# Modern Physics NOTES.

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Modern Physics NOTES

Relativity in Classical Physics
Galileo and Newton dealt with the issue of relativity The issue deals with observing nature in different reference frames, that is, with different coordinate systems We have always tried to pick a coordinate system to ease calculations

Relativity and Classical Physics
We defined something called an inertial reference frame This was a coordinate system in which Newton’s First Law was valid An object, not subjected to forces, moves at constant velocity (constant speed in a straight line) or sits still

Relativity and Classical Physics
Coordinate systems that rotate or accelerate are NOT inertial reference frames A coordinate system that moves at constant velocity with respect to an inertial reference frame is also an inertial reference frame

Moving Reference Frames
While the motion of a dropped coin looks different in the two systems, the laws of physics remain the same!

Classical Relativity The relativity principle is that the basic laws of physics are the same in all inertial reference frames Galilean/Newtonian Relativity rests on certain unprovable assumptions Rather like Euclid’s Axioms and Postulates

Classical Assumptions
The lengths of objects are the same in all inertial reference frames Time passes at the same rate in all inertial reference frames Time and space are absolute and unchanging in all inertial reference frames Masses and Forces are the same in all inertial reference frames

Measurements of Variables
When we measure positions in different inertial reference frames, we get different results When we measure velocities in different inertial reference frames, we get different results When we measure accelerations in different inertial reference frames, we get the SAME results The change in velocity and the change in time are identical

Classical Relativity Since accelerations and forces and time are the same in all inertial reference frames, we say that Newton’s Second Law, F = ma satisfies the relativity principle All inertial reference frames are equivalent for the description of mechanical phenomena

Classical Relativity Think of the constant acceleration situation
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?

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”

Einstein’s Special Theory
In 1905 Einstein proposed the solution we accept today He may not even have known about the M-M result He visualized what it would look like riding an EM wave at the speed of light Concluded that what he imagined violated Maxwell’s Equations Something was seriously wrong

Special Theory of Relativity
The laws of physics have the same form in all inertial reference frames. Light propagates through empty space (no ether) with a definite speed c independent of the speed of the source or observer. These postulates are the basis of Einstein’s Special Theory of Relativity

Gedanken Experiments Simultaneity Time Dilation
Length Contraction (Fitzgerald & Lorentz)

Simultaneity

Simultaneity

Simultaneity Time is NOT absolute!!

Time Dilation

Time Dilation

Time Dilation Clocks moving relative to an observer are measured by that observer to run more slowly compared to clocks at rest by an amount

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

Wave-Particle Duality
Last time we discussed several situations in which we had to conclude that light behaves as a particle called a photon with energy equal to hf Earlier, we discussed interference and diffraction which could only be explained by concluding that light is a wave Which conclusion is correct?

Wave-Particle Duality
The answer is that both are correct!! How can this be??? In order for our minds to grasp concepts we build models These models are necessarily based on things we observe in the macroscopic world When we deal with light, we are moving into the microscopic world and talking about electrons and atoms and molecules

Wave-Particle Duality
There is no good reason to expect that what we observe in the microscopic world will exactly correspond with the macroscopic world We must embrace Niels Bohr’s Principle of Complementarity which says we must use either the wave or particle approach to understand a phenomenon, but not both!

Wave-Particle Duality
Bohr says the two approaches complement each other and both are necessary for a full understanding The notion of saying that the energy of a particle of light is hf is itself an expression of complementarity since it links a property of a particle to a wave property

Wave -Particle Duality
Why must we restrict this principle to light alone? Might microscopic particles like electrons or protons or neutrons exhibit wave properties as well as particle properties? The answer is a resounding YES!!!

