the atomic nucleus - NPAC

THE ATOMIC
NUCLEUS
1
2
The Atomic Nucleus
First Semester
Master 2 - NPAC Orsay
Marcella Grasso (1) and Miguel Marqués (2)
(1)
Institut de Physique Nucléaire, Orsay
01 69 15 62 49
[email protected]
(2)
Laboratoire de Physique Corpusculaire, Caen
02 31 45 29 67
[email protected]
3
10 Lectures (30 hours): 5 Lectures are focused on the experimental aspects in nuclear physics
(Miguel Marqués) and 5 Lectures are devoted to several theoretical issues related to the
nuclear many-body problem (Marcella Grasso).
Each group of 3 hours is a ‘Cours Magistral’ + (eventually) some exercises.
This manuscript contains five Chapters corresponding to the Lectures 2, 4, 5, 8 and 10
(Lectures by Marcella Grasso).
-----------------
Lectures Outline
1. Miguel Marqués
-
Introduction to nuclear physics. Basic concepts
Simples models: the liquid drop model and the Fermi gas
Quantum behavior in nuclei: shell effects
Exercise: simple calculations with a liquid drop
2. (Chapter 1) Marcella Grasso
- The properties of the nuclear interaction
- Realistic interactions
- Links with quantum chromodynamics established through effective field
theories
- Effective interactions for mean-field models: phenomenological
interactions
- Nuclear matter: concepts, properties and the equation of state
3. Miguel Marqués
-
Production and detection of nuclei
Exotic nuclei and the drip lines
Measurements on exotic nuclei in France: Ganil, Spiral et Spiral2
Other facilities
4. (Chapter 2) Marcella Grasso
-
Introduction to the nuclear many-body problem
Light Nuclei: ab initio methods
The shell model
Medium-mass and heavy nuclei: mean-field models, links with the
density functional theory
4
5. (Chapter 3) Marcella Grasso
- The mean-field approach as a variational procedure. The Hartree-Fock
equations
- Time-dependent Hartree-Fock theory
- Small-amplitude excitation modes: the random-phase approximation
- Examples of results obtained with mean-field-based approaches
6 and 7. Miguel Marqués (two Lectures)
-
Introduction to reactions
Unstable nuclei, limits of stability and super-heavy nuclei
Modification of the shell structure far from stability and deformations
Halos and clusters of neutrons
8. (Chapter 4) Marcella Grasso
- Superfluidity in nuclear systems
- Pairing correlations (Cooper pairs of nucleons) in the mean-field
framework: the Hartree-Fock-Bogoliubov method
- Very exotic nuclear systems: neutron stars
9. Miguel Marqués
- Nuclear astrophysics and nucleosynthesis
10. (Chapter 5) Marcella Grasso
-
Utility of symmetries in quantum many-body physics
Some elements of group theory
Symmetries in quantum mechanics
Isospin symmetry in nuclei
5
Contents of the manuscript
- Chapter 1 – p. 7
- Chapter 2 – p. 27
- Chapter 3 – p. 49
- Chapter 4 – p. 71
- Chapter 5 – p. 85
- References – p. 100
6
Chapter 1
Marcella Grasso
1.1 The properties of the nuclear interaction 8
1.2 Realistic interactions 10
1.3 Links with quantum chromodynamics established through effective
field theories 16
1.4 Effective interactions for mean-field models: phenomenological interactions 20
1.5 Nuclear matter: concepts, properties and the equation of state 22
7
Useful constants
- Average mass of the nucleon: 938.9 MeV
- h c: 197.3 MeV fm
- Fine structure constant: e2/(4πε0 h c): 1/137.03
where e is the elementary charge and ε0 the electric constant
------Our life is impacted by nuclear physics in several ways. Let us enumerate some examples:
Nuclear radiation is used in materials characterization and cancer therapy; Radioactive
isotopes are used as tracers and diagnostic tools in medicine; Medical imaging employs
nuclear magnetic resonance and positron emission tomography; Nuclear fission is used in
nuclear power reactors.
Nuclei are the cores of atoms. They have a radius of about 10-14 m, which is extremely small
if compared with the radius of an atom (about 10-9 m). In spite of this huge difference, the
mass of the nucleus is almost all the mass of the atom (99.97%). This means that, in terms of
mass, ~ 99.97% of matter is composed by ‘nuclear matter’.
Light and medium-mass nuclei are created in stars by nuclear reactions (stellar
nucleosynthesis). Heavier nuclei (heavier than iron) are created by other nucleosynthesis
processes, which are believed to probably take place in supernova explosions (end of life of
massive stars).
Nuclear physics is a very exciting and rich domain where quantum many-body physics is
necessary to provide a satisfactory theoretical description of a huge variety of nuclei with
different shapes, properties and numbers of protons and neutrons.
To date, 117 chemical elements have been identified. An element may have more than one
isotopic form (number of neutrons). Approximately 3000 isotopes have been identified. A
total of 7000 are believed to be sufficiently long-lived for their identification with current
techniques. Next-generation radioactive ion beam (RIB) facilities which are being constructed
(for example, Spiral2 at Ganil in France and FAIR at GSI in Germany) will allow nuclear
physicists to produce with sufficiently high intensities (so that measurements become
possible) many unknown unstable isotopes.
1.1 The properties of the nuclear interaction
The basic concepts on which nuclear physics is based are introduced in Lecture n. 1 by
Miguel Marqués. We summarize some of them in what follows:
-
The comprehension of a quantum many-body system is one of the most difficult
problems in physics. Many-body physics is an interdisciplinary domain where links
and connections may be established among several fields such as, for instance, solidstate physics, atomic physics, nuclear physics and quantum chemistry. Our world is
8
composed by many-body systems which are different one from the other for their
typical energy and length scales, for their composition and their properties: nuclei,
atoms, molecules, atomic systems, nanoparticles, clusters, solids, systems of planets,
galaxies,… In principle, a huge number of degrees of freedom should be taken into
account in the analysis of the properties of these systems, but this would lead to very
difficult theoretical models. These models become extremely complicated when the
quantum nature of the system cannot be neglected to provide a correct interpretation of
its properties and has to be explicitly included in the theoretical description.
-
The nucleus is a quantum system composed by A particles, neutrons and protons,
called nucleons. Nucleons are baryons and thus fermions. Their behavior is dictated
by the Pauli principle and this feature is reflected in the antisymmetrization properties
of the wave functions with respect to the exchange of identical particles. In lowenergy nuclear physics (scales of ~ 10 ÷ 100 MeV) the degrees of freedom that are
taken into account in the nuclear quantum many-body problem are the nucleons.
Internal degrees of freedom are typically not considered and nucleons are treated as
elementary point-like particles.
-
The nucleus is a self-bound system and the force acting on the nucleons is the strong
interaction. This interaction acts on hadrons (baryons and mesons).
-
In principle, starting from quantum chromodynamics (QCD) and using quarks and
gluons as degrees of freedom, it should be possible to analyze the properties of the
nucleons in nuclei.
-
In practice, in low-energy nuclear physics the typical energies are low enough to allow
one to neglect some degrees of freedom of the strong interaction. It is reasonable to
imagine that the strong interaction does not show all its complexity. Two possible
directions may be followed. One possibility is to treat the nucleus as a system
composed by A nucleons. The other degrees of freedom are not explicitly included and
they are taken into account in an approximate way by the introduction of effective
forces. The second possibility is to include more explicitly the other degrees of
freedom. This is achieved, for instance, when effective field theories are employed
(Sec.1.3).
-
Some processes that are observed in nuclei, like the β decay, are the effect of the weak
interaction. These processes are very strongly related to the stability properties of a
nucleus (Lectures 6 and 7 by Miguel Marqués).
-
In nuclear many-body systems, the A nucleons interact among themselves by two-,
three-,…, A-body forces (too complicated problem!). It turns out that 2-body forces
(and eventually 3-body) describe quite well the nucleus and this allows us to
considerably simplify the nuclear many-body problem. The main properties of the 2(and 3-) body forces are accessible by analyzing free nucleon-nucleon scattering
processes, the ground state of the deuteron and the spectroscopy of bound few-nucleon
systems.
-
To summarize, the nuclear many-body problem is very complicated owing to two
main aspects: (i) the complexity of the nuclear force; (ii) the huge number of degrees
of freedom involved in the observed phenomena.
9
-
Symmetry properties are extremely helpful and are used in many cases to simplify
the problem (Chapter 5 - Lecture 10).
-
If one describes the nuclear force with a phenomenological potential which
reproduces the data of nucleon-nucleon scattering, deuteron structure and
spectroscopy of few-nucleon systems, the main properties may be summarized:
attractive force;
short-range (some fm);
spin-dependent force. A spin-orbit term is necessary;
charge independent force (example of symmetry of the strong interaction: isobaric
symmetry or invariance). In a nucleus, the isobaric invariance is violated by the action
of the electromagnetic interaction;
necessity of considering an exchange term (Paui principle);
presence of the hard core (strong repulsion at short distances, rC ≈ 0.4 – 0.5 fm);
in some nuclei, correlations related to superfluidity (formation of Cooper pairs of
nucleons in open-shell superfluid nuclei) contribute to the binding energy.
a)
b)
c)
d)
e)
f)
g)
Deriving the nucleon-nucleon interaction from QCD is extremely difficult and all the attempts
done so far have not provided quantitative results. To treat the nucleon-nucleon interaction,
three main directions are actually followed to obtain quantitative information:
1) Theories of nucleons interacting via the exchange of mesons (realistic interactions);
2) Effective field theories (deep links with QCD);
3) Phenomenological interactions.
1.2 Realistic Interactions
The starting point in the development of all the so-called nuclear realistic interactions is the
work made by Yukawa in the 30s. The electromagnetic field is quantized by Born,
Heisenberg and Jordan in the 20s: the electromagnetic interaction is mediated by the exchange
of photons. At the beginning of the 30s, quantum electrodynamics is already formulated.
In 1932, Chadwick discovers the neutron (Nobel Prize in 1935). Slightly later, Yukawa makes
a hypothesis on the existence of a particle with mass between that of the proton and that of the
electron. He denotes this particle by ‘U’ (mU ≈ 200 me ; proton mass ≈ 1800 me). Yukawa
suggests that this particle is the mediator of the strong interaction (similarly to the photon
which is known to be the mediator of the electromagnetic interaction). The difference with the
photon is that the Yukawa boson has a mass different from zero. In 1947 and 1948 the
Yukawa meson is identified with the π meson (π mesons are pseudoscalar mesons, i.e., of
negative parity and spin 0 1, of mass ≈ 140 MeV/c2), which has been observed in cosmic rays
and has been produced at Berkeley. The Yukawa particle is recognized in the π meson for two
reasons: the π meson interacts strongly with the nucleons and has the correct mass that leads
to the correct observed range of the nuclear force. In 1949 the Nobel Prize in Physics is
attributed to Yukawa.
1
The quarks that compose the π meson have opposite spin. Scalar mesons have also spin equal to 0 but positive
parity. Vector and pseudovector mesons have spin equal to 1 (quarks have parallel spin). Vector mesons have
negative parity and pseudovector mesons have positive parity.
10
In Fig. 1.1, the first page of an article published by Yukawa on the particle ‘U’ is shown.
Yukawa has demonstrated, by using simple arguments, that the exchange of mesons of mass
m can be described by the potential
V(r)= −g 2
e −mr
,
r
(1.1)
where g is the coupling constant between the fermionic and the mesonic fields. The minus
sign indicates that the force is attractive. If the mass of the pion was zero, the potential would
be equivalent to a Coulomb potential 1/r. It is important to emphasize that the pion determines
only the long-range part of the nuclear potential (it should be noted that the Compton wave
length of the pion is λπ = h mπ c ≈ 1.5 fm). In Fig. 1.2, a Yukawa potential is plotted and
compared with the Coulomb potential.
Fig. 1.1 First page of an article of Yukawa on the particle ‘U’.
11
Fig. 1.2 Example of radial profile of a Yukawa central potential associated to the nuclear
interaction (comparison with the Coulomb potential). The Compton wave length of the pion
is λπ = h mπ c ≈ 1.5 fm.
Fig. 1.3 From an article by Taketani and collaborators
12
In the 50s, Taketani formulates a hypothesis about the existence of two-pion exchange
processes in the radial region 0.7 fm ≤ r ≤ 1.5 fm (extremely difficult calculations). A page
from an article by Taketani and collaborators is shown in Fig. 1.3.
The earliest two-nucleon realistic potentials (introduced in the 60s) are of ‘one-pion exchange
potential’ (OPEP) type with adjustable parameters to reproduce low-energy nucleon-nucleon
scattering data (phase shifts) plus a repulsive hard core interaction at ~ 0.45 fm, necessary to
describe high-energy scattering data.
After Yukawa’s idea and the discovery of the pion, other heavier mesons are found in the 60s
(ω, ρ, mass ≈ 800 MeV) and the ‘one-boson exchange potentials’ (OBEP) are introduced.
In the 70s and 80s, well-known realistic interactions are constructed (Lacombe et al. introduce
the Paris potential; Machleidt et al. introduce the Bonn potential); in the 90s, high-precision
realistic potentials are introduced (CD-Bonn, Argonne V18, Nijmegen I and II, Reid93). The
parameters that are adjusted in these potentials are around 50.
It has been found that all the potentials between pairs of nucleons underbind almost all the
nuclei and overestimate the equilibrium density of nuclear matter (see Sec. 1.5 for the
definition of nuclear matter and its equilibrium density). This suggests that 3-body (and
maybe 4-body) interactions are important in nuclei.
The fine adjustments in the high-precision realistic potentials are done to reproduce the phase
shifts (free nucleon-nucleon scattering) and the spectroscopy of few-body nuclear systems
(deuteron, 3H, 3He). This allows one to take into account both two- and three-body effects. In
general, realistic interactions contain a long-range part, which describes one-pion-exchange
processes, an intermediate-range part, which is more phenomenological (adjustment of
parameters) and a short-range part, which describes the hard core. These two latter parts
eventually contain the exchange of different real mesons like ω and ρ or virtual mesons like σ
(mass ∼ 550 MeV; it does not exist in nature).
In which channels the phase shift is measured in nucleon-nucleon scattering? Isospin,
spin and spatial states. Possible combinations for a pair of nucleons.
The idea of isospin was introduced by Heisenberg in the 30s by making a parallel between
two orientations of spin ½ and two kinds of nucleons, neutrons and protons. Let us introduce
the spinor wave function for isospin as follows,
ϕ1/2,1/2
=>
neutron isospin state,
ϕ1/2,-1/2
=>
proton isospin state.
These wave functions combine with the spin and spatial wave functions to provide the total
wave function. For a two-nucleon wave function the total isospin can be T = 0 and 1.
- T= 0 => T0 = 0, where T0 = 1/2(N-Z).
- T = 1 => T0
antisymmetric neutron-proton pair (singlet)
1 two-neutron pair
0 symmetric neutron-proton pair
-1 two-proton pair
(triplet)
We introduce the following spectroscopic notation for a two-nucleon state:
L => relative orbital angular momentum
S => combined spin
13
J => total angular momentum
The angular momentum state of a pair is denoted by 2S+1LJ.
The states with L = 0, 1, 2, 3, 4,… are called S, P, D, F, G, … states, respectively (S, P, D and
F stand for Sharp, Principal, Diffuse and Fundamental; these names have been chosen at the
beginning of atomic spectroscopy. After F, the alphabetic order is followed).
-
The symmetry of the spatial state is given by (-1)L;
For the spin state, S = 0 is an antisymmetric state and S = 1 is a symmetric state.
For a pair of particles, the combination of isospin, spin and spatial wave functions must be
antisymmetric. It has be seen that for a pair of like particles (two neutrons or two protons) the
total isospin can be only T = 1 (symmetric state). This means that the combination of spin and
spatial states must be antisymmetric:
Pair of like particles: T = 1
- L = 0 => S = 0;
- L = 1 => S = 1;
- L = 2 => S = 0;
etc.
1
S0 (singlet S state);
P0, 3P1, 3P2 (triplet P state);
1
D2 (singlet D state);
3
Neutron-proton pair (T = 0 and 1)
The space-spin wave function can be symmetric or antisymmetric. We have two cases:
- T = 1 (T0 = 0) => the space-spin wave function must be antisymmetric (we have the
same states as for the pair of like particles)
- T = 0 (T0 = 0) => the space-spin wave function must be symmetric. The possible states
are
3
S1
- L = 0 => S = 1;
1
- L = 1 => S = 0;
P1
3
- L = 2 => S = 1;
D1, 3D2, 3D3
etc.
To construct a realistic interaction, the adjustments of the parameters to reproduce the phase
shift (as a function of the energy) are done channel by channel.
From the bare interaction to well-behaved interactions.
The main difficulty in using bare realistic interactions for nucleons in a medium (nucleons in
nuclei or nuclear matter, not free nucleons) is related to the existence of the infinite repulsive
hard core.
One direction: perturbative approach.
The basic observation is that, even if the bare interaction has a infinite repulsive hard core, the
scattering of pairs of nucleons from an initial to a final state in a nucleus is well defined.
Starting from this observation, Brueckner has proposed to construct an effective interaction by
replacing the interaction with the reaction matrix associated to the scattering of pairs of
nucleons in a medium, the G matrix or the Brueckner matrix. This matrix is an extension of
14
the scattering matrix T, which describes the scattering of free nucleons. A Bethe-Goldstonetype equation (analogous to the Lippmann-Schwinger equation for the scattering matrix T) is
solved for G.
The T-matrix is defined by the equation T φ = v ψ , where v is the antisymmetrized
interaction ( v abcd ≡ v abcd − v abdc is the antisymmetrized matrix element of the
interaction), φ is an initial non-interacting state of energy E of two non-interacting
nucleons and ψ is the interacting state. T depends on the energy E. The LippmannSchwinger equation for the scattering matrix T of two free particles is written as
1
1
T kE k ,k ' k ' = v k1 k 2 ,k1 ' k 2 ' +
v k1 k 2 , p1 p 2
T pE p ,k ' k '
∑
2
2
1 2 1 2
2 p p
E − p1 /2m − p 2 /2m + iη 1 2 1 2
1 2
, (1.2)
(
) (
)
where k1 and k2 are the initial moments of the particles, k1’ and k2’ are the final moments and
p1 and p2 are the moments of intermediate states over which the sum runs; E is the total
energy. The generalization to the scattering of two particles in a medium (the states are not
described by plane waves in this case and the kinetic energies have to be replaced by the
single-particle energies of the states) is the Bethe-Goldstone equation,
E
G ab,
cd = v ab,cd +
1
1
E
v ab, mn
G mn,
∑
cd ,
2 ε , ε >E
E − ε m − ε n + iη
m n
F
(1.3)
where the indices a, b, c, d, m and n denote the single-particle states and EF is the Fermi
energy. The sum runs over the intermediate states m and n which are not occupied (the
energies are larger than the Fermi energy): The diffusion toward occupied states (i.e., states
with energy smaller than the Fermi energy) is not allowed because of the Pauli principle. Eq.
(1.3) can be written in a compact form as,
G = v +v
QF
G ,
E − H0
(1.4)
where H0 is an independent-particle Hamiltonian for the two nucleons (it is given by the sum
of a kinetic term and a mean-field term where the mean field is constructed by all the
nucleons present in the medium) and QF is a projector that excludes all the occupied
intermediate states:
QF =
∑
mn mn .
m <n
unoccupied
(1.5)
The formal solution of Eq. (1.4) is:
G =
v
QF
1 −v
E − H0
.
(1.6)
15
We do not enter into the technical details. However, already looking at this schematic
expression, one can realize that the value of G may remain finite even if the interaction
diverges. The hard-core divergence can be thus eliminated. The Bethe-Goldstone equation is
however extremely difficult to solve.
A second possibility: Vlow-k
An advantageous alternative to the use of a G-matrix to replace the bare interaction by a wellbehaved effective interaction has been recently proposed and is developed in a momentum
representation. In momentum representation the too strongly repulsive interaction is seen at
high relative momenta. It is important to stress that in low-energy nuclear physics the typical
relative momenta are much smaller than those where the strong repulsion manifests itself. It is
thus reasonable to introduce, starting from the bare interaction, an effective interaction where
the high-momentum components are integrated out by some transformations. The resulting
interaction is called Vlow-k and well describes low-energy scattering data. Furthermore, it turns
out (as it can be expected) that Vlow-k interactions derived from different bare interactions are
very similar one to the other.
1.3 Links with quantum chromodynamics established through
effective field theories
The most fundamental way of treating nuclei should be done by starting from the underlying
theory of the strong interaction, QCD. When dealing with the realistic interactions which have
been introduced in Sec. 1.2, the link with QCD is not very clearly established. Many efforts
are indeed presently devoted to develop theories where the derivation of the nuclear
interaction is more deeply linked with QCD.
QCD theory has been formulated in the 70s within the Standard Model. It is thus a quite well
established theory. In spite of this, very few things are presently known about the way of
describing low-energy nuclear physics starting from the Lagrangian of this theory.
Fig. 1.4 Internal degrees of freedom of the nucleon
Very active actors in the formulation of QCD, Gross, Politzer and Wilczeck, obtained the
Nobel Prize in 2004 for their investigations on the asymptotic freedom: when two quarks get
closer, they interact less (asymptotic freedom). This effect is related to the vacuum
16
polarization in QCD: a charged quark can be viewed as immersed in a cloud of virtual
charged gluons (the quark charge is diluted in space). When they get farther one from the
other, they interact more strongly because the integrated charge increases (this effect is
responsible for the confinement).
In the figure contained in the picture below, the asymptotic freedom (WEAK) and the
increasing of the interaction intensity with the distance (STRONG) are illustrated.
In the following picture, a schematic illustration of QCD, the associated Lagrangian and a
table with the properties of the six quarks are displayed.
17
Six flavors of quarks exist and each quark is characterized by a mass, an electric charge and a
color charge (three possible colors, red, green and blue with the three associated anti-colors).
The gluons are the mediators of the strong interaction and the quarks interact with the gluons
with a coupling constant g which depends on the energy.
18
The QCD Lagrangian preserves chiral 1 symmetry (parity inversion) in the case where the
masses of quarks are put equal to zero. In the case of chiral symmetry the chiral partners
(particles that differ only for the parity) are degenerate, i.e., they have the same mass.
However, this symmetry is spontaneously broken (for example, all nucleons have positive
parity and chiral partners are not degenerate). It turns out that the Goldstone bosons (see
Goldstone theorem) that correspond to this spontaneously broken symmetry are the pions2.
Chiral symmetry thus plays a fundamental role in the determination of the long-range part of
the nuclear force (exchange of one pion –> see Yukawa model). Pions should have zero mass
according to the Goldstone theorem. However, it turns out that, in practice, chiral symmetry is
not only spontaneously broken. It is explicitly broken owing to the fact that the mass of
quarks is not zero (and thus the mass of pions is not zero). If one considers only the quarks u,d
and s (small masses), ona can say that the chiral symmetry is approximately preserved. The
energy scale for the symmetry breaking is 1 GeV (mass of quark c). In some cases, the effects
of this explicit symmetry breaking can be analyzed within a perturbative theory.
- A perturbative expansion of QCD in powers of the coupling constant is not possible at the
energy scales of low-energy nuclear physics, where the strong coupling constant is ≈ 1.
- Effective field theories for low-energy QCD. At all distance and energy scales, one can
identify for each system the physical phenomena and degrees of freedom that are relevant at
these scales. One can thus fix a kind of hierarchy and separation of scales. At each scale,
some given processes are more important than others. This means that, at each scale, one can
introduce an effective theory governed by the relevant degrees of freedom.
Effective field theories in nuclear physics are non perturbative approaches that are introduced
to describe the low-energy behavior of nuclear systems by using a scale separation. In the
framework of these theories, one builds the so-called chiral potentials. The scenario of lowenergy QCD is characterized by pions and nucleons interacting via a force governed by
spontaneously broken approximate chiral symmetry.
Effective field theories allow one to fix a hierarchy of contributions to the nuclear force based
on a scale separation. If Q represents the typical energy scale in the process under
consideration (soft scale) and Λ represents the so-called hard scale (for example, the scale
corresponding to the chiral symmetry breaking, ∼ 1 GeV), the dominant contributions to the
effective Hamiltonian are of the order of (Q/Λ) (leading order, LO). It turns out that these
contributions are 2-body (2-nucleon) forces (2N) (see Fig. 1.5) (exchange of one pion et
contact forces).
1
‘I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror,
ideally realized, cannot be brought to coincide with itself’. Lord Kelvin
2
The pions are identified as the Goldstone bosons associated to this symmetry breaking because they have the
correct quantum numbers and because they are the lightest hadrons (they are the closest to the case of zero mass).
19
Figure 1.5 See legend on the left of the figure
The corrections at the next-to-leading order (NLO) are of the order (Q/Λ)2 (2N forces also in
this case; the exchange of two pions can also be seen in Fig. 1.5). At the next-to-next-toleading order (NNLO) (∼(Q/Λ)3), 3-body forces (3N) appear, and so on. The effective field
theory thus allows one to establish in a natural way a hierarchy for the 2-, 3-, 4-,…body
forces: the 2-body contribution is the most important; the 3-body contributions are more
important that 4-body, etc.
1.4 Effective interactions for mean-field models:
phenomenological interactions
The advantage of using phenomenological interactions is that they allow one to perform more
easily sophisticated numerical calculations for medium-mass and heavy nuclei.
Phenomenological effective interactions, like the Gogny and Skyrme forces, are actually very
currently employed in the framework of mean-field theories and mean-field models are
suitable approaches employed to describe medium-mass and heavy nuclei. Phenomenological
interactions are constructed through the adjustment of some parameters (typically around 10
parameters), which are fitted to reproduce ground-state properties (binding energies and radii)
of some selected nuclei and the main properties of nuclear matter (see Sec. 1.5).
The phenomenological effective interactions used in the framework of non-relativistic meanfield models are divided into two main groups, the finite-range interactions, like the Gogny
force (Gaussian terms), and the zero-range interactions like the Skyrme forces. Both
interactions are density dependent.
The Gogny force has a finite-range part given by the sum of two Gaussians, a zero-range
density-dependent part and a zero-range spin-orbit part. The number of parameters to adjust is
equal to 14 (a few parametrizations exist).
20
The Skyrme force is the sum of four zero-range parts: a central part, a velocity-dependent part,
a density-dependent term and a spin-orbit term. The parameters are 10. Many
parametrizations exist.
The Skyrme force was first introduced by Skyrme in the 50s as the sum of a zero-range twobody and a zero-range three-body term. Starting from the works of Vautherin and Brink in the
70s, the three-body part has been replaced by a two-body density-dependent term and the first
systematic numerical calculations have been done.
The Gogny force was introduced in the 70s.
To give an example of phenomenological interaction, we write in what follows the standard
expression of the Skyrme force,
r r
V (r1 , r2 ) = r
t0 (1 + x 0 Pσ )δ (r )
r2
r
r
r r
r r
1
+ t1 (1 + x1 Pσ ) P' δ (r ) + δ (r )P 2 + t 2 (1 + x 2 Pσ )P' ⋅δ (r )P
2
r α r
1
+ t3 (1 + x 3 Pσ ) ρ R δ (r )
6
r
r r
+ iW0 σ ⋅ P' ×δ (r )P
[
]
,
(1.7)
[ ( )]
[
]
where t0, x0, t1, x1, t2, x2, t3, x3, α and W0 are the parameters. The first term is the central term
(t0 term). The second and third terms constitute the velocity-dependent part (t1-t2 terms); the
fourth term is the density-dependent term (t3 term) and the last term is the spin-orbit part. The
following notation holds:
r r r
r = r1 − r2 ;
(1.8)
r 1 r r
R = (r1 + r2 ) ;
2
(1.9)
r
r
v 1
P =
(∇1 − ∇ 2 ) ; P' cc of P acting on the left;
2i
(1.10)
Pσ is the spin-exchange operator given by
Pσ =
1
(1 + σ1 ⋅ σ 2 ) = +−11
2
{
S =1 .
S =0
(1.11)
In the above expression, σ = σ1 + σ 2 ; σ ix expressed in terms of the spin operator, σ = 2s,
with σ z = ±1 . The spin matrices are written as
σ x = 0
1
1  σ = 0
i
y
0 

