FK5024 - Nuclear & particle physics, astrophysics and cosmology

• Course information

• Review of special relativity

• Units

Theoretical background

Particle and nuclear physics rests on special relativity and quantum mechanics (review today SR)

Cosmology rests on general relativity and thermodynamics. Short-cuts used and GR avoided in this course.

From the small to the large

Object Size (m)

Quark <10-18

Proton 10-15

Nucleus 10-14

Star 109

Galaxy 1021

Galaxy cluster 1023

The observable universe

1026

During the course physical processes and phenomena spanning more than 44 orders of magnitude in size will be studied.

Course information All relevant information to be found here: http://staff.fysik.su.se/~lbe/FK5024/FK5024.htm Topics : • Particle physics • Nuclear physics • Astrophysics and cosmology

• Lectures on each topic • 11 tutorials • 2 seminars • 3 sets of hand-in problems

Examination

Minimum requirements for passing: • Participating in the two compulsory seminars

and doing satisfactorily the preparatory work (three sets of hand-in problems)

• Completing at least 50% on the written exam • The final grade will be determined based on

the result on the written exam, with the possibility of a bonus point for well done hand-ins

Schedule

Important dates

• Thursday, September 20: Deadline for hand-in problems, set I

• Tuesday, October 9: Seminar I

• Tuesday, October 9: Deadline for hand-in problems, set II

• Thursday, October 18: Deadline for hand-in problems, set III

• Thursday, October 25: Seminar II

• Thursday, November 1: Exam

Texts

Jan Conrad (nuclear)

David Milstead (particle)

Lars Bergström (astro)

Lecturers

Exercises and tutorials

Anthony Bonfils Franceso Torsello

Teachers

Special relativity Two postulates

(1) The laws of physics are the same in all inertial frames

of reference.

(2) The speed of light in free space has the same value in all

inertial frames of refrence.

c

Two simple but extremely powerful assumptions.

First postulate

What do we mean by the statement that the laws of physics are invariant to a change of inertial reference frame ?

,

, ,

.

0

A photon bounces between two mirrors

Separation=

Observer at rest wrt apparatus :

Observer measures the time for light to bounce:

Difference in start and end position:

Time f

A B

L

S x y z

t

A B A

x

2 2 2

2 2

2:

4

or

Define

LA B A t

c

s c t x

s L

x

Second postulate

12 2

2

22 2

2 2 2 22

2 2 2

12

22 1' 4 0

2

4

Observer moving to the left wrt apparatus with speed :

The quantity is a Lorentz invar

v

x v t

L xD

t c t L xc c

c t x L c t x

s c t x

iant

x

v

x v t

Second postulate

2 2

.

Postulating that the speed of light is constant in all

inertial frames

Instead of an invariant time or space interval there

is an invariant space-time : s c t x

x

ct

x

ct

Photon leaves A

tt

Photon returns to A

Minkowksi diagram

x

2 2 2 2

's c t x c t

22 2

12 2

1222 2

12 22 2

2

2

24

12

42'

'

1

If the speed of light is the same for each frame:

Time dilation.

Lt c t L

c

L v tL v tD

tc c c

c t v t tt t

c v

c

Time dilation

Lorentz transformations

2

2

, , ,

1

1

Boost in -direction.

If an observer in records an event at then an

observer in records the same event with coordinates:

=relative speed between frames in -direction.

x

S ct x y z

S

v xv

c

t

2 (time dilation) , (length contraction)

,

vxt x x vt

c

y y z z

Proper time

'

'

''

Consider frame moving at speed wrt a particle.

Frame is the particle rest frame

time interval in frame

time interval in ="proper" time

Time dilation

For infin

S v

S

t S

t S

tt t

t

'itesimal intervals

dt

dt

Nature’s speed limit

2 2 2 2 2

2

22 2 2

2

.

: , ' : ', '

0 0

0

' '

Consider the motion of the particle over a space-time interval

,

,

Particle speed in

N

s

S c t x S c t x

s c t x c t x

x c t

xc t x c

t

xS v c

t

o particle can move faster than light!

This violates causality.

The speed of light and causality

If information can be sent faster than light a message

can be sent to the past paradox!

Eg you can arrange for your father to be killed before he met

your mother. But then how would you be present

to send the message ?

Causality cannot be violated - the cause must precede the effect!

Can nature’s speed limit be beaten ?

Yes.

Eg dot from a laser pointer, a shadow...

Quantum mechanics allows it within the

Copenhagen interpretation.

However, it is impossible to send a message

faster than light (we think).× 𝟏𝟎𝟕 𝐦𝐬−𝟏

Space-time

2 2 2

2

2

0

0 ( )

Particle interacts (event) as it moves

through space-time.

Space-time between events:

( ) Time-like

Light-like

Interactions are causal.

Particle confined to t

s c t x

v x

c c t

s v c

s v c

he light cone.

worldline of a particle with v<c.

E

Light cone for past, present and future of an object (eg particle).

World-line follows particle's path.

2

0

Event , non-causal eg decay of another particle.

space-like

E

s

Four vectors

, , , ,

, .

