{"id":326,"date":"2018-11-05T09:24:51","date_gmt":"2018-11-05T09:24:51","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=326"},"modified":"2019-04-29T06:04:49","modified_gmt":"2019-04-29T06:04:49","slug":"elastic-scattering-scattering-cross-section","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/elastic-scattering-scattering-cross-section\/","title":{"rendered":"Elastic Scattering : Scattering Cross-Section"},"content":{"raw":"
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  1. Introduction<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    Analysis of data obtained through scattering of particles on fixed or moving targets has emerged as one of the most powerful tools used by particle physicist to unravel the nature of fundamental interactions and the existence of hitherto unknown particles. Large Hadron Collider at CERN (GENEVA) is presently the most powerful machine which proved the existence of Higgs Bosons. Historically Rutherford\u2019s experiments on the scattering of \u03b1-particles on gold foil established the Atomic Model with a tiny nucleus where practically the entire mass of the atom is concentrated. Existence of the constituents of hadrons namely, quarks interacting through gluons which carry a new quantum number called \u2018colour\u2019 was discovered at the SLAC National Accelerator Laboratory in the electron-proton collision. In this and the next unit we will briefly discuss the kinematics of elastic collision and the concept of scattering cross-section.<\/p>\r\n\r\n

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    1. Elastic Scattering<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      We will consider elastic scattering between two particles of mass \u00a0and \u00a0respectively. By elastic scattering is meant that the internal energy of the particle if any remains unchanged. In elastic collision thus, the linear momentum and mechanical energy is conserved. In two body problems it is convenient to define the coordinates of the particle with respect to their centre of mass as follows:<\/p>\r\n \r\n\r\n\"\"\r\n\r\nThe coordinates of the particle with respect to\u00a0 M are given by\r\n\r\n \r\n\r\n\"\"\r\n

      If we define by \u00a0and \u00a0the initial velocities of the two particle with respect to the coordinates system fixed at O and \u00a0and \u00a0the velocities in the CM system, then from equation (12.1) and (12.2) by differentiation with respect to time, we get<\/p>\r\n \r\n\r\n\"\"\r\n

      where \u00a0is the velocity of the CM and \u00a0and \u00a0are the velocities of \u00a0and \u00a0in the CM frame.The Laboratory frame is defined as one in which the particle \u00a0strikes a stationary target . Thus in the Lab Frame:\u00a0is the velocity of the \u00a0and \u00a0and \u00a0are final velocities i.e. velocity after the scattering of \u00a0and \u00a0respectively.<\/p>\r\n \r\n

      CM Frame:<\/strong><\/p>\r\n \r\n

      In the CM frame \u00a0and \u00a0are the initial and \u00a0and \u00a0the final velocities of \u00a0and \u00a0respectively.In the centre of mass frame \u00a0which implies =0 i.e. the two particles have equal and opposite momentum in the CM frame.The scattering between the two particles in the Lab and CM frame would look like the following. In the Lab frame, the particle \u00a0moving with velocity strikes \u00a0at rest. After the scattering \u00a0and\u00a0m<\/em>2 have velocities \u00a0and , \u00a0scatters by an angle \u00a0with respect to the initial direction of motion and \u00a0scatters by an angle \u00a0as shown<\/p>\r\n \r\n\r\n\"\"\r\n\r\nScattering in the Lab Frame.<\/strong>\r\n\r\n \r\n

      In the CM frame, the particles \u00a0and \u00a0approach each other with equal momenta and after scattering go in opposite directions with equal momenta such that total initial and final momenta are zero.<\/p>\r\n \r\n\r\n\"\"\r\n\r\nScattering in the CM Frame<\/strong>\r\n\r\n \r\n

      The conservation of linear momentum and kinetic energy implies that the CM velocities of and \u00a0before and after the collision are equal<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

      Thus the scattering angle in the Lab and CM frames are approximately equal. In this case the particle \u00a0essentially scatters from a fixed scattering centre.<\/p>\r\n\"\"\r\n

      and the scattering angle in the Lab frame is equal to half the CM scattering angle. Further since the maximum value of the scattering angle \u00a0in the CM frame is 1800 , the scattering angle in the Lab frame for the case \u00a0can not be greater than 900 . There is no back scattering. Let us now look at the scattering of the second particle i.e. of the recoil particle , \u00a0scatters by an angle \u00a0which satisfies<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

      The energy expressions in the elastic scattering and the relationship between the expressions in the Lab and CM frame can be easily obtained. We will state the results and leave it as a problem to the done by you. The initial energy of the particles is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

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      1. Scattering Cross Section<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        In a scattering experiment when the article interactions are known in terms of a force-field, it is possible to predict the scattering outcome in terms of the final energies and scattering angles of the particles. We will now investigate the scattering process in the presence of force field. Let us consider the collision in the Laboratory system in the presence of repulsive force between the particles\u00a0 M1 and M2.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n

        Particle \u00a0approaches particle \u00a0such that if no force exists between them, the particle will pass through in a straight line with an impact parameter b. The impact parameter being defined as the distance of closest approach between the two particles. If the particle is moving with an initial velocity , the angular momentum of \u00a0about \u00a0is<\/p>\r\n\"\"\r\n

        Thus if the force field is known that is of the interaction between the particles is specified,then for a given initial energy and impact parameter, the scattering angle and the final energies are calculable. In actual practice we have a beam of particles of mass \u00a0and energy \u00a0moving towards a collection of particle of mass \u00a0each and which are at rest in the Lab. Frame<\/p>\r\n\"\"\r\n

        The negative sign indicates that the scattering angle decrease with increasing impact parameter.<\/p>\r\n\"\"\r\n\r\n \r\n

        We can write down the change in the scattering angle moving in a central field from the earlier expression obtained in unit 11 namely<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n4. Summary<\/strong>\r\n