Transcript Modern Physics

Modern Physics
NOTES
Relativity in Classical Physics
• Galileo and Newton dealt with the issue of
relativity
• The issue deals with observing nature in
different reference frames, that is, with
different coordinate systems
• We have always tried to pick a coordinate
system to ease calculations
Relativity and Classical Physics
• We defined something called an inertial
reference frame
• This was a coordinate system in which
Newton’s First Law was valid
• An object, not subjected to forces, moves at
constant velocity (constant speed in a straight
line) or sits still
Relativity and Classical Physics
• Coordinate systems that rotate or accelerate
are NOT inertial reference frames
• A coordinate system that moves at constant
velocity with respect to an inertial reference
frame is also an inertial reference frame
Moving Reference Frames
• While the motion of a dropped coin looks
different in the two systems, the laws of
physics remain the same!
Classical Relativity
• The relativity principle is that the basic laws of
physics are the same in all inertial reference
frames
• Galilean/Newtonian Relativity rests on certain
unprovable assumptions
• Rather like Euclid’s Axioms and Postulates
Classical Assumptions
• The lengths of objects are the same in all
inertial reference frames
• Time passes at the same rate in all inertial
reference frames
• Time and space are absolute and unchanging
in all inertial reference frames
• Masses and Forces are the same in all inertial
reference frames
Measurements of Variables
• When we measure positions in different inertial
reference frames, we get different results
• When we measure velocities in different inertial
reference frames, we get different results
• When we measure accelerations in different
inertial reference frames, we get the SAME
results
• The change in velocity and the change in time are
identical
Classical Relativity
• Since accelerations and forces and time are
the same in all inertial reference frames, we
say that Newton’s Second Law, F = ma satisfies
the relativity principle
• All inertial reference frames are equivalent for
the description of mechanical phenomena
Classical Relativity
• Think of the constant acceleration situation
1 2
x  x0  v0t  at
2
v  v0  at
Changing to a new moving
coordinate system means we just
need to change the initial values.
We make a “coordinate
transformation.”
The Problem!!!
• Maxwell’s Equations predict the velocity of
light to be 3 x 108 m/s
• The question is, “In what coordinate system
do we measure it?”
• If you fly in an airplane at 500 mph and have a
200 mph tailwind in the jet stream, your
ground speed is 700 mph
• If something emitting light is moving at 1 x 108
m/s, does this means that that particular light
moves at 4 x 108 m/s?
The Problem!!
• Maxwell’s Equations have no way to account
for a relative velocity
• They say that
c
1/


0
0 a medium, the
• Waves in water move through
water
• Same for waves in air
• What medium do EM waves move in?
The Ether
• It was presumed that the medium in which
light moved permeated all space and was
called the ether
• It was also presumed that the velocity of light
was measured relative to this ether
• Maxwell’s Equations then would only be true
in the reference frame where the ether is at
rest since Maxwell’s Equations didn’t translate
to other frames
The Ether
• Unlike Newton’s Laws of Mechanics, Maxwell’s
Equations singled out a unique reference
frame
• In this frame the ether is absolutely at rest
• So, try an experiment to determine the speed
of the earth with respect to the ether
• This was the Michelson-Morley Experiment
Michelson-Morley
• Use an interferometer to measure the speed
of light at different times of the year
• Since the earth rotates on its axis and revolves
around the sun, we have all kinds of chances
to observe different motions of the earth w.r.t.
the ether
Michelson-Morley
We get an interference pattern by
adding the horizontal path light to
the vertical path light.
If the apparatus moves w.r.t. the
ether, then assume the speed of light
in the horizontal direction is
modified. Then rotate the apparatus
and the fringes will shift.
Michelson-Morley
• Calculation in the text
• Upshot is that no fringe shift was seen so the
light had the same speed regardless of
presumed earth motion w.r.t. the ether
• Independently, Fitzgerald and Lorentz
proposed length contraction in the direction
of motion through the ether to account for
the null result of the M-M experiment
2 2
• Found a factor that worked 1  v / c
• Scientists call this a “kludge”
Einstein’s Special Theory
• In 1905 Einstein proposed the solution we
accept today
• He may not even have known about the M-M
result
• He visualized what it would look like riding an
EM wave at the speed of light
• Concluded that what he imagined violated
Maxwell’s Equations
• Something was seriously wrong
Special Theory of Relativity
• The laws of physics have the same form in all
inertial reference frames.
• Light propagates through empty space (no
ether) with a definite speed c independent of
the speed of the source or observer.
• These postulates are the basis of Einstein’s
Special Theory of Relativity
Gedanken Experiments
• Simultaneity
• Time Dilation
• Length Contraction (Fitzgerald & Lorentz)
Simultaneity
Simultaneity
Simultaneity
• Time is NOT absolute!!
Time Dilation
Time Dilation
Time Dilation
Clocks moving relative to an observer are measured by that observer to run more
slowly compared to clocks at rest by an amount
2
2
1 v /c
Length Contraction
• A moving object’s length is measured to be
shorter in the direction of motion by an
amount
2 2
1 v /c
Wave-Particle Duality
• Last time we discussed several situations in
which we had to conclude that light behaves
as a particle called a photon with energy equal
to hf
• Earlier, we discussed interference and
diffraction which could only be explained by
concluding that light is a wave
• Which conclusion is correct?
Wave-Particle Duality
• The answer is that both are correct!!
• How can this be???
• In order for our minds to grasp concepts we
build models
• These models are necessarily based on things
we observe in the macroscopic world
• When we deal with light, we are moving into
the microscopic world and talking about
electrons and atoms and molecules
Wave-Particle Duality
• There is no good reason to expect that what
we observe in the microscopic world will
exactly correspond with the macroscopic
world
• We must embrace Niels Bohr’s Principle of
Complementarity which says we must use
either the wave or particle approach to
understand a phenomenon, but not both!
Wave-Particle Duality
• Bohr says the two approaches complement
each other and both are necessary for a full
understanding
• The notion of saying that the energy of a
particle of light is hf is itself an expression of
complementarity since it links a property of a
particle to a wave property
Wave -Particle Duality
• Why must we restrict this principle to light
alone?
• Might microscopic particles like electrons or
protons or neutrons exhibit wave properties
as well as particle properties?
• The answer is a resounding YES!!!
Wave Nature of Matter
• Louis de Broglie proposed that particles could
also have wave properties and just as light had
a momentum related to wavelength, so
particles should exhibit a wavelength related
to momentum
h

