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These are lecture notes for a course on nuclear structure given at the 2005 summer school of the Center Nuclear for Study at RIKEN (Wako). The notes cover the self-consistent mean-field theory, Hartree-Fock and DFT, computer programs, and elementary applications. They also cover projections, RPA and QRPA, generator coordinate method, and auxiliary field methods. useful for students studying nuclear physics and related fields.
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G.F. Bertsch University of Washington
These are notes for a lecture course given at the 2005 summer school of the Center Nuclear for Study at RIKEN (Wako). gene:japan2005/riken3.tex
- PART I: THE STATIC MEAN FIELD - I. Introduction - A. Self-consistent mean-field theory versus density-functional theory - B. The old and the new in SCMF - II. Fundamental equations - A. Hartree-Fock and DFT - B. HF-BCS - C. HFB - III. Computer programs - A. Choice of basis - B. Method of solution - C. An example: 48 Ca - IV. Elementary Applications - A. Separation energies - B. Particle orbitals - C. Particle-hole gaps - D. Shell closures - E. Deformations - F. Binding and separation energy systematics - G. Energy landscapes, intruder states, and fission barriers - PART II: BEYOND THE SCMF - V. Projections - VI. RPA and QRPA TABLE I: Atomization energies of selected molecules Li 2 C 2 H 2 20 simple molecules (mean absolute error) Experimental 1.04 eV 17.6 eV - Hartree-Fock [37] -0.94 -4.9 3. LDA [31] -0.05 2.4 1. GGA [38] -0.2 0.4 0. τ [39] -0.05 -0.2 0.
a Hamiltonian theory which interaction energy expressible through second quantized op- erators that depend on density through the combination ˆρ(r) = a† rar. The SCMF is an approximate treatment of the Hamiltonian that can possibly be improved. The mean-field energy may be augmented by additional terms, for example terms that restore the effects of broken symmetry or terms that account for long-range correlations in the wave function.
B. The old and the new in SCMF
Self-consistent mean field theory has a long history in nuclear physics, going back to the pioneering calculations of Brink and his collaborators in the mid 1960’s. With the limited computer resources available at the time, one could only treat magic nuclei such as 16 O. An important advance was the introduction of the Skyrme parameterization of the energy functional, which effectively decoupled the problem of what the nuclear interaction is, from the problem of describing the structure with some fixed interaction. The Skyrme parameterization simplified the interaction to a contact form. It continues to be used up to the present, but now there are also comprehensive calculations available for SCMF theories with relativistic interactions [7] and with finite-range interactions [8]. Most of the formal development of SCMF and its extensions can be found in the textbook by Ring and Schuck [2]. The present status of SCMF has been reviewed by Bender, Heenen, and Reinhard [1].
II. FUNDAMENTAL EQUATIONS
A. Hartree-Fock and DFT
The fundamental basis of all self-consistent mean-field models is Hartree-Fock theory. Hartree-Fock theory provides a computationally tractable approximation to the general N - particle wave function and can be readily extended as in the Kohn-Sham density functional theory. I shall derive the equations here using second-quantized notation for the HF but ordinary functions for the DFT. I should mention that the second quantized notation is not really needed for the Hartree theory. But the formalism is unavoidable for BCS and the more sophisticated extensions of the theory, and this is a good a place as any to introduce it. The Hartree-Fock approximation approximates the many-particle wave function by the product form |Φ〉 = a† k 1... a† kN |〉.
where a† k creates a particle in orbital k. In first quantized notation, |Φ〉 is the same as the Slater determinant made up of the orbitals k. The relation between the orbital operator and the field operator Ψ†^ is a† k = ∫^ d^3 r φk(r)Ψ†, where φk is the orbital wave function. In Kohn-Sham DFT one simply takes the orbital wave functions for the occupied or- bitals φk as variational parameters. However, one requires that the orbitals be orthonormal, φ∗ kφk′ d^3 r = δkk′. Consider now a Hamiltonian made of a kinetic energy and a two-body potential
H =
∫ d^3 r ¯h
2 2 m∇Ψ
†(r) · ∇Ψ(r) +^1 2
∫ d^3 r
∫ d^3 r′V (r, r′)Ψ†(r)Ψ†(r′)Ψ(r′)Ψ(r).
