Tutorial 1: SubAtomicPhysics: Nuclear Physics

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1 Tutorial 1: SubAtomicPhysics: Nuclear Physics Key questions 1) In the experiment of Rutherford, Geiger and Marsden a particles of energy 7.7 MeV were incident on a gold nucleus 197 Au (Z=79). Estimate the distance of closest approach of the alpha particles to the nucleus (ie. a head on collision). Later experiments used the same energy of alpha particles on 27 Al(Z=13) and noted the sudden onset of deviations from the coulomb scattering expectations for backscattered alpha particles at ~180 o. Propose why this is the case and use the given information to obtain an estimate of the size of the aluminium nucleus. What approximations may cause inaccuracies in this estimate? Compare your value with that obtained from R ~ R 0 A 1/3 fm. [ε 0 = x10-12 fm -1, e = x10-19 C] Equate kinetic energy of alpha with the colulomb potential energy of alpha+nucleus (assuming non relativistic kinematics). At distance of closest approach (d 0 ) kinetic energy of incident alpha completely converted to potential energy. 197 Au: T= q1 q 2 /4πε 0 : Therefore d 0 =(q 1 q 2 /4πε 0 T α ) = ( (2e)(79e) / ( 4π( x10-12 )(7.7x10 6) e) ) = 29.5 fm. 27 Al: similar calculation yields d0 =4.86fm, As you are told the coulomb picture breaks down for this case you can assume that this is the distance at which the alpha feels the strong force i.e is the potential radius of the nucleus. Therefore can estimate size of nucleus ~0.7 fm less than this = ~4.2fm. Approximations used don t account for finite size alpha or nucleus, simple approximation for average range of nuclear force, Size of Al nucleus from R=R 0 A 1/3 =1.3 x 27 1/3 = 3.9 fm. 2) Use the uncertainty relation to estimate (in MeV) the minimum kinetic energy possible for an electron confined to a sphere of diameter 2 fm [Hint assume electrons are fully relativistic (kinetic energy ~ pc)]. What are the implications of this energy when deciding whether electrons exist in the nucleus? p x>ħ : p> ħ / x > x / 2 x = 5.27x10-20 Jsm -1 T~pc~(5.27 x )(3x10 8 ) ~ 1.58 x J = 98.7 MeV Electrons interact via the coulomb force - this force is not sufficiently strong to confine electrons to the nucleus (compare with typical atomic binding eneries which are of the order of ev) 3) If a spherical drop of water (radius=0.15cm) were to have the same density as nuclear matter (~10 17 kg/m 3 ), what would its mass be? Assume drop has constant density and is spherical : M = ρv = (4/3xπx0.15x10-2 ))3 = 1.4x10 9 kg!! 4) Calculate the ratio of the gravitational potential energy to coulomb potential energy between two protons in the nucleus. 5) What is the parity of a nucleon arising from its orbital motion if it has a) orbital angular momentum l=1? b) orbital angular momentum l=2? Briefly discuss what is meant by parity and how it relates to the wavefunction properties.

