Examination cover sheet

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1 Student name: Student number: Examination cover sheet (to be completed by the examiner) Course name: Radiation Physics Course code: 8CM10/8N120 Date: Start time: 13:30 End time : 16:30 Number of pages: 6 Number of questions: 6 Maximum number of points/distribution of points over questions:100 Method of determining final grade: divide total of points by 10 Answering style: open questions Exam inspection: With H. ten Eikelder or D. Bosnacki Other remarks: The answers may be given in English or in Dutch Instructions for students and invigilators Permitted examination aids (to be supplied by students): Notebook Calculator Graphic calculator Lecture notes/book One A4 sheet of annotations Dictionar(y)(ies). If yes, please specify: Other: Important: examinees are only permitted to visit the toilets under supervision it is not permitted to leave the examination room within 15 minutes of the start and within the final 15 minutes of the examination, unless stated otherwise examination scripts (fully completed examination paper, stating name, student number, etc.) must always be handed in the house rules must be observed during the examination the instructions of examiners and invigilators must be followed no pencil cases are permitted on desks examinees are not permitted to share examination aids or lend them to each other During written examinations, the following actions will in any case be deemed to constitute fraud or attempted fraud: using another person s proof of identity/campus card (student identity card) having a mobile telephone or any other type of media-carrying device on your desk or in your clothes using, or attempting to use, unauthorized resources and aids, such as the internet, a mobile telephone, etc. using a clicker that does not belong to you having any paper at hand other than that provided by TU/e, unless stated otherwise visiting the toilet (or going outside) without permission or supervision Associated with the Central Examination Regulations

2 Examination Radiation Physics - 8CM10/8N January 2016, 13:30-16:30 General remarks: There is a table on page 4 listing the maximum number of points that can be achieved in each assignment. You can use the appended list of constants and formulas during the exam. All answers should be formulated and motivated clearly. 1. Consider an alpha particle and a beta particle, both with a kinetic energy of 1 MeV. (a) What is the velocity and the De Broglie wavelength of the alpha particle? (b) What is the velocity and the De Broglie wavelength of the beta particle? Consider a gamma particle with an energy of 1 MeV. (c) What is the wavelength corresponding to the gamma particle? 2. Consider the Compton effect, where incoming X-rays with a wavelength of nm interact with not moving, free electrons. The trajectory of a scattered photon has an angle θ = 45 with respect to the incoming beam. (a) Compute the corresponding wavelength of the scattered photon. (b) Compute the energy of the scattered photon. (c) Compute the energy of the electron after this interaction. (d) Compute the angle φ between the trajectory of the scattered electron and the incoming beam. We now investigate the hypothetical variant of the Compton effect, where after the interaction of an incident photon with an electron at rest only a scattered electron occurs, so there is no scattered photon. The goal of this part of the exercise is to show that such an interaction cannot exist, since a photon cannot give up all its momentum and energy to a free electron. Let E > 0 be the energy of the incoming photon, m e the mass of the electron and E e the total energy of the electron after the interaction. Let p be the momentum of the photon before the interaction and p e the momentum of the electron after the interaction. (e) Formulate energy and momentum conservation for the described reaction. (f) Show that these two conditions cannot hold simultaneously. Hint: Use the relativistic energy and momentum relation for the electron to compute its energy from its momentum. 3. In this exercise we consider a particle with mass m in an infinitely deep, one-dimensional potential well ("particle in box") with width L. Inside the well the potential is 0, outside. (a) Formulate the Schrödinger equation for the wave function Ψ(x) for this case. 2

3 (b) Which boundary conditions on Ψ(x) must hold in this case? (c) Derive the general (not normalized) solutions of the Schrödinger equation for the considered case and the corresponding values of the energy E. (d) Give the corresponding normalized solutions of the Schrödinger equation. Consider the macroscopic case of a ball with mass m = 0.1 kg that can move in a one-dimensional region of width L = 1 m. (e) What is the minimal value of the energie E in joules for this situation? (f) What is the velocity of a classical ball with mass m that has the minimal energy as kinetic energy? 4. The radial part of the wave function of the electron in the ground state of a hydrogen atom is R 1,0 (r) = 2 e r/a0 with a a 3/2 0 = 4πε 0 2 m 0 e q. 2 In this formula a 0 is the Bohr radius. In this exercise we use q for the elementary electric charge to avoid confusion with the mathematical constant e. The corresponding radial probability density is P (r) = r 2 R 1,0 (r) 2. (a) What is the most likely distance of an electron to the nucleus in the ground state? The energy of a classical electron that moves in a circular orbit with radius r around the nucleus is given by E = 1 q 2. 8πε 0 r (b) Compute the expected value of the energy of the electron if the probability of the distances r is given by P (r). You may use that 0 e x x k dx = k!. (c) What is the relation between the energy of ground state computed in this way and the energy E 1 in the Bohr model? 5. We consider an X-ray tube with a Thungsten anode. The acceleration voltage V AC of the tube is gradually increased. If V AC reaches 69.5 kv the characteristic K α and K β X-rays appear. The characteristic K α X-ray has a wavelength of nm. (a) What is the binding energy of electrons in the K-shell (n = 1) of Thungsten? (b) What is the binding energy of electrons in the L-shell (n = 2) of Thungsten? Figure 1: Example of X-ray spectrum with characteristic X-rays 3

6. The Iodine isotope I 131 53 has several medical applications. It is used in the treatment of thyrotoxicosis (hyperthyroidism) and some types of thyroid cancer that absorb Iodine.

