Transcript For a “black body” - University of Sheffield

PHY 102: Waves & Quanta
Topic 11
EM Radiation from atoms
John Cockburn (j.cockburn@... Room E15)
•Broadband Thermal Radiation
•“Blackbody” spectrum
•Resolution of “ultraviolet catastrophe”
•Atomic line spectra
•Structure of the atom: Rutherford scattering
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….
Thermal Radiation
•Unlike convection and conduction, transfer of heat by
thermal radiation doesn’t 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 Stefan’s Law
P  AeT 4
= “Stefan’s constant” = 5.67 x 10-8 Wm-2K-4
e = “emissivity” ; 0< e < 1, depending on nature of surface
For a “black body” (perfect emitter/absorber), e=1
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)
 pT  constant  2.90 103 m.K
“Wien’s displacement law”
Modelling the black body spectrum
•Rayleigh attempted to calculate the black body spectra from
solids by assuming the material to consist of an assembly of
classical oscillators, with each “normal mode” of vibration
having energy kBT
•Result:
2ck BT
I ( ) 
4

•Agrees OK at long wavelengths, but I at short
wavelengths: “ultraviolet catastrophe”
•Max Planck sorted this out in 1900 with the introduction
of…….
QUANTUM THEORY
Modelling the black body spectrum
Classical (Rayleigh) picture:
•Oscillators have continuous spread of energies
•Average energy of oscillator at temperature T = kT
Quantum (Planck) picture:
•Oscillator only allowed to have energy in integer
multiples of some constant times the oscillator
frequency: E = nhf
•Average energy of oscillator at temperature T:
hf
E
hf
e
kT
1
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:
2hc 2
I ( )  5 hc kT
 e
1


•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…………………………
2hc 2
I ( )  5 hc kT
 e
1

2.50E+10
1.50E+10
1.00E+10
5.00E+09
Wavelength/m
9.8
9.2
8.6
8
7.4
6.8
6.2
5
5.6
4.4
3.8
3.2
2.6
2
1.4
0.8
0.00E+00
0.2
I(W/m3)
2.00E+10

Line spectra
•“Hot” solids and liquids display the continuous emission spectra
described above
•“excited” gases display something completely different: LINE SPECTRA
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???
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
The atom: a brief (incomplete) history
So, by the 19th century, it was universally accepted that matter was composed
of atoms. But we still haven’t answered the question. What is an atom?
1897: JJ Thomson discovers electron, measures ratio e/m
1907: Millikan measures charge on electron
~1910: Thomson’s “plum pudding” model of the atom
1910-1911: Rutherford, Geiger and Marsden clarify internal structure of
atom by scattering of positively charged -particles…………..
Rutherford Scattering
Most particles
pass straight
through, or
deflected
only slightly
Some particles
deflected back
through large
angles
Rutherford Scattering
To explain results of the Rutherford scattering :
1) Atom must be mostly empty space
2) Positive charge must be concentrated in a small volume occupying a
very small fraction of the total volume of the atom…………
Christmas pudding model
doesn’t work
Nuclear model does work
Atomic radius ~ 10-10m
Nuclear radius ~ 10-14m