherent wave. This coherent wave causes two more stimulated emission. This results in 4 photon strong coherent beam. In this way the coherent beam exponentially multiplies and becomes strong enough to emerge out of one of the partially reflecting surfaces. This is known as Light Amplification by Stimulated Emission Radiation.
In a LASER device first energy from DC power source is utilized to cause the population inversion. All the ground state atoms are brought to an excited metastable state through irradiation by suitable light source. Metastable means that these atoms will remain in excited state for periods of milliseconds and then if no stimulation takes place they will spontaneously settle down to ground state radiating off photons as incoherent light.
In a LASER the first photon is obtained through spontaneous emission. This photon causes stimulated emission of in-phase and same optical frequency photon.
The two photons become 4 photon in-phase , same frequency beam. When a transition to lasing mode occurs, all atoms cooperate and emit in synchronism- producing giant coherent wave. A spontaneous emission changes into stimulated emission. A chaotic system self organizes itself into an orderly system. This beam geometrically grows into a powerful coherent beam which can cause miracles. This is illustrated in Figure 1.81.
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