Lecture 4 Nuclear models: Collective Nuclear Models (part 2) WS2012/13: Introduction to Nuclear and Particle Physics,, Part I 1
Reminder : cf. Lecture 3 Collective 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(θ,φ,t) denotes the nuclear radius in the direction (θ,φ) at time t, and R 0 is the radius of the spherical nucleus, which is realized when all α λµ =0. The time-dependent amplitudes α λµ (t) describe the vibrations of the nucleus with different multipolarity around the ground state and thus serve as collective coordinates (tensor). 2 λµ (1)
Collective excitations of nuclei I. vibrations II. rotations 3
Collective coordinates Properties of the coefficients α λµ λµ : - Complex conjugation: the nuclear radius must be real, i.e., R(θ,φ,t)=R*(θ,φ,t). (2) Applying (2) to (1) and using the property of the spherical harmonics (3) (1) one finds that the α λµ have to fulfill the condition: λµ (4) - The dynamical collective coordinates α λµ (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: λ=0,1,2,. 4
I. Types of Multipole Deformations Groundstate The monopole mode, λ = 0. 1 α Y 00 = 00 R( ϑ, φ,t ) = R 0(1+ α00y00 ) = R 0(1+ ) 4π 4π The spherical harmonic Y 00 is constant, so that a nonvanishing value of α 00 corresponds to a change of the radius of the sphere. Monopole mode λ=0 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 α 00 can be used to cancel the overall density change present as a side effect in the other multipole deformations. The dipole mode, λ = 1. Y 10 cosθ Dipole deformations, λ = 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. 5
Types of Multipole Deformations The quadrupole mode, λ = 2 The quadrupole deformations - the most important collective low energy excitations of the nucleus. The octupole mode, λ = 3 The octupole deformations are the principal asymmetric modes of the nucleus associated with negative-parity bands. The hexadecupole mode, λ = 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. 6
Types of Multipole Deformations 7
Quadrupole deformations The quadrupole deformations are the most important vibrational degrees of freedom of the nucleus. For the case of pure quadrupole deformation (λ = 2) the nuclear surface is given by (5) Consider the different components of the quadrupole deformation tensor α 2µ The parameters α 2µ are not independent - cf. (4): From (4): (6) => α 20 is real (since α 20 = α 20 ) ; and we are left with five independent real degrees of freedom: α 20 and the real and imaginary parts of α 21 and α 22 20 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 (θ,φ) : 21 (6) 22 (7) 2µ 8
From spherical to cartesian coordinates Spherical coordinates (r,θ,φ) Cartesian coordinates (x,y,z)=>(ζ,ξ,η) r The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) 9
Cartesian coordinates Cartesian coordinates fulfil subsidiary conditions (8) (9) Substitute (9) in (5): (10) where the cartesian components of the deformation are related to the spherical ones by (11) 10
Cartesian coordinates 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(θ,φ) fulfills (12) Subs. (10) into (12) and accounting that we obtain: 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: α 20 describes a stretching of the z axis with respect to the у and x axes, α 22, α 2 2 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, α 21, α 2 1 indicate an oblique deformation of the z axis. 21 (12) 11
Now - continue! Principal axis system 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 α 2µ. 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 (13) 2µ (14) We get for the spherical components: Note: z symmetry axis (15) 12
* Euler angles 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). 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. 13
Principal coordinates There are still five independent real parameters, but now with more clearer geometrical significance: a 0 indicating the stretching of the z' axis with respect to the x' and y' axes; a 2 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 a 0 and a 2. Note also that a 2 =0 describes a shape with equal axis lengths in the x andу directions, i.e., one with axial symmetry around the z axis. 14
(β,γ) coordinates There is also another set of parameters introduced by Aage Niels Bohr (β, γ). It corresponds to something like polar coordinates in the space of (a 0,a 2 ) and is defined via (16) Thus, (17) This particular sum (17) over the components of α 2µ is rotationally invariant, i.e. it has the same value in the laboratory and the principal axis systems Consider the nuclear shapes in the principal axis system (x',y',z'), i.e. calculate the cartesian components as a function of γ for fixed β: 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) 15
(β,γ) coordinates Cartesian deformation components indicate the stretching of the nuclear axis in that direction. Using the new notation δr k for these, where к = 1,2,3 corresponds to the x',y' and z' directions, respectively, one may combine these results into one equation: к = 1,2,3 (i.e. x',y' z' ) (19) At γ = 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 γ, 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 γ = 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 x z y Prolate (x=y) (z=x) Oblate (x=y) (z=x) 16
