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Chapter 29
Atomic Physics
Importance of
the Hydrogen Atom


The hydrogen atom is the only atomic
system that can be solved exactly
Much of what was learned about the
hydrogen atom, with its single electron,
can be extended to such single-electron
ions as He+ and Li2+
More Reasons the
Hydrogen Atom is Important

The hydrogen atom proved to be an ideal
system for performing precision tests of
theory against experiment
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Also for improving our understanding of atomic
structure
The quantum numbers that are used to
characterize the allowed states of hydrogen
can also be used to investigate more complex
atoms

This allows us to understand the periodic table
Final Reason 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 structures and the electronic
structure of solids
Early Models of the Atom –
Newton’s Time
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The atom was a tiny, hard indestructible
sphere
It was a particle model that ignored any
internal structure
The model was a good basis for the
kinetic theory of gases
Early Models of the Atom –
JJ Thomson
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J. J. Thomson
established the charge
to mass ratio for
electrons
His model of the atom
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A volume of positive
charge
Electrons embedded
throughout the volume
Rutherford’s Thin Foil
Experiment
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Experiments done in
1911
A beam of positively
charged alpha particles
hit and are scattered
from a thin foil target
Large deflections could
not be explained by
Thomson’s model
Early Models of the Atom –
Rutherford
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Rutherford
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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
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The Rutherford model is unable to explain this
phenomena
Rutherford’s electrons are undergoing a
centripetal acceleration

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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 doesn’t
The Bohr Theory of Hydrogen
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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
He applied Planck’s ideas of quantized
energy levels to orbiting electrons
Bohr’s Theory, cont
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In this model, the electrons are
generally confined to stable,
nonradiating orbits called stationary
states
Used Einstein’s concept of the photon
to arrive at an expression for the
frequency of radiation emitted when the
atom makes a transition
Problem’s With Bohr’s Model
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Improved spectroscopic techniques
showed many of the single spectral
lines were actually a group of closely
spaced lines
Single spectral lines could be split into
three closely spaced lines when the
atom was placed in a magnetic field
Mathematical Description
of Hydrogen
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The solution to Schrödinger’s equation
as applied to the hydrogen atom gives a
complete description of the atom’s
properties
Apply the quantum particle under
boundary conditions and determine the
allowed wave functions and energies of
the atom
Mathematical
Description, cont
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For a three-dimensional system, the
boundary conditions will generate three
quantum numbers
A fourth quantum number is needed
for spin
The potential energy function for the
hydrogen atom is
Mathematical
Description, final
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The energies of the allowed states
becomes
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ao is the Bohr radius
The allowed energy levels depend only
on the principle quantum number, n
Quantum Numbers, General
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The imposition of boundary conditions
also leads to two new quantum
numbers
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Orbital quantum number,
Orbital magnetic quantum number, m
Principle Quantum Number
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The first quantum number is associated
with the radial function
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It is called the principle 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
Orbital and Orbital
Magnetic Quantum Numbers
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The orbital quantum number is symbolized
by
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It is associated with the orbital angular momentum
of the electron
It is an integer
The orbital magnetic quantum number is
symbolized by ml
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It is also associated with the angular orbital
momentum of the electron and is an integer
Quantum Numbers,
Summary of Allowed Values
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The values of n can range from 1 to 
The values of can range from 0 to n-1
The values of m can range from – to
Example:
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If n = 1, then only = 0 and ml = 0 are permitted
If n = 2, then = 0 or 1
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If
If
= 0 then m = 0
= 1 then m may be –1, 0, or 1
Quantum Numbers,
Summary Table
Shells
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Historically, all states having the same
principle quantum number are said to
form a shell
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Shells are identified by letters K, L, M …
All states having the same values of n
and are said to form a subshell
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The letters s, p,d, f, g, h, .. are used to
designate the subshells for which = 0, 1,
2, 3, …
Shell and Subshell
Notation, Summary Table
Quantum Numbers, final
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State with quantum numbers that violate
the previous rules cannot exist
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They would not satisfy the boundary
conditions on the wave function of the
system
Wave Functions for Hydrogen
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The simplest wave function for hydrogen is
the one that describes the 1s state and is
designated y1s (r)
As y1s (r) approaches zero, r approaches 
and is normalized as presented
y1s (r) is also spherically symmetric

