Chapter 12 Special Relativity and Elementary Particles

Chapter 12

Special Relativity and Elementary Particles

2

Special Relativity

Imagine sitting in the back of a truck as a friend throws a ball to you. You are moving away from your friend at

20 km/h. The ball is thrown at 50 km/h relative to the

ground.

3

Special Relativity, cont’d

The speed of the ball, relative to you, depends on how you are moving relative to the ball. If you were standing, the ball would approach

you at 50 km/h. Since you are moving away, the ball appears

to approach you at 30 km/h.

4


Now imagine traveling away from you friend in a spaceship as he shines a light at you. You travel at 200,000 km/s away from your

friend. The light travels away from your friend at

300,000 km/s.

5


The speed of the light, relative to you, does not depend on you are moving relative to your light. You measure the light at 300,000 km/s from

the spaceship. If standing still, you still measure

300,000 km/s.

6


So, the speed of light does not depend upon the motion of the observer.

Light, behaving as Maxwell predicted, does not act as we would expect. Based on the physics we’ve studied so far.

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The postulates of special relativity Einstein recognized the contradiction

between the predictions of classical mechanics and electromagnetism. Classical mechanics says the speed of light

depends on the motion of the observer. Electromagnetism says the speed of light does

not depend on the motion of the observer. He put forth two postulates to overcome this

discrepancy.

8

The postulates of special relativity, cont’d

The 1st postulate of relativity states: The speed of light, c = 300,000 km/s, is the

same for all observers, regardless of their motion.

The speed of light is a fundamental constant of nature. Much like the gravitational constant, G.

Prior to Einstein, several experiments were unsuccessful at demonstrating different values of the speed of light based on the observer’s motion.

9


The 2nd postulate of relativity states: The laws of physics are the same for all

observers moving uniformly. Uniformly means at a constant velocity.

This is known as the principle of relativity. This means if two observers traveling toward one

another at a constant acceleration perform the same experiment, they obtain identical results.

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These two postulates constitute Einstein’s special theory of relativity. It is “special” because it is restricted to uniform

motion.

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Predications of special relativity These two postulates have some unusual

implications. Time dilation — you observe the clock of someone

moving, relative to you, to run slower than your clock.

Length contraction — you observe the length of an object moving, relative to you, to be less than an identical object at rest, relative to you.

In the direction of the object’s motion.

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Time dilation

Consider an experiment. Create a clock out of a flashbulb and a photo-

detector. The bulb sends out a flash of light. The flash reflects off the mirror and triggers

the detector.

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Time dilation, cont’d

You synchronize your clock with a friend’s clock in a spaceship traveling at a speed v.

The question is: will the clocks keep the same time?

14


For your clock: The distance the light must travel from the

bulb to the detector is d. The speed at which light travels is c. The amount of time between clicks is:

2dtc

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For your friend’s clock (as seen by you): Since the spaceship is traveling with speed v,

you think the light must travel farther than the distance 2d.

From your observation, let the time interval between “ticks” of your friend’s clock is t′.

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The distance the light must travel during a “click” is

Since this is the distance light travels during a “click,” we also have

222 / 2d v t

c t

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So we can equate these two:

After some algebra, we can solve for t′.

222 / 2c t d v t

22

2

2

21 v dtc c

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Recall that from your clock, you know:

So you can relate the two time intervals:

2dtc

2 21

ttv c

19


So what does this formula tell us?

t is the interval between clicks of your clock. t′ is the interval between clicks of your friend’s

clock as observed by you. You see your friend’s clock run more slowly

than your click. Even though they are synchronized to “click” at

the same rate when side-by-side at rest.

2 21

ttv c

20


How significant is this effect? For “everyday” speeds (v < 0.5c), the effect is

negligible. For higher

speeds, the effect can become very significant.

Even doubling or tripling how slow his clock appears to run.

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Is this real? Muons are subatomic particles created high in the

atmosphere through collision with atmospheric molecules and energetic particles from space.

They are created about 10 km or more above the Earth’s surface.

They have an average lifetime of 0.000 002 s. Since they die so quickly, they should not reach

the surface in any great quantity. So, why do we detected such a large number at

the Earth’s surface?

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ExampleExample 12.1What is the mean lifetime of a muon as

measured in the laboratory if it is traveling at 0.90c with respect to the laboratory? The mean lifetime of a muon at rest is 2.2×10-6 seconds.

