Download LECTURE 31 SHELL MODEL PHY492 Nuclear and Elementary Particle Physics

LECTURE 31
SHELL MODEL
PHY492 Nuclear and Elementary Particle Physics
Fermi Energy and Momentum (last lecture)
n(p) dp =
4πV
(2πh)3 If N = Z = A/2 assumed;
A/2 = N (Z)
=
PF p2 dp n = 2
= N(P) VPF 3
3π2h3 n(p) dp ( 2: spin )
0 VPF3
3π2h3 N(P) PF 3
4
=
πR03A × 2 3 3 3π h
V h 9π
1/3
PF = PF = R0 ( 8 ) ≈ 250MeV/c N(P) EF = PF2/2M ≈ 33 MeV V0 = EF + B/A ≈ 40 MeV V0 Coulomb force March 31, 2014 PHY492, Lecture 31 2 Relation with the Mass Formula
The average kinetic energy per nucleon is given,
The total kinetic energy of one nucleus is given or With the parameterization of
the total energy is approximated by relation to the mass formula : (contribute to Volume term) (Asymmetry term) March 31, 2014 PHY492, Lecture 31 3 Shell Model
Atomic theory based on the shell model
has provided remarkable clarification of
the complicated details of atomic structure
(magic number, energy levels…)
Central potential is given by Coulomb force
analogy Atom
eNuclei
Nuclei
Nuclear shell model can also provide
detailed predictions of nuclear properties
Central potential? Where does it come from?
the nucleons move in a potential that they themselves create
March 31, 2014 PHY492, Lecture 31 4 Shell structure of atoms
Atomic shell structure is based on the central Coulomb potential
The energy levels are characterized by
principal quantum number ( n )
n = 1, 2, 3, 4, …
orbital angular momentum ( l )
l = 0, 1, 2, 3, 4, …, (n-1)
(s) (p) (d) (f) (g) ….
magnetic quantum number ( m )
m = -l, -l+1, .., 0, 1, …, l-1, l
spin quantum number ( ms )
ms = +1/2, -1/2 (up,down) For any n, there are degenerate energy states with (l,ml,ms),
number of such states is given nd = 2 × ∑ (2l+1) = 2n2
magic numbers
2 (He), 10 (Ne),
18 (Ar), 36 (Kr), … March 31, 2014 PHY492, Lecture 31 5 Magic numbers of nuclei
In nuclei, one can also find
magic numbers both for
proton number (Z) and
neutron number (N),
from the enhanced values
of the binging energies
With respect to the
semi-empirical mass
formula.
N = 20 28
50
Z = 20 28
82
50
126
82 Magic numbers of nuclei N = 2, 8, 20, 28, 50, 82, 126
Z = 2, 8, 20, 28, 50, 82 Magic numbers of atoms 2, 10, 18, 36, 54 …
March 31, 2014 PHY492, Lecture 31 6 Evidence for magicity
Other phenomena
Magic nuclei have
(1) more stable isotopes than other nuclei
(2) small electric quarupole moment,
the first excited states at high energies
( spherical )
(3) hindered neutron capture cross sections
(4) sharp changes in separation energies 2 4 3 Magic numbers N = 2, 8, 20, 28, 50, 82, 126
Z = 2, 8, 20, 28, 50, 82 March 31, 2014 PHY492, Lecture 31 7 Woods-Saxon Potential
atomic shell structure … Coulomb potential VCoulomb = e/4πε0r nuclear shell structure … Woods-Saxon potential (density distribution)
R - V0
V(r) = 1 + e (r-R)/a 0
a - V0
shell structure with different potential shapes
magic numbers of nuclei can be explained only up to 20 harmonic oscillator March 31, 2014 Woods-Saxon pot.
PHY492, Lecture 31 Infinite well
8 LS force
To explain magic numbers of nuclei, in 1949, Mayer and Jensen
proposed a spin-orbit term ( ls force ) in the potential;
V = Vcenter + Vls(r) (L·S) l-1/2 ΔEls the ls splitting between
is given
2l + 1
ΔEls =
l+1/2 March 31, 2014 PHY492, Lecture 31 2 l-1/2 l+1/2 states
h2 < Vls > With the LS term,
shell model can
successfully
explain nuclear
magic numbers
9 Shell model configuration
The shell model configuration is described using the notation (nlj)k
for each sub-shell, where k is the occupancy of the sub-shell.
2s1/2 For example,
configuration of 17O is written as
1d5/2 1p1/2 for protons
1p3/2 (1s1/2)2(1p3/2)4(1p1/2)2
1s1/2 for neutrons
(1s1/2)2(1p3/2)4(1p1/2)2(1d5/2)1
March 31, 2014 PHY492, Lecture 31 π
(proton)
ν
(neutron) 10 Spins and parities of the ground states
The shell model can predict spins and parities of nuclear states
Spins
even-even nuclei : ground states have always spin J = 0
even-odd nuclei : spin of the ground states is determined by
a spin of valence proton (or neutron)
in case of 17O, J = 5/2
odd-odd nuclei : coupling of valence proton and neutron
Parities determined by the “l” value
of the valence nucleon (s)
17O
2s1/2 1d5/2 1p1/2 Parity transformation for
the spherical harmonic function
1p3/2 P Ylm(θ,φ) = (-1)l Ylm(θ,φ)
(HW1-2)
in case of 17O one particle in 1d5/2 (l=2) … parity = 1 (+)
March 31, 2014 PHY492, Lecture 31 1s1/2 π
ν
11 Excited states
The shell model can also predict spins and parities of excited states
17O
ground state Jπ = 5/2+
2s1/2 1d5/2 1p1/2 1p3/2 1s1/2 17O
Excitation energies
3 < 2 < 1 excited states (1) π(1d5/2)1(1p1/2)-1
×ν(1d5/2)1 March 31, 2014 (2) Jπ = 1/2- ν(1p1/2)-1 (3) Jπ = 1/2+ or 3/2+ ν(2s1/2)1 or ν(1d3/2)1 2s1/2
or 1d3/2 PHY492, Lecture 31 12 Magnetic moments
Nuclei have magnetic moments µ = gj j [µN] (if spin not 0)
gj is given by j(j+1) + l(l+1) – s(s+1)
2j(j+1)
gl + (l:orbital) j(j+1) – l(l+1) + s(s+1)
2j(j+1)
gs (s:intrinsic spin) and simplified as
(l=1±1/2) With the values of
gl = 1, gs = +5.6 for protons
gl = 0, gs = -3.8 for neutrons,
one can obtain the Schmidt values
(valid for simplest configuration)
proton
neutron
j + 2.3 (l+1/2)
-1.9 (l+1/2)
j – 2.3j/(j+1) (l-1/2) 1.9j/(j+1) (l-1/2)
March 31, 2014 PHY492, Lecture 31 13