Transcript Atomic Physics

Chapter 28
Atomic Physics
What energy photon is needed to “see”
a proton of radius 1 fm?
10
1.
2.
3.
4.
5.
6.
1
1
1
1
1
1
eV
keV
MeV
GeV
TeV
PeV
(100 eV)
(103 eV)
(106 eV)
(109 eV)
(1012 eV)
(1015 eV)
17%
17%
1
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
2
11
12
13
17%
17%
3
4
14
General Physics
15
17%
17%
5
16
17
6
18
19
20
Atom Physics
Sections 1–4
General Physics
Emission Spectra


When a high voltage is applied to a gas at low pressure, it
emits light characteristic of the gas
When the emitted light is analyzed with a spectrometer, a
series of discrete bright lines is observed - emission spectrum

Each line has a different wavelength and color
General Physics
Spectral Lines of Hydrogen


The Balmer Series has
lines whose wavelengths
are given by the
preceding equation
Examples of spectral lines




n
n
n
n
=
=
=
=
3, λ
4, λ
5, λ
6, λ
General Physics
=
=
=
=
656.3 nm
486.1 nm
434.1 nm
410.2 nm
Emission Spectrum of Hydrogen
– Equation

The wavelengths of hydrogen’s spectral lines
experimentally agree with the equation
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 = 3, 4, 5, 6, …
The spectral lines correspond to different values of n
General Physics
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
General Physics
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
General Physics
Importance of Hydrogen Atom




Hydrogen is the simplest atom
Enables us to understand the periodic table
Ideal system for performing precise
comparisons of theory with experiment
Much of what we know about the hydrogen
atom can be extended to other single-electron
ions

For example, He+ and Li2+
General Physics
Sir Joseph John Thomson






“J. J.” Thomson
1856 - 1940
Developed model of the atom
Discovered the electron
Did extensive work with
cathode ray deflections
1906 Nobel Prize for
discovery of electron
General Physics
Early Models of the Atom

Newton’s model


J.J. Thomson’s model



tiny, hard, indestructible sphere
A volume of positive charge
Electrons embedded throughout
the volume
Vibrational “modes”


responsible for spectral lines
Didn’t work!
General Physics
Rutherford’s Scattering Experiments



The source was a
naturally radioactive
material that produced
alpha particles (He++)
Most of the alpha
particles passed
though the gold foil
A few deflected from
their original paths

Some even reversed
their direction of travel
Active Figure: Rutherford Scattering
General Physics
Rutherford Model of the Atom

Rutherford, 1911



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
General Physics
Rutherford Model, Problems

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
at a frequency related to their orbital speed



The radius should steadily decrease and the speed should steadily
increase as this radiation is given off
The electron should eventually spiral into the nucleus, but it doesn’t
The radiation frequency should steadily increase – should observe a
continuous spectrum of radiation at progressively shorter and
shorter wavelengths, but you don’t
General Physics
Neils Bohr




1885 – 1962
Participated in the early
development of quantum
mechanics
Headed Institute in
Copenhagen
1922 Nobel Prize for
structure of atoms and
radiation from atoms
General Physics
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 nonclassical ideas
His model included an attempt to explain
why the atom was stable
General Physics
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
Only certain electron orbits
are stable

These are the orbits in which
the atom does not emit energy
in the form of electromagnetic
radiation
General Physics
Bohr’s Assumptions, cont

Radiation is emitted by the atom when the electron
“jumps” 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
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
 Ln = me v r = n ħ where n = 1, 2, 3, …
General Physics
Bohr Radius

The radii of the Bohr orbits are quantized
n2 2
rn 
n  1, 2, 3,
2
me kee

This is based on the assumption 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
General Physics
Quantized Energies

The total energy of the atom


2
2
k
e
1
e
E  KE  PE  me v 2  ke   e
2
r
2r
Using the radius equation for the allowed Bohr orbits
me ke e 4  1 
En  
 2  n  1, 2, 3...
2
2  n 
2



When n = 1, the orbit has the lowest energy, called the
ground state energy
E1 = -13.6 eV (the ionization energy)
General Physics
Radii and Energy of Orbits



A general expression for
the radius of any orbit in a
hydrogen atom is
 r n = n2 ao
The energy of any orbit is
 En = - E0 / n2
Bohr radius and energy:
 ao = 0.529 Å
 E0 = 13.6 eV
General Physics
Energy Level Diagram & Equation

Whenever a transition occurs from a
state, ni to another state, nf (where
ni > nf), a photon is emitted

The photon is emitted with energy
hf = (Ei – Ef) with a wavelength λ
given by
 1

1
 RH  2  2 
n


n
i 
 f
1



For the Paschen series, nf = 3
For the Balmer series, nf = 2
For the Lyman series, nf = 1
Active Figure: Bohr's Model of the Hydrogen Atom
General Physics
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
General Physics
Successes of the Bohr Theory

Explained several features of the hydrogen spectrum






Accounts for Balmer and other series
Predicts a value for RH that agrees with experiment
Gives an expression for the radius of the atom
Predicts energy levels of hydrogen
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
General Physics
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



Determines the energy of the allowed states
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
General Physics
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
General Physics
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
There are two directions for the spin


Spin up, ms = ½
Spin down, ms = -½
General Physics
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
General Physics
Wolfgang Pauli


1900 – 1958
Contributions include






Major review of relativity
Exclusion Principle
Connect between electron
spin and statistics
Theories of relativistic
quantum electrodynamics
Neutrino hypothesis
Nuclear spin hypothesis
General Physics
The Pauli Exclusion Principle


No two electrons in an atom 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
General Physics
Number of Electrons in Filled Subshells and Shells (First Three Shells)
Shell principle
quantum
number
Subshell orbital
quantum
number
orbital magnetic
quantum number
spin magnetic
quantum
number
Electrons
in Filled
Subshell
Electrons
in Filled
Shell
K(n=1)
s(ℓ=0)
mℓ = 0
ms = +½, −½
2
2
s(ℓ=0)
mℓ = 0
ms = +½, −½
2
L(n=2)
M(n=3)
8
p(ℓ=1)
mℓ = +1, 0, −1
ms = +½, −½
6
s(ℓ=0)
mℓ = 0
ms = +½, −½
2
p(ℓ=1)
mℓ = +1, 0, −1
ms = +½, −½
6
d(ℓ=2)
mℓ = +2, +1, 0, −1, −2
ms = +½, −½
10
General Physics
18

Atomic Orbitals
General Physics
Periodic Table
1s2; 2s2 2p6; 3s2 3p6; 4s2 3d10 4p6; 5s2 4d10 5p6; 6s2 4f14 5d10 6p6; 7s2 5f14 6d10 7p6
General Physics
General Physics
Atomic Transitions –
Stimulated Absorption

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
General Physics
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
General Physics
Atomic Transitions –
Stimulated Emission



An atom is in an excited stated
and a photon is incident on it
The incoming photon stimulates
the excited atom to return to
the ground state
There are two emitted photons,
the incident one and the emitted
one

The emitted photon has the
same wavelength and is in
phase with the incident photon
Active Figure: Spontaneous and Stimulated Emission
General Physics
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
General Physics
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


More atoms in an excited state than the ground 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
General Physics
Laser Beam – He Ne Example




The energy level diagram for Ne
in a He-Ne laser
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
from E*3 in Ne, a 632.8 nm
photon is emitted
General Physics
Production of a Laser Beam
General Physics
Holography




Holography is the production of
three-dimensional 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
General Physics
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
General Physics