Nuclear Models Basic Concepts in Nuclear Theory Joachim A. Maruhn
Topics Foundations Collective models Single-particle models: phenomenological and self-consistent The Fermi-gas model Literature W. Greiner and J. A. Maruhn, Nuclear Models, Springer P. Ring and P. Schuck, The Nuclear Many-body Problem, Springer. J. A. Maruhn, P.-G. Reinhard, and E. Suraud, Simple Models of Many-Fermion Systems, Springer
Basic facts for nuclear models Constant binding: saturating interaction Decrease for heavy nuclei indicates increasing importance of Coulomb force. Nuclear radii go as R r A, r 1. fm 3 0 0 showing constant density on the average independent of A. 0.15 0.17 fm 3
Basic facts () The density falls off rapidly at the surface: some models assume a sharp surface. Heavier nuclei deviate from N=Z: again effect of Coulomb. Otherwise there is a tendency to symmetry.
The liquid-drop model The Bethe-Weizsäcker formula describes the binding energy of spherical nuclei with mass number A, charge number Z, and neutron number N. Z ( N Z) 3 EB a V A a S A ac a 3 Sym A A volume term surface term Coulomb term omitting smaller corrections. Symmetry term a 16 MeV a 0 MeV a 0.75MeV a 1MeV V S C Sym This simple formula already provides an understanding of the binding properties going from light to heavy nuclei the location of the b-stable line the possibility of fusion and fission the potential energy involved in nuclear deformation (unfortunately not the kinetic part)
Fission in the liquid drop
Angular momentum Two wave functions (or operators) with angular momentum eigenvalues can becoupled to a resulting angular momentum using J M ( j j J m m M ) j m j m mm 1 1 1 1 1 where M m1 m, j1 j J j1 j and ( j j J m m M) is a Clebsch-Gordan coefficient. 1 1 Most important case: coupling to a scalar of two operators: ( jj0 m m0) ( 1) jm j 1 ( 1) 1 jm * ( jj0 m m0) TjmTj m TjmTj m TjmTjm m m j1 j1 m with the usual Condon-Shortlex phase convention T j m ( 1) j m T * jm
Nuclear deformation Assuming a time-dependent sharp surface located at it can be expanded as with the The parity is given by ( 1) and the collective deformation coordinates. The lowest values of describe the simplest deformation modes: R R t Y * (, ) 0(1 ( ) (, )), () t 0: monopole or breathing mode: higher energy 1: translation, not an excitation : quadrupole, most important deformation 3: octupole, asymmetric deformation *, ( 1) 4: hexadecupole: important for heavy nuclei R(, )
The spherical vibrator model Assume a spherical ground state and a harmonic potential and consider the quadrupole only 1 V( ) C For the kinetic energy use collective velocities and a harmonic form leading to the Lagrangian This corresponds to five harmonic oscillators for, 1,0, 1, all with an energy quantum of C B Second quantization can be done as usual leading to boson operators 1 1 1 L B C T B ˆ ˆ ˆ b ˆ ˆ and a total energy, b,n bb 5 E N Thus an equidistant spectrum is generated. N is the phonon number.
Spherical vibrator spectrum The multiplets for a given phonon number are determined by angular momentum coupling and Bose symmetry. Each phonon carries an angular momentum of. Transition probabilities via gamma emission can also be calculated. The model describes a limiting case that is of very limited applicability. Example: 114 Cd
The rigid rotor In the surface deformation model, a nucleus cannot rotate around an axis of symmetry z, so that J ' 0 z For a deformed axially symmetric nucleus the Hamiltonian is with the angular momenta in the comoving frame (see classical mechanics). Now we have Jˆ Jˆ ' Jˆ ' ˆ x J ' y so that the rotational energy with the quantized lab-frame angular momentum becomes Hˆ Jˆ' Jˆ' x y E J J( J 1) leading to the famous rotational band structure. Because of symmetry only even angular momenta occur.
