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PHY 102: Quantum Physics

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PHY 102: Quantum Physics Topic 2 EM Radiation from atoms – PowerPoint PPT presentation

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Title: PHY 102: Quantum Physics


1
PHY 102 Quantum Physics Topic 2 EM Radiation
from atoms
2
  • Broadband Thermal Radiation
  • Blackbody spectrum
  • Resolution of ultraviolet catastrophe
  • Atomic line spectra
  • Structure of the atom Rutherford scattering

3
Thermal Radiation
  • Heat is associated with vibrational thermal
    motion of atoms/molecules
  • General principle accelerating charged particles
    generate electromagnetic radiation (examples
    generation of radio waves by moving electrons in
    antenna, generation of continuous X-ray spectrum
    by electrons decelerated by interaction with
    atoms of metal target)
  • So, e.m. radiation is generated by the thermally
    induced motion of atoms/molecules THERMAL
    RADIATION.

4
Thermal Radiation
  • Unlike convection and conduction, transfer of
    heat by thermal radiation doesnt require a
    medium
  • So, for example, heat can reach Earth from the
    Sun through millions of kilometres of empty
    space.
  • Rate at which an object, surface area A,
    temperature T, radiates energy is given by
    Stefans Law
  • Stefans constant 5.67 x 10-8 Wm-2K-4
  • e emissivity 0lt e lt 1, depending on nature
    of surface
  • For a black body (perfect emitter/absorber), e1

5
Spectrum of emitted radiation
Black body emission spectrum for various
temperatures
  • Peak wavelength decreases with increasing
    temperature
  • Area under curve (total emitted power increases
    with increasing temperature
  • Experimentally, the dependence of peak wavelength
    on temperature is found to be given by

? (?m)
Wiens displacement law
6
Modelling the black body spectrum
  • Rayleigh attempted to calculate the black body
    spectra from solids by assuming the radiation to
    originate from classical EM standing waves
    (normal modes) within the object
  • Standing wavelength in one dimension for cubic
    box of side L
  • wavenumber
  • n 1,2,3,..

7
Modelling the black body spectrum
  • Rayleigh attempted to calculate the black body
    spectra from solids by assuming the radiation to
    originate from classical EM standing waves
    (normal modes) within the object
  • Standing wavelength in one dimension for cubic
    box of side L
  • wavenumber
  • n 1,2,3,..

8
Modelling the black body spectrum
  • In 3 dimensions, ktotal given by
  • for cube of side L
  • In k-space each allowed state occupies a
    volume p3/L3

9
k-space
10
How much energy is emitted?
  • energy emitted in a wavenumber range between k
    and k dk,
  • Number of modes in wavenumber range x energy per
    mode

11
Number of modes
Volume of spherical shell between k and k dk?
Volume occupied by each mode in k-space?
Factor of 2 for two different independent
polarizations per mode, VL3
12
Energy per mode?
According to classical equipartition theory, each
mode (classical oscillator) has energy E
kbT So.
13
Changing variables to wavelength
14
Spectral Intensity (Rayleigh prediction)
  • Defined as radiated power per unit wavelength
    volume per unit area per unit time
  • Rayleigh-Jeans result

15
Spectrum of emitted radiation
Black body emission spectrum for various
temperatures
  • Peak wavelength decreases with increasing
    temperature
  • Area under curve (total emitted power increases
    with increasing temperature
  • Experimentally, the dependence of peak wavelength
    on temperature is found to be given by

? (?m)
Wiens displacement law
16
ULTRAVIOLET CATASTROPHE. Rayleigh
classical theory doesnt work.
17
Classical vs Quantum
  • Classical (Rayleigh-Jeans) picture
  • EM modes have continuous spread of energies
  • Average energy of oscillator at temperature T
    kT
  • Quantum (Planck) picture
  • EM modes only allowed to have energy in integer
    multiples of some constant times the oscillator
    frequency E nhf
  • Average energy of oscillator at temperature T

18
Modelling the black body spectrum
Obtain expression for spectral intensity by
taking product of average energy per oscillator
and number of oscillator modes per unit
volume. Planck result
  • This model predicts the form of the blackbody
    spectrum perfectly, no UV catastrophe
  • First experimental anomaly to be explained by
    the need for a quantum theory (1900)
  • h originally introduced by Planck purely as an
    empirical constant to fit data

19
I(W/m3)
Wavelength/?m
Quantum theory gives excellent agreement with
experiment.
20
Line spectra
  • Hot solids and liquids display the continuous
    emission spectra described above
  • excited gases display something completely
    different LINE SPECTRA

21
Line spectra
  • Line spectrum of a gas of atoms/molecules is
    reproducible, and is a unique fingerprint of
    the gas
  • Suggests that the spectrum is somehow related to
    the internal structure of the atom.
  • So, what is an atom???

