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Transcript
Modern Physics (PHY 251) Lecture 19 Joanna Kiryluk Fall Semester Lectures Department of Physics and Astronomy, Stony Brook University • Models of an Atom Reminder Models of the Atom Ernest Rutherford – “the father of nuclear physics” § 1911: Rutherford proposed the existence of a massive nucleus as a small central part of an atom J.J. Thompson’s Plum Pudding Model (1904) Rutherford’s Planetary Model (1911) 1 Reminder Rutherford experiment § Rutherford designed an experiment to use a particles as atomic bullets (then new technology). § The “gold-foil experiment” was performed in 1909 by Hans Geiger and Ernest Marsden (under Rutherford’s direction) 2 Reminder Rutherford scattering - Differential Cross Section Here a nucleus = point like particle (i.e. no structure) dN dσ = Nα ⋅ n, dΩ dΩ 2 2 &2 # & # dσ 1 zZe 1 =% × ( ×% ( dΩ $ 4πε 0 ' $ 2Mv 2 ' sin 4 (θ 2 ) Coulomb interaction dA ( r sin θ ) × dΩ = 2 = r dΩ = 2π sin θ dθ In the spherical coordinate system 3 Reminder Result: Rutherford model of Atom Rutherford type of experiment has been applied with higher energy beams to study structure of nuclei and nucleons 4 Probing Nuclei Structure with High Energy Electron Beams Elastic scattering: e− + A p → e− + pA R. Hofstadter, Department of Physics, Stanford University, Stanford, CA, "Electron Scattering and Nuclear Structure", Rev. Mod. Phys. 28, 214–254 (1956) Robert Hofstadter Nobel Prize 1961 Electrons are point like (elementary), sensitive to charge only, can be accelerated to high energies. http://www.nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf 5 Form Factors and Charge Density Robert Hofstadter Results of electron scattering off various nuclei charge density distributions. * " dσ % " dσ % 2 2 $ ' =$ ' ⋅ F (q ) # dΩ &exp # dΩ &Mott Charge distribution (nuclei or nucleon) ρ ( x ) = Ze ⋅ f ( x ) and its Fourier transform F (q ) = ∫e iq⋅ x/ http://www.nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf f ( x) d 3x 6 Even higher electron beams in scattering experiments revealed that nucleons (protons, neutrons) are not point-like: they consist of point like particles (quarks, and gluons)! Nucleon Structure nucleon Small distances …. 9 How? Scattering experiment (Rutherford, again …) Use high energy beam of particles to look inside the nucleon Extra Even higher electron beams in scattering experiments revealed that nucleons are not point-like: they consist of point like particles (quarks, and gluons)! Parameterization of the measured lepton-nucleon cross section by ”structure functions” / “form factors” and learning about nucleon structure from these structure functions. Nobel Prize 1990 Extra Even higher electron beams in scattering experiments revealed that nucleons are not point-like: they consist of point like particles (quarks, and gluons)! Form factors, Extra Extra Number of quarks: e+e- scattering http://hyperphysics.phy-astr.gsu.edu/hbase/particles/qevid.html Extra Standard Model of Fundamental Interactions (Elementary Particle Physics) electron u u d quarks proton Hydrogen atom Extra Next step: Electron-Ion Collider (EIC) Research done at Stony Brook U. EIC R&D prof. A. Deshpande, prof. T. Hemmick (Exp. Nuclear Physics) https://www.bnl.gov/rhic/eic.asp 16 Models of the Atom Ernest Rutherford – “the father of nuclear physics” § 1911: Rutherford proposed the existence of a massive nucleus as a small central part of an atom J.J. Thompson’s Plum Pudding Model (1904) Rutherford’s Planetary Model (1911) 13 The Problem of Stability of Atoms Size of the atoms Continuous spectrum of radiation (will come back to this later) 14 De’Broglie to rescue: Source: C. Rogers 15 De’Broglie to rescue: Source: C. Rogers 16 De’Broglie to rescue: A. B. C. D. E. 1 5 10 20 Cannot determine from picture Source: C. Rogers 17 De’Broglie to rescue: Source: C. Rogers 18 Next Source: C. Rogers 19 The Problem of Stability of Atoms Size of the atoms Continuous spectrum of radiation Atomic Spectra Atomic Spectra A