Wave Nature of Matter Louis de Broglie proposed that particles could also have wave properties and just as light had a momentum related to wavelength, so particles should exhibit a wavelength related to momentum

Wave Nature of Matter For macroscopic objects, the wavelengths are terrifically short Since we only see wave behavior when the wavelengths correspond to the size of structures (like slits) we can’t build structures small enough to detect the wavelengths of macroscopic objects

Wave Nature of Matter Electrons have wavelengths comparable to atomic spacings in molecules when their energies are several electron-volts (eV) Shoot electrons at metal foils and amazing diffraction patterns appear which confirm de Broglie’s hypothesis

Wave Nature of Matter So, what is an electron? Particle? Wave?
The answer is BOTH Just as with light, for some situations we need to consider the particle properties of electrons and for others we need to consider the wave properties The two aspects are complementary An electron is neither a particle nor a wave, it just is!

Electron Microscopes

Models of the Atom It is clear that electrons are components of atoms
That must mean there is some positive charge somewhere inside the atom so that atoms remain neutral The earliest model was called the “plum pudding” model

Plum Pudding Model We have a blob of positive charge and the electrons are embedded in the blob like currants in a plum pudding. However, people thought that the electrons couldn’t just sit still inside the blob. Electrostatic forces would cause accelerations. How could it work?

Rutherford Scattering
Ernest Rutherford undertook experiments to find out what atoms must be like He wanted to slam some particle into an atom to see how it reacted You can determine the size and shape of an object by throwing ping-pong balls at the object and watching how they bounce off Is the object flat or round? You can tell!

Rutherford Scattering
Rutherford used alpha particles which are the nuclei of helium atoms and are emitted from some radioactive materials He shot alphas into gold foils and observed the alphas as they bounced off If the plum pudding model was correct, you would expect to see a series of slight deviations as the alphas slipped through the positive pudding

Rutherford Scattering
Instead, what was observed was alphas were scattered in all directions

Rutherford Scattering
In fact, some alphas scattered through very large angles, coming right back at the source!!! He concluded that there had to be a small massive nucleus from which the alphas bounced off He did a simple collision model conserving energy and momentum

Rutherford Scattering
The model predicted how many alphas should be scattered at each possible angle Consider the impact parameter

Rutherford Scattering
Rutherford’s model allowed calculating the radius of the seat of positive charge in order to produce the observed angular distribution of rebounding alpha particles Remarkably, the size of the seat of positive charge turned out to be about meters Atomic spacings were about meters in solids, so atoms are mostly empty space

Rutherford Scattering
From the edge of the atom, the nucleus appears to be 1 meter across from a distance of 105 meters or 10 km. Translating sizes a bit, the nucleus appears as an orange viewed from a distance of just over three miles!!! This is TINY!!!

Rutherford Scattering
Rutherford assumed the electrons must be in some kind of orbits around the nucleus that extended out to the size of the atom. Major problem is that electrons would be undergoing centripetal acceleration and should emit EM waves, lose energy and spiral into the nucleus! Not very satisfactory situation!

Light from Atoms Atoms don’t routinely emit continuous spectra
Their spectra consists of a series of discrete wavelengths or frequencies Set up atoms in a discharge tube and make the atoms glow Different atoms glow with different colors

Atomic Spectra Hydrogen spectrum has a pattern!

Atomic Spectra Balmer showed that the relationship is

Atomic Spectra Lyman Series Balmer Series Paschen Series

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

Atomic Spectra Electrons can behave as waves
Rutherford scattering shows tiny nucleus Planetary model cannot be stable classically What produces the spectral lines of isolated atoms? Why the regularity of hydrogen spectra? The answers will be revealed next time!!!

Summary of 2nd lecture electron was identified as particle emitted in photoelectric effect Einstein’s explanation of p.e. effect lends further credence to quantum idea Geiger, Marsden, Rutherford experiment disproves Thomson’s atom model Planetary model of Rutherford not stable by classical electrodynamics Bohr atom model with de Broglie waves gives some qualitative understanding of atoms, but only semiquantitative no explanation for missing transition lines angular momentum in ground state = 0 (1 ) spin??

