PHY 102: Waves & Quanta

Topic 11EM 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

4AeTP

= “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.K 1090.2constant 3Tp

“Wien’s displacement law”

(m)

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:

4

2)(

Tck

I B

•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:

1

kThf

e

hfE

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:

1

2)(

5

2

kThce

hcI

•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…………………………

1

2)(

5

2

kThce

hcI

0.00E+00

5.00E+09

1.00E+10

1.50E+10

2.00E+10

2.50E+10

0.2

0.8

1.4 2

2.6

3.2

3.8

4.4 5

5.6

6.2

6.8

7.4 8

8.6

9.2

9.8

Wavelength/m

I(W

/m3 )

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) historyLeucippus 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 ~ 10-10mNuclear radius ~ 10-14m