Transcript Chapter 28

Chapter 28
Atomic Physics
Plum Pudding Model of the Atom
• J. J. Thomson’s “Plum Pudding” model of the atom:
• Electrons embedded throughout the a volume of
positive charge
• A change from Newton’s model of the atom as a tiny,
hard, indestructible sphere
Sir Joseph John Thomson
1856 – 1940
Scattering Experiments
Ernest Rutherford
1871 – 1937
• The source was a naturally radioactive material that
produced alpha particles
• Most of the alpha particles passed though the foil
• A few deflected from their original paths
• Some even reversed their direction of travel
Planetary Model of the Atom
• Based on results of thin foil scattering experiments,
Rutherford’s Planetary model of the atom:
• Positive charge is concentrated in the center of the
atom, called the nucleus
• Electrons orbit the nucleus like planets orbit the sun
Difficulties with the Rutherford Model
• Atoms emit certain discrete characteristic frequencies
of electromagnetic radiation but the Rutherford model
is unable to explain this phenomena
• Rutherford’s electrons are undergoing a centripetal
acceleration and so 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,
but it doesn’t
Emission Spectra
• A gas at low pressure and a voltage applied to it emits
light characteristic of the gas
• When the emitted light is analyzed with a
spectrometer, a series of discrete bright lines –
emission spectrum – is observed
• Each line has a different wavelength and color
Emission Spectrum of Hydrogen
• The wavelengths of hydrogen’s spectral lines can be
found from
1
1
 1
 RH  2  2 

2 n 
• RH = 1.097 373 2 x 107 m-1 is the Rydberg constant and
n is an integer, n = 3, 4, 5, …
• The spectral lines correspond to different values of n
• n = 3, λ = 656.3 nm
• n = 4, λ = 486.1 nm
Johannes Robert Rydberg
1854 – 1919
Absorption Spectra
• An element can also absorb light at specific
wavelengths
• An absorption spectrum can be obtained by passing a
continuous radiation spectrum through a vapor of the
gas
• Such spectrum consists of a series of dark lines
superimposed on the otherwise continuous spectrum
• The dark lines of the absorption spectrum coincide
with the bright lines of the emission spectrum
The Bohr Theory of Hydrogen
• In 1913 Bohr provided an explanation of atomic
spectra that includes some features of the currently
accepted theory
• His model was an attempt to explain why the atom was
stable and included both classical and non-classical
ideas
Niels Henrik David Bohr
1885 – 1962
The Bohr Theory of Hydrogen
• The electron moves in circular orbits around the
proton under the influence of the Coulomb force of
attraction, which produces the centripetal acceleration
• Only certain electron orbits are stable
• In these orbits electrons do not emit energy in the form
of electromagnetic radiation
• Therefore, the energy of the atom
remains constant and classical
mechanics can be used to describe
the electron’s motion
The Bohr Theory of Hydrogen
• Radiation is emitted when the electrons “jump” (not in
a classical sense) from a more energetic initial state to
a lower state
• The frequency emitted in the “jump” is related to the
change in the atom’s energy: Ei – Ef = h ƒ
• The size of the allowed electron orbits is determined
by a quantization condition imposed on the electron’s
orbital angular momentum:
me v r = n ħ where n = 1, 2, 3, …; ħ = h / 2 π
Radii and Energies of Orbits
2
q1q2
(e)e
e
PE  k e
 ke
 ke
r
r
r
2
2
me v
e
 ke
E  KE  PE 
2
r
2
2
v
e
k e 2  F = me ac  me
r
r
2
2
2
2
e
e
m
v
1
e
2
ke
 me v
E  ke
ke
 e
r
2r
2 r
2
Radii and Energies of Orbits
me vr  n
n  1,2,3,...
 n 
e

ke
 me 
r
 me r 
2 2
n
r
2
me ke e
2
e
2
ke
 me v
r
2
2
2
2
n
v
me r
2
e
n
ke
=
2
r me r
2
4
e me k e
E
2 2
2n 
2
e
E  ke
2r
Radii and Energies of Orbits
2
4
2

e me ke 1
2
rn 
n
En  
2
2
2
me ke e
2
n
n = 1,2,3,...
• The radii of the Bohr orbits are
quantized
• When n = 1, the orbit has the
smallest radius, called the Bohr
radius, ao = 0.0529 nm
• A general expression for the
radius of any orbit in a
hydrogen atom is rn = n2 ao
Radii and Energies of Orbits
2
4
2

e me ke 1
2
rn 
n
En  
2
2
2
me ke e
2
n
n = 1,2,3,...
• The lowest energy state (n = 1)
is called the ground state, with
energy of –13.6 eV
• The next energy level (n = 2)
has an energy of –3.40 eV
• The energies can be compiled
in an energy level diagram with
the energy of any orbit of En = 13.6 eV / n2
Energy Level Diagram
2

