Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

1. 13.2.1 Explain how the radii of nuclei may be estimated from charges particle scattering experiments. 13.2.2 Describe how the masses of nuclei may be determined using a Bainbridge mass spectrometer. 13.2.3 Describe one piece of evidence for the existence of nuclear energy level. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

2. Explain how the radii of nuclei may be estimated from charged particle scattering experiments. As you may recall from Topic 7, Rutherford determined that the bulk of the atom’s positive charge (and mass) is located in a very small central nucleus have a radius of about 10-15m. If we analyze a head-on collision between an alpha particle and a nucleus, we can obtain a rough value for the diameter of a nucleus. A nucleus has a charge of positive Ze and an alpha particle has a charge of positive 2e. As the alpha particle approaches the nucleus it will be slowed down, stopped (for an instant), and reversed by the Coulomb force. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics FYI Review Topic 7 for much of what is in this option.

3. rmax nuclear radius rmax = 2kZe2/EK,0 Explain how the radii of nuclei may be estimated from charged particle scattering experiments. The alpha particle feels a coulomb potential caused by the nucleus given by V = kQ/r or V = kZe/r. If the alpha particle approaches from infinity, the work needed to stop it at rmax is given by W = qV = 2eV = 2kZe2/rmax. From the work-kinetic energy theorem (W = ∆EK) we get W = EK-EK,0. Since at rmax, EK= 0, W = EK,0. where EK,0 is the initial kinetic energy of the  particle. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

4. rmax nuclear radius rmax = 2kZe2/EK,0 Explain how the radii of nuclei may be estimated from charged particle scattering experiments. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics EXAMPLE: Alpha particles having a kinetic energy of EK = 5 MeV bombard a gold nucleus and rebound straight back. Estimate the radius of the gold nucleus. SOLUTION: Gold (Au) has Z = 79. The alpha particle’s kinetic energy must be converted to joules: EK,0 = (5106 eV)(1.610-19 J/eV)= 810-13 J. Then rmax = 2kZe2/EK,0 = 2(9109)(79)(1.610-19)2/810-13 = 510-14 m. Why is the actual radius of the nucleus less than rmax.

5. Describe how the masses of nuclei may be determined using a Bainbridge mass spectrometer. Recall the mass spectrometer in which an atom is stripped of its elec- trons and accelerated through a voltage into a magnetic field. Scientists determined through the use of such a device that hydrogen nuclei came in three different masses: Since the charge of the hydrogen nu-cleus is e, scientists postulated the existence of a neutral particle called the neutron. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

6. mass and radius r = mv/(qB) Describe how the masses of nuclei may be determined using a Bainbridge mass spectrometer. While in the magnetic field the charged particle feels a centripetal force caused by the magnetic field of Fc = qvBsin. But the angle between v and B is 90° so that sin = 1. Since Fc = mv2/r then mv2/r = qvB so that Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

7. B-field Source A B X C D Describe how the masses of nuclei may be determined using a Bainbridge mass spectrometer. PRACTICE: Track X shows the deflection of a singly-charged carbon-12 ion in the deflection chamber of a mass spectrometer. Which path best shows the deflection of a singly-charged carbon-14 ion? Assume both ions travel at the same speed. SOLUTION: Since carbon-14 is heavier, it will have a bigger radius than carbon-12. Since its mass is NOT twice the mass of carbon-12, it will NOT have twice the radius. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

8. Describe how the masses of nuclei may be determined using a Bainbridge mass spectrometer. EXAMPLE: A hydrogen ion (proton) is accelerated through a potential difference of V = 475 V and projected into a mass spectrometer having a magnetic field strength of 0.250 T. (a) What is the velocity of the proton after its acceleration? (b) What is its radius of curvature in the spec? SOLUTION: (a) EK = qV = (1.610-19)(475) = 7.610-17J. Then 7.610-17 = (1/2)mv2 = (1/2)(1.6710-27)v2 and v = 3.0105 ms-1. (b) r = mv/(qB) = (1.6710-27)(3.0105)/[(1.610-19)(0.250)] = 0.013 m (1.3 cm) Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

9. Describe one piece of evidence for the existence of nuclear energy levels. Recall that electrons in an atom moving from an excited state to a de-excited state release a photon. The emission spectra of de-exciting atoms show the existence of atomic energy levels. In exactly the same way, de-exciting nuclei also release photons which also produce spectra - only with very high energy photons called gamma  rays: 234Pu*  234Pu +  Topic 13: Quantum and nuclear physics 13.2 Nuclear physics Atomic spectral lines. Nuclear spectral lines.

