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modern physicis

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  • Light and its nature have caused a lot of ink to flow during these last decades. Its dual behavior is partly explained by (1)Double-slit experiment of Thomas Young - who represents the photon’s motion as a wave - and also by (2)the Photoelectric effect in which the photon is considered as a particle. A Revolution: SALEH THEORY solves this ambiguity and this difficulty presenting a three-dimensional trajectory for the photon's motion and a new formula to calculate its energy. More information on
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  1. 1. AP Phys B Test Review Modern Physics 5/9/2008
  2. 2. Overview  Basics  Photoelectric Effect  Bohr Model of the atom • Energy Transitions  Nuclear Physics
  3. 3. Basics  Quantization: the idea that light and matter come in discreet, indivisible packets • Wave-particle duality in light and matter • Matter behaves both as a wave and as a particle.
  4. 4. Energy of a photon  Blackbody radiation • Ultraviolet catastrophe • Planck came up with the idea that light is emitted by certain discreet resonators that emit energy packets called photons • This energy is given by: E h= ν
  5. 5. Photoelectric Effect Schematic  When light strikes E, photoelectrons are emitted  Electrons collected at C and passing through the ammeter are a current in the circuit  C is maintained at a positive potential by the power supply
  6. 6. Photoelectric Current/Voltage Graph  The current increases with intensity, but reaches a saturation level for large ΔV’s  No current flows for voltages less than or equal to –ΔVs, the stopping potential • The stopping potential is independent of the radiation intensity
  7. 7. Features Not Explained by Classical Physics/Wave Theory  No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated  The maximum kinetic energy of the photoelectrons is independent of the light intensity  The maximum kinetic energy of the photoelectrons increases with increasing light frequency  Electrons are emitted from the surface almost instantaneously, even at low intensities
  8. 8. Einstein’s Explanation  A tiny packet of light energy, called a photon, would be emitted when a quantized oscillator jumped from one energy level to the next lower one • Extended Planck’s idea of quantization to electromagnetic radiation  The photon’s energy would be E = hƒ  Each photon can give all its energy to an electron in the metal  The maximum kinetic energy of the liberated photoelectron is KE = hƒ – Φ
  9. 9. Explanation of Classical “Problems”  The effect is not observed below a certain cutoff frequency since the photon energy must be greater than or equal to the work function • Without this, electrons are not emitted, regardless of the intensity of the light  The maximum KE depends only on the frequency and the work function, not on the intensity  The maximum KE increases with increasing frequency  The effect is instantaneous since there is a one-to- one interaction between the photon and the electron
  10. 10. Verification of Einstein’s Theory  Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory  The x-intercept is the cutoff frequency cf h Φ =
  11. 11. 27.4 X-Rays  Electromagnetic radiation with short wavelengths • Wavelengths less than for ultraviolet • Wavelengths are typically about 0.1 nm • X-rays have the ability to penetrate most materials with relative ease  Discovered and named by Roentgen in 1895
  12. 12. Production of X-rays  X-rays are produced when high-speed electrons are suddenly slowed down • Can be caused by the electron striking a metal target  A current in the filament causes electrons to be emitted
  13. 13. Production of X-rays  An electron passes near a target nucleus  The electron is deflected from its path by its attraction to the nucleus  It will emit electromagnetic radiation when it is accelerated
  14. 14. 27.8 Photons and Electromagnetic Waves  Light has a dual nature. It exhibits both wave and particle characteristics • Applies to all electromagnetic radiation  The photoelectric effect and Compton scattering offer evidence for the particle nature of light • When light and matter interact, light behaves as if it were composed of particles  Interference and diffraction offer evidence of the wave nature of light
  15. 15. 28.9 Wave Properties of Particles  In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties  The de Broglie wavelength of a particle is  The frequency of matter waves is mv h =λ h E =ƒ
  16. 16. The Davisson-Germer Experiment  They scattered low-energy electrons from a nickel target  The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength  This confirmed the wave nature of electrons  Other experimenters have confirmed the wave nature of other particles
  17. 17. 27.10 The Wave Function  In 1926 Schrödinger proposed a wave equation that describes the manner in which matter waves change in space and time  Schrödinger’s wave equation is a key element in quantum mechanics  Schrödinger’s wave equation is generally solved for the wave function, Ψ i H t ∆Ψ = Ψ ∆
  18. 18. The Wave Function  The wave function depends on the particle’s position and the time  The value of |Ψ|2 at some location at a given time is proportional to the probability of finding the particle at that location at that time
  19. 19. 27.11 The Uncertainty Principle  When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements • Classical mechanics offers no fundamental barrier to ultimate refinements in measurements • Classical mechanics would allow for measurements with arbitrarily small uncertainties
  20. 20. The Uncertainty Principle  Quantum mechanics predicts that a barrier to measurements with ultimately small uncertainties does exist  In 1927 Heisenberg introduced the uncertainty principle • If a measurement of position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δp, then the product of the two uncertainties can never be smaller than h/4π
  21. 21. The Uncertainty Principle  Mathematically,  It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a particle  Another form of the principle deals with energy and time: π ≥∆∆ 4 h px x π ≥∆∆ 4 h tE
  22. 22. Early Models of the Atom  Rutherford’s model • Planetary model • Based on results of thin foil experiments • Positive charge is concentrated in the center of the atom, called the nucleus • Electrons orbit the nucleus like planets orbit the sun
  23. 23. Experimental tests Expect: 1. Mostly small angle scattering 2. No backward scattering events Results: 1. Mostly small scattering events 2. Several backward scatterings!!!