− i  σ = 1
0
z
0 

0 .
− 1 
(1.12)
As an illustration, the 10 parameters of the Skyrme interaction are shown in Table 1.1 for two
currently used parametrizations, SKM* and SLy4.
21
Force
t0 (MeV fm3)
t1 (MeV fm5)
t2 (MeV fm5)
t3 (MeV fm3+3αα)
x0
x1
x2
x3
α
W0 (MeV fm5)
SKM*
SLy4
-2645.00
-2488.91
410.00
486.82
-135.00
-546.39
15595.00
13777.00
0.09
0.834
0.00
-0.344
0.00
-1.00
0.00
1.354
1/6
1/6
130.00
123.00
Table 1.1 Two Skyrme parametrizations
1.4 Nuclear matter: concepts, properties and the equation of
state
Nuclear matter is an ideal infinite system (surface effects are absent) composed by nucleons.
This ideal system is a Fermi gas and is a very helpful framework used to better understand
some properties of the real nuclear systems.
In symmetric matter (equal densities of neutrons and protons), the neutron/proton
asymmetry parameter I=(N-Z)/A is equal to zero (N, Z, and A are the number of neutrons,
protons and nucleons, respectively).
The equilibrium density of symmetric matter, ρ0, called saturation density, corresponds to
the minimum point of the energy per particle as a function of the density (see Fig. 1.6). The
energy per particle E/A as a function of the density is called equation of state. It turns out that
the saturation density corresponds in practice to the central density of all heavy nuclei
(independently of the nucleus). This feature is dictated by the saturation properties of the
nuclear interaction, i.e., the fact that a nucleon in a medium does not interact with all the other
nucleons, but with a limited number of close nucleons (these saturation properties are
generated by the combination of the Pauli and the uncertainty principles). Even if the number
of nucleons is not the same, the central density in heavy nuclei is always very similar to the
saturation density owing to the saturation properties of the nuclear interaction.
Each theoretical model provides a prediction for the equation of state of nuclear matter. In Fig.
1.6, three examples of equations of state calculated for symmetric nuclear matter with three
different theoretical models are displayed. In particular, the green dashed line is obtained
with a relativistic mean-field model whereas the dotted blue line is obtained with a Skyrme
parametrization.
22
Figure 1.6 Three examples of equations of state for symmetric nuclear matter.
Systematic measurements of electron scattering and/or the use of microscopic mass formulae
(much more sophisticated than the liquid drop model), have provided an estimation for the
saturation point (ρ0 ≈ 0.16 fm-3, E/A(ρ0) ≈ -16 MeV). This value of ρ0 corresponds to the
Fermi momentum kF ≈ 1.34 fm-1. In a Fermi gas, the two quantities are related one to the
other by the relation
ρ0 =
2
3π 2
k F3 .
(1.13)
The distance r0 between the nucleons is given by the following expression
4
πr 3 ρ0 = 1 .
3 0
(1.14)
The distance between nucleons corresponding to ρ0 ≈ 0.16 fm-3 results r0 ≈ 1.14 fm.
In Fig. 1.7, the radial profiles of the total densities of two Ni isotopes, 56Ni et 72Ni, calculated
with a Skyrme-mean-field model, are shown. Spherical symmetry is assumed in these
calculations. One observes that the central densities are practically the same, ≈ 0.16 fm-3, for
the two nuclei even if the number of nucleons is different.
23
0,2
56
Ni
Ni
72
Density
0,15
0,1
0,05
0
0
2
4
r
6
8
10
Figure 1.7 Total densities of two Ni isotopes calculated with a mean-field model.
As already mentioned, phenomenological interactions are adjusted to reproduce some
properties of finite nuclei and to provide a reasonable equation of state for nuclear matter. A
special attention is devoted to the description of the saturation point.
Other properties that characterize nuclear matter are for example the compressibility K∞ and
the effective mass m*.
The compressibility is related to the second derivative of the equation of state at the
saturation density, i.e., to the curvature of the equation of state. In particular,
 d 2 E (ρ ) 
2

K ∞ = 9ρ0 
.
 dρ 2 A 

 ρ = ρ0
(1.15)
From theoretical and experimental systematic studies (on monopole giant resonances) the
currently accepted value of the compressibility is K∞ = 210 ± 30 MeV.
Effective mass. The energy of a particle with momentum p in infinite matter is written as
εp =
p2
p2
+ Σ (p, ε p ) =
.
2m
2m *
(1.16)
Eq. (1.16) is self-consistent for εp. The term Σ, called self-energy, describes the interaction of
the particle with the medium (the particle is not in vacuum). Hence, the effective mass m* is
different from the bare mass m because the nucleons are not isolated but in a medium. The
empirical accepted value for the effective mass is m*/m = 0.8 ÷ 0.9.
24
In asymmetric matter (I =(N-Z)/A ≠ 0) a very important quantity is the symmetry energy,
that is given by the coefficient aI of the development introduced in the liquid drop model (see
first Lecture):
−1
−4
E
= aV + aI I 2 + a s A 3 + aCoul Z 2 A 3 + ...
A
(1.17)
Starting from the equation of state for asymmetric matter, the symmetry energy aI is given by
1 ∂2 E A
aI =
2 ∂I 2
I =0
.
(1.18)
The empirical accepted value of the symmetry energy (obtained from the values of masses of
nuclei and from models) is 28 MeV ≤ aI ≤ 32 MeV.
Since E/A is a function of the density, the symmetry energy is also density dependent. It turns
out that the density dependence of the symmetry energy has very important implications for
some properties of nuclei and for applications to nuclear astrophysics (second semester
Lectures).
As an illustration, we write explicitly in what follows the equations of state for symmetric and
asymmetric matter which are provided by the Skyrme interaction.
Skyrme equation of state for symmetric nuclear matter.
2
2
2


3
3 h  3π 
E
(ρ ) =
A
10 m  2
2
3
3  3π 2
ρ 3 + t0 ρ +

8
80  2

2
5
3
 Θs ρ 3 + 1 t3 ρ α +1 ,

16

(1.19)
where: Θs = 3t1 + t 2 (5 + 4x 2 ) .
Skyrme equation of state for asymmetric nuclear matter.
2
2
2


3
3 h  3π 
2
1
ρ 3 F5 3 + t0 ρ [2 (2 + x 0 ) − (1 + 2x 0 )F2 ]

8

2
5
2


3
1
3  3π 
1


α
+1
+
t3 ρ
2(2 + x 3 )− (1 + 2x 3 )F2 +
ρ 3 Θv F5 3 + (Θs − 2Θv )F8 3 


48
40
2
2


E
(I; ρ ) =
A
10 m  2
[
]