2
rn 
n
2
me ke e
4
2
e me ke 1
En  
2
2
2
n
2 
4
e me ke  1
1 
hc
 2
 hf  Ei  E f 
2
2
2  n f

ni 
2 
4
1 e me ke  1
1 

 2
2
2
 2 hc  n f
ni 
 1

1
1
 RH  2  2 
n


n
i 
 f
Energy Level Diagram
• The value of RH from Bohr’s analysis is in excellent
agreement with the experimental value of the Rydberg
constant
• A more generalized equation can be
used to find the wavelengths of any
spectral lines
2
4
e e
2
2
2
f
i
e m k hc  1
1


n

2
n

 1

1
1
 RH  2  2 
n


n
i 
 f
1




Energy Level Diagram
• The uppermost level corresponds to E = 0 and n  
• The ionization energy: energy needed to completely
remove the electron from the atom
• The ionization energy for hydrogen
is 13.6 eV
e me ke hc  1
1 

 2
2
2
n


2
n
i 
 f
 1

1
1
 RH  2  2 
n


n
i 
 f
1
4
2
Modifications of the Bohr Theory –
Elliptical Orbits
• Sommerfeld extended the results to include elliptical
orbits
• Retained the principal quantum number, n, which
determines the energy of the allowed states
• Added the orbital quantum number, ℓ, ranging from 0 to
n-1 in integer steps
• All states with the same principal quantum
number are said to form a shell, whereas the
states with given values of n and ℓ are said
Arnold Johannes
to form a subshell
Wilhelm Sommerfeld
1868 – 1951
Modifications of the Bohr Theory –
Elliptical Orbits
Arnold Johannes
Wilhelm Sommerfeld
1868 – 1951
Modifications of the Bohr Theory –
Zeeman Effect
• Another modification was needed to
account for the Zeeman effect: splitting of
spectral lines in a strong magnetic field,
indicating that the energy of an electron is
slightly modified when the atom is
immersed in a magnetic field
• A new quantum number, m ℓ, called the
orbital magnetic quantum number, had to
be introduced
• m ℓ can vary from - ℓ to + ℓ in integer steps
Pieter Zeeman
1865 – 1943
Quantum Number Summary
• The values of n can range from 1 to  in integer steps
• The values of ℓ can range from 0 to n-1 in integer steps
• The values of m ℓ can range from -ℓ to ℓ in integer steps
Modifications of the Bohr Theory – Fine
Structure
• High resolution spectrometers
show that spectral lines are, in
fact, two very closely spaced
lines, even in the absence of a
magnetic field
• This splitting is called fine
structure
• Another quantum number, ms,
called the spin magnetic quantum
number, was introduced to explain
the fine structure
Spin Magnetic Quantum Number
• It is convenient to think of the electron as
spinning on its axis (the electron is not
physically spinning)
• There are two directions for the spin:
spin up, ms = ½; spin down, ms = - ½
• There is a slight energy difference
between the two spins and this accounts
for the doublet in some lines
• A classical description of electron spin is
incorrect: the electron cannot be located
precisely in space, thus it cannot be
considered to be a spinning solid object
de Broglie Waves in the Hydrogen Atom
• One of Bohr’s postulates was the angular momentum
of the electron is quantized, but there was no
explanation why the restriction occurred
• de Broglie assumed that the electron orbit would be
stable only if it contained an integral number of
electron wavelengths
nh
n  1,2,...
2r  n 
me v
h
h
h
 
