8/31/2018. PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103

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1 PHY 7 Classical Mechanics and Mathematical Methods 0-0:50 AM MWF Olin 03 Plan for Lecture :. Brief comment on quiz. Particle interactions 3. Notion of center of mass reference fame 4. Introduction to scattering theory PHY 7 Fall Lecture PHY 7 Fall Lecture PHY 7 Fall Lecture 3

2 Schedule additional office hours by office: Olin 300 8/7/08 PHY 7 Fall Lecture 4 Comment on quiz questions t gt () x tdx t dg d x t 0 dt x dt dt. 0 0 t dt t t t t t xt i i i dz e id Suppose that z e dz e id i i z 0 e 3. df f f( x) Ae x f( x) A dx N N N n a a n N 4. a Let S a Note that as S a a a n n PHY 7 Fall Lecture 5 Scattering theory: detector PHY 7 Fall Lecture 6

3 Example: Diagram of Rutherford scattering experiment PHY 7 Fall Lecture 7 Differential cross section d Number of detected particles at per target particle d Number of incident particles per unit area Area of incident beam that is scattered into detector at angle b dbdb d d b db b db d d sin d sin d Figure from Marion & Thorton, Classical Dynamics PHY 7 Fall Lecture 8 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. b dbdb Figure from Marion & Thorton, Classical Dynamics d d b db b db d d sin d sin d Note: We are assuming that the process is isotropic in PHY 7 Fall Lecture 9 3

4 Simple example collision of hard spheres d b db d sin d Microscopic view: b b? Dsin d D d 4 PHY 7 Fall Lecture 0 Simple example collision of hard spheres -- continued Total scattering cross section: d d d Hard sphere: d D d 4 D PHY 7 Fall Lecture Relationship of scattering cross-section to particle interactions -- Classical mechanics of a conservative -particle system. dp F dt dp F dt F V r r r r E mv m v V PHY 7 Fall Lecture 4

5 Typical two-particle interactions r r r r Central potential: V V V r r a Hard sphere: Vr 0 r a K Coulomb or gravitational: Vr r A B Lennard-Jones: Vr 6 r r PHY 7 Fall Lecture 3 Relationship between center of mass and laboratory frames of reference Definition of center of mass R mr m r m m R PHY 7 Fall Lecture 4 mr m r m m R m m V E mv mvv r r mm where: m m m m V v v V r r Classical mechanics of a conservative -particle system -- continued E m mv vv V rr For central potentials: V r r = V r r V r Relative angular momentum is also conserved: L r v L E m m V v V r Simpler notation: r E m V V r m r r PHY 7 Fall Lecture 5 5

6 Note: The following analysis will be carried out in the center of mass frame of reference. In laboratory frame: In center-of-mass frame: V m mm μ m m r μv target target m target v r origin v Also note: We are assuming that the interaction between particle and target V(r) conserves energy and angular momentum. PHY 7 Fall Lecture 6 For a continuous potential interaction in center of mass reference frame: Erel r V r ( r) r V r V(r) Need to relate these parameters to differential cross section to be discussed on Friday E rel r min PHY 7 Fall Lecture 7 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 m m u u =0 Center of mass reference frame: Before m m PHY 7 Fall Lecture 8 v v U U V V After 6

7 Relationship between center of mass and laboratory frames of reference -- continued Since m is initially at rest : m V u u U V m m v v V V V V u U V m U m m m U u m m m u V m V PHY 7 Fall Lecture 9 Relationship between center of mass and laboratory frames of reference V V v v V V v sin V sin v cos V cos V sin tan cos V / V sin cos m / m PHY 7 Fall Lecture 0 For elastic scattering Digression elastic scattering m U Also note: m U m U m U m U V m U So that : V Also note that : m V 0 V and U m U /V V m m V m m V PHY 7 Fall Lecture m V V U m V m V V /U m /m m U V 0 7

8 Relationship between center of mass and laboratory frames of reference continued (elastic scattering) V v V V v sin V sin v cos V cos V sin tan cos V / V V v sin cos m / m Also : cos cos m / m m / m cos m / m PHY 7 Fall Lecture Differential cross sections in different reference frames d d d d d d d sin d d cos d sin d d cos Using : cos d cos d cos cos m / m m / m cos m / m m / m cos m / m cos m / m 3/ PHY 7 Fall Lecture 3 Differential cross sections in different reference frames continued: d d d d d d d cos d cos d m / m cos m / m d m / m cos 3/ where : sin tan cos m m / PHY 7 Fall Lecture 4 8

9 d d where : d m / m cos m / m d m / m cos sin tan cos m m Example: suppose m = m sin In this case : tan cos note that 0 d d / d d PHY 7 Fall Lecture 5 4 cos 3/ 9