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Atom, Nucleus, and Radiation Lec 2 of Intro Rad SciMarch

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Wave viewpoint Electromagnetic Radiation Wave viewpoint Changing B induces E Changing E induces B The inextricable exchange causes E and B fields to propagate outward at the speed of light c = 3 × 10 8 m/s in vacuum (courtesy Dr. Naqvi)

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Electromagnetic Spectrum keV- MeV ~ eV range

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Quantum viewpoint Electromagnetic Radiation Quantum viewpoint A photon is a “packet” or “quantum” of EM radiation The photon energy, h, is proportional to the frequency, and hence inversely proportional to the wavelength, A photon has zero rest mass (m 0 c 2 =0), and can therefore travel, and hence according to relativity, can travel only at the speed of light, c E = h h = Planck’s constant = 6.63 × J/Hz (courtesy Dr. Naqvi)

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Plum Pudding Model In mid-nineteenth century, optical spectroscopy –Balmer’s empirical formula Eq. (2.1) for visible spectra of H was derived theoretically by Bohr in 1913 –Eq. (2.1) was visible, and by replacing 2 2 by 1 2 or 3 2 (and 4 2, …) was ultraviolet or infrared, respectively J. J. Thomson in 1897 –Charge-to-mass ratio of cathode rays (only ~1/1700 of H) –Atom ~ plum pudding model –Ionization by radiation

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Rutherford Nuclear Atom In 1909, large-angle deflection of α-ptls (as probes) was evidence for the existence of a very small & massive nucleus of + charge –Planetary model with mostly empty space –Light e- move rapidly about the nucleus Nuclear force vs. Atomic force –Saturate within ~ m vs. not saturate –i.e., a given nucleon interacts with only a few others vs. all pairs of charges interact with one another –Radius of nucleus ≈ 1.3·A 1/3 × m vs. Atomic size of all elements is more or less the same (~ m)

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Bohr’s Theory of Hydrogen Atom An accelerated charge emits EM radiation, but Bohr’s theory –w/o radiating only in certain discrete orbits about the nucleus (2.3) –transition of e- from one orbit to another → emission or absorption of a photon of orbital energy lost or gained by e- (2.4) Some definitions –Bohr radius –Fine-structure constant (1/137) –Ionization potential –Rydberg constant, R M & R ∞

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Energy Levels of Hydrogen Atom Ionization Continuum n = 1 n = 2 n = 3 n = eV 0 eV Lyman Series (ultraviolet) Balmer Series (visible) The normal condition of the atom, or ground state, is the state with n = 1 The atom is in it’s lowest possible energy state and it’s most stable condition

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Problem with Bohr’s model & classical mechanics Only correct for the energy levels of H & He + Semi-classical mechanics, i.e., mixing classical mechanic w/ quantizing certain variables + relativistic models de Broglie’s wave/particle dualism –X-ray diffraction vs. Compton scattering –e- diffraction in Ni-crystal –Optical microscopy vs. electron microscopy (SEM, TEM)

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Quantum Mechanics Heisenberg’s uncertainty principle (matrix mechanics) Δp·Δx ≥ ħ & ΔE·Δt ≥ ħ in 1925 –Uncertainties in momentum of e - in atomic orbit ( m) and nucleus ( m) vs. position: a few hundred MeV vs. a few eV –But, betas from nuclei ~ a few MeV → neutron in 1932 Schroedinger’s wave mechanics, 2πr = nλ n = 1, 2, 3... –Linear differential equ (2 nd order in space & 1 st order in time) > superposition > wave packets > ptl –Boundary condition > eigenvalue > discrete energy –Dirac’s 1 st order in space & time for relativistic motion

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Atomic Structure Bohr vs. Modern Quantum Models

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Find the energy gained by an electron (in eV) when accelerated through a potential difference of 50 kV in an x-ray tube -+ Application: x-ray tube 50 kV (x-ray tube picture Courtesy of Hyperphysics) Cathode Anode

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Bremsstrahlung X-rays

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Characteristic X-rays

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Bremsstrahlung and Characteristic X-rays

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Nucleus of Atom, Assemblage of neutrons and protons clustered in a nucleus and surrounded by electrons whirling in a variety of orbits Atomic number, Z = No. of protons Mass number, A = No. of nucleons (protons, Z plus neutrons, N = A – Z)

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Isotopes, Isomers, Isobars Isotopes = elements having the same Z but different A, e.g., 131 I, 125 I, 127 I Isomers = identical elements, but different nuclear energy states, e.g., Isobars = elements having the same A but different Z Isotones = elements having the same N

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Nuclear Structure and Forces TUG-of-WAR between the ATTRACTIVE STRONG NUCLEAR FORCE and the REPULSIVE ELECTROMAGNETIC FORCE

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Binding Energy The nucleons (protons & neutrons) are bound together by a net force which NUCLEAR ATTRACTION forces exceed the ELECTROSTATIC (COULOMB) REPULSION forces. Associated with the net force is a POTENTIAL ENERGY of BINDING In order to separate the nucleus into its component nucleons, energy must be supplied from the outside Binding Energy (BE) = total mass of separate particles - mass of the atom

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Binding Energy

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Natural Radioactive Series

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Auger electron [1/3] Physical phenomenon in which the transition of an e - in an atom filling in an inner-shell vacancy causes the emission of another e - Releasing an energy equal to the difference in binding energies, E K – E LI. As the alternative to photon emission, this energy can be transferred to an L III e -, ejecting it from the atom with a K.E. = E K – E LI – E LIII Emission of an Auger e - increases the number of vacancies in the atomic shells by one unit Vacancy by P.E., internal conversion, PIXE, or orbital e - capture