The principal-axes frame The quadrupole deformations because of contain five real parameters ( 1) *, Expressing the trigonometric functions through Cartesian coordinates z x y cos sin cos sin sin r r r we get Cartesian deformation coefficients via R R (1 Y ) * 0 R 0(1 ) Inserting the definitions of the spherical harmonics leads to 1 8 8 1 8 0 ( ), 1 ( i ), ( i ) 6 15 15 15 Selecting the axes along the principal axes requires 0
The coordinates b and This leads to 0, a, a with a and a real 1 0 0 0 The five degrees of freedom are now the intrinsic deformation coordinates a 0 and a plus the Euler angles giving the orientation. a 1 8 1 8 ( ) a ( ) 6 15 15 0 Bohr and Mottelson used a 1 bcos, a bsin 0 chosen such that b The elongation becomes b cos, b cos( ), cos( ) 5 5 5 4 4 4 3 4 3
Symmetries in the intrinsic frame The arbitrariness of choosing the intrinsic axes leads to symmetry requirements for the collective wave function that restricts certain quantum numbers. prolate oblate
Types of collective behavior
The rotation-vibration model We assume a well-deformed axially-symmetric nucleus executing small vibrations around its equilibrium shape at a b, a 0 Assume dynamic deviations given by and a harmonic potential V (, ) C C To lowest order, a Hamiltonian can then be set up as Note that the dynamic deformation makes rotations about the z-axis possible. This contains harmonic b-vibrations in, -vibrations in modified by the dynamic coupling to the rotation, plus rotational excitations. Originally proposed by Bohr and Mottelson. 1 a 0 0 0 b, a 0 0 1 1 ˆ ˆ ˆ ˆ J J J H C C B 16B z z
The spectrum The resulting energy formula is 1 1 E b nb n K 1 J ( J 1) K yielding the rotational-vibrational spectrum. Note the coupling of the - vibrations to rotation via the eigenvalue K of J z This structure is quite well realized in nature, but the interpretation of bands higher than the b-band remains controversial. The selection of even J only is caused by the symmetries for some bands.
-unstable nuclei (Wilets-Jean) 4 C b0 V( b, ) b b 0 D 8 b
Generalizations Using a potential with higher powers of the allows for describing more complicated behavior (Gneuss & Greiner) Contains a harmonic kinetic energy plus polynomial to sixth order of coupled to scalar. All of these models are purely phenomenological: their parameters have to be fitted for each nucleus. proton-neutron scissor mode using p, n (V. Maruhn-Rezwani et al., 1975)
The interacting boson approximation proposed by Arima and Iachello Constructs the collective excitations through two types of bosons: scalar s-bosons and J= d-bosons: operators These are interpreted as combinations of valence nucleons and their total number should be constant: N n n s s d d s d A Hamiltonian is set up consisting of the bosons energies times their number plus couplings between the bosons, again all with fitted coefficients. L H n n C ( d d ) ( dd) s s d d L L 0 V ( d d ) ( ds) ( d s ) ( dd) V ( d d ) ( ss) ( s s ) ( dd) 0 0 0 0 0 0 U ( d s ) ( ds) U ( s s ss) Here (...) L means coupling to angular momentum L L 0 0 0 s, s, d d,
The IBA () An attractive feature is that several limiting cases can be solved exactly using group theory: SU(3), SU(5), O(6). A transition between different structures in neighboring nuclei appears more clearly. The fixed boson number should lead to a cutoff in angular momenta, which is not seen. The model had to add additional bosons. The types of spectra described now appears to be quite similar to that of the geometric model. Other difference to geometric model: kinetic energy is as complicated as potential one; transition operators are not constrained by the geometric interpretation. Odd nuclei can be described in the Boson-Fermion model, where singleparticle operators are added.
IBA Applications
Single-particle models: Motivation Experimental evidence for shell structure: for magic numbers of protons or neutrons larger total binding energy larger separation energy for removing single nucleon larger energy of lowest excited state larger number of isotones or isotopes, respectively nuclei tend to be spherical Magic numbers are:, 8, 0, 8, 50, and 8 for bith protons and neutrons 16 for neutrons 114 or 16 for protons (predicted for superheavy elements) Analogous to noble gases in atomic structure? Shell structure! These effects describe small perturbations on the liquid-drop binding energy but are crucial for the description of excitations
Magic numbers
Extrapolation to superheavies
Single-particle potentials There is no external potential like in atoms: the single-particle structure: it must be mitvated as a mean field by Hartree-Fock theory The magic numbers can be explained by phenomenological potentials Woods-Saxon: V Vr () 1 e Harmonic oscillator Square well 0 ( r R) a V R A a 3 0 50MeV, 1. fm, 0,5fm 1 V () r m r 41MeV V 0 r R Vr () r R Woods-Saxon is most realistic, but harmonic oscillator is often preferred because of analytic eigenfunctions
Shell structure for simple potentials Higher magic numbers are not correct with any of these potentials.