Outline more on photons
Compton scattering Double slit experiment double slit experiment with photons and matter particles interpretation Copenhagen interpretation of quantum mechanics spin of the electron Stern-Gerlach experiment spin hypothesis (Goudsmit, Uhlenbeck) Summary

Photon properties Relativistic relationship between a particle’s momentum and energy: E2 = p2c2 + m02c4 For massless (i.e. restmass = 0) particles propagating at the speed of light: E2 = p2c2 For photon, E = h = ħω angular frequency ω = 2π momentum of photon = h/c = h/ = ħk wave vector k = 2π/ (moving) mass of a photon: E=mc2  m = E/c2 m = h/c2 = ħω/c2

Compton scattering 1 Expectation from classical electrodynamics:
Scattering of X-rays on free electrons; Electrons supplied by graphite target; Outermost electrons in C loosely bound; binding energy << X ray energy  electrons “quasi-free” Expectation from classical electrodynamics: radiation incident on free electrons  electrons oscillate at frequency of incident radiation  emit light of same frequency  light scattered in all directions electrons don’t gain energy no change in frequency of light

Compton scattering 2 Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles. Nobel prize 1927. X-ray source 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 (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 electron Incident light wave Emitted light wave θ Before After Electron Incoming photon scattered photon scattered electron

Compton scattering 4 Before After θ Electron Incoming photon
scattered photon scattered electron Conservation of energy Conservation of momentum 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 mN >> me

WAVE-PARTICLE DUALITY OF LIGHT
Einstein (1924) : “There are therefore now two theories of light, both indispensable, and … without any logical connection.” evidence for wave-nature of light: diffraction interference evidence for particle-nature of light: photoelectric effect Compton effect Light exhibits diffraction and interference phenomena that are only explicable in terms of wave properties Light is always detected as packets (photons); we never observe half a photon Number of photons proportional to energy density (i.e. to square of electromagnetic field strength)

Double slit experiment
Originally performed by Young (1801) to demonstrate the wave-nature of light. Has now been done with electrons, neutrons, He atoms,… Alternative method of detection: scan a detector across the plane and record number of arrivals at each point y d Detecting screen D Expectation: two peaks for particles, interference pattern for waves

Fringe spacing in double slit experiment
θ D y Maxima when: D >> d  use small angle approximation Position on screen: So separation between adjacent maxima:

Double slit experiment -- interpretation
classical: two slits are coherent sources of light interference due to superposition of secondary waves on screen intensity minima and maxima governed by optical path differences light intensity I  A2, A = total amplitude amplitude A at a point on the screen A2 = A12 + A22 + 2A1 A2 cosφ, φ = phase difference between A1 and A2 at the point maxima for φ = 2nπ minima for φ = (2n+1)π φ depends on optical path difference δ: φ = 2πδ/ interference only for coherent light sources; two independent light sources: no interference since not coherent (random phase differences)

Double slit experiment: low intensity
Taylor’s experiment (1908): double slit experiment with very dim light: interference pattern emerged after waiting for few weeks interference cannot be due to interaction between photons, i.e. cannot be outcome of destructive or constructive combination of photons  interference pattern is due to some inherent property of each photon – it “interferes with itself” while passing from source to screen photons don’t “split” – light detectors always show signals of same intensity slits open alternatingly: get two overlapping single-slit diffraction patterns – no two-slit interference add detector to determine through which slit photon goes:  no interference interference pattern only appears when experiment provides no means of determining through which slit photon passes

double slit experiment with very low intensity , i. e
double slit experiment with very low intensity , i.e. one photon or atom at a time: get still interference pattern if we wait long enough

Double slit experiment – QM interpretation
patterns on screen are result of distribution of photons no way of anticipating where particular photon will strike impossible to tell which path photon took – cannot assign specific trajectory to photon cannot suppose that half went through one slit and half through other can only predict how photons will be distributed on screen (or over detector(s)) interference and diffraction are statistical phenomena associated with probability that, in a given experimental setup, a photon will strike a certain point high probability  bright fringes low probability  dark fringes

