Examination Radiation Physics - 8N0, November 0-4.00-7.00 Four general remarks: This exam consists of 6 assignments on a total of pages. There is a table on page listing the maximum number of that can be achieved in each assignment. You are not allowed to use books, notes, lecture notes, notebooks, telephones, tablets etc during the exam. It is allowed to use a ordinary calculator. You can use the appended list of constants and formulas during the exam. All answers should be formulated and motivated clearly.. (a) What is the energy in electron-volt of a photon with a wavelengths of 550 nm? (b) How many photons are emitted each second by a 5 watt LED lamp, that emits only photons of 550 nm? You may assume that all the energy used by the lamp is transferred to the photons.. Consider the Compton effect where an incoming photon with an energy of.0 MeV interacts with a free electron. The trajectory of the scattered photon has an angle θ = 80 with respect to the trajectory of the incoming photon. (a) Compute the wavelength of the incoming photon. (b) Compute the wavelength and energy of the scattered photon. Next we consider the Compton effect for an incoming photon with an energy of 0.0 MeV, that is again scattered by an angle θ = 80 with respect to the trajectory of the incoming photon. (c) Compute the wavelength of the incoming photon. (d) Compute the wavelength and energy of the scattered photon. The effect where an incoming photon is scattered over an angle of (approximately) 80 is called Compton backscattering. (e) What is in the case of Compton backscattering the maximal energy of the scattered photon? Hint: Look at incoming photons with very high energy.. Suppose an electron is trapped in a region ( the hydrogen atom ) of width x = a 0, where a 0 = 0.059 nm is the Bohr radius. (a) What is the uncertainty of the velocity in the x direction of this electron? (b) What is in the Bohr model the velocity of an electron in the ground state of hydrogen? Hint: For a circular orbit mv = E, where E is the energy of the electron.
4. In this exercise we consider a one-dimensional particle with mass m that can only move along the positive x-axis (so 0 < x < ), in a potential U(x) = q x, where q is a positive constant. This potential occurs for instance if the particle has an electric charge and is attracted by another electric charge that is fixed in x = 0. (a) Give the Schrödinger equation for this case. Given is that a solution of the Schrödinger equation for this case is of the form Ψ(x) = A x e ax. (b) Compute the first and second derivative of this wave function. (c) Substitute Ψ(x) and the second derivative of Ψ(x) in the Schrödinger equation. By considering the coefficients of e ax and x e ax we obtain two equations, from which a and E can be found. Give these two equations. (d) Solve these two equations and give the corresponding energy value E and the corresponding wave function Ψ(x). (e) The constant A is not yet known. Which condition for the wave function Ψ(x) can be used to find A? (f) Compute the value of A. You may use that 0 x n e bx dx = n!/b n+, for integer n 0 and b > 0. 5. Consider the α emission A ZX A 4 Z Y + α. (a) Give the Q-value of this expression, expressed in nuclear masses. Use A ˆM Z for the nuclear mass and A ZM for the corresponding atomic mass. (b) Show how this Q-value can be rewritten in terms of atomic masses. (c) Assuming that the mother X has no kinetic energy before the reaction and that all of the decay energy is distributed between the α-particle and daughter nucleus Y, derive the formulas that gives the distribution of Q between α-particle and daughter. Give the actual kinetic energies of the α-particle and of the daughter nucleus. 6. Consider the part of the nuclides chart in Figure. (a) Give the three possible beta decay reaction from 98 Au and compute the Q-values for these three forms of beta decay. (b) Which of these three beta decays are energetically possible? (c) Following the information on the chart of nuclides, only one of these beta decay reactions is observed experimentally. Which reaction is that? (d) Explain which combinations of successive decay processes on the nuclides chart corresponds approximately with this beta decay reaction. (Note : use the right part of the 98 Au entry in the chart; the left part describes excited states of 98 Au. Note : on the nuclides chart the energies of alpha and beta decay are indicted in MeV, the energies of gamma decays are in kev.)
Figuur : part of the nuclides chart for assignment 6 Points: (total 60) a: b: 5a: 5b: 5c: 4a: 4b: 4c: 4d: 4e: 4f: point point a: b: c: d: e: 6 a: b: 4 4 6a: 6b: 6c: 6d: point
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Radiation Physics - 8N0 Formulas and constants - 0/0 physical constants: h = 6.66 0 4 Js (Planck s constant) k B =.8 0 J/K (Boltzmann s constant) c =.998 0 8 m/s (speed of light) e =.60 0 9 C (elementary charge) σ = 5.67099 0 8 W/(m K 4 ) (Stefan-Boltzmann constant) N a = 6.0 0 mole (Avogadro s constant) V m =.4 liter (at atm, 0 C) (Molair volume R =.097 0 7 m (Rydberg constant) µ B = 9.74 0 4 J/T (Bohr magneton) ɛ 0 = 8.854 0 F/m (vacuum permittivity) a 0 = 0.059 0 9 m = 0.059 nm (Bohr radius) hydrogen ionization energy=.6 ev unit conversions: ev =.60 0 9 J (electron-volt to Joule) J = 6.4 0 8 ev (Joule to electron-volt) u=.66056 0 7 kg = 9.5 MeV/c (atomair mass unit to kg to energy) hc = 40 ev nm (hc in electronvolt nanometer) masses: m e = 9.09 0 kg =0.000549 u =5 kev/c (mass of electron) m p =.67 0 7 kg =.007766 u m n =.675 0 7 kg =.0086654 u (mass of proton) (mass of neutron) M( H) = M =.00785 u (mass of hydrogen atom) M( 4 He) = 4 M = 4.00605 u (mass of helium atom) M( 98 80 M( 98 79 M( 98 78 Hg) = 98 80 M = 97.966769 u (mass of mercury isotope) Au) = 98 79 M = 97.9684 u (mass of gold isotope) Pt) = 98 78 Photons and radiation laws: M = 97.96789 u (mass of platinium isotope) λ ν = c E = hν λ max T =.898 0 mk R(λ) = 8π λ 4 kt c 4 R(λ) = c 8π hc 4 λ 4 λ I = σt 4 e hc/(λk BT ) (relation wave length frequency) (energy of photon) (Wien s displacement law) (radiation law of Rayleigh Jeans) (Planck s radiation law) (law of Stefan Boltzmann)
Fotoelectric effect and Compton effect: hf = ev s + φ λ λ = h ( cos(θ)) m e c (foto electric equation) (Compton scattering) Relativistic energy and impuls: E = (mc ) + (p c) p = mv v /c Atom model of Bohr: f = cr ( n m ) (Rydberg formula) r n = a 0 n = 4π h m e e E n = (4πɛ 0 ) me 4 h n =.6 n ev (radius nth orbit) (energy of nth orbit of hydrogen E n =.6Z n ev (idem, for other atoms (ions) with one electron) Wave mechanics: λ = h p h d Ψ + UΨ = EΨ m dx x p x h E t h (De Broglie wave length) (-dim. Schrödinger equation) (uncertainty relationship) (uncertainty relationship) Nuclear physics: R = r 0 A / (radius on nucleus) B tot (A, Z) = ZM H + (A Z)M n A ZM (binding energy) Q = (m initial m final ) c (Q-value) Some codings on the Chart of Nuclids: β + : beta plus decay, ɛ : electron capture, β : beta minus decay, α : alpha decay