Transcript The simple shell model and ab initio calculations

The Nuclear Shell Model –
Past and Present
Igal Talmi
The Weizmann Institute of Science
Rehovot Israel
The main success of the
nuclear shell model
• Nuclei with “magic numbers” of protons and
neutrons exhibit extra stability.
• In the shell model these are interpreted as the
numbers of protons and neutrons in closed
shells, where nucleons are moving
independently in closely lying orbits in a
common central potential well.
• In 1949, Maria G.Mayer and independently
Haxel, Jensen and Suess introduced a strong
spin-orbit interaction. This gave rise to the
observed magic numbers 28, 50, 82 and 126 as
well as to 8 and 20 which were easier to accept.
The need of an effective interaction
in the shell model
• In the Mayer-Jensen shell model, wave functions
of magic nuclei are well determined. So are
states with one valence nucleon or hole.
• States with several valence nucleons are
degenerate in the single nucleon Hamiltonian.
Mutual interactions remove degeneracies and
determine wave functions and energies of
states.
• In the early days, the rather mild potentials used
for the interaction between free nucleons, were
used in shell model calculations. The results
were only qualitative at best.
Theoretical derivation of the
effective interaction
A litle later, the bare interaction turned out to be too
singular for use with shell model wave functions. It
should be renormalized to obtain the effective
interaction. More than 50 years ago, Brueckner
introduced the G-matrix and was followed by many
authors who refined the nuclear many-body theory for
application to finite nuclei.
• Starting from the shell model the aim is to calculate from
the bare interaction the effective interaction between
valence nucleons. Also other operators like
electromagnetic moments should be renormalized (e.g.
to obtain the neutron effective charge!)
• Only recently this effort seems to yield some reliable
results. This does not solve the major problem - how
independent nucleon motion can be reconciled with the
strong and short ranged bare interaction.
The simple shell model
• In the absence of reliable theoretical
calculations, matrix elements of the effective
interaction were determined from experimental
data in a consistent way.
• Restriction to two-body interactions leads to
matrix elements between n-nucleon states which
are linear combinations of two-nucleon matrix
elements.
• Nuclear energies may be calculated by using a
smaller set of two-nucleon matrix elements
determined consistently from experimental data.
The first successful calculation –
low lying levels of 40K and 38Cl
• In the simplest shell model configurations of these nuclei, the 12
neutrons outside the 16O core, completely fill the 1d5/2, 2s1/2, 1d3/2
orbits while the proton 1d5/2 and 2s1/2 orbits are also closed.
• The valence nucleons are
in 38Cl: one 1d3/2 proton and a 1f7/2 neutron
in 40K: (1d3/2)3Jp=3/2 proton configuration and
a 1f7/2 neutron.
• In each nucleus there are states with J=2, 3, 4, 5
• Using levels of 38Cl, the 40K levels may be calculated and vice versa.
Comparison with 1954 data
only the spin 2- agreed with our prediction
We were disappointed but not
surprized. Why?
• The assumption that the states belong to rather
pure jj-coupling configurations may have been
far fetched. Also the restriction to two-body
interactions could not be justified a priori.
• There was no evidence that values of matrix
elements do not appreciably change when going
from one nucleus to the next.
• Naturally, we did not publish our results
but then…
Comparison of our
predictions with
experiment in 1955
Conclusions from the relation
between the 38Cl and 40K levels
• The restriction to two-body effective interaction
is in good agreement with some experimental data.
• Matrix elements (or differences) do not change
appreciably when going from one nucleus to its
neighbors (Nature has been kind to us).
• Some shell model configurations in nuclei are very
simple. It may be stated that Z=16 is a proton magic
number (as long as the neutron number is N=21).
Effective interactions, no longer restricted by
the bare interactions, have been adopted
• This was the first successful calculation and it
was followed by more complicated ones carried
out in the same way.
• The complete p-shell, p3/2 and p1/2 orbits (Cohen
and Kurath), Zr isotopes (mostly the Argonne
group), the complete d5/2,d3/2,s1/2 shell
(Wildenthal, Alex Brown et al) and others.