mv
Wave Nature of Matter
• For macroscopic objects, the wavelengths are
terrifically short
• Since we only see wave behavior when the
wavelengths correspond to the size of
structures (like slits) we can’t build structures
small enough to detect the wavelengths of
macroscopic objects
Wave Nature of Matter
• Electrons have wavelengths comparable to
atomic spacings in molecules when their
energies are several electron-volts (eV)
• Shoot electrons at metal foils and amazing
diffraction patterns appear which confirm de
Broglie’s hypothesis
Wave Nature of Matter
• So, what is an electron? Particle? Wave?
• The answer is BOTH
• Just as with light, for some situations we need
to consider the particle properties of electrons
and for others we need to consider the wave
properties
• The two aspects are complementary
• An electron is neither a particle nor a wave, it
just is!
Electron Microscopes
Models of the Atom
• It is clear that electrons are components of
atoms
• That must mean there is some positive charge
somewhere inside the atom so that atoms
remain neutral
• The earliest model was called the “plum
pudding” model
Plum Pudding Model
We have a blob of positive charge and the
electrons are embedded in the blob like
currants in a plum pudding.
However, people thought that the electrons
couldn’t just sit still inside the blob.
Electrostatic forces would cause accelerations.
How could it work?
Rutherford Scattering
• Ernest Rutherford undertook experiments to
find out what atoms must be like
• He wanted to slam some particle into an atom
to see how it reacted
• You can determine the size and shape of an
object by throwing ping-pong balls at the
object and watching how they bounce off
• Is the object flat or round? You can tell!
Rutherford Scattering
• Rutherford used alpha particles which are the
nuclei of helium atoms and are emitted from
some radioactive materials
• He shot alphas into gold foils and observed
the alphas as they bounced off
• If the plum pudding model was correct, you
would expect to see a series of slight
deviations as the alphas slipped through the
positive pudding
Rutherford Scattering
• Instead, what was observed was alphas were
scattered in all directions
Rutherford Scattering
• In fact, some alphas scattered through very
large angles, coming right back at the
source!!!
• He concluded that there had to be a small
massive nucleus from which the alphas
bounced off
• He did a simple collision model conserving
energy and momentum
Rutherford Scattering
• The model predicted how many alphas should
be scattered at each possible angle
• Consider the impact parameter
Rutherford Scattering
• Rutherford’s model allowed calculating the
radius of the seat of positive charge in order
to produce the observed angular distribution
of rebounding alpha particles
• Remarkably, the size of the seat of positive
charge turned out to be about 10-15 meters
• Atomic spacings were about 10-10 meters in
solids, so atoms are mostly empty space
Rutherford Scattering
From the edge of the atom, the nucleus appears to
be 1 meter across from a distance of 105 meters or
10 km.
Translating sizes a bit, the nucleus appears as an
orange viewed from a distance of just over three
miles!!!
This is TINY!!!
Rutherford Scattering
Rutherford assumed the electrons must be in some
kind of orbits around the nucleus that extended
out to the size of the atom.
Major problem is that electrons would be
undergoing centripetal acceleration and should
emit EM waves, lose energy and spiral into the
nucleus!
Not very satisfactory situation!
Light from Atoms
• Atoms don’t routinely emit continuous spectra
• Their spectra consists of a series of discrete
wavelengths or frequencies
• Set up atoms in a discharge tube and make
the atoms glow
• Different atoms glow with different colors
Atomic Spectra
• Hydrogen spectrum has a pattern!
Atomic Spectra
• Balmer showed that the relationship is
 1 1 
 R 2  2  for n  3, 4,5,...

2 n 
1
Atomic Spectra
• Lyman Series
• Balmer Series
• Paschen Series
 1 1 
 R 2  2 for n  2,3,4,...

1 n 
 1 1 
1
 R 2  2  for n  3, 4,5,...

2 n 
 1 1 
1
 R 2  2  for n  4,5,6,...

3 n 
1
Atomic Spectra
•
•
•
•
•
Lyman Series
 1 1 
1
 R 2  2 for n  2,3,4,...

1 n 
Balmer Series
 1 1 
1
 R 2  2  for n  3, 4,5,...

2 n 
Paschen Series
 1 1 
1
 R 2  2  for n  4,5,6,...

