Nuclear and Particle Physics Lecture #2 Mayda M. Velasco Fall 2011
Constituents of Matter
The Nucleus Atoms consist of a nucleus and an electron shell. A nucleus consists of nucleons: protons and neutrons. Nucleon mass ~ 2000 mass e- 1 fm (femtometer, Fermi) = 10 m -15 is the typical length scale of nuclear physics 1 MeV (Mega-electron volt) = 1.602J is the typical energy scale of nuclear physics
To study structure of hadrons we take advantage of relation between momentum scales & position resolution P Δx > ħc ~ 200 MeV fm
Nuclear Theory and Experiment Atomic physics has a single consistent theory Quantum Electro Dynamics (QED). This is unfortunately not true for nuclear physics: There is a fundamental theory of the strong interaction Quantum Chromo Dynamics (QCD) - but it describes the interactions between quarks, not nucleons. The energies involved in nuclear decays are of the order of 1-10 MeV, less than 0.1 % of the mass of the nucleus. As a result nonrelativistic QM can be used to describe the nucleus.
Nuclear Theory and Experiment This is not true for the study of the structure of the nucleon, where the incident beam energy in a scattering experiment may be 100 times the proton mass equivalent. Both nuclei and nucleons are complex systems involving many constituents. The theories and models that describe them are therefore often phenomenological in nature and nuclear physics is rather led by experiment than by theory.
Nucleons consists of 3 quarks (and gluons) Electron Electron
Nuclides A nuclide is a specific combination of a number of protons and neutrons. Z A X N is the complete symbol for a nuclide, but the information is redundant and A X is sufficient. X is the chemical symbol of the element Z is the atomic number, giving the number of protons in the nucleus (and electrons in the shell) N is the number of neutrons A = Z + N is the mass number
Nuclides Nuclides with the same atomic number Z are called isotopes E.g. 235 U and 238 U same N isotones E.g. 2 H(d) and 3 He same A isobars Mirror Nuclei: Two nuclei with odd A in which the number of protons in one nucleus is equal to the number of neutrons in the other and vice versa.
Nuclide Chart nuclides can be put onto a chart. e.g. Z vs N different radioactive decays can be connected with movement in the chart, e.g. α-decay this allows to visualize entire decay chains it also allows to visualize other properties, e.g. lifetime or date of first detection
Nuclide Chart Lifetime Experimental Chart of Nuclides 2000 2975 isotopes Dominant Decay type: α Dominant Decay type: β+ 82 8 20 28 20 28 50 50 82 Dominant Decay type: β- 126 Half-life Range Unknown <0.1 s 0.1-5 s 5-100 s 100 s - 1 h 1 h - 1 y 1 y - 1 Gy Stable 2 2 8
Evolution of the Table of Isotopes Publication Year 1940 1944 1948 1953 1958 1967 1978 1995 1997 Naturally Abundant 33 As 32 Ge 31 Ga 30 Zn 29 Cu 28 Ni 27 Co 26 Fe 25 Mn 24 Cr 23 V 22 Ti 21 Sc 20 Ca 19 K 18 Ar 17 Cl 16 S 15 P 14 Si 13 Al 12 Mg 26 11 Na 10 Ne 24 9 F 22 8 O 20 7 N 18 6 C 5 B 16 4 Be 12 14 3 Li 10 2 He 1 H 6 8 2 4 28 38 Sr 37 Rb 36 Kr 35 Br 34 Se 30 32 34 59 Pr 58 Ce 57 La 56 Ba 55 Cs 54 Xe 53 I 52 Te 51 Sb 50 Sn 49 In 48 Cd 47 Ag 46 Pd 45 Rh 44 Ru 43 Tc 42 Mo 41 Nb 40 Zr 39 Y 36 38 40 42 44 46 48 50 52 54 83 Bi 56 58 60 86 Rn 85 At 84 Po 71 Lu 70 Yb 69 Tm 68 Er 67 Ho 66 Dy 65 Tb 64 Gd 63 Eu 62 Sm 61 Pm 60 Nd 62 64 66 88 Ra 87 Fr 68 70 72 74 91 Pa 90 Th 89 Ac 82 Pb 81 Tl 80 Hg 79 Au 78 Pt 77 Ir 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 76 92 U 78 80 76 Os 75 Re 74 W 73 Ta 72 Hf 82 128 126 124 84 86 88 90 92 94 96 110110 109 Mt 108 Hs 107 Ns 106 Sg 105 Ha 104 Rf 103 Lr 102 No 101 Md 100 Fm 99 Es 98 Cf 97 Bk 96 Cm 95 Am 154 156 94 Pu 93 Np 152 130 132 98 100 146 144 140 142 136 138 134 102 148 104 150 106 108 114 112 110 111111 158 160 116 112112 118 120
Stability
Final comment on Stability
Important Property Nuclear Masses Atomic masses (actually, ionic masses) can be determined with high precision using mass spectrometers. Because the electron mass is know very precisely this allows to determine the mass of the concerned nucleus. Mass spectrometers use a combination of electric and magnetic fields to measure the Q/M ratio and thus the mass M.