  24. 24. Difficulties with the Rutherford Model  Atoms emit certain discrete characteristic frequencies of electromagnetic radiation • The Rutherford model is unable to explain this phenomena  Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency • The radius should steadily decrease as this radiation is given off • The electron should eventually spiral into the nucleus
  25. 25. 28.2 Emission Spectra  A gas at low pressure has a voltage applied to it  When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed • Each line has a different wavelength and color
  26. 26. Emission Spectrum of Hydrogen  The wavelengths of hydrogen’s spectral lines can be found from • RH is the Rydberg constant • RH = 1.0973732 x 107 m-1 • n is an integer, n = 1, 2, 3, … • The spectral lines correspond to different values of n  A.k.a. Balmer series       −= λ 22H n 1 2 1 R 1
  27. 27. Absorption Spectra  An element can also absorb light at specific wavelengths  An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas  The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum • The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum
  28. 28. 28.3 The Bohr Theory of Hydrogen  In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory  His model includes both classical and non- classical ideas  His model included an attempt to explain why the atom was stable
  29. 29. Bohr’s Assumptions for Hydrogen  The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction  Only certain electron orbits are stable • These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation • Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion  Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state • The “jump” cannot be treated classically i fE E hf− =
  30. 30. Bohr’s Assumptions  More on the electron’s “jump”: • The frequency emitted in the “jump” is related to the change in the atom’s energy • It is generally not the same as the frequency of the electron’s orbital motion  The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital i fE E hf− = , 1,2,3,... 2 e h m vr n n π   = = ÷  
  31. 31. Results  The total energy of the atom •  Newton’s law  This can be used to rewrite kinetic energy as 2 21 2 e e e E KE PE m v k r = + = − r2 ek E 2 e −= 2 2 2e e e e v F m a or k m r r = = 2 2 2 2 e mv e KE k r ≡ =
  32. 32. Bohr Radius  The radii of the Bohr orbits are quantized • This shows that the electron can only exist in certain allowed orbits determined by the integer n •When n = 1, the orbit has the smallest radius, called the Bohr radius, ao •ao = 0.0529 nm   ,3,2,1n ekm n r 2 ee 22 n == 2 h π=
  33. 33. Radii and Energy of Orbits  A general expression for the radius of any orbit in a hydrogen atom is • rn = n2 ao  The energy of any orbit is • En = - 13.6 eV/ n2  The lowest energy state is called the ground state • This corresponds to n = 1 • Energy is –13.6 eV  The next energy level has an energy of – 3.40 eV  The ionization energy is the energy needed to completely remove the electron from the atom
  34. 34. Energy Level Diagram  The value of RH from Bohr’s analysis is in excellent agreement with the experimental value  A more generalized equation can be used to find the wavelengths of any spectral lines • For the Balmer series, nf = 2 • For the Lyman series, nf = 1  Whenever a transition occurs between a state, ni and another state, nf (where ni > nf), a photon is emitted       −= λ 2 i 2 f H n 1 n 1 R 1
  35. 35. Quantum Number Summary  The values of n can increase from 1 in integer steps  The values of ℓ can range from 0 to n-1 in integer steps  The values of mℓ can range from -ℓ to ℓ in integer steps
  36. 36. Atomic Transitions – Energy Levels  An atom may have many possible energy levels  At ordinary temperatures, most of the atoms in a sample are in the ground state  Only photons with energies corresponding to differences between energy levels can be absorbed
  37. 37. Atomic Transitions – Stimulated Absorption  The blue dots represent electrons  When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level • These higher levels are called excited states • ΔE = hƒ = E2 – E1
  38. 38. Atomic Transitions – Spontaneous Emission  Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon  This process is called spontaneous emission
  39. 39. Atomic Transitions – Stimulated Emission  An atom is in an excited stated and a photon is incident on it  The incoming photon increases the probability that the excited atom will return to the ground state  There are two emitted photons, the incident one and the emitted one
  40. 40. 29.1 Some Properties of Nuclei  All nuclei are composed of protons and neutrons • Exception is ordinary hydrogen with just a proton  The atomic number, Z, equals the number of protons in the nucleus  The neutron number, N, is the number of neutrons in the nucleus  The mass number, A, is the number of nucleons in the nucleus • A = Z + N • Nucleon is a generic term used to refer to either a proton or a neutron • The mass number is not the same as the mass
  41. 41. Charge and mass Charge:  The electron has a single negative charge, -e (e = 1.60217733 x 10-19 C)  The proton has a single positive charge, +e • Thus, charge of a nucleus is equal to Ze  The neutron has no charge • Makes it difficult to detect Mass:  It is convenient to use atomic mass units, u, to express masses • 1 u = 1.660559 x 10-27 kg  Mass can also be expressed in MeV/c2 • 1 u = 931.494 MeV/c2