single element makes emission and absorption lines at the same wavelength Spectral lines are fingerprints of atoms (astronomy students: Kirchhoff’s laws) Extra material A single element makes emission and absorption lines at the same wavelength Spectral lines are fingerprints of atoms 28 Atomic Spectra Source: Electric discharge passing through a monoatomic gas, for example hydrogen Atomic Spectra Emission spectrum l (wavelength) e.g. hydrogen spectrum Electromagnetic radiation emitted by free atoms is concentrated at a number of discrete wavelengths. Application: spectroscopy Atomic Spectra e.g. hydrogen spectrum Electromagnetic radiation emitted by free atoms is concentrated at a number of discrete wavelengths. Balmer (1885) Hydrogen: o series limit: λ = 3646 A κ = 1 λ = RH (1 2 2 −1 n 2 ) reciprocal wavelength RH = Rydberg constant for hydrogen (H) Hα : n = 3 Hβ : n = 4 Hγ : n = 5 etc n = 3, 4, 5,.... n2 " o % λ = 3646 2 A' $ n − 4# & http://en.wikipedia.org/wiki/Physical_constant Atomic Spectra RH = κ = 1 λ = RH (1 m 2 −1 n 2 ) n = m +1, m + 2,.... Atomic Spectra For alkali element atoms (e.g. Li, Na, K) " 1 1 % ' κ = 1 λ = R$ − 2 2 $# ( m − a ) ( n − b) '& For the particular series: m − fixed, n − variable a, b − constants 33 Rydberg constant for the particular element Next we will show how Bohr’s model of the atom explains discrete emmission/absorption lines § Frequency of the electromagnetic radiation emitted when the electron makes a transition from the quantum state ni to the quantum state nf Bohr’s Postulates Bohr § postulated that classical radiation theory did not hold for atomic systems § applied Planck’s ideas of quantized energy levels to orbiting electrons and postulated that: electrons in atoms are confined to stable, non-radiating energy levels and orbits (stationary states) § applied Einstein’s concept of the photon to arrive at an expressions for the frequency of the light emitted when electron jumps from one stationary state (i) to another (f) ΔE = Ei − E f = hν § postulated that the electron orbital momentum is quantized L = n Bohr’s Model of the Atom § Justification of Bohr’s postulates: comparison with experimental observations § Assumptions: Single electron atom of charge Ze , me << M § Stability of the electron (classical) 1 Ze 2 v2 F= × 2 = me = ma 4πε 0 r r § Orbital angular momentum Centripetal acceleration keeps the electron in its circular orbit "π % L = L = r × p = mervsin $ ' = merv #2& L = merv=n e #2& e L = merv=n 1 Ze 2 v2 F= × 2 = me = ma 4πε 0 r r centripetal acceleration Optional (classical mechanics) F = ma a) planar motion φ dθφ • ω= =φ θ dt φ φ d 2φ θ •• α = 2 = θφ dt Bohr’s Model of the Atom § Bohr orbits 2 4 πε 2 0 r ≡ rn = n 2 × ≡ n a0 2 me Ze Ze 2 1 v ≡ vn = × 4πε 0 n where n =1,2,3, …. – quantum number Z = 1, n = 1: c ≈ 197eV ⋅ nm = 197MeV ⋅ fm e2 1 α em ≡ ≈ 4πε 0 c 137 o r1 ≡ a0 ~ 0.053nm = 0.53 A v1 ~ 2.2 ×10 6 m / s ~ 0.01c non-relativistic Bohr’s Model of the Atom § Total energy of an electron moving in one of the orbits E = Ekin + E pot ∞ Ze 2 Ze 2 E pot = − ∫ dr = − 2 4πε 0 r r 4πε 0 r (E pot #r→∞ ## → 0) me Z 2 e 4 1 E ≡ En = − × 2 2 2 ( 4πε0 ) 2 n Quantization of the orbital angular momentum of the electron leads to a quantization of its total energy Hydrogen atom 13.6 Z = 1 ⇒ En = − 2 eV n Bohr’s Model of the Atom § Electron energy levels 13.6 Z = 1 ⇒ En = − 2 eV n Excited states: n >1 http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html Math Reminder Spherical coordinate system r ∈ [0, ∞) φ ∈ [0, 2π], θ ∈ [0, π] Transformations between the cartesian and spherical coordinates: Math Spherical coordinates, solid angle Reminder A Ω≡ 2 r W is the same for these three surfaces: The differential solid angle: dA ( r sin θ ) × ( rdθ ) × dφ dΩ = 2 = r r2 dΩ = sin θ dθ dφ rsinq r rdq (rsinq)df Math Spherical coordinates, solid angle Reminder A Ω≡ 2 r dΩ dΩ r dΩ The differential solid angle: dA ( r sin θ ) × ( rdθ ) × 2π azimuthally symmetric case dΩ = 2 = integrated over the azimuthal angle f r r2 dΩ = 2π sin θ dθ