Honors Physics III Lecture 21: Nuclear Fission and Fusion (Chapter 1)

Transcription

1 Honors Physics III Lecture 21: Nuclear Fission and Fusion (Chapter 1) Weida Wu

2 Properties of nuclei and the strong force 1) The force is very strong, attractive, and definitely not 1/r 2. 2) The force is very short range. The force abruptly drops to zero for separations > ~2 fm. F(r) repulsive core r (fm) Strong, attractive force short range 2

3 Properties of nuclei and the strong force 3) The strong force is independent of the electric charge. F pp =F nn =F pn =F np Protons and neutrons are collectively called nucleons. 4) Nucleons are subject to the strong force. Electrons do not feel it. (No electrons inside nuclei!) 3

4 Properties of nuclei and the 5) Nuclear charges: strong force Z = atomic number = number of protons N = number of neutrons A = Z + N = mass number = number of nucleons Two nuclei with the same Z but different A are called isotopes. Two nuclei with the same N but different Z are called isotones. Two nuclei with the same A but different Z are called isobars. 4

5 Properties of nuclei and the 6) Nuclear Masses: strong force The mass of the proton is M p = MeV/c 2. The mass of the neutron is M n = MeV/c 2. The mass of the deuteron (a proton neutron bound state, 2 1H) is M deuteron = MeV/c 2. But = MeV/c 2. There is a 2.2 MeV/c 2 mass difference! Q: Is binding energy the depth of potential well? This mass difference is known as the nuclear binding energy. It is what keeps the nucleus stable. Recall that the ionization energy for the hydrogen atom is ev. 5

6 Atomic Masses Commonly use atomic masses rather than nuclear masses (see Appendix 8 in textbook). Atomic mass unit 1u = MeV/c 2 (Based on setting unit to Carbon: 12 6C: has a mass of 12u exactly) M H = u (not the proton mass) M n = u 6

7 Properties of nuclei and the strong force 7) Nuclear Binding Energies: 7

8 Properties of nuclei and the strong force 7) Nuclear Binding Energies: protons neutrons nucleus Important things to notice: Electron masses cancel in this formula M H is the hydrogen mass, not the proton mass M atom is the mass of the atom, not the nucleus 8

9 Example Calculate binding energy of Oxygen-16, 16 8O (M atom = u) In fact, it is more common to calculate the binding energy per nucleon: B/A B/A = MeV/16 = 7.98 MeV Calculating B/A for all nuclides gives the following... 9

10 Binding Energy per Nucleon 10

11 Binding Energy per Nucleon Fission Fusion 11

12 Properties of nuclei and the strong force 8) For A>20 or so, B/A is roughly 8 MeV for all nuclides. This is because of the short range nature of the nuclear force. Qualitatively, each nucleon only interacts with its nearest neighbors (within ~2 fm or so) and not with those further away. For a small nucleus, all nucleons still bind each other together. But for a large nucleus (A>20), only nearest neighbors do. This is another manifestation of the short-range nature of the strong force. 10 fm 12

13 Nuclear Stability The line representing the stable nuclides is the line of stability. All stable and unstable nuclei that are longlived enough to be observed. It appears that for A 40, nature prefers the number of protons and neutrons in the nucleus to be about the same Z N. However, for A 40, there is a decided preference for N > Z because the nuclear force is independent of electric charge, i.e. independent of of whether the particles are nn, np, or pp. As the number of protons increases, the Coulomb force between all the protons becomes stronger until it eventually affects the binding significantly. Q: How strong is the Coulomb interaction between protons? How does the total Coulomb interaction increase with A?

14 Properties of nuclei and the strong force 10) Nuclear Magnetic Moments: Just replace m e with m p to get the nuclear magneton: Gives rise to the hyperfine structure. The nuclear magneton is μ B (because the mass of the proton is so much more than the mass of the electron). However, there s something odd: μ proton = μ N μ neutron = μ N The g factor here is not 1 (again). This points to the fact that protons and neutrons are not fundamental particles! 14

15 Practice Question 1 Which of the following statements are true about the nuclear force? I. It is charge-independent (F pp =F pn =F nn ) II. It s strength goes as 1/r 2 where r is the distance between two nucleons III. It is weaker than the gravitational force between two nucleons separated by 0.5 fm A. I and II are true; III is false B. I is true; II and III are false C. I and III are true; II is false D. all are true E. all are false 15

