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Lecture 4Lecture 4

Nuclear models:Nuclear models:

CollectiveCollective Nuclear Models Nuclear Models

(part 2)(part 2)

WS2012/13WS2012/13: : ‚‚Introduction to Nuclear and Particle PhysicsIntroduction to Nuclear and Particle Physics‘‘, Part I, Part I

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Collective excitations of nucleiCollective excitations of nuclei

�The single-particle shell model can not properly describe the excited states of

nuclei: the excitation spectra of even-even nuclei show characteristic band

structures which can be interpreted as vibrations and rotations of the nuclear

surface

low energy excitations have a collective origin !

�The liquid drop model is used for the description of collective excitations of

nuclei: the interior structure, i.e., the existence of individual nucleons, is neglected

in favor of the picture of a homogeneous fluid-like nuclear matter.

� The moving nuclear surface may be described quite generally by an expansion

in spherical harmonics with time-dependent shape parameters as coefficients:

where R(θθθθ,φφφφ,t) denotes the nuclear radius in the direction (θθθθ,φφφφ) at time t, and R0 is

the radius of the spherical nucleus, which is realized when all ααααλµ λµ λµ λµ =0.

The time-dependent amplitudes ααααλµλµλµλµ(t) describe the vibrations of the nucleus with

different multipolarity around the ground state and thus serve as collective

coordinates (tensor).

(1)

Reminder :

Reminder :

cf. Lecture 3

cf. Lecture 3

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Collective excitations of nucleiCollective excitations of nuclei

I. I. vibrationsvibrations II. II. rotationsrotations

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Properties of the coefficients ααααλµ λµ λµ λµ ::::

- Complex conjugation: the nuclear radius must be real, i.e.,

R(θθθθ,φφφφ,t)=R*(θθθθ,φφφφ,t).

Applying (2) to (1) and using the property of the spherical harmonics

one finds that the ααααλµλµλµλµ have to fulfill the condition:

Collective coordinates

(2)

(3)

(4)

- The dynamical collective coordinates ααααλµλµλµλµ (tensors!) define the distortion -

vibrations - of the nuclear surface relative to the groundstate.

- The general expansion of the nuclear surface in (1) allows for arbitrary

distortions: λλλλ=0,1,2,….

(1)

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I. Types of Multipole Deformations I. Types of Multipole Deformations

�The monopole mode, , , , λλλλ = 0.

The spherical harmonic Y00 is constant, so that

a nonvanishing value of αααα00 corresponds to a change of the radius of the sphere.

The associated excitation is the so-called breathing mode of the nucleus. Because

of the large amount of energy needed for the compression of nuclear matter, this

mode is far too high in energy to be important for the low-energy spectra

discussed here. The deformation parameter αααα00 can be used to cancel the overall

density change present as a side effect in the other multipole deformations.

ππππ4

100 ====Y

Groundstate

�The dipole mode, λλλλ = 1.

Dipole deformations, λλλλ = 1 to lowest order, really do not

correspond to a deformation of the nucleus but rather to a

shift of the center of mass, i.e. a translation of the nucleus, and

should be disregarded for nuclear excitations since

translational shifts are spurious.

θθθθcos10 ≈≈≈≈Y

)4

1(R)Y1(R)t,,(R 00000000 ππππ

ααααααααφφφφϑϑϑϑ ++++====++++====

Monopole

mode λλλλ=0

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Types of Multipole Deformations Types of Multipole Deformations

�The quadrupole mode, , , , λλλλ = 2

The quadrupole deformations - the most important

collective low energy excitations of the nucleus.

�The octupole mode, , , , λλλλ = 3

The octupole deformations are the principal asymmetric

modes of the nucleus associated with negative-parity bands.

�The hexadecupole mode, , , , λλλλ = 4

The hexadecupole deformations: this is the highest angular

momentum that has been of any importance in nuclear

theory. While there is no evidence for pure hexadecupole

excitations in the spectra, it seems to play an important role

as an admixture to quadrupole excitations and for the

groundstate shape of heavy nuclei.

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Types of Multipole Deformations Types of Multipole Deformations

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Quadrupole deformations

(5)

� The quadrupole deformations are the most important vibrational degrees of

freedom of the nucleus.