(β,γ) coordinates Figure: The (β,γ β,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,γ β,γ) plane are repeated every 60. Because the axis orientations are different, the associated Euler angles also differ. In conclusion, the same physical shape (including its orientation in space) can be represented by different sets of deformation parameters (β,γ β,γ) and Euler angles! 17
Description of the quadrupole deformation Thus, the quadrupole deformation may be described: either in a laboratory-fixed reference frame through the spherical tensor α 2µ, or, alternatively, by giving the deformation of the nucleus with respect to the principal axis frame using the parameters (a 0,a 2 ) or β,γ) (β,γ and the Euler angles (θ 1,θ 2,θ 3 ) indicating the instantaneous orientation of the body-fixed frame. Both cases require different treatments of rotational symmetry. 18
Surface vibration model Describe the nuclei deformations vibrations - in the laboratory-fixed reference frame through the spherical tensor α λµ (t). From rotational invariants quadratic in α λµ and velocities α& λµ terms restrictions on the structure of the potential V and kinetic T energies of the Lagrangian (dictated by symmetry): kinetic energy: potential energy: (20) B λ - the collective mass parameters C λ - the stiffness coefficients for the potential Each single mode (characterized by λ and µ) behaves like a harmonic oscillator with both the mass parameters and the stiffness coefficients depending on the angular momentum. Lagrangian for the quadrupole deformations (λ=2): (21) 19
Surface vibration model Introduce the conjugate momentum: π µν = L & α λµ (22) λ=2 (23) Hamiltonian for a harmonic oscillator : (24) there are 5 harmonic oscillators (for λ=2): µ = -2,-1,0,1,2 Quantization is done by imposing the boson commutator relations : (25) 20
Surface vibration model Introduce creation and annihilation operators: (26) where The pseudoparticles - that are created and annihilated by these operators - are called phonons in analogy to the quanta of vibrations in solids. 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 µ, which can be excited independently and have a zero-point energy of 1 hω each. 2 21
Surface vibration model N counts the total number of quanta present in the system. Additional quantum numbers are the angular momentum λ and its projection µ, 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 2hω. They should couple to good total angular momentum: 22
* Coupling of Angular Momenta The system of two particles with angular momenta Total angular momentum Eigenfunctions of : j 1,m 1 > and j 2,m 2 > The basis for the system of two particles: are Clebsch-Gordon-coefficients j + 1 j2 J j1 j2 m m = 1 + 2 M 23
Surface vibration model Consider two-phonon states: Angular-momentum selection rules allow for the values of λ = 0,1,2,3,4. However, it turns out that not all of these values are possible. Exchanging µ' and µ'' in the Clebsch-Gordan coefficient and using a symmetry property of the Clebsch- Gordan coefficients to symmetrize the expression we get because the operators commute. Consequently, the wave functions for odd values of λ 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). 24
Spherical vibrator Figure: Comparison of the spherical vibrator model with experimental data for 114 Cd. The energy levels are in MeV, while the B(E2) values, indicated next to the transition arrows, are given in e 2 fm 4. Qualitatively reasonable agreement with the experimental data Quantitative differences due to higher order effects (not accounted) in the harmonic oscillator vibrator model 25
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! E.g: 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. 26
Rotating nuclei: Rigid rotator The Hamiltonian for a rigid rotor with moments of inertia Θ : The last term is dropped for nuclei with axial symmetry about the z-axis: J Z =0. J denotes the rotation about a body-fixed axis, J is the rotation about a stationary axis. Quantum numbers for the rotor will be generated by the space-fixed operators J 2 (and J z ). Since the energy of the nucleus does not depend on its orientation in space: Hamiltonian: Make quantization, considering H and J as a operators 27
Rotating nuclei: Rigid rotator Figure: Lowest experimental bands for the nucleus 238 U 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 e 2 b 2. 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 28
Rotation-vibration model Bohr-Mottelson-model: using deformation parameters (β,γ β,γ) and Euler angles Fässler-Greiner-model: using cartesian coordinates (ζ,ξ,η) : Hamiltonian: Energy spectra: β-state band γ-state band ground-state band Here K is eigenvalue of J Z Figure: Structure of the spectrum of the rotation-vibration model. The bands are characterized by a given set of (К, n β,n γ ) and follow the J(J + 1) rule of the rigid rotor. 29
Literature Walter Greiner Joachim A. Maruhn NUCLEAR MODELS (Springer) 30