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					          Modern Physics

             Assistant Professor

    Relativity in Classical Physics
• Galileo and Newton dealt with the issue of
• 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
• 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
• The change in velocity and the change in time are
            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
                                       Changing to a new moving

x  x0  v0t  at
                                       coordinate system means we just
                                       need to change the initial values.

              2                        We make a “coordinate

v  v0  at

              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
                           0 0
                    c 1/through a medium, the
• Waves in water move
• 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
• 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

• 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


                    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.
• 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
• 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
• 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)

• 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
                    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
• 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
• 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
• 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

         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
• 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
• 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
• 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
      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
      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 1 
          R 2  2  for n  3, 4,5,...
           2   n 

             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,...
• Paschen Series         2   n 
                     1     1 1 
                        R 2  2  for n  4,5,6,...
                         3 n 

                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,...
• Paschen Series         2   n 
                    1      1 1 
                        R 2  2  for n  4,5,6,...
•                    here???
  So what is going on 3 n 
• This regularity must have some fundamental
• 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
• 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
• 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
• 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??
• more on photons
  – Compton scattering
  – Double slit experiment
• double slit experiment with photons and matter
  – 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;
                                               • Expectation from classical
Outermost electrons in C loosely bound;          electrodynamics:
binding energy << X ray energy
                                                  – radiation incident on free
 electrons “quasi-free”
                                                    electrons  electrons
                                                    oscillate at frequency of
                                                    incident radiation  emit
                                                    light of same frequency 
                                                    light scattered in all
                                                  – electrons don’t gain energy
                                                  – no change in frequency of

                         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           Crystal (selects
                         (selects angle)      wavelength)


Result: peak in scattered radiation shifts to longer wavelength
than source. Amount depends on θ (but not on the target
                                                                 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
                              pe         scattered electron
                           Compton scattering 4
 Before                                          After                      p 
                                                                         scattered photon
           Incoming photon
                                                           pe          scattered electron

      Conservation of energy                             Conservation of momentum

 h  me c  h    p c  m c              
                                      2 4 1/ 2                      hˆ
            2               2 2
                                                                p  i  p   pe
                            e         e

From this derive change in wavelength:

                                        1  cos 
                                        me c
                                      c 1  cos    0
                c  Compton wavelength 
                       2.4  10
                                               me c
               Compton scattering 5

• unshifted peaks come from
   collision between the X-ray
   photon and the nucleus of the

• ’ -  = (h/mNc)(1 - cos)  0
     since mN >> me

• 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,…

                                                                             Alternative method
                                                                             of detection: scan a
                                                                 y           detector across the
                                                                             plane and record
                                                                             number of arrivals
                     d                                                       at each point


Expectation: two peaks for particles, interference pattern for waves
           Fringe spacing in double slit experiment

 Maxima when:          d sin   n
D >> d  use small angle approximation

                                                                                y
                                                      d
                                                                      θ
                                                                d sin 
 Position on screen:     y  D tan   D
So separation between adjacent maxima:                                        D

                   y  D
                    y 
       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

            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

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                               quantum theory

• pattern of fringes:                   • pattern of fringes:
  – Intensity bands due to                – Intensity bands due to
     variations in square of                variations in probability, P, of
     amplitude, A2, of resultant            a photon striking points on
     wave on each point on                  screen
• role of the slits:                    • role of the slits:
  – to provide two coherent               – to present two potential
     sources of the secondary                routes by which photon can
     waves that interfere on the             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

                                                                     He atoms: O Carnal and J Mlynek
                                                                     Physical Review Letters 66 2689-2692

                               C60 molecules: M
                               Arndt et al. Nature                                        Fringe visibility
                                                                                          decreases as
                               401, 680-682 (1999)
                                                                                          molecules are
                                                                                          heated. L.
                                                                                          Hackermüller et
                               With multiple-slit                                         al. , Nature 427
                               grating                                                    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.
                                                                Need ph < d to distinguish
                                                                the slits.
•The photon momentum is:
                                                              Diffraction is significant only
                                                              when the aperture is ~ the
•The electron momentum is:                                    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
          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                             
       = m μB Bz                                    L
       (m = z –comp of L)                     
                                           A              I

       Splitting of atomic energy levels
        B0                                      B0

                                                                       m = +1

                                                                       m = -1

(2l+1) states with same               B ≠ 0: (2l+1) states with distinct
energy: m=-l,…+l                      energies
                                                                           (Hence the name
                                                                           “magnetic quantum
   Predictions: should always get an odd number of levels.                 number” for m.)
   An s state (such as the ground state of hydrogen, n=1,
   l=0, m=0) should not be split.
   Splitting was observed by
           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                         S       mag.
•   Predictions:
     – Beam should split into an odd number of                          N       parts
     – 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.
Stern-Gerlach experiment (1921)
                                               N                       x

                                                             Ag beam
 Ag-vapor         Ag

       N                                        # Ag atoms
           Ag beam
                          
               B  Bz z  e z
       S       non-uniform
                            0               z
       Stern-Gerlach experiment - 3 =0)) in
    beam of Ag atoms (with electron in s-state (l
  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
Radiation: The process of emitting
energy in the form of waves or

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!