For the case of pure quadrupole deformation (λλλλ = 2) the nuclear surface is given by

� Consider the different components of the quadrupole deformation tensor αααα2µ2µ2µ2µ

The parameters αααα2µ 2µ 2µ 2µ are not independent - cf. (4):

From (4):

(6) => αααα20202020 is real (since αααα20202020==== αααα∗∗∗∗20202020)))) ; and we are left with five independent real

degrees of freedom: αααα20202020 and the real and imaginary parts of αααα21212121 and αααα22222222

(6)

�To investigate the actual form of the nucleus, it is best to express this in

cartesian coordinates by rewriting the spherical harmonics in terms of the

cartesian components of the unit vector in the direction (θθθθ,φφφφ) :

(7)

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From spherical to cartesian coordinates

Spherical coordinates (r,θ,φθ,φθ,φθ,φ) Cartesian coordinates (x,y,z)=>((((ζ,ξ,η)ζ,ξ,η)ζ,ξ,η)ζ,ξ,η)

r

The invention of Cartesian coordinates in

the 17th century by René Descartes

(Latinized name: Cartesius)

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Cartesian coordinates

(8)Cartesian coordinates fulfil subsidiary conditions

(9)

Substitute (9) in (5):

where the cartesian components of the deformation are related to the spherical

ones by

(10)

(11)

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Subs. (10) into (12) and accounting that

we obtain:

In (11) six independent cartesian components appear (all real) , compared to the

five degrees of freedom contained in the spherical components. However, the

function R(θθθθ,φφφφ) fulfills

Cartesian coordinates

(12)

(12)

���� 5 independent cartesian components

As the cartesian deformations are directly related to the streching (or

contraction) of the nucleus in the appropriate direction, we can read off that:

� αααα20202020 describes a stretching of the z axis with respect to the у and x axes,

� αααα22222222, , , , αααα2222−−−−2222 describes the relative length of the x axis compared to the у axis (real

part), as well as an oblique deformation in the x-y plane,

� αααα21212121, , , , αααα2222−−−−1111 indicate an oblique deformation of the z axis.

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� The problem with cartesian parameters is that the symmetry axis of the

nucleus (if there are any) can still have an arbitrary orientation in space, so that

the shape of the nucleus and its orientation are somehow mixed in the αααα2µ2µ2µ2µ .

The geometry of the situation becomes clearer if this orientation is separated by

going into the principal axis system which is rotated by Euler angules

with respect to the laboratory-fixed frame

� If we denote this new coordinate frame by primed quantities, the cartesian

deformation tensor must be diagonal, so that

Principal axis system

(13)

(14)

(15)

���� We get for the spherical components: Note: z‘ || symmetry axis

Now Now --

continue!

continue!

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* * Euler angles

The definition is Static. Given a reference frame

and the one whose orientation we want to

describe, first we define the line of nodes (N) as

the intersection of the xy and the XY coordinate

planes (in other words, line of nodes is the line

perpendicular to both z and Z axis). Then we

define its Euler angles as:

�α (or ψ) is the angle between the x-axis and

the line of nodes.

�β (or θ) is the angle between the z-axis and

the Z-axis.

�γ (or φ) is the angle between the line of

nodes and the X-axis.

Euler angles are a means of representing the spatial orientation of any frame

(coordinate system) as a composition of rotations from a frame of reference

(coordinate system). In the following the fixed system is denoted in lower case

(x,y,z) and the rotated system is denoted in upper case letters (X,Y,Z).

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Principal coordinates

There are still five independent real parameters, but now with more clearer

geometrical significance:

� a0 indicating the stretching of the z' axis with respect to the x' and y' axes;

� a2 which determines the difference in length between the x' and y' axes;

� three Euler angles , which determine the orientation of the

principal axis system (x',y',z') with respect to the laboratory-fixed frame (x,y,z).

���� The advantage of the principal axis system is that rotation and shape

vibration are clearly separated:

�a change in the Euler angles denotes a pure rotation of the nucleus without

any change in its shape,

�a change in shape – vibration -is only determined by a0 and a2.

Note also that a2=0 describes a shape with equal axis lengths in the x and у

directions, i.e., one with axial symmetry around the z axis.

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(β,γ)β,γ)β,γ)β,γ) coordinates

There is also another set of parameters introduced by Aage Niels Bohr – (β, γ)(β, γ)(β, γ)(β, γ)....It corresponds to something like polar coordinates in the space of (a0 ,a2) and is

defined via (16)

This particular sum (17) over the components of αααα2µ2µ2µ2µ is rotationally invariant,

i.e. it has the same value in the laboratory and the principal axis systems

Thus, (17)

� Consider the nuclear shapes in the principal axis system (x',y',z'), i.e.

calculate the cartesian components as a function of γγγγ for fixed ββββ:

Using (12,15,16) =>

(18)

Here the principal axis system (x',y',z') is rotated by Euler angules

with respect to the laboratory-fixed frame (x,y,z)

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Cartesian deformation components indicate the stretching of the nuclear axis in that

direction. Using the new notation δδδδRk for these, where к = 1,2,3 corresponds to the x',y'‚

and z' directions, respectively, one may combine these results into

one equation:

(β,γ)β,γ)β,γ)β,γ) coordinates

(19)

�At γγγγ = 0° the nucleus is elongated along the z' axis,

but the x' and y' axes are equal. This axially

symmetric type of shape is reminiscent of a cigar and

is called prolate (for x=y).