Double slit expt. -- wave vs quantum
wave theory quantum theory pattern of fringes: Intensity bands due to variations in square of amplitude, A2, of resultant wave on each point on screen role of the slits: to provide two coherent sources of the secondary waves that interfere on the screen pattern of fringes: Intensity bands due to variations in probability, P, of a photon striking points on screen role of the slits: to present two potential routes by which photon can pass from source to screen

double slit expt., wave function
light intensity at a point on screen I depends on number of photons striking the point number of photons  probability P of finding photon there, i.e I  P = |ψ|2, ψ = wave function probability to find photon at a point on the screen : P = |ψ|2 = |ψ1 + ψ2|2 = |ψ1|2 + |ψ2|2 + 2 |ψ1| |ψ2| cosφ; 2 |ψ1| |ψ2| cosφ is “interference term”; factor cosφ due to fact that ψs are complex functions wave function changes when experimental setup is changed by opening only one slit at a time by adding detector to determine which path photon took by introducing anything which makes paths distinguishable

Waves or Particles? Young’s double-slit diffraction experiment demonstrates the wave property of light. However, dimming the light results in single flashes on the screen representative of particles.

Electron Double-Slit Experiment
C. Jönsson (Tübingen, Germany, 1961) showed double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen. experiment demonstrates that precisely the same behavior occurs for both light (waves) and electrons (particles).

Results on matter wave interference
Neutrons, A Zeilinger et al. Reviews of Modern Physics (1988) He atoms: O Carnal and J Mlynek Physical Review Letters (1991) C60 molecules: M Arndt et al. Nature 401, (1999) Fringe visibility decreases as molecules are heated. L. Hackermüller et al. , Nature (2004) With multiple-slit grating Without grating Interference patterns can not be explained classically - clear demonstration of matter waves

Which slit? Try to determine which slit the electron went through.
Shine light on the double slit and observe with a microscope. After the electron passes through one of the slits, light bounces off it; observing the reflected light, we determine which slit the electron went through. The photon momentum is: The electron momentum is: 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) e I A

Splitting of atomic energy levels
(2l+1) states with same energy: m=-l,…+l B ≠ 0: (2l+1) states with distinct energies (Hence the name “magnetic quantum number” for m.) 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

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

N S N S z Stern-Gerlach experiment (1921) Oven Ag Ag-vapor collim.
screen z x Ag beam N S Magnet z # Ag atoms B  0 B↗ B↗↗ N S Ag beam non-uniform

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

Radiation Radiation: The process of emitting energy in the form of waves or particles. Where does radiation come from? Radiation is generally produced when particles interact or decay. A large contribution of the radiation on earth is from the sun (solar) or from radioactive isotopes of the elements (terrestrial). Radiation is going through you at this very moment!

Isotopes What’s an isotope?
Two or more varieties of an element having the same number of protons but different number of neutrons. Certain isotopes are “unstable” and decay to lighter isotopes or elements. Deuterium and tritium are isotopes of hydrogen. In addition to the 1 proton, they have 1 and 2 additional neutrons in the nucleus respectively*. Another prime example is Uranium 238, or just 238U. To be more clear, deuterium contains 1 proton and 1 neutron in the nucleus, and tritium contains 1 proton and 2 neutrons in its nucleus. Both isotopes behave similarly to ordinary hydrogen, as this chemical behavior is mostly driven by the atomic electrons.

Radioactivity By the end of the 1800s, it was known that certain isotopes emit penetrating rays. Three types of radiation were known: Alpha particles (a) Beta particles (b) 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 R226 Radon Rn222 + p n n p a (4He) 88 protons 138 neutrons 86 protons 136 neutrons 2 protons 2 neutrons Note: The 226 refers to the atomic weight, which is the equal to the number of protons plus neutrons The alpha-particle (a) is a Helium nucleus. It’s the same as the element Helium, with the electrons stripped off !

Yes, the same neutrino we saw previously
Beta Particles (b) Carbon C14 Nitrogen N14 + e- 6 protons 8 neutrons 7 protons 7 neutrons 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 ( + n ) Note that in beta decay, the atomic mass not change, since the neutron and proton have nearly the same mass… Yes, the same neutrino we saw previously

Gamma particles (g) In much the same way that electrons in atoms can be in an excited state, so can a nucleus. Neon Ne20 Neon Ne20 + 10 protons 10 neutrons (in excited state) 10 protons 10 neutrons (lowest energy state) gamma 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 Neon Ne20 Neon Ne20 +
The gamma from nuclear decay is in the X-ray/ Gamma ray part of the EM spectrum (very energetic!)