The spin-orbit coupling Suggestion (Mayer & Jensen): add a spin-orbit force V V Cl s with l and s now coupling to Since j ( l s), l, s are good quantum numbers, the effect of te coupling can be calculated via and the splitting becomes j l 1 1 l s j l s j j l l s s ( ) ( 1) ( 1) ( 1) 1 1 1 3 1 1 1 ( )( ) ( )( ) ( ) E 1 E 1 C l l l l C l jl jl Note that C turns out to be negative. Question: origin of the spin-orbit force?
Shells with spin-orbit coupling The magic numbers now come out correct. In many cases, the angular momentum of the single-particle states also explains the nuclear angular momentum near magic nuclei. There are large deviations for nuclei between magic shells: nuclear deformation has to be added. The spin-orbit force can now be seen as a relativistic effect (see relativistic mean-field model)
Deformed nuclei: the Nilsson model The oscillator potential can simply be generalized as m V ( r) ( x x y y z z ) and the single-particle Hamiltonian becomes (maintaining axial symmetry) ˆ 1 h m 0r b0m 0r Y0 (, ) ( l s l ) m The deformation parameter is b 0 It turns out that, contrary to expectations, gaps in the spectrum appear also for deformed shapes, leading to deformed ground states and fission isomers The behavior of the total energy is wrong, however: needs shell corrections
The Nilsson single-particle level scheme
Strutinsky shell corrections The total energy in a single-particle model should be given by E( b ) ( b ) Coulomb k occupied k This leads to huge fluctuations. The shell correction idea is based on E( b) E ( b) du( b) LDM where du is calculated by subtracting a smoothed part from the sum of single-particle energies. The resulting shell correction is typically a few MeV, negative near shell closures, positive otherwise. This is the base of the modern microscopic-macroscopic (mic-mac) model, which uses more advanced versions of the phenomenological single-particle model and the droplet model. It is very successful for describing binding energies and fission barriers. Note that there are still conceptual problems: neutrons and protons are decoupled largely and the parameters are fitted, making extrapolation risky. Hartree-Fock models do not have a problem with total energy but are not as accurate at present.
Two-center models The Nilsson model does not correctly describe the transition to fission (it always produces ellipsoids, not two fragments). Two-center models try to cure this and are therefore better for the description of fission and heavy-ion interactions. Example: the two-center oscillator It uses two oscillator potentials centered in the fragments and interpolated smoothly Shape parameters: Separation Asymmetry Neck z A A d z 1 A A 1 1 Deformations bi a i The breakup of the neck poses prolems d
Hierarchy of Models Models with prescribed potentials: square-well, harmonic oscillator, Woods-Saxon, Yukawa+Exponential (YPE), two-center models based on these. In their latest versions these are still quantitatively superior. Self-consistent models based on Hartree or Hartree-Fock. Most widespread are: Skyrme-force, zero-range nonrelativistic Gogny force, finite range nonrelativistic Relativistic Meson-Field Theory (a.k.a. Walecka model). Interaction mediated by relativistic mesons Point-coupling model. Interaction through relativistic point interaction terms. Density functionals: appear as intermediate step in self-consistent models but can be more general since they need not derive from a force model. Recently source of hype. The number of parameters is generally similar: 6-1 The coexistence of many approaches shows the richness of nuclear theory
Hartree-Fock We start with a general Hamiltonian with -body forces p 1 H v( r r ) A i i j i1 m ij (dependence on spin, isospin, and momenta can be added). The many-body state can be expanded in Slater determinant states c aˆ ˆa. ˆa 0 k k 1 A k k k k k 1 A 1 A with the k i a selction from a complete set of single-particle states. The Hartree-Fock approximation consists in replacing this expansion by a single Slater determinant (SSD), but with the single-particle wave functions determined from a variational principle d Hˆ 0 d Hˆ 0 The SSD is varied by changing the occupation of states aˆ aˆ... aˆ 0, d aˆ aˆ with m A, i A 1 A m i This is a particle-hole (ph) excitation.
