PHY492: Nuclear & Particle Physics. Lecture 5 Angular momentum Nucleon magnetic moments Nuclear models

Transcription

1 PHY492: Nuclear & Particle Physics Lecture 5 Angular momentum Nucleon magnetic moments Nuclear models

2 eigenfunctions & eigenvalues: Classical: L = r p; Spherical Harmonics: Orbital angular momentum Orbital angular momentum, L L 2 ψ m = ( + 1) 2 ψ m L z ψ m = mψ m m ψ m = Y m Quantum: p i, L z = i x y y x L = ( + 1), an integer L z takes ( + 1) >, therefore L > L z = m L 2,m = + 1 ( ) 2, m L z ( θ,φ) = f lm ( θ)e imφ,m = m,m (odd number of) ( ) possible values Coordinate representation if angular distributions are desired eigenstates & eigenvalues: L z = ih φ Matrix representation if only eigenvalues are of interest January 24, 2007 Carl Bromberg - Prof. of Physics 2

3 S 2 s,m s = s s + 1 ( ) 2 s,m s Spin angular momentum Many elementary particles, e.g., electron and neutrino, carry one half unit ( ) of angular momentum known as spin. With no spatial dimension, nothing is spinning or orbiting. Protons and neutrons are spin 1/2, therefore, nuclei can have s = 1/2 integer. Matrix representation Fermi-Dirac statistics s = 1, m 2 s = ± 1 2 s = 1 2,1, 3 2 ; s m s s Nuclear energy levels have an L S component: Eigenvalues s j + s J 2,s, j,m j = j( j + 1) 2, s, j, m j L 2,s, j,m j = ( + 1) 2, s, j,m j S 2,s, j,m j = s( s + 1) 2, s, j,m j J z,s, j,m j = m j,s, j, m j S z s,m s = m s s, m s Angular momentum coupling J = L + S J 2 = L 2 + S 2 + 2L S L S = 1 2 J 2 L 2 S 2 =,s, j,m j 1 2 = 2 2 ( ) Expectation value of L S, s, j,m j L S, s, j,m j J 2 L 2 S 2 ( ), s, j, m j { j( j + 1) ( + 1) s( s + 1) } January 24, 2007 Carl Bromberg - Prof. of Physics 3

4 Nucleon spin and magnetic moments Classical energy of magnetic dipole in a magnetic field: µ = IA = qv 2πr πr 2 ( ) = q 2m mvr = q 2m L Orbital angular momentum: L = n, n = 0,1,2 Quantum mechanics has only two values of for the projection of the magnetic dipole on the B axis, ±1. U = µ B U = µb µ = I A c I = qv 2πr ; A = πr 2 cgs Bohr magneton: Nuclear magneton: µ B = e 2m e (F = qvb) SI µ B = e 2m e c F = qv B c cgs January 24, 2007 Carl Bromberg - Prof. of Physics 4 SI: µ N = e 2m p = m e m p µ B 2000 times smaller than Bohr magneton µ B = e 2m e = ec 2m e c ( ) ( )( m/s) = ( C) MeV m kg = MeV/T = A m 2 (MKS) Magnetic dipole moment associated with point-like particles with spin µ = e 2m g S Landé g factor: gs = S = 2 g = 2 A spin 1/2 particle with a magnetic moment (Landé) g-factor not close to 2, implies internal structure.

5 Anomalous magnetic moments of proton and neutron Spin of proton and neutron are 1/2 Prediction for point-like particles Measured values µ p = µ N = MeV/T µ n = 0 (no charge) Quark model of nucleons u-quark charge = +2/3 e d-quark charge = -1/3 e µ p = 2.79 µ N µ n = 1.91 µ N Mighty anomalous! Nucleon quark content proton = 2u + 1d neutron = 1u + 2d Assuming magnetic moments of the u and d quarks are u-quark d-quark µ u = +1.86µ N µ d = 0.93µ N Predictions of quark model µ p = 4 3 µ u 1 3 µ d = 2.79 µ n = 4 3 µ d 1 3 µ u = 1.86 January 24, 2007 Carl Bromberg - Prof. of Physics 5

6 Nuclear phenomenology Strong vs. Coulomb forces Liquid drop model Binding energy parameterization Fermi-gas effects Justification for terms in Liquid Drop, and Shell models Shell model potential Infinite square well Harmonic oscillator Spin-Orbit Collective model Heavy nuclei January 24, 2007 Carl Bromberg - Prof. of Physics 6

7 The nuclear potential Character of the nuclear potential Coulomb force due to exchange of a γ V ( r) 1 r Long range Yukawa (strong nuclear) potential V r ( ) e mc r r Short range: down by 1/e at the nucleon surface E(eV) = Coulomb Potential energy of two protons 1 fm apart e 4πε 0 r = ( Nm 2 / C 2 ) C m = ev = 1.4 MeV Strong nuclear force results from the exchange of a massive particle mc 2 = c r = 200 MeV m m 160 MeV January 24, 2007 Carl Bromberg - Prof. of Physics 7

8 Liquid drop model Five terms (+ means weaker binding) in a prediction of the B.E. r ~A 1/3, Binding is short ranged, depending only on nearest neighbors. This leads to a B.E. term proportional to A: a 1 A. The surface nucleons are not surrounded by others. This leads to a term proportional to A 2/3 that weakens the B.E. : +a 2 A 2/3. Coulomb repulsion of the protons. This leads to a term proportional to Z 2 /r that weakens the binding energy: +a 3 (Z 2 /A 1/3 ). Orderly mix of p and n favors equal number of nucleons, but dilutes at big A. This leads to a B.E. term: +a 4 (N-Z) 2 /A. Spin effects favor even numbers of protons or neutrons, but dilutes at big A. This leads to a term: ±a 5 1/A 3/4 (Z,N) + (odd,odd), for (even,even), 0 for (even,odd) or (odd,even) B.E.(A, Z) = a 1 A + a 2 A a 3 Z 2 A a 4 ( 2Z A) 2 A 1 ± a 5 A 3 4 in MeV: a , a , a , a , a 5 34 January 24, 2007 Carl Bromberg - Prof. of Physics 8

9 Binding energy per nucleon B/A vs. A Most stable: 56 Fe, 8.8 MeV/ nucleon gradual decrease at large A due to Coulomb repulsion very sharp rise at small A tabularized binding energies and masses - January 24, 2007 Carl Bromberg - Prof. of Physics 9