, , , , , ,

Four vector:

Consider two four vectors:

;

Eg space-time of production of a particle in frame .

space-time of decay of a particle

A B

A A A A A B B B B B

A

B

X ct x y z ct r

X X

X ct x y z X ct x y z

X S

X

, , ,

, , ,

in frame

Four vectors from sum/subtraction:

C A B A B A B A B A B

D A B A B A B A B A B

S

X X X c t t x x y y z z

X X X c t t x x y y z z

Four vectors

2

, , , , , ,

Scalar product:

The scalar product is a lorentz invariant:

Use this property to derive earlier result:

A B A B A B A B A B

A B A BS S

A B A B

A B A B A B A B A B A B A B

X X c t t x x y y z z

X X X X

X X X X

c t t x x y y z z c t t x x y y z

2 2 2 2 2

invariant.

A Bz

c t x y z s

Four momentum

22

2

2

•

• , • ,

•

, 0

•

Invariant scalar product

Evaluate for object with four-momentum in an arbitrary frame

But can also evaluated for frame when object is at rest:

P P P

E E EP P p p p

c c c

P P

E mc p

P P m

2 2

22 2 2

2

2 22 2

,0 • ,0

c mc m c

Em c p

c

E mc pc

Velocity, momentum, energy

2

, , ,

, , ,

, , , ,

Energy invariant rest mass

Momentum

proper time ;

Four velocity

E mc m

p mv

dt

d

dX cdt dx dy dz

U

dX cdt dx dy dzU

d d d d d

dt dt dx dt dy dt dz dt dxU c c

dt d dt d dt d dt d dt

, , ,

, ,Four momentum

dy dzc v

dt dt

EP Um mc mv p

c

Lorentz transformations

2

- 0 0

- 0 0,

0 0 1 0

0

(time dilation) , (length contraction)

Write in matrix formulation

vxt t x x vt

c

v

ct c

x vX

cy

z

,

0 0

- 0 0

- 0 0

0 0 1 0

0 0 0 1

ct

xX

y

z

v

ct ctc

x v xX X

cy y

z

z

Lorentz transformations

- 0 0

- 0 0,

0 0 1 0

0 0 0 1

General result for 4-vectors, eg also for 4-momentum

x

y

z

vE

cc

vpP

cp

p

'

,

- 0 0

- 0 0

0 0 1 0

0 0 0 1

x

y

z

x x

y y

z z

E

c

pP

p

p

vE E

cc c

vp pP P

cp p

p p

Frames of reference

0

Two main frames used:

(1) Centre-of-mass frame i.e. total

Eg Process: ; particle or system of particles.

What is the centre-of-mass energy of the system ?

Rest mass of or system.

CM

p

A B X X

A B X

E

2 22 2 4

22 2 2 2 2 2 2 2 2 2

22

,

(2) Fixed target frame. Eg is at rest before the interaaction.

A B X X

A BCM A B A B A B A B A B

cP cP cP m c

E EE c P P c P P P P c m c m c p p

c

A B X B

Usefulness of working in a fixed frame Eg many different experiments measure the ratio of the processes:

(a) (b)

Some experiments are fixed target (electron beam on stationary

proton) and the others take place in th

R

e p e p n n e p e p

e centre-of-mass (colliding beams).

The beam energies are often different between the experiments.

To compare the experiments' results, plot their results as a function of

centre-of-mass energy.

Could similarly choose a frame in which the target proton is at rest

and plot the results as a function of pion beam energy. Just fix a frame!

R

ECM

R

Ee

centre-of-mass

experiments

Fixed target

experiments

Formulae

2

2

2

1

1

,

,

Lorentz transformation:

(time dilation) , (length contraction)

,

Space-time four vector:

Velocity four-vector:

Energy, momen

dt

dv

c

vxt t x x vt

c

y y z z

X ct r

U c v

2 22 2 2

,

tum:

; ;

Momentum four-vector

E mc E mc pc p mv

EP p

c

Units

19

3 6

9

15

1.602 10

1 10

Typical units used in this course.

Energy: 1 eV= J :

1 keV =10 eV (ionisation energy) 1 MeV=10 eV (nuclear binding energy) ,

1 GeV=10 GeV (accelerator energy)

Distance fm = m (s

2

ize of a proton)

eV eVMomentum units : Mass units : c c

Conversion

1923 1 27 1

8 1

234

27

1.602 10

3 10

9.36 10 93.6

:

Convert from MKS-SI units :

JeVEg Momentum 5 10 kgms = 0.534 10 kgmsms

5 10 eV keV = 0.534 10

Convert to MKS-SI units

GeVEg mass = 1.2

X X Xc

Xc c

c

9 1927

22 8

10 1.602 101.2 2.136 10

3 10

2 -2

J kg

m s

Natural units

222 2 2 2 2

1, 1

,

You may encounter natural units in some texts.

Convention

Eg

Counter intuitive eg frequency in units of GeV.

Conversion table:

c

E E E pc mc E p m

Quantity Natural units SI

Energy 1 GeV 1.602 × 10−10J

Mass 1 GeV 1.78 × 10−27kg

Momentum 1 GeV 5.34 × 10−19kgms-1

Time 1 GeV-1 6.58 × 10−25s

Frequency 1 GeV 1.52 × 1024s-1

Distance 1 GeV-1 1.97 × 10−16m

Summary

• Course information given

• Special relativity

– Two postulates

– Lorentz transformations

– Four vectors

• Units