• This series culminated in more detailed
calculations including millions of shell model
states, with only two-body forces (Strasbourg
and Tokyo). Not all matrix elements determined.
• Only simple examples will be shown.
General features of the effective
interaction extracted even from
simple cases
• The T=1 interaction is strong and attractive
in J=0 states.
• The T=1 interactions in other states are weak
and their average is repulsive.
•
It leads to a seniority type spectrum.
• The average T=0 interaction, between protons
and neutrons, is strong and attractive.
•
It breaks seniority in a major way.
Consequences of these features
• The potential well of the shell model is created
by the attractive proton-neutron interaction
which determines its depth and its shape.
• Hence, energies of proton orbits are determined
by the occupation numbers of neutrons and vice versa.
• These conclusions were published in 1960 addressing
11Be, and in more detail in a review article in 1962. It will
be considered later but first let us look at identical
valence nucleons.
.
Seniority in jn Configurations
• The seniority scheme is the set of states which
diagonalizes the pairing interaction
VJ=<j2JM|V|j2JM>=0
unless J=0, M=0
• States of identical nucleons are characterized by v, the
seniority which in a certain sense, is the number of
unpaired particles. In states in with seniority v in the
jv configuration, no two particles are coupled to states
with J=0, i.e. zero pairing. From such a state, with given
value of J, a seniority v state may be obtained by adding
(n-v)/2 pairs with J=0 and antisymmetrizing. The state, in
the jn configuration thus obtained, has the same value
of J.
Seniority in Nuclei
Seniority was introduced by Racah in 1943
for classifying states of electrons in atoms.
There is strong attraction in T=1, J=0 states
of nucleons. Hence, seniority is much more
useful in nuclei.
The lower the seniority the higher the
amount of pairing.
Ground states of semi-magic nuclei are
expected to have lowest seniorities, v=0 in
even nuclei and v=1 in odd ones.
Seniority in jn configurations of
identical nucleons
This is the theory of J=0 pairing
S†=½Σ(−1)j-majm†aj,−m†
Acting on the state with no nucleons
creates a J=0 pair, S=(S†)† annihilates it.
HS†|0>=V0S†|0>
Seniority v is defined by S|jvvJ>=0.
In j n configuration, seniority v states are
(S†)(n−v)/2|jvvJ>
These states are eigenstates of the
pairing interaction 2S†S
A single nucleon state is ajm†|0>. It has seniority v=1.
If H is a two-body interaction,
Hajm†|0>=0. This state with n=3 is S†ajm†|0>.
If H is diagonal in seniority
HS† ajm†|0>=[H,S†]a jm†|0>+ S†Hajm†|0>=
[[H,S†],a jm†]|0>+ajm†[H,S†|0>=
[[H,S†],ajm†]|0>+a jm†HS†|0>=
[[H,S†],ajm†]|0>+ajm†V0 S†|0>
If it is an eigenstate then
[[H,S†],ajm†]|0>=½WS†ajm†|0>
Actually, it is an operator equation
[[H,S†],ajm†]=½WS†a jm†
From this follows
[[H,S†],S†]=W(S†)2
States with seniority v=0. with maximum pairing are given by (S†)n|0>.
A useful
lemmalemma
A useful
B(S†)N|1>=(S†)NB|1>+N(S†)N−1[B,S†]|1>+
½N(N−1)(S†)N−2[[B,S†],S†]|1>+…
If B is a sum of one body and two-body operators, the
series expansion stops.
H(S†)(n−v)/2|jvvJ>=(S†)(n−v)/2H|jvvJ>+
½(n−v)(S†)(n−v)/ 2−1[H,S†]|jvvJ>+
(1/8)(n−v)(n−v−2)(S†)(n−v−4)/2[[H,S†],S†]| jvvJ>
The energy of the jv state is denoted by
H|jvvJ>=E(jvvJ)|jvvJ>
For [H,S†]|jvvJ>, |jvvJ> is a linear
combination of products of v creation
operators ajm† with various m-values,
denoted by C†, acting on |0>. Commuting
[H,S†] through the ajm† the result is
[H,S†]|jvvJ>= [H,S†]C†|0>=½WvC†S†|0>+
C† [H,S†]|0>={½Wv+V0}S†|jvvJ>
Eigenvalues of H diagonal in seniority
Collecting all terms, the result is
H(S†)(n−v)/2|jvvJ>=E(jvvJ)(S†)(n−v)/2|jvvJ>+
{n(n−1)/2−v(v−1)/2}(W/4)(S†)(n−v)/2|jvvJ>+
½(n−v)(V0−W/4)(S†)(n−v)/2|jvvJ>
Thus, the double commutator conditions
make the states above, eigenstates of H.