10. Radioactive decay 13.2.4 Describe + decay, including the existence of the neutrino. 13.2.5 State that the radioactive decay law is an exponential function and define the decay constant. 13.2.6 Derive the relationship between the decay constant and the half-life. 13.2.7 Outline methods for measuring the half-life of an isotope. 13.2.8 Solve problems involving radioactive half-life. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

11. Radioactive decay Describe beta plus (+) decay including the existence of the neutrino. There are two types of beta () particle decay: In - decay, a neutron becomes a proton and an electron is emitted from the nucleus. 14C  14N +  + e- In + decay, a proton becomes a neutron and a positron is emitted from the nucleus. 10C  10B +  + e+ In short, a beta particle is either an electron or an anti-electron. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics - +

12. Slow Medium Fast Medium Radioactive decay Describe beta plus (+) decay including the existence of the neutrino. In contrast to the alpha particle, it was discovered that beta particles could have a large variety of kinetic energies. In order to conserve energy it was postulated that another particle called a neutrino was created to carry the additional EKneeded to balance the energy. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics Same total energy

13. population decay N = Noe-t Radioactive decay State the radioactive decay law as an exponential function and define the decay constant. The higher the initial popu- lation of a radioactive ma- terial, the more decays there will be in a time interval. But each decay decreases the population. Hence the decay rate decreases over time for a fixed sample and it is an exponential decrease. where N0 is the initial population, N is the new one, t is the time, and  is the decay constant. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics Radioactive material remaining Time in half-lives

14. T1/2 = ln2/ decay constant and half-life Radioactive decay Derive the relationship between decay constant and half-life. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics EXAMPLE: Show that the relationship between half-life and decay constant is given by T1/2 = ln2/. SOLUTION: Use N=Noe-t.Then N=N0/2 when t=T1/2. ThenN = Noe-t N0/2 = Noe-T (1/2) = e-T ln(1/2) = -T1/2 -ln(1/2) = T1/2 ln2 = T1/2 Exponential decay function. Substitution. Cancel N0. lnx and exare inverses. Multiply by -1. -ln(1/x) = +lnx.

15. Radioactive decay Solve problems involving radioactive half-life. EXAMPLE: The half-life of U-238 is 4.51010 y and for I-123 is 13.3 h. Find the decay constant for each radioactive nuclide. SOLUTION: Use T1/2 = ln2/. Then  = ln2/T1/2. For U-238 we have  = ln2/T1/2 = 0.693/4.51010 y = 1.510-11 y-1. For I-123 we have  = ln2/T1/2 = 0.693/13.3 h = 0.052 h-1. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics FYI The decay constant is the probability of decay of a nucleus per unit time.

16. 1 Becquerel 1 Bq = 1 decay / second Radioactive decay Outline methods for measuring the half-life of an isotope. Rather than measuring the amount of remaining radioactive nuclide there is in a sample (which is hard to do) we measure instead the decay rate (which is much easier). Decay rates are measured using various devices, most commonly the Geiger-Mueller counter. Decay rates are measured in Becquerels (Bq). Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

17. decay rate or activity A = -∆N/∆t = N = Noe-t Radioactive decay Outline methods for measuring the half-life of an isotope. The decay rate A is given by The ∆N is the change in the number of nuclei, and is negative (the radioactive sample loses population with each decay). The negative sign is in A = -∆N/∆tto make the activity A positive. A =N shows that the activity is proportional to the remaining population of radioactive nuclei, which we learned in Topic 7. Since N = Noe-t the last equation A = Noe-t is true. Topic 13: Quantum and nuclear physics 13.2 Nuclear physics

18. Radioactive decay Solve problems involving radioactive half-life. PRACTICE: Topic 13: Quantum and nuclear physics 13.2 Nuclear physics Remember that the mass of the material does not change appreciatively during radioactive decay. Nuclei are just transmuted.

19. 1p 0e 1 1 Radioactive decay Solve problems involving radioactive half-life. PRACTICE: Topic 13: Quantum and nuclear physics 13.2 Nuclear physics It is a proton. If you look at the lower numbers you see that we are short a positive charge on the right: The only two particles with a positive charge (that we have studied) are the beta+ and the proton. Looking at the nucleon number we see that it must be the proton.

20. Radioactive decay Solve problems involving radioactive half-life. PRACTICE: Topic 13: Quantum and nuclear physics 13.2 Nuclear physics The CO2 in the atmosphere has a specific percentage of carbon-14. The moment the wood dies, the carbon-14 is NOT replenished. Since the carbon-14 is always disintegrating and is NOT being replenished in the dead wood, its activity will decrease over time.

21. Radioactive decay Solve problems involving radioactive half-life. PRACTICE: Topic 13: Quantum and nuclear physics 13.2 Nuclear physics From Thalf = ln 2/ we get = ln 2/Thalf or = 0.693/5500 = 0.00013 y-1. From A= N we see that in the beginning 9.6 = N0 and now 2.1 = N. Thus N= N0e-t becomes 2.1 = 9.6e-t so that 2.1/9.6 = e-t. ln(2.1/9.6) = ln(e-t) -1.5198 = -t t = 1.5198/0.00013 = 12000 y

22. Radioactive decay Solve problems involving radioactive half-life. PRACTICE: Topic 13: Quantum and nuclear physics 13.2 Nuclear physics The activity would be too small to be reliable. For this sample A = 9.1e-tbecomes A = 9.1e-0.00013(20000)= 0.68 decay min-1.