2 π = (-1) l if l=1 parity is negative. If l=2 parity is positive. See notes for last part 6) Calculate the following masses in mass atomic units (u) for a) electron (m e = x10-31 kg) b) proton (m p = x kg) c) neutron (m n =1.6749x10-27 kg). Hence convert these values into units of MeV/c 2 m e =9.109 x / x = x 10-4 u c 2 (As 1u = MeV/c 2 therefore m e = MeV/c 2 Similarly m p = MeV/c 2 and m n = MeV/c 2 7) Using the relationship between nuclear radii and atomic number (R ~ R 0 A 1/3 fm) estimate the density of nuclear matter and the average separation of nucleons in the nucleus. Neutron stars can be thought of as a large assembly of neutrons packed with a density close to that of nuclear matter estimate the radius of a neutron star of mass 2 solar masses ( mass of sun = 1.988x10 30 kg) ρ = M/V = AM p /[(4/3)π(1.25.A 1/3 ) 3 ] = M p /[(4/3)π(1.25) 3 ] = 1.67 x / 8.18 x = 2 x kg/m 3 Assume neutron star is spherical: Volume = M star / ρ = 2(1.988 x )/2x10 17 = 1.99 x m 3 = 4/3πR Ν 3 R Ν =17km 8) Why are electrons a good probe of the nuclear charge distribution? Estimate the electron beam energy required to probe the nuclear charge distribution with a spatial resolution of 0.2 fm. Using this beam energy estimate the angle of the first diffraction minimum for scattering from a target of 27 Al(Z=13) stating any assumptions you make. Electrons are a good probe of the nuclear charge distribution as i) their interaction is by the electromagnetic force which is well understood and ii) The interaction strength not so strong, therefore electrons can probe the entire nuclear volume (This is not the case for strongly interacting probes like protons / neutrons which mainly interact near the nuclear surface) To probe with resolution of 0.2fm require the reduced wavelength (λ/2π) < 0.2fm. (Usually written as lambda bar) E=hbar c/ lamda(bar) = 197.3/0.2 = MeV 1 st diffraction minimum sin(θ)=1.22 λ / 2R = 1.22 (0.2 x 2π) / 2(1.3 x28 1/3 ) Therefore θ =11.2 o Tutorial 2: SubAtomicPhysics: Nuclear Physics 1. Sketch the form of the nuclear potential. What does the potential radius and charge radius relate to? Estimate the size of the Coulomb potential barrier faced by an incident proton on i) 238 U(Z=92) and ii) 12 C (assume the potential radius is ~0.7 fm larger than the nuclear radius). The fusion of a proton with 12 C is one of the first steps of the CNO cycle in stellar burning estimate what the calculated barrier height corresponds to in terms of temperature.

3 R=R 0 A 1/3 : radius of 238 U = 1.2(238) 1/3 ~7.44fm radius of 12 C = 1.3(12) 1/3 = 2.97fm. Barrier height = zze 2 /4πε 0 R where R is the potential radius. Take the potential radius as 0.7fm larger than the values calculated above to account for the range of the strong force. This yields a barrier height for 238 U = 16.3 MeV and for 12 C a height of 4.69 MeV. Assuming this ia a monoatomic gas we can take E ~ 3/2kT. Therefore T~3.6x10 10 K. The centre of the sun has a temperature of about 1.3x10 7 K. The fusion process relies on fast nuclei near to the tail of the Maxwell-Boltzmann distribution and quantum tunnelling through the coulomb barrier in order to proceed (even with these effects you have to wait for the sun to heat up some more (as it will in its later life) to get significant probability for this fusion reaction) 2. A contribution to the short range part of the nucleon-nucleon interaction is thought to arise from the exchange of virtual ω mesons. Estimate the typical range over which the force would operate (M ω =782.7 MeV/c 2 ) From lecture r ~ ħc/mc 2 ~ (MeV fm) / MeV/c 2 ~ 0.25 fm 3. In the nuclear reaction Calculate the possible values of the orbital angular momentum quantum number for the 2 final state nuclei. You may assume the initial reaction occurs in s-wave. J π ( 10 B)=3 +, J π ( 4 He)=0 +, J π ( 13 N)=1/2 - Reaction occurs in s-wave so relative orbital angular momenta of Boron and helium nuclei (l=0) Therefore in entrance channel have total angular momentum J=3. As both Boron and Helium have even parity and l=0 (even parity from (-1) l ) therefore total parity in entrance channel is even (remember that parity is multiplicative) As the reaction is a strong force process then expect angular momentum and parity are conserved. Therefore final state should be J=3 and even parity. Total J of final state nuclei (neglecting orbital ang mom) is 0 or 1. Therefore the relative orbital ang mom of final state particles must be l=3 with the first case and l=2 or 3 with the latter (to allow a total ang mom of final state of 3). However as the parity of the nuclei in the final state is negative then only the l=3 case is allowed to ensure the total parity of the final state is positive. 