nuclear models: collective nuclear models (part...
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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)