3 n 
So what is going on here???
This regularity must have some fundamental
explanation
• Reminiscent of notes on a guitar string
Atomic Spectra
•
•
•
•
Electrons can behave as waves
Rutherford scattering shows tiny nucleus
Planetary model cannot be stable classically
What produces the spectral lines of isolated
atoms?
• Why the regularity of hydrogen spectra?
• The answers will be revealed next time!!!
Summary of 2nd lecture
•
•
•
•
•
electron was identified as particle emitted in photoelectric
effect
Einstein’s explanation of p.e. effect lends further credence to
quantum idea
Geiger, Marsden, Rutherford experiment disproves
Thomson’s atom model
Planetary model of Rutherford not stable by classical
electrodynamics
Bohr atom model with de Broglie waves gives some
qualitative understanding of atoms, but
– only semiquantitative
– no explanation for missing transition lines
– angular momentum in ground state = 0 (1 )
– spin??
Outline
• more on photons
– Compton scattering
– Double slit experiment
• double slit experiment with photons and matter
particles
– interpretation
– Copenhagen interpretation of quantum mechanics
• spin of the electron
– Stern-Gerlach experiment
– spin hypothesis (Goudsmit, Uhlenbeck)
• Summary
Photon properties
• Relativistic relationship between a particle’s
momentum and energy: E2 = p2c2 + m02c4
• For massless (i.e. restmass = 0) particles
propagating at the speed of light: E2 = p2c2
• For photon, E = h = ħω
• angular frequency ω = 2π
• momentum of photon = h/c = h/ = ħk
• wave vector k = 2π/
• (moving) mass of a photon: E=mc2  m = E/c2
2
2
Compton scattering 1
Scattering of X-rays on free electrons;
Electrons supplied by graphite target;
Outermost electrons in C loosely bound;
binding energy << X ray energy
 electrons “quasi-free”
• Expectation from classical
electrodynamics:
– radiation incident on free
electrons  electrons
oscillate at frequency of
incident radiation  emit
light of same frequency 
light scattered in all
directions
– electrons don’t gain energy
– no change in frequency of
light
Compton scattering 2
Compton (1923) measured intensity of scattered X-rays
from solid target, as function of wavelength for
different angles. Nobel prize 1927.
X-ray source
Collimator
(selects angle)
Crystal (selects
wavelength)

Target
Result: peak in scattered radiation shifts to longer wavelength
than source. Amount depends on θ (but not on the target
material).
A.H. Compton, Phys. Rev. 22 409 (1923)
•
Compton scattering 3
Classical picture: oscillating electromagnetic field causes oscillations in positions
of charged particles, which re-radiate in all directions at same frequency as incident
radiation. No change in wavelength of scattered light is expected
Incident light wave
•
Oscillating electron
Emitted light wave
Compton’s explanation: collisions between particles of light (X-ray photons) and
electrons in the material
Before
After
p 
scattered photon
Incoming photon
p
θ
Electron
pe
scattered electron
Compton scattering 4
Before
After
p 
scattered photon
Incoming photon
p
θ
Electron
pe
Conservation of energy
Conservation of momentum
h  me c  h    p c  m c
2
2 2
e
scattered electron

2 4 1/ 2
e
p 
hˆ
i  p   p e

From this derive change in wavelength:
   
h
1  cos 
me c
 c 1  cos   0
h
c  Compton wavelength 
 2.4 1012 m
me c
Compton scattering 5
•
unshifted peaks come from
collision between the X-ray
photon and the nucleus of the
atom
• ’ -  = (h/mNc)(1 - cos)  0
since mN >> me
WAVE-PARTICLE DUALITY OF LIGHT
• Einstein (1924) : “There are therefore now two theories of light, both
indispensable, and … without any logical connection.”
• evidence for wave-nature of light:
– diffraction
– interference
• evidence for particle-nature of light:
– photoelectric effect
– Compton effect
• Light exhibits diffraction and interference phenomena that are only explicable in
terms of wave properties
• Light is always detected as packets (photons); we never observe half a photon
• Number of photons proportional to energy density (i.e. to square of
electromagnetic field strength)
Double slit experiment
Originally performed by Young (1801) to demonstrate the wave-nature of light. Has
now been done with electrons, neutrons, He atoms,…
y
d
Detecting
screen
D
Expectation: two peaks for particles, interference pattern for waves
Alternative method
of detection: scan a
detector across the
plane and record
number of arrivals
at each point
Fringe spacing in double slit experiment
d sin   n
Maxima when:
D >> d  use small angle approximation
n