4. A nucleus with J π =0 + undergoes d-wave neutron emission. Specify the spins and parities of nuclear energy levels which can be populated in the final state (A-1) nucleus. How will this change when the initial nucleus has J π =1/2 -? From similar arguments as Qn above energy states of final nuclei following A (A-1) + neutron with l=2 orbital ang mom between neutron and (A-1) nucleus are J π =3/2 +, 5/2 +. For 1/2 - nucleus the allowed states are J π =1 -, How much energy is necessary to split up an alpha particle into its constituent nucleons (M n = u, M H = u, M( 4 He)= u). Binding energy = [(2M H +2M n ) M( 4 He)]c 2 = [ u]c 2 as 1u = MeV/c 2 therefore c 2 = MeV/u Therefore : Binding Energy = ( ) ( ) = 28.3 MeV 6. How much energy is required to remove a proton from 56 Fe(Z=26)? Calculate the binding energy per nucleon for 56 Fe and compare with your value for the proton separation energy. Can you explain any differences? ( M( 56 Fe) = u, M( 55 Mn) = u) S p = [M(A-1,Z) M(A,Z) +M H ]c 2 = [ u u u]c 2 = [ u]c 2 As c 2 = MeV/u S p =[ u] (931,393 MeV/u) = MeV BE = [ZM H + (A-Z)M n - M(A,Z)]c 2 = [ u] ( MeV/u) = MeV BE/A = /56 = 8.79 MeV S P larger than BE/A as average BE/A for 55 Mn must be smaller than 56 Fe (expect this as 56 Fe is the most tightly bound nucleus)

4 7. The atomic mass of a nucleus is predicted by the semi-empirical mass formula to be given by M(N,Z)c 2 = ZM H c 2 + NM n c 2 - a v A + a s A 2/3 + a c Z 2 /A 1/3 + a sym (N-Z) 2 /A + δ Where M H = MeV/c 2, M n = MeV/c 2, a v = 15.8 MeV, a s = 18.0 MeV, a c = 0.7 MeV and a sym = 25.3 MeV. Mirror nuclei are nuclei with the same mass number A and interchanged values of N and Z. Which terms in the above formula give rise to differences in atomic mass between a pair of mirror nuclei? Calculate the predicted atomic mass difference between 11 C(Z=6) and 11 B(Z=5) in units of MeV/c 2. From this information and any general considerations, what can you deduce about the possible decay modes for these two nuclei? Mirror nuclei have the same value of A but interchanged values for N,Z. The only terms in the SEMF which survive when taking the mass difference between mirror nuclei is the ZM H c 2, NM n c 2 and a c Z 2 /A 1/3. Mc 2 = M( 11 B)-M( 11 C) = (Z 1 -Z 2 )M H c 2 + (N 1 -N 2 )M n c 2 + a c /A 1/3 (Z 1 2 -Z 2 2 ) Using given values of Z aand masses gives Mc 2 ) = MeV Difference shows 11 B more stable than 11 C. 11 C will decay to 11 B by β + emission 8. In the semi-empirical mass formula what experimental evidence does the pairing term δ account for? For a given value of the mass number A, nuclear masses can be expressed as a quadratic function in Z. Using the minimum isobar formula (as derived in the lecture notes) determine whether the 142 Xe(Z=54) is β + or β - unstable. Pairing term accounts for the observational evidence that even-even nuclei are more stable than odd-odd nuclei. It has a quantum mechanical origin and reflects the extra binding felt by paired nucleons (with opposite spins) which can share the same spatial states, increasing their probability to be in close proximity to each other and feel the strong force. From formula derived in the notes Z stable = 142 / (142) 2/3 = 60 Therefore 142 Xe(Z=54) will decay by β - towards the stable nucleus at Z=60 Tutorial 3: SubAtomicPhysics: Nuclear Physics 1) A neutron star can be crudely approximated as a large assembly (>~10 55 ) of neutrons. Use the semi empirical mass formula to estimate the binding energy per