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ExampleExample 12.1ANSWER:The problem gives us:

So the lifetime as seen from the lab is

6

0.90

2.2 10 s

v c

t

6

2 2 2 2

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2.2 10 s

1 1 0.90

2.2 10 s 5.0 10 s0.19

ttv c c c

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ExampleExample 12.1DISCUSSION:So the muon “thinks” it only exists for 2.2 s (as

measured by its moving clock).

But we on the ground think the muon exists for twice as long (as measured by our stationary clocks): 5.0 s.

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Length contraction

Another phenomenon predicted by special relativity is length contraction. This is the apparent shortening of moving

objects in the direction of their motion. So how does this happen?

One method to measure the length of an object by timing how long it takes light to traverse the object.

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Length contraction, cont’d

But we just discovered that moving clocks appear to run more slowly than stationary clocks.

Since the moving clock appears to take less time to measure the traversal of light, it predicts that the length should be shorter. Recall that So a shorter time interval indicates a shorter

length.

distance speed time

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As for time dilation, length contraction is not observed at “normal” speeds.

But it is noticeable in particle accelerators. These are scientific devices used to

accelerate particles to very high speed.

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Our muon story also demonstrates length contraction.

Even though the muon is 10 km or more above the surface, they reach the surface.

From the muon’s perspective, their lifetime is too short to reach the Earth’s surface.

But since they do, the muon’s must measure the distance to be less than 10 km.

The speed and lifetime are fixed. The distance is the only thing that can be changed in

order for them to accomplish this.

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Rest energy

Our definitions of kinetic energy and momentum no longer work in the realm of special relativity.

In order for conservation of kinetic energy and momentum to hold, we must include a rest energy, E0, in our calculations.

The rest energy is2

0E mc

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Rest energy, cont’d

The total relativistic energy of a particle has two terms. the rest energy, and the kinetic energy.

22

rel rel 2 21

mcE KE mcv c

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Rest energy, cont’d

The relativistic kinetic energy is then

This reduces to our usual formula for very low speeds:

22

rel 2 21

mcKE mcv c

212KE mv

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ExampleExample 12.2In an x-ray tube, an electron with mass m = 9.1×10-31 kg is accelerated to a speed of 1.8×108 m/s. How much energy does the electron possess? Give the answer in joules and MeVs (million electron-Volts).

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The energy is

The speed ratio is

31

8

9.1 10 kg

1.8 10 m/s

m

v

22

rel rel 2 21

mcE KE mcv c

8

8

1.8 10 m/s 0.60 0.603.0 10 m/s

v v cc

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ExampleExample 12.2ANSWER:The relativistic energy is then

231 8

rel 2

14

13

9.1 10 kg 3 10 m/s

1 0.60

8.2 10 J0.80

1.0 10 J

E

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ExampleExample 12.2ANSWER:To express this in MeV

13rel

1319

1.0 10 J1eV1.0 10 J

1.6 10 J640,000 eV0.640 MeV

E

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ExampleExample 12.2DISCUSSION:Notice that the rest energy of an electron is

20

231 8

14

9.1 10 kg 3 10 m/s

8.2 10 J512,000 eV 0.512 MeV

E mc

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ExampleExample 12.2DISCUSSION:The classical result for the energy is

Too small by almost 30%.

212

231 812

14

9.1 10 kg 1.8 10 m/s

1.5 10 J0.092 MeV.

KE mv

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The four forces

There are four fundamental forces.

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The four forces, cont’d

Every interaction in our environment is due to these forces.

We initially defined a force as a push or pull. This is acceptable in classical physics.

It needs to be expanded to deal with subatomic particles. It has to include every process a particle can

undergo. disintegration, annihilation, reaction, creation, …

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The four forces, cont’d

So we typically use the term four basic interactions instead of four basic forces. An interaction means the mutual action or

influence of one or more particles on another.

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The four forces — gravity

Gravity is the most familiar of the four basic interactions. We investigated it in Chapter 2.

In the world of particle physics, it is the least important. Its strength is so feeble compared to the other

interactions so it can be ignored. It is 10-38 times weaker than the

electromagnetic interaction.

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The four forces — gravitycont’d

Gravity is probably the most important for large-scale interactions. Since most objects are electrically neutral, the

gravitational interaction is more important than the electromagnetic interaction.