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Auger electron [2/3] K fluorescence yield = No. of K X-ray photons emitted per vacancy in K shell Auger cascades can occur in relatively heavy atoms, as inner-shell vacancies are successively filled by the Auger process, with simultaneous ejection of more loosely bound atomic e - ’s An original, singly charged ion with one inner-shell vacancy can thus be converted into a highly charged ion by an Auger cascade Auger e - Yield

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Auger electron [3/3] 125 I decays by electron capture. The ensuing cascade can release some 20 e - s, depositing a large amount of energy (~1 keV) within a few nanometers A highly charged 125 Te ion is left behind; DNA strand breaks, chromatid aberrations, mutations, bacteriophage inactivation, and cell killing

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Gamma Emission vs. Internal Conversion Excited daughter nucleus decays to the stable nucleus via either -emission or internal conversion – -emission: isomeric (Z & A unchanged) discrete in -spectrum –Internal Conversion (IC): process in which the energy of an excited nuclear state is transferred to an atomic e -, most likely a K- or L-shell e - E e = E * -E B atomic inner-shell vacancies and thus emits characteristic X-ray isomeric (Z & A unchanged) dominant in heavy nuclei with low-lying excited state (small E * )

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Gamma Emission vs. Internal Conversion 10% IC E avg of emitting beta = 1/3 E max A long-lived excited nuclear state is termed metastable and is designated by the symbol m: e.g.,

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Orbital Electron Capture

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Radioactive Decay N, No of unstable nuclei left at time, t A, Activity (Bq or Ci) at time, t = decay constant [s -1 ] N 0 = initial No of unstable nuclei

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Relation between half-life and decay constant 2 -t/ e - t HALF-LIFE (T) REPRESENTATION DECAY CONSTANT ( ) REPRESENTATION is the time taken for 50% of the atoms to survive 1/ is the time taken for the fraction 1/e (37%) of the atoms to survive (i.e., mean-life time, ). T 1/2 = ln(2) / = /

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Point Source in Vacuum N [photons/s] (1/1 2 ) ∙ I 0 = N/A [photons/s/cm 2 ] (1/3) 2 ∙ I 0 = N/(3 2 A )[photons/s/cm 2 ] (1/2 2 ) ∙ I 0 = N/(2 2 A) [photons/s/cm 2 ]

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Photon Beam Attenuation = σ N = linear attenuation coefficient [cm -1 ] Source r [cm ] dr I(r)I(r) I + dI I 0 [photons/cm 2 ] N [atoms or electrons/cm 3 ] Collimator Detector 0

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Photon Fluence in Matter Point source in matter (collimated) Point source in matter (Broad)

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Relation between half-value layer and attn coefficient 2 -x/HVL e-xe-x HALF-VALUE LAYER (HVL) REPRESENTATION ATTENUATION COEFFICIENT ( ) REPRESENTATION HVL is the thickness taken for 50% of the photons to survive 1/ is the thickness taken for the fraction 1/e (37%) of the photons to survive (i.e., mean-free path, x m ). HVL = ln (2) /

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Linear vs. Semi-log Plotting of e -µx or e - t Mono-energetic photons, µ = constant and, thus HVL = constant LinearSemi-log

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Beam Hardening: Selective Absorption of Low-Energy Photons Absorber Thickness (mm AL) Transmittance (%) 1st HVL = 0.99mm 2nd HVL = 1.99 mm 3rd HVL = 2 mm 1 st HVL < 2 nd HVL < 3 rd HVL Ē 1st < Ē 2nd <Ē 3rd

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Parallels between nuclear decay and photon attenuation

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Parallels of Exponential Behavior Note: (i) The exponent of exp, e.g. D/D 0, µx, t, should be dimensionless

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e.g., λ, T 1/2, μ, and HVL Which of the following expressions is most appropriate? a)A =A 0 ·e -λ/t, I =I 0 ·e -μ·x b)A =A 0 ·2 -λ·t, I =I 0 ·2 -x/HVL c)A =A 0 ·e -t/T 1/2, I =I 0 ·2 -μ·x d)A =A 0 ·e -ln2·t/T 1/2, I =I 0 ·2 -ln2·x/HVL e)A =A 0 ·2 -t/T 1/2, I =I 0 ·e -ln2·x/HVL

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e.g., Mass and Atomic No. An atom of Zn-65 has a mass number of 65. A.Number of protons in the zinc atom 1) 302) 353) 65 B.Number of neutrons in the zinc atom 1) 302) 353) 65 C. What is the mass number of a zinc isotope with 37 neutrons? 1) 372) 653) 67

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H.W. #1 Calculate Q-value for 10 B(n,α) 7 Li reaction Calculate the atomic density of sodium, 0.97 g/cm 3 A sample contains 1.0 GBq of 90 Sr and 0.62 GBq of 90 Y. –What will be the total activity of the sample 10 days later? –What will be the total activity of the sample years later? A high-energy e - strikes a lead atom and eject one of K-e - ’s from the atom. –What wavelength radiation is emitted when an outer e - drops into the vacancy? –What is the probability for Auger e - emitted? Calculate the recoil energy of the technetium atom as a result of photon emission in the isomeric transition Find the binding energy of the nuclide 24 Na Turner Chap. 2: 11, 12, 13, 14, 15, 18, 19, 36, 37, 43, 53, 56 Turner Chap. 3: 3, 4, 8, 11, 17, 29

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