d
  
Position on screen:
y

d
d
θ
d sin 
y  D tan   D
So separation between adjacent maxima:
y  D
D
 y 
d
D
Double slit experiment -- interpretation
• classical:
–
–
–
–
–
–
–
–
–
two slits are coherent sources of light
interference due to superposition of secondary waves on screen
intensity minima and maxima governed by optical path differences
light intensity I  A2, A = total amplitude
amplitude A at a point on the screen A2 = A12 + A22 + 2A1 A2 cosφ, φ = phase
difference between A1 and A2 at the point
maxima for φ = 2nπ
minima for φ = (2n+1)π
φ depends on optical path difference δ: φ = 2πδ/
interference only for coherent light sources;
two
independent light sources: no interference since not coherent (random phase
differences)
Double slit experiment: low intensity
– Taylor’s experiment (1908): double slit experiment with very dim light:
interference pattern emerged after waiting for few weeks
– interference cannot be due to interaction between photons, i.e. cannot be
outcome of destructive or constructive combination of photons
–  interference pattern is due to some inherent property of each photon – it
“interferes with itself” while passing from source to screen
– photons don’t “split” – light detectors always show signals of same intensity
– slits open alternatingly: get two overlapping single-slit diffraction patterns – no
two-slit interference
– add detector to determine through which slit photon goes:  no interference
– interference pattern only appears when experiment provides no means of
determining through which slit photon passes
• double slit experiment with very low intensity ,
i.e. one photon or atom at a time:
get still interference pattern if we wait long
enough
Double slit experiment – QM interpretation
– patterns on screen are result of distribution of photons
– no way of anticipating where particular photon will strike
– impossible to tell which path photon took – cannot assign
specific trajectory to photon
– cannot suppose that half went through one slit and half
through other
– can only predict how photons will be distributed on
screen (or over detector(s))
– interference and diffraction are statistical phenomena
associated with probability that, in a given experimental
setup, a photon will strike a certain point
– high probability  bright fringes
– low probability  dark fringes
Double slit expt. -- wave vs quantum
wave theory
•
•
pattern of fringes:
– Intensity bands due to
variations in square of
amplitude, A2, of resultant
wave on each point on
screen
role of the slits:
– to provide two coherent
sources of the secondary
waves that interfere on the
screen
quantum theory
•
pattern of fringes:
– Intensity bands due to
variations in probability, P, of
a photon striking points on
screen
•
role of the slits:
– to present two potential
routes by which photon can
pass from source to screen
double slit expt., wave function
–
light intensity at a point on screen I depends on number of photons
striking the point
number of photons  probability P of finding photon there, i.e
I  P = |ψ|2, ψ = wave function
– probability to find photon at a point on the screen :
P = |ψ|2 = |ψ1 + ψ2|2 = |ψ1|2 + |ψ2|2 + 2 |ψ1| |ψ2| cosφ;
– 2 |ψ1| |ψ2| cosφ is “interference term”; factor cosφ due to fact that
ψs are complex functions
– wave function changes when experimental setup is changed
• by opening only one slit at a time
• by adding detector to determine which path photon took
• by introducing anything which makes paths distinguishable
Waves or Particles? • Young’s double-slit
diffraction experiment
demonstrates the wave
property of light.
• However, dimming the
light results in single flashes
on the screen
representative of particles.
Electron Double-Slit Experiment
• C. Jönsson (Tübingen, Germany,
1961) showed double-slit
interference effects for electrons
by constructing very narrow slits
and using relatively large
distances between the slits and
the observation screen.
• experiment demonstrates that
precisely the same behavior
occurs for both light (waves) and
electrons (particles).
Results on matter wave interference
Neutrons, A Zeilinger et
al. Reviews of Modern
Physics 60 1067-1073
(1988)
He atoms: O Carnal and J Mlynek
Physical Review Letters 66 2689-2692
(1991)
C60 molecules: M
Arndt et al. Nature
401, 680-682 (1999)
With multiple-slit
grating
Fringe visibility
decreases as
molecules are
heated. L.
Hackermüller et
al. , Nature 427
711-714 (2004)
Without grating
Interference patterns can not be explained classically - clear demonstration of matter waves
Which slit?
• Try to determine which slit the electron went through.
• Shine light on the double slit and observe with a microscope. After the electron
passes through one of the slits, light bounces off it; observing the reflected light, we
determine which slit the electron went through.
•The photon momentum is:
•The electron momentum is:
Need ph < d to distinguish
the slits.
Diffraction is significant only
when the aperture is ~ the
wavelength of the wave.
•The momentum of the photons used to determine which slit the electron went
through is enough to strongly modify the momentum of the electron itself—changing
the direction of the electron! The attempt to identify which slit the electron passes
through will in itself change the diffraction pattern!
Discussion/interpretation of double slit experiment
•
Reduce flux of particles arriving at the slits so that only one particle
arrives at a time. -- still interference fringes observed!
– Wave-behavior can be shown by a single atom or photon.
– Each particle goes through both slits at once.
– A matter wave can interfere with itself.
•
Wavelength of matter wave unconnected to any internal size of
particle -- determined by the momentum
• If we try to find out which slit the particle goes through the
interference pattern vanishes!
– We cannot see the wave and particle nature at the same time.
– If we know which path the particle takes, we lose the fringes .
Richard Feynman about two-slit experiment: “…a phenomenon which is impossible,
absolutely impossible, to explain in any classical way, and which has in it the heart
of quantum mechanics. In reality it contains the only mystery.”
Wave – particle - duality
•
So, everything is both a particle and a wave -- disturbing!??
•
“Solution”: Bohr’s Principle of Complementarity:
– It is not possible to describe physical observables
simultaneously in terms of both particles and waves
– Physical observables:
• quantities that can be experimentally measured. (e.g. position,
velocity, momentum, and energy..)
• in any given instance we must use either the particle description
or the wave description
– When we’re trying to measure particle properties, things
behave like particles; when we’re not, they behave like
waves.
Probability, Wave Functions, and the
Copenhagen Interpretation
• Particles are also waves -- described by wave function
• The wave function determines the probability of finding a particle
at a particular position in space at a given time.
• The total probability of finding the particle is 1. Forcing this
condition on the wave function is called normalization.
The Copenhagen Interpretation
• Bohr’s interpretation of the wave function
consisted of three principles:
–
Born’s statistical interpretation, based on probabilities
determined by the wave function
– Heisenberg’s uncertainty principle
– Bohr’s complementarity principle
• Together these three concepts form a logical interpretation of the
physical meaning of quantum theory. In the Copenhagen
interpretation, physics describes only the results of measurements.
Atoms in magnetic field
•
orbiting electron behaves like current loop  magnetic moment
interaction energy = μ · B (both vectors!)
– loop current = -ev/(2πr)
– magnetic moment μ = current x area = - μB L/ħ
μB
= e ħ/2me = Bohr magneton
– interaction energy

L
= m μB Bz

(m = z –comp of L)
n
A
I

r
e
Splitting of atomic energy levels
B0
B0
m = +1
m=0
m = -1
(2l+1) states with same
energy: m=-l,…+l
B ≠ 0: (2l+1) states with distinct
energies
Predictions: should always get an odd number of levels.
An s state (such as the ground state of hydrogen, n=1,
l=0, m=0) should not be split.
Splitting was observed by Zeeman
(Hence the name
“magnetic quantum
number” for m.)
Stern - Gerlach experiment - 1
•
•
•
•
•
magnetic dipole moment associated with angular momentum
magnetic dipole moment of atoms and quantization of angular momentum
direction anticipated from Bohr-Sommerfeld atom model
magnetic dipole in uniform field magnetic field feels torque,but no net force
in non-uniform field there will be net force  deflection
extent of deflection depends on
– non-uniformity of field
– particle’s magnetic dipole moment
– orientation of dipole moment relative to
mag.
field
Predictions:
– Beam should split into an odd number of
parts
(2l+1)
– A beam of atoms in an s state
(e.g. the ground state of hydrogen,
n = 1, l
= 0, m = 0) should not be split.
S
N
•
Stern-Gerlach experiment (1921)
z
Magnet
Oven
N
x
0
S
Ag-vapor
Ag
Ag beam
collim.
screen

