The Nuclear Shell ModelSimplicity from Complexity
Igal Talmi
The Weizmann Institute of Science
Rehovot Israel
To our dear
friend and
colleague
Aldo Covello
with our very
best wishes
Various approaches to nuclear structure physics
The simple
Derivation of the
Ab initio
shell model
effective interaction calculations with
with effective
of the shell model
only the bare
interactions
from the bare one
interaction
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.
• Only recently this effort seems to yield some reliable
results like those obtained by Aldo Covello et al.
• This does not address the major problem - how
independent nucleon motion can be reconciled with the
strong and short ranged bare interaction.
Ab initio calculations of nuclear
many body problem
• Ab initio calculations are of great
importance. Their input is the bare
interaction. IF it is sufficiently correct and
IF 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.
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.
General features of the effective
interaction extracted 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.
Properties of the T=1 interactions, are evident in
jn-configuations of identical nucleons. An old case is of
neutrons in calcium isotopes from 40Ca to 48Ca which
occupy the 1f7/2 orbit. The jn ground states have
maximum pairing and hence, lowest seniorities, v=0
(J=0) for even n values and v=1 (J=j) for odd ones.
Binding energies are given for any two-body
interaction by the general formula
BE(jn)=BE(n=0)+nCj+αn(n-1)/2+β[n/2]
It includes the binding energy of the nucleus with
closed shells and no j-nucleons and n single j-nucleon
energies. The mutual interaction of the n valence
nucleons is expressed by a quadratic term and a
pairing term in n. The coefficients of these terms are
linear
combinations
of
two-body
energies,
VJ=<j2J|V|j2J>.
Neutron separation energies
from calcium isotopes
Three-body interactionswhere are they?
There is no evidence of three-body interactions
between valence nucleons. They could be weak,
state independent or both.
The two-body interactions extracted from
experiment, may well include contributions from
polarization of the core by valence nucleons.
They could also include contributions of threebody interactions between a core nucleon with
two valence ones. Single nucleon energies may
include contributions from three-body interaction
between a valence nucleon and two ones in the
core.
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.
Which nuclei are magic?
In magic nuclei, energies of first excited
states are rather high.
.
Shell closure may be demonstrated by
a large drop in separation energies (no
stronger binding of the closed shells!)
+
J=2
First excited
levels in some
calcium isotopes
Beyond 48Ca, configurations are
determined by the single nucleon
levels in 49Ca. Thus, it is expected
that valence neutrons will occupy
the 2p3/2 orbit. Relevant diagonal
and non-diagonal matrix elements
were determined from Ni isotopes.
Mixing with other configurations
may be obtained as second order
perturbation. Hence, they may be
absorbed into the two-body effective
interaction.
Neutron separation energies
from calcium isotopes
The position of the first excited
+
30
state with J=2 in Ca agrees with
binding energies. For j=3/2, the
0-2 spacing is related to the
coefficient β by the seniority
scheme formula
β =(2j+2)(V0-V2)/(2j+1)
Thus, the 0-2 spacing is equal here
to
V2-V0=4β/5=1.36*0.8=1.088 MeV.
The measured value is 1.027 MeV.
+
J=2
Positions of
levels in nickel
and calcium isotopes
The difference between low lying levels
of 57Ni and 49Ca is due to the attractive
interaction between neutrons and 1f7/2
protons. It is stronger between
1f7/2 protons and 1f5/2 neutrons.
The positions of J=2+ levels in nickel isotopes exhibit strong
mixing of configurations of 2p3/2, 1f/5/2 and 2p1/2 neutrons .
Generalization of seniority gives a fair description of 0 – 2
spacings as well as of binding energies. In a single j-orbit, the
seniority scheme is obtained by using the pair creation
operator
Sj+=∑(-1)j-majm†aj,-m†
In generalized seniority a linear combination is used
S+=∑αjSj+
S+ creates an eigenstate of the shell model Hamiltonian H,
HS+|0>=VS+|0>. If also (S+)2|0> is an eigenstate with eigenvalue
2V+W, in which case [[H,S+],S+]=W(S+)2 , all analogs of seniority
v=0 states are eigenstates
H(S+)N|0>={NV+WN(N-1)/2}(S+)N|0>
Pair separation energy is linear in N with no breaks due to subshell closures..
H(S+)N|0>={NV+WN(N-1)/2}(S+)N|0>
Pair separation energy is linear in N with no breaks which would
indicate sub-shell closures. These conditions do not guarantee a
similar generalization for odd-even nuclei. The latter is obtained
only if all coefficients αj are equal. This is not the case for Ni
isotopes.
