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Chapter 42

Atomic Physics

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Importance of the

Hydrogen Atom The hydrogen atom is the only atomic

system that can be solved exactly

Much of what was learned in the

twentieth century about the hydrogen

atom, with its single electron, can be

extended to such single-electron ionsas He+ and Li2+

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More Reasons the Hydrogen

Atom is Important The hydrogen atom proved to be an ideal

system for performing precision tests of

theory against experiment Also for improving our understanding of atomicstructure

The quantum numbers that are used tocharacterize the allowed states of hydrogencan also be used to investigate more complexatoms This allows us to understand the periodic table

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Final Reasons for the Importance

of the Hydrogen Atom The basic ideas about atomic structure must

be well understood before we attempt to deal

with the complexities of molecular structuresand the electronic structure of solids

The full mathematical solution of the

Schrdinger equation applied to the hydrogen

atom gives a complete and beautifuldescription of the atoms properties

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Atomic Spectra A discrete line spectrum is observed

when a low-pressure gas is subjected to

an electric discharge Observation and analysis of these

spectral lines is called emissionspectroscopy

The simplest line spectrum is that foratomic hydrogen

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Uniqueness of Atomic Spectra Other atoms exhibit completely different

line spectra

Because no two elements have the

same line spectrum, the phenomena

represents a practical and sensitive

technique for identifying the elementspresent in unknown samples

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Emission Spectra Examples

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Absorption Spectroscopy An absorption spectrum is obtained

by passing white light from a continuous

source through a gas or a dilutesolution of the element being analyzed

The absorption spectrum consists of a

series of dark lines superimposed onthe continuous spectrum of the light

source

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Absorption Spectrum,

Example

A practical example is the continuous

spectrum emitted by the sun The radiation must pass through the cooler

gases of the solar atmosphere and throughthe Earths atmosphere

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Balmer Series In 1885, Johann Balmer found an

empirical equation that correctly

predicted the four visible emission linesof hydrogen H is red, = 656.3 nm

H is green, = 486.1 nm

H is blue, = 434.1 nm

H is violet, = 410.2 nm

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Emission Spectrum of

Hydrogen Equation The wavelengths of hydrogens spectral lines

can be found from

RH is the Rydberg constant

RH

= 1.097 373 2 x 107 m-1

n is an integer, n = 3, 4, 5,

The spectral lines correspond to different values ofn

H 2 21 1 1

2R

n

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Other Hydrogen Series Other series were also discovered and

their wavelengths can be calculated

Lyman series:

Paschen series:

Brackett series:

H 21 11 2 3 4 , , ,R n n

K

H 2 2

1 1 14 5 6

3

, , ,R n

n

K

H 2 2

1 1 15 6 7

4 , , ,R n

n

K

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Joseph John Thomson 1856 1940

Received Nobel Prize in

1906 Usually considered the

discoverer of the

electron

Worked with thedeflection of cathode

rays in an electric field

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Early Models of the Atom J. J. Thomson

established the charge

to mass ratio for

electrons

His model of the atom A volume of positive

charge

Electrons embeddedthroughout the volume

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Early Models of the Atom, 2 Rutherford

Planetary model

Based on results ofthin foil experiments Positive charge is

concentrated in thecenter of the atom,

called the nucleus Electrons orbit the

nucleus like planetsorbit the sun

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Rutherfords Thin Foil

Experiment Experiments done in

1911

A beam of positivelycharged alpha particles

hit and are scattered

from a thin foil target

Large deflections couldnot be explained by

Thomsons model

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Difficulties with the

Rutherford Model Atoms emit certain discrete characteristic

frequencies of electromagnetic radiation The Rutherford model is unable to explain this

phenomena Rutherfords electrons are undergoing a

centripetal acceleration It should radiate electromagnetic waves of the same

frequency The radius should steadily decrease as this radiation is

given off The electron should eventually spiral into the nucleus

It doesnt

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Niels Bohr 1885 1962 An active participant in

the early development

of quantum mechanics Headed the Institute for

Advanced Studies inCopenhagen

Awarded the 1922Nobel Prize in physics For structure of atoms

and the radiationemanating from them

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The Bohr Theory of Hydrogen In 1913 Bohr provided an explanation of

atomic spectra that includes some

features of the currently acceptedtheory

His model includes both classical andnon-classical ideas

He applied Plancks ideas of quantizedenergy levels to orbiting electrons

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Bohrs Theory, cont. This model is now considered obsolete

It has been replaced by a probabilistic

quantum-mechanical theory

The model can still be used to develop

ideas of energy quantization and

angular momentum quantization asapplied to atomic-sized systems

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Bohrs Assumptions for

Hydrogen, 1 The electron moves

in circular orbits

around the protonunder the electric

force of attraction The Coulomb force

produces thecentripetal

acceleration

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Bohrs Assumptions, 2 Only certain electron orbits are stable