How do these particles differ ?
Mass* (MeV/c2) Charge Gamma (g) Beta (b) ~0.5 -1 Alpha (a) ~3752 +2 * m = E / c2

Rate of Decay Beyond knowing the types of particles which are emitted when an isotope decays, we also are interested in how frequently one of the atoms emits this radiation. A very important point here is that we cannot predict when a particular entity will decay. We do know though, that if we had a large sample of a radioactive substance, some number will decay after a given amount of time. Some radioactive substances have a very high “rate of decay”, while others have a very low decay rate. To differentiate different radioactive substances, we look to quantify this idea of “decay rate”

Half-Life The “half-life” (h) is the time it takes for half the atoms of a radioactive substance to decay. For example, suppose we had 20,000 atoms of a radioactive substance. If the half-life is 1 hour, how many atoms of that substance would be left after: Time #atoms remaining % of atoms remaining 1 hour (one lifetime) ? 10, (50%) 2 hours (two lifetimes) ? 5, (25%) 3 hours (three lifetimes) ? 2, (12.5%)

Lifetime (t) The “lifetime” of a particle is an alternate definition of the rate of decay, one which we prefer. It is just another way of expressing how fast the substance decays.. It is simply: 1.44 x h, and one often associates the letter “t” to it. The lifetime of a “free” neutron is 14.7 minutes {t (neutron)=14.7 min.} Let’s use this a bit to become comfortable with it…

If I had 1000 free neutrons in a box, after 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 particles t = particle’s lifetime This is the “exponential”. It’s value is 2.718, and is a very useful number. Can you find it on your calculator?

Fraction of remaining neutrons
Lifetime (II) Note by slight rearrangement of this formula: Fraction of particles which did not decay: N / N0 = e-t/t # lifetimes Time (min) Fraction of remaining neutrons 0t 1.0 1t 14.7 0.368 2t 29.4 0.135 3t 44.1 0.050 4t 58.8 0.018 5t 73.5 0.007 So, lifetime is just another measure of how quickly the particles will decay away. If the lifetime is short, the particles will decay away quickly. If the lifetime is long (like some U-238 isotopes), it will be around for a very long time! After 4-5 lifetimes, almost all of the unstable particles have decayed away!

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. * In the context of talking about the lifetime, we are implying that we have a large sample of the substance containing many radioactive atoms. The lifetime represents the fraction pf atoms which will have decayed. Unfortunately, we cannot say exactly which ones will have decayed…

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 Note: The number “e” is very common in math and physics. It has the value: e = 2.718

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!

Decay Probability = 1.0 – Survival Probability (Percent)
Lifetime (IV) What if we only have 1 particle before us? What can we say about it? Survival Probability = N / N0 = e-t/t Decay Probability = 1.0 – (Survival Probability) # lifetimes Survival Probability (percent) Decay Probability = 1.0 – Survival Probability (Percent) 1 37% 63% 2 14% 86% 3 5% 95% 4 2% 98% 5 0.7% 99.3% But, what if we only have 1 particle before us? What can be said about it’s decay? In this case, the radioactive decay law gives the probability that this particle will have NOT decayed (I.e., it survived without decaying) after some time. Survival Probability = N / N0 = e-t/t So, the probability that a single unstable particle will survive after 1 lifetime is 37%; 5% chance it’ll be around after 2 lifetimes; 2% chance it’ll be around after 3 lifetimes, and so on… Now, sometimes, we want to know the probability for a certain particle to decay. This is simply obtained by saying: Decay Probability = 1.0 – (Survival Probability)

Summary Certain particles are radioactive and undergo decay.
Radiation in nuclear decay consists of a, b, and g particles The rate of decay is give by the radioactive decay law: Survival Probability = (N/N0)e-t/t After 5 lifetimes more than 99% of the initial particles have decayed away. Some elements have lifetimes ~billions of years. Subatomic particles usually have lifetimes which are fractions of a second… We’ll come back to this!

