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1 Lecture Notes in Nuclear Structure Physics B. Alex Brown November 1 Lecture Notes in Nuclear Structure Physics B. Alex Brown November 2005 National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy Michigan State University, E. Lansing, MI 48824 CONTENTS 2 Contents 1 Nuclear masses 6 1.1 Masses and binding energies . 6 1.2 Q valuesandseparationenergies . 10 1.3 Theliquid-dropmodel .......................... 18 2 Rms charge radii 25 3 Charge densities and form factors 31 4 Overview of nuclear decays 40 4.1 Decaywidthsandlifetimes. 41 4.2 Alphaandclusterdecay ......................... 42 4.3 Betadecay................................. 51 4.3.1 BetadecayQvalues ....................... 52 4.3.2 Allowedbetadecay ........................ 53 4.3.3 Phase-space for allowed beta decay . 57 4.3.4 Weak-interaction coupling constants . 59 4.3.5 Doublebetadecay ........................ 59 4.4 Gammadecay............................... 61 4.4.1 Reduced transition probabilities for gamma decay . .... 62 4.4.2 Weisskopf units for gamma decay . 65 5 The Fermi gas model 68 6 Overview of the nuclear shell model 71 7 The one-body potential 77 7.1 Generalproperties ............................ 77 7.2 Theharmonic-oscillatorpotential . ... 79 7.3 Separation of intrinsic and center-of-mass motion . ....... 81 7.3.1 Thekineticenergy ........................ 81 7.3.2 Theharmonic-oscillator . 83 8 The Woods-Saxon potential 87 8.1 Generalform ............................... 87 8.2 Computer program for the Woods-Saxon potential . .... 91 8.2.1 Exampleforboundstates . 92 8.2.2 Changingthepotentialparameters . 93 8.2.3 Widthofanunboundstateresonance . 94 8.2.4 Width of an unbound state resonance at a fixed energy . 95 9 The general many-body problem for fermions 97 CONTENTS 3 10 Conserved quantum numbers 101 10.1 Angularmomentum. 101 10.2Parity ................................... 101 10.3Isospin................................... 102 11 Quantum numbers for the two nucleon system 107 11.1 Twoneutronsortwoprotons . 107 11.2Proton-neutron .............................. 108 11.3Twonucleons ............................... 108 12 The Hartree-Fock approximation 110 12.1 Properties of single Slater determinants . ...... 110 12.2 Derivation of the Hartree-Fock equations . ..... 111 12.3 Examples of single-particle energies . ..... 113 12.4 Results with the Skyrme hamiltonian . 115 12.4.1 Bindingenergies . 118 12.4.2 Single-particleenergies . 121 12.4.3 Rms charge radii and charge densities . 122 12.4.4 Displacementenergies . 129 13 Angular Momentum and Tensor Algebra 136 13.1 Angularmomentumcoupling . 136 13.1.1 Coupling of Two Angular Momenta . 136 13.1.2 Coupling of Three Angular Momenta . 138 13.1.3 Coupling of Four Angular Momenta . 140 13.2 Tensors and reduced matrix elements . 142 13.2.1 Special reduced matrix elements . 143 13.2.2 Productsoftensoroperators . 144 13.2.3 Other convensions for reduced matrix elements . .... 146 14 Single-particle electromagnetic moments 148 14.1 Generalresultsandnotation . 148 14.2 General results for closed-shell and single-particle configurations . 149 14.3 Magneticmoments . .. .. 151 14.4 Electricquadrupolemoments . 154 15 The Creation Operator Method 159 15.1Introduction................................ 159 15.2 Creation Operatorsand Wavefunctions . ... 159 15.3Operators ................................. 162 15.4 Examples of a+ and a MatrixElements . 162 CONTENTS 4 16 Many-Body Wavefunctions 165 16.1 Wavefunctions in the m-scheme ..................... 165 16.2 Wavefunctions in the J-scheme ..................... 170 16.3 Angularmomentumprojection. 173 16.3.1 Examples for (j =5/2)n configurations . 176 16.4 General form for the matrix elements in the J scheme. 179 17 The Two-Body Hamiltonian 181 17.1Introduction................................ 181 17.2 Formofthetwo-bodymatrixelements . 182 17.3 Generaltypesofinteractions . 183 17.4 Transformation from relative to center-of-mass coordinates . 185 17.5 Simplepotentials ............................. 186 17.6 Onebosonexchangepotentials. 187 17.7 Numericalexamples. 193 17.8 Density-dependent interactions . ... 193 18 Applications of the Two-Body Interactions 197 18.1 Interaction energies for closed shell, one-particle and one-hole configu- rations................................... 197 18.2 Interaction energies for diagonal two-particle configurations. 201 18.3 Interaction energies for diagonal two-hole configurations. .. 203 18.4 Interaction energies for particle-hole configurations .......... 204 19 Configuration mixing 206 19.1 Two-particleconfigurations . 206 19.2 Application to 18O ............................ 209 19.3 Many-particleconfigurations . 210 20 One-particle transfer 216 20.1 Fractionalparentagecoefficients . ... 216 20.1.1 Oneorbit ............................. 