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PHY303 Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2014-2015) NUCLEAR PHYSICS 2 HOURS Answer question ONE (Compulsory) and TWO other questions. All questions are marked out of twenty. The breakdown on the right-hand side of the paper is meant as a guide to the marks that can be obtained from each part. Please clearly indicate the question numbers on which you would like to be examined on the front cover of your answer book. Cross through any work that you do not wish to be examined. 1 PHY303 TURN OVER PHY303 COMPULSORY 1 (a) Write down the equations for the energy release Q for the three processes of electron emission, positron emission and electron capture, carefully defining the symbols that you use. [4] (b) Briefly outline the means by which nuclei may come to have a nuclear magnetic [4] moment and an electric quadrupole moment. (c) Briefly explain the term nuclear force. Include a sketch to illustrate the force [4] experienced between two nucleons as a function of distance. (d) Explain briefly the Saxon-Woods form of charge distribution in nuclei. Include [4] suitable sketches and state how the distribution can be experimentally determined. (e) Outline the concept of an isospin doublet in the context of nuclear physics. 2 PHY303 CONTINUED [4] PHY303 2 Many interesting properties of nuclei are revealed by the chart of nuclides and the curve of average binding energy per nucleon as a function of atomic mass number A. (a) Sketch a plot of the chart of nuclides. Define the terms isotones, isotopes, isobars and island of stability, using your plot to illustrate each of these. [6] (b) Isodiaphers are nuclides with the same difference between neutrons and protons. [2] Illustrate isodiaphers on your plot. (c) Sketch a plot of the average binding energy per nucleon vs. atomic mass number A. Show on this plot how fission and fusion occurs and state which isotope has the [4] maximum average binding energy. (d) The semi-empirical formula for the binding energy of a nucleus can be written B(A,Z) av A as A2 / 3 ac Z(Z 1)A1/ 3 aa (A 2Z)2 A1 ap A1/ 2 where values for the constants av, as, ac, aa and ap, can be taken as 14.0, 13.0, 0.6, 19.0 and 12.0 MeV. Although this equation is useful for estimating the binding energy of most nuclei there is a class of nuclei for which it underestimates B. Explain the characteristics of these nuclei and where in general they lie on the chart [2] of nuclides. (e) If we consider nuclei with the same odd value of A, then the equation for the binding energy B is a parabola as a function of Z. Find an expression for the Z of the single stable isobar that occurs in this situation. Find the Z for the stable isobar with [6] A = 127. 3 PHY303 TURN OVER PHY303 3 9 12 The isotopes 4 Be and 6 C are commonly used in the nuclear industry. Like all nuclei they show properties that are compatible with a so-called liquid drop model, in which the nucleons involved behave collectively. However, they also behave as if the nucleons involved fill specific energy levels, compatible with the so-called shell model. (a) Briefly describe, with the aid of suitable plots or sketches, six pieces of empirical evidence that support the shell model of nuclei. [6] (b) Write a simple expression for the radius R of a nucleus in terms of the nucleon radius and the number of nucleons contained. State what assumption is made in this expression that is compatible with the liquid drop model. Using this expression give 9 12 estimates of the radii of 4 Be and 6 C . [4] (c) In the shell model the energy levels are assigned in order of increasing energy as, [6] 1s1/ 2,1p3 / 2,1p1/ 2,1d5 / 2,2s1/ 2,1d3 / 2,1f j ,..... Use this sequence, adding further levels as required, to determine the ground state 9 12 27 43 spin and parity of the following nuclei, 4 Be , 6 C , 13 Al , 21 Sc . (d) On the basis of the shell model for nuclei that have an unpaired nucleon, a first excited state can be produced by excitation of the unpaired nucleon into the next sub9 [2] shell. Determine the spin and parity for the first excited state of 4 Be . 