Ψ(r) is the field operator; it creates a nucleon at position r. Using the wave functions φk(r) corresponding to the creation operators a† k
{Ψ(r), a† k} = φk(r).
The expectation value of this Hamiltonian for the Hartree-Fock state Φ is
EΦ = 〈Φ|H|Φ〉 = ∑^ N i
¯h^2 2 m
∫ d^3 r∇φ∗ k(r) · ∇φk(r) + ∑ a<b
∫ d^3 r
∫ d^3 r′V (r, r′)|φa(r)|^2 |φb(r)|^2 (1)
− ∑ a<b
∫ d^3 r
∫ d^3 r′V (r, r′)φa∗(r)φa(r′)φb∗(r′)φb(r).
B. HF-BCS
A very important generalization of the Hartree-Fock theory is to pairing theory. It may viewed as a mean-field theory includes both particle creation and annihilation operators in the definition of the mean-field orbitals. The general equations governing the orbitals are the Hartree-Fock-Bogoliubov equations, called the Bogoliubov-deGenne equations in condensed matter physics. I will present the equations in the next subsection; here I will present the simpler BCS theory [3]. In the BCS theory as carried out in nuclear physics, we first solve the HF equations to get a set of orbitals k. In favorable cases such as in dealing with even-even nuclei, the HF equations are invariant under time reversal. This implies that each orbital k has its time-reversed parter k¯, and the two have the same single-particle energy ≤k. Now we come to the basic BCS Ansatz, to take the many-particle wave function to have the form
|BCS〉 = ∏ k> 0
(uk + vka† ka†k ¯)|〉.
The uk and vk are parameters are to be determined by minimizing the expectation value of the Hamiltonian. Normalization of the state requires
|uk|^2 + |vk|^2 = 1.
The BCS state is not an eigenstate of the particle number operator. However, by augmenting the Hamiltonian with a Lagrange multiplier term,
H → H − λ N ,ˆ
the particle number expectation value can at least be fixed to the desired value N ,
〈BCS| Nˆ |BCS〉 = 2 ∑ k> 0
v k^2 = N.
I now give two examples of BCS pairing that shows the range of pairing effects in nuclei. The first is a Hamiltonian with degenerate orbitals and a constant pairing interaction. Let us suppose there are 2Ω orbitals including the time-reversed partners and Np particles (either neutrons or protons). First consider an even number of particles. Then the vk amplitudes are all equal and are determined by fixing the number of particles to Np. Namely,
v^2 k = N 2Ωp.
There is no single-particle energy in the Hamiltonian so its expection value is just that of the interaction. Let us take it to be −g for all orbitals. Then we have
〈BCS|v|BCS〉 = − (^12) ∑^ Ω k
∑^ Ω k′
Np 2Ω
( 1 − N 2Ωp
)
= −g 8 Np(2Ω − Np). (even)
If there are an odd number of particles, one will be alone in some particular orbital blocking the parter orbital. Thus the pairing will now occur among the Np − 1 remaining particles occupying the Ω − 1 available orbitals. Thus the energy in this case is
〈BCS|v|BCS〉 = g 8 (Np − 1)(2Ω − Np − 1) (odd).
If you try out these formulas with not too small a value for Ω, you will see that the even systems have lower energies that the average of the neighboring odd systems. The precise amount of this pairing varies, but with the orbitals half filled it is about gΩ/4. Another simple model is called the picket fence Hamiltonian. The orbitals (together with their degenerate time-reversed partners have an equal energy spacing. Let’s call the spacing ≤ 0. Again we assume that the interaction has equal matrix elements for all the orbitals. In this case, we need to impose a cutoff on the single-particle spectrum to get a solution of the BCS equations. Let us suppose that the cutoff is the same above and below the Fermi energy, and that it is large compared to ≤. If the pairing strength is such that ∆ comes out large compared to ≤ 0 but small compare to the cutoff energy, the total energy (compared to that of the same system without the pairing interaction is given by
E = − ∆
2 2 ≤ 0 ,
and the difference in energy between even and nearby odd systems is
Eodd − Eeven = ∆.