B
Ag beam
S
# Ag atoms
N
B0
B↗
B↗↗


B  Bz z  e z
non-uniform
0
z
•
Stern-Gerlach
experiment
3
beam of Ag atoms (with electron in s-state (l =0)) in
non-uniform magnetic field
• force on atoms: F = z· Bz/z
• results show two groups of atoms, deflected in
opposite directions, with magnetic moments
z =   B
• Conundrum:
– classical physics would predict a continuous distribution
of μ
– quantum mechanics à la Bohr-Sommerfeld predicts an
odd number (2 l +1) of groups, i.e. just one for an s state
The concept of spin
•
•
•
•
•
•
•
Stern-Gerlach results cannot be explained by interaction of
magnetic moment from orbital angular momentum
must be due to some additional internal source of angular
momentum that does not require motion of the electron.
internal angular momentum of electron (“spin”) was
suggested in 1925 by Goudsmit and Uhlenbeck building
on
an idea of Pauli.
Spin is a relativistic effect and comes out directly
from
Dirac’s theory of the electron (1928)
spin has mathematical analogies with angular momentum,
but is not to be understood as actual rotation of electron
electrons have “half-integer” spin, i.e. ħ/2
Fermions vs Bosons
Radioactivity
Radiation
Radiation: The process of emitting
energy in the form of waves or
particles.
Where does radiation come from?
Radiation is generally produced
when particles interact or decay.
A large contribution of the radiation
on earth is from the sun (solar) or
from radioactive isotopes of the
elements (terrestrial).
Radiation is going through you at
this very moment!
http://www.atral.com/U238.html
Isotopes
What’s an isotope?
Two or more varieties of an element
having the same number of protons but
different number of neutrons. Certain
isotopes are “unstable” and decay to
lighter isotopes or elements.
Deuterium and tritium are isotopes of hydrogen. In
addition to the 1 proton, they have 1 and 2
additional neutrons in the nucleus respectively*.
Another prime example is Uranium 238, or just
238U.
Radioactivity
By the end of the 1800s, it was known that certain
isotopes emit penetrating rays. Three types of radiation
were known:
1)
Alpha particles (a)
2)
Beta particles (b)
3)
Gamma-rays
(g)
Where do these particles come from ?
These particles generally come
from the nuclei of atomic isotopes
which are not stable.
 The decay chain of Uranium
produces all three of these forms
of radiation.
 Let’s look at them in more detail…
Note: This is the
atomic weight, which
is the number of
protons plus neutrons
Alpha Particles (a)
Radium
Radon
R226
88 protons
138 neutrons
Rn222
86 protons
136 neutrons
The alpha-particle a is a Helium nucleus.
It’s the same as the element Helium, with the
electrons stripped off !
+
n p
p n
a 4He)
2 protons
2 neutrons
Beta Particles (b)
Carbon
C14
Nitrogen
N14
6 protons
8 neutrons
7 protons
7 neutrons
+
eelectron
(beta-particle)
We see that one of the neutrons from the C14 nucleus
“converted” into a proton, and an electron was ejected.
The remaining nucleus contains 7p and 7n, which is a nitrogen
nucleus. In symbolic notation, the following process occurred:
n  p + e ( + 
Yes, the same neutrino
we saw previously
Gamma particles (g)
In much the same way that electrons in atoms can be in an
excited state, so can a nucleus.
Neon
Ne20
10 protons
10 neutrons
(in excited state)
Neon
Ne20
10 protons
10 neutrons
(lowest energy state)
A gamma is a high energy light particle.
It is NOT visible by your naked eye because it is not in
the visible part of the EM spectrum.
+
gamma
Gamma Rays
Neon
Ne20
Neon
Ne20
+
The gamma from nuclear decay
is in the X-ray/ Gamma ray
part of the EM spectrum
(very energetic!)
How do these particles differ ?
Particle
Mass*
(MeV/c2)
Charge
Gamma (g)
0
0
Beta (b)
~0.5
-1
Alpha (a)
~3752
+2
* m = E / c2
Rate of Decay
Beyond knowing the types of particles which are emitted
when an isotope decays, we also are interested in how frequently
one of the atoms emits this radiation.
 A very important point here is that we cannot predict when a
particular entity will decay.
 We do know though, that if we had a large sample of a radioactive
substance, some number will decay after a given amount of time.
 Some radioactive substances have a very high “rate of decay”,
while others have a very low decay rate.
 To differentiate different radioactive substances, we look to
quantify this idea of “decay rate”
Half-Life
 The “half-life” (h) is the time it takes for half the atoms of a
radioactive substance to decay.
 For example, suppose we had 20,000 atoms of a radioactive
substance. If the half-life is 1 hour, how many atoms of that
substance would be left after:
Time
#atoms
remaining
% of atoms
remaining
1 hour (one lifetime) ?
10,000
(50%)
2 hours (two lifetimes) ?
5,000
(25%)
3 hours (three lifetimes) ?
2,500
(12.5%)
Lifetime (t)
 The “lifetime” of a particle is an alternate definition of
the rate of decay, one which we prefer.
 It is just another way of expressing how fast the substance
decays..
 It is simply: 1.44 x h, and one often associates the
letter “t” to it.
 The lifetime of a “free” neutron is 14.7 minutes
{tneutron=14.7 min.}
 Let’s use this a bit to become comfortable with it…
Lifetime (I)
 The lifetime of a free neutron is 14.7 minutes.
 If I had 1000 free neutrons in a box, after 14.7
minutes some number of them will have decayed.
 The number remaining after some time is given by the
radioactive decay law
N  N0e
t /t
N0 = starting number of
particles
t = particle’s lifetime
This is the “exponential”. It’s
value is 2.718, and is a very useful
number. Can you find it on your
calculator?
Lifetime (II)
N  N0e
Note by slight rearrangement of this formula:
Fraction of particles which did not decay:
t /t
N / N0 = e-t/t
1.20
0t
1t
2t
0
14.7
29.4
3t
4t
44.1
58.8
5t
73.5
Fraction of
remaining
neutrons
1.0
0.368
0.135
0.050
0.018
0.007
1.00
Fraction Survived
#
Time
lifetimes (min)
0.80
0.60
0.40
0.20
0.00
0
2
4
6
Lifetimes
After 4-5 lifetimes, almost all of the
unstable particles have decayed away!
8
10
Lifetime (III)
 Not all particles have the same lifetime.
 Uranium-238 has a lifetime of about 6 billion
(6x109) years !
 Some subatomic particles have lifetimes that are
less than 1x10-12 sec !
 Given a batch of unstable particles, we cannot
say which one will decay.
 The process of decay is statistical. That is, we can
only talk about either,
1) the lifetime of a radioactive substance*, or
2) the “probability” that a given particle will decay.
Lifetime (IV)
 Given a batch of 1 species of particles, some will decay
within 1 lifetime (1t, some within 2t, some within 3t,and
so on…
 We CANNOT say “Particle 44 will decay at t =22 min”.
You just can’t !
 All we can say is that:
 After 1 lifetime, there will be (37%) remaining
 After 2 lifetimes, there will be (14%) remaining
 After 3 lifetimes, there will be (5%) remaining
 After 4 lifetimes, there will be (2%) remaining, etc
Lifetime (V)
 If the particle’s lifetime is very short, the particles decay away very quickly.
 When we get to subatomic particles, the lifetimes
are typically only a small fraction of a second!
 If the lifetime is long (like 238U) it will hang around for a very long time!
Lifetime (IV)
