lezione_4 - particle physics - uniud

8/12/2019 lezione_4 - Particle physics - uniud

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Scattering processes

Scattering experiments:- to study details of the interactions between particles- to obtain information about the internal structure ofatomic nuclei and their constituents

In a typical scattering experiment:- target object to be studied, bombarded with a beam

of particles with (mostly) well-defined energy- e.g.:

Target projectile reaction products

a + b c + d


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Targets and Beams

Target- Solid, liquid or gas- In collider experiments, another beam of particle mayserve as a target (e.g.: the e-p storage ring LEP atCERN in Geneva, the p-p storage ring Tevatron at FNALin the USA)

Beams:- Possible to produce beams of a broad variety ofparticles (e,p,n,heavy ions..).- Beam energies vary between 10-3 eV for cold neutronsup to 1012 eV for p


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Elastic Scattering

a + b a + b

- Same particles present before and after the scattering- Target b remains in its ground state, absorbing therecoil momentum and changing its kinetic energy

- Scattering angle and energy of a AND production angle

and energy of b unambigously correlated- To resolve small target structures, larger beamenergies required


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Largest wavelenght which can resolve structures oflinear extension x:

From Heisemberg s uncertainty principle, the

corresponding particle momentum is:

Nuclei (few fm radius) beam of 10-100 MeV/cNucleons (~0.8 fm radius) beam above 100 MeV/c

Quarks beam of many GeV/c

xD

x

fmMeV

x

cpcp

200,

hh


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Inelastic Scattering

a + b a + b*c+d

- Part of the kinetic energy transferred from a to thetarget b excites it into a higher energy state b*

- The excited state will return to the ground state byemitting a light particle OR it may decay into two ormore different particles

- A measurement of a reaction in which only thescattered particle a is observed is called exclusivemeasurement

- If all reactions products are detected inclusive

measurement


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The interaction cross-section

- The reaction rates measured, together with the energyspectra and angular distributions of the reaction productsgive information about the shape of the interaction

-Most important quantity for description and interpretationof these reactions is the cross-section (which gives theprobability of a reaction between 2 colliding particles)


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Geometrical cross-section

Consider an idealised experiment:d = thickness of scattering targetNb = scattering centre bnb = particle density

A = beam areaNr. Particles/

Unit area

na

Cross sectional area bdensity: nb

d

Each target particle has a cross-sectional area bTarget is bombarded with a monoenergetic beam of

point-like particles a


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- Reaction: whenever a particle hits a target particle(does not matter here whether eleastic or inelastic)

-Total reaction rate N = nr of reactions per unit time= difference in Na upstream and downstream the target

-Nr. of particles hitting target/ unit area / unit time =

(flux [area x time]-1)

-Nr. of target particles within the beam area =

-Reaction rate(if no overlap between scattering centres)

aaa

a vnA

N ==

AdnN bb=

bbaNN =


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-The area presented by a scattering centre to the incomingprojectile a is the geometric reaction cross-section

This definition assumes a homogeneous constant beam

(e.g. neutrons from a reactor)

centresscatteringxareaunitpertimeunitperparticlestimeunitperreactionofnr

NN

ba

b ==


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The interaction cross-section

- Cross-section = `area` denoted by [L2

]- Independent of the specific experimental design

- 1 barn = 1 b = 10-28 m2

1 millibarn = 1 mb = 10-31 m2

atom a20, a0 = 1 A nucleon a2N, aN 1fm

31 mb 3.1 b

-But typical total cross sections at a beam energy of 10 Ge

pp (10 GeV) ~40 mb

p (10 GeV) ~70 fb


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Cross Sections

- In high energy p-p scattering, the geometric extent ofthe particle is comparable to their interaction range

- However reaction probability for two particles can be very

different to what geometric considerations can imply.e.g.: a strong energy dependence is also observed

-Shape, strength and range of the interaction potential,primarily determine the effective cross-sectional area

areaunitpercentresscatteringxtimeunitperparticlesbeam

timeunitperreactionsofnrtot=

ineleltot +=


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Luminosity

-L

= a Nb is called luminosity. Dimension: [(areaxtime)-1

]L = a Nb = Na nb d = na va Nb

- Analogous equation for the case of two particle beamscolliding in a storage ring:

- Assume j particle packets, each of Na or Nb particlesinjected into a ring of circumference U.The two packet types travels with velocity v in opposite

direction. Steered by magnetic fields, they collide at aninteraction point jv/U times per unit time

L =

=

A

UvjNN ba /


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- For a Gaussian distribution of the beam particles aroundthe beam centres:

A = 4xy

With x and y horizontal and vertical standard deviations

-To achieve a high luminosity, beam must be focused atthe interaction point into the smallest possible ATypical beam diamaters: tenths of mm

- In storage ring the integrated luminosity is used

Nr. of reactions =

e.g.: = 1 nb, = 100 pb-1 105 reactions expected

L

L

L


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Differential Cross Section

- Only a fraction of all the reactions are measured.A detector of area AD is placed at a distance r and at anangle with respect to the beam direction

- Rate of reactions seen by detector:

ADr

=AD/r2

=d

EN )(E,d

),,(

L


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- If the detector can determine the energy E of thescattered particle the doubly differential cross section:

-The total cross section tot is given by:

AD

r

=AD/r2

'

),E'(E,d2

dEd

''

),',()(

'max

0 4

2

dEddEd

EEdE

E

tot =


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The Golden Rule

- Cross sections can be determined from reaction rate Nin experiment. And in theory?- N depends on the properties of the interaction potential

described by the Hamilton operator Hint.

Hint transforms the initial state wave-function i into thefinal-state wave function f.The transition matrix element is:

also called probability amplitude,includes coupling constants involved, propagator termsand any angular dependence of the reaction rate.

dVifif int*

intfi || HHM ==


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- Reaction rate N will depend also from final statesavailable to the reaction

- According to uncertainty principle, each particleoccupies a volume: in phase space(six-dimensional space of momentum and position)

- Consider a particle scattered into a volume V and into amomentum interval between pand p+dpIn momentum space: a spherical shell with inner radiusp and thickness dp V = 4p2dp

- Excluding processes where the spin changes, the nr offinal states available is:

33 )2( h=h

( ) '

2

'4)'(

3

2

dppV

pdnh

=


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- Energy and momentum of a particle are connected by:

- Hence, density of final states in the interval dE

- Connection between reaction rate, transition matrixelement and density of final state, expressed byFermis second golden rule:

( )3

2

2'

'4

'

)'()'(

h

==v

pV

dE

EdnE

''' dpvdE=

)'(2 2

EW if M

h=

W l d th t daN NN =


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- We saw already that: and

-Then

V = Na/na = spatial volume occupied by beam particles

-The cross section is:

-The Golden rule applies to both scattering andspectroscopic processes (e.g.: decay, excitation ofparticle resonances etc..)

- In this case W = 1/ (W can be derived from

lifetime or read off from the energy width )

aaa

a vnA

N ==

V

v

NN

ENW a

ab

==

)(

bbaNN =

VEv

ifa

)'(2 2

M

=h

/h=E


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Is the energy density dN/dE of final states, i.e., the numbeof states in phase space available to the product particles,

per unit interval of the total energy

W measures a rate per unit time

Neglecting any spin effect, the nr. of states in phase spacedirected into a solid angle d and enclosed in a physicalvolume V is:

Final states will contain the productFor each i we need to insert a normalization factorV and 1/V simplify. Same for the initial state

( )

= dpdpV

dN 23

2 h

dcf =

V

1

In the cms: dpMWWd2

12


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In the cms:

( )

dc

dcf

f

f

i

if

ii

EEE

p

dE

dpp

v

M

v

WW

d

d

+=

==

===

0

0

2

32

12

pp

hh

Conservation of energy gives:

Or:

Finally:

0

2222 Empmp dfcf =+++

ff

dcf

vpE

EE

dE

dp 1

00

==

fi

f

ifvv

pMdcba

d

d 2

2

424

1)(

h

=++

Spin


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Spin

uppose the initial-state particles are unpolarised.otal number of final spin substates available is:

gf = (2sc+1)(2sd+1)otal number of initial spin substates: gi = (2sa+1)(2sb+1)

ne has to average the transition probability over all possibl

nitial states, all equally probable, and sum over all final stateMultiply by factor gf /gi

All the so-called crossed reactions areallowed as well, and described by thesame matrix-elements (but differentkinematic constraints)

badc

dcba

bcda

dbca

dcba

++

++

++

++

++

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