nucleon of a neutron star (you may neglect the δ term). Comment on the physical implications of the value obtained. Which additional term should also be included in the mass formula if it is to be extrapolated to neutron stars? Propose a form for this extra term. Yhe expression for the binding energy per nucleon (note you get this by dividing the formula for the binding energy (B) given in the notes by A!!) B/A = a v a s A -1/3 a c Z 2 /A 4/3 a s (A-2Z) 2 /A 2 As A is very large the surface and coulomb terms are negligible. The B/A is determined by the difference between the volume term and the asymmetry term. The latter approaches if we assume the star is dominantly neutrons i.e. A>>Z B/A = a v - a s = MeV = -7.7 MeV. The expression is negative implying the system is unbound. Therefore to bind a star other forces contribute. For a large object such as this the gravitational effects cannot be neglected. An attractive gravitational potential energy term should be added to the SEMF. The form will be similar to the derivation of the coulomb potential term in the lectures. i.e. (3/5) (G/r o ) (M 2 /A 1/3 ). Using this term you can go on to calculate the minimum size needed for a stable star!! 2) Use the semi empirical mass formula to calculate the energy released when 238 U fissions symmetrically. Propose why fission products are often unstable.

5 Fission process total nucleons remain the same. Terms dependent on A only will not contribute. Mc 2 = [M( 238 U)- 2*M( 119 Pd)]c 2 = [ZM H + NMn B( 238 U)/c 2 2*(Z 1 M H + N 1 Mn B( 119 Pd)/c 2 ) In symmetric fission Z 1 =Z/2. Therefore using the SEMF. = 2* B( 119 Pd)/c 2 - B( 238 U)/c 2 = [2*( ) ( ) = 181 MeV Note the differences in the SEMF terms for the 2 nuclei. Fission products are often unstable as you have gone from a heavy nucleus where the most stable configuration favours a large neutron excess, to a lighter nucleus where the most stable configuration has smaller (or zero) neutron excess. Therefore the fission fragments tend to have an excess of neutrons compared to the stable configuration and are unstable. 3) Work out the spin and parity of the ground states of a) 16 O(Z=8) b) 15 N(Z=7) c) 23 Na (Z=11) in terms of the nuclear shell model. 4) Give possible shell model nucleon configurations which could produce the 5/2 +, 1/2 +, 1/2 -, 5/2 -, 3/2 - and 3/2 + excited states of 17 O(Z=8) 5) 208 Pb(Z=82) has a closed 3p 1/2 sub-shell of neutrons and a closed 3s 1/2 shell of protons. In terms of the shell model what spin and parity is expected for ce ground state of 207 Pb and 207 Tl(Z=81). 207 Pb 1 unpaired neutron in the 3p 1/2 level. Therefore J π =1/2-207 Tl 1 unpaired proton in the 3s 1/2 level. Therefore J π =1/2

6 6) In odd-odd nuclei, an interaction between the last odd neutron and the last odd proton must be taken into account in order to explain the ground state spins. The coupling favours parallel spins of the odd proton and the odd neutron. On this basis determine the ground state spin and parity for the 14 N (Z=7) nucleus, giving a sketch of the occupation of levels as prescribed by the shell model. 7) The rotational model for permanently deformed nuclei predicts excitation energies given by E(J)=J(J+1)ħ 2 / 2I with J=the spin of the nuclear state and I=moment of inertia of the nucleus about the axis of rotation. The energies and spins of the first two excited states of 180 Hf are Energy (MeV) J Determine whether these values agree with the rotational model and if so a) Predict the energy of the third excited state (J=6) and b) Calculate the moment of inertia for the nucleus (ħ =6.582x10-22 MeVs) Tutorial 4: SubAtomicPhysics: Nuclear Physics 1) Calculate the Q value for 228 Th(Z=90) (mass= u) to emit a) An α particle b) A proton For the first case calculate the kinetic energies of the alpha particle and the daughter nucleus. 224 Ra(Z=88) = u; 227 Ac(Z=89) = u] α emission Q α = [ M(A,Z) M(A-4,Z-2) - M( 4 He) ] c 2 = [0.059u]c 2 As 1u = MeV/c 2 Q α =[0.059u] ( MeV/u = MeV Q value is energy released in α decay of 228 Th. This will be split between the daughter and the alpha.