(1.20)
with: Θs = t1 (2 + x1) + t 2 (2 + x 2 ) and Fm (I ) =
1
(1 + I )m + (1 − I )m .
2
[
]
25
26
Chapter 2
Marcella Grasso
2.1 Introduction to the nuclear many-body problem 28
2.2 Light nuclei: ab initio methods 36
2.3 The shell model 39
2.4 Medium-mass and heavy nuclei: mean-field models, links with the densityfunctional theory 41
27
2.1 Introduction to the nuclear many-body problem
In principle, the wave function Ψ that describes a many-body system composed by N particles
and which contains all the information about its properties, can be determined by solving the
Schroedinger equation,
HΨ
= EΨ
,
(2.1)
where the Hamiltonian is the sum of a kinetic term T and an interaction term V,
H = T +V
.
(2.2)
V is the interaction and should in principle contain two-, three-, …N-body contributions. It
turns out that, in most cases, the interaction is well enough described by the 2-body (and
eventually the 3-body) terms.
H ≈
N p2 1 N
+
∑Vij .
2 i ≠ j =1
i =1
i
∑ 2m
(2.3)
Even with this simplification, the problem is still too complicated and the solution of the
Schroedinger equation cannot be obtained without adopting some approximations.
The semiclassical approaches strongly simplify the problem to treat. This can be achieved by
neglecting (completely or partially) the quantum nature of the system. In some cases, it is
reasonable to fix energy, length, mass… scales that allow one to establish a hierarchy between
the degrees of freedom and to reduce the number of constituents for which a quantum
treatment is necessary. The other constituents can be described by semiclassical approaches.
This is the case, for instance, within the adiabatic Born-Oppenheimer approximation which is
currently used in the physics of many-electron systems like the physics of metal clusters (the
huge difference between the mass of the ions and the mass of the electrons, and thus the huge
difference between the typical velocities of the two types of particles, allows one to treat only
the valence electrons with a quantum approach).
Quantum many-body physics has been founded in the first years of quantum physics, in the
first half of last century. In what follows, we mention some examples of methods that are very
often employed in the framework of many-body physics, namely the microscopic mean-field
Hartree-Fock (HF) model and the density functional theory (DFT).
In 1927, in Cambridge, Hartree introduced for the first time the concept of mean field which
was based on the so-called independent-particle approximation. In the same years, Slater and
Fock completed this variational model by making an ansatz on the wave function of the
ground state for fermionic systems, which correctly accounted for the Pauli principle. The
ground state was chosen as an anti-symmetrized product of single-particle wave functions, a
Slater determinant (Slater, 1929; Fock, 1930). The first numerical applications were done
much later, only in the 50s. This method, that had been introduced to treat electronic systems,
was applied later also to other domains of physics. In the 70s, the first systematic numerical
applications were done in the framework of nuclear physics by using the effective
phenomenological interactions available at that time.
28
In the 60s, DFT was formulated in the framework of the physics of many-electron systems.
This variational approach is based on two theorems demonstrated by Hohenberg and Kohn in
1964. Shortly later, Kohn-Sham equations were introduced. In the currently used density
functionals there are two terms: a direct Hartree term and an exchange-correlation term that
describes both the exchange part (usually not exactly calculated, but evaluated with some
approximations) and the presence of some correlations in the ground state. DFT is applied to
calculate for instance the binding energies of molecules in chemistry and the band structure of
solids in solid-state physics.
In the last years, many efforts have been concentrated in nuclear physics to the extension and
application of DFT concepts and theorems to nuclear-structure studies. Density functionals
are indeed very well known in nuclear physics. In the framework of non-relativistic meanfield approaches with Skyrme interactions, for instance, the Hamiltonian density H (which is
derived in a variational way from the Skyrme interaction and which provides the ground-state
energy when integrated) is a functional of the local density (Skyrme functional). Since these
functionals allow one to calculate the ground-state energy, the associated theoretical models
are often called energy density functional (EDF) theories.
Also in the framework of relativistic theories many efforts are devoted to the investigation of
EDF. Relativistic EDF models inspired from low-energy QCD have been recently introduced.
In general, the efforts are devoted to find (if possible) a kind of universal functional that can
reproduce reasonably well the ground-state properties of all nuclei, stable and unstable, up to
the nucleonic drip lines3.
The choice of the theoretical model which is employed to analyze the nucleus under study
depends first of all on the number of nucleons present in the system. The so-called ab initio
methods are suitable approaches for very light nuclei (few-body systems, A up to 12). Within
these methods, the Schroedinger equation is solved directly. Mean-field models, on the other
side, are well adapted to investigate the properties of medium-mass and heavy nuclei. The
shell model is located between the two methods, as it can be seen in Fig. 2.1 (the so-called no
core shell model belongs to the category of ab initio methods).
Figure 2.1 Nuclear landscape and available models.
3
Drip lines define in the nuclear chart the limits of existence of nuclei. These lines are so far determined almost
entirely by theoretical predictions. Experimental drip lines have been attained only for light nuclei.
29
Some words about these models will be spent in the following three Sections. We illustrate
before some concepts and tools which are very important in many-body physics.
Independent-particle approximation.
Both the shell model and the mean-field model are based on the independent-particle
approximation. This approximation is valid in those systems where the mean free path of the
particles is comparable with or larger than the size of the system. In these cases, it is
reasonable to approximate the interaction between the particle i and the other N-1 particles
with an average potential that acts on the particle i and describes in an effective way the
presence of the other N-1 particles. The problem is equivalent to that of a system of
independent particles with an external potential (the external potential is in this case
constructed with the constituents of the system). This approximation is in general quite good
in nuclei (the existence of shells and magic numbers is a proof of this). Indeed, the mean free
path in the nucleus of a nucleon having 10 MeV-kinetic energy is around 15 fm (larger than
the size of the nucleus). The Pauli principle is responsible for the fact that the nucleons do not
collide very frequently one with the others in a nucleus.
Second quantization
The use of creation and annihilation operators for creating and annihilating particles is
commonly called second quantization. When this terminology was introduced (in the 20s), it
was common to consider standard quantum mechanics, where the classical dynamics of
particles is replaced by wave dynamics, as first quantization. On the other hand, the
quantization of electromagnetic radiation, done by replacing the concept of electromagnetic
waves by photons, was called second quantization.
Second quantization simplifies the formal description of a many-body system owing to the
introduction of creation and annihilation operators (see harmonic oscillator). The properties of
these operators describe the correct Bose or Fermi statistics of the particles under study. All
the operators, and in particular the Hamiltonian of the system, can be expressed in terms of
these creation and annihilation operators.
Let us consider a system composed by N identical fermions and introduce the Hilbert space
HN for N identical particles as the space that contains all the functions satisfying the property
∫ ψ (r1 ,..., rN )
r
r
2 d 3 rr ...d 3 rr < ∞ ,
1
N
(2.4)
the sub-space FN for fermions is defined as the sub-space of HN that contains only the
antisymmetrized functions with respect to the exchange of the indices of two particles, i.e.,
the functions associated to the states of the type
α1 α 2 ...α N = − α 2 α1 ...α N .
(2.5)
Let us introduce the creation operators aα+ (which act from the sub-space FN to the sub-space
FN+1) and the annihilation operators aα (which act from the sub-space FN to the sub-space
i
i
FN-1). Since we deal with fermions, it holds:
30
- aα+ α1 ,..., α i ,...α N = 0 , because the state αi is already occupied and the Pauli principle
i
does not allow one to occupy it twice;
- aα α1 ,.., α i −1 , α i +1 ..α N = 0 , if the state αi is already empty because in this case there
i
are no particles to be destroyed.
Let us write:
aα+ α1 ,...α N = α i , α1 ,...α N .
(2.6)
i
If |0> is the vacuum (no particles), it holds
aα+ 0 = α .
(2.7)
( )+
Let us define the operators aα ≡ aα+ . One can show that
1) aα α1 ,...α i ,...α N = (− 1 )i −1 α1 ,...α i −1 ; α i +1 ,...α N
i
(the sign is related to the number of permutations necessary to move αi to the left up to the
first place, so that it can be annihilated)
a + , a + } = {a , a } = 0
{
2)
{a , a + } = δ
i
j
i
i
j
j
ij
It is possible to show that these anticommutation relations contain the Fermi statistics for
fermions.
The Hamiltonian for a system of N fermions with a two-body interaction term
H =
N
∑ T (x k ) +
k =1
1
2
N
∑V (x k , x l ) ,
k ≠ l =1
(2.8)
is written in second quantization as
H =
∑Tαβ aα+a β
αβ
+
1
Vαβγδ aα+a β+ aδ aγ .
∑
2 αβγδ
(2.9)
where Tαβ and Vαβγδ are the matrix elements between the states indicated by α, β, γ and δ.
Green’s functions
A possible approach for the many-body problem is based on perturbation theory where
Green’s function and Feynman diagrams are used.
31
In quantum mechanics, three possible representations exist to describe the dynamics of a
system: the Heisenberg (H), the Schroedinger (S) and the Interaction (I) representation.
In the Heisenberg representation, the operators are time-dependent whereas the states are not
time-dependent. All the dynamics is thus contained in the operators. Let us consider the
creation and annihilation operators Ψˆ + (x,t) et Ψˆ (x,t) in the Heisenberg representation.
Hα
Hα
By construction, they have the following time evolution,
Ψˆ
(x, t ) = e Ψˆ (x, t = 0 )e −
ˆ
iHt
Hα
ˆ
iHt
H
,
(2.10)
where Hˆ is the Hamiltonian of the system and the operator ΨˆH (x, t = 0 ) , at the time t=0,
corresponds by construction to the operator Ψˆ (x ) in the Schroedinger representation (where
S
all the dynamics is contained in the state vectors which are time-dependent, whereas the
operators are not time-dependent).
Let us write the Hamiltonian Hˆ as the sum of a term Hˆ0 (for which the solution of the
Schroedinger equation is known) plus a term Vˆ , Hˆ = Hˆ0 + Vˆ . When the Hamiltonian is
written in this way, one can introduce the so-called interaction representation. In the
interaction representation, both the operators and the state vectors are time-dependent. In
particular, the time evolution for the operators is given by the relation
Ψˆ
Iα
(x, t ) = e
ˆ
0 tΨˆ (x)e − iH 0 t
ˆ
iH
S
,
(2.11)
where ΨˆS (x ) is the operator in the Schroedinger representation.
Let us denote with ψ H0 the ground state of Hˆ in the Heisenberg representation. The
+
quantity ψ H0 ΨˆHα (x, t )ΨˆHβ
(x' , t' ) ψH0
contains information about the probability to
find a particle in x at time t if we have created it in x’ at time t’. Thus, this quantity contains
information about the dynamics of the system.
Let us define the time ordering of two operators as follows
 Ψˆ (x, t )ΨˆH + (x' , t' ) t > t'
.
T ΨˆH (x, t )ΨˆH + (x' , t' ) ≡  H +
− ΨˆH (x' , t' )ΨˆH (x, t ) t < t'
[
]
(2.12)
The one-body Green’s function (or propagator) is defined as
iGαβ (x, t, x' t' ) ≡
+
ψ H0 T ΨˆHα (x, t )Ψ Hβ
(x' t' ) ψH0
[
]
ψH0 ψ H0
.
(2.13)
Two main advantages of using Green’s function in treating a many-body problem are:
1) In the framework of the perturbation theory, the contribution of order n (Feynman
rules) can be written more easily for G than for any other combination of operators;
32
2) Green’s functions provide very important information about the properties of a system.
In particular, they allow one to calculate the mean value in the ground state of any
one-body operator, the ground-state energy and the excitation spectrum.
Feynman diagrams. Self-energy, proper self-energy
By using a theorem known as Wick theorem and the interaction representation, the exact
)
)
Green’s function can be expressed with a perturbative expansion (expansion in V , where V
is the interaction in the Hamiltonian, Hˆ = Hˆ + Vˆ ) in terms of the unperturbed or free
0
Green’s function (G0) associated to Hˆ0 .
In particular, it is possible to write the following perturbative development
iGαβ (x, y ) =
+
ψ H0 T ΨˆHα(x) ΨˆHβ
(y)ψH0
[
]
ψH0 ψH0
+∞


−
i
Vˆ(t' )dt' 
∫

,
+
Φ0 T ΨˆIα(x) ΨˆIβ
(y)e − ∞
 Φ0





=
ˆ
Φ SΦ
I
0
(2.14)
0
where, for simplicity, the coordinates x and y contain also the time dependence and Sˆ is
defined as
+∞

−
i
∫V(t)dt

ˆ
−
S ≡ T e ∞


I


 ;


