Nuclear Physics Instructor: Golam Dastegir Al-Quaderi Nuclear Shell Model • Both the infinite square-well potential and the harmonic oscillator potentials reproduce a few of the magic numbers. • Recall that the nuclear magic numbers are: 𝑁𝑁 = 2, 8, 20, 28, 50, 82, 126 𝑍𝑍 = 2, 8, 20, 28, 50, 82 • Infinite square-well potential gives closed shells at 𝑁𝑁 or 𝑍𝑍 equal to 2,8,20,34,40,58, … • Harmonic oscillator potential gives closed shells for proton or neutron numbers of 2, 8, 20, 40, 70, etc. Nuclear Shell Model • Shell structures: • The capacity of each level is indicated to its right. • The circled numbers indicate the total number of nucleons. Nuclear Shell Model • How can we modify the potential to give the proper magic numbers? • We certainly cannot make a radical change in the potential, because we do not want to destroy the physical content of the model. • It was fairly clear by the 1940s that a central potential could not reproduce all the magic numbers. • Many unsuccessful attempts were made at finding the needed correction. Nuclear Shell Model • The crucial breakthrough came in 1949 when Maria Goeppert Mayer and Hans Jensen suggested - once again following the lead from atomic physics - that inside the nucleus, in addition to the central potential, there is a strong spin-orbit interaction. • The inclusion of a spin-orbit potential could give the proper separation of the subshells. • The spin-orbit interaction occurs because of the electromagnetic interaction of the electron's magnetic moment (spin part) with the magnetic field generated by its motion about the nucleus (orbital part). Nuclear Shell Model • In atomic physics, a spin-orbit interaction splits the two degenerate 𝑗𝑗 = 𝑙𝑙 ± 1/2 energy levels and produces a fine structure. • The effects are typically very small, one part in 105 in the spacing of atomic levels (hence the name fine). • No such electromagnetic interaction would be strong enough to give the substantial changes in the nuclear level spacing needed to generate the observed magic numbers. Nuclear Shell Model • We adopt the concept of a nuclear spin-orbit force of the same form as the atomic spin-orbit force but certainly not electromagnetic in origin. • The total potential sensed by a nucleon has the form: 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟 = 𝑉𝑉 𝑟𝑟 − 𝑓𝑓 𝑟𝑟 𝐿𝐿. 𝑆𝑆⃗ where 𝐿𝐿 and 𝑆𝑆⃗ are the orbital and the spin angular momentum operators for a nucleon, 𝑓𝑓(𝑟𝑟) is an arbitrary function of the radial coordinates Nuclear Shell Model • The sign of this interaction must be chosen to be consistent with the data, so that the state with 𝑗𝑗 = 𝑙𝑙 + 1/2 can have a lower energy than the state with 𝑗𝑗 = 𝑙𝑙 − 1/2 which is opposite to what happens in atoms. • Total angular momentum is: 𝐽𝐽⃗ = 𝐿𝐿 + 𝑆𝑆⃗ and hence, 𝐽𝐽⃗. 𝐽𝐽⃗ = 𝐿𝐿2 + 𝑆𝑆⃗2 + 2𝐿𝐿. 𝑆𝑆⃗ 1 2 𝐿𝐿. 𝑆𝑆⃗ = 𝐽𝐽⃗ − 𝐿𝐿2 − 𝑆𝑆⃗2 2 where we use: 𝐿𝐿, 𝑆𝑆⃗ = 0. Nuclear Shell Model • A quantum state can thus be labelled by : 𝑙𝑙, 𝑚𝑚𝑙𝑙 , 𝑠𝑠, 𝑚𝑚𝑠𝑠 𝑙𝑙, 𝑠𝑠, 𝑗𝑗, 𝑚𝑚𝑗𝑗 where we use the second choice of bases. • Now, 𝐿𝐿. 𝑆𝑆⃗ = 1 2 𝐽𝐽⃗2 − 𝐿𝐿2 − 𝑆𝑆⃗2 ℏ2 = 𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 − 𝑠𝑠(𝑠𝑠 + 1) 2 ℏ2 3 = 𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 − 4 2 ℏ2 /2 for 𝑗𝑗 = 𝑙𝑙 + 1/2 = � ℏ2 − (𝑙𝑙 + 1) for 𝑗𝑗 = 𝑙𝑙 − 1/2 2 Nuclear Shell Model • The change of energies from their degenerate values are: 𝑛𝑛𝑛𝑛𝑛𝑛𝑚𝑚𝑗𝑗 𝑉𝑉�𝑆𝑆𝑆𝑆 𝑛𝑛𝑛𝑛𝑛𝑛𝑚𝑚𝑗𝑗 2 3 ℏ 2 = 𝑓𝑓 𝑟𝑟 𝑛𝑛𝑛𝑛 𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 − 4 2 ℏ2 /2 for 𝑗𝑗 = 𝑙𝑙 + 1/2 = � 𝑑𝑑3 𝑟𝑟 𝜓𝜓𝑛𝑛𝑛𝑛 𝑟𝑟⃗ 2 𝑓𝑓(𝑟𝑟) � ℏ2 − (𝑙𝑙 + 1) for 𝑗𝑗 = 𝑙𝑙 − 1/2 2 Nuclear Shell Model • The total splitting between the two levels becomes: 1 1 − Δ𝐸𝐸𝑛𝑛𝑛𝑛 𝑗𝑗 = 𝑙𝑙 + Δ𝑆𝑆𝑆𝑆 = Δ𝐸𝐸𝑛𝑛𝑛𝑛 𝑗𝑗 = 𝑙𝑙 − 2 2 1 2 = ℏ 𝑙𝑙 + � 𝑑𝑑 3 𝑟𝑟 𝜓𝜓𝑛𝑛𝑛𝑛 𝑟𝑟⃗ 2 𝑓𝑓(𝑟𝑟) 2 • Hence, the splitting due to the spin-orbit interaction is larger for higher values of orbital angular momentum. Nuclear