PHY 142! Assignment 11! Summer 2018
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- Hester Holland
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1 Reading: Modern Physics 1, 2 Key concepts: Bohr model of hydrogen; photoelectric effect; debroglie wavelength; uncertainty principle; nuclear decays; nuclear binding energy. 1.! Comment on these early assumptions about atomic structure, indicating which one(s) survived when quantum mechanics was discovered in the 1920 s. Electrons circulate about the nucleus in definite orbits. Radiation by an atom results from acceleration of the electrons. The frequency of radiation is the same as the frequency of revolution of an electron in its orbital motion. The energy carried off by radiation is equal to the loss of energy by the radiating atom. 2.! Comment on the following statements about the restrictions imposed by the uncertainty principle. In principle, it renders the program of Newtonian mechanics impossible. It forbids a system to have a precise value of the angular momentum. It poses no problem to the practical use of classical physics for systems of macroscopic size. It means that the Bohr model s use of precise orbits for electrons in atoms cannot be valid. 3.! Compton s analysis of the effect named for him assumed that the process is an elastic collision between an incoming X-ray photon and a free (conduction) electron on a metal surface, both treated as particles. Because of momentum conservation the electron must recoil after the collision, taking away some of the photon s energy with it. This loss of energy is what makes the frequency of the scattered photons lower, by the equation E = hf.! But many of the X-rays in Compton s experiment came off the metal with essentially unchanged frequency. Explain this in terms of collisions between photons and electrons that are bound tightly to their atoms, so that the whole atom recoils. [In an elastic collision between particles of different masses, how does the energy lost by the lighter one vary with the mass of the heavier one?]
2 4.! Some numerical comparisons. What is the energy of a visible photon of wavelength 500 nm, in ev? What is the wavelength of a 10 MeV γ -ray? When two molecules of hydrogen combine with one of oxygen to form two molecules of water, about 6 ev of energy is released as thermal energy. The rest energy mc 2 for a proton or neutron is about 1 GeV. Count up the number of protons and neutrons involved in the reaction and determine the fraction of the initial mass that is converted to energy. [The electron masses are negligible.] Fission of U produces about 200 MeV of released energy. What is the fraction of the initial mass converted to energy in this case? 5.! Our supply of helium gas on earth comes mainly from natural gas wells. The organic gases in those wells presumably originated in plant and animal materials on the earth s surface at some early time. But helium was never part of those materials. Where did the helium underground originate? 6.! Shown is a rough sketch of the data on binding energy per nucleon as a function of nucleon number. E b /A Use it to explain why nuclei more massive than Fe are not generated in normal stars, which get their energy from nuclear fusion A How can we explain the presence of the more massive atoms on earth?
3 7.! The same curve can be used to understand how we can obtain large amounts of usable energy from nuclear transformations. Explain how fusion of light nuclei into heavier ones releases energy. Explain how fission of very heavy nuclei into lighter ones releases energy. As an example of the former, consider deuterium ( 1 2 H ) and helium ( 2 4 He ). The binding energy per nucleon of 1 2 H is about 1.1 Mev; for 2 4 He it is about 7.1 Mev. Find the energy released in fusion of two deuterium nuclei into a helium nucleus. What fraction of the rest energy of the deuterium nuclei has been converted into energy? Take a nucleon to have rest energy 1 GeV. 8.! A bead of mass m is constrained to slide back and forth on a frictionless wire of length L, attached at each end to a rigid wall. The wire runs along the x-axis, and collisions with the walls are elastic. Write the kinetic energy of the bead in terms of its momentum p. Ans: E = p 2 /2m. Estimate the uncertainty Δx in the bead s position? Ans: Δx L. Using this for Δx, find the minimum value of Δp x. Ans: Δp x!/2l. For a cycle of the motion p x = 0 so the average of p 2 is equal to (Δp x ) 2. Use this to estimate the minimum average energy. Ans: E!2 8mL 2. 9.! A particle confined to a certain space must have a minimum amount of kinetic energy, so it must be in motion. In the lowest possible energy state, use p x = Δp x, use Δx Δp x!, and use! = (SI units) in the numerical calculations below. Consider a particle of mass m confined between x = 0 and x = L. What is its minimum average speed v in the lowest energy state? Evaluate this speed for a classical situation with m = 1 g and L = 10 cm. Evaluate the average speed of an electron ( m kg) confined to a space of atomic dimensions, L 1 nm.!