(2.15)
Φ0 is the ground state of Hˆ0 .
The six contributions of the first order to the numerator of Eq. (2.14) are given by Eq. (9.1) at
page 34 (taken from the book Fetter andWalecka, Quantum Theory of Many-Particle
Systems). The six associated Feynman diagrams are shown in Fig. 9.1 of the same page. In
the shown diagrams, the straight lines represent the unperturbed Green’s function GO and the
wavy lines represent the interaction.
33
34
The exact Green’s function G, represented by a straight thick line (to be distinguished from
the unperturbed Green’s function) is the solution of the equation
=
The compact diagram
indicated with Σ.
+
contains several diagrams, is called self-energy and
The proper self-energy
(denoted with Σ*) is composed by a class of diagrams of Σ ,
those that cannot be separated in two parts by cutting only a straight line (i.e., a line that
represents a free Green’s function G0).
The equation for G, written in terms of the proper self-energy, is called Dyson equation:
=
+
By using Feynman diagrams, the independent-particle approximation corresponds to
considering only the first-order terms in the proper self-energy,
~ Σ*(1) =
+
,
where now the dashed line represents the interaction (in Fetter-Walecka book the interaction
is represented by wavy lines). The first diagram in the right-hand side describes the direct
term and the second the exchange term. The existence of an exchange term depends on the
fact that the particles are fermions and have thus to satisfy the Fermi-Dirac statistics.
35
2.2 Light Nuclei: ab initio methods
Light nuclei are very special systems for several reasons: experimentally, it is easier for these
systems to approach the drip lines and, in some cases, to attain them (this means that very
unstable light nuclei can be produced and observed); complex correlations manifest
themselves in these nuclei giving rise to very particular scenarios, like the existence of halo
nuclei, of borromean systems and of nuclei with a cluster structure. In the so-called halo
nuclei, one can quite easily separate a core internal region and an external ‘halo’ region which
is very diffuse and where the halo nucleons are located. Radii are thus extremely large for
these systems. In Fig. 2.2 one can see a schema where halo nuclei are represented: oneneutron (blue) and proton (green) halo nuclei, borromean nuclei (two-neutron halo systems)
and nuclei with a core + 4 neutrons are shown.
Figure 2.2 Several examples of halo nuclei.
Borromean nuclei are halo nuclei which have the same structure of borromean rings (Fig. 2.3).
They are bound three-body systems characterized by the fact that all the two-body
combinations of their particles that one can imagine to create are not bound.
Figure 2.3 Borromean rings.
36
An example of halo nucleus is 11Li = 9Li + n + n (two-neutron halo nucleus). The neutron and
proton density profiles calculated with a theoretical model for this nucleus are displayed in
Fig. 2.4.
Figure 2.4 Proton and neutron density profiles of 11Li.
In the theoretical treatment of light nuclei, it is numerically possible to use very sophisticated
and computationally demanding few-body methods were realistic interactions are employed.
These approaches are called ab initio models. In these models, one computes the properties of
an A-nucleon system with a realistic interaction (next generation models based on chiral
effective field theories promise a closer connection to QCD). Different ab initio models are
available. In the last 15 years, there has been much progress in three approaches to the nuclear
many-body problem for light nuclei: no-core shell model (NCSM) for nuclei with A up to ≈
12, Green’s function Montecarlo (GFMC) for nuclei with A up to ≈ 12, and coupled cluster
expansion (CCE) which can be applied also to heavier nuclei.
The NCSM is based on a new variation of the well-known shell model for nuclei (see Sec.2.3
for some details about the shell model). Historically, shell-model calculations have been made
assuming a closed, inert core of nucleons with only a few active valence nucleons. With the
development of the NCSM in the 90s, all nucleons in the nucleus are treated as active
particles. The computational problem is extremely difficult (diagonalization of huge matrices)
and much effort in the recent past has been devoted to the development of parallelizable
procedures to handle the computations on massively parallel computers.
The first application of Monte Carlo methods to nuclei interacting with realistic potentials was
a variational (VMC) calculation done in the 80s. A VMC calculation finds un upper bound, ET,
to an eigenvalue of the Hamiltonian H, by evaluating the expectation value of H using a trial
wave function ψT. The parameters in ψT are varied to minimize ET and the lowest value is
taken as the approximate energy. Over the years, rather sophisticated ψT have been developed.
GFMC projects out the lowest-energy eigenstate from the VMC ψT by using
ψ0 = lim ψ (τ ) = lim exp [− (H − E 0 )τ ]ψT ,
τ →∞
τ →∞
(2.16)
37
where τ is an imaginary time. If sufficiently large τ is reached, the eigenvalue E0 is calculated
exactly. The evaluation of exp [− (H − E 0 )τ ] is made by introducing a small time step,
∆τ=τ/n,
ψ (τ ) = {exp [− (H − E 0 )∆τ ]}n ψT = G n ψT ,
(2.17)
where G is called short-time Green’s function. In coordinate space, this leads to a
multidimensional integral which is done by Monte Carlo.
In Fig. 2.4, GFMC results for some energy levels are compared with experimental values. The
calculations AV18 use only a NN potential, whereas the results denoted with IL2 are obtained
by introducing also a 3N potential.
Figure 2.4 Examples of GFMC results for light nuclei (energy levels).
The CCE method was developed in the 60s but practical applications to nuclei have been done
much later. The CCE formalism relies on expansions of the nuclear wave functions. The
expansion coefficients represent the nuclear correlations. Let us consider first the ground state
ψ . We make an ansatz for this ground state,
ψ = eS φ ,
+
(2.18)
where φ represents the physical vacuum defined such that ah+ φ = a p φ = 0 . The cluster
correlation operator is introduced as
S+ =
1
∑ n! SnOn+ ,
(2.19)
38
with: On+ = 1, a p+ ah1 , a p+ a p+ ah1 ah2 ,... . The ground-state energy E and the amplitudes Sn
1
2
1
are obtained by solving a set of coupled non linear equations,
{
}
E = φ e S He −S φ
.
0 = φ e S He −S On+ φ , n ≤ A
(2.20)
Truncations of the above set of equations involve decisions regarding contributions to the
2p2h, 3p3h, 4p4h, … sectors.
The excited-state spectrum is subsequently described relying on the ground state as a new
vacuum.
2.3 The shell model
More details about the shell model are given in the Lectures of second semester of NPAC. In
this model, valence nucleons (outside closed shells) move in a mean-field potential and
interact via a residual nuclear force.
The model (in its simplest non-interacting version) was introduced more than 50 years ago by
Göppert-Mayer1 and by Haxel, Jensen and Suess. It has proven very successful in describing
the properties of nuclei with few valence nucleons, including energy levels, magnetic and
quadrupole moments, electromagnetic transition probabilities, beta-decay rates and reaction
cross-sections. It has also been used as the theoretical basis for several algebraic nuclear
models.
If one introduces a harmonic oscillator potential plus a centrifugal and a spin-orbit term,
r r
r r
mω 2 r 2
U(r)=
+α l ⋅l + β l ⋅s ,
2
(
)
(
)
(2.21)
like done in the first models (Göppert-Mayer, 1949; Jensen, Haxel, Suess, 1949), the
associated energy levels are plotted in Fig. 2.5.
1
‘The shell model has initiated a large field of research. It has served as the starting point for more refined
calculations. There are enough nuclei to investigate so that the shell modellists will not soon be unemployed.’
Maria Göppert-Mayer, Nobel Lecture, December 1963.
39
Figure 2.5 Energy levels for the potential (2.21).
The essential strategy of present shell-model approaches is to separate the many-body
Hamiltonian into an independent-particle Hamiltonian and a residual effective interaction,
A p2
A
H = ∑ k + ∑v (i, j ) + ... = H0 + Vres ,
k =1 2mk i < j =1
(2.22)
where k denotes a set of quantum numbers for a single-particle state and H0 is a single-particle
Hamiltonian,
A p2
A
H0 = ∑ k + ∑ U (k ) .
k =1 2mk k =1
(2.23)
Vres is the so-called residual interaction. By considering only a two-body force in the manybody Hamiltonian,
V
res
∑v (i, j ) − ∑ U(i).
=
i
<j
(2.24)
i
40
The strategy for shell-model calculations is based on the following scheme:
1) Constructing a basis (single-particle states associated to H0);
2) Defining a valence space (to simplify the numerical problem: one defines a core and
an active space. In no-core shell models all the nucleons are treated as active particles
and the matrices to diagonalize become very big)
3) Introducing an effective interaction
4) Diagonalizing the Hamiltonian matrix (where the matrix elements are calculated
between the valence states which are expressed in terms of the basis functions).
Multiparticle-multihole configurations are thus constructed when the effective
Hamiltonian is diagonalized.
The choice of the effective interaction is very important. One can choose different attitudes.
In present calculations the employed interactions are either empirical (with many parameters
to adjust) or more microscopic (derived from a bare force)
2.4 Medium-mass and heavy nuclei: mean-field models, links with
the density-functional theory
The mean-field method is based on the independent-particle approximation and the interaction
between the nucleons is replaced by the interaction of each nucleon with a mean field. The HF
theory is a mean-field model that provides a way to define the independent-particle (onebody) Hamiltonian as a first step in a many-body theory of ground-state energies in which the
correlation energy due to the residual interaction is calculated in perturbation theory (HF
corresponds to the first-order diagrams).
Mean-field methods can be carried out in huge spaces of single-particle wave functions. They
predict ground-state properties of nuclei with increasing accuracy as 1/A → 0. Thus, they are
particularly useful for describing masses, radii, shapes and stabilities of relatively heavy
nuclei. In this sense, the mean-field model and the shell model are complementary one with
respect to the other.
The time-dependent mean-field approximations (Chapter 3) provide an elegant technique to
describe the dynamics of the nucleus and its excitation spectrum. The small-amplitude timedependent HF theory is equivalent to the so-called random-phase approximation (RPA)
(Chapter 3) and allows one to describe small-amplitude oscillations.
The one-body HF average potential (built by all the nucleons) is obtained with a variational
procedure. For a nucleus with A nucleons the Slater determinant is written as
41
ψ HF (x1 ,...x A ) =
1
det ϕα1 (x1 )ϕα 2 (x 2 )...ϕα A (x A ) .
A!
{
}
(2.25)
It is constructed as an antisymmetrized product of single-particle wave functions ϕ of the
nucleons. In second quantization, by introducing the annihilation and creation operators a and
a+ which anticommute, the Slater determinant is written as
ψ HF =
A
∏ ai+ 0
i =1
,
(2.26)
where 0 is the vacuum state (no particles).
The single-particle wave functions are obtained by minimizing the total energy of the nucleus
(variational procedure),
E HF =
ψHF H ψ HF
.
ψ HF ψHF
(2.27)
The procedure to construct the mean potential is self-consistent and the problem is solved
iteratively. This means that the proper self-energy is expressed by the following first-order
diagrams,
~
+
,
where the thick lines represent the exact Green’s function G. The Dyson equation, solved with
this proper self-energy, can be written as
 h 2
r 
r
r
rr
r
r
Δ + Γ H( r )ϕk( r )+ ∫ dr ' Γ Ex( r , r ' )ϕk( r ' )= ε k ϕk( r ),
−
 2m

(2.28)
where
r
r rr
r
Γ H( r )= ∫ dr ' v (r , r ' )ρ (r ' ) ,
(2.29)
rr
rr
rr
Γ Ex (r , r ' ) = −v (r , r ' )ρ (r , r ' ) ;
(2.30)
and
r
rr
ρ (r ) and ρ (r , r ' ) are the local and non-local densities. Equations (2.28) are the HF equations
and ΓH and ΓEx are the direct and exchange potential, respectively. For simplicity, Eqs. (2.28)
42
are written for the case of a local two-body potential that does not depend on the spin and
isospin. The HF equations are derived later in Chapter 3.
****
In the HF ground state, which is a Slater determinant, correlations are not included by
construction. A first natural extension of HF is the so-called Hartree-Fock-Bogoliubov (HFB)
or Bogoliubov-de Gennes method where pairing correlations are introduced in the selfconsistent scheme. These correlations, responsible for the formation of Cooper pairs in
superfluid systems, are taken into account in the formal scheme by introducing the so-called
quasiparticles through the unitary Bogoliubov transformations. With these transformations,
the quasiparticle operators β are defined as linear combinations of operators γ+ and γ of
creation and annihilation of particles,
βk ↑ =
∑ (u nk γ n ↑ − v *nk γ n+ ↓ ),
βk ↓ =
∑ (u nk γ n ↓ + v *nk γ n+↑ ),
n
(2.31)
n
where the arrows ↑ and ↓ indicate two time-reversed2 states.
The ground state is not a Slater determinant of independent particles; it is a Slater determinant
of independent quasiparticles. This extension allows one to include in the ground state the
correlations associated to the superfluidity. Like the HF model, this method neglects all the
other types of correlations and is based on a variational procedure. The variational approach
leads to an effective average potential by replacing the two-body interaction in the
Hamiltonian with a one-body term. The HFB equations, that describe the ground state, have
the following compact form:
W
Δ
Δ u  = E u  ,
v 
− W v 
 
(2.32)
where W contain the kinetic term, the mean field and an eventual one-body external potential
and ∆ represents the pairing field (we suppose that it is real). The solutions of these equations
are the quasiparticle energies E and the two components (upper and lower) of the wave
functions u and v, introduced by the Bogoliubov transformations. These equations are derived
in a simplified case in Chapter 4.
If one supposes that the pairing field couples only time-reversed states, a simplified approach
with respect to HFB is obtained that corresponds to the BCS model introduced by Bardeen,
Cooper and Schrieffer to describe the Cooper pairs of electrons in superconductors. Within
this simplification, the Bogoliubov quasiparticles are expressed as
βn ↑ = u nnγ n ↑ − v *nnγ +
n ↓,
*
βn ↓ = u nnγ n ↓ + v nnγ + .
(2.33)
n↑
2
The classical equations of motion governing a system of interacting particles are invariant with respect to the
direction of time, that is, with respect to a transformation that reverses the motion of all the components of the
system. For a quantum system, such a transformation (that we indicate with τ) is characterized by: r’k = τ rkτ-1 =
rk; p’k = τ pkτ-1 = -pk; s’k = τ skτ-1 = -sk
43
****
The study of the dynamics of many-body systems may be done by using the timedependent formalism of mean-field approaches. Time-dependent HF or HFB (TDHF or
TDHFB) equations are solved in this case. These equations can be simplified in the
framework of the linear response theory. In this case the equations of motion describe smallamplitude oscillations of the system. This formal scheme is called RPA. The RPA (QRPA if
one includes pairing correlations) equations have been formalized some decades ago and their
first applications have been done in the 50s by Bohm and Pines to study the oscillations of a
gas of electrons. The (Q)RPA equations can be derived from the TDHF(B) equations by
adding a weak external perturbation to the mean field and by considering variations of the
density ρ (HF) or of the generalized density R (HFB) which are linear in the external
perturbation. Time-dependent methods and the RPA will be illustrated in more detail in
Chapter 3.
Links with DFT
DFT is very currently employed to study quantum many-body systems in physics and in
chemistry. It is based on very general concepts and ideas and this aspect explains its large
diffusion and its easy applicability to different scientific domains. In 1998, the Nobel prize for
chemistry has been attributed to Walter Kohn, the father of this theory in the 60s, and to John
Pople, who gave in the following years very important contributions in the numerical
applications of this theory.
DFT has been formalized for a system in an external potential. The Schroedinger equation to
solve is
 N  h2 ∇2
i + v( rr ) + ∑ U( rr , rr
 ∑  −
i 
i j
2m
i =1 
 i <j
 r r
r r
)ψ (r1 ,...rN ) = Eψ (r1 ,...rN ) ,

(2.34)
where v is the external potential which acts on the system and U describes the interaction.
Given v and U, one could in principle solve the Schroedinger equation and obtain the wave
function ψ (but, for a many-body system, the direct solution of the Schroedinger equation
without approximations is too difficult).
When one knows the wave function of a system, the mean values of the operators (the
observables) can be evaluated. The density is one of these mean values. In the framework of
DFT, it turns out the density is not considered as an observable like the others. It plays a very
special role. The formal procedure on which DFT is based goes, from a conceptual point of
view, in the opposite direction with respect to the usual procedure adopted to solve the
Schroedinger equation: in DFT the starting point is the density (a mean value of operators).
Using this density, the external potential v can be determined.
Functional
A functional F of a function f(x), F[f], is a rule that allows one to associate a numerical value
to a function.
For a given x,
f(x)
function
F[f]
numerical value
44
Variation of a functional
For a given x, δF is the variation of the functional generated by a variation of the function f(x),
δf(x) (and not by a variation of the argument x).
Derivative of a functional
It is defined by the following relation,
F [f(x)+ δf(x)] = F [f(x)] + ∫ s (x' )δf (x' )dx' +O(δ f 2 ) .
(2.35)
The derivative of the functional is defined as the function s(x), that is the coefficient
multiplying δf(x) in the integral,
δF [f ]
≡ s(x).
δf (x )
(2.36)
The integral in Eq. (2.35) is introduced because the variation of the functional is generated by
the variations of the function at every point (the spatial coordinate must be integrated).
Hohenberg-Kohn theorem
It is demonstrated for a system of particles interacting in an external potential v. The manybody problem is replaced by an energy minimization.
The Hohenberg-Kohn (HK) theorem is often divided in different pieces. Here we consider
two parts.
The important points of the Theorem 1 are:
- For a system of interacting particles (U is the interaction) in an external potential (v), if the
ground-state density ρ0(r) is known, it is possible, using this density, to determine the external
potential (up to an additive constant) and, consequently, the ground-state wave function ψ0;
- ψ0 is thus a functional of ρ0. This means that all the observables associated to the ground
state (that are calculated as mean values of the operators in the ground state) are a functional
of ρ0.
- For each external potential a functional E[ρ] of the density exists, which is minimized by ρ0
and thus by ψ0. The ground-state energy E0 is equal to E0=E[ρ0].
- The ground state must be non degenerate. If the ground state is degenerate, a unique
functional ψ[ρ] does not exist because the different degenerate wave functions provide the
same density. However, a unique functional E[ρ] still exists also in this case.
Theorem 2.
For a density ρ≠ρ0, the associated wave function ψ is different from ψ0 and the value E[ρ] is
E[ρ] ≥ E0.
The original proof of the HK theorem, that was published in the article of 1964 (P. Hohenberg,
W. Kohn, Phys. Rev. 136 B 864 (1964)) is a proof by contradiction: one assumes that ψ0 is
not determined in a unique way by ρ0 and one shows that this leads to a contradiction with the
variational principle. Afterwards, several other different proofs of this theorem have been
proposed.
45
Kohn-Sham equations
The Kohn-Sham (KS) equations (W. Kohn and L.J. Sham, Phys. Rev. 140 A 1133 (1965))
allow one to replace the energy-minimization problem of the DFT in the solution of a
Schroedinger equation for independent particles (one-body problem).
The energy E, is written as a functional of the density (HK theorem),
E [ρ ] = T [ρ ] + U [ρ ] + V [ρ ] .
(2.37)
Kin. Interact. Functional related to the
part
external potential (see Eq. (4.31))
r r r
V [ρ ] = ∫ d 3 r ρ (r )v (r )
For the kinetic part:
T [ρ ] = TS [ρ ] + TC [ρ ] ,
(2.38)
where TS[ρ] is the mean value of the operator Tˆ calculated with the Slater determinant
associated to ρ. Since the wave function is a Slater determinant, the exchange contribution is
automatically taken into account. It follows that the term TC must contain only correlations.
For the orbitals ϕi that constitute the Slater determinant,
r r
r
h2 N
TS [ρ ] = −
d 3 r ϕ*i (r )∇ 2 φi (r ) .
∑
∫
2m i =1
(2.39)
All the functions ϕi are functionals of ρ (HK theorem). Hence:
T
S
[ρ ] = T [{ϕ [ρ ]}].
S
i
(2.40)
The exact functional is written as,
E [ρ ] = TS [{ϕi [ρ ]}] + U Hartree [ρ ] + E xc [ρ ] + V [ρ ] .
(2.41)
Exc contains the quantities T-TS and U-UHartree (this term is typically much less important than
TS and UHartree). The functional Exc is not known (the HK theorem tells us only that it is a
functional of ρ). Exc is called exchange-correlation term and is often written as the sum of an
exchange term Ex and a correlation term EC. Different approximations are used to treat this
term (in some cases the exchange part is calculated exactly but the correlation part is always
evaluated by choosing some approximation).
The minimization that generates the KS equations is
0 =
r
r
r
δE [ρ ] δTS [ρ ] δV [ρ ] δU Hartree [ρ ] δE xc [ρ ] δTS [ρ ]
+
r =
r +
r +
r
r =
r + v (r ) + v Hartree (r ) + v xc( r )
δρ (r )
δρ (r )
δρ (r )
δρ (r )
δρ (r )
δρ (r )
(2.42)
46
δE xc
can be calculated explicitly only when an approximation for Exc is chosen.
δρ
Let us consider a system of independent particles in an external potential vS. The minimization
condition does not contain the Hartree and the exchange-correlation terms in this case. It
reads:
0
=
[ ] = δTS [ρ ] + δVS [ρ ] = δTS [ρ ] + v (rr ) .
r
r
r
r
S
δρ (r )
δρ (r )
δρ (r )
δρ (r )
δE ρ
(2.43)
By comparing Eqs. (2.42) and (2.43), we conclude that the two minimizations lead to the
same result if
r
r
r
r
v S (r ) = v (r ) + v Hartree( r )+ v xc( r ).
(2.44)
The Schroedinger equation to solve is thus,
 h ∇
−
 2m
2
2
+v
r
S
(r )ϕ (r ) = ε