The Hartree-Fock conditions Using the second-quantized version of the Hamiltonian 1 H t aˆ aˆ v aˆ aˆ aˆ aˆ T V, ˆ ˆ ˆ ij i j k1k k3k4 k1 k k4 k 3 ij k k k k with 1 3 4 * 3 3 3 * * tkl l d r and vk ( ) ( ) ( ) ( ) k 1kk3k d r d r 4 k r 1 k r v k r 3 k r 4 m the variational equation ˆ ˆ ˆ * 0 d H aiamdc H. ˆ leads to the Hartree-Fock conditions A t ( v v ) 0, with m A and i A mi mjij mjji j1 which imply that the single-particle Hamiltonian A hˆ t ( v v ), with k and l unrestricted kl kl kjlj kjjl j1 must have vanishing ph matrix elements
The Hartree-Fock equations This can be achieved by choosing the single-particle states as eigenstates A h t ( v v ) d kl kl kjlj kjjl k kl j1 defining also the single-particle energies. In coordinate space the HF equations take the form A 3 * k r U r k r j r d r v r r j r k r k k r m j1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). with the mean field U r d r v r r r r 3 * ( ) ' ( ') j( ) j( ') j This is the simplest case without spin and isospin dependence. The HF equations are a self-consistent problem that has to solved iteratively. Practical solution for heavy nuclei still requires simplified interactions like the Skyrme or Gogny forces.
Features of Hartree-Fock Skyrme force: a typical effective force which simulates many-body effects through density dependence introduced via a three-body force. It is easy to handle because of zero range. 1 v1 t0(1 x0p ) d ( r1 r ) t 1 d ( r1 r ) k k ' d ( r1 r ) tk d ( r1 r ) k i W0 ( 1 ) k ' d ( r1 r ) k v13 t3d( r1 r ) d( r r3 ) with k=( ) / i and k'=-( ) / i 1 1 The interactions are normally fitted only to a very small number of spherical nuclei and provide impressive predicitive power. Hartree-Fock calculations generally describe bulk properties of nuclei quite well, but details of the spectra do not come out as well. Constraints are needed to calculate properties for non-equilibrium deformations.
The Skyrme Energy Functional E x 3 E d x E x ECoulomb Epair Ecm b b m b 0 0 q b1 b1 q q q q 3 qq q b4 q 3 q 1 1 b 4 q Jq t x t x J t t J 16 1 q b b J 1 1 1 q 6 q Note: b 4 was introduced to make spin-orbit coupling similar to that in RMFT x0 1 1 x1 x 4 x1 x t1 x1 t x b t1 t x3 3 t1 x1 t x, b t3 1, b t 1, b t x, b t 1 t 1, 0 0 0 0 0 1 1 b, 3 1 1, 1 1 1 1 1 4 8 b 1 1 1 1 8 3 4 3 b t x, b t b b (standard choice) 1 1 1 4 3 3 4 4, 4 4
The RMFT Lagrangian (most common variant) L L L L L L L nucleons nucleons mesons coupling i m m m 1 1 mesons 1 1 g ( ) b d g ( ) 1 3 1 4 coupling 3 4 1 1 g R A e ( ) (1 ) with A A A B R R m R R A A 0
Fitting Strategies -fit to experimental data: binding energies (all forces) diffraction radii (NL-Z, NL-3, P-F1, SkIx) surface thicknesses (NL-Z, NL-3, P-F1, SkIx) r.m.s. radii (NL3, NL-Z, P-LA, P-F1) spin-orbit splitting (Skyrme forces, P-LA) isotope shift in Pb (SkIx) neutron radii (NL3) nuclear matter properties (NL3) The selected (semi-) magic nuclei are: 16 O, 40,48 Ca, 56,58 Ni, 88 Sr, 90 Zr, 100,11,10,14,13 Sn, 136 Xe, 144 Sm, 0,08,14 Pb These are highly unrepresentative nuclei!