Defining α=W/4 β= V0−W/4 the
eigenvalues are equal to
E(jvvJ)+α{n(n−1)/2−v(v−1)/2}+ ½(n−v)β
Seniority binding energy formula
For ground state energies of semi-magic nuclei,
v=0 states of even-even nuclei and v=1 states of
odd-even ones, these expressions are simpler.
Using them, binding energies are expressed by
BE(jn)=BE(n=0)+nCj+αn(n−1)/2+½(n−v)β=
BE(n=0)+nCj+αn(n−1)/2+[n/2]β
A linear term, a quadratic term and a pairing term.
Properties of the seniority scheme
ground states
• If a two-body interaction is diagonal in the
seniority scheme, its eigenvalues in states with
v=0 and v=1 are given by
αn(n-1)/2 + β(n-v)/2
where α and β are linear combinations of twobody energies VJ=<j2J|V|j2J>.
This leads to a binding energy expression (mass
formula) for semi-magic nuclei
B.E.(jn)=B.E.(n=0)+nCj+αn(n-1)/2+ β(n-v)/2
It includes a linear, quadratic and pairing terms.
Neutron separation energies
from calcium isotopes
A direct result of this behavior is
that magic nuclei are not more
tightly bound than their preceding
even-even neighbors.
Their magic properties (stability etc.) are
due to the fact that nuclei beyond them
are less tightly bound.
Seniority: level spacings
independent of n
From the expression of eigenvalues
E(jvvJ)+α{n(n−1)/2−v(v−1)/2}+ ½(n−v)β
follows an important feature of seniority.
The derivation holds also for states with
seniority v’<v, it uses only the number v of
particles. Hence, energies of |j nv’J> states
with v’≤v are equal to energies of |jvv’J>
states plus the same energy. Their
differences are equal.
Properties of the seniority scheme
excited states
• If the two-body interaction is diagonal in
the seniority scheme in jn configurations,
then level spacings are constant,
independent of n.
Levels of even
(1h11/2)n configurations
Levels of odd
(1h11/2)n configurations
Properties of the seniority scheme
moments and transitions
• Odd tensor operators are diagonal and
their matrix elements in jn configurations
are independent of n.
• Matrix elements of even tensor operators
are functions of n and seniorities. Between
states with the same v, matrix elements
are equal to those for n=v multiplied by
(2j+1-2n)/(2j+1-2v)
Simple results follow for E2 transtions.
E2 matrix elements in
(1h11/2)n configurations
B(E2) values in even
(1h9/2)n configurations
• The rates of E2 transitions usually
increase as more valence nucleons are
added. Here, the opposite behavior is
evident.
Generalized seniority
There are semi-magic nuclei where there
is evidence that valence nucleons occupy
several orbits. Still, they exhibit seniority
features, in binding energies and constant
0 - 2 spacings. A generalization of seniority
to mixed configurations is needed.
Generalized seniority
Pair creation operators in different orbits
Sj’†, Sj”†,… may be added to S†=Σ Sj† which
has, with its hermitean conjugate the same
commutation relations as in a single j-orbit,
with 2j+1 replaced by 2Ω=Σ(2j+1). A more
realistic pair creation operator is
S†=ΣαjSj†
If not all coefficients are equal the
algebra is more difficult. Still,
Generalized seniority:ground states
There are states with seniority-like features.