Determining the Mass of an Atom If a charged particle moves at a constant speed through a magnetic field it will experience a force (=Bqv) that provides a centripetal force (=mv 2 /r)to make it move through circular motion: Therefore, B q v = m v 2 r m = B q r v 1 If velocity, charge and flux density are constant then the mass of the particle is proportional to the radius of the circle it moves through.
Bainbridge Mass Spectrometer In a Bainbridge mass spectrometer, positive ions are created and injected into the velocity selector. Here the positive ions are subjected to both magnetic and electric fields. These create forces that act in opposite directions.
Photographic plate
If the two forces are balanced the particle will travel in a straight line through the exit slit (s 2 in the diagram)... B q v = E q so the only particles to exit the velocity selector are those with velocity equal to the ratio of the Electric field strength (E) to the magnetic flux density (B)... v = E B Sub into Eq. 1 gives: m = B 2 q r E
Nuclear Masses The mass reference is not the proton or the hydrogen atom, but the isotope 12 C. Carbon and it s many compounds are always present in a spectrometer and are well suited for a mass calibration. An atomic mass unit u is therefore defined as 1/12 of the mass of the 12 C nuclide: For comparison, the proton mass is 938.272 MeV/c 2.
Compare u with GeV/c 2 Early indication of Neutron composition
Nuclear Abundance One application of nuclear mass spectroscopy is the study of relative isotope abundances in the solar system. (see figure, normalized to Si). They are generally the same throughout the solar system deuterium and helium : fusion in the first minutes after the big bang, nuclei up to Fe : stars, heavier nuclei in supernovae.
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Nuclear Binding Energy The binding energy B of a nucleus is the difference in in mass energy between a nucleus A Z X N and it s constituent Z protons and N neutrons: If B > 0 stability to go down Where m( A X ) is the atomic mass of A X. The binding energy is determined from atomic masses, since they can be measured much more precisely than nuclear masses. Grouping the Z proton and electron masses into Z neutral hydrogen atoms, we can re-write this as: With the masses generally given in atomic mass units, it is convenient to include the conversion factor in, thus c 2 = 931.481 MeV/u
Nuclear Binding Energy Fission favored Saturation This agrees with experiment, which suggests that the radius of a nucleus scales with the 1/3rd power of A, R RMS 1.1 A 1/3 fm
B/A vs A plot curve is relatively constant, with the exception of very light nuclei. The average binding energy of most nuclei is about 8 MeV per nucleon. curve reaches a peak around A=60, to be precise at 56 Fe. This suggests that light nuclei, below 56 Fe can gain energy by fusion into heavier nuclei. Heavy nuclei above 56 Fe can release energy by fission into lighter nuclei. This is already the basic argument why only nuclides up to 56 Fe can be formed in normal stars. 4 He and 12 C appear to be off the curve. We will get back to this when we study the shell model.