This symmetry exists for all s states
Probability Density
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The probability density for the 1s state is
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The radial probability density function, P(r), is
the probability per unit radial length of finding
the electron in a spherical shell of radius r
and thickness dr
Radial Probability Density
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A spherical shell of
radius r and
thickness dr has a
volume of 4 p r2 dr
The radial
probability function
is P(r) = 4 p r2 |y|2
P(r) for 1s State of Hydrogen
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The radial probability
density function for the
hydrogen atom in its
ground state is
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The peak indicates the
most probable location
The peak occurs at the
Bohr radius
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P(r) for 1s
State of Hydrogen, cont
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The average value of r for the ground
state of hydrogen is 3/2 ao
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The graph shows asymmetry, with much
more area to the right of the peak
According to quantum mechanics, the
atom has no sharply defined boundary
as suggested by the Bohr theory
Electron Clouds
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The charge of the
electron is extended
throughout a diffuse
region of space,
commonly called an
electron cloud
This shows the
probability density as a
function of position in
the xy plane
The darkest area, r =
ao, corresponds to the
most probable region
Electron Clouds, cont
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The electron cloud model is quite different
from the Bohr model
The electron cloud structure remains the
same, on the average, over time
The atom does not radiate when it is in one
particular quantum state
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This removes the problem of the Rutherford model
Radiation occurs when a transition is made,
causing the structure to change in time
Wave Function of the 2s state
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The next-simplest wave function for hydrogen
is for the 2s state
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n = 2; l = 0
The wave function is
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y2s depends only on r and is spherically symmetric
Comparison of
1s and 2s States
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The plot of the radial
probability density
for the 2s state has
two peaks
The highest value of
P corresponds to the
most probable value
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In this case, r 5ao
Physical Interpretation of
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The magnitude of the angular
momentum of an electron moving in a
circle of radius r is L = me v r
The direction of is perpendicular to
the plane of the circle
In the Bohr model, the angular
momentum of the electron is restricted
to integer multiples of
Physical Interpretation
of , cont
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According to quantum mechanics, an atom in a
state whose principle quantum number is n can
take on the following discrete values of the
magnitude of the orbital angular momentum:
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That L can equal zero causes great difficulty when
attempting to apply classical mechanics to this
system
In the quantum mechanical interpretation, the
electron cloud for the L = 0 state is spherically
symmetrical with no fundamental axis of rotation
Physical Interpretation of m
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The atom possesses an orbital angular
momentum
Because angular momentum is a vector,
its direction must be specified
An orbiting electron can be considered
an effective current loop with a
corresponding magnetic moment
Physical Interpretation of m , 2
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The direction of the angular momentum
vector relative to an axis is quantized
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Once an axis is specified, the angular
momentum vector can only point in certain
directions with respect to this axis
Therefore, Lz, the projection of along
the z axis, can have only discrete
values
Physical Interpretation of m , 3
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The orbital magnetic quantum number
m specifies the allowed values of the z
component of orbital angular
momentum
Lz = m
The quantization of the possible
orientations of with respect to an
external magnetic field is often referred
to as space quantization
Physical Interpretation of m , 4
does not point in a specific direction
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Even though its z-component is fixed
Knowing all the components is
inconsistent with the Uncertainty
Principle
Image that must lie anywhere on the
surface of a cone that makes an angle q
with the z axis
Physical Interpretation
of m , final
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q is also quantized
Its values are specified
through
m is never greater than
, therefore q can never
be zero

can never be parallel
to the z-axis
Spin Quantum Number, ms
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Electron spin does not come from the
Schrödinger equation
Additional quantum states can be
explained by requiring a fourth quantum
number for each state
This fourth quantum number is the spin
magnetic quantum number, ms
Electron Spins
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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 spin directions and
this produces doublets
in spectra of certain
gases
Electron Spins, cont
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The concept of a spinning electron is
conceptually useful
The electron is a point particle, without any
spatial extent
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Therefore the electron cannot be considered to be
actually spinning
The experimental evidence supports the
electron having some intrinsic angular
momentum that can be described by ms
Sommerfeld and Dirac showed this results
from the relativistic properties of the electron
Spin Angular Momentum
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The total angular momentum of a particular
electron state contains both an orbital
contribution and a spin contribution
Electron spin can be described by a single
quantum number s, whose value can only be
s = 1/2
The spin angular momentum of the electron
never changes
Spin Angular Momentum, cont
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The magnitude of the spin angular
momentum is
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The spin angular momentum can have two
orientations relative to a z axis, specified by
the spin quantum number ms = 1/2
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ms =+ 1/2 corresponds to the spin up case
ms = - 1/2 corresponds to the spin down case
Spin Angular Momentum, final
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The allowed values
of the z component
of the spin angular
momentum is
Sz = ms = 1/2
Spin angular
moment S is
quantized
Spin Magnetic Moment
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The spin magnetic moment
is
related to the spin angular momentum
by

