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Chapter 28
Atomic Physics
Importance of Hydrogen Atom
Hydrogen is the simplest atom
 The quantum numbers used to
characterize the allowed states of
hydrogen can also be used to describe
(approximately) the allowed states of
more complex atoms


This enables us to understand the periodic
table
More Reasons the Hydrogen
Atom is so Important

The hydrogen atom is an ideal system for
performing precise comparisons of theory and
experiment


Also for improving our understanding of atomic
structure
Much of what we know about the hydrogen
atom can be extended to other singleelectron ions

For example, He+ and Li2+
Early Models of the Atom

J.J. Thomson’s model of
the atom



A volume of positive
charge
Electrons embedded
throughout the volume
A change from Newton’s
model of the atom as a
tiny, hard, indestructible
sphere
Early Models of the Atom, 2

Rutherford




Planetary model
Based on results of
thin foil experiments
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


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

It doesn’t
Emission Spectra
A gas at low pressure has a voltage applied
to it
 A gas emits light characteristic of the gas
 When the emitted light is analyzed with a
spectrometer, a series of discrete bright lines
is observed



Each line has a different wavelength and color
This series of lines is called an emission spectrum
Examples of Spectra
Emission Spectrum of
Hydrogen – Equation

The wavelengths of hydrogen’s spectral lines
can be found from
1
 1 1
 RH  2  2 

2 n 

RH is the Rydberg constant



RH = 1.0973732 x 107 m-1
n is an integer, n = 1, 2, 3, …
The spectral lines correspond to different values of
n
Spectral Lines of Hydrogen
The Balmer Series
has lines whose
wavelengths are
given by the
preceding equation
 Examples of spectral
lines



n = 3, λ = 656.3 nm
n = 4, λ = 486.1 nm
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
 The absorption 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
Applications of Absorption
Spectrum

The continuous spectrum emitted by
the Sun passes through the cooler
gases of the Sun’s atmosphere
The various absorption lines can be used to
identify elements in the solar atmosphere
 Led to the discovery of helium

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 includes both classical and
non-classical ideas
 His model included an attempt to
explain why the atom was stable

Bohr’s Assumptions for
Hydrogen

The electron moves
in circular orbits
around the proton
under the influence
of the Coulomb
force of attraction

The Coulomb force
produces the
centripetal
acceleration
Bohr’s Assumptions, cont

Only certain electron orbits are stable



These are the orbits in which the atom does 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
Radiation is emitted by the atom when the
electron “jumps” from a more energetic initial
state to a lower state

The “jump” cannot be treated classically
Bohr’s Assumptions, final

The electron’s “jump,” continued
The frequency emitted in the “jump” is
related to the change in the atom’s energy
 It is generally not the same as the

frequency of the electron’s orbital motion

The size of the allowed electron orbits is
determined by a condition imposed on
the electron’s orbital angular
momentum
Mathematics of Bohr’s
Assumptions and Results

Electron’s orbital angular momentum


me v r = n ħ where n = 1, 2, 3, …
The total energy of the atom
2
1
e
2
 E  KE  PE  me v  k e
2
r

The energy can also be expressed as

k ee2
E
2r
Bohr Radius

The radii of the Bohr orbits are
quantized
n2  2
rn 
n  1, 2, 3, 
2
m ek e e
 This shows that the electron can only exist
in certain allowed orbits determined by the
integer n
When n = 1, the orbit has the smallest radius,
called the Bohr radius, ao
 ao = 0.0529 nm

Radii and Energy of Orbits

A general expression
for the radius of any
orbit in a hydrogen
atom is


rn = n2 ao
The energy of any
orbit is

En = - 13.6 eV/ n2
Specific Energy Levels

The lowest energy state is called the ground
state



The next energy level has an energy of –3.40
eV


This corresponds to n = 1
Energy is –13.6 eV
The energies can be compiled in an energy level
diagram
The ionization energy is the energy needed to
completely remove the electron from the
atom

The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram
The value of RH from
Bohr’s analysis is in
excellent agreement
with the
experimental value
 A more generalized
equation can be
used to find the
wavelengths of any
spectral lines

Generalized Equation
 1 1
1
 RH  2  2 

 nf ni 



For the Balmer series, nf = 2
For the Lyman series, nf = 1
Whenever an transition occurs between a
state, ni to another state, nf (where ni > nf ),
a photon is emitted

The photon has a frequency f = (Ei – Ef)/h and
wavelength λ
Bohr’s Correspondence
Principle

