Wave-Particle Duality

Section Contents1. What was Newton’s corpuscular theory of light?2. What did corpuscular theory predict for Young’s double slit

experiment?3. What does Young’s experiment tell us about the nature of light?4. What are electromagnetic waves?5. What did Maxwell prove about the speed of EM waves?6. How were radio waves first produced and detected?7. Why is wave theory unable to explain photoelectricity?8. What is meant by “stopping potential” in the photoelectric effect?9. What is the significance of Einstein’s explanation of

photoelectricity?10. Do matter particles have a dual wave=particle nature?11. Can matter particles be diffracted?12. Why is an electron microscope more powerful than an optical one?

Newton’s Corpuscular Theory of Light

• Having achieved so much success with his work in mechanics, it must have been tempting to use that to explain everything.

• His studies in light seemed to him to suggest he look for a basis in particles to explain its behaviour.

• Not just him, but it took a lot of persuading to bring the majority of people in line with a Wave Theory of light. After all it was Newton’s idea.

Reflection, Refraction and Momentum

His idea was:• Light is a stream of particles

– emitted by shining materials, moving in straight lines.

– can behave elastically in some circumstances i.e they can bounce off mirrors without loss of speed.

– can go through certain materials unhindered, though they might experience a force .

Is this more believable than waves in fact?

Newton’s Corpuscles

Reflection • Elastic collision with the plane mirror reverses direction of vN but leaves vP unchanged.

• Magnitude of the resultant velocity before and after reflection is unchanged.

• Direction changed so that i = r

Refraction

Refraction • Newton said the particles are attracted into a transparent substance

• So they travel faster than in air

• The component of velocity perpendicular to the boundary increases as the corpuscle crosses it

• The component of velocity parallel to the boundary is unchanged.

Huygens’ Wavelets

Detail follows, but in brief• Each point on one

wavefront starts a new wave/wavelet

• The new waves build up to form a new wavefront

• The wavefront and its velocity are perpendicular

Huygen’s Principle

• Each point on an advancing wave front can be considered to be the (secondary) source of a new set of circular waves

• The new waves build up to make a wavefront moving at right angles to the wavefront.

• The advancing wave front as a whole can be regarded as the sum of all the secondary waves arising from points in the medium already traversed

• Applet 1 shows reflection and refraction using Huygen’s principle (as does applet 2)

Huygens theory’s consequences

• His theory meant that waves would travel more slowly in the denser medium – and speed up when they emerged from the other side.

• He thought this required a heavy all pervading medium that had so far avoided detection – the Aether.

Newton vs Huygens• Both theories accounted for reflection and refraction

• It was not possible to measure the speed of light in water at that time, though they got close to the speed of light in empty space

• Thomas Young (1803) – double slits required wave nature for an explanation – but no huge impact

• Fizeau (1849) measured speed over 12 miles,

• Foucault (1862) over a shorter distance making it possible to measure speed through water

• Michelson and Morley (1887) brought strong evidence that the Aether did not exist, along with an improvement on Foucault’s method

Newton vs Huygens• Newton b. 1642 had the

stronger scientific reputation

• Already had a broader success with mechanics although Huygens had explored many of the areas Newton looked at.

• No Aether needed

• particles were a familiar idea

• Required to travel faster in glass.

• Huygens b. 1629, prominent in Holland but less well-known in England

• The idea of waves did not take root easily – it was “new”

• He thought there was an Aether that they travelled in.

• Required to travel slower in glass

• Light waves were understood in terms of longitudinal waves so did not explain polarisation

Young’s Double Slit Experiment (1)

• Coherent source of monochromatic light shone on two slits

• Two waves are said to be coherent if they have a constant relative phase.

• If light were a stream of particles there would be two fringes opposite the two fringes.

Young’s Double Slit Experiment (2)• The pattern of fringes actually

seen requires– a coherent source

• laser or

• single slit in front of source

– a wave explanation

• Bright fringes where waves arrive in phase and the interference is constructive

• Dark fringes where waves arrive 180⁰ out of phase and interference is destructive

• Path difference = nλ for Bright fringe, or (n+½)λ for dark

Newton vs Huygens

Single Slit Diffraction Patterns• Broad central fringe, fainter outer fringes• Where two patterns overlap, interference fringes are

seen on a smaller scale inside the diffraction fringes

Narrow slit

Wider slit

Single slit diffraction – Intensity Graph

Double Slit Diffraction – Interference Graph

• At a great distance from the slits, the two diffraction patterns more or less overlap

• so the “central broad, bright fringe/lesser other fringes” pattern is superimposed on the smaller scale interference pattern

Young’s Double Slit Experiment (3)

• Fringe spacing depends on how far apart the slits are– The further apart the slits, the smaller the fringe

spacing (narrower fringes)• The number of visible fringes depends on the

individual slit width– Narrower slit produces a wider diffraction pattern

so more overlap and more fringes seen for a given slit separation

Conclusive?