What if we only have 1 particle before us? What can we say
about it?
Survival Probability = N / N0 = e-t/t
Decay Probability = 1.0 – (Survival Probability)
# lifetimes Survival Probability
(percent)
1
2
3
37%
14%
5%
4
5
2%
0.7%
Decay Probability =
1.0 – Survival Probability
(Percent)
63%
86%
95%
98%
99.3%
Summary
 Certain particles are radioactive and undergo decay.
 Radiation in nuclear decay consists of a, b, and g particles
 The rate of decay is give by the radioactive decay law:
Survival Probability = (N/N0)e-t/t
 After 5 lifetimes more than 99% of the initial particles
have decayed away.
 Some elements have lifetimes ~billions of years.
 Subatomic particles usually have lifetimes which are
fractions of a second… We’ll come back to this!
Ionization sensors (detectors)
• In an ionization sensor, the radiation passing
through a medium (gas or solid) creates
electron-proton pairs
• Their density and energy depends on the
energy of the ionizing radiation.
• These charges can then be attracted to
electrodes and measured or they may be
accelerated through the use of magnetic
fields for further use.
• The simplest and oldest type of sensor is the
Ionization chamber
•
•
•
•
The chamber is a gas filled chamber
Usually at low pressure
Has predictable response to radiation.
In most gases, the ionization energy for the outer electrons
is fairly small – 10 to 20 eV.
• A somewhat higher energy is required since some energy
may be absorbed without releasing charged pairs (by
moving electrons into higher energy bands within the
atom).
• For sensing, the important quantity is the W value.
• It is an average energy transferred per ion pair generated.
Table 9.1 gives the W values for a few gases used in ion
chambers.
W values for gases
Table 9.1. W va lues for var ious gases used in ionization chambers (eV/ion pair)
Gas
Electro ns (fast)
Alpha particles
Argo n (A)
27.0
25.9
Helium (He)
32.5
31.7
Nitroge n (N2)
35.8
36.0
Air
35.0
35.2
CH4
30.2
29.0
Ionization chamber
• Clearly ion pairs can also recombine.
• The current generated is due to an average
rate of ion generation.
• The principle is shown in Figure 9.1.
• When no ionization occurs, there is no
current as the gas has negligible resistance.
• The voltage across the cell is relatively high
and attracts the charges, reducing
recombination.
• Under these conditions, the steady state
Ionization chamber
Ionization chamber
• The chamber operates in the saturation region
of the I-V curve.
• The higher the radiation frequency and the
higher the voltage across the chamber
electrodes the higher the current across the
chamber.
• The chamber in Figure 9.1. is sufficient for
high energy radiation
• For low energy X-rays, a better approach is
needed.
Ionization chamber - applications
• The most common use for ionization
chambers is in smoke detectors.
• The chamber is open to the air and ionization
occurs in air.
• A small radioactive source (usually Americum
241) ionizes the air at a constant rate
• This causes a small, constant ionization
current between the anode and cathode of
the chamber.
• Combustion products such as smoke enter the
chamber
Ionization chamber - applications
• Smoke particles are much larger and heavier than air
• They form centers around which positive and negative
charges recombine.
• This reduces the ionization current and triggers an alarm.
• In most smoke detectors, there are two chambers.
• One is as described above. It can be triggered by humidity,
dust and even by pressure differences or small insects, a
second, reference chamber is provided
• In it the openings to air are too small to allow the large smoke
particles but will allow humidity.
• The trigger is now based on the difference between these two
currents.
Ionization chambers in a residential
smoke detector
Ionization chambers - application
• Fabric density sensor (see figure).
• The lower part contains a low energy radioactive isotope
(Krypton 85)
• The upper part is an ionization chamber.
• The fabric passes between them.
• The ionization current is calibrated in terms of density (i.e.
weight per unit area).
• Similar devices are calibrated in terms of thickness (rubber for
example) or other quantities that affect the amount of
radiation that passes through such as moisture
A nuclear fabric density sensor
Proportional chamber
• A proportional chamber is a gas ionization
chamber but:
• The potential across the electrodes is high
enough to produce an electric field in excess
of 106 V/m.
• The electrons are accelerated, process collide
with atoms releasing additional electrons (and
protons) in a process called the Townsend
avalanche.
• These charges are collected by the anode and
because of this multiplication effect can be
Proportional chamber
• The device is also called a proportional
counter or multiplier.
• If the electric field is increased further, the
output becomes nonlinear due to protons
which cannot move as fast as electrons
causing a space charge.
• Figure 9.2 shows the region of operation of
the various types of gas chambers.
Operation of ionization chambers
Geiger-Muller counters
• An ionization chamber
• Voltage across an ionization chamber is very
high
• The output is not dependent on the ionization
energy but rather is a function of the electric
field in the chamber.
• Because of this, the GM counter can “count”
single particles whereas this would be
insufficient to trigger a proportional chamber.
• This very high voltage can also trigger a false
reading immediately after a valid reading.
Geiger-Muller counters
• To prevent this, a quenching gas is added to the noble gas
that fills the counter chamber.
• The G-M counter is made as a tube, up to 10-15cm long
and about 3cm in diameter.
• A window is provided to allow penetration of radiation.
• The tube is filled with argon or helium with about 5-10%
alcohol (Ethyl alcohol) to quench triggering.
• The operation relies heavily on the avalanche effect
• UV radiation is released which, in itself adds to the
avalanche process.
• The output is about the same no matter what the
ionization energy of the input radiation is.
Geiger-Muller counters
• Because of the very high voltage, a single
particle can release 109 to 1010 ion pairs.
• This means that a G-M counter is
essentially guaranteed to detect any
radiation through it.
• The efficiency of all ionization chambers
depends on the type of radiation.
• The cathodes also influence this efficiency
• High atomic number cathodes are used for
higher energy radiation (g rays) and lower
atomic number cathodes to lower energy
Geiger-Muller sensor
Scintillation sensors
• Takes advantage of the radiation to light
conversion (scintillation) that occurs in certain
materials.
• The light intensity generated is then a
measure of the radiation’s kinetic energy.
• Some scintillation sensors are used as
detectors in which the exact relationship to
radiation is not critical.
• In others it is important that a linear relation
exists and that the light conversion be
efficient.
Scintillation sensors
• Materials used should exhibit fast light decay
following irradiation (photoluminescence) to
allow fast response of the detector.
• The most common material used for this
