Transcript SCE 18 – Part 6

Electrons in Metals and Insulators
Metals: Electrons in filled orbitals
occupy the lowest level: “valence band”.
Electrons in unfilled shells occupy the
bottom of the “conduction band”.
These are then able to move to higher,
unfilled states with a small energy jump.
Insulators: All the electrons are in the
valence band. To reach higher un-filled
states requires a high energy jump over the
“Band Gap”.
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How fast do electrons move?
• When a voltage difference is applied across a metal, electrons
can move very quickly.
• In copper the instantaneous speed is around 1.6 million metres
per second.
• But electrons meet resistance from
Grain boundaries
Thermal vibrations
• These cause the electron ”gas” to be scattered – like the
scattering of molecules in a gas – “Brownian Motion”.
• Actual speed – “drift velocity” is 3/100th mm/sec.
• But this “resistance” does cause heating in a lamp filament.
What does travel along a metallic wire?
In mechanics,
Work done = Force applied X Distance moved
In electricity,
Work done = Potential x Electric charge moved
Electrons in atoms at the pole of the battery are at an enhanced energy level.
When a wire is connected, electrons in atoms in the wire are excited to this level.
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Battery
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Wire
The Potential is then passed along the wire
• Atoms at one end of the wire pass on the potential to the next set of
atoms - and so on - to the other terminal of the battery.
• If the connection is for a short time – as in a telegraph key – then a
pulse of potential passes along the wire.
• This is similar to a pulse of light – both are Electro – Magnetic waves.
• So what travels along the wire is a wave of electric potential.
• Travelling at the velocity of an E-M wave in, say, copper.
• That is, almost instantaneously.
Electrical currents - a simple analogy
• One simple analogy is a garden hose-pipe connected to a tap.
• Turn on the tap and water exits at the other end.
• We could think of the flow of water modelling electrons moving
down a metal wire.
• But it’s a poor model. Because electrons move so slowly, (drift
velocity less than 3/100 mm per second) the electrical signal
would take too long to cross the street, let alone the Atlantic.
• A better model is a hose-pipe already filled with water.
• Then, when the tap is turned on, a pulse of pressure moves
rapidly along the pipe and water, already at the end comes out
of the other end is forced out almost immediately.
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What happens in insulators?
An insulator is placed between two
metallic plates with a potential difference
between them.
This produces an Electric (Force) Field acting
across the insulator, which causes a shift of electric charge at an atomic level.
But no current flows because insulators have no mobile electrons – unless a
Very High voltage is applied – in which case the insulator may “Break Down”.
Metals, Insulators - and Semi-conductors
In a semi-conductor,
the filled valence band
and the empty conduction band are separated
by a small Band Gap.
Heating can “promote”
a few electrons - allowing a small current to flow.
Alternatively, a semiconductor can be “doped” to give an
excess of electron states,
or a deficiency - “electron holes”.
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What might a photon look like?
• Quanta of light - photons have characteristics of
• BOTH particles AND waves.
• Diagrammatically, a photon can be represented as a
wave packet:
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Uncertainty and other marvels
• We have seen that quantum phenomena need to be treated in
terms of particles, that also act as waves.
• The wave amplitude, when squared, gives the wave intensityessentially the (probable) energy.
• Peaks in wave intensity show the relative probability of finding
the particle at any point. (Born).
• But is there a limit on the accuracy?
• We need the probability of position, and velocity.
• Heisenberg suggested that accuracy is limited.
• We cannot establish the exact position, x, and the
• exact momentum, p, (= mass times velocity).
• There will (certainly) be an uncertainty ∆(x) in position and ∆(p)
in momentum such that:
• ∆(p) times ∆(x) is greater than h/4π.
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Heisenberg: Uncertainty Principle
• Heisenberg realised this is a fundamental limitation & his
name is attached to the principle (if not the detail).
• This is not a problem of sensitivity of detection.
• It is a matter of principle. The wave-function is expressed in
terms of position and momentum.
• Determining one to high accuracy limits the accuracy of the
other.
• Similar limitations are found in other types of wave
functions.
• Sound: a note, well-defined in frequency, is extended in
time.
• A staccato note, sounding for a very narrow range of times,
consists of a wide range of tones (frequencies).
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Uncertainty: Energy and Time
• There is similar uncertainty surrounding energy and time: ∆E
times ∆t cannot be less than h/4π.
• One result of this is that energy can never be zero!
If E were zero, then its uncertainty would also be zero &
Heisenberg’s uncertainty principle would not be obeyed.
• Even at the lowest temperature energy must exist.
• This is usually called zero-point energy.
• One result is that a vacuum cannot be completely devoid of
energy.
• “A vacuum is what remains when everything has been removed
that can be removed.” James Clerk Maxwell.
• Energy can thus be “borrowed” from the vacuum provided it’s a
short term loan:
• ∆E can be large if ∆t is very small.
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“Virtual” Particles
• This leads to energy being released as particles - from the
vacuum!
• For example: an electron and a positron can be released from
the vacuum by a short-term loan of energy:
• E = 2 mc2
Provided E is repaid quickly by particle annihilation.
• These are “virtual” particles, and though the vacuum is pictured
as “swimming with them”, they cannot be harnessed.
• But they are detectable through the “Casimir effect”.
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Energy of empty space
Hendrik Casimir (1948).
Two uncharged metal plates in vacuo, near zero Kelvin with
radiation from outside excluded.
• “Zero point” waves must have zero amplitude at
each surface and fit between the plates.
• So only certain waves can exist between the plates
• But there is no restriction outside the plates – so more waves
are outside - which press the plates together.
• Casimir found force was similar to weight of a fly’s wing.
Sailing ships need Casimir
Ships can be drawn together by the action
of (light) waves) – even in Harbours and masts
and rigging may become entangled.
L’Album du Marin 1836
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Superposition : Now you see it.....
• The amplitude of the wave function can be almost equally
high at two or more places.
• So the associated particle has a probability of being at
more than one place - or of moving with more than one
momentum.
• This is Quantum “Superposition”.
• An extreme: a quantum particle is faced with a barrier that
could not be passed by a classical particle.
• The quantum particle has a probability of passing
through:
• by Quantum tunnelling.
• While the probability may be small it is important, even
vital in many cases.
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Quantum tunnelling
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Hydrogen bonding
• Hydrogen bonds are weaker than covalent or ionic bonds
• but have an immense effect on properties relevant to life.
• Important in forming the (secondary) structures of
proteins and nucleic acids.
• Vital for the DNA structure and properties.
• H-bonds only hold together the two branches of the
double helix
• and allow them to be separated and copied in
reproduction.
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Hydrogen bonding between water molecules
H-Bonding in water and ice.
• In water, H20 molecules pack together (almost) randomly
but efficiently.
• On freezing, forming ice crystals, H-bonds couple molecules
to form a symmetric structure,
• But the ordered crystalline structure has a lower density
than (liquid) water.
• Ice therefore forms at the surface and floats above the liquid
water.
• Which is very unusual - most liquids freeze to give a more
dense crystalline structure.
• This allows marine life to survive over winters.
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Density of water
at different temperatures
Ice (below 0C)
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Did you know?
Wool, being a protein fibre is held together by hydrogen bonds,
causing wool to recoil when stretched.
However, washing at high temperatures can permanently break
the hydrogen bonds and a garment may permanently lose its shap
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