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
νj
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 = 4w ; 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