Generalized seniority is not a complete scheme if not all
coefficients are equal. Still it is possible to define for every J>0,
even, a state which is analogous to v=2 states. Consider a J=2+
state created by DM+=∑βjj’DMjj’+ where
DMjj’+=[1+δjj’]-1/2∑(jmj’m’|jj’J=2M)ajm†aj’m’†
We start from HDM+|0>=V2DM+|0>. If also [[H,S+],DM+]=WS+DM+
then it follows for any N
H(S+)N-1DM+|0>={(N-1)V+V2+WN(N-1)/2}(S+)N-1DM+|0>
The positions of these eigenstates above the ground states are
constant, V2-V, independent of N.
Two neutron separation
energies from even Ni isotopes
In odd Ni isotopes level
spacings change appreciably
with the neutron number
How real are shell model
wave functions?
• In view of the apparent complicated
calculations of the effective interaction,
shell model wave functions, could be just
model wave functions. Since they do not
include the short range correlations they
could be very different from the real ones.
• Still, the “wounds” inflicted on the single
nucleon wave functions may not change
them too much. The correlations are
important for short range observables, but
may be less for other (“long range”) ones.
Shell model wave functions are
real to an appreciable extent
• Some evidence is in the halo of the
11Be nucleus, due to an extended
2s1/2 valence neutron wave function.
• This is also evident from pick-up and
stripping reactions.
• Other evidence is offered by
differences of charge distributions.
Charge distribution due to a
1h9/2 proton wave function
Other evidence is offered by
Coulomb energy differences like
between 1d5/2 and 2s1/2 orbits
In states of s nucleons
the Coulomb energy
differences are smaller
than in states of
nucleons in
d- or p-orbits.
Levels of 14C and 14N
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 as a
good approximation? 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.
This rather surprizing consequence may still make
sense. In the case of Sn, generalized seniority seems to
give a good description of ground states and the first
excited J=2+ states. Spacings of low levels change with n
indicating that not all coefficients in the pair creation
operator are equal. It seems that the coefficients of the
2d5/2 and 1g7/2 orbits are larger than those of the
others. These orbits are filled at a faster rate in the
beginning of the major shell. This seems to be the
situation also in Cd isotopes. It is possible that the
smaller coefficients of the 1h11/2, 2d3/2 and 1s1/2 orbits
may be approximately equal. In fact, in Cd isotopes the
spacings of the lowest 11/2-, 3/2+ and ½+ are rather
small and do not change noticeably for neutron
numbers higher than 64.
Binding energies of isotopes from 112Cd to 124Cd agree
fairly well with the seniority scheme formula (there is
no experimental information about heavier isotopes).
• 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.
Neutron separation energies
from cadmium isotopes
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.
In the Sn case, not all coefficients in the pair
operator are equal. In Cd isotopes with
50<n<82, it may be different. Quadrupole
moments of Cd isotopes were recently
measured. The experimental values lie on a
straight line. This is expected in a pure (1h11/2)n
configuration, according to the seniority
scheme formula
n
Q(j J=j)=Q(j)(2j+1-2n)/(2j+1-2v)
v=1.
The straight line should cross zero at
n=(2j+1)/2. Experimentally, the nucleus with an
almost vanishing moment has 9 holes in the
closed shell of n=82. The authors attribute this
behavior to filling of the 1h11/2 orbit being
smeared over the range n=64 to n=82.
Experimental information on
p3/2p1/2 and p3/2s1/2 interactions in 12B7
Proton separation energies from
N=28 isotones
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
In semi-magic nuclei, with only
valence protons or neutrons
experiment shows features of
generalized seniority
Constant 0-2 spacings in Sn isotopes
Levels of Sm (Z=56) isotopes
Drastic reduction of 0-2 spacings
In the Sn case, not all coefficients in the pair operator are
equal. The situation in Cd isotopes with 50<n<82 may be
different. Quadrupole moments of Cd isotopes were
recently measured. The experimental values lie on a
straight line. Such dependence is expected in a pure
(1h11/2)n configuration, according to the seniority
scheme formula
n
Q(j J=j)=Q(j)(2j+1-2n)/(2j+1-2v)
v=1.
The straight line should cross zero at n=(2j+1)/2.
Experimentally, the nucleus with an almost vanishing
moment has 9 holes in the closed shell of n=82. The
authors attribute this behavior to filling of the 1h11/2 orbit
being smeared over the range n=64 to n=82.