Assume that (S†)N|0> are exact eigenstates
H(S†)N|0>=EN(S†)N|0>
HS†|0>=VS†|0>
In fact, using the lemma, it follows,
H(S†)N|0>=(S†)NH|0>+N(S†)N−1[H,S†]|0>+
½N(N−1)(S†)N−2[[H,S†],S†]|0>=
{NV+½N(N−1)W}(S†)N|0>
Generalized seniority mass formula
Here, V is the (lowest) eigenvalue of the two
nucleon J=0 states. It is obtained by
diagonalization of the Hamiltonian matrix
for N=2, J=0, including single nucleon
energies and two-body interactions. The
formula for binding energies of even-even
nuclei is
BE(N)=BE(N=0)+NV+½N(N−1)W
Two-nucleon separation energies should lie
on a straight line.
Generalized seniority in Sn isotopes
Binding energies of tin isotopes were measured
in all isotopes from N=50 to N=82. Two
neutron separation energies lie on a smooth
curve and no breaks due to possible subshells are present. Still, the line has a slight
curvature indicating a possible small cubic
term in binding energies. Including a term
proportional to N(N−1)(N−2)/6 led to very
good agreement. Such a term could be due
to weak 3-body effective interactions or due
to polarization of the core by valence
nucleons.

Generalized seniority: excited states
When not all αj coefficients are equal, it is no longer true that all
level spacings are constant. Still, there is one state for any
given J>0 which may lie at a constant spacing above the J=0
ground state. Consider the operator
DJ†=(1+δjj’)−½Σ(jmj’m’|jj’JM)ajm†aj’m’†
which creates an eigenstate of H in a two particle system,
HDJ†|0>=VJDJ†|0>.
If HS†DJ†|0>=[H,S†]DJ†|0>+S†HDJ†|0>=
[[H,S†],DJ†]|0>+(V+VJ)S†DJ†|0> is an eigenstate, the
double commutator condition must be satisfied,
[[H,S†],DJ†]=WS†DJ†
from which follows
H(S†)N−1DJ†|0>={(N−1)V+VJ+½N(N−1)W(S†)N−1DJ†|0>+
{EN+V2−V}(S†)N−1DJ†|0>
This means that V2−V spacings are independent of N
In semi-magic nuclei, with only
valence protons or neutrons
experiment shows features of
generalized seniority
Constant 0-2 spacings in Sn isotopes
Yrast states of odd Sn
isotopes
Proton-neutron interactionslevel order and spacings
•
•
From a 1959 experiment it was “concluded
that the assignment J= ½- for 11Be, as
expected from the shell model is possible
but cannot be established firmly on the
basis of present evidence.”
We did not think so.
•
In 13C the first excited state, 3 MeV above
the ½- g.s. has spin ½+ attributed to a 2s1/2
valence neutron. The g.s.of 11Be is
obtained from 13C by removing
two 1p3/2 protons.
•
Their interaction with a 1p1/2 neutron is
expected to be stronger than their
interaction with a 2s1/2 neutron, hence, the
latter’s orbit may become lower in 11Be.
Experimental information on
p3/2p1/2 and p3/2s1/2 interactions in 12B7
How should this information be
used?
• Removing two j-protons coupled to J=0 reduces
the interaction with a j’-neutron by TWICE the
average jj’ interaction (the monopole part).
• The average interaction is determined by the
position of the center-of-mass of the
jj’ levels,
V(jj’)=SUMJ(2J+1)<jj’J|V|jj’J>/SUMJ(2J+1)
Shell model prediction of the
11Be ground state and excited level
Comments on the 11Be
shell model calculation
•
Last figure is not an “extrapolation”. It is a graphic solution of an
exact shell model calculation in a rather limited space.
•
The ground state is an “intruder” from a higher major shell. It can
be said that here the neutron number 8 is no longer a magic number.
•
The calculated separation energy agreed fairly with a subsequent
measurement. It is rather small and yet, we used matrix elements
which were determined from stable nuclei.
•
We failed to see that the s neutron wave function should be
appreciably extended and 11Be should be a “halo nucleus”.
Shell model wave functions do not include the short range correlations which
are due to the strong bare interaction. Thus, they cannot be the real wave
functions of the nucleus. Still, some states of actual nuclei demonstrate
features of shell model states. The large radius of 11Be, implied by the shell
model, is a real effect.
