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Basic Theoretical Physics: A Concise Overview P6

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Basic Theoretical Physics: A Concise Overview P6. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 42 7 The Rutherford Scattering Cross-section where we note the fact that a solid-angle element in spherical coordinates can be written as dH 2n sin 0d0 . The differential cross-section is defined as the ratio dTT d 2 b 0 db 0 dH sin 0d0 . This expression is often complicated but its meaning can be visualized as follows. Consider a stream of particles with current density j0 per crosssectional area flowing towards the target and being scattered by the potential V. At a large distance beyond the target a fraction of the particles enters a counter where they are recorded. The number of counts in a time At is given by AN 2 jo At AH. dH The aperture of the counter corresponds to scattering angles in the interval 0 0 d0 i.e. to the corresponding solid angle element AH 2n sin 0A0 . The differential scattering cross-section is essentially the missing proportionality factor in the relation AN x j0 At AH and 7.1 should only be used for evaluation of this quantity. The description of these relations is supported by Fig. 7.1. For A r-potentials the differential cross-section can be evaluated exactly with the result d 2 A2 1 dH 16E2 sin4 2 which is called the Rutherford scattering cross-section. This result was obtained by Rutherford in Cambridge U.K. at the beginning of the twentieth century. At the same time he was able to confirm this formula motivated by his ground-breaking scintillation experiments with o-particles. In this way he discovered that atoms consist of a negatively-charged electron shell with a radius of the order of 10 8 cm and a much smaller positively-charged nucleus with a radius of the order of 10 _ 13 cm. In fact the differential cross-sections for atomic nuclei are of the order of 10 _26 cm2 i.e. for o-particles the space between the nuclei is almost empty. -20-15-10-5 0 5 10 15 Fig. 7.1. Schematic diagram on differential scattering cross-sections. A particle enters the diagram from the left on a path parallel to the x-axis at a perpendicular distance b .