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
an(n-1)/2 + b(n-v)/2
where a and b are linear combinations of two-body
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+an(n-1)/2+b(n-v)/2
It includes a linear, quadratic and pairing terms.
NCSM calculations and experiment
Simple shell model calculations in
the p-shell
• Configurations of 1p3/2 and 1p1/2 nucleons
were considered by Cohen and Kurath.
They determined matrix elements of a
two-body interaction from energies of
some states and calculated the others.
• In 14C the neutron p-orbit is closed and
states are due to p-protons only.
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 well understood.
Ab initio calculations
• 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 just as a complete
scheme.
Comparison with 1954 data
only the spin 2- agreed with our prediction
Experimental and p-shell levels in
14C
14C
level schemes
Excitations of simple shell model
states in the p-shell are reasonably
well reproduced by NCSM
• States not considered by Cohen and
Kurath are missed by NCSM.
• In the simple shell model they are
“intruder” states in which two p1/2 neutrons
are raised into s1/2 and d5/2 orbits.
• Odd parity intruder states are due to
excitations of one p1/2 nucleon into
s1/2, d5/2 orbits.
Single neutron levels in
13C
States of two neutrons in
s1/2, d5/2 intruder orbits
• J=0, 2 states, (s1/2)2 and (d5/2)2
• J=2, 2 states, s1/2d5/2 and (d5/2)2
• J=3, 1 state, s1/2d5/2
• To calculate their energies, diagonal as
well as non-diagonal matrix elements are
needed, between them and also with
J=0,2 states of the p-shell.
Analog (quantum) computer for
intruder states
The 0+ position in 14C, calculated from 15C,16C is 6.38 Mev, rather close to the
measured value of 6.59 MeV
14C
levels and T=1 levels in 14N
NCSM calculations and experiment
Why does NCSM miss
intruder states?
Why does NCSM miss
intruder states?
• The kinetic energy of states in different oscillator major
shells is different. The difference between neighbor
shells is HALF of hw, 7 MeV between the p-shell and the
s,d shell. This is why intruder states lie rather high in
NCSM. As we saw, in the simple shell model, single
nucleon level spacings are taken from experiment.
• Still, oscillator functions are a complete set and any
state, like intruders, may be expanded in them. NCSM
wave functions, however, are limited by computational
complexities. The A=14 calculations make use of only 6
oscillator shells.
• It is indeed a challenge for NCSM to overcome this
difficulty. This concerns not only the A=14 levels but also
positive parity levels in 13C and the 11Be ground state.
More about 11Be
• In an experimental paper it was explained that “the parity
inversion (is) ascribed to a combined effect of core
excitation to the first 2+ state, pairing blocking and
proton-neutron monopole interaction.
• Percent admixtures of this excitation are then quoted
between 10% and 40%, in contrast to experimental data.
• In another experimental paper we are assured that the
“interpretation of this level inversion is far from trivial, but
it can be understood in the framework of shell-model
configuration mixing involving a combination of shellquenching quadrupole deformation and pairing-energy
effects”.
Subsequent theories on 11Be
• The “anomalous ½+ state” is “bound due to
dynamical coupling between the core and the
loosely bound neutron which oscillates between
the 2s1/2 and the 1d5/2 orbits”.
• “The proton-neutron monopole interaction has a
large effect…but not enough to explain the
inverted spectrum.” “the effects of quadrupole
core excitation and pairing blocking are equally
important”.
Nuclei with a valence p-shell
• Due to computational complexities, only
light nuclei were calculated by NCSM.
• The case of 11Be is a good starting point
for comparison of those calculations and
experiment. Results of NCSM calculations
of A=14 nuclei were published and
compared with new experimental data
(39 authors).
The simple shell model and
ab initio calculations
Igal Talmi
The Weizmann Institute of Science
Rehovot Israel
Prehistory
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 more than 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.
Levels of even
(1h11/2)n configurations
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.
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.
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)
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.
In spite of their complexity, nuclei
exhibit some simple regularities. An
important one is expressed in the shell
model in which nucleons seem to move
independently. This behavior cannot be
simply reconciled with the strong and
singular interaction between free
nucleons.
Some way to understand this situation
was pioneered by Keith Brueckner and
was followed and improved by many
people. In some way, the complexity is
transformed onto the Hamiltonian.
Shell model wave functions may be
used for calculation of energies and
other operators, provided the interaction
(and other operators) are strongly
renormalized. Such calculations were
successful even though important
matrix elements of the effective
interaction were taken from
experimental energies of nuclei.
Shell model prediction of the
11Be ground state and excited level
Quadrupole moments of
cadmium isotopes