Nuclear Physics II Mr. Jean IB Physics 12 Nuclear Physics II Mr. Jean

Atomic mass unit: 1 u = 1.6606 x 10-27 kg One atomic mass unit (1 u) is equal to one-twelfth of the mass of the most abundant form of the carbon atom--carbon-12. Atomic mass unit: 1 u = 1.6606 x 10-27 kg Common atomic masses: Proton: 1.007276 u Neutron: 1.008665 u Electron: 0.00055 u Hydrogen: 1.007825 u

Example 2: The average atomic mass of Boron-11 is 11. 009305 u Example 2: The average atomic mass of Boron-11 is 11.009305 u. What is the mass of the nucleus of one boron atom in kg? = 11.009305 Electron: 0.00055 u The mass of the nucleus is the atomic mass less the mass of Z = 5 electrons: Mass = 11.009305 u – 5(0.00055 u) 1 boron nucleus = 11.00656 u m = 1.83 x 10-26 kg

Mass and Energy Recall Einstein’s equivalency formula for m and E: The energy of a mass of 1 u can be found: E = (1 u)c2 = (1.66 x 10-27 kg)(3 x 108 m/s)2 E = 1.49 x 10-10 J Or E = 931.5 MeV When converting amu to energy:

Example 3: What is the rest mass energy of a proton (1.007276 u)? E = mc2 = (1.00726 u)(931.5 MeV/u) Proton: E = 938.3 MeV Similar conversions show other rest mass energies: Neutron: E = 939.6 MeV Electron: E = 0.511 MeV

The Mass Defect The mass defect is the difference between the rest mass of a nucleus and the sum of the rest masses of its constituent nucleons. The whole is less than the sum of the parts! Consider the carbon-12 atom (12.00000 u): Nuclear mass = Mass of atom – Electron masses = 12.00000 u – 6(0.00055 u) = 11.996706 u The nucleus of the carbon-12 atom has this mass. (Continued . . .)

Mass Defect (Continued) Mass of carbon-12 nucleus: 11.996706 Proton: 1.007276 u Neutron: 1.008665 u The nucleus contains 6 protons and 6 neutrons: 6 p = 6(1.007276 u) = 6.043656 u 6 n = 6(1.008665 u) = 6.051990 u Total mass of parts: = 12.095646 u Mass defect mD = 12.095646 u – 11.996706 u mD = 0.098940 u

The binding energy for the carbon-12 example is: The binding energy EB of a nucleus is the energy required to separate a nucleus into its constituent parts. EB = mDc2 where c2 = 931.5 MeV/u The binding energy for the carbon-12 example is: EB = (0.098940 u)(931.5 MeV/u) EB = 92.2 MeV Binding EB for C-12:

Binding Energy per Nucleon An important way of comparing the nuclei of atoms is finding their binding energy per nucleon: Binding energy per nucleon For our C-12 example A = 12 and:

Formula for Mass Defect The following formula is useful for mass defect: Mass defect mD mH = 1.007825 u; mn = 1.008665 u Z is atomic number; N is neutron number; M is mass of atom (including electrons). By using the mass of the hydrogen atom, you avoid the necessity of subtracting electron masses.

M = 4.002603 u (From nuclide tables) Example 4: Find the mass defect for the nucleus of helium-4. (M = 4.002603 u) Mass defect mD ZmH = (2)(1.007825 u) = 2.015650 u Nmn = (2)(1.008665 u) = 2.017330 u M = 4.002603 u (From nuclide tables) mD = (2.015650 u + 2.017330 u) - 4.002603 u mD = 0.030377 u

Since there are four nucleons, we find that Example 4 (Cont.) Find the binding energy per nucleon for helium-4. (mD = 0.030377 u) EB = mDc2 where c2 = 931.5 MeV/u EB = (0.030377 u)(931.5 MeV/u) = 28.3 MeV A total of 28.3 MeV is required To tear apart the nucleons from the He-4 atom. Since there are four nucleons, we find that

Binding Energy Vs. Mass Number Curve shows that EB increases with A and peaks at A = 60. Heavier nuclei are less stable. Mass number A Binding Energy per nucleon 50 100 150 250 200 2 6 8 4 Green region is for most stable atoms. For heavier nuclei, energy is released when they break up (fission). For lighter nuclei, energy is released when they fuse together (fusion).