Shell Model • Level Crossing: As the splitting between previously degenerate energy levels due to spin-orbit coupling increases as the orbital angular momentum increases, we can have level crossing. • Energy level with lower 𝑙𝑙 can be split to cross the energy level with higher angular momentum level. • For large 𝑙𝑙 , the splitting of any two neighboring 1 degenerate levels can shift the 𝑗𝑗 = 𝑙𝑙 − state of the 2 1 2 initially lower level to lie above the 𝑗𝑗 = 𝑙𝑙 + state of the previously higher level. Nuclear Shell Model • For an appropriately chosen 𝑓𝑓(𝑟𝑟), the energy levels for a central potential (like the infinite square-well potential or the harmonic oscillator potential) can split upon the addition of a spin-orbit interaction. • Including a spin-orbit interaction we can reproduce the desired magic numbers, and thereby accommodate a shell-like structure in the nuclei. Nuclear Shell Model • The energy levels are labeled according to the spectroscopic notation of atomic physics, as: 𝑛𝑛𝐿𝐿𝑗𝑗 1𝑆𝑆1 , 1𝑃𝑃1/2 , 1𝑃𝑃3/2 , 1𝐷𝐷5/2 , 2𝑆𝑆1 , … 2 2 • The multiplicity of any final level is given, as usual, by (2𝑗𝑗 + 1). • Adding spin-orbit interaction can thus have energy level splitting of both the infinite squarewell potential as well as that of the harmonic oscillator potential. Nuclear Shell Model • Infinite square-well +spin-orbit coupling: Nuclear Shell Model • HO+SO: Nuclear Shell Model • The predicted magic numbers correspond to orbitals with a large gap separating them from the next highest orbital. • For the lowest levels, the spin-orbit splitting is sufficiently small that the original magic numbers, 2, 8, and 20, are retained. • For the higher levels, the splitting becomes important and the gaps now appear at the numbers 28, 50, 82 and 126. Predictions of the Shell Model • 1. Magic Numbers: The shell model correctly predicts the magic numbers of the nuclei. • 2. Spin-Parity: The shell model also correctly predicts the spins and parities of many nuclear states. • According to the model, the proton and the neutron levels fill up independently and, due to the Pauli exclusion principle, only two neutrons or two protons can occupy any given level (with their intrinsic spins anti-parallel). Predictions of the Shell Model • Nucleons pair off in every filled level, yielding zero total angular momentum. • Thus the last unpaired nucleon must determine the spin-parity of the ground state. • As a result, the ground states of even–even nuclei are expected to be 0+ because all nucleons are paired with a partner of opposite angular momentum (i.e. the ground states of all even-even nuclei have zero spin). Predictions of the Shell Model • The ground states of odd–even nuclei should then take the quantum numbers of the one unpaired nucleon. • Example-1: 17𝐹𝐹8 ( 17𝐹𝐹 9 ) and 17𝑂𝑂9 ( 17𝑂𝑂8 ) have one unpaired nucleon outside a doubly magic 16𝑂𝑂8 core. • The nine neutrons for 17𝑂𝑂8 and nine protons for 17 9 𝐹𝐹 will fill the following levels 1𝑆𝑆1 2 , 1𝑃𝑃3 4 , 1𝑃𝑃1 2 , 1𝐷𝐷5 2 2 2 2 1 Predictions of the Shell Model • The total angular momentum of the last unpaired nucleon, is in 𝑙𝑙 = 2, 𝑗𝑗 = 5/2. 5+ 2 • The spin parity of the nucleus is predicted to be since the parity is −1 𝑙𝑙 . • This agrees with observation. • Example-2:The first excited states of 17𝐹𝐹8 and 17𝑂𝑂9 , corresponding to raising the unpaired nucleon to the next higher orbital, are predicted to be 1+ . 2 • Once again in agreement with observation. Predictions of the Shell Model • Example-3: Consider the isobars 13𝐶𝐶 6 and 13𝑁𝑁7 . (Note that these are, in fact, mirror nuclei.) • The six protons in 12𝐶𝐶 and the six neutrons in 13𝑁𝑁 should be completely paired off. • The remaining seven nucleons in both cases should fill the following shells: 1𝑆𝑆1 2 , 1𝑃𝑃3 4 , 1𝑃𝑃1 2 2 2 1 • The last unpaired nucleon - a neutron for 13𝐶𝐶 6 and a 1 13 7 proton for 𝑁𝑁 - has total angular momentum 𝑗𝑗 = 2 and orbital angular momentum 𝑙𝑙 = 1. Predictions of the Shell Model • Hence, according to the shell model, the spin-parity of 1− the ground state for these nuclei is expected to be . • Example-4: For 2 33𝑆𝑆 , 16 the measured value of the , 1𝑃𝑃 2 ground state spin-parity is 3 + . 