22
The atom a brief (incomplete) history
Leucippus of Miletus, Democritus
(450BC) Suggest universe composed of hard,
uniform, indivisible particles and the space
between them (atom cannot be cut) Pierre
Gassendi (1592-1655), Robert Boyle
(1627-1691) Matter composed of rigid,
indestructible atoms, varied size and form,
different elements composed of different atoms,
atoms can combine to form molecules. Joseph
Louis Proust (1754-1826), John Dalton
(1766-1844) Law of definite proportions,
atomic picture of chemical processes,
stoichiometry Lothar Meyer (1830-95), Dmitry
Mendeleev (1834-1907) Significance of atomic
weights, Periodic Table of the elements
23
The atom a brief (incomplete) history
So, by the 19th century, it was universally
accepted that matter was composed of atoms. But
we still havent answered the question. What is
an atom?
1897 JJ Thomson discovers electron, measures
ratio e/m 1907 Millikan measures charge on
electron 1910 Thomsons plum pudding model
of the atom 1910-1911 Rutherford, Geiger and
Marsden clarify internal structure of atom by
scattering of positively charged ?-particles..
24
Rutherford Scattering
Most ?-particles pass straight through, or
deflected only slightly
Some ?-particles deflected back through large
angles
25
Rutherford Scattering
  • To explain results of the Rutherford scattering
  • Atom must be mostly empty space
  • Positive charge must be concentrated in a small
    volume occupying a very small fraction of the
    total volume of the atom

Nuclear model does work
Christmas pudding model doesnt work
Atomic radius 10-10m Nuclear radius 10-14m
26
The Rutherford/Bohr Model
27
  • More on line spectra
  • Orbital model of the hydrogen atom
  • Failure of classical model
  • Quantisation of orbital angular momentum
    stationary states
  • Successes and failures of the Bohr Model

28
Line Spectrum of hydrogen
  • Hydrogen has line spectrum ranging in wavelength
    from the UV to the infrared
  • Balmer (1885) found that the wavelengths of the
    spectral lines in the visible region of the
    spectrum could be EMPIRICALLY fitted to the
    relationship

(The group of hydrogen spectral lines in the
visible region still known as the Balmer Series)
29
Line Spectrum of hydrogen
  • Rydberg and Ritz subsequently obtained a more
    general expression which applies to ALL hydrogen
    spectral lines (not just visible), and also to
    certain elements (eg alkaline metals)

n2, n1 integers, n2 lt n1
  • R is called the Rydberg constant, which changes
    slightly from element to element.
  • For hydrogen, RH 1.097776 x 107 m-1
  • Can a model of the atom be developed thats
    consistent with this nice, elegant formula??

30
Rutherford Scattering
  • To explain results of the Rutherford scattering
  • Atom must be mostly empty space
  • Positive charge must be concentrated in a small
    volume occupying a very small fraction of the
    total volume of the atom

Nuclear model does work
Christmas pudding model doesnt work
Atomic radius 10-10m Nuclear radius 10-14m
31
Rutherford planetary model
Basic idea electrons in an atom orbit the
positively-charged nucleus, in a similar way to
planets orbiting the Sun (but centripetal force
provided by electrostatic attraction rather that
gravitation) Hydrogen atom single electron
orbiting positive nucleus of charge Ze, where Z
1
32
Rutherford Model electron energy
From electrostatics, the potential energy of the
electron is given by
33
Rutherford Model electron energy
Centripetal force equation
Kinetic energy of electron
34
Total energy of electron P.E. K.E
But this classical treatment leaves us with a big
problem
35
Failure of the Classical model
The orbiting electron is an accelerating
charge. Accelerating charges emit
electromagnetic waves and therefore lose
energy Classical physics predicts electron
should spiral in to the nucleus emitting
continuous spectrum of radiation as the atom
collapses CLASSICAL PHYSICS CANT GIVE US
STABLE ATOMS..
36
Bohrs postulates
  • Only certain electron orbits are allowed, in
    which the electron does not emit em radiation
    (STATIONARY STATES)
  • An atom emits radiation only when an electron
    makes a transition from one stationary state to
    another.
  • The frequency of the radiation emitted when an
    electron makes a transition from a stationary
    state with energy E2 to one with energy E1 is
    given by

37
Transition energies
Suppose an electron is initially in stationary
state with energy E1, orbital radius r1. It then
undergoes a transition to a lower energy state
E2, with (smaller) radius r2
If Bohrs postulates are correct, then the
frequency of the radiation emitted in the
transition is given by
38
Rydberg-Ritz Revisited
c f?
Bohr result
Looks promising, if we can make the connection
that r is somehow proportional to integer
squared.
39
Quantisation of angular momentum
Bohr now makes the bold assumption that the
orbital angular momentum of the electron is
quantised Since v is perpendicular to r, the
orbital angular momentum is just given by L
mvr. Bohr suggested that this is quantised, so
that
IMPLICATIONS???...................................
.......................................
40
Kinetic energy (earlier slide)
quantisation of A.M. (last slide)
41
Bohr radius
So, introduction of the idea that angular
momentum is quantised has the desired effect
r?n2. Simplifying the expression for r a bit (Z1
for hydrogen)
a0, the radius of the n1 orbit, is called the
BOHR RADIUS
42
We conclude that in the Bohr model only certain
orbital radii (and electron velocities) are
allowed.
Rydberg-Ritz
R1.07 x 107 m-1 How nice.
43
Origins of hydrogen spectral lines
44
Bohr Model Shortcomings
  • The Bohr model does an excellent job of
    explaining the gross features of hydrogen line
    spectrum

BUT
  • Doesnt work well for many-electron atoms (even
    helium)
  • Cant explain fine structure of spectral lines
    observed at high resolution, or relative
    intensities of spectral lines
  • Cant explain effect of magnetic field on
    spectral lines (Zeeman effects), although
    Sommerfelds modifications (elliptical orbits,
    varying orientations) help to some extent
  • Is fundamentally inconsistent with Heisenbergs
    uncertainty principle

THE BOHR MODEL IS WRONG
45
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