modern physics notes relativity in classical physics galileo and newton dealt with the issue of...

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


• 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


• Think of the constant acceleration situation

x x0 v0t 12

at2

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 • Waves in water move through a medium, the

water• Same for waves in air• What medium do EM waves move in?

c 1/ 00

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

• Found a factor that worked• Scientists call this a “kludge”

1 v2 /c2

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

• Time is NOT absolute!!

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

1 v2 /c2

Length Contraction

• A moving object’s length is measured to be shorter in the direction of motion by an amount

1 v2 /c2

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?


• 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


• 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!


• 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


• 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


• 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


• 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 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


• Instead, what was observed was alphas were scattered in all directions


• 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


• The model predicted how many alphas should be scattered at each possible angle

• Consider the impact parameter


• 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


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 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

1R

1

22

1

n2

for n 3,4,5,...

Atomic Spectra

• Lyman Series• Balmer Series• Paschen Series

1R

1

22

1

n2

for n 3,4,5,...

1R

1

12

1

n2

for n2,3,4,...

1R

1

32

1

n2

for n 4,5,6,...

Atomic Spectra

• Lyman Series• Balmer Series• Paschen Series• So what is going on here???• This regularity must have some fundamental

explanation• Reminiscent of notes on a guitar string

1R

1

22

1

n2

for n 3,4,5,...

1R

1

12

1

n2

for n2,3,4,...

1R

1

32

1

n2

for n 4,5,6,...

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 + m0

2c4

• 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

m = h/c2 = ħω/c2

Compton scattering 1• 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

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”

Compton scattering 2Compton (1923) measured intensity of scattered X-rays

from solid target, as function of wavelength for different angles. Nobel prize 1927.

X-ray source

Target

Crystal (selects wavelength)

Collimator (selects angle)

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

• Compton’s explanation: collisions between particles of light (X-ray photons) and electrons in the material

Oscillating electronIncident light wave Emitted light wave

θ

ep

pBefore After

Electron

Incoming photon

p

scattered photon

scattered electron

Compton scattering 4

θ

ep

pBefore After

Electron

Incoming photon

p

scattered photon

scattered electron

Conservation of energy Conservation of momentum

1/ 22 2 2 2 4e e eh m c h p c m c ˆ

e

h p i p p

1 cos

1 cos 0e

c

h

m c

12 Compton wavelength 2.4 10 mce

h

m c

From this derive change in wavelength:

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 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 experimentOriginally performed by Young (1801) to demonstrate the wave-nature of light. Has

now been done with electrons, neutrons, He atoms,…

D

d

Detecting screen

y

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

Expectation: two peaks for particles, interference pattern for waves

Maxima when: sind n

y D

Dy

d

Position on screen: tany D D

n

d

d

D >> d use small angle approximation

So separation between adjacent maxima:

Fringe spacing in double slit experiment

dθ

sind

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 = A1

2 + 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

• 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

• 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

wave theory quantum theory

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).

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

Without grating

Results on matter wave interferenceResults on matter wave interference

Interference patterns can not be explained classically - clear demonstration of matter waves

Fringe visibility decreases as molecules are heated. L. Hackermüller et al. , Nature 427 711-714 (2004)

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:

•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!

Need ph < d to distinguish the slits.

Diffraction is significant only when the aperture is ~ the wavelength of the wave.

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= m μB Bz

(m = z –comp of L)

er IA

L

n

Splitting of atomic energy levels

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

(2l+1) states with same energy: m=-l,…+l

(Hence the name “magnetic quantum number” for m.)

0B 0B

B ≠ 0: (2l+1) states with distinct energies

m = 0

m = -1

m = +1

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.

N

S

Stern-Gerlach experiment (1921)

Oven

AgAg-vaporcollim.

screen

z

x

Ag beam

N

S

Magnet

0

B

N

S

Ag beam

zz ezBB

non-uniform

z0

# Ag

ato

ms

B 0

B↗B↗↗

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

RadiationRadiation: 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 radiationon earth is from the sun (solar) or from radioactive isotopes of the elements (terrestrial).

Radiation is going through you atthis very moment!

http://www.atral.com/U238.html

IsotopesWhat’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.

RadioactivityBy the end of the 1800s, it was known that certain isotopes emit penetrating rays. Three types of radiation were known:

1) Alpha particles ()

2) Beta particles ()

3) Gamma-rays ()

By the end of the 1800s, it was known that certain isotopes emit penetrating rays. Three types of radiation were known:

1) Alpha particles ()

2) Beta particles ()

3) Gamma-rays ()

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 formsof radiation.

Let’s look at them in more detail…

Alpha Particles ()

Radium

R226

88 protons138 neutrons

Radon

Rn222

Note: This is theatomic weight, whichis the number ofprotons plus neutrons


+ nnp

p

He)

2 protons2 neutrons

The alpha-particle is a Helium nucleus.

It’s the same as the element Helium, with the electrons stripped off !

Beta Particles ()

CarbonC14

6 protons8 neutrons

NitrogenN14

7 protons7 neutrons

+ e-

electron(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 ()In much the same way that electrons in atoms can be in an excited state, so can a nucleus.

NeonNe20


(in excited state)


(lowest energy state)

+

gamma

NeonNe20

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.

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

NeonNe20 +

The gamma from nuclear decayis in the X-ray/ Gamma ray

part of the EM spectrum(very energetic!)

NeonNe20

How do these particles differ ?

ParticleMass*

(MeV/c2)Charge

Gamma () 0 0

Beta () ~0.5 -1

Alpha () ~3752 +2

* m = E / c2* m = E / c2

Rate of DecayBeyond knowing the types of particles which are emittedwhen an isotope decays, we also are interested in how frequentlyone of the atoms emits this radiation.