9 (e) 4 Be is used in the nuclear industry as a neutron reflector and can also be used as a source of neutrons by bombardment with alpha particles. Write out the nuclear [2] interaction equations that describe these two processes. 4 PHY303 CONTINUED PHY303 4 The study of nuclear reactions provides an important tool for our understanding of the excited states of nuclei. (a) Briefly describe the characteristics of so-called compound nuclear reactions and of direct reactions. Include a description of the independence hypothesis in compound interactions. Give an example of a compound reaction. [6] (b) Consider the following reaction involving protons and nickel nuclei, p 64 Ni p 64 Ni* State what specific type of direct reaction this is and explain the meaning of the [2] notation used. (c) The figure below shows the excited states of 64Ni. These states can be produced by bombarding 64Ni with a beam of protons if sufficient energy is transferred from the incident protons. Under the assumption that all the kinetic energy of the interaction is taken by the proton and that the nickel nucleus is excited into the first excited state relative to the ground state, what is the kinetic energy of the proton after the interaction if the initial proton energy is 11 MeV? [2] (d) Now use conservation of momentum to estimate the momentum and kinetic energy of the recoiling 64Ni nucleus for the situation where the protons recoil at 600 from the beam direction. Re-estimate the proton energy by imposing energy conservation taking account the recoil of the 64Ni nucleus. [6] (e) As can be seen in the figure in part (c) the gap between the ground state and first excited state is quite large. Briefly explain why this is the case and why the gaps [4] between the energy levels above about 3 MeV are much smaller. 5 PHY303 TURN OVER PHY303 5 Nuclear power can be generated either by the process of fission or, hopefully in the future, fusion reactions. (a) Consider the fusion of two deuterium nuclei with release of a single neutron. How much energy is released per fusion? (note: the nuclear masses of deuterium and 3 [5] He are 3.445 × 1027 kg and 5.008 × 1027 kg respectively). (b) Briefly describe the process of neutron-induced nuclear fission of 235U and its use in power generation. Include in your discussion explanations of the terms moderator, [5] prompt and delayed neutrons, chain reaction and fission fragments. (c) Consider neutrons of energy 0.1 eV incident on a piece of natural uranium. The cross section for fission of 235U at that energy is 250 barns. The amount of 235U in natural uranium is 0.72% and the density of uranium is 19 g cm3. Estimate the mean free path length in cm for the neutrons, assuming this is dominated by fission [4] of 235U. (d) Taking the flux of neutrons in part (c) to be 1012 neutrons s1 cm2 and given that each fission produces 165 MeV, estimate the initial nuclear power produced in watts if the piece of uranium is 1 cm3. State what assumptions you make about the geometry of the piece of uranium. [4] (e) Using your answer to part (a) and (d) estimate the mass of deuterium that must undergo fusion per second to match the power output for the fission reaction. [2] END OF EXAMINATION PAPER 6 PHY303 CONTINUED PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physical Constants electron charge electron mass proton mass neutron mass Planck’s constant Dirac’s