purpose is Sodium-Iodine (other of the alkali
halide crystals may be used and activation
materials such as thalium are added)
• There are also organic materials and plastics
that may be used for this purpose. Many of
these have faster responses than the inorganic
crystals.
Scintillation sensors
• The light conversion is fairly weak because it
involves inefficient processes.
• Light obtained in these scintillating materials
is of light intensity and requires
“amplification” to be detectable.
• A photomultiplier can be used as the detector
mechanism as shown in Figure 9.5 to increase
sensitivity.
• The large gain of photomultipliers is critical in
the success of these devices.
Scintillation sensors
• The reading is a function of many parameters.
• First, the energy of the particles and the efficiency of
conversion (about 10%) defines how many photons are
generated.
• Part of this number, say k, reaches the cathode of the
photomultiplier.
• The cathode of the photomultiplier has a quantuum efficiency
(about 20-25%).
• This number, say k1 is now multiplied by the gain of the
photomultiplier G which can be of the order of 106 to 108.
Scintillation sensor
Semiconductor radiation detectors
• Light radiation can be detected in
semiconductors through release of charges
across the band gap
• Higher energy radiation can be expected do so
at much higher efficiencies.
• Any semiconductor light sensor will also be
sensitive to higher energy radiation
• In practice there are a few issues that have to
be resolved.
Semiconductor radiation detectors
• First, because the energy is high, the lower bandgap materials
are not useful since they would produce currents that are too
high.
• Second, high energy radiation can easily penetrate through
the semiconductor without releasing charges.
• Thicker devices and heavier materials are needed.
• Also, in detection of low radiation levels, the background
noise, due to the “dark” current (current from thermal
sources) can seriously interfere with the detector.
• Because of this, some semiconducting radiation sensors can
only be used at cryogenic temperatures.
Semiconductor radiation detectors
• When an energetic particle penetrates into a
semiconductor, it initiates a process which
releases electrons (and holes)
– through direct interaction with the crystal
– through secondary emissions by the primary electrons
• To produce a hole-electron pair energy is
required:
– Called ionization energy - 3-5 eV (Table 9.2).
– This is only about 1/10 of the energy required to release
an ion pair in gases
• The basic sensitivity of semiconductor sensors
is an order of magnitude higher than in gases.
Properties of semiconductors
Table 9.2. Properties of some common semiconductors
Material
Operating Atomic
Band gap [eV]
temp [K] number
Silicon (Si)
300
14
1.12
Germanium (Ge)
77
32
0.74
Cadmium- teluride
300
48, 52
1.47
(CdTe)
Mercury- Iodine (HgI2) 300
80, 53
2.13
Gallium-Ars enide
300
31, 33
1.43
(GaAs)
Energy per electronhole pair [eV]
3.61
2.98
4.43
6.5
4.2
Semiconductor radiation detectors
• Semiconductor radiation sensors are
essentially diodes in reverse bias.
• This ensures a small (ideally negligible)
background (dark) current.
• The reverse current produced by radiation is
then a measure of the kinetic energy of the
radiation.
• The diode must be thick to ensure absorption
of the energy due to fast particles.
• The most common construction is similar to
the PIN diode and is shown in Figure 9.6.
Semiconductor radiation sensor
Semiconductor radiation detectors
• In this construction, a normal diode is built
but with a much thicker intrinsic region.
• This region is doped with balanced impurities
so that it resembles an intrinsic material.
• To accomplish that and to avoid the tendency
of drift towards either an n or p behavior, an
ion-drifting process is employed by diffusing a
compensating material throughout the layer.
• Lithium is the material of choice for this
purpose.
Semiconductor radiation detectors
• Additional restrictions must be imposed:
• Germanium can be used at cryogenic
temperatures
• Silicon can be used at room temperature but:
• Silicon is a light material (atomic number 14)
• It is therefore very inefficient for energetic
radiation such as g rays.
• For this purpose, cadmium telluride (CdTe) is
the most often used because it combines
heavy materials (atomic numbers 48 and 52)
with relatively high bandgap energies.
Semiconductor radiation detectors
• Other materials that can be used are the mercuric iodine
(HgI2) and gallium arsenide (GaAs).
• Higher atomic number materials may also be used as a simple
intrinsic material detector (not a diode) because the
background current is very small (see chapter 3).
• The surface area of these devices can be quite large (some as
high as 50mm in diameter) or very small (1mm in diameter)
depending on applications.
• Resistivity under dark conditions is of the order of 108 to 1010
.cm depending on the construction and on doping, if any
(intrinsic materials have higher resistivity).
• .
Semiconductor radiation detectors
- notes
• The idea of avalanche can be used to increase
sensitivity of semiconductor radiation
detectors, especially at lower energy
radiation.
• These are called avalanche detectors and
operate similarly to the proportional detectors
• While this can increase the sensitivity by
about two orders of magnitude it is important
to use these only for low energies or the
barrier can be easily breached and the sensor
Semiconductor radiation detectors
- notes
• Semiconducting radiation sensors are the most sensitive and
most versatile radiation sensors
• They suffer from a number of limitations.
• Damage can occur when exposed to radiation over time.
• Damage can occur in the semiconductor lattice, in the
package or in the metal layers and connectors.
• Prolonged radiation may also increase the leakage (dark)
current and result in a loss of energy resolution of the sensor.
• The temperature limits of the sensor must be taken into
account (unless a cooled sensor is used).
History of Constituents of
Matter
AD
•In Nuclear Reactions momentum and mass-energy is
conserved – for a closed system the total momentum
and energy of the particles present after the reaction is
equal to the total momentum and energy of the
particles before the reaction
•In the case where an alpha particle is released from an
unstable nucleus the momentum of the alpha particle
and the new nucleus is the same as the momentum of
the original unstable nucleus
__
Wolfgang Pauli
1
1
0
0
n0  p1e 1  0
•Large variations in the emission velocities of the b particle
seemed to indicate that both energy and momentum were not
conserved.
•This led to the proposal by Wolfgang Pauli of another particle,
the neutrino, being emitted in b decay to carry away the
missing mass and momentum.
•The neutrino (little neutral one) was discovered in 1956.
1
1
0
0
n0  p1e 1  0
__
1.008665 u
1.007825 u
1u=
1J=
0.0005486 u
1.660 10 27
kg
1.6 10 19
eV
Mass difference
 1.008665  (1.007825  0.0005486 )
 0.0002914 u
 0.0002914 1.660 10 27
 4.83724 10
31
kg
kg
E  mc
2
 (4.837241031)(3.0 108 ) 2J
 4.353516 10 14