These are the orbits in which the atom does not

emit energy in the form of electromagneticradiation

Therefore, the energy of the atom remains

constant and classical mechanics can be used to

describe the electrons motion

This representation claims the centripetally

accelerated electron does not emit energy and

eventually spirals into the nucleus

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Bohrs Assumptions, 3 Radiation is emitted by the atom when the

electron makes a transition from a moreenergetic initial state to a lower-energy orbit The transition cannot be treated classically The frequency emitted in the transition is related

to the change in the atoms energy The frequency is independent of frequencyof the

electrons orbital motion The frequency of the emitted radiation is given by

Ei Ef = h

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Bohrs Assumptions, 4 The size of the allowed electron orbits is

determined by a condition imposed on

the electrons orbital angularmomentum

The allowed orbits are those for which

the electrons orbital angularmomentum about the nucleus is

quantized and equal to an integral

multiple of

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Bohr Radius The radii of the Bohr orbits are

quantized

This shows that the radii of the allowedorbits have discrete valuesthey are

quantized When n = 1, the orbit has the smallest radius,

called the Bohr radius, ao ao = 0.0529 nm

2 2

21 2 3 , , ,n

e e

nr nm k e

h K

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Radii and Energy of Orbits A general expression for

the radius of any orbit ina hydrogen atom is rn = n

2ao The energy of any orbit

is

This becomes

En = - 13.606 eV/ n2

2

2

11 2 3,

2, ,en

o

k eE n

a n

K

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Specific Energy Levels Only energies satisfying the previous

equation are allowed The lowest energy state is called the ground

state This corresponds to n = 1 with E= 13.606 eV

The ionization energy is the energy neededto completely remove the electron from theground state in the atom The ionization energy for hydrogen is 13.6 eV

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Energy Level Diagram Quantum numbers are

given on the left and

energies on the right The uppermost level,

E= 0, represents the

state for which theelectron is removed

from the atom

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Frequency of Emitted Photons The frequency of the photon emitted

when the electron makes a transition

from an outer orbit to an inner orbit is

It is convenient to look at the

wavelength instead

2

2 2

1 1

2 i f e

o f i

E E k e

h a h n n

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Wavelength of Emitted

Photons The wavelengths are found by

The value ofRH from Bohrs analysis isin excellent agreement with the

experimental value

2

2 2 2 21 1 1 1 1

2 e

H

o f i f i

k e R c a hc n n n n

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Difficulties with the Bohr

Model Improved spectroscopic techniques

found that many of the spectral lines of

hydrogen were not single lines Each line was actually a group of lines

spaced very close together

Certain single spectral lines split intothree closely spaced lines when the

atoms were placed in a magnetic field

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Bohrs Correspondence

Principle Bohrscorrespondence principle states

that quantum physics agrees with

classical physics when the differencesbetween quantized levels become

vanishingly small

Similar to having Newtonian mechanics bea special case of relativistic mechanics

when v

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The Quantum Model of the

Hydrogen Atom The potential energy function for the

hydrogen atom is

ke is the Coulomb constant

ris the radial distance from the proton tothe electron The proton is situated at r= 0

2

( ) e eU r kr

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Quantum Model, cont. The formal procedure to solve the

hydrogen atom is to substitute U(r) into

the Schrdinger equation and find theappropriate solutions to the equations

Because it is a three-dimensional

problem, it is easier to solve if therectangular coordinates are converted

to spherical polar coordinates

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Quantum Model, final (x, y, z) is converted to

(r, , )

Then, the space variables

can be separated:

(r, , ) = R(r), (), g()

When the full set of

boundary conditions are

applied, we are led to three

different quantum numbers

for each allowed state

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Quantum Numbers, General The three different quantum numbers

are restricted to integer values

They correspond to three degrees offreedom Three space dimensions

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Principal Quantum Number The first quantum number is associated

with the radial function R(r) It is called the principal quantum number It is symbolized by n

The potential energy function depends

only on the radial coordinate r The energies of the allowed states in

the hydrogen atom are the same En

values found from the Bohr theory

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Quantum Numbers,

Summary Table

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Shells Historically, all states having the same

principle quantum number are said to

form a shell Shells are identified by letters K, L, M,

All states having the same values ofnand are said to form a subshell The letters s, p, d, f, g, h, .. are used to

designate the subshells for which = 0, 1,2, 3,

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Shell and Subshell Notation,

Summary Table

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Wave Functions for Hydrogen The simplest wave function for hydrogen is

the one that describes the 1s state and isdesignated 1s(r)