• Light has to be a wave – but this was not widely accepted

• Until the speed of light in water/other media could be measured and shown to be slower than in air (not faster as particles required)

• Light has to be a transverse wave to account for polarisation.

But is it?

• Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame; superposition of 200; then 1,000 and 500,000 frames.

• Individual photons (“particles”?) arriving individually, building up an interference pattern

• . . . . . . . . ?

Radio Waves

Electromagnetic waves

• Maxwell brought together electric and magnetic fields in a series of equations (1864).

• One solution is a travelling wave whose speed is

• √(1/εμ)

– ε is the permittivity of the medium

– μ is the permeability of the medium

• In free space, ε takes on the value ε0, 8.85x10-12 Fm-1, and in any other medium it is higher.

• In free space, μ takes on the value μ0, 4πx10-7 TmA-1 (Hm-1), and is higher in any other medium except diamagnets and superconductors.

In free space• The value of √(1/ε0μ 0) is:

299 792 458 ms-1

• In effect, an a.c. in a conductor creates an alternating magnetic field that creates an alternating electric field, which generates an alternating magnetic field further away, etc.

• Waves of changing electric and magnetic fields radiate away from the wire

• They were expected to be transverse but in phase, and mutually perpendicular.

© Nick Strobel

Predictions ...

• There was no restriction on frequency or wavelength, so other electromagnetic waves were predicted, beyond the IR, visible and UV known at that time.

• 20 years later, Radio Waves• X-Rays 1896• THz etc

Heinrich Hertz

• Hertz used a cell, capacitor, and an induction coil to produce sparks at a small gap.

• A simple loop of wire with its own gap some metres away, sparked in response

Spark Gap Transmitter

• Hertz produced a high voltage spark jumping across a gap

• Radio waves produced by the spark spread out and pass through the detector loop

• The changing magnetic field of the wave in the loop induces a voltage in it resulting in a spark jumping across the detector gap.

Why Waves?• A metal sheet in front of the loop stopped the sparks.

• An insulating sheet in front did not

• A concave metal sheet behind the detector made the sparks stronger because it reflected radio waves back to the detector

• Moving a flat zinc sheet as reflector, he found that at certain positions the sparks were not produced

• Recognising these as nodes in standing waves, he worked out a wavelength of 4 m. (book says 66 cm)

Why are aerials made like this?

• Can you think of a problem for TV reception?• Look at the shape – no loops.

Transverse waves• Rather more

sophisticated than the detector loop was the invention of a dipole

• Two metal rods aligned with each other at the centre of curvature of a concave reflector

• Waves are focused onto the dipole and the oscillating E-Field creates an alternating pd between the rods causing sparks.

• Spark gap at transmitter and dipole have to be parallel for a strong result

• Detector signal decreases when rotated to 90o

Maximum vs minimum signal detection

Hertz’ verdict on his discovery?

“It's of no use whatsoever[...] this is just an experiment that proves Maestro Maxwell was right - we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there.”

Asked about the ramifications of his discoveries, Hertz replied,

"Nothing, I guess."

Hertz and his Radio Waves

• Photoelectric emission was first discovered by Hertz when he was investigating radio waves using a spark gap detector.

• He observed that the sparks were much stronger when ultraviolet radiation was directed at the spark gap contacts.

The secret life of the radio

Controlled experiments

• not

• Charge an electroscope• Shine UV light• Leaf falls immediately

• Millikan investigated different frequencies of light vs energies of photoelectrons

• Light below a certain frequency emits no photoelectrons• The intensity of radiation did not affect this graph, i.e. their KE

Wave Theory vs Observation

• According to wave theory, light of any frequency should cause photoelectric emission.

• Wave theory predicted that the lower the frequency of the light, the longer the time taken by electrons in the metal to gain sufficient kinetic energy to escape from the metal.

But• photoelectric emission

occurs at the instant that light of a suitably high frequency is incident on the metal surface.