constant (~ = h/2π) Boltzmann’s constant speed of light in free space permittivity of free space permeability of free space Avogadro’s constant gas constant ideal gas volume (STP) gravitational constant Rydberg constant Rydberg energy of hydrogen Bohr radius Bohr magneton fine structure constant Wien displacement law constant Stefan’s constant radiation density constant mass of the Sun radius of the Sun luminosity of the Sun mass of the Earth radius of the Earth e = 1.60×10−19 C me = 9.11×10−31 kg = 0.511 MeV c−2 mp = 1.673×10−27 kg = 938.3 MeV c−2 mn = 1.675×10−27 kg = 939.6 MeV c−2 h = 6.63×10−34 J s ~ = 1.05×10−34 J s kB = 1.38×10−23 J K−1 = 8.62×10−5 eV K−1 c = 299 792 458 m s−1 ≈ 3.00×108 m s−1 ε0 = 8.85×10−12 F m−1 µ0 = 4π×10−7 H m−1 NA = 6.02×1023 mol−1 R = 8.314 J mol−1 K−1 V0 = 22.4 l mol−1 G = 6.67×10−11 N m2 kg−2 R∞ = 1.10×107 m−1 RH = 13.6 eV a0 = 0.529×10−10 m µB = 9.27×10−24 J T−1 α ≈ 1/137 b = 2.898×10−3 m K σ = 5.67×10−8 W m−2 K−4 a = 7.55×10−16 J m−3 K−4 M = 1.99×1030 kg R = 6.96×108 m L = 3.85×1026 W M⊕ = 6.0×1024 kg R⊕ = 6.4×106 m Conversion Factors 1 u (atomic mass unit) = 1.66×10−27 kg = 931.5 MeV c−2 1 astronomical unit = 1.50×1011 m 1 eV = 1.60×10−19 J 1 atmosphere = 1.01×105 Pa 1 Å (angstrom) = 10−10 m 1 g (gravity) = 9.81 m s−2 1 parsec = 3.08×1016 m 1 year = 3.16×107 s Polar Coordinates x = r cos θ y = r sin θ ∂ 1 ∂2 1 ∂ 2 r + 2 2 ∇ = r ∂r ∂r r ∂θ dA = r dr dθ Spherical Coordinates x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dV = r2 sin θ dr dθ dφ 1 ∂ 1 ∂2 1 ∂ ∂ 2 2 ∂ ∇ = 2 r + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 Calculus f (x) f 0 (x) f (x) f 0 (x) xn ex nxn−1 ex tan x sin−1 ln x = loge x 1 x sin x cos x cos x − sin x cosh x sinh x sinh x cosh x a cos−1 xa tan−1 xa sinh−1 xa cosh−1 xa tanh−1 xa cosec x −cosec x cot x uv sec x sec x tan x u/v sec2 x x √ 1 a2 −x2 − √a21−x2 a a2 +x2 √ 1 x2 +a2 √ 1 x2 −a2 a a2 −x2 0 0 u v + uv u0 v−uv 0 v2 Definite Integrals Z ∞ xn e−ax dx = 0 Z +∞ n! an+1 r (n ≥ 0 and a > 0) π a −∞ r Z +∞ 1 π 2 −ax2 xe dx = 2 a3 −∞ Z b b Z b du(x) dv(x) Integration by Parts: u(x) dx = u(x)v(x) − v(x) dx dx dx a a a −ax2 e dx = Series Expansions (x − a) 0 (x − a)2 00 (x − a)3 000 f (a) + f (a) + f (a) + · · · 1! 2! 3! n X n n−k k n n! n Binomial expansion: (x + y) = x y and = (n − k)!k! k k k=0 Taylor series: f (x) = f (a) + (1 + x)n = 1 + nx + ex = 1 + x + n(n − 1) 2 x + ··· 2! x 2 x3 + +· · · , 2! 3! sin x = x − ln(1 + x) = loge (1 + x) = x − Geometric series: n X rk = k=0 Stirling’s formula: (|x| < 1) x3 x5 + −· · · 3! 5! x2 x3 + − ··· 2 3 and cos x = 1 − x2 x4 + −· · · 2! 4! (|x| < 1) 1 − rn+1 1−r loge N ! = N loge N − N or ln N ! = N ln N − N Trigonometry sin(a ± b) = sin a cos b ± cos a sin b cos(a ± b) = cos a cos b ∓ sin a sin b tan a ± tan b 1 ∓ tan a tan b sin 2a = 2 sin a cos a tan(a ± b) = cos 2a = cos2 a − sin2 a = 2 cos2 a − 1 = 1 − 2 sin2 a sin a + sin b = 2 sin 21 (a + b) cos 12 (a − b) sin a − sin b = 2 cos 12 (a + b) sin 12 (a − b) cos a + cos b = 2 cos 12 (a + b) cos 12 (a − b) cos a − cos b = −2 sin 12 (a + b) sin 21 (a − b) eiθ = cos θ + i sin θ 1 iθ 1 iθ cos θ = e + e−iθ and sin θ = e − e−iθ 2 2i 1 θ 1 θ cosh θ = e + e−θ and sinh θ = e − e−θ 2 2 sin a sin b sin c Spherical geometry: = = and cos a = cos b cos c+sin b sin c cos A sin A sin B sin C Vector Calculus A · B = Ax Bx + Ay By + Az Bz = Aj Bj A×B = (Ay Bz − Az By ) î + (Az Bx − Ax Bz ) ĵ + (Ax By − Ay Bx ) k̂ = ijk Aj Bk A×(B×C) = (A · C)B − (A · B)C A · (B×C) = B · (C×A) = C · (A×B) grad φ = ∇φ = ∂ j φ = ∂φ ∂φ ∂φ î + ĵ + k̂ ∂x ∂y ∂z ∂Ax ∂Ay ∂Az + + ∂x ∂y ∂z ∂Ax ∂Az ∂Ay ∂Ax ∂Az ∂Ay − î + − ĵ + − k̂ curl A = ∇×A = ijk ∂ j Ak = ∂y ∂z ∂z ∂x ∂x ∂y div A = ∇ · A = ∂ j Aj = ∇ · ∇φ = ∇2 φ = ∂ 2φ ∂ 2φ ∂ 2φ + + 2 ∂x2 ∂y 2 ∂z ∇×(∇φ) = 0 and ∇ · (∇×A) = 0 ∇×(∇×A) = ∇(∇ · A) − ∇2 A