Scattering Theory in Classical Mechanics: Lecture 3 Highlights
Explore the fundamentals of scattering theory in classical mechanics, covering topics such as center of mass reference frame, analytical evaluation of differential scattering cross section, and experimental setups like those at CERN. Understand the significance of cross sections and their application in various areas of physics. Dive into examples like collision of hard spheres and gain valuable insights into particle interactions.
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PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Online or (occasionally) in Olin 103 Plan for Lecture 3: -- Chap. 1 of F&W Scattering theory 1. Background and motivation 2. Center of mass reference fame and its relationship to laboratory reference frame 3. Next time -- Analytical evaluation of the differential scattering cross section in general and for Rutherford scattering 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 1
8/31/2020 PHY 711 Fall 2020 -- Lecture 3 2
8/31/2020 PHY 711 Fall 2020 -- Lecture 3 3
Scattering theory: detector 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 4
Can you think of examples of such an experimental setup? Other experimental designs At CERN https://home.cern/science/experiments/totem the study of highly energetic proton-proton scattering is designed in the center of mass frame of reference by accelerating two proton beams focused to collide head on in the Large Hadron Collider LHC facility. Figure from CERN website 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 5
Differenti d cross al section d Number of detected particles at per target particle = Number of incident particles per unit area Area = of scattered is that beam incident detector into angle at d bdb b d d d b db b db = = d d d sin sin Figure from Marion & Thorton, Classical Dynamics 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 6
Note: The notion of cross section is common to many areas of physics including classical mechanics, quantum mechanics, optics, etc. Only in the classical mechanics can we calculate it from a knowledge of the particle trajectory as it relates to the scattering geometry. d bdb b Figure from Marion & Thorton, Classical Dynamics d d d b db b db Note: We are assuming that the process is isotropic in = = d d d sin sin 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 7
Simple example collision of hard spheres = d d b db d sin ( ) b = ( ) = ? Microscopic view: sin b D 2 2 2 = d d D 4 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 8
Simple example collision of hard spheres -- continued Total scattering cross section: d d d = Hard sphere: 2 = d d D 4 = 2 D 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 9
Now consider the more general case of particle interactions and the corresponding scattering analysis. 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 10
Relationship of scattering cross-section to particle interactions -- Classical mechanics of a conservative 2-particle system. p p d d = = F F 1 2 12 21 1 dt dt 2 1 2 1 2 ( ) ( ) = = + + 2 2 F r r r r V E m 1 1 v m 2 2 v V 12 1 1 2 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 11
Typical two-particle interactions Central potential: ( ) ( ) ( ) = r r r r V V V r 1 2 1 a a 2 r r ( ) = Hard sphere: V r 0 K r ( ) = Coulomb or gravitational: V r A r B r ( ) = Lennard-Jones: V r 12 6 Scattering theory can help us analyze the interaction potential V(r). First we need to simply the number of variables. 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 12
Relationship between center of mass and laboratory frames of reference. At and time t, the following relationships apply -- R Definition of center of mass m m m + = + = CM ( ( ) ) + + 1 1 r r 2 2 r r R m 1 2 CM ( ) = + R V m m m m m m 1 1 2 2 1 2 1 2 CM CM r d dt r Note that 1 2 1 2 1 2 ( ) = + + 2 2 r r E m 1 1 v m 2 2 v V 1 2 1 2 ( ) ( ) 2 = + + + 2 v v r r m m V V 1 2 1 2 1 2 CM m m where: 1 + 2 m m 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 13
Classical mechanics of a conservative 2-particle system -- continued ( ) 1 2 2 2 1 1 ( ) 2 = + + + 2 v v r r E m m V V 1 2 1 2 CM ( ) ( ) ( ) r r r r For central potentials: = V V V 12 r 1 2 1 2 Relative angular momentum is also conserved: L ( ) 1 2 2 Simpler notation: 1 2 r v 12 12 1 2 12 2 12 r 1 L ( ) = + + + + 2 2 12 E m m V v V 12 r CM 2 2 12 2 1 2 ( ) ( ) r = + + + + 2 2 E m m V r V 1 2 CM 2 2 r 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 14
Simpler notation: 1 2 2 1 2 ( ) ( ) r = + + + + 2 2 E m m V r V 1 2 CM 2 2 r For scattering analysis only need to know trajectory before and after the collision. 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 15