4 10.! According to the Bohr model of hydrogen, the electron in the nth stationary state is in a circular orbit around the proton with radius and velocity!! r n =! αmc n2, v n = αc n.! Here α = ke 2 /!c is a dimensionless number which is almost exactly 1/137. You are to use the Correspondence Principle to derive the formula for f 0. Find the frequency of radiation emitted in a transition from the (n + 1) th state to the nth state, in terms of the frequency f 0. Find the classical frequency of revolution of the electron in the nth state orbit. [Classical physics says this should the be frequency of the radiation.] Use the Correspondence Principle, setting these frequencies equal for very large n, to evaluate f 0, and show that E 1 = 1 2 α 2 mc ! Many materials used in buildings contain uranium, which decays into radium, among other products. Radium decays ( τ 2300 yr) into radon, which decays ( τ 6 d) into polonium emitting an α -particle. Radon, an inert gas, collects in closed spaces; if breathed in its decay inside the body can damage cells.! In such a closed space let there be N 1 radium atoms at t = 0. We seek N 2 (t), the number of of radon atoms at time t. Call the average lifetimesτ 1 and τ 2, respectively. Show that the rate of radon buildup is given by dn 2 /dt = N 1 /τ 1 N 2 /τ 2. Because τ for radium is so long, we can treat N 1 as essentially constant. With that assumption, show that the above equation, and the condition that N 2 = 0 at t = 0, are satisfied by N 2 (t) = τ 2 τ 1 N 1 (1 e t /τ 2 ). What is the amount of radon in the space after time long compared to τ 2?
5 12.! The isotope 14 6C ( τ = 8270 yr) is formed in the upper atmosphere by interactions with cosmic rays. Like the stable isotope 12 6C, it combines with oxygen to form CO 2. The known small fraction f 0 of 14 6C in atmospheric carbon remains approximately constant over time. Plants use both forms of CO 2 equally, so 14 6 C constitutes a fraction f 0 of the carbon in living plants. When plant material is used in non-living forms (wood for furniture, etc.), replenishment by the atmosphere of decayed 14 6 C in the material stops.! You take a sample of pure carbon from a piece of wood which stopped growing T years ago. It contains N atoms, of which N 14 are 14 6C. You measure the decay rate R of these atoms, i.e., R = dn 14 /dt. Express N 14 in terms of τ and R. Express N 14 in terms of f 0, τ, N and the age T of the wood. Express T in terms of τ, N, f 0 and R. Ans: T = τ ln f 0 N τr.! [Knowing N and measuring R, one determines T. This is carbon dating.] 13.! We introduced τ as the average lifetime of a decaying object, for which the number remaining at time t is N(t) = N 0 e t/τ. An alternate measure of lifetime is the half-life, which is the time it takes for half the sample to decay. First, show that τ really is the average time that a member of the sample lives. The probability of the object being alive at time t is N(t)/N 0 = e t/τ, so the probability of its decay between t and t + dt is proportional to this: P(t)dt = Ae t/τ dt. The total probability of decay at any time (between 0 and ) must be 1. Set P(t)dt = 1 and determine A. 0 The average time at which the object decays will be the average of t over this probability distribution: t = t P(t)dt. Show that this gives t = τ. 0 Now relate this to the half-life. The half-life τ 1 2 is the time at which half of the objects have decayed. Show that τ 1 2 = ln 2 τ τ.
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