r
i
i
r
ϕ (r ) .
i
(2.45)
The solution of this equation allows one to construct the density,
r
ρ (r ) =
N
∑ f i ϕi (r )
i =1
r
2,
(2.46)
where the coefficients f are the occupation probabilities of each state. Eqs. (2.44)-(2.46) are
the non-linear KS equations that are solved iteratively up to the self-consistent solution.
IMPORTANT : Only the density and the total energy have a physical meaning. The singleparticle energies do not have a precise physical meaning and do not correspond, in general, to
the physical single-particle spectrum.
DIFFERENCES BETWEEN DFT-KOHN-SHAM AND HF:
1) In HF the exchange term is always evaluated exactly and the correlation term is
always missing.
2) The HF single-particle energies have a physical meaning and actually describe the
single-particle spectrum. The Kohn-Sham single-particle energies do not have a
physical meaning.
3) KS equations are simpler to solve (the exchange term is not explicitly included; the
integral in Eq. (2.28) is thus not calculated).
4) In HF the starting point is the Hamiltonian. In DFT the starting point is the functional.
5) DFT is formalized for systems in an external potential. Recent works try to extend the
DFT theorems to the case of nuclear physics (nuclei are self-bound systems. The
external potential is missing).
47
Many efforts are devoted to find a kind of universal functional for nuclei. If one accepts the
idea that a unique functional valid for all the nuclei should exist, the question is: which
aspects and properties should be taken into account and included to construct this functional?
48
Chapter 3
Marcella Grasso
3.1 The mean-field approach as a variational procedure. The Hartree-Fock
equations 50
3.2 Time-dependent Hartree-Fock theory 54
3.3 Small-amplitude excitation modes: the random-phase approximation
55
3.4 Examples of results obtained with mean-field-based approaches 60
49
3.1 The mean-field approach as a variational procedure. The
Hartree-Fock equations
A brief description about the philosophy of the HF method has been presented in Chapter 2.
We derive here in more detail the HF equations.
HF theory is a self-consistent mean-field theory. It can also be viewed in terms of a
variational principle as an optimal independent-particle approximation. As such, its essential
approximation is to neglect the effects of two-body correlations (the residual interaction,
which is treated in the shell model, is neglected in HF).
Let us start with the Schroedinger equation for our system of A particles,
H Ψ =E Ψ .
(3.1)
We write the total energy as
E =
Ψ EΨ
= E [Ψ ] .
Ψ Ψ
(3.2)
The ground-state energy is determined by a minimization procedure of the energy (variational
procedure), i.e.,
δE [Ψ ] = 0 .
(3.3)
In the HF theory one simplifies the Hamiltonian by assuming that it is reasonable to replace it
with an effective single-particle Hamiltonian. The effective Hamiltonian is written as
H eff = h1 − body ,
(3.4)
where h1-body is a single-particle Hamiltonian. In HF, the minimization procedure is done by
using only Slater determinants as wave functions for the ground state. As already mentioned,
a Slater determinant is an antisymmetrized product of single-particle wave functions
(uncorrelated ground state),
Ψ =
A
∏ ai+ 0
i =1
,
(3.5)
where the state |0> represents the vacuum of particles. The operators a+i and ai correspond to
the single-particle wave functions ϕ(i) associated to the eigenstates of the one-body
Hamiltonian h1-body:
h1 - body ϕ
(i)= ε i ϕ
(i).
(3.6)
In Eqs. (3.4)-(3.6) i denotes a set of quantum number characterizing a single-particle state.
The antisymmetrized product can also be written as
50
ϕν1 (ε1 ) ϕν1 (ε 2 )
1 ϕν 2 (ε1 )
...
Ψν (ε1 ,..., ε A ) =
...
...
A!
ϕν A (ε1 )
...
...
...
...
...
ϕν1 (ε A )
ϕν 2 (ε A )
.
...
ϕν A (ε A )
(3.7)
The HF state |HF> is the Slater determinant that minimizes the total energy, i.e., the Slater
determinant where the A lowest-energy single-particle levels are occupied. By convention, the
A occupied single-particle states are called hole (h) states (occupation numbers equal to 1)
whereas the higher-energy empty states are called particle (p) states (occupation numbers
equal to 0).
Empty states (particle states)
Fermi surface
A occupied states (hole states)
If |ψ> is a Slater determinant, we define its associated one-body density matrix ρ (one can
show that there is a one-to-one correspondence between the Slater determinant and ρ) as:
ρ ≡ Ψ a +a Ψ ,
ij
j
i
(3.8)
The property tr ρ = A holds, where A is the total number of particles. In particular, owing to
the form of the Slater determinant |ψ>, Eq. (3.5), it holds ρij ≡ Ψ a +j ai Ψ = δ ij ni , where
ni is the occupation number of the state i (the density matrix is diagonal and the diagonal
matrix elements are equal to the occupation numbers). Hence, in the basis defined by the
operators a (i.e., the basis defined by the eigenstates ϕ of h), ρ is diagonal with eigenvalues
equal to 1 (occupied states) and 0 (empty states). This means that
ρ2 = ρ .
(3.9)
To apply the variational procedure, we impose that the energy must be stationary when the
density is modified (we use the fact that there is a one-to-one correspondence between the
Slater determinants and the one-body density matrices). We need to write the energy as a
function of the density:
E =
Ψ HΨ
= E [Ψ ] = E [ρ ] .
Ψ Ψ
(3.10)
The many-body Hamiltonian with a two-body interaction is written in second quantization as
51
H =
∑ tαβ cα+c β
+
αβ
1
∑Vαβγδ cα+c β+cδ cγ .
4 αβγδ
(3.11)
The matrix elements of the interaction are antisymmetrized, that is,
Vαβγδ = Vαβγδ − Vαβδγ .
(3.12)
Eq. (3.11) is equivalent to Eq. (2.9) because
Vαβγδ =
1
1
(
Vαβγδ + Vαβδγ ) + (Vαβγδ − Vαβδγ ).
2
2
(3.13)
In Eq. (3.13), the sum of the first two terms is symmetric for the exchange γ ↔ δ while the
sum of the last two terms is antisymmetric for the exchange γ ↔ δ. Since c+α c+β cδ cγ is antisymmetric for the exchange γ ↔ δ, the only part of Eq. (3.13) that contributes is the antisymmetric part (the product must be symmetric).
A theorem called Wick theorem (see, for example, David J. Rowe, Nuclear Collective
Motion) allows one to express the energy E as a functional of ρ (called energy density
functional or Hamiltonian density),
E [ρ] =
1
∑Vαβγδ Ψ cα+c β+cδ cγ Ψ
4
αβ
αβγδ
.
1
= ∑ t αβ ρ βα +
ρ
V
ρ
∑ γα αβγδ δβ
2 αβγδ
αβ
∑ t αβ
Ψ c α+c β Ψ +
(3.14)
In the HF basis, where ρ is diagonal with eigenvalues 0 and 1, one can write in particular the
HF energy,
E HF =
A
∑ t hh
h =1
+
1 A
∑Vhh' ,hh' .
2 h,h' =1
(3.15)
We impose now the variational principle. The variation on the density δρ is done by assuming
that the final density ρ+δρ is also associated to an uncorrelated Slater determinant. This is
equivalent to impose
(ρ + δρ )2 = ρ + δρ .
(3.16)
We also assume that the variation δρ is small. We thus take only the linear contribution in δρ,
ρ 2 + ρδρ + δρρ + (δρ )2 = ρ + δρ .
(3.17)
If in the left-hand side we neglect the quadratic term and if we use the property ρ2=ρ, we end
up with the equation
δρ = ρδρ + δρρ .
(3.18)
52
In the HF basis, where the density is diagonal, Eq. (3.18) is verified only if the hh and pp
elements of δρ are zero. One can easily see this property by writing:
(δρ )αα' = ∑ (ραβ δρ βα' ) + ∑ (δραβ ρ βα' ),
β
(3.19)
β
and by evaluating this expression for all kinds of matrix elements.
Hence, only the ph and hp elements of δρ are different from zero, that leads to
δE = E [ρ + δρ ] − E [ρ ] =
=
∂E [ρ ]
δρ βα
αβ ∂ρ βα
∑
∂E [ρ ]
∑ ∂ρ δρhp + c.c.
hp
ph
.
(3.20)
Let us introduce the quantity hαβ (derivative of the energy density functional with respect to
the density),
hαβ ≡
∂E [ρ ]
,
∂ρ βα
which defines the one-body HF effective Hamiltonian: h1 − body =
(3.21)
∑ h αβa α+ a β .
αβ
By considering the expression for the energy (density functional), Eq. (3.14), one can write
hαβ = tαβ + Γ αβ ,
(3.22)
where:
Γ αβ =
∑Vαbβa ρab .
(3.23)
αβ
If we impose δE=0 (where δE is given by Eq. (3.20)), since in general it is δρph, δρhp ≠0, it
must be hph=hhp=0 in the basis where ρ is diagonal: one says that h does not mix particle and
hole states of ρ. If the minimization condition δE=0 leads to hph=0, this means that h and ρ
can be diagonalized simultaneously,
[h, ρ ] = [t
+ Γ [ρ ], ρ ] = 0 .
(3.24)
Among the basis that diagonalize the density, one can choose the basis that diagonalize also h.
The minimization procedure becomes an eigenvalue problem,
hkk' = t kk' +
A
∑Vkik' i
i =1
= ε k δ kk' ,
(3.25)
53
where the quantities ε k correspond actually to the Lagrange multipliers associated to the
minimization. The above equations are the HF equations. If one makes a basis transformation,
one can write the equations in the coordinate representation. They read in this case
A
r
r rr
r
h2
−
Δϕk (r ) + ∑ ∫ d 3 r ' v (r , r ' )ϕ*j (r ' )
2m
.
j =1
r
r
r
r
r
ϕ j (r ' )ϕk (r ) − ϕ j (r )ϕk (r ' ) = ε k ϕk (r )
{
(3.26)
}
For simplicity, we omit the spin and isospin variables. Let us define the Hartree or direct
potential,
r
r rr A
r 2
r r r
r
Γ H (r ) ≡ ∫ d 3 r ' v (r , r ' ) ∑ ϕ j (r ' ) = ∫ d 3 r ' v (r , r ' )ρ (r ' ) ,
j =1
(3.27)
and the Fock or exchange potential,
rr
rr A
r
r
rr
rr
Γ EX (r , r ' ) ≡ −v (r , r ' ) ∑ ϕ*j (r ' )ϕ j (r ) = −v (r , r ' )ρ (r , r ' ) ,
j =1
(3.28)
which is a non-local term (to be noticed that the non-locality is related to the range of the
interaction).
The HF equations can thus be written as
 h 2
r 
r
r
rr
r
r
Δ + Γ H (r )ϕk (r ) + ∫ d 3 r ' Γ EX (r , r ' )ϕk (r ' ) = ε k ϕk (r ) .
−
 2m

(3.29)
The HF equations define a non-linear problem because the potentials depend on the density
which has to be constructed with the single-particle wave functions, solutions of the equations.
This non-linear problem is solved iteratively up to convergence when the self-consistent
solution is found.
In the second semester Lectures, the HF equations will be derived again and the energy
density functional, Eq. (3.14), will be shown for the case of the Skyrme zero-range
phenomenological interaction in nuclear physics.
3.2 Time-dependent Hartree-Fock theory
We consider a state |ψ(t)> at time t. The time evolution is governed by
ψ (t ) = e −
iHt/
h
ψ (0 ) ,
(3.30)
where |ψ(0)> is the state at t = 0. Let us define the associated one-body density at time t,
54
ρkl(t)= ψ (t ) c l+c k ψ (t ) .
(3.31)
To obtain an equation of motion, we calculate its time derivative,
ih ρ kl(t)= ψ (t ) c l+c k , H ψ (t ) .
⋅
[
]
(3.32)
For a two-body Hamiltonian this can be explicitly written as
ih ρ kl − ∑ (t kp ρ pl − ρkp t pl ) =
⋅
p
1
Vkprs ρrslp (2 ) − Vrslp ρkprs (2 ) ,
∑
2 prs
(
)
(3.33)
where in the right-hand side we have introduced the two-body density matrix
) ≡ ψ (t ) c +c +c c ψ (t ) .
ρklpq (2(t)
p q l k
(3.34)
We can derive an equation of motion for this quantity in the same way as we have done for
the one-body density. It connects the two-body density with the three-body density, and so on.
This system of equations is exact but not closed. To close it, we have to introduce some
approximations. Let us define the correlation function g(2) as the correlated part of the twobody density matrix, that is
ρklpq (2 ) = ρkp ρlq − ρkq ρ lp + g klpq (2 ) .
(3.35)
In the approximation where the correlated part g is equal to zero, the two-body density
factorizes and is expressed only by products of one-body densities. This means that the state
|ψ(t)> stays a Slater determinant for all times. In the approximation g = 0 it can be shown that,
starting from Eq. (3.33), one ends up with the time-dependent HF (TDHF) equation,
ih ρ kl = [h, ρ ]kl ,
⋅
(3.36)
where h is the same mean-field Hamiltonian as in the stationary HF equations, Eqs. (3.24).
3.3 Small-amplitude
approximation
excitation
modes:
the
random-phase
Let us first derive the RPA equations by following a procedure introduced by Rowe and
called equations-of-motion method.
Elementary excitations (particle-hole excitations) can be created by applying the pair of
operators a+pah to the HF ground state.
55
particle
a+pah
hole
The energy of these excitation modes is εp - εh where εi are the HF single-particle energies.
These excitations are individual or single-particle excitations and for this reason they cannot
describe collective excited states which correspond to the coherent motion of several particles
and which are observed in many-body systems like nuclei. These collective states can be
described by defining an excitation operator Q+n that creates the excited state |n> when it acts
on the ground state,
Qn+ gs = n .
(3.37)
In RPA this operator is written as a superposition of particle-hole elementary excitations,
Qn+ =
n a +a
∑ (X ph
p h
ph
n a +a ,
− Y ph
h p
)
(3.38)
where the p and h particle and hole indices are defined as usual with respect to the HF ground
state of the system. In the RPA approximation, the operator Qn+ acts on the RPA ground state
|RPA> to create the excited state |n>. The state |RPA> is defined as the vacuum for the
operators Qn
Qn RPA = 0 .
(3.39)
If one starts with the stationary Schroedinger equation,
H n = En n ,
(3.40)
one can write
HQn+ RPA = H n = E nQn+ RPA .
(3.41)
Thus,
HQn+ RPA − Qn+H RPA = (E n − E 0 )Qn+ RPA ,
(3.42)
where E0 is the ground-state energy. This can be written as
[H,Qn+ ] RPA = (E n − E 0 )Qn+ RPA .
(3.43)
Let us multiply on the left by an arbitrary state <RPA|δQ, and take the commutator,
RPA δQ, H,Qn+ RPA = (E n − E 0 ) RPA δQ,Qn+ RPA .
[ [
]
[
]
(3.44)
56
If we choose a special form of δQ, i.e., a linear combination of particle-hole configurations
(as for the operators Q), after some manipulations we can write the equations
RPA ah+a p , H, Qn+ RPA = (E n − E 0 ) RPA ah+a p , Qn+ RPA
.
RPA a p+ah , H, Qn+ RPA = (E n − E 0 ) RPA a p+ah , Qn+ RPA
[
[
[
[
]
]
[
[
]
]
(3.45)
The matrix elements in Eqs. (3.45) are very difficult to calculate because the Hamiltonian
contains also two-body terms. A simplification which is done in standard RPA is the so-called
quasiboson approximation (QBA), that corresponds in practice to replace the ground state
|RPA> (which in principle is correlated) with the uncorrelated HF ground state. It can be
shown that this approximation leads to some violations of the Pauli principle (see book of
Ring and Schuck, page 303).
Let us define the RPA matrices A and B,
+
Aphp' h' ≡ HF ah+a p , H, a p'
ah' HF = (ε p − ε h )δ pp' δ hh' + V ph' hp'
[
[
]]
B php' h' ≡ − HF ah a p , H, ah'+ a p' HF = V pp' hh'
[
+
[
]
.
(3.46)
By manipulating Eqs. (3.45) one can finally write the RPA equations. In their compact form
they read
 A
B *
B  X n  = (E − E )1
n
0 0
A * Y n 
0  X n  .
− 1 Y n 
(3.47)
The RPA equations look like a diagonalization in a basis with a metric tensor 1 0 which
0 −1
is not positive definite. A consequence of this is that the eigenvalues are not necessarily real
(when they are imaginary one says that RPA collapses). A theorem demonstrated by Thouless
says that, if the HF solution is a minimum in the energy surface (and not a saddle point or a
maximum), then the RPA equations provide only real frequencies. The opposite is not
necessarily true.
(
)
If in Eq. (3.38) one puts the amplitudes Y equal to zero, the so-called Tamm-Dancoff
approximation (TDA) is obtained. The corresponding equations are equal to Eqs. (3.47) with
Y = 0. This approximation corresponds to work with an uncorrelated ground state. Indeed,
when the ground state is the uncorrelated HF state, then the second term of Eq. (3.38) does
not contribute because ah+a p HF = 0 (particle states are empty in the HF state). This
explains why one usually says that the amplitudes Y are a kind of estimation of the presence
of correlations in the RPA scheme where the ground state is a priori correlated (even if the
QBA approximation is used to simplify the equations).
Energy Weighted Sum Rules
Sum rules are very important quantities often used to check the validity of the adopted
approximations. The Energy Weighted Sum Rules (EWSRs) are defined by the following
identity,
57
1
− E gs ) n F gs 2 =
gs [F, [H, F ]] gs .
(3.48)
2
n
where F is a hermitian single-particle operator. This relation holds for a set of exact
eigenstates |n> (with energies En) of H. When approximations are used for the eigenstates and
eigenvalues of H, the EWSRs are a test of the validity of these approximations.
Thouless showed that the EWSRs are satisfied if the left-hand side of Eq. (3.48) is evaluated
with the RPA wave functions and energies and the right-hand side is calculated with the HF
ground state.
An important aspect is the following: if the operator F corresponds to a symmetry of the
Hamiltonian, it commutes with H. This implies that the right-hand side of Eq. (3.48) is equal
to zero. Consequently, the excitation mode associated to the operator F must have excitation
energy equal to zero (this mode is called spurious mode). For example, the isoscalar (neutrons
and protons move together) dipole excitation in nuclei simply corresponds to a translation of
the center of mass and is predicted at excitation energy equal to zero (if the adopted
approximation method is good).
∑ (E n
Another very useful way to derive the RPA equations is done within the so-called linear
response theory. RPA equations are found as the small-amplitude limit of TDHF equations.
Let us consider the influence on the system of an external time-dependent field,
F (t ) = Fe − ω + F +e ω ,
it
it
(3.49)
where we assume that F is a one-body operator,
F (t ) =
∑ fαβ (t )aα+a β
.
(3.50)
αβ
We also assume that the field is weak and it introduces only small modifications of the
nuclear density, which we can treat in linear order. The nuclear density oscillates with the
external field and one obtains the resonances when the frequency corresponds to one
excitation energy of the system.
The equations of motion are
ih ρ = [h [ρ ] + f(t), ρ ] .
⋅
(3.51)
Since the external field is weak, it introduces only oscillations with small amplitude around
the stationary density ρ(0), which satisfies the stationary HF equations h ρ (0 ) , ρ (0 ) = 0 .
The density can thus be written as
[[ ]
ρ (t ) = ρ (0 ) + δρ (t ) ,
]
(3.52)
where
δρ (t ) = ρ ( )e −
1
itω
+ ρ ( )e
1
itω
(3.53)
58
is linear in the external field. Let us work in the HF basis (where ρ(0) and h[ρ(0)] are diagonal).
Thus,
0 particles
ρkl(0 ) = δ kl ρ k(0 ) = 
1
holes .

