PHY 102: Quantum Physics

Topic 2EM 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 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 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,………………..€

=2L

n

€

k =2π

λ=

πn

L

•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,………………..€

=2L

n

€

k =2π

λ=

πn

L

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” π3/L3

€

ktotal2 = kx

2 + ky2 + kz

2

€

kx =n1π

L;ky =

n2π

L;kz =

n3π

L

Modelling the black body spectrum

“k-space”

• energy emitted in a wavenumber range between k and k + dk,

Number of modes in wavenumber range x energy per mode

How much energy is emitted?

Number of modes

“Volume” of spherical shell betweenk and k + dk?

“Volume” occupied by each mode in k-space?

€

dN(k) =4πk 2dk8π 3

L3

× 2 =Vk 2

π 2 dk

Factor of 2 for two different independent polarizations per mode,V=L3

Energy per mode?

According to classical equipartition theory,each mode (classical oscillator) has energy

E = kbT

So…………….

€

dE(k) =Vk 2kbT

π 2 dk

€

dE(λ ) =Vk 2kbT

π 2

dk

dλdλ =

V 4π 2kbT

λ2π 2 ×2π

λ2 dλ

Changing variables to wavelength:

€

dE(k) =Vk 2kbT

π 2 dk

€

=8VπkbT

λ4 dλ ⇒dE(λ )

dλ=

8πVkbT

λ4

Spectral Intensity (Rayleigh prediction)

• Defined as radiated power per unit wavelength volume per unit area per unit time:

• Rayleigh-Jeans result:

€

I(λ ) =1

V

dE(λ )

dλ× c × geometrical factors

€

I(λ ) =2πckT

λ4

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)

“ULTRAVIOLET CATASTROPHE”…………….

Rayleigh classical theory doesn’t 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:

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

π

•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(λ ) =2πhc 2

λ5 ehc λkT −1( )

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 )

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) 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

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:

5......... 4, 3, m :4

6.364(nm)2

2

m

m

(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):

21

22

111

nnR

n2, n1 integers, n2 < 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 that’s consistent with this nice, elegant formula??

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

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:

r

v

F+Ze

-e

r

v

F+Ze

-e

Rutherford Model: electron energy

From electrostatics, the potential energy of the electron is given by:

r

Ze

r

eZe

r

qqU

0

2

00

21

44

)()(

4 πππ

r

v

F+Ze

-e

Rutherford Model: electron energy

Centripetal force equation:

20

22

4 r

Ze

r

vme

π

Kinetic energy of electron:

r

Zevme

0

22

82 π

r

Zevme

0

22

82 π

Total energy of electron = P.E. + K.E:

r

Ze

r

eZe

r

qqU

0

2

00

21

44

)()(

4 πππ

r

Ze

r

Ze

r

Ze

0

2

0

2

0

2

884energy Total

πππ

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 CAN’T GIVE US STABLE ATOMS………………..

Bohr’s 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:

h

EEf 12

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:

120

2

20

2

10

2

21

11

888 rr

Ze

r

Ze

r

ZeEE

πππ

If Bohr’s postulates are correct, then the frequency of the radiation emitted in the transition is given by:

120

221 11

8 rrh

Ze

h

EEf

π

Rydberg-Ritz Revisited

21

22

111

nnR

21

22

11

nncRf

c = fλ

120

2 11

8 rrh

Zef

π

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:

nnh

mvr π2

IMPLICATIONS???..........................................................................

r

Zevme

0

22

82 π

Kinetic energy (earlier slide)

rm

Zev

e0

22

4π

nnh

mvr π2

quantisation of A.M. (last slide)

22

222

mr

nv

22

22

0

2

4 mr

n

rm

Ze

e

π2

022 4

Zem

nr

e

π

Bohr radius

So, introduction of the idea that angular momentum is quantised has the desired effect: rn2. Simplifying the expression for r a bit (Z=1 for hydrogen):

02

20

22

anrem

hnr n

e

π

a0, the radius of the n=1 orbit, is called the BOHR RADIUS

20

2

0 em

ha

eπ

We conclude that in the Bohr model only certain orbital radii (and electron velocities) are allowed.

120

2 11

8 rrh

ef

π

21

2200

2 11

8 nnah

ef

π

21

22

20

3

4 11

8 nnh

mef e

21

22

11

nncRf

Rydberg-Ritz

R=1.07 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•Doesn’t work well for many-electron atoms (even helium)

•Can’t explain fine structure of spectral lines observed at high resolution, or relative intensities of spectral lines

•Can’t explain effect of magnetic field on spectral lines (Zeeman effects), although Sommerfeld’s modifications (elliptical orbits, varying orientations) help to some extent

•Is fundamentally inconsistent with Heisenberg’s uncertainty principle

THE BOHR MODEL IS WRONG