Bohr’s Correspondence Principle states
that quantum mechanics is in
agreement with classical physics when
the energy differences between
quantized levels are very small

Similar to having Newtonian Mechanics be
a special case of relativistic mechanics
when v << c
Successes of the Bohr Theory

Explained several features of the hydrogen
spectrum



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

Accounts for Balmer and other series
Predicts a value for RH that agrees with the
experimental value
Gives an expression for the radius of the atom
Predicts energy levels of hydrogen
Gives a model of what the atom looks like and how it
behaves
Can be extended to “hydrogen-like” atoms


Those with one electron
Ze2 needs to be substituted for e2 in equations

Z is the atomic number of the element
Modifications of the Bohr
Theory – Elliptical Orbits

Sommerfeld extended the results to
include elliptical orbits
Retained the principle quantum number, n
 Added the orbital quantum number, ℓ


ℓ ranges from 0 to n-1 in integer steps
All states with the same principle quantum
number are said to form a shell
 The states with given values of n and ℓ are
said to form a subshell

Modifications of the Bohr
Theory – Zeeman Effect

Another modification was needed to account
for the Zeeman effect



The Zeeman effect is the splitting of spectral lines
in a strong magnetic field
This indicates 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
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

de Broglie Waves
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

de Broglie Waves in the
Hydrogen Atom


In this example, three
complete wavelengths
are contained in the
circumference of the
orbit
In general, the
circumference must
equal some integer
number of wavelengths

2  r = n λ n = 1, 2, …
de Broglie Waves in the
Hydrogen Atom, cont

The expression for the de Broglie wavelength
can be included in the circumference
calculation


me v r = n 
This is the same quantization of angular
momentum that Bohr imposed in his original
theory
This was the first convincing argument that
the wave nature of matter was at the heart of
the behavior of atomic systems
 Schrödinger’s wave equation was
subsequently applied to atomic systems

QUICK QUIZ 28.1
In an analysis relating Bohr's theory to the de Broglie
wavelength of electrons, when an electron moves
from the n = 1 level to the n = 3 level, the
circumference of its orbit becomes 9 times greater.
This occurs because (a) there are 3 times as many
wavelengths in the new orbit, (b) there are 3 times as
many wavelengths and each wavelength is 3 times as
long, (c) the wavelength of the electron becomes 9
times as long, or (d) the electron is moving 9 times as
fast.
QUICK QUIZ 28.1 ANSWER
(b). The circumference of the orbit is n times the
de Broglie wavelength (2πr = nλ), so there are
three times as many wavelengths in the n = 3
level as in the n = 1 level. Also, by combining
Equations 28.4, 28.6 and the defining equation
for the de Broglie wavelength (λ = h/mv), one
can show that the wavelength in the n = 3 level
is three times as long.
Quantum Mechanics and the
Hydrogen Atom
One of the first great achievements of
quantum mechanics was the solution of the
wave equation for the hydrogen atom
 The significance of quantum mechanics is
that 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

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


Also see Table 28.2
QUICK QUIZ 28.2
How many possible orbital states are there for
(a) the n = 3 level of hydrogen? (b) the n = 4 level?
QUICK QUIZ 28.2 ANSWER
The quantum numbers associated with orbital states
are n, , and m . For a specified value of n, the allowed
values of  range from 0 to n – 1. For each value of ,
there are (2  + 1) possible values of m.
(a) If n = 3, then  = 0, 1, or 2. The number of possible
orbital states is then
[2(0) + 1] + [2(1) + 1] + [2(2) + 1] = 1 + 3 + 5 = 9.
(b) If n = 4, one additional value of  is allowed ( = 3)
so the number of possible orbital states is now
9 + [2(3) + 1] = 9 + 7 = 16
QUICK QUIZ 28.3
When the principal quantum number is n =
5, how many different values of (a)  and
(b) m are possible?
QUICK QUIZ 28.3 ANSWER
(a) For n = 5, there are 5 allowed values of ,
namely  = 0, 1, 2, 3, and 4.
(b) Since m ranges from – to + in integer
steps, the largest allowed value of  ( = 4 in
this case) permits the greatest range of values
for m. For n = 5, there are 9 possible values
for m: -4, -3, -2, –1, 0, +1, +2, +3, and +4.
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 Zeeman
effect
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, cont

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 can ever
be in the same quantum state


In other words, no two electrons in the
same atom can have exactly the same
values for 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
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
explained by quantum numbers and Pauli’s
Exclusion Principle explains the configuration