As 1s(r) approaches zero, rapproaches and is normalized as presented

1s(r) is also spherically symmetric

This symmetry exists for all s states

1 3

1( ) or as

o

r ea

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Radial Probability Density A spherical shell of

radius rand

thickness drhas avolume of 4r2dr

The radial

probability function

is P(r) = 4r2||2

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Wave Function of the 2s state The next-simplest wave function for the

hydrogen atom is for the 2s state

n = 2; = 0 The wave function is

2s depends only on rand is spherically symmetric

32

2

2

1 1( ) 2

4 2

or a

s

o o

r r e

a a

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Comparison of 1s and 2s

States The plot of the radial

probability density

for the 2s state hastwo peaks

The highest value of

Pcorresponds to the

most probable value In this case, r 5ao

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Active Figure 42.13

(SLIDESHOW MODE ONLY)

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Physical Interpretation of The magnitude of the angular

momentum of an electron moving in a

circle of radius ris L = mevr The direction ofL is perpendicular to

the plane of the circle

In the Bohr model, the angularmomentum of the electron is restrictedto multiples of

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Physical Interpretation

of, cont. According to quantum mechanics, an atom in

a state whose principle quantum number is ncan take on the following discrete values of

the magnitude of the orbital angularmomentum:

L can equal zero, which causes great difficultywhen attempting to apply classical mechanics tothis system

1 0 1 2 1 , , ,L n l l l Kh

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Physical Interpretation of m

, 2

Because the magnetic momentof the

atom can be related to the angular

momentum vector, L, the discretedirection oftranslates into the fact that

the direction ofL is quantized

Therefore, Lz, the projection ofL alongthe zaxis, can have only discrete

values

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Physical Interpretation of m, 3

The orbital magnetic quantum numbermspecifies the allowed values of the z

component of orbital angularmomentum Lz = m

The quantization of the possibleorientations ofL with respect to anexternal magnetic field is often referredto as space quantization

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Physical Interpretation of m, 4

L does not point in a specific direction Even though its z-component is fixed

Knowing all the components is inconsistentwith the uncertainty principle

Imagine that L must lie anywhere on the

surface of a cone that makes an angle with the zaxis

Ph i l I t t ti f

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Physical Interpretation of m,

final

is also quantized

Its values are

specified through

mis never greaterthan , therefore

can never be zero

cos

1

zL m L

l

l l

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Electron Spins

Only two directions exist for

electron spins

The electron can have spin

up (a) or spin down (b) In the presence of a

magnetic field, the energy

of the electron is slightly

different for the two spindirections and this

produces doublets in

spectra of certain gases

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Electron Spins, cont.

The concept of a spinning electron isconceptually useful

The electron is a point particle, without anyspatial extent Therefore the electron cannot be considered to be

actually spinning

The experimental evidence supports theelectron having some intrinsic angularmomentum that can be described by ms

Dirac showed this results from the relativistic

properties of the electron

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Spin Angular Momentum

The total angular momentum of a particular

electron state contains both an orbital

contribution L and a spin contribution S Electron spin can be described by a single

quantum numbers, whose value can only be

s =

The spin angular momentum of the electronnever changes

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Spin Angular Momentum, cont

The magnitude of the spin angularmomentum is

The spin angular momentum can have twoorientations relative to a zaxis, specified by

the spin quantum numberms = ms = + corresponds to the spin up case

ms = - corresponds to the spin down case

3( 1)2

S s s h h

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Spin Angular Momentum, final

