nuclear models: from spherical -shell model to deformed -shell...

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

symmetry): φφφφµµµµ

µµµµππππ

φφφφηηηηi

e2

1)( ====

(9)

(10)

(11)

),,(),,(ˆ φφφφρρρρψψψψφφφφρρρρψψψψ zEzH ====

),,()(),,()ˆˆˆ( φφφφρρρρψψψψφφφφρρρρψψψψ φφφφρρρρφφφφρρρρ zEEEzHHH zz ++++++++====++++++++


z-dependent part: eigenenergy

ρρρρ-dependent part: look at the asymptotic behavior of the wave functions:

- For ρρρρ0 the equation becomes

Solved with (the negative sign has to be discarded

since the wave function must go to zero at the origin).

- For the equation approaches

a trial with

In this limit we get

)exp()( 2βρβρβρβρρρρρχχχχ −−−−====

(12)

(13)

(14)

(15)


Introduce

trial function

should fulfill the orthogonality condition :

The appropriate orthogonal polynomials are Laguerre polynomials:

The final trial function:

Insert (19) into (10) =>

where

(16)

(17)

(18)

(19)

(20)


Compare (20) with the differential equation for the generalized Laguerre

polynomials

with the condition for A:

The new quantum number n (or denoted as nρ ρ ρ ρ )))) can take values from

Finally: the eigenfunctions of the harmonic oscillator in cylindrical coordinates

are given by

where N is the normalization constant, .

The energy of the levels is given by

(21)

(22)

(23)

(24)


The degeneracy of the levels is independent of the coordinate system used, and the

same for the principal quantum number N which can be split up as

Thus,

The various coordinate systems are useful in different situations:

• the spherical basis makes the spin-orbit coupling diagonal,

• while deformed nuclei are often treated more simply in the cylindrical basis.

(25))2

3( ++++==== NE ωωωωh

µµµµ=0, ±1, ±2,...

nρρρρ=0,1,2,3,…

nz=0,1,2,3,…

Introduce the spin-orbit interaction Vls – a coupling of the spin and the orbital

angular momentum. The additional term in the single-particle potential is thus

In the spherical case this term is diagonal and its value is

the splitting of the two levels with

The The spinspin--orbit interactionorbit interaction

(26)

(27)

(28)

slCVlsˆˆˆ ⋅⋅⋅⋅====

The coefficient C can be r-dependent. Usually a constant is assumed: typical

values for C are in the range of 0.3 to 0.6

Experimentally one finds that the state with j = l+1/2 is lower in energy then

j = l-1/2 , so that the spin-orbit coupling term must have a negative sign, i.e.

to be attractive.

The Harmonic Oscillator with SpinThe Harmonic Oscillator with Spin--Orbit CouplingOrbit Coupling

The energy levels are labeled as nlj :

by the radial quantum number n, orbital angular momentum l, and total angular

momentum j.

The nlj level is (2j + 1) times degenerate

with projections

Shell structure:

One can distinguish two types of shell closures:

Closed j shell: if all the projections Ω Ω Ω Ω belonging to the same j state are filled.

Major shell closure: if there is a larger gap in the level scheme to the next

unfilled j shell, i.e., if a magic number is reached.

The closed shell can be only for even-even nuclei the magic numbers are even!

In both cases the angular momentum of the nucleus should be zero.

(29)

The Harmonic Oscillator with SpinThe Harmonic Oscillator with Spin--Orbit CouplingOrbit Coupling

The magic numbers are now described correctly!

Level scheme for a

harmonic oscillator

with spin-orbit

coupling:

N=2n+l

(2, 8, 20, 28, 50, 82, 126)

N nl nlj (2j+1) shell closed shell

[126]

The DeformedThe Deformed--Shell ModelShell Model

The generalization of the phenomenological shell model to deformed nuclear

shapes was first given by S. G. Nilsson in 1955, so this version is often referred to

the Nilsson model.

