Modern PhysicsLecture IX

The Quantum Hypothesis In this lecture we examine the evidence for In this lecture we examine the evidence for

“light quanta” and the implications of their “light quanta” and the implications of their existenceexistence Waves as ParticlesWaves as Particles

The photoelectric effectThe photoelectric effect Compton scatteringCompton scattering

Particles as WavesParticles as Waves Electron diffractionElectron diffraction

The Double Slit RevisitedThe Double Slit Revisited

Photoelectric effect

When light strikes the When light strikes the cathodecathode, electrons , electrons are emittedare emittedElectrons moving between the two plates Electrons moving between the two plates constitute a currentconstitute a current

Photoelectric Effect

Properties of the photoelectric Properties of the photoelectric effecteffect

Electrons are only emitted above a Electrons are only emitted above a certain “cut-off” frequencycertain “cut-off” frequency

This frequency is different for different This frequency is different for different materialsmaterials

It is called the “work function”It is called the “work function”

Below the “work function” no electrons Below the “work function” no electrons are emitted no matter how intense the light are emitted no matter how intense the light isis

The maximum energy of the ejected The maximum energy of the ejected electron is Kelectron is Kmaxmax=e=eVVss

Photoelectric Effect Properties Properties

No photoelectrons are emitted if the No photoelectrons are emitted if the frequency falls below some cut-off frequency falls below some cut-off frequency frequency ffcc

The maximum energy of the photons is The maximum energy of the photons is independent of the light intensityindependent of the light intensity

The maximum kinetic energy of the The maximum kinetic energy of the photoelectrons increases with increasing photoelectrons increases with increasing frequencyfrequency

Photoelectrons are emitted almost Photoelectrons are emitted almost instantaneously from the surfaceinstantaneously from the surface

Photoelectric Effect ExplanationExplanation

Einstein extended Planck’s explanation for blackbody Einstein extended Planck’s explanation for blackbody radiation to suggest that in fact the quanta of energy used in radiation to suggest that in fact the quanta of energy used in blackbody radiation are in fact localised “particle like” blackbody radiation are in fact localised “particle like” energy packets energy packets

Each having an energy given by Each having an energy given by hf hf Emitted electrons will have an energy given byEmitted electrons will have an energy given by

Where Where is known as the “ is known as the “work function”work function” of the of the materialmaterial

hfKmax

Photoelectric Effect Quantum interpretationQuantum interpretation

If the energy of a photon is less than the work function If the energy of a photon is less than the work function , the , the photon cannot give enough energy to the electron to leave photon cannot give enough energy to the electron to leave the surfacethe surface

KKmaxmax does not depend on light intensity, because doubling the does not depend on light intensity, because doubling the

number of photons would only double the number of number of photons would only double the number of electrons and not double their energyelectrons and not double their energy

KKmaxmax increases with frequency because energy and frequency increases with frequency because energy and frequency

are relatedare related If light is particle-like, then all of the energy will be If light is particle-like, then all of the energy will be

delivered instantaneously thus liberating an electron with no delivered instantaneously thus liberating an electron with no time delay between the light hitting the surface and the time delay between the light hitting the surface and the electron escapingelectron escaping

Inverse photoelectric effect - Production of X-rays

1. Photons are absorbed in whole - electrons can transfer part of their energy.

2. Brehmsstrahlung - electrons decelerate in electromagnetic field of nuclei: Ef = Ei - h , wide distribution

3. Maximal frequency - minimal wavelength (observed empirically first) eV = h max = h c / min

4. Also discrete spectrum (atomic levels)

X-ray photons

Electrons

High voltage

Roentgen Lamp

minmax

2

2v

hc

hfeVm

ACe

Tungsten - wolframTungsten - wolfram

Compton Scattering If light is like a particle does it have momentum?If light is like a particle does it have momentum? In Compton scattering x-rays impart momentum to matter, In Compton scattering x-rays impart momentum to matter,

scattering electrons like billiard balls scattering electrons like billiard balls Thus photons also have momentum. The momentum of a photon Thus photons also have momentum. The momentum of a photon

is given byis given by

IncidentPhoton, 0

ScatteredPhoton, ’

Recoiling electron

)cos1(0 cm

h

e

h

c

hf

c

Ep

Photons and Electromagnetic Waves

How can light be considered a photon (particle) How can light be considered a photon (particle) when we know it is a wavewhen we know it is a wave

Light has a dual nature: it exhibits both wave and Light has a dual nature: it exhibits both wave and particle characteristicsparticle characteristics There is a smooth transition of these properties across the There is a smooth transition of these properties across the

electromagnetic spectrumelectromagnetic spectrum At low frequencies (radio waves) photons have a vanishingly At low frequencies (radio waves) photons have a vanishingly

small energy and the wave properties dominatesmall energy and the wave properties dominate At high frequencies (x-rays, At high frequencies (x-rays, -rays) it is the particle properties that -rays) it is the particle properties that

dominate dominate

But…But…

Louis de Broglie1892 - 1987

Wave Properties of Matter In 1923 Louis de Broglie postulated that perhaps matter In 1923 Louis de Broglie postulated that perhaps matter

exhibits the same “duality” that light exhibitsexhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as wellPerhaps all matter has both characteristics as well Previously we saw that, for photons,Previously we saw that, for photons,

h

c

hf

c

Ep

mv

h

p

h

Which says that the wavelength of light is related to its Which says that the wavelength of light is related to its momentummomentum