The z component of the spin magnetic
moment can have values
Bohr Magneton
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The quantity e /2me is called the Bohr
magneton
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It has a numerical value of 9.274 x 10-24
J/T
Stern-Gerlach Experiment
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A beam of neutral silver atoms is split into two
components by a nonuniform magnetic field
The atoms experienced a force due to their magnetic
moments
The beam had two distinct components in contrast to
the classical prediction
Stern-Gerlach
Experimental Results
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The experiment provided two important
results:
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It verified the concept of space
quanitization
It showed that spin angular momentum
exists even though the property was not
recognized until long after the experiments
were performed
Wolfgang Pauli
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1900 – 1958
Important review article on
relativity
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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
The Exclusion Principle
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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 in an atom can ever be in the same
quantum state
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Therefore, no two electrons in the same atom can
have the same set of quantum numbers
Filling Subshells
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Once a subshell is filled, the next
electron goes into the lowest-energy
vacant state
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If the atom were not in the lowest-energy
state available to it, it would radiate energy
until it reached this state
Orbitals
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An orbital is 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
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One electron will have ms = 1/2 and one
will have ms = -1/2
Allowed Quantum States,
Example
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The arrows represent spin
The n=1 shell can accommodate only two electrons,
since only one orbital is allowed
In general, each shell can accommodate up to 2n2
electrons
Hund’s Rule
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Hund’s Rule states that when an atom
has orbitals of equal energy, the order in
which they are filled by electrons is such
that a maximum number of electrons
have unpaired spins
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Some exceptions to the rule occur in
elements having subshells that are close to
being filled or half-filled
Periodic Table
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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 only
because the elements had not yet been
discovered
Periodic Table, cont
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By noting the columns in which some missing
elements should be located, he was able to
make rough predictions about their chemical
properties
Within 20 years of the predictions, most of the
elements were discovered
The elements in the periodic table are
arranged so that all those in a column have
similar chemical properties
Periodic Table, Explained
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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 the
filled shell and the next available shell
Hydrogen Energy Level
Diagram Revisited
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The allowed values of
are separated
Transitions in which
does not change are
very unlikely to occur
and are called forbidden
transitions
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Such transitions actually
can occur, but their
probability is very low
compared to allowed
transitions
Selection Rules
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The selection rules for allowed transitions are
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D =1
Dm = 0or 1
The angular momentum of the atom-photon
system must be conserved
Therefore, the photon involved in the process
must carry angular momentum
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The photon has angular momentum equivalent to
that of a particle with spin 1
A photon has energy, linear momentum and
angular momentum
Multielectron Atoms
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For multielectron atoms, the positive
nuclear charge Ze is largely shielded by
the negative charge of the inner shell
electrons
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The outer electrons interact with a net
charge that is smaller than the nuclear
charge
Allowed energies are

Zeff depends on n and
X-Ray Spectra
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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
X-Ray Spectra, cont
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The discrete lines are called
characteristic x-rays
These are created when

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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
X-Ray Spectra, final
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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
wavelengths in the range of 0.01 nm
to 1 nm
Moseley Plot


Henry G. J. Moseley
plotted the values of
atoms as shown
l is the wavelength of the
Ka line of each element


The Ka line refers to the
photon emitted when an
electron falls from the L
to the K shell
From this plot, Moseley
developed a periodic chart
in agreement with the one
based on chemical
properties
Hydrogen Atoms in Space

Hydrogen is the most abundant element
in space


Its role in astronomy and cosmology is very
important
Assume a cloud of hydrogen near a hot
star

The high-energy photons from the star can
interact with the hydrogen atoms

Either raises them to a high energy state or
ionizes them
Hydrogen Atoms
in Space, cont


As the excited electrons fall to lower
energy levels, many atoms emit Balmer
series of wavelengths
These atoms provide red, green, blue,
and violet colors to nebulae

These correspond to the colors in the
hydrogen spectrum
Emission Nebulae

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Emission nebulae
are near hot stars
The hydrogen atoms
are excited by light
from the star
The light is
dominated by
discrete emission
spectral lines and
contains colors
Reflection Nebulae

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Reflection nebulae are
near cool stars
Most of the light from
the nebula is starlight
reflected from larger
grains of material in the
nebula rather than
emitted by excited
atoms
The spectrum of light
from the nebula is the
same as from the star
Reflection Nebulae, cont
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
The spectrum is an absorption spectrum
with dark lines corresponding to atoms
and ions in the outer regions of the star
The light from these nebulae tends to
appear white
Dark Nebulae
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Dark nebulae are not
near a star
Little radiation is
available to excite
atoms or reflect from
grains of dust
The material in the
nebulae screen out light
from stars beyond them
They appear as black
patches against the
brightness of more
distant stars
Other Gases
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In addition to hydrogen, other atoms
and ions are excited by the photons
from stars
Some of the prominent colors are

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Violet (l= 373 nm) from O+ ions
Green (l= 496 nm and 501 nm) from
O++ ions
Also contributing are helium and nitrogen
21 cm Radiation

States where proton spin and electron
spin are parallel are slightly higher than
when they are antiparallel

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Since they are not at the same energy,
transitions can occur between them
The energy difference is 5.9meV
The wavelength of this photon is 21 cm
21 cm Radiation, cont
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
By looking for this radiation, you can
detect hydrogen in space
If the wavelength has been Doppler
shifted, the shift can be used to
measure the relative speed of the
source