The principal idea is to make the oscillator constants different in the different

spatial directions:

Associated nuclear shape:

define a geometric nuclear surface consisting of all the points (x,y,z)

where is the oscillator constant for the equivalent

spherical nucleus. This describes an ellipsoid with axes X, У, and Z and given by .

The condition of incompressibility of nuclear matter requires that the volume of

the ellipsoid should be the same as that of the sphere, implying R3 = XYZ,

and this imposes a condition on the oscillator frequencies:

(30)

(31)

(33)

Now assume axial symmetry around the z axis, i.e., ωωωωx = ωωωωy , and a small deviation

from the spherical shape given by a small parameter δδδδ.

Define

which fulfils the volume conservation condition of (33) to first order

with

Volume conservation can be fulfilled to second order using

leading in second order to

In spherical coordinates:

Using the explicit expression for the spherical harmonic Y20:

one can write the potential as

where


(34)

(35)

(36)

(37)

2

2222

20

1)2(

16

5)1cos3(

16

5),(

ryxzY −−−−−−−−====−−−−====

ππππθθθθ

ππππφφφφθθθθ

(38)


• The l2 term is introduced phenomenologically to lower the energy of the single-

particle states closer to the nuclear surface in order to correct for the steep rise in

the harmonic-oscillator potential there.

• к and µ µ µ µ may be different for protons and neutrons and also depend on the

nucleon number; the values are of the order of 0.05 for к and 0.3 for µµµµ.

The Hamiltonian may be diagonalized in the basis of the harmonic oscillator

using either spherical or cylindrical coordinates depending on the application:

In spherical coordinates the spin-orbit and l2 terms are diagonal, but the

deformed oscillator potential not (since the Y20 term couples orbital angular

momenta differing by ±2)

In cylindrical coordinates the deformed oscillator potential is diagonal and the

angular-momentum terms must be diagonalized numerically.

Thus, the Hamiltonian of the deformed-shell model reads (spherical coordinates) :

spin-orbit term

deformed oscillator potential

(40)


Consider the quantum numbers resulting from both basis

The energy levels in the spherical basis are given by

with the principal quantum number

where nr is the radial quantum number, l is angular-momentum quantum

number with projection m

In the cylindrical basis

(41)

(42)

where nz is the number of quanta in the z direction, nρρρρ is that of radial excitations,

and m is the angular momentum projection on the z axis.

For the spherical shape the levels are grouped according to the principal

quantum number N (with the splitting by the spin-orbit force determined through

the total angular momentum j), but the behavior with deformation depends on

how much of the excitation is in the z direction.

label the single-particle levels with the set

where the projection of total angular momentum Ω Ω Ω Ω , and the parity ππππ are good

quantum numbers while N, nz and m are only approximate and may be determined

for a given level only by looking at its behavior near the spherical state


For prolate deformation, the potential becomes small in z direction, and

the energy contributed by nz excitations decreases. The cylindrical

quantum numbers are thus helpful in understanding the splitting for

small deformations.

For very large deformations the influence of the spin-orbit and l2 terms

becomes less important and one may classify the levels according to the

cylindrical quantum numbers.

Note: the deformed shapes obtained in the deformed shell-model are different

from those in the collective model, where the radius and not, as here, the

potential, was expanded in spherical harmonics the shapes in the deformed-

shell model always remain ellipsoidal, even with arbitrarily long stretching,

whereas in the collective model e.g. a fission shape is possible


Lowest part of the level diagram

(Nilsson diagram) for the

deformed shell model.

The single-particle energies are

plotted as functions of

deformation ββββ0 and are given in

units of

The quantum numbers Ω Ω Ω Ω ππππ for

the individual levels and lj,

for the spherical ones are

indicated as well as the magic

numbers for the spherical

shape.

ΩΩΩΩππππljΩΩΩΩππππ

nuclear models: from spherical -shell model to deformed -shell...

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