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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