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PHY 102 Quantum Physics Topic 2 EM Radiation

from atoms

- 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 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

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

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,..

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,..

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

k-space

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

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

Energy per mode?

According to classical equipartition theory, each

mode (classical oscillator) has energy E

kbT So.

Changing variables to wavelength

Spectral Intensity (Rayleigh prediction)

- Defined as radiated power per unit wavelength

volume per unit area per unit time - Rayleigh-Jeans result

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

ULTRAVIOLET CATASTROPHE. Rayleigh

classical theory doesnt work.

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

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

I(W/m3)

Wavelength/?m

Quantum theory gives excellent agreement with

experiment.

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 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..

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
- 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

The Rutherford/Bohr Model

- 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

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)

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??

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

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

Rutherford Model electron energy

From electrostatics, the potential energy of the

electron is given by

Rutherford Model electron energy

Centripetal force equation

Kinetic energy of electron

Total energy of electron P.E. K.E

But this classical treatment leaves us with a big

problem

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..

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

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

Rydberg-Ritz Revisited

c f?

Bohr result

Looks promising, if we can make the connection

that r is somehow proportional to integer

squared.

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???...................................

.......................................

Kinetic energy (earlier slide)

quantisation of A.M. (last slide)

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

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.

Origins of hydrogen spectral lines

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

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