Stability Curve Nuclear particles are held together by a nuclear strong force. Atomic number Z Neutron number N Stable nuclei Z = N 20 40 60 80 100 140 120 A stable nucleus remains forever, but as the ratio of N/Z gets larger, the atoms decay. Elements with Z > 82 are all unstable.

Radioactivity As the heavier atoms become more unstable, particles and photons are emitted from the nucleus and it is said to be radioactive. All elements with A > 82 are radioactive. a b- b+ g Examples are: Alpha particles a b- particles (electrons) Gamma rays g b+ particles (positrons)

Types of Radioactivity: There are three major types of radiation that the nuclear physicist is concerned with: alpha, beta, and gamma. Alpha radiation consists of particles, alpha particles. The alpha particles are actually helium nuclei. Beta radiation is also made up of particles – electrons. Gamma radiation is made up of very short wavelength electromagnetic waves. The reason for the odd names is a simple one. The types of radiation were discovered before the particles were. So Ernest Rutherford discovered alpha particles before anyone knew anything about helium nuclei.

Parent nucleus Cm-244. The daughter isotope is Pu-240 Alpha Decay Parent nucleus Cm-244. The daughter isotope is Pu-240 96Cm244 94Pu240

The Alpha Particle An alpha particle a is the nucleus of a helium atom consisting of two protons and two neutrons tightly bound. Charge = +2e- = 3.2 x 10-19 C Mass = 4.001506 u Relatively low speeds ( 0.1c ) Not very penetrating

Why alpha particle instead of other light nuclei Energy Q associated with the emission of various particles from a 235U nucleus.

There are always two questions that can be asked about any decay in atomic, nuclear or particle physics: (i) How much kinetic energy was released? and (ii) How quickly did it happen? (i.e. Energy? and Time?). Lets look at energy and then we will deal with time at the end for all types of decays.

The Energy of the α-particle, Tα Mass of X Q Mass of Y +  particle And the energy released in the decay is simply given by energy

X is parent atom and Y is daughter atom Radioactive Decay As discussed, when the ratio of N/Z gets very large, the nucleus becomes unstable and often particles and/or photons are emitted. Alpha decay results in the loss of two protons and two neutrons from the nucleus. X is parent atom and Y is daughter atom The energy is carried away primarily by the K.E. of the alpha particle.

Radium-226 decays into radon-222. Example: Write the reaction that occurs when radium-226 decays by alpha emission. From tables, we find Z and A for nuclides. The daughter atom: Z = 86, A = 222 Radium-226 decays into radon-222.

The Beta-minus Particle A beta-minus particle b- is simply an electron that has been expelled from the nucleus. Charge = e- = -1.6 x 10-19 C - Mass = 0.00055 u - High speeds (near c) - Very penetrating -

Beta decay Positive: The Positron A beta positive particle b+ is essentially an electron with positive charge. The mass and speeds are similar. Charge = +e- = 1.6 x 10-19 C + Mass = 0.00055 u + High speeds (near c) + Very penetrating +

X is parent atom and Y is daughter atom Beta-minus Decay Beta-minus b- decay results when a neutron decays into a proton and an electron. Thus, the Z-number increases by one. X is parent atom and Y is daughter atom The energy is carried away primarily by the K.E. of the electron. -

X is parent atom and Y is daughter atom Beta-plus Decay Beta-plus b+ decay results when a proton decays into a neutron and a positron. Thus, the Z-number decreases by one. X is parent atom and Y is daughter atom The energy is carried away primarily by the K.E. of the positron. +

X is parent atom and Y is daughter atom Beta-plus Decay Beta-plus b+ decay results when a proton decays into a neutron and a positron. Thus, the Z-number decreases by one. X is parent atom and Y is daughter atom The energy is carried away primarily by the K.E. of the positron. +