me vr  n
p me v
2
de Broglie Waves in the Hydrogen Atom
• This was the first convincing argument that the wave
nature of matter was at the heart of the behavior of
atomic systems
• By applying wave theory to the electrons in an atom,
de Broglie was able to explain the appearance of
integers in Bohr’s equations as a natural consequence
of standing wave patterns
h
me vr  n
2
Quantum Mechanics and the Hydrogen
Atom
• Schrödinger’s wave equation was subsequently
applied to hydrogen and other atomic systems - one of
the first great achievements of quantum mechanics
• The quantum numbers and the restrictions placed on
their values arise directly from the mathematics and
not from any assumptions made to make the theory
agree with experiments
Electron Clouds
• The graph shows the solution to
the wave equation for hydrogen
in the ground state
• The curve peaks at the Bohr
radius
• The electron is not confined to a
particular orbital distance from
the nucleus
• The probability of finding the
electron at the Bohr radius is a
maximum
Electron Clouds
• The wave function for hydrogen
in the ground state is symmetric
• The electron can be found in a
spherical region surrounding the
nucleus
• The result is interpreted by
viewing the electron as a cloud
surrounding the nucleus
• The densest regions of the cloud
represent the highest probability
for finding the electron
The Pauli Exclusion Principle
• No two electrons in an atom or in the same location can
ever have the same set of values of the quantum
numbers n, ℓ, m ℓ, and ms
• This explains the electronic structure of complex atoms
as a succession of filled energy levels with different
quantum numbers
Wolfgang Ernst Pauli
1900 – 1958
Filling Shells
• As a general rule, the order that electrons fill an
atom’s subshell is:
• 1) Once one subshell is filled, the next electron goes
into the vacant subshell that is lowest in energy
• 2) Otherwise, the electron would radiate energy until
it reached the subshell with the lowest energy
• 3) A subshell is filled when it holds 2(2ℓ+1) electrons
Filling Shells
The Periodic Table
• The outermost electrons are primarily responsible
for the chemical properties of the atom
• Mendeleev arranged the elements according to their
atomic masses and chemical similarities
• The electronic configuration of the elements is
explained by quantum numbers and Pauli’s
Exclusion Principle
Dmitriy Ivanovich
Mendeleyev
1834 – 1907
The Periodic Table
Explanation of Characteristic X-Rays
• The details of atomic structure
can be used to explain
characteristic x-rays
• A bombarding electron collides
with an electron in the target
metal that is in an inner shell
• If there is sufficient energy, the
electron is removed from the
target atom
Explanation of Characteristic X-Rays
• The vacancy created by the lost
electron is filled by an electron
falling to the vacancy from a
higher energy level
• The transition is accompanied
by the emission of a photon
whose energy is equal to the
difference between the two
levels
Energy Bands in Solids
• In solids, the discrete energy levels of isolated atoms
broaden into allowed energy bands separated by
forbidden gaps
• The separation and the electron population of the
highest bands determine whether the solid is a
conductor, an insulator, or a semiconductor
Energy Bands in Solids
• Sodium example
• Blue represents energy bands
occupied by the sodium
electrons when the atoms are in
their ground states, gold
represents energy bands that are
empty, and white represents
energy gaps
• Electrons can have any energy
within the allowed bands and
cannot have energies in the gaps
Energy Level Definitions
• The valence band is the highest filled band
• The conduction band is the next higher empty band
• The energy gap has an energy, Eg, equal to the
difference in energy between the top of the valence
band and the bottom of the conduction band
Conductors
• When a voltage is applied to a conductor, the electrons
accelerate and gain energy
• In quantum terms, electron energies increase if there
are a high number of unoccupied energy levels for the
electron to jump to
• For example, it takes very little
energy for electrons to jump
from the partially filled to one of
the nearby empty states
Insulators
• The valence band is completely
full of electrons
• A large band gap separates the
valence and conduction bands
• A large amount of energy is
needed for an electron to be able
to jump from the valence to the
conduction band
• The minimum required energy is
Eg
Semiconductors
• A semiconductor has a small energy
gap
• Thermally excited electrons have
enough energy to cross the band gap
• The resistivity of semiconductors
decreases with increases in
temperature
• The light-color area in the valence
band represents holes – empty states
in the valence band created by
electrons that have jumped to the
conduction band
Semiconductors
• Some electrons in the valence
band move to fill the holes and
therefore also carry current
• The valence electrons that fill the
holes leave behind other holes
• It is common to view the
conduction process in the valence
band as a flow of positive holes
toward the negative electrode
applied to the semiconductor
Semiconductors
• An external voltage is supplied
• Electrons move toward the positive electrode
• Holes move toward the negative electrode
• There is a symmetrical current process in a
semiconductor
Doping in Semiconductors
• Doping is the adding of impurities to a
semiconductor (generally about 1 impurity atom per
107 semiconductor atoms)
• Doping results in both the band structure and the
resistivity being changed
n-type Semiconductors
• Donor atoms are doping materials that contain one
more electron than the semiconductor material
• This creates an essentially free electron with an energy
level in the energy gap, just below the conduction
band
• Only a small amount of thermal energy is needed to
cause this electron to move into the conduction band
p-type Semiconductors
• Acceptor atoms are doping materials that contain one
less electron than the semiconductor material
• A hole is left where the missing electron would be
• The energy level of the hole lies in the energy gap, just
above the valence band
• An electron from the valence band has enough
thermal energy to fill this impurity level, leaving
behind a hole in the valence band
A p-n Junction
• A p-n junction is formed when a p-type semiconductor
is joined to an n-type
• Three distinct regions exist: a p region, an n region,
and a depletion region
• Mobile donor electrons from the n side nearest the
junction diffuse to the p side, leaving behind immobile
positive ions
A p-n Junction
• At the same time, holes from the p side nearest the
junction diffuse to the n side and leave behind a region
of fixed negative ions
• The resulting depletion region is depleted of mobile
charge carriers
• There is also an electric field in this region that sweeps
out mobile charge carriers to keep the region truly
depleted
Diode Action
• The p-n junction has the
ability to pass current in only
one direction
• When the p-side is
connected to a positive
terminal, the device is
forward biased and current
flows
• When the n-side is
connected to the positive
terminal, the device is
reverse biased and a very
small reverse current results
Applications of Semiconductor Devices
• Rectifiers: change AC voltage to DC voltage
• Transistors: may be used to amplify small signals
• Integrated circuits: a collection of interconnected
transistors, diodes, resistors and capacitors
fabricated on a single piece of silicon