16 Practice Question 1 Which of the following statements are true about the nuclear force? I. It is charge-independent (F pp =F pn =F nn ) II. III. It s strength goes as 1/r 2 where r is the distance between two nucleons It is weaker than the gravitational force between two nucleons separated by 0.5 fm A. I and II are true; III is false B. I is true; II and III are false C. I and III are true; II is false D. all are true Strength is approximately constant. Much stronger than gravity. Recall gravitational potential energy is ~ MeV E. all are false 16

17 Practice Question 2 Which of the following best describes nuclear forces? A. They are short ranged. B. They go to zero at r = 0. C. They are long ranged like the Coulomb force. D. They are charge dependent. E. All of the above 17

18 Practice Question 2 Which of the following best describes nuclear forces? A. They are short ranged. B. They go to zero at r = 0. C. They are long ranged like the Coulomb force. D. They are charge dependent. E. All of the above 18

19 Radioactivity Many nuclides are stable, but many are not. There are three (common) types of radioactive decay: 1) Alpha decay: a Helium nucleus is emitted 2) Beta decay: an electron or positron is emitted 3) Gamma decay: a photon is emitted We will consider the physics of each of these processes. 19

20 Radioactive Decay For each kind of radioactive isotope, we define decay constant λ as the probability of decay per second. If we have N nuclei at some instant, the average number that will decay in time interval dt is λndt so dn = - λndt 20

21 Radioactive decay law We can solve this differential equation by integrating: N 0 =N(0)=the initial number of nuclides. This is the Radioactive decay law The number of surviving nuclei follows an exponential decay with time. The number that decayed is 21

22 Half-life We can also define the half-live t 1/2 as the time for half of the original sample of nuclei to decay: Therefore, so that Behavior of half-lives: At t=t 1/2, N=N 0 /2 At t=2t 1/2, N=N 0 /4 At t=3t 1/2, N=N 0 /8 etc... N t = N 0 2 t t ( ) / 1/2 Note decay constant λ has units of inverse time, i.e. s -1 or min -1 or yr -1 or 22

23 Example The half-life of U-235 is years. What fraction will survive after 1 billion years (1 Gyr)? 23

24 Activity A Geiger counter measures the number of decays per second. This quantity is called the activity R. Notice that: the half-life t 1/2 and decay constant λ are constant in time. But the activity decreases with time, and it depends on the amount of the substance (i.e. how many nuclei there are). So, in summary activity is: A common unit of radioactivity: 1 Becquerel (1 Bq) = 1 decay per second. 24

25 Example A sample of 2 grams of Ra has an activity of 7.4 x Bq. Find the half-life of Ra. First find N, the number of nuclei of Ra. So: Finally: 25

26 Carbon Dating Carbon in atmospheric CO 2 is mostly Carbon-12, 12 6C. Cosmic ray neutrons strike the atmosphere, turning Nitrogen- 14 into Carbon-14 (which is radioactive): This happens at a steady rate. Plants absorb CO 2, and animals ingest plants. So all living organisms have the same ratio of C-14 to C-12. And they have the same activity, R=0.255 Bq per gram of Carbon. When an organism dies, it stops absorbing CO 2. But its C-14 keeps decaying, so the C-14/C-12 ratio drops, and the activity R drops. Measuring the activity determines how long ago the organism died. 26

27 Example A A dead organism has activity today of 0.03 Bq per gram of Carbon. How long ago did it die? The half-life of carbon is 5740 years. original activity: current activity: Solve for T: 4 27

28 Example B What fraction of Carbon in living organisms is C-14? From before, we know that the time constant is: And the activity is for a living organism is: Solving for N (or N 14 ) gives: 14 But Carbon is mostly C-12, so the number of carbon atoms per gram is: So the ratio is: 14 28

29 Practice Question 1 A certain radioactive substance has a half-life of 8 hours. After 6 hours, what will be the half-life of a sample of this substance A. 2 hours B. 8 hours C. 14 hours 29

30 Practice Question 1 A certain radioactive substance has a half-life of 8 hours. After 6 hours, what will be the half-life of a sample of this substance A. 2 hours B. 8 hours C. 14 hours Half-life characterizes the particular substance, and doesn t change with time. The amount of substance does decrease with time. 30

31 The 3 types of radioactive decay Ι. γ-decay Just as the (hydrogen) atom can have excited states, the nucleus can also have excited states. When a nucleus in an excited state decays to a lower energy level, it will emit a photon (γ-ray): All nuclei have many excited states. Another example: Cobalt-60 decays via beta decay (see later) to an excited Nickel state, which then decays to the groundstate Nickel and two photons: 31