The Gamma Photon A gamma ray g has very high electromagnetic radiation carrying energy away from the nucleus. Charge = Zero (0) g Mass = zero (0) g Speed = c (3 x 108 m/s) g Most penetrating radiation g

Gamma and Beta decays are similar A Family of Four Force Carriers β decay γ decay N Unlike α decay, β and γ decays are closely related (e.g. like cousins). They often occur together as in the typical decay scheme (i.e. 198Au) They just involved changes in nucleon states (p n, n p, p p) They involve the same basic force (γ, W± ) carrier but in different state But β decays are generally much slower (~100,000) than γ decays (produced by EM force) because the Ws are heavy particles (which makes force weaker)

Gamma and Beta decays are very similar Decay Name of process Interaction Out Channel Nucleon + Zero Leptons Gamma Decay EM Internal Conversion EM Nucleon + One Lepton Electron Capture weak Pair Internal Conversion EM Nucleon + Two Leptons β+ Decay weak weak β- Decay

Feynman Diagrams - Similarity OUT CHANNEL ---- One nucleon + 2 leptons TIME p n p p’ p n BETA MINUS DECAY BETA PLUS DECAY PAIR INTERNAL CONVERSION All these decay types are similar in structure They all have a 4 point vertex They all have 3 particles in the final state The fact that the Q of the decay is shared between 3 particles means that the outgoing observed particle [ie. electron or positron] has a spectrum of energies in the range (0 to Q).

Feynman Diagrams - Similarity OUT CHANNEL ---- One nucleon + 1 lepton TIME n p’ p’ p p p  GAMMA DECAY INTERNAL CONVERSION ELECTRON CAPTURE [Mono-energetic photons ] [Mono-energetic electrons ] [Mono-energetic neutrinos] All these decays have only two particles in their output state. The Q of the decay is shared between only 2 particles Conservation of Energy: The emitted particle (γ , e-, νe) is monoenergetic.

Quark level Feynman Diagrams - Similarity TIME d u u d u d d u u d u d d u u d u u BETA MINUS DECAY BETA PLUS DECAY PAIR INTERNAL CONVERSION The proton is made of 3 quarks – uud (up, up, down) The neutron is made also of 3 quarks - udd (up, down, down) We see the very close similarity of pattern between reactions through W and γ particles. NOTE: only vertices of 3 particles are now seen (makes sense)

Quark level Feynman Diagrams - Similarity TIME d u d d u u d u u d u u d u u d u u  GAMMA DECAY INTERNAL CONVERSION ELECTRON CAPTURE Again we see that there are ONLY 3 PARTICLE – VERTICES We see the similarity of the decays are propogated through the intermedicate “Force” particles (W and γ). Remember in INTERNAL CONV. And ELECTRON CAPTURE the electron comes from the core electron orbitals of THE ATOM.

Radioactive Materials The rate of decay for radioactive substances is expressed in terms of the activity R, given by: Activity N = Number of undecayed nuclei One becquerel (Bq) is an activity equal to one disintegration per second (1 s-1). One curie (Ci) is the activity of a radioactive material that decays at the rate of 3.7 x 1010 Bq or 3.7 x 1010 disintegrations per second.

The Half-Life The half-life T1/2 of an isotope is the time in which one-half of its unstable nuclei will decay. No Number of Half-lives Number Undecayed Nuclei 1 4 3 2 Where n is number of half-lives

Example: Radium-226 has a half-life of 1620 years. This is shown in the graph below. One kg of radium-226 begins the thing. After one half-life (1620years) only half of the sample remains – the other half has decayed into some other element. After two half-lives only one fourth would remain and so on.

Half-Life (Cont.) The same reasoning will apply to activity R or to amount of material. In general, the following three equations can be applied to radioactivity: Nuclei Remaining Activity R Mass Remaining Number of Half-lives:

First we determine the number of half-lives: Example 6: A sample of iodine-131 has an initial activity of 5 mCi. The half-life of I-131 is 8 days. What is the activity of the sample 32 days later? First we determine the number of half-lives: n = 4 half-lives R = 0.313 mCi There would also be 1/16 remaining of the mass and 1/16 of the number of nuclei.