SubAtomic Physics Nuclear Physics

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1 SubAtomic Physics Nuclear Physics Lecture 4 Main points of Lecture 3 Strong and coulomb interaction processes conserve angular momentum and parity External properties 1) Charge number of protons (ze) 2) Mass - measured in u (1/12 of mass 12 C) 3) Size (charge radius electron scattering potential radius neutron scattering) 3 rd Year Junior Honours Course Dr Daniel Watts R = r 0 A 1/3 Important relation!! Assuming constant nuclear density Started to discuss internal properties spin and parity Will continue this in the coming lecture

2 Notes Notes

3 Internal properties continued.. Magnetic moment µ: due to (and aligned with!) the nuclear spin Familiar with the magnetic moment of the electron electron: µ e ~ -1.0 µ B For a point like nucleon expect: µ p ~ 1.0µ N µ n ~ 0 Experimental measurements: proton: µ p = 2.79 µ N neutron: µ n = µ N eh µ = B Bohr magneton 2m µ N = eh 2m p nuclear magneton (~1/2000 of µ b ) Surprise! the neutron, although uncharged has a non-zero magnetic moment & proton magnetic moment is anomalously large consequence of quark sub-structure (consisting of charged components) p = 2 up + 1 down u = + 2/3e n = 1 up + 2 down d = - 1/3 e Aside I: Important applications of nuclear moments are magnetic resonance imaging, nuclear magnetic resonance (see Lilley, ch. 9) Aside II: The observation that the proton has excited states (see figure) also indicates it has.internal structure e Total photon absorption cross section for the proton γ + p Anything resonant structures correspond to nucleon excited states Nuclear magnetic moments arise from the sum of the spin and orbital magnetic moments of the component nucleons Magnetic moments of nuclear ground states lie in the range 2 µ < < 6 Therefore the magnetic moments of the component nucleons must tend to cancel to a large extent in nuclei we will see the reason for this later when we introduce the shell model Aside: Recap on magnetic moments Magnetic moment due to Orbital motion: v Classically arise from eg. Motion of current in a loop Magnetic moment QM get analogous result N µ Nuclear µ N Magnetic moment due to Intrinsic spin Magnetic moment operator due to intrinsic spin of a particle: e µ s = g s S 2m g s = 2 e h µ s = g s 2m 2 For electron For proton µ = µ = + s µ B s µ N r ev µ l = ia = πr 2πr e µ l = L 2m e gl µ l = g l L 2m g 2 = = 1 = 0 epr 2m charged neutral from Dirac theory, Rel. QM for point like spin ½ particle) Bohr magneton (µ B ) =eħ/2m e Nuclear magneton (µ N ) =eħ/2m N l

4 Notes Notes

5 Excited states E,J: nuclei can exist in different excited energy states, each corresponding to a given configuration of nucleons Excitation level scheme of a real nucleus 16 O excited states have a finite LIFETIME (τ) and WIDTH (Γ) Remember τ Γ ~ h Heisenberg s uncertainty relation at low energies levels are DISCRETE at higher energies level widths Γ become increasingly larger they eventually overlap forming a CONTINUUM continuum high energy low energy nucleon (or cluster) gamma ray de-excitation through: ground state 1) γ emission (Eγ = E) with lifetime τ h /Γ 2) particle emission (if energetically allowed) each state characterized by a total angular momentum J = L + S L = total orbital angular momentum S = total spin most nucleons combine in pairs all nuclei with even N even Z have J = 0 Even N, Even Z nucleus 8 Spin and parity assignments of the nuclear excited states Collectively called a nuclear level scheme see: for more examples!

6 Notes Notes

7 Excitation level scheme of a real nucleus 16 O Zoom out and it gets more complicated! Mechanism of the nuclear force - I Complicated!! e.g. one possible Feynman diagram for proton proton scattering at the quark level Quarks held together by strong interaction arising from the exchange of gluons and other quarks Calculations of nuclear force in terms of quark gluon field still in infancy Extract empirical information on nucleon-nucleon force from experimental data Nucleon-nucleon scattering Wealth of scattering data available from proton, neutron beam facilities particularly at energies applicable to nuclear physics Properties of deuteron, Binding energy, magnetic moment, electric quadropole moment Curve of binding energy per nucleon Mass number dependence