Making the same comparison for matter we find…Making the same comparison for matter we find…

de Broglie Wavelength of Electrons We now calculate the wavelength of a charged particle accelerated through potential We now calculate the wavelength of a charged particle accelerated through potential VV Assume that the particles have mass Assume that the particles have mass mm and charge and charge qq Equate kinetic energy of the particles with the electrostatic energyEquate kinetic energy of the particles with the electrostatic energy

KK = = m v m v 22/2 =/2 = q V q Vmomentummomentum pp = = m v m v

We can express kinetic energy in terms of momentumWe can express kinetic energy in terms of momentumKK = = p p 22/(2/(2 m m) =) = q V q V

Reorganise to getReorganise to getpp = (2 = (2 m q V m q V ))1/21/2

de Broglie’s hypothesis gives de Broglie’s hypothesis gives = = h / ph / p

Substitute for Substitute for pp to getto get

1/22 Vqm

hλ

Does Matter Really Have a Wavelength

The wavelength of matter waves is very small. This is The wavelength of matter waves is very small. This is why we do not see them in our every day experiencewhy we do not see them in our every day experience

To see diffraction a grating a very small slit width is To see diffraction a grating a very small slit width is required (eg the space between two atoms in a crystal)required (eg the space between two atoms in a crystal)

This is exactly how electron diffraction was first This is exactly how electron diffraction was first found!found! G. P. Thompson of Scotland and Davisson and Germer from the USA G. P. Thompson of Scotland and Davisson and Germer from the USA

used the close spacing between atoms in a crystal lattice to diffract used the close spacing between atoms in a crystal lattice to diffract electron waves thus proving that matter can also exhibit diffraction and electron waves thus proving that matter can also exhibit diffraction and interference interference

Sir Joseph John (JJ) Thomson

dNi=0.215nmdiffraction

nm165.0sin d

de Broglie

m

peVba 2

2

nm167.02

bameV

h

p

h

C.J.Davisson and L.G.Germer

Example of Measuring the Lattice Spacing Consider an electron accelerated to Consider an electron accelerated to V V == 5050 V scattered through V scattered through

angle angle . Note that 2. Note that 2==180180°, i.e. °, i.e. 9090°-°-/2/2

• The condition for constructive interference is2 d sin = n n integer

Fig

s. f

rom

R. Eis

berg

& R

. R

esn

ick Q

uantu

m P

hysi

cs

Example Electron scattering in nickelElectron scattering in nickel Electrons are accelerated through Electrons are accelerated through VV = = 54V.54V. The maximum of scattering is found to be at The maximum of scattering is found to be at == 6565 ° (° ( == 5050 °)°) Calculate the lattice spacing for nickelCalculate the lattice spacing for nickel

22 dd = = nn sinsin Verify that Verify that d d == 0.0920.092 nmnm

1/22 Vqm

h

Fig

s. f

rom

R. Eis

berg

& R

. R

esn

ick Q

uantu

m P

hysi

cs

a, b, c – computer simulation

d - experiment

Electron interference

Electron Microscope

Electron Waves Electrons with 20ev energy, Electrons with 20ev energy,

have a wavelength of about have a wavelength of about 0.27 nm0.27 nm

This is around the same size as This is around the same size as the average spacing of atoms in the average spacing of atoms in a crystal latticea crystal lattice

These atoms will therefore These atoms will therefore form a diffraction grating for form a diffraction grating for electron “waves”electron “waves”

Several pictures are shown left Several pictures are shown left (see the web links on the course (see the web links on the course home page)home page)

http://www.chem.qmw.ac.uk/surfaces/scc/scat6_2.htm

from this lecture… The solution to the blackbody spectrum leads to the concept of photons, and to a solution The solution to the blackbody spectrum leads to the concept of photons, and to a solution

for the photoelectric effectfor the photoelectric effect The maximum excess energy of a photoelectron isThe maximum excess energy of a photoelectron is

The particle nature of light is also shown by Compton scattering of electrons by photonsThe particle nature of light is also shown by Compton scattering of electrons by photons

Scattering shows that photons have momentum given byScattering shows that photons have momentum given by

This implies that matter also has wavelike properties given by the de Broglie formulaThis implies that matter also has wavelike properties given by the de Broglie formula

The de Broglie wavelength leads to phenomena such as electron diffraction. A common The de Broglie wavelength leads to phenomena such as electron diffraction. A common tool in modern crystallographytool in modern crystallography

hfKmax

)cos1(0 cm

h

e

h

c

hf

c

Ep

mv

h

p

h