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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.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Solid state physics and devices-the harbinger of third wave of civilization. OpenStax CNX. Sep 15, 2014 Download for free at http://legacy.cnx.org/content/col11170/1.89
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