Ionization sensors (detectors)
In an ionization sensor, the radiation passing through a medium (gas or solid) creates electron-proton pairs Their density and energy depends on the energy of the ionizing radiation. These charges can then be attracted to electrodes and measured or they may be accelerated through the use of magnetic fields for further use. The simplest and oldest type of sensor is the ionization chamber.

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

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 Fig 9.1

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
Fig 9.1x

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
Fig 9.1y

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
Fig 9.2

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 Fig 9.3

Scintillation sensors
Takes advantage of the radiation to light conversion (scintillation) that occurs in certain materials. The light intensity generated is then a measure of the radiation’s kinetic energy. Some scintillation sensors are used as detectors in which the exact relationship to radiation is not critical. In others it is important that a linear relation exists and that the light conversion be efficient.

Scintillation sensors
Materials used should exhibit fast light decay following irradiation (photoluminescence) to allow fast response of the detector. The most common material used for this purpose is Sodium-Iodine (other of the alkali halide crystals may be used and activation materials such as thalium are added) There are also organic materials and plastics that may be used for this purpose. Many of these have faster responses than the inorganic crystals.

Scintillation sensors
The light conversion is fairly weak because it involves inefficient processes. Light obtained in these scintillating materials is of light intensity and requires “amplification” to be detectable. A photomultiplier can be used as the detector mechanism as shown in Figure 9.5 to increase sensitivity. The large gain of photomultipliers is critical in the success of these devices.

Scintillation sensors
The reading is a function of many parameters. First, the energy of the particles and the efficiency of conversion (about 10%) defines how many photons are generated. Part of this number, say k, reaches the cathode of the photomultiplier. The cathode of the photomultiplier has a quantuum efficiency (about 20-25%). This number, say k1 is now multiplied by the gain of the photomultiplier G which can be of the order of 106 to 108.

Scintillation sensor Fig 9.5

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 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

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.

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). .

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.

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

Conservation of energy and momentum in nuclear reactions

Conservation Laws 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

conservation of energy and momentum
Neutrino must be present to account for conservation of energy and momentum __ Wolfgang Pauli 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.

released in the reaction
Calculate the energy released in the reaction __ u u u kg 1 u = 1 J = eV

Calculation Mass difference u kg kg

Calculation J J eV MeV It has been found by experiment that the emitted beta particle has less energy than MeV Neutrino accounts for the ‘missing’ energy

History of search for basic building blocks of nature Ancient Greeks:
Earth, Air, Fire, Water By 1900, nearly 100 elements By 1936, back to three particles: proton, neutron, electron