• photoelectric emission does not occur if the frequency of the incident light is below a threshold frequency,

Wave Theory vs Observation

• Over time an electron could absorb varying amounts of energy so have varying kinetic energy

But• the photoelectrons

have a range of kinetic energies from zero up to a maximum value that depends on the type of metal and the frequency of the incident light – but not the intensity

• The number of photoelectrons emitted from the metal surface per second is proportional to the intensity of the incident radiation (ie the light energy per second incident on the surface).

Measuring KE of Photoelectrons• Top half shows a circuit to

measure current due to emitted photoelectrons

• Bottom half is simply a potential divider to alter the pd applied.

• The plate is positive with respect to Q in order to make it harder for photoelectrons to leave the irradiated plate.

Stopping Potential Graph

• For a given intensity of light, there is a steady photocurrent until the opposing voltage saps the energy of the photoelectrons so they can no longer leave the metal.

Intensity vs applied PD

• Zero applied PD– some electrons leave and make their way across the gap hitting the other electrode by chance.

• Forward bias pd encourages electrons to reach the other electrode until they are all swept up and current maxes out.

• Reverse bias: the PD opposes the release of photoelectrons until the stopping potential is released and no current flows.

Planck’s idea for “Quanta” of energy

• Planck had already used the idea of energy “quanta” to explain other how vibrating (at frequency f) atoms emitted radiation:– Intensity of emitted

radiation should become infinite at decreasing wavelengths

– Solved if energy can only be emitted in multiples of a basic amount, E = hf

• Einstein knew the KE of conduction electrons was far too small to allow them to escape a metal surface

• Thermionic emission: a heater provides thermal energy for the electrons to escape

Einstein’s version of “quanta” and escape energy

• EM radiation consists of wave packets of electromagnetic energy that he called “photons”.

• Each photon carries energy given by • E=hf

– f is the frequency– h is Planck’s constant

• The minimum energy a conduction electron needs to escape the surface of a metal is called the work function

Einstein’s Explanation 1905

• For a conduction electron to escape, he assumed it needs to absorb a single whole photon and gain energy hf.

• Use energy equal to the work function φ of the metal to be able to escape

• So hf = φ + EK

• In fact the photon limits the KE to a maximum value.

• Deeper electrons would need more than φ to escape if indeed they could and have less KE once emitted

This accounts for

This assumes:• One electron absorbs one

photon only• One photon cannot be

spread across several electrons

This accounts for• The immediate start for the

photocurrent• The minimum frequency (hf

= φ; f = φ/h) before emission could begin

• The increase in EKmax with frequency

• The shapes of the graphs

eVstopping = EKmax = hf - φ

Vstopping = (h/e)f – φ/e

Duality

• Light consists of photons – wave packets of EM radiation of energy = hf

• No smaller amounts of energy can be exchanged by EM Radiation.

• If asked to behave like a wave, it does so: Young’s slits

• If asked to behave like a particle, it does so: Photoelectric effect.

• Can objects we consider particles behave like waves?

Waves can behave like particles ...

... so can particles behave like waves?

Louis de Broglie (say “broy”)

Thought of photons as particles of light• said hf ≡ mc2

• played around: mc = hf/c = h/λ; mc x λ = h• got the idea that if waves can be particles,

why can’t particles be waves withMomentum x wavelength = h

• Or λ = h/mv or h/p

Part of Nobel Prize acceptance speech:

• “As in my conversations with my brother we always arrived at the conclusion that in the case of X-rays one had both waves and corpuscles, thus suddenly - ... it was certain in the course of summer 1923 - I got the idea that one had to extend this duality to material particles, especially to electrons.”

• “Atoms of light” brought him some success in explaining some thermal physics

• Matter waves remained a hypothesis until ...• (Diffraction gratings and filters)

Electron diffraction

• Fire a beam of electrons at a very thin metal foil• Pattern of concentric rings obtained• Pattern appears very like that for X-ray diffraction by

crystals.

• As in wave (X-ray) diffraction:

d = Dλ/(separation between planes of C atoms)

• ring radius and electron wavelength varies with accelerating voltage

θ

Tube screen

Graphite Target

d

D

• Electrons behaving like waves– Bragg diffraction: – Increasing electron speed decreases their de Broglie

wavelength and the angle / rings to decrease

• Confirms de Broglie’s hypothesis

The relevant maths ...