Note: The following analysis will be carried out in the center of mass frame of reference. In laboratory frame: In center-of-mass frame: v1 V1 m1 r vCM origin m m + 1 target = mtarget m m 1 target = r 1 v Also note: We are assuming that the interaction between particle and target V(r) conserves energy and angular momentum. 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 16
It is often convenient to analyze the scattering cross section in the center of mass reference frame. Relationship between normal laboratory reference and center of mass: Laboratory reference frame: Before After m1 m2 v1 v2 u1 u2=0 Center of mass reference frame: Before After m1 m2 V1 U1 U2 V2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 17
Relationship between center of mass and laboratory frames of reference -- continued Since m initially is at rest : 2 m + m + m = = + = = V u u U V U u V 1 2 2 1 1 1 1 1 CM CM CM m m m m m 1 2 1 2 1 m + = + = = u U V U u V 1 2 2 2 1 CM CM m m 1 2 = + v V V + 1 1 CM V = v V 2 2 CM 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 18
Relationship between center of mass and laboratory frames of reference for the scattering particle 1 VCM V1 v1 = + = = v V V 1 1 CM + sin cos sin cos sin v v V V 1 1 + V 1 1 CM sin + = = tan For elastic scattering cos / cos / V V m m 1 1 2 CM 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 19
Digression elastic scattering ( ) + + + 2 2 2 2 m U m U m m V 1 1 1 1 1 2 m 1 2 CM m 2 Also note: 2 2 ( ) = + + + 2 2 2 V m V m V 1 1 1 1 1 2 2 1 2 CM 2 2 2 + = + = U U V V 0 0 m m m m 1 1 2 2 1 1 2 2 m = = U V U V 2 1 2 CM CM m 1 = = = U V U V V and 1 1 2 2 CM = U U Also note that : m m 1 2 1 2 = = So that : V /V V /U m /m 1 1 1 2 CM CM 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 20
Relationship between center of mass and laboratory frames of reference continued (elastic scattering) VCM V1 = + = = v V V v1 1 1 CM + sin cos sin cos sin v v V V 1 1 + V 1 1 CM sin + = = tan cos / cos / V V m m 1 1 2 CM + cos / m m = Also: cos 1 2 m ( ) 2 1 2 + + / cos / m m m 1 2 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 21
Differential cross sections in different reference frames ( ) ( ) LAB CM d d d d d d = CM LAB CM d LAB sin sin cos cos d d d d = = CM d LAB Using: + cos / m m = cos 1 2 ( ) ( ) 2 1 2 + + / cos / m m m m 1 2 m 1 2 ( ) + / cos 1 m cos cos d d = 1 2 ( ) 3/2 ( ) ( ) 2 1 2 + + / cos / m m m m 1 2 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 22
Differential cross sections in different reference frames continued: ( ) ( ) = d d cos cos d d LAB CM d d LAB CM ( ) 3/2 ( ) 2 1 2 + + ( ) ( ) / cos / m m m m = d d 1 2 1 + 2 LAB CM ( ) / cos 1 d d m m 1 2 LAB CM sin + = where: tan cos / m m 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 23
( ) 3/2 ( ) 2 1 2 + + ( ) ( ) / cos / m m m m = d d 1 2 1 + 2 LAB CM ( ) / cos 1 d d m m = 1 2 LAB CM sin where: tan + cos / m m 1 2 Example: suppose m1 = m2 sin = = In this case: tan + cos 1 2 note that 0 2 ) ( ) ( = 2 d d LAB CM d 4cos d LAB CM 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 24
Summary -- Differential cross sections in different reference frames continued: ( ) ( ) = d d cos cos d d LAB CM d d LAB CM ( ) 3/2 ( ) 2 1 2 + + ( ) ( ) / cos / m m m m = d d 1 2 1 + 2 LAB CM ( ) / cos 1 d d m m 1 2 LAB CM For elastic scattering sin + = where: tan cos / m m 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 25
Hard sphere example continued m1=m2 Center of mass frame Lab frame ( ) = = ( 2 ) d d D = 2cos LAB CM D 2 d 4 d LAB CM ( ) d ( ) d = d = lab d CM lab CM d d lab CM / 2 2 D 2 2 = 2 2 co s sin = D d D 4 D 4 0 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 26
Scattering cross section for hard sphere in lab frame for various mass ratios: m m = 1 1 2 m m = 0.8 1 ( ) d LAB 2 d m m LAB = 0.5 1 2 m m = 0 1 m m 2 = 0.1 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 27
For visualization, is convenient to make a "parametric" plot of d d ( LAB CM d d d d sin where: tan cos m + ( ) ( ) vs LAB ) 3/2 ( ) 2 1 2 + + ( ) ( ) / cos / m m m m = 1 2 1 + 2 ( ) / cos 1 m m 1 2 LAB CM = / m 1 2 Maple syntax: 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 28
For a continuous potential interaction in center of mass reference frame: E 2 1 2 ( ) r = + + 2 r V rel 2 2 r 2 + ( ) V r 2 2 r Need to relate these parameters to differential cross section d d ( ) V r CM CM Erel 2 2 r 2 m m rmin 1 + 2 =angular momentum m m 1 2 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 29
Focusing on the center of mass frame of reference: Typical two-particle interactions ( ) ( ) ( ) = r r r r Central potential: V V V r 1 2 1 D D 2 r r ( ) X = Hard sphere: V r 0 K r ( ) = Coulomb or gravitational: V r A r B r ( ) = Lennard-Jones: V r 12 6 8/31/2020 PHY 711 Fall 2020 -- Lecture 3 30
