PHYS 3446 – Lecture #5Scattering angles in Lab and CMDifferential cross sections in Lab and CMSpecial Relativity VariablesRelativistic VariablesRelativistic Variables-Special CasesSlide 7Slide 8Useful Invariant Scalar VariablesFeynman DiagramSlide 11Nuclear PhenomenologyNucleus LabelingTypes of NucleiSlide 15Slide 16Slide 17Slide 18Slide 19Slide 20Assignment 3 (Weds. Sep. 29)Wednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 1PHYS 3446 – Lecture #5Wednesday, Sep. 22 (+Fri 24th) 2010Dr. Brandt***Labs are due every 2 weeks at next lab 1st late =-10 2nd late=-50 no third late (Late buys you weekend)***Everyone should go to lab Friday 24 and Fri Oct. 1 as we will have a half lecture before each lab1. Relativistic Variables2. Invariant Scalars3. Feynman Diagram2• CM represents the change in the direction of the relative position vector r as a result of the collision in CM frame•Thus, it must be identical to the scattering angle for a particle with reduced mass, .•Z components of the velocities of scattered particle with m1 in lab and CM are:•The perpendicular components of the velocities are: (boost is only in the z direction)•Thus, the angles are related (for elastic scattering only) as: with Scattering angles in Lab and CMcosLab CMv vq - =sinLabv q =1sintancosCMLabCM CMv vqqq= =+%1 2sincosCMCMm mqq=+sincosCMCMqq z+1cosCMv q%1sinCMv q%1v =%2 11 2m vm m+1 11 2m vm m+CMv =(Later we use a version of this equation solving for cos)Wednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 3Differential cross sections in Lab and CM•The particles that scatter in lab at an angle Lab into solid angle dLab scatter at CM into solid angle dCM in CM.•Since is invariant, dLab = dCM.–Why?– is perpendicular to the direction of boost, thus is invariant.•Thus, the differential cross section becomes:( )sinLab Lab LabLabdddsq q q =W( )LabLabddsq =WreorganizeUsing Eq. 1.53( )LabLabddsq =W( )sinCM CM CMCMdddsq q qW( )CMCMddsqW( )( )coscosCMLabddqq( )CMCMddsqW( )3/ 221 2 cos1 cosCMCMz q zz q+ ++Wednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 4Special Relativity Variables •Fractional velocity •Lorentz factor•Relative momentum and the total energy of the particle moving at a velocity is•Square of four momentum P=(E/c,p), rest mass EP =rE =v cb =rrv cb=rr211gb=-Mvg =rM cg br2 T +Re2stE =2 2 2 4P c M c+ =2McgAppendix 1 reviewWednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 5Relativistic Variables•Velocity of CM in the scattering of two particles with rest mass m1 and m2 is: •If m1 is the mass of the projectile and m2 is that of the target, for a fixed target we obtainCMvc=rCMb =r12 2 2 4 21 1 2PcP c m c m c+ +rCMb =r( )1 21 2P P cE E++r r121 2PcE m c=+rWednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 6Relativistic Variables-Special Cases•At very low energies where m1c2>>P1c, the velocity reduces to:•At very high energies where m1c2<<P1c and m2c2<<P1c , the velocity can be written as:CMb =rCM CMb b= =r22 21 21 111m c m cPc Pc�� �+ +� �� �22 11 1112m c m cP P� �- -� �� �1 12 21 2m v cm c m c=+r( )1 11 2m vm m c+rExpansionWednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 7Relativistic Variables•For high energies, if m1~m2, • CM becomes:•In general, for fixed target we showed •Thus CM becomes211CMm cPb � -CMg =21CMb- =( )21/ 221 22 4 2 2 41 1 2 212CMCME m cm c E m c m cg b-+= - =+ +Invariant Scalar: s( )1/ 221CMb-- =( ) ( )1/ 21 1CM CMb b-� �- + �� �1 2212m cP-� �� �=� �� �� �� �� �122Pm c( )2 4 2 2 41 1 2 2221 22m c E m c m cE m c+ ++CMb =r121 2PcE m c+rWednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 8Relativistic Variables•The invariant scalar, s, is defined as:•So what is this in the CM frame?•Thus, represents the total available energy in the CM; At the LHC eventually (root-s is now 7 TeV)( )21 2s P P= + =2 4 2 4 21 2 1 22m c m c E m c= + +2 4 2 4 21 2 1 22s m c m c E m c= + +( )( )2221 2 1 2CM CM CM CME E P P c= + - +r r( )21 2CM CME E= +( )2CMToTE=s0( )( )2221 2 1 2E E P P c+ - +r rEvaluate in lab frame where p2=0 E2=m2c214TeVs =Wednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 9Useful Invariant Scalar Variables•Another invariant scalar, t, the momentum transfer (difference in four momenta), is useful for scattering:•Since momentum and total energy are conserved in all collisions, t can be expressed in terms of target variables•In CM frame for an elastic scattering, where PiCM=PfCM=PCM and EiCM=EfCM:( )2f it P P= - =( ) ( )2 222 2 2 2f i f it E E P P c= - - -r r( )2 2 22f i f iCM CM CM CMt P P P P c=- + - � =r r( )2 22 1 cos .CM CMP c q- -( ) ( )2 221 1 1 1f i f iE E P P c- - -r rWednesday, Sep. 22, 2010 PHYS 3446, Fall 2010 Andrew Brandt 10Feynman Diagram•The variable t is always negative for elastic scattering•The variable t could be viewed as the square of the invariant mass of a particle with and exchanged in the scattering•While the virtual particle cannot be detected in the scattering, the consequence of its exchange can be calculated and observed!!!–A virtual particle is a particle whose mass is less than the rest mass of an equivalent free particle2 2f iE E-2 2f iP P-r rTimet-channel diagramMomentum of the carrier is the difference between the two particles.Wednesday, Sep. 22, 2010PHYS 3446, Fall 2010 Andrew Brandt11Useful Invariant Scalar Variables•For convenience we define a variable q2,•In the lab frame, , thus we obtain:•In the non-relativistic limit:•q2 represents “hardness of the collision”. High q2 is more violent collision Small CM corresponds to small q2.• Divergence at q2~0 characteristic of a Coulomb field2 2q c t=-2 2q c =20iLabP =r( )2 22 2 22fLabm c E m c= -22 22fLabm c T=2fLabT �( ) ( )2 222 2 2f fLab LabE m c P c� �- - -� �� �22 212m v2ddqs=( )222 44 '1kZZ ev qpMonday, Sept. 15, 2008 PHYS 3446, Fall 2008Andrew Brandt12Nuclear Phenomenology•What did Rutherford scattering experiment do?–Demonstrated the existence of a positively charged central core in an atom–The formula did not quite work for high energy particles (E>25MeV), especially for low Z target nuclei.•In 1920’s, James Chadwick found –Serious discrepancies between Coulomb scattering expectation and the elastic scattering of
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