(
0)
hρ
kl = δ kl ε k
(3.54)
( [ ])
We have already shown that the condition ρ2 = ρ implies that only ph and hp matrix elements
of δρ are non-vanishing. We insert Eq. (3.52) in Eq. (3.51) and expand up to linear order,
⋅
 δh

ih δρ = h ρ (0 ) , δρ +  δρ, ρ (0 )  + f, ρ (0 ) ,
δρ

[[ ] ]
[
]
(3.55)
where
 ∂h

δh
∂h
δρ = ∑ 
δρ ph +
δρhp  .


δρ
∂ρhp
ph  ∂ρ ph ρ = ρ (0 )
ρ = ρ (0 )

(3.56)
It is possible to show that the pp and hh matrix elements of Eq. (3.55) are equal to zero (book
of Ring and Schuck). Using Eq. (3.53) one obtains the linear response equations for the ph
and hp elements, which can be written in the following compact form,
{(BA*
(1)ph
0  ρ
− 1  ρ(1)hp

B − hω 1
A*
0
)
(
)}

 ph 
 = − f hp  ,
f


(3.57)
where
Aphp' h' = (ε p − ε h )δ pp' δ hh' +
B php' h' =
∂h ph
∂h ph
∂ρ p' h'
.
(3.58)
∂ρhp
These matrices corresponds to the already introduced RPA matrices A and B if one takes
V psqr =
∂h pq
∂ρ rs
.
(3.59)
The linear response equations are inhomogeneous. By inverting the matrix on the left-hand
side one can write an expression for the density modification
(1 ) =
ραβ
∑ Rαβγδ (ω )fγδ .
(3.60)
γδ
59
The function R is called response function. It depends on the frequency of the external field. It
has poles at the excitation energies of the system where already an infinitesimal field is
sufficient to excite the modes. To find these resonances (ω = Ωn), the homogeneous equations
obtained from Eqs. (3.57) (assuming a vanishing external field) have to be solved. These
equations are identical to the RPA equations with hΩ n = E n − E 0 .
3.4 Examples
approaches
of
results
obtained
with
mean-field-based
The following results will be presented and commented during the Lectures.
60
61
62
63
64
Excitation modes.
Quadrupole mode. QRPA in particle-hole channel.
Response function for 22O
65
Two-neutron addition mode (pair transfer reaction). QRPA in
particle-particle channel.
Response function for 124Sn
66
Evolution of excitation modes in exotic neutron-riche nuclei.
PYGMY RESONANCES
67
67
68
69
70
Chapter 4
Marcella Grasso
4.1 Superfluidity in nuclear systems 72
4.2 Pairing correlations (Cooper pairs of nucleons) in the mean-field
framework: the Hartree-Fock-Bogoliubov method 73
4.3 Very exotic nuclear systems: neutron stars 76
71
4.1 Superfluidity in nuclear systems
The tendency of Fermi particles to pair has been recognized in several systems. It was
understood in atomic physics many years ago. Its importance in nuclear physics was also
recognized by Maria Göppert-Mayer (M.G. Mayer, Phys. Rev. 78, 16 (1950)) in 1950 to
explain why the ground state of even-even nuclei have spin zero while that of odd-mass nuclei
have the spin of the last unpaired nucleon. A major advance in the understanding of pairing
phenomena came with the theory of superconductivity. For some time the close similarity
between the properties of superfluid 4He and superconducting metals was a mystery. It was
understood that helium owed its superfluid properties to the bosonic nature of the 4He atom.
At the end of last century, the Bose-Einstein condensation (BEC) (superfluid phase of bosons)
has been observed experimentally for ultra-cold gases of bosonic atoms trapped optically
and/or magnetically. In Fig. 4.1, a picture of the BEC of a gas of 87Rb atoms is shown. The
typical temperatures were around 100 nK. For these experiments, the Nobel prize was
attributed in 2001 to Cornell, Ketterle, and Wieman.
But the electrons in a superconducting metal are fermions. The connection between
superfluidity of bosons and superconductivity of fermions was established by Cooper who
showed, in 1956, that fermions of opposite spin in a metallic lattice have an attractive
effective interaction and form correlated boson-like pairs, subsequently referred to as Cooper
pairs (Bardee, Cooper, Schrieffer, Phys. Rev. 108 (1957), 1175).
Figure 4.1 BEC of a gas of 87Rb ultra-cold atoms. The experiment has been done at the
JILA-NIST laboratory. The corresponding temperatures are 400 nK (left), 200 nK (middle),
and 50 nK (right).
In nuclei, the superfluid behavior is also due to the formation of Cooper pairs of fermions, the
nucleons. The formation of Cooper pairs composed by nucleons is a phenomenon which is
well-known in nuclei and confirmed by several experimental observations. This kind of
superfluidity manifests itself in many open-shell nuclei. Indeed, even if a conventional
interpretation of superfluidity is not really applicable to nuclei because these systems are
composed by a relatively small number of constituents, a typical superfluid behavior has been
found by some observations, for instance by moment-of-inertia measurements in rotating
deformed nuclei, the odd-even mass staggering along isotopic chains, and the excitation
spectra of even nuclei. An energy gap is found in the excitation spectra of even nuclei and it is
72
explained as due to the fact that Cooper pairs have to be broken (some energy has to be
provided to break them) before the nucleus can be excited. An illustration of the odd-even
mass staggering is plotted in Fig. 4.2 where the quantity that is displayed as a function of the
number of neutrons is the one-neutron separation energy Sn defined as Sn = E(N,Z) – E(N1,Z). E(N,Z) is the total binding energy (i.e., the mass) of the nucleus with N neutrons and Z
protons. Fig. 4.2 refers to No isotopes (Z = 102).
Figure 4.2 Odd-even mass staggering of No isotopes.
The oscillations are explained (at least partially) as due to the fact that even nuclei are more
bound (because of pairing correlations) than their neighbouring odd nuclei.
In the community, the question whether superfluidity in nuclei could be interpreted in an
analogous way as superconductivity in electronic systems is debated. Within this
interpretation, the exchange of phonons should be an important ingredient for the
comprehension of pairing correlations in nuclei (phonons in nuclei are the nuclear excited
states).
In the context of the mean-field theory, pairing correlations can be included by a simple
extension of the HF model through the introduction of the quasiparticles. When quasiparticles
are introduced, the number of particles is not conserved and this symmetry breaking allows
one to describe pairing correlations. The derivation of the corresponding HFB or Bogoliubovde Gennes equations may be found in different books (for example, de Gennes,
Superconductivity of Metals and Alloys; Ring and Shuck, The Nuclear Many-Body Problem).
4.2 Pairing correlations (Cooper pairs of nucleons) in the meanfield framework: the Hartree-Fock-Bogoliubov method
We sketch here the main points of a derivation of the HFB equations inspired by the book of
de Gennes. This derivation is developed for a simple case: a spherical system, two spin states
schematically indicated by ↑ and ↓, a central zero-range interaction acting only between
opposite spins. If we call V the strength of the interaction and if we suppose that the
interaction is well enough described by a two-body term, the Hamiltonian of the system is
given by the sum of a one-body term,
r
r
r
H1 = ∫ d 3 r ∑ Ψ + (r α )U1Ψ (r α ) ,
α
(4.1)
73
and a two-body term:
H2 =
r
r
r
r
r
1
V ∫ d 3 r ∑ Ψ + (r α )Ψ + (r β )Ψ (r β )Ψ (r α ) ,
2
αβ
(4.2)
where α and β represent the two spin states and U1 is the sum of the kinetic term and of an
eventual one-body external potential. In the spirit of the mean field, an effective one-body
Hamiltonian is introduced,
r
r
r
r
r
r
r
H eff = ∫ d 3 r ∑Ψ + (r α )H e( r )Ψ (r α ) + W (r )Ψ + (r α )Ψ (r α ) +
,
α
r + r
r
r
r
+ r
Δ(r )Ψ (r ↑ )Ψ (r ↓ ) + Δ * (r )Ψ (r ↓ )Ψ (r ↑ )
(4.3)
}
where He is defined by the relation
H1 − λN =
∑ ∫ d 3 r Ψ + (r α )H eΨ (r α ) ;
r
r
r
α
(4.4)
The operator N is called number of particles and is defined as
N ≡
∑ ∫ d 3 r Ψ + (r α )Ψ (r α ) ,
r
r
r
(4.5)
α
and λ is the chemical potential.
Since Heff is a quadratic form in the operators Ψ and Ψ+, it can be diagonalized by a unitary
transformation such as the Bogoliubov transformations:
r
Ψ (r ↑ ) =
∑ (u n (r )γ n ↑ − v *n (r )γ n+ ↓ )
r
Ψ (r ↓ ) =
∑ (u n (r )γ n ↓ + v *n (r )γ n+↑ )
r
n
r
r
r
.
(4.6)
n
These transformations define the quasiparticles. They diagonalize Heff, that means:
H eff = ε0 +
+
γ nα
∑ ε nγ nα
,
nα
(4.7)
where ε0 and εn are the energies of the ground state |0> and of the excited states |n>,
respectively. The previous equation may be alternatively written as:
[H eff ,γ nα ] = −ε nγ nα
.
(4.8)
[H eff ,γ nα+ ] = ε nγ nα+
If one calculates [Ψ,Heff] by using Eq. (4.3) for Heff and the anticommutation relations for the
operators Ψ, one obtains
74
[Ψ (rr ↑), H eff ] = (H e + W( rr))Ψ (rr ↑) + Δ(rr )Ψ + (rr ↓) .
[Ψ (rr ↓), H eff ] = (H e + W( rr))Ψ (rr ↓) − Δ * (rr )Ψ + (rr ↑)
(4.9)
If one uses now the Bogoliubov transformations and Eqs. (4.8), one obtains the HFB
equations:
r
r
r
r r
εu (r ) = [H e + W (r )]u (r ) + Δ(r )v (r )
r
r
r
r r .
εv (r ) = −[H * e +W (r )]v (r ) + Δ * (r )u (r )
(4.10)
In the above equations, W and ∆ are the mean field and the pairing field, respectively. They
may be derived by applying the variational principle. Let us impose that the free energy is
stationary:
0 = δF = δ H − Tδδ
(4.11)
,
where the mean value of H is defined as:
H ≡
∑φ
φ H φ exp (− βE φ )
∑φ exp (− βE φ )
,
β =
1
K BT
.
(4.12)
By evaluating <H>=<H1+H2> with the Wick theorem, one can write an expression for δF.
Then, if one supposes that also F1=<Heff>-TS (calculated with the states which diagonalize
Heff) is stationary, an expression for δF1 can be obtained. By comparing it with the expression
of δF, one concludes that the free energy is stationary if
r
r
r
r
r
W (r ) = V Ψ + (r ↑ )Ψ (r ↑ ) = V Ψ + (r ↓ )Ψ (r ↓ )
.
r
r
r
r
r
Δ(r ) = V Ψ (r ↓ )Ψ (r ↑ ) = −V Ψ (r ↑ )Ψ (r ↓ )
(4.13)
We can thus define the particle ( ρ ) and the pairing or anomalous ( ρ~ ) densities as follows
r
r
ρ (r ) ≡ Ψ + (r α )Ψ (rα ) =
∑ [(1 − f n )v n2 (r ) + f n u n2 (r )],
r
r
(4.14)
n
r
r
r
r
~
ρ( r ) ≡ Ψ (r ↓ )Ψ (r ↑ ) = −∑ (1 − 2f n )v *n ( r )u n ( r ) ,
[
]
(4.15)
n
where fn is a Fermi function providing the temperature dependence,
fn =
1
.
exp ( βε n ) + 1
(4.16)
In the nuclear case, these equations are usually used in the simple case of zero temperature.
However, the expressions for the interaction are much more complicated than the one used in
this simple derivation (see Sec. IV) leading to more complicated expressions for the potentials.
75
If one imposes that the pairing field couples only time-reversed states, a simplification of the
HFB equations is obtained corresponding to a simplified form of the Bogoliubov
transformations and leading to the model of Bardeen, Cooper and Schrieffer which has been
introduced in the 50s to describe Cooper pairs of electrons in superconductors. In nuclear
physics, this approximation of the HFB model is called HF+BCS.
It has been shown that RPA equations may be derived on top of HF equations. In an
analogous way, it is possible to derive the so-called quasiparticle RPA (QRPA) equations (an
extension of the RPA equations obtained by replacing particles with quasiparticles) as the
small-amplitude limit of time-dependent HFB (TDHFB) equations, where the ground state is
a Slater determinant of single-quasiparticle states. QRPA equations are a very useful tool to
describe the excitation spectra of open-shell nuclei (where pairing correlations are active).
4.3 Very exotic nuclear systems: neutron stars
Neutron stars, the most compact stellar objects after black holes, are the remnants of core
collapse supernovae. They are formed after the supernova explosion (end of life of a star) of
massive stars with typical masses ≥ 8 solar masses. The mass of a neutron star is about 1.4
solar masses and the radius is ≈ 10 km, which shows how much compact these objects are. In
Fig. 4.3, some measured masses of compact objects are illustrated (measurements are done in
binary systems).
76
Figure 4.3 Measured masses of compact objects (most of them are neutron stars).
In Fig. 4.4, a schema about the general structure of a neutron star is shown. The density
increases going from the surface to the interior of the star.
Figure 4.4 Schematic picture of the neutron star structure. See text for more details.
Different regions of a neutron star may be distinguished, an external ‘atmosphere’, a surface,
a crust (divided in outer and inner crust) and a core. The crust size is of some hundreds of m
(about 1 km).
Neutron stars are formed from matter that has been fully processed by nuclear combustion, so
that all available energy has been extracted. At zero pressure (surface) such matter is
composed of atoms of the most strongly bound nucleus, 56Fe, arranged as in ordinary solid
iron. At increasing density (going toward the interior of the star) a crystal structure still exists
but with different types of atoms. The atoms become progressively more ionized, and the
electrons fill the interstices. A Coulomb lattice arrangement of nuclei in the electron gas
minimizes the energy. As the density increases from that at the stellar surface, electrons
77
become increasingly relativistic. A lower energy state is achieved through the capture of
energetic electrons by nuclei (inverse beta decay). Any neutrinos or photons produced diffuse
out of the star, thus lowering its energy. With increasing density, nuclei become increasingly
neutron-rich by this neutronization process. Neutronization sets in at a density for which the
electron chemical potential equals the threshold for the electron capture reaction.
At higher density, ≈ 4 × 1011 g/cm3, the neutron drip is reached. The most weakly bound
neutrons drip out of nuclei, and a gas of neutrons and electrons occupies the interstices in
equilibrium with the nuclei. This drip density separates the outer crust from the inner crust. In
the inner crust, the nuclear systems that occupy the lattice sites are not strictly speaking nuclei.
These systems are beyond the drip lines, they would be unbound if isolated and can exist only
in this environment because of the gravitation.
To have a geometrical model of the crust region, Wigner-Seitz (WS) cells are used. WS cells
are geometrical structures commonly used to characterize solids (see Fig. 4.5). To construct a
WS cell, one draws the line segments that connect a lattice site with all the neighbouring sites.
The WS cell is the geometrical region delimitated by the planes that cross these line segments
by dividing them in two equal parts. The geometrical shape of a WS cell depends on the
geometry of the crystal.
Figure 4.5 Examples of Wigner-Seitz cells of crystals.
At still much higher density, above the saturation density of nuclear matter (2.51 × 1014 g/cm3
in these units) nuclei disassemble into a uniform charge-neutral mixture of baryons and
leptons. This is the superdense regime (the core of the star, which is called neutron star
matter). Nuclear matter is in its lowest energy state at each density under the constraint of
charge neutrality. Hyperons (baryons with one or two strange quarks) may form an important
part of the population of superdense regime. This regime, besides being rich in baryon species,
may even be composed of the baryon constituents (quarks) at high density (the central density
of a neutron star may be up to 1015 g/cm3). In this case stars are called quark or hybrid stars.
As previously illustrated, the baryonic density in the stars change very much from the surface
and the outer crust (sub-nuclear densities) up to the centre of the star where the density may
be one order of magnitude higher than the saturation density. To compute stellar properties,
we thus need an equation of state valid from the highest density down to the surface density.
This is a challenging task because equations of state may be easily adjusted (to reproduce
some properties of nuclei) only around the saturation density.
78
The physics of neutron stars is a multidisciplinary domain where nuclear physics plays a very
important role. The description of the neutron-star properties which are not directly linked to
the observations such as, for instance, the internal composition, requires the development of
theoretical models where the ingredients coming from nuclear physics are very important.
Furthermore, the comprehension of several properties of neutron stars may be obtained from
the study of the properties of atomic nuclei. In a natural way, the link between neutron stars
and nuclei is made via the nuclear matter: it turns out that asymmetric nuclear matter is rather
similar to the matter found in neutron stars, neutron star matter. Recently, more direct
relations between neutron-rich nuclei and neutron star matter have been proposed.
Isospin asymmetry is a common feature characterizing both neutron-rich nuclei and neutron
stars; this asymmetry can be ~ 0.95 in the interior of the star. The dependence of several
properties of nuclear systems on the isospin asymmetry may be related to the symmetry
energy, already defined in Chpater 1 (Sec. 1.5).
It turns out that the pressure of neutron matter (which is not so different from neutron star
matter) is related to ρ2∂Esym/ ∂ρ. The density dependence of the symmetry energy is thus very
important in determining all the properties associated to the pressure. Due to this, some
relations between neutron-rich nuclei and neutron-star properties have been found. To
mention some examples: (a) Lattimer and Prakash found the existence of correlations between
the neutron star radius and the pressure evaluated at densities above the equilibrium density;
(b) Typel and Brown pointed out the existence of correlations between the neutron skin
thickness in nuclei and the pressure of neutron matter evaluated at sub-nuclear densities; (c)
Horowitz and Piekarewicz found a relation between the neutron skin thickness in nuclei and
the neutron star radius.
What is shown in Fig. 4.6 is the calculated neutron skin thickness δR of nuclei as a function
of the radius of stars of mass equal to 1.4 solar masses (left panel) and as a function of the
radius of maximum-mass stars (right panel). The figure is extracted from a work of Steiner
and collaborators. Symbols represent results of mean-field calculations and provide thus an
example of applications of the mean-field theory used to calculate astrophysical quantities.
Figure 4.6
Calculated neutron skin thickness of nuclei versus the radius of stars of mass equal to 1.4
solar masses (left) and versus the radius of maximum-mass stars (right).
79
Another very direct link exists between nuclei and neutron stars: some exotic neutron-rich
nuclei produced in nuclear facilities are actually also located in the outer crust of neutron stars
surrounded by electrons (while the inner crust is composed by beyond-drip-line nuclear
systems surrounded by a neutron gas and by electrons). Thus, the comprehension of the
properties of these nuclei (for instance, the description of pairing correlations) has a direct
impact on the comprehension of neutron star properties.
Neutron stars evolve with time and new phenomena occur. Neutron stars do not produce
energy but lose the gravitational energy gained during the core collapse by neutrino emission.
This cooling determines the slowing down of the rotation motion in pulsars. Neutrinos and
anti-neutrinos mainly produced by beta decay and inverse beta decay carry out the energy of
the core. This is called the URCA 1 process. Actually, the URCA process is strongly
suppressed by energy and momentum conservation unless a minimum amount of protons,
around 11% of the baryonic density, is present. This minimal amount is strongly related to the
symmetry energy. It turns out that:
μe = μ μ = μ n − μ p = −∂E/∂x ≈ 4(1 − 2x)E sym ( ρ ) ,
(4.17)
where x = ρ p /ρ is the proton fraction (ρp is the proton density and ρ is the total density).
Eq. (4.17) is valid for beta-equilibrated matter, where µe, µµ, µn and µp are the electron, muon,
neutron and proton chemical potentials, respectively. Eq. (4.17), together with the charge
neutrality condition ρp = ρe + ρµ, allows determining the equilibrium proton fraction. This
shows how the proton fraction is related to the symmetry energy. However, URCA process is
too efficient to explain the cooling with time which is observed for several neutron stars. It
has been for instance suggested that superfluidity leading to the presence of a neutron pairing
gap may quench cooling from the URCA process. A modified URCA process is also
considered where adding an additional neutron as a spectator of the process allows
momentum and energy conservation. It should be noted that the specific heat of the crust is an
important ingredient in the cooling modelization. Specific heat actually depends on the
excitation spectrum (vibrations in the crust) which is different in the superfluid and in the
normal phases (pairing correlations would affect the specific heat and, consequently, the
cooling time).
Negele and Vautherin were the first, in the 70s, who applied mean-field calculations to the
crust of neutron stars. By modelling it with WS cells, they found the most favourable energy
configurations (they approximated the WS cells with spherical cells for simplicity; they found
the most favourable configuration at each baryonic density by varying the radius of the cell as
well as the numbers of neutrons and protons). For these configurations, they made the
approximation of treating each cell as an isolated many-body system and they made meanfield calculations to determine, for instance, the density profiles of protons and neutrons in the
cell. The found configurations are shown in Fig. 4.7 as a function of the baryonic density in
units of number of baryons per cm3. This figure is extracted from the article of Negele and
Vautherin of 1973. The neutron-rich nucleus 118Kr is just barely bound. The corresponding
baryonic density in this configuration is almost the drip density. Nuclei on the right with
respect to 118Kr are unbound (inner crust). For simplicity, we keep using the same convention
1
URCA is the name of a casino existing in the 50s in Rio de Janeiro, known by the promoter of this process,
Gamow. According to him, the efficiency of this casino in spoiling the money of gamblers was comparable to
this process in cooling down the stars.
80
as for real nuclei to denote these unbound nuclear systems. The number A is the total number
of protons plus neutrons in the WS cell.
The dashed line represents the energy of a beta-stable uniform gas of electrons, protons and
neutrons (one observes that the lattice configuration is energetically more favourable).
Figure 4.7
The configurations found by Negele and Vautherin for the crust of a neutron star.
In Fig. 4.8, radial proton and neutron density profiles calculated by Negele and Vautherin are
plotted for several WS cells.
Figure 4.8
Neutron and proton radial density profiles for some WS cells.
81
One observes the formation of a neutron gas between the crystal sites when the baryonic
density increases (from top to bottom).
The configurations found by Negele and Vautherin are still used nowadays but the
calculations that are presently done are more sophisticated (modern nuclear interactions are
used and pairing correlations are also included to take into account the superfludid properties
of the nucleons in the star).
On top of mean-field calculations for the ground state, QRPA calculations may also be done
to evaluate the excitation spectrum in each WS cell.
A typical example of excitation spectra calculated in an inner-crust WS cell is displayed in
Fig. 4.9. This is another example of applications of mean-field-based models to neutron star
physics: the vibrations in the crust are described with the HFB + QRPA approaches and the
quadrupole excitation spectrum shown in the figure is calculated for the WS cell having a
radius of 19.6 fm, a number of protons Z = 40 and a number of neutrons N = 1460. A very
collective low-energy excitation mode is found. Similar results have also been found for other
WS cells at different baryonic densities in the crust of the star.
In Fig. 4.10 the transition density associated to this collective low-lying mode is displayed. A
transition density shows the radial location of the particles that are mostly participating in the
excitation mode. Fig. 4.10 tells us that the strongest contribution to this excitation comes from
the free gas of neutrons located outside the central nuclear cluster of the WS cell with Z = 40
protons.
Figure 4.9
Neutron quadrupole QRPA strength distributions for the Wigner-Seitz cell with Z = 40 and N
= 1460 (inner crust of a neutron star)
82
Figure 4.10
Neutron transition density in units of fm−3 for the low-energy mode in the Wigner-Seitz cell
described in Fig. 4.9
The mass and the radius of a neutron star are determined by solving the hydrostatic
equilibrium equation. The equation of state is required for this. It is important noticing once
again that equations of state are typically constrained around the saturation density because
the observables used for fitting the parameters are measured in nuclei. In neutron stars, the
density ranges from subnuclear densities in the crust up to several times the saturation density
in the core. Furthermore, the isospin asymmetry can be as large as 0.95 in the interior of the
star. This means that the equations of state are used at densities and isospin asymmetries
which are very different from the values where the fits are done. This is a delicate point
related to the predictive power of the models. Several equations of state have been checked
(and some of them rejected). In Fig. 4.11 a mass-radius diagram for different equations of
state is shown.
Figure 4.11
Mass-radius diagram for different equations of state.
83
84
Chapter 5 (from Piet van Isacker (Ganil) notes)
Marcella Grasso
5.1 Utility of symmetries in quantum many-body physics 86
5.2 Some elements of group theory 87
5.3 Symmetries in quantum mechanics 95
5.4 Isospin symmetry in nuclei 97
85
5.1 Utility of symmetries in quantum many-body physics
The word ‘symmetry’ means ‘right correspondence of parts, quality, harmony, balance
between parts’. The word is derived from Greek (‘syn’: with; ‘metron’: measure).
Physicists often attempt to extract some regularities from observations of complex systems
and phenomena. When it is possible, it means that some symmetries are present in the laws of
physics that govern these systems and phenomena.
Symmetries in quantum many-body systems manifest themselves in the existence of some
regular behavior in the experimental results. They allow one a strong simplification in the
description of the system as well as a deeper comprehension of its properties. In many cases,
the type of symmetries that exists in a system is a very helpful guide in the choice of the most
suitable theoretical model to be employed.
Of course symmetries were already known and employed in classical mechanics. At that time,
however, all the used symmetries were of ‘geometrical’ type. For example, the physical laws
that govern an isolated system in classical physics are independent of the choice of the origin
and orientation of the coordinate system as well as of the choice of the origin of time. The
symmetries with respect to these coordinate transformations correspond to three conservation
laws: of momentum, of angular momentum and of energy.
In quantum mechanics the concept of symmetry is extended to transformations in abstract
spaces; these transformations are associated to intrinsic variables such as for instance the
isospin.
The mathematical tool that is adopted to formalize the existence of symmetries is the group
theory. Before starting with some elements of theory group, let us show two examples of
manifestation of symmetries in the properties of quantum systems. Degenerate spectra are
very often related to some symmetries. One example is the energy spectrum of the hydrogen
atom (Fig. 5.1),
Figure 5.1
Energy spectrum of the hydrogen atom.
where n = 1, 2, …; l = 0, 1, …, n – 1; m = -l, -l+1, …, +l; E n = −
me e 4
R
≡ − H ; RH is
2 h2 n 2
n2
the Rydberg constant.
Another example, that concerns nuclear physics, is the excitation spectrum in mirror nuclei
(mirror nuclei are nuclei where the number of neutrons and protons is interchanged). 49Cr (N
86
= 25, Z = 24) and 49Mn (N = 24, Z = 25) are mirror nuclei. The energy spectra are displayed
in Fig. 5.2, relative to the ground state of 49Cr. One observes that the symmetry concerns the
excitation energies (and not the absolute values) of the two nuclei. These two symmetries will
be analyzed more in detail in what follows.
49
Figure 5.2
Energy spectra of the mirror nuclei Cr and 49Mn relative to the ground state of the first
nucleus. Levels are labelled by their angular momentum and parity Jπ. The inset shows the
difference in excitation energy Ex(Cr; J) – Ex(Mn; J) as a function of 2J.
5.2 Some elements of group theory
Definition 1. A group G is a set of elements { Ĝ α } (where the index α enumerates the
elements of the set) together with an inner operation • called multiplication which satisfies:
- Closure: For all elements Ĝ α and Ĝβ belonging to G, the product Ĝ α • Ĝβ also
-
belongs to G;
Associativity: Every triplet of elements Ĝ α , Ĝβ and Ĝγ in G satisfies
Ĝ α •( Ĝβ • Ĝγ )=( Ĝ α • Ĝβ )• Ĝγ ;
-
Existence of the identity: The set G contains an element Ĝ 0 such that every element
Ĝ α in G satisfies Ĝ 0 • Ĝ α = Ĝ α • Ĝ 0 = Ĝ α ;
87
-
Existence of inverse: Every element Ĝ α in G has an inverse Ĝ α −1 such that
Ĝ α • Ĝ α −1 = Ĝ α −1 • Ĝ α = Ĝ 0 .
Note that the identity must be unique; also, a given element Ĝ α has a unique inverse.
Definition 2. A group G is called abelian if the following property is valid (in addition to
those enumerated in Definition 1):
- Commutativity: Every pair of elements Ĝ α and Ĝβ in G satisfies Ĝ α • Ĝβ = Ĝβ • Ĝ α .
The simplest example of groups consists of numbers with the ordinary multiplication as inner
operation.
Example. The group Sn of permutations of n objects. Given n objects which are labelled by 1,
2, …, n, any permutation of these objects can be represented as
1 2 ... n 
Pˆ = 
,
 p1 p 2 ... pn 
where p1, p2, …pn, is a reordering of the n objects. It can be verified that this set of n!
elements satisfies the group axioms. Let us consider the case of n = 3. One may introduce the
notation
1
Pˆ0 = 
1
2
2
3  ˆ 1 2
,P =
3  1 3 1
3 ˆ
1 2 3
, P2 = 