• The monopole part of the T=0 interaction
affects positions of single nucleon
energies.
• The quadrupole-quadrupole part in the
T=0 interaction breaks seniority in a major
way.
• In nuclei with valence protons and valence
neutrons it leads to strong reduction of the
0-2 spacings – a clear signature of the
transition to rotational spectra and nuclear
deformation.
Levels of Ba (Z=56) isotopes
Drastic reduction of 0-2 spacings
Ab initio calculations of nuclear
many body problem
• Ab initio calculations of nuclear states and
energies are of great importance. Their input is
the bare interaction. If it is sufficiently correct
and the approximations made are satisfactory,
accurate energies of nuclear states, also of
nuclei inaccessible experimentally, will be
obtained.
• The knowledge of the real nuclear wave
functions will enable the calculation of various
moments and transitions, electromagnetic and
weak ones, including double beta decay.
• There are still many difficulties to overcome.
Shell model wave functions are
real to an appreciable extent
• We saw some evidence in the
halo nucleus 11Be and in
• E2 transitions. This is also evident
from pick-up and stripping
reactions.
• Other evidence is offered by
Coulomb energy differences.
Coulomb energy differences of
1d5/2 and 2s1/2 orbits
Will simplicity emerge
from complexity?
• Shell model wave functions, of individual nucleons,
cannot be the real ones. They do not include short range
and other correlations due to the strong bare interaction.
Yet the simple shell model seems to have some reality
and considerable predictive power.
• The bare interaction is the basis of ab initio calculations
leading to admixtures of states with excitations to several
major shells, the higher the better…
• Will the shell model emerge from these calculations? It
has been so simple, useful and elegant and it would be
illogical to give it up. Reliable many-body theory should
explain why it works so well, which interactions lead to it
and where it becomes useless.
Three-body interactionswhere are they?
• There is no evidence of three-body interactions
between valence nucleons.
They could be weak, state independent or both.
Still, three-body interactions with core nucleons
could contribute to effective two-body
interactions between valence nucleons and to
single nucleon energies.
Which nuclei are magic?
In magic nuclei, energies of first excited
states are rather high as in 36S (Z=16, N=20)
where the first excited state is at 3.3 MeV,
considerably higher than in its neighbors.
Shell closure may be concluded if
valence nucleons occupy higher orbits
like 1d3/2 protons in the 38Cl, 40K case.
Shell closure may be demonstrated by
a large drop in separation energies (no
stronger binding of the closed shells!)
Two inaccurate statements
are often made
• The first is that the realization “that magic
numbers are actually not immutable occurred
only during the past 10 years”.
This was realized almost 50 years ago.
• The other is that “the shell structure may
change” ONLY “for nuclei away from stability”,
“nuclear orbitals change their ordering in regions
far away from stability”.
Such changes occur all over the nuclear
landscape.
Various approaches to nuclear
structure physics
The simple
Ab initio
shell model Theoretical derivation calculations
of the
The original effective interaction
Recent
approach
developments
Ab initio calculations
new developements
• A typical theory of this kind is the No Core Shell
Model (NCSM). The input of all ab initio
calculations is the bare interaction between free
nucleons. After 60 years of intensive work, there
are several prescriptions which reproduce rather
well, results of scattering experiments but none
has a solid theoretical foundation.
• No core is assumed and harmonic oscillator
wave functions are used as a complete scheme.
• Some results of NCSM will be presented and
discussed in the following, showing the present
status and some problems.
Merits of ab initio Calculations
• Ab initio calculations of nuclear states and
energies are of great importance. If the input, the
bare interaction, is sufficiently correct and the
approximations made are satisfactory, accurate
energies of nuclear states, also of nuclei
inaccessible experimentally, will be obtained.
• More important would be the knowledge of the
real nuclear wave functions. This will enable the
calculation of various moments and transitions,
electromagnetic and weak ones, including
double beta decay.
• There are still many difficulties to overcome.
Proton separation energies from
N=28 isotones