Forces, Potentials, and the Shell Model Recall the Infinite Square Well (1D)
Solve Shroedinger’s equation: Hψ = Eψ d2 ψ − Vψ = Eψ dx 2
Result: Consideration of boundary conditions (the behavior of the wavefunction at the walls) results in quantization. Both wavefunctions and eigenstates (energy levels) n2h2 En = 8mL2 Notice the dependence of the energy levels on the size of the box, and on the principal quantum number. Harmonic oscillator (1D) Hooke’s law : F = −k ( x − x0 ) If x = x0 , the system is at equilibrium because there is no force. However if x is different from x0 there is a force which acts to restore the position to the equilibrium value (Notice the negative sign.) dV F =− dx Integrating we get, 1 V = k ( x − x0 ) 2 2 Now solve Schrodinger’s equation using this potential. Solution: Wavefunctions and eigenvalues 1 Eigenvalues: E n = (n + )ω 2
where ω =
k m
Notice the energy spacing for the harmonic oscillator. What is the minimum energy of the harmonic oscillator?
V.
Nuclear Shell Model
A.
Quantum Properties of Nuclei 1.
Discrete Energy Levels
2.
Nuclear Spin − I a. Experimental Summary e-e : I = 0 ALWAYS n , where n is an odd integer (1/2, 3/2, ...) 2 I = n , where n is an integer (0, 1, 2 ...) WE'LL USE = 1 for our spins
e-o, o-e: I = o-o :
b. Implication e-e result implies strong pairing is energetically favorable ∴ spins must cancel c. Reason: Nuclear Force is attractive ; in contrast spins are unpaired in a atomic orbitals due to e-e repulsion (Pauli exclusion principle) 3. Closed Shells – Unusual Stability a. Magic Numbers 2, 8, 20, 28, 50, 82, 126 (neutrons) b. Energetics: (MLD – M), Bp, Bn, Bα c. Lifetimes: 208 Pb 82 126 STABLE
209 Pb127 82
22y
210 Po126 84
138d
212 Po120 84
10−7s
∴ Z=82 & N=126 appear to be stable
4. Magnetic Moments Moving Charge created a magnetic field with moment µ µ = a. Expect
e f(I) 2Mc
; µN = nuclear magneton (M = Mp)
µp = µN µn = 0
Observe :
µp = 2.793 µN µn = −1.913 µN
µe agrees with expectations
b. Implication: nucleon has substructure, since one observes charge on periphery of particle. e.g., proton +2/3, −1/3, +2/3 ; neutron −1/3, +2/3, −1/3 c. Effect on Chemical Environment • I = 1/2 for 1H, 13C, 57Fe • For a nucleus with spin, the magnetic field around the nucleus interacts with the electric field of its electronic environment. NOTE: µN << µe; ∴ all e−s must be paired. • Interaction is very sensitive to e− orbital distribution and ∴ is different for every chemical bond • Supply rf energy to induce transitions ( ↑→↓ ) and get resonance. NMR ≡ MRI d. Result for nuclei: e-e : o-e : e-o : o-o :
µ = 0 ALWAYS µ ≈ µp µ ≈ µn µ ≈ µp + µn
AGAIN, SUPPORTS STRONG PAIRING ARGUMENT
B.
Shell Model:
Quantum Mechanical Solution ( a la hydrogen atom).
1. Schroedinger Equation a. H = Hamiltonian: Summarizes forces acting on particles H = T + V(r) = kinetic + potential energy b. Ψ = Wave function: Describes properties of particles in system; i.e., Probability distributions in space and time (orbitals) Ψi = f(x, y, z, t, s, ...) (1)
Pauli Exclusion Principle For Fermions (particle with half-integer spins) Ψi ≠ Ψj (atoms and molecules also) i.e., all particles must have different wave functions
(2)
Parity − π Definition: a mathematical operator that reverses coordinates π Ψ(x) = Ψ (−x) = ± Ψ Example:EVEN Parity: x2, x4, x6, cos x, s, d, g, ... ODD Parity: x, x3, x5, sin x, p, f, h. ...
c. E Discrete energy states: Quantum states Produced by action of forces on particles
2.
Qualitative Expectations for orbitals of the same energy
Pairing: Shapes Spin-orbit: REASON
Atoms Weak (Hund's Rule) Diffuse (low preferred) (high preferred) Weak (re << ratom) (rnucleon ≲ rnucleus) e-e force REPULSIVE ATTRACTIVE
Nuclei Strong Compact Strong N-N force
Bottom Line: Atomic and Nuclear Shell structure should differ 3. Potential Models: V(r) T = 1/2 Mv2 ( or relativistic equivalent – differential equation)
Solvable approxim to the nuclear pot can't solve Fermi function
Harmonic Oscillator (HO) Square Well (SW) o ←r→ a. Square Well:
R = r0A1/3
V(r) = −V0, r ≤ R V(r) = 0 ,
r>R
b. Harmonic Oscillator: V(r) = −V0 [1 − r2/R2] 4.