,
2
2 3 1 
1
Pˆ3 = 
1
2
3
3 ˆ
1 2 3 ˆ
1 2
, P4 = 
, P5 = 


2
3 2 1 
2 1
3
.
3 
Group multiplication is now easy to work out; for example,
1 2
Pˆ1 • Pˆ2 = 
3 1
3  1 2 3   2 3 1  1 2 3  1
•
=
•
=
2   2 3 1  1 2 3   2 3 1  1
2
2
3
= Pˆ0 .
3 
To obtain this result, we have permuted first the objects according to P̂2 and subsequently
according to P̂1 . The following multiplication table can be constructed from which all the
properties of the group can be deduced.
88
Example. The group C3h of rotations and reflexions of an equilateral triangle. With reference
to Fig. 5.3, three rotations and three reflexions can be identified which satisfy the condition of
invariance.
Figure 5.3
Rotations and reflexions of an equilateral triangle.
- Rˆ0 ≡ Eˆ : no rotation;
- Rˆ1 ≡ Cˆ3 : clockwise rotation about the z axis through 2π/3;
- Rˆ2 ≡ Cˆ32 : clockwise rotation about the z axis through 4π/3;
- Rˆ3 ≡ σˆ1 : reflection with respect to the plane through 1 and the z axis;
- Rˆ ≡ σˆ : reflection with respect to the plane through 2 and the z axis;
4
2
- Rˆ5 ≡ σˆ3 : reflection with respect to the plane through 3 and the z axis.
The combination of two operations defines their inner multiplication. For example,
1

Rˆ1 • Rˆ2  2 Δ3


 3
 = Rˆ1 1 Δ2


1

 = 2 Δ3 ,

which implies that Rˆ1 • Rˆ2 = Rˆ0 . The multiplication table thus constructed for C3h is
identical to the one for S3 if one makes the association Pˆ ↔ Rˆ . This example of groups
with identical multiplication tables leads us to the following definition of isomorphic groups.
i
i
Definition 3. Two groups are isomorphic if a one-to-one correspondence between their
elements exists that preserves the inner multiplication structure.
A very important type of groups is formed by the n × n matrices. Associativity is satisfied
through the laws of matrix multiplication. The set of matrices must be chosen such that
closure is assured; it should include the identity matrix and every matrix must have a non-zero
determinant such that the existence of its inverse is guaranteed.
Let us enumerate some types of groups of matrices.
89
Linear groups. No restriction is imposed on the form of the matrices. One distinguishes
between the linear groups GL(n,R) (real matrix elements) and the linear groups GL(n,C)
(complex matrix elements). If one imposes the condition that the determinant of every matrix
is equal to +1, the special linear groups SL(n,R) and SL(n,C) are obtained.
Unitary groups. If we call aij the elements of a generic matrix belonging to the group and if
one imposes that, for each matrix,
∑a
n
i
a
*
ij
ik
=δ
jk
,
=1
the group is called unitary. In unitary groups, matrices are thus unitary and |aij|2 ≤ 1. These
groups are denoted as U(n,C). Special versions of the unitary groups can be defined through
SU (n, C ) = U (n, C ) ∩ SL (n, C ) .
Orthogonal groups. The following condition must hold,
∑a
n
i
ij
a
=δ
ik
jk
=1
that is, the matrix must be orthogonal. These groups are denoted by O(n,C) (or O(n,R)). Note
that the unitary groups U(n,R) correspond to O(n,R).
The orthogonality condition implies that the square of the determinant of every matrix is
equal to 1 but it does not determine its sign. Consequently, O(n,C) consists of two
disconnected pieces with matrices with determinant +1 and -1, respectively. Again, special
versions of orthogonal groups are defined as
SO (n, C ) = O (n, C ) ∩ SL (n, C )
SO (n, R ) = O (n, R ) ∩ SL (n, R ) .
In the following, only unitary matrix groups in complex space and orthogonal matrix groups
in real space are considered. To simplify the notation, reference to space will be dropped;
hence:
(S )U(n)=(S)U(n, C)
(S )O(n)=(S)O(n, R).
-
If the number of its elements is finite, a group G is said to be finite.
If a group is finite, it is also discrete (one can enumerate its elements).
If its number of elements is infinite, but denumerable, the group is called infinite
discrete. Discrete groups constitute an important class of groups: for instance, they
classify the point symmetries of quantum mechanical systems such as crystals and
molecules.
90
-
A group is said to be continuous if its elements form a continuum in a topological
sense and if, in addition, the function which defines the multiplication operation over
the group has the appropriate continuity conditions. This leads to the definition of a
Lie group.
Definition 4. A Lie group is the component of a continuous group which is connected with the
identity element and for which the multiplication function is analytic.
The essential idea of a Lie group is that all its elements can be constructed in terms of those
close to the identity. This can be done only if each element and the identity can be joined by a
line, hence the condition of connectivity. In addition, a Taylor expansion of the multiplication
function around the identity is required which explains the condition of analyticity.
Example. The rotation group in two dimensions.
There exists two rotation groups in real two-dimensional space: O(2) and SO(2). The second
consists of rotations through an angle α while the first includes reflexion as well. In terms of
matrices, SO(2) has the elements
Gˆα ≡ cosα
− sinα
(
sinα ,
cosα
)
which all have determinant +1. It is clear that a continuous path can be defined from any
element Gˆα and the identity Gˆ0 . The complete orthogonal group O(2) contains in addition
the elements
(−cosα
sinα
sinα
- cosα
)
which evidently cannot be continuously connected to Gˆ0 . Thus, SO(2) is connected while
O(2) is not.
The elements of a Lie group are characterized by a set of r real variables α ≡ (α1 , α 2 ,..., α r )
with each αi defined over a certain domain. The multiplication function then becomes a multivalued function of multi-valued parameters Φ (α , β ) .
Let us now turn to the central result of the theory of Lie groups which attempts to interpret
properties of the infinite number of elements of a Lie group in terms of a finite number of
entities that are defined from elements infinitesimally close to the identity. In particular, the
multiplication rule for group elements is replaced by a commutation rule for those entities.
The analyticity of the multiplication function implies that a Taylor expansion around the
identity can be proposed and hence, for small ε ,
r
Gˆε ≈ Gˆ0 +
∑ εi
i
=1
∂Gˆα
∂α i
r
≡ Gˆ0 +
α
=0
∑ ε i gˆi
i
=1
where the definition of an (infinitesimal group) generator
91
gˆi ≡
∂Gˆα
∂α i α =0
has been introduced.
By calculating the commutators of two infinitesimal elements Gˆε and by using the analyticity
of the multiplication function, it is possible to find the following relation for the generators,
r
[gˆ , gˆ ] = ∑ c
i
j
k
k
ij
gˆk ,
(5.1)
=1
with
 ∂ 2 Φk (α , β ) ∂ 2 Φk (α , β )
.
c ijk = 
−