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
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
                                                               +      n p
                                                                       p n
        R226                                           Rn222
                                                                     a 4He)
 88 protons                                  86 protons            2 protons
 138 neutrons                                136 neutrons          2 neutrons

         The alpha-particle a is a Helium nucleus.

         It’s the same as the element Helium, with the
         electrons stripped off !
                          Beta Particles (b)
     Carbon                                       Nitrogen       +
       C14                                           N14                    e-

 6 protons                                    7 protons                    electron
 8 neutrons                                   7 neutrons                (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                                          Neon
        Ne20                                          Ne20             +

       10 protons                                  10 protons
      10 neutrons                                  10 neutrons
    (in excited state)                         (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 Rays


          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                  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”
 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:

                                                     #atoms             % of atoms
                     Time                           remaining           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

 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

                                                    N0 = starting number of
                        t /t
  N  N 0e
                                                    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
                                 Lifetime (II)
                                                                                                          t /t
Note by slight rearrangement of this formula:                                                 N  N 0e
Fraction of particles which did not decay:   N / N0 = e-t/t

     #    Time           Fraction of
lifetimes (min)          remaining

                                                       Fraction Survived
                          neutrons                                         0.80

    0t           0           1.0                                           0.60

    1t         14.7          0.368                                         0.40

    2t         29.4          0.135                                         0.20

    3t         44.1          0.050                                         0.00
                                                                                  0   2   4       6   8       10

    4t         58.8          0.018
    5t         73.5          0.007              After 4-5 lifetimes, almost all of the
                                                unstable particles have decayed away!
                                 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                    Decay Probability =
                 (percent)                        1.0 – Survival Probability
     1                    37%                               63%
     2                    14%                               86%
     3                     5%                                 95%
     4                     2%                                 98%
     5                    0.7%                               99.3%
 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
• 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
                     W values for gases

Table 9.1. W values for various gases used in ionization chambers (eV/ion pair)
Gas                           Electrons (fast)              Alpha particles
Argon (A)                     27.0                          25.9
Helium (He)                   32.5                          31.7
Nitrogen (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
• 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
• The chamber in Figure 9.1. is sufficient for
  high energy radiation
• For low energy X-rays, a better approach is
 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
  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
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
• 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
• 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
• 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
          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
           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
• 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
• Part of this number, say k, reaches the cathode of the
• 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
• 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
  – 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]   Energy per electron-
                           temp [K] number                    hole pair [eV]
Silicon (Si)               300       14        1.12            3.61
Germanium (Ge)             77        32        0.74            2.98
Cadmium-teluride           300       48, 52    1.47            4.43
Mercury-Iodine (HgI2) 300            80, 53    2.13            6.5
Gallium-Arsenide           300       31, 33    1.43            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
• 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
Semiconductor radiation detectors
• Additional restrictions must be imposed:
• Germanium can be used at cryogenic
• 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
• 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


•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
                     n 0  p1 e 1  0
                       1    1     0    0

      •Large variations in the emission velocities of the b particle
      seemed to indicate that both energy and momentum were not
      •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.

    n 0  p1 e 1  0
      1    1     0    0
1.008665 u         1.007825 u          0.0005486 u

             1u=          1.6601027                kg

             1J=          1.6 1019                 eV

Mass difference    1.008665 (1.007825 0.0005486)
                   0.0002914 u
                    0.00029141.6601027                 kg

                    4.8372410                       kg

       E  mc       2

                  (4.837241031 )(3.0 108 ) 2J

                  4.3535161014                  J

                        4.353516 1014
                                                           271755
                         1.602  1019
                  0.272       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,

The Four Fundamental Forces


                            Weak               Strong             Gravity
       atoms                beta                                  falling
       molecules            decay              nuclei                  objects
       optics                                                     planet
       electronics          solar              particles               orbits
       telecom.             fusion                                stars

       inverse               short               short             inverse
       square law            range               range             square law

       photon                W , Z0              gluon             graviton

Institute of Physics
                       Peter Kalmus       Particles and the Universe
                           m       2

                          Particle zoo

                                       Feel weak force
    Neutrinos                          “predicted”  later discovered
                                       100,000,000,000,000 per second pass
                                       through each person from the Sun

                                       Equal and opposite properties
    Antiparticles                      “predicted”  later discovered
                                       Annihilate with normal particles
                                       Now used in PET scans

                                       Many new particles created
    1950s, 1960s                       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


27 Co  28 Ni  1 e 0 
60      60       0    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

In principle an antiworld can be built from
Produced only in accelerators and
in cosmic rays

                     
g rays  e  e

   
e  e  2hf

      3                             Q  1

      1                             Q0

                   James Joyce

Murray Gell-Mann
  1                         2
                         
  3                         3
  1                         2
                         
  3                         3
  1                         2
                         
  3                         3

Today’s building blocks
                                                           2   2   1
                                                                   1
                                                           3   3   3
Leptons                      Quarks                        proton = u u d
(do not feel strong force)   (feel strong force)         2
                                                        +2/3 1  1  =
                                                        +2/3 -1/3 0 +1
                                                   2     3 3 3
electron           e-    -1 up          u      +2/33
                                                           neutron = u d d
e-neutrino                0 down        d      -1/3
                                                           +2/3 -1/3 -1/3 = 0

4 particles         very simple         First generation

multiply by 3 (generations)
multiply by 2 (antiparticles)

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