Uniform density sphere
Parabola
HO Solution: Energy Levels a. Eν = [2(ν−1) + ] ħω
(c.f. Atoms: En α −
1 ) n2
where
ν = 1, 2, 3 ... Principal Quantum Number (QN) = 0, 1, 2 ... Orbital Angular Momentum QN (s, p, d, f, g ...) NOTE: is INDEPENDENT of ν (unlike atoms) Magnetic Substate (2 + 1) m = ± , ± (−1) ... 0 s = ± 1/2 Intrinsic Spin b. Notation
ν
; e.g. ν = 2, = 4 is 2 g
Energies for all m and s states are same for same ν and c. Pauli Exclusion Principle Each nucleon must have a UNIQUE set of QNs (ν, , m and s) NOTE: p & n are different particles (i.e., electric charge QN is +1,0) ∴ they can have the same ν, , m and s) d. Compare with Magic Numbers: DOESN'T WORK
5.
Square-Well Solution -- Doesn't work either ∴ need ADDITIONAL PHYSICS
6. Empirical Correction 1963 Maria Goeppert Mayer and Hans Jensen – Nobel Prize ASSUMPTION (1949): Attractive Nuclear Force will lead to a strong interaction between particle spin and its orbit (e.g., same would be true of moon and earth – if closer/tides)
a. Result: NEW QNs j = TOTAL ANGULAR MOMENTUM j = + s = ± 1/2
b. New QN Notation ν same same j = ± 1/2 mj = +j, (j−1) ... ~j
νj 2j + 1 values/j
c. Example ν = 1, = 2 ⇒ 1dj j = 2 ± 1/2 = 3/2, 5/2 ⇒ 1 d3/2 & 1 d5/2 for j = 3/2, mj = 3/2, 1/2, −1/2, −3/2 = 2j + 1 = 4 possible values j = 5/2, mj = 5/2, 3/2, 1/2, −1/2, −3/2, −5/2 = 2J + 1 = 6 possible values Total d states = 10 possible values d. Effect on Energy States (Levels) (1) RULE 1:
For same oscillator energy, E HO = [2(ν−1)+]w, v E < E-2 < E−4 ...
MORE COMPACT ORBITALS (high ) PERMIT STRONGER ATTRACTION (opposite of atoms)
Example: EHO = 4w ; comes from 1g, 2d, 3s states – all same energy Rule says (2) (3)
4 w
3s 2d 1g
RULE 2: E +1/2 < E −1/2 for same ν ; RULE 3 ∆Ej α
e. Rearranged Level Order Now matches experimental magic numbers. This is the same trick you play with Bohr model for atoms, except low preferred to keep electrons as far apart as possible. This is what we will use for predictive purposes.
C. Prediction of spins and Parities: 1. Even-Even Nuclei
GROUND RULES
Iπ=0+
RULE: All nucleon orbitals are filled pairwise, i.e., ν. , j, mj state followed by ν, , j, −mj state spin parity NO EXCEPTIONS
2. Odd-A Nuclei INDEPENDENT PARTICLE ASSUMPTION Nucleons fill orbitals pairwise up to last odd nucleon. RULE: Last odd nucleon determines quantum properties of entire nucleus Result: a. A−1X core is e-e; ∴ 0+ b. Last particle Iπ given by HO model with strong spin-orbit coupling; c. Total Nucleus I = I (core) + I (last nucleon) = 0+j=j π = π (core) × π (last nucleon = +•±=± NOTE: On figure of energy levels with spin-orbit coupling, parity alternates from shell to shell (ν → ν + 1) Filling levels: same as doing electron configurations in Bohr atom 3. Odd-Odd Nuclei Must couple last odd proton to last odd neutron. I = j n + j p = (jn + jp), ... 1 jn − jp NOT COVERED: difficult angular momentum (vector additions). 4.
Examples: a. {12C, 28O, 184Pb, 298114} All Iπ = 0+
b. 119 In = 118 Cd + p 49 48 0+ figure of energy levels with th spin-orbit coupling: 49 proton in level is:
1 g9/2 ; j = 9/2 ; g state: π = + =4 Predict
Iπ = 9/2 + observed
c. 47 Ca : 46 Ca + 1 n 20 20 0 figure of energy levels with th
spin-orbit coupling: 27 neutron is 1 f7/2 Iπ = 7/2−
Predict:
∴ j = 7/2, = 3, π = −
5. Bottom Line: Same counting game as in atoms (1s 2s 2p 3s ...) Works near closed shells ; deviations away from them. 2
2
6
D.
1/2 3/2
2
Excited States 1. Particles and Relative Energies
+
Given by level scheme on p. 38: 2 s1/2
+
5/2+
e.g. 15 O 8 7 (7) (6) (2)
E3
1 d3/2
E2
1 d5/2
E1
↿0
1 p1/2
↿⇂
1 s1/2
↿⇂↿⇂ 1 p3/2
2. Rotational and Vibrational States also exist Due to collective motion of nucleus, superimposed on singleparticle state.
single particle state
GROUND STATE
E.
The Shell Model and the Real World 1. Closed Shells Correct
2. systematically
Spins, Parities and Magnetic Moments – described a. b.
e-e: Always right o-A: usually correct for spherical nuclei (near
closed shells). less accurate in between c. 3.
o-o: difficult – horseshoes
Low-lying energy levels – also correct near closed shells.
VIII. Unified Model Combines LD and Shell models; allows for deformed shapes – changes order of levels between shells, but not magic numbers. e.g.,
a x2 b y2 c z2 V(r) = V0 (1 − 2 + + R2 R 2 R