FUNDAMENTAL FORCES OF NATURE
Familiar Forces Tension Forces Ask a student to hold one end of a piece of string in their hand, while you pull on the other end. Test your strength on a Newton spring balance. The tension (stretching) force is along the string and away from the support point. Compression Forces Push (gently) against the palm of someone's hand with a ruler. The compression (squashing) force is along the ruler and towards the support point. When a ruler is flexed so that it curves downward at its midpoint, the timber fibres on the ‘inside’ of the curve will be in compression. The fibres on the ‘outside’ of the curve will be in tension. The same thing happens in a concrete bridge or the lintel over a window or door, even though it is not obvious to the eye. That’s why it’s necessary to place reinforcing steel bars in the section of a concrete beam which is in tension, since concrete is weak in tension but reasonably strong in compression. Friction Forces Everyone is familiar with how difficult it is to walk on icy surfaces. Most people, at some time or other, have slipped at the kitchen sink because of water spillage. Many have experienced a nasty fright when the car in which they were travelling skidded. Try pushing the computer mouse pad along the table. Friction is a contact force between surfaces whose critical importance becomes obvious only when it’s absent. Reaction Forces When you push against a wall, the wall pushes back. When a lift travels from the top storey of a tall building, you experience a mild version of weightlessness, as the upward reaction exerted by the lift floor on you is momentarily reduced. On the other hand, you experience a momentary weight increase when the lift takes off from the ground. Seatbelts are worn in cars at all times and in aeroplanes at take-off and landing to provide reaction forces against the forces arising from accelerations. The Four Fundamental Forces of Nature The Gravitational Force When a baby starts to play by dropping objects out of its pram, it has begun its journey as an experimental physicist. Familiarity hides the wondrous and unusual nature of this force from our close scrutiny. This force intrigued the ancient Greeks, who claimed that heavier objects fell towards the ground faster than lighter ones. It is claimed that Galileo showed by experiment that two objects, regardless of their weights, would hit the ground simultaneously if dropped from identical heights. A careful reading of Galileo's experiments shows that he was well aware of the effects of air resistance on falling objects. Based on the classical wisdom of the Greek Philosophers, especially Plato and Aristotle, the Earth was the place where change occurred. In contrast the Heavens were eternal and unchanging. When Newton observed the 'apple fall from the tree', he had a brilliant insight. In his own words, 'I began to think of gravity extending to ye orb of the moon', Newton proceeded to show by calculation that the gravitational force which caused the apple to fall to the ground was the same as the force that caused the moon to accelerate towards the Earth. He showed, on the basis of known measurements, that , where is the acceleration experienced by the moon due to the Earth's gravitational pull and g is the acceleration due to gravity at the Earth's surface. He then proceeded to compute the ratio on the basis of the 'inverse square law' and obtained the same answer. The hammer blow to the classical view was his derivation of the elliptical orbits of the planets around the Sun from the same inverse square law of gravitational force. Every particle of matter in the universe attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of their distances apart. In symbols F is the gravitational force; G is the universal gravitational constant; m1 and m2 are the particle masses and d is the distance between their centres of mass. Newton believed the force was proportional to the mass of each particle, because the force on a falling body is proportional to its mass. This relationship is known as Newton’s law of gravitation. The law applies to particles or objects whose dimensions are very small compared with the other distances involved. Newton was able to show that even an object as big as the Earth could be viewed as a uniform sphere with all the mass concentrated at its centre point. The gravitational force is a very small force. It is a very difficult force to detect between two 1 kg masses 1 m apart. In this case it is in fact numerically equal to G, with a value of N. Some appreciation of just how tiny this is can be gauged by comparison with the force on a falling apple, which is roughly 1 N. It is so small that it can be ignored inside atoms. However, it dominates everyday life due to the close proximity of the huge mass of the Earth and because it is only attractive. Its range is infinite. Newton's law of gravitation explains how a body falls and how the planets move around the Sun, but leaves unexplained why these events happen as they do. The gravitational force pulls objects towards each other, even though they are not in physical contact. Modern physics interprets this action at a distance as arising from an exchange of particles between the objects experiencing the force. In the case of the gravitational force the exchange particle is called the graviton. The graviton is postulated to exist, but has not been discovered. The gravitational force is a fundamental force because it operates between any two elementary particles. The Electromagnetic Force Experiments show that, sometimes, after any two different materials are rubbed together they exert forces on each other. Each has acquired an 'electric charge'. Furthermore, experiments show that there are two kinds of charge. The two kinds tend to cancel one another out and in this respect are opposite. Hence one kind is called positive and the other kind is called negative. Polythene rubbed with wool acquires a negative