The zcomponent of

spin angular

momentum is Sz=ms =

Spin angular

moment S is

quantized

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Spin Magnetic Moment

The spin magnetic momentspin is

related to the spin angular momentum

by

The zcomponent of the spin magnetic

moment can have values

spin

e

em

S

spin2

, z

e

e

m

h

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Wolfgang Pauli

1900 1958 Important review article on

relativity

At age 21 Discovery of the exclusion

principle Explanation of the connection

between particle spin and

statistics Relativistic quantum

electrodynamics Neutrino hypothesis Hypotheses of nuclear spin

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The Exclusion Principle

The four quantum numbers discussed so far

can be used to describe all the electronic

states of an atom regardless of the number of

electrons in its structure

The exclusion principle states that no two

electrons can ever be in the same quantum

state Therefore, no two electrons in the same atom can

have the same set of quantum numbers

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Filling Subshells

Once a subshell is filled, the next

electron goes into the lowest-energy

vacant state If the atom were not in the lowest-energy

state available to it, it would radiate energy

until it reached this state

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Orbitals

An orbitalis defined as the atomic state

characterized by the quantum numbers

n, and m From the exclusion principle, it can be

seen that only two electrons can be

present in any orbital One electron will have spin up and one

spin down

All d Q t St t

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Allowed Quantum States,

Example

In general, each shell can accommodate up

to 2n2 electrons

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Hunds Rule

Hunds Rule states that when an atom

has orbitals of equal energy, the order

in which they are filled by electrons issuch that a maximum number of

electrons have unpaired spins

Some exceptions to the rule occur inelements having subshells that are close to

being filled or half-filled

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Periodic Table

Dmitri Mendeleev made an early attempt at

finding some order among the chemical

elements

He arranged the elements according to their

atomic masses and chemical similarities

The first table contained many blank spaces

and he stated that the gaps were there onlybecause the elements had not yet been

discovered

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Periodic Table, Explained

The chemical behavior of an element

depends on the outermost shell that

contains electrons For example, the inert gases (last

column) have filled subshells and a

wide energy gap occurs between thefilled shell and the next available shell

Hydrogen Energy Level

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Hydrogen Energy Level

Diagram Revisited

The allowed values of

are separated

Transitions in which

does not change arevery unlikely to occur

and are called forbidden

transitions

Such transitions actuallycan occur, but their

probability is very low

compared to allowed

transitions

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Selection Rules

The selection rules for allowed transitions are = 1 m = 0, 1

The angular momentum of theatom-photonsystem must be conserved

Therefore, the photon involved in the process

must carry angular momentum The photon has angular momentum equivalent tothat of a particle with spin 1

A photon has energy, linear momentum andangular momentum

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Multielectron Atoms

For multielectron atoms, the positive

nuclear charge Ze is largely shielded by

the negative charge of the inner shellelectrons The outer electrons interact with a net

charge that is smaller than the nuclear

charge

Allowed energies are2

eff

2

136.n

ZE eV

n

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X-Ray Spectra

These x-rays are a result of

the slowing down of high

energy electrons as they

strike a metal target The kinetic energy lost can

be anywhere from 0 to all of

the kinetic energy of the

electron The continuous spectrum is

called bremsstrahlung, the

German word for braking

radiation

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X-Ray Spectra, cont.

The discrete lines are calledcharacteristic x-rays

These are created when A bombarding electron collides with a

target atom The electron removes an inner-shell

electron from orbit An electron from a higher orbit drops down

to fill the vacancy

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X-Ray Spectra, final

The photon emitted during this

transition has an energy equal to the

energy difference between the levels Typically, the energy is greater than

1000 eV

The emitted photons have wavelengthsin the range of 0.01 nm to 1 nm

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Moseley Plot

Henry G. J. Moseley plotted

the values of atoms as shown

is the wavelength of the K

line of each element The K line refers to the

photon emitted when an

electron falls from the L to

the K shell

From this plot, Moseley

developed a periodic table in

agreement with the one

based on chemical properties

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Active Figure 42.24

(SLIDESHOW MODE ONLY)

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Spontaneous Emission

Once an atom is inan excited state, theexcited atom can

make a transition toa lower energy level

Because thisprocess happens

naturally, it is knownas spontaneousemission

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Stimulated Emission

In addition to

spontaneous emission,

stimulated emission

may also occur

Stimulated emission

may occur when the

excited state is ametastable state

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Stimulated Emission, cont.

A metastable state is a state whose lifetime ismuch longer than the typical 10-8 s

An incident photon can cause the atom toreturn to the ground state without beingabsorbed

Therefore, you have two photons withidentical energy, the emitted photon and theincident photon They both are in phase and travel in the same

direction

Lasers Properties of

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Lasers Properties of

Laser Light

Laser light is coherent The individual rays in a laser beam

maintain a fixed phase relationship witheach other

There is no destructive interference

Laser light is monochromatic The light has a very narrow range of

wavelengths

Lasers Properties of

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Lasers Properties of

Laser Light, cont.

Laser light has a small angle of

divergence

The beam spreads out very little, even overlong distances

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Lasers Operation

It is equally probable that an incident photon

would cause atomic transitions upward or

downward Stimulated absorption or stimulated emission

If a situation can be caused where there are

more electrons in excited states than in the

ground state, a net emission of photons canresult This condition is called population inversion

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Lasers Operation, cont.

The photons can stimulate other atoms

to emit photons in a chain of similar

processes The many photons produced in this

manner are the source of the intense,

coherent light in a laser

Conditions for Build-Up of

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Conditions for Build-Up of

Photons

The system must be in a state of populationinversion

The excited state of the system must be a

metastable state In this case, the population inversion can be

established and stimulated emission is likely tooccur before spontaneous emission

The emitted photons must be confined in thesystem long enough to enable them tostimulate further emission This is achieved by using reflecting mirrors

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Laser Design Schematic

The tube contains the atoms that are the activemedium An external source of energy pumps the atoms to

excited states The mirrors confine the photons to the tube

Mirror 2 is only partially reflective

Energy-Level Diagram for

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Energy Level Diagram for

Neon in a Helium-Neon Laser

The atoms emit

632.8-nm photons

through stimulated

emission

The transition is E3*

to E2 * indicates a

metastable state

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Laser Applications

Applications include: Medical and surgical procedures

Precision surveying and lengthmeasurements

Precision cutting of metals and other

materials

Telephone communications