Realize that both are “infinite-range” forces. It is responsible for the distribution of matter in

the Universe. Newton’s law of gravity has been replaced by

Einstein’s general theory of relativity.

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The four forces — electromagnetic The next most-familiar interaction is the

electromagnetic. We discussed it in Chapter 8.

The electromagnetic force can be attractive, repulsive or irrelevant. Gravity is always attractive.

It is also an “infinite-range” interaction. But if objects are electrically neutral, there is

no significant interaction.

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The four forces — electromagneticcont’d It can affect certain, non-charged particles.

The photon is the “force carrier” for the electromagnetic interaction.

The photon has no electrically charge. But it does involve an electric and magnetic

field. So the electromagnetic interaction influences the

fields associated with a photon.

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The four forces — weak

The interaction intermediate to the EM and gravitational interaction is the weak nuclear interaction. It is a nuclear interaction.

It only influences nuclear processes. So we never encounter it in daily life.

It is “weak” since it is 10-5 times weaker than the EM interaction.

But still 1031 times stronger than gravity. It only acts over ranges of 10-18 meters.

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The four forces —weakcont’d

It is also “weak” because it is unlikely that processes involving this interaction occur.

It plays a critical role in the generation of energy in the Sun and in building-up heavy elements.

Another aspect of it being “weak” is that it occurs on longer time scales than the electromagnetic force.

About a million times “slower.”

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The four forces —weakcont’d

There is a very close relationship between the electromagnetic and weak interactions. Each can be responsible for the same types of

interactions. But the EM interaction is much more likely to be

responsible. These two theories have been joined into an

electroweak interaction. This is similar to electricity and magnetism

being joined into electromagnetism.

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The four forces — strong

There is another nuclear force, called the strong interaction. It is about 100 times stronger than the EM

interaction. But it operates within very small ranges.

Although the EM interaction is an “infinite-range” interaction, the strong exerts influence only over a range of 10-15 meters.

It is always attractive and does not depend upon electric charge.

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The four forces — strongcont’d

The strong force is responsible for overcoming the electric repulsion of protons in the nucleus.

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Force carriers

Since we have to change our definition of force, we have a new impression of how forces cause particles to interact.

Each force has its own type of force carrier. These are particles that are exchanged due a

certain interaction and the type of particle depends on the interaction.

Gravity has the graviton.

Electromagnetic has the photon.

Weak has the Z0 and W± particles.

Strong has gluons.

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Force carriers This figure indicates the modern

representation of interactions. The top shows a neutron decaying into

a proton and W- boson. The W- boson then decays into an

electron and electron neutrino. The bottom shows two electrons

interacting through the exchange of a photon.

The photon transfers momentum to the electrons and exerts a “force” on the electrons.

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Classification schemes for particles Chapter 4 classified matter into solid, liquid,

gas and plasma phases. We can also classify the elementary particles. An elementary particle is the basic,

indivisible building blocks of the universe. The fundamental constituents from which all

matter, antimatter, and their interactions derive.

They are believe to be true “point” particles, devoid of internal structure or measurable size.

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Classification schemes for particles, cont’d

An antiparticle is a charge-reversed version of an ordinary particle. A particle of the same mass (and spin) but of

opposite electric charge (and certain other quantum mechanical “charges”).

Every known particle has a corresponding antiparticle. Electrons have anti-electrons, i.e., positrons. Protons have anti-protons, etc.

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When matter and antimatter meet, mutual annihilation occurs. Both particles are destroyed. Gamma rays are emitted.

The gamma rays have to have the same total energy of the two particles.

Antiparticles are denoted by the appropriate particle symbol with a bar over the symbol.

proton : anti-proton :

electron : positron :

p p

e e

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Each elementary particle has a spin. Spin is an intrinsic property of a particle and

measures the amount of angular momentum carried by the particle.

It is an intrinsic property like charge and mass. Recall that spin only comes in quantized

amounts of h/2. Electrons have spins of ½ × h/2. Photons have spins of 1× h/2.

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Stephen Hawking gives a useful method to think of spin. A spin-0 particle is like a period.

It looks the same from any direction. A spin-1 particle is like an arrow.

It takes rotating it 360º to look the same. A spin-2 particle is like a double-ended arrow.

It takes rotating it 180º to look the same. A spin- ½ particle requires two complete

revolutions to return to its original perspective.

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Here are examples of Hawking’s perspective on spin.

