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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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