C. HFB
In the last section, the variation to find the pairing amplitudes uk, vk was performed after the HF variation to set the orbitals k. Can one do better doing both variations at the same time? The answer is yes, at least in principle. An elegant formulation of the problem is given
assumed that all the orbitals have a reflection symmetry with respect to the three coordinate axes; this allows the meshed space to be one eighth of the total. Thus, the length of a side of the box is Lx = 2Nx∆x. We shall go through an example below for the nucleus 48 Ca, for which the parameters have values ∆x = 0.8 fm and Nx = 14. Thus the distance from the center of the nuclei to a wall of the box is 11.4 fm. This should be compared with the nominal radius of the nucleus, R = 1. 2 A^1 /^3 = 4.4 fm. The size of a real vector needed to represent an orbital is the number of mesh points times two for the two spin states times an additional factor of two for the real and imaginary parts of a complex amplitude. Thus for the 48 Ca example, the vector size is 14^3 × 4 ≈ 11 , 000. An important property of the mesh representation is that the single-particle Hamiltonian is a sparse matrix. The kinetic energy operator may be approximated by difference formulae that only involve nearby mesh points. Or, if one wants to treat the kinetic energy more exactly and is willing to accept periodic boundary conditions on the lattice, one can use the Fast Fourier Transform. That derives its speed from a sparse matrix representation. Concerning the potential energy, due to the short range of the nuclear interaction it only requires amplitudes on nearby mesh points as well. In the harmonic oscillator representation, the important numerical parameters are the oscillator frequencies ωx, ωy, ωz and the number of oscillator states included. Usually, there is a maximum for number of oscillator quanta N , where N = nx + ny + nz.
B. Method of solution
Given the single-particle Hamiltonian in some representation, one can solve for the or- bitals in eq. (2) by direct diagonalization if the dimensions are not too large. This is the case for the harmonic oscillator representation, but not for the lattice representation. In the latter case, one uses iterative techniques. They all have in common that the solution is built out of repeated applications of the Hamiltonian matrix to a vector. These matrix-on-vector operations are at the heart of the well-known Lanczos method of matrix diagonalization. However, for the problem at hand, many eigenfunctions are needed–one for each occupied orbital–and other methods are more robust. A common method is to (approximately) ex- ponentiate the single-particle Hamiltonian to filter the ground orbital out of mixed wave
function, φf = exp(−βh 0 )φi
The component of an eigenstate k with energy ≤k is suppressed by a factorexp(−β≤k) which is relatively larger the higher the energy of the component. When this procedure is used, one starts with a crude approximation to the Np distinct orbitals. At each stage of the filtering, the orbitals are orthonormalized by the Gram-Schmidt procedure. The numerical parameters here are β and the number of iterations Mβ needed for the desired accuracy. The self-consistency of the SCMF is achieved by iterating over solutions of the single- particle Hamiltonian, updating it by the improved information about the densities at each stage. To make the procedure stable and convergent, one usually updates by the replacement ρold → αρnew + (1 − α)ρold where α is a numerical parameter. A final numerical parameter is the number of HF iterations MHF needed for satisfactory convergence.
C. An example: 48 Ca
In this section, we will go through a specific example, just to see how ev8 works from a user’s perspective. First, log on to the computer, make a directory for yourself, and run the program using a script to do all the work. The actual steps from a terminal window are:
ssh riken@gene.phys.washington.edu password: ev8.f mkdir
*********** Ecm v2 *************** neutron = 6.215248 MeV *
Let’s look at the input file that was constructed to run this case. It is called data, and it reads:
TABLE II: Binding energy of 48 Ca. Source Binding Energy SLy4 [10] 417. hfbtho 417. ev8 418. exp. 415.