Liquid Drop Model Treats nucleus as a dense, incompressible, spherical liquid drop Constant density B/A~constant (except for low-a)
Nuclear Models
Semi-Empirical Mass Formula (SEMF) Liquid Drop Model
Binding Energy vs Mass Formula For most nuclei (nuclides) with A > 20 the binding energy is well reproduced by a semi- empirical formula based on the idea the the nucleus can be thought of as a liquid drop.
Details of various components Nucleon- Nucleon interactions Long and short range Spin dependent
Volume term: Each nucleon has a binding energy which binds it to the nucleus. Therefore we get a term proportional to the volume i.e. proportional to A. This term reflects the short-range nature of the strong forces. If a nucleon interacted with all other nucleons we would expect an energy term of proportional to A(A 1), but the fact that it turns out to be proportional to A indicates that a nucleon only interact with its nearest neighbors. Surface term: The nucleons at the surface of the liquid drop only interact with other nucleons inside the nucleus, so that their binding energy is reduced. This leads to a reduction of the binding energy proportional to the surface area of the drop, i.e. proportional to A 2/3
Coulomb term: Although the binding energy is mainly due to the strong nuclear force, the binding energy is reduced owing to the Coulomb repulsion between the protons. We expect this to be proportional to the square of the nuclear charge, Z, ( the electromagnetic force is long-range so each proton interact with all the others), and by Coulomb s law it is expected to be inversely proportional to the nuclear radius, (the Coulomb energy of a charged sphere of radius R and charge Q is 3Q 2 /(20π 0 R)). The Coulomb term is therefore proportional to A -1/3
Asymmetry term: This is a quantum effect arising from the Pauli exclusion principle which only allows two protons or two neutrons (with opposite spin direction) in each energy state. Ex. If we exchange one of the neutrons by a proton then that proton would be required by the exclusion principle to occupy a higher energy state, since all the ones below it are already occupied.
The upshot of this is that nuclides with Z = N = (A Z) have a higher binding energy, whereas for nuclei with different numbers of protons and neutrons (for fixed A) the binding energy decreases as the square of the number difference. The spacing between energy levels is inversely proportional to the volume of the nucleus - this can be seen by treating the nucleus as a three-dimensional potential well- and therefore inversely proportional to A. Thus we get a term 4
Pairing term: It is found experimentally that two protons or two neutrons bind more strongly than one proton and one neutron. In order to account for this experimentally observed phenomenon we add a term to the binding energy if number of protons and number of neutrons are both even, we subtract the same term if these are both odd, and do nothing if one is odd and the other is even. Bohr and Mottelson showed that this term was inversely proportional to the square root of the atomic mass number.
Volume term -- Summary For A 30 we observe and assume B / A Const. Vol R 3 A B V =a V A, where a V ~ 8 MeV A nucleon attracts its closest neighbors
Summary
Surface term -- Summary A nucleon near the surface has fewer neighbors As A, the surface area so the number of surface nucleons Need to reduce Binding Energy Area R 2 A ⅔ B S = a S A ⅔
Coulomb repulsion term -- Summary Proton-proton repulsion makes nucleons less tightly bound. Long-distance Coulomb repulsion Need to reduce Binding Energy Z protons: How many p p pairs? Z(Z 1)/2 Coulomb energy R -1 A ⅓ B C = a C Z(Z 1) A ⅓
Asymmetry term -- Summary Light, stable nuclei tend to have Z ~ N ~ A/2. As A there is more p p Coulomb repulsion so N > Z B Sym = a Sym (A 2Z ) 2 A -1 Imbalance between N and Z Reduces the effect as A
Pairing term -- Summary Even Even nuclei are particularly stable Z=even & N=even. Odd Odd nuclei tend to be unstable Later : Shell Model B pair =± Even Even + Odd Odd Even Odd 0 B pair A -¾ B pair =a pair A -¾
Isospin (I) I 3 = ½ proton and I 3 = - ½ neutron Neutrons and Protons are the same from the point of view of nuclear (strong) interactions Mirror nuclei 14 6 C 8 and 14 8 O 6 versus 14 7 N 7