QUICK QUIZ 28.4
Krypton (atomic number 36) has how many
electrons in its next to outer shell (n = 3)?
(a) 2
(c) 8
(b) 4
(d) 18
QUICK QUIZ 28.4 ANSWER
(d). Krypton has a closed configuration
consisting of filled n=1, n=2, and n=3
shells as well as filled 4s and 4p
subshells. The filled n=3 shell (the next to
outer shell in Krypton) has a total of 18
electrons, 2 in the 3s subshell, 6 in the 3p
subshell and 10 in the 3d subshell.
Characteristic X-Rays
When a metal target is
bombarded by highenergy electrons, x-rays
are emitted
 The x-ray spectrum
typically consists of a
broad continuous
spectrum and a series
of sharp lines



The lines are dependent
on the metal
The lines are called
characteristic x-rays
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
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
Moseley Plot

λ is the wavelength of the
K line


K is the line that is
produced by an electron
falling from the L shell to
the K shell
From this plot, Moseley
was able to determine the
Z values of other
elements and produce a
periodic chart in excellent
agreement with the
known chemical
properties of the elements
Atomic Transitions – Energy
Levels



An atom may have
many possible energy
levels
At ordinary
temperatures, most of
the atoms in a sample
are in the ground state
Only photons with
energies corresponding
to differences between
energy levels can be
absorbed
Atomic Transitions –
Stimulated Absorption


The blue dots
represent electrons
When a photon with
energy ΔE is
absorbed, one
electron jumps to a
higher energy level



These higher levels are
called excited states
ΔE = hƒ = E2 – E1
In general, ΔE can be
the difference between
any two energy levels
Atomic Transitions –
Spontaneous Emission
Once an atom is in
an excited state,
there is a constant
probability that it
will jump back to a
lower state by
emitting a photon
 This process is
called spontaneous

emission
Atomic Transitions –
Stimulated Emission
An atom is in an excited
stated and a photon is
incident on it
 The incoming photon
increases the probability
that the excited atom will
return to the ground state
 There are two emitted
photons, the incident one
and the emitted one


The emitted photon is in
exactly in phase with the
incident photon
Population Inversion
When light is incident on a system of atoms,
both stimulated absorption and stimulated
emission are equally probable
 Generally, a net absorption occurs since most
atoms are in the ground state
 If you can cause more atoms to be in excited
states, a net emission of photons can result


This situation is called a population inversion
Lasers

To achieve laser action, three conditions must
be met


The system must be in a state of population
inversion
The excited state of the system must be a
metastable state


Its lifetime must be long compared to the normal lifetime
of an excited state
The emitted photons must be confined in the
system long enough to allow them to stimulate
further emission from other excited atoms

This is achieved by using reflecting mirrors
Production of a Laser Beam
Laser Beam – He Ne Example




The energy level diagram
for Ne
The mixture of helium and
neon is confined to a glass
tube sealed at the ends by
mirrors
A high voltage applied
causes electrons to sweep
through the tube,
producing excited states
When the electron falls to
E2 in Ne, a 632.8 nm
photon is emitted
Holography
Holography is the
production of threedimensional images of
an object
 Light from a laser is
split at B
 One beam reflects off
the object and onto a
photographic plate
 The other beam is
diverged by Lens 2 and
reflected by the mirrors
before striking the film

Holography, cont

The two beams form a complex interference
pattern on the photographic film


It can be produced only if the phase relationship
of the two waves remains constant
This is accomplished by using a laser
The hologram records the intensity of the
light and the phase difference between the
reference beam and the scattered beam
 The image formed has a three-dimensional
perspective

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, Detail






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
White represents energy gaps
Electrons can have any energy
within the allowed bands
Electrons 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 white area in the
valence band represents
holes
Semiconductors, cont

Holes are empty states in the valence band
created by electrons that have jumped to the
conduction band
 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
Current Process in
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
A depletion region
The Depletion Region
Mobile donor electrons from the n side
nearest the junction diffuse to the p side,
leaving behind immobile positive ions
 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
Diodes

Rectifiers




Transistors


Change AC voltage to DC voltage
A half-wave rectifier allows current to flow during half
the AC cycle
A full-wave rectifier rectifies both halves of the AC
cycle
May be used to amplify small signals
Integrated circuit

A collection of interconnected transistors, diodes,
resistors and capacitors fabricated on a single piece of
silicon