32 The 3 types of radioactive decay ΙΙ. α-decay Only occurs in heavy nuclei: Z > 82 A Helium nucleus 4 2He, a.k.a. an alpha (α) particle, composed of 2 protons+2 neutrons, is emitted. Decay of parent into daughter: Energy conservation: The total net kinematic energy released is the disintegration energy or Q-value: 32

33 Alpha decay will only occur if and only if Q>0 for any particular parent-daughter possibility. This turns out to be true only for Z 82 isotopes (Pb and heavier). There are a few exceptions. Why don t we ever get radioactive decay with the emission of a proton, a neutron, or even a deuteron? See Example in book! Let s calculate the kinetic energy of an alpha particle in the decay P D+α. Start from the rest frame of the parent. From momentum conservation: We can replace the velocity with the kinetic energy: This means that the disintegration energy is: 33

34 Alternatively we can write where you can notice that this is an exact expression. K α is fixed for a given α-decay because it s a 2-body decay. Because mass is approximately proportional to the mass number A, we get where A is the mass number of the parent nucleus. Since A 4, K Q, which means that most of the kinetic energy is carried away by the α particle, very little by the daughter nucleus. α-decay occurs only for Z>82 and A>

35 Find K α and K D for: Example From the masses we get Q = 6.21 MeV. Daughter nucleus only carries: 35

36 Alpha Decay Example: Smoke Detectors The source of ionizing radiation is a minute quantity of americium-241 (~ 1/5000th of a gram), which is a source of alpha particles. The ionization chamber consists of two plates separated by about a centimeter. The battery applies a voltage to the plates, charging one plate positive and the other plate negative. Alpha particles constantly released by the americium knock electrons off of the atoms in the air, ionizing the oxygen and nitrogen atoms in the chamber. The positively-charged oxygen and nitrogen atoms are attracted to the negative plate and the electrons are attracted to the positive plate, generating a small, continuous electric current. When smoke enters the ionization chamber, the smoke particles attach to the ions and neutralize them, so they do not reach the plate. The drop in current between the plates triggers the alarm.

37 The 3 types of radioactive decay ΙΙΙ. β-decay: An electron and a neutrino are emitted: Here, a neutron in the parent turned into a proton, and electron, and a neutrino, but only the electron and neutrino are emitted. The neutrino has no electric charge. It has (almost) zero mass. It (almost) never interacts with anything. How do we know it exists? 37

38 Suppose that we don t know about the neutrino. So: By analogy with our discussion of alpha decay, we have: Q= Kinetic energy released = K D + K e If m P and m D are atomic masses, then let s calculate Q in terms of atomic masses m P and m D (not nuclear masses). Beta decay will occur whenever m P >m D. It is very common, and occurs for both light and heavy nuclei. 38

39 What is the kinetic energy of the electron? By analogy with α-decay we might expect: If there is no neutrino, red is what is expected. Blue was what was observed. So do we have non-conservation of energy? 39

40 Pauli s Hypothesis (1930) A neutrino is also emitted in β-decay, with zero mass, zero charge and spin ½ Then all conservation laws are preserved. There are 3 types of β-decay: electron emission positron emission 1) β - : We already showed that 2) β + : 1) Electron capture: Again, it s important to remember that m P and m D here are the atomic masses (i.e. they include the mass of the electrons). 40

41 Consider the isotopes: Example What are the β-decay possibilities? 1) β - possibilities: Violates conservation of energy. 2) β + possibilities: 3) Electron capture possibilities: 41

42 Practice Question 1 When 60 27Co undergoes β - decay, what is the daughter nucleus? A Fe B Fe C Ni D Ni 42

43 Practice Question 1 When 60 27Co undergoes β - decay, what is the daughter nucleus? A. B. C. D Fe Fe Ni Ni (note: total charge is conserved) 43

44 Alpha: Summary: Radiation Range is less than a tenth of a millimeter inside the body Its main radiation hazard comes when it is ingested into the body It has great destructive power within its short range In contact with fast-growing membranes and living cells, it is positioned for maximum damage. Not suitable for radiation therapy Beta: High energy electrons have greater range of penetration than alpha particles, but much less than gamma rays The radiation hazard from betas is greatest if they are ingested Gamma: Most gamma rays are high energy and very penetrating It is the most useful type of radiation for medical purposes But the most dangerous Ability to penetrate large thicknesses of material

45 Announcement Next lecture: Fission and Fusion, particle physics and Cosmology Follow by review session Final exam content: ch2-13, emphasis on chapters after 2 nd midterm Final exam time: TBD. Exam format: multiple-choices + open-end 45