�If we increase γγγγ, the x' axis grows at the expense of

the y' and z' axes through a region of triaxial shapes

with three unequal axis, until axial symmetry is again

reached at γγγγ = 60°, but now with the z' and x' axis

equal in length. These two axes are longer than the y'

axis: the nucleus has a flat, pancake-like shape, which

is called oblate (for x=z).

� This pattern is repeated: every 60° axial symmetry

repeated and prolate and oblate shapes alternate, but

with the axis permuted in their relative length ���� the

axis orientations are different; the associated Euler

angles also differ

Prolate

(x=y) (z=x)

Oblate

(x=y) (z=x)

x

z

y

к = 1,2,3 (i.e. x',y'‚ z' )

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(β,γ)β,γ)β,γ)β,γ) coordinates

� In conclusion, the same physical shape (including its

orientation in space) can be represented by different sets of

deformation parameters (β,γβ,γβ,γβ,γ) and Euler angles!

Figure: The (β,γβ,γβ,γβ,γ) plane is divided into six

equivalent parts by the symmetries:

the sector between 0° and 60° contains all

shapes uniquely, i.e. triaxial shapes

the types of shapes encountered along the

axis: e.g., prolate x=y implies prolate

shapes with the z‘ axis as the long axis

and the two other axis x‘ and y‘ equal.

���� various nuclear shapes – prolate or oblate - in the (β,γβ,γβ,γβ,γ)

plane are repeated every 60°. Because the axis orientations

are different, the associated Euler angles also differ.

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Description of the quadrupole deformation

Thus, the quadrupole deformation may be described:

�either in a laboratory-fixed reference frame through the spherical

tensor αααα2µ2µ2µ2µ ,

or, alternatively,

�by giving the deformation of the nucleus with respect to the principal

axis frame using the parameters (a0 ,a2) or (β,γβ,γβ,γβ,γ) and the Euler angles

(θθθθ1,θθθθ2,θθθθ3) indicating the instantaneous orientation of the body-fixed frame.

Both cases require different treatments of rotational symmetry.

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Surface vibration model

Describe the nuclei deformations – vibrations - in the laboratory-fixed reference

frame through the spherical tensor ααααλµλµλµλµ(t) .

� From rotational invariants ���� quadratic in ααααλµ λµ λµ λµ and velocities terms

���� restrictions on the structure of the potential V and kinetic T energies of the

Lagrangian (dictated by symmetry):

λµα&

kinetic energy:

potential energy:

Bλ λ λ λ - the collective mass parameters

Cλ λ λ λ - the stiffness coefficients for the potential

���� Each single mode (characterized by λλλλ and µµµµ) behaves like a harmonic

oscillator with both the mass parameters and the stiffness coefficients

depending on the angular momentum.

� Lagrangian for the quadrupole deformations (λλλλ=2):

(20)

(21)

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Surface vibration model

� Introduce the conjugate momentum:(22)

(23)

(24)

Hamiltonian for a harmonic oscillator :

there are 5 harmonic oscillators (for λλλλ=2): µ µ µ µ = -2,-1,0,1,2

Quantization is done by imposing the boson commutator relations :

(25)

λ=2λ=2λ=2λ=2

λµλµλµλµ

µνµνµνµν ααααππππ

&∂∂∂∂

∂∂∂∂====

L

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Surface vibration model

(26)

� Introduce creation and annihilation operators:

�The pseudoparticles - that are created and annihilated by these operators -

are called phonons in analogy to the quanta of vibrations in solids.

where

Commutation relations - like for bosons:

(27)

Number of particles: (28)

Hamiltonian for a harmonic oscillator : (29)

as we are effectively dealing with five oscillators, corresponding to the different

magnetic quantum numbers µµµµ, which can be excited independently and have a

zero-point energy of each. ωωωωh2

1

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N counts the total number of quanta present in the system. Additional

quantum numbers are the angular momentum λλλλ and its projection µµµµ, so

that the states can be labeled provisionally by

The lowest-lying states are as follows:

