Lecture 6 Lecture 6 Nuclear models: Nuclear models: from from Spherical Spherical - - Shell Model Shell Model to Deformed to Deformed - - Shell Model Shell Model WS2012/13 WS2012/13 : : ‚ ‚ Introduction to Nuclear and Particle Physics Introduction to Nuclear and Particle Physics ‘ ‘ , Part I , Part I
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Lecture 6Lecture 6
Nuclear models: Nuclear models:
from from SphericalSpherical--Shell ModelShell Model
to Deformedto Deformed--Shell ModelShell Model
WS2012/13WS2012/13: : ‚‚Introduction to Nuclear and Particle PhysicsIntroduction to Nuclear and Particle Physics‘‘, Part I, Part I
The SphericalThe Spherical--Shell ModelShell Model
The SphericalThe Spherical--Shell ModelShell Model belongs to the phenomenological single particle
models (‚independent particle models‘), i.e. a type of model of noninteracting
particles in a mean-field potential
Experimental evidance for shell structure is the existance of magic nuclei:
(2, 8, 20, 28, 50, 82, 126)
a larger total binding energy of the nucleus,
a larger energy required to separate a single nucleon,
a higher energy of the lowest excited states, and
a large number of isotopes or isotones with the same magic number for
protons (neutrons)
A phenomenological shell model is based on the Schrodinger equation for
the single-particle levels i :
Eigenstates: ψψψψi(r) - wave function; eigenvalues: εεεεi – energy; V(r) is a nuclear
potential – spherically symmetric
(1)
cf. Lecture 3
cf. Lecture 3
with
Problems: it goes to infinity instead of zero at large distances, it clearly does
not produce the correct large-distance behavior of the wave functions
The nuclear potentialsThe nuclear potentials
The type of a nuclear potential (spherically symmetric) :
the Woods-Saxon potential
Typical values for the parameters are: depth V0 = 50MeV, radius R =1.1 fmA1/3,
and surface thickness a=0.5 fm. This potential follows a similar form as the
experimental nuclear density distribution.
the harmonic-oscillator potential
the square-well potential
the wave functions vanish for r > R and are thus not realistic in this region.
(2)
(3)
(4)
The nuclear potentialsThe nuclear potentials
Sketch of the functional form of three popular phenomenological shell-
model potentials: Woods-Saxon, harmonic oscillator, and the square well.
The parameters are applicable to 208Pb.
This corresponds to the given number of excitations in the respective coordinate
direction and an energy of
The Harmonic Oscillator The Harmonic Oscillator
I. Determine the eigenfunctions of the harmonic oscillator in Cartesian coordinates.
In Cartesian coordinates the Schrödinger equation is
As the Hamiltonian in this case is simply the sum of three one-dimensional
harmonic-oscillator Hamiltonians, the solution is a product of one-dimensional
harmonic-oscillator wave functions:
with oscillator quantum numbers nx ,ny and nz,
All states with the principal quantum number N = nx + ny + nz are degenerate, i.e.
the states with the same N (but different combinations of nx,ny,nz) have the same
energy E.
Hn(x) are the Hermite polynomials:
(5)
(6)
(7)
(8)
The Harmonic OscillatorThe Harmonic Oscillator
II. Determine the eigenfunctions of the harmonic oscillator in cylindrical coordinates.
In cylindrical coordinates the Schrödinger equation is
The separation of variables
leads to
Here µµµµ and A are separation constants (to be defined).
φφφφ-dependent part: the azimuthal quantum number µµµµ=0, ±1, ±2,... (due to axial