Obviously, this is the 0s 1 / 2 -orbital. Now, can you find some of the other shell orbitals? The nucleus is spherical, so there should be a energy degeneracy for different jz -states for a given j. Note that only one of the orbitals in a k,k¯ is listed, and that the jz of the listed orbital can be either positive or negative. 48 Ca is supposed to be a magic nucleus. One characteristic is a gap in the single-particle levels between occupied and unoccupied levels. Verify that the proton gap is between the f 7 / 2 and the d 3 / 2 shells with a value of 5.7 MeV, and the neutron gap is between the f 7 / 2 and the p 3 / 2 shells with a value of 4.3 MeV. Finally, the total energy of the nucleus is given toward the bottom of the output file, in the line total energy (from functional) -418.480 energy per nucleon 8. We compare this value with ones obtained from other sources in Table II. At this point I should say something about the energy functional underlying the calcula- tion. It was done with the ”SLy4” parameterization [10] of the Skyrme energy functional, and in fact the binding energy of this nuclei was one of properties considering in constructing it. Their number, obtained with a spherical code, is shown on the top row. The code hfbtho is also quite accurate although it is a 3-dimensional code. We can see that these number agree for the purposes on nuclear mass tables (several hundred KeV). While program ev8.f is somewhat less accurate, for other properties such as single-particle spectra and separation
TABLE III:. Energy of the f 7 / 2 neutron orbital in 48 Ca by various methods. All energies are in MeV. Source ≤(f 7 / 2 ) (E(^48 Ca)-E(^46 Ca))/ ev8 -9.8 -8. hfbtho -9.8 -8. Exp. -8.
energies the results are virtually identical to other code outputs. The theory misses the experimental value by 2 MeV, which is rather typical.
IV. ELEMENTARY APPLICATIONS
Let us first examine the orbital energies of the SCMF, taking the example of 48 Ca.
A. Separation energies
The neutron and proton separation energies are defined by the difference in binding energies Sn = E(N, Z) − E(N − 1 , Z) ; Sp = E(N, Z) − E(N, Z − 1)
Naively, we might identify the orbital energies ≤k as separation energies, but that is only an approximation based on the assumption that the other occupied orbitals remain the same when a nucleon is removed from the nucleus. More rigorously, the separation energy should be calculated by taking energy differences between the nuclei with and without that particle. The two methods are compared in Table III. Note that the difference energy is more accurate than the orbital value. This is due to changes in the wave function such as a pairing-nopairing transition from one nucleus to the other. These contributions, called rearrangement energies, are not included in the ≤j. Still, the naive identification has some value as the orbital rearrangement effects are not large. This is not the case for electronic systems. There the orbital energies can be quite misleading.
FIG. 1: Spectra of light- and medium-mass doubly magic nuclei, calculated in the SCMF with the SLy4 energy functional.
in the excitation spectrum. Experimentally, the first excited state of 48 Ca is a J = 2 even parity state (2+) at excitation energy of 3.8 MeV. The state could not come from the proton shell gap, because the parities of the particle and hole states are different. It could come from the neutrons, however, exciting an f 7 / 2 neutron to the p 3 / 2 shell. The angular momentum of the state can be anywhere in the range of |jp − jh| to jp + jh, so the quantum number 2+ is permitted. The orbital gap energy is 4.3 MeV, somewhat larger than the observed value. In fact, effects of the residual interaction and of wave function mixing will be strong in such cases, so the naive energy should only be considered qualitative at best. We will consider the excitations in much more detail when we come to the extensions of SCMF.
D. Shell closures
Much experimental research has been directed to finding new shell effects in nuclei far from stability. With the SCMF and computer resources one now has available, theorists can see if there is an overall global consistency and provide predictions any region of the mass table. The main signatures of shell closures are:
Binding energies. Shell effects can be seen in the trends of nuclear binding energies when the smooth parts of the binding are subtracted out. This is shown very graphically in
the plot of residuals with respect to the liquid model [13]. A version of that plot with experimental data from the latest compilation [17] is Fig. 2. The residuals are plotted as a function of neutron number, with data for the same proton number connected by lines. The shell closures at N = 50, 82 , and 126 are very prominent. There are no obviously binding effects of shell closures in light nuclei.