1. The nuclear ground state is the phonon vacuum

Its energy is the zero-point energy:

2. The first excited state is the multiplet (one-phonon state) with angular

momentum 2, i.e. 2+ state:

3. The second set of excited states is given by the two-phonon states with an

excitation energy of . They should couple to good total angular

momentum:

Surface vibration model

ωωωωh2

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** Coupling of Angular Momenta Coupling of Angular Momenta

The system of two particles with angular momenta

Total angular momentum

Eigenfunctions of : |j1,m1> and |j2,m2 >

The basis for the system of two particles:

are Clebsch-Gordon-coefficients

2121 jjJjj ++++≤≤≤≤≤≤≤≤−−−−

Mmm ====++++ 21

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Surface vibration model

because the operators commute. Consequently, the wave functions for odd

values of λλλλ vanish: such states do not exist !

�The two-phonon states are thus restricted to angular momenta 0, 2, and 4,

forming the two-phonon triplet.

This effect is an example of the interplay of angular-momentum coupling and

symmetrization (or, for fermions, antisymmetrization).

Angular-momentum selection rules allow for the values of λλλλ = 0,1,2,3,4.

� However, it turns out that not all of these values are possible. Exchanging

µµµµ' and µµµµ'' in the Clebsch-Gordan coefficient and using a symmetry property

of the Clebsch- Gordan coefficients

to symmetrize the expression we get

Consider two-phonon states:

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Spherical vibrator

Figure: Comparison of the

spherical vibrator model

with experimental data for114Cd. The energy levels are

in MeV, while the B(E2)

values, indicated next to the

transition arrows, are given

in e2 fm4.

�Qualitatively reasonable agreement with the experimental data

�Quantitative differences – due to higher order effects (not accounted) in the

harmonic oscillator vibrator model

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II. Rotating nuclei: Rigid rotator

As known from classical mechanics, the degrees of freedom of a rigid rotor are

the three Euler angles, which describe the orientation of the body-fixed axes in

space

A classical rotor can rotate about any of its axis.

In quantum mechanics, however, the case is different, i.e if the nucleus has

rotational symmetries and no internal structure. For example, a spherical

nucleus cannot rotate, because any rotation leaves the surface invariant and

thus by definition does not change the quantum-mechanical state (and energy):

� a spherical nucleus has no rotational excitations at all !

� a nucleus with axial symmetry cannot rotate around the axis of symmetry!

Note: the final decision about the validity of these statements

has to come from experiment, of course; it will depend on

whether other degrees of freedom are involved. We shall see

that rotations about a symmetry axis are made possible by

simultaneous dynamic deviations from axial symmetry.

E.g:

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Quantum numbers for the rotor will be generated by the space-fixed operators

J2 (and Jz). Since the energy of the nucleus does not depend on its orientation in

space:

Rotating nuclei: Rigid rotator

The Hamiltonian for a rigid rotor with moments of inertia Θ Θ Θ Θ :

The last term is dropped for nuclei with axial symmetry about the z-axis: JZ=0.

J’ denotes the rotation about a body-fixed axis, J is the rotation about a

stationary axis.

Hamiltonian:

Make quantization, considering H and J as a operators

����

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Rotating nuclei: Rigid rotator

Figure: Lowest experimental bands

for the nucleus 238U with selected

transition probabilities.

The energies written next to the

levels are in MeV and the B(E2)

values (next to the transition

arrows) in e2 b2.

Note that the arrows indicate the

transition direction for the B(E2)

values.

�The spectra are proportional to J(J+1), i.e. the spectrum of a rotator.

� Reasonable agreement with experimental data

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Rotation-vibration model

�Bohr-Mottelson-model: using deformation parameters (β,γβ,γβ,γβ,γ) and Euler angles

�Fässler-Greiner-model: using cartesian coordinates (ζ,ξ,η) :(ζ,ξ,η) :(ζ,ξ,η) :(ζ,ξ,η) :

Hamiltonian:

Energy spectra:

Here K is eigenvalue of JZ

Figure: Structure of the spectrum of the

rotation-vibration model.

The bands are characterized by a given set

of (К, nββββ,nγ γ γ γ ) and follow the J(J + 1) rule of

the rigid rotor.

groundground--state bandstate bandββββββββ--state bandstate band γγγγγγγγ--state bandstate band

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LiteratureLiterature

Walter Greiner • Joachim A. Maruhn

NUCLEAR MODELS

(Springer)