Introductory Nuclear Physics

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PHY431 Lecture 2 1 2/6/2000 2. Measuring the Nuclear Charge Distribution The nuclear charge distribution tells us where in the nucleus the protons live, at least on average. The density of protons gives us information about the strong force. Several methods exist to study the internal structure of the atomic nucleus, and indeed of the protons and neutrons that make up the nucleus themselves. With 15 MeV α -particles scattering off a Z=13 Aluminum foil, one notices a departure from the classical Rutherford scattering formula at large scattering angles; indeed, alpha particles of high energies penetrate the nucleus at small impact parameters. With studies of this type the nuclear boundary can be measured for many elements. A second type of measurement is with high-energy electrons as projectiles . Of course, electrons at energies of a MeV or higher are quite relativistic ( E e > m e ). In that case, the classical Rutherford derivation is no longer valid: at relativistic speeds the spin of the electron plays a role in the scattering as well. The description of the process of electron-pointlike-nucleus scattering is being done using the Dirac equation. The resulting cross section formula is the Mott formula : 222 2 2 42 2 22 2 2 cosec 1 sin 1 sin 2 2 Mott Rutherford dz Z v d v dv p c d c σα θ σ  =− = ×  ΩΩ  (1.1) where p is the incident momentum of the projectile, and v its velocity. Note that for v << c the two formulae are equivalent. The formula can be re-written as function of the momentum-transfer squared variable q 2 , where the q = p ’- p , whence q = | q | = 2 p sin / 2 . We then use (integrating over the azimuthal variable φ ): 2 2 2 4s i n s i nc o s 2 2 sin s i o s 2 2 pd dq p ddd d θθ π == = (1.2) Then, using sin / 2 = q /(2 p ): 4 2 2 2 2 2 2 4 2 4 2 . 16 4 11 1 44 4 4 Mott Ruth d d d z Z p vq d zZ dq dq d p v p q c p v q c p dq c p σσ α =×= = = (1.3) Electrons can be accelerated easily to high energies, and are, as far as we know, truly pointlike particles, ideal probes for the study of nuclear charge distributions. A third method for study of the nuclear matter is the use of stopping negative muons . Muons are copiously produced in cosmic ray interactions in the upper atmosphere. High-energy solar and galactic protons impinging on molecules in the upper atmosphere produce showers of hadrons, mostly pions and Kaons. Both pions and Kaons are short-lived and decay ultimately in muons, electrons and neutrinos. Although muons have a mean life of only 2.1 µ s, because they move with relativistic speeds, they profit from time dilation and are able to reach sea level in great numbers ( 100 m -2 s -1 ). By capturing negative muons in materials, the muon first looses energy in electromagnetic collisions and eventually slows down enough to get captured in atomic orbits. Because of its much larger mass than the mass of the electron, m µ = 105 MeV 200 m e , the muon energy levels ( E n =( 1 / 2 n 2 ) Z 2 α 2 mc 2 ) are in the X-ray regime, and the (most probable) orbital radii ( r n = n 2 ħ c / Z α mc 2 ) are much smaller than electron orbits. Thus, the radius of the lowest orbit in a
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This note was uploaded on 02/05/2008 for the course PHY 431 taught by Professor Rijssenbeek during the Spring '01 term at SUNY Stony Brook.

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Lecture 02 - PHY431 Lecture 2 1 2/6/2000 2. Measuring the...

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