Spring 2012, P627, YK Friday, January 27, 2012

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Leo Nathaniel Walker
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1 7 Spring 1, P67, YK Friday, January 7, 1

2 Topic Software Talk date Kubra PC+VM Rachel Dark Matter PC+VM Joel Mac+? Chris PC+VM Kyle Linux +VM Possible project talk dates: pril 18,,3,5,7 Projects: Proton decay, Muon g,, Sterile neutrinos, Hidden photons, axions, WISP (sub ev), Dark matter detection, bar with spallation target, eutron and atomic EDM, Measurement of CPV phase in oscillations, Collider experiment,

3 Links for Root and Geant4 on Windows OS Required for both: us/products/1 editions/visual cpp express Root: version 53 (VC++ 1 MSI Installer) Geant4: Set environment variables: System >dvanced >Environment Variables (Just point to unzipped folder of data set) Officially Supported Platforms: Free Micro$oft programs for.edu students: In case of further questions to Eric Flumerfelt [eflumerf@utk.edu]

4 umber of essential kinematical parameters of the reaction. n particles For generic process with participating n particles: n 4-vectors = 4 n numbers - n masses Ei - pi = mi (if masses are known) - 4 energy and momentum conservation eqs. - 3 arbitrary xyz,, shifts in 3D momentum space - 3 arbitrary rotations in 3D momentum space # = 4n-n = 3n-1 E.g. n = 3 (1 decay) no free kinematical variables (everything is determined by masses) If n = 4 (13 decay, or scattering) # = (e.g. energy of incoming particle in Lab + scatt. angle) If n= 5 (e.g. n+ p p+ + p) # = 5 and reaction needs to be analyzed in 5-D space p -

5 The image part with relationship ID rid11 was not found in the file. Transformation of distribution functions. If f( p) -some practical experimental "distribution function" in 3D space 3 d = f ( p) dp dp dp = f ( p) d p x y z # of particles in d p of momentum space f( p ) will be transformed by Lorentz transformation d remains invariant when p and f ( p) are transformed to another coordinate system 3 See Landau - II 1 for the prove of invariance of d p E d invarinat = f ( p) E must be invariant 3 d p E invariant 3 f ( p) d -invariant dp E f ( p ) = f ( p) when p = L( v) p E Lorentz transform

6 inv 3 d p dv p p dp dw p E de dw p de dw = = = = = E E E c E c d = p + m c p dp= by differentiating E E E c c 3 3 Phase-space distribution: d = f ( r, p) d p d x when f ( r, p ) = L( v) f( r, p) invariant Summary of Lorentz transformation invariants: invariant see LL-II 1 invariant 3 d p E f( p) E p de dw f ( r, p)

7 Cross section in potential scattering (like Coulomb field) symmetric trajectory radial turning point Scattering angle in CMS is determined: (a) by particular potential (interact. dynamics) (b) particle starting velocity at ( ) (c) by relative position of particles in collision (impact parameter) c = p -j Potential U(r) is provided by the scattering center located at point O (at the center of inertia of two particles) rmin=o can be determined from radicand = Ur () M + = mr E infinite motion; see LL1 (14.7) E mv = T = > ; (conserved) M = mrv m - reduced mass r - impact parameter

8 Scattering of beam of particles on the scattering center d particles with v n Beam with area scattering center U(r) d surface density particle in the beam; [ n] d dc = dc n dsc = n pr dr dc 1 cm é cc+ cù «érr+ drù ë û ë û is monotonic one-value function of :, d, drc ( ) dsc = pr dr = pr( c) dc dc dsc dsc or dc called effective differential scattering cross section at angle c (per interval d c) in CMS

9 The image part with relationship ID rid7 was not found in the file. Let one particle be incident on the target with surface density of scattering centers n n = n l V s = p( r + r ) 1 Probability for 1 particle to interact is n (if n is small) then the beam of particles is incident on a target (assume that transversally target is larger than the beam area) int = sn For particles in the beam: D = - sn d d =- snvdx =-snvdx n - is volume density V of scattering centers

10 d =-sn dx V l ln s V =- n x =-sn ln =-sn = e -sn passed target with no interactions -sn -sn int = - = - e (1 e )» (1-1 + sn) = sn int Similar for x section for some particular final state or scattering at angle in interval d

11 Differential and total x-sections ds Lab Lab ç 3 çèmv cos q ø Lab æ a ö dw = ç ds æ a ö = dw ç çèmv 3 Lab ø Lab cos 1 q differential cross section d d d d d de d d de Can total X section be? total cross section

F O R SOCI AL WORK RESE ARCH

7 TH EUROPE AN CONFERENCE F O R SOCI AL WORK RESE ARCH C h a l l e n g e s i n s o c i a l w o r k r e s e a r c h c o n f l i c t s, b a r r i e r s a n d p o s s i b i l i t i e s i n r e l a t i o n