J
4.353516  10 14
1.602  10 19
 0.272
 271755eV
MeV
It has been found by experiment that the emitted beta particle
has less energy than 0.272 MeV
Neutrino accounts for the ‘missing’ energy
• Ancient Greeks:
Earth, Air, Fire, Water
• By 1900, nearly 100 elements
• By 1936, back to three
particles: proton, neutron,
electron
The Four Fundamental Forces
20
Forces
Electromagnetic
Weak
atoms
molecules
optics
electronics
telecom.
beta
decay
solar
fusion
particles
inverse
square law
short
range
short
range
inverse
square law
photon
W , Z0
±
gluon
graviton
Institute of Physics
Peter Kalmus
Strong
nuclei
Gravity
falling
objects
planet
orbits
stars
galaxies
Particles and the Universe
m
E
c
2
Particle zoo
11
Feel weak force
“predicted”  later discovered
Neutrinos
100,000,000,000,000 per second pass
through each person from the Sun
Antiparticles
Equal and opposite properties
“predicted”  later discovered
Annihilate with normal particles
Now used in PET scans
1950s, 1960s
Many new particles created
in high energy collisions
Convert energy to mass. Einstein E = mc2
> 200 new “elementary” (?) particles
Institute of Physics
Peter Kalmus
Particles and the Universe
Classification of Particle
Thomson (1897): Discovers electron
1x10 10 m
1x10 15 m
0.7 x10 15 m
 0.7 x10 18 m
_
60
60
0
0
27 Co28 Ni  1e 0 
Q = -1e almost all trapped in atoms
Q= 0 all freely moving through universe
Just as the equation x2=4 can have two possible solutions (x=2 OR
x=-2), so Dirac's equation could have two solutions, one for an
electron with positive energy, and one for an electron with
negative energy.
Dirac interpreted this to mean that for every particle that exists
there is a corresponding antiparticle, exactly matching the
particle but with opposite charge. For the electron, for instance,
there should be an "antielectron" called the positron identical in
every way but with a positive electric charge.

g e e

1928 Dirac predicted existence of antimatter
1932 antielectrons (positrons) found in conversion of energy into matter
1995 antihydrogen consisting of antiprotons and positrons produced at
CERN
In principle an antiworld can be built from
antimatter
Produced only in accelerators and
in cosmic rays
g rays  e   e 


e  e  2hf
Q
2
3
Q
1
3
Q  1
Q0
James Joyce
Murray Gell-Mann
1
3
1

3
1

3

2

3
2

3
2

3
12
Today’s building blocks

Leptons
Quarks
(do not feel strong force)
(feel strong force)
electron
e-neutrino
4 particles
ee
-1 up
0 down
very simple
multiply by 3 (generations)
multiply by 2 (antiparticles)
u
d

2
2
1


 1
3
3
3
proton = u u d
2
3
+2/3
1
-1/3

3
2 +2/3
1 1
= +1
+2/3
  -1/3
0
3
3 3
neutron = u d d
+2/3 -1/3 -1/3 = 0
First generation