• Anode, accelerating voltage in gun: VA

• Electrons of mass m and charge e

• ½mv2 = eVA if v<<c

• m2v2 = 2meVA

• mv = √(2meVA)

• De Broglie wavelength:• λ = h/mv = h/ √(2meVA)• Increase speed,

> decreases wavelength> decrease angle/ring radius

1. Find the wavelength of an electron of mass9.00 10-31 kg moving at 3.00 107 m s-1. – λ = h/p = [6.63 10-34] / [9.00 10-31 3.00 107]

= 6.63 10-34 / 2.7010-23

= 2.4610-11 m = 0.025 nm– This is comparable to atomic spacing, and explains why

electrons can be diffracted by graphite.

2. Find the wavelength of a cricket ball of mass 0.15 kg moving at 30 m s-1.– λ = h/p = [6.63 10-34] / [0.15 30]

= 1.4710-34 m = 1.5 10-34 J s (to 2 s.f.)

– This is a very small number, and explains why a cricket ball is not diffracted as it passes near to the stumps.

3. It is also desirable to be able to calculate the wavelength associated with an electron when the accelerating voltage is known. There are 3 steps in the calculation.

• Calculate the wavelength of an electron accelerated through a potential difference of 10 kV.– Step 1: Kinetic energy

Ek = eV = 1.6 10-19 10000

= 1.6 10-15 J

– Step 2: EK = ½ mv2 = (mv) 2 /(2m)

= p2 / 2m then rearrange

p = √(2mEk)

= √(2 9.1 10-31 1.6 10-15 )= 5.4 10-23 kg m s-1

– Step 3: λ = h / p = 6.63 10-34 / 5.4 10-23 = 1.2 10-11 m = 0.012 nm.

• Electrons behaving like waves means they can be used to form images like “electron photos” but in real time on a screen – just like at Hogwarts

Transmission Electron

Microscope

• Electron gun• Electromagnetic

“lenses” to collimate, focus an image then another to magnify further

• Fluorescent screen

• I spy the ... “electron gun”

• I spy the ... “fluorescent screen”

• I spy a pair of binoculars• I spy a load of gubbins

in between

• Human Flea• This needs a

scanning electron microscope

Cultured muscle cellsUsing a transmission electron microscope

TEM: the gubbins• 150V >> λ of 0.1 nm

• One lens to make beam parallel (???)

• Thin specimen

• Objective lens to produce a magnified image

• Projector lens to enlarge again for the final image on the

• screen

Notes:-Resolving power is

• the least separation between two objects that can just be seen apart.

Affected by

• the amount of diffraction – less diffraction from shorter wavelength – better resolution (also for light)

Set by

• the anode (accelerating) voltage

• higher voltage gives larger clearer image EK ∝ 1/λ

Limits are set by

• Slightly different speeds of electrons from the gun

• Sample thickness slowing down the electrons

• Lenses are not perfect. Edge focus might differ from centre focus.

...all of which cause blurring.

nanotubes

Wave absorption in a medium

• On the laboratory scale, waves are absorbed in passing through a substance and die away exponentially with depth.

• A thin barrier leaves some proportion of the waves getting through.

Tunnelling • If electrons are in a metal tip, with a certain work function, they cannot escape without extra energy

• If the metal tip is put very near another substance, some electrons do cross the gap despite the work function.

• Their de Broglie wavelength is long enough “to stretch across the gap”

• ..particles showing wave property.

STM: Scanning Tunnelling

Microscope

• Place a tip (at pd of -1V) near (~1 nm) the surface (at a pd of 0V) to be inspected,

• Tunnelling current increases if the gap is smaller (and the reverse).

• Moving the tip across the surface gives a varying current that can “map” the surface in considerable detail.

• The probe can be moved in three dimensions sideways x 2 and up and down

Mapping a surface:(in a vacuum to prevent contamination)

Constant height mode

• Tip scans across in a fixed plane, no up and down movement

• Varying current maps height via a computer on a screen

• If gap too big, no current• If gap too small, tip might

hit peaks

Constant current mode

• Gap width kept constant by feeding back the current to adjust the height

• High spot > increased current > raise the tip

• Height plotted on screen

Controlling position

• Piezo crystals produce a pd when stretched or compressed

• If a pd is applied they change their length• Such small changes allow the tunnelling

current to control a piezo crystal which adjusts the height on a very fine scale.

71 Angstrom diameter "quantum corral"

48 iron atoms in a circular ring in order to "corral" some surface state electrons

• Scanning tunnelling microscopy (STM)of a (105) facetted Ge hut cluster formed by 5.8 monolayer (ML) of Ge on a Si (100) surface (whatever that means)