Earlier Predictions for the Next Magic Superheavy magic proton number Nilsson Model (oscillator) Z=114 J. Grumann, U. Mosel, B. Fink, W. Greiner Z. Physik 8 (1969) 1 Nilsson-Strutinksy Z=114 S. G. Nilsson, C. F. Tsang, A. Sobiczewski, P. Möller, Nucl. Phys. A131 (1969) 1. Skyrme III Z=114, 10, 138 M. Beiner, H. Flocard, M. Vénéroni, P. Quentin, Phys. Scripta 10A (1974) 84. YPE and Folded Yukawa Z=114 P. Möller, J. R. Nix, G. A. Leander, Z. Physik A33 (1986) 41. YPE + Woods-Saxon Z=114 Z. Patyk, A. Sobiczewski, Nucl. Phys. A533 (1991) 13. RMF (NL-SH) Z=114 C. A. Lalazissis, M. M. Sharma, P. Ring, Y. K. Gambhir, Nucl. Phys. A608 (1996) 0. Skyrme SkP, Sly7 Z=16 S. Cwiok, J. Dobaczewski, P. H. Heenen, P. Magierski, W. Nazarewicz, Nucl. Phys. A611 (1996) 11.
Magic Numbers for 3 Forces
Single-Particle Levels for 114X184
Single-Particle Levels for 10X17
Density of 10 Central depression discovered independently by J. Déchargé, J. F. Berger, K. Dietrich, and M. S. Weiss, Phys. Lett. B451, 75 (1999).
Systematics of density distributions
Fission barriers Uncertainties are still quite large. In any case, the barriers are quite narrow and lead to short fission liftimes. In experiment, the main problem is still how to get sufficiently many neutrons into the system: radioactive beams?
Collectivity from single-particle models Based on phenomenological or self-consistent models, collective vibrations can be calculated as coherent excitations of many nuclei (RPA, TDHF). This works well for higher-lying states. It is also possible to calculate potentialsv(b,) to predict surface vibrations in a nonlinear description. This it was long possible to predict deformations, and moments of inertia, but not the vibrational excitations. The problem is the calculation of the kinetic energy. Only recently a reasonable description of the lowest vibrational states was achieved. The same holds for very large-scale motion like fission and heavy-ion reactions. For the latter a transition from the Hartree-Fock regime to a collisional system is expected.
The Fermi gas model (1) The nuclear potential is approximated by an infinite well of cubic shape V ( x, y, z) V, 0,, 0 x y z a, otherwise The eigenfunctions are x y z nx ny nz ( x, y, z) N sin x sin y sin z, n, n, n 1,, a a a n n n x y z and eigenenergies (occuopied up to the Fermi energy n, (,, ) x n y n n n n n n z m a x y z Going to spherical coordinates we get (for degeneraxy factor g) 3 3 a m N g 3 3 F 3 d n dnxdnydnz n dnd F k m F
The Fermi gas model () The density of particles is N N g m 3 V a 3 3 3 F leading to the relation between the Fermi momentum and the density k F 6 3 1.41fm g 1 (relatively constant throughout the periodic table.) The total energy density (excluding potential energy) becomes 3 m 1 e g 5 5 F mean per particle For different Z and N it behaves as 5 5 giving one reason for 3 N 3 Z 3 e ~ 5 the symmetry energy 5 A 3 e 3 e 5 F A100, Z 0100
Further Developments Treatment of correlations beyond the mean field Derivation from underlying QCD Special topics not addressed above, e.g.: pairing! giant resonances high-spin states b decay cluster models nuclear matter theory Nuclear reactions low-energy: models f many different kinds: TDHF, coupled channels, trajectory methods, hydrodynamics high energy: mostly thermodynamics and statistical mechanics Nuclear Theory remains a very rich field with two fundamental directions: small Fermion systems and the interplay of collectivity and single-particle aspects the search for the underlying interaction Many methods and models of quite different levels of sophistication coexist!