∂α j ∂βi 
 ∂α i ∂β j
α = β =0
(5.2)
This is a remarkable result: The entire structure of a Lie group can be expressed in terms of
commutation relations among the generators. These are finite in number, they form a closed
algebraic structure which can be formulated in terms of the coefficients c ijk , the structure
constants. The following identities hold,
[gˆ , gˆ ] = −[gˆ
i
[[gˆ , gˆ ], gˆ ] + [[gˆ
i
j
k
j
j
j
, gˆ ] ,
(5.3)
i
, gˆ ], gˆ ] + [[gˆ , gˆ ], gˆ
k
i
k
i
j
]=0 .
(5.4)
These identities lead to the following properties for the structure constants,
c ijk = −c kji ,
(5.5)
and
r
l
∑ (c ijm c mk
l
m l
+ cm
jk c mi + c ki c mj = 0 .
)
(5.6)
m =1
In summary, we have shown that with each Lie group is associated a Lie algebra, defined (up
to a change of basis) by its structure constants which should satisfy the relations (5.5) and
(5.6). The correspondence works also in the other direction: It can be shown that a Lie algebra
uniquely defines the associated Lie group.
For most applications in physics, symmetry properties can be cast in terms of a Lie algebra
rather that its associated group.
Example. The generator of SO(2).
This one-parameter Lie group gives rise to a single generator. For small values of the rotation
angle, the transformation matrix becomes
92
Gˆε = cosε
− sinε
(
sinε ≈ 1
cosε
−ε
ε =1
0
1
) ( 10 ) + ε (−01 10 ),
) (
which shows that the generator of SO(2) is
gˆ = 0 1 .
−1 0
)
(
Since the algebra contains a single element, its single structure constant is equal to zero.
Example. The generators of SU(2).
Let us denote a general 2 × 2 matrix close to the identity as
ε12 
1 + ε11
,
 ε
1 + ε 22 
 21
where εij are infinitesimal complex numbers. Unitarity of the matrix imposes the condition
ε12
1 + ε11
 ε
1 + ε 22
 21
*
ε*21
1 + ε11
 *
 ε12
1 + ε*22
 1
=
 0

( 10 )
which, by expanding up to first order in the εij, implies the relations
*
ε11 + ε11
= ε 22 + ε*22 = ε12 + ε*21 = 0
An additional condition is found from the special character of the matrix (determinant equal
to +1),
ε11 + ε 22 = 0
A special unitary 2 × 2 matrix close to the identity can thus be parameterized as
1
 1 − 1 iε
− i(ε1 − iε 2

3
2
2

1
1
 − i(ε1 + iε 2 )
1 + iε3
 2
2
)



which includes a conventional -1/2i factor and where the εi are infinitesimal real numbers.
This can be rewritten in terms of the Pauli spin matrices σi as
(10 10 ) − 12 iε1 (10 10 ) − 12 iε 2 (0i
With the identification gˆ = −
i
− i − 1 iε 1
30
0
2
)
(
0 .
−1
)
1
iσ , the commutation relations for the generators are
2
i
3
[gˆi , gˆ j ] = ∑ ε ijk gˆk .
k =1
93
Definition 5. Two algebras G and G’ are isomorphic, G ≈ G’, if a one-to-one correspondence
between their generators can be established that preserves their commutation relations.
Two Lie groups that are isomorphic necessarily define isomorphic Lie algebras. The converse
is not valid in general: the Lie algebras SO(3) and SU(2) are identical but the Lie groups are
not (there exists a two-to-one correspondence between the Lie groups SO(3) and SU(2)).
r
Definition 6. A Lie algebra G satisfying the relations [gˆi , gˆ j ] =
∑ c ijk gˆk
k
has an associated
=1
r
metric tensor defined as g ij =
∑ c ikl c kjl
k, l
.
=1
The metric tensor is symmetric gij=gji. Note that gij (which will be denoted in matrix form as
G) is not an invariant of the Lie algebra but it is modified by a change of basis. Its
determinant, however, is an invariant and, in addition, we have the following definition.
Definition 7. A Lie algebra G is semi-simple if and only if it has a non-singular metric tensor.
The metric tensor of a semi-simple Lie algebra is non-singular and hence its inverse exists.
Given the elements gij of a non-singular metric tensor G, the elements of its inverse G-1 shall
be denoted as gij. Hence
r
∑ g ij g
j
jk
r
=
=1
∑ g ij g jk
j
= δ ik .
=1
Since gij is symmetric, so is gij.
Definition 8. Given a semi-simple Lie algebra G with structure constants c ijk and an inverse
metric tensor gij, the operator
Cˆm [G ] =
r
∑
r
∑
1 2 ,..., i m =1 j1 , j 2 ,..., j m =1
i ,i
j
j
j
c i 2j c i 3 j ...c i 1
11 2 2
m m
j
gˆ i1 gˆ i 2 ...gˆ i m
with
gˆ i =
r
∑ g ij gˆ j
j
,
=1
is called the Casimir operator of G of order m.
94
5.3 Symmetries in quantum mechanics
No distinction is made in the following between geometric and quantum-mechanical
transformations: all elements ĝ will be taken as operators acting on the Hilbert space of
quantum-mechanical states.
A time-independent Hamiltonian Hˆ which commutes with the generators ĝ that form a Lie
algebra G,
i
i
∀gˆ ∈ G : Hˆ, gˆ = 0 ,
[
i
i
]
(5.7)
is said to have a symmetry G or, alternatively, to be invariant under G.
The determination of operators ĝ that leave invariant the Hamiltonian of a given physical
system is central to any quantum-mechanical description. The reasons for this are profound
and can be understood from the correspondence between geometrical and quantummechanical transformations. It can be shown that the transformations ĝ with the symmetry
property (5.7) are induced by geometrical transformations that leave unchanged the
corresponding classical Hamiltonian. In this way, the classical notion of a conserved quantity
is transcribed in quantum mechanics in the form of the symmetry property (5.7) of the timeindependent Hamiltonian.
A well-known consequence of a symmetry is the occurrence of degeneracies in the
eigenspectrum of the Hamiltonian.
i
i
Example. The hydrogen atom.
The Hamiltonian for a particle of charge –e and mass me in a Coulomb potential e/r is given
by
p2
e2
h2
e2
HˆH =
−
=−
∇2 −
.
2me
r
2me
r
(5.8)
This we take as a model Hamiltonian for the hydrogen atom which is exact if the spin-orbit
interaction is neglected. In this approximation, the Hamiltonian is independent of the spin of
the electron which leads to a two-fold degeneracy of all states corresponding to spin-up and
spin-down. In the following we shall ignore electron spin and study the symmetry properties
of the orbital (or spatial) part of the electron wave function.
r
r
The solutions of the associated Schroedinger equation, Hˆ Φ (r ) = EΦ (r ) , are well known
H
from standard quantum mechanics. The energies of the stationary states are
En = −
me e 4
R
≡− H ,
2 h2 n 2
n2
(5.9)
where RH is the Rydberg constant and n the so-called principal quantum number. The electron
wave functions (ignoring spin) are
Φ
nlm
(r, θ, ϕ) = R (r )Y (θ, ϕ) ,
nl
lm
(5.10)
95
with R and Y known functions. The Y (θ, ϕ) are spherical harmonics which are identical to
those occurring in the problem of the harmonic oscillator (and in any problem of a central
potential with spherical symmetry). The Rnl are of course different from the radial functions
r
r
of the harmonic oscillator. The solution of the differential equation Hˆ Φ (r ) = EΦ (r ) also
lm
H
leads to the conditions
n = 1, 2,…, l = 0, 1, …, n-1, m = -l, -l+1,…, +l.
The energy spectrum of the hydrogen atom has been shown in Fig. 5.1. The energy
eigenvalues En only depend on n and not on l or m. A given level with energy En is thus n2fold degenerate since
n −1
∑ (2l
l =0
+1 ) = n 2 .
Using symmetry arguments it is possible to explain the nature of this degeneracy. In addition,
it is possible to show that the entire spectrum can be determined with algebraic methods
without recourse to boundary conditions of differential equations.
We are going to explain part of the observed degeneracy, namely, the fact that levels with a
given l are (2l + 1)-fold degenerate. First, we note that the Hamiltonian of the hydrogen atom
is rotationally (or SO(3)) invariant. This is obvious on intuitive grounds: the properties of the
hydrogen atom do not change under rotation. Formally, it follows from the following
commutation property,
[HˆH , Lˆ] = 0 ,
where L̂ is the angular momentum operator, Lˆ = r × p = −ih(r × ∇ ) . It is of interest to look
more closely at the origin of the vanishing commutator between the Hamiltonian and L̂ . The
Hamiltonian of the hydrogen atom consists of two parts, kinetic and potential, and it can be
easily seen that both commute with L̂ because
[∇2 , Lˆ] = 0 [r −1 , Lˆ] = 0 .
Since the components of L̂ form an SO(3) algebra,
3
[Lˆk , Lˆl ] = ih ∑ ε klm Lˆm ,
m =1
and since L̂ commutes with the Hamiltonian, we conclude that the Hamiltonian of the
hydrogen atom has an SO(3) symmetry. This explains the degeneracy 2l+1 of each level with
a given l. But the degeneracies observed in the energy spectrum are higher than what obtained
from just rotational invariance. This means that a larger symmetry group exists (that actually
is SO(4)), but we do not enter in these details.
96
5.4 Isospin symmetry in nuclei
After the discovery of the neutron by Chadwick, Heisenberg realised that the mathematical
apparatus of the Pauli spin matrices could be applied to the labelling of the two nucleonic
charge states, the neutron and the proton. In this way he laid the foundation of an important
development in physics, namely the use of symmetry transformations in abstract spaces. The
starting point is the observation that the masses of the neutron and proton are very similar,
mnc2 = 939.55 MeV and mpc2 = 938.26 MeV, and that both have a spin of ½. Furthermore,
experiment shows that, if one neglects the contribution of the electromagnetic interaction, the
forces between two neutrons are about the same as those between two protons. More precisely,
the strong nuclear force between two nucleons with anti-parallel spin is found to be
(approximately) independent of whether they are neutrons or protons. This indicates the
existence of a symmetry of the strong interaction, and isospin is the appropriate formalism to
explore the consequences of that symmetry in nuclei. We stress that the equality of the masses
and the spins of the nucleons is not sufficient for isospin symmetry to be valid and that the
charge independence of the nuclear force is equally important. This point was emphasised by
Wigner who defined isospin for complex nuclei as we know it today and who also coined the
name of ‘isotopic spin’.
Because of the near-equality of the masses and of the interactions between nucleons, the
Hamiltonian of the nucleus is (approximately) invariant with respect to transformations
between neutron and proton states. For one nucleon, these can be defined by introducing the
abstract space spanned by the two vectors
n = 1 ,
0
()
p = 0.
1
()
The most general transformation among these states (which conserves their normalisation) is
a unitary 2 × 2 matrix. If we represent a matrix close to the identity as
ε12 
1 + ε11
 ε
1
+
ε 22 
 21
where the εij are infinitesimal complex numbers, unitarity imposes the relations
*
ε11 + ε11
= ε 22 + ε*22 = ε12 + ε*21 = 0 .
An additional condition is found by requiring the determinant of the unitary matrix to be equal
to +1,
ε11 + ε 22 = 0 ,
which removes the freedom to make a simultaneous and identical change of phase for the
neutron and the proton. We conclude that an infinitesimal, physical transformation between a
neutron and a proton can be parameterised as
97
1
1

− i(ε x − iε y
1 − iε z

2
2

1
1
 − i(ε x + iε y )
1 + iε z
 2
2
)
,


which includes a conventional factor –i/2 and where the {εx, εy, εz} are infinitesimal real
numbers. This can be rewritten in terms of the Pauli spin matrices as
(10 10 ) − 12 iε (10 10 ) − 12 iε (0i
x
y
− i − 1 iε 1
z 0
0
2
)
0 .
−1
)
(
The infinitesimal transformations between a neutron and a proton can thus be written in terms
of the three operators
t̂ x ≡
1 0
2 1
( 10 ),
t̂ y ≡
1 0
2 i
(
- i , tˆ ≡ 1 1
z
0
2 0
)
(
0 ,
-1
)
which satisfy exactly the same commutation relations which are valid for the angular
momentum operator. The action of the tˆμ operators on a nucleon state is easily found from its
matrix representation. For example,
1
tˆz n ≡ 1
2 0
(
0 1 =1 n ,
−1 0
2
1
tˆz p ≡ 1
2 0
)( )
0 0 = −1 p ,
−1 1
2
)( )
(
which shows that e 1 − 2tˆz /2 is the charge operator. Also, the combinations tˆ± ≡ tˆx ± itˆy
can be introduced, which satisfy the commutation relations
(
)
[tˆz , tˆ± ] = ±tˆ± , [tˆ+ , tˆ− ] = 2tˆz ,
and play the role of raising and lowering operators since
tˆ− n = p ,
tˆ+ n = 0 ,
tˆ− p = 0 ,
tˆ+ p = n .
This proves the formal equivalence between spin and isospin, and all results familiar from
angular momentum can now be readily transported to the isospin algebra. For a many-nucleon
system (such as a nucleus) a total isospin T and its z projection MT can be defined which
results from the coupling of the individual isospins, just as this can be done for the nucleon
spins. The appropriate isospin operators are
Tˆμ =
∑ tˆμ (k ) ,
k
where the sum is over all the nucleons in the nucleus.
If, in first approximation, the Coulomb interaction between the protons is neglected and,
furthermore, if it is assumed that the strong interaction does not distinguish between neutrons
98
and protons, the resulting nuclear Hamiltonian is isospin invariant. Explicitly, invariance
under the isospin algebra SU(2)≡ Tˆ ,Tˆ follows from
{ z ±}
[Hˆ,Tˆz ] = [Hˆ,Tˆ± ] = 0 .
As a consequence of these commutation relations, the many-particle eigenstates of the
Hamiltonian have good isospin symmetry. They can be classified as ηTMT , where T is the
total isospin of the nucleus obtained from the coupling of the individual isospins of all
nucleons, MT is its projection on the z axis in isospin space, MT=(N-Z)/2, and η denotes all
additional quantum numbers.
If isospin was a true symmetry, all states ηTMT with MT = -T,-T+1,…, +T, and with the
same T (and identical other quantum numbers), would be degenerate in energy; for example,
neutron and proton would have exactly the same mass. States with the same η T but different
MT (and hence in different nuclei) are referred to as isobaric analogue states.
The assumption of exact isospin symmetry is actually too strong. Thus one cannot expect that
isobaric analogue states have the same absolute energy. But, to a good approximation, they
have the same relative energies. As a result, the excitation spectra of two mirror nuclei should
be the same although the binding energy of their ground stats differs. This relation has been
observed in many cases. See Fig. 5.2 for an illustration for the nuclei 49Cr and 49Mn. The
spectra are not identical, as is clear from the inset of the figure where the difference in
excitation energy is plotted as a function of the angular momentum. The deviations from zero
signal a breakdown of the symmetry.
99
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100