Two-nucleon gap A more sensitive measure of shell effects comes from differences in bind- ing energies. The two-nucleon separation energy is less sensitive to pairing effects than the ordinary separation. It is given by S 2 n = E(N, Z) − E(N − 2 , Z) for the case of neutrons. The separation energy should be larger at a magic number than above the magic number, because the nucleons come from different shells. Thus the energy difference δ 2 n(N, Z) = S 2 n(N + 2, Z) − S 2 n(N, Z) should be an measure of the shell closure. It is called the two-nucleon gap. As an exercise, let us find the nuclei with largest values of δ 2 n(N, Z) and see what shells they mark. Because the shell spacing are larger for smaller nuclei, we should make a criterion that scales with nuclei size according to the single-particle level spacing. Let us make the criterion be δ 2 n(N, Z) > (^) A(^161 / 3 MeV Experimentally, there are 30 nuclei that satisfy this criterion. Their characteristics are shown in Table VI. One sees very clearly the magic numbers 28,50,82 and 126. The other even-even nuclei that have large gaps turn out to be lighter nuclei on the N = Z line. There is thus an enhanced stability of the so-called “alpha-particle nuclei”. The predictions of SCMF with SLy4 are qualitatively similar but not identical. There are more magic nuclei and relatively fewer alpha-particle nuclei passing its two-neutron gap test. From the table, one sees that N = 20 qualifies as a magic number in the SCMF but not experimentally.
Excited state gap. The most sensitive measure of shell effects is the systematic trend of the energy gap in the excitation spectra. The first excited state of most even-even
0
1
2
3
4
5
0 20 40 60 80 100
2+ Energy (MeV)
Proton Number
Experiment
FIG. 3: Systematics of 2+^ excitation energies.
for 162 Dy is Q 0 ∼ 1800 fm^2 What is the physical meaning of this number? The ground state of 162 Dy as angular momentum zero and thus no quadrupole moment. But in an important sense it is indeed physical, describing a the intrinsic state of the ground state rotational band. Thus to compare with experiment one has to examine band properties. The easiest one to use is the reduced quadrupole transition rate B(E2) connecting the excited 2+^ state in the band to the 0+ ground state. A simple formula can be derived in the rigid rotor model to relate it to Qp, the intrinsic quadrupole moment of the proton distribution in the intrinsic state. The formula is [18, eq. 4-68b]:
B(E2; 0 → 2) =^5 e
2 16 π Q
(^2) p ≈ 5 Z^2 e^2 16 πA^2 Q
(^20)
. We can get the experimental value from the compilation by Raman et al. [22], B(E2; 0 → 2) = 5.3 e^2 bn^2. The deduced value of Qp and Q 0 are then 7.3 bn and 17. bn respectively. The agreement between theory and experiment is perfect within the ac- curacies of the experimental measurement and the numerical calculation. We can’t expect to do so well in general. It is common to express the deformation in terms of the shape parameter β 2. Although in principle β 2 has a geometric definition, in practice it is often defined by the formula Q 0 = √^15 π β 2 A^5 /^3 1 .44 fm^2.
With this definition, β 2 = 0.34 for 162 Dy. The importance of deformation in the SCMF energies is illustrated in Fig. 4, showing
FIG. 4:
the energy differences between the SCMF with and without breaking spherical symmetry. We see that the deformation can contribution up to 20 MeV in the total energy.
F. Binding and separation energy systematics
Computer resources are now adequate to perform the SCMF calculations for all nuclei of interest at once, even varying the parameters of the energy functionals. Thus, it is now possible to construct theoretical mass tables using SCMF. We can then begin to duplicate the program of quantum chemistry where the DFT was improved to the point of being useful to predict the properties of unknown systems. The first mass theory was the liquid drop model, whose predictions were shown in Fig. 2. The rms residual of theoretical and experimental binding energies is about 3 MeV. A theory including shell effects that has become the benchmark for binding energies is the finite-range droplet model (FRDM) of M¨oller et al. [13]. It has between 10 and 20 adjustable parameters, achieving a fit to the known masses having an rms residual of about 600-700 keV. A guiding principle in that theory was to try to simulate SCMF to correct the liquid drop formula for shell effects.