A very important point here is that we cannot predict when aparticular 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 toquantify 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:

10,000 (50%)

5,000 (25%)

2,500 (12.5%)

1 hour (one lifetime) ?

2 hours (two lifetimes) ?

3 hours (three lifetimes) ?

Time #atoms

remaining% of atomsremaining

Lifetime () The “lifetime” of a particle is an alternate definition ofthe rate of decay, one which we prefer.

It is just another way of expressing how fast the substancedecays..

It is simply: 1.44 x h, and one often associates the letter “” to it.

The lifetime of a “free” neutron is 14.7 minutes {neutron=14.7 min.}

Let’s use this a bit to become comfortable with it…

The “lifetime” of a particle is an alternate definition ofthe rate of decay, one which we prefer.

It is just another way of expressing how fast the substancedecays..

It is simply: 1.44 x h, and one often associates the letter “” to it.

The lifetime of a “free” neutron is 14.7 minutes {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

/0

tN N e /0

tN N e N0 = starting number of particles = particle’s lifetime

This is the “exponential”. It’s value is 2.718, and is a very usefulnumber. Can you find it on yourcalculator?

Lifetime (II)/

0tN N e Note by slight rearrangement of this formula:

Fraction of particles which did not decay: N / N0 = e-t/

# lifetimes

Time

(min)

Fraction of remainingneutrons

0 0 1.0

1 14.7 0.368

2 29.4 0.135

3 44.1 0.050

4 58.8 0.018

5 73.5 0.007

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

Lifetimes

Fra

cti

on

Su

rviv

ed

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

Lifetimes

Fra

cti

on

Su

rviv

ed

After 4-5 lifetimes, almost all of theunstable particles have decayed away!

After 4-5 lifetimes, almost all of theunstable 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 cannotsay 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*, or2) the “probability” that a given particle will decay.

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 cannotsay 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*, or2) the “probability” that a given particle will decay.

Lifetime (IV) Given a batch of 1 species of particles, some will decay within 1 lifetime (1, some within 2, some within 3and 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 lifetimesare typically only a small fraction of a second!

If the lifetime is long (like 238U) it will hang around for a very long time!

If the particle’s lifetime is very short, the particles decay away very quickly.

When we get to subatomic particles, the lifetimesare 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 sayabout it?

Survival Probability = N / N0 = e-t/

Decay Probability = 1.0 – (Survival Probability)

# lifetimes Survival Probability

(percent)

Decay Probability = 1.0 – Survival Probability

(Percent)

1 37% 63%

2 14% 86%

3 5% 95%

4 2% 98%

5 0.7% 99.3%

Summary Certain particles are radioactive and undergo decay.

Radiation in nuclear decay consists of , , and particles

The rate of decay is give by the radioactive decay law:

Survival Probability = (N/N0)e-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!

Certain particles are radioactive and undergo decay.

Radiation in nuclear decay consists of , , and particles

The rate of decay is give by the radioactive decay law:

Survival Probability = (N/N0)e-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.

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 values for various gases used in ionization chambers (eV/ion pair)Gas Electrons (fast) Alpha particlesArgon (A) 27.0 25.9Helium (He) 32.5 31.7Nitrogen (N2) 35.8 36.0Air 35.0 35.2CH4 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 current is

a good measure of the ionization rate.

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 used to detect lower intensity radiation.

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 ( rays) and lower atomic number cathodes to lower energy radiation.

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.


• 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.


• 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.


• 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.


• 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 semiconductorsMaterial Operating

temp [K]Atomicnumber

Band gap [eV] Energy per electron-hole pair [eV]

Silicon (Si) 300 14 1.12 3.61Germanium (Ge) 77 32 0.74 2.98Cadmium-teluride(CdTe)

300 48, 52 1.47 4.43

Mercury-Iodine (HgI2) 300 80, 53 2.13 6.5Gallium-Arsenide(GaAs)

300 31, 33 1.43 4.2


• 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


• 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.


• 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 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.


• 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 destroyed.

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

00

01

11

10 epn

•Large variations in the emission velocities of the 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 decay to carry away the missing mass and momentum.

•The neutrino (little neutral one) was discovered in 1956.

Wolfgang Pauli

__

00

01

11

10 epn

1.008665 u 1.007825 u 0.0005486 u

2710660.1 1 u =

1 J = 19106.1

__

kg

eV

0002914.0

)0005486.0007825.1(008665.1

Mass difference

2710660.10002914.0 311083724.4

kg

kg

u

It has been found by experiment that the emitted beta particle has less energy than 0.272 MeVNeutrino accounts for the ‘missing’ energy

2mcE 2831 )100.3)(1083724.4(

1410353516.4

27175510602.1

10353516.419

14

J

J

eV

272.0 MeV

• Ancient Greeks: Earth, Air, Fire,

Water• By 1900, nearly 100 elements• By 1936, back to three

particles: proton, neutron, electron

The Four Fundamental Forces

Particle zoo

2c

Em

Classification of Particle

mx 10101

mx 15101

mx 15107.0

mx 18107.0

Thomson (1897): Discovers electron

00

01

6028

6027 eNiCo

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.

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

ee

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

ee rays

hfee 2

3

1Q

3

2Q 1Q

0Q

James Joyce

Murray Gell-Mann

http://membres.lycos.fr/geryon/bio.htm

3

1

3

1

3

1

3

2

3

2

3

2

13

1

3

2

3

2

03

1

3

1

3

2

3

2

3

1

modern physics notes relativity in classical physics galileo and newton dealt with the issue of...

Documents

inertial reference framestime

inertial reference framethis

different reference

different coordinate

different resultswhen

velocity of light

coordinate transformation

new moving coordinate