4 6. The Iodine isotope I has several medical applications. It is used in the treatment of thyrotoxicosis (hyperthyroidism) and some types of thyroid cancer that absorb Iodine. In Figure 2 the part of the chart of nuclides related to I is shown. Figure 2: Part of the nuclides chart for question 6 (a) The chart shows a possible radioactive decay from I to Xe What type of decay is this? (b) Compute the corresponding Q-value. (c) Describe two ways in which the decay of I to the ground state of Xe can proceed. Hint: The white rectangle in the square for Xe means that there also exists a meta stable state of Xe , i.e. an excited state with a relatively long halftime. That meta stable state can go to the ground state by a 164 kev gamma emission (indicated by lγ on the chart). Hint: alpha and beta energies on the chart of nuclides are in MeV, gamma energy is in kev. Note that energies are not always very precise. (d) The square for chart I also mentions a 637 kev gamma emission. If this gamma emission occurs as part of a two step decay from I to the ground state of Xe , what is the other decay step and what is its energy? Points: (total 100) Question 1a: 4 points Question 2a: 4 points Question 3a: 4 points Question 1b: 4 points Question 2b: 4 points Question 3b: 4 points Question 1c: 4 points Question 2c: 4 points Question 3c: 4 points Question 2d: 5 points Question 3d: 4 points Question 2e: 5 points Question 3e: 3 points Question 2f: 5 points Question 3f: 3 points Question 4a: 4 points Question 5a: 5 points Question 6a: 4 points Question 4b: 5 points Question 5b: 5 points Question 6b: 4 points Question 4c: 3 points Question 6c: 5 points Question 6d: 4 points 4

5 Radiation Physics - 8CM10 Formulas and constants /2016 physical constants: h = Js = ev s (Planck s constant) k B = J/K (Boltzmann s constant) c = m/s (speed of light) e = C (elementary charge) σ = W/(m 2 K 4 ) (Stefan-Boltzmann constant) N a = mole 1 (Avogadro s constant) V m = 22.4 liter (at 1 atm, 0 C) (Molair volume R = m 1 (Rydberg constant) µ B = J/T (Bohr magneton) ɛ 0 = F/m (vacuum permittivity) a 0 = m = nm (Bohr radius) hydrogen ionization energy= 13.6 ev unit conversions: 1 ev = J (electron-volt to Joule) 1 J = ev (Joule to electron-volt) 1 u= kg = MeV/c 2 (atomair mass unit to kg to energy) hc = 1240 ev nm (hc in electronvolt nanometer) masses: m e = kg = u =511 kev/c 2 m p = kg = u m n = kg = u (mass of electron) (mass of proton) (mass of neutron) M( 1 1H) = 1 1M = u (mass of hydrogen atom) M( 4 2He) = 4 2M = u (mass of helium atom) M( I) = M = u (mass of iodine isotope) M( Xe) = M = u (mass of xenon isotope) Photons and radiation laws: λ ν = c E = hν λ max T = mk R(λ) = 8π λ kt c 4 4 R(λ) = c 8π hc 1 4 λ 4 λ e hc/(λk BT ) 1 I = σt 4 Fotoelectric effect and Compton effect: (relation wave length frequency) (energy of photon) (Wien s displacement law) (radiation law of Rayleigh Jeans) (Planck s radiation law) (law of Stefan Boltzmann) hf = ev s + φ (foto electric equation) 5

6 λ λ = h (1 cos(θ)) m e c (Compton scattering) Relativistic energy and momentum (E is the total energy, including the rest energy): E 2 = (mc 2 ) 2 + (p c) 2 E = p = mc 2 1 v2 /c 2 mv 1 v2 /c 2 Atom model of Bohr: f = cr ( 1 n 2 1 m 2 ) (Rydberg formula) r n = a 0 n 2 = 4πɛ 0 2 m e e 2 n2 (radius nth orbit) E n = 1 m e e 4 1 (4πɛ 0 ) n = 13.6 ev (energy of nth orbit of hydrogen 2 n2 E n = 13.6Z2 n 2 ev (idem, for other atoms (ions) with one electron) Wave mechanics: λ = h p 2 d 2 Ψ + UΨ = EΨ 2m dx2 x p x E t (De Broglie wave length) (1-dim. Schrödinger equation) (uncertainty relationship) (uncertainty relationship) Nuclear physics: R = r 0 A 1/3 (radius of nucleus) B tot (A, Z) = (ZM H + (A Z)M n A ZM)c 2 (binding energy) Q = (m initial m final ) c 2 T α = A 4 A (Q-value) Q (kinetic energy of alpha particle) Some codings on the Chart of Nuclides: β + : beta plus decay, ɛ : electron capture, β : beta minus decay, α : alpha decay γ : gamma decay, e : internal conversion 6

Examination Radiation Physics - 8N120 5 November 2014, 13:30-16:30

Examination Radiation Physics - 8N120 5 November 2014, 13:30-16:30 Four general remarks: This exam consists of 8 assignments on a total of 3 pages. There is a table on page 4 listing the maximum number