2 • According to the shell model, the seventeen neutrons will fill up the levels as follows: 1𝑆𝑆 1 2 2 , 1𝑃𝑃 3 2 4 1 2 6 , 1𝐷𝐷5 , 2𝑆𝑆 2 1 2 2 , 1𝐷𝐷 3 2 1 once again leading to a prediction consistent with experiment Predictions of the Shell Model • Example-5: 15𝑁𝑁7 and 15𝑂𝑂7 have one “hole” in their 16𝑂𝑂8 core. • The ground state quantum numbers should then be the quantum numbers of the hole 1 which are 𝑙𝑙 = 1, 𝑗𝑗 = . 2 • The quantum numbers of the ground state are 1− then predicted to be , in agreement with 2 observation. Predictions of the Shell Model • 3. Nuclear Magnetic Moment: The shell model also makes predictions for nuclear magnetic moments. • As for the total angular momentum, the magnetic moments results from a combination of the spin and orbital angular momentum. • The weighting is different because the gyromagnetic ratio of the spin differs from that of the orbital angular momentum. Predictions of the Shell Model • As measurements show, the proton and the neutron have intrinsic dipole moments of 2.79 𝜇𝜇𝑁𝑁 and −1.91𝜇𝜇𝑁𝑁 , respectively. • Thus, we expect the intrinsic magnetic moment of any unpaired nucleon to contribute to the total magnetic moment of the nucleus. • In addition, since protons are charged, the orbital motion of any unpaired proton can also contribute to the magnetic moment of the nucleus. Predictions of the Shell Model • Example-1: For the deuteron, for example, if we assume that the proton and the neutron are in 1𝑆𝑆1/2 states. • Then without orbital angular momentum for the proton 𝑙𝑙 = 0 , we expect the magnetic moment of the deuteron to be the sum of the intrinsic dipole moments of the proton and the neutron: 𝜇𝜇𝑑𝑑 = 2.79 𝜇𝜇𝑁𝑁 − 1.91 𝜇𝜇𝑁𝑁 = 0.88 𝜇𝜇𝑁𝑁 • The observed magnetic moment of the deuteron is 0.88 𝜇𝜇𝑁𝑁 - in good agreement with expectation. Predictions of the Shell Model • Example-2: The nucleus of tritium 3𝐻𝐻1 has two neutrons and one proton, all in the 1𝑆𝑆1/2 state. • Since the neutrons are paired, they should not contribute to the magnetic moment. • The unpaired proton, having 𝑙𝑙 = 0, will have no contribution from its orbital motion. • Consequently, the total magnetic moment of 3𝐻𝐻1 should be the same as that of the unpaired proton i.e. 2.79 𝜇𝜇𝑁𝑁 . • The measured value is 2.98 𝜇𝜇𝑁𝑁 . Predictions of the Shell Model • Example-3: For 3𝐻𝐻𝐻𝐻 2 , the unpaired nucleon is a neutron in a 1𝑆𝑆1/2 state. • Consequently, the total magnetic moment should be the same as that of the neutron, −1.91 𝜇𝜇𝑁𝑁 (the measured value is −2.13 𝜇𝜇𝑁𝑁 ). • Example-4: 4𝐻𝐻𝐻𝐻 2 (𝛼𝛼-particle) has a closed shell structure (in fact, it is doubly magic). • The shell model would therefore predict no spin and no magnetic moment, which is indeed experimentally correct. Predictions of the Shell Model • Example-5: In 10𝐵𝐵 5 , the five protons and the five neutrons have the same level structure, namely, 2 1𝑆𝑆1 , 1𝑃𝑃3 2 2 3 • Thus, there is one unpaired proton and one unpaired neutron. • The unpaired proton will be in an 𝑙𝑙 = 1 state, and therefore the orbital motion will contribute: 2ℏ 𝜇𝜇 = 𝑙𝑙 = 𝜇𝜇𝑁𝑁 2𝑚𝑚𝑁𝑁 𝑐𝑐 Predictions of the Shell Model • Thus the total magnetic moment will be: 2.79 𝜇𝜇𝑁𝑁 − 1.91 𝜇𝜇𝑁𝑁 + 𝜇𝜇𝑁𝑁 = 1.88 𝜇𝜇𝑁𝑁 comparing quite well with the measured value of 1.80 𝜇𝜇𝑁𝑁 . • We see therefore that the shell model, in addition to providing the known magic numbers, also describes other important properties of light nuclei. • For heavy nuclei, however, there is marked difference between the predictions of the shell model and the measured quantities.