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There are two basic “families” of particles based on there spin. Fermions are particles that have half-integral

spins. Bosons are particles that have integral spins.

There are two distinct spin states for each value of angular momentum.

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Particles can have: spin-up: so it looks like the particle is rotating

counterclockwise when viewed from above. spin-down: so it looks like the particle is

rotating clockwise when viewed from above.

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Elementary particle lexicon

There are four groups of particles. Particles are split according to their spin.

Fermion or boson. Within this, they are split whether they

participate in strong interactions. baryons, mesons, leptons, and gauge bosons.

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Elementary particle lexicon, cont’d

There are two groups of particles that are fermions (half-integral spin): Baryons.

means heavy particles. interact through the strong interaction.

And through other interactions. includes nucleons (protons & neutrons) & others.

Leptons. means light particles. do not interact through the strong interaction. includes electrons & others.

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There are two groups of particles that are bosons (integral spin): Mesons.

means medium particles. interact through the strong interaction.

And through other interactions. includes pions, kaons, & others.

Gauge bosons. are force mediators. do not interact through the strong interaction. includes photons, Z0, W±, and gravitons.

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Baryons and mesons are typically grouped together as hadrons. Means “thick” or “strong.”

They participate in strong interactions. Note that the term hadron includes fermions.

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Conservation laws

In Chapter 3 we introduced conservation laws. Statements that some physical quantity does

not change during an interaction. We discussed:

conservation of mass, conservation of energy, conservation of linear momentum, and conservation of angular momentum.

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Conservation laws, cont’d

These conservation laws still hold for elementary particles.

If an electron annihilates a positron: Energy must be conserved.

The energy of the emitted gamma rays must equal the total energy of the two particles.

Momentum must be conserved. If there was no net momentum of the particles, the gamma

rays must also have no net momentum. Electric charge is conserved.

The negative electron and position positron have no net charge and the two photons have no net charge.

66


There are new conservation laws that apply to elementary particles that we don’t experience in everyday events. We must conserve baryon number.

If two baryons interact, we must have two baryons after the interaction.

If a baryon and its antiparticle interact, this implies no net baryon number so we do not need two baryons after the interaction.

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Consider a collision between two protons.

This process does not violate any classical conservation laws.

But this process is never seen. The reason is we do not conserve baryon

number. There are two baryons going in but none

coming out.

0p p

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This collision is seen:

Law of Conservation of Baryon Number: In a particle interaction, the baryon number (B) must remain constant. The number of baryons going into the reaction

must equal the number emerging. Baryons are given: Anti-baryons are given:

0p p

1B 1B

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A similar law holds for lepton number. Consider the process of a neutron decaying

into a proton.

There is one baryon going in and NO leptons going in. So one baryon must emerge:

the proton. A lepton emerges:

the electron. So an anti-lepton must emerge:

an electron antineutrino.

en p e

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To complicate issues for leptons, there is a specific lepton number for each “family” of leptons. The electron family:

the electron and the electron neutrino:

The tau family: the tau and the tau neutrino:

The muon family: the muon and the muon neutrino:

ee and

and

and

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Law of Conservation of Lepton Number: In particle interactions, lepton number (L) is conserved. Within each lepton family (electron, muon, and tau) the value

of the lepton number at the start of a reaction must equal its value at the end.

This means this reaction is invalid:

It does not conserve muon/electron lepton number. But this reaction is observed:

e

ee

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In the early 1950s, certain particles were discovered with strange properties. They were always observed in pairs.

They are the kaons (K), lambdas (), and sigmas (). Here are some illustrative reactions:

But this reaction is not observed:

0 0p K 0 0p p K K

0p p p

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It was proposed that certain possess another type of “charge” or quantum number termed strangeness (S).

Strangeness: In strong and electromagnetic interactions, strangeness (S) is conserved; in weak interactions, strangeness may change

by ±1 unit. Strangeness is a partially conserved quantity.

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ExampleExample 12.3Identify the conservation law(s) that would be

violated in each of the following reactions.a)

b)

c)

e

0 0p K p

0

75


On the left side, we have a particle that participates in strong interactions.

On the right we do not.So strangeness conservation is violated.

We also violate lepton number since there are no leptons on the left.

e

76

ExampleExample 12.3ANSWER:The problem gives us:Electric charge is not conserved.

The left side is neutral but the right side of +1 because of the proton.

The kaon and pi-meson are neutral.Baryon number is conserved.

