lezione_4 - particle physics - uniud
TRANSCRIPT
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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
++
++
++
++
++