2. The Particle-like Properties Of Electromagnetic Radiation

2.1 Photoelectric effect and Einstein’s theory 2.2 Black body radiation 2.3 Compton effect 2.4 Bremsstrahlung and pair production 2.5 The photon

Photoelectric effect and Einstein’s theory

V

A

light

electron

collectoremitter

i

The voltage V is increased gradually until no current pass through the outer circuit. The voltage in this case called stopping potential VS .The energy used to stop this electron is eVS .This value is equal to Kmax the maximum energy required to overcome the electric potential energy acquired by an electron.

Smax eVE

Classical PostulatesElectrons are released from the metal surface if

the energy of the incident light exceeds the binding energy of the electron to the metal surface. This value is called work function f.

The maximum kinetic energy Kmax should be proportional to the intensity of the radiation I.

(it is thought that as the intensity of the incident light increased more energy is delivered to the surface of the metal).

Classical Postulates (continued)

• The photoelectric effect should occur for light of any frequency or wavelength. ( as long as light intensity is enough)

• The first electrons should be emitted in a time interval of the order of seconds after radiation first strikes the surface.

Experimental ResultsComparison to the classical postulates

1. The maximum kinetic energy is totally independent of the intensity of the light source.

2. The photoelectric effect does not occur at all if the frequency of the light source is below a certain value (the cutoff frequency nC ) any light source of frequency above this value may cause emission of photoelectrons.

3. The first photoelectrons are emitted virtually instantaneously (within 10-9 s) after the light source is turned on.

Kmax (VS) and Intensity

Stopping potential is independent on the intensity

Einstein Theory

• The energy of light wave is not continuously distributed over the wave front , but instead is concentrated in localized bundles (photons).

• The energy of each photon is given by

nhchE

Einstein Theory (continued)

• Since photons travel with the electromagnetic waves at the speed of light, they must obey the relativistic relation

• Therefore,

• Like other particles, photons carry linear momentum as well as energy.

cEp

h

c

hcp

Einstein Theory (continued)

• Despite the rest mass of photon, according to the theory of relativity, is zero and photon vanishes at speed lower than that of light, its energy is still given by

2mcE

• If the photon energy is greater than the work function of the metal surface, photoelectron is released, or photoelectric effect doesn’t occur.

fnhkmax

• In this equation the intensity I of the light source doesn’t appear.

• if the photon energy is hardly equal to the work function, the photon frequency in this case is called cutoff frequency and is given by

hcf

n

Work function and Planck’s constant

fnhkmax

f

Example 3.3

• What are the energy and momentum of a red light photon of wavelength 650 nm?

• What is the wavelength of a photon of energy 2.4 eV?

Example 3.4

• The work function for tungsten metal is 4.52 eV. What is the cutoff frequency and wavelength? What is the maximum kinetic energy of the electrons when radiation of wavelength 198 nm is used? What is the stopping potential in this case?

Black Body Radiation

I is the total intensity of electromagnetic radiation emitted at all wavelengthsThe intensity dI in the wavelength interval between and + D is given by dI = R() dR() is the radiancy : which is the intensity per unit wavelength interval.

Stefan’s Radiation LawThe total intensity I is given by the area under the radiancy curve.which is proportional to the absolute temperature of the body T.

4

This is called Stefan's law and s is the Stefan-Boltzmann constant s 5.67 X 10-8 W/m2 K4 .

Wein’s Displacement Law

• It is noticed from the spectrum figure that the wavelength max at which the radiancy reaches it maximum value is inversely proportional to the temperature T.

• max a 1/T

• max T = 2.898 X 10-3 m.K

Example 3.5(a) At what wavelength does a room-temperature (T=20 OC) object emit the maximum thermal radiation? (b) To what temperature must we heat it until its peak thermal radiation is in the red region of the spectrum?(c) How many times as much thermal radiation does it emit at the higher temperature?

(a) Since max T = 2.898 X 10-3 m.KThereforemax = 2.898 X 10-3 m.K/293 K= 9.89 X10-6 m.(b) The wavelength for red is 650 nm.Therefore T=2.898 X 10-3 m.K/650 X 10-9 m = 4460 K(c)

Rayleigh-Jeans Formula

• Energy density = no. of standing waves per unite volume X energy per standing wave

The radiancy is then given by (Rayleigh-Jeans Formula)

This formula is built on classical theories of electromagnetism and thermodynamics.

Comparison between the experimental data and Rayleigh-Jeans formula:AT long wavelengths R() approaches the experimental data, but at short wavelengths, the classical theory fails. This failure is called ultraviolet catastrophe.

Planck’s Theory and Radiation Law• He suggested that an oscillating atom can absorb or reemit

energy only in discrete bundles (quanta). No indevidual wave could contain more than kT of energy, therefore no standing wave could exist whose energy quantum was larger than kT.

• Each oscillator can emit or absorb energy only in quantities that are integer multiples of e,

where n=1,2,3,……..• The energy of each quanta is determined by its frequency n

where h is Planck’s constant

Now the radiancy is given by

This law agreed perfectly with the experiment.

The relation between Stefan-Boltzmann constant and Planck’s constant

• h= 6.56 X 10-34 J.s

The Compton EffectRadiation scatter from nearly loosely bound electrons. The incident radiation gives part of its energy to the electron; which is released from the atom, and the remainder of this energy is reradiated as electromagnetic radiation.

The energy of the incedent photon Its linear momentum is

Electron total energy

Electron total energy after collision

Compton scattering formula

Since E=hc/Therefore

Example 3.6X-rays of wavelength 0.2400 nm are Compton-scattered, and the scattered beam is observed at an angle of 60o relative to the incident beam. Find:(a) the wavelength of the scattered X-rays.(b) the energy of the scattered X-rays(c) the kinetic energy of the scattered electrons(d) the direction of travel of the scattered electrons.

(a) Use (b) E’ can be calculated using (c) Conservation of energy requires • =

(d)Conservation of momentum requires

And in y axis

Dividing both equations we get

Bremsstrahlung and X-Ray Production

Electron=electron(with less energy) +photonThe X Ray photon has energy which is given by K is the incedent electron energy and K’ is the energy left for the electron after collision with the target. Some of the electrons loose all of thier kinetick energy K, then the X-Ray photon would have obtained maximum energy hnmax which correspnds to minimum wavelength min . Each accelerating potential has its own min which is inversly proportional.

The accelerating potential V is related to the electron kinetic energy as

Pair Production• Interaction of photons with matter (atoms) is illustrated in

X-ray production, Compton scattering and pair production.

• In pair production, photon looses all of its energy, provided that it is ≥ 1.02 MeV, and pair of identical particles but with opposite charges are produced.

• The produced particles are electron and positron (an electron with positive charge).

• The photons of such high energy are in the region of nuclear gamma rays of the electromagnetic spectrum.

Electron-Positron Annihilation• The opposite of pair production is the electron-positron

annihilation.

Conservation of momentum requires that the two photon to have the same energy 0.51 MeV.This process occur when the positron and electron essentially meet when they have about zero kinetic energies.