One proton on each side.Strangeness is not conserved.

The kaon has strangeness but nothing else does.

0 0p K p

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ExampleExample 12.3ANSWER:The problem gives us:Electric charge is conserved.The lambda baryon decays into two non-baryon

particles.So baryon number is not conserved.

No particles interact via the strong interaction so strangeness is not involved.

0

78

ExampleExample 12.4Each of the reactions below is missing a single

particle. Figure out what is must be if these interactions are permitted.

a) b) c)

p p n p n

0p p p

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ExampleExample 12.4ANSWER:

On the left, we have:Q = 0 for electric charge,B = 0 for baryon number, andS = 0 for strangeness.

+1 for the p and -1 for the p-bar for Q & B while S = 0 for both p and p-bar.

On the right, we have:Q = 0, B = +1 and S = 0.

p p n

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So we need a particle that is electrically neutral and has a negative baryon number.

Looking at Table 12.3, the only possibility is an anti-neutron.

p p n

p p n n

81


On the left, we have:L = -1 for lepton number (electron anti-neutrino),Q = +1 for electric charge (proton),B = +1 for baryon number (proton), andS = 0 for strangeness.

On the right, we have:Q = 0, B = +1 and S = 0.

p n

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So we need a particle that hasL = -1 (muon family),Q = +1, andS = 0.

From Table 12.4, we need a anti-muon.

p n

p n

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On the left, we have:Q = +2 for electric charge (2 protons),B = +2 for baryon number (2 protons), andS = 0 for strangeness.

On the right, we have:Q = 1, B = +2 and S = -1.

0p p p

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So we need a particle that hasQ = +1,B = 0, andS = +1.

From Table 12.4, we need a kaon.

0p p p

0p p p K

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Quarks

There was a terrific proliferation of “elementary” particles during the 1960s and early 1970s.

Scientists began to question whether all of the fundamental particles were indeed fundamental.

It was proposed that all hadrons were formed from three fundamental particles, called quarks.

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Quarks, cont’d

The quarks are designated as: u for “up,” d for “down,” and s for “strange.”

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Quarks, cont’d

Notice that the quarks do not have integral electric charge. They occur in multiples of 1/3 of the charge of

a proton. All quarks have a baryon number of +1/3.

Anti-quarks have -1/3. Only the strange quark has a strange

quantum number. The other quarks have S = 0.

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Quarks, cont’d

These three particles, and their anti-particles, allow all hadrons to be constructed.

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ExampleExample 12.5To what hadron does the combination of a d

and a ū quark correspond? Assume the spins of the quarks are anti-parallel.

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ExampleExample 12.5ANSWER:Looking at the table, we have:

d → Q = -1/3 and B = +1/3.ū → Q = -2/3 and B = -1/3.

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ExampleExample 12.5ANSWER:So we have a particle with Q = -1, B = 0, and

S = 0.

From Table 12.3, it cannot be a baryon so it must be a meson.

Checking the table, the candidate is an anti-pion, -.

92

ExampleExample 12.6Give the quark combination associated with the

xi minus, -, baryon.

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ExampleExample 12.6ANSWER:From Table 12.3, the xi-minus has

Q = -1, B = +1, and S = -2.We can’t use any anti-quarks.

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ExampleExample 12.6ANSWER: (Q = -1, B = +1, and S = -2.)To get S = -2, we need two s quarks.This gives Q = -2/3 and B = +1/3.If we add a d quark to this, we get the answer.

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The standard model and GUTs

Quarks are fermions. The have half-integral spin.

So they must obey the Pauli exclusion principle.

But the - particle seems to violate this. It is composed of three strange quarks.

To overcome this, scientists realized that quarks must have an additional property that avoids violating the exclusion principle.

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The standard model and GUTs, cont’d

Each quark comes in three colors. This additional

property is called color.

It is believe that the color charge is responsible for the inter-quark force.

97


The force carriers of the color force are called gluons. Gluons are electrically neutral, zero mass and

have spin 1. There are eight gluons. The emission/absorption of a gluon by a

quark can change the color of the quark. The theory of quark color and gluons is called

quantum chromodyanmics.

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Quarks actually come in three flavors. Not counting the anti-flavors. Up & down, Charm & strange, and Top & bottom.

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Here are the properties of the quarks.

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Our basic model can be summarized as:

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Chapter 12 Special Relativity and Elementary Particles