Collective excitations in nuclei

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Collective excitations in nuclei

• Dipole, quadrupole, scissor, twist modes

2

• J.Kvasil, N. Lo Iudice, Ch.Stoyanov, P.Alexa,
• 
  J.Phys. G Nucl.Phys. 29 (2003) 752.
• J.Kvasil, N. Lo Iudice, V.O.Nesterenko,
  M.Kopal,
• 
  Phys. Rev. C58 (1998) 209.
• V.O.Nesterenko, J.Kvasil, P.-G.Reinhard,
• 
  Phys. Rev. C66 (2002) 044307.
• J.Kvasil, N.Lo Iudice, V.O.Nesterenko,
  A.Macková, P.Alexa,
• 
  Phys. Rev. C63 (2001) 054305.
• J.Kvasil, R.G.Nazmitdinov, A.Tsvetkov, P.Alexa,
• 
  Phys. Rev. C63 (2001) 061305.
• A.Tsvetkov, J.Kvasil, R.G.Nazmitdinov,
• 
  J.Phys. G Nucl.Phys. 28 (2002) 2187
• L.M.Fraile, M.J.Borge,, J.Kvasil, . ,
  O.Tengblad,
• Nucl.
  Phys. A703 (2002) 45.
• K.Gulda, J.Kvasil, W.Kurcewicz, .,
  O.Tengblad,
• Nucl.
  Phys. A686 (2002) 71.

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Nucleus Hamiltonian (mean field approach)
A
A
å
å
å











V
h
H
V
t
H
res
i
ij
i


lt
i
i
j
i
1
1
- kinetic energy of i-th nucleon

• effective n-n interaction (strong interaction
  of short
• range (several fm) two-particle
  character)

- nuclear mean field (single-particle character)
- mean potential felt by i-th nucleon
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• Extraction of nuclear mean field in the
  Hamiltonian allows
• one to recognize following degrees of freedom of
  nuclear
• motion
• single-particle degrees of freedom
• collective vibration motion
• rotational motion (if mean field is deformed)

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Single-particle degrees of freedom Nucleons fill
eigen-states of mean field
ground state
lowest excited states (1p-1h configurations)
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phenomenological (harmonic oscillator with
and terms, Wood-Saxon field)
nuclear mean field
microscopical (Hartree-Fock method) (starting
from effective n-n interaction)
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• Advantages of phenomenological fields
• parameters of field are determined from
  experimental
• systematics of s.p. states for a broad
  region of nuclei
• good s.p. basis for the
  following investigation
• of collective modes
• technically simple
• Drawbacks of phenomenological fields
• using phenomenological mean field we should
  add
• phenomenological residual interaction (
  )
• and the selfconsistency between
• and is lost

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• Advantages of microscopical mean fields
• mean field is obtained selfconsistently from
  effective
• n-n interaction residual
  interactions are also
• selfconsistent
• theoretically more fundamental
• Disadvantages of microscopical mean fields
• parameters of effective n-n interactions are
  determined from
• bulk properties of nuclei (bind energy,
  radius, compressibility
• of the nuclear matter) but s.p.
  characteristics of nuclei are
• described badly microscopical HF
  or HFB approaches
• are not so good for the description of
  low-lying collective
• states where good s.p. spectrum is required
• technically complicated

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Mean field determines global characteristics of
nuclei (radius, shape, magic numbers etc.)
spherical
symmetric
(equipotential surfaces are spheres)



mean field
deformed
(in the ground states
equipotential
surfaces are ellipsoids rare earth region

actinide region
)


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Residual interactions (
)
- residual interaction of long range (several fm)

• very short range residual attractive interaction
• (almost contact) it leads to pairs
  of nucleons of
• one type (n-n or p-p)

- time reversed state
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Pairing interaction is taken into account by HFB
or BCS theory using Bogoliubov transformation
from particle to quasiparticle system
1p-1h configuration two-quasiparticle
configuration
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Collective vibrational degrees of freedom. long
range residual interactions
correlated motion of many nucleons which manifest
itself as the vibrations of the nuclear shape or
of the mass density (collective vibrations)
if small amplitude vibrations oscillator
approximation
Random Phase Approximation
- phonon creation operator
- phonon energy
- phonon annihilation operator
- enumerates solutions
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The solving of the RPA eq. with
corresponding to general two-particle effective
n-n interaction is complicated
is decomposed into multipoles
where
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Multipole and spin-multipole operators above are
symmetrized operators with good signature
Every term in the decomposition of
(with given or ) is
responsible for collective excittations with
given multipolarity and parity.
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Velocity distribution
neutrons
protons
E0
E1
neutrons
protons
E2
M1
M2
scissor
twist
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Strength of electromagnetic transitions. (electrom
agnetic excitation) Every collective state of
E? or M? type in the low-energy region can be
described as one-phonon state
- probability of
el.mag. excitation of e.-e. nucleus from its
ground state into the excited states 2 (?
enumerates the solution of the RPA equation with
ang. moment 2)
giant quadrupole resonance
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Electromagnetic transition operators Interaction
Hamiltonian of the microsystem (with
current density ) with external
el.mag. field
Probability of el.mag. transition of the
microsystem from the state i to the state f
Vector potential of the external el.mag. field,
can be expanded into spherical waves
and this leads to the el.mag. transition
operators (see textbooks)
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where
is the energy of absorbed photon
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In nuclear physics (similarly as in atomic
physics) one can use long wave limit
current density and density operators
with gyromagnetic ratios
1 (prot) 0 (neut)
5.58 (prot) -3.82 (neut)
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So in the long wave limit
where
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From experiment (e.g. photoabsorption, (e,e)) so
called strength function of el.mag excitation can
be extracted
with
with X E, M, tor
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Numerical calculations
phenomenological mean field

• Nilsson type (def. oscillator l-s coupling
  l-l term)
• Saxon-Woods type

parameters of mean field from the systematics of
exp. s.p. spectra of all possible nuclei
monopole pairing
parameters were
determined from the mass of even-even and odd
nuclei
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multipole-multipole spinmultipole-spinmultipole
terms
terms with
were involved in the Hamiltonian

• Parameters from
  theoretical requirements
• restoration of Hamiltonian symmetries violated
  by deformed
• mean field

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velocity field
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Rotational degrees of freedom
particle-rotor models
cranking models
statistical approaches
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particle-rotor models adiabatic approach

• moment inertia with
• respect to i-th axis

• i-th component of
• total ang. moment

cranking models stable rotation axis (high
rotational
frequency )

• lab. system

yrast line state for given O
- intrinsic rotating system
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Dependence of deformation parameters ß and ? on
rotational ang. moment I of yrast line states
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Dependence of total ang. Momentum of yrast line
states, I ltO I1 Ogt, on rotational frequency O
(backbending effect)
O (MeV)
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Dependence of kinematical moment of inertia FI/O
on rotational frequency O
O (MeV)
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• Explanation of backbending effect
• the energy of the lowest excited state above
  the yrast line
• goes to zero with increasing rotational
  frequency O

changing of the structure of yrast line

• pairing gap decreases with rotational frequency

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Dependence of energies of the lowest vibrational
exc. states above the yrast line (ß band and ?
band) on the rotational frequency
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Strength of el.mag. excitation from the yrast
line to excited bands
projection on axis 1
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Summary

• to clear up the connection between n-n effective
  interaction
• and s.p. mean field scheme
• to involve into the collective RPA excitations
  also 4 and
• more quasiparticle configurations (not only 2)
• to analyze the dependence on the rotational
  frequency also
• other than E2 and M1 el.mag. excitations