Lecture [Nuclear Parameters] (Chapter 3 in E&V) (Chapter 2 in L,M &S)
NUCLEAR PARAMETERS
I – Nuclear spin
–Parity
– Magnetic moment
Q – Quadrupole moment
– lifetime (chapter 5 E&V) (chap. 3 L,M&S)
A. Nuclear spin (I): Total angular momentum spin
(due to intrinsic spin of nucleons
and their orbital motion in nucleus)
1. Background
Classical approach
Angular momentum (single particle)
linear mom =
ang. mom. = l =
1
Angular momentum of non-uniform shape
ang. mom. =
Total ang. mom. =
Moment of inertia = I
Kinetic energy =
vector - L
2. Quantum Representation
Orbital motion of single nucleon
mvr = where l – projection of L
on some axis
L = ħ[l(l + 1)]1/2 BUT
e.g.
2
Coupling of 2 nucleons/orbital motion only
Given 2 nucleons with orbital ang. mom.
Then use
3. Spin – intrinsic angular momentum
nucleon can spin on its own axis
two allowed directions (up, down)
S = ħ[s(s + 1)]1/2 ; s = ħms ; ms = ½
4. Total angular momentum of a nucleon (j)
coupling of orbital and spin motion
j = l + s; mj = ml ½
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5. Coupling of two nucleons/total
j1 + j 2 =
6. Total Angular Momentum of Nucleus =
Nuclear Spin - I
total ang. mom (spin and orbital) of all nucleons
I=
Odd A nuclei
Even A nuclei
Special case: Ig.s (e-e) =
4
B. Magnetic Moment -
nucleon is a charged particle which
generates a magnetic field
= (pole strength) x distance between poles
= current x area = i R2
ang.mom
= = = 2 mass
e = constant x l
p = 2.793 nuclear magnetons (proton spin)
n = - 1.913 nm (neutron spin ???)
(nucleus) = (orbital and spin of all protons
and spin of neutron)
In general nucleons do form pairs and individua
moments cancel out.
Spin and magnetic moment of e-e = 0
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C. Parity -
- reflects the symmetry of the wave function
- reflection through the origin
- not a physical parameter
Even parity - +
(x, y, z) = (-x, -y, -z)
Odd parity -
(x, y, z) = (-x, -y, -z)
Simple example:
Representation : I e.g. 0+
Statistics :
Bose-einstein : If (system) = (system) when
coordinates changed.
Fermi-dirac: If (system) = - (system) when
coordinates changed. (Pauli)
fermions: leptons, baryons; bosons: mesons, photons
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D. Quadrupole moments – Q
- reflects deviation from spherical shape
- moment ~ 2/5 Z (a2 – b2)
- R=
- Moment can be high ( )
- For nuclei with I = 0,1, there is no
observable quadrupole moment.
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Selection Rules for Decay Transitions
A. Alpha emission
B. Beta decay
Type I log ft
superallowed
allowed
1st forbidden
2nd forbidden
C. Gamma decay
- electric(E) and magnetic(M) transitions
- lifetime prop. ( )
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