  42. 42. The Size of the Nucleus  First investigated by Rutherford in scattering experiments  The KE of the particle must be completely converted to PE 2 2 4 ek Ze d mv = ( ) ( )2 1 2 21 2 e e e Zeq q mv k k r d = = or
  43. 43. Size of Nucleus  Since the time of Rutherford, many other experiments have concluded the following • Most nuclei are approximately spherical 3 1 oArr =
  44. 44. Density of Nuclei  The volume of the nucleus (assumed to be spherical) is directly proportional to the total number of nucleons  This suggests that all nuclei have nearly the same density  Nucleons combine to form a nucleus as though they were tightly packed spheres
  45. 45. Nuclear Stability  There are very large repulsive electrostatic forces between protons • These forces should cause the nucleus to fly apart  The nuclei are stable because of the presence of another, short-range force, called the nuclear (or strong) force • This is an attractive force that acts between all nuclear particles • The nuclear attractive force is stronger than the Coulomb repulsive force at the short ranges within the nucleus
  46. 46. Nuclear Stability chart  Light nuclei are most stable if N = Z  Heavy nuclei are most stable when N > Z • As the number of protons increase, the Coulomb force increases and so more nucleons are needed to keep the nucleus stable  No nuclei are stable when Z > 83
  47. 47. Isotopes  The nuclei of all atoms of a particular element must contain the same number of protons  They may contain varying numbers of neutrons • Isotopes of an element have the same Z but differing N and A values C11 6 C14 6C13 6C12 6
  48. 48. 29.2 Binding Energy  The total energy of the bound system (the nucleus) is less than the combined energy of the separated nucleons • This difference in energy is called the binding energy of the nucleus • It can be thought of as the amount of energy Binding Energy per NucleonBinding Energy per Nucleon
  49. 49. Binding Energy Notes  Except for light nuclei, the binding energy is about 8 MeV per nucleon  The curve peaks in the vicinity of A = 60 • Nuclei with mass numbers greater than or less than 60 are not as strongly bound as those near the middle of the periodic table  The curve is slowly varying at A > 40 • This suggests that the nuclear force saturates • A particular nucleon can interact with only a limited number of other nucleons
  50. 50. 29.3 Radioactivity  Radioactivity is the spontaneous emission of radiation  Experiments suggested that radioactivity was the result of the decay, or disintegration, of unstable nuclei  Three types of radiation can be emitted • Alpha particles • The particles are 4 He nuclei • Beta particles • The particles are either electrons or positrons
  51. 51. Distinguishing Types of Radiation  The gamma particles carry no charge  The alpha particles are deflected upward  The beta particles are deflected downward • A positron would be deflected upward
  52. 52. Penetrating Ability of Particles  Alpha particles • Barely penetrate a piece of paper  Beta particles • Can penetrate a few mm of aluminum  Gamma rays • Can penetrate several cm of lead
  53. 53. The Decay Constant  The number of particles that decay in a given time is proportional to the total number of particles in a radioactive sample • λ is called the decay constant and determines the rate at which the material will decay  The decay rate or activity, R, of a sample is defined as the number of decays per second N R N t λ ∆ = = ∆ ( )N N tλ∆ = − ∆
  54. 54. Decay Curve  The decay curve follows the equation  The half-life is also a useful parameter λ = λ = 693.02ln T 21 0 t N N e λ− =
  55. 55. Units  The unit of activity, R, is the Curie, Ci • 1 Ci = 3.7 x 1010 decays/second  The SI unit of activity is the Becquerel, Bq • 1 Bq = 1 decay / second • Therefore, 1 Ci = 3.7 x 1010 Bq  The most commonly used units of activity are the mCi and the µCi
  56. 56. Alpha Decay  When a nucleus emits an alpha particle it loses two protons and two neutrons • N decreases by 2 • Z decreases by 2 • A decreases by 4 HeYX 4 2 4A 2Z A Z +→ − −
  57. 57. Beta Decay  During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is one less  In addition, an electron (positron) was observed  The emission of the electron is from the nucleus • The nucleus contains protons and neutrons • The process occurs when a neutron is transformed into a proton and an electron • Energy must be conserved
  58. 58. Beta Decay – Electron Energy  The energy released in the decay process should almost all go to kinetic energy of the electron  Experiments showed that few electrons had this amount of kinetic energy  To account for this “missing” energy, in 1930 Pauli proposed the existence of another particle  Enrico Fermi later named this particle the neutrino  Properties of the neutrino • Zero electrical charge • Mass much smaller than the electron, probably not zero • Spin of ½ • Very weak interaction with matter
  59. 59. Gamma Decay  Gamma rays are given off when an excited nucleus “falls” to a lower energy state • Similar to the process of electron “jumps” to lower energy states and giving off photons  The excited nuclear states result from “jumps” made by a proton or neutron  The excited nuclear states may be the result of violent collision or more likely of an alpha or beta emission  Example of a decay sequence • The first decay is a beta emission • The second step is a gamma emission γ+→ ν++→ − C*C e*CB 12 6 12 6 12 6 12 5