charge, whereas perspex (cellulose acetate) rubbed with wool acquires a positive charge. The force between two point charges is proportional to the product of the charges and is inversely proportional to the square of their distance apart. In symbols where F is the force, Q1 and Q2 are the charges and d is the separation distance. for air or vacuum. This relationship is called Coulomb’s law. Moving charges experience a force in a magnetic field and also create (induce) magnetic fields. The combined effect (if applicable) of the magnetic force and the coulomb electrostatic force is called the electromagnetic force. We do not directly experience the strength of the electrostatic force as individuals. The delicate balance between the negative electrons and the positive protons in our constituent atoms prevents such an experience. Suppose however that 0.1% of someone's electrons were transferred to someone else. The consequent force of attraction that these people would feel at a distance of 1 m apart can be found by applying Coulomb’s law. For simplicity, suppose the mass of each person is 50 kg and that each person is composed entirely of C-12 atoms. Now 12 grams of carbon contains electrons [No. of electrons in a carbon atom × No. of carbon atoms in one mole]. Hence the total number of electrons in each person is . Thus the number of electrons moved from one person to the other is . The force of attraction is N. This force is approximately equal to a thousandth part of the weight of the earth. It is also instructive to compare the eleectrostatic and gravitational attractions between a proton (charge +e and mass kg) and an electron (charge -e C and mass kg) placed 1 metre apart. Fe = electric attraction N Fg = gravitational attraction N Hence . The electromagnetic force acts between all charged particles. Its range is infinite. It is the force that binds atoms and molecules together. It is responsible for tension, compression, friction and reaction forces at the atomic level. Like gravity, it acts at a distance, with the photon acting as the exchange particle. The Strong Nuclear Force This is the very strong attractive force between nucleons, which holds the atomic nucleus together against the repulsive electrostatic forces between protons. It is also called the strong interaction. Its existence was confirmed by the discovery of the neutron. The strong force acts over a very short range. If its effects went much outside the nuclear surface, it would not be possible to explain Rutherford's alpha-particle scattering experiment solely in terms of electrostatic repulsion. In the range of internucleon separation of about 1 to 3 fm it is strongly attractive, but more or less disappears beyond 3 fm. (m) [1 fm = 1 femtometre] At distances of less than 1 fm the force must be sufficiently repulsive to prevent the nucleus collapsing. The strong nuclear force acts at a distance, as was the case with the gravitational and electrostatic forces. Imagine the nucleons as a group of dancers. If they form a ring by interlocking hands around their waists, they can continue to dance quite comfortably provided they stay within limits. If they try to pull apart, the 'force' holding them together gets stronger; if they get too close together they can no longer dance comfortably. The Weak Nuclear Force (The Weak Interaction) In 1930, on the basis of energy and momentum conservation, Pauli proposed the existence of a third particle to explain the range of energies shown by the electrons in beta emission. He offered a crate of champagne to the first person to prove the existence of this particle, which was christened the neutrino by the Italian physicist Enrico Fermi in a jocular response to a journalist's question about Chadwick's discovery of the neutron. The neutrino proved extremely elusive. Cowan and Reines finally found it in Its existence implies that there is a fourth distinct force in nature. Its interaction with matter is so rare and tenuous that this interaction cannot be explained in terms of any of the other three fundamental forces. This fourth force is called the weak nuclear force or the weak interaction. It is intermediate in strength between the gravitational and electromagnetic forces. It has a range of less than 10-2 fm. This force also acts at a distance. This weak interaction, or force, is involved when a neutron decays to a proton, electron, and an antineutrino in the process called beta decay. Comparison of the Four Fundamental Forces Force Relative Strength Range Action Gravitational 1 all particles Weak Nuclear 1023 10-18 m Electromagnetic 1036 charged particles Strong Nuclear 1038 10-15 m protons, neutrons

The Four Fundamental Forces

Families of particles

Particle zoo Mass of particles comes from energy of the reaction
The larger the energy the greater the variety of particles

Particle Zoo

Classification of Particle

Thomson (1897): Discovers electron

Indivisible point objects Not subject to the strong force
Leptons Indivisible point objects Not subject to the strong force produced in radioactive decay _ Q = -1e almost all trapped in atoms Q= 0 all freely moving through universe

Baryons Mesons Subject to all forces Subject to all forces mass between electron and proton e.g. protons, neutrons and heavier particles Composed of three quarks Composed of quark-antiquark pair

Antimatter

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.

History of Antimatter 1928 Dirac predicted existence of antimatter
1932 antielectrons (positrons) found in conversion of energy into matter 1995 antihydrogen consisting of antiprotons and positrons produced at CERN In principle an antiworld can be built from antimatter Produced only in accelerators and in cosmic rays

Pair Production

Annihilation

Quark model

Quarks Fundamental building block of baryons and mesons

Naming of Quark Three Quarks for Muster Mark James Joyce
Murray Gell-Mann

The six quarks

Thank You

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