216 20.2Manyorbits ................................ 221 20.3 Spectroscopicfactors . 222 20.3.1 Basic definitions and sum-rules . 222 20.3.2 Isospindependence . 223 20.3.3 Simplesituations . 226 20.3.4 Center-of-masscorrections . 227 20.3.5 Computation of shell-model spectroscopic factors . ...... 228 20.4 Overlapfunctions ............................. 229 20.4.1 Definitionandproperties . 229 20.4.2 Asymptoticproperties . 230 20.4.3 The well-depth prescription . 231 CONTENTS 5 20.4.4 Beyond the well-depth prescription . 232 21 Experiments related to spectroscopic factors 236 21.1 Experimental results for specific nuclei . ..... 236 21.1.1 51V 50Ti ............................ 236 21.1.2 (3He,d)inthesdshell→ ...................... 237 21.1.3 (e,e′p) ............................... 242 21.1.4 Protondecay ........................... 245 21.1.5 Radioactivebeams . 247 21.2 Structure models for specific nuclei . ... 247 21.2.1 7Li................................. 248 21.2.2 12C................................. 249 21.2.3 16O................................. 249 21.2.4 Thesd-shell ............................ 250 21.2.5 40Ca and 48Ca........................... 251 21.2.6 51V................................. 251 21.2.7 208Pb................................ 252 21.3 Short-rangecorrelations . 252 21.4Conclusions ................................ 254 22 One-body transition operators and the OBTD 258 22.1 Isospin andproton-neutron formalism . .... 259 22.2 OBTD for a single-orbital configuration . .... 260 22.3 OBTDforatwo-orbitalconfiguration . 260 22.4 Scalarone-bodymatrixelements . 262 23 Two-particle transfer operators 265 24 Two-body transition operators and the TBTD 269 24.1 Scalartwo-bodyoperators . 270 24.2 Scalar TBTD for a single-orbital configuration . ...... 271 24.3 Scalar TBTD for a two-orbital configuration . .... 271 24.4 Sample calculations for interaction energies . ........ 273 25 Electromagnetic transitions 275 25.1 Operatorsandtransitionrates . 275 25.2 Moments in terms of electromagnetic operators . ..... 277 25.3 Nuclearmatrixelements . 277 25.4 Applications to simple situations . ... 279 25.4.1 Closed shell plus one particle . 279 25.4.2 Single-orbitconfigurations . 280 CONTENTS 6 26 Allowed beta decay 282 26.1 Formulationofallowedbetadecay. 282 26.2 Operatorsforallowedbetadecay . 283 26.2.1 Fermidecay ............................ 283 26.2.2 Gamow-Tellerdecay . 285 26.3Sumrules ................................. 287 26.4 Effective operators for Gamow-Teller matrix elements . ........ 288 1 NUCLEAR MASSES 6 1 Nuclear masses 1.1 Masses and binding energies A basic quantity which can be measured for the ground states of nuclei is the atomic mass M(N,Z) of the neutral atom with atomic mass number A and charge Z. Atomic masses are usually tabulated in terms of the mass excess defined by ∆(N,Z) M(N,Z) uA, (1.1) ≡ − where u is the Atomic Mass Unit defined by u = M(12C)/12 = 931.49386 MeV/c2. 1 I will use the data from the 2003 compilation of Audi, Wapstra and Thibault [1]. Fig. (1.1) shows the position on the nuclear chart for these measured masses together with the experimental error. There are 2127 nuclei measured with an accuracy of 0.2 MeV or better and 101 nuclei measured with an accuracy of greater than 0.2 MeV. For heavy nuclei one observes several chains of nuclei with a constant N Z value whose masses are obtained from the alpha-decay Q values. − Nuclear binding energy is defined as the energy required to break up a given nucleus into its constituent parts of N neutrons and Z protons. In terms of the atomic masses M(N,Z) the binding energy B(N,Z) 2 is defined by: B(N,Z)= ZM c2 + NM c2 M(N,Z)c2, (1.2) H n − 3 where MH is the mass of the hydrogen atom and Mn is the mass of the neutron. In terms of the mass excess the binding energy is given by: B(N,Z)= Z∆ c2 + N∆ c2 ∆(N,Z)c2, (1.3) H n − 2 2 where ∆H c =7.2890 MeV and ∆nc =8.0713 MeV. How do we know that nuclei are made up of protons and neutrons? In the 1920’s when it was observed that nuclei decay by the emission of alpha particles, protons and electrons one tried to make nuclear models out of constituent protons and electrons. 4 However, after the discovery of the neutron in 1932, it was observed that the 1This and other constants can be found on the website: http://physics.nist.gov/cuu/Constants/index.html 2The binding energy will also be denoted by BE. 3This binding energy also includes contribution from the Coulomb interaction between electrons which is approximately given by 1.43x10−5 Z2.39 MeV. On the scale of nuclear binding this can usually be ignored. It is most important− for heavy nuclei where, for example, for Z = 120 the electronic contribution is 1.34 MeV. 4“Just because barks come− out of dogs does not mean that dogs are made of barks.” (Denys Wilkinson). 1 NUCLEAR MASSES 7 Nuclei with measured
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