Scattering Experiment Monoenergetic particle beam Beam impinges on a target Particles are scattered...
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Transcript of Scattering Experiment Monoenergetic particle beam Beam impinges on a target Particles are scattered...
Manuel Calderón de la Barca Sánchez
Intro. Relativistic Heavy Ion Collisions
Cross Sections and Collision Geometry
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The cross section: Experimental Meaning
Scattering Experiment
Monoenergetic particle beam
Beam impinges on a target
Particles are scattered by target
Final state particles are observed by detector at q.
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Beam characteristics: Flux
Flux :Number of particles/ unit area / unit timeArea: perpendicular to beam
For a uniform beam: particle density
Number of particles / unit volume
Consider box in Figure.Box has cross sectional area a.Particles move at speed v with respect to target.Make length of box
a particle entering left face just manages to cross right face in time Dt.
Volume of Box:
So Flux
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Target: Number of Scattering Centers
How many targets are illuminated by the beam?
Multiple nuclear targets within area a
Target Density,Number of targets per kg:
Recall: 1 mol of a nuclear species A will weigh A grams. i.e. the atomic mass unit and Avogadro’s number are inverses:(NA x u) = 1 g/mol
So:
L
Area a
Density r
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Incident Flux and Scattering Rate
Scattered rate: Proportional to
Incident Flux, Nt
size (and position) of detector
For a perfect detector :
Constant of proportionality:Dimensional analysis:Must have units of Area. Cross Section
L
Area a
Density r
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Differential Scattering cross section
For a detector subtending solid angle dW
If the detector is at an angle q from the beam, with the origin at the target location:
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Physical Meaning of stot.Compute:
Fraction of particles that are scatteredArea a contains Nt scattering centers
Total number of incident particles (per unit time)
Ni=Fa
Total number of scattered particles (per unit time)
Ns=F Nt stot
So Fraction of particles scattered is:Ns/Ni =F Nt stot / (F a) = Nt stot / a
Cross section: effective area of scatteringLorentz invariant: it is the same in CM or Lab.
For colliders, Luminosity:Rate:
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Interaction Cross Section: Theory
Quantum Mechanics: Fermi’s (2nd) Golden Rule
Calculation of transition ratesIn simplest form: QM perturbation theory
Golden Rule: particles from an initial state a scatter to a final state b due to an interaction Hamiltonian Hint with a rate given by:
sdNL
dt
Quantum Case: Yukawa PotentialQuantum theory of interaction between nucleons
1949 Nobel Prize
Limit m → ∞.Treat scattering of particle as interaction with static potential.
Interaction is spin dependentFirst, simple case: spin-0 boson exchange
Klein-Gordon Equation
Static case (time-independent):
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Observables: From theory to experiment
Steps to calculating and observable:Amplitude: f = Probability ~ |f|2 .
Example:Non-Relativistic quantum mechanicsAssume a is small.
Perturbative expansion in powers of a.
Problem: Find the amplitude for a particle in state with momentum qi to be scattered to final state with momentum qf by a potential Hint(x)=V(x).
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Propagator: Origins of QFT.
q = momentum transferq = qi - qf
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Structure of propagator
QFT case, recover similar form of propagator!Applies to single particle exchange
Lowest order in perturbation theory.Additional orders: additional powers of a.
Numerator:product of the couplings at each vertex.
g2, or a.
Denominator:Mass of exchanged particle.Momentum transfer squared: q2.
In relativistic case: 4-momentum transfer squared qmqm=q2.
Plug into Fermi’s 2nd Golden Rule:Obtain cross sections
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Cross Section in Nuclear Collisions
Nuclear forces are short rangeRange for Yukawa Potential R~1/Mx
Exchanged particles are pions: R~1/(140 MeV)~1.4 fm
Nuclei interact when their edges are ~ 1fm apart0th Order: Hard sphere
Bradt & Peters formula b decreases with increasing Amin
J.P. Vary’s formula: Last term: curvature effects on nuclear surfaces
R2R1
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Cross Sections at Bevalac
So:
Bevalac DataFixed TargetBeam: ~few Gev/A AGS, SPS: works too
Bonus question:Intercept: 7mb½
What is r0?Hints: 1 b = 100 fm2, √0.1=0.316, √π=1.772
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Colliders: Van der Meer Scan
Vernier ScanInvented by S. van der Meer
Sweep the beams across each other, monitor the counting rateObtain a Gaussian curve, peak at smallest displacementDoing horizontal and vertical sweeps:
zero-in on maximum rate at zero displacement
Luminosity for two beams with Gaussian profile
1,2 : blue, yellow beamNi: number of particles per bunch
Assumes all bunches have equal intensity
Exponential: Applies when beams are displaced by d
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RHIC Results: BBC X-section
van der Meer Scan. A. Drees et al., Conf.Proc. C030512 (2003) 1688
Cross Section:
STAR:
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Total and Elastic Cross Sections
World Data on pp total and elastic cross sectionPDG: http://pdg.arsip.lipi.go.id/2009/hadronic-xsections/hadron.html
RHIC, 200 GeV
tot~50 mb
el~8 mb
nsd=42 mb
LHC, 7 TeV
tot=98.3±2.8 mb
el=24.8±1.2 mb
nsd=73.5 +1.8 – 1.3 mb (TOTEM, Europhys.Lett. 96 (2011) 21002)
CERN-HERA Parameterization
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Important Facts on Cross Sections
Froissart Bound, Phys. Rev. 123, 1053–1057 (1961)
Marcel Froissart: Unitarity, Analiticityrequire the strong interaction cross sections to grow at most as for
Particles and AntiparticlesCross sections converge for
Simple relation between pion-nucleon and nucleon-nucleon cross sections
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Nuclear Cross Sections: Glauber Model
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Nuclear Charge Densities
Charge densities: similar to a hard sphere.Edge is “fuzzy”.
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For the Pb nucleus (used at LHC)
Woods-Saxon density: R = 1.07 fm * A 1/3
a =0.54 fmA = 208 nucleons
Probability :
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Nuclei: A bunch of nucleons
Each nucleon is distributed with:
Angular probabilities:Flat in f, flat in cos(q).
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Impact parameter distribution
Like hitting a target:
Rings have more area
Area of ring of radius b, thickness db:
Area proportional to probability
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Collision:2 Nuclei colliding
Red: nucleons from nucleus A
Blue: nucleons from nucleus B
M.L.Miller, et al. Annu. Rev. Nucl. Part. Sci. 2007.57:205-243
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Interaction Probability vs. Impact Parameter, b
After 100,000 events
Beyond b~2R Nuclei miss each other
Note fuzzy edge
Largest probability:Collision at b~12-14 fm
Head on collisions:b~0: Small probability
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Binary Collisions, Number of participants
If two nucleons get closer than d< / s p they collide.Each colliding nucleon is a “participant” (Dark colors)Count number of binary collisions.Count number of participants
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Find Npart, Ncoll, b distributions
Nuclear Collisions
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From Glauber to Measurements
Multiplicty Distributions in STAR
MCBS, Ph.D ThesisPhys.Rev.Lett. 87 (2001) 112303
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Comparing to Experimental data:CMS example
Each nucleon-nucleon collision produces particles.
Particle production: negative binomial distribution.
Particles can be measured: tracks, energy in a detector.CMS: Energy deposited by Hadrons in “Forward” region
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Centrality Table in CMS
From CMS MC Glauber model. CMS: HIN-10-001,
JHEP 08 (2011) 141