7 T α = (A-4/A)Q α = (228-4/228)Q α = 5.4 MeV Proton emission Q p = [ M(A,Z) M(A-1,Z-1) - M( 1 H) ] c 2 = MeV Q p value negative so energy required to remove a proton from 228Th. For many heavy nucei alpha particles are preferentially emitted over protons, detuterons, tritons as the 2n-2p configuration os very tightly bound (e.g. See BE/A curve). Therefore, as can be seen in the above example, alpha emission can proceed spontaneously with a release of energy whereas the other decay modes would require an input of energy. 2) Can 57 Co(Z=27) spontaneously decay by β + emission? [masses: 57 Co= u; 57 Fe = u; 2m e = u] Q β + = [ M(A,Z) M(A,Z-1) 2m e ] c 2 = [ u u-2( )u]c 2 = MeV Q<0. Therefore cannot decay spontaneously. 3) The half life of Radon ( 222 Ra) is days. What fraction of freshly separated sample decays in a) 1day? b) 10 days? λ=ln2/τ 1/2 = day 1 1 day N(t)/N 0 = e λt = e = (16.6% decayed). Similarly after 10 days 53.7% decayed 4) There are an estimated kg of uranium in the top 20km of the earth s crust. 5.5x10-3 percent are 234 U, 0.72 percent are 235 U and the rest is 238 U. The decay constants for the three isotopes are 8.97x10-14 s -1, 3.12x10-17 s -1 and 4.92x10-18 s -1 respectively. Calculate the half lives of the isotopes in years. Calculate the activity of the total amount of uranium in the earth s crust in Curies [Atomic mass of Uranium = g/mol)] The previous existence of a natural thermal fission reactor based on Uranium deposits and moderated by groundwater has been suggested in the Gabon. Estimate how long ago such a thermal reactor could have operated. No. of uranium nuclei = x 6.02 x10 23 / = 2.53 x N( 234 U) = (5.5 x10-5 )(2.53x10 41 ) = 1.39 x Nuclei ; halflife = ln2/lamda = ln2/8.97x10-14 ) = 2.45 x 10 5 years Activity(234) = N x lambda = 1.39 x ( 8.97x10-14 ) = 1.25 x Bq = 3.4 x Ci Similarly A(235) = 1.5 x Ci ; A(238) = 3.3 x Ci Total activity = Activity(234)+Activity(235)+Activity(238)= 6.9 x Ci. This is a large amount of radiation luckily we are shielded from this! Note: Activity U(234) is larger than Activity(238) even though have 10 4 time less in the earth s crust because of large decay constant For fission reactor to be moderated by water then fissile component of uranium must be 5%. (see notes). Therefore can calculate how long ago this condition was reached.

8 5) The ground state of 8 4Be is unstable and decays into two alpha particles ( 4 2He) with a half life τ 1/2 =1.0x10-16 s. Determine the relative orbital angular momentum of the two alpha particles and estimate the particle decay width (ev) of the 8 4Be ground state. 6) The nuclide 12 N(Z=7) decays by β + emission with a maximum positron energy of MeV. Calculate the neutral atomic mass of the parent atom. 7) Which of the following heavy nuclei would you expect to have large cross sections for reactions with thermal neutrons? a) 251 Cf (Z=98) b) 255 Fm (Z=100) c) 250 Bk (Z=97) d) 238 U(Z=92) Fissile nuclei have large thermal neutron cross sections. These are even-odd or odd-odd. The addition of the neutron to a fissile nucleus results in an energy benefit from the pairing energy which allows even thermal energy neutrons to have an appreciable probability to induce fission. Nuclei a) b) and c) are fissile