8 Notes Notes

9 Mechanism of the nuclear force II Important properties of the nuclear force NN force is complicated but for energies and nucleon separations applicable in nuclear physics (below ~100 MeV) the NN force can adequately be described as the exchange of virtual mesons between nucleons. Theoretical framework developed by Yukawa (1935) nucleon-nucleon P.E. repulsive ω attractive Heavier meson exchange equilibrium position π distance Pion exchange Heisenberg Uncertainty Principle - borrow energy to create particle if energy ( E=mc 2 ) repaid within time ( t) where E t > ħ mc 2 m ~ h ~ ~ t h rc hc r (Assumption: Particle velocity ~ c) e.g. r=1.4fm m~140mev/c 2 Particle of mass appropriate to range of nuclear forces discovered in cosmic ray interactions pion (J π =0 - ) Three types : m(π + )=m(π - )=139.6MeV/c 2 m(π 0 )=135 MeV/c 2 Short range part? - heavier mesons, multiple pion, quark? short range : Strongly attractive component over short range repulsive core repulsive component at very short distance (<0.5 fm) average separation between nucleons SATURATION of nuclear force Force is spin dependent Depends on intrinsic spins of nucleons repulsive attractive equilibrium position distance deuteron only stable if spins parallel p n p n S=1 force stronger than S=0 & no pp, or nn states (S=0 only Pauli exclusion principle) also spin-orbit interaction ~ L S Depends on relative orientation of total spin and angular momentum (attractive force if parallel) nucleon-nucleon P.E. also tensor force Energetically more preferable to have spins parallel to line joining nucleons Force is charge independent S=1 S=0 nuclear force is charge symmetric light nuclei (A < 40) : coulomb repulsion << binding energy stable nuclei tend to have N ~ Z (see nuclear chart)

10 Notes Notes

11 Nuclear masses and binding energies The nucleus and its structure Presently no complete theory to fully describe structure and behaviour of nuclei based solely on knowledge of force between nucleons (Although tremendous progress in past few years for A<12!) Properties M n (A,Z) < Zm p + Nm n Mc 2 = [Zm H + Nm n ]c 2 - M a (A,Z)c 2 BINDING ENERGY (B) energy required to break nucleus into constituents energy released to create nucleus from constituents A useful quantity is to divide the total nuclear binding energy by the number of nucleons to give the binding energy felt per nucleon Positive for all nuclei nuclear force is attractive (on average) Peaks at A=n 4 and Z=A/2 multiple of 2n-2p configuration particularly stable B/A ~ constant for A>20 Saturation of nuclear force Curve of binding energy per nucleon Slow decrease of B/A for A>60 ( 56 Fe most stable nucleus) Coulomb repulsion slowly prevails limited number of nuclei Even-even nuclei more stable than odd-odd nuclei Pairing effect between like nucleons with opposite spins use MODELS: simplifying assumptions give reasonable account of observed properties make predictions LIQUID-DROP MODEL nucleus regarded as collection of neutrons and protons forming a droplet of incompressible fluid good description of overall trend of binding energy per nucleon fails to account for magic numbers or give any prediction for J π SHELL MODEL neutrons and protons arranged in stable quantum states in common potential well accounts for ground-state properties (e.g. J π ) and magic numbers does not predict many of the observed nuclear excited states COLLECTIVE MODELS neutrons and protons show collective motions give rise to vibrational and rotational states accounts for properties of non-spherical nuclei fails to reproduce other features

12 Notes Notes

13 How to explain these properties? LIQUID DROP MODEL nucleus regarded as collection of neutrons and protons forming a droplet of incompressible fluid Bethe and von Weizsäcker (1935): Remember Warning!! Total Binding Not B/A! B(A,Z) pairing term Semi Empirical Mass Formula (SEMF) 2 2/3 = ava asa a a C + δ (A,Z) 1/3 a δ > 0 for even N even Z δ < 0 for odd N odd Z δ = 0 for odd A Contributions from various terms to binding energy per nucleon Z A 2 (A 2Z) A δ ~12/A 1/2 a C a v Z A A 2 / 3 a s A 2 1/ 3 Volume energy: Each nucleon only feels interaction of close neighbours due to short range of nuclear force Gives a positive binding energy which is roughly the same for each nucleon Surface correction: Nucleons near surface of nucleus surrounded by fewer nucleons and will therefore experience less attractive potential energy than those inside the nucleus. Compensate with a reduction in binding energy proportional to number of nucleons in nuclear surface Coulomb energy: Nucleus has total charge Ze confined to a sphere of radius R. The resultant potential 2 3 ( Ze) energy given by electrostatic theory = 5 4πε R 0 ( A 2Z) A a a 2 Symmetry term: Stable light nuclei have N~Z (i.e. A~2Z). If A deviates from 2Z then binding energy is reduced. From fits to experimental binding energies: a v = 15.6 MeV a s = 17.2 MeV a a = 23.3 MeV a C = 0.70 MeV + δ (A,Z) Pairing term: Most stable nuclei have Z even and N even and therefore A even (even-even nuclei). Increases binding for eveneven nuclei and reduces for odd-odd. δ =12/A 1/2

14 Notes Notes

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